Datasets:
internlm
/
Lean-Github

Dataset card Data Studio Files Community
1
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_1_left
[654, 1]
[678, 9]
apply IsDeduct.mp_ P
case h1.h1.h1 P Q : Formula v : VarName ⊒ IsDeduct {forall_ v P, forall_ v ((P.imp_ Q).and_ (Q.imp_ P))} Q
case h1.h1.h1.a P Q : Formula v : VarName ⊒ IsDeduct {forall_ v P, forall_ v ((P.imp_ Q).and_ (Q.imp_ P))} (P.imp_ Q) case h1.h1.h1.a P Q : Formula v : VarName ⊒ IsDeduct {forall_ v P, forall_ v ((P.imp_ Q).and_ (Q.imp_ P))} P
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/NV/Margaris/Fol.lean
FOL.NV.T_18_1_left
[654, 1]
[678, 9]
apply IsDeduct.mp_ ((P.imp_ Q).and_ (Q.imp_ P))
case h1.h1.h1.a P Q : Formula v : VarName ⊒ IsDeduct {forall_ v P, forall_ v ((P.imp_ Q).and_ (Q.imp_ P))} (P.imp_ Q)
case h1.h1.h1.a.a P Q : Formula v : VarName ⊒ IsDeduct {forall_ v P, forall_ v ((P.imp_ Q).and_ (Q.imp_ P))} (((P.imp_ Q).and_ (Q.imp_ P)).imp_ (P.imp_ Q)) case h1.h1.h1.a.a P Q : Formula v : VarName ⊒ IsDeduct {forall_ v P, forall_ v ((P.imp_ Q).and_ (Q.imp_ P))} ((P.imp_ Q).and_ (Q.imp_ P))
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/NV/Margaris/Fol.lean
FOL.NV.T_18_1_left
[654, 1]
[678, 9]
simp only [def_and_]
case h1.h1.h1.a.a P Q : Formula v : VarName ⊒ IsDeduct {forall_ v P, forall_ v ((P.imp_ Q).and_ (Q.imp_ P))} (((P.imp_ Q).and_ (Q.imp_ P)).imp_ (P.imp_ Q))
case h1.h1.h1.a.a P Q : Formula v : VarName ⊒ IsDeduct {forall_ v P, forall_ v ((P.imp_ Q).imp_ (Q.imp_ P).not_).not_} (((P.imp_ Q).imp_ (Q.imp_ P).not_).not_.imp_ (P.imp_ Q))
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/NV/Margaris/Fol.lean
FOL.NV.T_18_1_left
[654, 1]
[678, 9]
SC
case h1.h1.h1.a.a P Q : Formula v : VarName ⊒ IsDeduct {forall_ v P, forall_ v ((P.imp_ Q).imp_ (Q.imp_ P).not_).not_} (((P.imp_ Q).imp_ (Q.imp_ P).not_).not_.imp_ (P.imp_ Q))
no goals
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/NV/Margaris/Fol.lean
FOL.NV.T_18_1_left
[654, 1]
[678, 9]
apply specId v
case h1.h1.h1.a.a P Q : Formula v : VarName ⊒ IsDeduct {forall_ v P, forall_ v ((P.imp_ Q).and_ (Q.imp_ P))} ((P.imp_ Q).and_ (Q.imp_ P))
case h1.h1.h1.a.a.h1 P Q : Formula v : VarName ⊒ IsDeduct {forall_ v P, forall_ v ((P.imp_ Q).and_ (Q.imp_ P))} (forall_ v ((P.imp_ Q).and_ (Q.imp_ P)))
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/NV/Margaris/Fol.lean
FOL.NV.T_18_1_left
[654, 1]
[678, 9]
apply IsDeduct.assume_
case h1.h1.h1.a.a.h1 P Q : Formula v : VarName ⊒ IsDeduct {forall_ v P, forall_ v ((P.imp_ Q).and_ (Q.imp_ P))} (forall_ v ((P.imp_ Q).and_ (Q.imp_ P)))
case h1.h1.h1.a.a.h1.a P Q : Formula v : VarName ⊒ forall_ v ((P.imp_ Q).and_ (Q.imp_ P)) ∈ {forall_ v P, forall_ v ((P.imp_ Q).and_ (Q.imp_ P))}
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/NV/Margaris/Fol.lean
FOL.NV.T_18_1_left
[654, 1]
[678, 9]
simp
case h1.h1.h1.a.a.h1.a P Q : Formula v : VarName ⊒ forall_ v ((P.imp_ Q).and_ (Q.imp_ P)) ∈ {forall_ v P, forall_ v ((P.imp_ Q).and_ (Q.imp_ P))}
no goals
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/NV/Margaris/Fol.lean
FOL.NV.T_18_1_left
[654, 1]
[678, 9]
apply specId v
case h1.h1.h1.a P Q : Formula v : VarName ⊒ IsDeduct {forall_ v P, forall_ v ((P.imp_ Q).and_ (Q.imp_ P))} P
case h1.h1.h1.a.h1 P Q : Formula v : VarName ⊒ IsDeduct {forall_ v P, forall_ v ((P.imp_ Q).and_ (Q.imp_ P))} (forall_ v P)
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/NV/Margaris/Fol.lean
FOL.NV.T_18_1_left
[654, 1]
[678, 9]
apply IsDeduct.assume_
case h1.h1.h1.a.h1 P Q : Formula v : VarName ⊒ IsDeduct {forall_ v P, forall_ v ((P.imp_ Q).and_ (Q.imp_ P))} (forall_ v P)
case h1.h1.h1.a.h1.a P Q : Formula v : VarName ⊒ forall_ v P ∈ {forall_ v P, forall_ v ((P.imp_ Q).and_ (Q.imp_ P))}
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/NV/Margaris/Fol.lean
FOL.NV.T_18_1_left
[654, 1]
[678, 9]
simp
case h1.h1.h1.a.h1.a P Q : Formula v : VarName ⊒ forall_ v P ∈ {forall_ v P, forall_ v ((P.imp_ Q).and_ (Q.imp_ P))}
no goals
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/NV/Margaris/Fol.lean
FOL.NV.T_18_1_left
[654, 1]
[678, 9]
simp
case h1.h1.h2 P Q : Formula v : VarName ⊒ βˆ€ H ∈ {forall_ v P, forall_ v ((P.imp_ Q).and_ (Q.imp_ P))}, Β¬isFreeIn v H
case h1.h1.h2 P Q : Formula v : VarName ⊒ ¬isFreeIn v (forall_ v P) ∧ ¬isFreeIn v (forall_ v ((P.imp_ Q).and_ (Q.imp_ P)))
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/NV/Margaris/Fol.lean
FOL.NV.T_18_1_left
[654, 1]
[678, 9]
simp only [isFreeIn]
case h1.h1.h2 P Q : Formula v : VarName ⊒ ¬isFreeIn v (forall_ v P) ∧ ¬isFreeIn v (forall_ v ((P.imp_ Q).and_ (Q.imp_ P)))
case h1.h1.h2 P Q : Formula v : VarName ⊒ ¬(¬True ∧ isFreeIn v P) ∧ ¬(¬True ∧ ((isFreeIn v P ∨ isFreeIn v Q) ∨ isFreeIn v Q ∨ isFreeIn v P))
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/NV/Margaris/Fol.lean
FOL.NV.T_18_1_left
[654, 1]
[678, 9]
simp
case h1.h1.h2 P Q : Formula v : VarName ⊒ ¬(¬True ∧ isFreeIn v P) ∧ ¬(¬True ∧ ((isFreeIn v P ∨ isFreeIn v Q) ∨ isFreeIn v Q ∨ isFreeIn v P))
no goals
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/NV/Margaris/Fol.lean
FOL.NV.T_18_1_right
[681, 1]
[705, 9]
simp only [def_iff_]
P Q : Formula v : VarName ⊒ IsProof ((forall_ v (P.iff_ Q)).imp_ ((forall_ v Q).imp_ (forall_ v P)))
P Q : Formula v : VarName ⊒ IsProof ((forall_ v ((P.imp_ Q).and_ (Q.imp_ P))).imp_ ((forall_ v Q).imp_ (forall_ v P)))
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/NV/Margaris/Fol.lean
FOL.NV.T_18_1_right
[681, 1]
[705, 9]
apply deduction_theorem
P Q : Formula v : VarName ⊒ IsProof ((forall_ v ((P.imp_ Q).and_ (Q.imp_ P))).imp_ ((forall_ v Q).imp_ (forall_ v P)))
case h1 P Q : Formula v : VarName ⊒ IsDeduct (βˆ… βˆͺ {forall_ v ((P.imp_ Q).and_ (Q.imp_ P))}) ((forall_ v Q).imp_ (forall_ v P))
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/NV/Margaris/Fol.lean
FOL.NV.T_18_1_right
[681, 1]
[705, 9]
apply deduction_theorem
case h1 P Q : Formula v : VarName ⊒ IsDeduct (βˆ… βˆͺ {forall_ v ((P.imp_ Q).and_ (Q.imp_ P))}) ((forall_ v Q).imp_ (forall_ v P))
case h1.h1 P Q : Formula v : VarName ⊒ IsDeduct (βˆ… βˆͺ {forall_ v ((P.imp_ Q).and_ (Q.imp_ P))} βˆͺ {forall_ v Q}) (forall_ v P)
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/NV/Margaris/Fol.lean
FOL.NV.T_18_1_right
[681, 1]
[705, 9]
simp
case h1.h1 P Q : Formula v : VarName ⊒ IsDeduct (βˆ… βˆͺ {forall_ v ((P.imp_ Q).and_ (Q.imp_ P))} βˆͺ {forall_ v Q}) (forall_ v P)
case h1.h1 P Q : Formula v : VarName ⊒ IsDeduct {forall_ v Q, forall_ v ((P.imp_ Q).and_ (Q.imp_ P))} (forall_ v P)
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/NV/Margaris/Fol.lean
FOL.NV.T_18_1_right
[681, 1]
[705, 9]
apply generalization
