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Theorem algcvg 11522
Description: One way to prove that an algorithm halts is to construct a countdown function 𝐶:𝑆⟶ℕ0 whose value is guaranteed to decrease for each iteration of 𝐹 until it reaches 0. That is, if 𝑋𝑆 is not a fixed point of 𝐹, then (𝐶‘(𝐹𝑋)) < (𝐶𝑋).

If 𝐶 is a countdown function for algorithm 𝐹, the sequence (𝐶‘(𝑅𝑘)) 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:𝑆𝑆
algcvg.2 𝑅 = seq0((𝐹 ∘ 1st ), (ℕ0 × {𝐴}))
algcvg.3 𝐶:𝑆⟶ℕ0
algcvg.4 (𝑧𝑆 → ((𝐶‘(𝐹𝑧)) ≠ 0 → (𝐶‘(𝐹𝑧)) < (𝐶𝑧)))
algcvg.5 𝑁 = (𝐶𝐴)
Assertion
Ref Expression
algcvg (𝐴𝑆 → (𝐶‘(𝑅𝑁)) = 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.
StepHypRef Expression
1 nn0uz 9210 . . . 4 0 = (ℤ‘0)
2 algcvg.2 . . . 4 𝑅 = seq0((𝐹 ∘ 1st ), (ℕ0 × {𝐴}))
3 0zd 8918 . . . 4 (𝐴𝑆 → 0 ∈ ℤ)
4 id 19 . . . 4 (𝐴𝑆𝐴𝑆)
5 algcvg.1 . . . . 5 𝐹:𝑆𝑆
65a1i 9 . . . 4 (𝐴𝑆𝐹:𝑆𝑆)
71, 2, 3, 4, 6algrf 11519 . . 3 (𝐴𝑆𝑅:ℕ0𝑆)
8 algcvg.5 . . . 4 𝑁 = (𝐶𝐴)
9 algcvg.3 . . . . 5 𝐶:𝑆⟶ℕ0
109ffvelrni 5486 . . . 4 (𝐴𝑆 → (𝐶𝐴) ∈ ℕ0)
118, 10syl5eqel 2186 . . 3 (𝐴𝑆𝑁 ∈ ℕ0)
12 fvco3 5424 . . 3 ((𝑅:ℕ0𝑆𝑁 ∈ ℕ0) → ((𝐶𝑅)‘𝑁) = (𝐶‘(𝑅𝑁)))
137, 11, 12syl2anc 406 . 2 (𝐴𝑆 → ((𝐶𝑅)‘𝑁) = (𝐶‘(𝑅𝑁)))
14 fco 5224 . . . 4 ((𝐶:𝑆⟶ℕ0𝑅:ℕ0𝑆) → (𝐶𝑅):ℕ0⟶ℕ0)
159, 7, 14sylancr 408 . . 3 (𝐴𝑆 → (𝐶𝑅):ℕ0⟶ℕ0)
16 0nn0 8844 . . . . . 6 0 ∈ ℕ0
17 fvco3 5424 . . . . . 6 ((𝑅:ℕ0𝑆 ∧ 0 ∈ ℕ0) → ((𝐶𝑅)‘0) = (𝐶‘(𝑅‘0)))
187, 16, 17sylancl 407 . . . . 5 (𝐴𝑆 → ((𝐶𝑅)‘0) = (𝐶‘(𝑅‘0)))
191, 2, 3, 4, 6ialgr0 11518 . . . . . 6 (𝐴𝑆 → (𝑅‘0) = 𝐴)
2019fveq2d 5357 . . . . 5 (𝐴𝑆 → (𝐶‘(𝑅‘0)) = (𝐶𝐴))
2118, 20eqtrd 2132 . . . 4 (𝐴𝑆 → ((𝐶𝑅)‘0) = (𝐶𝐴))
2221, 8syl6reqr 2151 . . 3 (𝐴𝑆𝑁 = ((𝐶𝑅)‘0))
237ffvelrnda 5487 . . . . 5 ((𝐴𝑆𝑘 ∈ ℕ0) → (𝑅𝑘) ∈ 𝑆)
24 2fveq3 5358 . . . . . . . 8 (𝑧 = (𝑅𝑘) → (𝐶‘(𝐹𝑧)) = (𝐶‘(𝐹‘(𝑅𝑘))))
2524neeq1d 2285 . . . . . . 7 (𝑧 = (𝑅𝑘) → ((𝐶‘(𝐹𝑧)) ≠ 0 ↔ (𝐶‘(𝐹‘(𝑅𝑘))) ≠ 0))
