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_21_1
[1345, 1]
[1370, 15]
apply generalization
case h1 x y : VarName ⊢ IsDeduct ∅ (forall_ y ((eq_ x y).imp_ (eq_ y x)))
case h1.h1 x y : VarName ⊢ IsDeduct ∅ ((eq_ x y).imp_ (eq_ y x)) case h1.h2 x y : VarName ⊢ ∀ H ∈ ∅, ¬isFreeIn y H
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/NV/Margaris/Fol.lean
FOL.NV.T_21_1
[1345, 1]
[1370, 15]
apply IsDeduct.mp_ (eq_ y y)
case h1.h1 x y : VarName ⊢ IsDeduct ∅ ((eq_ x y).imp_ (eq_ y x))
case h1.h1.a x y : VarName ⊢ IsDeduct ∅ ((eq_ y y).imp_ ((eq_ x y).imp_ (eq_ y x))) case h1.h1.a x y : VarName ⊢ IsDeduct ∅ (eq_ y y)
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/NV/Margaris/Fol.lean
FOL.NV.T_21_1
[1345, 1]
[1370, 15]
apply IsDeduct.mp_ (((eq_ y y).and_ (eq_ x y)).imp_ ((eq_ y x).iff_ (eq_ y y)))
case h1.h1.a x y : VarName ⊢ IsDeduct ∅ ((eq_ y y).imp_ ((eq_ x y).imp_ (eq_ y x)))
case h1.h1.a.a x y : VarName ⊢ IsDeduct ∅ ((((eq_ y y).and_ (eq_ x y)).imp_ ((eq_ y x).iff_ (eq_ y y))).imp_ ((eq_ y y).imp_ ((eq_ x y).imp_ (eq_ y x)))) case h1.h1.a.a x y : VarName ⊢ IsDeduct ∅ (((eq_ y y).and_ (eq_ x y)).imp_ ((eq_ y x).iff_ (eq_ y y)))
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/NV/Margaris/Fol.lean
FOL.NV.T_21_1
[1345, 1]
[1370, 15]
simp only [def_iff_]
case h1.h1.a.a x y : VarName ⊢ IsDeduct ∅ ((((eq_ y y).and_ (eq_ x y)).imp_ ((eq_ y x).iff_ (eq_ y y))).imp_ ((eq_ y y).imp_ ((eq_ x y).imp_ (eq_ y x))))
case h1.h1.a.a x y : VarName ⊢ IsDeduct ∅ ((((eq_ y y).and_ (eq_ x y)).imp_ (((eq_ y x).imp_ (eq_ y y)).and_ ((eq_ y y).imp_ (eq_ y x)))).imp_ ((eq_ y y).imp_ ((eq_ x y).imp_ (eq_ y x))))
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/NV/Margaris/Fol.lean
FOL.NV.T_21_1
[1345, 1]
[1370, 15]
simp only [def_and_]
case h1.h1.a.a x y : VarName ⊢ IsDeduct ∅ ((((eq_ y y).and_ (eq_ x y)).imp_ (((eq_ y x).imp_ (eq_ y y)).and_ ((eq_ y y).imp_ (eq_ y x)))).imp_ ((eq_ y y).imp_ ((eq_ x y).imp_ (eq_ y x))))
case h1.h1.a.a x y : VarName ⊢ IsDeduct ∅ ((((eq_ y y).imp_ (eq_ x y).not_).not_.imp_ (((eq_ y x).imp_ (eq_ y y)).imp_ ((eq_ y y).imp_ (eq_ y x)).not_).not_).imp_ ((eq_ y y).imp_ ((eq_ x y).imp_ (eq_ y x))))
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/NV/Margaris/Fol.lean
FOL.NV.T_21_1
[1345, 1]
[1370, 15]
SC
case h1.h1.a.a x y : VarName ⊢ IsDeduct ∅ ((((eq_ y y).imp_ (eq_ x y).not_).not_.imp_ (((eq_ y x).imp_ (eq_ y y)).imp_ ((eq_ y y).imp_ (eq_ y x)).not_).not_).imp_ ((eq_ y y).imp_ ((eq_ x y).imp_ (eq_ y x))))
no goals
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/NV/Margaris/Fol.lean
FOL.NV.T_21_1
[1345, 1]
[1370, 15]
apply specId y
case h1.h1.a.a x y : VarName ⊢ IsDeduct ∅ (((eq_ y y).and_ (eq_ x y)).imp_ ((eq_ y x).iff_ (eq_ y y)))
case h1.h1.a.a.h1 x y : VarName ⊢ IsDeduct ∅ (forall_ y (((eq_ y y).and_ (eq_ x y)).imp_ ((eq_ y x).iff_ (eq_ y y))))
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/NV/Margaris/Fol.lean
FOL.NV.T_21_1
[1345, 1]
[1370, 15]
apply specId y
case h1.h1.a.a.h1 x y : VarName ⊢ IsDeduct ∅ (forall_ y (((eq_ y y).and_ (eq_ x y)).imp_ ((eq_ y x).iff_ (eq_ y y))))
case h1.h1.a.a.h1.h1 x y : VarName ⊢ IsDeduct ∅ (forall_ y (forall_ y (((eq_ y y).and_ (eq_ x y)).imp_ ((eq_ y x).iff_ (eq_ y y)))))
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/NV/Margaris/Fol.lean
FOL.NV.T_21_1
[1345, 1]
[1370, 15]
apply specId x
case h1.h1.a.a.h1.h1 x y : VarName ⊢ IsDeduct ∅ (forall_ y (forall_ y (((eq_ y y).and_ (eq_ x y)).imp_ ((eq_ y x).iff_ (eq_ y y)))))
case h1.h1.a.a.h1.h1.h1 x y : VarName ⊢ IsDeduct ∅ (forall_ x (forall_ y (forall_ y (((eq_ y y).and_ (eq_ x y)).imp_ ((eq_ y x).iff_ (eq_ y y))))))
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/NV/Margaris/Fol.lean
FOL.NV.T_21_1
[1345, 1]
[1370, 15]
apply specId y
case h1.h1.a.a.h1.h1.h1 x y : VarName ⊢ IsDeduct ∅ (forall_ x (forall_ y (forall_ y (((eq_ y y).and_ (eq_ x y)).imp_ ((eq_ y x).iff_ (eq_ y y))))))
case h1.h1.a.a.h1.h1.h1.h1 x y : VarName ⊢ IsDeduct ∅ (forall_ y (forall_ x (forall_ y (forall_ y (((eq_ y y).and_ (eq_ x y)).imp_ ((eq_ y x).iff_ (eq_ y y)))))))
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/NV/Margaris/Fol.lean
FOL.NV.T_21_1
[1345, 1]
[1370, 15]
apply IsDeduct.axiom_
case h1.h1.a.a.h1.h1.h1.h1 x y : VarName ⊢ IsDeduct ∅ (forall_ y (forall_ x (forall_ y (forall_ y (((eq_ y y).and_ (eq_ x y)).imp_ ((eq_ y x).iff_ (eq_ y y)))))))
case h1.h1.a.a.h1.h1.h1.h1.a x y : VarName ⊢ IsAxiom (forall_ y (forall_ x (forall_ y (forall_ y (((eq_ y y).and_ (eq_ x y)).imp_ ((eq_ y x).iff_ (eq_ y y)))))))
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/NV/Margaris/Fol.lean
FOL.NV.T_21_1
[1345, 1]
[1370, 15]
exact IsAxiom.eq_2_eq_ y x y y
case h1.h1.a.a.h1.h1.h1.h1.a x y : VarName ⊢ IsAxiom (forall_ y (forall_ x (forall_ y (forall_ y (((eq_ y y).and_ (eq_ x y)).imp_ ((eq_ y x).iff_ (eq_ y y)))))))
no goals
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/NV/Margaris/Fol.lean
FOL.NV.T_21_1
[1345, 1]
[1370, 15]
apply specId y
case h1.h1.a x y : VarName ⊢ IsDeduct ∅ (eq_ y y)
case h1.h1.a.h1 x y : VarName ⊢ IsDeduct ∅ (forall_ y (eq_ y y))
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/NV/Margaris/Fol.lean
FOL.NV.T_21_1
[1345, 1]
[1370, 15]
apply IsDeduct.axiom_
case h1.h1.a.h1 x y : VarName ⊢ IsDeduct ∅ (forall_ y (eq_ y y))
case h1.h1.a.h1.a x y : VarName ⊢ IsAxiom (forall_ y (eq_ y y))
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/NV/Margaris/Fol.lean
FOL.NV.T_21_1
[1345, 1]
[1370, 15]
exact IsAxiom.eq_1_ y
case h1.h1.a.h1.a x y : VarName ⊢ IsAxiom (forall_ y (eq_ y y))
no goals
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/NV/Margaris/Fol.lean
FOL.NV.T_21_1
[1345, 1]
[1370, 15]
intro H a1
case h1.h2 x y : VarName ⊢ ∀ H ∈ ∅, ¬isFreeIn y H
case h1.h2 x y : VarName H : Formula a1 : H ∈ ∅ ⊢ ¬isFreeIn y H
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/NV/Margaris/Fol.lean
FOL.NV.T_21_1
[1345, 1]
[1370, 15]
simp at a1
case h1.h2 x y : VarName H : Formula a1 : H ∈ ∅ ⊢ ¬isFreeIn y H
no goals
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/NV/Margaris/Fol.lean
FOL.NV.T_21_1
[1345, 1]
[1370, 15]
intro H a1
case h2 x y : VarName ⊢ ∀ H ∈ ∅, ¬isFreeIn x H
case h2 x y : VarName H : Formula a1 : H ∈ ∅ ⊢ ¬isFreeIn x H
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/NV/Margaris/Fol.lean
FOL.NV.T_21_1
[1345, 1]
[1370, 15]
simp at a1
case h2 x y : VarName H : Formula a1 : H ∈ ∅ ⊢ ¬isFreeIn x H
no goals
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/NV/Margaris/Fol.lean
FOL.NV.T_21_2
[1373, 1]
[1401, 15]
apply generalization
x y z : VarName ⊢ IsProof (forall_ x (forall_ y (forall_ z (((eq_ x y).and_ (eq_ y z)).imp_ (eq_ x z)))))
case h1 x y z : VarName ⊢ IsDeduct ∅ (forall_ y (forall_ z (((eq_ x y).and_ (eq_ y z)).imp_ (eq_ x z)))) case h2 x y z : VarName ⊢ ∀ H ∈ ∅, ¬isFreeIn x H
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/NV/Margaris/Fol.lean
FOL.NV.T_21_2
[1373, 1]
[1401, 15]
apply generalization
case h1 x y z : VarName ⊢ IsDeduct ∅ (forall_ y (forall_ z (((eq_ x y).and_ (eq_ y z)).imp_ (eq_ x z))))
case h1.h1 x y z : VarName ⊢ IsDeduct ∅ (forall_ z (((eq_ x y).and_ (eq_ y z)).imp_ (eq_ x z))) case h1.h2 x y z : VarName ⊢ ∀ H ∈ ∅, ¬isFreeIn y H
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/NV/Margaris/Fol.lean
FOL.NV.T_21_2
[1373, 1]
[1401, 15]
apply generalization
case h1.h1 x y z : VarName ⊢ IsDeduct ∅ (forall_ z (((eq_ x y).and_ (eq_ y z)).imp_ (eq_ x z)))
case h1.h1.h1 x y z : VarName ⊢ IsDeduct ∅ (((eq_ x y).and_ (eq_ y z)).imp_ (eq_ x z)) case h1.h1.h2 x y z : VarName ⊢ ∀ H ∈ ∅, ¬isFreeIn z H
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/NV/Margaris/Fol.lean
FOL.NV.T_21_2
[1373, 1]
[1401, 15]
apply IsDeduct.mp_ (eq_ z z)
case h1.h1.h1 x y z : VarName ⊢ IsDeduct ∅ (((eq_ x y).and_ (eq_ y z)).imp_ (eq_ x z))
case h1.h1.h1.a x y z : VarName ⊢ IsDeduct ∅ ((eq_ z z).imp_ (((eq_ x y).and_ (eq_ y z)).imp_ (eq_ x z))) case h1.h1.h1.a x y z : VarName ⊢ IsDeduct ∅ (eq_ z z)
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/NV/Margaris/Fol.lean
FOL.NV.T_21_2
[1373, 1]
[1401, 15]
apply IsDeduct.mp_ (((eq_ x y).and_ (eq_ z z)).imp_ ((eq_ x z).iff_ (eq_ y z)))
case h1.h1.h1.a x y z : VarName ⊢ IsDeduct ∅ ((eq_ z z).imp_ (((eq_ x y).and_ (eq_ y z)).imp_ (eq_ x z)))
case h1.h1.h1.a.a x y z : VarName ⊢ IsDeduct ∅ ((((eq_ x y).and_ (eq_ z z)).imp_ ((eq_ x z).iff_ (eq_ y z))).imp_ ((eq_ z z).imp_ (((eq_ x y).and_ (eq_ y z)).imp_ (eq_ x z)))) case h1.h1.h1.a.a x y z : VarName ⊢ IsDeduct ∅ (((eq_ x y).and_ (eq_ z z)).imp_ ((eq_ x z).iff_ (eq_ y z)))
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/NV/Margaris/Fol.lean
FOL.NV.T_21_2
[1373, 1]
[1401, 15]
simp only [def_iff_]
case h1.h1.h1.a.a x y z : VarName ⊢ IsDeduct ∅ ((((eq_ x y).and_ (eq_ z z)).imp_ ((eq_ x z).iff_ (eq_ y z))).imp_ ((eq_ z z).imp_ (((eq_ x y).and_ (eq_ y z)).imp_ (eq_ x z))))
case h1.h1.h1.a.a x y z : VarName ⊢ IsDeduct ∅ ((((eq_ x y).and_ (eq_ z z)).imp_ (((eq_ x z).imp_ (eq_ y z)).and_ ((eq_ y z).imp_ (eq_ x z)))).imp_ ((eq_ z z).imp_ (((eq_ x y).and_ (eq_ y z)).imp_ (eq_ x z))))