case h1.h1 P Q : Formula v : VarName ⊒ IsDeduct {forall_ v Q, forall_ v ((P.imp_ Q).and_ (Q.imp_ P))} (forall_ v P)
case h1.h1.h1 P Q : Formula v : VarName ⊒ IsDeduct {forall_ v Q, forall_ v ((P.imp_ Q).and_ (Q.imp_ P))} P case h1.h1.h2 P Q : Formula v : VarName ⊒ βˆ€ H ∈ {forall_ v Q, forall_ v ((P.imp_ Q).and_ (Q.imp_ P))}, Β¬isFreeIn v H
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/NV/Margaris/Fol.lean
FOL.NV.T_18_1_right
[681, 1]
[705, 9]
apply IsDeduct.mp_ Q
case h1.h1.h1 P Q : Formula v : VarName ⊒ IsDeduct {forall_ v Q, forall_ v ((P.imp_ Q).and_ (Q.imp_ P))} P
case h1.h1.h1.a P Q : Formula v : VarName ⊒ IsDeduct {forall_ v Q, forall_ v ((P.imp_ Q).and_ (Q.imp_ P))} (Q.imp_ P) case h1.h1.h1.a P Q : Formula v : VarName ⊒ IsDeduct {forall_ v Q, forall_ v ((P.imp_ Q).and_ (Q.imp_ P))} Q
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/NV/Margaris/Fol.lean
FOL.NV.T_18_1_right
[681, 1]
[705, 9]
apply IsDeduct.mp_ ((P.imp_ Q).and_ (Q.imp_ P))
case h1.h1.h1.a P Q : Formula v : VarName ⊒ IsDeduct {forall_ v Q, forall_ v ((P.imp_ Q).and_ (Q.imp_ P))} (Q.imp_ P)
case h1.h1.h1.a.a P Q : Formula v : VarName ⊒ IsDeduct {forall_ v Q, forall_ v ((P.imp_ Q).and_ (Q.imp_ P))} (((P.imp_ Q).and_ (Q.imp_ P)).imp_ (Q.imp_ P)) case h1.h1.h1.a.a P Q : Formula v : VarName ⊒ IsDeduct {forall_ v Q, forall_ v ((P.imp_ Q).and_ (Q.imp_ P))} ((P.imp_ Q).and_ (Q.imp_ P))
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/NV/Margaris/Fol.lean
FOL.NV.T_18_1_right
[681, 1]
[705, 9]
simp only [def_and_]
case h1.h1.h1.a.a P Q : Formula v : VarName ⊒ IsDeduct {forall_ v Q, forall_ v ((P.imp_ Q).and_ (Q.imp_ P))} (((P.imp_ Q).and_ (Q.imp_ P)).imp_ (Q.imp_ P))
case h1.h1.h1.a.a P Q : Formula v : VarName ⊒ IsDeduct {forall_ v Q, forall_ v ((P.imp_ Q).imp_ (Q.imp_ P).not_).not_} (((P.imp_ Q).imp_ (Q.imp_ P).not_).not_.imp_ (Q.imp_ P))
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/NV/Margaris/Fol.lean
FOL.NV.T_18_1_right
[681, 1]
[705, 9]
SC
case h1.h1.h1.a.a P Q : Formula v : VarName ⊒ IsDeduct {forall_ v Q, forall_ v ((P.imp_ Q).imp_ (Q.imp_ P).not_).not_} (((P.imp_ Q).imp_ (Q.imp_ P).not_).not_.imp_ (Q.imp_ P))
no goals
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/NV/Margaris/Fol.lean
FOL.NV.T_18_1_right
[681, 1]
[705, 9]
apply specId v
case h1.h1.h1.a.a P Q : Formula v : VarName ⊒ IsDeduct {forall_ v Q, forall_ v ((P.imp_ Q).and_ (Q.imp_ P))} ((P.imp_ Q).and_ (Q.imp_ P))
case h1.h1.h1.a.a.h1 P Q : Formula v : VarName ⊒ IsDeduct {forall_ v Q, forall_ v ((P.imp_ Q).and_ (Q.imp_ P))} (forall_ v ((P.imp_ Q).and_ (Q.imp_ P)))
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/NV/Margaris/Fol.lean
FOL.NV.T_18_1_right
[681, 1]
[705, 9]
apply IsDeduct.assume_
case h1.h1.h1.a.a.h1 P Q : Formula v : VarName ⊒ IsDeduct {forall_ v Q, forall_ v ((P.imp_ Q).and_ (Q.imp_ P))} (forall_ v ((P.imp_ Q).and_ (Q.imp_ P)))
case h1.h1.h1.a.a.h1.a P Q : Formula v : VarName ⊒ forall_ v ((P.imp_ Q).and_ (Q.imp_ P)) ∈ {forall_ v Q, forall_ v ((P.imp_ Q).and_ (Q.imp_ P))}
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/NV/Margaris/Fol.lean
FOL.NV.T_18_1_right
[681, 1]
[705, 9]
simp
case h1.h1.h1.a.a.h1.a P Q : Formula v : VarName ⊒ forall_ v ((P.imp_ Q).and_ (Q.imp_ P)) ∈ {forall_ v Q, forall_ v ((P.imp_ Q).and_ (Q.imp_ P))}
no goals
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/NV/Margaris/Fol.lean
FOL.NV.T_18_1_right
[681, 1]
[705, 9]
apply specId v
case h1.h1.h1.a P Q : Formula v : VarName ⊒ IsDeduct {forall_ v Q, forall_ v ((P.imp_ Q).and_ (Q.imp_ P))} Q
case h1.h1.h1.a.h1 P Q : Formula v : VarName ⊒ IsDeduct {forall_ v Q, forall_ v ((P.imp_ Q).and_ (Q.imp_ P))} (forall_ v Q)
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/NV/Margaris/Fol.lean
FOL.NV.T_18_1_right
[681, 1]
[705, 9]
apply IsDeduct.assume_
case h1.h1.h1.a.h1 P Q : Formula v : VarName ⊒ IsDeduct {forall_ v Q, forall_ v ((P.imp_ Q).and_ (Q.imp_ P))} (forall_ v Q)
case h1.h1.h1.a.h1.a P Q : Formula v : VarName ⊒ forall_ v Q ∈ {forall_ v Q, forall_ v ((P.imp_ Q).and_ (Q.imp_ P))}
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/NV/Margaris/Fol.lean
FOL.NV.T_18_1_right
[681, 1]
[705, 9]
simp
case h1.h1.h1.a.h1.a P Q : Formula v : VarName ⊒ forall_ v Q ∈ {forall_ v Q, forall_ v ((P.imp_ Q).and_ (Q.imp_ P))}
no goals
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/NV/Margaris/Fol.lean
FOL.NV.T_18_1_right
[681, 1]
[705, 9]
simp
case h1.h1.h2 P Q : Formula v : VarName ⊒ βˆ€ H ∈ {forall_ v Q, forall_ v ((P.imp_ Q).and_ (Q.imp_ P))}, Β¬isFreeIn v H
case h1.h1.h2 P Q : Formula v : VarName ⊒ ¬isFreeIn v (forall_ v Q) ∧ ¬isFreeIn v (forall_ v ((P.imp_ Q).and_ (Q.imp_ P)))
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/NV/Margaris/Fol.lean
FOL.NV.T_18_1_right
[681, 1]
[705, 9]
simp only [isFreeIn]
case h1.h1.h2 P Q : Formula v : VarName ⊒ ¬isFreeIn v (forall_ v Q) ∧ ¬isFreeIn v (forall_ v ((P.imp_ Q).and_ (Q.imp_ P)))
case h1.h1.h2 P Q : Formula v : VarName ⊒ ¬(¬True ∧ isFreeIn v Q) ∧ ¬(¬True ∧ ((isFreeIn v P ∨ isFreeIn v Q) ∨ isFreeIn v Q ∨ isFreeIn v P))
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/NV/Margaris/Fol.lean
FOL.NV.T_18_1_right
[681, 1]
[705, 9]
simp
case h1.h1.h2 P Q : Formula v : VarName ⊒ ¬(¬True ∧ isFreeIn v Q) ∧ ¬(¬True ∧ ((isFreeIn v P ∨ isFreeIn v Q) ∨ isFreeIn v Q ∨ isFreeIn v P))
no goals
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/NV/Margaris/Fol.lean
FOL.NV.T_18_1
[708, 1]
[721, 23]
apply IsDeduct.mp_ ((forall_ v (P.iff_ Q)).imp_ ((forall_ v Q).imp_ (forall_ v P)))
P Q : Formula v : VarName ⊒ IsProof ((forall_ v (P.iff_ Q)).imp_ ((forall_ v P).iff_ (forall_ v Q)))
case a P Q : Formula v : VarName ⊒ IsDeduct βˆ… (((forall_ v (P.iff_ Q)).imp_ ((forall_ v Q).imp_ (forall_ v P))).imp_ ((forall_ v (P.iff_ Q)).imp_ ((forall_ v P).iff_ (forall_ v Q)))) case a P Q : Formula v : VarName ⊒ IsDeduct βˆ… ((forall_ v (P.iff_ Q)).imp_ ((forall_ v Q).imp_ (forall_ v P)))
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/NV/Margaris/Fol.lean
FOL.NV.T_18_1
[708, 1]
[721, 23]
apply IsDeduct.mp_ ((forall_ v (P.iff_ Q)).imp_ ((forall_ v P).imp_ (forall_ v Q)))
case a P Q : Formula v : VarName ⊒ IsDeduct βˆ… (((forall_ v (P.iff_ Q)).imp_ ((forall_ v Q).imp_ (forall_ v P))).imp_ ((forall_ v (P.iff_ Q)).imp_ ((forall_ v P).iff_ (forall_ v Q))))
case a.a P Q : Formula v : VarName ⊒ IsDeduct βˆ… (((forall_ v (P.iff_ Q)).imp_ ((forall_ v P).imp_ (forall_ v Q))).imp_ (((forall_ v (P.iff_ Q)).imp_ ((forall_ v Q).imp_ (forall_ v P))).imp_ ((forall_ v (P.iff_ Q)).imp_ ((forall_ v P).iff_ (forall_ v Q))))) case a.a P Q : Formula v : VarName ⊒ IsDeduct βˆ… ((forall_ v (P.iff_ Q)).imp_ ((forall_ v P).imp_ (forall_ v Q)))
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/NV/Margaris/Fol.lean
