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 nn1 : forall a : Term, SysPrf NN (notH (equal (Succ a) Zero)).