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Beating the Productivity Checker Using Embedded Languages
"... Abstract. Some total languages, like Agda and Coq, allow the use of guarded corecursion to construct infinite values and proofs. Guarded corecursion is a form of recursion in which arbitrary recursive calls are allowed, as long as they are guarded by a coinductive constructor. Guardedness ensures th ..."
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Abstract. Some total languages, like Agda and Coq, allow the use of guarded corecursion to construct infinite values and proofs. Guarded corecursion is a form of recursion in which arbitrary recursive calls are allowed, as long as they are guarded by a coinductive constructor. Guardedness ensures that programs are productive, i.e. that every finite prefix of an infinite value can be computed in finite time. However, many productive programs are not guarded, and it can be nontrivial to put them in guarded form. This paper gives a method for turning a productive program into a guarded program. The method amounts to defining a problemspecific language as a data type, writing the program in the problemspecific language, and writing a guarded interpreter for this language. 1
Subtyping, Declaratively An Exercise in Mixed Induction and Coinduction
"... Abstract. It is natural to present subtyping for recursive types coinductively. However, Gapeyev, Levin and Pierce have noted that there is a problem with coinductive definitions of nontrivial transitive inference systems: they cannot be “declarative”—as opposed to “algorithmic ” or syntaxdirected ..."
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Abstract. It is natural to present subtyping for recursive types coinductively. However, Gapeyev, Levin and Pierce have noted that there is a problem with coinductive definitions of nontrivial transitive inference systems: they cannot be “declarative”—as opposed to “algorithmic ” or syntaxdirected—because coinductive inference systems with an explicit rule of transitivity are trivial. We propose a solution to this problem. By using mixed induction and coinduction we define an inference system for subtyping which combines the advantages of coinduction with the convenience of an explicit rule of transitivity. The definition uses coinduction for the structural rules, and induction for the rule of transitivity. We also discuss under what conditions this technique can be used when defining other inference systems. The developments presented in the paper have been mechanised using Agda, a dependently typed programming language and proof assistant. 1