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Theorem algcvg 15336
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 11760 . . . 4 0 = (ℤ‘0)
2 algcvg.2 . . . 4 𝑅 = seq0((𝐹 ∘ 1st ), (ℕ0 × {𝐴}))
3 0zd 11427 . . . 4 (𝐴𝑆 → 0 ∈ ℤ)
4 id 22 . . . 4 (𝐴𝑆𝐴𝑆)
5 algcvg.1 . . . . 5 𝐹:𝑆𝑆
65a1i 11 . . . 4 (𝐴𝑆𝐹:𝑆𝑆)
71, 2, 3, 4, 6algrf 15333 . . 3 (𝐴𝑆𝑅:ℕ0𝑆)
8 algcvg.5 . . . 4 𝑁 = (𝐶𝐴)
9 algcvg.3 . . . . 5 𝐶:𝑆⟶ℕ0
109ffvelrni 6398 . . . 4 (𝐴𝑆 → (𝐶𝐴) ∈ ℕ0)
118, 10syl5eqel 2734 . . 3 (𝐴𝑆𝑁 ∈ ℕ0)
12 fvco3 6314 . . 3 ((𝑅:ℕ0𝑆𝑁 ∈ ℕ0) → ((𝐶𝑅)‘𝑁) = (𝐶‘(𝑅𝑁)))
137, 11, 12syl2anc 694 . 2 (𝐴𝑆 → ((𝐶𝑅)‘𝑁) = (𝐶‘(𝑅𝑁)))
14 fco 6096 . . . 4 ((𝐶:𝑆⟶ℕ0𝑅:ℕ0𝑆) → (𝐶𝑅):ℕ0⟶ℕ0)
159, 7, 14sylancr 696 . . 3 (𝐴𝑆 → (𝐶𝑅):ℕ0⟶ℕ0)
16 0nn0 11345 . . . . . 6 0 ∈ ℕ0
17 fvco3 6314 . . . . . 6 ((𝑅:ℕ0𝑆 ∧ 0 ∈ ℕ0) → ((𝐶𝑅)‘0) = (𝐶‘(𝑅‘0)))
187, 16, 17sylancl 695 . . . . 5 (𝐴𝑆 → ((𝐶𝑅)‘0) = (𝐶‘(𝑅‘0)))
191, 2, 3, 4algr0 15332 . . . . . 6 (𝐴𝑆 → (𝑅‘0) = 𝐴)
2019fveq2d 6233 . . . . 5 (𝐴𝑆 → (𝐶‘(𝑅‘0)) = (𝐶𝐴))
2118, 20eqtrd 2685 . . . 4 (𝐴𝑆 → ((𝐶𝑅)‘0) = (𝐶𝐴))
2221, 8syl6reqr 2704 . . 3 (𝐴𝑆𝑁 = ((𝐶𝑅)‘0))
237ffvelrnda 6399 . . . . 5 ((𝐴𝑆𝑘 ∈ ℕ0) → (𝑅𝑘) ∈ 𝑆)
24 fveq2 6229 . . . . . . . . 9 (𝑧 = (𝑅𝑘) → (𝐹𝑧) = (𝐹‘(𝑅𝑘)))
2524fveq2d 6233 . . . . . . . 8 (𝑧 = (𝑅𝑘) → (𝐶‘(𝐹𝑧)) = (𝐶‘(𝐹‘(𝑅𝑘))))
2625neeq1d 2882 . . . . . . 7 (𝑧 = (𝑅𝑘) → ((𝐶‘(𝐹𝑧)) ≠ 0 ↔ (𝐶‘(𝐹‘(𝑅𝑘))) ≠ 0))
27 fveq2 6229 . . . . . . . 8 (𝑧 = (𝑅𝑘) → (𝐶𝑧) = (𝐶‘(𝑅𝑘)))
2825, 27breq12d 4698 . . . . . . 7 (𝑧 = (𝑅𝑘) → ((𝐶‘(𝐹𝑧)) < (𝐶𝑧) ↔ (𝐶‘(𝐹‘(𝑅𝑘))) < (𝐶‘(𝑅𝑘))))
2926, 28imbi12d 333 . . . . . 6 (𝑧 = (𝑅𝑘) → (((𝐶‘(𝐹𝑧)) ≠ 0 → (𝐶‘(𝐹𝑧)) < (𝐶𝑧)) ↔ ((𝐶‘(𝐹‘(𝑅𝑘))) ≠ 0 → (𝐶‘(𝐹‘(𝑅𝑘))) < (𝐶‘(𝑅𝑘)))))
30 algcvg.4 . . . . . 6 (𝑧𝑆 → ((𝐶‘(𝐹𝑧)) ≠ 0 → (𝐶‘(𝐹𝑧)) < (𝐶𝑧)))
3129, 30vtoclga 3303 . . . . 5 ((𝑅𝑘) ∈ 𝑆 → ((𝐶‘(𝐹‘(𝑅𝑘))) ≠ 0 → (𝐶‘(𝐹‘(𝑅𝑘))) < (𝐶‘(𝑅𝑘))))
3223, 31syl 17 . . . 4 ((𝐴𝑆𝑘 ∈ ℕ0) → ((𝐶‘(𝐹‘(𝑅𝑘))) ≠ 0 → (𝐶‘(𝐹‘(𝑅𝑘))) < (𝐶‘(𝑅𝑘))))
33 peano2nn0 11371 . . . . . . 7 (𝑘 ∈ ℕ0 → (𝑘 + 1) ∈ ℕ0)
34 fvco3 6314 . . . . . . 7 ((𝑅:ℕ0𝑆 ∧ (𝑘 + 1) ∈ ℕ0) → ((𝐶𝑅)‘(𝑘 + 1)) = (𝐶‘(𝑅‘(𝑘 + 1))))
357, 33, 34syl2an 493 . . . . . 6 ((𝐴𝑆𝑘 ∈ ℕ0) → ((𝐶𝑅)‘(𝑘 + 1)) = (𝐶‘(𝑅‘(𝑘 + 1))))
361, 2, 3, 4, 6algrp1 15334 . . . . . . 7 ((𝐴𝑆𝑘 ∈ ℕ0) → (𝑅‘(𝑘 + 1)) = (𝐹‘(𝑅𝑘)))
3736fveq2d 6233 . . . . . 6 ((𝐴𝑆𝑘 ∈ ℕ0) → (𝐶‘(𝑅‘(𝑘 + 1))) = (𝐶‘(𝐹‘(𝑅𝑘))))
3835, 37eqtrd 2685 . . . . 5 ((𝐴𝑆𝑘 ∈ ℕ0) → ((𝐶𝑅)‘(𝑘 + 1)) = (𝐶‘(𝐹‘(𝑅𝑘))))
3938neeq1d 2882 . . . 4 ((𝐴𝑆𝑘 ∈ ℕ0) → (((𝐶𝑅)‘(𝑘 + 1)) ≠ 0 ↔ (𝐶‘(𝐹‘(𝑅𝑘))) ≠ 0))
