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".
7 include "basics/core_notation/napart_2.ma".
9 (************************* notation for minimization *****************************)
10 notation "μ_{ ident i < n } p"
11 with precedence 80 for @{min $n 0 (λ${ident i}.$p)}.
13 notation "μ_{ ident i ≤ n } p"
14 with precedence 80 for @{min (S $n) 0 (λ${ident i}.$p)}.
16 notation "μ_{ ident i ∈ [a,b[ } p"
17 with precedence 80 for @{min ($b-$a) $a (λ${ident i}.$p)}.
19 notation "μ_{ ident i ∈ [a,b] } p"
20 with precedence 80 for @{min (S $b-$a) $a (λ${ident i}.$p)}.
22 (************************************ MAX *************************************)
23 notation "Max_{ ident i < n | p } f"
25 for @{'bigop $n max 0 (λ${ident i}. $p) (λ${ident i}. $f)}.
27 notation "Max_{ ident i < n } f"
29 for @{'bigop $n max 0 (λ${ident i}.true) (λ${ident i}. $f)}.
31 notation "Max_{ ident j ∈ [a,b[ } f"
33 for @{'bigop ($b-$a) max 0 (λ${ident j}.((λ${ident j}.true) (${ident j}+$a)))
34 (λ${ident j}.((λ${ident j}.$f)(${ident j}+$a)))}.
36 notation "Max_{ ident j ∈ [a,b[ | p } f"
38 for @{'bigop ($b-$a) max 0 (λ${ident j}.((λ${ident j}.$p) (${ident j}+$a)))
39 (λ${ident j}.((λ${ident j}.$f)(${ident j}+$a)))}.
41 lemma Max_assoc: ∀a,b,c. max (max a b) c = max a (max b c).
42 #a #b #c normalize cases (true_or_false (leb a b)) #leab >leab normalize
43 [cases (true_or_false (leb b c )) #lebc >lebc normalize
44 [>(le_to_leb_true a c) // @(transitive_le ? b) @leb_true_to_le //
47 |cases (true_or_false (leb b c )) #lebc >lebc normalize //
48 >leab normalize >(not_le_to_leb_false a c) // @lt_to_not_le
49 @(transitive_lt ? b) @not_le_to_lt @leb_false_to_not_le //
53 lemma Max0 : ∀n. max 0 n = n.
56 lemma Max0r : ∀n. max n 0 = n.
57 #n >commutative_max //
61 mk_Aop nat 0 max Max0 Max0r (λa,b,c.sym_eq … (Max_assoc a b c)).
63 definition MaxAC ≝ mk_ACop nat 0 MaxA commutative_max.
65 lemma le_Max: ∀f,p,n,a. a < n → p a = true →
66 f a ≤ Max_{i < n | p i}(f i).
68 >(bigop_diff p ? 0 MaxAC f a n) // @(le_maxl … (le_n ?))
71 lemma le_MaxI: ∀f,p,n,m,a. m ≤ a → a < n → p a = true →
72 f a ≤ Max_{i ∈ [m,n[ | p i}(f i).
73 #f #p #n #m #a #lema #ltan #pa
74 >(bigop_diff ? ? 0 MaxAC (λi.f (i+m)) (a-m) (n-m))
75 [<plus_minus_m_m // @(le_maxl … (le_n ?))
77 |/2 by monotonic_lt_minus_l/
81 lemma Max_le: ∀f,p,n,b.
82 (∀a.a < n → p a = true → f a ≤ b) → Max_{i < n | p i}(f i) ≤ b.
83 #f #p #n elim n #b #H //
84 #b0 #H1 cases (true_or_false (p b)) #Hb
85 [>bigop_Strue [2:@Hb] @to_max [@H1 // | @H #a #ltab #pa @H1 // @le_S //]
86 |>bigop_Sfalse [2:@Hb] @H #a #ltab #pa @H1 // @le_S //
90 (********************************** pairing ***********************************)
91 axiom pair: nat → nat → nat.
92 axiom fst : nat → nat.
93 axiom snd : nat → nat.
95 interpretation "abstract pair" 'pair f g = (pair f g).
97 axiom fst_pair: ∀a,b. fst 〈a,b〉 = a.
98 axiom snd_pair: ∀a,b. snd 〈a,b〉 = b.
99 axiom surj_pair: ∀x. ∃a,b. x = 〈a,b〉.
101 axiom le_fst : ∀p. fst p ≤ p.
102 axiom le_snd : ∀p. snd p ≤ p.
103 axiom le_pair: ∀a,a1,b,b1. a ≤ a1 → b ≤ b1 → 〈a,b〉 ≤ 〈a1,b1〉.
105 (************************************* U **************************************)
106 axiom U: nat → nat →nat → option nat.
108 axiom monotonic_U: ∀i,x,n,m,y.n ≤m →
109 U i x n = Some ? y → U i x m = Some ? y.
111 lemma unique_U: ∀i,x,n,m,yn,ym.
112 U i x n = Some ? yn → U i x m = Some ? ym → yn = ym.
113 #i #x #n #m #yn #ym #Hn #Hm cases (decidable_le n m)
114 [#lenm lapply (monotonic_U … lenm Hn) >Hm #HS destruct (HS) //
115 |#ltmn lapply (monotonic_U … n … Hm) [@lt_to_le @not_le_to_lt //]
116 >Hn #HS destruct (HS) //
120 definition code_for ≝ λf,i.∀x.
121 ∃n.∀m. n ≤ m → U i x m = f x.
123 definition terminate ≝ λi,x,r. ∃y. U i x r = Some ? y.
125 notation "{i ⊙ x} ↓ r" with precedence 60 for @{terminate $i $x $r}.
127 lemma terminate_dec: ∀i,x,n. {i ⊙ x} ↓ n ∨ ¬ {i ⊙ x} ↓ n.
128 #i #x #n normalize cases (U i x n)
129 [%2 % * #y #H destruct|#y %1 %{y} //]
132 lemma monotonic_terminate: ∀i,x,n,m.
133 n ≤ m → {i ⊙ x} ↓ n → {i ⊙ x} ↓ m.
134 #i #x #n #m #lenm * #z #H %{z} @(monotonic_U … H) //
137 definition termb ≝ λi,x,t.
