Theorem algcvg 12625

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 9791	. . . 4 |- NN0 = ( ZZ>= ` 0 )
2		algcvg.2	. . . 4 |- R = seq 0 ( ( F o. 1st ) , ( NN0 X. { A } ) )
3		0zd 9491	. . . 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 12622	. . 3 |- ( A e. S -> R : NN0 --> S )
8		algcvg.5	. . . 4 |- N = ( C ` A )
9		algcvg.3	. . . . 5 |- C : S --> NN0
10	9	ffvelcdmi 5781	. . . 4 |- ( A e. S -> ( C ` A ) e. NN0 )
11	8, 10	eqeltrid 2318	. . 3 |- ( A e. S -> N e. NN0 )
12		fvco3 5717	. . 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 5500	. . . 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 9417	. . . . . 6 |- 0 e. NN0
17		fvco3 5717	. . . . . 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 12621	. . . . . 6 |- ( A e. S -> ( R ` 0 ) = A )
20	19	fveq2d 5643	. . . . 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 5782	. . . . 5 |- ( ( A e. S /\ k e. NN0 ) -> ( R ` k ) e. S )
24		2fveq3 5644	. . . . . . . 8 |- ( z = ( R ` k ) -> ( C ` ( F ` z ) ) = ( C ` ( F ` ( R ` k ) ) ) )
25	24	neeq1d 2420	. . . . . . 7 |- ( z = ( R ` k ) -> ( ( C ` ( F ` z ) ) =/= 0 <-> ( C ` ( F ` ( R ` k ) ) ) =/= 0 ) )
26		fveq2 5639	. . . . . . . 8 |- ( z = ( R ` k ) -> ( C ` z ) = ( C ` ( R ` k ) ) )
27	24, 26	breq12d 4101	. . . . . . 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 2870	. . . . 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 9442	. . . . . . 7 |- ( k e. NN0 -> ( k + 1 ) e. NN0 )
33		fvco3 5717	. . . . . . 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 12623	. . . . . . 7 |- ( ( A e. S /\ k e. NN0 ) -> ( R ` ( k + 1 ) ) = ( F ` ( R ` k ) ) )
36	35	fveq2d 5643	. . . . . 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 2420	. . . 4 |- ( ( A e. S /\ k e. NN0 ) -> ( ( ( C o. R ) ` ( k + 1 ) ) =/= 0 <-> ( C ` ( F ` ( R ` k ) ) ) =/= 0 ) )
39		fvco3 5717	. . . . . 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 4101	. . . 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 12618	. 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 1397   e. wcel 2202   =/= wne 2402   {csn 3669   class class class wbr 4088   X. cxp 4723   o. ccom 4729   -->wf 5322   ` cfv 5326  (class class class)co 6018   1stc1st 6301   0cc0 8032   1c1 8033   + caddc 8035   < clt 8214   NN0cn0 9402   seqcseq 10710

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 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-13 2204  ax-14 2205  ax-ext 2213  ax-coll 4204  ax-sep 4207  ax-nul 4215  ax-pow 4264  ax-pr 4299  ax-un 4530  ax-setind 4635  ax-iinf 4686  ax-cnex 8123  ax-resscn 8124  ax-1cn 8125  ax-1re 8126  ax-icn 8127  ax-addcl 8128  ax-addrcl 8129  ax-mulcl 8130  ax-addcom 8132  ax-addass 8134  ax-distr 8136  ax-i2m1 8137  ax-0lt1 8138  ax-0id 8140  ax-rnegex 8141  ax-cnre 8143  ax-pre-ltirr 8144  ax-pre-ltwlin 8145  ax-pre-lttrn 8146  ax-pre-apti 8147  ax-pre-ltadd 8148

This theorem depends on definitions:  df-bi 117  df-dc 842  df-3or 1005  df-3an 1006  df-tru 1400  df-fal 1403  df-nf 1509  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-ne 2403  df-nel 2498  df-ral 2515  df-rex 2516  df-reu 2517  df-rab 2519  df-v 2804  df-sbc 3032  df-csb 3128  df-dif 3202  df-un 3204  df-in 3206  df-ss 3213  df-nul 3495  df-pw 3654  df-sn 3675  df-pr 3676  df-op 3678  df-uni 3894  df-int 3929  df-iun 3972  df-br 4089  df-opab 4151  df-mpt 4152  df-tr 4188  df-id 4390  df-iord 4463  df-on 4465  df-ilim 4466  df-suc 4468  df-iom 4689  df-xp 4731  df-rel 4732  df-cnv 4733  df-co 4734  df-dm 4735  df-rn 4736  df-res 4737  df-ima 4738  df-iota 5286  df-fun 5328  df-fn 5329  df-f 5330  df-f1 5331  df-fo 5332  df-f1o 5333  df-fv 5334  df-riota 5971  df-ov 6021  df-oprab 6022  df-mpo 6023  df-1st 6303  df-2nd 6304  df-recs 6471  df-frec 6557  df-pnf 8216  df-mnf 8217  df-xr 8218  df-ltxr 8219  df-le 8220  df-sub 8352  df-neg 8353  df-inn 9144  df-n0 9403  df-z 9480  df-uz 9756  df-seqfrec 10711

This theorem is referenced by:  algcvga  12628