1 include "basics/types.ma".
2 include "arithmetics/minimization.ma".
3 include "arithmetics/bigops.ma".
4 include "arithmetics/sigma_pi.ma".
5 include "arithmetics/bounded_quantifiers.ma".
6 include "reverse_complexity/big_O.ma".
8 (************************* notation for minimization *****************************)
9 notation "μ_{ ident i < n } p"
10 with precedence 80 for @{min $n 0 (λ${ident i}.$p)}.
12 notation "μ_{ ident i ≤ n } p"
13 with precedence 80 for @{min (S $n) 0 (λ${ident i}.$p)}.
15 notation "μ_{ ident i ∈ [a,b[ } p"
16 with precedence 80 for @{min ($b-$a) $a (λ${ident i}.$p)}.
18 notation "μ_{ ident i ∈ [a,b] } p"
19 with precedence 80 for @{min (S $b-$a) $a (λ${ident i}.$p)}.
21 (************************************ MAX *************************************)
22 notation "Max_{ ident i < n | p } f"
24 for @{'bigop $n max 0 (λ${ident i}. $p) (λ${ident i}. $f)}.
26 notation "Max_{ ident i < n } f"
28 for @{'bigop $n max 0 (λ${ident i}.true) (λ${ident i}. $f)}.
30 notation "Max_{ ident j ∈ [a,b[ } f"
32 for @{'bigop ($b-$a) max 0 (λ${ident j}.((λ${ident j}.true) (${ident j}+$a)))
33 (λ${ident j}.((λ${ident j}.$f)(${ident j}+$a)))}.
35 notation "Max_{ ident j ∈ [a,b[ | p } f"
37 for @{'bigop ($b-$a) max 0 (λ${ident j}.((λ${ident j}.$p) (${ident j}+$a)))
38 (λ${ident j}.((λ${ident j}.$f)(${ident j}+$a)))}.
40 lemma Max_assoc: ∀a,b,c. max (max a b) c = max a (max b c).
41 #a #b #c normalize cases (true_or_false (leb a b)) #leab >leab normalize
42 [cases (true_or_false (leb b c )) #lebc >lebc normalize
43 [>(le_to_leb_true a c) // @(transitive_le ? b) @leb_true_to_le //
46 |cases (true_or_false (leb b c )) #lebc >lebc normalize //
47 >leab normalize >(not_le_to_leb_false a c) // @lt_to_not_le
48 @(transitive_lt ? b) @not_le_to_lt @leb_false_to_not_le //
52 lemma Max0 : ∀n. max 0 n = n.
55 lemma Max0r : ∀n. max n 0 = n.
56 #n >commutative_max //
60 mk_Aop nat 0 max Max0 Max0r (λa,b,c.sym_eq … (Max_assoc a b c)).
62 definition MaxAC ≝ mk_ACop nat 0 MaxA commutative_max.
64 lemma le_Max: ∀f,p,n,a. a < n → p a = true →
65 f a ≤ Max_{i < n | p i}(f i).
67 >(bigop_diff p ? 0 MaxAC f a n) // @(le_maxl … (le_n ?))
70 lemma le_MaxI: ∀f,p,n,m,a. m ≤ a → a < n → p a = true →
71 f a ≤ Max_{i ∈ [m,n[ | p i}(f i).
72 #f #p #n #m #a #lema #ltan #pa
73 >(bigop_diff ? ? 0 MaxAC (λi.f (i+m)) (a-m) (n-m))
74 [<plus_minus_m_m // @(le_maxl … (le_n ?))
76 |/2 by monotonic_lt_minus_l/
80 lemma Max_le: ∀f,p,n,b.
81 (∀a.a < n → p a = true → f a ≤ b) → Max_{i < n | p i}(f i) ≤ b.
82 #f #p #n elim n #b #H //
83 #b0 #H1 cases (true_or_false (p b)) #Hb
84 [>bigop_Strue [2:@Hb] @to_max [@H1 // | @H #a #ltab #pa @H1 // @le_S //]
85 |>bigop_Sfalse [2:@Hb] @H #a #ltab #pa @H1 // @le_S //
89 (********************************** pairing ***********************************)
90 axiom pair: nat → nat → nat.
91 axiom fst : nat → nat.
92 axiom snd : nat → nat.
94 interpretation "abstract pair" 'pair f g = (pair f g).
96 axiom fst_pair: ∀a,b. fst 〈a,b〉 = a.
97 axiom snd_pair: ∀a,b. snd 〈a,b〉 = b.
98 axiom surj_pair: ∀x. ∃a,b. x = 〈a,b〉.
100 axiom le_fst : ∀p. fst p ≤ p.
101 axiom le_snd : ∀p. snd p ≤ p.
102 axiom le_pair: ∀a,a1,b,b1. a ≤ a1 → b ≤ b1 → 〈a,b〉 ≤ 〈a1,b1〉.
104 (************************************* U **************************************)
105 axiom U: nat → nat →nat → option nat.
107 axiom monotonic_U: ∀i,x,n,m,y.n ≤m →
108 U i x n = Some ? y → U i x m = Some ? y.
110 lemma unique_U: ∀i,x,n,m,yn,ym.
111 U i x n = Some ? yn → U i x m = Some ? ym → yn = ym.
112 #i #x #n #m #yn #ym #Hn #Hm cases (decidable_le n m)
113 [#lenm lapply (monotonic_U … lenm Hn) >Hm #HS destruct (HS) //
114 |#ltmn lapply (monotonic_U … n … Hm) [@lt_to_le @not_le_to_lt //]
115 >Hn #HS destruct (HS) //
119 definition code_for ≝ λf,i.∀x.
120 ∃n.∀m. n ≤ m → U i x m = f x.
122 definition terminate ≝ λi,x,r. ∃y. U i x r = Some ? y.
124 notation "{i ⊙ x} ↓ r" with precedence 60 for @{terminate $i $x $r}.
126 lemma terminate_dec: ∀i,x,n. {i ⊙ x} ↓ n ∨ ¬ {i ⊙ x} ↓ n.
127 #i #x #n normalize cases (U i x n)
128 [%2 % * #y #H destruct|#y %1 %{y} //]
131 lemma monotonic_terminate: ∀i,x,n,m.
132 n ≤ m → {i ⊙ x} ↓ n → {i ⊙ x} ↓ m.