138 match U i x t with [None ⇒ false |Some y ⇒ true].
140 lemma termb_true_to_term: ∀i,x,t. termb i x t = true → {i ⊙ x} ↓ t.
141 #i #x #t normalize cases (U i x t) normalize [#H destruct | #y #_ %{y} //]
144 lemma term_to_termb_true: ∀i,x,t. {i ⊙ x} ↓ t → termb i x t = true.
145 #i #x #t * #y #H normalize >H //
148 definition out ≝ λi,x,r.
149 match U i x r with [ None ⇒ 0 | Some z ⇒ z].
151 definition bool_to_nat: bool → nat ≝
152 λb. match b with [true ⇒ 1 | false ⇒ 0].
154 coercion bool_to_nat.
156 definition pU : nat → nat → nat → nat ≝ λi,x,r.〈termb i x r,out i x r〉.
158 lemma pU_vs_U_Some : ∀i,x,r,y. pU i x r = 〈1,y〉 ↔ U i x r = Some ? y.
159 #i #x #r #y % normalize
160 [cases (U i x r) normalize
161 [#H cut (0=1) [lapply (eq_f … fst … H) >fst_pair >fst_pair #H @H]
163 |#a #H cut (a=y) [lapply (eq_f … snd … H) >snd_pair >snd_pair #H1 @H1]
169 lemma pU_vs_U_None : ∀i,x,r. pU i x r = 〈0,0〉 ↔ U i x r = None ?.
171 [cases (U i x r) normalize //
172 #a #H cut (1=0) [lapply (eq_f … fst … H) >fst_pair >fst_pair #H1 @H1]
177 lemma fst_pU: ∀i,x,r. fst (pU i x r) = termb i x r.
178 #i #x #r normalize cases (U i x r) normalize >fst_pair //
181 lemma snd_pU: ∀i,x,r. snd (pU i x r) = out i x r.
182 #i #x #r normalize cases (U i x r) normalize >snd_pair //
185 (********************************* the speedup ********************************)
187 definition min_input ≝ λh,i,x. μ_{y ∈ [S i,x] } (termb i y (h (S i) y)).
189 lemma min_input_def : ∀h,i,x.
190 min_input h i x = min (x -i) (S i) (λy.termb i y (h (S i) y)).
193 lemma min_input_i: ∀h,i,x. x ≤ i → min_input h i x = S i.
194 #h #i #x #lexi >min_input_def
195 cut (x - i = 0) [@sym_eq /2 by eq_minus_O/] #Hcut //
198 lemma min_input_to_terminate: ∀h,i,x.
199 min_input h i x = x → {i ⊙ x} ↓ (h (S i) x).
201 cases (decidable_le (S i) x) #Hix
202 [cases (true_or_false (termb i x (h (S i) x))) #Hcase
203 [@termb_true_to_term //
204 |<Hminx in Hcase; #H lapply (fmin_false (λx.termb i x (h (S i) x)) (x-i) (S i) H)
205 >min_input_def in Hminx; #Hminx >Hminx in ⊢ (%→?);
206 <plus_n_Sm <plus_minus_m_m [2: @lt_to_le //]
209 |@False_ind >min_input_i in Hminx;
210 [#eqix >eqix in Hix; * /2/ | @le_S_S_to_le @not_le_to_lt //]
214 lemma min_input_to_lt: ∀h,i,x.
215 min_input h i x = x → i < x.
216 #h #i #x #Hminx cases (decidable_le (S i) x) //
217 #ltxi @False_ind >min_input_i in Hminx;
218 [#eqix >eqix in ltxi; * /2/ | @le_S_S_to_le @not_le_to_lt //]
221 lemma le_to_min_input: ∀h,i,x,x1. x ≤ x1 →
222 min_input h i x = x → min_input h i x1 = x.
223 #h #i #x #x1 #lex #Hminx @(min_exists … (le_S_S … lex))
224 [@(fmin_true … (sym_eq … Hminx)) //
225 |@(min_input_to_lt … Hminx)
226 |#j #H1 <Hminx @lt_min_to_false //
227 |@plus_minus_m_m @le_S_S @(transitive_le … lex) @lt_to_le
228 @(min_input_to_lt … Hminx)
232 definition g ≝ λh,u,x.
233 S (max_{i ∈[u,x[ | eqb (min_input h i x) x} (out i x (h (S i) x))).
235 lemma g_def : ∀h,u,x. g h u x =
236 S (max_{i ∈[u,x[ | eqb (min_input h i x) x} (out i x (h (S i) x))).
239 lemma le_u_to_g_1 : ∀h,u,x. x ≤ u → g h u x = 1.
240 #h #u #x #lexu >g_def cut (x-u = 0) [/2 by minus_le_minus_minus_comm/]
241 #eq0 >eq0 normalize // qed.
243 lemma g_lt : ∀h,i,x. min_input h i x = x →
244 out i x (h (S i) x) < g h 0 x.
245 #h #i #x #H @le_S_S @(le_MaxI … i) /2 by min_input_to_lt/
248 lemma max_neq0 : ∀a,b. max a b ≠ 0 → a ≠ 0 ∨ b ≠ 0.
249 #a #b whd in match (max a b); cases (true_or_false (leb a b)) #Hcase >Hcase
250 [#H %2 @H | #H %1 @H]
253 definition almost_equal ≝ λf,g:nat → nat. ¬ ∀nu.∃x. nu < x ∧ f x ≠ g x.
254 interpretation "almost equal" 'napart f g = (almost_equal f g).
256 lemma eventually_cancelled: ∀h,u.¬∀nu.∃x. nu < x ∧
257 max_{i ∈ [0,u[ | eqb (min_input h i x) x} (out i x (h (S i) x)) ≠ 0.
