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Theorem algrf 14213
Description: An algorithm is step a function  F : S --> S on a state space  S. An algorithm acts on an initial state  A  e.  S by iteratively applying  F to give  A,  ( F `
 A ),  ( F `  ( F `
 A ) ) and so on. An algorithm is said to halt if a fixed point of  F is reached after a finite number of iterations.

The algorithm iterator  R : NN0 --> S "runs" the algorithm  F so that  ( R `  k ) is the state after  k iterations of  F on the initial state  A.

Domain and codomain of the algorithm iterator  R. (Contributed by Paul Chapman, 31-Mar-2011.) (Revised by Mario Carneiro, 28-May-2014.)

Hypotheses
Ref Expression
algrf.1  |-  Z  =  ( ZZ>= `  M )
algrf.2  |-  R  =  seq M ( ( F  o.  1st ) ,  ( Z  X.  { A } ) )
algrf.3  |-  ( ph  ->  M  e.  ZZ )
algrf.4  |-  ( ph  ->  A  e.  S )
algrf.5  |-  ( ph  ->  F : S --> S )
Assertion
Ref Expression
algrf  |-  ( ph  ->  R : Z --> S )

Proof of Theorem algrf
Dummy variables  x  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 algrf.1 . . 3  |-  Z  =  ( ZZ>= `  M )
2 algrf.3 . . 3  |-  ( ph  ->  M  e.  ZZ )
3 algrf.4 . . . . 5  |-  ( ph  ->  A  e.  S )
4 fvconst2g 6126 . . . . 5  |-  ( ( A  e.  S  /\  x  e.  Z )  ->  ( ( Z  X.  { A } ) `  x )  =  A )
53, 4sylan 471 . . . 4  |-  ( (
ph  /\  x  e.  Z )  ->  (
( Z  X.  { A } ) `  x
)  =  A )
63adantr 465 . . . 4  |-  ( (
ph  /\  x  e.  Z )  ->  A  e.  S )
75, 6eqeltrd 2545 . . 3  |-  ( (
ph  /\  x  e.  Z )  ->  (
( Z  X.  { A } ) `  x
)  e.  S )
8 vex 3112 . . . . 5  |-  x  e. 
_V
9 vex 3112 . . . . 5  |-  y  e. 
_V
108, 9algrflem 6908 . . . 4  |-  ( x ( F  o.  1st ) y )  =  ( F `  x
)
11 algrf.5 . . . . 5  |-  ( ph  ->  F : S --> S )
12 simpl 457 . . . . 5  |-  ( ( x  e.  S  /\  y  e.  S )  ->  x  e.  S )
13 ffvelrn 6030 . . . . 5  |-  ( ( F : S --> S  /\  x  e.  S )  ->  ( F `  x
)  e.  S )
1411, 12, 13syl2an 477 . . . 4  |-  ( (
ph  /\  ( x  e.  S  /\  y  e.  S ) )  -> 
( F `  x
)  e.  S )
1510, 14syl5eqel 2549 . . 3  |-  ( (
ph  /\  ( x  e.  S  /\  y  e.  S ) )  -> 
( x ( F  o.  1st ) y )  e.  S )
161, 2, 7, 15seqf 12130 . 2  |-  ( ph  ->  seq M ( ( F  o.  1st ) ,  ( Z  X.  { A } ) ) : Z --> S )
17 algrf.2 . . 3  |-  R  =  seq M ( ( F  o.  1st ) ,  ( Z  X.  { A } ) )
1817feq1i 5729 . 2  |-  ( R : Z --> S  <->  seq M ( ( F  o.  1st ) ,  ( Z  X.  { A } ) ) : Z --> S )
1916, 18sylibr 212 1  |-  ( ph  ->  R : Z --> S )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1395    e. wcel 1819   {csn 4032    X. cxp 5006    o. ccom 5012   -->wf 5590   ` cfv 5594  (class class class)co 6296   1stc1st 6797   ZZcz 10885   ZZ>=cuz 11106    seqcseq 12109
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-8 1821  ax-9 1823  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435  ax-sep 4578  ax-nul 4586  ax-pow 4634  ax-pr 4695  ax-un 6591  ax-cnex 9565  ax-resscn 9566  ax-1cn 9567  ax-icn 9568  ax-addcl 9569  ax-addrcl 9570  ax-mulcl 9571  ax-mulrcl 9572  ax-mulcom 9573  ax-addass 9574  ax-mulass 9575  ax-distr 9576  ax-i2m1 9577  ax-1ne0 9578  ax-1rid 9579  ax-rnegex 9580  ax-rrecex 9581  ax-cnre 9582  ax-pre-lttri 9583  ax-pre-lttrn 9584  ax-pre-ltadd 9585  ax-pre-mulgt0 9586
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-eu 2287  df-mo 2288  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-nel 2655  df-ral 2812  df-rex 2813  df-reu 2814  df-rab 2816  df-v 3111  df-sbc 3328  df-csb 3431  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-pss 3487  df-nul 3794  df-if 3945  df-pw 4017  df-sn 4033  df-pr 4035  df-tp 4037  df-op 4039  df-uni 4252  df-iun 4334  df-br 4457  df-opab 4516  df-mpt 4517  df-tr 4551  df-eprel 4800  df-id 4804  df-po 4809  df-so 4810  df-fr 4847  df-we 4849  df-ord 4890  df-on 4891  df-lim 4892  df-suc 4893  df-xp 5014  df-rel 5015  df-cnv 5016  df-co 5017  df-dm 5018  df-rn 5019  df-res 5020  df-ima 5021  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-riota 6258  df-ov 6299  df-oprab 6300  df-mpt2 6301  df-om 6700  df-1st 6799  df-2nd 6800  df-recs 7060  df-rdg 7094  df-er 7329  df-en 7536  df-dom 7537  df-sdom 7538  df-pnf 9647  df-mnf 9648  df-xr 9649  df-ltxr 9650  df-le 9651  df-sub 9826  df-neg 9827  df-nn 10557  df-n0 10817  df-z 10886  df-uz 11107  df-fz 11698  df-seq 12110
This theorem is referenced by:  alginv  14215  algcvg  14216  algcvga  14219  algfx  14220  eucalgcvga  14226  eucalg  14227  ovolicc2lem2  22054  ovolicc2lem3  22055  ovolicc2lem4  22056  bfplem1  30480  bfplem2  30481
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