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Definition df-rdg 8408
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 8395 and 𝐺 in tz7.44-1 8404 into one definition. This rather amazing operation allows 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 8369 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 8509, from which we prove the recursive textbook definition as Theorems oa0 8514, oasuc 8522, and oalim 8530 (with the help of Theorems rdg0 8419, rdgsuc 8422, and rdglim2a 8431). We can also restrict the rec operation to define otherwise recursive functions on the natural numbers ω; see fr0g 8434 and frsuc 8435. 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 4524) 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 13970 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 14237 and integer powers df-exp 14031.

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 8407 . 2 class rec(𝐹, 𝐼)
4 vg . . . 4 setvar 𝑔
5 cvv 3468 . . . 4 class V
64cv 1532 . . . . . 6 class 𝑔
7 c0 4317 . . . . . 6 class
86, 7wceq 1533 . . . . 5 wff 𝑔 = ∅
96cdm 5669 . . . . . . 7 class dom 𝑔
109wlim 6358 . . . . . 6 wff Lim dom 𝑔
116crn 5670 . . . . . . 7 class ran 𝑔
1211cuni 4902 . . . . . 6 class ran 𝑔
139cuni 4902 . . . . . . . 8 class dom 𝑔
1413, 6cfv 6536 . . . . . . 7 class (𝑔 dom 𝑔)
1514, 1cfv 6536 . . . . . 6 class (𝐹‘(𝑔 dom 𝑔))
1610, 12, 15cif 4523 . . . . 5 class if(Lim dom 𝑔, ran 𝑔, (𝐹‘(𝑔 dom 𝑔)))
178, 2, 16cif 4523 . . . 4 class if(𝑔 = ∅, 𝐼, if(Lim dom 𝑔, ran 𝑔, (𝐹‘(𝑔 dom 𝑔))))
184, 5, 17cmpt 5224 . . 3 class (𝑔 ∈ V ↦ if(𝑔 = ∅, 𝐼, if(Lim dom 𝑔, ran 𝑔, (𝐹‘(𝑔 dom 𝑔)))))
1918crecs 8368 . 2 class recs((𝑔 ∈ V ↦ if(𝑔 = ∅, 𝐼, if(Lim dom 𝑔, ran 𝑔, (𝐹‘(𝑔 dom 𝑔))))))
203, 19wceq 1533 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  8409  rdgeq2  8410  nfrdg  8412  rdgfun  8414  rdgdmlim  8415  rdgfnon  8416  rdgvalg  8417  rdgval  8418  rdgseg  8420  rdg0n  8432  dfrdg2  35300  csbrdgg  36717
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