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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 9508 | . . . 4 ⊢ ℕ0 = (ℤ≥‘0) | |
2 | algcvg.2 | . . . 4 ⊢ 𝑅 = seq0((𝐹 ∘ 1st ), (ℕ0 × {𝐴})) | |
3 | 0zd 9211 | . . . 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 11986 | . . 3 ⊢ (𝐴 ∈ 𝑆 → 𝑅:ℕ0⟶𝑆) |
8 | algcvg.5 | . . . 4 ⊢ 𝑁 = (𝐶‘𝐴) | |
9 | algcvg.3 | . . . . 5 ⊢ 𝐶:𝑆⟶ℕ0 | |
10 | 9 | ffvelrni 5627 | . . . 4 ⊢ (𝐴 ∈ 𝑆 → (𝐶‘𝐴) ∈ ℕ0) |
11 | 8, 10 | eqeltrid 2257 | . . 3 ⊢ (𝐴 ∈ 𝑆 → 𝑁 ∈ ℕ0) |
12 | fvco3 5565 | . . 3 ⊢ ((𝑅:ℕ0⟶𝑆 ∧ 𝑁 ∈ ℕ0) → ((𝐶 ∘ 𝑅)‘𝑁) = (𝐶‘(𝑅‘𝑁))) | |
13 | 7, 11, 12 | syl2anc 409 | . 2 ⊢ (𝐴 ∈ 𝑆 → ((𝐶 ∘ 𝑅)‘𝑁) = (𝐶‘(𝑅‘𝑁))) |
14 | fco 5361 | . . . 4 ⊢ ((𝐶:𝑆⟶ℕ0 ∧ 𝑅:ℕ0⟶𝑆) → (𝐶 ∘ 𝑅):ℕ0⟶ℕ0) | |
15 | 9, 7, 14 | sylancr 412 | . . 3 ⊢ (𝐴 ∈ 𝑆 → (𝐶 ∘ 𝑅):ℕ0⟶ℕ0) |
16 | 0nn0 9137 | . . . . . 6 ⊢ 0 ∈ ℕ0 | |
17 | fvco3 5565 | . . . . . 6 ⊢ ((𝑅:ℕ0⟶𝑆 ∧ 0 ∈ ℕ0) → ((𝐶 ∘ 𝑅)‘0) = (𝐶‘(𝑅‘0))) | |
18 | 7, 16, 17 | sylancl 411 | . . . . 5 ⊢ (𝐴 ∈ 𝑆 → ((𝐶 ∘ 𝑅)‘0) = (𝐶‘(𝑅‘0))) |
19 | 1, 2, 3, 4, 6 | ialgr0 11985 | . . . . . 6 ⊢ (𝐴 ∈ 𝑆 → (𝑅‘0) = 𝐴) |
20 | 19 | fveq2d 5498 | . . . . 5 ⊢ (𝐴 ∈ 𝑆 → (𝐶‘(𝑅‘0)) = (𝐶‘𝐴)) |
21 | 18, 20 | eqtrd 2203 | . . . 4 ⊢ (𝐴 ∈ 𝑆 → ((𝐶 ∘ 𝑅)‘0) = (𝐶‘𝐴)) |
22 | 8, 21 | eqtr4id 2222 | . . 3 ⊢ (𝐴 ∈ 𝑆 → 𝑁 = ((𝐶 ∘ 𝑅)‘0)) |
23 | 7 | ffvelrnda 5628 | . . . . 5 ⊢ ((𝐴 ∈ 𝑆 ∧ 𝑘 ∈ ℕ0) → (𝑅‘𝑘) ∈ 𝑆) |
24 | 2fveq3 5499 | . . . . . . . 8 ⊢ (𝑧 = (𝑅‘𝑘) → (𝐶‘(𝐹‘𝑧)) = (𝐶‘(𝐹‘(𝑅‘𝑘)))) | |
25 | 24 | neeq1d 2358 | . . . . . . 7 ⊢ (𝑧 = (𝑅‘𝑘) → ((𝐶‘(𝐹‘𝑧)) ≠ 0 ↔ (𝐶‘(𝐹‘(𝑅‘𝑘))) ≠ 0)) |
26 | fveq2 5494 | . . . . . . . 8 ⊢ (𝑧 = (𝑅‘𝑘) → (𝐶‘𝑧) = (𝐶‘(𝑅‘𝑘))) | |
27 | 24, 26 | breq12d 4000 | . . . . . . 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 2796 | . . . . 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 9162 | . . . . . . 7 ⊢ (𝑘 ∈ ℕ0 → (𝑘 + 1) ∈ ℕ0) | |
33 | fvco3 5565 | . . . . . . 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 11987 | . . . . . . 7 ⊢ ((𝐴 ∈ 𝑆 ∧ 𝑘 ∈ ℕ0) → (𝑅‘(𝑘 + 1)) = (𝐹‘(𝑅‘𝑘))) |
36 | 35 | fveq2d 5498 | . . . . . 6 ⊢ ((𝐴 ∈ 𝑆 ∧ 𝑘 ∈ ℕ0) → (𝐶‘(𝑅‘(𝑘 + 1))) = (𝐶‘(𝐹‘(𝑅‘𝑘)))) |
37 | 34, 36 | eqtrd 2203 | . . . . 5 ⊢ ((𝐴 ∈ 𝑆 ∧ 𝑘 ∈ ℕ0) → ((𝐶 ∘ 𝑅)‘(𝑘 + 1)) = (𝐶‘(𝐹‘(𝑅‘𝑘)))) |
38 | 37 | neeq1d 2358 | . . . 4 ⊢ ((𝐴 ∈ 𝑆 ∧ 𝑘 ∈ ℕ0) → (((𝐶 ∘ 𝑅)‘(𝑘 + 1)) ≠ 0 ↔ (𝐶‘(𝐹‘(𝑅‘𝑘))) ≠ 0)) |
