Results

**11 - 13**of**13**### Reductivity

, 1995

"... ing from these examples we propose the following definition: Definition 17 The relation R is said to be F-well-founded if and only if, for all relations S , the equation 19 X:: X = S ffl F:X ffl R has a unique solution. 2 By design, R is well-founded (in the conventional sense) if and only ..."

Abstract
- Add to MetaCart

ing from these examples we propose the following definition: Definition 17 The relation R is said to be F-well-founded if and only if, for all relations S , the equation 19 X:: X = S ffl F:X ffl R has a unique solution. 2 By design, R is well-founded (in the conventional sense) if and only if it is id-well-founded, where id is the identity relator. Moreover, the converse of any initial F -algebra is F-well-founded. A stronger statement can be made: Theorem 18 Suppose R is an F -coalgebra that is a bijection between F:R? and R? . Then the following are all equivalent: (a) R is F-well-founded, (b) R is F -reductive, (c) R[ is an initial F -algebra. 2 (The equivalence between (b) and (c) has already been observed.) One of the fundamental properties of reductivity is that it implies well-foundedness. This is theorem 7 of [3]. The converse is not true. Let R be a nonempty, well-founded relation (for example the relation succ[ on natural numbers) . Then it is easy to show th...

### Structured Formal Development with Quotient Types in Isabelle/HOL

"... Abstract. General purpose theorem provers provide sophisticated proof methods, but lack some of the advanced structuring mechanisms found in specification languages. This paper builds on previous work extending the theorem prover Isabelle with such mechanisms. A way to build the quotient type over a ..."

Abstract
- Add to MetaCart

Abstract. General purpose theorem provers provide sophisticated proof methods, but lack some of the advanced structuring mechanisms found in specification languages. This paper builds on previous work extending the theorem prover Isabelle with such mechanisms. A way to build the quotient type over a given base type and an equivalence relation on it, and a generalised notion of folding over quotiented types is given as a formalised high-level step called a design tactic. The core of this paper are four axiomatic theories capturing the design tactic. The applicability is demonstrated by derivations of implementations for finite multisets and finite sets from lists in Isabelle. 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 ..."

Abstract
- Add to MetaCart

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 non-trivial examples are presented: an optimisation problem, and a derivation of quicksort where well-founded recursion is used to model terminating hylomorphisms in a language with inductive types. 1