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Definition df-rdg 7370
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 7357 and 𝐺 in tz7.44-1 7366 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 7332 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 7455, from which we prove the recursive textbook definition as theorems oa0 7460, oasuc 7468, and oalim 7476 (with the help of theorems rdg0 7381, rdgsuc 7384, and rdglim2a 7393). We can also restrict the rec operation to define otherwise recursive functions on the natural numbers ω; see fr0g 7395 and frsuc 7396. 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 4036) 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 12619 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 12878 and integer powers df-exp 12678.

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 7369 . 2 class rec(𝐹, 𝐼)
4 vg . . . 4 setvar 𝑔
5 cvv 3172 . . . 4 class V
64cv 1473 . . . . . 6 class 𝑔
7 c0 3873 . . . . . 6 class
86, 7wceq 1474 . . . . 5 wff 𝑔 = ∅
96cdm 5028 . . . . . . 7 class dom 𝑔
109wlim 5627 . . . . . 6 wff Lim dom 𝑔
116crn 5029 . . . . . . 7 class ran 𝑔
1211cuni 4366 . . . . . 6 class ran 𝑔
139cuni 4366 . . . . . . . 8 class dom 𝑔
1413, 6cfv 5790 . . . . . . 7 class (𝑔 dom 𝑔)
1514, 1cfv 5790 . . . . . 6 class (𝐹‘(𝑔 dom 𝑔))
1610, 12, 15cif 4035 . . . . 5 class if(Lim dom 𝑔, ran 𝑔, (𝐹‘(𝑔 dom 𝑔)))
178, 2, 16cif 4035 . . . 4 class if(𝑔 = ∅, 𝐼, if(Lim dom 𝑔, ran 𝑔, (𝐹‘(𝑔 dom 𝑔))))
184, 5, 17cmpt 4637 . . 3 class (𝑔 ∈ V ↦ if(𝑔 = ∅, 𝐼, if(Lim dom 𝑔, ran 𝑔, (𝐹‘(𝑔 dom 𝑔)))))
1918crecs 7331 . 2 class recs((𝑔 ∈ V ↦ if(𝑔 = ∅, 𝐼, if(Lim dom 𝑔, ran 𝑔, (𝐹‘(𝑔 dom 𝑔))))))
203, 19wceq 1474 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  7371  rdgeq2  7372  nfrdg  7374  rdgfun  7376  rdgdmlim  7377  rdgfnon  7378  rdgvalg  7379  rdgval  7380  rdgseg  7382  dfrdg2  30751  csbrdgg  32147
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