(* This code is copyrighted by its authors; it is distributed under  *)
(* the terms of the LGPL license (see LICENSE and description files) *)
(* Contribution to the Coq Library   V6.3 (July 1999)                    *)
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(*                   Solange Coupet-Grimal & Line Jakubiec                  *)
(*                                                                          *)
(*                                                                          *)
(*              Laboratoire d'Informatique de Marseille                     *)
(*               CMI-Technopole de Chateau-Gombert                          *)
(*                   39, Rue F. Joliot Curie                                *)
(*                   13453 MARSEILLE Cedex 13                               *)
(*           e-mail:{Solange.Coupet,Line.Jakubiec}@lim.univ-mrs.fr          *)
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(*                                Coq V5.10                                 *)
(*                              May 30th 1996                               *)
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(*                            Factorization.v                               *)
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Require Export Numerals.
Section factorization.
Variable A : Set.
Variable BASE : BT.
Let b := base BASE.
Let Num := num BASE.
Let Digit := digit BASE.
Let Tl := tl Digit.
Let Cons := cons Digit.
Let Nil := nil Digit.
Let Val_bound := val_bound BASE.
Section Definitions_for_Relations.
Definition Diveucl (a b q r : nat) : Prop := a = b * q + r /\ r < b.
Definition Zero : inf 1 := Val_bound 0 Nil.
Variable R : forall n : nat, A -> inf n -> inf n -> A -> Prop.
Definition factorizable : Prop :=
forall (m n : nat) (q q' : inf m) (r r' : inf n)
(a a1 a' : A) (x x' : inf (m * n)),
Diveucl (val_inf (m * n) x) n (val_inf m q) (val_inf n r) ->
Diveucl (val_inf (m * n) x') n (val_inf m q') (val_inf n r') ->
R m a q q' a1 -> R n a1 r r' a' -> R (m * n) a x x' a'.
Definition proper : Prop := forall a : A, R 1 a Zero Zero a.
End Definitions_for_Relations.
Section Three_inputs.
Variable FR : forall n : nat, A -> inf n -> inf n -> A.
Let R (n : nat) (a : A) (x y : inf n) (a' : A) : Prop := a' = FR n a x y.

Lemma prop_Rel :
proper R ->
forall (X Y : Num 0) (a : A), R 1 a (Val_bound 0 X) (Val_bound 0 Y) a.