Definition PA1 := forallH 0 (notH (equal (Succ (var 0)) Zero)).Definition PA2 :=
  forallH 1
    (forallH 0
       (impH (equal (Succ (var 0)) (Succ (var 1))) (equal (var 0) (var 1)))).Definition PA3 := forallH 0 (equal (Plus (var 0) Zero) (var 0)).Definition PA4 :=
  forallH 1
    (forallH 0
       (equal (Plus (var 0) (Succ (var 1))) (Succ (Plus (var 0) (var 1))))).Definition PA5 := forallH 0 (equal (Times (var 0) Zero) Zero).Definition PA6 :=
  forallH 1
    (forallH 0
       (equal (Times (var 0) (Succ (var 1)))
          (Plus (Times (var 0) (var 1)) (var 0)))).Definition PA7 (f : Formula) (v : nat) : Formula :=
  close LNT
    (impH (substituteFormula LNT f v Zero)
       (impH (forallH v (impH f (substituteFormula LNT f v (Succ (var v)))))
          (forallH v f))).Definition InductionSchema (f : Formula) : Prop :=
  exists g : Formula, (exists v : nat, f = PA7 g v).Definition PA :=
  Ensembles.Add _
    (Ensembles.Add _ (Ensembles.Add _ (Ensembles.Add _ (Ensembles.Add _ (Ensembles.Add _ InductionSchema PA1) PA2) PA3) PA4)
       PA5) PA6.Definition open :=
  Formula_rec LNT (fun _ => Formula) (fun t t0 : Term => equal t t0)
    (fun (r : Relations LNT) (ts : Terms (arity LNT (inl (Functions LNT) r))) =>
     atomic LNT r ts) (fun (f : Formula) _ (f0 : Formula) _ => impH f f0)
    (fun (f : Formula) _ => notH f)
    (fun (n : nat) _ (recf : Formula) => recf).

Lemma closedPA : forall v : nat, ~ In_freeVarSys LNT v PA.