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_18_5
|
[938, 1]
|
[980, 7]
|
SC
|
case h3
P : Formula
v : VarName
⊢ IsDeduct ∅ (((forall_ v P).imp_ (forall_ v P).not_.not_).imp_ ((forall_ v P).not_.not_.imp_ (forall_ v P)).not_).not_
|
no goals
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Margaris/Fol.lean
|
FOL.NV.T_18_6
|
[983, 1]
|
[1036, 36]
|
simp only [Similar] at h1
|
P_u P_v : Formula
u v : VarName
h1 : Similar P_u P_v u v
⊢ IsProof ((forall_ u P_u).iff_ (forall_ v P_v))
|
P_u P_v : Formula
u v : VarName
h1 :
¬isFreeIn v P_u ∧
¬isFreeIn u P_v ∧
fastAdmits u v P_u ∧ fastAdmits v u P_v ∧ P_v = fastReplaceFree u v P_u ∧ P_u = fastReplaceFree v u P_v
⊢ IsProof ((forall_ u P_u).iff_ (forall_ v P_v))
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Margaris/Fol.lean
|
FOL.NV.T_18_6
|
[983, 1]
|
[1036, 36]
|
cases h1
|
P_u P_v : Formula
u v : VarName
h1 :
¬isFreeIn v P_u ∧
¬isFreeIn u P_v ∧
fastAdmits u v P_u ∧ fastAdmits v u P_v ∧ P_v = fastReplaceFree u v P_u ∧ P_u = fastReplaceFree v u P_v
⊢ IsProof ((forall_ u P_u).iff_ (forall_ v P_v))
|
case intro
P_u P_v : Formula
u v : VarName
left✝ : ¬isFreeIn v P_u
right✝ :
¬isFreeIn u P_v ∧
fastAdmits u v P_u ∧ fastAdmits v u P_v ∧ P_v = fastReplaceFree u v P_u ∧ P_u = fastReplaceFree v u P_v
⊢ IsProof ((forall_ u P_u).iff_ (forall_ v P_v))
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Margaris/Fol.lean
|
FOL.NV.T_18_6
|
[983, 1]
|
[1036, 36]
|
cases h1_right
|
P_u P_v : Formula
u v : VarName
h1_left : ¬isFreeIn v P_u
h1_right :
¬isFreeIn u P_v ∧
fastAdmits u v P_u ∧ fastAdmits v u P_v ∧ P_v = fastReplaceFree u v P_u ∧ P_u = fastReplaceFree v u P_v
⊢ IsProof ((forall_ u P_u).iff_ (forall_ v P_v))
|
case intro
P_u P_v : Formula
u v : VarName
h1_left : ¬isFreeIn v P_u
left✝ : ¬isFreeIn u P_v
right✝ : fastAdmits u v P_u ∧ fastAdmits v u P_v ∧ P_v = fastReplaceFree u v P_u ∧ P_u = fastReplaceFree v u P_v
⊢ IsProof ((forall_ u P_u).iff_ (forall_ v P_v))
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Margaris/Fol.lean
|
FOL.NV.T_18_6
|
[983, 1]
|
[1036, 36]
|
cases h1_right_right
|
P_u P_v : Formula
u v : VarName
h1_left : ¬isFreeIn v P_u
h1_right_left : ¬isFreeIn u P_v
h1_right_right : fastAdmits u v P_u ∧ fastAdmits v u P_v ∧ P_v = fastReplaceFree u v P_u ∧ P_u = fastReplaceFree v u P_v
⊢ IsProof ((forall_ u P_u).iff_ (forall_ v P_v))
|
case intro
P_u P_v : Formula
u v : VarName
h1_left : ¬isFreeIn v P_u
h1_right_left : ¬isFreeIn u P_v
left✝ : fastAdmits u v P_u
right✝ : fastAdmits v u P_v ∧ P_v = fastReplaceFree u v P_u ∧ P_u = fastReplaceFree v u P_v
⊢ IsProof ((forall_ u P_u).iff_ (forall_ v P_v))
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Margaris/Fol.lean
|
FOL.NV.T_18_6
|
[983, 1]
|
[1036, 36]
|
cases h1_right_right_right
|
P_u P_v : Formula
u v : VarName
h1_left : ¬isFreeIn v P_u
h1_right_left : ¬isFreeIn u P_v
h1_right_right_left : fastAdmits u v P_u
h1_right_right_right : fastAdmits v u P_v ∧ P_v = fastReplaceFree u v P_u ∧ P_u = fastReplaceFree v u P_v
⊢ IsProof ((forall_ u P_u).iff_ (forall_ v P_v))
|
case intro
P_u P_v : Formula
u v : VarName
h1_left : ¬isFreeIn v P_u
h1_right_left : ¬isFreeIn u P_v
h1_right_right_left : fastAdmits u v P_u
left✝ : fastAdmits v u P_v
right✝ : P_v = fastReplaceFree u v P_u ∧ P_u = fastReplaceFree v u P_v
⊢ IsProof ((forall_ u P_u).iff_ (forall_ v P_v))
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Margaris/Fol.lean
|
FOL.NV.T_18_6
|
[983, 1]
|
[1036, 36]
|
cases h1_right_right_right_right
|
P_u P_v : Formula
u v : VarName
h1_left : ¬isFreeIn v P_u
h1_right_left : ¬isFreeIn u P_v
h1_right_right_left : fastAdmits u v P_u
h1_right_right_right_left : fastAdmits v u P_v
h1_right_right_right_right : P_v = fastReplaceFree u v P_u ∧ P_u = fastReplaceFree v u P_v
⊢ IsProof ((forall_ u P_u).iff_ (forall_ v P_v))
|
case intro
P_u P_v : Formula
u v : VarName
h1_left : ¬isFreeIn v P_u
h1_right_left : ¬isFreeIn u P_v
h1_right_right_left : fastAdmits u v P_u
h1_right_right_right_left : fastAdmits v u P_v
left✝ : P_v = fastReplaceFree u v P_u
right✝ : P_u = fastReplaceFree v u P_v
⊢ IsProof ((forall_ u P_u).iff_ (forall_ v P_v))
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Margaris/Fol.lean
|
FOL.NV.T_18_6
|
[983, 1]
|
[1036, 36]
|
apply IsDeduct.mp_ ((forall_ v P_v).imp_ (forall_ u P_u))
|
P_u P_v : Formula
u v : VarName
h1_left : ¬isFreeIn v P_u
h1_right_left : ¬isFreeIn u P_v
h1_right_right_left : fastAdmits u v P_u
h1_right_right_right_left : fastAdmits v u P_v
h1_right_right_right_right_left : P_v = fastReplaceFree u v P_u
h1_right_right_right_right_right : P_u = fastReplaceFree v u P_v
⊢ IsProof ((forall_ u P_u).iff_ (forall_ v P_v))
|
case a
P_u P_v : Formula
u v : VarName
h1_left : ¬isFreeIn v P_u
h1_right_left : ¬isFreeIn u P_v
h1_right_right_left : fastAdmits u v P_u
h1_right_right_right_left : fastAdmits v u P_v
h1_right_right_right_right_left : P_v = fastReplaceFree u v P_u
h1_right_right_right_right_right : P_u = fastReplaceFree v u P_v
⊢ IsDeduct ∅ (((forall_ v P_v).imp_ (forall_ u P_u)).imp_ ((forall_ u P_u).iff_ (forall_ v P_v)))
case a
P_u P_v : Formula
u v : VarName
h1_left : ¬isFreeIn v P_u
h1_right_left : ¬isFreeIn u P_v
h1_right_right_left : fastAdmits u v P_u
h1_right_right_right_left : fastAdmits v u P_v
h1_right_right_right_right_left : P_v = fastReplaceFree u v P_u
h1_right_right_right_right_right : P_u = fastReplaceFree v u P_v
⊢ IsDeduct ∅ ((forall_ v P_v).imp_ (forall_ u P_u))
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Margaris/Fol.lean
|
FOL.NV.T_18_6
|
[983, 1]
|
[1036, 36]
|
apply IsDeduct.mp_ ((forall_ u P_u).imp_ (forall_ v P_v))
|
case a
P_u P_v : Formula
u v : VarName
h1_left : ¬isFreeIn v P_u
h1_right_left : ¬isFreeIn u P_v
h1_right_right_left : fastAdmits u v P_u
h1_right_right_right_left : fastAdmits v u P_v
h1_right_right_right_right_left : P_v = fastReplaceFree u v P_u
h1_right_right_right_right_right : P_u = fastReplaceFree v u P_v
⊢ IsDeduct ∅ (((forall_ v P_v).imp_ (forall_ u P_u)).imp_ ((forall_ u P_u).iff_ (forall_ v P_v)))
|
case a.a
P_u P_v : Formula
u v : VarName
h1_left : ¬isFreeIn v P_u
h1_right_left : ¬isFreeIn u P_v
h1_right_right_left : fastAdmits u v P_u
h1_right_right_right_left : fastAdmits v u P_v
h1_right_right_right_right_left : P_v = fastReplaceFree u v P_u
h1_right_right_right_right_right : P_u = fastReplaceFree v u P_v
⊢ IsDeduct ∅
(((forall_ u P_u).imp_ (forall_ v P_v)).imp_
(((forall_ v P_v).imp_ (forall_ u P_u)).imp_ ((forall_ u P_u).iff_ (forall_ v P_v))))
case a.a
P_u P_v : Formula
u v : VarName
h1_left : ¬isFreeIn v P_u
h1_right_left : ¬isFreeIn u P_v
h1_right_right_left : fastAdmits u v P_u
h1_right_right_right_left : fastAdmits v u P_v
h1_right_right_right_right_left : P_v = fastReplaceFree u v P_u
h1_right_right_right_right_right : P_u = fastReplaceFree v u P_v
⊢ IsDeduct ∅ ((forall_ u P_u).imp_ (forall_ v P_v))
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Margaris/Fol.lean
|
FOL.NV.T_18_6
|
[983, 1]
|
[1036, 36]
|
simp only [def_iff_]
|
case a.a
P_u P_v : Formula
u v : VarName
h1_left : ¬isFreeIn v P_u
h1_right_left : ¬isFreeIn u P_v
h1_right_right_left : fastAdmits u v P_u
h1_right_right_right_left : fastAdmits v u P_v
h1_right_right_right_right_left : P_v = fastReplaceFree u v P_u
h1_right_right_right_right_right : P_u = fastReplaceFree v u P_v
⊢ IsDeduct ∅
(((forall_ u P_u).imp_ (forall_ v P_v)).imp_
(((forall_ v P_v).imp_ (forall_ u P_u)).imp_ ((forall_ u P_u).iff_ (forall_ v P_v))))
|
case a.a
P_u P_v : Formula
u v : VarName
h1_left : ¬isFreeIn v P_u
h1_right_left : ¬isFreeIn u P_v
