2 include "arithmetics/nat.ma".
3 include "arithmetics/log.ma".
4 include "arithmetics/bigops.ma".
5 include "arithmetics/bounded_quantifiers.ma".
6 include "arithmetics/pidgeon_hole.ma".
7 include "basics/sets.ma".
8 include "basics/types.ma".
10 (************************************ MAX *************************************)
11 notation "Max_{ ident i < n | p } f"
13 for @{'bigop $n max 0 (λ${ident i}. $p) (λ${ident i}. $f)}.
15 notation "Max_{ ident i < n } f"
17 for @{'bigop $n max 0 (λ${ident i}.true) (λ${ident i}. $f)}.
19 notation "Max_{ ident j ∈ [a,b[ } f"
21 for @{'bigop ($b-$a) max 0 (λ${ident j}.((λ${ident j}.true) (${ident j}+$a)))
22 (λ${ident j}.((λ${ident j}.$f)(${ident j}+$a)))}.
24 notation "Max_{ ident j ∈ [a,b[ | p } f"
26 for @{'bigop ($b-$a) max 0 (λ${ident j}.((λ${ident j}.$p) (${ident j}+$a)))
27 (λ${ident j}.((λ${ident j}.$f)(${ident j}+$a)))}.
29 lemma Max_assoc: ∀a,b,c. max (max a b) c = max a (max b c).
30 #a #b #c normalize cases (true_or_false (leb a b)) #leab >leab normalize
31 [cases (true_or_false (leb b c )) #lebc >lebc normalize
32 [>(le_to_leb_true a c) // @(transitive_le ? b) @leb_true_to_le //
35 |cases (true_or_false (leb b c )) #lebc >lebc normalize //
36 >leab normalize >(not_le_to_leb_false a c) // @lt_to_not_le
37 @(transitive_lt ? b) @not_le_to_lt @leb_false_to_not_le //
41 lemma Max0 : ∀n. max 0 n = n.
44 lemma Max0r : ∀n. max n 0 = n.
45 #n >commutative_max //
48 alias id "max" = "cic:/matita/arithmetics/nat/max#def:2".
49 alias id "mk_Aop" = "cic:/matita/arithmetics/bigops/Aop#con:0:1:2".
51 mk_Aop nat 0 max Max0 Max0r (λa,b,c.sym_eq … (Max_assoc a b c)).
53 definition MaxAC ≝ mk_ACop nat 0 MaxA commutative_max.
55 lemma le_Max: ∀f,p,n,a. a < n → p a = true →
56 f a ≤ Max_{i < n | p i}(f i).
58 >(bigop_diff p ? 0 MaxAC f a n) // @(le_maxl … (le_n ?))
61 lemma Max_le: ∀f,p,n,b.
62 (∀a.a < n → p a = true → f a ≤ b) → Max_{i < n | p i}(f i) ≤ b.
63 #f #p #n elim n #b #H //
64 #b0 #H1 cases (true_or_false (p b)) #Hb
65 [>bigop_Strue [2:@Hb] @to_max [@H1 // | @H #a #ltab #pa @H1 // @le_S //]
66 |>bigop_Sfalse [2:@Hb] @H #a #ltab #pa @H1 // @le_S //
70 (******************************** big O notation ******************************)
72 (* O f g means g ∈ O(f) *)
73 definition O: relation (nat→nat) ≝
74 λf,g. ∃c.∃n0.∀n. n0 ≤ n → g n ≤ c* (f n).
76 lemma O_refl: ∀s. O s s.
77 #s %{1} %{0} #n #_ >commutative_times <times_n_1 @le_n qed.
79 lemma O_trans: ∀s1,s2,s3. O s2 s1 → O s3 s2 → O s3 s1.
80 #s1 #s2 #s3 * #c1 * #n1 #H1 * #c2 * # n2 #H2 %{(c1*c2)}
81 %{(max n1 n2)} #n #Hmax
82 @(transitive_le … (H1 ??)) [@(le_maxl … Hmax)]
83 >associative_times @le_times [//|@H2 @(le_maxr … Hmax)]
86 lemma sub_O_to_O: ∀s1,s2. O s1 ⊆ O s2 → O s2 s1.
89 lemma O_to_sub_O: ∀s1,s2. O s2 s1 → O s1 ⊆ O s2.
90 #s1 #s2 #H #g #Hg @(O_trans … H) // qed.
92 definition sum_f ≝ λf,g:nat→nat.λn.f n + g n.
93 interpretation "function sum" 'plus f g = (sum_f f g).
95 lemma O_plus: ∀f,g,s. O s f → O s g → O s (f+g).
