number-theory

3A Principal types and their history313A Principal types and their history3A1 Definition (Type-substitutions) A (type-) substitution s is any expression[at lal,... , Qnl an],where a1,..., an are distinct type-variables and 61, ... , an are any types. For any Tdefine s(T) to be the type obtained by simultaneously substituting Q1 for al, ... , anfor an throughout T. In more detail, define(i) s(a7) = a+,(ii) s(b) b if b is an atom {al,... an(iii) §(P-*a) =We call s(T) an instance of T.3A1.1 Notation Letters "r", "s", "t", "u", "d" will denote type-substitutions. Ifs - [at/al,... , Qn/an] a frequent alternative notation to s(T) will be[al /al, ... , Qnl an]T.Recall that the set of all variables occurring in a type T is called Vars(T). The sets ofall type-variables occurring in a finite sequence (T1,...,Tn) of types, or in a deductionA, are called respectivelyVars(TI...... ,),Vars(A).3A2 Definition The action of a substitution s is extended to finite sequences oftypes, to contexts and to TA,1-formulae thus:§((21,...92,,)) = (§(TI),.... §(Tn)),s(F) = {Xi:s(TI),...,Xm:§(2m)}s(F --> M:T) = s(F) --> M:s(T).tfF = {X1:T1,...,Xm:Tm},We also extend s to act on deductions A by defining s(A) to be the result ofapplying s to every TA2-formula in A. We call s(A) an instance of A. (Similarly forinstances of type-sequences, etc.)3A2.1 Notes (i) The consistency of F implies that of s(F).(ii) If A is a TA1-deduction then so is s(A), because the side-conditions in rules(-*E) and (-+I) in 2A8 remain true after s has been applied. HenceF E- M:T S(F) E- M:s(T),and the set of all types assigned to a term M is closed under substitution.3A2.2 Warning Two distinct concepts of substitution into deductions have nowbeen mentioned, for term-variables in 2B4 and for type-variables above. Note thatVars(A) is a set of type-variables not term-variables, and when s is applied to A theterms in A are completely unchanged.