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/NV/Margaris/Fol.lean
FOL.NV.T_21_2
[1373, 1]
[1401, 15]
simp only [def_and_]
case h1.h1.h1.a.a x y z : VarName ⊢ IsDeduct ∅ ((((eq_ x y).and_ (eq_ z z)).imp_ (((eq_ x z).imp_ (eq_ y z)).and_ ((eq_ y z).imp_ (eq_ x z)))).imp_ ((eq_ z z).imp_ (((eq_ x y).and_ (eq_ y z)).imp_ (eq_ x z))))
case h1.h1.h1.a.a x y z : VarName ⊢ IsDeduct ∅ ((((eq_ x y).imp_ (eq_ z z).not_).not_.imp_ (((eq_ x z).imp_ (eq_ y z)).imp_ ((eq_ y z).imp_ (eq_ x z)).not_).not_).imp_ ((eq_ z z).imp_ (((eq_ x y).imp_ (eq_ y z).not_).not_.imp_ (eq_ x z))))
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/NV/Margaris/Fol.lean
FOL.NV.T_21_2
[1373, 1]
[1401, 15]
SC
case h1.h1.h1.a.a x y z : VarName ⊢ IsDeduct ∅ ((((eq_ x y).imp_ (eq_ z z).not_).not_.imp_ (((eq_ x z).imp_ (eq_ y z)).imp_ ((eq_ y z).imp_ (eq_ x z)).not_).not_).imp_ ((eq_ z z).imp_ (((eq_ x y).imp_ (eq_ y z).not_).not_.imp_ (eq_ x z))))
no goals
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/NV/Margaris/Fol.lean
FOL.NV.T_21_2
[1373, 1]
[1401, 15]
apply specId z
case h1.h1.h1.a.a x y z : VarName ⊢ IsDeduct ∅ (((eq_ x y).and_ (eq_ z z)).imp_ ((eq_ x z).iff_ (eq_ y z)))
case h1.h1.h1.a.a.h1 x y z : VarName ⊢ IsDeduct ∅ (forall_ z (((eq_ x y).and_ (eq_ z z)).imp_ ((eq_ x z).iff_ (eq_ y z))))
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/NV/Margaris/Fol.lean
FOL.NV.T_21_2
[1373, 1]
[1401, 15]
apply specId y
case h1.h1.h1.a.a.h1 x y z : VarName ⊢ IsDeduct ∅ (forall_ z (((eq_ x y).and_ (eq_ z z)).imp_ ((eq_ x z).iff_ (eq_ y z))))
case h1.h1.h1.a.a.h1.h1 x y z : VarName ⊢ IsDeduct ∅ (forall_ y (forall_ z (((eq_ x y).and_ (eq_ z z)).imp_ ((eq_ x z).iff_ (eq_ y z)))))
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/NV/Margaris/Fol.lean
FOL.NV.T_21_2
[1373, 1]
[1401, 15]
apply specId z
case h1.h1.h1.a.a.h1.h1 x y z : VarName ⊢ IsDeduct ∅ (forall_ y (forall_ z (((eq_ x y).and_ (eq_ z z)).imp_ ((eq_ x z).iff_ (eq_ y z)))))
case h1.h1.h1.a.a.h1.h1.h1 x y z : VarName ⊢ IsDeduct ∅ (forall_ z (forall_ y (forall_ z (((eq_ x y).and_ (eq_ z z)).imp_ ((eq_ x z).iff_ (eq_ y z))))))
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/NV/Margaris/Fol.lean
FOL.NV.T_21_2
[1373, 1]
[1401, 15]
apply specId x
case h1.h1.h1.a.a.h1.h1.h1 x y z : VarName ⊢ IsDeduct ∅ (forall_ z (forall_ y (forall_ z (((eq_ x y).and_ (eq_ z z)).imp_ ((eq_ x z).iff_ (eq_ y z))))))
case h1.h1.h1.a.a.h1.h1.h1.h1 x y z : VarName ⊢ IsDeduct ∅ (forall_ x (forall_ z (forall_ y (forall_ z (((eq_ x y).and_ (eq_ z z)).imp_ ((eq_ x z).iff_ (eq_ y z)))))))
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/NV/Margaris/Fol.lean
FOL.NV.T_21_2
[1373, 1]
[1401, 15]
apply IsDeduct.axiom_
case h1.h1.h1.a.a.h1.h1.h1.h1 x y z : VarName ⊢ IsDeduct ∅ (forall_ x (forall_ z (forall_ y (forall_ z (((eq_ x y).and_ (eq_ z z)).imp_ ((eq_ x z).iff_ (eq_ y z)))))))
case h1.h1.h1.a.a.h1.h1.h1.h1.a x y z : VarName ⊢ IsAxiom (forall_ x (forall_ z (forall_ y (forall_ z (((eq_ x y).and_ (eq_ z z)).imp_ ((eq_ x z).iff_ (eq_ y z)))))))
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/NV/Margaris/Fol.lean
FOL.NV.T_21_2
[1373, 1]
[1401, 15]
exact IsAxiom.eq_2_eq_ x z y z
case h1.h1.h1.a.a.h1.h1.h1.h1.a x y z : VarName ⊢ IsAxiom (forall_ x (forall_ z (forall_ y (forall_ z (((eq_ x y).and_ (eq_ z z)).imp_ ((eq_ x z).iff_ (eq_ y z)))))))
no goals
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/NV/Margaris/Fol.lean
FOL.NV.T_21_2
[1373, 1]
[1401, 15]
apply specId z
case h1.h1.h1.a x y z : VarName ⊢ IsDeduct ∅ (eq_ z z)
case h1.h1.h1.a.h1 x y z : VarName ⊢ IsDeduct ∅ (forall_ z (eq_ z z))
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/NV/Margaris/Fol.lean
FOL.NV.T_21_2
[1373, 1]
[1401, 15]
apply IsDeduct.axiom_
case h1.h1.h1.a.h1 x y z : VarName ⊢ IsDeduct ∅ (forall_ z (eq_ z z))
case h1.h1.h1.a.h1.a x y z : VarName ⊢ IsAxiom (forall_ z (eq_ z z))
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/NV/Margaris/Fol.lean
FOL.NV.T_21_2
[1373, 1]
[1401, 15]
exact IsAxiom.eq_1_ z
case h1.h1.h1.a.h1.a x y z : VarName ⊢ IsAxiom (forall_ z (eq_ z z))
no goals
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/NV/Margaris/Fol.lean
FOL.NV.T_21_2
[1373, 1]
[1401, 15]
intro H a1
case h1.h1.h2 x y z : VarName ⊢ ∀ H ∈ ∅, ¬isFreeIn z H
case h1.h1.h2 x y z : VarName H : Formula a1 : H ∈ ∅ ⊢ ¬isFreeIn z H
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/NV/Margaris/Fol.lean
FOL.NV.T_21_2
[1373, 1]
[1401, 15]
simp at a1
case h1.h1.h2 x y z : VarName H : Formula a1 : H ∈ ∅ ⊢ ¬isFreeIn z H
no goals
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/NV/Margaris/Fol.lean
FOL.NV.T_21_2
[1373, 1]
[1401, 15]
intro H a1
case h1.h2 x y z : VarName ⊢ ∀ H ∈ ∅, ¬isFreeIn y H
case h1.h2 x y z : VarName H : Formula a1 : H ∈ ∅ ⊢ ¬isFreeIn y H
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/NV/Margaris/Fol.lean
FOL.NV.T_21_2
[1373, 1]
[1401, 15]
simp at a1
case h1.h2 x y z : VarName H : Formula a1 : H ∈ ∅ ⊢ ¬isFreeIn y H
no goals
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/NV/Margaris/Fol.lean
FOL.NV.T_21_2
[1373, 1]
[1401, 15]
intro H a1
case h2 x y z : VarName ⊢ ∀ H ∈ ∅, ¬isFreeIn x H
case h2 x y z : VarName H : Formula a1 : H ∈ ∅ ⊢ ¬isFreeIn x H
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/NV/Margaris/Fol.lean
FOL.NV.T_21_2
[1373, 1]
[1401, 15]
simp at a1
case h2 x y z : VarName H : Formula a1 : H ∈ ∅ ⊢ ¬isFreeIn x H
no goals
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/NV/Margaris/Fol.lean
FOL.NV.T_21_8
[1404, 1]
[1543, 10]
induction h1
P_r P_s : Formula r s : VarName h1 : IsReplOfVarInFormula r s P_r P_s h2 : ¬isBoundIn r P_r h3 : ¬isBoundIn s P_r ⊢ IsProof ((eq_ r s).imp_ (P_r.iff_ P_s))
case pred_const_ P_r P_s : Formula r s : VarName name✝ : PredName n✝ : ℕ args_u✝ args_v✝ : Fin n✝ → VarName a✝ : ∀ (i : Fin n✝), args_u✝ i = args_v✝ i ∨ args_u✝ i = r ∧ args_v✝ i = s h2 : ¬isBoundIn r (pred_const_ name✝ (List.ofFn args_u✝)) h3 : ¬isBoundIn s (pred_const_ name✝ (List.ofFn args_u✝)) ⊢ IsProof ((eq_ r s).imp_ ((pred_const_ name✝ (List.ofFn args_u✝)).iff_ (pred_const_ name✝ (List.ofFn args_v✝)))) case pred_var_ P_r P_s : Formula r s : VarName name✝ : PredName n✝ : ℕ args_u✝ args_v✝ : Fin n✝ → VarName a✝ : ∀ (i : Fin n✝), args_u✝ i = args_v✝ i ∨ args_u✝ i = r ∧ args_v✝ i = s h2 : ¬isBoundIn r (pred_var_ name✝ (List.ofFn args_u✝)) h3 : ¬isBoundIn s (pred_var_ name✝ (List.ofFn args_u✝)) ⊢ IsProof ((eq_ r s).imp_ ((pred_var_ name✝ (List.ofFn args_u✝)).iff_ (pred_var_ name✝ (List.ofFn args_v✝)))) case eq_ P_r P_s : Formula r s x_u✝ y_u✝ x_v✝ y_v✝ : VarName a✝¹ : x_u✝ = x_v✝ ∨ x_u✝ = r ∧ x_v✝ = s a✝ : y_u✝ = y_v✝ ∨ y_u✝ = r ∧ y_v✝ = s h2 : ¬isBoundIn r (eq_ x_u✝ y_u✝) h3 : ¬isBoundIn s (eq_ x_u✝ y_u✝) ⊢ IsProof ((eq_ r s).imp_ ((eq_ x_u✝ y_u✝).iff_ (eq_ x_v✝ y_v✝))) case true_ P_r P_s : Formula r s : VarName h2 : ¬isBoundIn r true_ h3 : ¬isBoundIn s true_ ⊢ IsProof ((eq_ r s).imp_ (true_.iff_ true_)) case false_ P_r P_s : Formula r s : VarName h2 : ¬isBoundIn r false_ h3 : ¬isBoundIn s false_ ⊢ IsProof ((eq_ r s).imp_ (false_.iff_ false_)) case not_ P_r P_s : Formula r s : VarName P_u✝ P_v✝ : Formula a✝ : IsReplOfVarInFormula r s P_u✝ P_v✝ a_ih✝ : ¬isBoundIn r P_u✝ → ¬isBoundIn s P_u✝ → IsProof ((eq_ r s).imp_ (P_u✝.iff_ P_v✝)) h2 : ¬isBoundIn r P_u✝.not_ h3 : ¬isBoundIn s P_u✝.not_ ⊢ IsProof ((eq_ r s).imp_ (P_u✝.not_.iff_ P_v✝.not_)) case imp_ P_r P_s : Formula r s : VarName P_u✝ Q_u✝ P_v✝ Q_v✝ : Formula a✝¹ : IsReplOfVarInFormula r s P_u✝ P_v✝ a✝ : IsReplOfVarInFormula r s Q_u✝ Q_v✝ a_ih✝¹ : ¬isBoundIn r P_u✝ → ¬isBoundIn s P_u✝ → IsProof ((eq_ r s).imp_ (P_u✝.iff_ P_v✝)) a_ih✝ : ¬isBoundIn r Q_u✝ → ¬isBoundIn s Q_u✝ → IsProof ((eq_ r s).imp_ (Q_u✝.iff_ Q_v✝)) h2 : ¬isBoundIn r (P_u✝.imp_ Q_u✝) h3 : ¬isBoundIn s (P_u✝.imp_ Q_u✝) ⊢ IsProof ((eq_ r s).imp_ ((P_u✝.imp_ Q_u✝).iff_ (P_v✝.imp_ Q_v✝))) case and_ P_r P_s : Formula r s : VarName P_u✝ Q_u✝ P_v✝ Q_v✝ : Formula a✝¹ : IsReplOfVarInFormula r s P_u✝ P_v✝ a✝ : IsReplOfVarInFormula r s Q_u✝ Q_v✝ a_ih✝¹ : ¬isBoundIn r P_u✝ → ¬isBoundIn s P_u✝ → IsProof ((eq_ r s).imp_ (P_u✝.iff_ P_v✝)) a_ih✝ : ¬isBoundIn r Q_u✝ → ¬isBoundIn s Q_u✝ → IsProof ((eq_ r s).imp_ (Q_u✝.iff_ Q_v✝)) h2 : ¬isBoundIn r (P_u✝.and_ Q_u✝) h3 : ¬isBoundIn s (P_u✝.and_ Q_u✝) ⊢ IsProof ((eq_ r s).imp_ ((P_u✝.and_ Q_u✝).iff_ (P_v✝.and_ Q_v✝))) case or_ P_r P_s : Formula r s : VarName P_u✝ Q_u✝ P_v✝ Q_v✝ : Formula a✝¹ : IsReplOfVarInFormula r s P_u✝ P_v✝ a✝ : IsReplOfVarInFormula r s Q_u✝ Q_v✝ a_ih✝¹ : ¬isBoundIn r P_u✝ → ¬isBoundIn s P_u✝ → IsProof ((eq_ r s).imp_ (P_u✝.iff_ P_v✝)) a_ih✝ : ¬isBoundIn r Q_u✝ → ¬isBoundIn s Q_u✝ → IsProof ((eq_ r s).imp_ (Q_u✝.iff_ Q_v✝)) h2 : ¬isBoundIn r (P_u✝.or_ Q_u✝) h3 : ¬isBoundIn s (P_u✝.or_ Q_u✝) ⊢ IsProof ((eq_ r s).imp_ ((P_u✝.or_ Q_u✝).iff_ (P_v✝.or_ Q_v✝))) case iff_ P_r P_s : Formula r s : VarName P_u✝ Q_u✝ P_v✝ Q_v✝ : Formula a✝¹ : IsReplOfVarInFormula r s P_u✝ P_v✝ a✝ : IsReplOfVarInFormula r s Q_u✝ Q_v✝ a_ih✝¹ : ¬isBoundIn r P_u✝ → ¬isBoundIn s P_u✝ → IsProof ((eq_ r s).imp_ (P_u✝.iff_ P_v✝)) a_ih✝ : ¬isBoundIn r Q_u✝ → ¬isBoundIn s Q_u✝ → IsProof ((eq_ r s).imp_ (Q_u✝.iff_ Q_v✝)) h2 : ¬isBoundIn r (P_u✝.iff_ Q_u✝) h3 : ¬isBoundIn s (P_u✝.iff_ Q_u✝) ⊢ IsProof ((eq_ r s).imp_ ((P_u✝.iff_ Q_u✝).iff_ (P_v✝.iff_ Q_v✝))) case forall_ P_r P_s : Formula r s x✝ : VarName P_u✝ P_v✝ : Formula a✝ : IsReplOfVarInFormula r s P_u✝ P_v✝ a_ih✝ : ¬isBoundIn r P_u✝ → ¬isBoundIn s P_u✝ → IsProof ((eq_ r s).imp_ (P_u✝.iff_ P_v✝)) h2 : ¬isBoundIn r (forall_ x✝ P_u✝) h3 : ¬isBoundIn s (forall_ x✝ P_u✝) ⊢ IsProof ((eq_ r s).imp_ ((forall_ x✝ P_u✝).iff_ (forall_ x✝ P_v✝))) case exists_ P_r P_s : Formula r s x✝ : VarName P_u✝ P_v✝ : Formula a✝ : IsReplOfVarInFormula r s P_u✝ P_v✝ a_ih✝ : ¬isBoundIn r P_u✝ → ¬isBoundIn s P_u✝ → IsProof ((eq_ r s).imp_ (P_u✝.iff_ P_v✝)) h2 : ¬isBoundIn r (exists_ x✝ P_u✝) h3 : ¬isBoundIn s (exists_ x✝ P_u✝) ⊢ IsProof ((eq_ r s).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_21_8