259 [normalize % #H cases (H u) #x * #_ * #H1 @H1 //
260 |#u0 @not_to_not #Hind #nu cases (Hind nu) #x * #ltx
261 cases (true_or_false (eqb (min_input h (u0+O) x) x)) #Hcase
262 [>bigop_Strue [2:@Hcase] #Hmax cases (max_neq0 … Hmax) -Hmax
263 [2: #H %{x} % // <minus_n_O @H]
264 #Hneq0 (* if x is not enough we retry with nu=x *)
265 cases (Hind x) #x1 * #ltx1
267 [#H %{x1} % [@transitive_lt //| <minus_n_O @H]
268 |@not_eq_to_eqb_false >(le_to_min_input … (eqb_true_to_eq … Hcase))
269 [@lt_to_not_eq @ltx1 | @lt_to_le @ltx1]
271 |>bigop_Sfalse [2:@Hcase] #H %{x} % // <minus_n_O @H
276 lemma condition_1: ∀h,u.g h 0 ≈ g h u.
277 #h #u @(not_to_not … (eventually_cancelled h u))
278 #H #nu cases (H (max u nu)) #x * #ltx #Hdiff
279 %{x} % [@(le_to_lt_to_lt … ltx) @(le_maxr … (le_n …))] @(not_to_not … Hdiff)
280 #H @(eq_f ?? S) >(bigop_sumI 0 u x (λi:ℕ.eqb (min_input h i x) x) nat 0 MaxA)
281 [>H // |@lt_to_le @(le_to_lt_to_lt …ltx) /2 by le_maxr/ |//]
284 (******************************** Condition 2 *********************************)
285 definition total ≝ λf.λx:nat. Some nat (f x).
287 lemma exists_to_exists_min: ∀h,i. (∃x. i < x ∧ {i ⊙ x} ↓ h (S i) x) → ∃y. min_input h i y = y.
288 #h #i * #x * #ltix #Hx %{(min_input h i x)} @min_spec_to_min @found //
289 [@(f_min_true (λy:ℕ.termb i y (h (S i) y))) %{x} % [% // | @term_to_termb_true //]
290 |#y #leiy #lty @(lt_min_to_false ????? lty) //
294 lemma condition_2: ∀h,i. code_for (total (g h 0)) i → ¬∃x. i<x ∧ {i ⊙ x} ↓ h (S i) x.
295 #h #i whd in ⊢(%→?); #H % #H1 cases (exists_to_exists_min … H1) #y #Hminy
296 lapply (g_lt … Hminy)
297 lapply (min_input_to_terminate … Hminy) * #r #termy
298 cases (H y) -H #ny #Hy
299 cut (r = g h 0 y) [@(unique_U … ny … termy) @Hy //] #Hr
300 whd in match (out ???); >termy >Hr
301 #H @(absurd ? H) @le_to_not_lt @le_n
305 (********************************* complexity *********************************)
307 (* We assume operations have a minimal structural complexity MSC.
308 For instance, for time complexity, MSC is equal to the size of input.
309 For space complexity, MSC is typically 0, since we only measure the
310 space required in addition to dimension of the input. *)
312 axiom MSC : nat → nat.
313 axiom MSC_le: ∀n. MSC n ≤ n.
314 axiom monotonic_MSC: monotonic ? le MSC.
315 axiom MSC_pair: ∀a,b. MSC 〈a,b〉 ≤ MSC a + MSC b.
317 (* C s i means i is running in O(s) *)
319 definition C ≝ λs,i.∃c.∃a.∀x.a ≤ x → ∃y.
320 U i x (c*(s x)) = Some ? y.
322 (* C f s means f ∈ O(s) where MSC ∈O(s) *)
323 definition CF ≝ λs,f.O s MSC ∧ ∃i.code_for (total f) i ∧ C s i.
325 lemma ext_CF : ∀f,g,s. (∀n. f n = g n) → CF s f → CF s g.
326 #f #g #s #Hext * #HO * #i * #Hcode #HC % // %{i} %
327 [#x cases (Hcode x) #a #H %{a} whd in match (total ??); <Hext @H | //]
330 lemma monotonic_CF: ∀s1,s2,f.(∀x. s1 x ≤ s2 x) → CF s1 f → CF s2 f.
331 #s1 #s2 #f #H * #HO * #i * #Hcode #Hs1 %
332 [cases HO #c * #a -HO #HO %{c} %{a} #n #lean @(transitive_le … (HO n lean))
334 |%{i} % [//] cases Hs1 #c * #a -Hs1 #Hs1 %{c} %{a} #n #lean
335 cases(Hs1 n lean) #y #Hy %{y} @(monotonic_U …Hy) @le_times //
339 lemma O_to_CF: ∀s1,s2,f.O s2 s1 → CF s1 f → CF s2 f.
340 #s1 #s2 #f #H * #HO * #i * #Hcode #Hs1 %
342 |%{i} % [//] cases Hs1 #c * #a -Hs1 #Hs1
343 cases H #c1 * #a1 #Ha1 %{(c*c1)} %{(a+a1)} #n #lean
344 cases(Hs1 n ?) [2:@(transitive_le … lean) //] #y #Hy %{y} @(monotonic_U …Hy)
345 >associative_times @le_times // @Ha1 @(transitive_le … lean) //
349 lemma timesc_CF: ∀s,f,c.CF (λx.c*s x) f → CF s f.
350 #s #f #c @O_to_CF @O_times_c
353 (********************************* composition ********************************)
354 axiom CF_comp: ∀f,g,sf,sg,sh. CF sg g → CF sf f →
355 O sh (λx. sg x + sf (g x)) → CF sh (f ∘ g).
357 lemma CF_comp_ext: ∀f,g,h,sh,sf,sg. CF sg g → CF sf f →
358 (∀x.f(g x) = h x) → O sh (λx. sg x + sf (g x)) → CF sh h.
359 #f #g #h #sh #sf #sg #Hg #Hf #Heq #H @(ext_CF (f ∘ g))
360 [#n normalize @Heq | @(CF_comp … H) //]
364 (**************************** primitive operations*****************************)
366 definition id ≝ λx:nat.x.
368 axiom CF_id: CF MSC id.
369 axiom CF_compS: ∀h,f. CF h f → CF h (S ∘ f).
370 axiom CF_comp_fst: ∀h,f. CF h f → CF h (fst ∘ f).
371 axiom CF_comp_snd: ∀h,f. CF h f → CF h (snd ∘ f).
372 axiom CF_comp_pair: ∀h,f,g. CF h f → CF h g → CF h (λx. 〈f x,g x〉).
374 lemma CF_fst: CF MSC fst.
375 @(ext_CF (fst ∘ id)) [#n //] @(CF_comp_fst … CF_id)
378 lemma CF_snd: CF MSC snd.