133 #i #x #n #m #lenm * #z #H %{z} @(monotonic_U … H) //
136 definition termb ≝ λi,x,t.
137 match U i x t with [None ⇒ false |Some y ⇒ true].
139 lemma termb_true_to_term: ∀i,x,t. termb i x t = true → {i ⊙ x} ↓ t.
140 #i #x #t normalize cases (U i x t) normalize [#H destruct | #y #_ %{y} //]
143 lemma term_to_termb_true: ∀i,x,t. {i ⊙ x} ↓ t → termb i x t = true.
144 #i #x #t * #y #H normalize >H //
147 definition out ≝ λi,x,r.
148 match U i x r with [ None ⇒ 0 | Some z ⇒ z].
150 definition bool_to_nat: bool → nat ≝
151 λb. match b with [true ⇒ 1 | false ⇒ 0].
153 coercion bool_to_nat.
155 definition pU : nat → nat → nat → nat ≝ λi,x,r.〈termb i x r,out i x r〉.
157 lemma pU_vs_U_Some : ∀i,x,r,y. pU i x r = 〈1,y〉 ↔ U i x r = Some ? y.
158 #i #x #r #y % normalize
159 [cases (U i x r) normalize
160 [#H cut (0=1) [lapply (eq_f … fst … H) >fst_pair >fst_pair #H @H]
162 |#a #H cut (a=y) [lapply (eq_f … snd … H) >snd_pair >snd_pair #H1 @H1]
168 lemma pU_vs_U_None : ∀i,x,r. pU i x r = 〈0,0〉 ↔ U i x r = None ?.
170 [cases (U i x r) normalize //
171 #a #H cut (1=0) [lapply (eq_f … fst … H) >fst_pair >fst_pair #H1 @H1]
176 lemma fst_pU: ∀i,x,r. fst (pU i x r) = termb i x r.
177 #i #x #r normalize cases (U i x r) normalize >fst_pair //
180 lemma snd_pU: ∀i,x,r. snd (pU i x r) = out i x r.
181 #i #x #r normalize cases (U i x r) normalize >snd_pair //
184 (********************************* the speedup ********************************)
186 definition min_input ≝ λh,i,x. μ_{y ∈ [S i,x] } (termb i y (h (S i) y)).
188 lemma min_input_def : ∀h,i,x.
189 min_input h i x = min (x -i) (S i) (λy.termb i y (h (S i) y)).
192 lemma min_input_i: ∀h,i,x. x ≤ i → min_input h i x = S i.
193 #h #i #x #lexi >min_input_def
194 cut (x - i = 0) [@sym_eq /2 by eq_minus_O/] #Hcut //
197 lemma min_input_to_terminate: ∀h,i,x.
198 min_input h i x = x → {i ⊙ x} ↓ (h (S i) x).
200 cases (decidable_le (S i) x) #Hix
201 [cases (true_or_false (termb i x (h (S i) x))) #Hcase
202 [@termb_true_to_term //
203 |<Hminx in Hcase; #H lapply (fmin_false (λx.termb i x (h (S i) x)) (x-i) (S i) H)
204 >min_input_def in Hminx; #Hminx >Hminx in ⊢ (%→?);
205 <plus_n_Sm <plus_minus_m_m [2: @lt_to_le //]
208 |@False_ind >min_input_i in Hminx;
209 [#eqix >eqix in Hix; * /2/ | @le_S_S_to_le @not_le_to_lt //]
213 lemma min_input_to_lt: ∀h,i,x.
214 min_input h i x = x → i < x.
215 #h #i #x #Hminx cases (decidable_le (S i) x) //
216 #ltxi @False_ind >min_input_i in Hminx;
217 [#eqix >eqix in ltxi; * /2/ | @le_S_S_to_le @not_le_to_lt //]
220 lemma le_to_min_input: ∀h,i,x,x1. x ≤ x1 →
221 min_input h i x = x → min_input h i x1 = x.
222 #h #i #x #x1 #lex #Hminx @(min_exists … (le_S_S … lex))
223 [@(fmin_true … (sym_eq … Hminx)) //
224 |@(min_input_to_lt … Hminx)
225 |#j #H1 <Hminx @lt_min_to_false //
226 |@plus_minus_m_m @le_S_S @(transitive_le … lex) @lt_to_le
227 @(min_input_to_lt … Hminx)
231 definition g ≝ λh,u,x.
232 S (max_{i ∈[u,x[ | eqb (min_input h i x) x} (out i x (h (S i) x))).
234 lemma g_def : ∀h,u,x. g h u x =
235 S (max_{i ∈[u,x[ | eqb (min_input h i x) x} (out i x (h (S i) x))).
238 lemma le_u_to_g_1 : ∀h,u,x. x ≤ u → g h u x = 1.
239 #h #u #x #lexu >g_def cut (x-u = 0) [/2 by minus_le_minus_minus_comm/]
240 #eq0 >eq0 normalize // qed.
242 lemma g_lt : ∀h,i,x. min_input h i x = x →
243 out i x (h (S i) x) < g h 0 x.
244 #h #i #x #H @le_S_S @(le_MaxI … i) /2 by min_input_to_lt/
247 lemma max_neq0 : ∀a,b. max a b ≠ 0 → a ≠ 0 ∨ b ≠ 0.
248 #a #b whd in match (max a b); cases (true_or_false (leb a b)) #Hcase >Hcase
249 [#H %2 @H | #H %1 @H]
252 definition almost_equal ≝ λf,g:nat → nat. ¬ ∀nu.∃x. nu < x ∧ f x ≠ g x.
253 interpretation "almost equal" 'napart f g = (almost_equal f g).
255 lemma eventually_cancelled: ∀h,u.¬∀nu.∃x. nu < x ∧
256 max_{i ∈ [0,u[ | eqb (min_input h i x) x} (out i x (h (S i) x)) ≠ 0.