h1_right_right_left : fastAdmits u v P_u
h1_right_right_right_left : fastAdmits v u P_v
h1_right_right_right_right_left : P_v = fastReplaceFree u v P_u
h1_right_right_right_right_right : P_u = fastReplaceFree v u P_v
⊢ IsDeduct ∅
(((forall_ u P_u).imp_ (forall_ v P_v)).imp_
(((forall_ v P_v).imp_ (forall_ u P_u)).imp_
(((forall_ u P_u).imp_ (forall_ v P_v)).and_ ((forall_ v P_v).imp_ (forall_ u P_u)))))
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Margaris/Fol.lean
|
FOL.NV.T_18_6
|
[983, 1]
|
[1036, 36]
|
simp only [def_and_]
|
case a.a
P_u P_v : Formula
u v : VarName
h1_left : ¬isFreeIn v P_u
h1_right_left : ¬isFreeIn u P_v
h1_right_right_left : fastAdmits u v P_u
h1_right_right_right_left : fastAdmits v u P_v
h1_right_right_right_right_left : P_v = fastReplaceFree u v P_u
h1_right_right_right_right_right : P_u = fastReplaceFree v u P_v
⊢ IsDeduct ∅
(((forall_ u P_u).imp_ (forall_ v P_v)).imp_
(((forall_ v P_v).imp_ (forall_ u P_u)).imp_
(((forall_ u P_u).imp_ (forall_ v P_v)).and_ ((forall_ v P_v).imp_ (forall_ u P_u)))))
|
case a.a
P_u P_v : Formula
u v : VarName
h1_left : ¬isFreeIn v P_u
h1_right_left : ¬isFreeIn u P_v
h1_right_right_left : fastAdmits u v P_u
h1_right_right_right_left : fastAdmits v u P_v
h1_right_right_right_right_left : P_v = fastReplaceFree u v P_u
h1_right_right_right_right_right : P_u = fastReplaceFree v u P_v
⊢ IsDeduct ∅
(((forall_ u P_u).imp_ (forall_ v P_v)).imp_
(((forall_ v P_v).imp_ (forall_ u P_u)).imp_
(((forall_ u P_u).imp_ (forall_ v P_v)).imp_ ((forall_ v P_v).imp_ (forall_ u P_u)).not_).not_))
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Margaris/Fol.lean
|
FOL.NV.T_18_6
|
[983, 1]
|
[1036, 36]
|
SC
|
case a.a
P_u P_v : Formula
u v : VarName
h1_left : ¬isFreeIn v P_u
h1_right_left : ¬isFreeIn u P_v
h1_right_right_left : fastAdmits u v P_u
h1_right_right_right_left : fastAdmits v u P_v
h1_right_right_right_right_left : P_v = fastReplaceFree u v P_u
h1_right_right_right_right_right : P_u = fastReplaceFree v u P_v
⊢ IsDeduct ∅
(((forall_ u P_u).imp_ (forall_ v P_v)).imp_
(((forall_ v P_v).imp_ (forall_ u P_u)).imp_
(((forall_ u P_u).imp_ (forall_ v P_v)).imp_ ((forall_ v P_v).imp_ (forall_ u P_u)).not_).not_))
|
no goals
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Margaris/Fol.lean
|
FOL.NV.T_18_6
|
[983, 1]
|
[1036, 36]
|
apply deduction_theorem
|
case a.a
P_u P_v : Formula
u v : VarName
h1_left : ¬isFreeIn v P_u
h1_right_left : ¬isFreeIn u P_v
h1_right_right_left : fastAdmits u v P_u
h1_right_right_right_left : fastAdmits v u P_v
h1_right_right_right_right_left : P_v = fastReplaceFree u v P_u
h1_right_right_right_right_right : P_u = fastReplaceFree v u P_v
⊢ IsDeduct ∅ ((forall_ u P_u).imp_ (forall_ v P_v))
|
case a.a.h1
P_u P_v : Formula
u v : VarName
h1_left : ¬isFreeIn v P_u
h1_right_left : ¬isFreeIn u P_v
h1_right_right_left : fastAdmits u v P_u
h1_right_right_right_left : fastAdmits v u P_v
h1_right_right_right_right_left : P_v = fastReplaceFree u v P_u
h1_right_right_right_right_right : P_u = fastReplaceFree v u P_v
⊢ IsDeduct (∅ ∪ {forall_ u P_u}) (forall_ v P_v)
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Margaris/Fol.lean
|
FOL.NV.T_18_6
|
[983, 1]
|
[1036, 36]
|
simp
|
case a.a.h1
P_u P_v : Formula
u v : VarName
h1_left : ¬isFreeIn v P_u
h1_right_left : ¬isFreeIn u P_v
h1_right_right_left : fastAdmits u v P_u
h1_right_right_right_left : fastAdmits v u P_v
h1_right_right_right_right_left : P_v = fastReplaceFree u v P_u
h1_right_right_right_right_right : P_u = fastReplaceFree v u P_v
⊢ IsDeduct (∅ ∪ {forall_ u P_u}) (forall_ v P_v)
|
case a.a.h1
P_u P_v : Formula
u v : VarName
h1_left : ¬isFreeIn v P_u
h1_right_left : ¬isFreeIn u P_v
h1_right_right_left : fastAdmits u v P_u
h1_right_right_right_left : fastAdmits v u P_v
h1_right_right_right_right_left : P_v = fastReplaceFree u v P_u
h1_right_right_right_right_right : P_u = fastReplaceFree v u P_v
⊢ IsDeduct {forall_ u P_u} (forall_ v P_v)
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Margaris/Fol.lean
|
FOL.NV.T_18_6
|
[983, 1]
|
[1036, 36]
|
apply generalization
|
case a.a.h1
P_u P_v : Formula
u v : VarName
h1_left : ¬isFreeIn v P_u
h1_right_left : ¬isFreeIn u P_v
h1_right_right_left : fastAdmits u v P_u
h1_right_right_right_left : fastAdmits v u P_v
h1_right_right_right_right_left : P_v = fastReplaceFree u v P_u
h1_right_right_right_right_right : P_u = fastReplaceFree v u P_v
⊢ IsDeduct {forall_ u P_u} (forall_ v P_v)
|
case a.a.h1.h1
P_u P_v : Formula
u v : VarName
h1_left : ¬isFreeIn v P_u
h1_right_left : ¬isFreeIn u P_v
h1_right_right_left : fastAdmits u v P_u
h1_right_right_right_left : fastAdmits v u P_v
h1_right_right_right_right_left : P_v = fastReplaceFree u v P_u
h1_right_right_right_right_right : P_u = fastReplaceFree v u P_v
⊢ IsDeduct {forall_ u P_u} P_v
case a.a.h1.h2
P_u P_v : Formula
u v : VarName
h1_left : ¬isFreeIn v P_u
h1_right_left : ¬isFreeIn u P_v
h1_right_right_left : fastAdmits u v P_u
h1_right_right_right_left : fastAdmits v u P_v
h1_right_right_right_right_left : P_v = fastReplaceFree u v P_u
h1_right_right_right_right_right : P_u = fastReplaceFree v u P_v
⊢ ∀ H ∈ {forall_ u P_u}, ¬isFreeIn v H
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Margaris/Fol.lean
|
FOL.NV.T_18_6
|
[983, 1]
|
[1036, 36]
|
subst h1_right_right_right_right_left
|
case a.a.h1.h1
P_u P_v : Formula
u v : VarName
h1_left : ¬isFreeIn v P_u
h1_right_left : ¬isFreeIn u P_v
h1_right_right_left : fastAdmits u v P_u
h1_right_right_right_left : fastAdmits v u P_v
h1_right_right_right_right_left : P_v = fastReplaceFree u v P_u
h1_right_right_right_right_right : P_u = fastReplaceFree v u P_v
⊢ IsDeduct {forall_ u P_u} P_v
|
case a.a.h1.h1
P_u : Formula
u v : VarName
h1_left : ¬isFreeIn v P_u
h1_right_right_left : fastAdmits u v P_u
h1_right_left : ¬isFreeIn u (fastReplaceFree u v P_u)
h1_right_right_right_left : fastAdmits v u (fastReplaceFree u v P_u)
h1_right_right_right_right_right : P_u = fastReplaceFree v u (fastReplaceFree u v P_u)
⊢ IsDeduct {forall_ u P_u} (fastReplaceFree u v P_u)
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Margaris/Fol.lean
|
FOL.NV.T_18_6
|
[983, 1]
|
[1036, 36]
|
apply spec
|
case a.a.h1.h1
P_u : Formula
u v : VarName
h1_left : ¬isFreeIn v P_u
h1_right_right_left : fastAdmits u v P_u
h1_right_left : ¬isFreeIn u (fastReplaceFree u v P_u)
h1_right_right_right_left : fastAdmits v u (fastReplaceFree u v P_u)
h1_right_right_right_right_right : P_u = fastReplaceFree v u (fastReplaceFree u v P_u)
⊢ IsDeduct {forall_ u P_u} (fastReplaceFree u v P_u)
|
case a.a.h1.h1.h1
P_u : Formula
u v : VarName
h1_left : ¬isFreeIn v P_u
h1_right_right_left : fastAdmits u v P_u
h1_right_left : ¬isFreeIn u (fastReplaceFree u v P_u)
h1_right_right_right_left : fastAdmits v u (fastReplaceFree u v P_u)
h1_right_right_right_right_right : P_u = fastReplaceFree v u (fastReplaceFree u v P_u)
⊢ IsDeduct {forall_ u P_u} (forall_ u P_u)
case a.a.h1.h1.h2
P_u : Formula
u v : VarName
h1_left : ¬isFreeIn v P_u
h1_right_right_left : fastAdmits u v P_u
h1_right_left : ¬isFreeIn u (fastReplaceFree u v P_u)
h1_right_right_right_left : fastAdmits v u (fastReplaceFree u v P_u)
h1_right_right_right_right_right : P_u = fastReplaceFree v u (fastReplaceFree u v P_u)
⊢ fastAdmits u v P_u
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Margaris/Fol.lean
|
FOL.NV.T_18_6
|
[983, 1]
|
[1036, 36]
|
apply IsDeduct.assume_
|
case a.a.h1.h1.h1
P_u : Formula
u v : VarName
h1_left : ¬isFreeIn v P_u
h1_right_right_left : fastAdmits u v P_u
h1_right_left : ¬isFreeIn u (fastReplaceFree u v P_u)
