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Definition df-rdg 8040
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 8027 and 𝐺 in tz7.44-1 8036 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 8002 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 8130, from which we prove the recursive textbook definition as theorems oa0 8135, oasuc 8143, and oalim 8151 (with the help of theorems rdg0 8051, rdgsuc 8054, and rdglim2a 8063). We can also restrict the rec operation to define otherwise recursive functions on the natural numbers ω; see fr0g 8065 and frsuc 8066. 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 4470) 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 13363 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 13627 and integer powers df-exp 13423.

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 8039 . 2 class rec(𝐹, 𝐼)
4 vg . . . 4 setvar 𝑔
5 cvv 3499 . . . 4 class V
64cv 1529 . . . . . 6 class 𝑔
7 c0 4294 . . . . . 6 class
86, 7wceq 1530 . . . . 5 wff 𝑔 = ∅
96cdm 5553 . . . . . . 7 class dom 𝑔
109wlim 6189 . . . . . 6 wff Lim dom 𝑔
116crn 5554 . . . . . . 7 class ran 𝑔
1211cuni 4836 . . . . . 6 class ran 𝑔
139cuni 4836 . . . . . . . 8 class dom 𝑔
1413, 6cfv 6351 . . . . . . 7 class (𝑔 dom 𝑔)
1514, 1cfv 6351 . . . . . 6 class (𝐹‘(𝑔 dom 𝑔))
1610, 12, 15cif 4469 . . . . 5 class if(Lim dom 𝑔, ran 𝑔, (𝐹‘(𝑔 dom 𝑔)))
178, 2, 16cif 4469 . . . 4 class if(𝑔 = ∅, 𝐼, if(Lim dom 𝑔, ran 𝑔, (𝐹‘(𝑔 dom 𝑔))))
184, 5, 17cmpt 5142 . . 3 class (𝑔 ∈ V ↦ if(𝑔 = ∅, 𝐼, if(Lim dom 𝑔, ran 𝑔, (𝐹‘(𝑔 dom 𝑔)))))
1918crecs 8001 . 2 class recs((𝑔 ∈ V ↦ if(𝑔 = ∅, 𝐼, if(Lim dom 𝑔, ran 𝑔, (𝐹‘(𝑔 dom 𝑔))))))
203, 19wceq 1530 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  8041  rdgeq2  8042  nfrdg  8044  rdgfun  8046  rdgdmlim  8047  rdgfnon  8048  rdgvalg  8049  rdgval  8050  rdgseg  8052  dfrdg2  32926  csbrdgg  34481
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