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Definition df-rdg 4231

Description: Define a recursive definition generator on

(the class of ordinal numbers) with characteristic function

and initial value

. This combines functions

in tfr1 4223 and

in tz7.44-1 4227 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 4284, from which we prove the recursive textbook definition as theorems oa0 4289, oasuc 4297, and oalim 4301 (with the help of theorems rdg0 4240, rdgsuc 4244, and rdglimi 4242). We can also restrict the

operation to define otherwise recursive functions on the natural numbers

; see fr0g 4251 and frsuc 4252. 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 2415) 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 6671 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 7133 and integer powers df-exp 6762.

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 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 4230	. 2
4		vf	. . . . . . . 8
5	4	cv 990	. . . . . . 7
6		vx	. . . . . . . 8
7	6	cv 990	. . . . . . 7
8	5, 7	wfn 3257	. . . . . 6
9		vy	. . . . . . . . . 10
10	9	cv 990	. . . . . . . . 9
11	10, 5	cfv 3262	. . . . . . . 8
12	5, 10	cres 3252	. . . . . . . . 9
13		vz	. . . . . . . . . . . 12
14	13	cv 990	. . . . . . . . . . 11
15		vg	. . . . . . . . . . . . . 14
16	15	cv 990	. . . . . . . . . . . . 13
17		c0 2331	. . . . . . . . . . . . 13
18	16, 17	wceq 991	. . . . . . . . . . . 12
19	16	cdm 3250	. . . . . . . . . . . . . 14
20	19	wlim 2975	. . . . . . . . . . . . 13
21	16	crn 3251	. . . . . . . . . . . . . 14
22	21	cuni 2568	. . . . . . . . . . . . 13
23	19	cuni 2568	. . . . . . . . . . . . . . 15
24	23, 16	cfv 3262	. . . . . . . . . . . . . 14
25	24, 1	cfv 3262	. . . . . . . . . . . . 13
26	20, 22, 25	cif 2414	. . . . . . . . . . . 12
27	18, 2, 26	cif 2414	. . . . . . . . . . 11
28	14, 27	wceq 991	. . . . . . . . . 10
29	28, 15, 13	copab 2739	. . . . . . . . 9
30	12, 29	cfv 3262	. . . . . . . 8
31	11, 30	wceq 991	. . . . . . 7
32	31, 9, 7	wral 1690	. . . . . 6
33	8, 32	wa 221	. . . . 5 $/\$
34		con0 2974	. . . . 5
35	33, 6, 34	wrex 1691	. . . 4 $/\$
36	35, 4	cab 1504	. . 3 $/\$
37	36	cuni 2568	. 2 $/\$
38	3, 37	wceq 991	1 $/\$

Colors of variables: wff set class

This definition is referenced by: dfrdg2 4232 rdgeq1 4233 rdgeq2 4234 hbrdg 4235 rdgfnon 4238