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Definition df-rdg 7663
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 7650 and 𝐺 in tz7.44-1 7659 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 7625 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 7748, from which we prove the recursive textbook definition as theorems oa0 7753, oasuc 7761, and oalim 7769 (with the help of theorems rdg0 7674, rdgsuc 7677, and rdglim2a 7686). We can also restrict the rec operation to define otherwise recursive functions on the natural numbers ω; see fr0g 7688 and frsuc 7689. 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 4219) 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 12967 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 13226 and integer powers df-exp 13026.

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 7662 . 2 class rec(𝐹, 𝐼)
4 vg . . . 4 setvar 𝑔
5 cvv 3328 . . . 4 class V
64cv 1619 . . . . . 6 class 𝑔
7 c0 4046 . . . . . 6 class
86, 7wceq 1620 . . . . 5 wff 𝑔 = ∅
96cdm 5254 . . . . . . 7 class dom 𝑔
109wlim 5873 . . . . . 6 wff Lim dom 𝑔
116crn 5255 . . . . . . 7 class ran 𝑔
1211cuni 4576 . . . . . 6 class ran 𝑔
139cuni 4576 . . . . . . . 8 class dom 𝑔
1413, 6cfv 6037 . . . . . . 7 class (𝑔 dom 𝑔)
1514, 1cfv 6037 . . . . . 6 class (𝐹‘(𝑔 dom 𝑔))
1610, 12, 15cif 4218 . . . . 5 class if(Lim dom 𝑔, ran 𝑔, (𝐹‘(𝑔 dom 𝑔)))
178, 2, 16cif 4218 . . . 4 class if(𝑔 = ∅, 𝐼, if(Lim dom 𝑔, ran 𝑔, (𝐹‘(𝑔 dom 𝑔))))
184, 5, 17cmpt 4869 . . 3 class (𝑔 ∈ V ↦ if(𝑔 = ∅, 𝐼, if(Lim dom 𝑔, ran 𝑔, (𝐹‘(𝑔 dom 𝑔)))))
1918crecs 7624 . 2 class recs((𝑔 ∈ V ↦ if(𝑔 = ∅, 𝐼, if(Lim dom 𝑔, ran 𝑔, (𝐹‘(𝑔 dom 𝑔))))))
203, 19wceq 1620 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  7664  rdgeq2  7665  nfrdg  7667  rdgfun  7669  rdgdmlim  7670  rdgfnon  7671  rdgvalg  7672  rdgval  7673  rdgseg  7675  dfrdg2  31977  csbrdgg  33457
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