Theorem algcvg 16731

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 12984	. . . 4 ⊢ ℕ0 = (ℤ≥‘0)
2		algcvg.2	. . . 4 ⊢ 𝑅 = seq0((𝐹 ∘ 1st ), (ℕ0 × {𝐴}))
3		0zd 12686	. . . 4 ⊢ (𝐴 ∈ 𝑆 → 0 ∈ ℤ)
4		id 23	. . . 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 16728	. . 3 ⊢ (𝐴 ∈ 𝑆 → 𝑅:ℕ0⟶𝑆)
8		algcvg.5	. . . 4 ⊢ 𝑁 = (𝐶‘𝐴)
9		algcvg.3	. . . . 5 ⊢ 𝐶:𝑆⟶ℕ0
10	9	ffvelcdmi 7075	. . . 4 ⊢ (𝐴 ∈ 𝑆 → (𝐶‘𝐴) ∈ ℕ0)
11	8, 10	eqeltrid 2865	. . 3 ⊢ (𝐴 ∈ 𝑆 → 𝑁 ∈ ℕ0)
12		fvco3 6977	. . 3 ⊢ ((𝑅:ℕ0⟶𝑆 ∧ 𝑁 ∈ ℕ0) → ((𝐶 ∘ 𝑅)‘𝑁) = (𝐶‘(𝑅‘𝑁)))
13	7, 11, 12	syl2anc 596	. 2 ⊢ (𝐴 ∈ 𝑆 → ((𝐶 ∘ 𝑅)‘𝑁) = (𝐶‘(𝑅‘𝑁)))
14		fco 6726	. . . 4 ⊢ ((𝐶:𝑆⟶ℕ0 ∧ 𝑅:ℕ0⟶𝑆) → (𝐶 ∘ 𝑅):ℕ0⟶ℕ0)
15	9, 7, 14	sylancr 599	. . 3 ⊢ (𝐴 ∈ 𝑆 → (𝐶 ∘ 𝑅):ℕ0⟶ℕ0)
16		0nn0 12602	. . . . . 6 ⊢ 0 ∈ ℕ0
17		fvco3 6977	. . . . . 6 ⊢ ((𝑅:ℕ0⟶𝑆 ∧ 0 ∈ ℕ0) → ((𝐶 ∘ 𝑅)‘0) = (𝐶‘(𝑅‘0)))
18	7, 16, 17	sylancl 598	. . . . 5 ⊢ (𝐴 ∈ 𝑆 → ((𝐶 ∘ 𝑅)‘0) = (𝐶‘(𝑅‘0)))
19	1, 2, 3, 4	algr0 16727	. . . . . 6 ⊢ (𝐴 ∈ 𝑆 → (𝑅‘0) = 𝐴)
20	19	fveq2d 6881	. . . . 5 ⊢ (𝐴 ∈ 𝑆 → (𝐶‘(𝑅‘0)) = (𝐶‘𝐴))
21	18, 20	eqtrd 2796	. . . 4 ⊢ (𝐴 ∈ 𝑆 → ((𝐶 ∘ 𝑅)‘0) = (𝐶‘𝐴))
22	8, 21	eqtr4id 2815	. . 3 ⊢ (𝐴 ∈ 𝑆 → 𝑁 = ((𝐶 ∘ 𝑅)‘0))
23	7	ffvelcdmda 7076	. . . . 5 ⊢ ((𝐴 ∈ 𝑆 ∧ 𝑘 ∈ ℕ0) → (𝑅‘𝑘) ∈ 𝑆)
24		2fveq3 6882	. . . . . . . 8 ⊢ (𝑧 = (𝑅‘𝑘) → (𝐶‘(𝐹‘𝑧)) = (𝐶‘(𝐹‘(𝑅‘𝑘))))
25	24	neeq1d 3015	. . . . . . 7 ⊢ (𝑧 = (𝑅‘𝑘) → ((𝐶‘(𝐹‘𝑧)) ≠ 0 ↔ (𝐶‘(𝐹‘(𝑅‘𝑘))) ≠ 0))
26		fveq2 6877	. . . . . . . 8 ⊢ (𝑧 = (𝑅‘𝑘) → (𝐶‘𝑧) = (𝐶‘(𝑅‘𝑘)))
27	24, 26	breq12d 5116	. . . . . . 7 ⊢ (𝑧 = (𝑅‘𝑘) → ((𝐶‘(𝐹‘𝑧)) < (𝐶‘𝑧) ↔ (𝐶‘(𝐹‘(𝑅‘𝑘))) < (𝐶‘(𝑅‘𝑘))))