h1_right_right_right_left : fastAdmits v u (fastReplaceFree u v P_u)
h1_right_right_right_right_right : P_u = fastReplaceFree v u (fastReplaceFree u v P_u)
⊢ IsDeduct {forall_ u P_u} (forall_ u P_u)
|
case a.a.h1.h1.h1.a
P_u : Formula
u v : VarName
h1_left : ¬isFreeIn v P_u
h1_right_right_left : fastAdmits u v P_u
h1_right_left : ¬isFreeIn u (fastReplaceFree u v P_u)
h1_right_right_right_left : fastAdmits v u (fastReplaceFree u v P_u)
h1_right_right_right_right_right : P_u = fastReplaceFree v u (fastReplaceFree u v P_u)
⊢ forall_ u P_u ∈ {forall_ u P_u}
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Margaris/Fol.lean
|
FOL.NV.T_18_6
|
[983, 1]
|
[1036, 36]
|
simp
|
case a.a.h1.h1.h1.a
P_u : Formula
u v : VarName
h1_left : ¬isFreeIn v P_u
h1_right_right_left : fastAdmits u v P_u
h1_right_left : ¬isFreeIn u (fastReplaceFree u v P_u)
h1_right_right_right_left : fastAdmits v u (fastReplaceFree u v P_u)
h1_right_right_right_right_right : P_u = fastReplaceFree v u (fastReplaceFree u v P_u)
⊢ forall_ u P_u ∈ {forall_ u P_u}
|
no goals
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Margaris/Fol.lean
|
FOL.NV.T_18_6
|
[983, 1]
|
[1036, 36]
|
exact h1_right_right_left
|
case a.a.h1.h1.h2
P_u : Formula
u v : VarName
h1_left : ¬isFreeIn v P_u
h1_right_right_left : fastAdmits u v P_u
h1_right_left : ¬isFreeIn u (fastReplaceFree u v P_u)
h1_right_right_right_left : fastAdmits v u (fastReplaceFree u v P_u)
h1_right_right_right_right_right : P_u = fastReplaceFree v u (fastReplaceFree u v P_u)
⊢ fastAdmits u v P_u
|
no goals
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Margaris/Fol.lean
|
FOL.NV.T_18_6
|
[983, 1]
|
[1036, 36]
|
intro H a1
|
case a.a.h1.h2
P_u P_v : Formula
u v : VarName
h1_left : ¬isFreeIn v P_u
h1_right_left : ¬isFreeIn u P_v
h1_right_right_left : fastAdmits u v P_u
h1_right_right_right_left : fastAdmits v u P_v
h1_right_right_right_right_left : P_v = fastReplaceFree u v P_u
h1_right_right_right_right_right : P_u = fastReplaceFree v u P_v
⊢ ∀ H ∈ {forall_ u P_u}, ¬isFreeIn v H
|
case a.a.h1.h2
P_u P_v : Formula
u v : VarName
h1_left : ¬isFreeIn v P_u
h1_right_left : ¬isFreeIn u P_v
h1_right_right_left : fastAdmits u v P_u
h1_right_right_right_left : fastAdmits v u P_v
h1_right_right_right_right_left : P_v = fastReplaceFree u v P_u
h1_right_right_right_right_right : P_u = fastReplaceFree v u P_v
H : Formula
a1 : H ∈ {forall_ u P_u}
⊢ ¬isFreeIn v H
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Margaris/Fol.lean
|
FOL.NV.T_18_6
|
[983, 1]
|
[1036, 36]
|
simp at a1
|
case a.a.h1.h2
P_u P_v : Formula
u v : VarName
h1_left : ¬isFreeIn v P_u
h1_right_left : ¬isFreeIn u P_v
h1_right_right_left : fastAdmits u v P_u
h1_right_right_right_left : fastAdmits v u P_v
h1_right_right_right_right_left : P_v = fastReplaceFree u v P_u
h1_right_right_right_right_right : P_u = fastReplaceFree v u P_v
H : Formula
a1 : H ∈ {forall_ u P_u}
⊢ ¬isFreeIn v H
|
case a.a.h1.h2
P_u P_v : Formula
u v : VarName
h1_left : ¬isFreeIn v P_u
h1_right_left : ¬isFreeIn u P_v
h1_right_right_left : fastAdmits u v P_u
h1_right_right_right_left : fastAdmits v u P_v
h1_right_right_right_right_left : P_v = fastReplaceFree u v P_u
h1_right_right_right_right_right : P_u = fastReplaceFree v u P_v
H : Formula
a1 : H = forall_ u P_u
⊢ ¬isFreeIn v H
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Margaris/Fol.lean
|
FOL.NV.T_18_6
|
[983, 1]
|
[1036, 36]
|
subst a1
|
case a.a.h1.h2
P_u P_v : Formula
u v : VarName
h1_left : ¬isFreeIn v P_u
h1_right_left : ¬isFreeIn u P_v
h1_right_right_left : fastAdmits u v P_u
h1_right_right_right_left : fastAdmits v u P_v
h1_right_right_right_right_left : P_v = fastReplaceFree u v P_u
h1_right_right_right_right_right : P_u = fastReplaceFree v u P_v
H : Formula
a1 : H = forall_ u P_u
⊢ ¬isFreeIn v H
|
case a.a.h1.h2
P_u P_v : Formula
u v : VarName
h1_left : ¬isFreeIn v P_u
h1_right_left : ¬isFreeIn u P_v
h1_right_right_left : fastAdmits u v P_u
h1_right_right_right_left : fastAdmits v u P_v
h1_right_right_right_right_left : P_v = fastReplaceFree u v P_u
h1_right_right_right_right_right : P_u = fastReplaceFree v u P_v
⊢ ¬isFreeIn v (forall_ u P_u)
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Margaris/Fol.lean
|
FOL.NV.T_18_6
|
[983, 1]
|
[1036, 36]
|
simp only [isFreeIn]
|
case a.a.h1.h2
P_u P_v : Formula
u v : VarName
h1_left : ¬isFreeIn v P_u
h1_right_left : ¬isFreeIn u P_v
h1_right_right_left : fastAdmits u v P_u
h1_right_right_right_left : fastAdmits v u P_v
h1_right_right_right_right_left : P_v = fastReplaceFree u v P_u
h1_right_right_right_right_right : P_u = fastReplaceFree v u P_v
⊢ ¬isFreeIn v (forall_ u P_u)
|
case a.a.h1.h2
P_u P_v : Formula
u v : VarName
h1_left : ¬isFreeIn v P_u
h1_right_left : ¬isFreeIn u P_v
h1_right_right_left : fastAdmits u v P_u
h1_right_right_right_left : fastAdmits v u P_v
h1_right_right_right_right_left : P_v = fastReplaceFree u v P_u
h1_right_right_right_right_right : P_u = fastReplaceFree v u P_v
⊢ ¬(¬v = u ∧ isFreeIn v P_u)
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Margaris/Fol.lean
|
FOL.NV.T_18_6
|
[983, 1]
|
[1036, 36]
|
simp
|
case a.a.h1.h2
P_u P_v : Formula
u v : VarName
h1_left : ¬isFreeIn v P_u
h1_right_left : ¬isFreeIn u P_v
h1_right_right_left : fastAdmits u v P_u
h1_right_right_right_left : fastAdmits v u P_v
h1_right_right_right_right_left : P_v = fastReplaceFree u v P_u
h1_right_right_right_right_right : P_u = fastReplaceFree v u P_v
⊢ ¬(¬v = u ∧ isFreeIn v P_u)
|
case a.a.h1.h2
P_u P_v : Formula
u v : VarName
h1_left : ¬isFreeIn v P_u
h1_right_left : ¬isFreeIn u P_v
h1_right_right_left : fastAdmits u v P_u
h1_right_right_right_left : fastAdmits v u P_v
h1_right_right_right_right_left : P_v = fastReplaceFree u v P_u
h1_right_right_right_right_right : P_u = fastReplaceFree v u P_v
⊢ ¬v = u → ¬isFreeIn v P_u
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Margaris/Fol.lean
|
FOL.NV.T_18_6
|
[983, 1]
|
[1036, 36]
|
intro _
|
case a.a.h1.h2
P_u P_v : Formula
u v : VarName
h1_left : ¬isFreeIn v P_u
h1_right_left : ¬isFreeIn u P_v
h1_right_right_left : fastAdmits u v P_u
h1_right_right_right_left : fastAdmits v u P_v
h1_right_right_right_right_left : P_v = fastReplaceFree u v P_u
h1_right_right_right_right_right : P_u = fastReplaceFree v u P_v
⊢ ¬v = u → ¬isFreeIn v P_u
|
case a.a.h1.h2
P_u P_v : Formula
u v : VarName
h1_left : ¬isFreeIn v P_u
h1_right_left : ¬isFreeIn u P_v
h1_right_right_left : fastAdmits u v P_u
h1_right_right_right_left : fastAdmits v u P_v
h1_right_right_right_right_left : P_v = fastReplaceFree u v P_u
h1_right_right_right_right_right : P_u = fastReplaceFree v u P_v
a✝ : ¬v = u
⊢ ¬isFreeIn v P_u
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Margaris/Fol.lean
|
FOL.NV.T_18_6
|
[983, 1]
|
[1036, 36]
|
exact h1_left
|
case a.a.h1.h2
P_u P_v : Formula
u v : VarName
h1_left : ¬isFreeIn v P_u
h1_right_left : ¬isFreeIn u P_v
h1_right_right_left : fastAdmits u v P_u
h1_right_right_right_left : fastAdmits v u P_v
h1_right_right_right_right_left : P_v = fastReplaceFree u v P_u
h1_right_right_right_right_right : P_u = fastReplaceFree v u P_v
a✝ : ¬v = u
⊢ ¬isFreeIn v P_u
|
no goals
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Margaris/Fol.lean
|
FOL.NV.T_18_6
|
[983, 1]
|
[1036, 36]
|
apply deduction_theorem
|
case a
P_u P_v : Formula
u v : VarName
h1_left : ¬isFreeIn v P_u
h1_right_left : ¬isFreeIn u P_v
h1_right_right_left : fastAdmits u v P_u
h1_right_right_right_left : fastAdmits v u P_v
h1_right_right_right_right_left : P_v = fastReplaceFree u v P_u
h1_right_right_right_right_right : P_u = fastReplaceFree v u P_v
