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Definition df-rdg 3929

Description: Define a recursive definition generator on

(the class of ordinal numbers) with characteristic function

and initial value

. This combines functions

in tfr1 3921 and

in tz7.44-1 3925 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 4147, from which we prove the recursive textbook definition as theorems oa0 4152, oasuc 4160, and oalim 4164 (with the help of theorems rdg0 3938, rdgsuc 3939, and rdglim 3940). We can also restrict the

operation to define otherwise recursive functions on the natural numbers

; see fr0t 3949 and frsuct 3950. 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

operators (see df-if 2360) 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 6263 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 6898 and integer powers df-exp 6519.

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 3928	. 2
4		vf	. . . . . . . 8
5	4	cv 954	. . . . . . 7
6		vx	. . . . . . . 8
7	6	cv 954	. . . . . . 7
8	5, 7	wfn 3174	. . . . . 6
9		vy	. . . . . . . . . 10
10	9	cv 954	. . . . . . . . 9
11	10, 5	cfv 3179	. . . . . . . 8
12	5, 10	cres 3169	. . . . . . . . 9
13		vz	. . . . . . . . . . . 12
14	13	cv 954	. . . . . . . . . . 11
15		vg	. . . . . . . . . . . . . 14
16	15	cv 954	. . . . . . . . . . . . 13
17		c0 2278	. . . . . . . . . . . . 13
18	16, 17	wceq 955	. . . . . . . . . . . 12
19	16	cdm 3167	. . . . . . . . . . . . . 14
20	19	wlim 2946	. . . . . . . . . . . . 13
21	16	crn 3168	. . . . . . . . . . . . . 14
22	21	cuni 2500	. . . . . . . . . . . . 13
23	19	cuni 2500	. . . . . . . . . . . . . . 15
24	23, 16	cfv 3179	. . . . . . . . . . . . . 14
25	24, 1	cfv 3179	. . . . . . . . . . . . 13
26	20, 22, 25	cif 2359	. . . . . . . . . . . 12
27	18, 2, 26	cif 2359	. . . . . . . . . . 11
28	14, 27	wceq 955	. . . . . . . . . 10
29	28, 15, 13	copab 2663	. . . . . . . . 9
30	12, 29	cfv 3179	. . . . . . . 8
31	11, 30	wceq 955	. . . . . . 7
32	31, 9, 7	wral 1644	. . . . . 6
33	8, 32	wa 223	. . . . 5 $/\$
34		con0 2945	. . . . 5
35	33, 6, 34	wrex 1645	. . . 4 $/\$
36	35, 4	cab 1463	. . 3 $/\$
37	36	cuni 2500	. 2 $/\$
38	3, 37	wceq 955	1 $/\$

Colors of variables: wff set class

This definition is referenced by: dfrdg2 3930 rdgeq1 3931 rdgeq2 3932 hbrdg 3933 rdgfnon 3936