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Theorem ialgcvg 9887
Description: One way to prove that an algorithm halts is to construct a countdown function  C : S --> NN0 whose value is guaranteed to decrease for each iteration of  F until it reaches  0. That is, if  X  e.  S is not a fixed point of  F, then  ( C `  ( F `  X ) )  <  ( C `
 X ).

If  C is a countdown function for algorithm  F, the sequence  ( C `  ( R `  k ) ) reaches  0 after at most  N steps, where  N is the value of  C for the initial state  A. (Contributed by Paul Chapman, 22-Jun-2011.)

Hypotheses
Ref Expression
algcvg.1  |-  F : S
--> S
algcvg.2  |-  R  =  seq 0 ( ( F  o.  1st ) ,  ( NN0  X.  { A } ) ,  S )
algcvg.3  |-  C : S
--> NN0
algcvg.4  |-  ( z  e.  S  ->  (
( C `  ( F `  z )
)  =/=  0  -> 
( C `  ( F `  z )
)  <  ( C `  z ) ) )
algcvg.5  |-  N  =  ( C `  A
)
ialgcvg.s  |-  S  e.  V
Assertion
Ref Expression
ialgcvg  |-  ( A  e.  S  ->  ( C `  ( R `  N ) )  =  0 )
Distinct variable groups:    z, C    z, F    z, R    z, S
Allowed substitution hints:    A( z)    N( z)    V( z)

Proof of Theorem ialgcvg
Dummy variable  k is distinct from all other variables.
StepHypRef Expression
1 nn0uz 8507 . . . 4  |-  NN0  =  ( ZZ>= `  0 )
2 algcvg.2 . . . 4  |-  R  =  seq 0 ( ( F  o.  1st ) ,  ( NN0  X.  { A } ) ,  S )
3 0zd 8257 . . . 4  |-  ( A  e.  S  ->  0  e.  ZZ )
4 id 19 . . . 4  |-  ( A  e.  S  ->  A  e.  S )
5 algcvg.1 . . . . 5  |-  F : S
--> S
65a1i 9 . . . 4  |-  ( A  e.  S  ->  F : S --> S )
7 ialgcvg.s . . . . 5  |-  S  e.  V
87a1i 9 . . . 4  |-  ( A  e.  S  ->  S  e.  V )
91, 2, 3, 4, 6, 8ialgrf 9884 . . 3  |-  ( A  e.  S  ->  R : NN0 --> S )
10 algcvg.5 . . . 4  |-  N  =  ( C `  A
)
11 algcvg.3 . . . . 5  |-  C : S
--> NN0
1211ffvelrni 5301 . . . 4  |-  ( A  e.  S  ->  ( C `  A )  e.  NN0 )
1310, 12syl5eqel 2124 . . 3  |-  ( A  e.  S  ->  N  e.  NN0 )
14 fvco3 5244 . . 3  |-  ( ( R : NN0 --> S  /\  N  e.  NN0 )  -> 
( ( C  o.  R ) `  N
)  =  ( C `
 ( R `  N ) ) )
159, 13, 14syl2anc 391 . 2  |-  ( A  e.  S  ->  (
( C  o.  R
) `  N )  =  ( C `  ( R `  N ) ) )
16 fco 5056 . . . 4  |-  ( ( C : S --> NN0  /\  R : NN0 --> S )  ->  ( C  o.  R ) : NN0 --> NN0 )
1711, 9, 16sylancr 393 . . 3  |-  ( A  e.  S  ->  ( C  o.  R ) : NN0 --> NN0 )
18 0nn0 8196 . . . . . 6  |-  0  e.  NN0
19 fvco3 5244 . . . . . 6  |-  ( ( R : NN0 --> S  /\  0  e.  NN0 )  -> 
( ( C  o.  R ) `  0
)  =  ( C `
 ( R ` 
0 ) ) )
209, 18, 19sylancl 392 . . . . 5  |-  ( A  e.  S  ->  (
( C  o.  R
) `  0 )  =  ( C `  ( R `  0 ) ) )
211, 2, 3, 4, 6, 8ialgr0 9883 . . . . . 6  |-  ( A  e.  S  ->  ( R `  0 )  =  A )
2221fveq2d 5182 . . . . 5  |-  ( A  e.  S  ->  ( C `  ( R `  0 ) )  =  ( C `  A ) )
2320, 22eqtrd 2072 . . . 4  |-  ( A  e.  S  ->  (
( C  o.  R
) `  0 )  =  ( C `  A ) )
2423, 10syl6reqr 2091 . . 3  |-  ( A  e.  S  ->  N  =  ( ( C  o.  R ) ` 
0 ) )