⊢ IsDeduct ∅ ((forall_ v P_v).imp_ (forall_ u P_u))
|
case a.h1
P_u P_v : Formula
u v : VarName
h1_left : ¬isFreeIn v P_u
h1_right_left : ¬isFreeIn u P_v
h1_right_right_left : fastAdmits u v P_u
h1_right_right_right_left : fastAdmits v u P_v
h1_right_right_right_right_left : P_v = fastReplaceFree u v P_u
h1_right_right_right_right_right : P_u = fastReplaceFree v u P_v
⊢ IsDeduct (∅ ∪ {forall_ v P_v}) (forall_ u P_u)
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Margaris/Fol.lean
|
FOL.NV.T_18_6
|
[983, 1]
|
[1036, 36]
|
simp
|
case a.h1
P_u P_v : Formula
u v : VarName
h1_left : ¬isFreeIn v P_u
h1_right_left : ¬isFreeIn u P_v
h1_right_right_left : fastAdmits u v P_u
h1_right_right_right_left : fastAdmits v u P_v
h1_right_right_right_right_left : P_v = fastReplaceFree u v P_u
h1_right_right_right_right_right : P_u = fastReplaceFree v u P_v
⊢ IsDeduct (∅ ∪ {forall_ v P_v}) (forall_ u P_u)
|
case a.h1
P_u P_v : Formula
u v : VarName
h1_left : ¬isFreeIn v P_u
h1_right_left : ¬isFreeIn u P_v
h1_right_right_left : fastAdmits u v P_u
h1_right_right_right_left : fastAdmits v u P_v
h1_right_right_right_right_left : P_v = fastReplaceFree u v P_u
h1_right_right_right_right_right : P_u = fastReplaceFree v u P_v
⊢ IsDeduct {forall_ v P_v} (forall_ u P_u)
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Margaris/Fol.lean
|
FOL.NV.T_18_6
|
[983, 1]
|
[1036, 36]
|
apply generalization
|
case a.h1
P_u P_v : Formula
u v : VarName
h1_left : ¬isFreeIn v P_u
h1_right_left : ¬isFreeIn u P_v
h1_right_right_left : fastAdmits u v P_u
h1_right_right_right_left : fastAdmits v u P_v
h1_right_right_right_right_left : P_v = fastReplaceFree u v P_u
h1_right_right_right_right_right : P_u = fastReplaceFree v u P_v
⊢ IsDeduct {forall_ v P_v} (forall_ u P_u)
|
case a.h1.h1
P_u P_v : Formula
u v : VarName
h1_left : ¬isFreeIn v P_u
h1_right_left : ¬isFreeIn u P_v
h1_right_right_left : fastAdmits u v P_u
h1_right_right_right_left : fastAdmits v u P_v
h1_right_right_right_right_left : P_v = fastReplaceFree u v P_u
h1_right_right_right_right_right : P_u = fastReplaceFree v u P_v
⊢ IsDeduct {forall_ v P_v} P_u
case a.h1.h2
P_u P_v : Formula
u v : VarName
h1_left : ¬isFreeIn v P_u
h1_right_left : ¬isFreeIn u P_v
h1_right_right_left : fastAdmits u v P_u
h1_right_right_right_left : fastAdmits v u P_v
h1_right_right_right_right_left : P_v = fastReplaceFree u v P_u
h1_right_right_right_right_right : P_u = fastReplaceFree v u P_v
⊢ ∀ H ∈ {forall_ v P_v}, ¬isFreeIn u H
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Margaris/Fol.lean
|
FOL.NV.T_18_6
|
[983, 1]
|
[1036, 36]
|
subst h1_right_right_right_right_right
|
case a.h1.h1
P_u P_v : Formula
u v : VarName
h1_left : ¬isFreeIn v P_u
h1_right_left : ¬isFreeIn u P_v
h1_right_right_left : fastAdmits u v P_u
h1_right_right_right_left : fastAdmits v u P_v
h1_right_right_right_right_left : P_v = fastReplaceFree u v P_u
h1_right_right_right_right_right : P_u = fastReplaceFree v u P_v
⊢ IsDeduct {forall_ v P_v} P_u
|
case a.h1.h1
P_v : Formula
u v : VarName
h1_right_left : ¬isFreeIn u P_v
h1_right_right_right_left : fastAdmits v u P_v
h1_left : ¬isFreeIn v (fastReplaceFree v u P_v)
h1_right_right_left : fastAdmits u v (fastReplaceFree v u P_v)
h1_right_right_right_right_left : P_v = fastReplaceFree u v (fastReplaceFree v u P_v)
⊢ IsDeduct {forall_ v P_v} (fastReplaceFree v u P_v)
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Margaris/Fol.lean
|
FOL.NV.T_18_6
|
[983, 1]
|
[1036, 36]
|
apply spec
|
case a.h1.h1
P_v : Formula
u v : VarName
h1_right_left : ¬isFreeIn u P_v
h1_right_right_right_left : fastAdmits v u P_v
h1_left : ¬isFreeIn v (fastReplaceFree v u P_v)
h1_right_right_left : fastAdmits u v (fastReplaceFree v u P_v)
h1_right_right_right_right_left : P_v = fastReplaceFree u v (fastReplaceFree v u P_v)
⊢ IsDeduct {forall_ v P_v} (fastReplaceFree v u P_v)
|
case a.h1.h1.h1
P_v : Formula
u v : VarName
h1_right_left : ¬isFreeIn u P_v
h1_right_right_right_left : fastAdmits v u P_v
h1_left : ¬isFreeIn v (fastReplaceFree v u P_v)
h1_right_right_left : fastAdmits u v (fastReplaceFree v u P_v)
h1_right_right_right_right_left : P_v = fastReplaceFree u v (fastReplaceFree v u P_v)
⊢ IsDeduct {forall_ v P_v} (forall_ v P_v)
case a.h1.h1.h2
P_v : Formula
u v : VarName
h1_right_left : ¬isFreeIn u P_v
h1_right_right_right_left : fastAdmits v u P_v
h1_left : ¬isFreeIn v (fastReplaceFree v u P_v)
h1_right_right_left : fastAdmits u v (fastReplaceFree v u P_v)
h1_right_right_right_right_left : P_v = fastReplaceFree u v (fastReplaceFree v u P_v)
⊢ fastAdmits v u P_v
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Margaris/Fol.lean
|
FOL.NV.T_18_6
|
[983, 1]
|
[1036, 36]
|
apply IsDeduct.assume_
|
case a.h1.h1.h1
P_v : Formula
u v : VarName
h1_right_left : ¬isFreeIn u P_v
h1_right_right_right_left : fastAdmits v u P_v
h1_left : ¬isFreeIn v (fastReplaceFree v u P_v)
h1_right_right_left : fastAdmits u v (fastReplaceFree v u P_v)
h1_right_right_right_right_left : P_v = fastReplaceFree u v (fastReplaceFree v u P_v)
⊢ IsDeduct {forall_ v P_v} (forall_ v P_v)
|
case a.h1.h1.h1.a
P_v : Formula
u v : VarName
h1_right_left : ¬isFreeIn u P_v
h1_right_right_right_left : fastAdmits v u P_v
h1_left : ¬isFreeIn v (fastReplaceFree v u P_v)
h1_right_right_left : fastAdmits u v (fastReplaceFree v u P_v)
h1_right_right_right_right_left : P_v = fastReplaceFree u v (fastReplaceFree v u P_v)
⊢ forall_ v P_v ∈ {forall_ v P_v}
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Margaris/Fol.lean
|
FOL.NV.T_18_6
|
[983, 1]
|
[1036, 36]
|
simp
|
case a.h1.h1.h1.a
P_v : Formula
u v : VarName
h1_right_left : ¬isFreeIn u P_v
h1_right_right_right_left : fastAdmits v u P_v
h1_left : ¬isFreeIn v (fastReplaceFree v u P_v)
h1_right_right_left : fastAdmits u v (fastReplaceFree v u P_v)
h1_right_right_right_right_left : P_v = fastReplaceFree u v (fastReplaceFree v u P_v)
⊢ forall_ v P_v ∈ {forall_ v P_v}
|
no goals
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Margaris/Fol.lean
|
FOL.NV.T_18_6
|
[983, 1]
|
[1036, 36]
|
exact h1_right_right_right_left
|
case a.h1.h1.h2
P_v : Formula
u v : VarName
h1_right_left : ¬isFreeIn u P_v
h1_right_right_right_left : fastAdmits v u P_v
h1_left : ¬isFreeIn v (fastReplaceFree v u P_v)
h1_right_right_left : fastAdmits u v (fastReplaceFree v u P_v)
h1_right_right_right_right_left : P_v = fastReplaceFree u v (fastReplaceFree v u P_v)
⊢ fastAdmits v u P_v
|
no goals
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Margaris/Fol.lean
|
FOL.NV.T_18_6
|
[983, 1]
|
[1036, 36]
|
intro H a1
|
case a.h1.h2
P_u P_v : Formula
u v : VarName
h1_left : ¬isFreeIn v P_u
h1_right_left : ¬isFreeIn u P_v
h1_right_right_left : fastAdmits u v P_u
h1_right_right_right_left : fastAdmits v u P_v
h1_right_right_right_right_left : P_v = fastReplaceFree u v P_u
h1_right_right_right_right_right : P_u = fastReplaceFree v u P_v
⊢ ∀ H ∈ {forall_ v P_v}, ¬isFreeIn u H
|
case a.h1.h2
P_u P_v : Formula
u v : VarName
h1_left : ¬isFreeIn v P_u
h1_right_left : ¬isFreeIn u P_v
h1_right_right_left : fastAdmits u v P_u
h1_right_right_right_left : fastAdmits v u P_v
h1_right_right_right_right_left : P_v = fastReplaceFree u v P_u
h1_right_right_right_right_right : P_u = fastReplaceFree v u P_v
H : Formula
a1 : H ∈ {forall_ v P_v}
⊢ ¬isFreeIn u H
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Margaris/Fol.lean
|
FOL.NV.T_18_6
|
[983, 1]
|
[1036, 36]
|
simp at a1
|
case a.h1.h2
P_u P_v : Formula
u v : VarName
h1_left : ¬isFreeIn v P_u
h1_right_left : ¬isFreeIn u P_v
h1_right_right_left : fastAdmits u v P_u