[1404, 1]
[1543, 10]
case true_ => simp only [def_iff_] simp only [def_and_] SC
P_r P_s : Formula r s : VarName h2 : ¬isBoundIn r true_ h3 : ¬isBoundIn s true_ ⊢ IsProof ((eq_ r s).imp_ (true_.iff_ true_))
no goals
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/NV/Margaris/Fol.lean
FOL.NV.T_21_8
[1404, 1]
[1543, 10]
all_goals sorry
case pred_var_ P_r P_s : Formula r s : VarName name✝ : PredName n✝ : ℕ args_u✝ args_v✝ : Fin n✝ → VarName a✝ : ∀ (i : Fin n✝), args_u✝ i = args_v✝ i ∨ args_u✝ i = r ∧ args_v✝ i = s h2 : ¬isBoundIn r (pred_var_ name✝ (List.ofFn args_u✝)) h3 : ¬isBoundIn s (pred_var_ name✝ (List.ofFn args_u✝)) ⊢ IsProof ((eq_ r s).imp_ ((pred_var_ name✝ (List.ofFn args_u✝)).iff_ (pred_var_ name✝ (List.ofFn args_v✝)))) case eq_ P_r P_s : Formula r s x_u✝ y_u✝ x_v✝ y_v✝ : VarName a✝¹ : x_u✝ = x_v✝ ∨ x_u✝ = r ∧ x_v✝ = s a✝ : y_u✝ = y_v✝ ∨ y_u✝ = r ∧ y_v✝ = s h2 : ¬isBoundIn r (eq_ x_u✝ y_u✝) h3 : ¬isBoundIn s (eq_ x_u✝ y_u✝) ⊢ IsProof ((eq_ r s).imp_ ((eq_ x_u✝ y_u✝).iff_ (eq_ x_v✝ y_v✝))) case false_ P_r P_s : Formula r s : VarName h2 : ¬isBoundIn r false_ h3 : ¬isBoundIn s false_ ⊢ IsProof ((eq_ r s).imp_ (false_.iff_ false_)) case and_ P_r P_s : Formula r s : VarName P_u✝ Q_u✝ P_v✝ Q_v✝ : Formula a✝¹ : IsReplOfVarInFormula r s P_u✝ P_v✝ a✝ : IsReplOfVarInFormula r s Q_u✝ Q_v✝ a_ih✝¹ : ¬isBoundIn r P_u✝ → ¬isBoundIn s P_u✝ → IsProof ((eq_ r s).imp_ (P_u✝.iff_ P_v✝)) a_ih✝ : ¬isBoundIn r Q_u✝ → ¬isBoundIn s Q_u✝ → IsProof ((eq_ r s).imp_ (Q_u✝.iff_ Q_v✝)) h2 : ¬isBoundIn r (P_u✝.and_ Q_u✝) h3 : ¬isBoundIn s (P_u✝.and_ Q_u✝) ⊢ IsProof ((eq_ r s).imp_ ((P_u✝.and_ Q_u✝).iff_ (P_v✝.and_ Q_v✝))) case or_ P_r P_s : Formula r s : VarName P_u✝ Q_u✝ P_v✝ Q_v✝ : Formula a✝¹ : IsReplOfVarInFormula r s P_u✝ P_v✝ a✝ : IsReplOfVarInFormula r s Q_u✝ Q_v✝ a_ih✝¹ : ¬isBoundIn r P_u✝ → ¬isBoundIn s P_u✝ → IsProof ((eq_ r s).imp_ (P_u✝.iff_ P_v✝)) a_ih✝ : ¬isBoundIn r Q_u✝ → ¬isBoundIn s Q_u✝ → IsProof ((eq_ r s).imp_ (Q_u✝.iff_ Q_v✝)) h2 : ¬isBoundIn r (P_u✝.or_ Q_u✝) h3 : ¬isBoundIn s (P_u✝.or_ Q_u✝) ⊢ IsProof ((eq_ r s).imp_ ((P_u✝.or_ Q_u✝).iff_ (P_v✝.or_ Q_v✝))) case iff_ P_r P_s : Formula r s : VarName P_u✝ Q_u✝ P_v✝ Q_v✝ : Formula a✝¹ : IsReplOfVarInFormula r s P_u✝ P_v✝ a✝ : IsReplOfVarInFormula r s Q_u✝ Q_v✝ a_ih✝¹ : ¬isBoundIn r P_u✝ → ¬isBoundIn s P_u✝ → IsProof ((eq_ r s).imp_ (P_u✝.iff_ P_v✝)) a_ih✝ : ¬isBoundIn r Q_u✝ → ¬isBoundIn s Q_u✝ → IsProof ((eq_ r s).imp_ (Q_u✝.iff_ Q_v✝)) h2 : ¬isBoundIn r (P_u✝.iff_ Q_u✝) h3 : ¬isBoundIn s (P_u✝.iff_ Q_u✝) ⊢ IsProof ((eq_ r s).imp_ ((P_u✝.iff_ Q_u✝).iff_ (P_v✝.iff_ Q_v✝))) case exists_ P_r P_s : Formula r s x✝ : VarName P_u✝ P_v✝ : Formula a✝ : IsReplOfVarInFormula r s P_u✝ P_v✝ a_ih✝ : ¬isBoundIn r P_u✝ → ¬isBoundIn s P_u✝ → IsProof ((eq_ r s).imp_ (P_u✝.iff_ P_v✝)) h2 : ¬isBoundIn r (exists_ x✝ P_u✝) h3 : ¬isBoundIn s (exists_ x✝ P_u✝) ⊢ IsProof ((eq_ r s).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_21_8
[1404, 1]
[1543, 10]
simp only [def_iff_]
P_r P_s : Formula r s : VarName h2 : ¬isBoundIn r true_ h3 : ¬isBoundIn s true_ ⊢ IsProof ((eq_ r s).imp_ (true_.iff_ true_))
P_r P_s : Formula r s : VarName h2 : ¬isBoundIn r true_ h3 : ¬isBoundIn s true_ ⊢ IsProof ((eq_ r s).imp_ ((true_.imp_ true_).and_ (true_.imp_ true_)))
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/NV/Margaris/Fol.lean
FOL.NV.T_21_8
[1404, 1]
[1543, 10]
simp only [def_and_]
P_r P_s : Formula r s : VarName h2 : ¬isBoundIn r true_ h3 : ¬isBoundIn s true_ ⊢ IsProof ((eq_ r s).imp_ ((true_.imp_ true_).and_ (true_.imp_ true_)))
P_r P_s : Formula r s : VarName h2 : ¬isBoundIn r true_ h3 : ¬isBoundIn s true_ ⊢ IsProof ((eq_ r s).imp_ ((true_.imp_ true_).imp_ (true_.imp_ true_).not_).not_)
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/NV/Margaris/Fol.lean
FOL.NV.T_21_8
[1404, 1]
[1543, 10]
SC
P_r P_s : Formula r s : VarName h2 : ¬isBoundIn r true_ h3 : ¬isBoundIn s true_ ⊢ IsProof ((eq_ r s).imp_ ((true_.imp_ true_).imp_ (true_.imp_ true_).not_).not_)
no goals
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/NV/Margaris/Fol.lean
FOL.NV.T_21_8
[1404, 1]
[1543, 10]
apply IsDeduct.mp_ ((eq_ r s).imp_ ((pred_const_ name (List.ofFn args_u)).iff_ (pred_const_ name (List.ofFn args_v))))
P_r P_s : Formula r s : VarName name : PredName n : ℕ args_u args_v : Fin n → VarName h1_1 : ∀ (i : Fin n), args_u i = args_v i ∨ args_u i = r ∧ args_v i = s h2 : ¬isBoundIn r (pred_const_ name (List.ofFn args_u)) h3 : ¬isBoundIn s (pred_const_ name (List.ofFn args_u)) ⊢ IsProof ((eq_ r s).imp_ ((pred_const_ name (List.ofFn args_u)).iff_ (pred_const_ name (List.ofFn args_v))))
case a P_r P_s : Formula r s : VarName name : PredName n : ℕ args_u args_v : Fin n → VarName h1_1 : ∀ (i : Fin n), args_u i = args_v i ∨ args_u i = r ∧ args_v i = s h2 : ¬isBoundIn r (pred_const_ name (List.ofFn args_u)) h3 : ¬isBoundIn s (pred_const_ name (List.ofFn args_u)) ⊢ IsDeduct ∅ (((eq_ r s).imp_ ((pred_const_ name (List.ofFn args_u)).iff_ (pred_const_ name (List.ofFn args_v)))).imp_ ((eq_ r s).imp_ ((pred_const_ name (List.ofFn args_u)).iff_ (pred_const_ name (List.ofFn args_v))))) case a P_r P_s : Formula r s : VarName name : PredName n : ℕ args_u args_v : Fin n → VarName h1_1 : ∀ (i : Fin n), args_u i = args_v i ∨ args_u i = r ∧ args_v i = s h2 : ¬isBoundIn r (pred_const_ name (List.ofFn args_u)) h3 : ¬isBoundIn s (pred_const_ name (List.ofFn args_u)) ⊢ IsDeduct ∅ ((eq_ r s).imp_ ((pred_const_ name (List.ofFn args_u)).iff_ (pred_const_ name (List.ofFn args_v))))
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/NV/Margaris/Fol.lean
FOL.NV.T_21_8
[1404, 1]
[1543, 10]
SC
case a P_r P_s : Formula r s : VarName name : PredName n : ℕ args_u args_v : Fin n → VarName h1_1 : ∀ (i : Fin n), args_u i = args_v i ∨ args_u i = r ∧ args_v i = s h2 : ¬isBoundIn r (pred_const_ name (List.ofFn args_u)) h3 : ¬isBoundIn s (pred_const_ name (List.ofFn args_u)) ⊢ IsDeduct ∅ (((eq_ r s).imp_ ((pred_const_ name (List.ofFn args_u)).iff_ (pred_const_ name (List.ofFn args_v)))).imp_ ((eq_ r s).imp_ ((pred_const_ name (List.ofFn args_u)).iff_ (pred_const_ name (List.ofFn args_v)))))
no goals
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/NV/Margaris/Fol.lean
FOL.NV.T_21_8
[1404, 1]
[1543, 10]
apply IsDeduct.mp_ ((eq_ r s).imp_ (And_ (List.ofFn fun (i : Fin n) => eq_ (args_u i) (args_v i))))
case a P_r P_s : Formula r s : VarName name : PredName n : ℕ args_u args_v : Fin n → VarName h1_1 : ∀ (i : Fin n), args_u i = args_v i ∨ args_u i = r ∧ args_v i = s h2 : ¬isBoundIn r (pred_const_ name (List.ofFn args_u)) h3 : ¬isBoundIn s (pred_const_ name (List.ofFn args_u)) ⊢ IsDeduct ∅ ((eq_ r s).imp_ ((pred_const_ name (List.ofFn args_u)).iff_ (pred_const_ name (List.ofFn args_v))))
case a.a P_r P_s : Formula r s : VarName name : PredName n : ℕ args_u args_v : Fin n → VarName h1_1 : ∀ (i : Fin n), args_u i = args_v i ∨ args_u i = r ∧ args_v i = s h2 : ¬isBoundIn r (pred_const_ name (List.ofFn args_u)) h3 : ¬isBoundIn s (pred_const_ name (List.ofFn args_u)) ⊢ IsDeduct ∅ (((eq_ r s).imp_ (And_ (List.ofFn fun i => eq_ (args_u i) (args_v i)))).imp_ ((eq_ r s).imp_ ((pred_const_ name (List.ofFn args_u)).iff_ (pred_const_ name (List.ofFn args_v))))) case a.a P_r P_s : Formula r s : VarName name : PredName n : ℕ args_u args_v : Fin n → VarName h1_1 : ∀ (i : Fin n), args_u i = args_v i ∨ args_u i = r ∧ args_v i = s h2 : ¬isBoundIn r (pred_const_ name (List.ofFn args_u)) h3 : ¬isBoundIn s (pred_const_ name (List.ofFn args_u)) ⊢ IsDeduct ∅ ((eq_ r s).imp_ (And_ (List.ofFn fun i => eq_ (args_u i) (args_v i))))
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/NV/Margaris/Fol.lean
FOL.NV.T_21_8
[1404, 1]
[1543, 10]
apply IsDeduct.mp_ ((And_ (List.ofFn fun (i : Fin n) => eq_ (args_u i) (args_v i))).imp_ ((pred_const_ name (List.ofFn args_u)).iff_ (pred_const_ name (List.ofFn args_v))))
