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https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Margaris/Fol.lean | FOL.NV.T_17_10 | [489, 1] | [507, 9] | simp | case h1.h1.h2
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_17_10 | [489, 1] | [507, 9] | simp | case h1.h2
P : Formula
u v : VarName
⊢ ∀ H ∈ {forall_ u (forall_ v P)}, ¬isFreeIn v H | case h1.h2
P : Formula
u v : VarName
⊢ ¬isFreeIn v (forall_ u (forall_ v P)) |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Margaris/Fol.lean | FOL.NV.T_17_10 | [489, 1] | [507, 9] | simp only [isFreeIn] | case h1.h2
P : Formula
u v : VarName
⊢ ¬isFreeIn v (forall_ u (forall_ v P)) | 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_17_10 | [489, 1] | [507, 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_17_11 | [510, 1] | [539, 13] | apply deduction_theorem | P Q : Formula
v : VarName
h1 : ¬isFreeIn v Q
⊢ IsProof ((forall_ v (P.imp_ Q)).imp_ ((exists_ v P).imp_ Q)) | case h1
P Q : Formula
v : VarName
h1 : ¬isFreeIn v Q
⊢ IsDeduct (∅ ∪ {forall_ v (P.imp_ Q)}) ((exists_ v P).imp_ Q) |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Margaris/Fol.lean | FOL.NV.T_17_11 | [510, 1] | [539, 13] | simp | case h1
P Q : Formula
v : VarName
h1 : ¬isFreeIn v Q
⊢ IsDeduct (∅ ∪ {forall_ v (P.imp_ Q)}) ((exists_ v P).imp_ Q) | case h1
P Q : Formula
v : VarName
h1 : ¬isFreeIn v Q
⊢ IsDeduct {forall_ v (P.imp_ Q)} ((exists_ v P).imp_ Q) |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Margaris/Fol.lean | FOL.NV.T_17_11 | [510, 1] | [539, 13] | simp only [def_exists_] | case h1
P Q : Formula
v : VarName
h1 : ¬isFreeIn v Q
⊢ IsDeduct {forall_ v (P.imp_ Q)} ((exists_ v P).imp_ Q) | case h1
P Q : Formula
v : VarName
h1 : ¬isFreeIn v Q
⊢ IsDeduct {forall_ v (P.imp_ Q)} ((forall_ v P.not_).not_.imp_ Q) |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Margaris/Fol.lean | FOL.NV.T_17_11 | [510, 1] | [539, 13] | apply IsDeduct.mp_ (Q.not_.imp_ (forall_ v Q.not_)) | case h1
P Q : Formula
v : VarName
h1 : ¬isFreeIn v Q
⊢ IsDeduct {forall_ v (P.imp_ Q)} ((forall_ v P.not_).not_.imp_ Q) | case h1.a
P Q : Formula
v : VarName
h1 : ¬isFreeIn v Q
⊢ IsDeduct {forall_ v (P.imp_ Q)} ((Q.not_.imp_ (forall_ v Q.not_)).imp_ ((forall_ v P.not_).not_.imp_ Q))
case h1.a
P Q : Formula
v : VarName
h1 : ¬isFreeIn v Q
⊢ IsDeduct {forall_ v (P.imp_ Q)} (Q.not_.imp_ (forall_ v Q.not_)) |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Margaris/Fol.lean | FOL.NV.T_17_11 | [510, 1] | [539, 13] | apply IsDeduct.mp_ ((forall_ v Q.not_).imp_ (forall_ v P.not_)) | case h1.a
P Q : Formula
v : VarName
h1 : ¬isFreeIn v Q
⊢ IsDeduct {forall_ v (P.imp_ Q)} ((Q.not_.imp_ (forall_ v Q.not_)).imp_ ((forall_ v P.not_).not_.imp_ Q)) | case h1.a.a
P Q : Formula
v : VarName
h1 : ¬isFreeIn v Q
⊢ IsDeduct {forall_ v (P.imp_ Q)}
(((forall_ v Q.not_).imp_ (forall_ v P.not_)).imp_
((Q.not_.imp_ (forall_ v Q.not_)).imp_ ((forall_ v P.not_).not_.imp_ Q)))
case h1.a.a
P Q : Formula
v : VarName
h1 : ¬isFreeIn v Q
⊢ IsDeduct {forall_ v (P.imp_ Q)} ((forall_ v Q.not_).imp_ (forall_ v P.not_)) |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Margaris/Fol.lean | FOL.NV.T_17_11 | [510, 1] | [539, 13] | SC | case h1.a.a