258 [normalize % #H cases (H u) #x * #_ * #H1 @H1 //
259 |#u0 @not_to_not #Hind #nu cases (Hind nu) #x * #ltx
260 cases (true_or_false (eqb (min_input h (u0+O) x) x)) #Hcase
261 [>bigop_Strue [2:@Hcase] #Hmax cases (max_neq0 … Hmax) -Hmax
262 [2: #H %{x} % // <minus_n_O @H]
263 #Hneq0 (* if x is not enough we retry with nu=x *)
264 cases (Hind x) #x1 * #ltx1
266 [#H %{x1} % [@transitive_lt //| <minus_n_O @H]
267 |@not_eq_to_eqb_false >(le_to_min_input … (eqb_true_to_eq … Hcase))
268 [@lt_to_not_eq @ltx1 | @lt_to_le @ltx1]
270 |>bigop_Sfalse [2:@Hcase] #H %{x} % // <minus_n_O @H
275 lemma condition_1: ∀h,u.g h 0 ≈ g h u.
276 #h #u @(not_to_not … (eventually_cancelled h u))
277 #H #nu cases (H (max u nu)) #x * #ltx #Hdiff
278 %{x} % [@(le_to_lt_to_lt … ltx) @(le_maxr … (le_n …))] @(not_to_not … Hdiff)
279 #H @(eq_f ?? S) >(bigop_sumI 0 u x (λi:ℕ.eqb (min_input h i x) x) nat 0 MaxA)
280 [>H // |@lt_to_le @(le_to_lt_to_lt …ltx) /2 by le_maxr/ |//]
283 (******************************** Condition 2 *********************************)
284 definition total ≝ λf.λx:nat. Some nat (f x).
286 lemma exists_to_exists_min: ∀h,i. (∃x. i < x ∧ {i ⊙ x} ↓ h (S i) x) → ∃y. min_input h i y = y.
287 #h #i * #x * #ltix #Hx %{(min_input h i x)} @min_spec_to_min @found //
288 [@(f_min_true (λy:ℕ.termb i y (h (S i) y))) %{x} % [% // | @term_to_termb_true //]
289 |#y #leiy #lty @(lt_min_to_false ????? lty) //
293 lemma condition_2: ∀h,i. code_for (total (g h 0)) i → ¬∃x. i<x ∧ {i ⊙ x} ↓ h (S i) x.
294 #h #i whd in ⊢(%→?); #H % #H1 cases (exists_to_exists_min … H1) #y #Hminy
295 lapply (g_lt … Hminy)
296 lapply (min_input_to_terminate … Hminy) * #r #termy
297 cases (H y) -H #ny #Hy
298 cut (r = g h 0 y) [@(unique_U … ny … termy) @Hy //] #Hr
299 whd in match (out ???); >termy >Hr
300 #H @(absurd ? H) @le_to_not_lt @le_n
304 (********************************* complexity *********************************)
306 (* We assume operations have a minimal structural complexity MSC.
307 For instance, for time complexity, MSC is equal to the size of input.
308 For space complexity, MSC is typically 0, since we only measure the
309 space required in addition to dimension of the input. *)
311 axiom MSC : nat → nat.
312 axiom MSC_le: ∀n. MSC n ≤ n.
313 axiom monotonic_MSC: monotonic ? le MSC.
314 axiom MSC_pair: ∀a,b. MSC 〈a,b〉 ≤ MSC a + MSC b.
316 (* C s i means i is running in O(s) *)
318 definition C ≝ λs,i.∃c.∃a.∀x.a ≤ x → ∃y.
319 U i x (c*(s x)) = Some ? y.
321 (* C f s means f ∈ O(s) where MSC ∈O(s) *)
322 definition CF ≝ λs,f.O s MSC ∧ ∃i.code_for (total f) i ∧ C s i.
324 lemma ext_CF : ∀f,g,s. (∀n. f n = g n) → CF s f → CF s g.
325 #f #g #s #Hext * #HO * #i * #Hcode #HC % // %{i} %
326 [#x cases (Hcode x) #a #H %{a} whd in match (total ??); <Hext @H | //]
329 lemma monotonic_CF: ∀s1,s2,f.(∀x. s1 x ≤ s2 x) → CF s1 f → CF s2 f.
330 #s1 #s2 #f #H * #HO * #i * #Hcode #Hs1 %
331 [cases HO #c * #a -HO #HO %{c} %{a} #n #lean @(transitive_le … (HO n lean))
333 |%{i} % [//] cases Hs1 #c * #a -Hs1 #Hs1 %{c} %{a} #n #lean
334 cases(Hs1 n lean) #y #Hy %{y} @(monotonic_U …Hy) @le_times //
338 lemma O_to_CF: ∀s1,s2,f.O s2 s1 → CF s1 f → CF s2 f.
339 #s1 #s2 #f #H * #HO * #i * #Hcode #Hs1 %
341 |%{i} % [//] cases Hs1 #c * #a -Hs1 #Hs1
342 cases H #c1 * #a1 #Ha1 %{(c*c1)} %{(a+a1)} #n #lean
343 cases(Hs1 n ?) [2:@(transitive_le … lean) //] #y #Hy %{y} @(monotonic_U …Hy)
344 >associative_times @le_times // @Ha1 @(transitive_le … lean) //
348 lemma timesc_CF: ∀s,f,c.CF (λx.c*s x) f → CF s f.
349 #s #f #c @O_to_CF @O_times_c
352 (********************************* composition ********************************)
353 axiom CF_comp: ∀f,g,sf,sg,sh. CF sg g → CF sf f →
354 O sh (λx. sg x + sf (g x)) → CF sh (f ∘ g).
356 lemma CF_comp_ext: ∀f,g,h,sh,sf,sg. CF sg g → CF sf f →
357 (∀x.f(g x) = h x) → O sh (λx. sg x + sf (g x)) → CF sh h.
358 #f #g #h #sh #sf #sg #Hg #Hf #Heq #H @(ext_CF (f ∘ g))
359 [#n normalize @Heq | @(CF_comp … H) //]
363 (**************************** primitive operations*****************************)
365 definition id ≝ λx:nat.x.
367 axiom CF_id: CF MSC id.
368 axiom CF_compS: ∀h,f. CF h f → CF h (S ∘ f).
369 axiom CF_comp_fst: ∀h,f. CF h f → CF h (fst ∘ f).
370 axiom CF_comp_snd: ∀h,f. CF h f → CF h (snd ∘ f).
371 axiom CF_comp_pair: ∀h,f,g. CF h f → CF h g → CF h (λx. 〈f x,g x〉).
373 lemma CF_fst: CF MSC fst.
374 @(ext_CF (fst ∘ id)) [#n //] @(CF_comp_fst … CF_id)
377 lemma CF_snd: CF MSC snd.
