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
| input
stringlengths 73
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
|
Please generate a tactic in lean4 to solve the state.
STATE:
case h1
x y : VarName
⊢ IsDeduct ∅ (forall_ y ((eq_ x y).imp_ (eq_ y x)))
TACTIC:
|
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)
|
Please generate a tactic in lean4 to solve the state.
STATE:
case h1.h1
x y : VarName
⊢ IsDeduct ∅ ((eq_ x y).imp_ (eq_ y x))
TACTIC:
|
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)))
|
Please generate a tactic in lean4 to solve the state.
STATE:
case h1.h1.a
x y : VarName
⊢ IsDeduct ∅ ((eq_ y y).imp_ ((eq_ x y).imp_ (eq_ y x)))
TACTIC:
|
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))))
|
Please generate a tactic in lean4 to solve the state.
STATE:
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))))
TACTIC:
|
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))))
|
Please generate a tactic in lean4 to solve the state.
STATE:
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))))
TACTIC:
|
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
|
Please generate a tactic in lean4 to solve the state.
STATE:
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))))
TACTIC:
|
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))))
|
Please generate a tactic in lean4 to solve the state.
STATE:
case h1.h1.a.a
x y : VarName
⊢ IsDeduct ∅ (((eq_ y y).and_ (eq_ x y)).imp_ ((eq_ y x).iff_ (eq_ y y)))
TACTIC:
|
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)))))
|
Please generate a tactic in lean4 to solve the state.
STATE:
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))))
TACTIC:
|
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))))))
|
Please generate a tactic in lean4 to solve the state.
STATE:
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)))))
TACTIC:
|
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)))))))
|
Please generate a tactic in lean4 to solve the state.
STATE:
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))))))
TACTIC:
|
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)))))))
|
Please generate a tactic in lean4 to solve the state.
STATE:
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)))))))
TACTIC:
|
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
|
Please generate a tactic in lean4 to solve the state.
STATE:
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)))))))
TACTIC:
|
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))
|
Please generate a tactic in lean4 to solve the state.
STATE:
case h1.h1.a
x y : VarName
⊢ IsDeduct ∅ (eq_ y y)
TACTIC:
|
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))
|
Please generate a tactic in lean4 to solve the state.
STATE:
case h1.h1.a.h1
x y : VarName
⊢ IsDeduct ∅ (forall_ y (eq_ y y))
TACTIC:
|
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
|
Please generate a tactic in lean4 to solve the state.
STATE:
case h1.h1.a.h1.a
x y : VarName
⊢ IsAxiom (forall_ y (eq_ y y))
TACTIC:
|
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
|
Please generate a tactic in lean4 to solve the state.
STATE:
case h1.h2
x y : VarName
⊢ ∀ H ∈ ∅, ¬isFreeIn y H
TACTIC:
|
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
|
Please generate a tactic in lean4 to solve the state.
STATE:
case h1.h2
x y : VarName
H : Formula
a1 : H ∈ ∅
⊢ ¬isFreeIn y H
TACTIC:
|
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
|
Please generate a tactic in lean4 to solve the state.
STATE:
case h2
x y : VarName
⊢ ∀ H ∈ ∅, ¬isFreeIn x H
TACTIC:
|
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
|
Please generate a tactic in lean4 to solve the state.
STATE:
case h2
x y : VarName
H : Formula
a1 : H ∈ ∅
⊢ ¬isFreeIn x H
TACTIC:
|
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
|
Please generate a tactic in lean4 to solve the state.
STATE:
x y z : VarName
⊢ IsProof (forall_ x (forall_ y (forall_ z (((eq_ x y).and_ (eq_ y z)).imp_ (eq_ x z)))))
TACTIC:
|
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
|
Please generate a tactic in lean4 to solve the state.
STATE:
case h1
x y z : VarName
⊢ IsDeduct ∅ (forall_ y (forall_ z (((eq_ x y).and_ (eq_ y z)).imp_ (eq_ x z))))
TACTIC:
|
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
|
Please generate a tactic in lean4 to solve the state.
STATE:
case h1.h1
x y z : VarName
⊢ IsDeduct ∅ (forall_ z (((eq_ x y).and_ (eq_ y z)).imp_ (eq_ x z)))
TACTIC:
|
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)
|
Please generate a tactic in lean4 to solve the state.
STATE:
case h1.h1.h1
x y z : VarName
⊢ IsDeduct ∅ (((eq_ x y).and_ (eq_ y z)).imp_ (eq_ x z))
TACTIC:
|
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)))
|
Please generate a tactic in lean4 to solve the state.
STATE:
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)))
TACTIC:
|
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))))
|
Please generate a tactic in lean4 to solve the state.
STATE:
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))))
TACTIC:
|
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))))
|
Please generate a tactic in lean4 to solve the state.