96 #f #g #s * #cf * #nf #Hf * #cg * #ng #Hg
97 %{(cf+cg)} %{(max nf ng)} #n #Hmax normalize
98 >distributive_times_plus_r @le_plus
99 [@Hf @(le_maxl … Hmax) |@Hg @(le_maxr … Hmax) ]
102 lemma O_plus_l: ∀f,s1,s2. O s1 f → O (s1+s2) f.
103 #f #s1 #s2 * #c * #a #Os1f %{c} %{a} #n #lean
104 @(transitive_le … (Os1f n lean)) @le_times //
107 lemma O_plus_r: ∀f,s1,s2. O s2 f → O (s1+s2) f.
108 #f #s1 #s2 * #c * #a #Os1f %{c} %{a} #n #lean
109 @(transitive_le … (Os1f n lean)) @le_times //
112 lemma O_absorbl: ∀f,g,s. O s f → O f g → O s (g+f).
113 #f #g #s #Osf #Ofg @(O_plus … Osf) @(O_trans … Osf) //
116 lemma O_absorbr: ∀f,g,s. O s f → O f g → O s (f+g).
117 #f #g #s #Osf #Ofg @(O_plus … Osf) @(O_trans … Osf) //
120 lemma O_ext2: ∀f,g,s. O s f → (∀x.f x = g x) → O s g.
121 #f #g #s * #c * #a #Osf #eqfg %{c} %{a} #n #lean <eqfg @Osf //
124 definition not_O ≝ λf,g.∀c,n0.∃n. n0 ≤ n ∧ c* (f n) < g n .
126 (******************************* small O notation *****************************)
128 (* o f g means g ∈ o(f) *)
129 definition o: relation (nat→nat) ≝
130 λf,g.∀c.∃n0.∀n. n0 ≤ n → c * (g n) < f n.
132 lemma o_irrefl: ∀s. ¬ o s s.
133 #s % #oss cases (oss 1) #n0 #H @(absurd ? (le_n (s n0)))
134 @lt_to_not_le >(times_n_1 (s n0)) in ⊢ (?%?); >commutative_times @H //
137 lemma o_trans: ∀s1,s2,s3. o s2 s1 → o s3 s2 → o s3 s1.
138 #s1 #s2 #s3 #H1 #H2 #c cases (H1 c) #n1 -H1 #H1 cases (H2 1) #n2 -H2 #H2
139 %{(max n1 n2)} #n #Hmax
140 @(transitive_lt … (H1 ??)) [@(le_maxl … Hmax)]
141 >(times_n_1 (s2 n)) in ⊢ (?%?); >commutative_times @H2 @(le_maxr … Hmax)
145 (*********************************** pairing **********************************)
147 axiom pair: nat →nat →nat.
148 axiom fst : nat → nat.
149 axiom snd : nat → nat.
150 axiom fst_pair: ∀a,b. fst (pair a b) = a.
151 axiom snd_pair: ∀a,b. snd (pair a b) = b.
153 interpretation "abstract pair" 'pair f g = (pair f g).
155 (************************ basic complexity notions ****************************)
157 axiom U: nat → nat → nat → option nat.
159 axiom monotonic_U: ∀i,x,n,m,y.n ≤m →
160 U i x n = Some ? y → U i x m = Some ? y.
162 lemma unique_U: ∀i,x,n,m,yn,ym.
163 U i x n = Some ? yn → U i x m = Some ? ym → yn = ym.
164 #i #x #n #m #yn #ym #Hn #Hm cases (decidable_le n m)
165 [#lenm lapply (monotonic_U … lenm Hn) >Hm #HS destruct (HS) //
166 |#ltmn lapply (monotonic_U … n … Hm) [@lt_to_le @not_le_to_lt //]
167 >Hn #HS destruct (HS) //
171 definition code_for ≝ λf,i.∀x.
172 ∃n.∀m. n ≤ m → U i x m = f x.
174 definition terminate ≝ λi,x,r. ∃y. U i x r = Some ? y.
175 notation "{i ⊙ x} ↓ r" with precedence 60 for @{terminate $i $x $r}.
177 definition lang ≝ λi,x.∃r,y. U i x r = Some ? y ∧ 0 < y.
179 lemma lang_cf :∀f,i,x. code_for f i →
180 lang i x ↔ ∃y.f x = Some ? y ∧ 0 < y.