FOL.NV.T_18_1
[708, 1]
[721, 23]
simp only [def_iff_]
case a.a P Q : Formula v : VarName ⊒ IsDeduct βˆ… (((forall_ v (P.iff_ Q)).imp_ ((forall_ v P).imp_ (forall_ v Q))).imp_ (((forall_ v (P.iff_ Q)).imp_ ((forall_ v Q).imp_ (forall_ v P))).imp_ ((forall_ v (P.iff_ Q)).imp_ ((forall_ v P).iff_ (forall_ v Q)))))
case a.a P Q : Formula v : VarName ⊒ IsDeduct βˆ… (((forall_ v ((P.imp_ Q).and_ (Q.imp_ P))).imp_ ((forall_ v P).imp_ (forall_ v Q))).imp_ (((forall_ v ((P.imp_ Q).and_ (Q.imp_ P))).imp_ ((forall_ v Q).imp_ (forall_ v P))).imp_ ((forall_ v ((P.imp_ Q).and_ (Q.imp_ P))).imp_ (((forall_ v P).imp_ (forall_ v Q)).and_ ((forall_ v Q).imp_ (forall_ v P))))))
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/NV/Margaris/Fol.lean
FOL.NV.T_18_1
[708, 1]
[721, 23]
simp only [def_and_]
case a.a P Q : Formula v : VarName ⊒ IsDeduct βˆ… (((forall_ v ((P.imp_ Q).and_ (Q.imp_ P))).imp_ ((forall_ v P).imp_ (forall_ v Q))).imp_ (((forall_ v ((P.imp_ Q).and_ (Q.imp_ P))).imp_ ((forall_ v Q).imp_ (forall_ v P))).imp_ ((forall_ v ((P.imp_ Q).and_ (Q.imp_ P))).imp_ (((forall_ v P).imp_ (forall_ v Q)).and_ ((forall_ v Q).imp_ (forall_ v P))))))
case a.a P Q : Formula v : VarName ⊒ IsDeduct βˆ… (((forall_ v ((P.imp_ Q).imp_ (Q.imp_ P).not_).not_).imp_ ((forall_ v P).imp_ (forall_ v Q))).imp_ (((forall_ v ((P.imp_ Q).imp_ (Q.imp_ P).not_).not_).imp_ ((forall_ v Q).imp_ (forall_ v P))).imp_ ((forall_ v ((P.imp_ Q).imp_ (Q.imp_ P).not_).not_).imp_ (((forall_ v P).imp_ (forall_ v Q)).imp_ ((forall_ v Q).imp_ (forall_ v P)).not_).not_)))
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/NV/Margaris/Fol.lean
FOL.NV.T_18_1
[708, 1]
[721, 23]
SC
case a.a P Q : Formula v : VarName ⊒ IsDeduct βˆ… (((forall_ v ((P.imp_ Q).imp_ (Q.imp_ P).not_).not_).imp_ ((forall_ v P).imp_ (forall_ v Q))).imp_ (((forall_ v ((P.imp_ Q).imp_ (Q.imp_ P).not_).not_).imp_ ((forall_ v Q).imp_ (forall_ v P))).imp_ ((forall_ v ((P.imp_ Q).imp_ (Q.imp_ P).not_).not_).imp_ (((forall_ v P).imp_ (forall_ v Q)).imp_ ((forall_ v Q).imp_ (forall_ v P)).not_).not_)))
no goals
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/NV/Margaris/Fol.lean
FOL.NV.T_18_1
[708, 1]
[721, 23]
apply T_18_1_left
case a.a P Q : Formula v : VarName ⊒ IsDeduct βˆ… ((forall_ v (P.iff_ Q)).imp_ ((forall_ v P).imp_ (forall_ v Q)))
no goals
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/NV/Margaris/Fol.lean
FOL.NV.T_18_1
[708, 1]
[721, 23]
apply T_18_1_right
case a P Q : Formula v : VarName ⊒ IsDeduct βˆ… ((forall_ v (P.iff_ Q)).imp_ ((forall_ v Q).imp_ (forall_ v P)))
no goals
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/NV/Margaris/Fol.lean
FOL.NV.Forall_spec_id
[724, 1]
[743, 8]
induction xs
xs : List VarName P : Formula ⊒ IsProof ((Forall_ xs P).imp_ P)
case nil P : Formula ⊒ IsProof ((Forall_ [] P).imp_ P) case cons P : Formula head✝ : VarName tail✝ : List VarName tail_ih✝ : IsProof ((Forall_ tail✝ P).imp_ P) ⊒ IsProof ((Forall_ (head✝ :: tail✝) P).imp_ P)
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/NV/Margaris/Fol.lean
FOL.NV.Forall_spec_id
[724, 1]
[743, 8]
case nil => simp only [Forall_] apply prop_id
P : Formula ⊒ IsProof ((Forall_ [] P).imp_ P)
no goals
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/NV/Margaris/Fol.lean
FOL.NV.Forall_spec_id
[724, 1]
[743, 8]
case cons xs_hd xs_tl xs_ih => simp only [Forall_] apply deduction_theorem simp apply IsDeduct.mp_ (Forall_ xs_tl P) apply proof_imp_deduct exact xs_ih apply specId xs_hd apply IsDeduct.assume_ simp rfl
P : Formula xs_hd : VarName xs_tl : List VarName xs_ih : IsProof ((Forall_ xs_tl P).imp_ P) ⊒ IsProof ((Forall_ (xs_hd :: xs_tl) P).imp_ P)
no goals
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/NV/Margaris/Fol.lean
FOL.NV.Forall_spec_id
[724, 1]
[743, 8]
simp only [Forall_]
P : Formula ⊒ IsProof ((Forall_ [] P).imp_ P)
P : Formula ⊒ IsProof ((List.foldr forall_ P []).imp_ P)
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/NV/Margaris/Fol.lean
FOL.NV.Forall_spec_id
[724, 1]
[743, 8]
apply prop_id
P : Formula ⊒ IsProof ((List.foldr forall_ P []).imp_ P)
no goals
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/NV/Margaris/Fol.lean
FOL.NV.Forall_spec_id
[724, 1]
[743, 8]
simp only [Forall_]
P : Formula xs_hd : VarName xs_tl : List VarName xs_ih : IsProof ((Forall_ xs_tl P).imp_ P) ⊒ IsProof ((Forall_ (xs_hd :: xs_tl) P).imp_ P)
P : Formula xs_hd : VarName xs_tl : List VarName xs_ih : IsProof ((Forall_ xs_tl P).imp_ P) ⊒ IsProof ((List.foldr forall_ P (xs_hd :: xs_tl)).imp_ P)
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/NV/Margaris/Fol.lean
FOL.NV.Forall_spec_id
[724, 1]
[743, 8]
apply deduction_theorem
P : Formula xs_hd : VarName xs_tl : List VarName xs_ih : IsProof ((Forall_ xs_tl P).imp_ P) ⊒ IsProof ((List.foldr forall_ P (xs_hd :: xs_tl)).imp_ P)
case h1 P : Formula xs_hd : VarName xs_tl : List VarName xs_ih : IsProof ((Forall_ xs_tl P).imp_ P) ⊒ IsDeduct (βˆ… βˆͺ {List.foldr forall_ P (xs_hd :: xs_tl)}) P
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/NV/Margaris/Fol.lean
FOL.NV.Forall_spec_id
[724, 1]
[743, 8]
simp
case h1 P : Formula xs_hd : VarName xs_tl : List VarName xs_ih : IsProof ((Forall_ xs_tl P).imp_ P) ⊒ IsDeduct (βˆ… βˆͺ {List.foldr forall_ P (xs_hd :: xs_tl)}) P
case h1 P : Formula xs_hd : VarName xs_tl : List VarName xs_ih : IsProof ((Forall_ xs_tl P).imp_ P) ⊒ IsDeduct {forall_ xs_hd (List.foldr forall_ P xs_tl)} P
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/NV/Margaris/Fol.lean
FOL.NV.Forall_spec_id
[724, 1]
[743, 8]
apply IsDeduct.mp_ (Forall_ xs_tl P)
case h1 P : Formula xs_hd : VarName xs_tl : List VarName xs_ih : IsProof ((Forall_ xs_tl P).imp_ P) ⊒ IsDeduct {forall_ xs_hd (List.foldr forall_ P xs_tl)} P
case h1.a P : Formula xs_hd : VarName xs_tl : List VarName xs_ih : IsProof ((Forall_ xs_tl P).imp_ P) ⊒ IsDeduct {forall_ xs_hd (List.foldr forall_ P xs_tl)} ((Forall_ xs_tl P).imp_ P) case h1.a P : Formula xs_hd : VarName xs_tl : List VarName xs_ih : IsProof ((Forall_ xs_tl P).imp_ P) ⊒ IsDeduct {forall_ xs_hd (List.foldr forall_ P xs_tl)} (Forall_ xs_tl P)
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/NV/Margaris/Fol.lean
FOL.NV.Forall_spec_id
[724, 1]
[743, 8]
apply proof_imp_deduct
case h1.a P : Formula xs_hd : VarName xs_tl : List VarName xs_ih : IsProof ((Forall_ xs_tl P).imp_ P) ⊒ IsDeduct {forall_ xs_hd (List.foldr forall_ P xs_tl)} ((Forall_ xs_tl P).imp_ P) case h1.a P : Formula xs_hd : VarName xs_tl : List VarName xs_ih : IsProof ((Forall_ xs_tl P).imp_ P) ⊒ IsDeduct {forall_ xs_hd (List.foldr forall_ P xs_tl)} (Forall_ xs_tl P)
