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(* GNU General Public License for more details.                       *)
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(* 02110-1301 USA                                                     *)
Require Export Arith.
Require Export Compare_dec.
Require Export ArithRing.
Require Export Omega.
Require Export ZArith.
Require Export ZArithRing.
Ltac CaseEq f := generalize (refl_equal f); pattern f at -1 in |- *; case f.
Inductive Qpositive : Set :=
| nR : Qpositive -> Qpositive
| dL : Qpositive -> Qpositive
| One : Qpositive.
Fixpoint Qpositive_i (w : Qpositive) : nat * nat :=
match w with
| One => (1, 1)
| nR w' => match Qpositive_i w' with
| (p, q) => (p + q, q)
end
| dL w' => match Qpositive_i w' with
| (p, q) => (p, p + q)
end
end.
Coercion Z_of_nat : nat >-> Z.
intros w; elim w; clear w.
intros w Hrec; elim Hrec; intros a Hrec2; elim Hrec2; intros b; simpl in |- *;
case (Qpositive_i w); simpl in |- *.
intros p q Heq; exists a; exists (b - a)%Z.
rewrite <- Heq; repeat rewrite Znat.inj_plus; ring.
intros w Hrec; elim Hrec; intros a Hrec2; elim Hrec2; intros b; simpl in |- *;
case (Qpositive_i w); simpl in |- *.
intros p q Heq; exists (a - b)%Z; exists b.
rewrite <- Heq; repeat rewrite Znat.inj_plus; ring.
exists 1%Z; exists 0%Z; simpl in |- *; omega.
Qed.
Fixpoint Qpositive_inv (w : Qpositive) : Qpositive :=
match w with
| One => One
| nR w' => dL (Qpositive_inv w')
| dL w' => nR (Qpositive_inv w')
end.

Theorem inv_correct :
forall (w : Qpositive) (p q : nat),
Qpositive_i w = (p, q) -> Qpositive_i (Qpositive_inv w) = (q, p).