MathClasses.implementations.polynomials

Require Import
  List
  abstract_algebra.

Section contents.
  Context R `{Ring R}.

  Definition poly := list R.

  Coercion poly_constant (c : R) : poly := [c].

  Global Instance poly_zero: Zero poly := [].
  Global Instance poly_one: One poly := poly_constant 1.

  Definition all (l: list Prop): Prop := fold_left and l True.

  Definition poly_eq_zero: poly → Prop := all ∘ map ((=) 0).

  Global Instance poly_eq: Equiv poly :=
    fix F p q :=
    match p, q with
    | [], _ ⇒ poly_eq_zero q
    | _, [] ⇒ poly_eq_zero p
    | h :: t, h' :: t' ⇒ h = h ∧ F t t'
    end.


  Instance: Reflexive poly_eq.
  Proof with intuition. repeat intro. induction x... split... Qed.

  Instance: Symmetric poly_eq.
  Admitted.

  Instance: Transitive poly_eq.
  Admitted.

  Global Instance: Setoid poly.
  Proof. split; try apply _. Qed.

  Global Instance: Plus poly := fix F p q :=
    match p, q with
    | [], _ ⇒ q
    | _, [] ⇒ p
    | h :: t, h' :: t' ⇒ h + h' :: F t t'
    end.

  Global Instance: Negate poly := map (-).

  Fixpoint poly_mult_cr (q: poly) (c: R): poly :=
    match q with
    | [] ⇒ 0
    | d :: q1 ⇒ c×d :: poly_mult_cr q1 c
    end.

  Global Instance: Mult poly := fix F p q :=
    match p with
    | [] ⇒ 0
    | c :: p1 ⇒ poly_mult_cr q c + (0 :: F p1 q)
    end.

  Global Instance: Ring poly.
   constructor.
  Admitted.

End contents.