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Definition df-rdg 5304

Description: Define a recursive definition generator on

(the class of ordinal numbers) with characteristic function

and initial value

. This combines functions

in tfr1 5296 and

in tz7.44-1 5300 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 5361, from which we prove the recursive textbook definition as theorems oa0 5366, oasuc 5374, and oalim 5378 (with the help of theorems rdg0 5313, rdgsuc 5317, and rdglim2a 5322). We can also restrict the

operation to define otherwise recursive functions on the natural numbers

; see fr0g 5324 and frsuc 5325. 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 3181) 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-seq1 8094 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 8569 and integer powers df-exp 8196.

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.

**Assertion**
Ref	Expression
df-rdg	$/\$

Distinct variable groups:

**Detailed syntax breakdown of Definition df-rdg**
Step	Hyp	Ref	Expression
1		cF	. . 3
2		cA	. . 3
3	1, 2	crdg 5303	. 2
4		vf	. . . . . . . 8
5	4	cv 1585	. . . . . . 7
6		vx	. . . . . . . 8
7	6	cv 1585	. . . . . . 7
8	5, 7	wfn 4126	. . . . . 6
9		vy	. . . . . . . . . 10
10	9	cv 1585	. . . . . . . . 9
11	10, 5	cfv 4131	. . . . . . . 8
12	5, 10	cres 4121	. . . . . . . . 9
13		vz	. . . . . . . . . . . 12
14	13	cv 1585	. . . . . . . . . . 11
15		vg	. . . . . . . . . . . . . 14
16	15	cv 1585	. . . . . . . . . . . . 13
17		c0 3082	. . . . . . . . . . . . 13
18	16, 17	wceq 1586	. . . . . . . . . . . 12
19	16	cdm 4119	. . . . . . . . . . . . . 14
20	19	wlim 3812	. . . . . . . . . . . . 13
21	16	crn 4120	. . . . . . . . . . . . . 14
22	21	cuni 3366	. . . . . . . . . . . . 13
23	19	cuni 3366	. . . . . . . . . . . . . . 15
24	23, 16	cfv 4131	. . . . . . . . . . . . . 14
25	24, 1	cfv 4131	. . . . . . . . . . . . 13
26	20, 22, 25	cif 3180	. . . . . . . . . . . 12
27	18, 2, 26	cif 3180	. . . . . . . . . . 11
28	14, 27	wceq 1586	. . . . . . . . . 10
29	28, 15, 13	copab 3565	. . . . . . . . 9
30	12, 29	cfv 4131	. . . . . . . 8
31	11, 30	wceq 1586	. . . . . . 7
32	31, 9, 7	wral 2355	. . . . . 6
33	8, 32	wa 337	. . . . 5 $/\$
34		con0 3811	. . . . 5
35	33, 6, 34	wrex 2356	. . . 4 $/\$
36	35, 4	cab 2128	. . 3 $/\$
37	36	cuni 3366	. 2 $/\$
38	3, 37	wceq 1586	1 $/\$

Colors of variables: wff set class

This definition is referenced by: dfrdg2 5305 rdgeq1 5306 rdgeq2 5307 hbrdg 5308 rdgfnon 5311