Require Import Qabs Qcanon.

Lemma Qred_abs (x : Q) : Qred (Qabs x) = Qabs (Qred x).

Lemma Qcabs_canon (x : Q) : Qred x = x -> Qred (Qabs x) = Qabs x.

Definition Qcabs (x:Qc) : Qc := {| canon := Qcabs_canon x (canon x) |}.
Notation "[ q ]" := (Qcabs q) : Qc_scope.

Ltac Qc_unfolds :=
  unfold Qcabs, Qcminus, Qcopp, Qcplus, Qcmult, Qcle, Q2Qc, this.

Lemma Qcabs_case (x:Qc) (P : Qc -> Type) :
  (0 <= x -> P x) -> (x <= 0 -> P (- x)) -> P [x].

Lemma Qcabs_pos x : 0 <= x -> [x] = x.

Lemma Qcabs_neg x : x <= 0 -> [x] = - x.

Lemma Qcabs_nonneg x : 0 <= [x].

Lemma Qcabs_opp x : [(-x)] = [x].

Lemma Qcabs_triangle x y : [x+y] <= [x] + [y].

Lemma Qcabs_Qcmult x y : [x*y] = [x]*[y].

Lemma Qcabs_Qcminus x y: [x-y] = [y-x].

Lemma Qcle_Qcabs x : x <= [x].

Lemma Qcabs_triangle_reverse x y : [x] - [y] <= [x - y].

Lemma Qcabs_Qcle_condition x y : [x] <= y <-> -y <= x <= y.

Lemma Qcabs_diff_Qcle_condition x y r : [x-y] <= r <-> x - r <= y <= x + r.

Lemma Qcabs_null x : [x] = 0 -> x = 0.