Theorem algcvg 13744

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	|- S e. _V
algcvg.2	|- F:S-->S
algcvg.3	|- R = ((F o. (1st |` (S X. S))) seq0 (NN0 X. {A}))
algcvg.4	|- C:S-->NN0
algcvg.5	|- (z e. S -> ((C` (F` z)) =/= 0 -> (C` (F` z)) < (C` z)))
algcvg.6	|- 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

Proof of Theorem algcvg

Step	Hyp	Ref	Expression
1		algcvg.1	. . . 4 |- S e. _V
2		algcvg.2	. . . 4 |- F:S-->S
3		algcvg.3	. . . 4 |- R = ((F o. (1st |` (S X. S))) seq0 (NN0 X. {A}))
4	1, 2, 3	algrf 13739	. . 3 |- (A e. S -> R:NN0-->S)
5		algcvg.4	. . . . 5 |- C:S-->NN0
6	5	ffvelrni 4788	. . . 4 |- (A e. S -> (C` A) e. NN0)
7		algcvg.6	. . . 4 |- N = (C` A)
8	6, 7	syl5eqel 1975	. . 3 |- (A e. S -> N e. NN0)
9		ffun 4565	. . . . 5 |- (C:S-->NN0 -> Fun C)
10	5, 9	ax-mp 7	. . . 4 |- Fun C
11		fvco3 4739	. . . 4 |- ((Fun C /\ R:NN0-->S /\ N e. NN0) -> ((C o. R)` N) = (C` (R` N)))
12	10, 11	mp3an1 1178	. . 3 |- ((R:NN0-->S /\ N e. NN0) -> ((C o. R)` N) = (C` (R` N)))
13	4, 8, 12	syl11anc 524	. 2 |- (A e. S -> ((C o. R)` N) = (C` (R` N)))
14		fco 4573	. . . . 5 |- ((C:S-->NN0 /\ R:NN0-->S) -> (C o. R):NN0-->NN0)
15	5, 14	mpan 759	. . . 4 |- (R:NN0-->S -> (C o. R):NN0-->NN0)
16	4, 15	syl 12	. . 3 |- (A e. S -> (C o. R):NN0-->NN0)
17		0nn0 7322	. . . . . . 7 |- 0 e. NN0
18		fvco3 4739	. . . . . . 7 |- ((Fun C /\ R:NN0-->S /\ 0 e. NN0) -> ((C o. R)` 0) = (C` (R` 0)))
19	10, 17, 18	mp3an13 1182	. . . . . 6 |- (R:NN0-->S -> ((C o. R)` 0) = (C` (R` 0)))
20	4, 19	syl 12	. . . . 5 |- (A e. S -> ((C o. R)` 0) = (C` (R` 0)))
21	1, 2, 3	algr0 13740	. . . . . 6 |- (A e. S -> (R` 0) = A)
22	21	fveq2d 4685	. . . . 5 |- (A e. S -> (C` (R` 0)) = (C` A))
23	20, 22	eqtrd 1925	. . . 4 |- (A e. S -> ((C o. R)` 0) = (C` A))
24	23, 7	syl6reqr 1947	. . 3 |- (A e. S -> N = ((C o. R)` 0))
25		ffvelrn 4787	. . . . . 6 |- ((R:NN0-->S /\ k e. NN0) -> (R` k) e. S)
26	25, 4	sylan 497	. . . . 5 |- ((A e. S /\ k e. NN0) -> (R` k) e. S)
27		fveq2 4681	. . . . . . . . 9 |- (z = (R` k) -> (F` z) = (F` (R` k)))
28	27	fveq2d 4685	. . . . . . . 8 |- (z = (R` k) -> (C` (F` z)) = (C` (F` (R` k))))
29	28	neeq1d 2028	. . . . . . 7 |- (z = (R` k) -> ((C` (F` z)) =/= 0 <-> (C` (F` (R` k))) =/= 0))
30		fveq2 4681	. . . . . . . 8 |- (z = (R` k) -> (C` z) = (C` (R` k)))
