Background info for Map and Reduce

2 Notation. Function composition is written as f · g , and application of f to a is written f a or f . a. Application written as a space has the highest priority and application written as a “low dot” has the lowest priority (as suggested by the wide space surrounding the dot). So, f ·g . x +y = f (g (x +y)). This convention saves parentheses, thus improving readability. In order to facilitate reasoning in the form of algebraic manipulation (that is, repeatedly replacing a part of an expression by a different but semantically equal part) we generally prefer to work on the function level (expressing a function as combination of other functions) instead of the point level (where the outcome of a function application is expressed as a combination of the outcomes of other function applications). Thus we prefer to say, for example, f = f1 · (+) · (f2 ∆ f3) · f4 over f (x ) = f1(f2(f4(x ))+ f3(f4(x ))), so that the replacement of (f2 ∆ f3) by (f3 ∆ f2) or by f ′ is easier to perform. The list of items a, b, c, . . ., in that order, is denoted [a, b, c, . . .]. Operation + is list concatenation (also called join), so that [a, b, c] + [d , e] = [a, b, c, d , e]. Function tip is the singleton list former: tip x = [x ].