P Q : Formula
v : VarName
h1 : ¬isFreeIn v Q
⊢ IsDeduct {forall_ v (P.imp_ Q)}
(((forall_ v Q.not_).imp_ (forall_ v P.not_)).imp_
((Q.not_.imp_ (forall_ v Q.not_)).imp_ ((forall_ v P.not_).not_.imp_ Q))) | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Margaris/Fol.lean | FOL.NV.T_17_11 | [510, 1] | [539, 13] | apply IsDeduct.mp_ (forall_ v (Q.not_.imp_ P.not_)) | case h1.a.a
P Q : Formula
v : VarName
h1 : ¬isFreeIn v Q
⊢ IsDeduct {forall_ v (P.imp_ Q)} ((forall_ v Q.not_).imp_ (forall_ v P.not_)) | case h1.a.a.a
P Q : Formula
v : VarName
h1 : ¬isFreeIn v Q
⊢ IsDeduct {forall_ v (P.imp_ Q)} ((forall_ v (Q.not_.imp_ P.not_)).imp_ ((forall_ v Q.not_).imp_ (forall_ v P.not_)))
case h1.a.a.a
P Q : Formula
v : VarName
h1 : ¬isFreeIn v Q
⊢ IsDeduct {forall_ v (P.imp_ Q)} (forall_ v (Q.not_.imp_ P.not_)) |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Margaris/Fol.lean | FOL.NV.T_17_11 | [510, 1] | [539, 13] | apply IsDeduct.axiom_ | case h1.a.a.a
P Q : Formula
v : VarName
h1 : ¬isFreeIn v Q
⊢ IsDeduct {forall_ v (P.imp_ Q)} ((forall_ v (Q.not_.imp_ P.not_)).imp_ ((forall_ v Q.not_).imp_ (forall_ v P.not_))) | case h1.a.a.a.a
P Q : Formula
v : VarName
h1 : ¬isFreeIn v Q
⊢ IsAxiom ((forall_ v (Q.not_.imp_ P.not_)).imp_ ((forall_ v Q.not_).imp_ (forall_ v P.not_))) |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Margaris/Fol.lean | FOL.NV.T_17_11 | [510, 1] | [539, 13] | apply IsAxiom.pred_1_ | case h1.a.a.a.a
P Q : Formula
v : VarName
h1 : ¬isFreeIn v Q
⊢ IsAxiom ((forall_ v (Q.not_.imp_ P.not_)).imp_ ((forall_ v Q.not_).imp_ (forall_ v P.not_))) | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Margaris/Fol.lean | FOL.NV.T_17_11 | [510, 1] | [539, 13] | apply generalization | case h1.a.a.a
P Q : Formula
v : VarName
h1 : ¬isFreeIn v Q
⊢ IsDeduct {forall_ v (P.imp_ Q)} (forall_ v (Q.not_.imp_ P.not_)) | case h1.a.a.a.h1
P Q : Formula
v : VarName
h1 : ¬isFreeIn v Q
⊢ IsDeduct {forall_ v (P.imp_ Q)} (Q.not_.imp_ P.not_)
case h1.a.a.a.h2
P Q : Formula
v : VarName
h1 : ¬isFreeIn v Q
⊢ ∀ H ∈ {forall_ v (P.imp_ Q)}, ¬isFreeIn v H |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Margaris/Fol.lean | FOL.NV.T_17_11 | [510, 1] | [539, 13] | apply IsDeduct.mp_ (P.imp_ Q) | case h1.a.a.a.h1
P Q : Formula
v : VarName
h1 : ¬isFreeIn v Q
⊢ IsDeduct {forall_ v (P.imp_ Q)} (Q.not_.imp_ P.not_) | case h1.a.a.a.h1.a
P Q : Formula
v : VarName
h1 : ¬isFreeIn v Q
⊢ IsDeduct {forall_ v (P.imp_ Q)} ((P.imp_ Q).imp_ (Q.not_.imp_ P.not_))
case h1.a.a.a.h1.a
P Q : Formula
v : VarName
h1 : ¬isFreeIn v Q
⊢ IsDeduct {forall_ v (P.imp_ Q)} (P.imp_ Q) |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Margaris/Fol.lean | FOL.NV.T_17_11 | [510, 1] | [539, 13] | apply proof_imp_deduct | case h1.a.a.a.h1.a
P Q : Formula
v : VarName
h1 : ¬isFreeIn v Q
⊢ IsDeduct {forall_ v (P.imp_ Q)} ((P.imp_ Q).imp_ (Q.not_.imp_ P.not_)) | case h1.a.a.a.h1.a.h1
P Q : Formula
v : VarName
h1 : ¬isFreeIn v Q