31	28, 30	breq12d 3351	. . . . . . 7 |- (z = (R` k) -> ((C` (F` z)) < (C` z) <-> (C` (F` (R` k))) < (C` (R` k))))
32	29, 31	imbi12d 688	. . . . . 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)))))
33		algcvg.5	. . . . . . 7 |- (z e. S -> ((C` (F` z)) =/= 0 -> (C` (F` z)) < (C` z)))
34	33	rgen 2159	. . . . . 6 |- A.z e. S ((C` (F` z)) =/= 0 -> (C` (F` z)) < (C` z))
35	32, 34	vtoclri 2360	. . . . 5 |- ((R` k) e. S -> ((C` (F` (R` k))) =/= 0 -> (C` (F` (R` k))) < (C` (R` k))))
36	26, 35	syl 12	. . . 4 |- ((A e. S /\ k e. NN0) -> ((C` (F` (R` k))) =/= 0 -> (C` (F` (R` k))) < (C` (R` k))))
37		fvco3 4739	. . . . . . . . 9 |- ((Fun C /\ R:NN0-->S /\ (k + 1) e. NN0) -> ((C o. R)` (k + 1)) = (C` (R` (k + 1))))
38	10, 37	mp3an1 1178	. . . . . . . 8 |- ((R:NN0-->S /\ (k + 1) e. NN0) -> ((C o. R)` (k + 1)) = (C` (R` (k + 1))))
39		peano2nn0 7333	. . . . . . . 8 |- (k e. NN0 -> (k + 1) e. NN0)
40	38, 4, 39	syl2an 503	. . . . . . 7 |- ((A e. S /\ k e. NN0) -> ((C o. R)` (k + 1)) = (C` (R` (k + 1))))
41	1, 2, 3	algrp1 13742	. . . . . . . 8 |- ((A e. S /\ k e. NN0) -> (R` (k + 1)) = (F` (R` k)))
42	41	fveq2d 4685	. . . . . . 7 |- ((A e. S /\ k e. NN0) -> (C` (R` (k + 1))) = (C` (F` (R` k))))
43	40, 42	eqtrd 1925	. . . . . 6 |- ((A e. S /\ k e. NN0) -> ((C o. R)` (k + 1)) = (C` (F` (R` k))))
44	43	neeq1d 2028	. . . . 5 |- ((A e. S /\ k e. NN0) -> (((C o. R)` (k + 1)) =/= 0 <-> (C` (F` (R` k))) =/= 0))
45		fvco3 4739	. . . . . . . 8 |- ((Fun C /\ R:NN0-->S /\ k e. NN0) -> ((C o. R)` k) = (C` (R` k)))
46	10, 45	mp3an1 1178	. . . . . . 7 |- ((R:NN0-->S /\ k e. NN0) -> ((C o. R)` k) = (C` (R` k)))
47	46, 4	sylan 497	. . . . . 6 |- ((A e. S /\ k e. NN0) -> ((C o. R)` k) = (C` (R` k)))
48	43, 47	breq12d 3351	. . . . 5 |- ((A e. S /\ k e. NN0) -> (((C o. R)` (k + 1)) < ((C o. R)` k) <-> (C` (F` (R` k))) < (C` (R` k))))
49	44, 48	imbi12d 688	. . . 4 |- ((A e. S /\ k e. NN0) -> ((((C o. R)` (k + 1)) =/= 0 -> ((C o. R)` (k + 1)) < ((C o. R)` k)) <-> ((C` (F` (R` k))) =/= 0 -> (C` (F` (R` k))) < (C` (R` k)))))
50	36, 49	mpbird 213	. . 3 |- ((A e. S /\ k e. NN0) -> (((C o. R)` (k + 1)) =/= 0 -> ((C o. R)` (k + 1)) < ((C o. R)` k)))
51	16, 24, 50	nn0seqcvgd 13659	. 2 |- (A e. S -> ((C o. R)` N) = 0)
52	13, 51	eqtr3d 1927	1 |- (A e. S -> (C` (R` N)) = 0)

