- p is bounds(D) complete for c iff p(d) ⊆ **conv(**c ∩ d).

- p is bounds(Z) complete for c iff p(d) ⊆ conv(c ∩ conv(d)). - p is bounds(R) complete for c iff p(d) ⊆ conv(cR ∩convR(d**)), where conv**R(d)
is the convex hull of d in R, and cR is c **relaxed to** R.
Bounds completeness of derived propagators. Theorem 4 states that propagators derived from domain complete propagators are domain complete. A similar theorem holds for bounds completeness, if views commute with conv(·**) in**
the following ways:
A view ϕ is interval injective iff ϕ
−(conv(c**)) = conv(**ϕ
−(c)) for all ϕ constraints c. It is interval bijective iff it is interval injective and ϕ(conv(d**)) =**
conv(ϕ(d**)) for all domains** d.

Proving bounds completeness of derived propagators is now similar to proving domain completeness. We only formulate bounds(Z**) completeness.** Theorem 5. If p is bounds(Z) complete for c and ϕ **is interval bijective, then**
ϕb(p) is bounds(Z**) complete for** ϕ(c). 

Proof. By monotonicity of ϕ and bounds(Z) completeness of p**, we know that**
ϕ
− ◦ p ◦ ϕ(d) ⊆ ϕ
−(conv(c ∩ conv(ϕ(d)))). We now use the fact that both ϕ and ϕ
− **commute with conv and intersection:**
ϕ(conv(c ∩ **conv(**ϕ
−1(d)))) = ϕ**(conv(**c ∩ ϕ
−1(conv(d**)))) =**
conv(ϕ(c ∩ ϕ
−1(conv(d**)))) = conv(**ϕ(c) ∩ ϕ(ϕ
−1(conv(d**)))) =**
conv(ϕ(c) ∩ **conv(**d))
The proof for bounds(D**) is analogous, but we only require interval injectivity**
for the view. With an interval injective view, one can also derive bounds(R) complete propagators from bounds(R) or bounds(Z**) complete propagators. Table 1**
summarizes how completeness depends on view bijectivity.

| propagator                                      | view      |           |        |
|-------------------------------------------------|-----------|-----------|--------|
| interval bijective interval injective arbitrary |           |           |        |
| domain                                          | domain    | domain    | domain |
| bounds(D)                                       | bounds(D) | bounds(D) | weakly |
| bounds(Z)                                       | bounds(Z) | bounds(R) | weakly |
| bounds(R)                                       | bounds(R) | bounds(R) | weakly |

The views for integer variables presented at the beginning of this section have the following properties: minus and offset views are interval bijective, whereas a scale view for a ∈ Z on x **is always interval injective and only interval bijective**
if a = 1 or a = −**1 (in which cases it coincides with the identity view or a minus**
view, respectively). An important consequence is that a bounds(Z**) complete**
propagator for the constraint Pi xi = c**, when instantiated with scale views for**
the xi, results in a bounds(R**) complete propagator for** Pi aixi = c.