proposition: Theorem spolyf_def : forall (a b : Term A n) (nZa : ~ zeroP (A:=A) A0 eqA (n:=n) a) (nZb : ~ zeroP (A:=A) A0 eqA (n:=n) b) (p q : list (Term A n)) (Cpxa : canonical A0 eqA ltM (pX a p)) (Cpxb : canonical A0 eqA ltM (pX b q)), eqP A eqA n (spolyf (pX a p) (pX b q) Cpxa Cpxb) (minuspf A A0 A1 eqA invA minusA multA eqA_dec n ltM ltM_dec (mults (A:=A) multA (n:=n) (divTerm (A:=A) (A0:=A0) (eqA:=eqA) divA (n:=n) (ppc (A:=A) A1 (n:=n) a b) (b:=a) nZa) p) (mults (A:=A) multA (n:=n) (divTerm (A:=A) (A0:=A0) (eqA:=eqA) divA (n:=n) (ppc (A:=A) A1 (n:=n) a b) (b:=b) nZb) q)).
proof: Proof. Proof using plusA os cs. intros a b nZa nZb p q Cpxa Cpxb; simpl in |- *; auto. cut (canonical A0 eqA ltM p); [ intros Op1 | auto ]; auto. cut (canonical A0 eqA ltM q); [ intros Op2 | auto ]; unfold LetP in |- *; simpl in |- *; auto. apply canonical_imp_canonical with (a := b); auto. apply canonical_imp_canonical with (a := a); auto. Qed.