h1_right_right_right_left : fastAdmits v u P_v
h1_right_right_right_right_left : P_v = fastReplaceFree u v P_u
h1_right_right_right_right_right : P_u = fastReplaceFree v u P_v
H : Formula
a1 : H ∈ {forall_ v P_v}
⊢ ¬isFreeIn u H
|
case a.h1.h2
P_u P_v : Formula
u v : VarName
h1_left : ¬isFreeIn v P_u
h1_right_left : ¬isFreeIn u P_v
h1_right_right_left : fastAdmits u v P_u
h1_right_right_right_left : fastAdmits v u P_v
h1_right_right_right_right_left : P_v = fastReplaceFree u v P_u
h1_right_right_right_right_right : P_u = fastReplaceFree v u P_v
H : Formula
a1 : H = forall_ v P_v
⊢ ¬isFreeIn u H
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Margaris/Fol.lean
|
FOL.NV.T_18_6
|
[983, 1]
|
[1036, 36]
|
subst a1
|
case a.h1.h2
P_u P_v : Formula
u v : VarName
h1_left : ¬isFreeIn v P_u
h1_right_left : ¬isFreeIn u P_v
h1_right_right_left : fastAdmits u v P_u
h1_right_right_right_left : fastAdmits v u P_v
h1_right_right_right_right_left : P_v = fastReplaceFree u v P_u
h1_right_right_right_right_right : P_u = fastReplaceFree v u P_v
H : Formula
a1 : H = forall_ v P_v
⊢ ¬isFreeIn u H
|
case a.h1.h2
P_u P_v : Formula
u v : VarName
h1_left : ¬isFreeIn v P_u
h1_right_left : ¬isFreeIn u P_v
h1_right_right_left : fastAdmits u v P_u
h1_right_right_right_left : fastAdmits v u P_v
h1_right_right_right_right_left : P_v = fastReplaceFree u v P_u
h1_right_right_right_right_right : P_u = fastReplaceFree v u P_v
⊢ ¬isFreeIn u (forall_ v P_v)
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Margaris/Fol.lean
|
FOL.NV.T_18_6
|
[983, 1]
|
[1036, 36]
|
simp only [isFreeIn]
|
case a.h1.h2
P_u P_v : Formula
u v : VarName
h1_left : ¬isFreeIn v P_u
h1_right_left : ¬isFreeIn u P_v
h1_right_right_left : fastAdmits u v P_u
h1_right_right_right_left : fastAdmits v u P_v
h1_right_right_right_right_left : P_v = fastReplaceFree u v P_u
h1_right_right_right_right_right : P_u = fastReplaceFree v u P_v
⊢ ¬isFreeIn u (forall_ v P_v)
|
case a.h1.h2
P_u P_v : Formula
u v : VarName
h1_left : ¬isFreeIn v P_u
h1_right_left : ¬isFreeIn u P_v
h1_right_right_left : fastAdmits u v P_u
h1_right_right_right_left : fastAdmits v u P_v
h1_right_right_right_right_left : P_v = fastReplaceFree u v P_u
h1_right_right_right_right_right : P_u = fastReplaceFree v u P_v
⊢ ¬(¬u = v ∧ isFreeIn u P_v)
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Margaris/Fol.lean
|
FOL.NV.T_18_6
|
[983, 1]
|
[1036, 36]
|
simp
|
case a.h1.h2
P_u P_v : Formula
u v : VarName
h1_left : ¬isFreeIn v P_u
h1_right_left : ¬isFreeIn u P_v
h1_right_right_left : fastAdmits u v P_u
h1_right_right_right_left : fastAdmits v u P_v
h1_right_right_right_right_left : P_v = fastReplaceFree u v P_u
h1_right_right_right_right_right : P_u = fastReplaceFree v u P_v
⊢ ¬(¬u = v ∧ isFreeIn u P_v)
|
case a.h1.h2
P_u P_v : Formula
u v : VarName
h1_left : ¬isFreeIn v P_u
h1_right_left : ¬isFreeIn u P_v
h1_right_right_left : fastAdmits u v P_u
h1_right_right_right_left : fastAdmits v u P_v
h1_right_right_right_right_left : P_v = fastReplaceFree u v P_u
h1_right_right_right_right_right : P_u = fastReplaceFree v u P_v
⊢ ¬u = v → ¬isFreeIn u P_v
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Margaris/Fol.lean
|
FOL.NV.T_18_6
|
[983, 1]
|
[1036, 36]
|
intro _
|
case a.h1.h2
P_u P_v : Formula
u v : VarName
h1_left : ¬isFreeIn v P_u
h1_right_left : ¬isFreeIn u P_v
h1_right_right_left : fastAdmits u v P_u
h1_right_right_right_left : fastAdmits v u P_v
h1_right_right_right_right_left : P_v = fastReplaceFree u v P_u
h1_right_right_right_right_right : P_u = fastReplaceFree v u P_v
⊢ ¬u = v → ¬isFreeIn u P_v
|
case a.h1.h2
P_u P_v : Formula
u v : VarName
h1_left : ¬isFreeIn v P_u
h1_right_left : ¬isFreeIn u P_v
h1_right_right_left : fastAdmits u v P_u
h1_right_right_right_left : fastAdmits v u P_v
h1_right_right_right_right_left : P_v = fastReplaceFree u v P_u
h1_right_right_right_right_right : P_u = fastReplaceFree v u P_v
a✝ : ¬u = v
⊢ ¬isFreeIn u P_v
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Margaris/Fol.lean
|
FOL.NV.T_18_6
|
[983, 1]
|
[1036, 36]
|
exact h1_right_left
|
case a.h1.h2
P_u P_v : Formula
u v : VarName
h1_left : ¬isFreeIn v P_u
h1_right_left : ¬isFreeIn u P_v
h1_right_right_left : fastAdmits u v P_u
h1_right_right_right_left : fastAdmits v u P_v
h1_right_right_right_right_left : P_v = fastReplaceFree u v P_u
h1_right_right_right_right_right : P_u = fastReplaceFree v u P_v
a✝ : ¬u = v
⊢ ¬isFreeIn u P_v
|
no goals
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Margaris/Fol.lean
|
FOL.NV.T_18_7
|
[1040, 1]
|
[1051, 13]
|
apply C_18_4 (forall_ u P_u) (forall_ v P_v) Q Q' Δ h2
|
P_u P_v Q Q' : Formula
u v : VarName
Δ : Set Formula
h1 : IsDeduct Δ Q
h2 : IsReplOfFormulaInFormula (forall_ u P_u) (forall_ v P_v) Q Q'
h3 : Similar P_u P_v u v
⊢ IsDeduct Δ Q'
|
case h2
P_u P_v Q Q' : Formula
u v : VarName
Δ : Set Formula
h1 : IsDeduct Δ Q
h2 : IsReplOfFormulaInFormula (forall_ u P_u) (forall_ v P_v) Q Q'
h3 : Similar P_u P_v u v
⊢ IsProof ((forall_ u P_u).iff_ (forall_ v P_v))
case h3
P_u P_v Q Q' : Formula
u v : VarName
Δ : Set Formula
h1 : IsDeduct Δ Q
h2 : IsReplOfFormulaInFormula (forall_ u P_u) (forall_ v P_v) Q Q'
h3 : Similar P_u P_v u v
⊢ IsDeduct Δ Q
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Margaris/Fol.lean
|
FOL.NV.T_18_7
|
[1040, 1]
|
[1051, 13]
|
apply T_18_6 P_u P_v u v h3
|
case h2
P_u P_v Q Q' : Formula
u v : VarName
Δ : Set Formula
h1 : IsDeduct Δ Q
h2 : IsReplOfFormulaInFormula (forall_ u P_u) (forall_ v P_v) Q Q'
h3 : Similar P_u P_v u v
⊢ IsProof ((forall_ u P_u).iff_ (forall_ v P_v))
|
no goals
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Margaris/Fol.lean
|
FOL.NV.T_18_7
|
[1040, 1]
|
[1051, 13]
|
exact h1
|
case h3
P_u P_v Q Q' : Formula
u v : VarName
Δ : Set Formula
h1 : IsDeduct Δ Q
h2 : IsReplOfFormulaInFormula (forall_ u P_u) (forall_ v P_v) Q Q'
h3 : Similar P_u P_v u v
⊢ IsDeduct Δ Q
|
no goals
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Margaris/Fol.lean
|
FOL.NV.similar_not
|
[1054, 1]
|
[1065, 8]
|
simp only [Similar] at *
|
P_u P_v : Formula
u v : VarName
h1 : Similar P_u P_v u v
⊢ Similar P_u.not_ P_v.not_ u v
|
P_u P_v : Formula
u v : VarName
h1 :
¬isFreeIn v P_u ∧
¬isFreeIn u P_v ∧
fastAdmits u v P_u ∧ fastAdmits v u P_v ∧ P_v = fastReplaceFree u v P_u ∧ P_u = fastReplaceFree v u P_v
⊢ ¬isFreeIn v P_u.not_ ∧
¬isFreeIn u P_v.not_ ∧
fastAdmits u v P_u.not_ ∧
fastAdmits v u P_v.not_ ∧ P_v.not_ = fastReplaceFree u v P_u.not_ ∧ P_u.not_ = fastReplaceFree v u P_v.not_
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Margaris/Fol.lean
|
FOL.NV.similar_not
|
[1054, 1]
|
[1065, 8]
|
simp only [isFreeIn] at *
|
P_u P_v : Formula
u v : VarName
h1 :
¬isFreeIn v P_u ∧
¬isFreeIn u P_v ∧
fastAdmits u v P_u ∧ fastAdmits v u P_v ∧ P_v = fastReplaceFree u v P_u ∧ P_u = fastReplaceFree v u P_v
⊢ ¬isFreeIn v P_u.not_ ∧
¬isFreeIn u P_v.not_ ∧
fastAdmits u v P_u.not_ ∧
fastAdmits v u P_v.not_ ∧ P_v.not_ = fastReplaceFree u v P_u.not_ ∧ P_u.not_ = fastReplaceFree v u P_v.not_
|
P_u P_v : Formula
u v : VarName
h1 :
¬isFreeIn v P_u ∧
¬isFreeIn u P_v ∧
fastAdmits u v P_u ∧ fastAdmits v u P_v ∧ P_v = fastReplaceFree u v P_u ∧ P_u = fastReplaceFree v u P_v
⊢ ¬isFreeIn v P_u ∧
¬isFreeIn u P_v ∧
fastAdmits u v P_u.not_ ∧
fastAdmits v u P_v.not_ ∧ P_v.not_ = fastReplaceFree u v P_u.not_ ∧ P_u.not_ = fastReplaceFree v u P_v.not_
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Margaris/Fol.lean
|
FOL.NV.similar_not
|
[1054, 1]
|
[1065, 8]
|
simp only [fastAdmits] at *
|
P_u P_v : Formula
u v : VarName
h1 :
¬isFreeIn v P_u ∧
¬isFreeIn u P_v ∧
