ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  algcvg GIF version

Theorem algcvg 12031
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 9551 . . . 4 0 = (ℤ‘0)
2 algcvg.2 . . . 4 𝑅 = seq0((𝐹 ∘ 1st ), (ℕ0 × {𝐴}))
3 0zd 9254 . . . 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 12028 . . 3 (𝐴𝑆𝑅:ℕ0𝑆)
8 algcvg.5 . . . 4 𝑁 = (𝐶𝐴)
9 algcvg.3 . . . . 5 𝐶:𝑆⟶ℕ0
109ffvelcdmi 5646 . . . 4 (𝐴𝑆 → (𝐶𝐴) ∈ ℕ0)
118, 10eqeltrid 2264 . . 3 (𝐴𝑆𝑁 ∈ ℕ0)
12 fvco3 5583 . . 3 ((𝑅:ℕ0𝑆𝑁 ∈ ℕ0) → ((𝐶𝑅)‘𝑁) = (𝐶‘(𝑅𝑁)))
137, 11, 12syl2anc 411 . 2 (𝐴𝑆 → ((𝐶𝑅)‘𝑁) = (𝐶‘(𝑅𝑁)))
14 fco 5377 . . . 4 ((𝐶:𝑆⟶ℕ0𝑅:ℕ0𝑆) → (𝐶𝑅):ℕ0⟶ℕ0)
159, 7, 14sylancr 414 . . 3 (𝐴𝑆 → (𝐶𝑅):ℕ0⟶ℕ0)
16 0nn0 9180 . . . . . 6 0 ∈ ℕ0
17 fvco3 5583 . . . . . 6 ((𝑅:ℕ0𝑆 ∧ 0 ∈ ℕ0) → ((𝐶𝑅)‘0) = (𝐶‘(𝑅‘0)))
187, 16, 17sylancl 413 . . . . 5 (𝐴𝑆 → ((𝐶𝑅)‘0) = (𝐶‘(𝑅‘0)))
191, 2, 3, 4, 6ialgr0 12027 . . . . . 6 (𝐴𝑆 → (𝑅‘0) = 𝐴)
2019fveq2d 5515 . . . . 5 (𝐴𝑆 → (𝐶‘(𝑅‘0)) = (𝐶𝐴))
2118, 20eqtrd 2210 . . . 4 (𝐴𝑆 → ((𝐶𝑅)‘0) = (𝐶𝐴))
228, 21eqtr4id 2229 . . 3 (𝐴𝑆𝑁 = ((𝐶𝑅)‘0))
237ffvelcdmda 5647 . . . . 5 ((𝐴𝑆𝑘 ∈ ℕ0) → (𝑅𝑘) ∈ 𝑆)
24 2fveq3 5516 . . . . . . . 8 (𝑧 = (𝑅𝑘) → (𝐶‘(𝐹𝑧)) = (𝐶‘(𝐹‘(𝑅𝑘))))
2524neeq1d 2365 . . . . . . 7 (𝑧 = (𝑅𝑘) → ((𝐶‘(𝐹𝑧)) ≠ 0 ↔ (𝐶‘(𝐹‘(𝑅𝑘))) ≠ 0))
26 fveq2 5511 . . . . . . . 8 (𝑧 = (𝑅𝑘) → (𝐶𝑧) = (𝐶‘(𝑅𝑘)))
2724, 26breq12d 4013 . . . . . . 7 (𝑧 = (𝑅𝑘) → ((𝐶‘(𝐹𝑧)) < (𝐶𝑧) ↔ (𝐶‘(𝐹‘(𝑅𝑘))) < (𝐶‘(𝑅𝑘))))
2825, 27imbi12d 234 . . . . . 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 2803 . . . . 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 9205 . . . . . . 7 (𝑘 ∈ ℕ0 → (𝑘 + 1) ∈ ℕ0)
33 fvco3 5583 . . . . . . 7 ((𝑅:ℕ0𝑆 ∧ (𝑘 + 1) ∈ ℕ0) → ((𝐶𝑅)‘(𝑘 + 1)) = (𝐶‘(𝑅‘(𝑘 + 1))))
347, 32, 33syl2an 289 . . . . . 6 ((𝐴𝑆𝑘 ∈ ℕ0) → ((𝐶𝑅)‘(𝑘 + 1)) = (𝐶‘(𝑅‘(𝑘 + 1))))
351, 2, 3, 4, 6algrp1 12029 . . . . . . 7 ((𝐴𝑆𝑘 ∈ ℕ0) → (𝑅‘(𝑘 + 1)) = (𝐹‘(𝑅𝑘)))
3635fveq2d 5515 . . . . . 6 ((𝐴𝑆𝑘 ∈ ℕ0) → (𝐶‘(𝑅‘(𝑘 + 1))) = (𝐶‘(𝐹‘(𝑅𝑘))))
3734, 36eqtrd 2210 . . . . 5 ((𝐴𝑆𝑘 ∈ ℕ0) → ((𝐶𝑅)‘(𝑘 + 1)) = (𝐶‘(𝐹‘(𝑅𝑘))))
3837neeq1d 2365 . . . 4 ((𝐴𝑆𝑘 ∈ ℕ0) → (((𝐶𝑅)‘(𝑘 + 1)) ≠ 0 ↔ (𝐶‘(𝐹‘(𝑅𝑘))) ≠ 0))
39 fvco3 5583 . . . . . 6 ((𝑅:ℕ0𝑆𝑘 ∈ ℕ0) → ((𝐶𝑅)‘𝑘) = (𝐶‘(𝑅𝑘)))
407, 39sylan 283 . . . . 5 ((𝐴𝑆𝑘 ∈ ℕ0) → ((𝐶𝑅)‘𝑘) = (𝐶‘(𝑅𝑘)))