case a.a P_r P_s : Formula r s : VarName name : PredName n : ℕ args_u args_v : Fin n → VarName h1_1 : ∀ (i : Fin n), args_u i = args_v i ∨ args_u i = r ∧ args_v i = s h2 : ¬isBoundIn r (pred_const_ name (List.ofFn args_u)) h3 : ¬isBoundIn s (pred_const_ name (List.ofFn args_u)) ⊢ IsDeduct ∅ (((eq_ r s).imp_ (And_ (List.ofFn fun i => eq_ (args_u i) (args_v i)))).imp_ ((eq_ r s).imp_ ((pred_const_ name (List.ofFn args_u)).iff_ (pred_const_ name (List.ofFn args_v)))))
case a.a.a P_r P_s : Formula r s : VarName name : PredName n : ℕ args_u args_v : Fin n → VarName h1_1 : ∀ (i : Fin n), args_u i = args_v i ∨ args_u i = r ∧ args_v i = s h2 : ¬isBoundIn r (pred_const_ name (List.ofFn args_u)) h3 : ¬isBoundIn s (pred_const_ name (List.ofFn args_u)) ⊢ IsDeduct ∅ (((And_ (List.ofFn fun i => eq_ (args_u i) (args_v i))).imp_ ((pred_const_ name (List.ofFn args_u)).iff_ (pred_const_ name (List.ofFn args_v)))).imp_ (((eq_ r s).imp_ (And_ (List.ofFn fun i => eq_ (args_u i) (args_v i)))).imp_ ((eq_ r s).imp_ ((pred_const_ name (List.ofFn args_u)).iff_ (pred_const_ name (List.ofFn args_v)))))) case a.a.a P_r P_s : Formula r s : VarName name : PredName n : ℕ args_u args_v : Fin n → VarName h1_1 : ∀ (i : Fin n), args_u i = args_v i ∨ args_u i = r ∧ args_v i = s h2 : ¬isBoundIn r (pred_const_ name (List.ofFn args_u)) h3 : ¬isBoundIn s (pred_const_ name (List.ofFn args_u)) ⊢ IsDeduct ∅ ((And_ (List.ofFn fun i => eq_ (args_u i) (args_v i))).imp_ ((pred_const_ name (List.ofFn args_u)).iff_ (pred_const_ name (List.ofFn args_v))))
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/NV/Margaris/Fol.lean
FOL.NV.T_21_8
[1404, 1]
[1543, 10]
simp only [def_iff_]
case a.a.a P_r P_s : Formula r s : VarName name : PredName n : ℕ args_u args_v : Fin n → VarName h1_1 : ∀ (i : Fin n), args_u i = args_v i ∨ args_u i = r ∧ args_v i = s h2 : ¬isBoundIn r (pred_const_ name (List.ofFn args_u)) h3 : ¬isBoundIn s (pred_const_ name (List.ofFn args_u)) ⊢ IsDeduct ∅ (((And_ (List.ofFn fun i => eq_ (args_u i) (args_v i))).imp_ ((pred_const_ name (List.ofFn args_u)).iff_ (pred_const_ name (List.ofFn args_v)))).imp_ (((eq_ r s).imp_ (And_ (List.ofFn fun i => eq_ (args_u i) (args_v i)))).imp_ ((eq_ r s).imp_ ((pred_const_ name (List.ofFn args_u)).iff_ (pred_const_ name (List.ofFn args_v))))))
case a.a.a P_r P_s : Formula r s : VarName name : PredName n : ℕ args_u args_v : Fin n → VarName h1_1 : ∀ (i : Fin n), args_u i = args_v i ∨ args_u i = r ∧ args_v i = s h2 : ¬isBoundIn r (pred_const_ name (List.ofFn args_u)) h3 : ¬isBoundIn s (pred_const_ name (List.ofFn args_u)) ⊢ IsDeduct ∅ (((And_ (List.ofFn fun i => eq_ (args_u i) (args_v i))).imp_ (((pred_const_ name (List.ofFn args_u)).imp_ (pred_const_ name (List.ofFn args_v))).and_ ((pred_const_ name (List.ofFn args_v)).imp_ (pred_const_ name (List.ofFn args_u))))).imp_ (((eq_ r s).imp_ (And_ (List.ofFn fun i => eq_ (args_u i) (args_v i)))).imp_ ((eq_ r s).imp_ (((pred_const_ name (List.ofFn args_u)).imp_ (pred_const_ name (List.ofFn args_v))).and_ ((pred_const_ name (List.ofFn args_v)).imp_ (pred_const_ name (List.ofFn args_u)))))))
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/NV/Margaris/Fol.lean
FOL.NV.T_21_8
[1404, 1]
[1543, 10]
simp only [def_and_]
case a.a.a P_r P_s : Formula r s : VarName name : PredName n : ℕ args_u args_v : Fin n → VarName h1_1 : ∀ (i : Fin n), args_u i = args_v i ∨ args_u i = r ∧ args_v i = s h2 : ¬isBoundIn r (pred_const_ name (List.ofFn args_u)) h3 : ¬isBoundIn s (pred_const_ name (List.ofFn args_u)) ⊢ IsDeduct ∅ (((And_ (List.ofFn fun i => eq_ (args_u i) (args_v i))).imp_ (((pred_const_ name (List.ofFn args_u)).imp_ (pred_const_ name (List.ofFn args_v))).and_ ((pred_const_ name (List.ofFn args_v)).imp_ (pred_const_ name (List.ofFn args_u))))).imp_ (((eq_ r s).imp_ (And_ (List.ofFn fun i => eq_ (args_u i) (args_v i)))).imp_ ((eq_ r s).imp_ (((pred_const_ name (List.ofFn args_u)).imp_ (pred_const_ name (List.ofFn args_v))).and_ ((pred_const_ name (List.ofFn args_v)).imp_ (pred_const_ name (List.ofFn args_u)))))))
case a.a.a P_r P_s : Formula r s : VarName name : PredName n : ℕ args_u args_v : Fin n → VarName h1_1 : ∀ (i : Fin n), args_u i = args_v i ∨ args_u i = r ∧ args_v i = s h2 : ¬isBoundIn r (pred_const_ name (List.ofFn args_u)) h3 : ¬isBoundIn s (pred_const_ name (List.ofFn args_u)) ⊢ IsDeduct ∅ (((And_ (List.ofFn fun i => eq_ (args_u i) (args_v i))).imp_ (((pred_const_ name (List.ofFn args_u)).imp_ (pred_const_ name (List.ofFn args_v))).imp_ ((pred_const_ name (List.ofFn args_v)).imp_ (pred_const_ name (List.ofFn args_u))).not_).not_).imp_ (((eq_ r s).imp_ (And_ (List.ofFn fun i => eq_ (args_u i) (args_v i)))).imp_ ((eq_ r s).imp_ (((pred_const_ name (List.ofFn args_u)).imp_ (pred_const_ name (List.ofFn args_v))).imp_ ((pred_const_ name (List.ofFn args_v)).imp_ (pred_const_ name (List.ofFn args_u))).not_).not_)))
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/NV/Margaris/Fol.lean
FOL.NV.T_21_8
[1404, 1]
[1543, 10]
SC
case a.a.a P_r P_s : Formula r s : VarName name : PredName n : ℕ args_u args_v : Fin n → VarName h1_1 : ∀ (i : Fin n), args_u i = args_v i ∨ args_u i = r ∧ args_v i = s h2 : ¬isBoundIn r (pred_const_ name (List.ofFn args_u)) h3 : ¬isBoundIn s (pred_const_ name (List.ofFn args_u)) ⊢ IsDeduct ∅ (((And_ (List.ofFn fun i => eq_ (args_u i) (args_v i))).imp_ (((pred_const_ name (List.ofFn args_u)).imp_ (pred_const_ name (List.ofFn args_v))).imp_ ((pred_const_ name (List.ofFn args_v)).imp_ (pred_const_ name (List.ofFn args_u))).not_).not_).imp_ (((eq_ r s).imp_ (And_ (List.ofFn fun i => eq_ (args_u i) (args_v i)))).imp_ ((eq_ r s).imp_ (((pred_const_ name (List.ofFn args_u)).imp_ (pred_const_ name (List.ofFn args_v))).imp_ ((pred_const_ name (List.ofFn args_v)).imp_ (pred_const_ name (List.ofFn args_u))).not_).not_)))
no goals
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/NV/Margaris/Fol.lean
FOL.NV.T_21_8
[1404, 1]
[1543, 10]
apply Forall_spec_id' (List.ofFn args_v)
case a.a.a P_r P_s : Formula r s : VarName name : PredName n : ℕ args_u args_v : Fin n → VarName h1_1 : ∀ (i : Fin n), args_u i = args_v i ∨ args_u i = r ∧ args_v i = s h2 : ¬isBoundIn r (pred_const_ name (List.ofFn args_u)) h3 : ¬isBoundIn s (pred_const_ name (List.ofFn args_u)) ⊢ IsDeduct ∅ ((And_ (List.ofFn fun i => eq_ (args_u i) (args_v i))).imp_ ((pred_const_ name (List.ofFn args_u)).iff_ (pred_const_ name (List.ofFn args_v))))
case a.a.a.h1 P_r P_s : Formula r s : VarName name : PredName n : ℕ args_u args_v : Fin n → VarName h1_1 : ∀ (i : Fin n), args_u i = args_v i ∨ args_u i = r ∧ args_v i = s h2 : ¬isBoundIn r (pred_const_ name (List.ofFn args_u)) h3 : ¬isBoundIn s (pred_const_ name (List.ofFn args_u)) ⊢ IsDeduct ∅ (Forall_ (List.ofFn args_v) ((And_ (List.ofFn fun i => eq_ (args_u i) (args_v i))).imp_ ((pred_const_ name (List.ofFn args_u)).iff_ (pred_const_ name (List.ofFn args_v)))))
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/NV/Margaris/Fol.lean
FOL.NV.T_21_8
[1404, 1]
[1543, 10]
apply Forall_spec_id' (List.ofFn args_u)
case a.a.a.h1 P_r P_s : Formula r s : VarName name : PredName n : ℕ args_u args_v : Fin n → VarName h1_1 : ∀ (i : Fin n), args_u i = args_v i ∨ args_u i = r ∧ args_v i = s h2 : ¬isBoundIn r (pred_const_ name (List.ofFn args_u)) h3 : ¬isBoundIn s (pred_const_ name (List.ofFn args_u)) ⊢ IsDeduct ∅ (Forall_ (List.ofFn args_v) ((And_ (List.ofFn fun i => eq_ (args_u i) (args_v i))).imp_ ((pred_const_ name (List.ofFn args_u)).iff_ (pred_const_ name (List.ofFn args_v)))))
case a.a.a.h1.h1 P_r P_s : Formula r s : VarName name : PredName n : ℕ args_u args_v : Fin n → VarName h1_1 : ∀ (i : Fin n), args_u i = args_v i ∨ args_u i = r ∧ args_v i = s h2 : ¬isBoundIn r (pred_const_ name (List.ofFn args_u)) h3 : ¬isBoundIn s (pred_const_ name (List.ofFn args_u)) ⊢ IsDeduct ∅ (Forall_ (List.ofFn args_u) (Forall_ (List.ofFn args_v) ((And_ (List.ofFn fun i => eq_ (args_u i) (args_v i))).imp_ ((pred_const_ name (List.ofFn args_u)).iff_ (pred_const_ name (List.ofFn args_v))))))
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/NV/Margaris/Fol.lean
FOL.NV.T_21_8
[1404, 1]
[1543, 10]
apply IsDeduct.axiom_
case a.a.a.h1.h1 P_r P_s : Formula r s : VarName name : PredName n : ℕ args_u args_v : Fin n → VarName h1_1 : ∀ (i : Fin n), args_u i = args_v i ∨ args_u i = r ∧ args_v i = s h2 : ¬isBoundIn r (pred_const_ name (List.ofFn args_u)) h3 : ¬isBoundIn s (pred_const_ name (List.ofFn args_u)) ⊢ IsDeduct ∅ (Forall_ (List.ofFn args_u) (Forall_ (List.ofFn args_v) ((And_ (List.ofFn fun i => eq_ (args_u i) (args_v i))).imp_ ((pred_const_ name (List.ofFn args_u)).iff_ (pred_const_ name (List.ofFn args_v))))))
case a.a.a.h1.h1.a P_r P_s : Formula r s : VarName name : PredName n : ℕ args_u args_v : Fin n → VarName h1_1 : ∀ (i : Fin n), args_u i = args_v i ∨ args_u i = r ∧ args_v i = s h2 : ¬isBoundIn r (pred_const_ name (List.ofFn args_u)) h3 : ¬isBoundIn s (pred_const_ name (List.ofFn args_u)) ⊢ IsAxiom (Forall_ (List.ofFn args_u) (Forall_ (List.ofFn args_v) ((And_ (List.ofFn fun i => eq_ (args_u i) (args_v i))).imp_ ((pred_const_ name (List.ofFn args_u)).iff_ (pred_const_ name (List.ofFn args_v))))))