⊢ IsProof ((P.imp_ Q).imp_ (Q.not_.imp_ P.not_)) |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Margaris/Fol.lean | FOL.NV.T_17_11 | [510, 1] | [539, 13] | apply T_14_7 | case h1.a.a.a.h1.a.h1
P Q : Formula
v : VarName
h1 : ¬isFreeIn v Q
⊢ IsProof ((P.imp_ Q).imp_ (Q.not_.imp_ P.not_)) | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Margaris/Fol.lean | FOL.NV.T_17_11 | [510, 1] | [539, 13] | apply specId v (P.imp_ Q) | case h1.a.a.a.h1.a
P Q : Formula
v : VarName
h1 : ¬isFreeIn v Q
⊢ IsDeduct {forall_ v (P.imp_ Q)} (P.imp_ Q) | case h1.a.a.a.h1.a.h1
P Q : Formula
v : VarName
h1 : ¬isFreeIn v Q
⊢ IsDeduct {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_17_11 | [510, 1] | [539, 13] | apply IsDeduct.assume_ | case h1.a.a.a.h1.a.h1
P Q : Formula
v : VarName
h1 : ¬isFreeIn v Q
⊢ IsDeduct {forall_ v (P.imp_ Q)} (forall_ v (P.imp_ Q)) | case h1.a.a.a.h1.a.h1.a
P Q : Formula
v : VarName
h1 : ¬isFreeIn v Q
⊢ 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_17_11 | [510, 1] | [539, 13] | simp | case h1.a.a.a.h1.a.h1.a
P Q : Formula
v : VarName
h1 : ¬isFreeIn v Q
⊢ 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_17_11 | [510, 1] | [539, 13] | simp | case h1.a.a.a.h2
P Q : Formula
v : VarName
h1 : ¬isFreeIn v Q
⊢ ∀ H ∈ {forall_ v (P.imp_ Q)}, ¬isFreeIn v H | case h1.a.a.a.h2
P Q : Formula
v : VarName
h1 : ¬isFreeIn v Q
⊢ ¬isFreeIn v (forall_ v (P.imp_ Q)) |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Margaris/Fol.lean | FOL.NV.T_17_11 | [510, 1] | [539, 13] | simp only [isFreeIn] | case h1.a.a.a.h2
P Q : Formula
v : VarName
h1 : ¬isFreeIn v Q
⊢ ¬isFreeIn v (forall_ v (P.imp_ Q)) | case h1.a.a.a.h2
P Q : Formula
v : VarName
h1 : ¬isFreeIn v Q
⊢ ¬(¬True ∧ (isFreeIn v P ∨ isFreeIn v Q)) |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Margaris/Fol.lean | FOL.NV.T_17_11 | [510, 1] | [539, 13] | simp | case h1.a.a.a.h2
P Q : Formula
v : VarName
h1 : ¬isFreeIn v Q
⊢ ¬(¬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_11 | [510, 1] | [539, 13] | apply IsDeduct.axiom_ | case h1.a
P Q : Formula
v : VarName
h1 : ¬isFreeIn v Q
⊢ IsDeduct {forall_ v (P.imp_ Q)} (Q.not_.imp_ (forall_ v Q.not_)) | case h1.a.a
P Q : Formula
v : VarName
h1 : ¬isFreeIn v Q
⊢ IsAxiom (Q.not_.imp_ (forall_ v Q.not_)) |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Margaris/Fol.lean | FOL.NV.T_17_11 | [510, 1] | [539, 13] | apply IsAxiom.pred_3_ | case h1.a.a
P Q : Formula
v : VarName
h1 : ¬isFreeIn v Q
⊢ IsAxiom (Q.not_.imp_ (forall_ v Q.not_)) | case h1.a.a.a
P Q : Formula
v : VarName
h1 : ¬isFreeIn v Q
⊢ ¬isFreeIn v Q.not_ |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Margaris/Fol.lean | FOL.NV.T_17_11 | [510, 1] | [539, 13] | simp only [isFreeIn] | case h1.a.a.a
P Q : Formula
v : VarName
h1 : ¬isFreeIn v Q
⊢ ¬isFreeIn v Q.not_ | case h1.a.a.a
P Q : Formula
v : VarName
h1 : ¬isFreeIn v Q
⊢ ¬isFreeIn v Q |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Margaris/Fol.lean | FOL.NV.T_17_11 | [510, 1] | [539, 13] | exact h1 | case h1.a.a.a
P Q : Formula
v : VarName
h1 : ¬isFreeIn v Q
⊢ ¬isFreeIn v Q | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Margaris/Fol.lean | FOL.NV.T_17_12 | [543, 1] | [561, 13] | apply IsDeduct.mp_ (exists_ v P) | P Q : Formula
