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Definition df-rdg 7451
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 7438 and 𝐺 in tz7.44-1 7447 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 7413 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 7536, from which we prove the recursive textbook definition as theorems oa0 7541, oasuc 7549, and oalim 7557 (with the help of theorems rdg0 7462, rdgsuc 7465, and rdglim2a 7474). We can also restrict the rec operation to define otherwise recursive functions on the natural numbers ω; see fr0g 7476 and frsuc 7477. 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 4059) 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 12742 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 13001 and integer powers df-exp 12801.

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 7450 . 2 class rec(𝐹, 𝐼)
4 vg . . . 4 setvar 𝑔
5 cvv 3186 . . . 4 class V
64cv 1479 . . . . . 6 class 𝑔
7 c0 3891 . . . . . 6 class
86, 7wceq 1480 . . . . 5 wff 𝑔 = ∅
96cdm 5074 . . . . . . 7 class dom 𝑔
109wlim 5683 . . . . . 6 wff Lim dom 𝑔
116crn 5075 . . . . . . 7 class ran 𝑔
1211cuni 4402 . . . . . 6 class ran 𝑔
139cuni 4402 . . . . . . . 8 class dom 𝑔
1413, 6cfv 5847 . . . . . . 7 class (𝑔 dom 𝑔)
1514, 1cfv 5847 . . . . . 6 class (𝐹‘(𝑔 dom 𝑔))
1610, 12, 15cif 4058 . . . . 5 class if(Lim dom 𝑔, ran 𝑔, (𝐹‘(𝑔 dom 𝑔)))
178, 2, 16cif 4058 . . . 4 class if(𝑔 = ∅, 𝐼, if(Lim dom 𝑔, ran 𝑔, (𝐹‘(𝑔 dom 𝑔))))
184, 5, 17cmpt 4673 . . 3 class (𝑔 ∈ V ↦ if(𝑔 = ∅, 𝐼, if(Lim dom 𝑔, ran 𝑔, (𝐹‘(𝑔 dom 𝑔)))))
1918crecs 7412 . 2 class recs((𝑔 ∈ V ↦ if(𝑔 = ∅, 𝐼, if(Lim dom 𝑔, ran 𝑔, (𝐹‘(𝑔 dom 𝑔))))))
203, 19wceq 1480 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  7452  rdgeq2  7453  nfrdg  7455  rdgfun  7457  rdgdmlim  7458  rdgfnon  7459  rdgvalg  7460  rdgval  7461  rdgseg  7463  dfrdg2  31399  csbrdgg  32804
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