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Theorem ialgrf 10805
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 } ) ,  S )
algrf.3  |-  ( ph  ->  M  e.  ZZ )
algrf.4  |-  ( ph  ->  A  e.  S )
algrf.5  |-  ( ph  ->  F : S --> S )
algrf.s  |-  ( ph  ->  S  e.  V )
Assertion
Ref Expression
ialgrf  |-  ( ph  ->  R : Z --> S )

Proof of Theorem ialgrf
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 )
31eleq2i 2149 . . . 4  |-  ( x  e.  Z  <->  x  e.  ( ZZ>= `  M )
)
4 algrf.4 . . . . 5  |-  ( ph  ->  A  e.  S )
51, 4ialgrlemconst 10803 . . . 4  |-  ( (
ph  /\  x  e.  ( ZZ>= `  M )
)  ->  ( ( Z  X.  { A }
) `  x )  e.  S )
63, 5sylan2b 281 . . 3  |-  ( (
ph  /\  x  e.  Z )  ->  (
( Z  X.  { A } ) `  x
)  e.  S )
7 algrf.5 . . . 4  |-  ( ph  ->  F : S --> S )
87ialgrlem1st 10802 . . 3  |-  ( (
ph  /\  ( x  e.  S  /\  y  e.  S ) )  -> 
( x ( F  o.  1st ) y )  e.  S )
91, 2, 6, 8iseqfcl 9752 . 2  |-  ( ph  ->  seq M ( ( F  o.  1st ) ,  ( Z  X.  { A } ) ,  S ) : Z --> S )
10 algrf.2 . . 3  |-  R  =  seq M ( ( F  o.  1st ) ,  ( Z  X.  { A } ) ,  S )
1110feq1i 5105 . 2  |-  ( R : Z --> S  <->  seq M ( ( F  o.  1st ) ,  ( Z  X.  { A } ) ,  S ) : Z --> S )
129, 11sylibr 132 1  |-  ( ph  ->  R : Z --> S )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1285    e. wcel 1434   {csn 3422    X. cxp 4397    o. ccom 4403   -->wf 4963   ` cfv 4967   1stc1st 5842   ZZcz 8644   ZZ>=cuz 8912    seqcseq 9738
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 577  ax-in2 578  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-13 1445  ax-14 1446  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2065  ax-coll 3919  ax-sep 3922  ax-nul 3930  ax-pow 3974  ax-pr 3999  ax-un 4223  ax-setind 4315  ax-iinf 4365  ax-cnex 7337  ax-resscn 7338  ax-1cn 7339  ax-1re 7340  ax-icn 7341  ax-addcl 7342  ax-addrcl 7343  ax-mulcl 7344  ax-addcom 7346  ax-addass 7348  ax-distr 7350  ax-i2m1 7351  ax-0lt1 7352  ax-0id 7354  ax-rnegex 7355  ax-cnre 7357  ax-pre-ltirr 7358  ax-pre-ltwlin 7359  ax-pre-lttrn 7360  ax-pre-ltadd 7362
This theorem depends on definitions:  df-bi 115  df-3or 921  df-3an 922  df-tru 1288  df-fal 1291  df-nf 1391  df-sb 1688  df-eu 1946  df-mo 1947  df-clab 2070  df-cleq 2076  df-clel 2079  df-nfc 2212  df-ne 2250  df-nel 2345  df-ral 2358  df-rex 2359  df-reu 2360  df-rab 2362  df-v 2614  df-sbc 2827  df-csb 2920  df-dif 2986  df-un 2988  df-in 2990  df-ss 2997  df-nul 3270  df-pw 3408  df-sn 3428  df-pr 3429  df-op 3431  df-uni 3628  df-int 3663  df-iun 3706  df-br 3812  df-opab 3866  df-mpt 3867  df-tr 3902  df-id 4083  df-iord 4156  df-on 4158  df-ilim 4159  df-suc 4161  df-iom 4368  df-xp 4405  df-rel 4406  df-cnv 4407  df-co 4408  df-dm 4409  df-rn 4410  df-res 4411  df-ima 4412  df-iota 4932  df-fun 4969  df-fn 4970  df-f 4971  df-f1 4972  df-fo 4973  df-f1o 4974  df-fv 4975  df-riota 5545  df-ov 5592  df-oprab 5593  df-mpt2 5594  df-1st 5844  df-2nd 5845  df-recs 6000  df-frec 6086  df-pnf 7425  df-mnf 7426  df-xr 7427  df-ltxr 7428  df-le 7429  df-sub 7556  df-neg 7557  df-inn 8315  df-n0 8564  df-z 8645  df-uz 8913  df-iseq 9739
This theorem is referenced by:  ialginv  10807  ialgcvg  10808  ialgcvga  10811  ialgfx  10812  eucialgcvga  10818  eucialg  10819
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