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/NV/Margaris/Fol.lean
FOL.NV.T_21_8
[1404, 1]
[1543, 10]
exact IsAxiom.eq_2_pred_const_ name n args_u args_v
case a.a.a.h1.h1.a P_r P_s : Formula r s : VarName name : PredName n : ℕ args_u args_v : Fin n → VarName h1_1 : ∀ (i : Fin n), args_u i = args_v i ∨ args_u i = r ∧ args_v i = s h2 : ¬isBoundIn r (pred_const_ name (List.ofFn args_u)) h3 : ¬isBoundIn s (pred_const_ name (List.ofFn args_u)) ⊢ IsAxiom (Forall_ (List.ofFn args_u) (Forall_ (List.ofFn args_v) ((And_ (List.ofFn fun i => eq_ (args_u i) (args_v i))).imp_ ((pred_const_ name (List.ofFn args_u)).iff_ (pred_const_ name (List.ofFn args_v))))))
no goals
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/NV/Margaris/Fol.lean
FOL.NV.T_21_8
[1404, 1]
[1543, 10]
clear h2
case a.a P_r P_s : Formula r s : VarName name : PredName n : ℕ args_u args_v : Fin n → VarName h1_1 : ∀ (i : Fin n), args_u i = args_v i ∨ args_u i = r ∧ args_v i = s h2 : ¬isBoundIn r (pred_const_ name (List.ofFn args_u)) h3 : ¬isBoundIn s (pred_const_ name (List.ofFn args_u)) ⊢ IsDeduct ∅ ((eq_ r s).imp_ (And_ (List.ofFn fun i => eq_ (args_u i) (args_v i))))
case a.a P_r P_s : Formula r s : VarName name : PredName n : ℕ args_u args_v : Fin n → VarName h1_1 : ∀ (i : Fin n), args_u i = args_v i ∨ args_u i = r ∧ args_v i = s h3 : ¬isBoundIn s (pred_const_ name (List.ofFn args_u)) ⊢ IsDeduct ∅ ((eq_ r s).imp_ (And_ (List.ofFn fun i => eq_ (args_u i) (args_v i))))
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/NV/Margaris/Fol.lean
FOL.NV.T_21_8
[1404, 1]
[1543, 10]
clear h3
case a.a P_r P_s : Formula r s : VarName name : PredName n : ℕ args_u args_v : Fin n → VarName h1_1 : ∀ (i : Fin n), args_u i = args_v i ∨ args_u i = r ∧ args_v i = s h3 : ¬isBoundIn s (pred_const_ name (List.ofFn args_u)) ⊢ IsDeduct ∅ ((eq_ r s).imp_ (And_ (List.ofFn fun i => eq_ (args_u i) (args_v i))))
case a.a P_r P_s : Formula r s : VarName name : PredName n : ℕ args_u args_v : Fin n → VarName h1_1 : ∀ (i : Fin n), args_u i = args_v i ∨ args_u i = r ∧ args_v i = s ⊢ IsDeduct ∅ ((eq_ r s).imp_ (And_ (List.ofFn fun i => eq_ (args_u i) (args_v i))))
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/NV/Margaris/Fol.lean
FOL.NV.T_21_8
[1404, 1]
[1543, 10]
simp only [And_]
case a.a P_r P_s : Formula r s : VarName name : PredName n : ℕ args_u args_v : Fin n → VarName h1_1 : ∀ (i : Fin n), args_u i = args_v i ∨ args_u i = r ∧ args_v i = s ⊢ IsDeduct ∅ ((eq_ r s).imp_ (And_ (List.ofFn fun i => eq_ (args_u i) (args_v i))))
case a.a P_r P_s : Formula r s : VarName name : PredName n : ℕ args_u args_v : Fin n → VarName h1_1 : ∀ (i : Fin n), args_u i = args_v i ∨ args_u i = r ∧ args_v i = s ⊢ IsDeduct ∅ ((eq_ r s).imp_ (List.foldr and_ true_ (List.ofFn fun i => eq_ (args_u i) (args_v i))))
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/NV/Margaris/Fol.lean
FOL.NV.T_21_8
[1404, 1]
[1543, 10]
induction n
case a.a P_r P_s : Formula r s : VarName name : PredName n : ℕ args_u args_v : Fin n → VarName h1_1 : ∀ (i : Fin n), args_u i = args_v i ∨ args_u i = r ∧ args_v i = s ⊢ IsDeduct ∅ ((eq_ r s).imp_ (List.foldr and_ true_ (List.ofFn fun i => eq_ (args_u i) (args_v i))))
case a.a.zero P_r P_s : Formula r s : VarName name : PredName args_u args_v : Fin 0 → VarName h1_1 : ∀ (i : Fin 0), args_u i = args_v i ∨ args_u i = r ∧ args_v i = s ⊢ IsDeduct ∅ ((eq_ r s).imp_ (List.foldr and_ true_ (List.ofFn fun i => eq_ (args_u i) (args_v i)))) case a.a.succ P_r P_s : Formula r s : VarName name : PredName n✝ : ℕ a✝ : ∀ (args_u args_v : Fin n✝ → VarName), (∀ (i : Fin n✝), args_u i = args_v i ∨ args_u i = r ∧ args_v i = s) → IsDeduct ∅ ((eq_ r s).imp_ (List.foldr and_ true_ (List.ofFn fun i => eq_ (args_u i) (args_v i)))) args_u args_v : Fin (n✝ + 1) → VarName h1_1 : ∀ (i : Fin (n✝ + 1)), args_u i = args_v i ∨ args_u i = r ∧ args_v i = s ⊢ IsDeduct ∅ ((eq_ r s).imp_ (List.foldr and_ true_ (List.ofFn fun i => eq_ (args_u i) (args_v i))))
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/NV/Margaris/Fol.lean
FOL.NV.T_21_8
[1404, 1]
[1543, 10]
case _ => simp SC
P_r P_s : Formula r s : VarName name : PredName args_u args_v : Fin 0 → VarName h1_1 : ∀ (i : Fin 0), args_u i = args_v i ∨ args_u i = r ∧ args_v i = s ⊢ IsDeduct ∅ ((eq_ r s).imp_ (List.foldr and_ true_ (List.ofFn fun i => eq_ (args_u i) (args_v i))))
no goals
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/NV/Margaris/Fol.lean
FOL.NV.T_21_8
[1404, 1]
[1543, 10]
simp
P_r P_s : Formula r s : VarName name : PredName args_u args_v : Fin 0 → VarName h1_1 : ∀ (i : Fin 0), args_u i = args_v i ∨ args_u i = r ∧ args_v i = s ⊢ IsDeduct ∅ ((eq_ r s).imp_ (List.foldr and_ true_ (List.ofFn fun i => eq_ (args_u i) (args_v i))))
P_r P_s : Formula r s : VarName name : PredName args_u args_v : Fin 0 → VarName h1_1 : ∀ (i : Fin 0), args_u i = args_v i ∨ args_u i = r ∧ args_v i = s ⊢ IsDeduct ∅ ((eq_ r s).imp_ true_)
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/NV/Margaris/Fol.lean
FOL.NV.T_21_8
[1404, 1]
[1543, 10]
SC
P_r P_s : Formula r s : VarName name : PredName args_u args_v : Fin 0 → VarName h1_1 : ∀ (i : Fin 0), args_u i = args_v i ∨ args_u i = r ∧ args_v i = s ⊢ IsDeduct ∅ ((eq_ r s).imp_ true_)
no goals
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/NV/Margaris/Fol.lean
FOL.NV.T_21_8
[1404, 1]
[1543, 10]
simp
P_r P_s : Formula r s : VarName name : PredName n : ℕ ih : ∀ (args_u args_v : Fin n → VarName), (∀ (i : Fin n), args_u i = args_v i ∨ args_u i = r ∧ args_v i = s) → IsDeduct ∅ ((eq_ r s).imp_ (List.foldr and_ true_ (List.ofFn fun i => eq_ (args_u i) (args_v i)))) args_u args_v : Fin (n + 1) → VarName h1_1 : ∀ (i : Fin (n + 1)), args_u i = args_v i ∨ args_u i = r ∧ args_v i = s ⊢ IsDeduct ∅ ((eq_ r s).imp_ (List.foldr and_ true_ (List.ofFn fun i => eq_ (args_u i) (args_v i))))
P_r P_s : Formula r s : VarName name : PredName n : ℕ ih : ∀ (args_u args_v : Fin n → VarName), (∀ (i : Fin n), args_u i = args_v i ∨ args_u i = r ∧ args_v i = s) → IsDeduct ∅ ((eq_ r s).imp_ (List.foldr and_ true_ (List.ofFn fun i => eq_ (args_u i) (args_v i)))) args_u args_v : Fin (n + 1) → VarName h1_1 : ∀ (i : Fin (n + 1)), args_u i = args_v i ∨ args_u i = r ∧ args_v i = s ⊢ IsDeduct ∅ ((eq_ r s).imp_ ((eq_ (args_u 0) (args_v 0)).and_ (List.foldr and_ true_ (List.ofFn fun i => eq_ (args_u i.succ) (args_v i.succ)))))
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/NV/Margaris/Fol.lean
FOL.NV.T_21_8
[1404, 1]
[1543, 10]
apply IsDeduct.mp_ ((eq_ r s).imp_ (List.foldr and_ true_ (List.ofFn fun (i : Fin n) => eq_ (args_u i.succ) (args_v i.succ))))
P_r P_s : Formula r s : VarName name : PredName n : ℕ ih : ∀ (args_u args_v : Fin n → VarName), (∀ (i : Fin n), args_u i = args_v i ∨ args_u i = r ∧ args_v i = s) → IsDeduct ∅ ((eq_ r s).imp_ (List.foldr and_ true_ (List.ofFn fun i => eq_ (args_u i) (args_v i)))) args_u args_v : Fin (n + 1) → VarName h1_1 : ∀ (i : Fin (n + 1)), args_u i = args_v i ∨ args_u i = r ∧ args_v i = s ⊢ IsDeduct ∅ ((eq_ r s).imp_ ((eq_ (args_u 0) (args_v 0)).and_ (List.foldr and_ true_ (List.ofFn fun i => eq_ (args_u i.succ) (args_v i.succ)))))
case a P_r P_s : Formula r s : VarName name : PredName n : ℕ ih : ∀ (args_u args_v : Fin n → VarName), (∀ (i : Fin n), args_u i = args_v i ∨ args_u i = r ∧ args_v i = s) → IsDeduct ∅ ((eq_ r s).imp_ (List.foldr and_ true_ (List.ofFn fun i => eq_ (args_u i) (args_v i)))) args_u args_v : Fin (n + 1) → VarName h1_1 : ∀ (i : Fin (n + 1)), args_u i = args_v i ∨ args_u i = r ∧ args_v i = s ⊢ IsDeduct ∅ (((eq_ r s).imp_ (List.foldr and_ true_ (List.ofFn fun i => eq_ (args_u i.succ) (args_v i.succ)))).imp_ ((eq_ r s).imp_ ((eq_ (args_u 0) (args_v 0)).and_ (List.foldr and_ true_ (List.ofFn fun i => eq_ (args_u i.succ) (args_v i.succ)))))) case a P_r P_s : Formula r s : VarName name : PredName n : ℕ ih : ∀ (args_u args_v : Fin n → VarName), (∀ (i : Fin n), args_u i = args_v i ∨ args_u i = r ∧ args_v i = s) → IsDeduct ∅ ((eq_ r s).imp_ (List.foldr and_ true_ (List.ofFn fun i => eq_ (args_u i) (args_v i)))) args_u args_v : Fin (n + 1) → VarName h1_1 : ∀ (i : Fin (n + 1)), args_u i = args_v i ∨ args_u i = r ∧ args_v i = s ⊢ IsDeduct ∅ ((eq_ r s).imp_ (List.foldr and_ true_ (List.ofFn fun i => eq_ (args_u i.succ) (args_v i.succ))))
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/NV/Margaris/Fol.lean
FOL.NV.T_21_8
[1404, 1]
[1543, 10]
apply IsDeduct.mp_ ((eq_ r s).imp_ (eq_ (args_u 0) (args_v 0)))
case a P_r P_s : Formula r s : VarName name : PredName n : ℕ ih : ∀ (args_u args_v : Fin n → VarName), (∀ (i : Fin n), args_u i = args_v i ∨ args_u i = r ∧ args_v i = s) → IsDeduct ∅ ((eq_ r s).imp_ (List.foldr and_ true_ (List.ofFn fun i => eq_ (args_u i) (args_v i)))) args_u args_v : Fin (n + 1) → VarName h1_1 : ∀ (i : Fin (n + 1)), args_u i = args_v i ∨ args_u i = r ∧ args_v i = s ⊢ IsDeduct ∅ (((eq_ r s).imp_ (List.foldr and_ true_ (List.ofFn fun i => eq_ (args_u i.succ) (args_v i.succ)))).imp_ ((eq_ r s).imp_ ((eq_ (args_u 0) (args_v 0)).and_ (List.foldr and_ true_ (List.ofFn fun i => eq_ (args_u i.succ) (args_v i.succ))))))