28	25, 27	imbi12d 347	. . . . . 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)))))
29		algcvg.4	. . . . . 6 ⊢ (𝑧 ∈ 𝑆 → ((𝐶‘(𝐹‘𝑧)) ≠ 0 → (𝐶‘(𝐹‘𝑧)) < (𝐶‘𝑧)))
30	28, 29	vtoclga 3537	. . . . 5 ⊢ ((𝑅‘𝑘) ∈ 𝑆 → ((𝐶‘(𝐹‘(𝑅‘𝑘))) ≠ 0 → (𝐶‘(𝐹‘(𝑅‘𝑘))) < (𝐶‘(𝑅‘𝑘))))
31	23, 30	syl 18	. . . 4 ⊢ ((𝐴 ∈ 𝑆 ∧ 𝑘 ∈ ℕ0) → ((𝐶‘(𝐹‘(𝑅‘𝑘))) ≠ 0 → (𝐶‘(𝐹‘(𝑅‘𝑘))) < (𝐶‘(𝑅‘𝑘))))
32		peano2nn0 12627	. . . . . . 7 ⊢ (𝑘 ∈ ℕ0 → (𝑘 + 1) ∈ ℕ0)
33		fvco3 6977	. . . . . . 7 ⊢ ((𝑅:ℕ0⟶𝑆 ∧ (𝑘 + 1) ∈ ℕ0) → ((𝐶 ∘ 𝑅)‘(𝑘 + 1)) = (𝐶‘(𝑅‘(𝑘 + 1))))
34	7, 32, 33	syl2an 608	. . . . . 6 ⊢ ((𝐴 ∈ 𝑆 ∧ 𝑘 ∈ ℕ0) → ((𝐶 ∘ 𝑅)‘(𝑘 + 1)) = (𝐶‘(𝑅‘(𝑘 + 1))))
35	1, 2, 3, 4, 6	algrp1 16729	. . . . . . 7 ⊢ ((𝐴 ∈ 𝑆 ∧ 𝑘 ∈ ℕ0) → (𝑅‘(𝑘 + 1)) = (𝐹‘(𝑅‘𝑘)))
36	35	fveq2d 6881	. . . . . 6 ⊢ ((𝐴 ∈ 𝑆 ∧ 𝑘 ∈ ℕ0) → (𝐶‘(𝑅‘(𝑘 + 1))) = (𝐶‘(𝐹‘(𝑅‘𝑘))))
37	34, 36	eqtrd 2796	. . . . 5 ⊢ ((𝐴 ∈ 𝑆 ∧ 𝑘 ∈ ℕ0) → ((𝐶 ∘ 𝑅)‘(𝑘 + 1)) = (𝐶‘(𝐹‘(𝑅‘𝑘))))
38	37	neeq1d 3015	. . . 4 ⊢ ((𝐴 ∈ 𝑆 ∧ 𝑘 ∈ ℕ0) → (((𝐶 ∘ 𝑅)‘(𝑘 + 1)) ≠ 0 ↔ (𝐶‘(𝐹‘(𝑅‘𝑘))) ≠ 0))
39		fvco3 6977	. . . . . 6 ⊢ ((𝑅:ℕ0⟶𝑆 ∧ 𝑘 ∈ ℕ0) → ((𝐶 ∘ 𝑅)‘𝑘) = (𝐶‘(𝑅‘𝑘)))
40	7, 39	sylan 592	. . . . 5 ⊢ ((𝐴 ∈ 𝑆 ∧ 𝑘 ∈ ℕ0) → ((𝐶 ∘ 𝑅)‘𝑘) = (𝐶‘(𝑅‘𝑘)))
41	37, 40	breq12d 5116	. . . 4 ⊢ ((𝐴 ∈ 𝑆 ∧ 𝑘 ∈ ℕ0) → (((𝐶 ∘ 𝑅)‘(𝑘 + 1)) < ((𝐶 ∘ 𝑅)‘𝑘) ↔ (𝐶‘(𝐹‘(𝑅‘𝑘))) < (𝐶‘(𝑅‘𝑘))))
42	31, 38, 41	3imtr4d 297	. . 3 ⊢ ((𝐴 ∈ 𝑆 ∧ 𝑘 ∈ ℕ0) → (((𝐶 ∘ 𝑅)‘(𝑘 + 1)) ≠ 0 → ((𝐶 ∘ 𝑅)‘(𝑘 + 1)) < ((𝐶 ∘ 𝑅)‘𝑘)))
43	15, 22, 42	nn0seqcvgd 16725	. 2 ⊢ (𝐴 ∈ 𝑆 → ((𝐶 ∘ 𝑅)‘𝑁) = 0)
44	13, 43	eqtr3d 2798	1 ⊢ (𝐴 ∈ 𝑆 → (𝐶‘(𝑅‘𝑁)) = 0)

