Theorem algcvg 12681

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 9834	. . . 4 |- NN0 = ( ZZ>= ` 0 )
2		algcvg.2	. . . 4 |- R = seq 0 ( ( F o. 1st ) , ( NN0 X. { A } ) )
3		0zd 9534	. . . 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 12678	. . 3 |- ( A e. S -> R : NN0 --> S )
8		algcvg.5	. . . 4 |- N = ( C ` A )
9		algcvg.3	. . . . 5 |- C : S --> NN0
10	9	ffvelcdmi 5789	. . . 4 |- ( A e. S -> ( C ` A ) e. NN0 )
11	8, 10	eqeltrid 2318	. . 3 |- ( A e. S -> N e. NN0 )
12		fvco3 5726	. . 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 5507	. . . 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 9460	. . . . . 6 |- 0 e. NN0
17		fvco3 5726	. . . . . 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 12677	. . . . . 6 |- ( A e. S -> ( R ` 0 ) = A )
20	19	fveq2d 5652	. . . . 5 |- ( A e. S -> ( C ` ( R ` 0 ) ) = ( C ` A ) )
21	18, 20	eqtrd 2264	. . . 4 |- ( A e. S -> ( ( C o. R ) ` 0 ) = ( C ` A ) )
22	8, 21	eqtr4id 2283	. . 3 |- ( A e. S -> N = ( ( C o. R ) ` 0 ) )
23	7	ffvelcdmda 5790	. . . . 5 |- ( ( A e. S /\ k e. NN0 ) -> ( R ` k ) e. S )
24		2fveq3 5653	. . . . . . . 8 |- ( z = ( R ` k ) -> ( C ` ( F ` z ) ) = ( C ` ( F ` ( R ` k ) ) ) )
25	24	neeq1d 2421	. . . . . . 7 |- ( z = ( R ` k ) -> ( ( C ` ( F ` z ) ) =/= 0 <-> ( C ` ( F ` ( R ` k ) ) ) =/= 0 ) )
26		fveq2 5648	. . . . . . . 8 |- ( z = ( R ` k ) -> ( C ` z ) = ( C ` ( R ` k ) ) )
27	24, 26	breq12d 4106	. . . . . . 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 2871	. . . . 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 9485	. . . . . . 7 |- ( k e. NN0 -> ( k + 1 ) e. NN0 )
33		fvco3 5726	. . . . . . 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 12679	. . . . . . 7 |- ( ( A e. S /\ k e. NN0 ) -> ( R ` ( k + 1 ) ) = ( F ` ( R ` k ) ) )
36	35	fveq2d 5652	. . . . . 6 |- ( ( A e. S /\ k e. NN0 ) -> ( C ` ( R ` ( k + 1 ) ) ) = ( C ` ( F ` ( R ` k ) ) ) )
37	34, 36	eqtrd 2264	. . . . 5 |- ( ( A e. S /\ k e. NN0 ) -> ( ( C o. R ) ` ( k + 1 ) ) = ( C ` ( F ` ( R ` k ) ) ) )
38	37	neeq1d 2421	. . . 4 |- ( ( A e. S /\ k e. NN0 ) -> ( ( ( C o. R ) ` ( k + 1 ) ) =/= 0 <-> ( C ` ( F ` ( R ` k ) ) ) =/= 0 ) )
39		fvco3 5726	. . . . . 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 4106	. . . 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 12674	. 2 |- ( A e. S -> ( ( C o. R ) ` N ) = 0 )
44	13, 43	eqtr3d 2266	1 |- ( A e. S -> ( C ` ( R ` N ) ) = 0 )

Syntax hints:   -> wi 4   /\ wa 104   = wceq 1398   e. wcel 2202   =/= wne 2403   {csn 3673   class class class wbr 4093   X. cxp 4729   o. ccom 4735   -->wf 5329   ` cfv 5333  (class class class)co 6028   1stc1st 6310   0cc0 8075   1c1 8076   + caddc 8078   < clt 8257   NN0cn0 9445   seqcseq 10753

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 619  ax-in2 620  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2204  ax-14 2205  ax-ext 2213  ax-coll 4209  ax-sep 4212  ax-nul 4220  ax-pow 4270  ax-pr 4305  ax-un 4536  ax-setind 4641  ax-iinf 4692  ax-cnex 8166  ax-resscn 8167  ax-1cn 8168  ax-1re 8169  ax-icn 8170  ax-addcl 8171  ax-addrcl 8172  ax-mulcl 8173  ax-addcom 8175  ax-addass 8177  ax-distr 8179  ax-i2m1 8180  ax-0lt1 8181  ax-0id 8183  ax-rnegex 8184  ax-cnre 8186  ax-pre-ltirr 8187  ax-pre-ltwlin 8188  ax-pre-lttrn 8189  ax-pre-apti 8190  ax-pre-ltadd 8191

This theorem depends on definitions:  df-bi 117  df-dc 843  df-3or 1006  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2364  df-ne 2404  df-nel 2499  df-ral 2516  df-rex 2517  df-reu 2518  df-rab 2520  df-v 2805  df-sbc 3033  df-csb 3129  df-dif 3203  df-un 3205  df-in 3207  df-ss 3214  df-nul 3497  df-pw 3658  df-sn 3679  df-pr 3680  df-op 3682  df-uni 3899  df-int 3934  df-iun 3977  df-br 4094  df-opab 4156  df-mpt 4157  df-tr 4193  df-id 4396  df-iord 4469  df-on 4471  df-ilim 4472  df-suc 4474  df-iom 4695  df-xp 4737  df-rel 4738  df-cnv 4739  df-co 4740  df-dm 4741  df-rn 4742  df-res 4743  df-ima 4744  df-iota 5293  df-fun 5335  df-fn 5336  df-f 5337  df-f1 5338  df-fo 5339  df-f1o 5340  df-fv 5341  df-riota 5981  df-ov 6031  df-oprab 6032  df-mpo 6033  df-1st 6312  df-2nd 6313  df-recs 6514  df-frec 6600  df-pnf 8259  df-mnf 8260  df-xr 8261  df-ltxr 8262  df-le 8263  df-sub 8395  df-neg 8396  df-inn 9187  df-n0 9446  df-z 9523  df-uz 9799  df-seqfrec 10754

This theorem is referenced by:  algcvga  12684