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Definition df-rdg 7205
Description: Define a recursive definition generator on On (the class of ordinal numbers) with characteristic function 𝐹 and initial value 𝐼. This combines functions 𝐹 in tfr1 7192 and 𝐺 in tz7.44-1 7201 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 rec operation (especially when df-recs 7167 that it is built on is also eliminated). But once we get past this hurdle, definitions that would otherwise be recursive become relatively simple, as in for example oav 7290, from which we prove the recursive textbook definition as theorems oa0 7295, oasuc 7303, and oalim 7311 (with the help of theorems rdg0 7216, rdgsuc 7219, and rdglim2a 7228). We can also restrict the rec operation to define otherwise recursive functions on the natural numbers ω; see fr0g 7230 and frsuc 7231. Our rec 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 if operations (see df-if 3909) select cases based on whether the domain of 𝑔 is zero, a successor, or a limit ordinal.

An important use of this definition is in the recursive sequence generator df-seq 12328 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 12574 and integer powers df-exp 12387.

Note: We introduce rec 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. (Contributed by NM, 9-Apr-1995.) (Revised by Mario Carneiro, 9-May-2015.)

Assertion
Ref Expression
df-rdg rec(𝐹, 𝐼) = recs((𝑔 ∈ V ↦ if(𝑔 = ∅, 𝐼, if(Lim dom 𝑔, ran 𝑔, (𝐹‘(𝑔 dom 𝑔))))))
Distinct variable groups:   𝑔,𝐹   𝑔,𝐼

Detailed syntax breakdown of Definition df-rdg
StepHypRef Expression
1 cF . . 3 class 𝐹
2 cI . . 3 class 𝐼
31, 2crdg 7204 . 2 class rec(𝐹, 𝐼)
4 vg . . . 4 setvar 𝑔
5 cvv 3066 . . . 4 class V
64cv 1467 . . . . . 6 class 𝑔
7 c0 3757 . . . . . 6 class
86, 7wceq 1468 . . . . 5 wff 𝑔 = ∅
96cdm 4880 . . . . . . 7 class dom 𝑔
109wlim 5475 . . . . . 6 wff Lim dom 𝑔
116crn 4881 . . . . . . 7 class ran 𝑔
1211cuni 4228 . . . . . 6 class ran 𝑔
139cuni 4228 . . . . . . . 8 class dom 𝑔
1413, 6cfv 5633 . . . . . . 7 class (𝑔 dom 𝑔)
1514, 1cfv 5633 . . . . . 6 class (𝐹‘(𝑔 dom 𝑔))
1610, 12, 15cif 3908 . . . . 5 class if(Lim dom 𝑔, ran 𝑔, (𝐹‘(𝑔 dom 𝑔)))
178, 2, 16cif 3908 . . . 4 class if(𝑔 = ∅, 𝐼, if(Lim dom 𝑔, ran 𝑔, (𝐹‘(𝑔 dom 𝑔))))
184, 5, 17cmpt 4493 . . 3 class (𝑔 ∈ V ↦ if(𝑔 = ∅, 𝐼, if(Lim dom 𝑔, ran 𝑔, (𝐹‘(𝑔 dom 𝑔)))))
1918crecs 7166 . 2 class recs((𝑔 ∈ V ↦ if(𝑔 = ∅, 𝐼, if(Lim dom 𝑔, ran 𝑔, (𝐹‘(𝑔 dom 𝑔))))))
203, 19wceq 1468 1 wff rec(𝐹, 𝐼) = recs((𝑔 ∈ V ↦ if(𝑔 = ∅, 𝐼, if(Lim dom 𝑔, ran 𝑔, (𝐹‘(𝑔 dom 𝑔))))))
Colors of variables: wff setvar class
This definition is referenced by:  rdgeq1  7206  rdgeq2  7207  nfrdg  7209  rdgfun  7211  rdgdmlim  7212  rdgfnon  7213  rdgvalg  7214  rdgval  7215  rdgseg  7217  dfrdg2  30593  csbrdgg  31951
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