ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  algrf GIF version

Theorem algrf 11519
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 5566 . . . . 5 ((𝐴𝑆𝑥𝑍) → ((𝑍 × {𝐴})‘𝑥) = 𝐴)
53, 4sylan 279 . . . 4 ((𝜑𝑥𝑍) → ((𝑍 × {𝐴})‘𝑥) = 𝐴)
63adantr 272 . . . 4 ((𝜑𝑥𝑍) → 𝐴𝑆)
75, 6eqeltrd 2176 . . 3 ((𝜑𝑥𝑍) → ((𝑍 × {𝐴})‘𝑥) ∈ 𝑆)
8 vex 2644 . . . . 5 𝑥 ∈ V
9 vex 2644 . . . . 5 𝑦 ∈ V
108, 9algrflem 6056 . . . 4 (𝑥(𝐹 ∘ 1st )𝑦) = (𝐹𝑥)
11 algrf.5 . . . . 5 (𝜑𝐹:𝑆𝑆)
12 simpl 108 . . . . 5 ((𝑥𝑆𝑦𝑆) → 𝑥𝑆)
13 ffvelrn 5485 . . . . 5 ((𝐹:𝑆𝑆𝑥𝑆) → (𝐹𝑥) ∈ 𝑆)
1411, 12, 13syl2an 285 . . . 4 ((𝜑 ∧ (𝑥𝑆𝑦𝑆)) → (𝐹𝑥) ∈ 𝑆)
1510, 14syl5eqel 2186 . . 3 ((𝜑 ∧ (𝑥𝑆𝑦𝑆)) → (𝑥(𝐹 ∘ 1st )𝑦) ∈ 𝑆)
161, 2, 7, 15seqf 10075 . 2 (𝜑 → seq𝑀((𝐹 ∘ 1st ), (𝑍 × {𝐴})):𝑍𝑆)
17 algrf.2 . . 3 𝑅 = seq𝑀((𝐹 ∘ 1st ), (𝑍 × {𝐴}))
1817feq1i 5201 . 2 (𝑅:𝑍𝑆 ↔ seq𝑀((𝐹 ∘ 1st ), (𝑍 × {𝐴})):𝑍𝑆)
1916, 18sylibr 133 1 (𝜑𝑅:𝑍𝑆)
Colors of variables: wff set class
Syntax hints:  wi 4  wa 103   = wceq 1299  wcel 1448  {csn 3474   × cxp 4475  ccom 4481  wf 5055  cfv 5059  (class class class)co 5706  1st c1st 5967  cz 8906  cuz 9176  seqcseq 10059
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 584  ax-in2 585  ax-io 671  ax-5 1391  ax-7 1392  ax-gen 1393  ax-ie1 1437  ax-ie2 1438  ax-8 1450  ax-10 1451  ax-11 1452  ax-i12 1453  ax-bndl 1454  ax-4 1455  ax-13 1459  ax-14 1460  ax-17 1474  ax-i9 1478  ax-ial 1482  ax-i5r 1483  ax-ext 2082  ax-coll 3983  ax-sep 3986  ax-nul 3994  ax-pow 4038  ax-pr 4069  ax-un 4293  ax-setind 4390  ax-iinf 4440  ax-cnex 7586  ax-resscn 7587  ax-1cn 7588  ax-1re 7589  ax-icn 7590  ax-addcl 7591  ax-addrcl 7592  ax-mulcl 7593  ax-addcom 7595  ax-addass 7597  ax-distr 7599  ax-i2m1 7600  ax-0lt1 7601  ax-0id 7603  ax-rnegex 7604  ax-cnre 7606  ax-pre-ltirr 7607  ax-pre-ltwlin 7608  ax-pre-lttrn 7609  ax-pre-ltadd 7611
This theorem depends on definitions:  df-bi 116  df-3or 931  df-3an 932  df-tru 1302  df-fal 1305  df-nf 1405  df-sb 1704  df-eu 1963  df-mo 1964  df-clab 2087  df-cleq 2093  df-clel 2096  df-nfc 2229  df-ne 2268  df-nel 2363  df-ral 2380  df-rex 2381  df-reu 2382  df-rab 2384  df-v 2643  df-sbc 2863  df-csb 2956  df-dif 3023  df-un 3025  df-in 3027  df-ss 3034  df-nul 3311  df-pw 3459  df-sn 3480  df-pr 3481  df-op 3483  df-uni 3684  df-int 3719  df-iun 3762  df-br 3876  df-opab 3930  df-mpt 3931  df-tr 3967  df-id 4153  df-iord 4226  df-on 4228  df-ilim 4229  df-suc 4231  df-iom 4443  df-xp 4483  df-rel 4484  df-cnv 4485  df-co 4486  df-dm 4487  df-rn 4488  df-res 4489  df-ima 4490  df-iota 5024  df-fun 5061  df-fn 5062  df-f 5063  df-f1 5064  df-fo 5065  df-f1o 5066  df-fv 5067  df-riota 5662  df-ov 5709  df-oprab 5710  df-mpo 5711  df-1st 5969  df-2nd 5970  df-recs 6132  df-frec 6218  df-pnf 7674  df-mnf 7675  df-xr 7676  df-ltxr 7677  df-le 7678  df-sub 7806  df-neg 7807  df-inn 8579  df-n0 8830  df-z 8907  df-uz 9177  df-seqfrec 10060
This theorem is referenced by:  algrp1  11520  alginv  11521  algcvg  11522  algcvga  11525  algfx  11526  eucalgcvga  11532  eucalg  11533
  Copyright terms: Public domain W3C validator