include "arithmetics/log.ma".
(* include "arithmetics/ord.ma". *)
include "arithmetics/bigops.ma".
+include "arithmetics/bounded_quantifiers.ma".
+include "arithmetics/pidgeon_hole.ma".
include "basics/sets.ma".
include "basics/types.ma".
@(transitive_le … (Os1f n lean)) @le_times //
qed.
+lemma O_plus_r: ∀f,s1,s2. O s2 f → O (s1+s2) f.
+#f #s1 #s2 * #c * #a #Os1f %{c} %{a} #n #lean
+@(transitive_le … (Os1f n lean)) @le_times //
+qed.
+
lemma O_absorbl: ∀f,g,s. O s f → O f g → O s (g+f).
#f #g #s #Osf #Ofg @(O_plus … Osf) @(O_trans … Osf) //
qed.
(* this is the only classical result *)
axiom not_O_def: ∀f,g. ¬ O f g → not_O f g.
+(******************************* small O notation *****************************)
+
+(* o f g means g ∈ o(f) *)
+definition o: relation (nat→nat) ≝
+ λf,g.∀c.∃n0.∀n. n0 ≤ n → c * (g n) < f n.
+
+lemma o_irrefl: ∀s. ¬ o s s.
+#s % #oss cases (oss 1) #n0 #H @(absurd ? (le_n (s n0)))
+@lt_to_not_le >(times_n_1 (s n0)) in ⊢ (?%?); >commutative_times @H //
+qed.
+
+lemma o_trans: ∀s1,s2,s3. o s2 s1 → o s3 s2 → o s3 s1.
+#s1 #s2 #s3 #H1 #H2 #c cases (H1 c) #n1 -H1 #H1 cases (H2 1) #n2 -H2 #H2
+%{(max n1 n2)} #n #Hmax
+@(transitive_lt … (H1 ??)) [@(le_maxl … Hmax)]
+>(times_n_1 (s2 n)) in ⊢ (?%?); >commutative_times @H2 @(le_maxr … Hmax)
+qed.
+
+
(*********************************** pairing **********************************)
axiom pair: nat →nat →nat.
(************************ basic complexity notions ****************************)
(* u is the deterministic configuration relation of the universal machine (one
- step) *)
+ step)
axiom u: nat → option nat.
[cases (u i) [/2/ | #c %2 /2/ ]]
*[#H >H normalize #H1 destruct (H1) // |* #c #H >H normalize @Hind ]
]
-qed.
+qed. *)
+axiom U: nat → nat → nat → option nat.
+(*
lemma monotonici_U: ∀y,n,m,i.
U i m = Some ? y → U i (n+m) = Some ? y.
#y #n #m elim m
[cases (u i) [/2/ | #c %2 /2/ ]]
*[#H1 >H1 normalize // |* #c #H >H normalize #H1 @Hind //]
]
-qed.
+qed. *)
-lemma monotonic_U: ∀i,n,m,y.n ≤m →
- U i n = Some ? y → U i m = Some ? y.
-#i #n #m #y #lenm #H >(plus_minus_m_m m n) // @monotonici_U //
-qed.
+axiom monotonic_U: ∀i,x,n,m,y.n ≤m →
+ U i x n = Some ? y → U i x m = Some ? y.
+(* #i #n #m #y #lenm #H >(plus_minus_m_m m n) // @monotonici_U //
+qed. *)
(* axiom U: nat → nat → option nat. *)
(* axiom monotonic_U: ∀i,n,m,y.n ≤m →
U i n = Some ? y → U i m = Some ? y. *)
-lemma unique_U: ∀i,n,m,yn,ym.
- U i n = Some ? yn → U i m = Some ? ym → yn = ym.
-#i #n #m #yn #ym #Hn #Hm cases (decidable_le n m)
+lemma unique_U: ∀i,x,n,m,yn,ym.
+ U i x n = Some ? yn → U i x m = Some ? ym → yn = ym.