v : VarName
Δ : Set Formula
h1 : IsDeduct Δ (exists_ v P)
h2 : IsDeduct (Δ ∪ {P}) Q
h3 : ∀ H ∈ Δ, ¬isFreeIn v H
h4 : ¬isFreeIn v Q
⊢ IsDeduct Δ Q | case a
P Q : Formula
v : VarName
Δ : Set Formula
h1 : IsDeduct Δ (exists_ v P)
h2 : IsDeduct (Δ ∪ {P}) Q
h3 : ∀ H ∈ Δ, ¬isFreeIn v H
h4 : ¬isFreeIn v Q
⊢ IsDeduct Δ ((exists_ v P).imp_ Q)
case a
P Q : Formula
v : VarName
Δ : Set Formula
h1 : IsDeduct Δ (exists_ v P)
h2 : IsDeduct (Δ ∪ {P}) Q
h3 : ∀ H ∈ Δ, ¬isFreeIn v H
h4 : ¬isFreeIn v Q
⊢ IsDeduct Δ (exists_ v P) |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Margaris/Fol.lean | FOL.NV.T_17_12 | [543, 1] | [561, 13] | apply IsDeduct.mp_ (forall_ v (P.imp_ Q)) | case a
P Q : Formula
v : VarName
Δ : Set Formula
h1 : IsDeduct Δ (exists_ v P)
h2 : IsDeduct (Δ ∪ {P}) Q
h3 : ∀ H ∈ Δ, ¬isFreeIn v H
h4 : ¬isFreeIn v Q
⊢ IsDeduct Δ ((exists_ v P).imp_ Q) | case a.a
P Q : Formula
v : VarName
Δ : Set Formula
h1 : IsDeduct Δ (exists_ v P)
h2 : IsDeduct (Δ ∪ {P}) Q
h3 : ∀ H ∈ Δ, ¬isFreeIn v H
h4 : ¬isFreeIn v Q
⊢ IsDeduct Δ ((forall_ v (P.imp_ Q)).imp_ ((exists_ v P).imp_ Q))
case a.a
P Q : Formula
v : VarName
Δ : Set Formula
h1 : IsDeduct Δ (exists_ v P)
h2 : IsDeduct (Δ ∪ {P}) Q
h3 : ∀ H ∈ Δ, ¬isFreeIn v H
h4 : ¬isFreeIn v Q
⊢ IsDeduct Δ (forall_ v (P.imp_ Q)) |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Margaris/Fol.lean | FOL.NV.T_17_12 | [543, 1] | [561, 13] | apply proof_imp_deduct | case a.a
P Q : Formula
v : VarName
Δ : Set Formula
h1 : IsDeduct Δ (exists_ v P)
h2 : IsDeduct (Δ ∪ {P}) Q
h3 : ∀ H ∈ Δ, ¬isFreeIn v H
h4 : ¬isFreeIn v Q
⊢ IsDeduct Δ ((forall_ v (P.imp_ Q)).imp_ ((exists_ v P).imp_ Q)) | case a.a.h1
P Q : Formula
v : VarName
Δ : Set Formula
h1 : IsDeduct Δ (exists_ v P)
h2 : IsDeduct (Δ ∪ {P}) Q
h3 : ∀ H ∈ Δ, ¬isFreeIn v H
h4 : ¬isFreeIn v Q
⊢ IsProof ((forall_ v (P.imp_ Q)).imp_ ((exists_ v P).imp_ Q)) |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Margaris/Fol.lean | FOL.NV.T_17_12 | [543, 1] | [561, 13] | exact T_17_11 P Q v h4 | case a.a.h1
P Q : Formula
v : VarName
Δ : Set Formula
h1 : IsDeduct Δ (exists_ v P)
h2 : IsDeduct (Δ ∪ {P}) Q
h3 : ∀ H ∈ Δ, ¬isFreeIn v H
h4 : ¬isFreeIn v Q
⊢ IsProof ((forall_ v (P.imp_ Q)).imp_ ((exists_ v P).imp_ Q)) | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Margaris/Fol.lean | FOL.NV.T_17_12 | [543, 1] | [561, 13] | apply generalization | case a.a
P Q : Formula
v : VarName
Δ : Set Formula
h1 : IsDeduct Δ (exists_ v P)
h2 : IsDeduct (Δ ∪ {P}) Q
h3 : ∀ H ∈ Δ, ¬isFreeIn v H
h4 : ¬isFreeIn v Q
⊢ IsDeduct Δ (forall_ v (P.imp_ Q)) | case a.a.h1
P Q : Formula
v : VarName
Δ : Set Formula
h1 : IsDeduct Δ (exists_ v P)
h2 : IsDeduct (Δ ∪ {P}) Q
h3 : ∀ H ∈ Δ, ¬isFreeIn v H
h4 : ¬isFreeIn v Q
⊢ IsDeduct Δ (P.imp_ Q)
case a.a.h2
P Q : Formula
v : VarName
Δ : Set Formula
h1 : IsDeduct Δ (exists_ v P)
h2 : IsDeduct (Δ ∪ {P}) Q
h3 : ∀ H ∈ Δ, ¬isFreeIn v H
h4 : ¬isFreeIn v Q
⊢ ∀ H ∈ Δ, ¬isFreeIn v H |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Margaris/Fol.lean | FOL.NV.T_17_12 | [543, 1] | [561, 13] | apply deduction_theorem | case a.a.h1
P Q : Formula
