Theorem algcvg 15289

Description: One way to prove that an algorithm halts is to construct a countdown function 𝐶:𝑆⟶ℕ₀ whose value is guaranteed to decrease for each iteration of 𝐹 until it reaches 0. That is, if 𝑋 ∈ 𝑆 is not a fixed point of 𝐹, then (𝐶‘(𝐹‘𝑋)) < (𝐶‘𝑋).

If 𝐶 is a countdown function for algorithm 𝐹, the sequence (𝐶‘(𝑅‘𝑘)) reaches 0 after at most 𝑁 steps, where 𝑁 is the value of 𝐶 for the initial state 𝐴. (Contributed by Paul Chapman, 22-Jun-2011.)

Hypotheses

Ref	Expression
algcvg.1	⊢ 𝐹:𝑆⟶𝑆
algcvg.2	⊢ 𝑅 = seq0((𝐹 ∘ 1st ), (ℕ0 × {𝐴}))
algcvg.3	⊢ 𝐶:𝑆⟶ℕ0
algcvg.4	⊢ (𝑧 ∈ 𝑆 → ((𝐶‘(𝐹‘𝑧)) ≠ 0 → (𝐶‘(𝐹‘𝑧)) < (𝐶‘𝑧)))
algcvg.5	⊢ 𝑁 = (𝐶‘𝐴)

Assertion

Ref	Expression
algcvg	⊢ (𝐴 ∈ 𝑆 → (𝐶‘(𝑅‘𝑁)) = 0)

Distinct variable groups:   𝑧,𝐶   𝑧,𝐹   𝑧,𝑅   𝑧,𝑆

Allowed substitution hints:   𝐴(𝑧)   𝑁(𝑧)