+#i #x #n #m #yn #ym #Hn #Hm cases (decidable_le n m)
[#lenm lapply (monotonic_U … lenm Hn) >Hm #HS destruct (HS) //
|#ltmn lapply (monotonic_U … n … Hm) [@lt_to_le @not_le_to_lt //]
>Hn #HS destruct (HS) //
qed.
definition code_for ≝ λf,i.∀x.
- ∃n.∀m. n ≤ m → U 〈i,x〉 m = f x.
+ ∃n.∀m. n ≤ m → U i x m = f x.
-definition terminate ≝ λc,r. ∃y. U c r = Some ? y.
+definition terminate ≝ λi,x,r. ∃y. U i x r = Some ? y.
+notation "[i,x] ↓ r" with precedence 60 for @{terminate $i $x $r}.
-interpretation "termination" 'fintersects c r = (terminate c r).
-
-definition lang ≝ λi,x.∃r,y. U 〈i,x〉 r = Some ? y ∧ 0 < y.
+definition lang ≝ λi,x.∃r,y. U i x r = Some ? y ∧ 0 < y.
lemma lang_cf :∀f,i,x. code_for f i →
lang i x ↔ ∃y.f x = Some ? y ∧ 0 < y.
interpretation "size" 'card n = (size n).
axiom size_of_size: ∀n. |of_size n| = n.
+axiom monotonic_size: monotonic ? le size.
+
axiom of_size_max: ∀i,n. |i| = n → i ≤ of_size n.
axiom size_fst : ∀n. |fst n| ≤ |n|.
(* C s i means that the complexity of i is O(s) *)
definition C ≝ λs,i.∃c.∃a.∀x.a ≤ |x| → ∃y.
- U 〈i,x〉 (c*(s(|x|))) = Some ? y.
+ U i x (c*(s(|x|))) = Some ? y.
definition CF ≝ λs,f.∃i.code_for f i ∧ C s i.
(* the diagonal language used for the hierarchy theorem *)
definition diag ≝ λs,i.
- U 〈fst i,i〉 (s (|i|)) = Some ? 0.
+ U (fst i) i (s (|i|)) = Some ? 0.
lemma equiv_diag: ∀s,i.
- diag s i ↔ 〈fst i, i〉 ↓ s (|i|) ∧ ¬lang (fst i) i.
+ diag s i ↔ [fst i,i] ↓ s (|i|) ∧ ¬lang (fst i) i.
#s #i %
[whd in ⊢ (%→?); #H % [%{0} //] % * #x * #y *
#H1 #Hy cut (0 = y) [@(unique_U … H H1)] #eqy /2/
it correctness *)
definition diag_cf ≝ λs,i.
- match U 〈fst i,i〉 (s (|i|)) with
+ match U (fst i) i (s (|i|)) with
[ None ⇒ None ?
| Some y ⇒ if (eqb y 0) then (Some ? 1) else (Some ? 0)].
#s #x %
[whd in ⊢ (%→?); #H %{1} % // whd in ⊢ (??%?); >H //
|* #y * whd in ⊢ (??%?→?→%);
- cases (U 〈fst x,x〉 (s (|x|))) normalize
+ cases (U (fst x) x (s (|x|))) normalize
[#H destruct
|#x cases (true_or_false (eqb x 0)) #Hx >Hx
[>(eqb_true_to_eq … Hx) //
qed.
(* axiom weak_pad : ∀a,∃a0.∀n. a0 < n → ∃b. |〈a,b〉| = n. *)
-axiom pad : ∀a,n. |a| < n → ∃b. |〈a,b〉| = n.
+lemma weak_pad1 :∀n,a.∃b. n ≤ 〈a,b〉.
+#n #a
+cut (∀i.decidable (〈a,i〉 < n))
+ [#i @decidable_le ]
+ #Hdec cases(decidable_forall (λb. 〈a,b〉 < n) Hdec n)
+ [#H cut (∀i. i < n → ∃b. b < n ∧ 〈a,b〉 = i)
+ [@(injective_to_exists … H) //]
+ #Hcut %{n} @not_lt_to_le % #Han
+ lapply(Hcut ? Han) * #x * #Hx #Hx2
+ cut (x = n) [//] #Hxn >Hxn in Hx; /2 by absurd/
+ |#H lapply(not_forall_to_exists … Hdec H)
+ * #b * #H1 #H2 %{b} @not_lt_to_le @H2
+ ]
+qed.
