Theorem algcvg 12189

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.

Step	Hyp	Ref	Expression
1		nn0uz 9630	. . . 4 |- NN0 = ( ZZ>= ` 0 )
2		algcvg.2	. . . 4 |- R = seq 0 ( ( F o. 1st ) , ( NN0 X. { A } ) )
3		0zd 9332	. . . 4 |- ( A e. S -> 0 e. ZZ )
4		id 19	. . . 4 |- ( A e. S -> A e. S )
5		algcvg.1	. . . . 5 |- F : S --> S
6	5	a1i 9	. . . 4 |- ( A e. S -> F : S --> S )
7	1, 2, 3, 4, 6	algrf 12186	. . 3 |- ( A e. S -> R : NN0 --> S )
8		algcvg.5	. . . 4 |- N = ( C ` A )
9		algcvg.3	. . . . 5 |- C : S --> NN0
10	9	ffvelcdmi 5693	. . . 4 |- ( A e. S -> ( C ` A ) e. NN0 )
11	8, 10	eqeltrid 2280	. . 3 |- ( A e. S -> N e. NN0 )
12		fvco3 5629	. . 3 |- ( ( R : NN0 --> S /\ N e. NN0 ) -> ( ( C o. R ) ` N ) = ( C ` ( R ` N ) ) )
13	7, 11, 12	syl2anc 411	. 2 |- ( A e. S -> ( ( C o. R ) ` N ) = ( C ` ( R ` N ) ) )
14		fco 5420	. . . 4 |- ( ( C : S --> NN0 /\ R : NN0 --> S ) -> ( C o. R ) : NN0 --> NN0 )
15	9, 7, 14	sylancr 414	. . 3 |- ( A e. S -> ( C o. R ) : NN0 --> NN0 )
16		0nn0 9258	. . . . . 6 |- 0 e. NN0
17		fvco3 5629	. . . . . 6 |- ( ( R : NN0 --> S /\ 0 e. NN0 ) -> ( ( C o. R ) ` 0 ) = ( C ` ( R ` 0 ) ) )
18	7, 16, 17	sylancl 413	. . . . 5 |- ( A e. S -> ( ( C o. R ) ` 0 ) = ( C ` ( R ` 0 ) ) )
19	1, 2, 3, 4, 6	ialgr0 12185	. . . . . 6 |- ( A e. S -> ( R ` 0 ) = A )
20	19	fveq2d 5559	. . . . 5 |- ( A e. S -> ( C ` ( R ` 0 ) ) = ( C ` A ) )
21	18, 20	eqtrd 2226	. . . 4 |- ( A e. S -> ( ( C o. R ) ` 0 ) = ( C ` A ) )
22	8, 21	eqtr4id 2245	. . 3 |- ( A e. S -> N = ( ( C o. R ) ` 0 ) )
23	7	ffvelcdmda 5694	. . . . 5 |- ( ( A e. S /\ k e. NN0 ) -> ( R ` k ) e. S )
24		2fveq3 5560	. . . . . . . 8 |- ( z = ( R ` k ) -> ( C ` ( F ` z ) ) = ( C ` ( F ` ( R ` k ) ) ) )
25	24	neeq1d 2382	. . . . . . 7 |- ( z = ( R ` k ) -> ( ( C ` ( F ` z ) ) =/= 0 <-> ( C ` ( F ` ( R ` k ) ) ) =/= 0 ) )
26		fveq2 5555	. . . . . . . 8 |- ( z = ( R ` k ) -> ( C ` z ) = ( C ` ( R ` k ) ) )
27	24, 26	breq12d 4043	. . . . . . 7 |- ( z = ( R ` k ) -> ( ( C ` ( F ` z ) ) < ( C ` z ) <-> ( C ` ( F ` ( R ` k ) ) ) < ( C ` ( R ` k ) ) ) )
28	25, 27	imbi12d 234	. . . . . 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 ) ) )
30	28, 29	vtoclga 2827	. . . . 5 |- ( ( R ` k ) e. S -> ( ( C ` ( F ` ( R ` k ) ) ) =/= 0 -> ( C ` ( F ` ( R ` k ) ) ) < ( C ` ( R ` k ) ) ) )
31	23, 30	syl 14	. . . 4 |- ( ( A e. S /\ k e. NN0 ) -> ( ( C ` ( F ` ( R ` k ) ) ) =/= 0 -> ( C ` ( F ` ( R ` k ) ) ) < ( C ` ( R ` k ) ) ) )
32		peano2nn0 9283	. . . . . . 7 |- ( k e. NN0 -> ( k + 1 ) e. NN0 )
33		fvco3 5629	. . . . . . 7 |- ( ( R : NN0 --> S /\ ( k + 1 ) e. NN0 ) -> ( ( C o. R ) ` ( k + 1 ) ) = ( C ` ( R ` ( k + 1 ) ) ) )
34	7, 32, 33	syl2an 289	. . . . . 6 |- ( ( A e. S /\ k e. NN0 ) -> ( ( C o. R ) ` ( k + 1 ) ) = ( C ` ( R ` ( k + 1 ) ) ) )
35	1, 2, 3, 4, 6	algrp1 12187	. . . . . . 7 |- ( ( A e. S /\ k e. NN0 ) -> ( R ` ( k + 1 ) ) = ( F ` ( R ` k ) ) )
36	35	fveq2d 5559	. . . . . 6 |- ( ( A e. S /\ k e. NN0 ) -> ( C ` ( R ` ( k + 1 ) ) ) = ( C ` ( F ` ( R ` k ) ) ) )
37	34, 36	eqtrd 2226	. . . . 5 |- ( ( A e. S /\ k e. NN0 ) -> ( ( C o. R ) ` ( k + 1 ) ) = ( C ` ( F ` ( R ` k ) ) ) )
38	37	neeq1d 2382	. . . 4 |- ( ( A e. S /\ k e. NN0 ) -> ( ( ( C o. R ) ` ( k + 1 ) ) =/= 0 <-> ( C ` ( F ` ( R ` k ) ) ) =/= 0 ) )
39		fvco3 5629	. . . . . 6 |- ( ( R : NN0 --> S /\ k e. NN0 ) -> ( ( C o. R ) ` k ) = ( C ` ( R ` k ) ) )
40	7, 39	sylan 283	. . . . 5 |- ( ( A e. S /\ k e. NN0 ) -> ( ( C o. R ) ` k ) = ( C ` ( R ` k ) ) )
41	37, 40	breq12d 4043	. . . 4 |- ( ( A e. S /\ k e. NN0 ) -> ( ( ( C o. R ) ` ( k + 1 ) ) < ( ( C o. R ) ` k ) <-> ( C ` ( F ` ( R ` k ) ) ) < ( C ` ( R ` k ) ) ) )
42	31, 38, 41	3imtr4d 203	. . 3 |- ( ( A e. S /\ k e. NN0 ) -> ( ( ( C o. R ) ` ( k + 1 ) ) =/= 0 -> ( ( C o. R ) ` ( k + 1 ) ) < ( ( C o. R ) ` k ) ) )
43	15, 22, 42	nn0seqcvgd 12182	. 2 |- ( A e. S -> ( ( C o. R ) ` N ) = 0 )
44	13, 43	eqtr3d 2228	1 |- ( A e. S -> ( C ` ( R ` N ) ) = 0 )