26 fveq2 5353 . . . . . . . 8 (𝑧 = (𝑅𝑘) → (𝐶𝑧) = (𝐶‘(𝑅𝑘)))
2724, 26breq12d 3888 . . . . . . 7 (𝑧 = (𝑅𝑘) → ((𝐶‘(𝐹𝑧)) < (𝐶𝑧) ↔ (𝐶‘(𝐹‘(𝑅𝑘))) < (𝐶‘(𝑅𝑘))))
2825, 27imbi12d 233 . . . . . 6 (𝑧 = (𝑅𝑘) → (((𝐶‘(𝐹𝑧)) ≠ 0 → (𝐶‘(𝐹𝑧)) < (𝐶𝑧)) ↔ ((𝐶‘(𝐹‘(𝑅𝑘))) ≠ 0 → (𝐶‘(𝐹‘(𝑅𝑘))) < (𝐶‘(𝑅𝑘)))))
29 algcvg.4 . . . . . 6 (𝑧𝑆 → ((𝐶‘(𝐹𝑧)) ≠ 0 → (𝐶‘(𝐹𝑧)) < (𝐶𝑧)))
3028, 29vtoclga 2707 . . . . 5 ((𝑅𝑘) ∈ 𝑆 → ((𝐶‘(𝐹‘(𝑅𝑘))) ≠ 0 → (𝐶‘(𝐹‘(𝑅𝑘))) < (𝐶‘(𝑅𝑘))))
3123, 30syl 14 . . . 4 ((𝐴𝑆𝑘 ∈ ℕ0) → ((𝐶‘(𝐹‘(𝑅𝑘))) ≠ 0 → (𝐶‘(𝐹‘(𝑅𝑘))) < (𝐶‘(𝑅𝑘))))
32 peano2nn0 8869 . . . . . . 7 (𝑘 ∈ ℕ0 → (𝑘 + 1) ∈ ℕ0)
33 fvco3 5424 . . . . . . 7 ((𝑅:ℕ0𝑆 ∧ (𝑘 + 1) ∈ ℕ0) → ((𝐶𝑅)‘(𝑘 + 1)) = (𝐶‘(𝑅‘(𝑘 + 1))))
347, 32, 33syl2an 285 . . . . . 6 ((𝐴𝑆𝑘 ∈ ℕ0) → ((𝐶𝑅)‘(𝑘 + 1)) = (𝐶‘(𝑅‘(𝑘 + 1))))
351, 2, 3, 4, 6algrp1 11520 . . . . . . 7 ((𝐴𝑆𝑘 ∈ ℕ0) → (𝑅‘(𝑘 + 1)) = (𝐹‘(𝑅𝑘)))
3635fveq2d 5357 . . . . . 6 ((𝐴𝑆𝑘 ∈ ℕ0) → (𝐶‘(𝑅‘(𝑘 + 1))) = (𝐶‘(𝐹‘(𝑅𝑘))))
3734, 36eqtrd 2132 . . . . 5 ((𝐴𝑆𝑘 ∈ ℕ0) → ((𝐶𝑅)‘(𝑘 + 1)) = (𝐶‘(𝐹‘(𝑅𝑘))))
3837neeq1d 2285 . . . 4 ((𝐴𝑆𝑘 ∈ ℕ0) → (((𝐶𝑅)‘(𝑘 + 1)) ≠ 0 ↔ (𝐶‘(𝐹‘(𝑅𝑘))) ≠ 0))
39 fvco3 5424 . . . . . 6 ((𝑅:ℕ0𝑆𝑘 ∈ ℕ0) → ((𝐶𝑅)‘𝑘) = (𝐶‘(𝑅𝑘)))
407, 39sylan 279 . . . . 5 ((𝐴𝑆𝑘 ∈ ℕ0) → ((𝐶𝑅)‘𝑘) = (𝐶‘(𝑅𝑘)))
4137, 40breq12d 3888 . . . 4 ((𝐴𝑆𝑘 ∈ ℕ0) → (((𝐶𝑅)‘(𝑘 + 1)) < ((𝐶𝑅)‘𝑘) ↔ (𝐶‘(𝐹‘(𝑅𝑘))) < (𝐶‘(𝑅𝑘))))
4231, 38, 413imtr4d 202 . . 3 ((𝐴𝑆𝑘 ∈ ℕ0) → (((𝐶𝑅)‘(𝑘 + 1)) ≠ 0 → ((𝐶𝑅)‘(𝑘 + 1)) < ((𝐶𝑅)‘𝑘)))
4315, 22, 42nn0seqcvgd 11515 . 2 (𝐴𝑆 → ((𝐶𝑅)‘𝑁) = 0)
4413, 43eqtr3d 2134 1 (𝐴𝑆 → (𝐶‘(𝑅𝑁)) = 0)
Colors of variables: wff set class
Syntax hints:  wi 4  wa 103   = wceq 1299  wcel 1448  wne 2267  {csn 3474   class class class wbr 3875   × cxp 4475  ccom 4481  wf 5055  cfv 5059  (class class class)co 5706  1st c1st 5967  0cc0 7500  1c1 7501   + caddc 7503   < clt 7672  0cn0 8829  seqcseq 10059