+
+lemma pad : ∀n,a. ∃b. n ≤ |〈a,b〉|.
+#n #a cases (weak_pad1 (of_size n) a) #b #Hb
+%{b} <(size_of_size n) @monotonic_size //
+qed.
-lemma not_included_ex: ∀s1,s2. not_O s2 s1 → ∀i. C s2 i →
- ∃b.〈i, 〈i,b〉〉 ↓ s1 (|〈i,b〉|).
+lemma o_to_ex: ∀s1,s2. o s1 s2 → ∀i. C s2 i →
+ ∃b.[i, 〈i,b〉] ↓ s1 (|〈i,b〉|).
#s1 #s2 #H #i * #c * #x0 #H1
-cases (H c (max (S(|i|)) x0)) #x1 * #H2 #H3 cases (pad i x1 ?)
- [#b #H4 %{b}
- cases (H1 〈i,b〉 ?)
- [#z >H4 #H5 %{z} @(monotonic_U … H5) @lt_to_le //
- |>H4 @(le_maxr … H2)
- ]
- |@(le_maxl … H2)
+cases (H c) #n0 #H2 cases (pad (max x0 n0) i) #b #Hmax
+%{b} cases (H1 〈i,b〉 ?)
+ [#z #H3 %{z} @(monotonic_U … H3) @lt_to_le @H2
+ @(le_maxr … Hmax)
+ |@(le_maxl … Hmax)
]
-qed.
+qed.
-lemma diag1_not_s1: ∀s1,s2. not_O s2 s1 → ¬ CF s2 (diag_cf s1).
+lemma diag1_not_s1: ∀s1,s2. o s1 s2 → ¬ CF s2 (diag_cf s1).
#s1 #s2 #H1 % * #i * #Hcode_i #Hs2_i
-cases (not_included_ex … H1 ? Hs2_i) #b #H2
+cases (o_to_ex … H1 ? Hs2_i) #b #H2
lapply (diag_spec … Hcode_i) #H3
@(absurd1 (lang i 〈i,b〉))
@(iff_trans … (H3 〈i,b〉))
qed.
(******************************************************************************)
-(* definition sumF ≝ λf,g:nat→nat.λn.f n + g n. *)
definition to_Some ≝ λf.λx:nat. Some nat (f x).
(* axiom CFU: CF sU (λx.U (fst x) (snd x)). *)
-axiom CFU: ∀h,g,f,s1,s2,s3.
+axiom CFU_new: ∀h,g,f,s.
+ CF s (to_Some h) → CF s (to_Some g) → CF s (to_Some f) →
+ O s (λx. sU (size_f h x) (size_f g x) (size_f f x)) →
+ CF s (λx.U (h x) (g x) (|f x|)).
+
+lemma CFU: ∀h,g,f,s1,s2,s3.
CF s1 (to_Some h) → CF s2 (to_Some g) → CF s3 (to_Some f) →
CF (λx. s1 x + s2 x + s3 x + sU (size_f h x) (size_f g x) (size_f f x))
- (λx.U 〈h x,g x〉 (|f x|)).
+ (λx.U (h x) (g x) (|f x|)).
+#h #g #f #s1 #s2 #s3 #Hh #Hg #Hf @CFU_new
+ [@(monotonic_CF … Hh) @O_plus_l @O_plus_l @O_plus_l //
+ |@(monotonic_CF … Hg) @O_plus_l @O_plus_l @O_plus_r //
+ |@(monotonic_CF … Hf) @O_plus_l @O_plus_r //
+ |@O_plus_r //
+ ]
+qed.
axiom monotonic_sU: ∀a1,a2,b1,b2,c1,c2. a1 ≤ a2 → b1 ≤ b2 → c1 ≤c2 →
sU a1 b1 c1 ≤ sU a2 b2 c2.
