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Combining de Bruijn indices and higherorder abstract syntax in Coq
 Proceedings of TYPES 2006, volume 4502 of Lecture Notes in Computer Science
, 2006
"... Abstract. The use of higherorder abstract syntax is an important approach for the representation of binding constructs in encodings of languages and logics in a logical framework. Formal metareasoning about such object languages is a particular challenge. We present a mechanism for such reasoning, ..."
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Cited by 12 (2 self)
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Abstract. The use of higherorder abstract syntax is an important approach for the representation of binding constructs in encodings of languages and logics in a logical framework. Formal metareasoning about such object languages is a particular challenge. We present a mechanism for such reasoning, formalized in Coq, inspired by the Hybrid tool in Isabelle. At the base level, we define a de Bruijn representation of terms with basic operations and a reasoning framework. At a higher level, we can represent languages and reason about them using higherorder syntax. We take advantage of Coq’s constructive logic by formulating many definitions as Coq programs. We illustrate the method on two examples: the untyped lambda calculus and quantified propositional logic. For each language, we can define recursion and induction principles that work directly on the higherorder syntax. 1
Biform theories in Chiron
 Towards Mechanized Mathematical Assistants, volume 4573 of Lecture Notes in Computer Science
, 2007
"... Abstract. An axiomatic theory represents mathematical knowledge declaratively as a set of axioms. An algorithmic theory represents mathematical knowledge procedurally as a set of algorithms. A biform theory is simultaneously an axiomatic theory and an algorithmic theory. It represents mathematical k ..."
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Cited by 8 (5 self)
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Abstract. An axiomatic theory represents mathematical knowledge declaratively as a set of axioms. An algorithmic theory represents mathematical knowledge procedurally as a set of algorithms. A biform theory is simultaneously an axiomatic theory and an algorithmic theory. It represents mathematical knowledge both declaratively and procedurally. Since the algorithms of algorithmic theories manipulate the syntax of expressions, biform theories—as well as algorithmic theories—are difficult to formalize in a traditional logic without the means to reason about syntax. Chiron is a derivative of vonNeumannBernaysGödel (nbg) set theory that is intended to be a practical, generalpurpose logic for mechanizing mathematics. It includes elements of type theory, a scheme for handling undefinedness, and a facility for reasoning about the syntax of expressions. It is an exceptionally wellsuited logic for formalizing biform theories. This paper defines the notion of a biform theory, gives an overview of Chiron, and illustrates how biform theories can be formalized in Chiron. 1
Formalizing type operations using the “Image” type constructor
 Workshop on Logic, Language, Information and Computation (WoLLIC
, 2006
"... In this paper we introduce a new approach to formalizing certain type operations in type theory. Traditionally, many type constructors in type theory are independently axiomatized and the correctness of these axioms is argued semantically. In this paper we introduce a notion of an “image ” of a give ..."
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Cited by 3 (1 self)
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In this paper we introduce a new approach to formalizing certain type operations in type theory. Traditionally, many type constructors in type theory are independently axiomatized and the correctness of these axioms is argued semantically. In this paper we introduce a notion of an “image ” of a given type under a mapping that captures the spirit of many of such semantical arguments. This allows us to use the new “image ” type to formalize within the type theory a large range of type constructors that were traditionally formalized via postulated axioms. We demonstrate the ability of the “image ” constructor to express “forgetful ” types by using it to formalize the “squash ” and “set ” type constructors. We also demonstrate its ability to handle types with nontrivial equality relations by using it to formalize the union type operator. We demonstrate the ability of the “image ” constructor to express certain inductive types by showing how the type of lists and a higherorder abstract syntax type can be naturally formalized using the new type constructor. The work presented in this paper have been implemented in the MetaPRL proof assistant and all the derivations checked by MetaPRL.
Higher Order Abstract Syntax in Type Theory
"... We develop a general tool to formalize higherorder languages and reason about them in a prooftool based on type theory (Coq). A language is specified by its signature, which consists of sets of sort and operation names and typing rules. These rules prescribe the sorts and bindings of each operat ..."
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Cited by 2 (1 self)
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We develop a general tool to formalize higherorder languages and reason about them in a prooftool based on type theory (Coq). A language is specified by its signature, which consists of sets of sort and operation names and typing rules. These rules prescribe the sorts and bindings of each operation. An algebra of terms is associated to a signature, using de Bruijn notation. Then a higherorder notation is built on top of the de Bruijn level, so that the user can work with metavariables instead of de Bruijn indices. We also provide recursion and induction principles formulated directly on the higherorder syntax. This generalizes work on the Hybrid approach to higherorder syntax in Isabelle and our earlier work on a constructive extension to Hybrid formalized in Coq. In particular, a large class of theorems that must be repeated for each object language in Hybrid is done once in our new approach and can be applied directly to each object language.
A Logic of Direct Evidence
"... Abstract. In the prooftheoretic study of logic, the notion of normal proof has been understood and investigated as a metalogical property. Usually we formulate a system of logic, identify a class of proofs as normal proofs, and show that every proof in the system reduces to a corresponding normal p ..."
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Abstract. In the prooftheoretic study of logic, the notion of normal proof has been understood and investigated as a metalogical property. Usually we formulate a system of logic, identify a class of proofs as normal proofs, and show that every proof in the system reduces to a corresponding normal proof. This paper develops a system of modal logic that is capable of expressing the notion of normal proof within the system itself, thereby making normal proofs an inherent property of the logic. Using a modality △ to express the existence of a normal proof, the system provides a means for both recognizing and manipulating its own normal proofs. We develop the system as a sequent calculus and prove the cut elimination theorem. From the sequent calculus, we derive a natural deduction system. 1