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Mirrors > Home > MPE Home > Th. List > algrf | Structured version Visualization version GIF version |
Description: An algorithm is a step
function 𝐹:𝑆⟶𝑆 on a state space 𝑆.
An algorithm acts on an initial state 𝐴 ∈ 𝑆 by iteratively applying
𝐹 to give 𝐴, (𝐹‘𝐴), (𝐹‘(𝐹‘𝐴)) and so
on. An algorithm is said to halt if a fixed point of 𝐹 is
reached
after a finite number of iterations.
The algorithm iterator 𝑅:ℕ0⟶𝑆 "runs" the algorithm 𝐹 so that (𝑅‘𝑘) is the state after 𝑘 iterations of 𝐹 on the initial state 𝐴. Domain and codomain of the algorithm iterator 𝑅. (Contributed by Paul Chapman, 31-Mar-2011.) (Revised by Mario Carneiro, 28-May-2014.) |
Ref | Expression |
---|---|
algrf.1 | ⊢ 𝑍 = (ℤ≥‘𝑀) |
algrf.2 | ⊢ 𝑅 = seq𝑀((𝐹 ∘ 1st ), (𝑍 × {𝐴})) |
algrf.3 | ⊢ (𝜑 → 𝑀 ∈ ℤ) |
algrf.4 | ⊢ (𝜑 → 𝐴 ∈ 𝑆) |
algrf.5 | ⊢ (𝜑 → 𝐹:𝑆⟶𝑆) |
Ref | Expression |
---|---|
algrf | ⊢ (𝜑 → 𝑅:𝑍⟶𝑆) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | algrf.1 | . . 3 ⊢ 𝑍 = (ℤ≥‘𝑀) | |
2 | algrf.3 | . . 3 ⊢ (𝜑 → 𝑀 ∈ ℤ) | |
3 | algrf.4 | . . . . 5 ⊢ (𝜑 → 𝐴 ∈ 𝑆) | |
4 | fvconst2g 7222 | . . . . 5 ⊢ ((𝐴 ∈ 𝑆 ∧ 𝑥 ∈ 𝑍) → ((𝑍 × {𝐴})‘𝑥) = 𝐴) | |
5 | 3, 4 | sylan 580 | . . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑍) → ((𝑍 × {𝐴})‘𝑥) = 𝐴) |
6 | 3 | adantr 480 | . . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑍) → 𝐴 ∈ 𝑆) |
7 | 5, 6 | eqeltrd 2839 | . . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑍) → ((𝑍 × {𝐴})‘𝑥) ∈ 𝑆) |
8 | vex 3482 | . . . . 5 ⊢ 𝑥 ∈ V | |
9 | vex 3482 | . . . . 5 ⊢ 𝑦 ∈ V | |
10 | 8, 9 | opco1i 8149 | . . . 4 ⊢ (𝑥(𝐹 ∘ 1st )𝑦) = (𝐹‘𝑥) |
11 | algrf.5 | . . . . 5 ⊢ (𝜑 → 𝐹:𝑆⟶𝑆) | |
12 | simpl 482 | . . . . 5 ⊢ ((𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆) → 𝑥 ∈ 𝑆) | |
13 | ffvelcdm 7101 | . . . . 5 ⊢ ((𝐹:𝑆⟶𝑆 ∧ 𝑥 ∈ 𝑆) → (𝐹‘𝑥) ∈ 𝑆) | |
14 | 11, 12, 13 | syl2an 596 | . . . 4 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → (𝐹‘𝑥) ∈ 𝑆) |
15 | 10, 14 | eqeltrid 2843 | . . 3 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → (𝑥(𝐹 ∘ 1st )𝑦) ∈ 𝑆) |
16 | 1, 2, 7, 15 | seqf 14061 | . 2 ⊢ (𝜑 → seq𝑀((𝐹 ∘ 1st ), (𝑍 × {𝐴})):𝑍⟶𝑆) |
17 | algrf.2 | . . 3 ⊢ 𝑅 = seq𝑀((𝐹 ∘ 1st ), (𝑍 × {𝐴})) | |
18 | 17 | feq1i 6728 | . 2 ⊢ (𝑅:𝑍⟶𝑆 ↔ seq𝑀((𝐹 ∘ 1st ), (𝑍 × {𝐴})):𝑍⟶𝑆) |
19 | 16, 18 | sylibr 234 | 1 ⊢ (𝜑 → 𝑅:𝑍⟶𝑆) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 395 = wceq 1537 ∈ wcel 2106 {csn 4631 × cxp 5687 ∘ ccom 5693 ⟶wf 6559 ‘cfv 6563 (class class class)co 7431 1st c1st 8011 ℤcz 12611 ℤ≥cuz 12876 seqcseq 14039 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-sep 5302 ax-nul 5312 ax-pow 5371 ax-pr 5438 ax-un 7754 ax-cnex 11209 ax-resscn 11210 ax-1cn 11211 ax-icn 11212 ax-addcl 11213 ax-addrcl 11214 ax-mulcl 11215 ax-mulrcl 11216 ax-mulcom 11217 ax-addass 11218 ax-mulass 11219 ax-distr 11220 ax-i2m1 11221 ax-1ne0 11222 ax-1rid 11223 ax-rnegex 11224 ax-rrecex 11225 ax-cnre 11226 ax-pre-lttri 11227 ax-pre-lttrn 11228 ax-pre-ltadd 11229 ax-pre-mulgt0 11230 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ne 2939 df-nel 3045 df-ral 3060 df-rex 3069 df-reu 3379 df-rab 3434 df-v 3480 df-sbc 3792 df-csb 3909 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-pss 3983 df-nul 4340 df-if 4532 df-pw 4607 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 df-iun 4998 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5583 df-eprel 5589 df-po 5597 df-so 5598 df-fr 5641 df-we 5643 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-rn 5700 df-res 5701 df-ima 5702 df-pred 6323 df-ord 6389 df-on 6390 df-lim 6391 df-suc 6392 df-iota 6516 df-fun 6565 df-fn 6566 df-f 6567 df-f1 6568 df-fo 6569 df-f1o 6570 df-fv 6571 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-om 7888 df-1st 8013 df-2nd 8014 df-frecs 8305 df-wrecs 8336 df-recs 8410 df-rdg 8449 df-er 8744 df-en 8985 df-dom 8986 df-sdom 8987 df-pnf 11295 df-mnf 11296 df-xr 11297 df-ltxr 11298 df-le 11299 df-sub 11492 df-neg 11493 df-nn 12265 df-n0 12525 df-z 12612 df-uz 12877 df-fz 13545 df-seq 14040 |
This theorem is referenced by: alginv 16609 algcvg 16610 algcvga 16613 algfx 16614 eucalgcvga 16620 eucalg 16621 ovolicc2lem2 25567 ovolicc2lem3 25568 ovolicc2lem4 25569 bfplem1 37809 bfplem2 37810 |
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