Syntax hints:   -> wi 3   /\ wa 240   = wceq 1298   e. wcel 1300   =/= wne 2017  _Vcvv 2292  {csn 3044   class class class wbr 3338   X. cxp 3984   |` cres 3988   o. ccom 3990  Fun wfun 3992  -->wf 3994  ` cfv 3998  (class class class)co 4884  1stc1st 5018  0cc0 6386  1c1 6387   + caddc 6389  NN0cn0 6450   < clt 6653   seq0 cseq0 7775

This theorem is referenced by:  algcvga 13747

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 1304  ax-gen 1305  ax-8 1306  ax-9 1307  ax-10 1308  ax-11 1309  ax-12 1310  ax-13 1311  ax-14 1312  ax-17 1317  ax-4 1319  ax-5o 1321  ax-6o 1324  ax-9o 1481  ax-10o 1500  ax-16 1580  ax-11o 1588  ax-ext 1865  ax-rep 3428  ax-sep 3438  ax-nul 3445  ax-pow 3481  ax-pr 3524  ax-un 3790  ax-inf2 5731

This theorem depends on definitions:  df-bi 164  df-or 241  df-an 242  df-3or 859  df-3an 860  df-ex 1327  df-sb 1536  df-eu 1775  df-mo 1776  df-clab 1872  df-cleq 1877  df-clel 1880  df-ne 2019  df-nel 2020  df-ral 2109  df-rex 2110  df-reu 2111  df-rab 2112  df-v 2294  df-sbc 2454  df-csb 2541  df-dif 2597  df-un 2600  df-in 2603  df-ss 2605  df-pss 2607  df-nul 2876  df-if 2983  df-pw 3035  df-sn 3049  df-pr 3050  df-tp 3052  df-op 3053  df-uni 3178  df-int 3215  df-iun 3257  df-br 3339  df-opab 3396  df-tr 3412  df-eprel 3583  df-id 3586  df-po 3591  df-so 3604  df-fr 3625  df-we 3644  df-ord 3660  df-on 3661  df-lim 3662  df-suc 3663  df-om 3950  df-xp 4000  df-rel 4001  df-cnv 4002  df-co 4003  df-dm 4004  df-rn 4005  df-res 4006  df-ima 4007  df-fun 4008  df-fn 4009  df-f 4010  df-f1 4011  df-fo 4012  df-f1o 4013  df-fv 4014  df-opr 4886  df-oprab 4887  df-mpt 5006  df-1st 5020  df-2nd 5021  df-iota 5089  df-rdg 5140  df-1o 5177  df-oadd 5179  df-omul 5180  df-er 5318  df-ec 5320  df-qs 5323  df-en 5427  df-dom 5428  df-sdom 5429  df-undef 5556  df-riota 5560  df-ni 6152  df-pli 6153  df-mi 6154  df-lti 6155  df-plpq 6187  df-mpq 6188  df-enq 6189  df-nq 6190  df-plq 6191  df-mq 6192  df-rq 6193  df-ltq 6194  df-1q 6195  df-np 6238  df-1p 6239  df-plp 6240  df-mp 6241  df-ltp 6242  df-plpr 6316  df-mpr 6317  df-enr 6318  df-nr 6319  df-plr 6320  df-mr 6321  df-ltr 6322  df-0r 6323  df-1r 6324  df-m1r 6325  df-c 6392  df-0 6393  df-1 6394  df-i 6395  df-r 6396  df-plus 6397  df-mul 6398  df-lt 6399  df-sub 6511  df-neg 6513  df-pnf 6654  df-mnf 6655  df-xr 6656  df-ltxr 6657  df-le 6658  df-n 7108  df-n0 7309  df-z 7345  df-uz 7587  df-seq1 7721  df-shft 7754  df-seqz 7776  df-seq0 7777

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