Proof of Theorem algcvg

Dummy variable 𝑘 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		nn0uz 11722	. . . 4 ⊢ ℕ0 = (ℤ≥‘0)
2		algcvg.2	. . . 4 ⊢ 𝑅 = seq0((𝐹 ∘ 1st ), (ℕ0 × {𝐴}))
3		0zd 11389	. . . 4 ⊢ (𝐴 ∈ 𝑆 → 0 ∈ ℤ)
4		id 22	. . . 4 ⊢ (𝐴 ∈ 𝑆 → 𝐴 ∈ 𝑆)
5		algcvg.1	. . . . 5 ⊢ 𝐹:𝑆⟶𝑆
6	5	a1i 11	. . . 4 ⊢ (𝐴 ∈ 𝑆 → 𝐹:𝑆⟶𝑆)
7	1, 2, 3, 4, 6	algrf 15286	. . 3 ⊢ (𝐴 ∈ 𝑆 → 𝑅:ℕ0⟶𝑆)
8		algcvg.5	. . . 4 ⊢ 𝑁 = (𝐶‘𝐴)
9		algcvg.3	. . . . 5 ⊢ 𝐶:𝑆⟶ℕ0
10	9	ffvelrni 6358	. . . 4 ⊢ (𝐴 ∈ 𝑆 → (𝐶‘𝐴) ∈ ℕ0)
11	8, 10	syl5eqel 2705	. . 3 ⊢ (𝐴 ∈ 𝑆 → 𝑁 ∈ ℕ0)
12		fvco3 6275	. . 3 ⊢ ((𝑅:ℕ0⟶𝑆 ∧ 𝑁 ∈ ℕ0) → ((𝐶 ∘ 𝑅)‘𝑁) = (𝐶‘(𝑅‘𝑁)))
13	7, 11, 12	syl2anc 693	. 2 ⊢ (𝐴 ∈ 𝑆 → ((𝐶 ∘ 𝑅)‘𝑁) = (𝐶‘(𝑅‘𝑁)))
14		fco 6058	. . . 4 ⊢ ((𝐶:𝑆⟶ℕ0 ∧ 𝑅:ℕ0⟶𝑆) → (𝐶 ∘ 𝑅):ℕ0⟶ℕ0)
15	9, 7, 14	sylancr 695	. . 3 ⊢ (𝐴 ∈ 𝑆 → (𝐶 ∘ 𝑅):ℕ0⟶ℕ0)
16		0nn0 11307	. . . . . 6 ⊢ 0 ∈ ℕ0
17		fvco3 6275	. . . . . 6 ⊢ ((𝑅:ℕ0⟶𝑆 ∧ 0 ∈ ℕ0) → ((𝐶 ∘ 𝑅)‘0) = (𝐶‘(𝑅‘0)))
18	7, 16, 17	sylancl 694	. . . . 5 ⊢ (𝐴 ∈ 𝑆 → ((𝐶 ∘ 𝑅)‘0) = (𝐶‘(𝑅‘0)))
19	1, 2, 3, 4	algr0 15285	. . . . . 6 ⊢ (𝐴 ∈ 𝑆 → (𝑅‘0) = 𝐴)
20	19	fveq2d 6195	. . . . 5 ⊢ (𝐴 ∈ 𝑆 → (𝐶‘(𝑅‘0)) = (𝐶‘𝐴))
21	18, 20	eqtrd 2656	. . . 4 ⊢ (𝐴 ∈ 𝑆 → ((𝐶 ∘ 𝑅)‘0) = (𝐶‘𝐴))
22	21, 8	syl6reqr 2675	. . 3 ⊢ (𝐴 ∈ 𝑆 → 𝑁 = ((𝐶 ∘ 𝑅)‘0))
23	7	ffvelrnda 6359	. . . . 5 ⊢ ((𝐴 ∈ 𝑆 ∧ 𝑘 ∈ ℕ0) → (𝑅‘𝑘) ∈ 𝑆)
24		fveq2 6191	. . . . . . . . 9 ⊢ (𝑧 = (𝑅‘𝑘) → (𝐹‘𝑧) = (𝐹‘(𝑅‘𝑘)))
25	24	fveq2d 6195	. . . . . . . 8 ⊢ (𝑧 = (𝑅‘𝑘) → (𝐶‘(𝐹‘𝑧)) = (𝐶‘(𝐹‘(𝑅‘𝑘))))
26	25	neeq1d 2853	. . . . . . 7 ⊢ (𝑧 = (𝑅‘𝑘) → ((𝐶‘(𝐹‘𝑧)) ≠ 0 ↔ (𝐶‘(𝐹‘(𝑅‘𝑘))) ≠ 0))
27		fveq2 6191	. . . . . . . 8 ⊢ (𝑧 = (𝑅‘𝑘) → (𝐶‘𝑧) = (𝐶‘(𝑅‘𝑘)))
28	25, 27	breq12d 4666	. . . . . . 7 ⊢ (𝑧 = (𝑅‘𝑘) → ((𝐶‘(𝐹‘𝑧)) < (𝐶‘𝑧) ↔ (𝐶‘(𝐹‘(𝑅‘𝑘))) < (𝐶‘(𝑅‘𝑘))))
29	26, 28	imbi12d 334	. . . . . 6 ⊢ (𝑧 = (𝑅‘𝑘) → (((𝐶‘(𝐹‘𝑧)) ≠ 0 → (𝐶‘(𝐹‘𝑧)) < (𝐶‘𝑧)) ↔ ((𝐶‘(𝐹‘(𝑅‘𝑘))) ≠ 0 → (𝐶‘(𝐹‘(𝑅‘𝑘))) < (𝐶‘(𝑅‘𝑘)))))
30		algcvg.4	. . . . . 6 ⊢ (𝑧 ∈ 𝑆 → ((𝐶‘(𝐹‘𝑧)) ≠ 0 → (𝐶‘(𝐹‘𝑧)) < (𝐶‘𝑧)))
31	29, 30	vtoclga 3272	. . . . 5 ⊢ ((𝑅‘𝑘) ∈ 𝑆 → ((𝐶‘(𝐹‘(𝑅‘𝑘))) ≠ 0 → (𝐶‘(𝐹‘(𝑅‘𝑘))) < (𝐶‘(𝑅‘𝑘))))
32	23, 31	syl 17	. . . 4 ⊢ ((𝐴 ∈ 𝑆 ∧ 𝑘 ∈ ℕ0) → ((𝐶‘(𝐹‘(𝑅‘𝑘))) ≠ 0 → (𝐶‘(𝐹‘(𝑅‘𝑘))) < (𝐶‘(𝑅‘𝑘))))
33		peano2nn0 11333	. . . . . . 7 ⊢ (𝑘 ∈ ℕ0 → (𝑘 + 1) ∈ ℕ0)
34		fvco3 6275	. . . . . . 7 ⊢ ((𝑅:ℕ0⟶𝑆 ∧ (𝑘 + 1) ∈ ℕ0) → ((𝐶 ∘ 𝑅)‘(𝑘 + 1)) = (𝐶‘(𝑅‘(𝑘 + 1))))
35	7, 33, 34	syl2an 494	. . . . . 6 ⊢ ((𝐴 ∈ 𝑆 ∧ 𝑘 ∈ ℕ0) → ((𝐶 ∘ 𝑅)‘(𝑘 + 1)) = (𝐶‘(𝑅‘(𝑘 + 1))))
36	1, 2, 3, 4, 6	algrp1 15287	. . . . . . 7 ⊢ ((𝐴 ∈ 𝑆 ∧ 𝑘 ∈ ℕ0) → (𝑅‘(𝑘 + 1)) = (𝐹‘(𝑅‘𝑘)))
37	36	fveq2d 6195	. . . . . 6 ⊢ ((𝐴 ∈ 𝑆 ∧ 𝑘 ∈ ℕ0) → (𝐶‘(𝑅‘(𝑘 + 1))) = (𝐶‘(𝐹‘(𝑅‘𝑘))))
38	35, 37	eqtrd 2656	. . . . 5 ⊢ ((𝐴 ∈ 𝑆 ∧ 𝑘 ∈ ℕ0) → ((𝐶 ∘ 𝑅)‘(𝑘 + 1)) = (𝐶‘(𝐹‘(𝑅‘𝑘))))
39	38	neeq1d 2853	. . . 4 ⊢ ((𝐴 ∈ 𝑆 ∧ 𝑘 ∈ ℕ0) → (((𝐶 ∘ 𝑅)‘(𝑘 + 1)) ≠ 0 ↔ (𝐶‘(𝐹‘(𝑅‘𝑘))) ≠ 0))
40		fvco3 6275	. . . . . 6 ⊢ ((𝑅:ℕ0⟶𝑆 ∧ 𝑘 ∈ ℕ0) → ((𝐶 ∘ 𝑅)‘𝑘) = (𝐶‘(𝑅‘𝑘)))
41	7, 40	sylan 488	. . . . 5 ⊢ ((𝐴 ∈ 𝑆 ∧ 𝑘 ∈ ℕ0) → ((𝐶 ∘ 𝑅)‘𝑘) = (𝐶‘(𝑅‘𝑘)))
42	38, 41	breq12d 4666	. . . 4 ⊢ ((𝐴 ∈ 𝑆 ∧ 𝑘 ∈ ℕ0) → (((𝐶 ∘ 𝑅)‘(𝑘 + 1)) < ((𝐶 ∘ 𝑅)‘𝑘) ↔ (𝐶‘(𝐹‘(𝑅‘𝑘))) < (𝐶‘(𝑅‘𝑘))))
43	32, 39, 42	3imtr4d 283	. . 3 ⊢ ((𝐴 ∈ 𝑆 ∧ 𝑘 ∈ ℕ0) → (((𝐶 ∘ 𝑅)‘(𝑘 + 1)) ≠ 0 → ((𝐶 ∘ 𝑅)‘(𝑘 + 1)) < ((𝐶 ∘ 𝑅)‘𝑘)))
44	15, 22, 43	nn0seqcvgd 15283	. 2 ⊢ (𝐴 ∈ 𝑆 → ((𝐶 ∘ 𝑅)‘𝑁) = 0)
45	13, 44	eqtr3d 2658	1 ⊢ (𝐴 ∈ 𝑆 → (𝐶‘(𝑅‘𝑁)) = 0)

