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Definition df-rdg 7491
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 7478 and 𝐺 in tz7.44-1 7487 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 7453 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 7576, from which we prove the recursive textbook definition as theorems oa0 7581, oasuc 7589, and oalim 7597 (with the help of theorems rdg0 7502, rdgsuc 7505, and rdglim2a 7514). We can also restrict the rec operation to define otherwise recursive functions on the natural numbers ω; see fr0g 7516 and frsuc 7517. 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 4078) 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 12785 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 13044 and integer powers df-exp 12844.

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 7490 . 2 class rec(𝐹, 𝐼)
4 vg . . . 4 setvar 𝑔
5 cvv 3195 . . . 4 class V
64cv 1480 . . . . . 6 class 𝑔
7 c0 3907 . . . . . 6 class
86, 7wceq 1481 . . . . 5 wff 𝑔 = ∅
96cdm 5104 . . . . . . 7 class dom 𝑔
109wlim 5712 . . . . . 6 wff Lim dom 𝑔
116crn 5105 . . . . . . 7 class ran 𝑔
1211cuni 4427 . . . . . 6 class ran 𝑔
139cuni 4427 . . . . . . . 8 class dom 𝑔
1413, 6cfv 5876 . . . . . . 7 class (𝑔 dom 𝑔)
1514, 1cfv 5876 . . . . . 6 class (𝐹‘(𝑔 dom 𝑔))
1610, 12, 15cif 4077 . . . . 5 class if(Lim dom 𝑔, ran 𝑔, (𝐹‘(𝑔 dom 𝑔)))
178, 2, 16cif 4077 . . . 4 class if(𝑔 = ∅, 𝐼, if(Lim dom 𝑔, ran 𝑔, (𝐹‘(𝑔 dom 𝑔))))
184, 5, 17cmpt 4720 . . 3 class (𝑔 ∈ V ↦ if(𝑔 = ∅, 𝐼, if(Lim dom 𝑔, ran 𝑔, (𝐹‘(𝑔 dom 𝑔)))))
1918crecs 7452 . 2 class recs((𝑔 ∈ V ↦ if(𝑔 = ∅, 𝐼, if(Lim dom 𝑔, ran 𝑔, (𝐹‘(𝑔 dom 𝑔))))))
203, 19wceq 1481 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  7492  rdgeq2  7493  nfrdg  7495  rdgfun  7497  rdgdmlim  7498  rdgfnon  7499  rdgvalg  7500  rdgval  7501  rdgseg  7503  dfrdg2  31675  csbrdgg  33146
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