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Mirrors > Home > MPE Home > Th. List > algcvg | Structured version Visualization version 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 12886 | . . . 4 ⊢ ℕ0 = (ℤ≥‘0) | |
2 | algcvg.2 | . . . 4 ⊢ 𝑅 = seq0((𝐹 ∘ 1st ), (ℕ0 × {𝐴})) | |
3 | 0zd 12592 | . . . 4 ⊢ (𝐴 ∈ 𝑆 → 0 ∈ ℤ) | |
4 | id 22 | . . . 4 ⊢ (𝐴 ∈ 𝑆 → 𝐴 ∈ 𝑆) | |
5 | algcvg.1 | . . . . 5 ⊢ 𝐹:𝑆⟶𝑆 | |
6 | 5 | a1i 11 | . . . 4 ⊢ (𝐴 ∈ 𝑆 → 𝐹:𝑆⟶𝑆) |
7 | 1, 2, 3, 4, 6 | algrf 16535 | . . 3 ⊢ (𝐴 ∈ 𝑆 → 𝑅:ℕ0⟶𝑆) |
8 | algcvg.5 | . . . 4 ⊢ 𝑁 = (𝐶‘𝐴) | |
9 | algcvg.3 | . . . . 5 ⊢ 𝐶:𝑆⟶ℕ0 | |
10 | 9 | ffvelcdmi 7087 | . . . 4 ⊢ (𝐴 ∈ 𝑆 → (𝐶‘𝐴) ∈ ℕ0) |
11 | 8, 10 | eqeltrid 2832 | . . 3 ⊢ (𝐴 ∈ 𝑆 → 𝑁 ∈ ℕ0) |
12 | fvco3 6991 | . . 3 ⊢ ((𝑅:ℕ0⟶𝑆 ∧ 𝑁 ∈ ℕ0) → ((𝐶 ∘ 𝑅)‘𝑁) = (𝐶‘(𝑅‘𝑁))) | |
13 | 7, 11, 12 | syl2anc 583 | . 2 ⊢ (𝐴 ∈ 𝑆 → ((𝐶 ∘ 𝑅)‘𝑁) = (𝐶‘(𝑅‘𝑁))) |
14 | fco 6741 | . . . 4 ⊢ ((𝐶:𝑆⟶ℕ0 ∧ 𝑅:ℕ0⟶𝑆) → (𝐶 ∘ 𝑅):ℕ0⟶ℕ0) | |
15 | 9, 7, 14 | sylancr 586 | . . 3 ⊢ (𝐴 ∈ 𝑆 → (𝐶 ∘ 𝑅):ℕ0⟶ℕ0) |
16 | 0nn0 12509 | . . . . . 6 ⊢ 0 ∈ ℕ0 | |
17 | fvco3 6991 | . . . . . 6 ⊢ ((𝑅:ℕ0⟶𝑆 ∧ 0 ∈ ℕ0) → ((𝐶 ∘ 𝑅)‘0) = (𝐶‘(𝑅‘0))) | |
18 | 7, 16, 17 | sylancl 585 | . . . . 5 ⊢ (𝐴 ∈ 𝑆 → ((𝐶 ∘ 𝑅)‘0) = (𝐶‘(𝑅‘0))) |
19 | 1, 2, 3, 4 | algr0 16534 | . . . . . 6 ⊢ (𝐴 ∈ 𝑆 → (𝑅‘0) = 𝐴) |
20 | 19 | fveq2d 6895 | . . . . 5 ⊢ (𝐴 ∈ 𝑆 → (𝐶‘(𝑅‘0)) = (𝐶‘𝐴)) |
21 | 18, 20 | eqtrd 2767 | . . . 4 ⊢ (𝐴 ∈ 𝑆 → ((𝐶 ∘ 𝑅)‘0) = (𝐶‘𝐴)) |
22 | 8, 21 | eqtr4id 2786 | . . 3 ⊢ (𝐴 ∈ 𝑆 → 𝑁 = ((𝐶 ∘ 𝑅)‘0)) |
23 | 7 | ffvelcdmda 7088 | . . . . 5 ⊢ ((𝐴 ∈ 𝑆 ∧ 𝑘 ∈ ℕ0) → (𝑅‘𝑘) ∈ 𝑆) |
24 | 2fveq3 6896 | . . . . . . . 8 ⊢ (𝑧 = (𝑅‘𝑘) → (𝐶‘(𝐹‘𝑧)) = (𝐶‘(𝐹‘(𝑅‘𝑘)))) | |
25 | 24 | neeq1d 2995 | . . . . . . 7 ⊢ (𝑧 = (𝑅‘𝑘) → ((𝐶‘(𝐹‘𝑧)) ≠ 0 ↔ (𝐶‘(𝐹‘(𝑅‘𝑘))) ≠ 0)) |
26 | fveq2 6891 | . . . . . . . 8 ⊢ (𝑧 = (𝑅‘𝑘) → (𝐶‘𝑧) = (𝐶‘(𝑅‘𝑘))) | |