v : VarName
Δ : Set Formula
h1 : IsDeduct Δ (exists_ v P)
h2 : IsDeduct (Δ ∪ {P}) Q
h3 : ∀ H ∈ Δ, ¬isFreeIn v H
h4 : ¬isFreeIn v Q
⊢ IsDeduct Δ (P.imp_ Q) | case a.a.h1.h1
P Q : Formula
v : VarName
Δ : Set Formula
h1 : IsDeduct Δ (exists_ v P)
h2 : IsDeduct (Δ ∪ {P}) Q
h3 : ∀ H ∈ Δ, ¬isFreeIn v H
h4 : ¬isFreeIn v Q
⊢ IsDeduct (Δ ∪ {P}) Q |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Margaris/Fol.lean | FOL.NV.T_17_12 | [543, 1] | [561, 13] | exact h2 | case a.a.h1.h1
P Q : Formula
v : VarName
Δ : Set Formula
h1 : IsDeduct Δ (exists_ v P)
h2 : IsDeduct (Δ ∪ {P}) Q
h3 : ∀ H ∈ Δ, ¬isFreeIn v H
h4 : ¬isFreeIn v Q
⊢ IsDeduct (Δ ∪ {P}) Q | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Margaris/Fol.lean | FOL.NV.T_17_12 | [543, 1] | [561, 13] | exact h3 | case a.a.h2
P Q : Formula
v : VarName
Δ : Set Formula
h1 : IsDeduct Δ (exists_ v P)
h2 : IsDeduct (Δ ∪ {P}) Q
h3 : ∀ H ∈ Δ, ¬isFreeIn v H
h4 : ¬isFreeIn v Q
⊢ ∀ H ∈ Δ, ¬isFreeIn v H | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Margaris/Fol.lean | FOL.NV.T_17_12 | [543, 1] | [561, 13] | exact h1 | case a
P Q : Formula
v : VarName
Δ : Set Formula
h1 : IsDeduct Δ (exists_ v P)
h2 : IsDeduct (Δ ∪ {P}) Q
h3 : ∀ H ∈ Δ, ¬isFreeIn v H
h4 : ¬isFreeIn v Q
⊢ IsDeduct Δ (exists_ v P) | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Margaris/Fol.lean | FOL.NV.existsElim | [567, 1] | [603, 17] | refine' rule_C (fastReplaceFree v t P) Q t Δ _ h2 h5 _ | 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
⊢ IsDeduct Δ Q | case refine'_1
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
⊢ IsDeduct Δ (exists_ t (fastReplaceFree v t P))
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 |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Margaris/Fol.lean | FOL.NV.existsElim | [567, 1] | [603, 17] | simp only [def_exists_] at h1 | case refine'_1
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
⊢ IsDeduct Δ (exists_ t (fastReplaceFree v t P)) | case refine'_1
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 (exists_ t (fastReplaceFree v t P)) |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Margaris/Fol.lean | FOL.NV.existsElim | [567, 1] | [603, 17] | simp only [def_exists_] | case refine'_1
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 (exists_ t (fastReplaceFree v t P)) | case refine'_1
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_ t (fastReplaceFree v t P).not_).not_ |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Margaris/Fol.lean | FOL.NV.existsElim | [567, 1] | [603, 17] | apply IsDeduct.mp_ (forall_ v P.not_).not_ | case refine'_1
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_ t (fastReplaceFree v t P).not_).not_ | 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_.imp_ (forall_ t (fastReplaceFree v t P).not_).not_)
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_ |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Margaris/Fol.lean | FOL.NV.existsElim | [567, 1] | [603, 17] | apply IsDeduct.mp_ ((forall_ t (fastReplaceFree v t P.not_)).imp_ (forall_ v P.not_)) | 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_.imp_ (forall_ t (fastReplaceFree v t P).not_).not_) | case refine'_1.a.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_ t (fastReplaceFree v t P.not_)).imp_ (forall_ v P.not_)).imp_
