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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 6964 | . . . . 5 ⊢ ((𝐴 ∈ 𝑆 ∧ 𝑥 ∈ 𝑍) → ((𝑍 × {𝐴})‘𝑥) = 𝐴) | |
5 | 3, 4 | sylan 582 | . . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑍) → ((𝑍 × {𝐴})‘𝑥) = 𝐴) |
6 | 3 | adantr 483 | . . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑍) → 𝐴 ∈ 𝑆) |
7 | 5, 6 | eqeltrd 2913 | . . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑍) → ((𝑍 × {𝐴})‘𝑥) ∈ 𝑆) |
8 | vex 3497 | . . . . 5 ⊢ 𝑥 ∈ V | |
9 | vex 3497 | . . . . 5 ⊢ 𝑦 ∈ V | |
10 | 8, 9 | algrflem 7819 | . . . 4 ⊢ (𝑥(𝐹 ∘ 1st )𝑦) = (𝐹‘𝑥) |
11 | algrf.5 | . . . . 5 ⊢ (𝜑 → 𝐹:𝑆⟶𝑆) | |
12 | simpl 485 | . . . . 5 ⊢ ((𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆) → 𝑥 ∈ 𝑆) | |
13 | ffvelrn 6849 | . . . . 5 ⊢ ((𝐹:𝑆⟶𝑆 ∧ 𝑥 ∈ 𝑆) → (𝐹‘𝑥) ∈ 𝑆) | |
14 | 11, 12, 13 | syl2an 597 | . . . 4 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → (𝐹‘𝑥) ∈ 𝑆) |
15 | 10, 14 | eqeltrid 2917 | . . 3 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → (𝑥(𝐹 ∘ 1st )𝑦) ∈ 𝑆) |
16 | 1, 2, 7, 15 | seqf 13392 | . 2 ⊢ (𝜑 → seq𝑀((𝐹 ∘ 1st ), (𝑍 × {𝐴})):𝑍⟶𝑆) |
17 | algrf.2 | . . 3 ⊢ 𝑅 = seq𝑀((𝐹 ∘ 1st ), (𝑍 × {𝐴})) | |
18 | 17 | feq1i 6505 | . 2 ⊢ (𝑅:𝑍⟶𝑆 ↔ seq𝑀((𝐹 ∘ 1st ), (𝑍 × {𝐴})):𝑍⟶𝑆) |
19 | 16, 18 | sylibr 236 | 1 ⊢ (𝜑 → 𝑅:𝑍⟶𝑆) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 = wceq 1537 ∈ wcel 2114 {csn 4567 × cxp 5553 ∘ ccom 5559 ⟶wf 6351 ‘cfv 6355 (class class class)co 7156 1st c1st 7687 ℤcz 11982 ℤ≥cuz 12244 seqcseq 13370 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 ax-sep 5203 ax-nul 5210 ax-pow 5266 ax-pr 5330 ax-un 7461 ax-cnex 10593 ax-resscn 10594 ax-1cn 10595 ax-icn 10596 ax-addcl 10597 ax-addrcl 10598 ax-mulcl 10599 ax-mulrcl 10600 ax-mulcom 10601 ax-addass 10602 ax-mulass 10603 ax-distr 10604 ax-i2m1 10605 ax-1ne0 10606 ax-1rid 10607 ax-rnegex 10608 ax-rrecex 10609 ax-cnre 10610 ax-pre-lttri 10611 ax-pre-lttrn 10612 ax-pre-ltadd 10613 ax-pre-mulgt0 10614 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-nel 3124 df-ral 3143 df-rex 3144 df-reu 3145 df-rab 3147 df-v 3496 df-sbc 3773 df-csb 3884 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-pss 3954 df-nul 4292 df-if 4468 df-pw 4541 df-sn 4568 df-pr 4570 df-tp 4572 df-op 4574 df-uni 4839 df-iun 4921 df-br 5067 df-opab 5129 df-mpt 5147 df-tr 5173 df-id 5460 df-eprel 5465 df-po 5474 df-so 5475 df-fr 5514 df-we 5516 df-xp 5561 df-rel 5562 df-cnv 5563 df-co 5564 df-dm 5565 df-rn 5566 df-res 5567 df-ima 5568 df-pred 6148 df-ord 6194 df-on 6195 df-lim 6196 df-suc 6197 df-iota 6314 df-fun 6357 df-fn 6358 df-f 6359 df-f1 6360 df-fo 6361 df-f1o 6362 df-fv 6363 df-riota 7114 df-ov 7159 df-oprab 7160 df-mpo 7161 df-om 7581 df-1st 7689 df-2nd 7690 df-wrecs 7947 df-recs 8008 df-rdg 8046 df-er 8289 df-en 8510 df-dom 8511 df-sdom 8512 df-pnf 10677 df-mnf 10678 df-xr 10679 df-ltxr 10680 df-le 10681 df-sub 10872 df-neg 10873 df-nn 11639 df-n0 11899 df-z 11983 df-uz 12245 df-fz 12894 df-seq 13371 |
This theorem is referenced by: alginv 15919 algcvg 15920 algcvga 15923 algfx 15924 eucalgcvga 15930 eucalg 15931 ovolicc2lem2 24119 ovolicc2lem3 24120 ovolicc2lem4 24121 bfplem1 35115 bfplem2 35116 |
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