378 @(ext_CF (snd ∘ id)) [#n //] @(CF_comp_snd … CF_id)
381 (************************************** eqb ***********************************)
383 axiom CF_eqb: ∀h,f,g.
384 CF h f → CF h g → CF h (λx.eqb (f x) (g x)).
386 (*********************************** maximum **********************************)
388 axiom CF_max: ∀a,b.∀p:nat →bool.∀f,ha,hb,hp,hf,s.
389 CF ha a → CF hb b → CF hp p → CF hf f →
390 O s (λx.ha x + hb x + ∑_{i ∈[a x ,b x[ }(hp 〈i,x〉 + hf 〈i,x〉)) →
391 CF s (λx.max_{i ∈[a x,b x[ | p 〈i,x〉 }(f 〈i,x〉)).
393 (******************************** minimization ********************************)
395 axiom CF_mu: ∀a,b.∀f:nat →bool.∀sa,sb,sf,s.
396 CF sa a → CF sb b → CF sf f →
397 O s (λx.sa x + sb x + ∑_{i ∈[a x ,S(b x)[ }(sf 〈i,x〉)) →
398 CF s (λx.μ_{i ∈[a x,b x] }(f 〈i,x〉)).
400 (************************************* smn ************************************)
401 axiom smn: ∀f,s. CF s f → ∀x. CF (λy.s 〈x,y〉) (λy.f 〈x,y〉).
403 (****************************** constructibility ******************************)
405 definition constructible ≝ λs. CF s s.
407 lemma constr_comp : ∀s1,s2. constructible s1 → constructible s2 →
408 (∀x. x ≤ s2 x) → constructible (s2 ∘ s1).
409 #s1 #s2 #Hs1 #Hs2 #Hle @(CF_comp … Hs1 Hs2) @O_plus @le_to_O #x [@Hle | //]
412 lemma ext_constr: ∀s1,s2. (∀x.s1 x = s2 x) →
413 constructible s1 → constructible s2.
414 #s1 #s2 #Hext #Hs1 @(ext_CF … Hext) @(monotonic_CF … Hs1) #x >Hext //
417 (********************************* simulation *********************************)
419 axiom sU : nat → nat.
421 axiom monotonic_sU: ∀i1,i2,x1,x2,s1,s2. i1 ≤ i2 → x1 ≤ x2 → s1 ≤ s2 →
422 sU 〈i1,〈x1,s1〉〉 ≤ sU 〈i2,〈x2,s2〉〉.
424 lemma monotonic_sU_aux : ∀x1,x2. fst x1 ≤ fst x2 → fst (snd x1) ≤ fst (snd x2) →
425 snd (snd x1) ≤ snd (snd x2) → sU x1 ≤ sU x2.
426 #x1 #x2 cases (surj_pair x1) #a1 * #y #eqx1 >eqx1 -eqx1 cases (surj_pair y)
427 #b1 * #c1 #eqy >eqy -eqy
428 cases (surj_pair x2) #a2 * #y2 #eqx2 >eqx2 -eqx2 cases (surj_pair y2)
429 #b2 * #c2 #eqy2 >eqy2 -eqy2 >fst_pair >snd_pair >fst_pair >snd_pair
430 >fst_pair >snd_pair >fst_pair >snd_pair @monotonic_sU
433 axiom sU_le: ∀i,x,s. s ≤ sU 〈i,〈x,s〉〉.
434 axiom sU_le_i: ∀i,x,s. MSC i ≤ sU 〈i,〈x,s〉〉.
435 axiom sU_le_x: ∀i,x,s. MSC x ≤ sU 〈i,〈x,s〉〉.
437 definition pU_unary ≝ λp. pU (fst p) (fst (snd p)) (snd (snd p)).
439 axiom CF_U : CF sU pU_unary.
441 definition termb_unary ≝ λx:ℕ.termb (fst x) (fst (snd x)) (snd (snd x)).
442 definition out_unary ≝ λx:ℕ.out (fst x) (fst (snd x)) (snd (snd x)).
444 lemma CF_termb: CF sU termb_unary.
445 @(ext_CF (fst ∘ pU_unary)) [2: @CF_comp_fst @CF_U]
446 #n whd in ⊢ (??%?); whd in ⊢ (??(?%)?); >fst_pair %
449 lemma CF_out: CF sU out_unary.
450 @(ext_CF (snd ∘ pU_unary)) [2: @CF_comp_snd @CF_U]
451 #n whd in ⊢ (??%?); whd in ⊢ (??(?%)?); >snd_pair %
455 (******************** complexity of g ********************)
457 definition unary_g ≝ λh.λux. g h (fst ux) (snd ux).
459 λh,ux. max_{i ∈[fst ux,snd ux[ | eqb (min_input h i (snd ux)) (snd ux)}
460 (out i (snd ux) (h (S i) (snd ux))).
462 lemma compl_g1 : ∀h,s. CF s (auxg h) → CF s (unary_g h).
463 #h #s #H1 @(CF_compS ? (auxg h) H1)
467 λh,ux. max_{i ∈[fst ux,snd ux[ | (λp. eqb (min_input h (fst p) (snd (snd p))) (snd (snd p))) 〈i,ux〉}
468 ((λp.out (fst p) (snd (snd p)) (h (S (fst p)) (snd (snd p)))) 〈i,ux〉).
470 lemma eq_aux : ∀h,x.aux1g h x = auxg h x.
472 [#n #_ >fst_pair >snd_pair // |#n #_ #_ >fst_pair >snd_pair //]
475 lemma compl_g2 : ∀h,s1,s2,s.
477 (λp:ℕ.eqb (min_input h (fst p) (snd (snd p))) (snd (snd p))) →
479 (λp:ℕ.out (fst p) (snd (snd p)) (h (S (fst p)) (snd (snd p)))) →
480 O s (λx.MSC x + ∑_{i ∈[fst x ,snd x[ }(s1 〈i,x〉+s2 〈i,x〉)) →
482 #h #s1 #s2 #s #Hs1 #Hs2 #HO @(ext_CF (aux1g h))
483 [#n whd in ⊢ (??%%); @eq_aux]
484 @(CF_max … CF_fst CF_snd Hs1 Hs2 …) @(O_trans … HO)
485 @O_plus [@O_plus @O_plus_l // | @O_plus_r //]
488 lemma compl_g3 : ∀h,s.
