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Definition df-rdg 4233

Description: Define a recursive definition generator on

(the class of ordinal numbers) with characteristic function

and initial value

. This combines functions

in tfr1 4225 and

in tz7.44-1 4229 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 4286, from which we prove the recursive textbook definition as theorems oa0 4291, oasuc 4299, and oalim 4303 (with the help of theorems rdg0 4242, rdgsuc 4246, and rdglimi 4244). We can also restrict the

operation to define otherwise recursive functions on the natural numbers

; see fr0g 4253 and frsuc 4254. 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 2416) 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 6673 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 7135 and integer powers df-exp 6764.

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 4232	. 2
4		vf	. . . . . . . 8
5	4	cv 991	. . . . . . 7
6		vx	. . . . . . . 8
7	6	cv 991	. . . . . . 7
8	5, 7	wfn 3258	. . . . . 6
9		vy	. . . . . . . . . 10
10	9	cv 991	. . . . . . . . 9
11	10, 5	cfv 3263	. . . . . . . 8
12	5, 10	cres 3253	. . . . . . . . 9
13		vz	. . . . . . . . . . . 12
14	13	cv 991	. . . . . . . . . . 11
15		vg	. . . . . . . . . . . . . 14
16	15	cv 991	. . . . . . . . . . . . 13
17		c0 2332	. . . . . . . . . . . . 13
18	16, 17	wceq 992	. . . . . . . . . . . 12
19	16	cdm 3251	. . . . . . . . . . . . . 14
20	19	wlim 2976	. . . . . . . . . . . . 13
21	16	crn 3252	. . . . . . . . . . . . . 14
22	21	cuni 2569	. . . . . . . . . . . . 13
23	19	cuni 2569	. . . . . . . . . . . . . . 15
24	23, 16	cfv 3263	. . . . . . . . . . . . . 14
25	24, 1	cfv 3263	. . . . . . . . . . . . 13
26	20, 22, 25	cif 2415	. . . . . . . . . . . 12
27	18, 2, 26	cif 2415	. . . . . . . . . . 11
28	14, 27	wceq 992	. . . . . . . . . 10
29	28, 15, 13	copab 2740	. . . . . . . . 9
30	12, 29	cfv 3263	. . . . . . . 8
31	11, 30	wceq 992	. . . . . . 7
32	31, 9, 7	wral 1691	. . . . . 6
33	8, 32	wa 221	. . . . 5 $/\$
34		con0 2975	. . . . 5
35	33, 6, 34	wrex 1692	. . . 4 $/\$
36	35, 4	cab 1505	. . 3 $/\$
37	36	cuni 2569	. 2 $/\$
38	3, 37	wceq 992	1 $/\$

Colors of variables: wff set class

This definition is referenced by: dfrdg2 4234 rdgeq1 4235 rdgeq2 4236 hbrdg 4237 rdgfnon 4240