Syntax hints:   -> wi 4   /\ wa 104   = wceq 1364   e. wcel 2164   =/= wne 2364   {csn 3619   class class class wbr 4030   X. cxp 4658   o. ccom 4664   -->wf 5251   ` cfv 5255  (class class class)co 5919   1stc1st 6193   0cc0 7874   1c1 7875   + caddc 7877   < clt 8056   NN0cn0 9243   seqcseq 10521

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 615  ax-in2 616  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2166  ax-14 2167  ax-ext 2175  ax-coll 4145  ax-sep 4148  ax-nul 4156  ax-pow 4204  ax-pr 4239  ax-un 4465  ax-setind 4570  ax-iinf 4621  ax-cnex 7965  ax-resscn 7966  ax-1cn 7967  ax-1re 7968  ax-icn 7969  ax-addcl 7970  ax-addrcl 7971  ax-mulcl 7972  ax-addcom 7974  ax-addass 7976  ax-distr 7978  ax-i2m1 7979  ax-0lt1 7980  ax-0id 7982  ax-rnegex 7983  ax-cnre 7985  ax-pre-ltirr 7986  ax-pre-ltwlin 7987  ax-pre-lttrn 7988  ax-pre-apti 7989  ax-pre-ltadd 7990

This theorem depends on definitions:  df-bi 117  df-dc 836  df-3or 981  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1472  df-sb 1774  df-eu 2045  df-mo 2046  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-ne 2365  df-nel 2460  df-ral 2477  df-rex 2478  df-reu 2479  df-rab 2481  df-v 2762  df-sbc 2987  df-csb 3082  df-dif 3156  df-un 3158  df-in 3160  df-ss 3167  df-nul 3448  df-pw 3604  df-sn 3625  df-pr 3626  df-op 3628  df-uni 3837  df-int 3872  df-iun 3915  df-br 4031  df-opab 4092  df-mpt 4093  df-tr 4129  df-id 4325  df-iord 4398  df-on 4400  df-ilim 4401  df-suc 4403  df-iom 4624  df-xp 4666  df-rel 4667  df-cnv 4668  df-co 4669  df-dm 4670  df-rn 4671  df-res 4672  df-ima 4673  df-iota 5216  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-riota 5874  df-ov 5922  df-oprab 5923  df-mpo 5924  df-1st 6195  df-2nd 6196  df-recs 6360  df-frec 6446  df-pnf 8058  df-mnf 8059  df-xr 8060  df-ltxr 8061  df-le 8062  df-sub 8194  df-neg 8195  df-inn 8985  df-n0 9244  df-z 9321  df-uz 9596  df-seqfrec 10522

This theorem is referenced by:  algcvga  12192