40 fvco3 6314 . . . . . 6 ((𝑅:ℕ0𝑆𝑘 ∈ ℕ0) → ((𝐶𝑅)‘𝑘) = (𝐶‘(𝑅𝑘)))
417, 40sylan 487 . . . . 5 ((𝐴𝑆𝑘 ∈ ℕ0) → ((𝐶𝑅)‘𝑘) = (𝐶‘(𝑅𝑘)))
4238, 41breq12d 4698 . . . 4 ((𝐴𝑆𝑘 ∈ ℕ0) → (((𝐶𝑅)‘(𝑘 + 1)) < ((𝐶𝑅)‘𝑘) ↔ (𝐶‘(𝐹‘(𝑅𝑘))) < (𝐶‘(𝑅𝑘))))
4332, 39, 423imtr4d 283 . . 3 ((𝐴𝑆𝑘 ∈ ℕ0) → (((𝐶𝑅)‘(𝑘 + 1)) ≠ 0 → ((𝐶𝑅)‘(𝑘 + 1)) < ((𝐶𝑅)‘𝑘)))
4415, 22, 43nn0seqcvgd 15330 . 2 (𝐴𝑆 → ((𝐶𝑅)‘𝑁) = 0)
4513, 44eqtr3d 2687 1 (𝐴𝑆 → (𝐶‘(𝑅𝑁)) = 0)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 383   = wceq 1523  wcel 2030  wne 2823  {csn 4210   class class class wbr 4685   × cxp 5141  ccom 5147  wf 5922  cfv 5926  (class class class)co 6690  1st c1st 7208  0cc0 9974  1c1 9975   + caddc 9977   < clt 10112  0cn0 11330  seqcseq 12841
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936  ax-un 6991  ax-cnex 10030  ax-resscn 10031  ax-1cn 10032  ax-icn 10033  ax-addcl 10034  ax-addrcl 10035  ax-mulcl 10036  ax-mulrcl 10037  ax-mulcom 10038  ax-addass 10039  ax-mulass 10040  ax-distr 10041  ax-i2m1 10042  ax-1ne0 10043  ax-1rid 10044  ax-rnegex 10045  ax-rrecex 10046  ax-cnre 10047  ax-pre-lttri 10048  ax-pre-lttrn 10049  ax-pre-ltadd 10050  ax-pre-mulgt0 10051
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1055  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-nel 2927  df-ral 2946  df-rex 2947  df-reu 2948  df-rab 2950  df-v 3233  df-sbc 3469  df-csb 3567  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-pss 3623  df-nul 3949  df-if 4120  df-pw 4193  df-sn 4211  df-pr 4213  df-tp 4215  df-op 4217  df-uni 4469  df-iun 4554  df-br 4686  df-opab 4746  df-mpt 4763  df-tr 4786  df-id 5053  df-eprel 5058  df-po 5064  df-so 5065  df-fr 5102  df-we 5104  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-res 5155  df-ima 5156  df-pred 5718  df-ord 5764  df-on 5765  df-lim 5766  df-suc 5767  df-iota 5889  df-fun 5928  df-fn 5929  df-f 5930  df-f1 5931  df-fo 5932  df-f1o 5933  df-fv 5934  df-riota 6651  df-ov 6693  df-oprab 6694  df-mpt2 6695  df-om 7108  df-1st 7210  df-2nd 7211  df-wrecs 7452  df-recs 7513  df-rdg 7551  df-er 7787  df-en 7998  df-dom 7999  df-sdom 8000  df-pnf 10114  df-mnf 10115  df-xr 10116  df-ltxr 10117  df-le 10118  df-sub 10306  df-neg 10307  df-nn 11059  df-n0 11331  df-z 11416  df-uz 11726  df-fz 12365  df-seq 12842
This theorem is referenced by:  algcvga  15339
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