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By Kripke Saul

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CONCUR 90, Theories of Concurrency: Unification and Extension, LNCS vol. 458, Springer-Verlag, Berlin, 1990, pp. , Vogler, W: Duality criteria for partial order semantics of place/transition nets. In: B. ) MFCS 90, LNCS vol. 452, SpringerVerlag, Berlin, 1990, pp. : A characterization of context-free languages. J. Comput. System Sci. 5 (1971), pp. : Viewing control structures as patterns of passing messages. Artificial Intelligence 8 (1977), pp. 323-364 [Hoa78] Hoare, C. A. : Communicating sequential processes.

Assume not, then we show a contradiction. (O < x). 1. (O < x). To show this, take 0 for x and show 0 < O. (0 < 0). 1. (O < x) by choosing 1 for x. So we must show 0 < 1, but this could be regarded as an axiom. ) Here is the proof obtained from the above argument which satisfies clause 1 of the theorem. We look at a formal proof for this which follows the informal argument above very closely. From this formal proof it is easy to build the inhabiting proof term. 1) -+1. (as required by clause 1 of the theorem).

F(x)(g(x)) : (A Check: Given f: A -+ (B -+ -+ (B e),g : (A -+ -+ e)) -+ (A -+ B), x: A Type: f(x)(g(x)) g(x) : B by AP f,x by AP g,x f(x){g(x)) : e AP f(x},g(x} f(x):B-+e End Now compare this type checking to the following proof. Theorem 1 (A:::} (B:::} e)):::} (A:::} B):::} (A:::} e). B) -+ (A -+ e). 43 Proof: Assume 1. A ~ (B ~ C) 2. (A ~ B) 3. A ShowC 4. (B ~ C) by MP 1,3 5. B by MP 2,3 6. C by MP 4,5 Qed. This Curry-Howard correspondence is seen to involve not only propositions and types, but rules of inference and typing rules.

### A Completeness Theorem in Modal Logic (paper) by Kripke Saul

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