4137, 40breq12d 4013 . . . 4 ((𝐴𝑆𝑘 ∈ ℕ0) → (((𝐶𝑅)‘(𝑘 + 1)) < ((𝐶𝑅)‘𝑘) ↔ (𝐶‘(𝐹‘(𝑅𝑘))) < (𝐶‘(𝑅𝑘))))
4231, 38, 413imtr4d 203 . . 3 ((𝐴𝑆𝑘 ∈ ℕ0) → (((𝐶𝑅)‘(𝑘 + 1)) ≠ 0 → ((𝐶𝑅)‘(𝑘 + 1)) < ((𝐶𝑅)‘𝑘)))
4315, 22, 42nn0seqcvgd 12024 . 2 (𝐴𝑆 → ((𝐶𝑅)‘𝑁) = 0)
4413, 43eqtr3d 2212 1 (𝐴𝑆 → (𝐶‘(𝑅𝑁)) = 0)
Colors of variables: wff set class
Syntax hints:  wi 4  wa 104   = wceq 1353  wcel 2148  wne 2347  {csn 3591   class class class wbr 4000   × cxp 4621  ccom 4627  wf 5208  cfv 5212  (class class class)co 5869  1st c1st 6133  0cc0 7802  1c1 7803   + caddc 7805   < clt 7982  0cn0 9165  seqcseq 10431
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 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-coll 4115  ax-sep 4118  ax-nul 4126  ax-pow 4171  ax-pr 4206  ax-un 4430  ax-setind 4533  ax-iinf 4584  ax-cnex 7893  ax-resscn 7894  ax-1cn 7895  ax-1re 7896  ax-icn 7897  ax-addcl 7898  ax-addrcl 7899  ax-mulcl 7900  ax-addcom 7902  ax-addass 7904  ax-distr 7906  ax-i2m1 7907  ax-0lt1 7908  ax-0id 7910  ax-rnegex 7911  ax-cnre 7913  ax-pre-ltirr 7914  ax-pre-ltwlin 7915  ax-pre-lttrn 7916  ax-pre-apti 7917  ax-pre-ltadd 7918
This theorem depends on definitions:  df-bi 117  df-dc 835  df-3or 979  df-3an 980  df-tru 1356  df-fal 1359  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ne 2348  df-nel 2443  df-ral 2460  df-rex 2461  df-reu 2462  df-rab 2464  df-v 2739  df-sbc 2963  df-csb 3058  df-dif 3131  df-un 3133  df-in 3135  df-ss 3142  df-nul 3423  df-pw 3576  df-sn 3597  df-pr 3598  df-op 3600  df-uni 3808  df-int 3843  df-iun 3886  df-br 4001  df-opab 4062  df-mpt 4063  df-tr 4099  df-id 4290  df-iord 4363  df-on 4365  df-ilim 4366  df-suc 4368  df-iom 4587  df-xp 4629  df-rel 4630  df-cnv 4631  df-co 4632  df-dm 4633  df-rn 4634  df-res 4635  df-ima 4636  df-iota 5174  df-fun 5214  df-fn 5215  df-f 5216  df-f1 5217  df-fo 5218  df-f1o 5219  df-fv 5220  df-riota 5825  df-ov 5872  df-oprab 5873  df-mpo 5874  df-1st 6135  df-2nd 6136  df-recs 6300  df-frec 6386  df-pnf 7984  df-mnf 7985  df-xr 7986  df-ltxr 7987  df-le 7988  df-sub 8120  df-neg 8121  df-inn 8909  df-n0 9166  df-z 9243  df-uz 9518  df-seqfrec 10432
This theorem is referenced by:  algcvga  12034
  Copyright terms: Public domain W3C validator