case h1.a.h1 P : Formula xs_hd : VarName xs_tl : List VarName xs_ih : IsProof ((Forall_ xs_tl P).imp_ P) ⊒ IsProof ((Forall_ xs_tl P).imp_ P) case h1.a P : Formula xs_hd : VarName xs_tl : List VarName xs_ih : IsProof ((Forall_ xs_tl P).imp_ P) ⊒ IsDeduct {forall_ xs_hd (List.foldr forall_ P xs_tl)} (Forall_ xs_tl P)
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/NV/Margaris/Fol.lean
FOL.NV.Forall_spec_id
[724, 1]
[743, 8]
exact xs_ih
case h1.a.h1 P : Formula xs_hd : VarName xs_tl : List VarName xs_ih : IsProof ((Forall_ xs_tl P).imp_ P) ⊒ IsProof ((Forall_ xs_tl P).imp_ P) case h1.a P : Formula xs_hd : VarName xs_tl : List VarName xs_ih : IsProof ((Forall_ xs_tl P).imp_ P) ⊒ IsDeduct {forall_ xs_hd (List.foldr forall_ P xs_tl)} (Forall_ xs_tl P)
case h1.a P : Formula xs_hd : VarName xs_tl : List VarName xs_ih : IsProof ((Forall_ xs_tl P).imp_ P) ⊒ IsDeduct {forall_ xs_hd (List.foldr forall_ P xs_tl)} (Forall_ xs_tl P)
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/NV/Margaris/Fol.lean
FOL.NV.Forall_spec_id
[724, 1]
[743, 8]
apply specId xs_hd
case h1.a P : Formula xs_hd : VarName xs_tl : List VarName xs_ih : IsProof ((Forall_ xs_tl P).imp_ P) ⊒ IsDeduct {forall_ xs_hd (List.foldr forall_ P xs_tl)} (Forall_ xs_tl P)
case h1.a.h1 P : Formula xs_hd : VarName xs_tl : List VarName xs_ih : IsProof ((Forall_ xs_tl P).imp_ P) ⊒ IsDeduct {forall_ xs_hd (List.foldr forall_ P xs_tl)} (forall_ xs_hd (Forall_ xs_tl P))
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/NV/Margaris/Fol.lean
FOL.NV.Forall_spec_id
[724, 1]
[743, 8]
apply IsDeduct.assume_
case h1.a.h1 P : Formula xs_hd : VarName xs_tl : List VarName xs_ih : IsProof ((Forall_ xs_tl P).imp_ P) ⊒ IsDeduct {forall_ xs_hd (List.foldr forall_ P xs_tl)} (forall_ xs_hd (Forall_ xs_tl P))
case h1.a.h1.a P : Formula xs_hd : VarName xs_tl : List VarName xs_ih : IsProof ((Forall_ xs_tl P).imp_ P) ⊒ forall_ xs_hd (Forall_ xs_tl P) ∈ {forall_ xs_hd (List.foldr forall_ P xs_tl)}
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/NV/Margaris/Fol.lean
FOL.NV.Forall_spec_id
[724, 1]
[743, 8]
simp
case h1.a.h1.a P : Formula xs_hd : VarName xs_tl : List VarName xs_ih : IsProof ((Forall_ xs_tl P).imp_ P) ⊒ forall_ xs_hd (Forall_ xs_tl P) ∈ {forall_ xs_hd (List.foldr forall_ P xs_tl)}
case h1.a.h1.a P : Formula xs_hd : VarName xs_tl : List VarName xs_ih : IsProof ((Forall_ xs_tl P).imp_ P) ⊒ Forall_ xs_tl P = List.foldr forall_ P xs_tl
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/NV/Margaris/Fol.lean
FOL.NV.Forall_spec_id
[724, 1]
[743, 8]
rfl
case h1.a.h1.a P : Formula xs_hd : VarName xs_tl : List VarName xs_ih : IsProof ((Forall_ xs_tl P).imp_ P) ⊒ Forall_ xs_tl P = List.foldr forall_ P xs_tl
no goals
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/NV/Margaris/Fol.lean
FOL.NV.Forall_spec_id'
[746, 1]
[764, 13]
induction xs
xs : List VarName P : Formula Ξ” : Set Formula h1 : IsDeduct Ξ” (Forall_ xs P) ⊒ IsDeduct Ξ” P
case nil P : Formula Ξ” : Set Formula h1 : IsDeduct Ξ” (Forall_ [] P) ⊒ IsDeduct Ξ” P case cons P : Formula Ξ” : Set Formula head✝ : VarName tail✝ : List VarName tail_ih✝ : IsDeduct Ξ” (Forall_ tail✝ P) β†’ IsDeduct Ξ” P h1 : IsDeduct Ξ” (Forall_ (head✝ :: tail✝) P) ⊒ IsDeduct Ξ” P
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/NV/Margaris/Fol.lean
FOL.NV.Forall_spec_id'
[746, 1]
[764, 13]
case nil => simp only [Forall_] at h1 simp at h1 exact h1
P : Formula Ξ” : Set Formula h1 : IsDeduct Ξ” (Forall_ [] P) ⊒ IsDeduct Ξ” P
no goals
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/NV/Margaris/Fol.lean
FOL.NV.Forall_spec_id'
[746, 1]
[764, 13]
case cons xs_hd xs_tl xs_ih => simp only [Forall_] at h1 simp at h1 apply xs_ih simp only [Forall_] apply specId xs_hd exact h1
P : Formula Ξ” : Set Formula xs_hd : VarName xs_tl : List VarName xs_ih : IsDeduct Ξ” (Forall_ xs_tl P) β†’ IsDeduct Ξ” P h1 : IsDeduct Ξ” (Forall_ (xs_hd :: xs_tl) P) ⊒ IsDeduct Ξ” P
no goals
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/NV/Margaris/Fol.lean
FOL.NV.Forall_spec_id'
[746, 1]
[764, 13]
simp only [Forall_] at h1
P : Formula Ξ” : Set Formula h1 : IsDeduct Ξ” (Forall_ [] P) ⊒ IsDeduct Ξ” P
P : Formula Ξ” : Set Formula h1 : IsDeduct Ξ” (List.foldr forall_ P []) ⊒ IsDeduct Ξ” P
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/NV/Margaris/Fol.lean
FOL.NV.Forall_spec_id'
[746, 1]
[764, 13]
simp at h1
P : Formula Ξ” : Set Formula h1 : IsDeduct Ξ” (List.foldr forall_ P []) ⊒ IsDeduct Ξ” P
P : Formula Ξ” : Set Formula h1 : IsDeduct Ξ” P ⊒ IsDeduct Ξ” P
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/NV/Margaris/Fol.lean
FOL.NV.Forall_spec_id'
[746, 1]
[764, 13]
exact h1
P : Formula Ξ” : Set Formula h1 : IsDeduct Ξ” P ⊒ IsDeduct Ξ” P
no goals
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/NV/Margaris/Fol.lean
FOL.NV.Forall_spec_id'
[746, 1]
[764, 13]
simp only [Forall_] at h1
P : Formula Ξ” : Set Formula xs_hd : VarName xs_tl : List VarName xs_ih : IsDeduct Ξ” (Forall_ xs_tl P) β†’ IsDeduct Ξ” P h1 : IsDeduct Ξ” (Forall_ (xs_hd :: xs_tl) P) ⊒ IsDeduct Ξ” P
P : Formula Ξ” : Set Formula xs_hd : VarName xs_tl : List VarName xs_ih : IsDeduct Ξ” (Forall_ xs_tl P) β†’ IsDeduct Ξ” P h1 : IsDeduct Ξ” (List.foldr forall_ P (xs_hd :: xs_tl)) ⊒ IsDeduct Ξ” P
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/NV/Margaris/Fol.lean
FOL.NV.Forall_spec_id'
[746, 1]
[764, 13]
simp at h1
P : Formula Ξ” : Set Formula xs_hd : VarName xs_tl : List VarName xs_ih : IsDeduct Ξ” (Forall_ xs_tl P) β†’ IsDeduct Ξ” P h1 : IsDeduct Ξ” (List.foldr forall_ P (xs_hd :: xs_tl)) ⊒ IsDeduct Ξ” P
P : Formula Ξ” : Set Formula xs_hd : VarName xs_tl : List VarName xs_ih : IsDeduct Ξ” (Forall_ xs_tl P) β†’ IsDeduct Ξ” P h1 : IsDeduct Ξ” (forall_ xs_hd (List.foldr forall_ P xs_tl)) ⊒ IsDeduct Ξ” P
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/NV/Margaris/Fol.lean
FOL.NV.Forall_spec_id'
[746, 1]
[764, 13]
apply xs_ih
P : Formula Ξ” : Set Formula xs_hd : VarName xs_tl : List VarName xs_ih : IsDeduct Ξ” (Forall_ xs_tl P) β†’ IsDeduct Ξ” P h1 : IsDeduct Ξ” (forall_ xs_hd (List.foldr forall_ P xs_tl)) ⊒ IsDeduct Ξ” P
P : Formula Ξ” : Set Formula xs_hd : VarName xs_tl : List VarName xs_ih : IsDeduct Ξ” (Forall_ xs_tl P) β†’ IsDeduct Ξ” P h1 : IsDeduct Ξ” (forall_ xs_hd (List.foldr forall_ P xs_tl)) ⊒ IsDeduct Ξ” (Forall_ xs_tl P)
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/NV/Margaris/Fol.lean
FOL.NV.Forall_spec_id'
[746, 1]
[764, 13]
simp only [Forall_]
P : Formula Ξ” : Set Formula xs_hd : VarName xs_tl : List VarName xs_ih : IsDeduct Ξ” (Forall_ xs_tl P) β†’ IsDeduct Ξ” P h1 : IsDeduct Ξ” (forall_ xs_hd (List.foldr forall_ P xs_tl)) ⊒ IsDeduct Ξ” (Forall_ xs_tl P)
P : Formula Ξ” : Set Formula xs_hd : VarName xs_tl : List VarName xs_ih : IsDeduct Ξ” (Forall_ xs_tl P) β†’ IsDeduct Ξ” P h1 : IsDeduct Ξ” (forall_ xs_hd (List.foldr forall_ P xs_tl)) ⊒ IsDeduct Ξ” (List.foldr forall_ P xs_tl)
