A coinductive monad for prop-bounded recursion

Abstract

This paper develops machinery necessary to mechanically import arbitrary functional programs into Coq's type theory, manually strengthen their specifications with additional proofs, and then mechanicaly re-extracting the newly-certified program in a form which is as efficient as the original program.In order to facilitate this goal, the coinductive technique of [Cap05] is modified to form a monad whose operators are the constructors of a coinductive type rather than functions defined over the type. The inductive invariant technique of [KM03] is extended to allow optional "after the fact" termination proofs. These proofs inhabit members of Prop, and therefore do not affect extracted code.Compared to [Cap05], the new monad makes it possible to directly represent unrestricted recursion without violating productivity requirements [Gim95], and it produces efficient code via Coq's extraction mechanism. The disadvantages of this technique include reliance on the JMeq axiom [McB00] and a significantly more complex notion of equality.The resulting technique is packaged as a Coq library, and is suitable for formalizing programs written in any side-effect-free functional language with call-by-value semantics.

Beating the Productivity Checker Using Embedded Languages

“…Unfortunately it can be inconvenient to use this definition of bind in systems based on guarded corecursion, because > > = is not a constructor. Megacz (2007) suggests (more or less) the following alternative definition:…”

Algebra of programming in Agda: Dependent types for relational program derivation

“…See Cousot (1990) for a tutorial. More recently, Megacz (2007) gave a nice survey of several alternative approaches to code terminating programs in a dependently typed language with inductive and coinductive types.…”

“…In computing science, Floyd (1967) proposed the use of well-ordering in programming language semantics (See Cousot 1990 for a tutorial). More recently, Megacz (2007) gave a nice survey of several alternative approaches to code terminating programs in a dependently typed language with inductive and coinductive types.…”

Fixed point semantics and partial recursion in Coq

“…There are analogous techniques that also do not require a termination proof. One such method is Prop-bounded recursion based on a coinductive monad [20,9]. In this method the coinductive type permits to represent infinitely long computations as (potentially) infinitely large objects of coinductive types.…”

“…One is also able to write a function that directly refers to itself, which would be incompatible with, e.g., Fixpoint due to syntactic restrictions. However, the coinductive methods require a more complex notion of equality due to the presence of infinite objects, such as, in the case of [20], the John Major equality which relies on the JMeq axiom of Coq.…”