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Definition df-rdg 8448
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 8435 and 𝐺 in tz7.44-1 8444 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 8409 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 8547, from which we prove the recursive textbook definition as Theorems oa0 8552, oasuc 8560, and oalim 8568 (with the help of Theorems rdg0 8459, rdgsuc 8462, and rdglim2a 8471). We can also restrict the rec operation to define otherwise recursive functions on the natural numbers ω; see fr0g 8474 and frsuc 8475. 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 4531) 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 14039 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 14309 and integer powers df-exp 14099.

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 8447 . 2 class rec(𝐹, 𝐼)
4 vg . . . 4 setvar 𝑔
5 cvv 3477 . . . 4 class V
64cv 1535 . . . . . 6 class 𝑔
7 c0 4338 . . . . . 6 class
86, 7wceq 1536 . . . . 5 wff 𝑔 = ∅
96cdm 5688 . . . . . . 7 class dom 𝑔
109wlim 6386 . . . . . 6 wff Lim dom 𝑔
116crn 5689 . . . . . . 7 class ran 𝑔
1211cuni 4911 . . . . . 6 class ran 𝑔
139cuni 4911 . . . . . . . 8 class dom 𝑔
1413, 6cfv 6562 . . . . . . 7 class (𝑔 dom 𝑔)
1514, 1cfv 6562 . . . . . 6 class (𝐹‘(𝑔 dom 𝑔))
1610, 12, 15cif 4530 . . . . 5 class if(Lim dom 𝑔, ran 𝑔, (𝐹‘(𝑔 dom 𝑔)))
178, 2, 16cif 4530 . . . 4 class if(𝑔 = ∅, 𝐼, if(Lim dom 𝑔, ran 𝑔, (𝐹‘(𝑔 dom 𝑔))))
184, 5, 17cmpt 5230 . . 3 class (𝑔 ∈ V ↦ if(𝑔 = ∅, 𝐼, if(Lim dom 𝑔, ran 𝑔, (𝐹‘(𝑔 dom 𝑔)))))
1918crecs 8408 . 2 class recs((𝑔 ∈ V ↦ if(𝑔 = ∅, 𝐼, if(Lim dom 𝑔, ran 𝑔, (𝐹‘(𝑔 dom 𝑔))))))
203, 19wceq 1536 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  8449  rdgeq2  8450  nfrdg  8452  rdgfun  8454  rdgdmlim  8455  rdgfnon  8456  rdgvalg  8457  rdgval  8458  rdgseg  8460  rdg0n  8472  dfrdg2  35776  csbrdgg  37311
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