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Definition df-rdg 5140

Description: Define a recursive definition generator on

(the class of ordinal numbers) with characteristic function

and initial value

. This combines functions

in tfr1 5132 and

in tz7.44-1 5136 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 5195, from which we prove the recursive textbook definition as theorems oa0 5200, oasuc 5208, and oalim 5212 (with the help of theorems rdg0 5149, rdgsuc 5153, and rdglimi 5151). We can also restrict the

operation to define otherwise recursive functions on the natural numbers

; see fr0g 5160 and frsuc 5161. 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 2983) 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 7721 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 8184 and integer powers df-exp 7812.

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 5139	. 2
4		vf	. . . . . . . 8
5	4	cv 1297	. . . . . . 7
6		vx	. . . . . . . 8
7	6	cv 1297	. . . . . . 7
8	5, 7	wfn 3993	. . . . . 6
9		vy	. . . . . . . . . 10
10	9	cv 1297	. . . . . . . . 9
11	10, 5	cfv 3998	. . . . . . . 8
12	5, 10	cres 3988	. . . . . . . . 9
13		vz	. . . . . . . . . . . 12
14	13	cv 1297	. . . . . . . . . . 11
15		vg	. . . . . . . . . . . . . 14
16	15	cv 1297	. . . . . . . . . . . . 13
17		c0 2875	. . . . . . . . . . . . 13
18	16, 17	wceq 1298	. . . . . . . . . . . 12
19	16	cdm 3986	. . . . . . . . . . . . . 14
20	19	wlim 3658	. . . . . . . . . . . . 13
21	16	crn 3987	. . . . . . . . . . . . . 14
22	21	cuni 3177	. . . . . . . . . . . . 13
23	19	cuni 3177	. . . . . . . . . . . . . . 15
24	23, 16	cfv 3998	. . . . . . . . . . . . . 14
25	24, 1	cfv 3998	. . . . . . . . . . . . 13
26	20, 22, 25	cif 2982	. . . . . . . . . . . 12
27	18, 2, 26	cif 2982	. . . . . . . . . . 11
28	14, 27	wceq 1298	. . . . . . . . . 10
29	28, 15, 13	copab 3395	. . . . . . . . 9
30	12, 29	cfv 3998	. . . . . . . 8
31	11, 30	wceq 1298	. . . . . . 7
32	31, 9, 7	wral 2105	. . . . . 6
33	8, 32	wa 240	. . . . 5 $/\$
34		con0 3657	. . . . 5
35	33, 6, 34	wrex 2106	. . . 4 $/\$
36	35, 4	cab 1871	. . 3 $/\$
37	36	cuni 3177	. 2 $/\$
38	3, 37	wceq 1298	1 $/\$

Colors of variables: wff set class

This definition is referenced by: dfrdg2 5141 rdgeq1 5142 rdgeq2 5143 hbrdg 5144 rdgfnon 5147