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Induction and coinduction in sequent calculus
 Postproceedings of TYPES 2003, number 3085 in LNCS
, 2003
"... Abstract. Proof search has been used to specify a wide range of computation systems. In order to build a framework for reasoning about such specifications, we make use of a sequent calculus involving induction and coinduction. These proof principles are based on a proof theoretic (rather than sett ..."
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Abstract. Proof search has been used to specify a wide range of computation systems. In order to build a framework for reasoning about such specifications, we make use of a sequent calculus involving induction and coinduction. These proof principles are based on a proof theoretic (rather than settheoretic) notion of definition [13, 20, 25, 51]. Definitions are akin to (stratified) logic programs, where the left and right rules for defined atoms allow one to view theories as “closed ” or defining fixed points. The use of definitions makes it possible to reason intensionally about syntax, in particular enforcing free equality via unification. We add in a consistent way rules for pre and post fixed points, thus allowing the user to reason inductively and coinductively about properties of computational system making full use of higherorder abstract syntax. Consistency is guaranteed via cutelimination, where we give the first, to our knowledge, cutelimination procedure in the presence of general inductive and coinductive definitions. 1
Reasoning about Objectbased Calculi in (Co)Inductive Type Theory and the Theory of Contexts ∗
"... Abstract. We illustrate a methodology for formalizing and reasoning about Abadi and Cardelli’s objectbased calculi, in (co)inductive type theory, such as the Calculus of (Co)Inductive Constructions, by taking advantage of Natural Deduction Semantics and coinduction in combination with weak HigherO ..."
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Abstract. We illustrate a methodology for formalizing and reasoning about Abadi and Cardelli’s objectbased calculi, in (co)inductive type theory, such as the Calculus of (Co)Inductive Constructions, by taking advantage of Natural Deduction Semantics and coinduction in combination with weak HigherOrder Abstract Syntax and the Theory of Contexts. Our methodology allows to implement smoothly the calculi in the target metalanguage; moreover, it suggests novel presentations of the calculi themselves. In detail, we present a compact formalization of the syntax and semantics for the functional and the imperative variants of the ςcalculus. Our approach simplifies the proof of Subject Reduction theorems, which are proved formally in the proof assistant Coq with a relatively small overhead.
Mechanized Operational Semantics via (Co)Induction
, 1999
"... We give a fully automated description of a small programming language in the theorem prover Isabelle98. The language syntax and semantics are encoded, and we formally verify a range of semantic properties. This is achieved via uniform (co)inductive methods. Keywords: automated deduction, tactical ..."
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We give a fully automated description of a small programming language in the theorem prover Isabelle98. The language syntax and semantics are encoded, and we formally verify a range of semantic properties. This is achieved via uniform (co)inductive methods. Keywords: automated deduction, tactical theorem proving (Isabelle), operational semantics, induction and coinduction, software specification and verification. Submitted for publication. 1 Introduction The design of new programming languages which are wellprincipled, reliable and expressive is an important part of Computer Science. In this paper we contribute towards the techniques for specification, design and development of programming languages by specifying and verifying properties of a core language, using tactical verification within the theorem prover Isabelle 98. We show how this can be done uniformly, so that our methodology readily adapts to new languages, using suitable variations of standard methods to ensure rapid m...
The Representational Adequacy of HYBRID
"... The Hybrid system (Ambler et al., 2002b), implemented within Isabelle/HOL, allows object logics to be represented using higher order abstract syntax (HOAS), and reasoned about using tactical theorem proving in general and principles of (co)induction in particular. The form of HOAS provided by Hybrid ..."
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The Hybrid system (Ambler et al., 2002b), implemented within Isabelle/HOL, allows object logics to be represented using higher order abstract syntax (HOAS), and reasoned about using tactical theorem proving in general and principles of (co)induction in particular. The form of HOAS provided by Hybrid is essentially a lambda calculus with constants. Of fundamental interest is the form of the lambda abstractions provided by Hybrid. The user has the convenience of writing lambda abstractions using names for the binding variables. However each abstraction is actually a definition of a de Bruijn expression, and Hybrid can unwind the user’s abstractions (written with names) to machine friendly de Bruijn expressions (without names). In this sense the formal system contains a hybrid of named and nameless bound variable notation. In this paper, we present a formal theory in a logical framework which can be viewed as a model of core Hybrid, and state and prove that the model is representationally adequate for HOAS. In particular, it is the canonical translation function from λexpressions to Hybrid that witnesses adequacy. We also prove two results that characterise how Hybrid represents certain classes of λexpressions. The Hybrid system contains a number of different syntactic classes of expression, and associated abstraction mechanisms. Hence this paper also aims to provide a selfcontained theoretical introduction to both the syntax and key ideas of the system; background in automated theorem proving is not essential, although this paper will be of considerable interest to those who wish to work with Hybrid in Isabelle/HOL.
Elementary Order Theory
"... We review ordered sets. An order, or “comparison”, can be used to generate equivalences. We discuss inductively, and coinductively defined sets. Such sets arise naturally when defining, and reasoning about, programs and processes. We define proof principles for such sets. We use the principle of coi ..."
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We review ordered sets. An order, or “comparison”, can be used to generate equivalences. We discuss inductively, and coinductively defined sets. Such sets arise naturally when defining, and reasoning about, programs and processes. We define proof principles for such sets. We use the principle of coinduction to validate equivalences, and discuss when this is possible.