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Observational Equality, Now!
 A SUBMISSION TO PLPV 2007
, 2007
"... This paper has something new and positive to say about propositional equality in programming and proof systems based on the CurryHoward correspondence between propositions and types. We have found a way to present a propositional equality type • which is substitutive, allowing us to reason by repla ..."
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Cited by 23 (8 self)
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This paper has something new and positive to say about propositional equality in programming and proof systems based on the CurryHoward correspondence between propositions and types. We have found a way to present a propositional equality type • which is substitutive, allowing us to reason by replacing equal for equal in propositions; • which reflects the observable behaviour of values rather than their construction: in particular, we have extensionality— functions are equal if they take equal inputs to equal outputs; • which retains strong normalisation, decidable typechecking and canonicity—the property that closed normal forms inhabiting datatypes have canonical constructors; • which allows inductive data structures to be expressed in terms of a standard characterisation of wellfounded trees; • which is presented syntactically—you can implement it directly, and we are doing so—this approach stands at the core of Epigram 2; • which you can play with now: we have simulated our system by a shallow embedding in Agda 2, shipping as part of the standard examples package for that system [20]. Until now, it has always been necessary to sacrifice some of these aspects. The closest attempt in the literature is Altenkirch’s construction of a setoidmodel for a system with canonicity and extensionality on top of an intensional type theory with proofirrelevant propositions [4]. Our new proposal simplifies Altenkirch’s construction by adopting McBride’s heterogeneous approach to equality [18].
The Gentle Art of Levitation
"... We present a closed dependent type theory whose inductive types are given not by a scheme for generative declarations, but by encoding in a universe. Each inductive datatype arises by interpreting its description—a firstclass value in a datatype of descriptions. Moreover, the latter itself has a de ..."
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Cited by 20 (4 self)
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We present a closed dependent type theory whose inductive types are given not by a scheme for generative declarations, but by encoding in a universe. Each inductive datatype arises by interpreting its description—a firstclass value in a datatype of descriptions. Moreover, the latter itself has a description. Datatypegeneric programming thus becomes ordinary programming. We show some of the resulting generic operations and deploy them in particular, useful ways on the datatype of datatype descriptions itself. Surprisingly this apparently selfsupporting setup is achievable without paradox or infinite regress. 1.
Phase distinctions in the compilation of Epigram
, 2005
"... Abstract. It is commonly believed that in dependently typed programming languages, the blurring of the distinction between types and values means that no type erasure is possible at runtime. In this paper, however, we propose an alternative phase distinction. Rather than distinguishing types and va ..."
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Cited by 1 (1 self)
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Abstract. It is commonly believed that in dependently typed programming languages, the blurring of the distinction between types and values means that no type erasure is possible at runtime. In this paper, however, we propose an alternative phase distinction. Rather than distinguishing types and values in the compilation of EPIGRAM, we distinguish compiletime and runtime evaluation, and show by a series of program transformations that values which are not required at runtime can be erased. 1
Under consideration for publication in J. Functional Programming 1 Algebra of Programming in Agda Dependent Types for Relational Program Derivation
, 2009
"... Relational program derivation is the technique of stepwise refining a relational specification to a program by algebraic rules. The program thus obtained is correct by construction. Meanwhile, dependent type theory is rich enough to express various correctness properties to be verified by the type c ..."
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Relational program derivation is the technique of stepwise refining a relational specification to a program by algebraic rules. The program thus obtained is correct by construction. Meanwhile, dependent type theory is rich enough to express various correctness properties to be verified by the type checker. We have developed a library, AoPA, to encode relational derivations in the dependently typed programming language Agda. A program is coupled with an algebraic derivation whose correctness is guaranteed by the type system. Two nontrivial examples are presented: an optimisation problem, and a derivation of quicksort where wellfounded recursion is used to model terminating hylomorphisms in a language with inductive types. 1
Abstract VeriML: A dependentlytyped, userextensible and languagecentric approach to proof assistants
, 2013
"... Software certification is a promising approach to producing programs which are virtually free of bugs. It requires the construction of a formal proof which establishes that the code in question will behave according to its specification – a higherlevel description of its functionality. The construc ..."
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Software certification is a promising approach to producing programs which are virtually free of bugs. It requires the construction of a formal proof which establishes that the code in question will behave according to its specification – a higherlevel description of its functionality. The construction of such formal proofs is carried out in tools called proof assistants. Advances in the current stateoftheart proof assistants have enabled the certification of a number of complex and realistic systems software. Despite such success stories, largescale proof development is an arcane art that requires significant manual effort and is extremely timeconsuming. The widely accepted best practice for limiting this effort is to develop domainspecific automation procedures to handle all but the most essential steps of proofs. Yet this practice is rarely followed or needs comparable development effort as well. This is due to a profound architectural shortcoming of existing proof assistants: developing automation procedures is currently overly complicated and errorprone. It involves the use of an amalgam of extension languages, each with a different programming model and a set of limitations, and with significant interfacing problems between them. This thesis posits that this situation can be significantly improved by designing a proof assistant with extensibility as the central focus. Towards that effect, I have designed a novel programming language called