Require Export Languages.
Require Export LNT.
Section PA.
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 PAdec : forall x : Formula, In _ PA x \/ ~ In _ PA x.
intros.