379 @(ext_CF (snd ∘ id)) [#n //] @(CF_comp_snd … CF_id)
382 (************************************** eqb ***********************************)
384 axiom CF_eqb: ∀h,f,g.
385 CF h f → CF h g → CF h (λx.eqb (f x) (g x)).
387 (*********************************** maximum **********************************)
389 axiom CF_max: ∀a,b.∀p:nat →bool.∀f,ha,hb,hp,hf,s.
390 CF ha a → CF hb b → CF hp p → CF hf f →
391 O s (λx.ha x + hb x + ∑_{i ∈[a x ,b x[ }(hp 〈i,x〉 + hf 〈i,x〉)) →
392 CF s (λx.max_{i ∈[a x,b x[ | p 〈i,x〉 }(f 〈i,x〉)).
394 (******************************** minimization ********************************)
396 axiom CF_mu: ∀a,b.∀f:nat →bool.∀sa,sb,sf,s.
397 CF sa a → CF sb b → CF sf f →
398 O s (λx.sa x + sb x + ∑_{i ∈[a x ,S(b x)[ }(sf 〈i,x〉)) →
399 CF s (λx.μ_{i ∈[a x,b x] }(f 〈i,x〉)).
401 (************************************* smn ************************************)
402 axiom smn: ∀f,s. CF s f → ∀x. CF (λy.s 〈x,y〉) (λy.f 〈x,y〉).
404 (****************************** constructibility ******************************)
406 definition constructible ≝ λs. CF s s.
408 lemma constr_comp : ∀s1,s2. constructible s1 → constructible s2 →
409 (∀x. x ≤ s2 x) → constructible (s2 ∘ s1).
410 #s1 #s2 #Hs1 #Hs2 #Hle @(CF_comp … Hs1 Hs2) @O_plus @le_to_O #x [@Hle | //]
413 lemma ext_constr: ∀s1,s2. (∀x.s1 x = s2 x) →
414 constructible s1 → constructible s2.
415 #s1 #s2 #Hext #Hs1 @(ext_CF … Hext) @(monotonic_CF … Hs1) #x >Hext //
418 (********************************* simulation *********************************)
420 axiom sU : nat → nat.
422 axiom monotonic_sU: ∀i1,i2,x1,x2,s1,s2. i1 ≤ i2 → x1 ≤ x2 → s1 ≤ s2 →
423 sU 〈i1,〈x1,s1〉〉 ≤ sU 〈i2,〈x2,s2〉〉.
425 lemma monotonic_sU_aux : ∀x1,x2. fst x1 ≤ fst x2 → fst (snd x1) ≤ fst (snd x2) →
426 snd (snd x1) ≤ snd (snd x2) → sU x1 ≤ sU x2.
427 #x1 #x2 cases (surj_pair x1) #a1 * #y #eqx1 >eqx1 -eqx1 cases (surj_pair y)
428 #b1 * #c1 #eqy >eqy -eqy
429 cases (surj_pair x2) #a2 * #y2 #eqx2 >eqx2 -eqx2 cases (surj_pair y2)
430 #b2 * #c2 #eqy2 >eqy2 -eqy2 >fst_pair >snd_pair >fst_pair >snd_pair
431 >fst_pair >snd_pair >fst_pair >snd_pair @monotonic_sU
434 axiom sU_le: ∀i,x,s. s ≤ sU 〈i,〈x,s〉〉.
435 axiom sU_le_i: ∀i,x,s. MSC i ≤ sU 〈i,〈x,s〉〉.
436 axiom sU_le_x: ∀i,x,s. MSC x ≤ sU 〈i,〈x,s〉〉.
438 definition pU_unary ≝ λp. pU (fst p) (fst (snd p)) (snd (snd p)).
440 axiom CF_U : CF sU pU_unary.
442 definition termb_unary ≝ λx:ℕ.termb (fst x) (fst (snd x)) (snd (snd x)).
443 definition out_unary ≝ λx:ℕ.out (fst x) (fst (snd x)) (snd (snd x)).
445 lemma CF_termb: CF sU termb_unary.
446 @(ext_CF (fst ∘ pU_unary)) [2: @CF_comp_fst @CF_U]
447 #n whd in ⊢ (??%?); whd in ⊢ (??(?%)?); >fst_pair %
450 lemma CF_out: CF sU out_unary.
451 @(ext_CF (snd ∘ pU_unary)) [2: @CF_comp_snd @CF_U]
452 #n whd in ⊢ (??%?); whd in ⊢ (??(?%)?); >snd_pair %
456 (******************** complexity of g ********************)
458 definition unary_g ≝ λh.λux. g h (fst ux) (snd ux).
460 λh,ux. max_{i ∈[fst ux,snd ux[ | eqb (min_input h i (snd ux)) (snd ux)}
461 (out i (snd ux) (h (S i) (snd ux))).
463 lemma compl_g1 : ∀h,s. CF s (auxg h) → CF s (unary_g h).
464 #h #s #H1 @(CF_compS ? (auxg h) H1)
468 λh,ux. max_{i ∈[fst ux,snd ux[ | (λp. eqb (min_input h (fst p) (snd (snd p))) (snd (snd p))) 〈i,ux〉}
469 ((λp.out (fst p) (snd (snd p)) (h (S (fst p)) (snd (snd p)))) 〈i,ux〉).
471 lemma eq_aux : ∀h,x.aux1g h x = auxg h x.
473 [#n #_ >fst_pair >snd_pair // |#n #_ #_ >fst_pair >snd_pair //]
476 lemma compl_g2 : ∀h,s1,s2,s.
478 (λp:ℕ.eqb (min_input h (fst p) (snd (snd p))) (snd (snd p))) →
480 (λp:ℕ.out (fst p) (snd (snd p)) (h (S (fst p)) (snd (snd p)))) →
481 O s (λx.MSC x + ∑_{i ∈[fst x ,snd x[ }(s1 〈i,x〉+s2 〈i,x〉)) →
483 #h #s1 #s2 #s #Hs1 #Hs2 #HO @(ext_CF (aux1g h))
484 [#n whd in ⊢ (??%%); @eq_aux]
485 @(CF_max … CF_fst CF_snd Hs1 Hs2 …) @(O_trans … HO)
486 @O_plus [@O_plus @O_plus_l // | @O_plus_r //]
489 lemma compl_g3 : ∀h,s.
