Distributive category theory is the study of categories with two monoidal structures, one of which “distributes” over the other in some manner. When these are the product and coproduct, this distribution is taken to be the law

which asserts that the obvious canonical map has an inverse. A distributive category is here taken to mean a category with finite products and binary coproducts such that this law is satisfied.

In any distributive category the coproduct of the final object with itself, 1 + 1, forms a boolean algebra. Thus, maps into 1 + 1 provide a boolean logic: if each such map recognizes a unique subobject, the category is a recognizable distributive category. If, furthermore, the category is such that these recognizers classify detachable subobjects (coproduct embeddings), it is an extensive distributive category.

Extensive distributive categories can be approached in various ways. For example, recognizable distributive categories, in which coproducts are disjoint or all preinitials are isomorphic, are extensive. Also, a category X having finite products and binary coproducts satisfying the slice equation (due to Schanuel and Lawvere) is extensive. This paper describes a series of embedding theorems. Any distributive category has a full faithful embedding into a recognizable distributive category. Any recognizable distributive category can be "solidified" faithfully to produce an extensive distributive category. Any extensive distributive category can be embedded into a topos.

A peculiar source of extensive distributive categories is the coproduct completion of categories with familial finite products. In particular, this includes the coproduct completion of cartesian categories, which is serendipitously, therefore, also the distributive completion. Familial distributive categories can be characterized as distributive categories for which every object has a finite decomposition into indecomposables.

Type theory allows us to extract from a constructive proof that a specification is satisfiable a program that satisfies the specification. Algorithms for optimization of such programs are currently the object of research.

In this paper we consider one such algorithm, which was described in Beeson (1985) and which we will call ‘Harrop’. This algorithm greatly simplifies programs extracted from proofs in the Pure Construction Calculus. We use a Partial Equivalence Relation model for higher order lambda calculus, to check that t and Harrop(t) return the same outputs from the same inputs, i.e. that they are extensionally equal.

As a corollary, we show that it is correct (and, of course, useful) to replace a program t with Harrop(t). Such a correctness result has already been proved by Möhring (Möhring 1989a, 1989b) using realizability semantics, but we obtain it as a corollary of a new result, the extensional equality between t and Harrop(t). Also the semantic method we use is interesting in its own right.

The study of type theory may offer a uniform language for modular programming, structured specification and logical reasoning. We develop an approach to program specification and data refinement in a type theory with a strong logical power and nice structural mechanisms to show that it provides an adequate formalism for modular development of programs and specifications. Specification of abstract data types is considered, and a notion of abstract implementation between specifications is defined in the type theory and studied as a basis for correct and modular development of programs by stepwise refinement. The higher-order structural mechanisms in the type theory provide useful and flexible tools (specification operations and parameterized specifications) for modular design and structured specification. Refinement maps (programs and design decisions) and proofs of implementation correctness can be developed by means of the existing proof development systems based on type theories.