Require Export Languages.
Require Export LNN.
Section NN.
Definition NN1 := forallH 0 (notH (equal (Succ (var 0)) Zero)).
Definition NN2 :=
forallH 1
(forallH 0
(impH (equal (Succ (var 0)) (Succ (var 1))) (equal (var 0) (var 1)))).
Definition NN3 := forallH 0 (equal (Plus (var 0) Zero) (var 0)).
Definition NN4 :=
forallH 1
(forallH 0
(equal (Plus (var 0) (Succ (var 1))) (Succ (Plus (var 0) (var 1))))).
Definition NN5 := forallH 0 (equal (Times (var 0) Zero) Zero).
Definition NN6 :=
forallH 1
(forallH 0
(equal (Times (var 0) (Succ (var 1)))
(Plus (Times (var 0) (var 1)) (var 0)))).
Definition NN7 := forallH 0 (notH (LT (var 0) Zero)).
Definition NN8 :=
forallH 1
(forallH 0
(impH (LT (var 0) (Succ (var 1)))
(orH (LT (var 0) (var 1)) (equal (var 0) (var 1))))).
Definition NN9 :=
forallH 1
(forallH 0
(orH (LT (var 0) (var 1))
(orH (equal (var 0) (var 1)) (LT (var 1) (var 0))))).
Definition NNStar :=
forallH 1
(forallH 0
(impH (orH (LT (var 0) (var 1)) (equal (var 0) (var 1)))
(LT (var 0) (Succ (var 1))))).
Definition NN :=
Ensembles.Add _
(Ensembles.Add _
(Ensembles.Add _
(Ensembles.Add _
(Ensembles.Add _ (Ensembles.Add _ (Ensembles.Add _ (Ensembles.Add _ (Singleton _ NN1) NN2) NN3) NN4) NN5)
NN6) NN7) NN8) NN9.

Lemma nn4 :
forall a b : Term, SysPrf NN (equal (Plus a (Succ b)) (Succ (Plus a b))).