181 #f #i #x normalize #H %
182 [* #n * #y * #H1 #posy %{y} % //
183 cases (H x) -H #m #H <(H (max n m)) [2:@(le_maxr … n) //]
184 @(monotonic_U … H1) @(le_maxl … m) //
185 |cases (H x) -H #m #Hm * #y #Hy %{m} %{y} >Hm //
189 (******************************* complexity classes ***************************)
191 axiom size: nat → nat.
192 axiom of_size: nat → nat.
194 interpretation "size" 'card n = (size n).
196 axiom size_of_size: ∀n. |of_size n| = n.
197 axiom monotonic_size: monotonic ? le size.
199 axiom of_size_max: ∀i,n. |i| = n → i ≤ of_size n.
201 axiom size_fst : ∀n. |fst n| ≤ |n|.
203 definition size_f ≝ λf,n.Max_{i < S (of_size n) | eqb (|i|) n}|(f i)|.
205 lemma size_f_def: ∀f,n. size_f f n =
206 Max_{i < S (of_size n) | eqb (|i|) n}|(f i)|.
209 lemma size_f_size : ∀f,n. size_f (f ∘ size) n = |(f n)|.
210 #f #n @le_to_le_to_eq
211 [@Max_le #a #lta #Ha normalize >(eqb_true_to_eq … Ha) //
212 |<(size_of_size n) in ⊢ (?%?); >size_f_def
213 @(le_Max (λi.|f (|i|)|) ? (S (of_size n)) (of_size n) ??)
214 [@le_S_S // | @eq_to_eqb_true //]
218 lemma size_f_id : ∀n. size_f (λx.x) n = n.
220 [@Max_le #a #lta #Ha >(eqb_true_to_eq … Ha) //
221 |<(size_of_size n) in ⊢ (?%?); >size_f_def
222 @(le_Max (λi.|i|) ? (S (of_size n)) (of_size n) ??)
223 [@le_S_S // | @eq_to_eqb_true //]
227 lemma size_f_fst : ∀n. size_f fst n ≤ n.
228 #n @Max_le #a #lta #Ha <(eqb_true_to_eq … Ha) //
231 (* C s i means that the complexity of i is O(s) *)
233 definition C ≝ λs,i.∃c.∃a.∀x.a ≤ |x| → {i ⊙ x} ↓ (c*(s(|x|))).
235 definition CF ≝ λs,f.∃i.code_for f i ∧ C s i.
237 lemma ext_CF : ∀f,g,s. (∀n. f n = g n) → CF s f → CF s g.
238 #f #g #s #Hext * #i * #Hcode #HC %{i} %
239 [#x cases (Hcode x) #a #H %{a} <Hext @H | //]
242 lemma monotonic_CF: ∀s1,s2,f. O s2 s1 → CF s1 f → CF s2 f.
243 #s1 #s2 #f * #c1 * #a #H * #i * #Hcodef #HCs1 %{i} % //
244 cases HCs1 #c2 * #b #H2 %{(c2*c1)} %{(max a b)}
245 #x #Hmax cases (H2 x ?) [2:@(le_maxr … Hmax)] #y #Hy
246 %{y} @(monotonic_U …Hy) >associative_times @le_times // @H @(le_maxl … Hmax)
249 (*********************** The hierachy theorem (left) **************************)
251 theorem hierarchy_theorem_left: ∀s1,s2:nat→nat.
252 O(s1) ⊆ O(s2) → CF s1 ⊆ CF s2.
253 #s1 #s2 #HO #f * #i * #Hcode * #c * #a #Hs1_i %{i} % //
254 cases (sub_O_to_O … HO) -HO #c1 * #b #Hs1s2
255 %{(c*c1)} %{(max a b)} #x #lemax
256 cases (Hs1_i x ?) [2: @(le_maxl …lemax)]
257 #y #Hy %{y} @(monotonic_U … Hy) >associative_times
258 @le_times // @Hs1s2 @(le_maxr … lemax)
261 (************************** The diagonal language *****************************)
263 (* the diagonal language used for the hierarchy theorem *)
265 definition diag ≝ λs,i.
266 U (fst i) i (s (|i|)) = Some ? 0.
268 lemma equiv_diag: ∀s,i.
269 diag s i ↔ {fst i ⊙ i} ↓ s(|i|) ∧ ¬lang (fst i) i.
271 [whd in ⊢ (%→?); #H % [%{0} //] % * #x * #y *
272 #H1 #Hy cut (0 = y) [@(unique_U … H H1)] #eqy /2/
274 #y0 #H * #H1 @False_ind @H1 -H1 whd %{(s (|i|))} %{(S y0)} % //
278 (* Let us define the characteristic function diag_cf for diag, and prove
281 definition diag_cf ≝ λs,i.