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/NV/Margaris/Fol.lean
FOL.NV.Forall_spec_id'
[746, 1]
[764, 13]
apply specId xs_hd
P : Formula Ξ” : Set Formula xs_hd : VarName xs_tl : List VarName xs_ih : IsDeduct Ξ” (Forall_ xs_tl P) β†’ IsDeduct Ξ” P h1 : IsDeduct Ξ” (forall_ xs_hd (List.foldr forall_ P xs_tl)) ⊒ IsDeduct Ξ” (List.foldr forall_ P xs_tl)
case h1 P : Formula Ξ” : Set Formula xs_hd : VarName xs_tl : List VarName xs_ih : IsDeduct Ξ” (Forall_ xs_tl P) β†’ IsDeduct Ξ” P h1 : IsDeduct Ξ” (forall_ xs_hd (List.foldr forall_ P xs_tl)) ⊒ IsDeduct Ξ” (forall_ xs_hd (List.foldr forall_ P xs_tl))
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/NV/Margaris/Fol.lean
FOL.NV.Forall_spec_id'
[746, 1]
[764, 13]
exact h1
case h1 P : Formula Ξ” : Set Formula xs_hd : VarName xs_tl : List VarName xs_ih : IsDeduct Ξ” (Forall_ xs_tl P) β†’ IsDeduct Ξ” P h1 : IsDeduct Ξ” (forall_ xs_hd (List.foldr forall_ P xs_tl)) ⊒ IsDeduct Ξ” (forall_ xs_hd (List.foldr forall_ P xs_tl))
no goals
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/NV/Margaris/Fol.lean
FOL.NV.Forall_isBoundIn
[767, 1]
[781, 10]
simp only [Formula.Forall_]
P : Formula xs : List VarName x : VarName ⊒ isBoundIn x (Forall_ xs P) ↔ x ∈ xs ∨ isBoundIn x P
P : Formula xs : List VarName x : VarName ⊒ isBoundIn x (List.foldr forall_ P xs) ↔ x ∈ xs ∨ isBoundIn x P
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/NV/Margaris/Fol.lean
FOL.NV.Forall_isBoundIn
[767, 1]
[781, 10]
induction xs
P : Formula xs : List VarName x : VarName ⊒ isBoundIn x (List.foldr forall_ P xs) ↔ x ∈ xs ∨ isBoundIn x P
case nil P : Formula x : VarName ⊒ isBoundIn x (List.foldr forall_ P []) ↔ x ∈ [] ∨ isBoundIn x P case cons P : Formula x head✝ : VarName tail✝ : List VarName tail_ih✝ : isBoundIn x (List.foldr forall_ P tail✝) ↔ x ∈ tail✝ ∨ isBoundIn x P ⊒ isBoundIn x (List.foldr forall_ P (head✝ :: tail✝)) ↔ x ∈ head✝ :: tail✝ ∨ isBoundIn x P
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/NV/Margaris/Fol.lean
FOL.NV.Forall_isBoundIn
[767, 1]
[781, 10]
case nil => simp
P : Formula x : VarName ⊒ isBoundIn x (List.foldr forall_ P []) ↔ x ∈ [] ∨ isBoundIn x P
no goals
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/NV/Margaris/Fol.lean
FOL.NV.Forall_isBoundIn
[767, 1]
[781, 10]
case cons xs_hd xs_tl xs_ih => simp simp only [isBoundIn] simp only [xs_ih] tauto
P : Formula x xs_hd : VarName xs_tl : List VarName xs_ih : isBoundIn x (List.foldr forall_ P xs_tl) ↔ x ∈ xs_tl ∨ isBoundIn x P ⊒ isBoundIn x (List.foldr forall_ P (xs_hd :: xs_tl)) ↔ x ∈ xs_hd :: xs_tl ∨ isBoundIn x P
no goals
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/NV/Margaris/Fol.lean
FOL.NV.Forall_isBoundIn
[767, 1]
[781, 10]
simp
P : Formula x : VarName ⊒ isBoundIn x (List.foldr forall_ P []) ↔ x ∈ [] ∨ isBoundIn x P
no goals
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/NV/Margaris/Fol.lean
FOL.NV.Forall_isBoundIn
[767, 1]
[781, 10]
simp
P : Formula x xs_hd : VarName xs_tl : List VarName xs_ih : isBoundIn x (List.foldr forall_ P xs_tl) ↔ x ∈ xs_tl ∨ isBoundIn x P ⊒ isBoundIn x (List.foldr forall_ P (xs_hd :: xs_tl)) ↔ x ∈ xs_hd :: xs_tl ∨ isBoundIn x P
P : Formula x xs_hd : VarName xs_tl : List VarName xs_ih : isBoundIn x (List.foldr forall_ P xs_tl) ↔ x ∈ xs_tl ∨ isBoundIn x P ⊒ isBoundIn x (forall_ xs_hd (List.foldr forall_ P xs_tl)) ↔ (x = xs_hd ∨ x ∈ xs_tl) ∨ isBoundIn x P
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/NV/Margaris/Fol.lean
FOL.NV.Forall_isBoundIn
[767, 1]
[781, 10]
simp only [isBoundIn]
P : Formula x xs_hd : VarName xs_tl : List VarName xs_ih : isBoundIn x (List.foldr forall_ P xs_tl) ↔ x ∈ xs_tl ∨ isBoundIn x P ⊒ isBoundIn x (forall_ xs_hd (List.foldr forall_ P xs_tl)) ↔ (x = xs_hd ∨ x ∈ xs_tl) ∨ isBoundIn x P
P : Formula x xs_hd : VarName xs_tl : List VarName xs_ih : isBoundIn x (List.foldr forall_ P xs_tl) ↔ x ∈ xs_tl ∨ isBoundIn x P ⊒ x = xs_hd ∨ isBoundIn x (List.foldr forall_ P xs_tl) ↔ (x = xs_hd ∨ x ∈ xs_tl) ∨ isBoundIn x P
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/NV/Margaris/Fol.lean
FOL.NV.Forall_isBoundIn
[767, 1]
[781, 10]
simp only [xs_ih]
P : Formula x xs_hd : VarName xs_tl : List VarName xs_ih : isBoundIn x (List.foldr forall_ P xs_tl) ↔ x ∈ xs_tl ∨ isBoundIn x P ⊒ x = xs_hd ∨ isBoundIn x (List.foldr forall_ P xs_tl) ↔ (x = xs_hd ∨ x ∈ xs_tl) ∨ isBoundIn x P
P : Formula x xs_hd : VarName xs_tl : List VarName xs_ih : isBoundIn x (List.foldr forall_ P xs_tl) ↔ x ∈ xs_tl ∨ isBoundIn x P ⊒ x = xs_hd ∨ x ∈ xs_tl ∨ isBoundIn x P ↔ (x = xs_hd ∨ x ∈ xs_tl) ∨ isBoundIn x P
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/NV/Margaris/Fol.lean
FOL.NV.Forall_isBoundIn
[767, 1]
[781, 10]
tauto
P : Formula x xs_hd : VarName xs_tl : List VarName xs_ih : isBoundIn x (List.foldr forall_ P xs_tl) ↔ x ∈ xs_tl ∨ isBoundIn x P ⊒ x = xs_hd ∨ x ∈ xs_tl ∨ isBoundIn x P ↔ (x = xs_hd ∨ x ∈ xs_tl) ∨ isBoundIn x P
no goals
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/NV/Margaris/Fol.lean
FOL.NV.Forall_isFreeIn
[784, 1]
[798, 10]
simp only [Formula.Forall_]
P : Formula xs : List VarName x : VarName ⊒ isFreeIn x (Forall_ xs P) ↔ x βˆ‰ xs ∧ isFreeIn x P
P : Formula xs : List VarName x : VarName ⊒ isFreeIn x (List.foldr forall_ P xs) ↔ x βˆ‰ xs ∧ isFreeIn x P
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/NV/Margaris/Fol.lean
FOL.NV.Forall_isFreeIn
[784, 1]
[798, 10]
induction xs
P : Formula xs : List VarName x : VarName ⊒ isFreeIn x (List.foldr forall_ P xs) ↔ x βˆ‰ xs ∧ isFreeIn x P
case nil P : Formula x : VarName ⊒ isFreeIn x (List.foldr forall_ P []) ↔ x βˆ‰ [] ∧ isFreeIn x P case cons P : Formula x head✝ : VarName tail✝ : List VarName tail_ih✝ : isFreeIn x (List.foldr forall_ P tail✝) ↔ x βˆ‰ tail✝ ∧ isFreeIn x P ⊒ isFreeIn x (List.foldr forall_ P (head✝ :: tail✝)) ↔ x βˆ‰ head✝ :: tail✝ ∧ isFreeIn x P
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/NV/Margaris/Fol.lean
FOL.NV.Forall_isFreeIn
[784, 1]
[798, 10]
case nil => simp
P : Formula x : VarName ⊒ isFreeIn x (List.foldr forall_ P []) ↔ x βˆ‰ [] ∧ isFreeIn x P
no goals
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/NV/Margaris/Fol.lean
FOL.NV.Forall_isFreeIn
[784, 1]
[798, 10]
case cons xs_hd xs_tl xs_ih => simp simp only [isFreeIn] simp only [xs_ih] tauto
P : Formula x xs_hd : VarName xs_tl : List VarName xs_ih : isFreeIn x (List.foldr forall_ P xs_tl) ↔ x βˆ‰ xs_tl ∧ isFreeIn x P ⊒ isFreeIn x (List.foldr forall_ P (xs_hd :: xs_tl)) ↔ x βˆ‰ xs_hd :: xs_tl ∧ isFreeIn x P
no goals
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/NV/Margaris/Fol.lean
FOL.NV.Forall_isFreeIn