27 | 24, 26 | breq12d 5155 | . . . . . . 7 ⊢ (𝑧 = (𝑅‘𝑘) → ((𝐶‘(𝐹‘𝑧)) < (𝐶‘𝑧) ↔ (𝐶‘(𝐹‘(𝑅‘𝑘))) < (𝐶‘(𝑅‘𝑘)))) |
28 | 25, 27 | imbi12d 344 | . . . . . 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 3561 | . . . . 5 ⊢ ((𝑅‘𝑘) ∈ 𝑆 → ((𝐶‘(𝐹‘(𝑅‘𝑘))) ≠ 0 → (𝐶‘(𝐹‘(𝑅‘𝑘))) < (𝐶‘(𝑅‘𝑘)))) |
31 | 23, 30 | syl 17 | . . . 4 ⊢ ((𝐴 ∈ 𝑆 ∧ 𝑘 ∈ ℕ0) → ((𝐶‘(𝐹‘(𝑅‘𝑘))) ≠ 0 → (𝐶‘(𝐹‘(𝑅‘𝑘))) < (𝐶‘(𝑅‘𝑘)))) |
32 | peano2nn0 12534 | . . . . . . 7 ⊢ (𝑘 ∈ ℕ0 → (𝑘 + 1) ∈ ℕ0) | |
33 | fvco3 6991 | . . . . . . 7 ⊢ ((𝑅:ℕ0⟶𝑆 ∧ (𝑘 + 1) ∈ ℕ0) → ((𝐶 ∘ 𝑅)‘(𝑘 + 1)) = (𝐶‘(𝑅‘(𝑘 + 1)))) | |
34 | 7, 32, 33 | syl2an 595 | . . . . . 6 ⊢ ((𝐴 ∈ 𝑆 ∧ 𝑘 ∈ ℕ0) → ((𝐶 ∘ 𝑅)‘(𝑘 + 1)) = (𝐶‘(𝑅‘(𝑘 + 1)))) |
35 | 1, 2, 3, 4, 6 | algrp1 16536 | . . . . . . 7 ⊢ ((𝐴 ∈ 𝑆 ∧ 𝑘 ∈ ℕ0) → (𝑅‘(𝑘 + 1)) = (𝐹‘(𝑅‘𝑘))) |
36 | 35 | fveq2d 6895 | . . . . . 6 ⊢ ((𝐴 ∈ 𝑆 ∧ 𝑘 ∈ ℕ0) → (𝐶‘(𝑅‘(𝑘 + 1))) = (𝐶‘(𝐹‘(𝑅‘𝑘)))) |
37 | 34, 36 | eqtrd 2767 | . . . . 5 ⊢ ((𝐴 ∈ 𝑆 ∧ 𝑘 ∈ ℕ0) → ((𝐶 ∘ 𝑅)‘(𝑘 + 1)) = (𝐶‘(𝐹‘(𝑅‘𝑘)))) |
38 | 37 | neeq1d 2995 | . . . 4 ⊢ ((𝐴 ∈ 𝑆 ∧ 𝑘 ∈ ℕ0) → (((𝐶 ∘ 𝑅)‘(𝑘 + 1)) ≠ 0 ↔ (𝐶‘(𝐹‘(𝑅‘𝑘))) ≠ 0)) |
39 | fvco3 6991 | . . . . . 6 ⊢ ((𝑅:ℕ0⟶𝑆 ∧ 𝑘 ∈ ℕ0) → ((𝐶 ∘ 𝑅)‘𝑘) = (𝐶‘(𝑅‘𝑘))) | |
40 | 7, 39 | sylan 579 | . . . . 5 ⊢ ((𝐴 ∈ 𝑆 ∧ 𝑘 ∈ ℕ0) → ((𝐶 ∘ 𝑅)‘𝑘) = (𝐶‘(𝑅‘𝑘))) |
41 | 37, 40 | breq12d 5155 | . . . 4 ⊢ ((𝐴 ∈ 𝑆 ∧ 𝑘 ∈ ℕ0) → (((𝐶 ∘ 𝑅)‘(𝑘 + 1)) < ((𝐶 ∘ 𝑅)‘𝑘) ↔ (𝐶‘(𝐹‘(𝑅‘𝑘))) < (𝐶‘(𝑅‘𝑘)))) |
42 | 31, 38, 41 | 3imtr4d 294 | . . 3 ⊢ ((𝐴 ∈ 𝑆 ∧ 𝑘 ∈ ℕ0) → (((𝐶 ∘ 𝑅)‘(𝑘 + 1)) ≠ 0 → ((𝐶 ∘ 𝑅)‘(𝑘 + 1)) < ((𝐶 ∘ 𝑅)‘𝑘))) |
43 | 15, 22, 42 | nn0seqcvgd 16532 | . 2 ⊢ (𝐴 ∈ 𝑆 → ((𝐶 ∘ 𝑅)‘𝑁) = 0) |