259ffvelrnda 5302 . . . . 5  |-  ( ( A  e.  S  /\  k  e.  NN0 )  -> 
( R `  k
)  e.  S )
26 fveq2 5178 . . . . . . . . 9  |-  ( z  =  ( R `  k )  ->  ( F `  z )  =  ( F `  ( R `  k ) ) )
2726fveq2d 5182 . . . . . . . 8  |-  ( z  =  ( R `  k )  ->  ( C `  ( F `  z ) )  =  ( C `  ( F `  ( R `  k ) ) ) )
2827neeq1d 2223 . . . . . . 7  |-  ( z  =  ( R `  k )  ->  (
( C `  ( F `  z )
)  =/=  0  <->  ( C `  ( F `  ( R `  k
) ) )  =/=  0 ) )
29 fveq2 5178 . . . . . . . 8  |-  ( z  =  ( R `  k )  ->  ( C `  z )  =  ( C `  ( R `  k ) ) )
3027, 29breq12d 3777 . . . . . . 7  |-  ( z  =  ( R `  k )  ->  (
( C `  ( F `  z )
)  <  ( C `  z )  <->  ( C `  ( F `  ( R `  k )
) )  <  ( C `  ( R `  k ) ) ) )
3128, 30imbi12d 223 . . . . . 6  |-  ( z  =  ( R `  k )  ->  (
( ( C `  ( F `  z ) )  =/=  0  -> 
( C `  ( F `  z )
)  <  ( C `  z ) )  <->  ( ( C `  ( F `  ( R `  k
) ) )  =/=  0  ->  ( C `  ( F `  ( R `  k )
) )  <  ( C `  ( R `  k ) ) ) ) )
32 algcvg.4 . . . . . 6  |-  ( z  e.  S  ->  (
( C `  ( F `  z )
)  =/=  0  -> 
( C `  ( F `  z )
)  <  ( C `  z ) ) )
3331, 32vtoclga 2619 . . . . 5  |-  ( ( R `  k )  e.  S  ->  (
( C `  ( F `  ( R `  k ) ) )  =/=  0  ->  ( C `  ( F `  ( R `  k
) ) )  < 
( C `  ( R `  k )
) ) )
3425, 33syl 14 . . . 4  |-  ( ( A  e.  S  /\  k  e.  NN0 )  -> 
( ( C `  ( F `  ( R `
 k ) ) )  =/=  0  -> 
( C `  ( F `  ( R `  k ) ) )  <  ( C `  ( R `  k ) ) ) )
35 peano2nn0 8222 . . . . . . 7  |-  ( k  e.  NN0  ->  ( k  +  1 )  e. 
NN0 )
36 fvco3 5244 . . . . . . 7  |-  ( ( R : NN0 --> S  /\  ( k  +  1 )  e.  NN0 )  ->  ( ( C  o.  R ) `  (
k  +  1 ) )  =  ( C `
 ( R `  ( k  +  1 ) ) ) )
379, 35, 36syl2an 273 . . . . . 6  |-  ( ( A  e.  S  /\  k  e.  NN0 )  -> 
( ( C  o.  R ) `  (
k  +  1 ) )  =  ( C `
 ( R `  ( k  +  1 ) ) ) )
381, 2, 3, 4, 6, 8ialgrp1 9885 . . . . . . 7  |-  ( ( A  e.  S  /\  k  e.  NN0 )  -> 
( R `  (
k  +  1 ) )  =  ( F `
 ( R `  k ) ) )
3938fveq2d 5182 . . . . . 6  |-  ( ( A  e.  S  /\  k  e.  NN0 )  -> 
( C `  ( R `  ( k  +  1 ) ) )  =  ( C `
 ( F `  ( R `  k ) ) ) )
4037, 39eqtrd 2072 . . . . 5  |-  ( ( A  e.  S  /\  k  e.  NN0 )  -> 
( ( C  o.  R ) `  (
k  +  1 ) )  =  ( C `
 ( F `  ( R `  k ) ) ) )
4140neeq1d 2223 . . . 4  |-  ( ( A  e.  S  /\  k  e.  NN0 )  -> 
( ( ( C  o.  R ) `  ( k  +  1 ) )  =/=  0  <->  ( C `  ( F `
 ( R `  k ) ) )  =/=  0 ) )
42 fvco3 5244 . . . . . 6  |-  ( ( R : NN0 --> S  /\  k  e.  NN0 )  -> 
( ( C  o.  R ) `  k
)  =  ( C `
 ( R `  k ) ) )
439, 42sylan 267 . . . . 5  |-  ( ( A  e.  S  /\  k  e.  NN0 )  -> 
( ( C  o.  R ) `  k
)  =  ( C `
 ( R `  k ) ) )
4440, 43breq12d 3777 . . . 4  |-  ( ( A  e.  S  /\  k  e.  NN0 )  -> 
( ( ( C  o.  R ) `  ( k  +  1 ) )  <  (
( C  o.  R
) `  k )  <->  ( C `  ( F `
 ( R `  k ) ) )  <  ( C `  ( R `  k ) ) ) )
4534, 41, 443imtr4d 192 . . 3  |-  ( ( A  e.  S  /\  k  e.  NN0 )  -> 
( ( ( C  o.  R ) `  ( k  +  1 ) )  =/=  0  ->  ( ( C  o.  R ) `  (
