number-theory

7B Identifications 957A3 Converse PT Theorem For Al (i) Every type of a closed AI-term is also a 21-PT.(ii) Further, there is an algorithm which accepts any typable closed Al-term M andany type r of M, and builds a closed AI-term M* whose PT is T.(iii) Furthermore, M* Pan M.7A3.1 Proof-note An algorithm and correctness proof will be given in 7C. Notethat (iii) is weaker than the corresponding result for AK in 7A2(iii): in AK we havet>p but in Al the proof will only give DRnThe first known algorithm for Al was devised in 1986 by Robert Meyer (Meyerand Bunder 1988 §9), and another was invented independently by Grigori Mintsand Tanel Tammet (Mints and Tammet 1991 §2). Both of these originated in thecontext of Hilbert-style axiom-based logics and although they can easily be translatedinto A-calculus they lose some of their directness when this is done. In 1990a direct and natural A-algorithm was described by Sachio Hirokawa (Hirokawa1992a §3).The most economical of these algorithms is the Meyer one, in the sense that itdoes not need all the Al-terms as building blocks but only a certain well-definedproper subset of them. So this algorithm is the one that will be described in7C.All three algorithms produce terms M* satisfying (iii) as well as (ii). Though (iii)was first proved for the Hirokawa algorithm (Hirokawa 1992a §3), a proof will begiven below for the Meyer algorithm and a careful reading of Mints and Tammet1991 shows that (iii) holds for that algorithm too.7A3.1 Question Can 7A3(iii) be strengthened to say "M* >0 M"?7A3.2 Exercise* If the answer to the above question is "yes" then as a consequencethere exist two /3-equal typable Al-terms with no types in common (cf. Corollary7A2.1 for AK). Show that this consequence holds even if the answer to theabove question is "no", by constructing two suitable Al-terms directly. (Hint (MartinBunder): consider the term P in 2A8.8.)7B IdentificationsThe construction of M* from M and T in the proof of 7A3 will depend on viewing Tas the result of applying certain substitutions to a particularly simple form of type.The present section introduces the notation needed for this.First recall the substitution notation introduced in 3B. In particular (from 3B7) avariables-for-variables substitution is one with forms = [blbal,...,bn/an]where bl,... , b are any variables; and a renaming in T (where T is a given type) isa variables-for-variables substitution such thatVars(T) and bl,...,bare distinct.