490 CF s (λp:ℕ.min_input h (fst p) (snd (snd p))) →
491 CF s (λp:ℕ.eqb (min_input h (fst p) (snd (snd p))) (snd (snd p))).
492 #h #s #H @(CF_eqb … H) @(CF_comp … CF_snd CF_snd) @(O_trans … (proj1 … H))
493 @O_plus // %{1} %{0} #n #_ >commutative_times <times_n_1 @monotonic_MSC //
496 definition min_input_aux ≝ λh,p.
497 μ_{y ∈ [S (fst p),snd (snd p)] }
498 ((λx.termb (fst (snd x)) (fst x) (h (S (fst (snd x))) (fst x))) 〈y,p〉).
500 lemma min_input_eq : ∀h,p.
502 min_input h (fst p) (snd (snd p)).
503 #h #p >min_input_def whd in ⊢ (??%?); >minus_S_S @min_f_g #i #_ #_
504 whd in ⊢ (??%%); >fst_pair >snd_pair //
507 definition termb_aux ≝ λh.
508 termb_unary ∘ λp.〈fst (snd p),〈fst p,h (S (fst (snd p))) (fst p)〉〉.
510 lemma compl_g4 : ∀h,s1,s.
512 (λp.termb (fst (snd p)) (fst p) (h (S (fst (snd p))) (fst p)))) →
513 (O s (λx.MSC x + ∑_{i ∈[S(fst x) ,S(snd (snd x))[ }(s1 〈i,x〉))) →
514 CF s (λp:ℕ.min_input h (fst p) (snd (snd p))).
515 #h #s1 #s #Hs1 #HO @(ext_CF (min_input_aux h))
516 [#n whd in ⊢ (??%%); @min_input_eq]
517 @(CF_mu … MSC MSC … Hs1)
519 |@CF_comp_snd @CF_snd
520 |@(O_trans … HO) @O_plus [@O_plus @O_plus_l // | @O_plus_r //]
523 (************************* a couple of technical lemmas ***********************)
524 lemma minus_to_0: ∀a,b. a ≤ b → minus a b = 0.
525 #a elim a // #n #Hind *
526 [#H @False_ind /2 by absurd/ | #b normalize #H @Hind @le_S_S_to_le /2/]
529 lemma sigma_bound: ∀h,a,b. monotonic nat le h →
530 ∑_{i ∈ [a,S b[ }(h i) ≤ (S b-a)*h b.
531 #h #a #b #H cases (decidable_le a b)
532 [#leab cut (b = pred (S b - a + a))
533 [<plus_minus_m_m // @le_S //] #Hb >Hb in match (h b);
534 generalize in match (S b -a);
537 |#m #Hind >bigop_Strue [2://] @le_plus
538 [@H @le_n |@(transitive_le … Hind) @le_times [//] @H //]
540 |#ltba lapply (not_le_to_lt … ltba) -ltba #ltba
541 cut (S b -a = 0) [@minus_to_0 //] #Hcut >Hcut //
545 lemma sigma_bound_decr: ∀h,a,b. (∀a1,a2. a1 ≤ a2 → a2 < b → h a2 ≤ h a1) →
546 ∑_{i ∈ [a,b[ }(h i) ≤ (b-a)*h a.
547 #h #a #b #H cases (decidable_le a b)
548 [#leab cut ((b -a) +a ≤ b) [/2 by le_minus_to_plus_r/] generalize in match (b -a);
551 |#m #Hind >bigop_Strue [2://] #Hm
552 cut (m+a ≤ b) [@(transitive_le … Hm) //] #Hm1
553 @le_plus [@H // |@(transitive_le … (Hind Hm1)) //]
555 |#ltba lapply (not_le_to_lt … ltba) -ltba #ltba
556 cut (b -a = 0) [@minus_to_0 @lt_to_le @ltba] #Hcut >Hcut //
560 lemma coroll: ∀s1:nat→nat. (∀n. monotonic ℕ le (λa:ℕ.s1 〈a,n〉)) →
561 O (λx.(snd (snd x)-fst x)*(s1 〈snd (snd x),x〉))
562 (λx.∑_{i ∈[S(fst x) ,S(snd (snd x))[ }(s1 〈i,x〉)).
563 #s1 #Hs1 %{1} %{0} #n #_ >commutative_times <times_n_1
564 @(transitive_le … (sigma_bound …)) [@Hs1|>minus_S_S //]
567 lemma coroll2: ∀s1:nat→nat. (∀n,a,b. a ≤ b → b < snd n → s1 〈b,n〉 ≤ s1 〈a,n〉) →
568 O (λx.(snd x - fst x)*s1 〈fst x,x〉) (λx.∑_{i ∈[fst x,snd x[ }(s1 〈i,x〉)).
569 #s1 #Hs1 %{1} %{0} #n #_ >commutative_times <times_n_1
570 @(transitive_le … (sigma_bound_decr …)) [2://] @Hs1
573 (**************************** end of technical lemmas *************************)
575 lemma compl_g5 : ∀h,s1.(∀n. monotonic ℕ le (λa:ℕ.s1 〈a,n〉)) →
577 (λp.termb (fst (snd p)) (fst p) (h (S (fst (snd p))) (fst p)))) →
578 CF (λx.MSC x + (snd (snd x)-fst x)*s1 〈snd (snd x),x〉)
579 (λp:ℕ.min_input h (fst p) (snd (snd p))).
580 #h #s1 #Hmono #Hs1 @(compl_g4 … Hs1) @O_plus
581 [@O_plus_l // |@O_plus_r @coroll @Hmono]
585 constructible (λx. h (fst x) (snd x)) →
586 (CF (λx. sU 〈max (fst (snd x)) (snd (snd x)),〈fst x,h (S (fst (snd x))) (fst x)〉〉)
587 (λp.termb (fst (snd p)) (fst p) (h (S (fst (snd p))) (fst p)))).
