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Definition df-rdg 6423
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 6413 and  G in tz7.44-1 6419 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 6510, from which we prove the recursive textbook definition as theorems oa0 6515, oasuc 6523, and oalim 6531 (with the help of theorems rdg0 6434, rdgsuc 6437, and rdglim2a 6446). We can also restrict the  rec operation to define otherwise recursive functions on the natural numbers  om; see fr0g 6448 and frsuc 6449. 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 3566) 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 11047 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 11289 and integer powers df-exp 11105.

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 6422 . 2  class  rec ( F ,  I )
4 vg . . . 4  set  g
5 cvv 2788 . . . 4  class  _V
64cv 1622 . . . . . 6  class  g
7 c0 3455 . . . . . 6  class  (/)
86, 7wceq 1623 . . . . 5  wff  g  =  (/)
96cdm 4689 . . . . . . 7  class  dom  g
109wlim 4393 . . . . . 6  wff  Lim  dom  g
116crn 4690 . . . . . . 7  class  ran  g
1211cuni 3827 . . . . . 6  class  U. ran  g
139cuni 3827 . . . . . . . 8  class  U. dom  g
1413, 6cfv 5255 . . . . . . 7  class  ( g `
 U. dom  g
)
1514, 1cfv 5255 . . . . . 6  class  ( F `
 ( g `  U. dom  g ) )
1610, 12, 15cif 3565 . . . . 5  class  if ( Lim  dom  g ,  U. ran  g ,  ( F `  ( g `
 U. dom  g
) ) )
178, 2, 16cif 3565 . . . 4  class  if ( g  =  (/) ,  I ,  if ( Lim  dom  g ,  U. ran  g ,  ( F `  ( g `  U. dom  g ) ) ) )
184, 5, 17cmpt 4077 . . 3  class  ( g  e.  _V  |->  if ( g  =  (/) ,  I ,  if ( Lim  dom  g ,  U. ran  g ,  ( F `  ( g `  U. dom  g ) ) ) ) )
1918crecs 6387 . 2  class recs ( ( g  e.  _V  |->  if ( g  =  (/) ,  I ,  if ( Lim  dom  g ,  U. ran  g ,  ( F `  ( g `
 U. dom  g
) ) ) ) ) )
203, 19wceq 1623 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  6424  rdgeq2  6425  nfrdg  6427  rdgfun  6429  rdgdmlim  6430  rdgfnon  6431  rdgvalg  6432  rdgval  6433  rdgseg  6435  dfrdg2  24152
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