An ML Editor based on Proofs-as-Programs

. C Y NTHIA is a novel editor for the functional programming language ML in which each function definition is represented as the proof of a simple specification. Users of C Y NTHIA edit programs by applying sequences of high-level editing commands to existing programs. These commands make changes to the proof representation from which a new program is then extracted. The use of proofs is a sound framework for analysing ML programs and giving useful feedback about errors. Amongst the properties analysed within C Y NTHIA at present is termination. C Y NTHIA has been successfully used in the teaching of ML in two courses at Napier University. 1 Introduction Current programming environments for novice functional programming (FP) are inadequate. This paper describes ways of using mechanised theorem proving to improve the situation, in the context of the language ML [9]. ML is a stronglytyped FP language with type inference [4]. ML incorporates extensive use of pattern match..

On choice rules in dependent type theory

In a dependent type theory satisfying the propositions as types correspondence together with the proofs-as-programs paradigm, the validity of the unique choice rule or even more of the choice rule says that the extraction of a computable witness from an existential statement under hypothesis can be performed within the same theory. Here we show that the unique choice rule, and hence the choice rule, are not valid both in Coquand\u2019s Calculus of Constructions with indexed sum types, list types and binary disjoint sums and in its predicative version implemented in the intensional level of the Minimalist Founda- tion. This means that in these theories the extraction of computational witnesses from existential statements must be performed in a more ex- pressive proofs-as-programs theory

Checking Zenon Modulo Proofs in Dedukti

Dedukti has been proposed as a universal proof checker. It is a logical framework based on the lambda Pi calculus modulo that is used as a backend to verify proofs coming from theorem provers, especially those implementing some form of rewriting. We present a shallow embedding into Dedukti of proofs produced by Zenon Modulo, an extension of the tableau-based first-order theorem prover Zenon to deduction modulo and typing. Zenon Modulo is applied to the verification of programs in both academic and industrial projects. The purpose of our embedding is to increase the confidence in automatically generated proofs by separating untrusted proof search from trusted proof verification.Comment: In Proceedings PxTP 2015, arXiv:1507.0837

Relational Rippling: a General Approach

We propose a new version of rippling, called relational rippling. Rippling is a heuristic for guiding proof search, especially in the step cases of inductive proofs. Relational rippling is designed for representations in which value passing is by shared existential variables, as opposed to function nesting. Thus relational rippling can be used to guide reasoning about logic programs or circuits represented as relations. We give an informal motivation and introduction to relational rippling. More details, including formal definitions and termination proofs can be found in the longer version of this paper, [Bundy and Lombart, 1995]

Proving uniformity and independence by self-composition and coupling

Proof by coupling is a classical proof technique for establishing probabilistic properties of two probabilistic processes, like stochastic dominance and rapid mixing of Markov chains. More recently, couplings have been investigated as a useful abstraction for formal reasoning about relational properties of probabilistic programs, in particular for modeling reduction-based cryptographic proofs and for verifying differential privacy. In this paper, we demonstrate that probabilistic couplings can be used for verifying non-relational probabilistic properties. Specifically, we show that the program logic pRHL---whose proofs are formal versions of proofs by coupling---can be used for formalizing uniformity and probabilistic independence. We formally verify our main examples using the EasyCrypt proof assistant