Description: Define a recursive
definition generator on (the class of ordinal
numbers) with characteristic function and initial value .
This combines functions in tfr1 6299 and in tz7.44-1 6305 into one
definition. This rather amazing operation allows us to define, with
compact direct definitions, functions that are usually defined in
textbooks only with indirect self-referencing recursive definitions. A
recursive definition requires advanced metalogic to justify - in
particular, eliminating a recursive definition is very difficult and
often not even shown in textbooks. On the other hand, the elimination
of a direct definition is a matter of simple mechanical substitution.
The price paid is the daunting complexity of our operation. But
once we get past this hurdle, otherwise recursive definitions become
relatively simple, as in for example oav 6396,
from which we prove the
recursive textbook definition as theorems oa0 6401,
oasuc 6409, and oalim 6417
(with the help of theorems rdg0 6320, rdgsuc 6323, and rdglim2a 6332). We can
also restrict the operation to define otherwise recursive
functions on the natural numbers ; see fr0g 6334 and frsuc 6335. Our
operation
apparently does not appear in published literature,
although closely related is Definition 25.2 of [Quine] p. 177, which he
uses to "turn...a recursion into a genuine or direct
definition" (p.
174). Note that the operations (see df-if 3471) select cases based
on whether the domain of is zero, a successor, or a limit ordinal.
An important use of this definition is in the recursive sequence
generator df-seq 10925 on the natural numbers (as a subset of the
complex
numbers), allowing us to define, with direct definitions, recursive
infinite sequences such as the factorial function df-fac 11167 and integer
powers df-exp 10983.
Note: We introduce with the philosophical goal of being able
to eliminate all definitions with direct mechanical substitution
and
to verify easily the soundness of definitions. Metamath itself has
no
built-in technical limitation that prevents multiple-part
recursive
definitions in the traditional textbook style. (Contributed by
NM,
9-Apr-1995.) (Revised by Mario Carneiro,
9-May-2015.) |