282 match U (fst i) i (s (|i|)) with
284 | Some y ⇒ if (eqb y 0) then (Some ? 1) else (Some ? 0)].
286 lemma diag_cf_OK: ∀s,x. diag s x ↔ ∃y.diag_cf s x = Some ? y ∧ 0 < y.
288 [whd in ⊢ (%→?); #H %{1} % // whd in ⊢ (??%?); >H //
289 |* #y * whd in ⊢ (??%?→?→%);
290 cases (U (fst x) x (s (|x|))) normalize
292 |#x cases (true_or_false (eqb x 0)) #Hx >Hx
293 [>(eqb_true_to_eq … Hx) //
294 |normalize #H destruct #H @False_ind @(absurd ? H) @lt_to_not_le //
300 lemma diag_spec: ∀s,i. code_for (diag_cf s) i → ∀x. lang i x ↔ diag s x.
301 #s #i #Hcode #x @(iff_trans … (lang_cf … Hcode)) @iff_sym @diag_cf_OK
304 (******************************************************************************)
306 lemma absurd1: ∀P. iff P (¬ P) →False.
307 #P * #H1 #H2 cut (¬P) [% #H2 @(absurd … H2) @H1 //]
308 #H3 @(absurd ?? H3) @H2 @H3
311 let rec f_img (f:nat →nat) n on n ≝
314 |S m ⇒ f m::f_img f m
317 (* a few lemma to prove injective_to_exists. This is a general result; a nice
318 example of the pidgeon hole pricniple *)
320 lemma f_img_to_exists:
321 ∀f.∀n,a. a ∈ f_img f n → ∃b. b < n ∧ f b = a.
322 #f #n #a elim n normalize [@False_ind]
324 [#Ha %{m} /2/ |#H cases(Hind H) #b * #Hb #Ha %{b} % // @le_S //]
327 lemma length_f_img: ∀f,n. |f_img f n| = n.
328 #f #n elim n // normalize //
331 lemma unique_f_img: ∀f,n. injective … f → unique ? (f_img f n).
332 #f #n #Hinj elim n normalize //
333 #m #Hind % // % #H lapply(f_img_to_exists …H) * #b * #ltbm
334 #eqbm @(absurd … ltbm) @le_to_not_lt >(Hinj… eqbm) //
337 lemma injective_to_exists: ∀f. injective nat nat f →
338 ∀n.(∀i.i < n → f i < n) → ∀a. a < n → ∃b. b<n ∧ f b = a.
339 #f #finj #n #H1 #a #ltan @(f_img_to_exists f n a)
340 @(eq_length_to_mem_all … (length_f_img …) (unique_f_img …finj …) …ltan)
341 #x #Hx cases(f_img_to_exists … Hx) #b * #ltbn #eqx <eqx @H1 //
344 lemma weak_pad1 :∀n,a.∃b. n ≤ 〈a,b〉.
346 cut (∀i.decidable (〈a,i〉 < n))
348 #Hdec cases(decidable_forall (λb. 〈a,b〉 < n) Hdec n)
349 [#H cut (∀i. i < n → ∃b. b < n ∧ 〈a,b〉 = i)
350 [@(injective_to_exists … H) //]
351 #Hcut %{n} @not_lt_to_le % #Han
352 lapply(Hcut ? Han) * #x * #Hx #Hx2
353 cut (x = n) [//] #Hxn >Hxn in Hx; /2 by absurd/
354 |#H lapply(not_forall_to_exists … Hdec H)
355 * #b * #H1 #H2 %{b} @not_lt_to_le @H2
359 lemma pad : ∀n,a. ∃b. n ≤ |〈a,b〉|.
360 #n #a cases (weak_pad1 (of_size n) a) #b #Hb
361 %{b} <(size_of_size n) @monotonic_size //
364 lemma o_to_ex: ∀s1,s2. o s1 s2 → ∀i. C s2 i →
365 ∃b.{i ⊙ 〈i,b〉} ↓ s1 (|〈i,b〉|).
366 #s1 #s2 #H #i * #c * #x0 #H1
367 cases (H c) #n0 #H2 cases (pad (max x0 n0) i) #b #Hmax
368 %{b} cases (H1 〈i,b〉 ?)
369 [#z #H3 %{z} @(monotonic_U … H3) @lt_to_le @H2
375 lemma diag1_not_s1: ∀s1,s2. o s1 s2 → ¬ CF s2 (diag_cf s1).
