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Structural Recursive Definitions in Type Theory
 Automata, Languages and Programming, 25th International Colloquium, ICALP’98
, 1998
"... We introduce an extension of the Calculus of Construction with inductive and coinductive types that preserves strong normalisation for a lazy computation relation. This extension considerably enlarges the expressiveness of the language, enabling a direct translation of recursive programs, while kee ..."
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We introduce an extension of the Calculus of Construction with inductive and coinductive types that preserves strong normalisation for a lazy computation relation. This extension considerably enlarges the expressiveness of the language, enabling a direct translation of recursive programs, while keeping a relatively simple collection of typing rules. 1 Introduction The last twenty five years have seen an increasing development of different proof environments based on type theory. Several type theories have been proposed as a foundation of such proof environments [15, 6, 16], trying to find an accurate compromise between two criteria. On the one hand, we search for extensions of type theory that preserve its conceptual simplicity of type theory (a few primitive constructions, a small number of typing rules) and metatheoretical properties ensuring its soundness and a direct mechanisation (strong normalisation, decidability of typechecking, etc). On the other hand, we would like to pro...
A Formalization of the Process Algebra CCS in Higher Order Logic
, 1992
"... : This paper describes a mechanization in higher order logic of the theory for a subset of Milner's ccs. The aim is to build a sound and effective tool to support verification and reasoning about process algebra specifications. To achieve this goal, the formal theory for pure ccs (no value passing) ..."
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: This paper describes a mechanization in higher order logic of the theory for a subset of Milner's ccs. The aim is to build a sound and effective tool to support verification and reasoning about process algebra specifications. To achieve this goal, the formal theory for pure ccs (no value passing) is defined in the interactive theorem prover hol, and a set of proof tools, based on the algebraic presentation of ccs, is provided. y Research supported by Consiglio Nazionale delle Ricerche (C.N.R.), Italy. Contents 1 Introduction 2 2 The HOL System 3 3 CCS 4 3.1 Syntax and Operational Semantics : : : : : : : : : : : : : : : : : : : : : : : 4 3.2 Observational Semantics : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 5 3.3 Axiomatic Characterization of Observational Congruence : : : : : : : : : : 6 3.4 A Modal Logic : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 7 4 Mechanization of CCS in HOL 8 4.1 The Syntax : : : : : : : : : : : : : : : : : : : : : ...
A Tactic Language for Ergo
 Formal Methods Pacific ’97
, 1997
"... A new version of the Ergo theorem prover is under development. It uses a single tactic language, based on Angel, for tactic programming, user interface, and proof representation. This paper describes the language as it is used in each of these cases, and explains the details of its implementation ..."
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A new version of the Ergo theorem prover is under development. It uses a single tactic language, based on Angel, for tactic programming, user interface, and proof representation. This paper describes the language as it is used in each of these cases, and explains the details of its implementation in QuProlog. An example from classical propositional calculus is included. 1 Introduction Ergo is an interactive proof tool that has been designed and implemented at the SVRC over the last ten years. It is implemented in QuProlog (Robinson and Hagen, 1997), and is designed to be extensible, so that users can add new theories, tactics and user interfaces. Ergo 5 is currently under development. Having no inbuilt object logic, it is a generic prover that can be instantiated by providing a collection of axiomatic and/or definitional theories. The core of Ergo 5 provides support for (uninterpreted) sequents with named tuples of arbitrary terms as antecedents and single terms as consequents...
MachineAssisted TheoremProving for Software Engineering
 Technical Monograph PRG121, ISBN 0902928953, Oxford University Computing LaboratoryWolfson Building, Parks Road
, 1994
"... The thesis describes the production of a large prototype proof system for Z, and a tactic language in which the proof tactics used in a wide range of systems (including the system described here) can be discussed. The details of the construction of the toolusing the W logic for Z, and implemented ..."
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Cited by 5 (1 self)
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The thesis describes the production of a large prototype proof system for Z, and a tactic language in which the proof tactics used in a wide range of systems (including the system described here) can be discussed. The details of the construction of the toolusing the W logic for Z, and implemented in 2OBJare presented, along with an account of some of the proof tactics which enable W to be applied to typical proofs in Z. A case study gives examples of such proofs. Special attention is paid to soundness concerns, since it is considerably easier to check that a program such as this one produces sound proofs, than to check that each of the impenetrable proofs which it creates is indeed sound. As the first such encoding of W, this helped to find bugs in the published presentations of W, and to demonstrate that W makes proof in Z tractable. The second part of the thesis presents a tactic language, with a formal semantics (independent of any particular tool) and a set of rules for reasoning about tactics written in this language. A small set of these rules is shown to be complete for the finite (nonrecursive)
Improving Angel's Parallel Operator: Gumtree's Approach
, 1997
"... We describe some features of the tactic language implemented in the theorem prover Ergo 5. This is a variant of the generic tactic language Angel. We have adapted the language by changing the semantics of its parallel composition operator, the operator by which different tactics are applied to di ..."
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We describe some features of the tactic language implemented in the theorem prover Ergo 5. This is a variant of the generic tactic language Angel. We have adapted the language by changing the semantics of its parallel composition operator, the operator by which different tactics are applied to different branches in a proof tree. The paper includes a denotational semantics for this operator, and a collection of derived tactics which use it, together with a collection of algebraic laws which they obey. Keywords Tactic, Tactical, Denotational Semantics, Algebraic Laws, Interactive theorem proving 1 Introduction Theoremproving tools have traditionally used their implementation language as a tactic language in which users can write procedures to assist in the discovery of proofs. In the LCF family of tools, this language is ML, with certain tactic combinators (tacticals) predefined. Various versions of Ergo (Nickson, Traynor, and Utting, 1996) have used QuProlog (Robinson and Ha...
A Monadic Interpretation of Tactics
, 2002
"... 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 theoremproveo activ5QbG) The semantics is parametrised by a monad (plus additional struc ..."
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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 theoremproveo 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[ arewiSfi6[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 thedi8Sfi[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#...
Formal Methods and Mechanical Verification applied to the development of a convergent distributed sorting program
, 1996
"... Gentle introductions to the programming logic UNITY, the theorem proving environment HOL, and the embedding of the first into the latter are presented. Equipped with this apparatus a methodology for designing distributed algorithms is described. ..."
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Gentle introductions to the programming logic UNITY, the theorem proving environment HOL, and the embedding of the first into the latter are presented. Equipped with this apparatus a methodology for designing distributed algorithms is described.