STATE:
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))))
TACTIC:
|
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
|
Please generate a tactic in lean4 to solve the state.
STATE:
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))))
TACTIC:
|
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))))
|
Please generate a tactic in lean4 to solve the state.
STATE:
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)))
TACTIC:
|
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)))))
|
Please generate a tactic in lean4 to solve the state.
STATE:
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))))
TACTIC:
|
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))))))
|
Please generate a tactic in lean4 to solve the state.
STATE:
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)))))
TACTIC:
|
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)))))))
|
Please generate a tactic in lean4 to solve the state.
STATE:
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))))))
TACTIC:
|
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)))))))
|
Please generate a tactic in lean4 to solve the state.
STATE:
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)))))))
TACTIC:
|
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
|
Please generate a tactic in lean4 to solve the state.
STATE:
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)))))))
TACTIC:
|
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))
|
Please generate a tactic in lean4 to solve the state.
STATE:
case h1.h1.h1.a
x y z : VarName
⊢ IsDeduct ∅ (eq_ z z)
TACTIC:
|
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))
|
Please generate a tactic in lean4 to solve the state.
STATE:
case h1.h1.h1.a.h1
x y z : VarName
⊢ IsDeduct ∅ (forall_ z (eq_ z z))
TACTIC:
|
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
|
Please generate a tactic in lean4 to solve the state.
STATE:
case h1.h1.h1.a.h1.a
x y z : VarName
⊢ IsAxiom (forall_ z (eq_ z z))
TACTIC:
|
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
|
Please generate a tactic in lean4 to solve the state.
STATE:
case h1.h1.h2
x y z : VarName
⊢ ∀ H ∈ ∅, ¬isFreeIn z H
TACTIC:
|
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
|
Please generate a tactic in lean4 to solve the state.
STATE:
case h1.h1.h2
x y z : VarName
H : Formula
a1 : H ∈ ∅
⊢ ¬isFreeIn z H
TACTIC:
|
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
|
Please generate a tactic in lean4 to solve the state.
STATE:
case h1.h2
x y z : VarName
⊢ ∀ H ∈ ∅, ¬isFreeIn y H
TACTIC:
|
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
|
Please generate a tactic in lean4 to solve the state.
STATE:
case h1.h2
x y z : VarName
H : Formula
a1 : H ∈ ∅
⊢ ¬isFreeIn y H
TACTIC:
|
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
|
Please generate a tactic in lean4 to solve the state.
STATE:
case h2
x y z : VarName
⊢ ∀ H ∈ ∅, ¬isFreeIn x H
TACTIC:
|
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
|
Please generate a tactic in lean4 to solve the state.
STATE:
case h2
x y z : VarName
H : Formula
a1 : H ∈ ∅
⊢ ¬isFreeIn x H
TACTIC:
|
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✝)))
|
Please generate a tactic in lean4 to solve the state.
STATE:
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))
TACTIC:
|
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
|
Please generate a tactic in lean4 to solve the state.
STATE:
P_r P_s : Formula
r s : VarName
h2 : ¬isBoundIn r true_
h3 : ¬isBoundIn s true_
⊢ IsProof ((eq_ r s).imp_ (true_.iff_ true_))
TACTIC:
|
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
|
Please generate a tactic in lean4 to solve the state.
STATE:
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✝)))
TACTIC:
|
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_)))
|
Please generate a tactic in lean4 to solve the state.
STATE:
P_r P_s : Formula
r s : VarName
h2 : ¬isBoundIn r true_
h3 : ¬isBoundIn s true_
⊢ IsProof ((eq_ r s).imp_ (true_.iff_ true_))
TACTIC:
|
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_)
|
Please generate a tactic in lean4 to solve the state.
STATE:
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_)))
TACTIC:
|
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
|
Please generate a tactic in lean4 to solve the state.
STATE:
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_)
TACTIC:
|
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))))
|
Please generate a tactic in lean4 to solve the state.
STATE:
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))))
TACTIC:
|
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
|
Please generate a tactic in lean4 to solve the state.
STATE:
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)))))
TACTIC:
|
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))))
|
Please generate a tactic in lean4 to solve the state.
STATE:
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))))
TACTIC:
|
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))))
|
Please generate a tactic in lean4 to solve the state.
STATE:
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)))))
TACTIC:
|
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)))))))
|
Please generate a tactic in lean4 to solve the state.
STATE:
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))))))
TACTIC:
|
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_)))
|
Please generate a tactic in lean4 to solve the state.
STATE:
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)))))))
TACTIC:
|
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
|
Please generate a tactic in lean4 to solve the state.
STATE:
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_)))
TACTIC:
|
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)))))
|
Please generate a tactic in lean4 to solve the state.
STATE:
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))))
TACTIC:
|
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))))))
|
Please generate a tactic in lean4 to solve the state.
