proposition: Theorem fspoly_Reduceplus_pO : forall (Q : list (poly A0 eqA ltM)) (a b : Term A n), eqT (ppc (A:=A) A1 (n:=n) a b) (multTerm (A:=A) multA (n:=n) a b) -> forall p q : list (Term A n), inPolySet A A0 eqA n ltM (pX b q) Q -> inPolySet A A0 eqA n ltM (pX a p) Q -> reduceplus A A0 A1 eqA invA minusA multA divA eqA_dec n ltM ltM_dec Q (minuspf A A0 A1 eqA invA minusA multA eqA_dec n ltM ltM_dec (mults (A:=A) multA (n:=n) b p) (mults (A:=A) multA (n:=n) a q)) (pO A n).
proof: Proof. Proof using plusA os cs. intros Q a b H' p q H'0 H'1. cut (canonical A0 eqA ltM (pX a p)); [ intros Op0 | apply inPolySet_imp_canonical with (L := Q) ]; auto. cut (canonical A0 eqA ltM (pX b q)); [ intros Op1 | apply inPolySet_imp_canonical with (L := Q) ]; auto. cut (canonical A0 eqA ltM p); [ intros Op2 | apply canonical_imp_canonical with (a := a) ]; auto. cut (canonical A0 eqA ltM q); [ intros Op3 | apply canonical_imp_canonical with (a := b) ]; auto. cut (~ zeroP (A:=A) A0 eqA (n:=n) a); [ intros Z0 | apply canonical_nzeroP with (ltM := ltM) (p := p) ]; auto. cut (~ zeroP (A:=A) A0 eqA (n:=n) b); [ intros Z1 | apply canonical_nzeroP with (ltM := ltM) (p := q) ]; auto. apply reduceplus_trans with (1 := cs) (y := mults (A:=A) multA (n:=n) (invTerm (A:=A) invA (n:=n) (T1 A1 n)) (multpf A A0 eqA plusA multA eqA_dec n ltM ltM_dec (pX a p) q)); auto. apply spoly_Reduce; auto. apply reduceplus_eqp_com with (1 := cs) (p := mults (A:=A) multA (n:=n) (invTerm (A:=A) invA (n:=n) (T1 A1 n)) (multpf A A0 eqA plusA multA eqA_dec n ltM ltM_dec q (pX a p))) (q := mults (A:=A) multA (n:=n) (invTerm (A:=A) invA (n:=n) (T1 A1 n)) (pO A n)); auto. apply reduceplus_mults with (1 := cs); auto. apply spoly_reduceplus_pO; auto. apply mults_comp with (1 := cs); auto. apply multpf_com with (1 := cs); auto. Qed.