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Mirrors > Home > ILE Home > Th. List > algrf | 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 5707 | . . . . 5 ⊢ ((𝐴 ∈ 𝑆 ∧ 𝑥 ∈ 𝑍) → ((𝑍 × {𝐴})‘𝑥) = 𝐴) | |
5 | 3, 4 | sylan 281 | . . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑍) → ((𝑍 × {𝐴})‘𝑥) = 𝐴) |
6 | 3 | adantr 274 | . . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑍) → 𝐴 ∈ 𝑆) |
7 | 5, 6 | eqeltrd 2247 | . . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑍) → ((𝑍 × {𝐴})‘𝑥) ∈ 𝑆) |
8 | vex 2733 | . . . . 5 ⊢ 𝑥 ∈ V | |
9 | vex 2733 | . . . . 5 ⊢ 𝑦 ∈ V | |
10 | 8, 9 | algrflem 6205 | . . . 4 ⊢ (𝑥(𝐹 ∘ 1st )𝑦) = (𝐹‘𝑥) |
11 | algrf.5 | . . . . 5 ⊢ (𝜑 → 𝐹:𝑆⟶𝑆) | |
12 | simpl 108 | . . . . 5 ⊢ ((𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆) → 𝑥 ∈ 𝑆) | |
13 | ffvelrn 5626 | . . . . 5 ⊢ ((𝐹:𝑆⟶𝑆 ∧ 𝑥 ∈ 𝑆) → (𝐹‘𝑥) ∈ 𝑆) | |
14 | 11, 12, 13 | syl2an 287 | . . . 4 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → (𝐹‘𝑥) ∈ 𝑆) |
15 | 10, 14 | eqeltrid 2257 | . . 3 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → (𝑥(𝐹 ∘ 1st )𝑦) ∈ 𝑆) |
16 | 1, 2, 7, 15 | seqf 10404 | . 2 ⊢ (𝜑 → seq𝑀((𝐹 ∘ 1st ), (𝑍 × {𝐴})):𝑍⟶𝑆) |
17 | algrf.2 | . . 3 ⊢ 𝑅 = seq𝑀((𝐹 ∘ 1st ), (𝑍 × {𝐴})) | |
18 | 17 | feq1i 5338 | . 2 ⊢ (𝑅:𝑍⟶𝑆 ↔ seq𝑀((𝐹 ∘ 1st ), (𝑍 × {𝐴})):𝑍⟶𝑆) |
19 | 16, 18 | sylibr 133 | 1 ⊢ (𝜑 → 𝑅:𝑍⟶𝑆) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 103 = wceq 1348 ∈ wcel 2141 {csn 3581 × cxp 4607 ∘ ccom 4613 ⟶wf 5192 ‘cfv 5196 (class class class)co 5850 1st c1st 6114 ℤcz 9199 ℤ≥cuz 9474 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-ltadd 7877 |
This theorem depends on definitions: df-bi 116 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: algrp1 11987 alginv 11988 algcvg 11989 algcvga 11992 algfx 11993 eucalgcvga 11999 eucalg 12000 |
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