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 584  ax-in2 585  ax-io 671  ax-5 1391  ax-7 1392  ax-gen 1393  ax-ie1 1437  ax-ie2 1438  ax-8 1450  ax-10 1451  ax-11 1452  ax-i12 1453  ax-bndl 1454  ax-4 1455  ax-13 1459  ax-14 1460  ax-17 1474  ax-i9 1478  ax-ial 1482  ax-i5r 1483  ax-ext 2082  ax-coll 3983  ax-sep 3986  ax-nul 3994  ax-pow 4038  ax-pr 4069  ax-un 4293  ax-setind 4390  ax-iinf 4440  ax-cnex 7586  ax-resscn 7587  ax-1cn 7588  ax-1re 7589  ax-icn 7590  ax-addcl 7591  ax-addrcl 7592  ax-mulcl 7593  ax-addcom 7595  ax-addass 7597  ax-distr 7599  ax-i2m1 7600  ax-0lt1 7601  ax-0id 7603  ax-rnegex 7604  ax-cnre 7606  ax-pre-ltirr 7607  ax-pre-ltwlin 7608  ax-pre-lttrn 7609  ax-pre-apti 7610  ax-pre-ltadd 7611
This theorem depends on definitions:  df-bi 116  df-dc 787  df-3or 931  df-3an 932  df-tru 1302  df-fal 1305  df-nf 1405  df-sb 1704  df-eu 1963  df-mo 1964  df-clab 2087  df-cleq 2093  df-clel 2096  df-nfc 2229  df-ne 2268  df-nel 2363  df-ral 2380  df-rex 2381  df-reu 2382  df-rab 2384  df-v 2643  df-sbc 2863  df-csb 2956  df-dif 3023  df-un 3025  df-in 3027  df-ss 3034  df-nul 3311  df-pw 3459  df-sn 3480  df-pr 3481  df-op 3483  df-uni 3684  df-int 3719  df-iun 3762  df-br 3876  df-opab 3930  df-mpt 3931  df-tr 3967  df-id 4153  df-iord 4226  df-on 4228  df-ilim 4229  df-suc 4231  df-iom 4443  df-xp 4483  df-rel 4484  df-cnv 4485  df-co 4486  df-dm 4487  df-rn 4488  df-res 4489  df-ima 4490  df-iota 5024  df-fun 5061  df-fn 5062  df-f 5063  df-f1 5064  df-fo 5065  df-f1o 5066  df-fv 5067  df-riota 5662  df-ov 5709  df-oprab 5710  df-mpo 5711  df-1st 5969  df-2nd 5970  df-recs 6132  df-frec 6218  df-pnf 7674  df-mnf 7675  df-xr 7676  df-ltxr 7677  df-le 7678  df-sub 7806  df-neg 7807  df-inn 8579  df-n0 8830  df-z 8907  df-uz 9177  df-seqfrec 10060
This theorem is referenced by:  algcvga  11525
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