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A construction of Type:Type in MartinLöf's partial type theory with one universe
"... ing on w and pairing with oe(p(c); (x)Ap(q(c); x) ! p(c)) in the first coordinate yields hoe(p(c);(x)Ap(q(c); x) ! p(c)); (w)(Ap(q(c); p(w)); (x)Ap(q(c); Ap(q(w); x)))i 2 PAR; i.e. s (c) 2 PAR. We define the operator that builds the universe (U 1 ; T 1 ) by putting f(c) := s (c) +hn 1 ; (x ..."
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ing on w and pairing with oe(p(c); (x)Ap(q(c); x) ! p(c)) in the first coordinate yields hoe(p(c);(x)Ap(q(c); x) ! p(c)); (w)(Ap(q(c); p(w)); (x)Ap(q(c); Ap(q(w); x)))i 2 PAR; i.e. s (c) 2 PAR. We define the operator that builds the universe (U 1 ; T 1 ) by putting f(c) := s (c) +hn 1 ; (x)R 1 (x; p(c))i; for c 2 PAR, and let e := fix((c)f(c)). Hence e 2 PAR is a fixed point of f , e = f(e). The right summand of f corresponds to the rules (2). We now interpret Type:Type. The universe (U 1 ; T 1 ) is defined by letting U 1 := T (p(e)) and T 1 (a) := T (Ap(q(e); a)); for a 2 U 1 . Thus the rules (1) are verified. Using the equality e = f(e) and the commutation of T with \Sigma, \Pi and + we get U 1 = T (p(e)) = T (p(f(e))) (4) = T (oe(p(e); (x)Ap(q(e); x) ! p(e))) + T (n 1 ) = (\Sigmax 2 T (p(e)))[T (Ap(q(e); x)) \Gamma! T (p(e))] +N 1 = (\Sigmax 2 U 1 )[T 1 (x) \Gamma! U 1 ] +N 1 and hence j(0 1 ) 2 U 1 . Furthermore we have T 1 (j(0 1 )) = T (Ap(q(...
Developing Theories of Types and Computability
, 1999
"... Introduction Domain Theory, type theory (both in the style of MartinLof [40, 41] and in the polymorphic style of Girard/Reynolds [23, 56]), and topos theory (both in the topological/sheaftheoretic treatments and in the realizability approach going back to the early work of Kleene) have attempted ..."
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Introduction Domain Theory, type theory (both in the style of MartinLof [40, 41] and in the polymorphic style of Girard/Reynolds [23, 56]), and topos theory (both in the topological/sheaftheoretic treatments and in the realizability approach going back to the early work of Kleene) have attempted to improve on set theory by providing a large suite of closure conditions on domains/types/objects as well as a farreaching logic of properties emphasizing the computable/constructive aspects of the definitions and qualities of functions. Scott's domain theory, (and the many variations proposed and studied; see [2] and [75] for recent introductions with references) has been especially successful in allowing for recursive definitions of types (i.e., solutions to domain equations) but at the expense of introducing a complex structure of "partial elements" in order to have solutions to fixedpoint equations in the domains. Moreover, the topological and e