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The calculus of constructions. (English) Zbl 0654.03045
This paper presents a calculus which is an impredicative extension of Martin-Löf type theory. It contains as a subsystem the functional system $$F\omega$$ developed by J. Y. Girard (1972) in order to extend Gödel’s Dialectica interpretation to higher-order arithmetic. It has been shown since then [in Ch. Paulin’s thesis (Paris VIII, 1989)] how to define a modified realizability interpretation from the present calculus in $$F\omega$$, and hence that the calculus of construction is conservative over $$F\omega$$.
The notation used is that of Automath and the system is suitable for implementation, and can be seen as a very general type system for functional programming language and/or mechanized mathematics. A realizability interpretation in untyped $$\lambda$$-calculus is described. This has been generalized to an extensional model of the calculus using the notion of $$\omega$$-sets of E. Moggi, in Th. Ehrhard’s thesis (Paris VII, 1988) and in Th. Streicher’s thesis (Passau, 1988).
Reviewer: T.Coquand

##### MSC:
 03F35 Second- and higher-order arithmetic and fragments 03B40 Combinatory logic and lambda calculus 68N01 General topics in the theory of software
Automath
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##### References:
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