k  +  1 ) )  <  ( ( C  o.  R ) `
 k ) ) )
4617, 24, 45nn0seqcvgd 9880 . 2  |-  ( A  e.  S  ->  (
( C  o.  R
) `  N )  =  0 )
4715, 46eqtr3d 2074 1  |-  ( A  e.  S  ->  ( C `  ( R `  N ) )  =  0 )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 97    = wceq 1243    e. wcel 1393    =/= wne 2204   {csn 3375   class class class wbr 3764    X. cxp 4343    o. ccom 4349   -->wf 4898   ` cfv 4902  (class class class)co 5512   1stc1st 5765   0cc0 6889   1c1 6890    + caddc 6892    < clt 7060   NN0cn0 8181    seqcseq 9211
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in1 544  ax-in2 545  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-13 1404  ax-14 1405  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022  ax-coll 3872  ax-sep 3875  ax-nul 3883  ax-pow 3927  ax-pr 3944  ax-un 4170  ax-setind 4262  ax-iinf 4311  ax-cnex 6975  ax-resscn 6976  ax-1cn 6977  ax-1re 6978  ax-icn 6979  ax-addcl 6980  ax-addrcl 6981  ax-mulcl 6982  ax-addcom 6984  ax-addass 6986  ax-distr 6988  ax-i2m1 6989  ax-0id 6992  ax-rnegex 6993  ax-cnre 6995  ax-pre-ltirr 6996  ax-pre-ltwlin 6997  ax-pre-lttrn 6998  ax-pre-apti 6999  ax-pre-ltadd 7000
This theorem depends on definitions:  df-bi 110  df-dc 743  df-3or 886  df-3an 887  df-tru 1246  df-fal 1249  df-nf 1350  df-sb 1646  df-eu 1903  df-mo 1904  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ne 2206  df-nel 2207  df-ral 2311  df-rex 2312  df-reu 2313  df-rab 2315  df-v 2559  df-sbc 2765  df-csb 2853  df-dif 2920  df-un 2922  df-in 2924  df-ss 2931  df-nul 3225  df-pw 3361  df-sn 3381  df-pr 3382  df-op 3384  df-uni 3581  df-int 3616  df-iun 3659  df-br 3765  df-opab 3819  df-mpt 3820  df-tr 3855  df-eprel 4026  df-id 4030  df-po 4033  df-iso 4034  df-iord 4103  df-on 4105  df-suc 4108  df-iom 4314  df-xp 4351  df-rel 4352  df-cnv 4353  df-co 4354  df-dm 4355  df-rn 4356  df-res 4357  df-ima 4358  df-iota 4867  df-fun 4904  df-fn 4905  df-f 4906  df-f1 4907  df-fo 4908  df-f1o 4909  df-fv 4910  df-riota 5468  df-ov 5515  df-oprab 5516  df-mpt2 5517  df-1st 5767  df-2nd 5768  df-recs 5920  df-irdg 5957  df-frec 5978  df-1o 6001  df-2o 6002  df-oadd 6005  df-omul 6006  df-er 6106  df-ec 6108  df-qs 6112  df-ni 6402  df-pli 6403  df-mi 6404  df-lti 6405  df-plpq 6442  df-mpq 6443  df-enq 6445  df-nqqs 6446  df-plqqs 6447  df-mqqs 6448  df-1nqqs 6449  df-rq 6450  df-ltnqqs 6451  df-enq0 6522  df-nq0 6523  df-0nq0 6524  df-plq0 6525  df-mq0 6526  df-inp 6564  df-i1p 6565  df-iplp 6566  df-iltp 6568  df-enr 6811  df-nr 6812  df-ltr 6815  df-0r 6816  df-1r 6817  df-0 6896  df-1 6897  df-r 6899  df-lt 6902  df-pnf 7062  df-mnf 7063  df-xr 7064  df-ltxr 7065  df-le 7066  df-sub 7184  df-neg 7185  df-inn 7915  df-n0 8182  df-z 8246  df-uz 8474  df-iseq 9212
This theorem is referenced by:  ialgcvga  9890
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