proposition: Theorem reduce_divp : forall (p q : poly A0 eqA ltM) (Q : list (poly A0 eqA ltM)), reduce A A0 A1 eqA invA minusA multA divA eqA_dec n ltM ltM_dec Q (s2p A A0 eqA n ltM p) (s2p A A0 eqA n ltM q) -> exists r : poly A0 eqA ltM, In r (q :: Q) /\ divp A A0 eqA multA divA n ltM p r /\ ~ zerop A A0 eqA n ltM r.
proof: Proof. Proof using plusA cs. intros p q; case p; case q; simpl in |- *. intros x H' x0 c Q H'0; inversion H'0. generalize c; rewrite <- H2; simpl in |- *; auto. case inPolySet_inv1 with (1 := H); auto. intros q1 E; case E; intros H'1 H'2; clear E. intros c0; exists q1; split; [ right | split ]; auto. generalize H'2; case q1; simpl in |- *; auto. intros x1; case x1; auto. intros H'4 H'5; discriminate H'5; auto. intros a0 l H'4 H'5; injection H'5; intros. rewrite <- H5; trivial. generalize H'1 H'2 c0; case q1; simpl in |- *; auto. intros x1; case x1; auto. intros c1 H'3 H'4; discriminate H'4. exists (exist (fun l : list (Term A n) => canonical A0 eqA ltM l) x H'); split; [ left | split ]; simpl in |- *; auto. generalize c; rewrite <- H1; unfold pX in |- *; auto. generalize H'; rewrite <- H2; unfold pX in |- *; auto. intros H'1 H'2. apply divP_eqTerm_comp with (a := b) (1 := cs); auto. cut (~ zeroP (A:=A) A0 eqA (n:=n) b); [ intros Z | idtac ]; auto. apply divTerm_def with (nZb := Z); auto. apply canonical_nzeroP with (ltM := ltM) (p := q0); auto. apply (eqTerm_sym _ _ _ _ _ _ _ _ _ cs n); auto. generalize c H'; rewrite <- H2; auto. Qed.
imports: Buchberger(Buch)