We present transformations for incrementally defining both inductive sum/variant types and coinductive product/record types in a formal refinement setting. Inductive types are built by incrementally accumulating constructors. Coinductive types are built by incrementally accumulating observers. In each case, when the developer decides that the constructor (resp. observer) set is complete, a transformation is applied that generates a canonical definition for the type. It also generates definitions for functions that have been characterized in terms of patterns over the constructors (resp. copatterns over the observers). Functions that input a possibly-recursive sum/variant type are defined inductively via patterns on the input data. Dually, functions that output a possibly-recursive record type are defined coinductively via copatterns on the function's output. The transformations have been implemented in the Specware system and have been used extensively in the automated synthesis of concurrent garbage collection algorithms and families of protocol-processing codes for distributed vehicle control.
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