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Mirrors > Home > ILE Home > Th. List > algcvg | GIF version |
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 𝑋 ∈ 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.) |
Ref | Expression |
---|---|
algcvg.1 | ⊢ 𝐹:𝑆⟶𝑆 |
algcvg.2 | ⊢ 𝑅 = seq0((𝐹 ∘ 1st ), (ℕ0 × {𝐴})) |
algcvg.3 | ⊢ 𝐶:𝑆⟶ℕ0 |
algcvg.4 | ⊢ (𝑧 ∈ 𝑆 → ((𝐶‘(𝐹‘𝑧)) ≠ 0 → (𝐶‘(𝐹‘𝑧)) < (𝐶‘𝑧))) |
algcvg.5 | ⊢ 𝑁 = (𝐶‘𝐴) |
Ref | Expression |
---|---|
algcvg | ⊢ (𝐴 ∈ 𝑆 → (𝐶‘(𝑅‘𝑁)) = 0) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nn0uz 9360 | . . . 4 ⊢ ℕ0 = (ℤ≥‘0) | |
2 | algcvg.2 | . . . 4 ⊢ 𝑅 = seq0((𝐹 ∘ 1st ), (ℕ0 × {𝐴})) | |
3 | 0zd 9066 | . . . 4 ⊢ (𝐴 ∈ 𝑆 → 0 ∈ ℤ) | |
4 | id 19 | . . . 4 ⊢ (𝐴 ∈ 𝑆 → 𝐴 ∈ 𝑆) | |
5 | algcvg.1 | . . . . 5 ⊢ 𝐹:𝑆⟶𝑆 | |
6 | 5 | a1i 9 | . . . 4 ⊢ (𝐴 ∈ 𝑆 → 𝐹:𝑆⟶𝑆) |
7 | 1, 2, 3, 4, 6 | algrf 11726 | . . 3 ⊢ (𝐴 ∈ 𝑆 → 𝑅:ℕ0⟶𝑆) |
8 | algcvg.5 | . . . 4 ⊢ 𝑁 = (𝐶‘𝐴) | |
9 | algcvg.3 | . . . . 5 ⊢ 𝐶:𝑆⟶ℕ0 | |
10 | 9 | ffvelrni 5554 | . . . 4 ⊢ (𝐴 ∈ 𝑆 → (𝐶‘𝐴) ∈ ℕ0) |
11 | 8, 10 | eqeltrid 2226 | . . 3 ⊢ (𝐴 ∈ 𝑆 → 𝑁 ∈ ℕ0) |
12 | fvco3 5492 | . . 3 ⊢ ((𝑅:ℕ0⟶𝑆 ∧ 𝑁 ∈ ℕ0) → ((𝐶 ∘ 𝑅)‘𝑁) = (𝐶‘(𝑅‘𝑁))) | |
13 | 7, 11, 12 | syl2anc 408 | . 2 ⊢ (𝐴 ∈ 𝑆 → ((𝐶 ∘ 𝑅)‘𝑁) = (𝐶‘(𝑅‘𝑁))) |
14 | fco 5288 | . . . 4 ⊢ ((𝐶:𝑆⟶ℕ0 ∧ 𝑅:ℕ0⟶𝑆) → (𝐶 ∘ 𝑅):ℕ0⟶ℕ0) | |
15 | 9, 7, 14 | sylancr 410 | . . 3 ⊢ (𝐴 ∈ 𝑆 → (𝐶 ∘ 𝑅):ℕ0⟶ℕ0) |
16 | 0nn0 8992 | . . . . . 6 ⊢ 0 ∈ ℕ0 | |
17 | fvco3 5492 | . . . . . 6 ⊢ ((𝑅:ℕ0⟶𝑆 ∧ 0 ∈ ℕ0) → ((𝐶 ∘ 𝑅)‘0) = (𝐶‘(𝑅‘0))) | |
18 | 7, 16, 17 | sylancl 409 | . . . . 5 ⊢ (𝐴 ∈ 𝑆 → ((𝐶 ∘ 𝑅)‘0) = (𝐶‘(𝑅‘0))) |
19 | 1, 2, 3, 4, 6 | ialgr0 11725 | . . . . . 6 ⊢ (𝐴 ∈ 𝑆 → (𝑅‘0) = 𝐴) |
20 | 19 | fveq2d 5425 | . . . . 5 ⊢ (𝐴 ∈ 𝑆 → (𝐶‘(𝑅‘0)) = (𝐶‘𝐴)) |
21 | 18, 20 | eqtrd 2172 | . . . 4 ⊢ (𝐴 ∈ 𝑆 → ((𝐶 ∘ 𝑅)‘0) = (𝐶‘𝐴)) |
22 | 21, 8 | syl6reqr 2191 | . . 3 ⊢ (𝐴 ∈ 𝑆 → 𝑁 = ((𝐶 ∘ 𝑅)‘0)) |
23 | 7 | ffvelrnda 5555 | . . . . 5 ⊢ ((𝐴 ∈ 𝑆 ∧ 𝑘 ∈ ℕ0) → (𝑅‘𝑘) ∈ 𝑆) |
24 | 2fveq3 5426 | . . . . . . . 8 ⊢ (𝑧 = (𝑅‘𝑘) → (𝐶‘(𝐹‘𝑧)) = (𝐶‘(𝐹‘(𝑅‘𝑘)))) | |