588 #h #hconstr @(ext_CF (termb_aux h))
589 [#n normalize >fst_pair >snd_pair >fst_pair >snd_pair // ]
590 @(CF_comp … (λx.MSC x + h (S (fst (snd x))) (fst x)) … CF_termb)
592 [@CF_comp_fst @(monotonic_CF … CF_snd) #x //
594 [@(monotonic_CF … CF_fst) #x //
595 |@(ext_CF ((λx.h (fst x) (snd x))∘(λx.〈S (fst (snd x)),fst x〉)))
596 [#n normalize >fst_pair >snd_pair %]
597 @(CF_comp … MSC …hconstr)
598 [@CF_comp_pair [@CF_compS @CF_comp_fst // |//]
599 |@O_plus @le_to_O [//|#n >fst_pair >snd_pair //]
605 [@(O_trans … (λx.MSC (fst x) + MSC (max (fst (snd x)) (snd (snd x)))))
606 [%{2} %{0} #x #_ cases (surj_pair x) #a * #b #eqx >eqx
607 >fst_pair >snd_pair @(transitive_le … (MSC_pair …))
608 >distributive_times_plus @le_plus [//]
609 cases (surj_pair b) #c * #d #eqb >eqb
610 >fst_pair >snd_pair @(transitive_le … (MSC_pair …))
611 whd in ⊢ (??%); @le_plus
612 [@monotonic_MSC @(le_maxl … (le_n …))
613 |>commutative_times <times_n_1 @monotonic_MSC @(le_maxr … (le_n …))
615 |@O_plus [@le_to_O #x @sU_le_x |@le_to_O #x @sU_le_i]
619 |@le_to_O #x @monotonic_sU // @(le_maxl … (le_n …)) ]
623 definition big : nat →nat ≝ λx.
624 let m ≝ max (fst x) (snd x) in 〈m,m〉.
626 lemma big_def : ∀a,b. big 〈a,b〉 = 〈max a b,max a b〉.
627 #a #b normalize >fst_pair >snd_pair // qed.
629 lemma le_big : ∀x. x ≤ big x.
630 #x cases (surj_pair x) #a * #b #eqx >eqx @le_pair >fst_pair >snd_pair
631 [@(le_maxl … (le_n …)) | @(le_maxr … (le_n …))]
634 definition faux2 ≝ λh.
635 (λx.MSC x + (snd (snd x)-fst x)*
636 (λx.sU 〈max (fst(snd x)) (snd(snd x)),〈fst x,h (S (fst (snd x))) (fst x)〉〉) 〈snd (snd x),x〉).
639 constructible (λx. h (fst x) (snd x)) →
640 (∀n. monotonic ? le (h n)) →
641 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))〉〉)
642 (λp:ℕ.min_input h (fst p) (snd (snd p))).
643 #h #hcostr #hmono @(monotonic_CF … (faux2 h))
644 [#n normalize >fst_pair >snd_pair //]
645 @compl_g5 [2:@(compl_g6 h hcostr)] #n #x #y #lexy >fst_pair >snd_pair
646 >fst_pair >snd_pair @monotonic_sU // @hmono @lexy
650 constructible (λx. h (fst x) (snd x)) →
651 (∀n. monotonic ? le (h n)) →
652 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))〉〉)
653 (λp:ℕ.min_input h (fst p) (snd (snd p))).
654 #h #hcostr #hmono @(monotonic_CF … (compl_g7 h hcostr hmono)) #x
655 @le_plus [@monotonic_MSC //]
656 cases (decidable_le (fst x) (snd(snd x)))
657 [#Hle @le_times // @monotonic_sU
658 |#Hlt >(minus_to_0 … (lt_to_le … )) [// | @not_le_to_lt @Hlt]
662 definition out_aux ≝ λh.
663 out_unary ∘ λp.〈fst p,〈snd(snd p),h (S (fst p)) (snd (snd p))〉〉.
666 constructible (λx. h (fst x) (snd x)) →
667 (CF (λx. sU 〈max (fst x) (snd x),〈snd(snd x),h (S (fst x)) (snd(snd x))〉〉)
668 (λp.out (fst p) (snd (snd p)) (h (S (fst p)) (snd (snd p))))).
669 #h #hconstr @(ext_CF (out_aux h))
670 [#n normalize >fst_pair >snd_pair >fst_pair >snd_pair // ]
671 @(CF_comp … (λx.h (S (fst x)) (snd(snd x)) + MSC x) … CF_out)
673 [@(monotonic_CF … CF_fst) #x //
675 [@CF_comp_snd @(monotonic_CF … CF_snd) #x //
676 |@(ext_CF ((λx.h (fst x) (snd x))∘(λx.〈S (fst x),snd(snd x)〉)))
677 [#n normalize >fst_pair >snd_pair %]
678 @(CF_comp … MSC …hconstr)
679 [@CF_comp_pair [@CF_compS // | @CF_comp_snd // ]
680 |@O_plus @le_to_O [//|#n >fst_pair >snd_pair //]
687 |@(O_trans … (λx.MSC (max (fst x) (snd x))))
688 [%{2} %{0} #x #_ cases (surj_pair x) #a * #b #eqx >eqx
689 >fst_pair >snd_pair @(transitive_le … (MSC_pair …))
690 whd in ⊢ (??%); @le_plus
691 [@monotonic_MSC @(le_maxl … (le_n …))
692 |>commutative_times <times_n_1 @monotonic_MSC @(le_maxr … (le_n …))
694 |@le_to_O #x @(transitive_le ???? (sU_le_i … )) //
697 |@le_to_O #x @monotonic_sU [@(le_maxl … (le_n …))|//|//]
702 constructible (λx. h (fst x) (snd x)) →
703 (∀n. monotonic ? le (h n)) →
704 (∀n,a,b. a ≤ b → b ≤ n → h b n ≤ h a n) →
705 CF (λx. (S (snd x-fst x))*MSC 〈x,x〉 +
706 (snd x-fst x)*(S(snd x-fst x))*sU 〈x,〈snd x,h (S (fst x)) (snd x)〉〉)
708 #h #hconstr #hmono #hantimono
709 @(compl_g2 h ??? (compl_g3 … (compl_g71 h hconstr hmono)) (compl_g8 h hconstr))
711 [@O_plus_l @le_to_O #x >(times_n_1 (MSC x)) >commutative_times @le_times
712 [// | @monotonic_MSC // ]]
713 @(O_trans … (coroll2 ??))
