Library Coq.QArith.Qring

Require Export Ring.
Require Export QArith_base.

A ring tactic for rational numbers


Definition Qeq_bool (x y : Q) :=
  if Qeq_dec x y then true else false.

Lemma Qeq_bool_correct : forall x y : Q, Qeq_bool x y = true -> x==y.
Proof.
  intros x y; unfold Qeq_bool in |- *; case (Qeq_dec x y); simpl in |- *; auto.
  intros _ H; inversion H.
Qed.

Definition Qsrt : ring_theory 0 1 Qplus Qmult Qminus Qopp Qeq.
Proof.
  constructor.
  exact Qplus_0_l.
  exact Qplus_comm.
  exact Qplus_assoc.
  exact Qmult_1_l.
  exact Qmult_comm.
  exact Qmult_assoc.
  exact Qmult_plus_distr_l.
  reflexivity.
  exact Qplus_opp_r.
Qed.

Ltac isQcst t :=
  match t with
  | inject_Z ?z => isZcst z
  | Qmake ?n ?d =>
    match isZcst n with
      true => isPcst d
    | _ => false
    end
  | _ => false
  end.

Ltac Qcst t :=
  match isQcst t with
    true => t
    | _ => NotConstant
  end.

Add Ring Qring : Qsrt (decidable Qeq_bool_correct, constants [Qcst]).
Exemple of use:

Section Examples.

Let ex1 : forall x y z : Q, (x+y)*z == (x*z)+(y*z).
  intros.
  ring.
Qed.

Let ex2 : forall x y : Q, x+y == y+x.
  intros.
  ring.
Qed.

Let ex3 : forall x y z : Q, (x+y)+z == x+(y+z).
  intros.
  ring.
Qed.

Let ex4 : (inject_Z 1)+(inject_Z 1)==(inject_Z 2).
 ring.
Qed.

Let ex5 : 1+1 == 2#1.
  ring.
Qed.

Let ex6 : (1#1)+(1#1) == 2#1.
  ring.
Qed.

Let ex7 : forall x : Q, x-x== 0#1.
  intro.
  ring.
Qed.

End Examples.

Lemma Qopp_plus : forall a b, -(a+b) == -a + -b.
Proof.
  intros; ring.
Qed.

Lemma Qopp_opp : forall q, - -q==q.
Proof.
  intros; ring.
Qed.