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Definition df-rdg 6377
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 6367 and  G in tz7.44-1 6373 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. But once we get past this hurdle, otherwise recursive definitions become relatively simple, as in for example oav 6464, from which we prove the recursive textbook definition as theorems oa0 6469, oasuc 6477, and oalim 6485 (with the help of theorems rdg0 6388, rdgsuc 6391, and rdglim2a 6400). We can also restrict the  rec operation to define otherwise recursive functions on the natural numbers  om; see fr0g 6402 and frsuc 6403. 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 3526) 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 10999 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 11241 and integer powers df-exp 11057.

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  e. 
_V  |->  if ( g  =  (/) ,  I ,  if ( Lim  dom  g ,  U. ran  g ,  ( F `  ( g `  U. 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 6376 . 2  class  rec ( F ,  I )
4 vg . . . 4  set  g
5 cvv 2757 . . . 4  class  _V
64cv 1618 . . . . . 6  class  g
7 c0 3416 . . . . . 6  class  (/)
86, 7wceq 1619 . . . . 5  wff  g  =  (/)
96cdm 4647 . . . . . . 7  class  dom  g
109wlim 4351 . . . . . 6  wff  Lim  dom  g
116crn 4648 . . . . . . 7  class  ran  g
1211cuni 3787 . . . . . 6  class  U. ran  g
139cuni 3787 . . . . . . . 8  class  U. dom  g
1413, 6cfv 4659 . . . . . . 7  class  ( g `
 U. dom  g
)
1514, 1cfv 4659 . . . . . 6  class  ( F `
 ( g `  U. dom  g ) )
1610, 12, 15cif 3525 . . . . 5  class  if ( Lim  dom  g ,  U. ran  g ,  ( F `  ( g `
 U. dom  g
) ) )
178, 2, 16cif 3525 . . . 4  class  if ( g  =  (/) ,  I ,  if ( Lim  dom  g ,  U. ran  g ,  ( F `  ( g `  U. dom  g ) ) ) )
184, 5, 17cmpt 4037 . . 3  class  ( g  e.  _V  |->  if ( g  =  (/) ,  I ,  if ( Lim  dom  g ,  U. ran  g ,  ( F `  ( g `  U. dom  g ) ) ) ) )
1918crecs 6341 . 2  class recs ( ( g  e.  _V  |->  if ( g  =  (/) ,  I ,  if ( Lim  dom  g ,  U. ran  g ,  ( F `  ( g `
 U. dom  g
) ) ) ) ) )
203, 19wceq 1619 1  wff  rec ( F ,  I )  = recs ( ( g  e. 
_V  |->  if ( g  =  (/) ,  I ,  if ( Lim  dom  g ,  U. ran  g ,  ( F `  ( g `  U. dom  g ) ) ) ) ) )
Colors of variables: wff set class
This definition is referenced by:  rdgeq1  6378  rdgeq2  6379  nfrdg  6381  rdgfun  6383  rdgdmlim  6384  rdgfnon  6385  rdgvalg  6386  rdgval  6387  rdgseg  6389  dfrdg2  23507
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