MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  df-rdg Structured version   Unicode version

Definition df-rdg 6671
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 6661 and  G in tz7.44-1 6667 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 6758, from which we prove the recursive textbook definition as theorems oa0 6763, oasuc 6771, and oalim 6779 (with the help of theorems rdg0 6682, rdgsuc 6685, and rdglim2a 6694). We can also restrict the  rec operation to define otherwise recursive functions on the natural numbers  om; see fr0g 6696 and frsuc 6697. 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 3742) 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 11329 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 11572 and integer powers df-exp 11388.

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 6670 . 2  class  rec ( F ,  I )
4 vg . . . 4  set  g
5 cvv 2958 . . . 4  class  _V
64cv 1652 . . . . . 6  class  g
7 c0 3630 . . . . . 6  class  (/)
86, 7wceq 1653 . . . . 5  wff  g  =  (/)
96cdm 4881 . . . . . . 7  class  dom  g
109wlim 4585 . . . . . 6  wff  Lim  dom  g
116crn 4882 . . . . . . 7  class  ran  g
1211cuni 4017 . . . . . 6  class  U. ran  g
139cuni 4017 . . . . . . . 8  class  U. dom  g
1413, 6cfv 5457 . . . . . . 7  class  ( g `
 U. dom  g
)
1514, 1cfv 5457 . . . . . 6  class  ( F `
 ( g `  U. dom  g ) )
1610, 12, 15cif 3741 . . . . 5  class  if ( Lim  dom  g ,  U. ran  g ,  ( F `  ( g `
 U. dom  g
) ) )
178, 2, 16cif 3741 . . . 4  class  if ( g  =  (/) ,  I ,  if ( Lim  dom  g ,  U. ran  g ,  ( F `  ( g `  U. dom  g ) ) ) )
184, 5, 17cmpt 4269 . . 3  class  ( g  e.  _V  |->  if ( g  =  (/) ,  I ,  if ( Lim  dom  g ,  U. ran  g ,  ( F `  ( g `  U. dom  g ) ) ) ) )
1918crecs 6635 . 2  class recs ( ( g  e.  _V  |->  if ( g  =  (/) ,  I ,  if ( Lim  dom  g ,  U. ran  g ,  ( F `  ( g `
 U. dom  g
) ) ) ) ) )
203, 19wceq 1653 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  6672  rdgeq2  6673  nfrdg  6675  rdgfun  6677  rdgdmlim  6678  rdgfnon  6679  rdgvalg  6680  rdgval  6681  rdgseg  6683  dfrdg2  25428
  Copyright terms: Public domain W3C validator