case a.a P_r P_s : Formula r s : VarName name : PredName n : ℕ ih : ∀ (args_u args_v : Fin n → VarName), (∀ (i : Fin n), args_u i = args_v i ∨ args_u i = r ∧ args_v i = s) → IsDeduct ∅ ((eq_ r s).imp_ (List.foldr and_ true_ (List.ofFn fun i => eq_ (args_u i) (args_v i)))) args_u args_v : Fin (n + 1) → VarName h1_1 : ∀ (i : Fin (n + 1)), args_u i = args_v i ∨ args_u i = r ∧ args_v i = s ⊢ IsDeduct ∅ (((eq_ r s).imp_ (eq_ (args_u 0) (args_v 0))).imp_ (((eq_ r s).imp_ (List.foldr and_ true_ (List.ofFn fun i => eq_ (args_u i.succ) (args_v i.succ)))).imp_ ((eq_ r s).imp_ ((eq_ (args_u 0) (args_v 0)).and_ (List.foldr and_ true_ (List.ofFn fun i => eq_ (args_u i.succ) (args_v i.succ))))))) case a.a P_r P_s : Formula r s : VarName name : PredName n : ℕ ih : ∀ (args_u args_v : Fin n → VarName), (∀ (i : Fin n), args_u i = args_v i ∨ args_u i = r ∧ args_v i = s) → IsDeduct ∅ ((eq_ r s).imp_ (List.foldr and_ true_ (List.ofFn fun i => eq_ (args_u i) (args_v i)))) args_u args_v : Fin (n + 1) → VarName h1_1 : ∀ (i : Fin (n + 1)), args_u i = args_v i ∨ args_u i = r ∧ args_v i = s ⊢ IsDeduct ∅ ((eq_ r s).imp_ (eq_ (args_u 0) (args_v 0)))
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/NV/Margaris/Fol.lean
FOL.NV.T_21_8
[1404, 1]
[1543, 10]
simp only [def_and_]
case a.a P_r P_s : Formula r s : VarName name : PredName n : ℕ ih : ∀ (args_u args_v : Fin n → VarName), (∀ (i : Fin n), args_u i = args_v i ∨ args_u i = r ∧ args_v i = s) → IsDeduct ∅ ((eq_ r s).imp_ (List.foldr and_ true_ (List.ofFn fun i => eq_ (args_u i) (args_v i)))) args_u args_v : Fin (n + 1) → VarName h1_1 : ∀ (i : Fin (n + 1)), args_u i = args_v i ∨ args_u i = r ∧ args_v i = s ⊢ IsDeduct ∅ (((eq_ r s).imp_ (eq_ (args_u 0) (args_v 0))).imp_ (((eq_ r s).imp_ (List.foldr and_ true_ (List.ofFn fun i => eq_ (args_u i.succ) (args_v i.succ)))).imp_ ((eq_ r s).imp_ ((eq_ (args_u 0) (args_v 0)).and_ (List.foldr and_ true_ (List.ofFn fun i => eq_ (args_u i.succ) (args_v i.succ)))))))
case a.a P_r P_s : Formula r s : VarName name : PredName n : ℕ ih : ∀ (args_u args_v : Fin n → VarName), (∀ (i : Fin n), args_u i = args_v i ∨ args_u i = r ∧ args_v i = s) → IsDeduct ∅ ((eq_ r s).imp_ (List.foldr and_ true_ (List.ofFn fun i => eq_ (args_u i) (args_v i)))) args_u args_v : Fin (n + 1) → VarName h1_1 : ∀ (i : Fin (n + 1)), args_u i = args_v i ∨ args_u i = r ∧ args_v i = s ⊢ IsDeduct ∅ (((eq_ r s).imp_ (eq_ (args_u 0) (args_v 0))).imp_ (((eq_ r s).imp_ (List.foldr and_ true_ (List.ofFn fun i => eq_ (args_u i.succ) (args_v i.succ)))).imp_ ((eq_ r s).imp_ ((eq_ (args_u 0) (args_v 0)).imp_ (List.foldr and_ true_ (List.ofFn fun i => eq_ (args_u i.succ) (args_v i.succ))).not_).not_)))
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/NV/Margaris/Fol.lean
FOL.NV.T_21_8
[1404, 1]
[1543, 10]
SC
case a.a P_r P_s : Formula r s : VarName name : PredName n : ℕ ih : ∀ (args_u args_v : Fin n → VarName), (∀ (i : Fin n), args_u i = args_v i ∨ args_u i = r ∧ args_v i = s) → IsDeduct ∅ ((eq_ r s).imp_ (List.foldr and_ true_ (List.ofFn fun i => eq_ (args_u i) (args_v i)))) args_u args_v : Fin (n + 1) → VarName h1_1 : ∀ (i : Fin (n + 1)), args_u i = args_v i ∨ args_u i = r ∧ args_v i = s ⊢ IsDeduct ∅ (((eq_ r s).imp_ (eq_ (args_u 0) (args_v 0))).imp_ (((eq_ r s).imp_ (List.foldr and_ true_ (List.ofFn fun i => eq_ (args_u i.succ) (args_v i.succ)))).imp_ ((eq_ r s).imp_ ((eq_ (args_u 0) (args_v 0)).imp_ (List.foldr and_ true_ (List.ofFn fun i => eq_ (args_u i.succ) (args_v i.succ))).not_).not_)))
no goals
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/NV/Margaris/Fol.lean
FOL.NV.T_21_8
[1404, 1]
[1543, 10]
specialize h1_1 0
case a.a P_r P_s : Formula r s : VarName name : PredName n : ℕ ih : ∀ (args_u args_v : Fin n → VarName), (∀ (i : Fin n), args_u i = args_v i ∨ args_u i = r ∧ args_v i = s) → IsDeduct ∅ ((eq_ r s).imp_ (List.foldr and_ true_ (List.ofFn fun i => eq_ (args_u i) (args_v i)))) args_u args_v : Fin (n + 1) → VarName h1_1 : ∀ (i : Fin (n + 1)), args_u i = args_v i ∨ args_u i = r ∧ args_v i = s ⊢ IsDeduct ∅ ((eq_ r s).imp_ (eq_ (args_u 0) (args_v 0)))
case a.a P_r P_s : Formula r s : VarName name : PredName n : ℕ ih : ∀ (args_u args_v : Fin n → VarName), (∀ (i : Fin n), args_u i = args_v i ∨ args_u i = r ∧ args_v i = s) → IsDeduct ∅ ((eq_ r s).imp_ (List.foldr and_ true_ (List.ofFn fun i => eq_ (args_u i) (args_v i)))) args_u args_v : Fin (n + 1) → VarName h1_1 : args_u 0 = args_v 0 ∨ args_u 0 = r ∧ args_v 0 = s ⊢ IsDeduct ∅ ((eq_ r s).imp_ (eq_ (args_u 0) (args_v 0)))
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/NV/Margaris/Fol.lean
FOL.NV.T_21_8
[1404, 1]
[1543, 10]
cases h1_1
case a.a P_r P_s : Formula r s : VarName name : PredName n : ℕ ih : ∀ (args_u args_v : Fin n → VarName), (∀ (i : Fin n), args_u i = args_v i ∨ args_u i = r ∧ args_v i = s) → IsDeduct ∅ ((eq_ r s).imp_ (List.foldr and_ true_ (List.ofFn fun i => eq_ (args_u i) (args_v i)))) args_u args_v : Fin (n + 1) → VarName h1_1 : args_u 0 = args_v 0 ∨ args_u 0 = r ∧ args_v 0 = s ⊢ IsDeduct ∅ ((eq_ r s).imp_ (eq_ (args_u 0) (args_v 0)))
case a.a.inl P_r P_s : Formula r s : VarName name : PredName n : ℕ ih : ∀ (args_u args_v : Fin n → VarName), (∀ (i : Fin n), args_u i = args_v i ∨ args_u i = r ∧ args_v i = s) → IsDeduct ∅ ((eq_ r s).imp_ (List.foldr and_ true_ (List.ofFn fun i => eq_ (args_u i) (args_v i)))) args_u args_v : Fin (n + 1) → VarName h✝ : args_u 0 = args_v 0 ⊢ IsDeduct ∅ ((eq_ r s).imp_ (eq_ (args_u 0) (args_v 0))) case a.a.inr P_r P_s : Formula r s : VarName name : PredName n : ℕ ih : ∀ (args_u args_v : Fin n → VarName), (∀ (i : Fin n), args_u i = args_v i ∨ args_u i = r ∧ args_v i = s) → IsDeduct ∅ ((eq_ r s).imp_ (List.foldr and_ true_ (List.ofFn fun i => eq_ (args_u i) (args_v i)))) args_u args_v : Fin (n + 1) → VarName h✝ : args_u 0 = r ∧ args_v 0 = s ⊢ IsDeduct ∅ ((eq_ r s).imp_ (eq_ (args_u 0) (args_v 0)))
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/NV/Margaris/Fol.lean
FOL.NV.T_21_8
[1404, 1]
[1543, 10]
case _ c1 => apply IsDeduct.mp_ (eq_ (args_u 0) (args_v 0)) case _ => SC case _ => simp only [c1] apply specId (args_v 0) apply IsDeduct.axiom_ apply IsAxiom.eq_1_
P_r P_s : Formula r s : VarName name : PredName n : ℕ ih : ∀ (args_u args_v : Fin n → VarName), (∀ (i : Fin n), args_u i = args_v i ∨ args_u i = r ∧ args_v i = s) → IsDeduct ∅ ((eq_ r s).imp_ (List.foldr and_ true_ (List.ofFn fun i => eq_ (args_u i) (args_v i)))) args_u args_v : Fin (n + 1) → VarName c1 : args_u 0 = args_v 0 ⊢ IsDeduct ∅ ((eq_ r s).imp_ (eq_ (args_u 0) (args_v 0)))
no goals
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/NV/Margaris/Fol.lean
FOL.NV.T_21_8
[1404, 1]
[1543, 10]
case _ c1 => cases c1 case _ c1_left c1_right => subst c1_left subst c1_right SC
P_r P_s : Formula r s : VarName name : PredName n : ℕ ih : ∀ (args_u args_v : Fin n → VarName), (∀ (i : Fin n), args_u i = args_v i ∨ args_u i = r ∧ args_v i = s) → IsDeduct ∅ ((eq_ r s).imp_ (List.foldr and_ true_ (List.ofFn fun i => eq_ (args_u i) (args_v i)))) args_u args_v : Fin (n + 1) → VarName c1 : args_u 0 = r ∧ args_v 0 = s ⊢ IsDeduct ∅ ((eq_ r s).imp_ (eq_ (args_u 0) (args_v 0)))
no goals
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/NV/Margaris/Fol.lean
FOL.NV.T_21_8
[1404, 1]
[1543, 10]
apply IsDeduct.mp_ (eq_ (args_u 0) (args_v 0))
P_r P_s : Formula r s : VarName name : PredName n : ℕ ih : ∀ (args_u args_v : Fin n → VarName), (∀ (i : Fin n), args_u i = args_v i ∨ args_u i = r ∧ args_v i = s) → IsDeduct ∅ ((eq_ r s).imp_ (List.foldr and_ true_ (List.ofFn fun i => eq_ (args_u i) (args_v i)))) args_u args_v : Fin (n + 1) → VarName c1 : args_u 0 = args_v 0 ⊢ IsDeduct ∅ ((eq_ r s).imp_ (eq_ (args_u 0) (args_v 0)))
case a P_r P_s : Formula r s : VarName name : PredName n : ℕ ih : ∀ (args_u args_v : Fin n → VarName), (∀ (i : Fin n), args_u i = args_v i ∨ args_u i = r ∧ args_v i = s) → IsDeduct ∅ ((eq_ r s).imp_ (List.foldr and_ true_ (List.ofFn fun i => eq_ (args_u i) (args_v i)))) args_u args_v : Fin (n + 1) → VarName c1 : args_u 0 = args_v 0 ⊢ IsDeduct ∅ ((eq_ (args_u 0) (args_v 0)).imp_ ((eq_ r s).imp_ (eq_ (args_u 0) (args_v 0)))) case a P_r P_s : Formula r s : VarName name : PredName n : ℕ ih : ∀ (args_u args_v : Fin n → VarName), (∀ (i : Fin n), args_u i = args_v i ∨ args_u i = r ∧ args_v i = s) → IsDeduct ∅ ((eq_ r s).imp_ (List.foldr and_ true_ (List.ofFn fun i => eq_ (args_u i) (args_v i)))) args_u args_v : Fin (n + 1) → VarName c1 : args_u 0 = args_v 0 ⊢ IsDeduct ∅ (eq_ (args_u 0) (args_v 0))
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/NV/Margaris/Fol.lean
FOL.NV.T_21_8
[1404, 1]
[1543, 10]
case _ => SC
