The Coq Proof Assistant

# Library Coq.QArith.Qcabs

# An absolute value for normalized rational numbers.

Contributed by Cédric Auger
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.