fastAdmits u v P_u ∧ fastAdmits v u P_v ∧ P_v = fastReplaceFree u v P_u ∧ P_u = fastReplaceFree v u P_v
⊢ ¬isFreeIn v P_u ∧
¬isFreeIn u P_v ∧
fastAdmits u v P_u.not_ ∧
fastAdmits v u P_v.not_ ∧ P_v.not_ = fastReplaceFree u v P_u.not_ ∧ P_u.not_ = fastReplaceFree v u P_v.not_
|
P_u P_v : Formula
u v : VarName
h1 :
¬isFreeIn v P_u ∧
¬isFreeIn u P_v ∧
fastAdmitsAux u v ∅ P_u ∧ fastAdmitsAux v u ∅ P_v ∧ P_v = fastReplaceFree u v P_u ∧ P_u = fastReplaceFree v u P_v
⊢ ¬isFreeIn v P_u ∧
¬isFreeIn u P_v ∧
fastAdmitsAux u v ∅ P_u.not_ ∧
fastAdmitsAux v u ∅ P_v.not_ ∧ P_v.not_ = fastReplaceFree u v P_u.not_ ∧ P_u.not_ = fastReplaceFree v u P_v.not_
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Margaris/Fol.lean
|
FOL.NV.similar_not
|
[1054, 1]
|
[1065, 8]
|
simp only [fastAdmitsAux] at *
|
P_u P_v : Formula
u v : VarName
h1 :
¬isFreeIn v P_u ∧
¬isFreeIn u P_v ∧
fastAdmitsAux u v ∅ P_u ∧ fastAdmitsAux v u ∅ P_v ∧ P_v = fastReplaceFree u v P_u ∧ P_u = fastReplaceFree v u P_v
⊢ ¬isFreeIn v P_u ∧
¬isFreeIn u P_v ∧
fastAdmitsAux u v ∅ P_u.not_ ∧
fastAdmitsAux v u ∅ P_v.not_ ∧ P_v.not_ = fastReplaceFree u v P_u.not_ ∧ P_u.not_ = fastReplaceFree v u P_v.not_
|
P_u P_v : Formula
u v : VarName
h1 :
¬isFreeIn v P_u ∧
¬isFreeIn u P_v ∧
fastAdmitsAux u v ∅ P_u ∧ fastAdmitsAux v u ∅ P_v ∧ P_v = fastReplaceFree u v P_u ∧ P_u = fastReplaceFree v u P_v
⊢ ¬isFreeIn v P_u ∧
¬isFreeIn u P_v ∧
fastAdmitsAux u v ∅ P_u ∧
fastAdmitsAux v u ∅ P_v ∧ P_v.not_ = fastReplaceFree u v P_u.not_ ∧ P_u.not_ = fastReplaceFree v u P_v.not_
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Margaris/Fol.lean
|
FOL.NV.similar_not
|
[1054, 1]
|
[1065, 8]
|
simp only [fastReplaceFree] at *
|
P_u P_v : Formula
u v : VarName
h1 :
¬isFreeIn v P_u ∧
¬isFreeIn u P_v ∧
fastAdmitsAux u v ∅ P_u ∧ fastAdmitsAux v u ∅ P_v ∧ P_v = fastReplaceFree u v P_u ∧ P_u = fastReplaceFree v u P_v
⊢ ¬isFreeIn v P_u ∧
¬isFreeIn u P_v ∧
fastAdmitsAux u v ∅ P_u ∧
fastAdmitsAux v u ∅ P_v ∧ P_v.not_ = fastReplaceFree u v P_u.not_ ∧ P_u.not_ = fastReplaceFree v u P_v.not_
|
P_u P_v : Formula
u v : VarName
h1 :
¬isFreeIn v P_u ∧
¬isFreeIn u P_v ∧
fastAdmitsAux u v ∅ P_u ∧ fastAdmitsAux v u ∅ P_v ∧ P_v = fastReplaceFree u v P_u ∧ P_u = fastReplaceFree v u P_v
⊢ ¬isFreeIn v P_u ∧
¬isFreeIn u P_v ∧
fastAdmitsAux u v ∅ P_u ∧
fastAdmitsAux v u ∅ P_v ∧ P_v.not_ = (fastReplaceFree u v P_u).not_ ∧ P_u.not_ = (fastReplaceFree v u P_v).not_
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Margaris/Fol.lean
|
FOL.NV.similar_not
|
[1054, 1]
|
[1065, 8]
|
tauto
|
P_u P_v : Formula
u v : VarName
h1 :
¬isFreeIn v P_u ∧
¬isFreeIn u P_v ∧
fastAdmitsAux u v ∅ P_u ∧ fastAdmitsAux v u ∅ P_v ∧ P_v = fastReplaceFree u v P_u ∧ P_u = fastReplaceFree v u P_v
⊢ ¬isFreeIn v P_u ∧
¬isFreeIn u P_v ∧
fastAdmitsAux u v ∅ P_u ∧
fastAdmitsAux v u ∅ P_v ∧ P_v.not_ = (fastReplaceFree u v P_u).not_ ∧ P_u.not_ = (fastReplaceFree v u P_v).not_
|
no goals
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Margaris/Fol.lean
|
FOL.NV.T_18_8
|
[1068, 1]
|
[1083, 37]
|
simp only [def_exists_]
|
P_u P_v : Formula
u v : VarName
h1 : Similar P_u P_v u v
⊢ IsProof ((exists_ u P_u).iff_ (exists_ v P_v))
|
P_u P_v : Formula
u v : VarName
h1 : Similar P_u P_v u v
⊢ IsProof ((forall_ u P_u.not_).not_.iff_ (forall_ v P_v.not_).not_)
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Margaris/Fol.lean
|
FOL.NV.T_18_8
|
[1068, 1]
|
[1083, 37]
|
apply IsDeduct.mp_ ((forall_ u P_u.not_).iff_ (forall_ v P_v.not_))
|
P_u P_v : Formula
u v : VarName
h1 : Similar P_u P_v u v
⊢ IsProof ((forall_ u P_u.not_).not_.iff_ (forall_ v P_v.not_).not_)
|
case a
P_u P_v : Formula
u v : VarName
h1 : Similar P_u P_v u v
⊢ IsDeduct ∅
(((forall_ u P_u.not_).iff_ (forall_ v P_v.not_)).imp_ ((forall_ u P_u.not_).not_.iff_ (forall_ v P_v.not_).not_))
case a
P_u P_v : Formula
u v : VarName
h1 : Similar P_u P_v u v
⊢ IsDeduct ∅ ((forall_ u P_u.not_).iff_ (forall_ v P_v.not_))
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Margaris/Fol.lean
|
FOL.NV.T_18_8
|
[1068, 1]
|
[1083, 37]
|
simp only [def_iff_]
|
case a
P_u P_v : Formula
u v : VarName
h1 : Similar P_u P_v u v
⊢ IsDeduct ∅
(((forall_ u P_u.not_).iff_ (forall_ v P_v.not_)).imp_ ((forall_ u P_u.not_).not_.iff_ (forall_ v P_v.not_).not_))
|
case a
P_u P_v : Formula
u v : VarName
h1 : Similar P_u P_v u v
⊢ IsDeduct ∅
((((forall_ u P_u.not_).imp_ (forall_ v P_v.not_)).and_ ((forall_ v P_v.not_).imp_ (forall_ u P_u.not_))).imp_
(((forall_ u P_u.not_).not_.imp_ (forall_ v P_v.not_).not_).and_
((forall_ v P_v.not_).not_.imp_ (forall_ u P_u.not_).not_)))
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Margaris/Fol.lean
|
FOL.NV.T_18_8
|
[1068, 1]
|
[1083, 37]
|
simp only [def_and_]
|
case a
P_u P_v : Formula
u v : VarName
h1 : Similar P_u P_v u v
⊢ IsDeduct ∅
((((forall_ u P_u.not_).imp_ (forall_ v P_v.not_)).and_ ((forall_ v P_v.not_).imp_ (forall_ u P_u.not_))).imp_
(((forall_ u P_u.not_).not_.imp_ (forall_ v P_v.not_).not_).and_
((forall_ v P_v.not_).not_.imp_ (forall_ u P_u.not_).not_)))
|
case a
P_u P_v : Formula
u v : VarName
h1 : Similar P_u P_v u v
⊢ IsDeduct ∅
((((forall_ u P_u.not_).imp_ (forall_ v P_v.not_)).imp_
((forall_ v P_v.not_).imp_ (forall_ u P_u.not_)).not_).not_.imp_
(((forall_ u P_u.not_).not_.imp_ (forall_ v P_v.not_).not_).imp_
((forall_ v P_v.not_).not_.imp_ (forall_ u P_u.not_).not_).not_).not_)
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Margaris/Fol.lean
|
FOL.NV.T_18_8
|
[1068, 1]
|
[1083, 37]
|
SC
|
case a
P_u P_v : Formula
u v : VarName
h1 : Similar P_u P_v u v
⊢ IsDeduct ∅
((((forall_ u P_u.not_).imp_ (forall_ v P_v.not_)).imp_
((forall_ v P_v.not_).imp_ (forall_ u P_u.not_)).not_).not_.imp_
(((forall_ u P_u.not_).not_.imp_ (forall_ v P_v.not_).not_).imp_
((forall_ v P_v.not_).not_.imp_ (forall_ u P_u.not_).not_).not_).not_)
|
no goals
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Margaris/Fol.lean
|
FOL.NV.T_18_8
|
[1068, 1]
|
[1083, 37]
|
apply T_18_6
|
case a
P_u P_v : Formula
u v : VarName
h1 : Similar P_u P_v u v
⊢ IsDeduct ∅ ((forall_ u P_u.not_).iff_ (forall_ v P_v.not_))
|
case a.h1
P_u P_v : Formula
u v : VarName
h1 : Similar P_u P_v u v
⊢ Similar P_u.not_ P_v.not_ u v
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Margaris/Fol.lean
|
FOL.NV.T_18_8
|
[1068, 1]
|
[1083, 37]
|
exact similar_not P_u P_v u v h1
|
case a.h1
P_u P_v : Formula
u v : VarName
h1 : Similar P_u P_v u v
⊢ Similar P_u.not_ P_v.not_ u v
|
no goals
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Margaris/Fol.lean
|
FOL.NV.T18_9
|
[1086, 1]
|
[1098, 13]
|
apply C_18_4 (exists_ u P_u) (exists_ v P_v) Q Q' Δ h2
|
Q Q' P_u P_v : Formula
u v : VarName
Δ : Set Formula
h1 : IsDeduct Δ Q
h2 : IsReplOfFormulaInFormula (exists_ u P_u) (exists_ v P_v) Q Q'
h3 : Similar P_u P_v u v
⊢ IsDeduct Δ Q'
|
case h2
Q Q' P_u P_v : Formula
u v : VarName
Δ : Set Formula
h1 : IsDeduct Δ Q
h2 : IsReplOfFormulaInFormula (exists_ u P_u) (exists_ v P_v) Q Q'
h3 : Similar P_u P_v u v
⊢ IsProof ((exists_ u P_u).iff_ (exists_ v P_v))
case h3
Q Q' P_u P_v : Formula
u v : VarName
Δ : Set Formula
h1 : IsDeduct Δ Q
h2 : IsReplOfFormulaInFormula (exists_ u P_u) (exists_ v P_v) Q Q'
h3 : Similar P_u P_v u v
⊢ IsDeduct Δ Q
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Margaris/Fol.lean
|
FOL.NV.T18_9
|
[1086, 1]
|
[1098, 13]
|
exact T_18_8 P_u P_v u v h3
|
case h2
Q Q' P_u P_v : Formula
u v : VarName
Δ : Set Formula
h1 : IsDeduct Δ Q
h2 : IsReplOfFormulaInFormula (exists_ u P_u) (exists_ v P_v) Q Q'
h3 : Similar P_u P_v u v
⊢ IsProof ((exists_ u P_u).iff_ (exists_ v P_v))
|
no goals
