proposition: Theorem ltP_reduce : forall (Q : list (poly A0 eqA ltM)) (p q : list (Term A n)), reduce Q p q -> canonical A0 eqA ltM p -> ltP (A:=A) (n:=n) ltM q p.
proof: Proof. Proof using plusA os cs. intros Q p q H'; elim H'; auto. 2: intros a b p0 q0 H'0 H'1 H'2 H'3; apply ltP_tl; auto. 2: apply (eqT_sym A n); apply (eqTerm_imp_eqT A eqA n); auto. 2: apply H'1; try apply canonical_imp_canonical with (a := a); auto. intros a b nZb p0; case p0; auto. intros q0 r H'0 H'1 H'2 H'3. change (ltP (A:=A) (n:=n) ltM r (pX a (pO A n))) in |- *. apply (canonical_pX_ltP A A0 eqA); auto. apply eqp_imp_canonical with (1 := cs) (p := pX a (spminusf A A0 A1 eqA invA minusA multA divA eqA_dec n ltM ltM_dec a b nZb (pO A n) q0)); auto. apply canonical_spminusf_full_t with (1 := cs); auto. apply inPolySet_imp_canonical with (L := Q); auto. intros t l q0 r H'0 H'1 H'2 H'3; apply ltP_trans with (y := pX a (pO A n)); auto. apply (canonical_pX_ltP A A0 eqA); auto. apply eqp_imp_canonical with (1 := cs) (p := pX a (spminusf A A0 A1 eqA invA minusA multA divA eqA_dec n ltM ltM_dec a b nZb (pX t l) q0)); auto. apply canonical_spminusf_full_t with (1 := cs); auto. apply inPolySet_imp_canonical with (L := Q); auto. change (ltP (A:=A) (n:=n) ltM (pX a (pO A n)) (pX a (pX t l))) in |- *; auto. Qed.