489 CF s (λp:ℕ.min_input h (fst p) (snd (snd p))) →
490 CF s (λp:ℕ.eqb (min_input h (fst p) (snd (snd p))) (snd (snd p))).
491 #h #s #H @(CF_eqb … H) @(CF_comp … CF_snd CF_snd) @(O_trans … (proj1 … H))
492 @O_plus // %{1} %{0} #n #_ >commutative_times <times_n_1 @monotonic_MSC //
495 definition min_input_aux ≝ λh,p.
496 μ_{y ∈ [S (fst p),snd (snd p)] }
497 ((λx.termb (fst (snd x)) (fst x) (h (S (fst (snd x))) (fst x))) 〈y,p〉).
499 lemma min_input_eq : ∀h,p.
501 min_input h (fst p) (snd (snd p)).
502 #h #p >min_input_def whd in ⊢ (??%?); >minus_S_S @min_f_g #i #_ #_
503 whd in ⊢ (??%%); >fst_pair >snd_pair //
506 definition termb_aux ≝ λh.
507 termb_unary ∘ λp.〈fst (snd p),〈fst p,h (S (fst (snd p))) (fst p)〉〉.
509 lemma compl_g4 : ∀h,s1,s.
511 (λp.termb (fst (snd p)) (fst p) (h (S (fst (snd p))) (fst p)))) →
512 (O s (λx.MSC x + ∑_{i ∈[S(fst x) ,S(snd (snd x))[ }(s1 〈i,x〉))) →
513 CF s (λp:ℕ.min_input h (fst p) (snd (snd p))).
514 #h #s1 #s #Hs1 #HO @(ext_CF (min_input_aux h))
515 [#n whd in ⊢ (??%%); @min_input_eq]
516 @(CF_mu … MSC MSC … Hs1)
518 |@CF_comp_snd @CF_snd
519 |@(O_trans … HO) @O_plus [@O_plus @O_plus_l // | @O_plus_r //]
522 (************************* a couple of technical lemmas ***********************)
523 lemma minus_to_0: ∀a,b. a ≤ b → minus a b = 0.
524 #a elim a // #n #Hind *
525 [#H @False_ind /2 by absurd/ | #b normalize #H @Hind @le_S_S_to_le /2/]
528 lemma sigma_bound: ∀h,a,b. monotonic nat le h →
529 ∑_{i ∈ [a,S b[ }(h i) ≤ (S b-a)*h b.
530 #h #a #b #H cases (decidable_le a b)
531 [#leab cut (b = pred (S b - a + a))
532 [<plus_minus_m_m // @le_S //] #Hb >Hb in match (h b);
533 generalize in match (S b -a);
536 |#m #Hind >bigop_Strue [2://] @le_plus
537 [@H @le_n |@(transitive_le … Hind) @le_times [//] @H //]
539 |#ltba lapply (not_le_to_lt … ltba) -ltba #ltba
540 cut (S b -a = 0) [@minus_to_0 //] #Hcut >Hcut //
544 lemma sigma_bound_decr: ∀h,a,b. (∀a1,a2. a1 ≤ a2 → a2 < b → h a2 ≤ h a1) →
545 ∑_{i ∈ [a,b[ }(h i) ≤ (b-a)*h a.
546 #h #a #b #H cases (decidable_le a b)
547 [#leab cut ((b -a) +a ≤ b) [/2 by le_minus_to_plus_r/] generalize in match (b -a);
550 |#m #Hind >bigop_Strue [2://] #Hm
551 cut (m+a ≤ b) [@(transitive_le … Hm) //] #Hm1
552 @le_plus [@H // |@(transitive_le … (Hind Hm1)) //]
554 |#ltba lapply (not_le_to_lt … ltba) -ltba #ltba
555 cut (b -a = 0) [@minus_to_0 @lt_to_le @ltba] #Hcut >Hcut //
559 lemma coroll: ∀s1:nat→nat. (∀n. monotonic ℕ le (λa:ℕ.s1 〈a,n〉)) →
560 O (λx.(snd (snd x)-fst x)*(s1 〈snd (snd x),x〉))
561 (λx.∑_{i ∈[S(fst x) ,S(snd (snd x))[ }(s1 〈i,x〉)).
562 #s1 #Hs1 %{1} %{0} #n #_ >commutative_times <times_n_1
563 @(transitive_le … (sigma_bound …)) [@Hs1|>minus_S_S //]
566 lemma coroll2: ∀s1:nat→nat. (∀n,a,b. a ≤ b → b < snd n → s1 〈b,n〉 ≤ s1 〈a,n〉) →
567 O (λx.(snd x - fst x)*s1 〈fst x,x〉) (λx.∑_{i ∈[fst x,snd x[ }(s1 〈i,x〉)).
568 #s1 #Hs1 %{1} %{0} #n #_ >commutative_times <times_n_1
569 @(transitive_le … (sigma_bound_decr …)) [2://] @Hs1
572 (**************************** end of technical lemmas *************************)
574 lemma compl_g5 : ∀h,s1.(∀n. monotonic ℕ le (λa:ℕ.s1 〈a,n〉)) →
576 (λp.termb (fst (snd p)) (fst p) (h (S (fst (snd p))) (fst p)))) →
577 CF (λx.MSC x + (snd (snd x)-fst x)*s1 〈snd (snd x),x〉)
578 (λp:ℕ.min_input h (fst p) (snd (snd p))).
579 #h #s1 #Hmono #Hs1 @(compl_g4 … Hs1) @O_plus
580 [@O_plus_l // |@O_plus_r @coroll @Hmono]
584 constructible (λx. h (fst x) (snd x)) →
585 (CF (λx. sU 〈max (fst (snd x)) (snd (snd x)),〈fst x,h (S (fst (snd x))) (fst x)〉〉)
586 (λp.termb (fst (snd p)) (fst p) (h (S (fst (snd p))) (fst p)))).
