Theorem algrf 12738

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 5897	. . . . 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 2309	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑍) → ((𝑍 × {𝐴})‘𝑥) ∈ 𝑆)
8		vex 2815	. . . . 5 ⊢ 𝑥 ∈ V
9		vex 2815	. . . . 5 ⊢ 𝑦 ∈ V
10	8, 9	algrflem 6424	. . . 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 5809	. . . . 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 2319	. . 3 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → (𝑥(𝐹 ∘ 1st )𝑦) ∈ 𝑆)
16	1, 2, 7, 15	seqf 10825	. 2 ⊢ (𝜑 → seq𝑀((𝐹 ∘ 1st ), (𝑍 × {𝐴})):𝑍⟶𝑆)
17		algrf.2	. . 3 ⊢ 𝑅 = seq𝑀((𝐹 ∘ 1st ), (𝑍 × {𝐴}))
18	17	feq1i 5500	. 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 1398   ∈ wcel 2203  {csn 3688   × cxp 4746   ∘ ccom 4752  ⟶wf 5347  ‘cfv 5351  (class class class)co 6049  1^st c1st 6331  ℤcz 9576  ℤ_≥cuz 9852  seqcseq 10808

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 619  ax-in2 620  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2205  ax-14 2206  ax-ext 2214  ax-coll 4224  ax-sep 4227  ax-nul 4235  ax-pow 4286  ax-pr 4321  ax-un 4553  ax-setind 4658  ax-iinf 4709  ax-cnex 8217  ax-resscn 8218  ax-1cn 8219  ax-1re 8220  ax-icn 8221  ax-addcl 8222  ax-addrcl 8223  ax-mulcl 8224  ax-addcom 8226  ax-addass 8228  ax-distr 8230  ax-i2m1 8231  ax-0lt1 8232  ax-0id 8234  ax-rnegex 8235  ax-cnre 8237  ax-pre-ltirr 8238  ax-pre-ltwlin 8239  ax-pre-lttrn 8240  ax-pre-ltadd 8242

This theorem depends on definitions:  df-bi 117  df-3or 1006  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1812  df-eu 2083  df-mo 2084  df-clab 2219  df-cleq 2225  df-clel 2228  df-nfc 2373  df-ne 2413  df-nel 2508  df-ral 2525  df-rex 2526  df-reu 2527  df-rab 2529  df-v 2814  df-sbc 3042  df-csb 3138  df-dif 3212  df-un 3214  df-in 3216  df-ss 3223  df-nul 3508  df-pw 3670  df-sn 3694  df-pr 3695  df-op 3697  df-uni 3914  df-int 3949  df-iun 3992  df-br 4109  df-opab 4171  df-mpt 4172  df-tr 4208  df-id 4413  df-iord 4486  df-on 4488  df-ilim 4489  df-suc 4491  df-iom 4712  df-xp 4754  df-rel 4755  df-cnv 4756  df-co 4757  df-dm 4758  df-rn 4759  df-res 4760  df-ima 4761  df-iota 5311  df-fun 5353  df-fn 5354  df-f 5355  df-f1 5356  df-fo 5357  df-f1o 5358  df-fv 5359  df-riota 6002  df-ov 6052  df-oprab 6053  df-mpo 6054  df-1st 6333  df-2nd 6334  df-recs 6535  df-frec 6621  df-pnf 8309  df-mnf 8310  df-xr 8311  df-ltxr 8312  df-le 8313  df-sub 8445  df-neg 8446  df-inn 9237  df-n0 9496  df-z 9577  df-uz 9853  df-seqfrec 10809

This theorem is referenced by:  algrp1  12739  alginv  12740  algcvg  12741  algcvga  12744  algfx  12745  eucalgcvga  12751  eucalg  12752