Remark lem_redln_cons_gen :
 forall (aP R Q : list (poly A0 eqA ltM)) (a b : poly A0 eqA ltM),
 In a aP ->
 In b aP ->
 redIn a b aP Q R ->
 forall (c : poly A0 eqA ltM) (Q1 : list (poly A0 eqA ltM)),
 incl
   (addEnd A A0 eqA n ltM
      (nf A A0 A1 eqA plusA invA minusA multA divA cs eqA_dec n ltM ltM_dec
         os c aP) aP) R ->
 Q = c :: Q1 ->
 redIn a b
   (addEnd A A0 eqA n ltM
      (nf A A0 A1 eqA plusA invA minusA multA divA cs eqA_dec n ltM ltM_dec
         os c aP) aP) Q1 R.

Theorem red_gen_in : forall (a : poly A0 eqA ltM) (aP R Q : list (poly A0 eqA ltM)), ~ BuchAux.zerop A A0 eqA n ltM (nf A A0 A1 eqA plusA invA minusA multA divA cs eqA_dec n ltM ltM_dec os a aP) -> OBuch (addEnd A A0 eqA n ltM (nf A A0 A1 eqA plusA invA minusA multA divA cs eqA_dec n ltM ltM_dec os a aP) aP) (genPcPf (nf A A0 A1 eqA plusA invA minusA multA divA cs eqA_dec n ltM ltM_dec os a aP) aP Q) R -> (forall b c : poly A0 eqA ltM, In b aP -> In c aP -> redIn b c aP (a :: Q) R) -> forall b : poly A0 eqA ltM, In b aP -> ~ BuchAux.zerop A A0 eqA n ltM b -> redIn (nf A A0 A1 eqA plusA invA minusA multA divA cs eqA_dec n ltM ltM_dec os a aP) b (addEnd A A0 eqA n ltM (nf A A0 A1 eqA plusA invA minusA multA divA cs eqA_dec n ltM ltM_dec os a aP) aP) (genPcPf (nf A A0 A1 eqA plusA invA minusA multA divA cs eqA_dec n ltM ltM_dec os a aP) aP Q) R.