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Definition df-rdg 6825
Description: Define a recursive definition generator on (the class of ordinal numbers) with characteristic function and initial value . This combines functions in tfr1 6815 and in tz7.44-1 6821 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 rec operation (especially when df-recs 6791 that it is built on is also eliminated). But once we get past this hurdle, definitions that would otherwise be recursive become relatively simple, as in for example oav 6912, from which we prove the recursive textbook definition as theorems oa0 6917, oasuc 6925, and oalim 6933 (with the help of theorems rdg0 6836, rdgsuc 6839, and rdglim2a 6848). We can also restrict the rec operation to define otherwise recursive functions on the natural numbers ; see fr0g 6850 and frsuc 6851. Our rec 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 if operations (see df-if 3769) 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 11748 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 11993 and integer powers df-exp 11807.

Note: We introduce rec 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.)

Assertion
Ref Expression
df-rdg
Distinct variable groups:   ,   ,I

Detailed syntax breakdown of Definition df-rdg
StepHypRef Expression
1 cF . . 3
2 cI . . 3
31, 2crdg 6824 . 2
4 vg . . . 4
5 cvv 2951 . . . 4
64cv 1686 . . . . . 6
7 c0 3614 . . . . . 6
86, 7wceq 1687 . . . . 5
96cdm 4811 . . . . . . 7
109wlim 4691 . . . . . 6
116crn 4812 . . . . . . 7
1211cuni 4066 . . . . . 6
139cuni 4066 . . . . . . . 8
1413, 6cfv 5390 . . . . . . 7
1514, 1cfv 5390 . . . . . 6
1610, 12, 15cif 3768 . . . . 5
178, 2, 16cif 3768 . . . 4
184, 5, 17cmpt 4325 . . 3
1918crecs 6790 . 2
203, 19wceq 1687 1
Colors of variables: wff setvar class
This definition is referenced by:  rdgeq1  6826  rdgeq2  6827  nfrdg  6829  rdgfun  6831  rdgdmlim  6832  rdgfnon  6833  rdgvalg  6834  rdgval  6835  rdgseg  6837  dfrdg2  27311
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