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Mirrors > Home > ILE Home > Th. List > algrf | Unicode version |
Description: An algorithm is a step
function on a state space .
An algorithm acts on an initial state by
iteratively applying
to give , , and so
on. An algorithm is said to halt if a fixed point of is reached
after a finite number of iterations.
The algorithm iterator "runs" the algorithm so that is the state after iterations of on the initial state . Domain and codomain of the algorithm iterator . (Contributed by Paul Chapman, 31-Mar-2011.) (Revised by Mario Carneiro, 28-May-2014.) |
Ref | Expression |
---|---|
algrf.1 | |
algrf.2 | |
algrf.3 | |
algrf.4 | |
algrf.5 |
Ref | Expression |
---|---|
algrf |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | algrf.1 | . . 3 | |
2 | algrf.3 | . . 3 | |
3 | algrf.4 | . . . . 5 | |
4 | fvconst2g 5627 | . . . . 5 | |
5 | 3, 4 | sylan 281 | . . . 4 |
6 | 3 | adantr 274 | . . . 4 |
7 | 5, 6 | eqeltrd 2214 | . . 3 |
8 | vex 2684 | . . . . 5 | |
9 | vex 2684 | . . . . 5 | |
10 | 8, 9 | algrflem 6119 | . . . 4 |
11 | algrf.5 | . . . . 5 | |
12 | simpl 108 | . . . . 5 | |
13 | ffvelrn 5546 | . . . . 5 | |
14 | 11, 12, 13 | syl2an 287 | . . . 4 |
15 | 10, 14 | eqeltrid 2224 | . . 3 |
16 | 1, 2, 7, 15 | seqf 10227 | . 2 |
17 | algrf.2 | . . 3 | |
18 | 17 | feq1i 5260 | . 2 |
19 | 16, 18 | sylibr 133 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 103 wceq 1331 wcel 1480 csn 3522 cxp 4532 ccom 4538 wf 5114 cfv 5118 (class class class)co 5767 c1st 6029 cz 9047 cuz 9319 cseq 10211 |
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 2119 ax-coll 4038 ax-sep 4041 ax-nul 4049 ax-pow 4093 ax-pr 4126 ax-un 4350 ax-setind 4447 ax-iinf 4497 ax-cnex 7704 ax-resscn 7705 ax-1cn 7706 ax-1re 7707 ax-icn 7708 ax-addcl 7709 ax-addrcl 7710 ax-mulcl 7711 ax-addcom 7713 ax-addass 7715 ax-distr 7717 ax-i2m1 7718 ax-0lt1 7719 ax-0id 7721 ax-rnegex 7722 ax-cnre 7724 ax-pre-ltirr 7725 ax-pre-ltwlin 7726 ax-pre-lttrn 7727 ax-pre-ltadd 7729 |
This theorem depends on definitions: df-bi 116 df-3or 963 df-3an 964 df-tru 1334 df-fal 1337 df-nf 1437 df-sb 1736 df-eu 2000 df-mo 2001 df-clab 2124 df-cleq 2130 df-clel 2133 df-nfc 2268 df-ne 2307 df-nel 2402 df-ral 2419 df-rex 2420 df-reu 2421 df-rab 2423 df-v 2683 df-sbc 2905 df-csb 2999 df-dif 3068 df-un 3070 df-in 3072 df-ss 3079 df-nul 3359 df-pw 3507 df-sn 3528 df-pr 3529 df-op 3531 df-uni 3732 df-int 3767 df-iun 3810 df-br 3925 df-opab 3985 df-mpt 3986 df-tr 4022 df-id 4210 df-iord 4283 df-on 4285 df-ilim 4286 df-suc 4288 df-iom 4500 df-xp 4540 df-rel 4541 df-cnv 4542 df-co 4543 df-dm 4544 df-rn 4545 df-res 4546 df-ima 4547 df-iota 5083 df-fun 5120 df-fn 5121 df-f 5122 df-f1 5123 df-fo 5124 df-f1o 5125 df-fv 5126 df-riota 5723 df-ov 5770 df-oprab 5771 df-mpo 5772 df-1st 6031 df-2nd 6032 df-recs 6195 df-frec 6281 df-pnf 7795 df-mnf 7796 df-xr 7797 df-ltxr 7798 df-le 7799 df-sub 7928 df-neg 7929 df-inn 8714 df-n0 8971 df-z 9048 df-uz 9320 df-seqfrec 10212 |
This theorem is referenced by: algrp1 11716 alginv 11717 algcvg 11718 algcvga 11721 algfx 11722 eucalgcvga 11728 eucalg 11729 |
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