587 #h #hconstr @(ext_CF (termb_aux h))
588 [#n normalize >fst_pair >snd_pair >fst_pair >snd_pair // ]
589 @(CF_comp … (λx.MSC x + h (S (fst (snd x))) (fst x)) … CF_termb)
591 [@CF_comp_fst @(monotonic_CF … CF_snd) #x //
593 [@(monotonic_CF … CF_fst) #x //
594 |@(ext_CF ((λx.h (fst x) (snd x))∘(λx.〈S (fst (snd x)),fst x〉)))
595 [#n normalize >fst_pair >snd_pair %]
596 @(CF_comp … MSC …hconstr)
597 [@CF_comp_pair [@CF_compS @CF_comp_fst // |//]
598 |@O_plus @le_to_O [//|#n >fst_pair >snd_pair //]
604 [@(O_trans … (λx.MSC (fst x) + MSC (max (fst (snd x)) (snd (snd x)))))
605 [%{2} %{0} #x #_ cases (surj_pair x) #a * #b #eqx >eqx
606 >fst_pair >snd_pair @(transitive_le … (MSC_pair …))
607 >distributive_times_plus @le_plus [//]
608 cases (surj_pair b) #c * #d #eqb >eqb
609 >fst_pair >snd_pair @(transitive_le … (MSC_pair …))
610 whd in ⊢ (??%); @le_plus
611 [@monotonic_MSC @(le_maxl … (le_n …))
612 |>commutative_times <times_n_1 @monotonic_MSC @(le_maxr … (le_n …))
614 |@O_plus [@le_to_O #x @sU_le_x |@le_to_O #x @sU_le_i]
618 |@le_to_O #x @monotonic_sU // @(le_maxl … (le_n …)) ]
622 definition big : nat →nat ≝ λx.
623 let m ≝ max (fst x) (snd x) in 〈m,m〉.
625 lemma big_def : ∀a,b. big 〈a,b〉 = 〈max a b,max a b〉.
626 #a #b normalize >fst_pair >snd_pair // qed.
628 lemma le_big : ∀x. x ≤ big x.
629 #x cases (surj_pair x) #a * #b #eqx >eqx @le_pair >fst_pair >snd_pair
630 [@(le_maxl … (le_n …)) | @(le_maxr … (le_n …))]
633 definition faux2 ≝ λh.
634 (λx.MSC x + (snd (snd x)-fst x)*
635 (λx.sU 〈max (fst(snd x)) (snd(snd x)),〈fst x,h (S (fst (snd x))) (fst x)〉〉) 〈snd (snd x),x〉).
638 constructible (λx. h (fst x) (snd x)) →
639 (∀n. monotonic ? le (h n)) →
640 CF (λx.MSC x + (snd (snd x)-fst x)*sU 〈max (fst x) (snd x),〈snd (snd x),h (S (fst x)) (snd (snd x))〉〉)
641 (λp:ℕ.min_input h (fst p) (snd (snd p))).
642 #h #hcostr #hmono @(monotonic_CF … (faux2 h))
643 [#n normalize >fst_pair >snd_pair //]
644 @compl_g5 [2:@(compl_g6 h hcostr)] #n #x #y #lexy >fst_pair >snd_pair
645 >fst_pair >snd_pair @monotonic_sU // @hmono @lexy
649 constructible (λx. h (fst x) (snd x)) →
650 (∀n. monotonic ? le (h n)) →
651 CF (λx.MSC (big x) + (snd (snd x)-fst x)*sU 〈max (fst x) (snd x),〈snd (snd x),h (S (fst x)) (snd (snd x))〉〉)
652 (λp:ℕ.min_input h (fst p) (snd (snd p))).
653 #h #hcostr #hmono @(monotonic_CF … (compl_g7 h hcostr hmono)) #x
654 @le_plus [@monotonic_MSC //]
655 cases (decidable_le (fst x) (snd(snd x)))
656 [#Hle @le_times // @monotonic_sU
657 |#Hlt >(minus_to_0 … (lt_to_le … )) [// | @not_le_to_lt @Hlt]
661 definition out_aux ≝ λh.
662 out_unary ∘ λp.〈fst p,〈snd(snd p),h (S (fst p)) (snd (snd p))〉〉.
665 constructible (λx. h (fst x) (snd x)) →
666 (CF (λx. sU 〈max (fst x) (snd x),〈snd(snd x),h (S (fst x)) (snd(snd x))〉〉)
667 (λp.out (fst p) (snd (snd p)) (h (S (fst p)) (snd (snd p))))).
668 #h #hconstr @(ext_CF (out_aux h))
669 [#n normalize >fst_pair >snd_pair >fst_pair >snd_pair // ]
670 @(CF_comp … (λx.h (S (fst x)) (snd(snd x)) + MSC x) … CF_out)
672 [@(monotonic_CF … CF_fst) #x //
674 [@CF_comp_snd @(monotonic_CF … CF_snd) #x //
675 |@(ext_CF ((λx.h (fst x) (snd x))∘(λx.〈S (fst x),snd(snd x)〉)))
676 [#n normalize >fst_pair >snd_pair %]
677 @(CF_comp … MSC …hconstr)
678 [@CF_comp_pair [@CF_compS // | @CF_comp_snd // ]
679 |@O_plus @le_to_O [//|#n >fst_pair >snd_pair //]
686 |@(O_trans … (λx.MSC (max (fst x) (snd x))))
687 [%{2} %{0} #x #_ cases (surj_pair x) #a * #b #eqx >eqx
688 >fst_pair >snd_pair @(transitive_le … (MSC_pair …))
689 whd in ⊢ (??%); @le_plus
690 [@monotonic_MSC @(le_maxl … (le_n …))
691 |>commutative_times <times_n_1 @monotonic_MSC @(le_maxr … (le_n …))
693 |@le_to_O #x @(transitive_le ???? (sU_le_i … )) //
696 |@le_to_O #x @monotonic_sU [@(le_maxl … (le_n …))|//|//]
701 constructible (λx. h (fst x) (snd x)) →
702 (∀n. monotonic ? le (h n)) →
703 (∀n,a,b. a ≤ b → b ≤ n → h b n ≤ h a n) →
704 CF (λx. (S (snd x-fst x))*MSC 〈x,x〉 +
705 (snd x-fst x)*(S(snd x-fst x))*sU 〈x,〈snd x,h (S (fst x)) (snd x)〉〉)
707 #h #hconstr #hmono #hantimono
708 @(compl_g2 h ??? (compl_g3 … (compl_g71 h hconstr hmono)) (compl_g8 h hconstr))
710 [@O_plus_l @le_to_O #x >(times_n_1 (MSC x)) >commutative_times @le_times
711 [// | @monotonic_MSC // ]]
712 @(O_trans … (coroll2 ??))
