Theorem algrf 12588

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 = (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)

Assertion

Ref	Expression
algrf	⊢ (𝜑 → 𝑅:𝑍⟶𝑆)

Proof of Theorem algrf

Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.

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 5860	. . . . 5 ⊢ ((𝐴 ∈ 𝑆 ∧ 𝑥 ∈ 𝑍) → ((𝑍 × {𝐴})‘𝑥) = 𝐴)
5	3, 4	sylan 283	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑍) → ((𝑍 × {𝐴})‘𝑥) = 𝐴)
6	3	adantr 276	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑍) → 𝐴 ∈ 𝑆)
7	5, 6	eqeltrd 2306	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑍) → ((𝑍 × {𝐴})‘𝑥) ∈ 𝑆)
8		vex 2802	. . . . 5 ⊢ 𝑥 ∈ V
9		vex 2802	. . . . 5 ⊢ 𝑦 ∈ V
10	8, 9	algrflem 6386	. . . 4 ⊢ (𝑥(𝐹 ∘ 1st )𝑦) = (𝐹‘𝑥)
11		algrf.5	. . . . 5 ⊢ (𝜑 → 𝐹:𝑆⟶𝑆)
12		simpl 109	. . . . 5 ⊢ ((𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆) → 𝑥 ∈ 𝑆)
13		ffvelcdm 5773	. . . . 5 ⊢ ((𝐹:𝑆⟶𝑆 ∧ 𝑥 ∈ 𝑆) → (𝐹‘𝑥) ∈ 𝑆)
14	11, 12, 13	syl2an 289	. . . 4 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → (𝐹‘𝑥) ∈ 𝑆)
15	10, 14	eqeltrid 2316	. . 3 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → (𝑥(𝐹 ∘ 1st )𝑦) ∈ 𝑆)
16	1, 2, 7, 15	seqf 10703	. 2 ⊢ (𝜑 → seq𝑀((𝐹 ∘ 1st ), (𝑍 × {𝐴})):𝑍⟶𝑆)
17		algrf.2	. . 3 ⊢ 𝑅 = seq𝑀((𝐹 ∘ 1st ), (𝑍 × {𝐴}))
18	17	feq1i 5469	. 2 ⊢ (𝑅:𝑍⟶𝑆 ↔ seq𝑀((𝐹 ∘ 1st ), (𝑍 × {𝐴})):𝑍⟶𝑆)
19	16, 18	sylibr 134	1 ⊢ (𝜑 → 𝑅:𝑍⟶𝑆)

Syntax hints:   → wi 4   ∧ wa 104   = wceq 1395   ∈ wcel 2200  {csn 3666   × cxp 4718   ∘ ccom 4724  ⟶wf 5317  ‘cfv 5321  (class class class)co 6010  1^st c1st 6293  ℤcz 9462  ℤ_≥cuz 9738  seqcseq 10686

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 617  ax-in2 618  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-13 2202  ax-14 2203  ax-ext 2211  ax-coll 4199  ax-sep 4202  ax-nul 4210  ax-pow 4259  ax-pr 4294  ax-un 4525  ax-setind 4630  ax-iinf 4681  ax-cnex 8106  ax-resscn 8107  ax-1cn 8108  ax-1re 8109  ax-icn 8110  ax-addcl 8111  ax-addrcl 8112  ax-mulcl 8113  ax-addcom 8115  ax-addass 8117  ax-distr 8119  ax-i2m1 8120  ax-0lt1 8121  ax-0id 8123  ax-rnegex 8124  ax-cnre 8126  ax-pre-ltirr 8127  ax-pre-ltwlin 8128  ax-pre-lttrn 8129  ax-pre-ltadd 8131

This theorem depends on definitions:  df-bi 117  df-3or 1003  df-3an 1004  df-tru 1398  df-fal 1401  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ne 2401  df-nel 2496  df-ral 2513  df-rex 2514  df-reu 2515  df-rab 2517  df-v 2801  df-sbc 3029  df-csb 3125  df-dif 3199  df-un 3201  df-in 3203  df-ss 3210  df-nul 3492  df-pw 3651  df-sn 3672  df-pr 3673  df-op 3675  df-uni 3889  df-int 3924  df-iun 3967  df-br 4084  df-opab 4146  df-mpt 4147  df-tr 4183  df-id 4385  df-iord 4458  df-on 4460  df-ilim 4461  df-suc 4463  df-iom 4684  df-xp 4726  df-rel 4727  df-cnv 4728  df-co 4729  df-dm 4730  df-rn 4731  df-res 4732  df-ima 4733  df-iota 5281  df-fun 5323  df-fn 5324  df-f 5325  df-f1 5326  df-fo 5327  df-f1o 5328  df-fv 5329  df-riota 5963  df-ov 6013  df-oprab 6014  df-mpo 6015  df-1st 6295  df-2nd 6296  df-recs 6462  df-frec 6548  df-pnf 8199  df-mnf 8200  df-xr 8201  df-ltxr 8202  df-le 8203  df-sub 8335  df-neg 8336  df-inn 9127  df-n0 9386  df-z 9463  df-uz 9739  df-seqfrec 10687

This theorem is referenced by:  algrp1  12589  alginv  12590  algcvg  12591  algcvga  12594  algfx  12595  eucalgcvga  12601  eucalg  12602