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15
Dependently Typed Functional Programs and their Proofs
, 1999
"... Research in dependent type theories [ML71a] has, in the past, concentrated on its use in the presentation of theorems and theoremproving. This thesis is concerned mainly with the exploitation of the computational aspects of type theory for programming, in a context where the properties of programs ..."
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Cited by 70 (13 self)
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Research in dependent type theories [ML71a] has, in the past, concentrated on its use in the presentation of theorems and theoremproving. This thesis is concerned mainly with the exploitation of the computational aspects of type theory for programming, in a context where the properties of programs may readily be specified and established. In particular, it develops technology for programming with dependent inductive families of datatypes and proving those programs correct. It demonstrates the considerable advantage to be gained by indexing data structures with pertinent characteristic information whose soundness is ensured by typechecking, rather than human effort. Type theory traditionally presents safe and terminating computation on inductive datatypes by means of elimination rules which serve as induction principles and, via their associated reduction behaviour, recursion operators [Dyb91]. In the programming language arena, these appear somewhat cumbersome and give rise to unappealing code, complicated by the inevitable interaction between case analysis on dependent types and equational reasoning on their indices which must appear explicitly in the terms. Thierry Coquand’s proposal [Coq92] to equip type theory directly with the kind of
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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Cited by 23 (8 self)
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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
A Full Formalisation of πCalculus Theory in the Calculus of Constructions
, 1997
"... A formalisation of picalculus in the Coq system is presented. Based on a de Bruijn notation for names, our... ..."
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Cited by 8 (0 self)
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A formalisation of picalculus in the Coq system is presented. Based on a de Bruijn notation for names, our...
Analysis of a Guard Condition in Type Theory
, 1997
"... We present a realizability interpretation of coinductive types based on partial equivalence relations (per's). We extract from the per's interpretation sound rules to type recursive definitions. These recursive definitions are needed to introduce "infinite" and "total" objects of coinductive type ..."
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Cited by 4 (0 self)
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We present a realizability interpretation of coinductive types based on partial equivalence relations (per's). We extract from the per's interpretation sound rules to type recursive definitions. These recursive definitions are needed to introduce "infinite" and "total" objects of coinductive type such as an infinite stream or a nonterminating process. We show that the proposed type system enjoys the basic syntactic properties of subject reduction and strong normalization with respect to a confluent rewriting system first studied by Gimenez. We also compare the proposed type system with those studied by Coquand and Gimenez. In particular, we provide a semantic reconstruction of Gimenez's system which suggests a rule to type nested recursive definitions.
Using structural recursion for corecursion
 In Types for Proofs and Programs, International Conference, TYPES 2008, volume 5497 of LNCS
, 2009
"... Abstract. We propose a (limited) solution to the problem of constructing stream values defined by recursive equations that do not respect the guardedness condition. The guardedness condition is imposed on definitions of corecursive functions in Coq, AGDA, and other higherorder proof assistants. In ..."
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Cited by 4 (0 self)
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Abstract. We propose a (limited) solution to the problem of constructing stream values defined by recursive equations that do not respect the guardedness condition. The guardedness condition is imposed on definitions of corecursive functions in Coq, AGDA, and other higherorder proof assistants. In this paper, we concentrate in particular on those nonguarded equations where recursive calls appear under functions. We use a correspondence between streams and functions over natural numbers to show that some classes of nonguarded definitions can be modelled through the encoding as structural recursive functions. In practice, this work extends the class of stream values that can be defined in a constructive type theorybased theorem prover with inductive and coinductive types, structural recursion and guarded corecursion.
Formalising exact arithmetic in type theory
 New Computational Paradigms: First Conference on Computability in Europe, CiE 2005
, 2005
"... Abstract. In this work we focus on a formalisation of the algorithms of lazy exact arithmetic à la Potts and Edalat [1]. We choose the constructive type theory as our formal verification tool. We discuss an extension of the constructive type theory with coinductive types that enables one to formalis ..."
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Cited by 3 (0 self)
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Abstract. In this work we focus on a formalisation of the algorithms of lazy exact arithmetic à la Potts and Edalat [1]. We choose the constructive type theory as our formal verification tool. We discuss an extension of the constructive type theory with coinductive types that enables one to formalise and reason about the infinite objects. We show examples of how infinite objects such as streams and expression trees can be formalised as coinductive types. We study the type theoretic notion of productivity which ensures the infiniteness of the outcome of the algorithms on infinite objects. Syntactical methods are not always strong enough to ensure the productivity. However, if some information about the complexity of a function is provided, one may be able to show the productivity of that function. In the case of the normalisation algorithm we show that such information can be obtained from the choice of real number representation that is used to represent the input and the output. 1
Practical Inference for TypedBased Termination in a Polymorphic Setting
"... We introduce a polymorphic #calculus that features inductive types and that enforces termination of recursive definitions through typing. Then, we define a sound and complete type inference algorithm that computes a set of constraints to be satisfied for terms to be typable. ..."
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Cited by 3 (0 self)
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We introduce a polymorphic #calculus that features inductive types and that enforces termination of recursive definitions through typing. Then, we define a sound and complete type inference algorithm that computes a set of constraints to be satisfied for terms to be typable.
Wellfounded Recursion with Copatterns A Unified Approach to Termination and Productivity
, 2013
"... In this paper, we study strong normalization of a core language based on System Fomega which supports programming with finite and infinite structures. Building on our prior work, finite data such as finite lists and trees are defined via constructors and manipulated via pattern matching, while infi ..."
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Cited by 3 (1 self)
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In this paper, we study strong normalization of a core language based on System Fomega which supports programming with finite and infinite structures. Building on our prior work, finite data such as finite lists and trees are defined via constructors and manipulated via pattern matching, while infinite data such as streams and infinite trees is defined by observations and synthesized via copattern matching. In this work, we take a typebased approach to strong normalization by tracking size information about finite and infinite data in the type. This guarantees compositionality. More importantly, the duality of pattern and copatterns provide a unifying semantic concept which allows us for the first time to elegantly and uniformly support both wellfounded induction and coinduction by mere rewriting. The strong normalization proof is structured around Girard’s reducibility candidates. As such our system allows for nondeterminism and does not rely on coverage. Since System Fomega is general enough that it can be the target of compilation for the Calculus of Constructions, this work is a significant step towards representing observationcentric infinite data in proof assistants such as Coq and Agda.
Cut elimination for a logic with induction and coinduction
 JOURNAL OF APPLIED LOGIC
, 2012
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