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BOOL TRUE FALSE Z none for none N N1 succ pred 0 1 ...

expression + expression expression expression expression expression. barcode pdf417 9.6.2 Peano axioms an d recursive de nitions The following predicates yield de nition of the boolean and arithmetic expressions: BOOL = {TRUE, FALSE} TRUE = FALSE 0 N succ Z pred = succ Z.

S 0 S ( n none for none n S succ(n) S) N S a a + 0 = a a a 0 = 0 a a 0 = succ(0) a, b a + succ(b) = succ(a + b) a, b a succ(b) = (a b) + a a, b a succ(b) = (a b) a. 9.6 Boolean and arithmetic language 9.6.3 Extension of th none none e arithmetic language We introduce the classical binary relations on numbers, the niteness predicate, the interval between two numbers, the subtraction, division, modulo, cardinal, maximum, and minimum constructs: .

.. predicate ::= .

.. expression expression expression < expression expression expression expression > expression nite(expression) .

.. expression .

. expression expression expression expression / expression expression mod expression card(expression) max(expression) min(expression). expression Operator smaller than none for none or equal smaller than greater than or equal greater than interval subtraction division modulo niteness cardinality. Predicate a b a<b a b a>b c a .. b c=a b c = a/b r = a mod b nite(s) n = card(s).

Rewritten c c N none for none b = a + c a b a=b (a < b) (a b) a c c b a=b+c r (r N r < b a = c b+r) a = (a/b) b + r n, f n N f 1 .. n f f 1 .

. n s s. Mathematical language Operator Predicate Rewritten n s ( x none for none x s x n) n s ( x x s x n). maximum n = max(s) n = min(s). minimum Division, modulo, car none for none dinal, minimum, and maximum are subjected to some wellde nedness conditions, which are the following:. Numeric expression Well-de nedness condition b=0 0 a b>0 nit e(s) s = x ( n n s x n) s = x ( n n s x n). a mod b card(s) max(s) min(s). 9.7 Advanced data str uctures In this section, we show how our basic mathematical language can still be extended to cope with some classical (advanced) data structures we shall use in subsequent chapters of the book, essentially strongly connected graphs, lists, rings, and trees. We present the axiomatic de nitions of these data structures together with some theorems.

We do not present proofs. In fact all such proofs have been done with the Rodin Platform..

9.7 Advanced data structures 9.7.1 Irre exive tran sitive closure We start with the de nition of the irre exive transitive closure of a relation, which is a very useful concept to be used in what follows.

Given a relation r from a set S to itself, the irre exive transitive closure of r, denoted by cl(r), is also a relation from S to S. The characteristic properties of cl(r) are:. (i) Relation r is inc none none luded in cl(r). (ii) The forward composition of cl(r) with r is included in cl(r). (iii) Relation cl(r) is the smallest relation dealing with (i) and (ii).

. This is illustrated i n Fig. 9.6.

It can be formalized as follows:. axm_1 : axm_2 : axm_3 none none : axm_4 : axm_5 :. r S S cl(r) S S r cl(r) cl(r) ; r cl(r) p r p p ; r p cl(r) p Fig. 9.6.

A relation none none (dashed) and its irre exive transitive closure (dashed and plain). Mathematical language The following theorem none for none s can be proved: thm_1 : thm_2 : thm_3 : thm_4 : thm_5 : cl(r) ; cl(r) cl(r) cl(r) = r r ; cl(r) cl(r) = r cl(r) ; r s r[s] s cl(r)[s] s cl(r 1 ) = cl(r) 1. These theorems are pr oved by nding some instantiations for the local variable p in the universally quanti ed axiom axm_5. In particular, the proof of thm_1 is handled by instantiating p with : { x y . cl(r) ; {x y} cl(r) }.. 9.7.2 Strongly connec ted graphs Given a set V and a binary relation r from V to itself, the graph representing this relation is said to be strongly connected if any two distinct points m and n in V are possibly connected by a path built on r.

This is illustrated in Fig. 9.7.

This can be.
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