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Margaris/Fol.lean
|
FOL.NV.T18_9
|
[1086, 1]
|
[1098, 13]
|
exact h1
|
case h3
Q Q' P_u P_v : Formula
u v : VarName
Δ : Set Formula
h1 : IsDeduct Δ Q
h2 : IsReplOfFormulaInFormula (exists_ u P_u) (exists_ v P_v) Q Q'
h3 : Similar P_u P_v u v
⊢ IsDeduct Δ Q
|
no goals
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Margaris/Fol.lean
|
FOL.NV.T_19_1
|
[1101, 1]
|
[1119, 33]
|
apply IsDeduct.mp_ ((forall_ v P).imp_ P)
|
P : Formula
v : VarName
h1 : ¬isFreeIn v P
⊢ IsProof ((forall_ v P).iff_ P)
|
case a
P : Formula
v : VarName
h1 : ¬isFreeIn v P
⊢ IsDeduct ∅ (((forall_ v P).imp_ P).imp_ ((forall_ v P).iff_ P))
case a
P : Formula
v : VarName
h1 : ¬isFreeIn v P
⊢ IsDeduct ∅ ((forall_ v P).imp_ P)
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Margaris/Fol.lean
|
FOL.NV.T_19_1
|
[1101, 1]
|
[1119, 33]
|
apply IsDeduct.mp_ (P.imp_ (forall_ v P))
|
case a
P : Formula
v : VarName
h1 : ¬isFreeIn v P
⊢ IsDeduct ∅ (((forall_ v P).imp_ P).imp_ ((forall_ v P).iff_ P))
|
case a.a
P : Formula
v : VarName
h1 : ¬isFreeIn v P
⊢ IsDeduct ∅ ((P.imp_ (forall_ v P)).imp_ (((forall_ v P).imp_ P).imp_ ((forall_ v P).iff_ P)))
case a.a
P : Formula
v : VarName
h1 : ¬isFreeIn v P
⊢ IsDeduct ∅ (P.imp_ (forall_ v P))
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Margaris/Fol.lean
|
FOL.NV.T_19_1
|
[1101, 1]
|
[1119, 33]
|
simp only [def_iff_]
|
case a.a
P : Formula
v : VarName
h1 : ¬isFreeIn v P
⊢ IsDeduct ∅ ((P.imp_ (forall_ v P)).imp_ (((forall_ v P).imp_ P).imp_ ((forall_ v P).iff_ P)))
|
case a.a
P : Formula
v : VarName
h1 : ¬isFreeIn v P
⊢ IsDeduct ∅
((P.imp_ (forall_ v P)).imp_ (((forall_ v P).imp_ P).imp_ (((forall_ v P).imp_ P).and_ (P.imp_ (forall_ v P)))))
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Margaris/Fol.lean
|
FOL.NV.T_19_1
|
[1101, 1]
|
[1119, 33]
|
simp only [def_and_]
|
case a.a
P : Formula
v : VarName
h1 : ¬isFreeIn v P
⊢ IsDeduct ∅
((P.imp_ (forall_ v P)).imp_ (((forall_ v P).imp_ P).imp_ (((forall_ v P).imp_ P).and_ (P.imp_ (forall_ v P)))))
|
case a.a
P : Formula
v : VarName
h1 : ¬isFreeIn v P
⊢ IsDeduct ∅
((P.imp_ (forall_ v P)).imp_
(((forall_ v P).imp_ P).imp_ (((forall_ v P).imp_ P).imp_ (P.imp_ (forall_ v P)).not_).not_))
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Margaris/Fol.lean
|
FOL.NV.T_19_1
|
[1101, 1]
|
[1119, 33]
|
SC
|
case a.a
P : Formula
v : VarName
h1 : ¬isFreeIn v P
⊢ IsDeduct ∅
((P.imp_ (forall_ v P)).imp_
(((forall_ v P).imp_ P).imp_ (((forall_ v P).imp_ P).imp_ (P.imp_ (forall_ v P)).not_).not_))
|
no goals
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Margaris/Fol.lean
|
FOL.NV.T_19_1
|
[1101, 1]
|
[1119, 33]
|
apply IsDeduct.axiom_
|
case a.a
P : Formula
v : VarName
h1 : ¬isFreeIn v P
⊢ IsDeduct ∅ (P.imp_ (forall_ v P))
|
case a.a.a
P : Formula
v : VarName
h1 : ¬isFreeIn v P
⊢ IsAxiom (P.imp_ (forall_ v P))
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Margaris/Fol.lean
|
FOL.NV.T_19_1
|
[1101, 1]
|
[1119, 33]
|
exact IsAxiom.pred_3_ v P h1
|
case a.a.a
P : Formula
v : VarName
h1 : ¬isFreeIn v P
⊢ IsAxiom (P.imp_ (forall_ v P))
|
no goals
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Margaris/Fol.lean
|
FOL.NV.T_19_1
|
[1101, 1]
|
[1119, 33]
|
apply IsDeduct.axiom_
|
case a
P : Formula
v : VarName
h1 : ¬isFreeIn v P
⊢ IsDeduct ∅ ((forall_ v P).imp_ P)
|
case a.a
P : Formula
v : VarName
h1 : ¬isFreeIn v P
⊢ IsAxiom ((forall_ v P).imp_ P)
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Margaris/Fol.lean
|
FOL.NV.T_19_1
|
[1101, 1]
|
[1119, 33]
|
apply IsAxiom.pred_2_ v v P P
|
case a.a
P : Formula
v : VarName
h1 : ¬isFreeIn v P
⊢ IsAxiom ((forall_ v P).imp_ P)
|
case a.a.a
P : Formula
v : VarName
h1 : ¬isFreeIn v P
⊢ fastAdmits v v P
case a.a.a
P : Formula
v : VarName
h1 : ¬isFreeIn v P
⊢ fastReplaceFree v v P = P
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Margaris/Fol.lean
|
FOL.NV.T_19_1
|
[1101, 1]
|
[1119, 33]
|
apply fastAdmits_self
|
case a.a.a
P : Formula
v : VarName
h1 : ¬isFreeIn v P
⊢ fastAdmits v v P
|
no goals
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Margaris/Fol.lean
|
FOL.NV.T_19_1
|
[1101, 1]
|
[1119, 33]
|
apply fastReplaceFree_self
|
case a.a.a
P : Formula
v : VarName
h1 : ¬isFreeIn v P
⊢ fastReplaceFree v v P = P
|
no goals
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Margaris/Fol.lean
|
FOL.NV.T_19_2
|
[1122, 1]
|
[1135, 18]
|
apply IsDeduct.mp_ ((forall_ u (forall_ v P)).imp_ (forall_ v (forall_ u P)))
|
P : Formula
u v : VarName
⊢ IsProof ((forall_ u (forall_ v P)).iff_ (forall_ v (forall_ u P)))
|
case a
P : Formula
u v : VarName
⊢ IsDeduct ∅
(((forall_ u (forall_ v P)).imp_ (forall_ v (forall_ u P))).imp_
((forall_ u (forall_ v P)).iff_ (forall_ v (forall_ u P))))
case a
P : Formula
u v : VarName
⊢ IsDeduct ∅ ((forall_ u (forall_ v P)).imp_ (forall_ v (forall_ u P)))
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Margaris/Fol.lean
|
FOL.NV.T_19_2
|
[1122, 1]
|
[1135, 18]
|
apply IsDeduct.mp_ ((forall_ v (forall_ u P)).imp_ (forall_ u (forall_ v P)))
|
case a
P : Formula
u v : VarName
⊢ IsDeduct ∅
(((forall_ u (forall_ v P)).imp_ (forall_ v (forall_ u P))).imp_
((forall_ u (forall_ v P)).iff_ (forall_ v (forall_ u P))))
|
case a.a
P : Formula
u v : VarName
⊢ IsDeduct ∅
(((forall_ v (forall_ u P)).imp_ (forall_ u (forall_ v P))).imp_
(((forall_ u (forall_ v P)).imp_ (forall_ v (forall_ u P))).imp_
((forall_ u (forall_ v P)).iff_ (forall_ v (forall_ u P)))))
case a.a
P : Formula
u v : VarName
⊢ IsDeduct ∅ ((forall_ v (forall_ u P)).imp_ (forall_ u (forall_ v P)))
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Margaris/Fol.lean
|
FOL.NV.T_19_2
|
[1122, 1]
|
[1135, 18]
|
simp only [def_iff_]
|
case a.a
P : Formula
u v : VarName
⊢ IsDeduct ∅
(((forall_ v (forall_ u P)).imp_ (forall_ u (forall_ v P))).imp_
(((forall_ u (forall_ v P)).imp_ (forall_ v (forall_ u P))).imp_
((forall_ u (forall_ v P)).iff_ (forall_ v (forall_ u P)))))
|
case a.a
P : Formula
u v : VarName
⊢ IsDeduct ∅
(((forall_ v (forall_ u P)).imp_ (forall_ u (forall_ v P))).imp_
(((forall_ u (forall_ v P)).imp_ (forall_ v (forall_ u P))).imp_
(((forall_ u (forall_ v P)).imp_ (forall_ v (forall_ u P))).and_
((forall_ v (forall_ u P)).imp_ (forall_ u (forall_ v P))))))
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Margaris/Fol.lean
|
FOL.NV.T_19_2
|
[1122, 1]
|
[1135, 18]
|
simp only [def_and_]
|
case a.a
P : Formula
u v : VarName
⊢ IsDeduct ∅
(((forall_ v (forall_ u P)).imp_ (forall_ u (forall_ v P))).imp_
(((forall_ u (forall_ v P)).imp_ (forall_ v (forall_ u P))).imp_
(((forall_ u (forall_ v P)).imp_ (forall_ v (forall_ u P))).and_
((forall_ v (forall_ u P)).imp_ (forall_ u (forall_ v P))))))
|
case a.a
P : Formula
u v : VarName
⊢ IsDeduct ∅
(((forall_ v (forall_ u P)).imp_ (forall_ u (forall_ v P))).imp_
(((forall_ u (forall_ v P)).imp_ (forall_ v (forall_ u P))).imp_
(((forall_ u (forall_ v P)).imp_ (forall_ v (forall_ u P))).imp_
((forall_ v (forall_ u P)).imp_ (forall_ u (forall_ v P))).not_).not_))
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Margaris/Fol.lean
|
FOL.NV.T_19_2
|
[1122, 1]
|
[1135, 18]
|
SC
|
case a.a
P : Formula
u v : VarName
⊢ IsDeduct ∅
(((forall_ v (forall_ u P)).imp_ (forall_ u (forall_ v P))).imp_
(((forall_ u (forall_ v P)).imp_ (forall_ v (forall_ u P))).imp_
(((forall_ u (forall_ v P)).imp_ (forall_ v (forall_ u P))).imp_
((forall_ v (forall_ u P)).imp_ (forall_ u (forall_ v P))).not_).not_))
|
no goals
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Margaris/Fol.lean
|
FOL.NV.T_19_2
|
[1122, 1]
|
[1135, 18]
|
apply T_17_10
|
case a.a
P : Formula
u v : VarName
⊢ IsDeduct ∅ ((forall_ v (forall_ u P)).imp_ (forall_ u (forall_ v P)))
|
no goals
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Margaris/Fol.lean
|
FOL.NV.T_19_2
|
[1122, 1]
|
[1135, 18]
|
apply T_17_10
|
case a
P : Formula
u v : VarName
⊢ IsDeduct ∅ ((forall_ u (forall_ v P)).imp_ (forall_ v (forall_ u P)))
|
no goals
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Margaris/Fol.lean
|
FOL.NV.T_19_3
|
[1138, 1]
|
[1149, 5]
|
simp only [def_exists_]
|
P : Formula
v : VarName
⊢ IsProof ((forall_ v P.not_).iff_ (exists_ v P).not_)
|
P : Formula
v : VarName
⊢ IsProof ((forall_ v P.not_).iff_ (forall_ v P.not_).not_.not_)
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Margaris/Fol.lean
|
FOL.NV.T_19_3
|
[1138, 1]
|
[1149, 5]
|
simp only [def_iff_]
|
P : Formula
v : VarName
⊢ IsProof ((forall_ v P.not_).iff_ (forall_ v P.not_).not_.not_)
|
P : Formula
v : VarName
⊢ IsProof
(((forall_ v P.not_).imp_ (forall_ v P.not_).not_.not_).and_ ((forall_ v P.not_).not_.not_.imp_ (forall_ v P.not_)))
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Margaris/Fol.lean
|
FOL.NV.T_19_3
|
[1138, 1]
|
[1149, 5]
|
simp only [def_and_]
|
P : Formula
v : VarName
⊢ IsProof
(((forall_ v P.not_).imp_ (forall_ v P.not_).not_.not_).and_ ((forall_ v P.not_).not_.not_.imp_ (forall_ v P.not_)))
|
P : Formula
v : VarName
⊢ IsProof
(((forall_ v P.not_).imp_ (forall_ v P.not_).not_.not_).imp_
((forall_ v P.not_).not_.not_.imp_ (forall_ v P.not_)).not_).not_
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Margaris/Fol.lean
|
FOL.NV.T_19_3
|
[1138, 1]
|
[1149, 5]
|
SC
|
P : Formula
v : VarName
⊢ IsProof
(((forall_ v P.not_).imp_ (forall_ v P.not_).not_.not_).imp_
((forall_ v P.not_).not_.not_.imp_ (forall_ v P.not_)).not_).not_
|
no goals
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Margaris/Fol.lean
|
FOL.NV.T_19_4
|
[1152, 1]
|
[1183, 9]
|
apply deduction_theorem
|
P : Formula
u v : VarName
⊢ IsProof ((exists_ u (forall_ v P)).imp_ (forall_ v (exists_ u P)))
|
case h1
P : Formula
u v : VarName
⊢ IsDeduct (∅ ∪ {exists_ u (forall_ v P)}) (forall_ v (exists_ u P))
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Margaris/Fol.lean
|
FOL.NV.T_19_4
|
[1152, 1]
|
[1183, 9]
|
simp
|
case h1
P : Formula
u v : VarName
⊢ IsDeduct (∅ ∪ {exists_ u (forall_ v P)}) (forall_ v (exists_ u P))
|
case h1
P : Formula
u v : VarName
⊢ IsDeduct {exists_ u (forall_ v P)} (forall_ v (exists_ u P))
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Margaris/Fol.lean
|
FOL.NV.T_19_4
|
[1152, 1]
|
[1183, 9]
|
apply generalization
|
case h1
P : Formula
u v : VarName
⊢ IsDeduct {exists_ u (forall_ v P)} (forall_ v (exists_ u P))
|
case h1.h1
P : Formula
u v : VarName
⊢ IsDeduct {exists_ u (forall_ v P)} (exists_ u P)
case h1.h2
P : Formula
u v : VarName
⊢ ∀ H ∈ {exists_ u (forall_ v P)}, ¬isFreeIn v H
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Margaris/Fol.lean
|
FOL.NV.T_19_4
|
[1152, 1]
|
[1183, 9]
|
apply rule_C (forall_ v P) (exists_ u P) u {exists_ u (forall_ v P)}
|
case h1.h1
P : Formula
u v : VarName
⊢ IsDeduct {exists_ u (forall_ v P)} (exists_ u P)
|
case h1.h1.h1
P : Formula
u v : VarName
⊢ IsDeduct {exists_ u (forall_ v P)} (exists_ u (forall_ v P))
case h1.h1.h2
P : Formula
u v : VarName
⊢ IsDeduct ({exists_ u (forall_ v P)} ∪ {forall_ v P}) (exists_ u P)
case h1.h1.h3
P : Formula
u v : VarName
⊢ ∀ H ∈ {exists_ u (forall_ v P)}, ¬isFreeIn u H
case h1.h1.h4
P : Formula
u v : VarName
⊢ ¬isFreeIn u (exists_ u P)
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Margaris/Fol.lean
|
FOL.NV.T_19_4
|
[1152, 1]
|
[1183, 9]
|
apply IsDeduct.assume_
|
case h1.h1.h1
P : Formula
u v : VarName
⊢ IsDeduct {exists_ u (forall_ v P)} (exists_ u (forall_ v P))
|
case h1.h1.h1.a
P : Formula
u v : VarName
⊢ exists_ u (forall_ v P) ∈ {exists_ u (forall_ v P)}
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Margaris/Fol.lean
|
FOL.NV.T_19_4
|
[1152, 1]
|
[1183, 9]
|
simp
|
case h1.h1.h1.a
P : Formula
u v : VarName
⊢ exists_ u (forall_ v P) ∈ {exists_ u (forall_ v P)}
|
no goals
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Margaris/Fol.lean
|
FOL.NV.T_19_4
|
[1152, 1]
|
[1183, 9]
|
apply exists_intro P u u
|
case h1.h1.h2
P : Formula
u v : VarName
⊢ IsDeduct ({exists_ u (forall_ v P)} ∪ {forall_ v P}) (exists_ u P)
|
case h1.h1.h2.h1
P : Formula
u v : VarName
⊢ fastAdmits u u P
case h1.h1.h2.h2
P : Formula
u v : VarName
⊢ IsDeduct ({exists_ u (forall_ v P)} ∪ {forall_ v P}) (fastReplaceFree u u P)
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Margaris/Fol.lean
|
FOL.NV.T_19_4
|
[1152, 1]
|
[1183, 9]
|
apply fastAdmits_self
|
case h1.h1.h2.h1
P : Formula
u v : VarName
⊢ fastAdmits u u P
|
no goals
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Margaris/Fol.lean
|
FOL.NV.T_19_4
|
[1152, 1]
|
[1183, 9]
|
simp only [fastReplaceFree_self]
|
case h1.h1.h2.h2
P : Formula
u v : VarName
⊢ IsDeduct ({exists_ u (forall_ v P)} ∪ {forall_ v P}) (fastReplaceFree u u P)
|
case h1.h1.h2.h2
P : Formula
u v : VarName
⊢ IsDeduct ({exists_ u (forall_ v P)} ∪ {forall_ v P}) P
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Margaris/Fol.lean
|
FOL.NV.T_19_4
|
[1152, 1]
|
[1183, 9]
|
apply specId v
|
case h1.h1.h2.h2
P : Formula
u v : VarName
⊢ IsDeduct ({exists_ u (forall_ v P)} ∪ {forall_ v P}) P
|
case h1.h1.h2.h2.h1
P : Formula
u v : VarName
⊢ IsDeduct ({exists_ u (forall_ v P)} ∪ {forall_ v P}) (forall_ v P)
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Margaris/Fol.lean
|
FOL.NV.T_19_4
|
[1152, 1]
|
[1183, 9]
|
apply IsDeduct.assume_
|
case h1.h1.h2.h2.h1
P : Formula
u v : VarName
⊢ IsDeduct ({exists_ u (forall_ v P)} ∪ {forall_ v P}) (forall_ v P)
|
case h1.h1.h2.h2.h1.a
P : Formula
u v : VarName
⊢ forall_ v P ∈ {exists_ u (forall_ v P)} ∪ {forall_ v P}
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Margaris/Fol.lean
|
FOL.NV.T_19_4
|
[1152, 1]
|
[1183, 9]
|
simp
|
case h1.h1.h2.h2.h1.a
P : Formula
u v : VarName
⊢ forall_ v P ∈ {exists_ u (forall_ v P)} ∪ {forall_ v P}
|
no goals
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Margaris/Fol.lean
|
FOL.NV.T_19_4
|
[1152, 1]
|
[1183, 9]
|
simp
|
case h1.h1.h3
P : Formula
u v : VarName
⊢ ∀ H ∈ {exists_ u (forall_ v P)}, ¬isFreeIn u H
|
case h1.h1.h3
P : Formula
u v : VarName
⊢ ¬isFreeIn u (exists_ u (forall_ v P))
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Margaris/Fol.lean
|
FOL.NV.T_19_4
|
[1152, 1]
|
[1183, 9]
|
simp only [def_exists_]
|
case h1.h1.h3
P : Formula
u v : VarName
⊢ ¬isFreeIn u (exists_ u (forall_ v P))
|
case h1.h1.h3
P : Formula
u v : VarName
⊢ ¬isFreeIn u (forall_ u (forall_ v P).not_).not_
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Margaris/Fol.lean
|
FOL.NV.T_19_4
|
[1152, 1]
|
[1183, 9]
|
simp only [isFreeIn]
|
case h1.h1.h3
P : Formula
u v : VarName
⊢ ¬isFreeIn u (forall_ u (forall_ v P).not_).not_
|
case h1.h1.h3
P : Formula
u v : VarName
⊢ ¬(¬True ∧ ¬u = v ∧ isFreeIn u P)
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Margaris/Fol.lean
|
FOL.NV.T_19_4
|
[1152, 1]
|
[1183, 9]
|
simp
|
case h1.h1.h3
P : Formula
u v : VarName
⊢ ¬(¬True ∧ ¬u = v ∧ isFreeIn u P)
|
no goals
|
https://github.com/pthomas505/FOL.git
|
097a4abea51b641d144539b9a0f7516f3b9d818c
|
FOL/NV/Margaris/Fol.lean
|
FOL.NV.T_19_4
|
[1152, 1]
|
[1183, 9]
|
simp only [def_exists_]
|
case h1.h1.h4
P : Formula
u v : VarName
⊢ ¬isFreeIn u (exists_ u P)
|
case h1.h1.h4
P : Formula
u v : VarName
⊢ ¬isFreeIn u (forall_ u P.not_).not_
|
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