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Theorem algcvg 11740
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 } ) )
algcvg.3  |-  C : S
--> NN0
algcvg.4  |-  ( z  e.  S  ->  (
( C `  ( F `  z )
)  =/=  0  -> 
( C `  ( F `  z )
)  <  ( C `  z ) ) )
algcvg.5  |-  N  =  ( C `  A
)
Assertion
Ref Expression
algcvg  |-  ( 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)

Proof of Theorem algcvg
Dummy variable  k is distinct from all other variables.
StepHypRef Expression
1 nn0uz 9372 . . . 4  |-  NN0  =  ( ZZ>= `  0 )
2 algcvg.2 . . . 4  |-  R  =  seq 0 ( ( F  o.  1st ) ,  ( NN0  X.  { A } ) )
3 0zd 9078 . . . 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 )
71, 2, 3, 4, 6algrf 11737 . . 3  |-  ( A  e.  S  ->  R : NN0 --> S )
8 algcvg.5 . . . 4  |-  N  =  ( C `  A
)
9 algcvg.3 . . . . 5  |-  C : S
--> NN0
109ffvelrni 5554 . . . 4  |-  ( A  e.  S  ->  ( C `  A )  e.  NN0 )
118, 10eqeltrid 2226 . . 3  |-  ( A  e.  S  ->  N  e.  NN0 )
12 fvco3 5492 . . 3  |-  ( ( R : NN0 --> S  /\  N  e.  NN0 )  -> 
( ( C  o.  R ) `  N
)  =  ( C `
 ( R `  N ) ) )
137, 11, 12syl2anc 408 . 2  |-  ( A  e.  S  ->  (
( C  o.  R
) `  N )  =  ( C `  ( R `  N ) ) )
14 fco 5288 . . . 4  |-  ( ( C : S --> NN0  /\  R : NN0 --> S )  ->  ( C  o.  R ) : NN0 --> NN0 )
159, 7, 14sylancr 410 . . 3  |-  ( A  e.  S  ->  ( C  o.  R ) : NN0 --> NN0 )
16 0nn0 9004 . . . . . 6  |-  0  e.  NN0
17 fvco3 5492 . . . . . 6  |-  ( ( R : NN0 --> S  /\  0  e.  NN0 )  -> 
( ( C  o.  R ) `  0
)  =  ( C `
 ( R ` 
0 ) ) )
187, 16, 17sylancl 409 . . . . 5  |-  ( A  e.  S  ->  (
( C  o.  R
) `  0 )  =  ( C `  ( R `  0 ) ) )
191, 2, 3, 4, 6ialgr0 11736 . . . . . 6  |-  ( A  e.  S  ->  ( R `  0 )  =  A )
2019fveq2d 5425 . . . . 5  |-  ( A  e.  S  ->  ( C `  ( R `  0 ) )  =  ( C `  A ) )
2118, 20eqtrd 2172 . . . 4  |-  ( A  e.  S  ->  (
( C  o.  R
) `  0 )  =  ( C `  A ) )
2221, 8syl6reqr 2191 . . 3  |-  ( A  e.  S  ->  N  =  ( ( C  o.  R ) ` 
0 ) )
237ffvelrnda 5555 . . . . 5  |-  ( ( A  e.  S  /\  k  e.  NN0 )  -> 
( R `  k
)  e.  S )
24 2fveq3 5426 . . . . . . . 8  |-  ( z  =  ( R `  k )  ->  ( C `  ( F `  z ) )  =  ( C `  ( F `  ( R `  k ) ) ) )
2524neeq1d 2326 . . . . . . 7  |-  ( z  =  ( R `  k )  ->  (
( C `  ( F `  z )
)  =/=  0  <->  ( C `  ( F `  ( R `  k
) ) )  =/=  0 ) )
26 fveq2 5421 . . . . . . . 8  |-  ( z  =  ( R `  k )  ->  ( C `  z )  =  ( C `  ( R `  k ) ) )
2724, 26breq12d 3942 . . . . . . 7  |-  ( z  =  ( R `  k )  ->  (
( C `  ( F `  z )
)  <  ( C `  z )  <->  ( C `  ( F `  ( R `  k )
) )  <  ( C `  ( R `  k ) ) ) )
2825, 27imbi12d 233 . . . . . 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 ) ) ) ) )
29 algcvg.4 . . . . . 6  |-  ( z  e.  S  ->  (
( C `  ( F `  z )
)  =/=  0  -> 
( C `  ( F `  z )
)  <  ( C `  z ) ) )
3028, 29vtoclga 2752 . . . . 5  |-  ( ( R `  k )  e.  S  ->  (
( C `  ( F `  ( R `  k ) ) )  =/=  0  ->  ( C `  ( F `  ( R `  k
) ) )  < 
( C `  ( R `  k )
) ) )
3123, 30syl 14 . . . 4  |-  ( ( A  e.  S  /\  k  e.  NN0 )  -> 
( ( C `  ( F `  ( R `
 k ) ) )  =/=  0  -> 
( C `  ( F `  ( R `  k ) ) )  <  ( C `  ( R `  k ) ) ) )
32 peano2nn0 9029 . . . . . . 7  |-  ( k  e.  NN0  ->  ( k  +  1 )  e. 
