Logical Analysis and Verification of Cryptographic Protocols - Loria

114 CHAPTER 5. SATURATED DEDUCTION SYSTEMS
Definition 46 (finite variant property) The pair of equational theories (H, H ′ ) has the
finite variant property if for every term t, we can effectively compute a finite complete
set of H-variants modulo H ′ . Sometimes and for simplicity, we will simply say variants
and complete set of variants when H and H ′ are clear from the context.
When R is a H ′ -convergent rewrite system generating H, we have that (R, H ′ )
satisfies the finite variant property if and only if (H, H ′ ) satisfies the finite variant
property.
Definition 47 (R, H ′ ) satisfies the finite variant property if for any term t, there is a
finite set of variants t1, . . . , tn, effectively computable, such that, for every substitution
σ, there is an index i and a substitution θ such that (tσ)↓ H ′ \R
=H ′ tiθ.
In [86], the authors showed that if (R, H ′ ) has the finite variant property, we
may not only compute in advance some instances ti of t such that (tσ)↓ is always
an instance modulo H ′ of some ti , but actually compute in advance substitutions
θi such that ti = (tθi)↓ is a complete set of variants and every normalised
substitution can be factorised through θi. This result is summarised by the following
lemma.
Lemma 40 (R, H ′ ) has the finite variant property if and only if for any term t, there is
a finite set of substitutions Σ(t) such that for any substitution σ, there exists a substitution
θ ∈ Σ(t), and a substitution τ verifying (σ)↓ =H ′ θτ and (tσ)↓ =H ′ (tθ)↓τ
In [86], S. Delaune and H. Comon-Lundh define the boundness property as follows:
Definition 48 (boundedness property) (R, H ′ ) satisfies the boundedness property if
for every term t, there exists an integer n such that for every normalised substitution σ,
the normal form of tσ is reachable by a derivation whose length can be bounded by n:
∀t, ∃n, ∀σ, t((σ)↓)
≤n
→H ′ \R (tσ)↓
and then, S. Delaune and H. Comon-Lundh showed the relationships between
the boundness property and the finite variant property by proving the following
theorem.
Theorem 9 (R, H ′ ) satisfies the boundedness property if and only if (R, H ′ ) satisfies
the finite variant property.
5.2.2 Equational theories having finite variant property
In [86], S. Delaune and H. Comon-Lundh showed that for any equational theory
H generated by a (∅-) convergent rewrite system R, if any R-basic narrowing