[784, 1]
[798, 10]
simp
P : Formula x : VarName ⊒ isFreeIn x (List.foldr forall_ P []) ↔ x βˆ‰ [] ∧ isFreeIn x P
no goals
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/NV/Margaris/Fol.lean
FOL.NV.Forall_isFreeIn
[784, 1]
[798, 10]
simp
P : Formula x xs_hd : VarName xs_tl : List VarName xs_ih : isFreeIn x (List.foldr forall_ P xs_tl) ↔ x βˆ‰ xs_tl ∧ isFreeIn x P ⊒ isFreeIn x (List.foldr forall_ P (xs_hd :: xs_tl)) ↔ x βˆ‰ xs_hd :: xs_tl ∧ isFreeIn x P
P : Formula x xs_hd : VarName xs_tl : List VarName xs_ih : isFreeIn x (List.foldr forall_ P xs_tl) ↔ x βˆ‰ xs_tl ∧ isFreeIn x P ⊒ isFreeIn x (forall_ xs_hd (List.foldr forall_ P xs_tl)) ↔ (Β¬x = xs_hd ∧ x βˆ‰ xs_tl) ∧ isFreeIn x P
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/NV/Margaris/Fol.lean
FOL.NV.Forall_isFreeIn
[784, 1]
[798, 10]
simp only [isFreeIn]
P : Formula x xs_hd : VarName xs_tl : List VarName xs_ih : isFreeIn x (List.foldr forall_ P xs_tl) ↔ x βˆ‰ xs_tl ∧ isFreeIn x P ⊒ isFreeIn x (forall_ xs_hd (List.foldr forall_ P xs_tl)) ↔ (Β¬x = xs_hd ∧ x βˆ‰ xs_tl) ∧ isFreeIn x P
P : Formula x xs_hd : VarName xs_tl : List VarName xs_ih : isFreeIn x (List.foldr forall_ P xs_tl) ↔ x βˆ‰ xs_tl ∧ isFreeIn x P ⊒ Β¬x = xs_hd ∧ isFreeIn x (List.foldr forall_ P xs_tl) ↔ (Β¬x = xs_hd ∧ x βˆ‰ xs_tl) ∧ isFreeIn x P
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/NV/Margaris/Fol.lean
FOL.NV.Forall_isFreeIn
[784, 1]
[798, 10]
simp only [xs_ih]
P : Formula x xs_hd : VarName xs_tl : List VarName xs_ih : isFreeIn x (List.foldr forall_ P xs_tl) ↔ x βˆ‰ xs_tl ∧ isFreeIn x P ⊒ Β¬x = xs_hd ∧ isFreeIn x (List.foldr forall_ P xs_tl) ↔ (Β¬x = xs_hd ∧ x βˆ‰ xs_tl) ∧ isFreeIn x P
P : Formula x xs_hd : VarName xs_tl : List VarName xs_ih : isFreeIn x (List.foldr forall_ P xs_tl) ↔ x βˆ‰ xs_tl ∧ isFreeIn x P ⊒ Β¬x = xs_hd ∧ x βˆ‰ xs_tl ∧ isFreeIn x P ↔ (Β¬x = xs_hd ∧ x βˆ‰ xs_tl) ∧ isFreeIn x P
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/NV/Margaris/Fol.lean
FOL.NV.Forall_isFreeIn
[784, 1]
[798, 10]
tauto
P : Formula x xs_hd : VarName xs_tl : List VarName xs_ih : isFreeIn x (List.foldr forall_ P xs_tl) ↔ x βˆ‰ xs_tl ∧ isFreeIn x P ⊒ Β¬x = xs_hd ∧ x βˆ‰ xs_tl ∧ isFreeIn x P ↔ (Β¬x = xs_hd ∧ x βˆ‰ xs_tl) ∧ isFreeIn x P
no goals
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/NV/Margaris/Fol.lean
FOL.NV.T_18_2
[802, 1]
[886, 10]
induction h1
U V P_U P_V : Formula l : List VarName h1 : IsReplOfFormulaInFormula U V P_U P_V h2 : βˆ€ (v : VarName), (isFreeIn v U ∨ isFreeIn v V) ∧ isBoundIn v P_U β†’ v ∈ l ⊒ IsProof ((Forall_ l (U.iff_ V)).imp_ (P_U.iff_ P_V))
case same_ U V P_U P_V : Formula l : List VarName P_u✝ P_v✝ : Formula a✝ : P_u✝ = P_v✝ h2 : βˆ€ (v : VarName), (isFreeIn v U ∨ isFreeIn v V) ∧ isBoundIn v P_u✝ β†’ v ∈ l ⊒ IsProof ((Forall_ l (U.iff_ V)).imp_ (P_u✝.iff_ P_v✝)) case diff_ U V P_U P_V : Formula l : List VarName P_u✝ P_v✝ : Formula a✝¹ : P_u✝ = U a✝ : P_v✝ = V h2 : βˆ€ (v : VarName), (isFreeIn v U ∨ isFreeIn v V) ∧ isBoundIn v P_u✝ β†’ v ∈ l ⊒ IsProof ((Forall_ l (U.iff_ V)).imp_ (P_u✝.iff_ P_v✝)) case not_ U V P_U P_V : Formula l : List VarName P_u✝ P_v✝ : Formula a✝ : IsReplOfFormulaInFormula U V P_u✝ P_v✝ a_ih✝ : (βˆ€ (v : VarName), (isFreeIn v U ∨ isFreeIn v V) ∧ isBoundIn v P_u✝ β†’ v ∈ l) β†’ IsProof ((Forall_ l (U.iff_ V)).imp_ (P_u✝.iff_ P_v✝)) h2 : βˆ€ (v : VarName), (isFreeIn v U ∨ isFreeIn v V) ∧ isBoundIn v P_u✝.not_ β†’ v ∈ l ⊒ IsProof ((Forall_ l (U.iff_ V)).imp_ (P_u✝.not_.iff_ P_v✝.not_)) case imp_ U V P_U P_V : Formula l : List VarName P_u✝ Q_u✝ P_v✝ Q_v✝ : Formula a✝¹ : IsReplOfFormulaInFormula U V P_u✝ P_v✝ a✝ : IsReplOfFormulaInFormula U V Q_u✝ Q_v✝ a_ih✝¹ : (βˆ€ (v : VarName), (isFreeIn v U ∨ isFreeIn v V) ∧ isBoundIn v P_u✝ β†’ v ∈ l) β†’ IsProof ((Forall_ l (U.iff_ V)).imp_ (P_u✝.iff_ P_v✝)) a_ih✝ : (βˆ€ (v : VarName), (isFreeIn v U ∨ isFreeIn v V) ∧ isBoundIn v Q_u✝ β†’ v ∈ l) β†’ IsProof ((Forall_ l (U.iff_ V)).imp_ (Q_u✝.iff_ Q_v✝)) h2 : βˆ€ (v : VarName), (isFreeIn v U ∨ isFreeIn v V) ∧ isBoundIn v (P_u✝.imp_ Q_u✝) β†’ v ∈ l ⊒ IsProof ((Forall_ l (U.iff_ V)).imp_ ((P_u✝.imp_ Q_u✝).iff_ (P_v✝.imp_ Q_v✝))) case and_ U V P_U P_V : Formula l : List VarName P_u✝ Q_u✝ P_v✝ Q_v✝ : Formula a✝¹ : IsReplOfFormulaInFormula U V P_u✝ P_v✝ a✝ : IsReplOfFormulaInFormula U V Q_u✝ Q_v✝ a_ih✝¹ : (βˆ€ (v : VarName), (isFreeIn v U ∨ isFreeIn v V) ∧ isBoundIn v P_u✝ β†’ v ∈ l) β†’ IsProof ((Forall_ l (U.iff_ V)).imp_ (P_u✝.iff_ P_v✝)) a_ih✝ : (βˆ€ (v : VarName), (isFreeIn v U ∨ isFreeIn v V) ∧ isBoundIn v Q_u✝ β†’ v ∈ l) β†’ IsProof ((Forall_ l (U.iff_ V)).imp_ (Q_u✝.iff_ Q_v✝)) h2 : βˆ€ (v : VarName), (isFreeIn v U ∨ isFreeIn v V) ∧ isBoundIn v (P_u✝.and_ Q_u✝) β†’ v ∈ l ⊒ IsProof ((Forall_ l (U.iff_ V)).imp_ ((P_u✝.and_ Q_u✝).iff_ (P_v✝.and_ Q_v✝))) case or_ U V P_U P_V : Formula l : List VarName P_u✝ Q_u✝ P_v✝ Q_v✝ : Formula a✝¹ : IsReplOfFormulaInFormula U V P_u✝ P_v✝ a✝ : IsReplOfFormulaInFormula U V Q_u✝ Q_v✝ a_ih✝¹ : (βˆ€ (v : VarName), (isFreeIn v U ∨ isFreeIn v V) ∧ isBoundIn v P_u✝ β†’ v ∈ l) β†’ IsProof ((Forall_ l (U.iff_ V)).imp_ (P_u✝.iff_ P_v✝)) a_ih✝ : (βˆ€ (v : VarName), (isFreeIn v U ∨ isFreeIn v V) ∧ isBoundIn v Q_u✝ β†’ v ∈ l) β†’ IsProof ((Forall_ l (U.iff_ V)).imp_ (Q_u✝.iff_ Q_v✝)) h2 : βˆ€ (v : VarName), (isFreeIn v U ∨ isFreeIn v V) ∧ isBoundIn v (P_u✝.or_ Q_u✝) β†’ v ∈ l ⊒ IsProof ((Forall_ l (U.iff_ V)).imp_ ((P_u✝.or_ Q_u✝).iff_ (P_v✝.or_ Q_v✝))) case iff_ U V P_U P_V : Formula l : List VarName P_u✝ Q_u✝ P_v✝ Q_v✝ : Formula a✝¹ : IsReplOfFormulaInFormula U V P_u✝ P_v✝ a✝ : IsReplOfFormulaInFormula U V Q_u✝ Q_v✝ a_ih✝¹ : (βˆ€ (v : VarName), (isFreeIn v U ∨ isFreeIn v V) ∧ isBoundIn v P_u✝ β†’ v ∈ l) β†’ IsProof ((Forall_ l (U.iff_ V)).imp_ (P_u✝.iff_ P_v✝)) a_ih✝ : (βˆ€ (v : VarName), (isFreeIn v U ∨ isFreeIn v V) ∧ isBoundIn v Q_u✝ β†’ v ∈ l) β†’ IsProof ((Forall_ l (U.iff_ V)).imp_ (Q_u✝.iff_ Q_v✝)) h2 : βˆ€ (v : VarName), (isFreeIn v U ∨ isFreeIn v V) ∧ isBoundIn v (P_u✝.iff_ Q_u✝) β†’ v ∈ l ⊒ IsProof ((Forall_ l (U.iff_ V)).imp_ ((P_u✝.iff_ Q_u✝).iff_ (P_v✝.iff_ Q_v✝))) case forall_ U V P_U P_V : Formula l : List VarName x✝ : VarName P_u✝ P_v✝ : Formula a✝ : IsReplOfFormulaInFormula U V P_u✝ P_v✝ a_ih✝ : (βˆ€ (v : VarName), (isFreeIn v U ∨ isFreeIn v V) ∧ isBoundIn v P_u✝ β†’ v ∈ l) β†’ IsProof ((Forall_ l (U.iff_ V)).imp_ (P_u✝.iff_ P_v✝)) h2 : βˆ€ (v : VarName), (isFreeIn v U ∨ isFreeIn v V) ∧ isBoundIn v (forall_ x✝ P_u✝) β†’ v ∈ l ⊒ IsProof ((Forall_ l (U.iff_ V)).imp_ ((forall_ x✝ P_u✝).iff_ (forall_ x✝ P_v✝))) case exists_ U V P_U P_V : Formula l : List VarName x✝ : VarName P_u✝ P_v✝ : Formula a✝ : IsReplOfFormulaInFormula U V P_u✝ P_v✝ a_ih✝ : (βˆ€ (v : VarName), (isFreeIn v U ∨ isFreeIn v V) ∧ isBoundIn v P_u✝ β†’ v ∈ l) β†’ IsProof ((Forall_ l (U.iff_ V)).imp_ (P_u✝.iff_ P_v✝)) h2 : βˆ€ (v : VarName), (isFreeIn v U ∨ isFreeIn v V) ∧ isBoundIn v (exists_ x✝ P_u✝) β†’ v ∈ l ⊒ IsProof ((Forall_ l (U.iff_ V)).imp_ ((exists_ x✝ P_u✝).iff_ (exists_ x✝ P_v✝)))