Syntax hints:   → wi 4   ∧ wa 384   = wceq 1483   ∈ wcel 1990   ≠ wne 2794  {csn 4177   class class class wbr 4653   × cxp 5112   ∘ ccom 5118  ⟶wf 5884  ‘cfv 5888  (class class class)co 6650  1^st c1st 7166  0cc0 9936  1c1 9937   + caddc 9939   < clt 10074  ℕ₀cn0 11292  seqcseq 12801

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-cnex 9992  ax-resscn 9993  ax-1cn 9994  ax-icn 9995  ax-addcl 9996  ax-addrcl 9997  ax-mulcl 9998  ax-mulrcl 9999  ax-mulcom 10000  ax-addass 10001  ax-mulass 10002  ax-distr 10003  ax-i2m1 10004  ax-1ne0 10005  ax-1rid 10006  ax-rnegex 10007  ax-rrecex 10008  ax-cnre 10009  ax-pre-lttri 10010  ax-pre-lttrn 10011  ax-pre-ltadd 10012  ax-pre-mulgt0 10013

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-nel 2898  df-ral 2917  df-rex 2918  df-reu 2919  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-om 7066  df-1st 7168  df-2nd 7169  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-er 7742  df-en 7956  df-dom 7957  df-sdom 7958  df-pnf 10076  df-mnf 10077  df-xr 10078  df-ltxr 10079  df-le 10080  df-sub 10268  df-neg 10269  df-nn 11021  df-n0 11293  df-z 11378  df-uz 11688  df-fz 12327  df-seq 12802

This theorem is referenced by:  algcvga  15292