714 [#n #a #b #leab #ltb >fst_pair >fst_pair >snd_pair >snd_pair
715 cut (b ≤ n) [@(transitive_le … (le_snd …)) @lt_to_le //] #lebn
717 [normalize >le_to_leb_true [//|@(transitive_le … leab lebn)]] #maxa
718 cut (max b n = n) [normalize >le_to_leb_true //] #maxb
720 [@le_plus [>big_def >big_def >maxa >maxb //]
722 [/2 by monotonic_le_minus_r/
723 |@monotonic_sU // @hantimono [@le_S_S // |@ltb]
725 |@monotonic_sU // @hantimono [@le_S_S // |@ltb]
727 |@le_to_O #n >fst_pair >snd_pair
728 cut (max (fst n) n = n) [normalize >le_to_leb_true //] #Hmax >Hmax
729 >associative_plus >distributive_times_plus
730 @le_plus [@le_times [@le_S // |>big_def >Hmax //] |//]
734 definition sg ≝ λh,x.
735 (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)〉〉.
737 lemma sg_def : ∀h,a,b.
738 sg h 〈a,b〉 = (S (b-a))*MSC 〈〈a,b〉,〈a,b〉〉 +
739 (b-a)*(S(b-a))*sU 〈〈a,b〉,〈b,h (S a) b〉〉.
740 #h #a #b whd in ⊢ (??%?); >fst_pair >snd_pair //
743 lemma compl_g11 : ∀h.
744 constructible (λx. h (fst x) (snd x)) →
745 (∀n. monotonic ? le (h n)) →
746 (∀n,a,b. a ≤ b → b ≤ n → h b n ≤ h a n) →
747 CF (sg h) (unary_g h).
748 #h #hconstr #Hm #Ham @compl_g1 @(compl_g9 h hconstr Hm Ham)
751 (**************************** closing the argument ****************************)
753 let rec h_of_aux (r:nat →nat) (c,d,b:nat) on d : nat ≝
756 | S d1 ⇒ (S d)*(MSC 〈〈b-d,b〉,〈b-d,b〉〉) +
757 d*(S d)*sU 〈〈b-d,b〉,〈b,r (h_of_aux r c d1 b)〉〉].
759 lemma h_of_aux_O: ∀r,c,b.
760 h_of_aux r c O b = c.
763 lemma h_of_aux_S : ∀r,c,d,b.
764 h_of_aux r c (S d) b =
765 (S (S d))*(MSC 〈〈b-(S d),b〉,〈b-(S d),b〉〉) +
766 (S d)*(S (S d))*sU 〈〈b-(S d),b〉,〈b,r(h_of_aux r c d b)〉〉.
769 definition h_of ≝ λr,p.
770 let m ≝ max (fst p) (snd p) in
771 h_of_aux r (MSC 〈〈m,m〉,〈m,m〉〉) (snd p - fst p) (snd p).
773 lemma h_of_O: ∀r,a,b. b ≤ a →
774 h_of r 〈a,b〉 = let m ≝ max a b in MSC 〈〈m,m〉,〈m,m〉〉.
775 #r #a #b #Hle normalize >fst_pair >snd_pair >(minus_to_0 … Hle) //
778 lemma h_of_def: ∀r,a,b.h_of r 〈a,b〉 =
780 h_of_aux r (MSC 〈〈m,m〉,〈m,m〉〉) (b - a) b.
781 #r #a #b normalize >fst_pair >snd_pair //
784 lemma mono_h_of_aux: ∀r.(∀x. x ≤ r x) → monotonic ? le r →
785 ∀d,d1,c,c1,b,b1.c ≤ c1 → d ≤ d1 → b ≤ b1 →
786 h_of_aux r c d b ≤ h_of_aux r c1 d1 b1.
787 #r #Hr #monor #d #d1 lapply d -d elim d1
788 [#d #c #c1 #b #b1 #Hc #Hd @(le_n_O_elim ? Hd) #leb
789 >h_of_aux_O >h_of_aux_O //
790 |#m #Hind #d #c #c1 #b #b1 #lec #led #leb cases (le_to_or_lt_eq … led)
791 [#ltd @(transitive_le … (Hind … lec ? leb)) [@le_S_S_to_le @ltd]
792 >h_of_aux_S @(transitive_le ???? (le_plus_n …))
793 >(times_n_1 (h_of_aux r c1 m b1)) in ⊢ (?%?);
794 >commutative_times @le_times [//|@(transitive_le … (Hr ?)) @sU_le]
795 |#Hd >Hd >h_of_aux_S >h_of_aux_S
796 cut (b-S m ≤ b1 - S m) [/2 by monotonic_le_minus_l/] #Hb1
797 @le_plus [@le_times //]
798 [@monotonic_MSC @le_pair @le_pair //
799 |@le_times [//] @monotonic_sU
800 [@le_pair // |// |@monor @Hind //]
806 lemma mono_h_of2: ∀r.(∀x. x ≤ r x) → monotonic ? le r →
807 ∀i,b,b1. b ≤ b1 → h_of r 〈i,b〉 ≤ h_of r 〈i,b1〉.
808 #r #Hr #Hmono #i #a #b #leab >h_of_def >h_of_def
809 cut (max i a ≤ max i b)
811 [@(le_maxl … (le_n …))|@(transitive_le … leab) @(le_maxr … (le_n …))]]
812 #Hmax @(mono_h_of_aux r Hr Hmono)
813 [@monotonic_MSC @le_pair @le_pair @Hmax |/2 by monotonic_le_minus_l/ |@leab]
816 axiom h_of_constr : ∀r:nat →nat.
817 (∀x. x ≤ r x) → monotonic ? le r → constructible r →
818 constructible (h_of r).
820 lemma speed_compl: ∀r:nat →nat.
