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Operationallybased theories of program equivalence
 Semantics and Logics of Computation
, 1997
"... ..."
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 ..."
Abstract

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.
MartinLöf's Type Theory As An OpenEnded Framework
, 2001
"... This paper treats MartinLöf's type theory as an openended framework composed of (i) flexibly extensible languages into which various forms of objects and types can be incorporated, (ii) their uniform, effectively given semantics, and (iii) persistently valid inference rules. The class of expr ..."
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This paper treats MartinLöf's type theory as an openended framework composed of (i) flexibly extensible languages into which various forms of objects and types can be incorporated, (ii) their uniform, effectively given semantics, and (iii) persistently valid inference rules. The class of expression systems is introduced here to define an openended body of languages underlying the theory. Each expression system consists of two parts: the computational part is a structured lazy evaluation system with a bisimulationlike program equivalence; the structural part is a system of strictly positive inductive definitions for type constructors in terms of partial equivalence relations. Types and their objects are uniformly and inductively constructed from a given expression system as a type system, which can provide a semantics of the theory. Building on these concepts, this paper presents two main results. First, all the inference rules of the theory are sound; that is, they remain valid in every type system built from an extension of an initial expression system. This result gives a characterization of the class of types that can be introduced into the theory. Second, each type system is complete with respect to the underlying bisimulationlike program equivalence. This result provides a useful form of typefree equational reasoning in the theory.
unknown title
"... the introduction rules for A may also refer to B. So we have formation rules A ∶ Set, B ∶ A → Set and typical introduction rules might take the form a ∶ A b ∶ B(a)... introA(a, b,...) ∶ A a0 ∶ A b ∶ B(a0) a1 ∶ A... ..."
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the introduction rules for A may also refer to B. So we have formation rules A ∶ Set, B ∶ A → Set and typical introduction rules might take the form a ∶ A b ∶ B(a)... introA(a, b,...) ∶ A a0 ∶ A b ∶ B(a0) a1 ∶ A...
unknown title
"... An intuitionistic theory of types with assumptions of higharity variables ..."
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An intuitionistic theory of types with assumptions of higharity variables
hangingUnder ∶ ((p ∶ Platform) × (b ∶ Building(p))) → Building(extension(⟨p, b⟩)).
"... the introduction rules for A may also refer to B. So we have formation rules A ∶ Set, B ∶ A → Set and typical introduction rules might take the form a ∶ A b ∶ B(a)... introA(a, b,...) ∶ A a0 ∶ A b ∶ B(a0) a1 ∶ A... ..."
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the introduction rules for A may also refer to B. So we have formation rules A ∶ Set, B ∶ A → Set and typical introduction rules might take the form a ∶ A b ∶ B(a)... introA(a, b,...) ∶ A a0 ∶ A b ∶ B(a0) a1 ∶ A...