Theorem ialgcvg 10430

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}), 𝑆)
algcvg.3	⊢ 𝐶:𝑆⟶ℕ0
algcvg.4	⊢ (𝑧 ∈ 𝑆 → ((𝐶‘(𝐹‘𝑧)) ≠ 0 → (𝐶‘(𝐹‘𝑧)) < (𝐶‘𝑧)))
algcvg.5	⊢ 𝑁 = (𝐶‘𝐴)
ialgcvg.s	⊢ 𝑆 ∈ 𝑉

Assertion

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

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

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

Proof of Theorem ialgcvg

Dummy variable 𝑘 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		nn0uz 8653	. . . 4 ⊢ ℕ0 = (ℤ≥‘0)
2		algcvg.2	. . . 4 ⊢ 𝑅 = seq0((𝐹 ∘ 1st ), (ℕ0 × {𝐴}), 𝑆)
3		0zd 8363	. . . 4 ⊢ (𝐴 ∈ 𝑆 → 0 ∈ ℤ)
4		id 19	. . . 4 ⊢ (𝐴 ∈ 𝑆 → 𝐴 ∈ 𝑆)
5		algcvg.1	. . . . 5 ⊢ 𝐹:𝑆⟶𝑆
6	5	a1i 9	. . . 4 ⊢ (𝐴 ∈ 𝑆 → 𝐹:𝑆⟶𝑆)
7		ialgcvg.s	. . . . 5 ⊢ 𝑆 ∈ 𝑉
8	7	a1i 9	. . . 4 ⊢ (𝐴 ∈ 𝑆 → 𝑆 ∈ 𝑉)
9	1, 2, 3, 4, 6, 8	ialgrf 10427	. . 3 ⊢ (𝐴 ∈ 𝑆 → 𝑅:ℕ0⟶𝑆)
10		algcvg.5	. . . 4 ⊢ 𝑁 = (𝐶‘𝐴)
11		algcvg.3	. . . . 5 ⊢ 𝐶:𝑆⟶ℕ0
12	11	ffvelrni 5322	. . . 4 ⊢ (𝐴 ∈ 𝑆 → (𝐶‘𝐴) ∈ ℕ0)
13	10, 12	syl5eqel 2165	. . 3 ⊢ (𝐴 ∈ 𝑆 → 𝑁 ∈ ℕ0)
14		fvco3 5265	. . 3 ⊢ ((𝑅:ℕ0⟶𝑆 ∧ 𝑁 ∈ ℕ0) → ((𝐶 ∘ 𝑅)‘𝑁) = (𝐶‘(𝑅‘𝑁)))
15	9, 13, 14	syl2anc 403	. 2 ⊢ (𝐴 ∈ 𝑆 → ((𝐶 ∘ 𝑅)‘𝑁) = (𝐶‘(𝑅‘𝑁)))
16		fco 5076	. . . 4 ⊢ ((𝐶:𝑆⟶ℕ0 ∧ 𝑅:ℕ0⟶𝑆) → (𝐶 ∘ 𝑅):ℕ0⟶ℕ0)
17	11, 9, 16	sylancr 405	. . 3 ⊢ (𝐴 ∈ 𝑆 → (𝐶 ∘ 𝑅):ℕ0⟶ℕ0)
18		0nn0 8303	. . . . . 6 ⊢ 0 ∈ ℕ0
19		fvco3 5265	. . . . . 6 ⊢ ((𝑅:ℕ0⟶𝑆 ∧ 0 ∈ ℕ0) → ((𝐶 ∘ 𝑅)‘0) = (𝐶‘(𝑅‘0)))
20	9, 18, 19	sylancl 404	. . . . 5 ⊢ (𝐴 ∈ 𝑆 → ((𝐶 ∘ 𝑅)‘0) = (𝐶‘(𝑅‘0)))
21	1, 2, 3, 4, 6, 8	ialgr0 10426	. . . . . 6 ⊢ (𝐴 ∈ 𝑆 → (𝑅‘0) = 𝐴)
22	21	fveq2d 5202	. . . . 5 ⊢ (𝐴 ∈ 𝑆 → (𝐶‘(𝑅‘0)) = (𝐶‘𝐴))
23	20, 22	eqtrd 2113	. . . 4 ⊢ (𝐴 ∈ 𝑆 → ((𝐶 ∘ 𝑅)‘0) = (𝐶‘𝐴))
24	23, 10	syl6reqr 2132	. . 3 ⊢ (𝐴 ∈ 𝑆 → 𝑁 = ((𝐶 ∘ 𝑅)‘0))
25	9	ffvelrnda 5323	. . . . 5 ⊢ ((𝐴 ∈ 𝑆 ∧ 𝑘 ∈ ℕ0) → (𝑅‘𝑘) ∈ 𝑆)
26		fveq2 5198	. . . . . . . . 9 ⊢ (𝑧 = (𝑅‘𝑘) → (𝐹‘𝑧) = (𝐹‘(𝑅‘𝑘)))