44 | 13, 43 | eqtr3d 2769 | 1 ⊢ (𝐴 ∈ 𝑆 → (𝐶‘(𝑅‘𝑁)) = 0) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 395 = wceq 1534 ∈ wcel 2099 ≠ wne 2935 {csn 4624 class class class wbr 5142 × cxp 5670 ∘ ccom 5676 ⟶wf 6538 ‘cfv 6542 (class class class)co 7414 1st c1st 7985 0cc0 11130 1c1 11131 + caddc 11133 < clt 11270 ℕ0cn0 12494 seqcseq 13990 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2164 ax-ext 2698 ax-sep 5293 ax-nul 5300 ax-pow 5359 ax-pr 5423 ax-un 7734 ax-cnex 11186 ax-resscn 11187 ax-1cn 11188 ax-icn 11189 ax-addcl 11190 ax-addrcl 11191 ax-mulcl 11192 ax-mulrcl 11193 ax-mulcom 11194 ax-addass 11195 ax-mulass 11196 ax-distr 11197 ax-i2m1 11198 ax-1ne0 11199 ax-1rid 11200 ax-rnegex 11201 ax-rrecex 11202 ax-cnre 11203 ax-pre-lttri 11204 ax-pre-lttrn 11205 ax-pre-ltadd 11206 ax-pre-mulgt0 11207 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3or 1086 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2529 df-eu 2558 df-clab 2705 df-cleq 2719 df-clel 2805 df-nfc 2880 df-ne 2936 df-nel 3042 df-ral 3057 df-rex 3066 df-reu 3372 df-rab 3428 df-v 3471 df-sbc 3775 df-csb 3890 df-dif 3947 df-un 3949 df-in 3951 df-ss 3961 df-pss 3963 df-nul 4319 df-if 4525 df-pw 4600 df-sn 4625 df-pr 4627 df-op 4631 df-uni 4904 df-iun 4993 df-br 5143 df-opab 5205 df-mpt 5226 df-tr 5260 df-id 5570 df-eprel 5576 df-po 5584 df-so 5585 df-fr 5627 df-we 5629 df-xp 5678 df-rel 5679 df-cnv 5680 df-co 5681 df-dm 5682 df-rn 5683 df-res 5684 df-ima 5685 df-pred 6299 df-ord 6366 df-on 6367 df-lim 6368 df-suc 6369 df-iota 6494 df-fun 6544 df-fn 6545 df-f 6546 df-f1 6547 df-fo 6548 df-f1o 6549 df-fv 6550 df-riota 7370 df-ov 7417 df-oprab 7418 df-mpo 7419 df-om 7865 df-1st 7987 df-2nd 7988 df-frecs 8280 df-wrecs 8311 df-recs 8385 df-rdg 8424 df-er 8718 df-en 8956 df-dom 8957 df-sdom 8958 df-pnf 11272 df-mnf 11273 df-xr 11274 df-ltxr 11275 df-le 11276 df-sub 11468 df-neg 11469 df-nn 12235 df-n0 12495 df-z 12581 df-uz 12845 df-fz 13509 df-seq 13991 |
This theorem is referenced by: algcvga 16541 |
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