P_r P_s : Formula r s : VarName name : PredName n : ℕ ih : ∀ (args_u args_v : Fin n → VarName), (∀ (i : Fin n), args_u i = args_v i ∨ args_u i = r ∧ args_v i = s) → IsDeduct ∅ ((eq_ r s).imp_ (List.foldr and_ true_ (List.ofFn fun i => eq_ (args_u i) (args_v i)))) args_u args_v : Fin (n + 1) → VarName c1 : args_u 0 = args_v 0 ⊢ IsDeduct ∅ ((eq_ (args_u 0) (args_v 0)).imp_ ((eq_ r s).imp_ (eq_ (args_u 0) (args_v 0))))
no goals
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/NV/Margaris/Fol.lean
FOL.NV.T_21_8
[1404, 1]
[1543, 10]
case _ => simp only [c1] apply specId (args_v 0) apply IsDeduct.axiom_ apply IsAxiom.eq_1_
P_r P_s : Formula r s : VarName name : PredName n : ℕ ih : ∀ (args_u args_v : Fin n → VarName), (∀ (i : Fin n), args_u i = args_v i ∨ args_u i = r ∧ args_v i = s) → IsDeduct ∅ ((eq_ r s).imp_ (List.foldr and_ true_ (List.ofFn fun i => eq_ (args_u i) (args_v i)))) args_u args_v : Fin (n + 1) → VarName c1 : args_u 0 = args_v 0 ⊢ IsDeduct ∅ (eq_ (args_u 0) (args_v 0))
no goals
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/NV/Margaris/Fol.lean
FOL.NV.T_21_8
[1404, 1]
[1543, 10]
SC
P_r P_s : Formula r s : VarName name : PredName n : ℕ ih : ∀ (args_u args_v : Fin n → VarName), (∀ (i : Fin n), args_u i = args_v i ∨ args_u i = r ∧ args_v i = s) → IsDeduct ∅ ((eq_ r s).imp_ (List.foldr and_ true_ (List.ofFn fun i => eq_ (args_u i) (args_v i)))) args_u args_v : Fin (n + 1) → VarName c1 : args_u 0 = args_v 0 ⊢ IsDeduct ∅ ((eq_ (args_u 0) (args_v 0)).imp_ ((eq_ r s).imp_ (eq_ (args_u 0) (args_v 0))))
no goals
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/NV/Margaris/Fol.lean
FOL.NV.T_21_8
[1404, 1]
[1543, 10]
simp only [c1]
P_r P_s : Formula r s : VarName name : PredName n : ℕ ih : ∀ (args_u args_v : Fin n → VarName), (∀ (i : Fin n), args_u i = args_v i ∨ args_u i = r ∧ args_v i = s) → IsDeduct ∅ ((eq_ r s).imp_ (List.foldr and_ true_ (List.ofFn fun i => eq_ (args_u i) (args_v i)))) args_u args_v : Fin (n + 1) → VarName c1 : args_u 0 = args_v 0 ⊢ IsDeduct ∅ (eq_ (args_u 0) (args_v 0))
P_r P_s : Formula r s : VarName name : PredName n : ℕ ih : ∀ (args_u args_v : Fin n → VarName), (∀ (i : Fin n), args_u i = args_v i ∨ args_u i = r ∧ args_v i = s) → IsDeduct ∅ ((eq_ r s).imp_ (List.foldr and_ true_ (List.ofFn fun i => eq_ (args_u i) (args_v i)))) args_u args_v : Fin (n + 1) → VarName c1 : args_u 0 = args_v 0 ⊢ IsDeduct ∅ (eq_ (args_v 0) (args_v 0))
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/NV/Margaris/Fol.lean
FOL.NV.T_21_8
[1404, 1]
[1543, 10]
apply specId (args_v 0)
P_r P_s : Formula r s : VarName name : PredName n : ℕ ih : ∀ (args_u args_v : Fin n → VarName), (∀ (i : Fin n), args_u i = args_v i ∨ args_u i = r ∧ args_v i = s) → IsDeduct ∅ ((eq_ r s).imp_ (List.foldr and_ true_ (List.ofFn fun i => eq_ (args_u i) (args_v i)))) args_u args_v : Fin (n + 1) → VarName c1 : args_u 0 = args_v 0 ⊢ IsDeduct ∅ (eq_ (args_v 0) (args_v 0))
case h1 P_r P_s : Formula r s : VarName name : PredName n : ℕ ih : ∀ (args_u args_v : Fin n → VarName), (∀ (i : Fin n), args_u i = args_v i ∨ args_u i = r ∧ args_v i = s) → IsDeduct ∅ ((eq_ r s).imp_ (List.foldr and_ true_ (List.ofFn fun i => eq_ (args_u i) (args_v i)))) args_u args_v : Fin (n + 1) → VarName c1 : args_u 0 = args_v 0 ⊢ IsDeduct ∅ (forall_ (args_v 0) (eq_ (args_v 0) (args_v 0)))
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/NV/Margaris/Fol.lean
FOL.NV.T_21_8
[1404, 1]
[1543, 10]
apply IsDeduct.axiom_
case h1 P_r P_s : Formula r s : VarName name : PredName n : ℕ ih : ∀ (args_u args_v : Fin n → VarName), (∀ (i : Fin n), args_u i = args_v i ∨ args_u i = r ∧ args_v i = s) → IsDeduct ∅ ((eq_ r s).imp_ (List.foldr and_ true_ (List.ofFn fun i => eq_ (args_u i) (args_v i)))) args_u args_v : Fin (n + 1) → VarName c1 : args_u 0 = args_v 0 ⊢ IsDeduct ∅ (forall_ (args_v 0) (eq_ (args_v 0) (args_v 0)))
case h1.a P_r P_s : Formula r s : VarName name : PredName n : ℕ ih : ∀ (args_u args_v : Fin n → VarName), (∀ (i : Fin n), args_u i = args_v i ∨ args_u i = r ∧ args_v i = s) → IsDeduct ∅ ((eq_ r s).imp_ (List.foldr and_ true_ (List.ofFn fun i => eq_ (args_u i) (args_v i)))) args_u args_v : Fin (n + 1) → VarName c1 : args_u 0 = args_v 0 ⊢ IsAxiom (forall_ (args_v 0) (eq_ (args_v 0) (args_v 0)))
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/NV/Margaris/Fol.lean
FOL.NV.T_21_8
[1404, 1]
[1543, 10]
apply IsAxiom.eq_1_
case h1.a P_r P_s : Formula r s : VarName name : PredName n : ℕ ih : ∀ (args_u args_v : Fin n → VarName), (∀ (i : Fin n), args_u i = args_v i ∨ args_u i = r ∧ args_v i = s) → IsDeduct ∅ ((eq_ r s).imp_ (List.foldr and_ true_ (List.ofFn fun i => eq_ (args_u i) (args_v i)))) args_u args_v : Fin (n + 1) → VarName c1 : args_u 0 = args_v 0 ⊢ IsAxiom (forall_ (args_v 0) (eq_ (args_v 0) (args_v 0)))
no goals
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/NV/Margaris/Fol.lean
FOL.NV.T_21_8
[1404, 1]
[1543, 10]
cases c1
P_r P_s : Formula r s : VarName name : PredName n : ℕ ih : ∀ (args_u args_v : Fin n → VarName), (∀ (i : Fin n), args_u i = args_v i ∨ args_u i = r ∧ args_v i = s) → IsDeduct ∅ ((eq_ r s).imp_ (List.foldr and_ true_ (List.ofFn fun i => eq_ (args_u i) (args_v i)))) args_u args_v : Fin (n + 1) → VarName c1 : args_u 0 = r ∧ args_v 0 = s ⊢ IsDeduct ∅ ((eq_ r s).imp_ (eq_ (args_u 0) (args_v 0)))
case intro P_r P_s : Formula r s : VarName name : PredName n : ℕ ih : ∀ (args_u args_v : Fin n → VarName), (∀ (i : Fin n), args_u i = args_v i ∨ args_u i = r ∧ args_v i = s) → IsDeduct ∅ ((eq_ r s).imp_ (List.foldr and_ true_ (List.ofFn fun i => eq_ (args_u i) (args_v i)))) args_u args_v : Fin (n + 1) → VarName left✝ : args_u 0 = r right✝ : args_v 0 = s ⊢ IsDeduct ∅ ((eq_ r s).imp_ (eq_ (args_u 0) (args_v 0)))
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/NV/Margaris/Fol.lean
FOL.NV.T_21_8
[1404, 1]
[1543, 10]
case _ c1_left c1_right => subst c1_left subst c1_right SC
P_r P_s : Formula r s : VarName name : PredName n : ℕ ih : ∀ (args_u args_v : Fin n → VarName), (∀ (i : Fin n), args_u i = args_v i ∨ args_u i = r ∧ args_v i = s) → IsDeduct ∅ ((eq_ r s).imp_ (List.foldr and_ true_ (List.ofFn fun i => eq_ (args_u i) (args_v i)))) args_u args_v : Fin (n + 1) → VarName c1_left : args_u 0 = r c1_right : args_v 0 = s ⊢ IsDeduct ∅ ((eq_ r s).imp_ (eq_ (args_u 0) (args_v 0)))
no goals
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/NV/Margaris/Fol.lean
FOL.NV.T_21_8
[1404, 1]
[1543, 10]
subst c1_left
P_r P_s : Formula r s : VarName name : PredName n : ℕ ih : ∀ (args_u args_v : Fin n → VarName), (∀ (i : Fin n), args_u i = args_v i ∨ args_u i = r ∧ args_v i = s) → IsDeduct ∅ ((eq_ r s).imp_ (List.foldr and_ true_ (List.ofFn fun i => eq_ (args_u i) (args_v i)))) args_u args_v : Fin (n + 1) → VarName c1_left : args_u 0 = r c1_right : args_v 0 = s ⊢ IsDeduct ∅ ((eq_ r s).imp_ (eq_ (args_u 0) (args_v 0)))
P_r P_s : Formula s : VarName name : PredName n : ℕ args_u args_v : Fin (n + 1) → VarName c1_right : args_v 0 = s ih : ∀ (args_u_1 args_v : Fin n → VarName), (∀ (i : Fin n), args_u_1 i = args_v i ∨ args_u_1 i = args_u 0 ∧ args_v i = s) → IsDeduct ∅ ((eq_ (args_u 0) s).imp_ (List.foldr and_ true_ (List.ofFn fun i => eq_ (args_u_1 i) (args_v i)))) ⊢ IsDeduct ∅ ((eq_ (args_u 0) s).imp_ (eq_ (args_u 0) (args_v 0)))
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/NV/Margaris/Fol.lean
FOL.NV.T_21_8
[1404, 1]
[1543, 10]
subst c1_right
P_r P_s : Formula s : VarName name : PredName n : ℕ args_u args_v : Fin (n + 1) → VarName c1_right : args_v 0 = s ih : ∀ (args_u_1 args_v : Fin n → VarName), (∀ (i : Fin n), args_u_1 i = args_v i ∨ args_u_1 i = args_u 0 ∧ args_v i = s) → IsDeduct ∅ ((eq_ (args_u 0) s).imp_ (List.foldr and_ true_ (List.ofFn fun i => eq_ (args_u_1 i) (args_v i)))) ⊢ IsDeduct ∅ ((eq_ (args_u 0) s).imp_ (eq_ (args_u 0) (args_v 0)))
P_r P_s : Formula name : PredName n : ℕ args_u args_v : Fin (n + 1) → VarName ih : ∀ (args_u_1 args_v_1 : Fin n → VarName), (∀ (i : Fin n), args_u_1 i = args_v_1 i ∨ args_u_1 i = args_u 0 ∧ args_v_1 i = args_v 0) → IsDeduct ∅ ((eq_ (args_u 0) (args_v 0)).imp_ (List.foldr and_ true_ (List.ofFn fun i => eq_ (args_u_1 i) (args_v_1 i)))) ⊢ IsDeduct ∅ ((eq_ (args_u 0) (args_v 0)).imp_ (eq_ (args_u 0) (args_v 0)))
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/NV/Margaris/Fol.lean
FOL.NV.T_21_8
[1404, 1]
[1543, 10]
SC
P_r P_s : Formula name : PredName n : ℕ args_u args_v : Fin (n + 1) → VarName ih : ∀ (args_u_1 args_v_1 : Fin n → VarName), (∀ (i : Fin n), args_u_1 i = args_v_1 i ∨ args_u_1 i = args_u 0 ∧ args_v_1 i = args_v 0) → IsDeduct ∅ ((eq_ (args_u 0) (args_v 0)).imp_ (List.foldr and_ true_ (List.ofFn fun i => eq_ (args_u_1 i) (args_v_1 i)))) ⊢ IsDeduct ∅ ((eq_ (args_u 0) (args_v 0)).imp_ (eq_ (args_u 0) (args_v 0)))
no goals
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/NV/Margaris/Fol.lean
FOL.NV.T_21_8
[1404, 1]
[1543, 10]
apply ih