((forall_ v P.not_).not_.imp_ (forall_ t (fastReplaceFree v t P).not_).not_))
case refine'_1.a.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_ t (fastReplaceFree v t P.not_)).imp_ (forall_ v P.not_)) |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Margaris/Fol.lean | FOL.NV.existsElim | [567, 1] | [603, 17] | SC | case refine'_1.a.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_ t (fastReplaceFree v t P.not_)).imp_ (forall_ v P.not_)).imp_
((forall_ v P.not_).not_.imp_ (forall_ t (fastReplaceFree v t P).not_).not_)) | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Margaris/Fol.lean | FOL.NV.existsElim | [567, 1] | [603, 17] | apply deduction_theorem | case refine'_1.a.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_ t (fastReplaceFree v t P.not_)).imp_ (forall_ v P.not_)) | case refine'_1.a.a.h1
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_ t (fastReplaceFree v t P.not_)}) (forall_ v P.not_) |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Margaris/Fol.lean | FOL.NV.existsElim | [567, 1] | [603, 17] | apply univIntro P.not_ v t _ h3 | case refine'_1.a.a.h1
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_ t (fastReplaceFree v t P.not_)}) (forall_ v P.not_) | case refine'_1.a.a.h1.h2
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_ t (fastReplaceFree v t P.not_)}) (fastReplaceFree v t P.not_)
case refine'_1.a.a.h1.h3
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_
⊢ ∀ H ∈ Δ ∪ {forall_ t (fastReplaceFree v t P.not_)}, ¬isFreeIn t H |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Margaris/Fol.lean | FOL.NV.existsElim | [567, 1] | [603, 17] | apply specId t | case refine'_1.a.a.h1.h2
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_ t (fastReplaceFree v t P.not_)}) (fastReplaceFree v t P.not_) | case refine'_1.a.a.h1.h2.h1
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_ t (fastReplaceFree v t P.not_)}) (forall_ t (fastReplaceFree v t P.not_)) |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Margaris/Fol.lean | FOL.NV.existsElim | [567, 1] | [603, 17] | apply IsDeduct.assume_ | case refine'_1.a.a.h1.h2.h1
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_ t (fastReplaceFree v t P.not_)}) (forall_ t (fastReplaceFree v t P.not_)) | case refine'_1.a.a.h1.h2.h1.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_
⊢ forall_ t (fastReplaceFree v t P.not_) ∈ Δ ∪ {forall_ t (fastReplaceFree v t P.not_)} |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Margaris/Fol.lean | FOL.NV.existsElim | [567, 1] | [603, 17] | simp | case refine'_1.a.a.h1.h2.h1.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_
⊢ forall_ t (fastReplaceFree v t P.not_) ∈ Δ ∪ {forall_ t (fastReplaceFree v t P.not_)} | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Margaris/Fol.lean | FOL.NV.existsElim | [567, 1] | [603, 17] | intro H a1 | case refine'_1.a.a.h1.h3
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_