25 | 24 | neeq1d 2326 | . . . . . . 7 ⊢ (𝑧 = (𝑅‘𝑘) → ((𝐶‘(𝐹‘𝑧)) ≠ 0 ↔ (𝐶‘(𝐹‘(𝑅‘𝑘))) ≠ 0)) |
26 | fveq2 5421 | . . . . . . . 8 ⊢ (𝑧 = (𝑅‘𝑘) → (𝐶‘𝑧) = (𝐶‘(𝑅‘𝑘))) | |
27 | 24, 26 | breq12d 3942 | . . . . . . 7 ⊢ (𝑧 = (𝑅‘𝑘) → ((𝐶‘(𝐹‘𝑧)) < (𝐶‘𝑧) ↔ (𝐶‘(𝐹‘(𝑅‘𝑘))) < (𝐶‘(𝑅‘𝑘)))) |
28 | 25, 27 | imbi12d 233 | . . . . . 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 ∈ 𝑆 → ((𝐶‘(𝐹‘𝑧)) ≠ 0 → (𝐶‘(𝐹‘𝑧)) < (𝐶‘𝑧))) | |
30 | 28, 29 | vtoclga 2752 | . . . . 5 ⊢ ((𝑅‘𝑘) ∈ 𝑆 → ((𝐶‘(𝐹‘(𝑅‘𝑘))) ≠ 0 → (𝐶‘(𝐹‘(𝑅‘𝑘))) < (𝐶‘(𝑅‘𝑘)))) |
31 | 23, 30 | syl 14 | . . . 4 ⊢ ((𝐴 ∈ 𝑆 ∧ 𝑘 ∈ ℕ0) → ((𝐶‘(𝐹‘(𝑅‘𝑘))) ≠ 0 → (𝐶‘(𝐹‘(𝑅‘𝑘))) < (𝐶‘(𝑅‘𝑘)))) |
32 | peano2nn0 9017 | . . . . . . 7 ⊢ (𝑘 ∈ ℕ0 → (𝑘 + 1) ∈ ℕ0) | |
33 | fvco3 5492 | . . . . . . 7 ⊢ ((𝑅:ℕ0⟶𝑆 ∧ (𝑘 + 1) ∈ ℕ0) → ((𝐶 ∘ 𝑅)‘(𝑘 + 1)) = (𝐶‘(𝑅‘(𝑘 + 1)))) | |
34 | 7, 32, 33 | syl2an 287 | . . . . . 6 ⊢ ((𝐴 ∈ 𝑆 ∧ 𝑘 ∈ ℕ0) → ((𝐶 ∘ 𝑅)‘(𝑘 + 1)) = (𝐶‘(𝑅‘(𝑘 + 1)))) |
35 | 1, 2, 3, 4, 6 | algrp1 11727 | . . . . . . 7 ⊢ ((𝐴 ∈ 𝑆 ∧ 𝑘 ∈ ℕ0) → (𝑅‘(𝑘 + 1)) = (𝐹‘(𝑅‘𝑘))) |
36 | 35 | fveq2d 5425 | . . . . . 6 ⊢ ((𝐴 ∈ 𝑆 ∧ 𝑘 ∈ ℕ0) → (𝐶‘(𝑅‘(𝑘 + 1))) = (𝐶‘(𝐹‘(𝑅‘𝑘)))) |
37 | 34, 36 | eqtrd 2172 | . . . . 5 ⊢ ((𝐴 ∈ 𝑆 ∧ 𝑘 ∈ ℕ0) → ((𝐶 ∘ 𝑅)‘(𝑘 + 1)) = (𝐶‘(𝐹‘(𝑅‘𝑘)))) |
38 | 37 | neeq1d 2326 | . . . 4 ⊢ ((𝐴 ∈ 𝑆 ∧ 𝑘 ∈ ℕ0) → (((𝐶 ∘ 𝑅)‘(𝑘 + 1)) ≠ 0 ↔ (𝐶‘(𝐹‘(𝑅‘𝑘))) ≠ 0)) |
39 | fvco3 5492 | . . . . . 6 ⊢ ((𝑅:ℕ0⟶𝑆 ∧ 𝑘 ∈ ℕ0) → ((𝐶 ∘ 𝑅)‘𝑘) = (𝐶‘(𝑅‘𝑘))) | |
40 | 7, 39 | sylan 281 | . . . . 5 ⊢ ((𝐴 ∈ 𝑆 ∧ 𝑘 ∈ ℕ0) → ((𝐶 ∘ 𝑅)‘𝑘) = (𝐶‘(𝑅‘𝑘))) |
41 | 37, 40 | breq12d 3942 | . . . 4 ⊢ ((𝐴 ∈ 𝑆 ∧ 𝑘 ∈ ℕ0) → (((𝐶 ∘ 𝑅)‘(𝑘 + 1)) < ((𝐶 ∘ 𝑅)‘𝑘) ↔ (𝐶‘(𝐹‘(𝑅‘𝑘))) < (𝐶‘(𝑅‘𝑘)))) |
42 | 31, 38, 41 | 3imtr4d 202 | . . 3 ⊢ ((𝐴 ∈ 𝑆 ∧ 𝑘 ∈ ℕ0) → (((𝐶 ∘ 𝑅)‘(𝑘 + 1)) ≠ 0 → ((𝐶 ∘ 𝑅)‘(𝑘 + 1)) < ((𝐶 ∘ 𝑅)‘𝑘))) |
43 | 15, 22, 42 | nn0seqcvgd 11722 | . 2 ⊢ (𝐴 ∈ 𝑆 → ((𝐶 ∘ 𝑅)‘𝑁) = 0) |
