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Definition df-rdg 7095
Description: Define a recursive definition generator on (the class of ordinal numbers) with characteristic function and initial value . This combines functions in tfr1 7085 and in tz7.44-1 7091 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 7061 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 7180, from which we prove the recursive textbook definition as theorems oa0 7185, oasuc 7193, and oalim 7201 (with the help of theorems rdg0 7106, rdgsuc 7109, and rdglim2a 7118). We can also restrict the rec operation to define otherwise recursive functions on the natural numbers ; see fr0g 7120 and frsuc 7121. 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 3942) 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 12108 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 12354 and integer powers df-exp 12167.

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 7094 . 2
4 vg . . . 4
5 cvv 3109 . . . 4
64cv 1394 . . . . . 6
7 c0 3784 . . . . . 6
86, 7wceq 1395 . . . . 5
96cdm 5004 . . . . . . 7
109wlim 4884 . . . . . 6
116crn 5005 . . . . . . 7
1211cuni 4249 . . . . . 6
139cuni 4249 . . . . . . . 8
1413, 6cfv 5593 . . . . . . 7
1514, 1cfv 5593 . . . . . 6
1610, 12, 15cif 3941 . . . . 5
178, 2, 16cif 3941 . . . 4
184, 5, 17cmpt 4510 . . 3
1918crecs 7060 . 2
203, 19wceq 1395 1
Colors of variables: wff setvar class
This definition is referenced by:  rdgeq1  7096  rdgeq2  7097  nfrdg  7099  rdgfun  7101  rdgdmlim  7102  rdgfnon  7103  rdgvalg  7104  rdgval  7105  rdgseg  7107  dfrdg2  29228
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