+axiom superlinear_sU: ∀i,x,r. r ≤ sU i x r.
+
definition sU_space ≝ λi,x,r.i+x+r.
definition sU_time ≝ λi,x,r.i+x+(i^2)*r*(log 2 r).
match b x with
[None ⇒ None ?
|Some n ⇒ if (eqb n 0) then f x else g x].
+
+axiom IF_CF_new: ∀b,f,g,s. CF s b → CF s f → CF s g → CF s (IF b f g).
-axiom IF_CF: ∀b,f,g,sb,sf,sg. CF sb b → CF sf f → CF sg g →
+lemma IF_CF: ∀b,f,g,sb,sf,sg. CF sb b → CF sf f → CF sg g →
CF (λn. sb n + sf n + sg n) (IF b f g).
+#b #f #g #sb #sf #sg #Hb #Hf #Hg @IF_CF_new
+ [@(monotonic_CF … Hb) @O_plus_l @O_plus_l //
+ |@(monotonic_CF … Hf) @O_plus_l @O_plus_r //
+ |@(monotonic_CF … Hg) @O_plus_r //
+ ]
+qed.
lemma diag_cf_def : ∀s.∀i.
diag_cf s i =
- IF (λi.U (pair (fst i) i) (|of_size (s (|i|))|)) (λi.Some ? 1) (λi.Some ? 0) i.
+ IF (λi.U (fst i) i (|of_size (s (|i|))|)) (λi.Some ? 1) (λi.Some ? 0) i.
#s #i normalize >size_of_size // qed.
(* and now ... *)
definition constructible ≝ λs. CF s (λx.Some ? (of_size (s (|x|)))).
+lemma diag_s: ∀s. minimal s → constructible s →
+ CF (λx.sU x x (s x)) (diag_cf s).
+#s * #Hs_id #Hs_c #Hs_constr
+cut (O (λx:ℕ.sU x x (s x)) s) [%{1} %{0} #n //]
+#O_sU_s @ext_CF [2: #n @sym_eq @diag_cf_def | skip]
+@IF_CF_new [2,3:@(monotonic_CF … (Hs_c ?)) // ]
+@CFU_new
+ [@CF_fst @(monotonic_CF … Hs_id) //
+ |@(monotonic_CF … Hs_id) //
+ |@(monotonic_CF … Hs_constr) //
+ |%{1} %{0} #n #_ >commutative_times <times_n_1
+ @monotonic_sU // >size_f_size >size_of_size //
+ ]
+qed.
+
(*
-lemma compl1: ∀s.
- CF s (to_Some fst) → CF s (to_Some (λx.x)) → CF s (to_Some (λx.(s (|x|)))) →
- CF (λx. s x + s x + s x + sU (size_f fst x) (size_f (λx.x) x) (size_f (λx.(s (|x|))) x))
- (λx.U 〈fst x,x〉 (|s (|x|)|)).
-#s #H1 #H2 #H3 @CFU //
-qed.
-
-lemma compl1: ∀s.
- CF s (to_Some fst) → CF s (to_Some (λx.x)) → CF s (to_Some (λx.(s (|x|)))) →
- CF (λx. s x + s x + s x + sU (size_f fst x) (size_f (λx.x) x) (|(s x)| ))
- (λx.U 〈fst x,x〉 (|s (|x|)|)).
-#s #H1 #H2 #H3 @monotonic_CF [3: @(CFU ??? s s s) @CFU //
-qed. *)
+lemma diag_s: ∀s. minimal s → constructible s →
+ CF (λx.s x + sU x x (s x)) (diag_cf s).
