Theorem ialgcvg 10430

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 } ) , S )
algcvg.3	|- C : S --> NN0
algcvg.4	|- ( z e. S -> ( ( C ` ( F ` z ) ) =/= 0 -> ( C ` ( F ` z ) ) < ( C ` z ) ) )
algcvg.5	|- N = ( C ` A )
ialgcvg.s	|- S e. V

Assertion

Ref	Expression
ialgcvg	|- ( 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)   V( z)

Proof of Theorem ialgcvg

Dummy variable k is distinct from all other variables.

Step	Hyp	Ref	Expression
1		nn0uz 8653	. . . 4 |- NN0 = ( ZZ>= ` 0 )
2		algcvg.2	. . . 4 |- R = seq 0 ( ( F o. 1st ) , ( NN0 X. { A } ) , S )
3		0zd 8363	. . . 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		ialgcvg.s	. . . . 5 |- S e. V
8	7	a1i 9	. . . 4 |- ( A e. S -> S e. V )
9	1, 2, 3, 4, 6, 8	ialgrf 10427	. . 3 |- ( A e. S -> R : NN0 --> S )
10		algcvg.5	. . . 4 |- N = ( C ` A )
11		algcvg.3	. . . . 5 |- C : S --> NN0
12	11	ffvelrni 5322	. . . 4 |- ( A e. S -> ( C ` A ) e. NN0 )
13	10, 12	syl5eqel 2165	. . 3 |- ( A e. S -> N e. NN0 )
14		fvco3 5265	. . 3 |- ( ( R : NN0 --> S /\ N e. NN0 ) -> ( ( C o. R ) ` N ) = ( C ` ( R ` N ) ) )
15	9, 13, 14	syl2anc 403	. 2 |- ( A e. S -> ( ( C o. R ) ` N ) = ( C ` ( R ` N ) ) )
16		fco 5076	. . . 4 |- ( ( C : S --> NN0 /\ R : NN0 --> S ) -> ( C o. R ) : NN0 --> NN0 )
17	11, 9, 16	sylancr 405	. . 3 |- ( A e. S -> ( C o. R ) : NN0 --> NN0 )
18		0nn0 8303	. . . . . 6 |- 0 e. NN0
19		fvco3 5265	. . . . . 6 |- ( ( R : NN0 --> S /\ 0 e. NN0 ) -> ( ( C o. R ) ` 0 ) = ( C ` ( R ` 0 ) ) )
20	9, 18, 19	sylancl 404	. . . . 5 |- ( A e. S -> ( ( C o. R ) ` 0 ) = ( C ` ( R ` 0 ) ) )
21	1, 2, 3, 4, 6, 8	ialgr0 10426	. . . . . 6 |- ( A e. S -> ( R ` 0 ) = A )
22	21	fveq2d 5202	. . . . 5 |- ( A e. S -> ( C ` ( R ` 0 ) ) = ( C ` A ) )
23	20, 22	eqtrd 2113	. . . 4 |- ( A e. S -> ( ( C o. R ) ` 0 ) = ( C ` A ) )
24	23, 10	syl6reqr 2132	. . 3 |- ( A e. S -> N = ( ( C o. R ) ` 0 ) )
25	9	ffvelrnda 5323	. . . . 5 |- ( ( A e. S /\ k e. NN0 ) -> ( R ` k ) e. S )
26		fveq2 5198	. . . . . . . . 9 |- ( z = ( R ` k ) -> ( F ` z ) = ( F ` ( R ` k ) ) )
27	26	fveq2d 5202	. . . . . . . 8 |- ( z = ( R ` k ) -> ( C ` ( F ` z ) ) = ( C ` ( F ` ( R ` k ) ) ) )
28	27	neeq1d 2263	. . . . . . 7 |- ( z = ( R ` k ) -> ( ( C ` ( F ` z ) ) =/= 0 <-> ( C ` ( F ` ( R ` k ) ) ) =/= 0 ) )
29		fveq2 5198	. . . . . . . 8 |- ( z = ( R ` k ) -> ( C ` z ) = ( C ` ( R ` k ) ) )
30	27, 29	breq12d 3798	. . . . . . 7 |- ( z = ( R ` k ) -> ( ( C ` ( F ` z ) ) < ( C ` z ) <-> ( C ` ( F ` ( R ` k ) ) ) < ( C ` ( R ` k ) ) ) )
31	28, 30	imbi12d 232	. . . . . 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 ) ) ) ) )
32		algcvg.4	. . . . . 6 |- ( z e. S -> ( ( C ` ( F ` z ) ) =/= 0 -> ( C ` ( F ` z ) ) < ( C ` z ) ) )
33	31, 32	vtoclga 2664	. . . . 5 |- ( ( R ` k ) e. S -> ( ( C ` ( F ` ( R ` k ) ) ) =/= 0 -> ( C ` ( F ` ( R ` k ) ) ) < ( C ` ( R ` k ) ) ) )
34	25, 33	syl 14	. . . 4 |- ( ( A e. S /\ k e. NN0 ) -> ( ( C ` ( F ` ( R ` k ) ) ) =/= 0 -> ( C ` ( F ` ( R ` k ) ) ) < ( C ` ( R ` k ) ) ) )
35		peano2nn0 8328	. . . . . . 7 |- ( k e. NN0 -> ( k + 1 ) e. NN0 )
36		fvco3 5265	. . . . . . 7 |- ( ( R : NN0 --> S /\ ( k + 1 ) e. NN0 ) -> ( ( C o. R ) ` ( k + 1 ) ) = ( C ` ( R ` ( k + 1 ) ) ) )
37	9, 35, 36	syl2an 283	. . . . . 6 |- ( ( A e. S /\ k e. NN0 ) -> ( ( C o. R ) ` ( k + 1 ) ) = ( C ` ( R ` ( k + 1 ) ) ) )
38	1, 2, 3, 4, 6, 8	ialgrp1 10428	. . . . . . 7 |- ( ( A e. S /\ k e. NN0 ) -> ( R ` ( k + 1 ) ) = ( F ` ( R ` k ) ) )
39	38	fveq2d 5202	. . . . . 6 |- ( ( A e. S /\ k e. NN0 ) -> ( C ` ( R ` ( k + 1 ) ) ) = ( C ` ( F ` ( R ` k ) ) ) )
40	37, 39	eqtrd 2113	. . . . 5 |- ( ( A e. S /\ k e. NN0 ) -> ( ( C o. R ) ` ( k + 1 ) ) = ( C ` ( F ` ( R ` k ) ) ) )
41	40	neeq1d 2263	. . . 4 |- ( ( A e. S /\ k e. NN0 ) -> ( ( ( C o. R ) ` ( k + 1 ) ) =/= 0 <-> ( C ` ( F ` ( R ` k ) ) ) =/= 0 ) )
42		fvco3 5265	. . . . . 6 |- ( ( R : NN0 --> S /\ k e. NN0 ) -> ( ( C o. R ) ` k ) = ( C ` ( R ` k ) ) )
43	9, 42	sylan 277	. . . . 5 |- ( ( A e. S /\ k e. NN0 ) -> ( ( C o. R ) ` k ) = ( C ` ( R ` k ) ) )
44	40, 43	breq12d 3798	. . . 4 |- ( ( A e. S /\ k e. NN0 ) -> ( ( ( C o. R ) ` ( k + 1 ) ) < ( ( C o. R ) ` k ) <-> ( C ` ( F ` ( R ` k ) ) ) < ( C ` ( R ` k ) ) ) )
45	34, 41, 44	3imtr4d 201	. . 3 |- ( ( A e. S /\ k e. NN0 ) -> ( ( ( C o. R ) ` ( k + 1 ) ) =/= 0 -> ( ( C o. R ) ` ( k + 1 ) ) < ( ( C o. R ) ` k ) ) )
46	17, 24, 45	nn0seqcvgd 10423	. 2 |- ( A e. S -> ( ( C o. R ) ` N ) = 0 )
47	15, 46	eqtr3d 2115	1 |- ( A e. S -> ( C ` ( R ` N ) ) = 0 )

