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By Jim Woodcock

This e-book comprises adequate mnaterial for 3 whole classes of research. It presents an creation to the realm of common sense, units and relatives. It explains using the Znotation within the specification of life like structures. It exhibits how Z requisites should be sophisticated to provide executable code; this can be verified in a variety of case reviews. The necessities of specification, refinement and evidence are coated, revealing strategies by no means formerly released. routines, ideas and set of Tranparencies can be found through http://www.comlab.ox.ac.uk/usingz.html

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Using Z.Specification,refinement,and proof

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Chapter 4 Equality and Definite Description In this chapter we extend our language of mathematics by adding a theory of equality between expressions. The language of predicate calculus with equality is strictly more expressive than without, since it allows us to assert the identity of two objects, or to distinguish between them. We provide inference rules to support the intuitive notion that expressions which are equal may be substituted one for the other, without affecting the truth of a statement, or the value of a larger expression.

1] p [⇒−intro[1] ] p∧q⇒p p∧q p .. [⇒−intro] p⇒p∧q [ −intro] The left-hand subtree may now be completed by conjunction elimination on the assumption. Turning now to the right-hand subtree, we should immediately introduce the implication: p p ∧ q [1] [∧−elim1] p [⇒−intro[1] ] p∧q⇒p p∧q p [2] .. 5 / Equivalence 19 Now, the major connective is a conjunction, so we introduce it: p p ∧ q [1] [∧−elim1] p [⇒−intro[1] ] p∧q⇒p p∧q p [2] .. p p [2] .. q [∧−intro] p∧q [⇒−intro[2] ] p⇒p∧q [ −intro] The left-most unfinished subtree can be closed easily, since we have to prove p from the assumption p: that is immediate.

Equalities form the atomic propositions in our logical language; the only other way of obtaining an atomic proposition is through set membership, described in Chapter 5. Everything is identical to itself: thus, if t is any expression, then t is equal to t . This principle is known as the law of reflection: t =t [eq-ref] It should be remarked that there are logics in which this principle does not hold. It is, however, an axiom of standard Z. 2 In basic arithmetic, everybody knows that 1+1 = 1+1 whatever the properties of numbers and addition.

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