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Definition df-rdg 6668
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 6658 and  G in tz7.44-1 6664 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 6755, from which we prove the recursive textbook definition as theorems oa0 6760, oasuc 6768, and oalim 6776 (with the help of theorems rdg0 6679, rdgsuc 6682, and rdglim2a 6691). We can also restrict the  rec operation to define otherwise recursive functions on the natural numbers  om; see fr0g 6693 and frsuc 6694. 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 3740) 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 11324 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 11567 and integer powers df-exp 11383.

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 6667 . 2  class  rec ( F ,  I )
4 vg . . . 4  set  g
5 cvv 2956 . . . 4  class  _V
64cv 1651 . . . . . 6  class  g
7 c0 3628 . . . . . 6  class  (/)
86, 7wceq 1652 . . . . 5  wff  g  =  (/)
96cdm 4878 . . . . . . 7  class  dom  g
109wlim 4582 . . . . . 6  wff  Lim  dom  g
116crn 4879 . . . . . . 7  class  ran  g
1211cuni 4015 . . . . . 6  class  U. ran  g
139cuni 4015 . . . . . . . 8  class  U. dom  g
1413, 6cfv 5454 . . . . . . 7  class  ( g `
 U. dom  g
)
1514, 1cfv 5454 . . . . . 6  class  ( F `
 ( g `  U. dom  g ) )
1610, 12, 15cif 3739 . . . . 5  class  if ( Lim  dom  g ,  U. ran  g ,  ( F `  ( g `
 U. dom  g
) ) )
178, 2, 16cif 3739 . . . 4  class  if ( g  =  (/) ,  I ,  if ( Lim  dom  g ,  U. ran  g ,  ( F `  ( g `  U. dom  g ) ) ) )
184, 5, 17cmpt 4266 . . 3  class  ( g  e.  _V  |->  if ( g  =  (/) ,  I ,  if ( Lim  dom  g ,  U. ran  g ,  ( F `  ( g `  U. dom  g ) ) ) ) )
1918crecs 6632 . 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  6669  rdgeq2  6670  nfrdg  6672  rdgfun  6674  rdgdmlim  6675  rdgfnon  6676  rdgvalg  6677  rdgval  6678  rdgseg  6680  dfrdg2  25423
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