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   = wceq 1570   ∈ wcel 2145   ≠ wne 2956  {csn 4584   class class class wbr 5103   × cxp 5649   ∘ ccom 5655  ⟶wf 6527  ‘cfv 6531  (class class class)co 7412  1^st c1st 7988  0cc0 11181  1c1 11182   + caddc 11184   < clt 11324  ℕ₀cn0 12587  seqcseq 14124

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2213  ax-ext 2733  ax-sep 5249  ax-nul 5260  ax-pow 5327  ax-pr 5391  ax-un 7740  ax-cnex 11237  ax-resscn 11238  ax-1cn 11239  ax-icn 11240  ax-addcl 11241  ax-addrcl 11242  ax-mulcl 11243  ax-mulrcl 11244  ax-mulcom 11245  ax-addass 11246  ax-mulass 11247  ax-distr 11248  ax-i2m1 11249  ax-1ne0 11250  ax-1rid 11251  ax-rnegex 11252  ax-rrecex 11253  ax-cnre 11254  ax-pre-lttri 11255  ax-pre-lttrn 11256  ax-pre-ltadd 11257  ax-pre-mulgt0 11258

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2565  df-eu 2595  df-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-ne 2957  df-nel 3063  df-ral 3078  df-rex 3088  df-reu 3367  df-rab 3414  df-v 3453  df-sbc 3740  df-csb 3848  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-pss 3919  df-nul 4280  df-if 4483  df-pw 4559  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4868  df-iun 4953  df-br 5104  df-opab 5168  df-mpt 5187  df-tr 5213  df-id 5546  df-eprel 5551  df-po 5559  df-so 5560  df-fr 5604  df-we 5606  df-xp 5657  df-rel 5658  df-cnv 5659  df-co 5660  df-dm 5661  df-rn 5662  df-res 5663  df-ima 5664  df-pred 6297  df-ord 6358  df-on 6359  df-lim 6360  df-suc 6361  df-iota 6487  df-fun 6533  df-fn 6534  df-f 6535  df-f1 6536  df-fo 6537  df-f1o 6538  df-fv 6539  df-riota 7369  df-ov 7415  df-oprab 7416  df-mpo 7417  df-om 7867  df-1st 7990  df-2nd 7991  df-frecs 8283  df-wrecs 8314  df-recs 8363  df-rdg 8402  df-er 8701  df-en 8958  df-dom 8959  df-sdom 8960  df-pnf 11326  df-mnf 11327  df-xr 11328  df-ltxr 11329  df-le 11330  df-sub 11524  df-neg 11525  df-nn 12317  df-n0 12588  df-z 12675  df-uz 12947  df-fz 13621  df-seq 14125

This theorem is used by:  algcvga  16734