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/NV/Margaris/Fol.lean
FOL.NV.T_18_2
[802, 1]
[886, 10]
case same_ h1_P h1_P' h1_1 => subst h1_1 simp only [def_iff_] simp only [def_and_] SC
U V P_U P_V : Formula l : List VarName h1_P h1_P' : Formula h1_1 : h1_P = h1_P' h2 : βˆ€ (v : VarName), (isFreeIn v U ∨ isFreeIn v V) ∧ isBoundIn v h1_P β†’ v ∈ l ⊒ IsProof ((Forall_ l (U.iff_ V)).imp_ (h1_P.iff_ h1_P'))
no goals
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/NV/Margaris/Fol.lean
FOL.NV.T_18_2
[802, 1]
[886, 10]
case diff_ h1_P h1_P' h1_1 h1_2 => subst h1_1 subst h1_2 apply Forall_spec_id
U V P_U P_V : Formula l : List VarName h1_P h1_P' : Formula h1_1 : h1_P = U h1_2 : h1_P' = V h2 : βˆ€ (v : VarName), (isFreeIn v U ∨ isFreeIn v V) ∧ isBoundIn v h1_P β†’ v ∈ l ⊒ IsProof ((Forall_ l (U.iff_ V)).imp_ (h1_P.iff_ h1_P'))
no goals
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/NV/Margaris/Fol.lean
FOL.NV.T_18_2
[802, 1]
[886, 10]
all_goals sorry
case and_ U V P_U P_V : Formula l : List VarName P_u✝ Q_u✝ P_v✝ Q_v✝ : Formula a✝¹ : IsReplOfFormulaInFormula U V P_u✝ P_v✝ a✝ : IsReplOfFormulaInFormula U V Q_u✝ Q_v✝ a_ih✝¹ : (βˆ€ (v : VarName), (isFreeIn v U ∨ isFreeIn v V) ∧ isBoundIn v P_u✝ β†’ v ∈ l) β†’ IsProof ((Forall_ l (U.iff_ V)).imp_ (P_u✝.iff_ P_v✝)) a_ih✝ : (βˆ€ (v : VarName), (isFreeIn v U ∨ isFreeIn v V) ∧ isBoundIn v Q_u✝ β†’ v ∈ l) β†’ IsProof ((Forall_ l (U.iff_ V)).imp_ (Q_u✝.iff_ Q_v✝)) h2 : βˆ€ (v : VarName), (isFreeIn v U ∨ isFreeIn v V) ∧ isBoundIn v (P_u✝.and_ Q_u✝) β†’ v ∈ l ⊒ IsProof ((Forall_ l (U.iff_ V)).imp_ ((P_u✝.and_ Q_u✝).iff_ (P_v✝.and_ Q_v✝))) case or_ U V P_U P_V : Formula l : List VarName P_u✝ Q_u✝ P_v✝ Q_v✝ : Formula a✝¹ : IsReplOfFormulaInFormula U V P_u✝ P_v✝ a✝ : IsReplOfFormulaInFormula U V Q_u✝ Q_v✝ a_ih✝¹ : (βˆ€ (v : VarName), (isFreeIn v U ∨ isFreeIn v V) ∧ isBoundIn v P_u✝ β†’ v ∈ l) β†’ IsProof ((Forall_ l (U.iff_ V)).imp_ (P_u✝.iff_ P_v✝)) a_ih✝ : (βˆ€ (v : VarName), (isFreeIn v U ∨ isFreeIn v V) ∧ isBoundIn v Q_u✝ β†’ v ∈ l) β†’ IsProof ((Forall_ l (U.iff_ V)).imp_ (Q_u✝.iff_ Q_v✝)) h2 : βˆ€ (v : VarName), (isFreeIn v U ∨ isFreeIn v V) ∧ isBoundIn v (P_u✝.or_ Q_u✝) β†’ v ∈ l ⊒ IsProof ((Forall_ l (U.iff_ V)).imp_ ((P_u✝.or_ Q_u✝).iff_ (P_v✝.or_ Q_v✝))) case iff_ U V P_U P_V : Formula l : List VarName P_u✝ Q_u✝ P_v✝ Q_v✝ : Formula a✝¹ : IsReplOfFormulaInFormula U V P_u✝ P_v✝ a✝ : IsReplOfFormulaInFormula U V Q_u✝ Q_v✝ a_ih✝¹ : (βˆ€ (v : VarName), (isFreeIn v U ∨ isFreeIn v V) ∧ isBoundIn v P_u✝ β†’ v ∈ l) β†’ IsProof ((Forall_ l (U.iff_ V)).imp_ (P_u✝.iff_ P_v✝)) a_ih✝ : (βˆ€ (v : VarName), (isFreeIn v U ∨ isFreeIn v V) ∧ isBoundIn v Q_u✝ β†’ v ∈ l) β†’ IsProof ((Forall_ l (U.iff_ V)).imp_ (Q_u✝.iff_ Q_v✝)) h2 : βˆ€ (v : VarName), (isFreeIn v U ∨ isFreeIn v V) ∧ isBoundIn v (P_u✝.iff_ Q_u✝) β†’ v ∈ l ⊒ IsProof ((Forall_ l (U.iff_ V)).imp_ ((P_u✝.iff_ Q_u✝).iff_ (P_v✝.iff_ Q_v✝))) case exists_ U V P_U P_V : Formula l : List VarName x✝ : VarName P_u✝ P_v✝ : Formula a✝ : IsReplOfFormulaInFormula U V P_u✝ P_v✝ a_ih✝ : (βˆ€ (v : VarName), (isFreeIn v U ∨ isFreeIn v V) ∧ isBoundIn v P_u✝ β†’ v ∈ l) β†’ IsProof ((Forall_ l (U.iff_ V)).imp_ (P_u✝.iff_ P_v✝)) h2 : βˆ€ (v : VarName), (isFreeIn v U ∨ isFreeIn v V) ∧ isBoundIn v (exists_ x✝ P_u✝) β†’ v ∈ l ⊒ IsProof ((Forall_ l (U.iff_ V)).imp_ ((exists_ x✝ P_u✝).iff_ (exists_ x✝ P_v✝)))
no goals
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/NV/Margaris/Fol.lean
FOL.NV.T_18_2
[802, 1]
[886, 10]
subst h1_1
U V P_U P_V : Formula l : List VarName h1_P h1_P' : Formula h1_1 : h1_P = h1_P' h2 : βˆ€ (v : VarName), (isFreeIn v U ∨ isFreeIn v V) ∧ isBoundIn v h1_P β†’ v ∈ l ⊒ IsProof ((Forall_ l (U.iff_ V)).imp_ (h1_P.iff_ h1_P'))
U V P_U P_V : Formula l : List VarName h1_P : Formula h2 : βˆ€ (v : VarName), (isFreeIn v U ∨ isFreeIn v V) ∧ isBoundIn v h1_P β†’ v ∈ l ⊒ IsProof ((Forall_ l (U.iff_ V)).imp_ (h1_P.iff_ h1_P))
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/NV/Margaris/Fol.lean
FOL.NV.T_18_2
[802, 1]
[886, 10]
simp only [def_iff_]
U V P_U P_V : Formula l : List VarName h1_P : Formula h2 : βˆ€ (v : VarName), (isFreeIn v U ∨ isFreeIn v V) ∧ isBoundIn v h1_P β†’ v ∈ l ⊒ IsProof ((Forall_ l (U.iff_ V)).imp_ (h1_P.iff_ h1_P))
U V P_U P_V : Formula l : List VarName h1_P : Formula h2 : βˆ€ (v : VarName), (isFreeIn v U ∨ isFreeIn v V) ∧ isBoundIn v h1_P β†’ v ∈ l ⊒ IsProof ((Forall_ l ((U.imp_ V).and_ (V.imp_ U))).imp_ ((h1_P.imp_ h1_P).and_ (h1_P.imp_ h1_P)))
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/NV/Margaris/Fol.lean
FOL.NV.T_18_2
[802, 1]
[886, 10]
simp only [def_and_]
U V P_U P_V : Formula l : List VarName h1_P : Formula h2 : βˆ€ (v : VarName), (isFreeIn v U ∨ isFreeIn v V) ∧ isBoundIn v h1_P β†’ v ∈ l ⊒ IsProof ((Forall_ l ((U.imp_ V).and_ (V.imp_ U))).imp_ ((h1_P.imp_ h1_P).and_ (h1_P.imp_ h1_P)))
U V P_U P_V : Formula l : List VarName h1_P : Formula h2 : βˆ€ (v : VarName), (isFreeIn v U ∨ isFreeIn v V) ∧ isBoundIn v h1_P β†’ v ∈ l ⊒ IsProof ((Forall_ l ((U.imp_ V).imp_ (V.imp_ U).not_).not_).imp_ ((h1_P.imp_ h1_P).imp_ (h1_P.imp_ h1_P).not_).not_)
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/NV/Margaris/Fol.lean
FOL.NV.T_18_2
[802, 1]
[886, 10]
SC
U V P_U P_V : Formula l : List VarName h1_P : Formula h2 : βˆ€ (v : VarName), (isFreeIn v U ∨ isFreeIn v V) ∧ isBoundIn v h1_P β†’ v ∈ l ⊒ IsProof ((Forall_ l ((U.imp_ V).imp_ (V.imp_ U).not_).not_).imp_ ((h1_P.imp_ h1_P).imp_ (h1_P.imp_ h1_P).not_).not_)
no goals
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/NV/Margaris/Fol.lean
FOL.NV.T_18_2
[802, 1]
[886, 10]
subst h1_1
U V P_U P_V : Formula l : List VarName h1_P h1_P' : Formula h1_1 : h1_P = U h1_2 : h1_P' = V h2 : βˆ€ (v : VarName), (isFreeIn v U ∨ isFreeIn v V) ∧ isBoundIn v h1_P β†’ v ∈ l ⊒ IsProof ((Forall_ l (U.iff_ V)).imp_ (h1_P.iff_ h1_P'))
V P_U P_V : Formula l : List VarName h1_P h1_P' : Formula h1_2 : h1_P' = V h2 : βˆ€ (v : VarName), (isFreeIn v h1_P ∨ isFreeIn v V) ∧ isBoundIn v h1_P β†’ v ∈ l ⊒ IsProof ((Forall_ l (h1_P.iff_ V)).imp_ (h1_P.iff_ h1_P'))
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/NV/Margaris/Fol.lean
FOL.NV.T_18_2
[802, 1]
[886, 10]
subst h1_2
V P_U P_V : Formula l : List VarName h1_P h1_P' : Formula h1_2 : h1_P' = V h2 : βˆ€ (v : VarName), (isFreeIn v h1_P ∨ isFreeIn v V) ∧ isBoundIn v h1_P β†’ v ∈ l ⊒ IsProof ((Forall_ l (h1_P.iff_ V)).imp_ (h1_P.iff_ h1_P'))