⊢ ∀ H ∈ Δ ∪ {forall_ t (fastReplaceFree v t P.not_)}, ¬isFreeIn t H | case refine'_1.a.a.h1.h3
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_
H : Formula
a1 : H ∈ Δ ∪ {forall_ t (fastReplaceFree v t P.not_)}
⊢ ¬isFreeIn t H |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Margaris/Fol.lean | FOL.NV.existsElim | [567, 1] | [603, 17] | simp at a1 | case refine'_1.a.a.h1.h3
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_
H : Formula
a1 : H ∈ Δ ∪ {forall_ t (fastReplaceFree v t P.not_)}
⊢ ¬isFreeIn t H | case refine'_1.a.a.h1.h3
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_
H : Formula
a1 : H = forall_ t (fastReplaceFree v t P.not_) ∨ H ∈ Δ
⊢ ¬isFreeIn t H |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Margaris/Fol.lean | FOL.NV.existsElim | [567, 1] | [603, 17] | cases a1 | case refine'_1.a.a.h1.h3
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_
H : Formula
a1 : H = forall_ t (fastReplaceFree v t P.not_) ∨ H ∈ Δ
⊢ ¬isFreeIn t H | case refine'_1.a.a.h1.h3.inl
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_
H : Formula
h✝ : H = forall_ t (fastReplaceFree v t P.not_)
⊢ ¬isFreeIn t H
case refine'_1.a.a.h1.h3.inr
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_
H : Formula
h✝ : H ∈ Δ
⊢ ¬isFreeIn t H |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Margaris/Fol.lean | FOL.NV.existsElim | [567, 1] | [603, 17] | case _ c1 =>
subst c1
simp only [isFreeIn]
simp | 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_
H : Formula
c1 : H = forall_ t (fastReplaceFree v t P.not_)
⊢ ¬isFreeIn t H | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Margaris/Fol.lean | FOL.NV.existsElim | [567, 1] | [603, 17] | case _ c1 =>
exact h5 H c1 | 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_
H : Formula
c1 : H ∈ Δ
⊢ ¬isFreeIn t H | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Margaris/Fol.lean | FOL.NV.existsElim | [567, 1] | [603, 17] | subst c1 | 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_
H : Formula
c1 : H = forall_ t (fastReplaceFree v t P.not_)
⊢ ¬isFreeIn t H | 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_
⊢ ¬isFreeIn t (forall_ t (fastReplaceFree v t P.not_)) |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Margaris/Fol.lean | FOL.NV.existsElim | [567, 1] | [603, 17] | simp only [isFreeIn] | 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_
⊢ ¬isFreeIn t (forall_ t (fastReplaceFree v t P.not_)) | 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_
⊢ ¬(¬True ∧ isFreeIn t (fastReplaceFree v t P)) |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Margaris/Fol.lean | FOL.NV.existsElim | [567, 1] | [603, 17] | simp | 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_
⊢ ¬(¬True ∧ isFreeIn t (fastReplaceFree v t P)) | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Margaris/Fol.lean | FOL.NV.existsElim | [567, 1] | [603, 17] | exact h5 H c1 | 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_
H : Formula
c1 : H ∈ Δ
⊢ ¬isFreeIn t H | no goals |
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 |
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