27	26	fveq2d 5202	. . . . . . . 8 ⊢ (𝑧 = (𝑅‘𝑘) → (𝐶‘(𝐹‘𝑧)) = (𝐶‘(𝐹‘(𝑅‘𝑘))))
28	27	neeq1d 2263	. . . . . . 7 ⊢ (𝑧 = (𝑅‘𝑘) → ((𝐶‘(𝐹‘𝑧)) ≠ 0 ↔ (𝐶‘(𝐹‘(𝑅‘𝑘))) ≠ 0))
29		fveq2 5198	. . . . . . . 8 ⊢ (𝑧 = (𝑅‘𝑘) → (𝐶‘𝑧) = (𝐶‘(𝑅‘𝑘)))
30	27, 29	breq12d 3798	. . . . . . 7 ⊢ (𝑧 = (𝑅‘𝑘) → ((𝐶‘(𝐹‘𝑧)) < (𝐶‘𝑧) ↔ (𝐶‘(𝐹‘(𝑅‘𝑘))) < (𝐶‘(𝑅‘𝑘))))
31	28, 30	imbi12d 232	. . . . . 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)))))
32		algcvg.4	. . . . . 6 ⊢ (𝑧 ∈ 𝑆 → ((𝐶‘(𝐹‘𝑧)) ≠ 0 → (𝐶‘(𝐹‘𝑧)) < (𝐶‘𝑧)))
33	31, 32	vtoclga 2664	. . . . 5 ⊢ ((𝑅‘𝑘) ∈ 𝑆 → ((𝐶‘(𝐹‘(𝑅‘𝑘))) ≠ 0 → (𝐶‘(𝐹‘(𝑅‘𝑘))) < (𝐶‘(𝑅‘𝑘))))
34	25, 33	syl 14	. . . 4 ⊢ ((𝐴 ∈ 𝑆 ∧ 𝑘 ∈ ℕ0) → ((𝐶‘(𝐹‘(𝑅‘𝑘))) ≠ 0 → (𝐶‘(𝐹‘(𝑅‘𝑘))) < (𝐶‘(𝑅‘𝑘))))
35		peano2nn0 8328	. . . . . . 7 ⊢ (𝑘 ∈ ℕ0 → (𝑘 + 1) ∈ ℕ0)
36		fvco3 5265	. . . . . . 7 ⊢ ((𝑅:ℕ0⟶𝑆 ∧ (𝑘 + 1) ∈ ℕ0) → ((𝐶 ∘ 𝑅)‘(𝑘 + 1)) = (𝐶‘(𝑅‘(𝑘 + 1))))
37	9, 35, 36	syl2an 283	. . . . . 6 ⊢ ((𝐴 ∈ 𝑆 ∧ 𝑘 ∈ ℕ0) → ((𝐶 ∘ 𝑅)‘(𝑘 + 1)) = (𝐶‘(𝑅‘(𝑘 + 1))))
38	1, 2, 3, 4, 6, 8	ialgrp1 10428	. . . . . . 7 ⊢ ((𝐴 ∈ 𝑆 ∧ 𝑘 ∈ ℕ0) → (𝑅‘(𝑘 + 1)) = (𝐹‘(𝑅‘𝑘)))
39	38	fveq2d 5202	. . . . . 6 ⊢ ((𝐴 ∈ 𝑆 ∧ 𝑘 ∈ ℕ0) → (𝐶‘(𝑅‘(𝑘 + 1))) = (𝐶‘(𝐹‘(𝑅‘𝑘))))
40	37, 39	eqtrd 2113	. . . . 5 ⊢ ((𝐴 ∈ 𝑆 ∧ 𝑘 ∈ ℕ0) → ((𝐶 ∘ 𝑅)‘(𝑘 + 1)) = (𝐶‘(𝐹‘(𝑅‘𝑘))))
41	40	neeq1d 2263	. . . 4 ⊢ ((𝐴 ∈ 𝑆 ∧ 𝑘 ∈ ℕ0) → (((𝐶 ∘ 𝑅)‘(𝑘 + 1)) ≠ 0 ↔ (𝐶‘(𝐹‘(𝑅‘𝑘))) ≠ 0))
42		fvco3 5265	. . . . . 6 ⊢ ((𝑅:ℕ0⟶𝑆 ∧ 𝑘 ∈ ℕ0) → ((𝐶 ∘ 𝑅)‘𝑘) = (𝐶‘(𝑅‘𝑘)))
43	9, 42	sylan 277	. . . . 5 ⊢ ((𝐴 ∈ 𝑆 ∧ 𝑘 ∈ ℕ0) → ((𝐶 ∘ 𝑅)‘𝑘) = (𝐶‘(𝑅‘𝑘)))
44	40, 43	breq12d 3798	. . . 4 ⊢ ((𝐴 ∈ 𝑆 ∧ 𝑘 ∈ ℕ0) → (((𝐶 ∘ 𝑅)‘(𝑘 + 1)) < ((𝐶 ∘ 𝑅)‘𝑘) ↔ (𝐶‘(𝐹‘(𝑅‘𝑘))) < (𝐶‘(𝑅‘𝑘))))
45	34, 41, 44	3imtr4d 201	. . 3 ⊢ ((𝐴 ∈ 𝑆 ∧ 𝑘 ∈ ℕ0) → (((𝐶 ∘ 𝑅)‘(𝑘 + 1)) ≠ 0 → ((𝐶 ∘ 𝑅)‘(𝑘 + 1)) < ((𝐶 ∘ 𝑅)‘𝑘)))
46	17, 24, 45	nn0seqcvgd 10423	. 2 ⊢ (𝐴 ∈ 𝑆 → ((𝐶 ∘ 𝑅)‘𝑁) = 0)
47	15, 46	eqtr3d 2115	1 ⊢ (𝐴 ∈ 𝑆 → (𝐶‘(𝑅‘𝑁)) = 0)

