H. Lauchli. An abstract notion of realizability for which intuitionistic predicate calculus is complete. In A. Kino, J. Myhill, and R. E. Vesley, editors, Intuitionism and proof theory, pages 227--234. North Holland, 1970.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....limits in general. In any case, we can conclude, using Proposition 11, that the exact completion of the category of G sets is a presheaf topos. Indeed, Sets G ) ex is equivalent to Sets Q op . Due to its connection with Lauchli s abstract notion of realizability and completeness result [14], it may be of interest to pay special attention to the exact completion of the topos of Z sets. In [19] see also [15] the hyperdoctrine that assigns to each object X of Sets Z the small Heyting algebra Prf (X) Sets Z X) is used to give an abstract account of Lauchli s completeness ....

H. Lauchli. An abstract notion of realizability for which intuitionistic predicate calculus is complete. In A. Kino, J. Myhill, and R. E. Vesley, editors, Intuitionism and proof theory, pages 227--234. North Holland, 1970.

....the type theory of the original category in the internal logic of the topos. Chapter 6 Boolean presheaf toposes In this very short chapter we characterize the presheaf toposes that have a generic proof. We also explain briefly the connection of these toposes with Lauchli s realizability [53, 62]. Moreover, the characterization above will also let us find other examples of Grothendieck toposes whose exact completions are themselves toposes. 6.1 Boolean presheaf toposes In [16] the observation that for the topos Set # # of irreflexive directed graphs, Prf (1) is not small is attributed ....

....that this fact is harder than the case of Set # # because the classes of proofs of Set # are small) This generalization is the main proof presented above. 6. 2 A word about Lauchli s realizability Due to its connection with Lauchli s abstract notion of realizability and completeness result [53], it may be of interest to pay special attention to the exact completion of the topos of Z sets. In [73] see also [62] the hyperdoctrine that assigns to each object X of Set Z the small Heyting algebra Prf (X) Set Z X) is used to give an abstract account of Lauchli s completeness ....

H. Lauchli. An abstract notion of realizability for which intuitionistic predicate calculus is complete. In A. Kino, J. Myhill, and R. E. Vesley, editors, Intuitionism and proof theory, pages 227--234. North Holland, 1970.

....space, under pointwise operations. We call it the space of proofs associated to the sequent F F 0 . Note that we identify formulas with definable functors. Before obtaining a full completeness theorem, we first obtained a traditional completeness theorem, which is analogous to the results of [25, 20]. Theorem 2.9 (Completeness) Let M N be a balanced binary sequent. If the unique cutfree proof structure associated to M N is not a proof net for the theory MLL MIX, then Dinat(M; N) is a zero dimensional vector space. The key lemma in extending this result to a full completeness theorem ....

H. Lauchli, An Abstract Notion of Realizability for which Intuitionistic Predicate Calculus is Complete, Intuitionism and Proof Theory, North-Holland (1970), pp. 227-234.

....functionality and the theory of implication in Curry s book with Feys [11, sec. 9E] from 1958. Eleven years later, the remark was developed in a widely circulated manuscript by Howard (which awaited actual publication [17] for another eleven years) In the meantime, Lauchli s completeness theorem [29] and automath, the very first logical framework [5, 6, x14] had already put the propositions as types at work. The two facets of this paradigm were recognized in the very successfull type systems due to Martin Lof on one side [36] and to Girard [13] and Reynolds [44] on the other. Synthesis of ....

H. Lauchli, An abstract notion of realizability for which intuitionistic predicate calculus is complete, in: Intuitionism and Proof Theory, A. Kino et al., eds. (North-Holland, 1970) 227--234

....1996 c fl IGPL 160 Maps II: Chasing Diagrams in Categorical Proof Theory sec. 9E] from 1958. Eleven years later, the remark was developed in a widely circulated manuscript by Howard (which awaited actual publication [17] for another eleven years) In the meantime, Lauchli s completeness theorem [29] and automath, the very first logical framework [5, 6, x14] had already put the propositions as types at work. The two facets of this paradigm were recognized in the very successful type systems due to Martin Lof on one side [36] and to Girard [13] and Reynolds [44] on the other. Synthesis of ....

H. Lauchli, An abstract notion of realizability for which intuitionistic predicate calculus is complete, in: Intuitionism and Proof Theory, A. Kino et al., eds. (North-Holland, 1970) 227--234

....For a long time the results on automath were not very accessible, since a lot of it could be found in internal reports and Ph.D. theses only; but recently all the interesting material on automath was brought together in one volume ( NGdV94] H. Lauchli Independently also, H. Lauchli ([Lau70]) used the ideas of formulas astypes for obtaining a completeness proof for intuitionistic predicate logic, relative to a notion which might be regarded as a version of realizability. Martin Lof s type theories Formulas as types in the form FAT(B) and FAT(C) was the guiding idea behind the ....

H. Lauchli. An abstract notion of realizability for which intuitionistic predicate calculus is complete. In A. Kino, J. R. Myhill, and R. E. Vesley, editors, Intuitionism and Proof Theory, pages 227--234. NorthHolland Publ. Co., Amsterdam, 1970.

.... rigorous mathematical framework for Heyting s ideas led the way to many fundamental discoveries, for example Kleene s Realizability, Godel s Dialectica Interpretation , and (more recently) the Curry Howard Isomorphism ( 21] However somewhat less familiar is Lauchli s seminal work in the 1960 s [30]: this was the first attempt to give both an abstract model of proof for intuitionistic logic and a Completeness Theorem for provability. Lauchli s viewpoint models a formula by a set: intuitively, by its set of (abstract) proofs. Ordinary sets, however, have insufficient structure to obtain a ....

....functor Gamma F : B(G) Set G hence the meaning map Gamma interprets f ; g proofs via hereditary permutations: a closed term M : 1 oe corresponds to an invariant lambda term in Sets G (now for the extended lambda calculus with binary coproduct types. This is the viewpoint of Lauchli [30]. The Lauchli Completeness Theorem is a converse to Soundness, for the case G = Z: Theorem 4.4 (Lauchli [30] A formula oe of intuitionistic propositional calculus is provable if and only if for every interpretation F of the base types, its Set Z interpretation oe F has an invariant element. ....

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H. Lauchli, An Abstract Notion of Realizability for which Intuitionistic Predicate Calculus is Complete, Intuitionism and Proof Theory, North-Holland,(1970), pp. 227-234.

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