End Internals.
Module PredInstances.
Canonical pos_mulrPred.
Canonical pos_divrPred.
Canonical nneg_addrPred.
Canonical nneg_mulrPred.
Canonical nneg_divrPred.
Canonical nneg_semiringPred.
Canonical real_addrPred.
Canonical real_opprPred.
Canonical real_zmodPred.
Canonical real_mulrPred.
Canonical real_smulrPred.
Canonical real_divrPred.
Canonical real_sdivrPred.
Canonical real_semiringPred.
Canonical real_subringPred.
Canonical real_divringPred.
End PredInstances.
Module Import ExtraDef.
Definition archi_bound {R} x := sval (sigW (@archi_bound_subproof R x)).
Definition sqrtr {R} x := s2val (sig2W (@sqrtr_subproof R x)).
End ExtraDef.
Notation bound := archi_bound.
Notation sqrt := sqrtr.
Module Theory.
Section NumIntegralDomainTheory.
Variable R : numDomainType.
Implicit Types x y z t : R.
(* Lemmas from the signature (reexported from internals). *)
Definition ler_norm_add x y : `|x + y| <= `|x| + `|y| := ler_norm_add x y.
Definition addr_gt0 x y : 0 < x -> 0 < y -> 0 < x + y := @addr_gt0 R x y.
Definition normr0_eq0 x : `|x| = 0 -> x = 0 := @normr0_eq0 R x.
Definition ger_leVge x y : 0 <= x -> 0 <= y -> (x <= y) || (y <= x) :=
@ger_leVge R x y.
Definition normrM : {morph normr : x y / x * y : R} := @normrM R.
Definition ler_def x y : (x <= y) = (`|y - x| == y - x) := @ler_def R x y.
Definition ltr_def x y : (x < y) = (y != x) && (x <= y) := @ltr_def R x y.
(* Predicate and relation definitions. *)

Lemma ler0_norm x : x <= 0 -> `|x| = - x.