713 [#n #a #b #leab #ltb >fst_pair >fst_pair >snd_pair >snd_pair
714 cut (b ≤ n) [@(transitive_le … (le_snd …)) @lt_to_le //] #lebn
716 [normalize >le_to_leb_true [//|@(transitive_le … leab lebn)]] #maxa
717 cut (max b n = n) [normalize >le_to_leb_true //] #maxb
719 [@le_plus [>big_def >big_def >maxa >maxb //]
721 [/2 by monotonic_le_minus_r/
722 |@monotonic_sU // @hantimono [@le_S_S // |@ltb]
724 |@monotonic_sU // @hantimono [@le_S_S // |@ltb]
726 |@le_to_O #n >fst_pair >snd_pair
727 cut (max (fst n) n = n) [normalize >le_to_leb_true //] #Hmax >Hmax
728 >associative_plus >distributive_times_plus
729 @le_plus [@le_times [@le_S // |>big_def >Hmax //] |//]
733 definition sg ≝ λh,x.
734 (S (snd x-fst x))*MSC 〈x,x〉 + (snd x-fst x)*(S(snd x-fst x))*sU 〈x,〈snd x,h (S (fst x)) (snd x)〉〉.
736 lemma sg_def : ∀h,a,b.
737 sg h 〈a,b〉 = (S (b-a))*MSC 〈〈a,b〉,〈a,b〉〉 +
738 (b-a)*(S(b-a))*sU 〈〈a,b〉,〈b,h (S a) b〉〉.
739 #h #a #b whd in ⊢ (??%?); >fst_pair >snd_pair //
742 lemma compl_g11 : ∀h.
743 constructible (λx. h (fst x) (snd x)) →
744 (∀n. monotonic ? le (h n)) →
745 (∀n,a,b. a ≤ b → b ≤ n → h b n ≤ h a n) →
746 CF (sg h) (unary_g h).
747 #h #hconstr #Hm #Ham @compl_g1 @(compl_g9 h hconstr Hm Ham)
750 (**************************** closing the argument ****************************)
752 let rec h_of_aux (r:nat →nat) (c,d,b:nat) on d : nat ≝
755 | S d1 ⇒ (S d)*(MSC 〈〈b-d,b〉,〈b-d,b〉〉) +
756 d*(S d)*sU 〈〈b-d,b〉,〈b,r (h_of_aux r c d1 b)〉〉].
758 lemma h_of_aux_O: ∀r,c,b.
759 h_of_aux r c O b = c.
762 lemma h_of_aux_S : ∀r,c,d,b.
763 h_of_aux r c (S d) b =
764 (S (S d))*(MSC 〈〈b-(S d),b〉,〈b-(S d),b〉〉) +
765 (S d)*(S (S d))*sU 〈〈b-(S d),b〉,〈b,r(h_of_aux r c d b)〉〉.
768 definition h_of ≝ λr,p.
769 let m ≝ max (fst p) (snd p) in
770 h_of_aux r (MSC 〈〈m,m〉,〈m,m〉〉) (snd p - fst p) (snd p).
772 lemma h_of_O: ∀r,a,b. b ≤ a →
773 h_of r 〈a,b〉 = let m ≝ max a b in MSC 〈〈m,m〉,〈m,m〉〉.
774 #r #a #b #Hle normalize >fst_pair >snd_pair >(minus_to_0 … Hle) //
777 lemma h_of_def: ∀r,a,b.h_of r 〈a,b〉 =
779 h_of_aux r (MSC 〈〈m,m〉,〈m,m〉〉) (b - a) b.
780 #r #a #b normalize >fst_pair >snd_pair //
783 lemma mono_h_of_aux: ∀r.(∀x. x ≤ r x) → monotonic ? le r →
784 ∀d,d1,c,c1,b,b1.c ≤ c1 → d ≤ d1 → b ≤ b1 →
785 h_of_aux r c d b ≤ h_of_aux r c1 d1 b1.
786 #r #Hr #monor #d #d1 lapply d -d elim d1
787 [#d #c #c1 #b #b1 #Hc #Hd @(le_n_O_elim ? Hd) #leb
788 >h_of_aux_O >h_of_aux_O //
789 |#m #Hind #d #c #c1 #b #b1 #lec #led #leb cases (le_to_or_lt_eq … led)
790 [#ltd @(transitive_le … (Hind … lec ? leb)) [@le_S_S_to_le @ltd]
791 >h_of_aux_S @(transitive_le ???? (le_plus_n …))
792 >(times_n_1 (h_of_aux r c1 m b1)) in ⊢ (?%?);
793 >commutative_times @le_times [//|@(transitive_le … (Hr ?)) @sU_le]
794 |#Hd >Hd >h_of_aux_S >h_of_aux_S
795 cut (b-S m ≤ b1 - S m) [/2 by monotonic_le_minus_l/] #Hb1
796 @le_plus [@le_times //]
797 [@monotonic_MSC @le_pair @le_pair //
798 |@le_times [//] @monotonic_sU
799 [@le_pair // |// |@monor @Hind //]
805 lemma mono_h_of2: ∀r.(∀x. x ≤ r x) → monotonic ? le r →
806 ∀i,b,b1. b ≤ b1 → h_of r 〈i,b〉 ≤ h_of r 〈i,b1〉.
807 #r #Hr #Hmono #i #a #b #leab >h_of_def >h_of_def
808 cut (max i a ≤ max i b)
810 [@(le_maxl … (le_n …))|@(transitive_le … leab) @(le_maxr … (le_n …))]]
811 #Hmax @(mono_h_of_aux r Hr Hmono)
812 [@monotonic_MSC @le_pair @le_pair @Hmax |/2 by monotonic_le_minus_l/ |@leab]
815 axiom h_of_constr : ∀r:nat →nat.
816 (∀x. x ≤ r x) → monotonic ? le r → constructible r →
817 constructible (h_of r).
819 lemma speed_compl: ∀r:nat →nat.
