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Mirrors > Home > ILE Home > Th. List > algcvg | Unicode version |
Description: One way to prove that an
algorithm halts is to construct a countdown
function whose value is guaranteed to decrease for
each iteration of until it reaches . That is, if
is not a fixed point of , then
.
If is a countdown function for algorithm , the sequence reaches after at most steps, where is the value of for the initial state . (Contributed by Paul Chapman, 22-Jun-2011.) |
Ref | Expression |
---|---|
algcvg.1 | |
algcvg.2 | |
algcvg.3 | |
algcvg.4 | |
algcvg.5 |
Ref | Expression |
---|---|
algcvg |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nn0uz 9360 | . . . 4 | |
2 | algcvg.2 | . . . 4 | |
3 | 0zd 9066 | . . . 4 | |
4 | id 19 | . . . 4 | |
5 | algcvg.1 | . . . . 5 | |
6 | 5 | a1i 9 | . . . 4 |
7 | 1, 2, 3, 4, 6 | algrf 11726 | . . 3 |
8 | algcvg.5 | . . . 4 | |
9 | algcvg.3 | . . . . 5 | |
10 | 9 | ffvelrni 5554 | . . . 4 |
11 | 8, 10 | eqeltrid 2226 | . . 3 |
12 | fvco3 5492 | . . 3 | |
13 | 7, 11, 12 | syl2anc 408 | . 2 |
14 | fco 5288 | . . . 4 | |
15 | 9, 7, 14 | sylancr 410 | . . 3 |
16 | 0nn0 8992 | . . . . . 6 | |
17 | fvco3 5492 | . . . . . 6 | |
18 | 7, 16, 17 | sylancl 409 | . . . . 5 |
19 | 1, 2, 3, 4, 6 | ialgr0 11725 | . . . . . 6 |
20 | 19 | fveq2d 5425 | . . . . 5 |
21 | 18, 20 | eqtrd 2172 | . . . 4 |
22 | 21, 8 | syl6reqr 2191 | . . 3 |
23 | 7 | ffvelrnda 5555 | . . . . 5 |
24 | 2fveq3 5426 | . . . . . . . 8 | |
25 | 24 | neeq1d 2326 | . . . . . . 7 |
26 | fveq2 5421 | . . . . . . . 8 | |
27 | 24, 26 | breq12d 3942 | . . . . . . 7 |
28 | 25, 27 | imbi12d 233 | . . . . . 6 |
29 | algcvg.4 | . . . . . 6 | |
30 | 28, 29 | vtoclga 2752 | . . . . 5 |
31 | 23, 30 | syl 14 | . . . 4 |
32 | peano2nn0 9017 | . . . . . . 7 | |
33 | fvco3 5492 | . . . . . . 7 | |
34 | 7, 32, 33 | syl2an 287 | . . . . . 6 |
35 | 1, 2, 3, 4, 6 | algrp1 11727 | . . . . . . 7 |
36 | 35 | fveq2d 5425 | . . . . . 6 |
37 | 34, 36 | eqtrd 2172 | . . . . 5 |
38 | 37 | neeq1d 2326 | . . . 4 |
39 | fvco3 5492 | . . . . . 6 | |
40 | 7, 39 | sylan 281 | . . . . 5 |
41 | 37, 40 | breq12d 3942 | . . . 4 |
42 | 31, 38, 41 | 3imtr4d 202 | . . 3 |
43 | 15, 22, 42 | nn0seqcvgd 11722 | . 2 |
44 | 13, 43 | eqtr3d 2174 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 103 wceq 1331 wcel 1480 wne 2308 csn 3527 class class class wbr 3929 cxp 4537 ccom 4543 wf 5119 cfv 5123 (class class class)co 5774 c1st 6036 cc0 7620 c1 7621 caddc 7623 clt 7800 cn0 8977 cseq 10218 |
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 7711 ax-resscn 7712 ax-1cn 7713 ax-1re 7714 ax-icn 7715 ax-addcl 7716 ax-addrcl 7717 ax-mulcl 7718 ax-addcom 7720 ax-addass 7722 ax-distr 7724 ax-i2m1 7725 ax-0lt1 7726 ax-0id 7728 ax-rnegex 7729 ax-cnre 7731 ax-pre-ltirr 7732 ax-pre-ltwlin 7733 ax-pre-lttrn 7734 ax-pre-apti 7735 ax-pre-ltadd 7736 |
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 7802 df-mnf 7803 df-xr 7804 df-ltxr 7805 df-le 7806 df-sub 7935 df-neg 7936 df-inn 8721 df-n0 8978 df-z 9055 df-uz 9327 df-seqfrec 10219 |
This theorem is referenced by: algcvga 11732 |
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