NN0 )
33 fvco3 5492 . . . . . . 7  |-  ( ( R : NN0 --> S  /\  ( k  +  1 )  e.  NN0 )  ->  ( ( C  o.  R ) `  (
k  +  1 ) )  =  ( C `
 ( R `  ( k  +  1 ) ) ) )
347, 32, 33syl2an 287 . . . . . 6  |-  ( ( A  e.  S  /\  k  e.  NN0 )  -> 
( ( C  o.  R ) `  (
k  +  1 ) )  =  ( C `
 ( R `  ( k  +  1 ) ) ) )
351, 2, 3, 4, 6algrp1 11738 . . . . . . 7  |-  ( ( A  e.  S  /\  k  e.  NN0 )  -> 
( R `  (
k  +  1 ) )  =  ( F `
 ( R `  k ) ) )
3635fveq2d 5425 . . . . . 6  |-  ( ( A  e.  S  /\  k  e.  NN0 )  -> 
( C `  ( R `  ( k  +  1 ) ) )  =  ( C `
 ( F `  ( R `  k ) ) ) )
3734, 36eqtrd 2172 . . . . 5  |-  ( ( A  e.  S  /\  k  e.  NN0 )  -> 
( ( C  o.  R ) `  (
k  +  1 ) )  =  ( C `
 ( F `  ( R `  k ) ) ) )
3837neeq1d 2326 . . . 4  |-  ( ( A  e.  S  /\  k  e.  NN0 )  -> 
( ( ( C  o.  R ) `  ( k  +  1 ) )  =/=  0  <->  ( C `  ( F `
 ( R `  k ) ) )  =/=  0 ) )
39 fvco3 5492 . . . . . 6  |-  ( ( R : NN0 --> S  /\  k  e.  NN0 )  -> 
( ( C  o.  R ) `  k
)  =  ( C `
 ( R `  k ) ) )
407, 39sylan 281 . . . . 5  |-  ( ( A  e.  S  /\  k  e.  NN0 )  -> 
( ( C  o.  R ) `  k
)  =  ( C `
 ( R `  k ) ) )
4137, 40breq12d 3942 . . . 4  |-  ( ( A  e.  S  /\  k  e.  NN0 )  -> 
( ( ( C  o.  R ) `  ( k  +  1 ) )  <  (
( C  o.  R
) `  k )  <->  ( C `  ( F `
 ( R `  k ) ) )  <  ( C `  ( R `  k ) ) ) )
4231, 38, 413imtr4d 202 . . 3  |-  ( ( A  e.  S  /\  k  e.  NN0 )  -> 
( ( ( C  o.  R ) `  ( k  +  1 ) )  =/=  0  ->  ( ( C  o.  R ) `  (
k  +  1 ) )  <  ( ( C  o.  R ) `
 k ) ) )
4315, 22, 42nn0seqcvgd 11733 . 2  |-  ( A  e.  S  ->  (
( C  o.  R
) `  N )  =  0 )
4413, 43eqtr3d 2174 1  |-  ( A  e.  S  ->  ( C `  ( R `  N ) )  =  0 )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 103    = wceq 1331    e. wcel 1480    =/= wne 2308   {csn 3527   class class class wbr 3929    X. cxp 4537    o. ccom 4543   -->wf 5119   ` cfv 5123  (class class class)co 5774   1stc1st 6036   0cc0 7632   1c1 7633    + caddc 7635    < clt 7812   NN0cn0 8989    seqcseq 10230
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 603  ax-in2 604  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-13 1491  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121  ax-coll 4043  ax-sep 4046  ax-nul 4054  ax-pow 4098  ax-pr 4131  ax-un 4355  ax-setind 4452  ax-iinf 4502  ax-cnex 7723  ax-resscn 7724  ax-1cn 7725  ax-1re 7726  ax-icn 7727  ax-addcl 7728  ax-addrcl 7729  ax-mulcl 7730  ax-addcom 7732  ax-addass 7734  ax-distr 7736  ax-i2m1 7737  ax-0lt1 7738  ax-0id 7740  ax-rnegex 7741  ax-cnre 7743  ax-pre-ltirr 7744  ax-pre-ltwlin 7745  ax-pre-lttrn 7746  ax-pre-apti 7747  ax-pre-ltadd 7748
This theorem depends on definitions:  df-bi 116  df-dc 820  df-3or 963  df-3an 964  df-tru 1334  df-fal 1337  df-nf 1437  df-sb 1736  df-eu 2002  df-mo 2003  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-ne 2309  df-nel 2404  df-ral 2421  df-rex 2422  df-reu 2423  df-rab 2425  df-v 2688  df-sbc 2910  df-csb 3004  df-dif 3073  df-un 3075  df-in 3077  df-ss 3084  df-nul 3364  df-pw 3512  df-sn 3533  df-pr 3534  df-op 3536  df-uni 3737  df-int 3772  df-iun 3815  df-br 3930  df-opab 3990  df-mpt 3991  df-tr 4027  df-id 4215  df-iord 4288  df-on 4290  df-ilim 4291  df-suc 4293  df-iom 4505  df-xp 4545  df-rel 4546  df-cnv 4547  df-co 4548  df-dm 4549  df-rn 4550  df-res 4551  df-ima 4552  df-iota 5088  df-fun 5125  df-fn 5126  df-f 5127  df-f1 5128  df-fo 5129  df-f1o 5130  df-fv 5131  df-riota 5730  df-ov 5777  df-oprab 5778  df-mpo 5779  df-1st 6038  df-2nd 6039  df-recs 6202  df-frec 6288  df-pnf 7814  df-mnf 7815  df-xr 7816  df-ltxr 7817  df-le 7818  df-sub 7947  df-neg 7948  df-inn 8733  df-n0 8990  df-z 9067  df-uz 9339  df-seqfrec 10231
This theorem is referenced by:  algcvga  11743
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