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Theorem algrf 11999
Description: An algorithm is a step 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 5710 . . . . 5  |-  ( ( A  e.  S  /\  x  e.  Z )  ->  ( ( Z  X.  { A } ) `  x )  =  A )
53, 4sylan 281 . . . 4  |-  ( (
ph  /\  x  e.  Z )  ->  (
( Z  X.  { A } ) `  x
)  =  A )
63adantr 274 . . . 4  |-  ( (
ph  /\  x  e.  Z )  ->  A  e.  S )
75, 6eqeltrd 2247 . . 3  |-  ( (
ph  /\  x  e.  Z )  ->  (
( Z  X.  { A } ) `  x
)  e.  S )
8 vex 2733 . . . . 5  |-  x  e. 
_V
9 vex 2733 . . . . 5  |-  y  e. 
_V
108, 9algrflem 6208 . . . 4  |-  ( x ( F  o.  1st ) y )  =  ( F `  x
)
11 algrf.5 . . . . 5  |-  ( ph  ->  F : S --> S )
12 simpl 108 . . . . 5  |-  ( ( x  e.  S  /\  y  e.  S )  ->  x  e.  S )
13 ffvelrn 5629 . . . . 5  |-  ( ( F : S --> S  /\  x  e.  S )  ->  ( F `  x
)  e.  S )
1411, 12, 13syl2an 287 . . . 4  |-  ( (
ph  /\  ( x  e.  S  /\  y  e.  S ) )  -> 
( F `  x
)  e.  S )
1510, 14eqeltrid 2257 . . 3  |-  ( (
ph  /\  ( x  e.  S  /\  y  e.  S ) )  -> 
( x ( F  o.  1st ) y )  e.  S )
161, 2, 7, 15seqf 10417 . 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 5340 . 2  |-  ( R : Z --> S  <->  seq M ( ( F  o.  1st ) ,  ( Z  X.  { A } ) ) : Z --> S )
1916, 18sylibr 133 1  |-  ( ph  ->  R : Z --> S )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 103    = wceq 1348    e. wcel 2141   {csn 3583    X. cxp 4609    o. ccom 4615   -->wf 5194   ` cfv 5198  (class class class)co 5853   1stc1st 6117   ZZcz 9212   ZZ>=cuz 9487    seqcseq 10401
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 609  ax-in2 610  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-13 2143  ax-14 2144  ax-ext 2152  ax-coll 4104  ax-sep 4107  ax-nul 4115  ax-pow 4160  ax-pr 4194  ax-un 4418  ax-setind 4521  ax-iinf 4572  ax-cnex 7865  ax-resscn 7866  ax-1cn 7867  ax-1re 7868  ax-icn 7869  ax-addcl 7870  ax-addrcl 7871  ax-mulcl 7872  ax-addcom 7874  ax-addass 7876  ax-distr 7878  ax-i2m1 7879  ax-0lt1 7880  ax-0id 7882  ax-rnegex 7883  ax-cnre 7885  ax-pre-ltirr 7886  ax-pre-ltwlin 7887  ax-pre-lttrn 7888  ax-pre-ltadd 7890
This theorem depends on definitions:  df-bi 116  df-3or 974  df-3an 975  df-tru 1351  df-fal 1354  df-nf 1454  df-sb 1756  df-eu 2022  df-mo 2023  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ne 2341  df-nel 2436  df-ral 2453  df-rex 2454  df-reu 2455  df-rab 2457  df-v 2732  df-sbc 2956  df-csb 3050  df-dif 3123  df-un 3125  df-in 3127  df-ss 3134  df-nul 3415  df-pw 3568  df-sn 3589  df-pr 3590  df-op 3592  df-uni 3797  df-int 3832  df-iun 3875  df-br 3990  df-opab 4051  df-mpt 4052  df-tr 4088  df-id 4278  df-iord 4351  df-on 4353  df-ilim 4354  df-suc 4356  df-iom 4575  df-xp 4617  df-rel 4618  df-cnv 4619  df-co 4620  df-dm 4621  df-rn 4622  df-res 4623  df-ima 4624  df-iota 5160  df-fun 5200  df-fn 5201  df-f 5202  df-f1 5203  df-fo 5204  df-f1o 5205  df-fv 5206  df-riota 5809  df-ov 5856  df-oprab 5857  df-mpo 5858  df-1st 6119  df-2nd 6120  df-recs 6284  df-frec 6370  df-pnf 7956  df-mnf 7957  df-xr 7958  df-ltxr 7959  df-le 7960  df-sub 8092  df-neg 8093  df-inn 8879  df-n0 9136  df-z 9213  df-uz 9488  df-seqfrec 10402
This theorem is referenced by:  algrp1  12000  alginv  12001  algcvg  12002  algcvga  12005  algfx  12006  eucalgcvga  12012  eucalg  12013
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