Syntax hints:   → wi 4   ∧ wa 102   = wceq 1284   ∈ wcel 1433   ≠ wne 2245  {csn 3398   class class class wbr 3785   × cxp 4361   ∘ ccom 4367  ⟶wf 4918  ‘cfv 4922  (class class class)co 5532  1^st c1st 5785  0cc0 6981  1c1 6982   + caddc 6984   < clt 7153  ℕ₀cn0 8288  seqcseq 9431

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 576  ax-in2 577  ax-io 662  ax-5 1376  ax-7 1377  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-8 1435  ax-10 1436  ax-11 1437  ax-i12 1438  ax-bndl 1439  ax-4 1440  ax-13 1444  ax-14 1445  ax-17 1459  ax-i9 1463  ax-ial 1467  ax-i5r 1468  ax-ext 2063  ax-coll 3893  ax-sep 3896  ax-nul 3904  ax-pow 3948  ax-pr 3964  ax-un 4188  ax-setind 4280  ax-iinf 4329  ax-cnex 7067  ax-resscn 7068  ax-1cn 7069  ax-1re 7070  ax-icn 7071  ax-addcl 7072  ax-addrcl 7073  ax-mulcl 7074  ax-addcom 7076  ax-addass 7078  ax-distr 7080  ax-i2m1 7081  ax-0lt1 7082  ax-0id 7084  ax-rnegex 7085  ax-cnre 7087  ax-pre-ltirr 7088  ax-pre-ltwlin 7089  ax-pre-lttrn 7090  ax-pre-apti 7091  ax-pre-ltadd 7092

This theorem depends on definitions:  df-bi 115  df-dc 776  df-3or 920  df-3an 921  df-tru 1287  df-fal 1290  df-nf 1390  df-sb 1686  df-eu 1944  df-mo 1945  df-clab 2068  df-cleq 2074  df-clel 2077  df-nfc 2208  df-ne 2246  df-nel 2340  df-ral 2353  df-rex 2354  df-reu 2355  df-rab 2357  df-v 2603  df-sbc 2816  df-csb 2909  df-dif 2975  df-un 2977  df-in 2979  df-ss 2986  df-nul 3252  df-pw 3384  df-sn 3404  df-pr 3405  df-op 3407  df-uni 3602  df-int 3637  df-iun 3680  df-br 3786  df-opab 3840  df-mpt 3841  df-tr 3876  df-id 4048  df-iord 4121  df-on 4123  df-suc 4126  df-iom 4332  df-xp 4369  df-rel 4370  df-cnv 4371  df-co 4372  df-dm 4373  df-rn 4374  df-res 4375  df-ima 4376  df-iota 4887  df-fun 4924  df-fn 4925  df-f 4926  df-f1 4927  df-fo 4928  df-f1o 4929  df-fv 4930  df-riota 5488  df-ov 5535  df-oprab 5536  df-mpt2 5537  df-1st 5787  df-2nd 5788  df-recs 5943  df-frec 6001  df-pnf 7155  df-mnf 7156  df-xr 7157  df-ltxr 7158  df-le 7159  df-sub 7281  df-neg 7282  df-inn 8040  df-n0 8289  df-z 8352  df-uz 8620  df-iseq 9432

This theorem is referenced by:  ialgcvga  10433