An early version of the Z Standard included the deductive system W for reasoning about Z specifications. Later versions contain a different deductive system. In this paper we sketch a proof that W is relatively sound with respect to this new deductive system. We do this by demonstrating a semantic basis for a correspondence between the two systems, then showing that each of the inference rules of W can be simulated as derived rules in the new system. These new rules are presented as tactics over the the inference rules of the new deductive system. 1 Introduction An important part of the Z Standardization activity has been the definition of a logical deductive system for Z. Whilst some have sought to provide support for reasoning about Z specifications by embedding the language in an existing well-understood framework (HOL, Eves, PVS, Isabelle, for example; [BG94,Jon92,Saa92,KSW96,ES94]), other research has attempted to provide support for reasoning within Z, making use of Z's type ...

s and compressed postscript files are available via http://svrc.it.uq.edu.au The Ergo 5 Generic Proof Engine Mark Utting Abstract This paper describes the design principles and the architecture of the latest version of the Ergo proof engine, Ergo 5. Ergo 5 is a generic interactive theorem prover, similar to Isabelle, but based on sequent calculus rather than natural deduction and with a quite different approach to handling variable scoping. An efficient implementation of Ergo 5, based on Qu-Prolog, is also described, together with some benchmark results. 1 Motivation The Software Verification Research Centre, a special research centre of the Australian Research Council, is developing a suite of tools for reasoning about Z specifications and verifying refinement of specifications to code. There are several different projects investigating various aspects and approaches. To gain synergy, we want a common proof tool for all the projects, even though they have differing requi...

Many proof tools use `tactic languages' as programs to direct their proofs. We present a simplified idealised tactic language, and describe its denotational semantics. The language has many applications outside theorem-proveo activ5QbG) The semantics is parametrised by a monad (plus additional structure). By instantiating this inv arious ways, the core semantics of a number of di#erent tactic languages is obtained. 1 Int roduct45 The notiB of a tactic as a program usedi the constructifi of a (machic[ assi46fi8 formal proof has become quie wie[S---#fifi[ Tacti# orifi---#z[ i the work of Gordon et al [GMW79] onEdi burgh LCF. The extent to whi h other`tacti4 based' systems istems[ t essentien[ the same style of programmifi faci---#---[I vari4 consi[I8#Bfi . InEdi burgh LCF, atacti does notit[8B construct a proof. Rather,i ti s usedi backwardreasoni[ to construct a vali#fiz[I functi[ whi h mayi46z8 prove thedesi6B property. Theoremhood i guarded by use of a `safe datatype', and only sound vali484[I functi[I may construct elements ofthi type. In other work, the type of theoremsi protected by havi8 the class oftacti--- icti protected, so thati i ia ossiSB tobui# unsound proofs. The account here tends towards the secondvion though the treatment oftacti6 i s actually so abstract that thi may not be an i[ edi---# t to i[ appli#[IS# i eipli sense. Whie. tacti[ arewiS---fi6[IS--- tacti programmi--- remai4 adiBfiBS task. Inthi paper, weconsi#[ abstractdescri[S#fi--- oftactifi[ wit the hope that modern algori------ desii techniSzS# such as thosedescri ed byBiS and de Moor [BdM97], can be brought to bear on thedi8S---fi[IS ontacti programmi#4 Earlia di#------S[ISS oftacti6 i n the abstract (wiract operati6z[ bii to any parti[ISS proof tool)i)[SS--- those by SchmiB [Sch84] and Mi4#...

The CADiZ documentation is organised as a hierarchy of pages, connected by hypertext links, for browsing on-line. This document collects those pages together into a book, with hypertext links replaced by section numbers. 1 Home page — release 4.3 CADiZ is a set of free software tools that assist use of a specification notation called Z. The source is not yet available.

Let |h | ∋ ∅. In [50], the authors characterized algebras. We show that there exists an almost everywhere sub-ordered, pseudo-normal, pairwise contravariant and maximal subalgebra. Therefore a useful survey of the subject can be found in [50]. It would be interesting to apply the techniques of [35, 34] to composite categories. 1