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Definition df-rdg 8431
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 8418 and 𝐺 in tz7.44-1 8427 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 8392 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 8532, from which we prove the recursive textbook definition as Theorems oa0 8537, oasuc 8545, and oalim 8553 (with the help of Theorems rdg0 8442, rdgsuc 8445, and rdglim2a 8454). We can also restrict the rec operation to define otherwise recursive functions on the natural numbers ω; see fr0g 8457 and frsuc 8458. 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 4530) 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 14000 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 14266 and integer powers df-exp 14060.

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 8430 . 2 class rec(𝐹, 𝐼)
4 vg . . . 4 setvar 𝑔
5 cvv 3471 . . . 4 class V
64cv 1533 . . . . . 6 class 𝑔
7 c0 4323 . . . . . 6 class
86, 7wceq 1534 . . . . 5 wff 𝑔 = ∅
96cdm 5678 . . . . . . 7 class dom 𝑔
109wlim 6370 . . . . . 6 wff Lim dom 𝑔
116crn 5679 . . . . . . 7 class ran 𝑔
1211cuni 4908 . . . . . 6 class ran 𝑔
139cuni 4908 . . . . . . . 8 class dom 𝑔
1413, 6cfv 6548 . . . . . . 7 class (𝑔 dom 𝑔)
1514, 1cfv 6548 . . . . . 6 class (𝐹‘(𝑔 dom 𝑔))
1610, 12, 15cif 4529 . . . . 5 class if(Lim dom 𝑔, ran 𝑔, (𝐹‘(𝑔 dom 𝑔)))
178, 2, 16cif 4529 . . . 4 class if(𝑔 = ∅, 𝐼, if(Lim dom 𝑔, ran 𝑔, (𝐹‘(𝑔 dom 𝑔))))
184, 5, 17cmpt 5231 . . 3 class (𝑔 ∈ V ↦ if(𝑔 = ∅, 𝐼, if(Lim dom 𝑔, ran 𝑔, (𝐹‘(𝑔 dom 𝑔)))))
1918crecs 8391 . 2 class recs((𝑔 ∈ V ↦ if(𝑔 = ∅, 𝐼, if(Lim dom 𝑔, ran 𝑔, (𝐹‘(𝑔 dom 𝑔))))))
203, 19wceq 1534 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  8432  rdgeq2  8433  nfrdg  8435  rdgfun  8437  rdgdmlim  8438  rdgfnon  8439  rdgvalg  8440  rdgval  8441  rdgseg  8443  rdg0n  8455  dfrdg2  35391  csbrdgg  36808
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