Metamath Proof Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >  df-rdg Structured version   Visualization version   GIF version

Definition df-rdg 7393
 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 7380 and 𝐺 in tz7.44-1 7389 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 7355 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 7478, from which we prove the recursive textbook definition as theorems oa0 7483, oasuc 7491, and oalim 7499 (with the help of theorems rdg0 7404, rdgsuc 7407, and rdglim2a 7416). We can also restrict the rec operation to define otherwise recursive functions on the natural numbers ω; see fr0g 7418 and frsuc 7419. 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 4037) 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 12664 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 12923 and integer powers df-exp 12723. 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 7392 . 2 class rec(𝐹, 𝐼)
4 vg . . . 4 setvar 𝑔
5 cvv 3173 . . . 4 class V
64cv 1474 . . . . . 6 class 𝑔
7 c0 3874 . . . . . 6 class
86, 7wceq 1475 . . . . 5 wff 𝑔 = ∅
96cdm 5038 . . . . . . 7 class dom 𝑔
109wlim 5641 . . . . . 6 wff Lim dom 𝑔
116crn 5039 . . . . . . 7 class ran 𝑔
1211cuni 4372 . . . . . 6 class ran 𝑔
139cuni 4372 . . . . . . . 8 class dom 𝑔
1413, 6cfv 5804 . . . . . . 7 class (𝑔 dom 𝑔)
1514, 1cfv 5804 . . . . . 6 class (𝐹‘(𝑔 dom 𝑔))
1610, 12, 15cif 4036 . . . . 5 class if(Lim dom 𝑔, ran 𝑔, (𝐹‘(𝑔 dom 𝑔)))
178, 2, 16cif 4036 . . . 4 class if(𝑔 = ∅, 𝐼, if(Lim dom 𝑔, ran 𝑔, (𝐹‘(𝑔 dom 𝑔))))
184, 5, 17cmpt 4643 . . 3 class (𝑔 ∈ V ↦ if(𝑔 = ∅, 𝐼, if(Lim dom 𝑔, ran 𝑔, (𝐹‘(𝑔 dom 𝑔)))))
1918crecs 7354 . 2 class recs((𝑔 ∈ V ↦ if(𝑔 = ∅, 𝐼, if(Lim dom 𝑔, ran 𝑔, (𝐹‘(𝑔 dom 𝑔))))))
203, 19wceq 1475 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  7394  rdgeq2  7395  nfrdg  7397  rdgfun  7399  rdgdmlim  7400  rdgfnon  7401  rdgvalg  7402  rdgval  7403  rdgseg  7405  dfrdg2  30945  csbrdgg  32351
 Copyright terms: Public domain W3C validator