An Introduction to Functional Programming Through Lambda Calculus

- 77 -def ISBOOL B = MAKE_BOOL (isbool B)5.8. Numbers and arithmeticWe will represent the number type as two:def numb_type = twoand a number will be a pair starting with numb_type:def MAKE_NUMB = make_obj numb_typeMAKE_NUMBs definition expands to:λvalue.λs.(s numb_type value)We need an error object for arithmetic type errors:def NUMB_ERROR = MAKE_ERROR numb_typewhich expands to:λs.(s error_type numb_type)We also need a type tester:def isnumb = istype numb_typewhich expands to:λobj.(equal (type obj) numb_type)from which we can define a typed type tester:def ISNUMB N = MAKE_BOOL (isnumb N)Next we can construct a typed zero:def 0 = MAKE_NUMB zerowhich expands as:λs.(s numb_type zero)We now need a typed successor function:def SUCC N =if isnumb Nthen MAKE_NUMB (succ (value N))else NUMB_ERRORto define numbers:def 1 = SUCC 0def 2 = SUCC 1