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Definition df-rdg 6659
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 6649 and  G in tz7.44-1 6655 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 6746, from which we prove the recursive textbook definition as theorems oa0 6751, oasuc 6759, and oalim 6767 (with the help of theorems rdg0 6670, rdgsuc 6673, and rdglim2a 6682). We can also restrict the  rec operation to define otherwise recursive functions on the natural numbers  om; see fr0g 6684 and frsuc 6685. 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 3732) 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 11312 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 11555 and integer powers df-exp 11371.

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 6658 . 2  class  rec ( F ,  I )
4 vg . . . 4  set  g
5 cvv 2948 . . . 4  class  _V
64cv 1651 . . . . . 6  class  g
7 c0 3620 . . . . . 6  class  (/)
86, 7wceq 1652 . . . . 5  wff  g  =  (/)
96cdm 4869 . . . . . . 7  class  dom  g
109wlim 4574 . . . . . 6  wff  Lim  dom  g
116crn 4870 . . . . . . 7  class  ran  g
1211cuni 4007 . . . . . 6  class  U. ran  g
139cuni 4007 . . . . . . . 8  class  U. dom  g
1413, 6cfv 5445 . . . . . . 7  class  ( g `
 U. dom  g
)
1514, 1cfv 5445 . . . . . 6  class  ( F `
 ( g `  U. dom  g ) )
1610, 12, 15cif 3731 . . . . 5  class  if ( Lim  dom  g ,  U. ran  g ,  ( F `  ( g `
 U. dom  g
) ) )
178, 2, 16cif 3731 . . . 4  class  if ( g  =  (/) ,  I ,  if ( Lim  dom  g ,  U. ran  g ,  ( F `  ( g `  U. dom  g ) ) ) )
184, 5, 17cmpt 4258 . . 3  class  ( g  e.  _V  |->  if ( g  =  (/) ,  I ,  if ( Lim  dom  g ,  U. ran  g ,  ( F `  ( g `  U. dom  g ) ) ) ) )
1918crecs 6623 . 2  class recs ( ( g  e.  _V  |->  if ( g  =  (/) ,  I ,  if ( Lim  dom  g ,  U. ran  g ,  ( F `  ( g `
 U. dom  g
) ) ) ) ) )
203, 19wceq 1652 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  6660  rdgeq2  6661  nfrdg  6663  rdgfun  6665  rdgdmlim  6666  rdgfnon  6667  rdgvalg  6668  rdgval  6669  rdgseg  6671  dfrdg2  25407
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