+#s * #Hs_id #Hs_c #Hs_constr
+@ext_CF [2: #n @sym_eq @diag_cf_def | skip]
+@IF_CF_new [2,3:@(monotonic_CF … (Hs_c ?)) @O_plus_l //]
+@CFU_new
+ [@CF_fst @(monotonic_CF … Hs_id) @O_plus_l //
+ |@(monotonic_CF … Hs_id) @O_plus_l //
+ |@(monotonic_CF … Hs_constr) @O_plus_l //
+ |@O_plus_r %{1} %{0} #n #_ >commutative_times <times_n_1
+ @monotonic_sU // >size_f_size >size_of_size //
+ ]
+qed. *)
+(* proof with old axioms
lemma diag_s: ∀s. minimal s → constructible s →
CF (λx.s x + sU x x (s x)) (diag_cf s).
#s * #Hs_id #Hs_c #Hs_constr
@(O_plus … (O_refl s)) //
]
qed.
+*)
-
(*************************** The hierachy theorem *****************************)
(*
--- /dev/null
+include "basics/types.ma".
+include "arithmetics/minimization.ma".
+include "arithmetics/bigops.ma".
+include "arithmetics/sigma_pi.ma".
+include "arithmetics/bounded_quantifiers.ma".
+include "reverse_complexity/big_O.ma".
+
+(************************* notation for minimization *****************************)
+notation "μ_{ ident i < n } p"
+ with precedence 80 for @{min $n 0 (λ${ident i}.$p)}.
+
+notation "μ_{ ident i ≤ n } p"
+ with precedence 80 for @{min (S $n) 0 (λ${ident i}.$p)}.
+
+notation "μ_{ ident i ∈ [a,b[ } p"
+ with precedence 80 for @{min ($b-$a) $a (λ${ident i}.$p)}.
+
+notation "μ_{ ident i ∈ [a,b] } p"
+ with precedence 80 for @{min (S $b-$a) $a (λ${ident i}.$p)}.
+
+(************************************ MAX *************************************)
+notation "Max_{ ident i < n | p } f"
+ with precedence 80
+for @{'bigop $n max 0 (λ${ident i}. $p) (λ${ident i}. $f)}.
+
+notation "Max_{ ident i < n } f"
+ with precedence 80
+for @{'bigop $n max 0 (λ${ident i}.true) (λ${ident i}. $f)}.
+
+notation "Max_{ ident j ∈ [a,b[ } f"
+ with precedence 80
+for @{'bigop ($b-$a) max 0 (λ${ident j}.((λ${ident j}.true) (${ident j}+$a)))
+ (λ${ident j}.((λ${ident j}.$f)(${ident j}+$a)))}.
+
+notation "Max_{ ident j ∈ [a,b[ | p } f"
+ with precedence 80
+for @{'bigop ($b-$a) max 0 (λ${ident j}.((λ${ident j}.$p) (${ident j}+$a)))
+ (λ${ident j}.((λ${ident j}.$f)(${ident j}+$a)))}.
+
+lemma Max_assoc: ∀a,b,c. max (max a b) c = max a (max b c).
+#a #b #c normalize cases (true_or_false (leb a b)) #leab >leab normalize
+ [cases (true_or_false (leb b c )) #lebc >lebc normalize
+ [>(le_to_leb_true a c) // @(transitive_le ? b) @leb_true_to_le //
+ |>leab //
+ ]
+ |cases (true_or_false (leb b c )) #lebc >lebc normalize //
+ >leab normalize >(not_le_to_leb_false a c) // @lt_to_not_le
+ @(transitive_lt ? b) @not_le_to_lt @leb_false_to_not_le //
+ ]
+qed.
+
+lemma Max0 : ∀n. max 0 n = n.
+// qed.
+
+lemma Max0r : ∀n. max n 0 = n.
+#n >commutative_max //
+qed.
+
+definition MaxA ≝
+ mk_Aop nat 0 max Max0 Max0r (λa,b,c.sym_eq … (Max_assoc a b c)).
+
+definition MaxAC ≝ mk_ACop nat 0 MaxA commutative_max.
+
+lemma le_Max: ∀f,p,n,a. a < n → p a = true →
+ f a ≤ Max_{i < n | p i}(f i).
+#f #p #n #a #ltan #pa
+>(bigop_diff p ? 0 MaxAC f a n) // @(le_maxl … (le_n ?))
+qed.