P_U P_V : Formula l : List VarName h1_P h1_P' : Formula h2 : βˆ€ (v : VarName), (isFreeIn v h1_P ∨ isFreeIn v h1_P') ∧ isBoundIn v h1_P β†’ v ∈ l ⊒ IsProof ((Forall_ l (h1_P.iff_ h1_P')).imp_ (h1_P.iff_ h1_P'))
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/NV/Margaris/Fol.lean
FOL.NV.T_18_2
[802, 1]
[886, 10]
apply Forall_spec_id
P_U P_V : Formula l : List VarName h1_P h1_P' : Formula h2 : βˆ€ (v : VarName), (isFreeIn v h1_P ∨ isFreeIn v h1_P') ∧ isBoundIn v h1_P β†’ v ∈ l ⊒ IsProof ((Forall_ l (h1_P.iff_ h1_P')).imp_ (h1_P.iff_ h1_P'))
no goals
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/NV/Margaris/Fol.lean
FOL.NV.T_18_2
[802, 1]
[886, 10]
simp only [isBoundIn] at h2
U V P_U P_V : Formula l : List VarName h1_P h1_P' : Formula h1_1 : IsReplOfFormulaInFormula U V h1_P h1_P' h1_ih : (βˆ€ (v : VarName), (isFreeIn v U ∨ isFreeIn v V) ∧ isBoundIn v h1_P β†’ v ∈ l) β†’ IsProof ((Forall_ l (U.iff_ V)).imp_ (h1_P.iff_ h1_P')) h2 : βˆ€ (v : VarName), (isFreeIn v U ∨ isFreeIn v V) ∧ isBoundIn v h1_P.not_ β†’ v ∈ l ⊒ IsProof ((Forall_ l (U.iff_ V)).imp_ (h1_P.not_.iff_ h1_P'.not_))
U V P_U P_V : Formula l : List VarName h1_P h1_P' : Formula h1_1 : IsReplOfFormulaInFormula U V h1_P h1_P' h1_ih : (βˆ€ (v : VarName), (isFreeIn v U ∨ isFreeIn v V) ∧ isBoundIn v h1_P β†’ v ∈ l) β†’ IsProof ((Forall_ l (U.iff_ V)).imp_ (h1_P.iff_ h1_P')) h2 : βˆ€ (v : VarName), (isFreeIn v U ∨ isFreeIn v V) ∧ isBoundIn v h1_P β†’ v ∈ l ⊒ IsProof ((Forall_ l (U.iff_ V)).imp_ (h1_P.not_.iff_ h1_P'.not_))
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/NV/Margaris/Fol.lean
FOL.NV.T_18_2
[802, 1]
[886, 10]
apply IsDeduct.mp_ ((Forall_ l (U.iff_ V)).imp_ (h1_P.iff_ h1_P'))
U V P_U P_V : Formula l : List VarName h1_P h1_P' : Formula h1_1 : IsReplOfFormulaInFormula U V h1_P h1_P' h1_ih : (βˆ€ (v : VarName), (isFreeIn v U ∨ isFreeIn v V) ∧ isBoundIn v h1_P β†’ v ∈ l) β†’ IsProof ((Forall_ l (U.iff_ V)).imp_ (h1_P.iff_ h1_P')) h2 : βˆ€ (v : VarName), (isFreeIn v U ∨ isFreeIn v V) ∧ isBoundIn v h1_P β†’ v ∈ l ⊒ IsProof ((Forall_ l (U.iff_ V)).imp_ (h1_P.not_.iff_ h1_P'.not_))
case a U V P_U P_V : Formula l : List VarName h1_P h1_P' : Formula h1_1 : IsReplOfFormulaInFormula U V h1_P h1_P' h1_ih : (βˆ€ (v : VarName), (isFreeIn v U ∨ isFreeIn v V) ∧ isBoundIn v h1_P β†’ v ∈ l) β†’ IsProof ((Forall_ l (U.iff_ V)).imp_ (h1_P.iff_ h1_P')) h2 : βˆ€ (v : VarName), (isFreeIn v U ∨ isFreeIn v V) ∧ isBoundIn v h1_P β†’ v ∈ l ⊒ IsDeduct βˆ… (((Forall_ l (U.iff_ V)).imp_ (h1_P.iff_ h1_P')).imp_ ((Forall_ l (U.iff_ V)).imp_ (h1_P.not_.iff_ h1_P'.not_))) case a U V P_U P_V : Formula l : List VarName h1_P h1_P' : Formula h1_1 : IsReplOfFormulaInFormula U V h1_P h1_P' h1_ih : (βˆ€ (v : VarName), (isFreeIn v U ∨ isFreeIn v V) ∧ isBoundIn v h1_P β†’ v ∈ l) β†’ IsProof ((Forall_ l (U.iff_ V)).imp_ (h1_P.iff_ h1_P')) h2 : βˆ€ (v : VarName), (isFreeIn v U ∨ isFreeIn v V) ∧ isBoundIn v h1_P β†’ v ∈ l ⊒ IsDeduct βˆ… ((Forall_ l (U.iff_ V)).imp_ (h1_P.iff_ h1_P'))
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/NV/Margaris/Fol.lean
FOL.NV.T_18_2
[802, 1]
[886, 10]
simp only [def_iff_]
case a U V P_U P_V : Formula l : List VarName h1_P h1_P' : Formula h1_1 : IsReplOfFormulaInFormula U V h1_P h1_P' h1_ih : (βˆ€ (v : VarName), (isFreeIn v U ∨ isFreeIn v V) ∧ isBoundIn v h1_P β†’ v ∈ l) β†’ IsProof ((Forall_ l (U.iff_ V)).imp_ (h1_P.iff_ h1_P')) h2 : βˆ€ (v : VarName), (isFreeIn v U ∨ isFreeIn v V) ∧ isBoundIn v h1_P β†’ v ∈ l ⊒ IsDeduct βˆ… (((Forall_ l (U.iff_ V)).imp_ (h1_P.iff_ h1_P')).imp_ ((Forall_ l (U.iff_ V)).imp_ (h1_P.not_.iff_ h1_P'.not_)))
case a U V P_U P_V : Formula l : List VarName h1_P h1_P' : Formula h1_1 : IsReplOfFormulaInFormula U V h1_P h1_P' h1_ih : (βˆ€ (v : VarName), (isFreeIn v U ∨ isFreeIn v V) ∧ isBoundIn v h1_P β†’ v ∈ l) β†’ IsProof ((Forall_ l (U.iff_ V)).imp_ (h1_P.iff_ h1_P')) h2 : βˆ€ (v : VarName), (isFreeIn v U ∨ isFreeIn v V) ∧ isBoundIn v h1_P β†’ v ∈ l ⊒ IsDeduct βˆ… (((Forall_ l ((U.imp_ V).and_ (V.imp_ U))).imp_ ((h1_P.imp_ h1_P').and_ (h1_P'.imp_ h1_P))).imp_ ((Forall_ l ((U.imp_ V).and_ (V.imp_ U))).imp_ ((h1_P.not_.imp_ h1_P'.not_).and_ (h1_P'.not_.imp_ h1_P.not_))))
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/NV/Margaris/Fol.lean
FOL.NV.T_18_2
[802, 1]
[886, 10]
simp only [def_and_]
case a U V P_U P_V : Formula l : List VarName h1_P h1_P' : Formula h1_1 : IsReplOfFormulaInFormula U V h1_P h1_P' h1_ih : (βˆ€ (v : VarName), (isFreeIn v U ∨ isFreeIn v V) ∧ isBoundIn v h1_P β†’ v ∈ l) β†’ IsProof ((Forall_ l (U.iff_ V)).imp_ (h1_P.iff_ h1_P')) h2 : βˆ€ (v : VarName), (isFreeIn v U ∨ isFreeIn v V) ∧ isBoundIn v h1_P β†’ v ∈ l ⊒ IsDeduct βˆ… (((Forall_ l ((U.imp_ V).and_ (V.imp_ U))).imp_ ((h1_P.imp_ h1_P').and_ (h1_P'.imp_ h1_P))).imp_ ((Forall_ l ((U.imp_ V).and_ (V.imp_ U))).imp_ ((h1_P.not_.imp_ h1_P'.not_).and_ (h1_P'.not_.imp_ h1_P.not_))))
case a U V P_U P_V : Formula l : List VarName h1_P h1_P' : Formula h1_1 : IsReplOfFormulaInFormula U V h1_P h1_P' h1_ih : (βˆ€ (v : VarName), (isFreeIn v U ∨ isFreeIn v V) ∧ isBoundIn v h1_P β†’ v ∈ l) β†’ IsProof ((Forall_ l (U.iff_ V)).imp_ (h1_P.iff_ h1_P')) h2 : βˆ€ (v : VarName), (isFreeIn v U ∨ isFreeIn v V) ∧ isBoundIn v h1_P β†’ v ∈ l ⊒ IsDeduct βˆ… (((Forall_ l ((U.imp_ V).imp_ (V.imp_ U).not_).not_).imp_ ((h1_P.imp_ h1_P').imp_ (h1_P'.imp_ h1_P).not_).not_).imp_ ((Forall_ l ((U.imp_ V).imp_ (V.imp_ U).not_).not_).imp_ ((h1_P.not_.imp_ h1_P'.not_).imp_ (h1_P'.not_.imp_ h1_P.not_).not_).not_))
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/NV/Margaris/Fol.lean
FOL.NV.T_18_2
[802, 1]
[886, 10]
SC
case a U V P_U P_V : Formula l : List VarName h1_P h1_P' : Formula h1_1 : IsReplOfFormulaInFormula U V h1_P h1_P' h1_ih : (βˆ€ (v : VarName), (isFreeIn v U ∨ isFreeIn v V) ∧ isBoundIn v h1_P β†’ v ∈ l) β†’ IsProof ((Forall_ l (U.iff_ V)).imp_ (h1_P.iff_ h1_P')) h2 : βˆ€ (v : VarName), (isFreeIn v U ∨ isFreeIn v V) ∧ isBoundIn v h1_P β†’ v ∈ l ⊒ IsDeduct βˆ… (((Forall_ l ((U.imp_ V).imp_ (V.imp_ U).not_).not_).imp_ ((h1_P.imp_ h1_P').imp_ (h1_P'.imp_ h1_P).not_).not_).imp_ ((Forall_ l ((U.imp_ V).imp_ (V.imp_ U).not_).not_).imp_ ((h1_P.not_.imp_ h1_P'.not_).imp_ (h1_P'.not_.imp_ h1_P.not_).not_).not_))
no goals
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/NV/Margaris/Fol.lean
FOL.NV.T_18_2
[802, 1]
[886, 10]
exact h1_ih h2
case a U V P_U P_V : Formula l : List VarName h1_P h1_P' : Formula h1_1 : IsReplOfFormulaInFormula U V h1_P h1_P' h1_ih : (βˆ€ (v : VarName), (isFreeIn v U ∨ isFreeIn v V) ∧ isBoundIn v h1_P β†’ v ∈ l) β†’ IsProof ((Forall_ l (U.iff_ V)).imp_ (h1_P.iff_ h1_P')) h2 : βˆ€ (v : VarName), (isFreeIn v U ∨ isFreeIn v V) ∧ isBoundIn v h1_P β†’ v ∈ l ⊒ IsDeduct βˆ… ((Forall_ l (U.iff_ V)).imp_ (h1_P.iff_ h1_P'))
no goals