Proof Principles for Datatypes with Iterated Recursion
نویسندگان
چکیده
Data types like trees which are nitely branching and of (possibly) innnite depth are described by iterating initial algebras and terminal coalgebras. We study proof principles for such data types in the context of categorical logic, following and extending the approach of 14, 15]. The technical contribution of this paper involves a description of initial algebras and terminal coalgebras in total categories of brations for lifted \datafunctors". These lifted functors are used to formulate our proof principles. We test these principles by proving some elementary results for four kinds of trees (with nite or innnite breadth or depth) using the proof tool pvs.
منابع مشابه
Foundational (Co)datatypes and (Co)recursion for Higher-Order Logic
We describe a line of work that started in 2011 towards enriching Isabelle/HOL’s language with coinductive datatypes, which allow infinite values, and with a more expressive notion of inductive datatype than previously supported by any system based on higher-order logic. These (co)datatypes are complemented by definitional principles for (co)recursive functions and reasoning principles for (co)...
متن کاملA Logic for Parametric Polymorphism
In this paper we introduce a logic for parametric polymorphism. Just as LCF is a logic for the simply-typed -calculus with recursion and arithmetic, our logic is a logic for System F. The logic permits the formal presentation and use of relational parametricity. Parametricity yields|for example|encodings of initial algebras, nal co-algebras and abstract datatypes, with corresponding proof princ...
متن کاملA Recursion Combinator for Nominal Datatypes Implemented in Isabelle/HOL
The nominal datatype package implements an infrastructure in Isabelle/HOL for defining languages involving binders and for reasoning conveniently about alpha-equivalence classes. Pitts stated some general conditions under which functions over alpha-equivalence classes can be defined by a form of structural recursion and gave a clever proof for the existence of a primitive-recursion combinator. ...
متن کاملInduction Rules, Reeection Principles, and Provably Recursive Functions
A well-known result of D. Leivant states that, over basic Kalmar elementary arithmetic EA, the induction schema for n formulas is equivalent to the uniform reeection principle for n+1 formulas. We show that fragments of arithmetic axiomatized by various forms of induction rules admit a precise axiomatization in terms of reeection principles as well. Thus, the closure of EA under the induction r...
متن کاملReasoning about modular datatypes with Mendler induction
In functional programming, datatypes à la carte provide a convenient modular representation of recursive datatypes, based on their initial algebra semantics. Unfortunately it is highly challenging to implement this technique in proof assistants that are based on type theory, like Coq. The reason is that it involves type definitions, such as those of type-level fixpoint operators, that are not s...
متن کامل