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Definition df-rdg 6439
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 6429 and  G in tz7.44-1 6435 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 6526, from which we prove the recursive textbook definition as theorems oa0 6531, oasuc 6539, and oalim 6547 (with the help of theorems rdg0 6450, rdgsuc 6453, and rdglim2a 6462). We can also restrict the  rec operation to define otherwise recursive functions on the natural numbers  om; see fr0g 6464 and frsuc 6465. 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 3579) 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 11063 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 11305 and integer powers df-exp 11121.

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 6438 . 2  class  rec ( F ,  I )
4 vg . . . 4  set  g
5 cvv 2801 . . . 4  class  _V
64cv 1631 . . . . . 6  class  g
7 c0 3468 . . . . . 6  class  (/)
86, 7wceq 1632 . . . . 5  wff  g  =  (/)
96cdm 4705 . . . . . . 7  class  dom  g
109wlim 4409 . . . . . 6  wff  Lim  dom  g
116crn 4706 . . . . . . 7  class  ran  g
1211cuni 3843 . . . . . 6  class  U. ran  g
139cuni 3843 . . . . . . . 8  class  U. dom  g
1413, 6cfv 5271 . . . . . . 7  class  ( g `
 U. dom  g
)
1514, 1cfv 5271 . . . . . 6  class  ( F `
 ( g `  U. dom  g ) )
1610, 12, 15cif 3578 . . . . 5  class  if ( Lim  dom  g ,  U. ran  g ,  ( F `  ( g `
 U. dom  g
) ) )
178, 2, 16cif 3578 . . . 4  class  if ( g  =  (/) ,  I ,  if ( Lim  dom  g ,  U. ran  g ,  ( F `  ( g `  U. dom  g ) ) ) )
184, 5, 17cmpt 4093 . . 3  class  ( g  e.  _V  |->  if ( g  =  (/) ,  I ,  if ( Lim  dom  g ,  U. ran  g ,  ( F `  ( g `  U. dom  g ) ) ) ) )
1918crecs 6403 . 2  class recs ( ( g  e.  _V  |->  if ( g  =  (/) ,  I ,  if ( Lim  dom  g ,  U. ran  g ,  ( F `  ( g `
 U. dom  g
) ) ) ) ) )
203, 19wceq 1632 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  6440  rdgeq2  6441  nfrdg  6443  rdgfun  6445  rdgdmlim  6446  rdgfnon  6447  rdgvalg  6448  rdgval  6449  rdgseg  6451  dfrdg2  24223
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