History
In his 1952 text Introduction to Metamathematics, Stephen Kleene formally proved that the CASE function (the IF-THEN-ELSE function being its simplest form) is a primitive recursive function, where he defines the notion definition by cases in the following manner:
- "#F. The function φ defined thus
- φ(x1, ..., xn ) =
-
- φ1(x1, ..., xn ) if Q1(x1, ..., xn ),
- . . . . . . . . . . . .
- φm(x1, ..., xn ) if Qm(x1, ..., xn ),
- φm+1(x1, ..., xn ) otherwise,
-
- φ(x1, ..., xn ) =
- where Q1, ..., Qm are mutually exclusive predicates (or φ(x1, ..., xn) shall have the value given by the first clause which applies) is primitive recursive in φ1, ..., φm+1, Q1, ..., Qm+1.
Kleene provides a proof of this in terms of the Boolean-like recursive functions "sign-of" sg( ) and "not sign of" ~sg( ) (Kleene 1952:222-223); the first returns 1 if its input is positive and −1 if its input is negative.
Boolos-Burgess-Jeffrey make the additional observation that "definition by cases" must be both mutually exclusive and collectively exhaustive. They too offer a proof of the primitive recursiveness of this function (Boolos-Burgess-Jeffrey 2002:74-75).
The IF-THEN-ELSE is the basis of the McCarthy formalism: its usage replaces both primitive recursion and the mu-operator.
Read more about this topic: Switch Statement
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