+
+lemma le_MaxI: ∀f,p,n,m,a. m ≤ a → a < n → p a = true →
+ f a ≤ Max_{i ∈ [m,n[ | p i}(f i).
+#f #p #n #m #a #lema #ltan #pa
+>(bigop_diff ? ? 0 MaxAC (λi.f (i+m)) (a-m) (n-m))
+ [<plus_minus_m_m // @(le_maxl … (le_n ?))
+ |<plus_minus_m_m //
+ |/2 by monotonic_lt_minus_l/
+ ]
+qed.
+
+lemma Max_le: ∀f,p,n,b.
+ (∀a.a < n → p a = true → f a ≤ b) → Max_{i < n | p i}(f i) ≤ b.
+#f #p #n elim n #b #H //
+#b0 #H1 cases (true_or_false (p b)) #Hb
+ [>bigop_Strue [2:@Hb] @to_max [@H1 // | @H #a #ltab #pa @H1 // @le_S //]
+ |>bigop_Sfalse [2:@Hb] @H #a #ltab #pa @H1 // @le_S //
+ ]
+qed.
+
+(********************************** pairing ***********************************)
+axiom pair: nat → nat → nat.
+axiom fst : nat → nat.
+axiom snd : nat → nat.
+
+interpretation "abstract pair" 'pair f g = (pair f g).
+
+axiom fst_pair: ∀a,b. fst 〈a,b〉 = a.
+axiom snd_pair: ∀a,b. snd 〈a,b〉 = b.
+axiom surj_pair: ∀x. ∃a,b. x = 〈a,b〉.
+
+axiom le_fst : ∀p. fst p ≤ p.
+axiom le_snd : ∀p. snd p ≤ p.
+axiom le_pair: ∀a,a1,b,b1. a ≤ a1 → b ≤ b1 → 〈a,b〉 ≤ 〈a1,b1〉.
+
+(************************************* U **************************************)
+axiom U: nat → nat →nat → option nat.
+
+axiom monotonic_U: ∀i,x,n,m,y.n ≤m →
+ U i x n = Some ? y → U i x m = Some ? y.
+
+lemma unique_U: ∀i,x,n,m,yn,ym.
+ U i x n = Some ? yn → U i x m = Some ? ym → yn = ym.
+#i #x #n #m #yn #ym #Hn #Hm cases (decidable_le n m)
+ [#lenm lapply (monotonic_U … lenm Hn) >Hm #HS destruct (HS) //
+ |#ltmn lapply (monotonic_U … n … Hm) [@lt_to_le @not_le_to_lt //]
+ >Hn #HS destruct (HS) //
+ ]
+qed.
+
+definition code_for ≝ λf,i.∀x.
+ ∃n.∀m. n ≤ m → U i x m = f x.
+
+definition terminate ≝ λi,x,r. ∃y. U i x r = Some ? y.
+
+notation "{i ⊙ x} ↓ r" with precedence 60 for @{terminate $i $x $r}.
+
+lemma terminate_dec: ∀i,x,n. {i ⊙ x} ↓ n ∨ ¬ {i ⊙ x} ↓ n.
+#i #x #n normalize cases (U i x n)
+ [%2 % * #y #H destruct|#y %1 %{y} //]
+qed.
+
+lemma monotonic_terminate: ∀i,x,n,m.
+ n ≤ m → {i ⊙ x} ↓ n → {i ⊙ x} ↓ m.
+#i #x #n #m #lenm * #z #H %{z} @(monotonic_U … H) //
+qed.
+
+definition termb ≝ λi,x,t.
+ match U i x t with [None ⇒ false |Some y ⇒ true].
+
+lemma termb_true_to_term: ∀i,x,t. termb i x t = true → {i ⊙ x} ↓ t.
+#i #x #t normalize cases (U i x t) normalize [#H destruct | #y #_ %{y} //]
+qed.
+
+lemma term_to_termb_true: ∀i,x,t. {i ⊙ x} ↓ t → termb i x t = true.
+#i #x #t * #y #H normalize >H //
+qed.
