url
stringclasses 147
values | commit
stringclasses 147
values | file_path
stringlengths 7
101
| full_name
stringlengths 1
94
| start
stringlengths 6
10
| end
stringlengths 6
11
| tactic
stringlengths 1
11.2k
| state_before
stringlengths 3
2.09M
| state_after
stringlengths 6
2.09M
|
|---|---|---|---|---|---|---|---|---|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Margaris/Fol.lean
|
FOL.NV.T_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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