By Jim Woodcock
This e-book includes adequate mnaterial for 3 entire classes of research. It presents an advent to the area of good judgment, units and kinfolk. It explains using the Znotation within the specification of practical structures. It exhibits how Z necessities should be subtle to provide executable code; this is often validated in a range of case stories. The necessities of specification, refinement and evidence are coated, revealing recommendations by no means formerly released. workouts, options and set of Tranparencies can be found through http://www.comlab.ox.ac.uk/usingz.html
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Extra resources for Using Z: Specification, Refinement, and Proof (Prentice-Hall International Series in Computer Science)
Of course, the negation of a contradiction is a tautology, and vice versa. 12 The following propositions are tautologies: p ∨ ¬p p⇒p p ⇒ (q ⇒ p) while the following are contradictions: p ∧ ¬p p ¬p ¬(p ⇒ (q ⇒ p)) 2 / Propositional Logic 24 To prove that a proposition is a tautology, we have only to produce a truth table and check that the major connective takes the value t for each combination of propositional variables. 13 We prove that ¬p ∨ q following table: p q ¬p ∨ q p ⇒ q is a tautology by exhibiting the p⇒q t t f t t t t f f f t f f t t t t t f f t t t t Tautologies involving equivalences are particularly useful in proofs; they can be used to rewrite goals and assumptions to facilitate the completion of an argument.
This is not the only way to fill a gap, however. We could also choose to put an object variable in the empty place above. The predicate ‘x > 5’ is still not a proposition; we cannot say whether it is true or false without knowing what x is. The use of object variables is a powerful technique, and holds the key to expressing the universal and existential properties described above. We can make a proposition out of ‘x > 5’ by adding a quantifier to the front of the expression. For example, we could state that ‘there is an x, which is a natural number, such that x > 5’.
The substitution has brought an unwanted change of meaning. To avoid such confusion, we may rename bound variables prior to substitution, choosing fresh variable names to avoid variable capture. We can give equivalences to explain the effect of substitution into quantified expressions. In the simplest case, the variable being substituted for has the same name as the one being quantified: (∀ x : a | p • q)[t / x] (∀ x : a[t / x] | p • q) (∃ x : a | p • q)[t / x] (∃ x : a[t / x] | p • q) In this case, the only part of the expression that may change is the range of the quantified variable.