820 (∀x. x ≤ r x) → monotonic ? le r → constructible r →
821 CF (h_of r) (unary_g (λi,x. r(h_of r 〈i,x〉))).
822 #r #Hr #Hmono #Hconstr @(monotonic_CF … (compl_g11 …))
823 [#x cases (surj_pair x) #a * #b #eqx >eqx
824 >sg_def cases (decidable_le b a)
825 [#leba >(minus_to_0 … leba) normalize in ⊢ (?%?);
826 <plus_n_O <plus_n_O >h_of_def
828 [normalize cases (le_to_or_lt_eq … leba)
829 [#ltba >(lt_to_leb_false … ltba) %
830 |#eqba <eqba >(le_to_leb_true … (le_n ?)) % ]]
831 #Hmax >Hmax normalize >(minus_to_0 … leba) normalize
832 @monotonic_MSC @le_pair @le_pair //
833 |#ltab >h_of_def >h_of_def
835 [normalize >(le_to_leb_true … ) [%] @lt_to_le @not_le_to_lt @ltab]
837 cut (max (S a) b = b)
838 [whd in ⊢ (??%?); >(le_to_leb_true … ) [%] @not_le_to_lt @ltab]
841 [%{(pred(b-a))} >S_pred [//] @lt_plus_to_minus_r @not_le_to_lt @ltab]
843 cut (b-S a = d) [//] #eqd1 >eqd1 >h_of_aux_S >eqd1
845 [@plus_to_minus >commutative_plus @minus_to_plus
846 [@lt_to_le @not_le_to_lt // | //]] #eqd2 >eqd2
849 |#n #a #b #leab #lebn >h_of_def >h_of_def
851 [normalize >le_to_leb_true [%|@(transitive_le … leab lebn)]] #Hmaxa
853 [normalize >(le_to_leb_true … lebn) %] #Hmaxb
854 >Hmaxa >Hmaxb @Hmono @(mono_h_of_aux r … Hr Hmono) // /2 by monotonic_le_minus_r/
855 |#n #a #b #leab @Hmono @(mono_h_of2 … Hr Hmono … leab)
856 |@(constr_comp … Hconstr Hr) @(ext_constr (h_of r))
857 [#x cases (surj_pair x) #a * #b #eqx >eqx >fst_pair >snd_pair //]
858 @(h_of_constr r Hr Hmono Hconstr)
862 lemma speed_compl_i: ∀r:nat →nat.
863 (∀x. x ≤ r x) → monotonic ? le r → constructible r →
864 ∀i. CF (λx.h_of r 〈i,x〉) (λx.g (λi,x. r(h_of r 〈i,x〉)) i x).
865 #r #Hr #Hmono #Hconstr #i
866 @(ext_CF (λx.unary_g (λi,x. r(h_of r 〈i,x〉)) 〈i,x〉))
867 [#n whd in ⊢ (??%%); @eq_f @sym_eq >fst_pair >snd_pair %]
868 @smn @(ext_CF … (speed_compl r Hr Hmono Hconstr)) #n //
871 (**************************** the speedup theorem *****************************)
872 theorem pseudo_speedup:
873 ∀r:nat →nat. (∀x. x ≤ r x) → monotonic ? le r → constructible r →
874 ∃f.∀sf. CF sf f → ∃g,sg. f ≈ g ∧ CF sg g ∧ O sf (r ∘ sg).
875 (* ∃m,a.∀n. a≤n → r(sg a) < m * sf n. *)
876 #r #Hr #Hmono #Hconstr
877 (* f is (g (λi,x. r(h_of r 〈i,x〉)) 0) *)
878 %{(g (λi,x. r(h_of r 〈i,x〉)) 0)} #sf * #H * #i *
880 (* g is (g (λi,x. r(h_of r 〈i,x〉)) (S i)) *)
881 %{(g (λi,x. r(h_of r 〈i,x〉)) (S i))}
882 (* sg is (λx.h_of r 〈i,x〉) *)
883 %{(λx. h_of r 〈S i,x〉)}
884 lapply (speed_compl_i … Hr Hmono Hconstr (S i)) #Hg
885 %[%[@condition_1 |@Hg]
886 |cases Hg #H1 * #j * #Hcodej #HCj
887 lapply (condition_2 … Hcodei) #Hcond2 (* @not_to_not *)
888 cases HCi #m * #a #Ha %{m} %{(max (S i) a)} #n #ltin @lt_to_le @not_le_to_lt
889 @(not_to_not … Hcond2) -Hcond2 #Hlesf %{n} %
890 [@(transitive_le … ltin) @(le_maxl … (le_n …))]
891 cases (Ha n ?) [2: @(transitive_le … ltin) @(le_maxr … (le_n …))]
892 #y #Uy %{y} @(monotonic_U … Uy) @(transitive_le … Hlesf) //
896 theorem pseudo_speedup':
897 ∀r:nat →nat. (∀x. x ≤ r x) → monotonic ? le r → constructible r →
898 ∃f.∀sf. CF sf f → ∃g,sg. f ≈ g ∧ CF sg g ∧
899 (* ¬ O (r ∘ sg) sf. *)
900 ∃m,a.∀n. a≤n → r(sg a) < m * sf n.
901 #r #Hr #Hmono #Hconstr
902 (* f is (g (λi,x. r(h_of r 〈i,x〉)) 0) *)
903 %{(g (λi,x. r(h_of r 〈i,x〉)) 0)} #sf * #H * #i *
905 (* g is (g (λi,x. r(h_of r 〈i,x〉)) (S i)) *)
906 %{(g (λi,x. r(h_of r 〈i,x〉)) (S i))}
907 (* sg is (λx.h_of r 〈i,x〉) *)
908 %{(λx. h_of r 〈S i,x〉)}
909 lapply (speed_compl_i … Hr Hmono Hconstr (S i)) #Hg
910 %[%[@condition_1 |@Hg]
911 |cases Hg #H1 * #j * #Hcodej #HCj
912 lapply (condition_2 … Hcodei) #Hcond2 (* @not_to_not *)
913 cases HCi #m * #a #Ha
914 %{m} %{(max (S i) a)} #n #ltin @not_le_to_lt @(not_to_not … Hcond2) -Hcond2 #Hlesf
915 %{n} % [@(transitive_le … ltin) @(le_maxl … (le_n …))]
916 cases (Ha n ?) [2: @(transitive_le … ltin) @(le_maxr … (le_n …))]
917 #y #Uy %{y} @(monotonic_U … Uy) @(transitive_le … Hlesf)
918 @Hmono @(mono_h_of2 … Hr Hmono … ltin)