Theorem algcvg 12770

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 9907	. . . 4 |- NN0 = ( ZZ>= ` 0 )
2		algcvg.2	. . . 4 |- R = seq 0 ( ( F o. 1st ) , ( NN0 X. { A } ) )
3		0zd 9606	. . . 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 12767	. . 3 |- ( A e. S -> R : NN0 --> S )
8		algcvg.5	. . . 4 |- N = ( C ` A )
9		algcvg.3	. . . . 5 |- C : S --> NN0
10	9	ffvelcdmi 5816	. . . 4 |- ( A e. S -> ( C ` A ) e. NN0 )
11	8, 10	eqeltrid 2321	. . 3 |- ( A e. S -> N e. NN0 )
12		fvco3 5753	. . 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 5532	. . . 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 9528	. . . . . 6 |- 0 e. NN0
17		fvco3 5753	. . . . . 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 12766	. . . . . 6 |- ( A e. S -> ( R ` 0 ) = A )
20	19	fveq2d 5679	. . . . 5 |- ( A e. S -> ( C ` ( R ` 0 ) ) = ( C ` A ) )
21	18, 20	eqtrd 2267	. . . 4 |- ( A e. S -> ( ( C o. R ) ` 0 ) = ( C ` A ) )
22	8, 21	eqtr4id 2286	. . 3 |- ( A e. S -> N = ( ( C o. R ) ` 0 ) )
23	7	ffvelcdmda 5817	. . . . 5 |- ( ( A e. S /\ k e. NN0 ) -> ( R ` k ) e. S )
24		2fveq3 5680	. . . . . . . 8 |- ( z = ( R ` k ) -> ( C ` ( F ` z ) ) = ( C ` ( F ` ( R ` k ) ) ) )
25	24	neeq1d 2432	. . . . . . 7 |- ( z = ( R ` k ) -> ( ( C ` ( F ` z ) ) =/= 0 <-> ( C ` ( F ` ( R ` k ) ) ) =/= 0 ) )
26		fveq2 5675	. . . . . . . 8 |- ( z = ( R ` k ) -> ( C ` z ) = ( C ` ( R ` k ) ) )
27	24, 26	breq12d 4127	. . . . . . 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 2883	. . . . 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 9553	. . . . . . 7 |- ( k e. NN0 -> ( k + 1 ) e. NN0 )
33		fvco3 5753	. . . . . . 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 12768	. . . . . . 7 |- ( ( A e. S /\ k e. NN0 ) -> ( R ` ( k + 1 ) ) = ( F ` ( R ` k ) ) )
36	35	fveq2d 5679	. . . . . 6 |- ( ( A e. S /\ k e. NN0 ) -> ( C ` ( R ` ( k + 1 ) ) ) = ( C ` ( F ` ( R ` k ) ) ) )
37	34, 36	eqtrd 2267	. . . . 5 |- ( ( A e. S /\ k e. NN0 ) -> ( ( C o. R ) ` ( k + 1 ) ) = ( C ` ( F ` ( R ` k ) ) ) )
38	37	neeq1d 2432	. . . 4 |- ( ( A e. S /\ k e. NN0 ) -> ( ( ( C o. R ) ` ( k + 1 ) ) =/= 0 <-> ( C ` ( F ` ( R ` k ) ) ) =/= 0 ) )
39		fvco3 5753	. . . . . 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 4127	. . . 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 12763	. 2 |- ( A e. S -> ( ( C o. R ) ` N ) = 0 )
44	13, 43	eqtr3d 2269	1 |- ( A e. S -> ( C ` ( R ` N ) ) = 0 )

Syntax hints:   -> wi 4   /\ wa 104   = wceq 1398   e. wcel 2205   =/= wne 2414   {csn 3694   class class class wbr 4114   X. cxp 4752   o. ccom 4758   -->wf 5353   ` cfv 5357  (class class class)co 6058   1stc1st 6345   0cc0 8143   1c1 8144   + caddc 8146   < clt 8324   NN0cn0 9513   seqcseq 10833

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 2207  ax-14 2208  ax-ext 2216  ax-coll 4230  ax-sep 4233  ax-nul 4241  ax-pow 4292  ax-pr 4327  ax-un 4559  ax-setind 4664  ax-iinf 4715  ax-cnex 8234  ax-resscn 8235  ax-1cn 8236  ax-1re 8237  ax-icn 8238  ax-addcl 8239  ax-addrcl 8240  ax-mulcl 8241  ax-addcom 8243  ax-addass 8245  ax-distr 8247  ax-i2m1 8248  ax-0lt1 8249  ax-0id 8251  ax-rnegex 8252  ax-cnre 8254  ax-pre-ltirr 8255  ax-pre-ltwlin 8256  ax-pre-lttrn 8257  ax-pre-apti 8258  ax-pre-ltadd 8259

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 1812  df-eu 2085  df-mo 2086  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-ne 2415  df-nel 2510  df-ral 2527  df-rex 2528  df-reu 2529  df-rab 2531  df-v 2817  df-sbc 3046  df-csb 3142  df-dif 3216  df-un 3218  df-in 3220  df-ss 3227  df-nul 3513  df-pw 3676  df-sn 3700  df-pr 3701  df-op 3703  df-uni 3920  df-int 3955  df-iun 3998  df-br 4115  df-opab 4177  df-mpt 4178  df-tr 4214  df-id 4419  df-iord 4492  df-on 4494  df-ilim 4495  df-suc 4497  df-iom 4718  df-xp 4760  df-rel 4761  df-cnv 4762  df-co 4763  df-dm 4764  df-rn 4765  df-res 4766  df-ima 4767  df-iota 5317  df-fun 5359  df-fn 5360  df-f 5361  df-f1 5362  df-fo 5363  df-f1o 5364  df-fv 5365  df-riota 6011  df-ov 6061  df-oprab 6062  df-mpo 6063  df-1st 6347  df-2nd 6348  df-recs 6549  df-frec 6635  df-pnf 8326  df-mnf 8327  df-xr 8328  df-ltxr 8329  df-le 8330  df-sub 8462  df-neg 8463  df-inn 9255  df-n0 9514  df-z 9595  df-uz 9872  df-seqfrec 10834

This theorem is referenced by:  algcvga  12773