case a P_r P_s : Formula r s : VarName name : PredName n : ℕ ih : ∀ (args_u args_v : Fin n → VarName), (∀ (i : Fin n), args_u i = args_v i ∨ args_u i = r ∧ args_v i = s) → IsDeduct ∅ ((eq_ r s).imp_ (List.foldr and_ true_ (List.ofFn fun i => eq_ (args_u i) (args_v i)))) args_u args_v : Fin (n + 1) → VarName h1_1 : ∀ (i : Fin (n + 1)), args_u i = args_v i ∨ args_u i = r ∧ args_v i = s ⊢ IsDeduct ∅ ((eq_ r s).imp_ (List.foldr and_ true_ (List.ofFn fun i => eq_ (args_u i.succ) (args_v i.succ))))
case a.h1_1 P_r P_s : Formula r s : VarName name : PredName n : ℕ ih : ∀ (args_u args_v : Fin n → VarName), (∀ (i : Fin n), args_u i = args_v i ∨ args_u i = r ∧ args_v i = s) → IsDeduct ∅ ((eq_ r s).imp_ (List.foldr and_ true_ (List.ofFn fun i => eq_ (args_u i) (args_v i)))) args_u args_v : Fin (n + 1) → VarName h1_1 : ∀ (i : Fin (n + 1)), args_u i = args_v i ∨ args_u i = r ∧ args_v i = s ⊢ ∀ (i : Fin n), args_u i.succ = args_v i.succ ∨ args_u i.succ = r ∧ args_v i.succ = s
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/NV/Margaris/Fol.lean
FOL.NV.T_21_8
[1404, 1]
[1543, 10]
intro i
case a.h1_1 P_r P_s : Formula r s : VarName name : PredName n : ℕ ih : ∀ (args_u args_v : Fin n → VarName), (∀ (i : Fin n), args_u i = args_v i ∨ args_u i = r ∧ args_v i = s) → IsDeduct ∅ ((eq_ r s).imp_ (List.foldr and_ true_ (List.ofFn fun i => eq_ (args_u i) (args_v i)))) args_u args_v : Fin (n + 1) → VarName h1_1 : ∀ (i : Fin (n + 1)), args_u i = args_v i ∨ args_u i = r ∧ args_v i = s ⊢ ∀ (i : Fin n), args_u i.succ = args_v i.succ ∨ args_u i.succ = r ∧ args_v i.succ = s
case a.h1_1 P_r P_s : Formula r s : VarName name : PredName n : ℕ ih : ∀ (args_u args_v : Fin n → VarName), (∀ (i : Fin n), args_u i = args_v i ∨ args_u i = r ∧ args_v i = s) → IsDeduct ∅ ((eq_ r s).imp_ (List.foldr and_ true_ (List.ofFn fun i => eq_ (args_u i) (args_v i)))) args_u args_v : Fin (n + 1) → VarName h1_1 : ∀ (i : Fin (n + 1)), args_u i = args_v i ∨ args_u i = r ∧ args_v i = s i : Fin n ⊢ args_u i.succ = args_v i.succ ∨ args_u i.succ = r ∧ args_v i.succ = s
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/NV/Margaris/Fol.lean
FOL.NV.T_21_8
[1404, 1]
[1543, 10]
apply h1_1
case a.h1_1 P_r P_s : Formula r s : VarName name : PredName n : ℕ ih : ∀ (args_u args_v : Fin n → VarName), (∀ (i : Fin n), args_u i = args_v i ∨ args_u i = r ∧ args_v i = s) → IsDeduct ∅ ((eq_ r s).imp_ (List.foldr and_ true_ (List.ofFn fun i => eq_ (args_u i) (args_v i)))) args_u args_v : Fin (n + 1) → VarName h1_1 : ∀ (i : Fin (n + 1)), args_u i = args_v i ∨ args_u i = r ∧ args_v i = s i : Fin n ⊢ args_u i.succ = args_v i.succ ∨ args_u i.succ = r ∧ args_v i.succ = s
no goals
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/NV/Margaris/Fol.lean
FOL.NV.T_21_8
[1404, 1]
[1543, 10]
simp only [isBoundIn] at h2
P_r P_s : Formula r s : VarName P_u P_v : Formula h1_1 : IsReplOfVarInFormula r s P_u P_v h1_ih : ¬isBoundIn r P_u → ¬isBoundIn s P_u → IsProof ((eq_ r s).imp_ (P_u.iff_ P_v)) h2 : ¬isBoundIn r P_u.not_ h3 : ¬isBoundIn s P_u.not_ ⊢ IsProof ((eq_ r s).imp_ (P_u.not_.iff_ P_v.not_))
P_r P_s : Formula r s : VarName P_u P_v : Formula h1_1 : IsReplOfVarInFormula r s P_u P_v h1_ih : ¬isBoundIn r P_u → ¬isBoundIn s P_u → IsProof ((eq_ r s).imp_ (P_u.iff_ P_v)) h2 : ¬isBoundIn r P_u h3 : ¬isBoundIn s P_u.not_ ⊢ IsProof ((eq_ r s).imp_ (P_u.not_.iff_ P_v.not_))
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/NV/Margaris/Fol.lean
FOL.NV.T_21_8
[1404, 1]
[1543, 10]
simp only [isBoundIn] at h3
P_r P_s : Formula r s : VarName P_u P_v : Formula h1_1 : IsReplOfVarInFormula r s P_u P_v h1_ih : ¬isBoundIn r P_u → ¬isBoundIn s P_u → IsProof ((eq_ r s).imp_ (P_u.iff_ P_v)) h2 : ¬isBoundIn r P_u h3 : ¬isBoundIn s P_u.not_ ⊢ IsProof ((eq_ r s).imp_ (P_u.not_.iff_ P_v.not_))
P_r P_s : Formula r s : VarName P_u P_v : Formula h1_1 : IsReplOfVarInFormula r s P_u P_v h1_ih : ¬isBoundIn r P_u → ¬isBoundIn s P_u → IsProof ((eq_ r s).imp_ (P_u.iff_ P_v)) h2 : ¬isBoundIn r P_u h3 : ¬isBoundIn s P_u ⊢ IsProof ((eq_ r s).imp_ (P_u.not_.iff_ P_v.not_))
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/NV/Margaris/Fol.lean
FOL.NV.T_21_8
[1404, 1]
[1543, 10]
specialize h1_ih h2 h3
P_r P_s : Formula r s : VarName P_u P_v : Formula h1_1 : IsReplOfVarInFormula r s P_u P_v h1_ih : ¬isBoundIn r P_u → ¬isBoundIn s P_u → IsProof ((eq_ r s).imp_ (P_u.iff_ P_v)) h2 : ¬isBoundIn r P_u h3 : ¬isBoundIn s P_u ⊢ IsProof ((eq_ r s).imp_ (P_u.not_.iff_ P_v.not_))
P_r P_s : Formula r s : VarName P_u P_v : Formula h1_1 : IsReplOfVarInFormula r s P_u P_v h2 : ¬isBoundIn r P_u h3 : ¬isBoundIn s P_u h1_ih : IsProof ((eq_ r s).imp_ (P_u.iff_ P_v)) ⊢ IsProof ((eq_ r s).imp_ (P_u.not_.iff_ P_v.not_))
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/NV/Margaris/Fol.lean
FOL.NV.T_21_8
[1404, 1]
[1543, 10]
apply IsDeduct.mp_ ((eq_ r s).imp_ (P_u.iff_ P_v))
P_r P_s : Formula r s : VarName P_u P_v : Formula h1_1 : IsReplOfVarInFormula r s P_u P_v h2 : ¬isBoundIn r P_u h3 : ¬isBoundIn s P_u h1_ih : IsProof ((eq_ r s).imp_ (P_u.iff_ P_v)) ⊢ IsProof ((eq_ r s).imp_ (P_u.not_.iff_ P_v.not_))
case a P_r P_s : Formula r s : VarName P_u P_v : Formula h1_1 : IsReplOfVarInFormula r s P_u P_v h2 : ¬isBoundIn r P_u h3 : ¬isBoundIn s P_u h1_ih : IsProof ((eq_ r s).imp_ (P_u.iff_ P_v)) ⊢ IsDeduct ∅ (((eq_ r s).imp_ (P_u.iff_ P_v)).imp_ ((eq_ r s).imp_ (P_u.not_.iff_ P_v.not_))) case a P_r P_s : Formula r s : VarName P_u P_v : Formula h1_1 : IsReplOfVarInFormula r s P_u P_v h2 : ¬isBoundIn r P_u h3 : ¬isBoundIn s P_u h1_ih : IsProof ((eq_ r s).imp_ (P_u.iff_ P_v)) ⊢ IsDeduct ∅ ((eq_ r s).imp_ (P_u.iff_ P_v))
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/NV/Margaris/Fol.lean
FOL.NV.T_21_8
[1404, 1]
[1543, 10]
simp only [def_iff_]
case a P_r P_s : Formula r s : VarName P_u P_v : Formula h1_1 : IsReplOfVarInFormula r s P_u P_v h2 : ¬isBoundIn r P_u h3 : ¬isBoundIn s P_u h1_ih : IsProof ((eq_ r s).imp_ (P_u.iff_ P_v)) ⊢ IsDeduct ∅ (((eq_ r s).imp_ (P_u.iff_ P_v)).imp_ ((eq_ r s).imp_ (P_u.not_.iff_ P_v.not_)))
case a P_r P_s : Formula r s : VarName P_u P_v : Formula h1_1 : IsReplOfVarInFormula r s P_u P_v h2 : ¬isBoundIn r P_u h3 : ¬isBoundIn s P_u h1_ih : IsProof ((eq_ r s).imp_ (P_u.iff_ P_v)) ⊢ IsDeduct ∅ (((eq_ r s).imp_ ((P_u.imp_ P_v).and_ (P_v.imp_ P_u))).imp_ ((eq_ r s).imp_ ((P_u.not_.imp_ P_v.not_).and_ (P_v.not_.imp_ P_u.not_))))
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/NV/Margaris/Fol.lean
FOL.NV.T_21_8
[1404, 1]
[1543, 10]
simp only [def_and_]
case a P_r P_s : Formula r s : VarName P_u P_v : Formula h1_1 : IsReplOfVarInFormula r s P_u P_v h2 : ¬isBoundIn r P_u h3 : ¬isBoundIn s P_u h1_ih : IsProof ((eq_ r s).imp_ (P_u.iff_ P_v)) ⊢ IsDeduct ∅ (((eq_ r s).imp_ ((P_u.imp_ P_v).and_ (P_v.imp_ P_u))).imp_ ((eq_ r s).imp_ ((P_u.not_.imp_ P_v.not_).and_ (P_v.not_.imp_ P_u.not_))))
case a P_r P_s : Formula r s : VarName P_u P_v : Formula h1_1 : IsReplOfVarInFormula r s P_u P_v h2 : ¬isBoundIn r P_u h3 : ¬isBoundIn s P_u h1_ih : IsProof ((eq_ r s).imp_ (P_u.iff_ P_v)) ⊢ IsDeduct ∅ (((eq_ r s).imp_ ((P_u.imp_ P_v).imp_ (P_v.imp_ P_u).not_).not_).imp_ ((eq_ r s).imp_ ((P_u.not_.imp_ P_v.not_).imp_ (P_v.not_.imp_ P_u.not_).not_).not_))
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/NV/Margaris/Fol.lean
FOL.NV.T_21_8
[1404, 1]
[1543, 10]
SC
case a P_r P_s : Formula r s : VarName P_u P_v : Formula h1_1 : IsReplOfVarInFormula r s P_u P_v h2 : ¬isBoundIn r P_u h3 : ¬isBoundIn s P_u h1_ih : IsProof ((eq_ r s).imp_ (P_u.iff_ P_v)) ⊢ IsDeduct ∅ (((eq_ r s).imp_ ((P_u.imp_ P_v).imp_ (P_v.imp_ P_u).not_).not_).imp_ ((eq_ r s).imp_ ((P_u.not_.imp_ P_v.not_).imp_ (P_v.not_.imp_ P_u.not_).not_).not_))
no goals
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/NV/Margaris/Fol.lean
FOL.NV.T_21_8
[1404, 1]
[1543, 10]
exact h1_ih
case a P_r P_s : Formula r s : VarName P_u P_v : Formula h1_1 : IsReplOfVarInFormula r s P_u P_v h2 : ¬isBoundIn r P_u h3 : ¬isBoundIn s P_u h1_ih : IsProof ((eq_ r s).imp_ (P_u.iff_ P_v)) ⊢ IsDeduct ∅ ((eq_ r s).imp_ (P_u.iff_ P_v))
no goals
https://github.com/pthomas505/FOL.git
097a4abea51b641d144539b9a0f7516f3b9d818c
FOL/NV/Margaris/Fol.lean
FOL.NV.T_21_8
[1404, 1]
[1543, 10]
simp only [isBoundIn] at h2
P_r P_s : Formula r s : VarName P_u Q_u P_v Q_v : Formula h1_1 : IsReplOfVarInFormula r s P_u P_v h1_2 : IsReplOfVarInFormula r s Q_u Q_v h1_ih_1 : ¬isBoundIn r P_u → ¬isBoundIn s P_u → IsProof ((eq_ r s).imp_ (P_u.iff_ P_v)) h1_ih_2 : ¬isBoundIn r Q_u → ¬isBoundIn s Q_u → IsProof ((eq_ r s).imp_ (Q_u.iff_ Q_v)) h2 : ¬isBoundIn r (P_u.imp_ Q_u) h3 : ¬isBoundIn s (P_u.imp_ Q_u) ⊢ IsProof ((eq_ r s).imp_ ((P_u.imp_ Q_u).iff_ (P_v.imp_ Q_v)))
P_r P_s : Formula r s : VarName P_u Q_u P_v Q_v : Formula h1_1 : IsReplOfVarInFormula r s P_u P_v h1_2 : IsReplOfVarInFormula r s Q_u Q_v h1_ih_1 : ¬isBoundIn r P_u → ¬isBoundIn s P_u → IsProof ((eq_ r s).imp_ (P_u.iff_ P_v)) h1_ih_2 : ¬isBoundIn r Q_u → ¬isBoundIn s Q_u → IsProof ((eq_ r s).imp_ (Q_u.iff_ Q_v)) h2 : ¬(isBoundIn r P_u ∨ isBoundIn r Q_u) h3 : ¬isBoundIn s (P_u.imp_ Q_u) ⊢ IsProof ((eq_ r s).imp_ ((P_u.imp_ Q_u).iff_ (P_v.imp_ Q_v)))