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Theorem ialgrf 10821
 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.)
Hypotheses
Ref Expression
algrf.1 𝑍 = (ℤ𝑀)
algrf.2 𝑅 = seq𝑀((𝐹 ∘ 1st ), (𝑍 × {𝐴}), 𝑆)
algrf.3 (𝜑𝑀 ∈ ℤ)
algrf.4 (𝜑𝐴𝑆)
algrf.5 (𝜑𝐹:𝑆𝑆)
algrf.s (𝜑𝑆𝑉)
Assertion
Ref Expression
ialgrf (𝜑𝑅:𝑍𝑆)

Proof of Theorem ialgrf
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)
31eleq2i 2151 . . . 4 (𝑥𝑍𝑥 ∈ (ℤ𝑀))
4 algrf.4 . . . . 5 (𝜑𝐴𝑆)
51, 4ialgrlemconst 10819 . . . 4 ((𝜑𝑥 ∈ (ℤ𝑀)) → ((𝑍 × {𝐴})‘𝑥) ∈ 𝑆)
63, 5sylan2b 281 . . 3 ((𝜑𝑥𝑍) → ((𝑍 × {𝐴})‘𝑥) ∈ 𝑆)
7 algrf.5 . . . 4 (𝜑𝐹:𝑆𝑆)
87ialgrlem1st 10818 . . 3 ((𝜑 ∧ (𝑥𝑆𝑦𝑆)) → (𝑥(𝐹 ∘ 1st )𝑦) ∈ 𝑆)
91, 2, 6, 8iseqfcl 9768 . 2 (𝜑 → seq𝑀((𝐹 ∘ 1st ), (𝑍 × {𝐴}), 𝑆):𝑍𝑆)
10 algrf.2 . . 3 𝑅 = seq𝑀((𝐹 ∘ 1st ), (𝑍 × {𝐴}), 𝑆)
1110feq1i 5110 . 2 (𝑅:𝑍𝑆 ↔ seq𝑀((𝐹 ∘ 1st ), (𝑍 × {𝐴}), 𝑆):𝑍𝑆)
129, 11sylibr 132 1 (𝜑𝑅:𝑍𝑆)
 Colors of variables: wff set class Syntax hints:   → wi 4   = wceq 1287   ∈ wcel 1436  {csn 3425   × cxp 4402   ∘ ccom 4408  ⟶wf 4968  ‘cfv 4972  1st c1st 5847  ℤcz 8660  ℤ≥cuz 8928  seqcseq 9754 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 577  ax-in2 578  ax-io 663  ax-5 1379  ax-7 1380  ax-gen 1381  ax-ie1 1425  ax-ie2 1426  ax-8 1438  ax-10 1439  ax-11 1440  ax-i12 1441  ax-bndl 1442  ax-4 1443  ax-13 1447  ax-14 1448  ax-17 1462  ax-i9 1466  ax-ial 1470  ax-i5r 1471  ax-ext 2067  ax-coll 3922  ax-sep 3925  ax-nul 3933  ax-pow 3977  ax-pr 4003  ax-un 4227  ax-setind 4319  ax-iinf 4369  ax-cnex 7357  ax-resscn 7358  ax-1cn 7359  ax-1re 7360  ax-icn 7361  ax-addcl 7362  ax-addrcl 7363  ax-mulcl 7364  ax-addcom 7366  ax-addass 7368  ax-distr 7370  ax-i2m1 7371  ax-0lt1 7372  ax-0id 7374  ax-rnegex 7375  ax-cnre 7377  ax-pre-ltirr 7378  ax-pre-ltwlin 7379  ax-pre-lttrn 7380  ax-pre-ltadd 7382 This theorem depends on definitions:  df-bi 115  df-3or 923  df-3an 924  df-tru 1290  df-fal 1293  df-nf 1393  df-sb 1690  df-eu 1948  df-mo 1949  df-clab 2072  df-cleq 2078  df-clel 2081  df-nfc 2214  df-ne 2252  df-nel 2347  df-ral 2360  df-rex 2361  df-reu 2362  df-rab 2364  df-v 2616  df-sbc 2829  df-csb 2922  df-dif 2988  df-un 2990  df-in 2992  df-ss 2999  df-nul 3273  df-pw 3411  df-sn 3431  df-pr 3432  df-op 3434  df-uni 3631  df-int 3666  df-iun 3709  df-br 3815  df-opab 3869  df-mpt 3870  df-tr 3905  df-id 4087  df-iord 4160  df-on 4162  df-ilim 4163  df-suc 4165  df-iom 4372  df-xp 4410  df-rel 4411  df-cnv 4412  df-co 4413  df-dm 4414  df-rn 4415  df-res 4416  df-ima 4417  df-iota 4937  df-fun 4974  df-fn 4975  df-f 4976  df-f1 4977  df-fo 4978  df-f1o 4979  df-fv 4980  df-riota 5550  df-ov 5597  df-oprab 5598  df-mpt2 5599  df-1st 5849  df-2nd 5850  df-recs 6005  df-frec 6091  df-pnf 7445  df-mnf 7446  df-xr 7447  df-ltxr 7448  df-le 7449  df-sub 7576  df-neg 7577  df-inn 8335  df-n0 8584  df-z 8661  df-uz 8929  df-iseq 9755 This theorem is referenced by:  ialginv  10823  ialgcvg  10824  ialgcvga  10827  ialgfx  10828  eucialgcvga  10834  eucialg  10835
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