Mathematics for Computer Science

“mcs” — 2017/3/3 — 11:21 — page 226 — #234
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Chapter 7
Recursive Data Types
3. the composition f ı g.
Prove by structural induction on this definition that if f .x/ is an Atrig, then so is
f 0 WWD df =dx.
Problem 7.13.
Definition. The set RAF of rational functions of one real variable is the set of
functions defined recursively as follows:
Base cases:
The identity function, id.r/ WWD r for r 2 R (the real numbers), is an RAF,
any constant function on R is an RAF.
Constructor cases: If f; g are RAF’s, then so is f ~ g, where ~ is one of the
operations
1. addition C,
2. multiplication or
3. division =.
(a) Describe how to construct functions e; f; g 2 RAF such that
e ı .f C g/ ¤ .e ı f / C .e ı g/: (7.32)
(b) Prove that for all real-valued functions e; f; g (not just those in RAF):
Hint: .e ~ f /.x/ WWD e.x/ ~ f .x/.
.e ~ f / ı g D .e ı g/ ~ .f ı g/; (7.33)
(c) Let predicate P.h/ be the following predicate on functions h 2 RAF:
P.h/ WWD 8g 2 RAF: h ı g 2 RAF:
Prove by structural induction on the definition of RAF that P.h/ holds for all h 2
RAF.
Make sure to indicate explicitly