44 | 13, 43 | eqtr3d 2174 | 1 ⊢ (𝐴 ∈ 𝑆 → (𝐶‘(𝑅‘𝑁)) = 0) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 103 = wceq 1331 ∈ wcel 1480 ≠ wne 2308 {csn 3527 class class class wbr 3929 × cxp 4537 ∘ ccom 4543 ⟶wf 5119 ‘cfv 5123 (class class class)co 5774 1st c1st 6036 0cc0 7620 1c1 7621 + caddc 7623 < clt 7800 ℕ0cn0 8977 seqcseq 10218 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 603 ax-in2 604 ax-io 698 ax-5 1423 ax-7 1424 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-8 1482 ax-10 1483 ax-11 1484 ax-i12 1485 ax-bndl 1486 ax-4 1487 ax-13 1491 ax-14 1492 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2121 ax-coll 4043 ax-sep 4046 ax-nul 4054 ax-pow 4098 ax-pr 4131 ax-un 4355 ax-setind 4452 ax-iinf 4502 ax-cnex 7711 ax-resscn 7712 ax-1cn 7713 ax-1re 7714 ax-icn 7715 ax-addcl 7716 ax-addrcl 7717 ax-mulcl 7718 ax-addcom 7720 ax-addass 7722 ax-distr 7724 ax-i2m1 7725 ax-0lt1 7726 ax-0id 7728 ax-rnegex 7729 ax-cnre 7731 ax-pre-ltirr 7732 ax-pre-ltwlin 7733 ax-pre-lttrn 7734 ax-pre-apti 7735 ax-pre-ltadd 7736 |
This theorem depends on definitions: df-bi 116 df-dc 820 df-3or 963 df-3an 964 df-tru 1334 df-fal 1337 df-nf 1437 df-sb 1736 df-eu 2002 df-mo 2003 df-clab 2126 df-cleq 2132 df-clel 2135 df-nfc 2270 df-ne 2309 df-nel 2404 df-ral 2421 df-rex 2422 df-reu 2423 df-rab 2425 df-v 2688 df-sbc 2910 df-csb 3004 df-dif 3073 df-un 3075 df-in 3077 df-ss 3084 df-nul 3364 df-pw 3512 df-sn 3533 df-pr 3534 df-op 3536 df-uni 3737 df-int 3772 df-iun 3815 df-br 3930 df-opab 3990 df-mpt 3991 df-tr 4027 df-id 4215 df-iord 4288 df-on 4290 df-ilim 4291 df-suc 4293 df-iom 4505 df-xp 4545 df-rel 4546 df-cnv 4547 df-co 4548 df-dm 4549 df-rn 4550 df-res 4551 df-ima 4552 df-iota 5088 df-fun 5125 df-fn 5126 df-f 5127 df-f1 5128 df-fo 5129 df-f1o 5130 df-fv 5131 df-riota 5730 df-ov 5777 df-oprab 5778 df-mpo 5779 df-1st 6038 df-2nd 6039 df-recs 6202 df-frec 6288 df-pnf 7802 df-mnf 7803 df-xr 7804 df-ltxr 7805 df-le 7806 df-sub 7935 df-neg 7936 df-inn 8721 df-n0 8978 df-z 9055 df-uz 9327 df-seqfrec 10219 |
This theorem is referenced by: algcvga 11732 |
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