821 (∀x. x ≤ r x) → monotonic ? le r → constructible r →
822 CF (h_of r) (unary_g (λi,x. r(h_of r 〈i,x〉))).
823 #r #Hr #Hmono #Hconstr @(monotonic_CF … (compl_g11 …))
824 [#x cases (surj_pair x) #a * #b #eqx >eqx
825 >sg_def cases (decidable_le b a)
826 [#leba >(minus_to_0 … leba) normalize in ⊢ (?%?);
827 <plus_n_O <plus_n_O >h_of_def
829 [normalize cases (le_to_or_lt_eq … leba)
830 [#ltba >(lt_to_leb_false … ltba) %
831 |#eqba <eqba >(le_to_leb_true … (le_n ?)) % ]]
832 #Hmax >Hmax normalize >(minus_to_0 … leba) normalize
833 @monotonic_MSC @le_pair @le_pair //
834 |#ltab >h_of_def >h_of_def
836 [normalize >(le_to_leb_true … ) [%] @lt_to_le @not_le_to_lt @ltab]
838 cut (max (S a) b = b)
839 [whd in ⊢ (??%?); >(le_to_leb_true … ) [%] @not_le_to_lt @ltab]
842 [%{(pred(b-a))} >S_pred [//] @lt_plus_to_minus_r @not_le_to_lt @ltab]
844 cut (b-S a = d) [//] #eqd1 >eqd1 >h_of_aux_S >eqd1
846 [@plus_to_minus >commutative_plus @minus_to_plus
847 [@lt_to_le @not_le_to_lt // | //]] #eqd2 >eqd2
850 |#n #a #b #leab #lebn >h_of_def >h_of_def
852 [normalize >le_to_leb_true [%|@(transitive_le … leab lebn)]] #Hmaxa
854 [normalize >(le_to_leb_true … lebn) %] #Hmaxb
855 >Hmaxa >Hmaxb @Hmono @(mono_h_of_aux r … Hr Hmono) // /2 by monotonic_le_minus_r/
856 |#n #a #b #leab @Hmono @(mono_h_of2 … Hr Hmono … leab)
857 |@(constr_comp … Hconstr Hr) @(ext_constr (h_of r))
858 [#x cases (surj_pair x) #a * #b #eqx >eqx >fst_pair >snd_pair //]
859 @(h_of_constr r Hr Hmono Hconstr)
863 lemma speed_compl_i: ∀r:nat →nat.
864 (∀x. x ≤ r x) → monotonic ? le r → constructible r →
865 ∀i. CF (λx.h_of r 〈i,x〉) (λx.g (λi,x. r(h_of r 〈i,x〉)) i x).
866 #r #Hr #Hmono #Hconstr #i
867 @(ext_CF (λx.unary_g (λi,x. r(h_of r 〈i,x〉)) 〈i,x〉))
868 [#n whd in ⊢ (??%%); @eq_f @sym_eq >fst_pair >snd_pair %]
869 @smn @(ext_CF … (speed_compl r Hr Hmono Hconstr)) #n //
872 (**************************** the speedup theorem *****************************)
873 theorem pseudo_speedup:
874 ∀r:nat →nat. (∀x. x ≤ r x) → monotonic ? le r → constructible r →
875 ∃f.∀sf. CF sf f → ∃g,sg. f ≈ g ∧ CF sg g ∧ O sf (r ∘ sg).
876 (* ∃m,a.∀n. a≤n → r(sg a) < m * sf n. *)
877 #r #Hr #Hmono #Hconstr
878 (* f is (g (λi,x. r(h_of r 〈i,x〉)) 0) *)
879 %{(g (λi,x. r(h_of r 〈i,x〉)) 0)} #sf * #H * #i *
881 (* g is (g (λi,x. r(h_of r 〈i,x〉)) (S i)) *)
882 %{(g (λi,x. r(h_of r 〈i,x〉)) (S i))}
883 (* sg is (λx.h_of r 〈i,x〉) *)
884 %{(λx. h_of r 〈S i,x〉)}
885 lapply (speed_compl_i … Hr Hmono Hconstr (S i)) #Hg
886 %[%[@condition_1 |@Hg]
887 |cases Hg #H1 * #j * #Hcodej #HCj
888 lapply (condition_2 … Hcodei) #Hcond2 (* @not_to_not *)
889 cases HCi #m * #a #Ha %{m} %{(max (S i) a)} #n #ltin @lt_to_le @not_le_to_lt
890 @(not_to_not … Hcond2) -Hcond2 #Hlesf %{n} %
891 [@(transitive_le … ltin) @(le_maxl … (le_n …))]
892 cases (Ha n ?) [2: @(transitive_le … ltin) @(le_maxr … (le_n …))]
893 #y #Uy %{y} @(monotonic_U … Uy) @(transitive_le … Hlesf) //
897 theorem pseudo_speedup':
898 ∀r:nat →nat. (∀x. x ≤ r x) → monotonic ? le r → constructible r →
899 ∃f.∀sf. CF sf f → ∃g,sg. f ≈ g ∧ CF sg g ∧
900 (* ¬ O (r ∘ sg) sf. *)
901 ∃m,a.∀n. a≤n → r(sg a) < m * sf n.
902 #r #Hr #Hmono #Hconstr
903 (* f is (g (λi,x. r(h_of r 〈i,x〉)) 0) *)
904 %{(g (λi,x. r(h_of r 〈i,x〉)) 0)} #sf * #H * #i *
906 (* g is (g (λi,x. r(h_of r 〈i,x〉)) (S i)) *)
907 %{(g (λi,x. r(h_of r 〈i,x〉)) (S i))}
908 (* sg is (λx.h_of r 〈i,x〉) *)
909 %{(λx. h_of r 〈S i,x〉)}
910 lapply (speed_compl_i … Hr Hmono Hconstr (S i)) #Hg
911 %[%[@condition_1 |@Hg]
912 |cases Hg #H1 * #j * #Hcodej #HCj
913 lapply (condition_2 … Hcodei) #Hcond2 (* @not_to_not *)
914 cases HCi #m * #a #Ha
915 %{m} %{(max (S i) a)} #n #ltin @not_le_to_lt @(not_to_not … Hcond2) -Hcond2 #Hlesf
916 %{n} % [@(transitive_le … ltin) @(le_maxl … (le_n …))]
917 cases (Ha n ?) [2: @(transitive_le … ltin) @(le_maxr … (le_n …))]
918 #y #Uy %{y} @(monotonic_U … Uy) @(transitive_le … Hlesf)
919 @Hmono @(mono_h_of2 … Hr Hmono … ltin)