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Theorem algrf 12072
Description: An algorithm is a step function 𝐹:𝑆𝑆 on a state space 𝑆. An algorithm acts on an initial state 𝐴𝑆 by iteratively applying 𝐹 to give 𝐴, (𝐹𝐴), (𝐹‘(𝐹𝐴)) 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:ℕ0𝑆 "runs" the algorithm 𝐹 so that (𝑅𝑘) 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.)

Hypotheses
Ref Expression
algrf.1 𝑍 = (ℤ𝑀)
algrf.2 𝑅 = seq𝑀((𝐹 ∘ 1st ), (𝑍 × {𝐴}))
algrf.3 (𝜑𝑀 ∈ ℤ)
algrf.4 (𝜑𝐴𝑆)
algrf.5 (𝜑𝐹:𝑆𝑆)
Assertion
Ref Expression
algrf (𝜑𝑅:𝑍𝑆)

Proof of Theorem algrf
Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 algrf.1 . . 3 𝑍 = (ℤ𝑀)
2 algrf.3 . . 3 (𝜑𝑀 ∈ ℤ)
3 algrf.4 . . . . 5 (𝜑𝐴𝑆)
4 fvconst2g 5747 . . . . 5 ((𝐴𝑆𝑥𝑍) → ((𝑍 × {𝐴})‘𝑥) = 𝐴)
53, 4sylan 283 . . . 4 ((𝜑𝑥𝑍) → ((𝑍 × {𝐴})‘𝑥) = 𝐴)
63adantr 276 . . . 4 ((𝜑𝑥𝑍) → 𝐴𝑆)
75, 6eqeltrd 2266 . . 3 ((𝜑𝑥𝑍) → ((𝑍 × {𝐴})‘𝑥) ∈ 𝑆)
8 vex 2755 . . . . 5 𝑥 ∈ V
9 vex 2755 . . . . 5 𝑦 ∈ V
108, 9algrflem 6249 . . . 4 (𝑥(𝐹 ∘ 1st )𝑦) = (𝐹𝑥)
11 algrf.5 . . . . 5 (𝜑𝐹:𝑆𝑆)
12 simpl 109 . . . . 5 ((𝑥𝑆𝑦𝑆) → 𝑥𝑆)
13 ffvelcdm 5666 . . . . 5 ((𝐹:𝑆𝑆𝑥𝑆) → (𝐹𝑥) ∈ 𝑆)
1411, 12, 13syl2an 289 . . . 4 ((𝜑 ∧ (𝑥𝑆𝑦𝑆)) → (𝐹𝑥) ∈ 𝑆)
1510, 14eqeltrid 2276 . . 3 ((𝜑 ∧ (𝑥𝑆𝑦𝑆)) → (𝑥(𝐹 ∘ 1st )𝑦) ∈ 𝑆)
161, 2, 7, 15seqf 10487 . 2 (𝜑 → seq𝑀((𝐹 ∘ 1st ), (𝑍 × {𝐴})):𝑍𝑆)
17 algrf.2 . . 3 𝑅 = seq𝑀((𝐹 ∘ 1st ), (𝑍 × {𝐴}))
1817feq1i 5374 . 2 (𝑅:𝑍𝑆 ↔ seq𝑀((𝐹 ∘ 1st ), (𝑍 × {𝐴})):𝑍𝑆)
1916, 18sylibr 134 1 (𝜑𝑅:𝑍𝑆)
Colors of variables: wff set class
Syntax hints:  wi 4  wa 104   = wceq 1364  wcel 2160  {csn 3607   × cxp 4639  ccom 4645  wf 5228  cfv 5232  (class class class)co 5892  1st c1st 6158  cz 9278  cuz 9553  seqcseq 10471
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 615  ax-in2 616  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2162  ax-14 2163  ax-ext 2171  ax-coll 4133  ax-sep 4136  ax-nul 4144  ax-pow 4189  ax-pr 4224  ax-un 4448  ax-setind 4551  ax-iinf 4602  ax-cnex 7927  ax-resscn 7928  ax-1cn 7929  ax-1re 7930  ax-icn 7931  ax-addcl 7932  ax-addrcl 7933  ax-mulcl 7934  ax-addcom 7936  ax-addass 7938  ax-distr 7940  ax-i2m1 7941  ax-0lt1 7942  ax-0id 7944  ax-rnegex 7945  ax-cnre 7947  ax-pre-ltirr 7948  ax-pre-ltwlin 7949  ax-pre-lttrn 7950  ax-pre-ltadd 7952
This theorem depends on definitions:  df-bi 117  df-3or 981  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1472  df-sb 1774  df-eu 2041  df-mo 2042  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-ne 2361  df-nel 2456  df-ral 2473  df-rex 2474  df-reu 2475  df-rab 2477  df-v 2754  df-sbc 2978  df-csb 3073  df-dif 3146  df-un 3148  df-in 3150  df-ss 3157  df-nul 3438  df-pw 3592  df-sn 3613  df-pr 3614  df-op 3616  df-uni 3825  df-int 3860  df-iun 3903  df-br 4019  df-opab 4080  df-mpt 4081  df-tr 4117  df-id 4308  df-iord 4381  df-on 4383  df-ilim 4384  df-suc 4386  df-iom 4605  df-xp 4647  df-rel 4648  df-cnv 4649  df-co 4650  df-dm 4651  df-rn 4652  df-res 4653  df-ima 4654  df-iota 5193  df-fun 5234  df-fn 5235  df-f 5236  df-f1 5237  df-fo 5238  df-f1o 5239  df-fv 5240  df-riota 5848  df-ov 5895  df-oprab 5896  df-mpo 5897  df-1st 6160  df-2nd 6161  df-recs 6325  df-frec 6411  df-pnf 8019  df-mnf 8020  df-xr 8021  df-ltxr 8022  df-le 8023  df-sub 8155  df-neg 8156  df-inn 8945  df-n0 9202  df-z 9279  df-uz 9554  df-seqfrec 10472
This theorem is referenced by:  algrp1  12073  alginv  12074  algcvg  12075  algcvga  12078  algfx  12079  eucalgcvga  12085  eucalg  12086
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