39 | fvco3 5565 | . . . . . 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 4000 | . . . 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 11982 | . 2 ⊢ (𝐴 ∈ 𝑆 → ((𝐶 ∘ 𝑅)‘𝑁) = 0) |
44 | 13, 43 | eqtr3d 2205 | 1 ⊢ (𝐴 ∈ 𝑆 → (𝐶‘(𝑅‘𝑁)) = 0) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 103 = wceq 1348 ∈ wcel 2141 ≠ wne 2340 {csn 3581 class class class wbr 3987 × cxp 4607 ∘ ccom 4613 ⟶wf 5192 ‘cfv 5196 (class class class)co 5850 1st c1st 6114 0cc0 7761 1c1 7762 + caddc 7764 < clt 7941 ℕ0cn0 9122 seqcseq 10388 |
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 609 ax-in2 610 ax-io 704 ax-5 1440 ax-7 1441 ax-gen 1442 ax-ie1 1486 ax-ie2 1487 ax-8 1497 ax-10 1498 ax-11 1499 ax-i12 1500 ax-bndl 1502 ax-4 1503 ax-17 1519 ax-i9 1523 ax-ial 1527 ax-i5r 1528 ax-13 2143 ax-14 2144 ax-ext 2152 ax-coll 4102 ax-sep 4105 ax-nul 4113 ax-pow 4158 ax-pr 4192 ax-un 4416 ax-setind 4519 ax-iinf 4570 ax-cnex 7852 ax-resscn 7853 ax-1cn 7854 ax-1re 7855 ax-icn 7856 ax-addcl 7857 ax-addrcl 7858 ax-mulcl 7859 ax-addcom 7861 ax-addass 7863 ax-distr 7865 ax-i2m1 7866 ax-0lt1 7867 ax-0id 7869 ax-rnegex 7870 ax-cnre 7872 ax-pre-ltirr 7873 ax-pre-ltwlin 7874 ax-pre-lttrn 7875 ax-pre-apti 7876 ax-pre-ltadd 7877 |
This theorem depends on definitions: df-bi 116 df-dc 830 df-3or 974 df-3an 975 df-tru 1351 df-fal 1354 df-nf 1454 df-sb 1756 df-eu 2022 df-mo 2023 df-clab 2157 df-cleq 2163 df-clel 2166 df-nfc 2301 df-ne 2341 df-nel 2436 df-ral 2453 df-rex 2454 df-reu 2455 df-rab 2457 df-v 2732 df-sbc 2956 df-csb 3050 df-dif 3123 df-un 3125 df-in 3127 df-ss 3134 df-nul 3415 df-pw 3566 df-sn 3587 df-pr 3588 df-op 3590 df-uni 3795 df-int 3830 df-iun 3873 df-br 3988 df-opab 4049 df-mpt 4050 df-tr 4086 df-id 4276 df-iord 4349 df-on 4351 df-ilim 4352 df-suc 4354 df-iom 4573 df-xp 4615 df-rel 4616 df-cnv 4617 df-co 4618 df-dm 4619 df-rn 4620 df-res 4621 df-ima 4622 df-iota 5158 df-fun 5198 df-fn 5199 df-f 5200 df-f1 5201 df-fo 5202 df-f1o 5203 df-fv 5204 df-riota 5806 df-ov 5853 df-oprab 5854 df-mpo 5855 df-1st 6116 df-2nd 6117 df-recs 6281 df-frec 6367 df-pnf 7943 df-mnf 7944 df-xr 7945 df-ltxr 7946 df-le 7947 df-sub 8079 df-neg 8080 df-inn 8866 df-n0 9123 df-z 9200 df-uz 9475 df-seqfrec 10389 |
This theorem is referenced by: algcvga 11992 |
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