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Theorem algrf 12743
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 5727 . . . . 5  |-  ( ( A  e.  S  /\  x  e.  Z )  ->  ( ( Z  X.  { A } ) `  x )  =  A )
53, 4sylan 457 . . . 4  |-  ( (
ph  /\  x  e.  Z )  ->  (
( Z  X.  { A } ) `  x
)  =  A )
63adantr 451 . . . 4  |-  ( (
ph  /\  x  e.  Z )  ->  A  e.  S )
75, 6eqeltrd 2357 . . 3  |-  ( (
ph  /\  x  e.  Z )  ->  (
( Z  X.  { A } ) `  x
)  e.  S )
8 vex 2791 . . . . 5  |-  x  e. 
_V
9 vex 2791 . . . . 5  |-  y  e. 
_V
108, 9algrflem 6224 . . . 4  |-  ( x ( F  o.  1st ) y )  =  ( F `  x
)
11 algrf.5 . . . . 5  |-  ( ph  ->  F : S --> S )
12 simpl 443 . . . . 5  |-  ( ( x  e.  S  /\  y  e.  S )  ->  x  e.  S )
13 ffvelrn 5663 . . . . 5  |-  ( ( F : S --> S  /\  x  e.  S )  ->  ( F `  x
)  e.  S )
1411, 12, 13syl2an 463 . . . 4  |-  ( (
ph  /\  ( x  e.  S  /\  y  e.  S ) )  -> 
( F `  x
)  e.  S )
1510, 14syl5eqel 2367 . . 3  |-  ( (
ph  /\  ( x  e.  S  /\  y  e.  S ) )  -> 
( x ( F  o.  1st ) y )  e.  S )
161, 2, 7, 15seqf 11067 . 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 5383 . 2  |-  ( R : Z --> S  <->  seq  M ( ( F  o.  1st ) ,  ( Z  X.  { A } ) ) : Z --> S )
1916, 18sylibr 203 1  |-  ( ph  ->  R : Z --> S )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    = wceq 1623    e. wcel 1684   {csn 3640    X. cxp 4687    o. ccom 4693   -->wf 5251   ` cfv 5255  (class class class)co 5858   1stc1st 6120   ZZcz 10024   ZZ>=cuz 10230    seq cseq 11046
This theorem is referenced by:  alginv  12745  algcvg  12746  algcvga  12749  algfx  12750  eucalgcvga  12756  eucalg  12757  ovolicc2lem2  18877  ovolicc2lem3  18878  ovolicc2lem4  18879  bfplem1  26546  bfplem2  26547
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512  ax-cnex 8793  ax-resscn 8794  ax-1cn 8795  ax-icn 8796  ax-addcl 8797  ax-addrcl 8798  ax-mulcl 8799  ax-mulrcl 8800  ax-mulcom 8801  ax-addass 8802  ax-mulass 8803  ax-distr 8804  ax-i2m1 8805  ax-1ne0 8806  ax-1rid 8807  ax-rnegex 8808  ax-rrecex 8809  ax-cnre 8810  ax-pre-lttri 8811  ax-pre-lttrn 8812  ax-pre-ltadd 8813  ax-pre-mulgt0 8814
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-nel 2449  df-ral 2548  df-rex 2549  df-reu 2550  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pss 3168  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-tp 3648  df-op 3649  df-uni 3828  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-tr 4114  df-eprel 4305  df-id 4309  df-po 4314  df-so 4315  df-fr 4352  df-we 4354  df-ord 4395  df-on 4396  df-lim 4397  df-suc 4398  df-om 4657  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-1st 6122  df-2nd 6123  df-riota 6304  df-recs 6388  df-rdg 6423  df-er 6660  df-en 6864  df-dom 6865  df-sdom 6866  df-pnf 8869  df-mnf 8870  df-xr 8871  df-ltxr 8872  df-le 8873  df-sub 9039  df-neg 9040  df-nn 9747  df-n0 9966  df-z 10025  df-uz 10231  df-fz 10783  df-seq 11047
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