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Theorem algrf 13746
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 5929 . . . . 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 2515 . . 3  |-  ( (
ph  /\  x  e.  Z )  ->  (
( Z  X.  { A } ) `  x
)  e.  S )
8 vex 2973 . . . . 5  |-  x  e. 
_V
9 vex 2973 . . . . 5  |-  y  e. 
_V
108, 9algrflem 6679 . . . 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 5839 . . . . 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 2525 . . 3  |-  ( (
ph  /\  ( x  e.  S  /\  y  e.  S ) )  -> 
( x ( F  o.  1st ) y )  e.  S )
161, 2, 7, 15seqf 11825 . 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 5549 . 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 1369    e. wcel 1756   {csn 3875    X. cxp 4836    o. ccom 4842   -->wf 5412   ` cfv 5416  (class class class)co 6089   1stc1st 6573   ZZcz 10644   ZZ>=cuz 10859    seqcseq 11804
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2422  ax-sep 4411  ax-nul 4419  ax-pow 4468  ax-pr 4529  ax-un 6370  ax-cnex 9336  ax-resscn 9337  ax-1cn 9338  ax-icn 9339  ax-addcl 9340  ax-addrcl 9341  ax-mulcl 9342  ax-mulrcl 9343  ax-mulcom 9344  ax-addass 9345  ax-mulass 9346  ax-distr 9347  ax-i2m1 9348  ax-1ne0 9349  ax-1rid 9350  ax-rnegex 9351  ax-rrecex 9352  ax-cnre 9353  ax-pre-lttri 9354  ax-pre-lttrn 9355  ax-pre-ltadd 9356  ax-pre-mulgt0 9357
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2428  df-cleq 2434  df-clel 2437  df-nfc 2566  df-ne 2606  df-nel 2607  df-ral 2718  df-rex 2719  df-reu 2720  df-rab 2722  df-v 2972  df-sbc 3185  df-csb 3287  df-dif 3329  df-un 3331  df-in 3333  df-ss 3340  df-pss 3342  df-nul 3636  df-if 3790  df-pw 3860  df-sn 3876  df-pr 3878  df-tp 3880  df-op 3882  df-uni 4090  df-iun 4171  df-br 4291  df-opab 4349  df-mpt 4350  df-tr 4384  df-eprel 4630  df-id 4634  df-po 4639  df-so 4640  df-fr 4677  df-we 4679  df-ord 4720  df-on 4721  df-lim 4722  df-suc 4723  df-xp 4844  df-rel 4845  df-cnv 4846  df-co 4847  df-dm 4848  df-rn 4849  df-res 4850  df-ima 4851  df-iota 5379  df-fun 5418  df-fn 5419  df-f 5420  df-f1 5421  df-fo 5422  df-f1o 5423  df-fv 5424  df-riota 6050  df-ov 6092  df-oprab 6093  df-mpt2 6094  df-om 6475  df-1st 6575  df-2nd 6576  df-recs 6830  df-rdg 6864  df-er 7099  df-en 7309  df-dom 7310  df-sdom 7311  df-pnf 9418  df-mnf 9419  df-xr 9420  df-ltxr 9421  df-le 9422  df-sub 9595  df-neg 9596  df-nn 10321  df-n0 10578  df-z 10645  df-uz 10860  df-fz 11436  df-seq 11805
This theorem is referenced by:  alginv  13748  algcvg  13749  algcvga  13752  algfx  13753  eucalgcvga  13759  eucalg  13760  ovolicc2lem2  20999  ovolicc2lem3  21000  ovolicc2lem4  21001  bfplem1  28718  bfplem2  28719
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