STATE:
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)))))
TACTIC:
|
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))))))
|
Please generate a tactic in lean4 to solve the state.
STATE:
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))))))
TACTIC:
|
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
|
Please generate a tactic in lean4 to solve the state.
STATE:
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))))))
TACTIC:
|
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))))
|
Please generate a tactic in lean4 to solve the state.
STATE:
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))))
TACTIC:
|
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))))
|
Please generate a tactic in lean4 to solve the state.
STATE:
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))))
TACTIC:
|
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))))
|
Please generate a tactic in lean4 to solve the state.
STATE:
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))))
TACTIC:
|
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))))
|
Please generate a tactic in lean4 to solve the state.
STATE:
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))))
TACTIC:
|
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
|
Please generate a tactic in lean4 to solve the state.
STATE:
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))))
TACTIC:
|
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_)
|
Please generate a tactic in lean4 to solve the state.
STATE:
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))))
TACTIC:
|
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
|
Please generate a tactic in lean4 to solve the state.
STATE:
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_)
TACTIC:
|
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)))))
|
Please generate a tactic in lean4 to solve the state.
STATE:
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))))
TACTIC:
|
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))))
|
Please generate a tactic in lean4 to solve the state.
STATE:
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)))))
TACTIC:
|
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)))
|
Please generate a tactic in lean4 to solve the state.
STATE:
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))))))
TACTIC:
|
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_)))
|
Please generate a tactic in lean4 to solve the state.
STATE:
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)))))))
TACTIC:
|
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
|
Please generate a tactic in lean4 to solve the state.
STATE:
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_)))
TACTIC:
|
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)))
|
Please generate a tactic in lean4 to solve the state.
STATE:
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)))
TACTIC:
|
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)))
|
Please generate a tactic in lean4 to solve the state.
STATE:
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)))
TACTIC:
|
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
|
Please generate a tactic in lean4 to solve the state.
STATE:
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)))
TACTIC:
|
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
|
Please generate a tactic in lean4 to solve the state.
STATE:
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)))
TACTIC:
|
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))
|
Please generate a tactic in lean4 to solve the state.
STATE:
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)))
TACTIC:
|
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
|
Please generate a tactic in lean4 to solve the state.
STATE:
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))))
TACTIC:
|
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
|
Please generate a tactic in lean4 to solve the state.
STATE:
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))
TACTIC:
|
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
|
Please generate a tactic in lean4 to solve the state.
STATE:
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))))
TACTIC:
|
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))
|
Please generate a tactic in lean4 to solve the state.
STATE:
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))
TACTIC:
|
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)))
|
Please generate a tactic in lean4 to solve the state.
STATE:
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))
TACTIC:
|
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)))
|
Please generate a tactic in lean4 to solve the state.
STATE:
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)))
TACTIC:
|
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
|
Please generate a tactic in lean4 to solve the state.
STATE:
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)))
TACTIC:
|
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)))
|
Please generate a tactic in lean4 to solve the state.
STATE:
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)))
TACTIC:
|
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
|
Please generate a tactic in lean4 to solve the state.
STATE:
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)))
TACTIC:
|
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)))
|
Please generate a tactic in lean4 to solve the state.
STATE:
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)))
TACTIC:
|
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)))
|
Please generate a tactic in lean4 to solve the state.
STATE:
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)))
TACTIC:
|
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
|
Please generate a tactic in lean4 to solve the state.
STATE:
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)))
TACTIC:
|
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
|
Please generate a tactic in lean4 to solve the state.
STATE:
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))))
TACTIC:
|
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
|
Please generate a tactic in lean4 to solve the state.
STATE:
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
TACTIC:
|
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
|
Please generate a tactic in lean4 to solve the state.
STATE:
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
TACTIC:
|
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_))
|
Please generate a tactic in lean4 to solve the state.
STATE:
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_))
TACTIC:
|
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_))
|
Please generate a tactic in lean4 to solve the state.
STATE:
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_))
TACTIC:
|
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_))
|
Please generate a tactic in lean4 to solve the state.
STATE:
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_))
TACTIC:
|
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))
|
Please generate a tactic in lean4 to solve the state.
STATE:
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_))
TACTIC:
|
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_))))
|
Please generate a tactic in lean4 to solve the state.
STATE:
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_)))
TACTIC:
|
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_))
|
Please generate a tactic in lean4 to solve the state.
STATE:
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_))))
TACTIC:
|
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
|
Please generate a tactic in lean4 to solve the state.
STATE:
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_))
TACTIC:
|
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
|
Please generate a tactic in lean4 to solve the state.
STATE:
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))
TACTIC:
|
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)))
|
Please generate a tactic in lean4 to solve the state.
STATE:
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)))
TACTIC:
|
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