ϕAsn. This is the case here, as the intersection with ϕ(hull(d)) can only contain such assignments. So in summary, we get ϕ
− ◦ p ◦ϕ(d) ⊆ hull(dom(ϕ
−(cp)∩hull(d))
which is the definition of ϕb(p) being bounds(Z)-complete.

Stronger notions of completeness. Similar theorems hold for domain completeness, range and bounds(Z) completeness. The theorems directly follow from the fact that any view ϕ is *domain-injective*, meaning that ϕ
−(dom(c)) = dom(ϕ
−(c)) for all constraints c. We split this statement into the following two lemmas.

Lemma 2 Given a constraint c, let d = dom(c). Then for all x ∈ X, we have v ∈
d(x) ⇔ ∃a ∈ c : a(x) = v. ∗
Proof. We prove both directions of the equivalence:

⇒ There must be such an assignment a because otherwise one can construct a strictly stronger d
′ ⊂ d with v ∈/ d
′(x) such that still c ⊆ d
′.

⇐ Each domain d
′in the intersection T{d
′ ∈ Dom | c ⊆ con(d
′)} must contain the value v ∈ d
′(x) as c ⊆ d
′. So for the result of the intersection d, v ∈ d(x).
Lemma 3 Any view ϕ is domain-injective. ∗
Proof. We have to show that ϕ
−(dom(c)) = dom(ϕ
−(c)) holds for any constraint c and any view ϕ. For clarity, we write the equation including the implicit con(·) operations:
ϕ
−(con(dom(c))) = con(dom(ϕ
−(c))). By definition of ϕ
− and con(·), we have ϕ
−(con(dom(c))) = {a ∈ Asn | ∀x ∈ X : ϕAsn(a)(x) ∈ dom(c)(x)}
= {a ∈ Asn | ∀x ∈ X ∃b ∈ c : ϕAsn(a)(x) = b(x)} (Lemma 2)
As ϕAsn is an injective function, we can find such a b that is in the range of ϕAsn, if and only if there is also a b
′ ∈ ϕ
−(c) such that ϕAsn(b
′) = b. Therefore, we get a ∈ Asn
 ∀x ∈ X ∃b
′ ∈ ϕ
−(c) : a(x) = b
′(x)
	
=
a ∈ Asn
 ∀x ∈ X : a(x) ∈ dom(ϕ
−(c))(x)
	
=con(dom(ϕ
−(c)))
The following three theorems express under which conditions the different notions of completeness are preserved when deriving propagators. The proofs for these theorems are analogous to the proof of Theorem 1, using Lemma 3.

Theorem 2 (Bounds(D) **completeness)** Let p be a bounds(D)-complete propagator.

For any hull-injective view ϕ, the propagator ϕb(p) is bounds(D)-complete. ∗
Theorem 3 (Range completeness) Let p be a range-complete propagator. For any hull-surjective view ϕ, the propagator ϕb(p) is range-complete. ∗
Theorem 4 (Domain completeness) Let p be a domain-complete propagator, and let ϕ be a view. Then ϕb(p) is domain-complete. ∗