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Definition df-rdg 7268
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 7255 and 𝐺 in tz7.44-1 7264 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 7230 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 7353, from which we prove the recursive textbook definition as theorems oa0 7358, oasuc 7366, and oalim 7374 (with the help of theorems rdg0 7279, rdgsuc 7282, and rdglim2a 7291). We can also restrict the rec operation to define otherwise recursive functions on the natural numbers ω; see fr0g 7293 and frsuc 7294. 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 3940) 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 12531 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 12790 and integer powers df-exp 12590.

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 7267 . 2 class rec(𝐹, 𝐼)
4 vg . . . 4 setvar 𝑔
5 cvv 3077 . . . 4 class V
64cv 1473 . . . . . 6 class 𝑔
7 c0 3777 . . . . . 6 class
86, 7wceq 1474 . . . . 5 wff 𝑔 = ∅
96cdm 4932 . . . . . . 7 class dom 𝑔
109wlim 5531 . . . . . 6 wff Lim dom 𝑔
116crn 4933 . . . . . . 7 class ran 𝑔
1211cuni 4270 . . . . . 6 class ran 𝑔
139cuni 4270 . . . . . . . 8 class dom 𝑔
1413, 6cfv 5689 . . . . . . 7 class (𝑔 dom 𝑔)
1514, 1cfv 5689 . . . . . 6 class (𝐹‘(𝑔 dom 𝑔))
1610, 12, 15cif 3939 . . . . 5 class if(Lim dom 𝑔, ran 𝑔, (𝐹‘(𝑔 dom 𝑔)))
178, 2, 16cif 3939 . . . 4 class if(𝑔 = ∅, 𝐼, if(Lim dom 𝑔, ran 𝑔, (𝐹‘(𝑔 dom 𝑔))))
184, 5, 17cmpt 4541 . . 3 class (𝑔 ∈ V ↦ if(𝑔 = ∅, 𝐼, if(Lim dom 𝑔, ran 𝑔, (𝐹‘(𝑔 dom 𝑔)))))
1918crecs 7229 . 2 class recs((𝑔 ∈ V ↦ if(𝑔 = ∅, 𝐼, if(Lim dom 𝑔, ran 𝑔, (𝐹‘(𝑔 dom 𝑔))))))
203, 19wceq 1474 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  7269  rdgeq2  7270  nfrdg  7272  rdgfun  7274  rdgdmlim  7275  rdgfnon  7276  rdgvalg  7277  rdgval  7278  rdgseg  7280  dfrdg2  30788  csbrdgg  32183
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