number-theory

96 7 The converse principal-type algorithm7B1 Definition For any type T, an identification in T (sometimes called a contractionin T) is any substitution [b/a] such that a, b both occur in T. A type a is obtainedfrom ,r by identifications ifv = [b./a/,](... ([bt/ai](T))...)where [bl/al] is an identification in T, [b2/a2] is an identification in [bl/at]T, etc.7B1.1 Example If T - (a-+b-+c)-+(a-+b)-+a-+c then [b/a] is an identification in T,and[b/a]T - (b-4b--+c)--+(b-+b)--+b--+c.7B1.2 Lemma Every variables-for-variables substitution can be performed by a renamingfollowed by a series of identifications. More precisely, if p - §(T) and s substitutesvariables. for variables, thenp = [bn/an](...([bl/al](v(T)))...)where n > 0, v is a renaming in T, and each [b;/a;] is an identification in[b1_i /a,_i](... (d(T)) ...).7B2 Definition A type T is skeletal if each variable in T occurs exactly once. (Theproperty of being skeletal is sometimes called the 1 -property.)7B3 Lemma Every type r can be obtained by identifications from a skeletal type T°(which will be called a skeleton of T).Proof If T is skeletal choose T° - T. If not, then for each variable b with two or moreoccurrences in r, replace all but one of these occurrences by distinct new variablesE7C The converse PT proofThis section contains a proof of Theorem 7A3 obtained by translating the proofin Meyer and Bunder 1988 §9 into A-calculus. The strategy will be to show firstthat every type of a closed 2I-term can be obtained from a AI-PT by a series ofidentifications, and then that identifications preserve the property of being a 2I-PT.But a preliminary step will be to prove the special case of the theorem in which Thas form 9-.0 where 0 is skeletal.7C1 Lemma Let 0 be any skeletal type. Thensuch thatis the PT of a closed RI-term IBProof We shall construct Io by induction on 101. If 0 is an atom, say 0 - a, choose(1) la - I.