Logical Analysis and Verification of Cryptographic Protocols - Loria

2.1. PRELIMINARIES 21
the function symbols, arity : F → N. The arity of a function symbol indicates
the number of arguments that it expects. We call signature the tuple (F, arity),
and for simplicity, in this document, we denote by signature only the set F.
We define the set of terms T (F, X ) over the signature F to be the smallest set
containing X , C , and such that if f is a function symbol in F with arity n ≥ 0,
and if t1, . . . , tn are terms in T (F, X ), then f(t1, . . . , tn) is a term in T (F, X ). We
define the set of ground terms T (F) over the signature F to be biggest subset of
the set of terms T (F, X ) such that T (F) does not contain variables. We denote
the set of variables (respectively set of free constants) occurring in a term t by
V ar(t) (respectively Cons(t)). A term without variables, i.e. a term in T (F),
is called ground term. If T, T ′ are two sets of terms, and t, t ′ are two terms, we
abbreviate T ∪ T ′ by T, T ′ , T ∪ {t} by T, t, T \ {t} by T \ t, and {t, t ′ } by t, t ′ .
2.1.3 Positions and subterms
The subterms of a term t are denoted by the set of terms Sub(t) which is defined
recursively as follows. If t ∈ X ∪ C then Sub(t) = {t}, and if t = g(t1, . . . , tn)
then Sub(t) = {t} ∪ ( �n i=1 Sub(ti)). The strict (or proper) subterms of a term t are
denoted by the set of terms SSub(t), and SSub(t) = Sub(t) \ t. We denote t[s] a
term t that admits s as subterm.
Example 4 Consider the term t = Sig(h(a), Sk(Bob)).
We have: Sub(t) = {t, h(a), Sk(Bob), a, Bob}, and SSub(t) =
{h(a), Sk(Bob), a, Bob}.
We call positions the elements of N ∗ . We say that a position p is smaller than a
position q, and we write p ≤ q, if p is a prefix of q, i.e. there exists a position p ′
such that p.p ′ = q. Given a term t, the set of positions of t, denoted by P os(t), is
defined recursively as follows:
• if t is a variable or a constant then P os(t) = {ɛ};
• if t = f(t1, . . . , tn) then P os(t) = {ɛ} ∪ ( �
1≤i≤n {i.p such that p ∈ P os(ti)}).
The position ɛ is called the root position of the term t. The dag-size of a term t,
denoted by �t�dag, is defined to be the number of distinct subterms of t.
The T op symbol of a term t, denoted by T op(t), is defined by T op : T (F, X ) →
F ∪ X ∪ C, with
• T op(t) = t if t ∈ X ∪ C
• T op(t) = f if t = f(t1, . . . , tn)
The subterm of t at position p ∈ P os(t), denoted by t|p, is defined recursively
as follows: