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Definition df-rdg 8450
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 8437 and 𝐺 in tz7.44-1 8446 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 8411 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 8549, from which we prove the recursive textbook definition as Theorems oa0 8554, oasuc 8562, and oalim 8570 (with the help of Theorems rdg0 8461, rdgsuc 8464, and rdglim2a 8473). We can also restrict the rec operation to define otherwise recursive functions on the natural numbers ω; see fr0g 8476 and frsuc 8477. 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 4526) 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 14043 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 14313 and integer powers df-exp 14103.

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 8449 . 2 class rec(𝐹, 𝐼)
4 vg . . . 4 setvar 𝑔
5 cvv 3480 . . . 4 class V
64cv 1539 . . . . . 6 class 𝑔
7 c0 4333 . . . . . 6 class
86, 7wceq 1540 . . . . 5 wff 𝑔 = ∅
96cdm 5685 . . . . . . 7 class dom 𝑔
109wlim 6385 . . . . . 6 wff Lim dom 𝑔
116crn 5686 . . . . . . 7 class ran 𝑔
1211cuni 4907 . . . . . 6 class ran 𝑔
139cuni 4907 . . . . . . . 8 class dom 𝑔
1413, 6cfv 6561 . . . . . . 7 class (𝑔 dom 𝑔)
1514, 1cfv 6561 . . . . . 6 class (𝐹‘(𝑔 dom 𝑔))
1610, 12, 15cif 4525 . . . . 5 class if(Lim dom 𝑔, ran 𝑔, (𝐹‘(𝑔 dom 𝑔)))
178, 2, 16cif 4525 . . . 4 class if(𝑔 = ∅, 𝐼, if(Lim dom 𝑔, ran 𝑔, (𝐹‘(𝑔 dom 𝑔))))
184, 5, 17cmpt 5225 . . 3 class (𝑔 ∈ V ↦ if(𝑔 = ∅, 𝐼, if(Lim dom 𝑔, ran 𝑔, (𝐹‘(𝑔 dom 𝑔)))))
1918crecs 8410 . 2 class recs((𝑔 ∈ V ↦ if(𝑔 = ∅, 𝐼, if(Lim dom 𝑔, ran 𝑔, (𝐹‘(𝑔 dom 𝑔))))))
203, 19wceq 1540 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  8451  rdgeq2  8452  nfrdg  8454  rdgfun  8456  rdgdmlim  8457  rdgfnon  8458  rdgvalg  8459  rdgval  8460  rdgseg  8462  rdg0n  8474  dfrdg2  35796  csbrdgg  37330
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