Syntax hints:   -> wi 4   /\ wa 102   = wceq 1284   e. wcel 1433   =/= wne 2245   {csn 3398   class class class wbr 3785   X. cxp 4361   o. ccom 4367   -->wf 4918   ` cfv 4922  (class class class)co 5532   1stc1st 5785   0cc0 6981   1c1 6982   + caddc 6984   < clt 7153   NN0cn0 8288   seqcseq 9431

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 576  ax-in2 577  ax-io 662  ax-5 1376  ax-7 1377  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-8 1435  ax-10 1436  ax-11 1437  ax-i12 1438  ax-bndl 1439  ax-4 1440  ax-13 1444  ax-14 1445  ax-17 1459  ax-i9 1463  ax-ial 1467  ax-i5r 1468  ax-ext 2063  ax-coll 3893  ax-sep 3896  ax-nul 3904  ax-pow 3948  ax-pr 3964  ax-un 4188  ax-setind 4280  ax-iinf 4329  ax-cnex 7067  ax-resscn 7068  ax-1cn 7069  ax-1re 7070  ax-icn 7071  ax-addcl 7072  ax-addrcl 7073  ax-mulcl 7074  ax-addcom 7076  ax-addass 7078  ax-distr 7080  ax-i2m1 7081  ax-0lt1 7082  ax-0id 7084  ax-rnegex 7085  ax-cnre 7087  ax-pre-ltirr 7088  ax-pre-ltwlin 7089  ax-pre-lttrn 7090  ax-pre-apti 7091  ax-pre-ltadd 7092

This theorem depends on definitions:  df-bi 115  df-dc 776  df-3or 920  df-3an 921  df-tru 1287  df-fal 1290  df-nf 1390  df-sb 1686  df-eu 1944  df-mo 1945  df-clab 2068  df-cleq 2074  df-clel 2077  df-nfc 2208  df-ne 2246  df-nel 2340  df-ral 2353  df-rex 2354  df-reu 2355  df-rab 2357  df-v 2603  df-sbc 2816  df-csb 2909  df-dif 2975  df-un 2977  df-in 2979  df-ss 2986  df-nul 3252  df-pw 3384  df-sn 3404  df-pr 3405  df-op 3407  df-uni 3602  df-int 3637  df-iun 3680  df-br 3786  df-opab 3840  df-mpt 3841  df-tr 3876  df-id 4048  df-iord 4121  df-on 4123  df-suc 4126  df-iom 4332  df-xp 4369  df-rel 4370  df-cnv 4371  df-co 4372  df-dm 4373  df-rn 4374  df-res 4375  df-ima 4376  df-iota 4887  df-fun 4924  df-fn 4925  df-f 4926  df-f1 4927  df-fo 4928  df-f1o 4929  df-fv 4930  df-riota 5488  df-ov 5535  df-oprab 5536  df-mpt2 5537  df-1st 5787  df-2nd 5788  df-recs 5943  df-frec 6001  df-pnf 7155  df-mnf 7156  df-xr 7157  df-ltxr 7158  df-le 7159  df-sub 7281  df-neg 7282  df-inn 8040  df-n0 8289  df-z 8352  df-uz 8620  df-iseq 9432

This theorem is referenced by:  ialgcvga  10433