(* This code is copyrighted by its authors; it is distributed under  *)
(* the terms of the LGPL license (see LICENSE and description files) *)
(****************************************************************************  Buchberger : multiplication by a scalar  Laurent Thery March 98 (revised Avril 01)  *************************************************************************** definition of Pmults*)
Require Export Pplus.
Section Pmults.
Load "hCoefStructure".
Load "hOrderStructure".
Load "hPlus".
Set Implicit Arguments.
Unset Strict Implicit.
Definition mults : Term A n -> list (Term A n) -> list (Term A n).
intros a p; elim p; clear p.
exact (pO A n).
intros b p1 p'1.
exact (pX (multTerm (A:=A) multA (n:=n) a b) p'1).
Defined.
Set Strict Implicit.
Unset Implicit Arguments.
Hint Resolve multTerm_eqT.
Hint Resolve invTerm_eqT.
Hint Resolve T1_is_min_ltT.

Lemma mults_order_l :
forall l m1 m2,
~ zeroP (A:=A) A0 eqA (n:=n) m1 ->
canonical A0 eqA ltM (pX m2 l) ->
canonical A0 eqA ltM (pX (multTerm (A:=A) multA (n:=n) m1 m2) (mults m1 l)).