Section NumIntegralDomainTheory.
Variable R : numDomainType.
Implicit Types (V : normedZmodType R) (x y z t : R).
(* Lemmas from the signature (reexported from internals). *)
Definition ler_norm_add V (x y : V) : `|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 V (x : V) : `|x| = 0 -> x = 0 := @normr0_eq0 R V x.
Definition ger_leVge x y : 0 <= x -> 0 <= y -> (x <= y) || (y <= x) :=
@ger_leVge R x y.
Definition normrM : {morph norm : x y / (x : R) * y} := @normrM R.
Definition ler_def x y : (x <= y) = (`|y - x| == y - x) := ler_def x y.
Definition normrMn V (x : V) n : `|x *+ n| = `|x| *+ n := normrMn x n.
Definition normrN V (x : V) : `|- x| = `|x| := normrN x.
(* Predicate definitions. *)
(* General properties of <= and < *)
Hint Resolve ler01 ltr01 ler0n : core.

Lemma big_real x0 op I (P : pred I) F (s : seq I) :
{in real &, forall x y, op x y \is real} -> x0 \is real ->
{in P, forall i, F i \is real} -> \big[op/x0]_(i <- s | P i) F i \is real.