376 #s1 #s2 #H1 % * #i * #Hcode_i #Hs2_i
377 cases (o_to_ex … H1 ? Hs2_i) #b #H2
378 lapply (diag_spec … Hcode_i) #H3
379 @(absurd1 (lang i 〈i,b〉))
380 @(iff_trans … (H3 〈i,b〉))
381 @(iff_trans … (equiv_diag …)) >fst_pair
382 %[* #_ // |#H6 % // ]
385 (******************************************************************************)
387 definition to_Some ≝ λf.λx:nat. Some nat (f x).
389 definition deopt ≝ λn. match n with
393 definition opt_comp ≝ λf,g:nat → option nat. λx.
398 axiom sU2: nat → nat → nat.
399 axiom sU: nat → nat → nat → nat.
401 axiom CFU_new: ∀h,g,f,s.
402 CF s (to_Some h) → CF s (to_Some g) → CF s (to_Some f) →
403 O s (λx. sU (size_f h x) (size_f g x) (size_f f x)) →
404 CF s (λx.U (h x) (g x) (|f x|)).
406 lemma CFU: ∀h,g,f,s1,s2,s3.
407 CF s1 (to_Some h) → CF s2 (to_Some g) → CF s3 (to_Some f) →
408 CF (λx. s1 x + s2 x + s3 x + sU (size_f h x) (size_f g x) (size_f f x))
409 (λx.U (h x) (g x) (|f x|)).
410 #h #g #f #s1 #s2 #s3 #Hh #Hg #Hf @CFU_new
411 [@(monotonic_CF … Hh) @O_plus_l @O_plus_l @O_plus_l //
412 |@(monotonic_CF … Hg) @O_plus_l @O_plus_l @O_plus_r //
413 |@(monotonic_CF … Hf) @O_plus_l @O_plus_r //
418 axiom monotonic_sU: ∀a1,a2,b1,b2,c1,c2. a1 ≤ a2 → b1 ≤ b2 → c1 ≤c2 →
419 sU a1 b1 c1 ≤ sU a2 b2 c2.
421 axiom superlinear_sU: ∀i,x,r. r ≤ sU i x r.
424 definition sU_space ≝ λi,x,r.i+x+r.
425 definition sU_time ≝ λi,x,r.i+x+(i^2)*r*(log 2 r).
427 definition IF ≝ λb,f,g:nat →option nat. λx.
430 |Some n ⇒ if (eqb n 0) then f x else g x].
432 axiom IF_CF_new: ∀b,f,g,s. CF s b → CF s f → CF s g → CF s (IF b f g).
434 lemma IF_CF: ∀b,f,g,sb,sf,sg. CF sb b → CF sf f → CF sg g →
435 CF (λn. sb n + sf n + sg n) (IF b f g).
436 #b #f #g #sb #sf #sg #Hb #Hf #Hg @IF_CF_new
437 [@(monotonic_CF … Hb) @O_plus_l @O_plus_l //
438 |@(monotonic_CF … Hf) @O_plus_l @O_plus_r //
439 |@(monotonic_CF … Hg) @O_plus_r //
443 lemma diag_cf_def : ∀s.∀i.
445 IF (λi.U (fst i) i (|of_size (s (|i|))|)) (λi.Some ? 1) (λi.Some ? 0) i.
446 #s #i normalize >size_of_size // qed.
449 axiom CF_pair: ∀f,g,s. CF s (λx.Some ? (f x)) → CF s (λx.Some ? (g x)) →
450 CF s (λx.Some ? (pair (f x) (g x))).
452 axiom CF_fst: ∀f,s. CF s (λx.Some ? (f x)) → CF s (λx.Some ? (fst (f x))).
454 definition sufficiently_large ≝ λs. CF s (λn. Some ? n) ∧ ∀c. CF s (λn. Some ? c).
456 definition constructible ≝ λs. CF s (λx.Some ? (of_size (s (|x|)))).
458 lemma diag_s: ∀s. sufficiently_large s → constructible s →
459 CF (λx.sU x x (s x)) (diag_cf s).
460 #s * #Hs_id #Hs_c #Hs_constr
461 cut (O (λx:ℕ.sU x x (s x)) s) [%{1} %{0} #n //]
462 #O_sU_s @ext_CF [2: #n @sym_eq @diag_cf_def | skip]
463 @IF_CF_new [2,3:@(monotonic_CF … (Hs_c ?)) // ]
465 [@CF_fst @(monotonic_CF … Hs_id) //
466 |@(monotonic_CF … Hs_id) //
467 |@(monotonic_CF … Hs_constr) //
468 |%{1} %{0} #n #_ >commutative_times <times_n_1
469 @monotonic_sU // >size_f_size >size_of_size //