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Definition df-rdg 8037
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 8024 and 𝐺 in tz7.44-1 8033 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 7999 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 8127, from which we prove the recursive textbook definition as theorems oa0 8132, oasuc 8140, and oalim 8148 (with the help of theorems rdg0 8048, rdgsuc 8051, and rdglim2a 8060). We can also restrict the rec operation to define otherwise recursive functions on the natural numbers ω; see fr0g 8062 and frsuc 8063. 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 4466) 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 13360 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 13624 and integer powers df-exp 13420.

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 8036 . 2 class rec(𝐹, 𝐼)
4 vg . . . 4 setvar 𝑔
5 cvv 3495 . . . 4 class V
64cv 1527 . . . . . 6 class 𝑔
7 c0 4290 . . . . . 6 class
86, 7wceq 1528 . . . . 5 wff 𝑔 = ∅
96cdm 5549 . . . . . . 7 class dom 𝑔
109wlim 6186 . . . . . 6 wff Lim dom 𝑔
116crn 5550 . . . . . . 7 class ran 𝑔
1211cuni 4832 . . . . . 6 class ran 𝑔
139cuni 4832 . . . . . . . 8 class dom 𝑔
1413, 6cfv 6349 . . . . . . 7 class (𝑔 dom 𝑔)
1514, 1cfv 6349 . . . . . 6 class (𝐹‘(𝑔 dom 𝑔))
1610, 12, 15cif 4465 . . . . 5 class if(Lim dom 𝑔, ran 𝑔, (𝐹‘(𝑔 dom 𝑔)))
178, 2, 16cif 4465 . . . 4 class if(𝑔 = ∅, 𝐼, if(Lim dom 𝑔, ran 𝑔, (𝐹‘(𝑔 dom 𝑔))))
184, 5, 17cmpt 5138 . . 3 class (𝑔 ∈ V ↦ if(𝑔 = ∅, 𝐼, if(Lim dom 𝑔, ran 𝑔, (𝐹‘(𝑔 dom 𝑔)))))
1918crecs 7998 . 2 class recs((𝑔 ∈ V ↦ if(𝑔 = ∅, 𝐼, if(Lim dom 𝑔, ran 𝑔, (𝐹‘(𝑔 dom 𝑔))))))
203, 19wceq 1528 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  8038  rdgeq2  8039  nfrdg  8041  rdgfun  8043  rdgdmlim  8044  rdgfnon  8045  rdgvalg  8046  rdgval  8047  rdgseg  8049  dfrdg2  32938  csbrdgg  34493
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