+
+definition out ≝ λi,x,r.
+ match U i x r with [ None ⇒ 0 | Some z ⇒ z].
+
+definition bool_to_nat: bool → nat ≝
+ λb. match b with [true ⇒ 1 | false ⇒ 0].
+
+coercion bool_to_nat.
+
+definition pU : nat → nat → nat → nat ≝ λi,x,r.〈termb i x r,out i x r〉.
+
+lemma pU_vs_U_Some : ∀i,x,r,y. pU i x r = 〈1,y〉 ↔ U i x r = Some ? y.
+#i #x #r #y % normalize
+ [cases (U i x r) normalize
+ [#H cut (0=1) [lapply (eq_f … fst … H) >fst_pair >fst_pair #H @H]
+ #H1 destruct
+ |#a #H cut (a=y) [lapply (eq_f … snd … H) >snd_pair >snd_pair #H1 @H1]
+ #H1 //
+ ]
+ |#H >H //]
+qed.
+
+lemma pU_vs_U_None : ∀i,x,r. pU i x r = 〈0,0〉 ↔ U i x r = None ?.
+#i #x #r % normalize
+ [cases (U i x r) normalize //
+ #a #H cut (1=0) [lapply (eq_f … fst … H) >fst_pair >fst_pair #H1 @H1]
+ #H1 destruct
+ |#H >H //]
+qed.
+
+(********************************* the speedup ********************************)
+
+definition min_input ≝ λh,i,x. μ_{y ∈ [S i,x] } (termb i y (h (S i) y)).
+
+lemma min_input_def : ∀h,i,x.
+ min_input h i x = min (x -i) (S i) (λy.termb i y (h (S i) y)).
+// qed.
+
+lemma min_input_i: ∀h,i,x. x ≤ i → min_input h i x = S i.
+#h #i #x #lexi >min_input_def
+cut (x - i = 0) [@sym_eq /2 by eq_minus_O/] #Hcut //
+qed.
+
+lemma min_input_to_terminate: ∀h,i,x.
+ min_input h i x = x → {i ⊙ x} ↓ (h (S i) x).
+#h #i #x #Hminx
+cases (decidable_le (S i) x) #Hix
+ [cases (true_or_false (termb i x (h (S i) x))) #Hcase
+ [@termb_true_to_term //
+ |<Hminx in Hcase; #H lapply (fmin_false (λx.termb i x (h (S i) x)) (x-i) (S i) H)
+ >min_input_def in Hminx; #Hminx >Hminx in ⊢ (%→?);
+ <plus_n_Sm <plus_minus_m_m [2: @lt_to_le //]
+ #Habs @False_ind /2/
+ ]
+ |@False_ind >min_input_i in Hminx;
+ [#eqix >eqix in Hix; * /2/ | @le_S_S_to_le @not_le_to_lt //]
+ ]
+qed.
+
+lemma min_input_to_lt: ∀h,i,x.
+ min_input h i x = x → i < x.
+#h #i #x #Hminx cases (decidable_le (S i) x) //
+#ltxi @False_ind >min_input_i in Hminx;
+ [#eqix >eqix in ltxi; * /2/ | @le_S_S_to_le @not_le_to_lt //]
+qed.
+
+lemma le_to_min_input: ∀h,i,x,x1. x ≤ x1 →
+ min_input h i x = x → min_input h i x1 = x.
+#h #i #x #x1 #lex #Hminx @(min_exists … (le_S_S … lex))
+ [@(fmin_true … (sym_eq … Hminx)) //
+ |@(min_input_to_lt … Hminx)
+ |#j #H1 <Hminx @lt_min_to_false //
+ |@plus_minus_m_m @le_S_S @(transitive_le … lex) @lt_to_le
+ @(min_input_to_lt … Hminx)
+ ]
+qed.
+
+definition g ≝ λh,u,x.
+ S (max_{i ∈[u,x[ | eqb (min_input h i x) x} (out i x (h (S i) x))).
+
+lemma g_def : ∀h,u,x. g h u x =
+ S (max_{i ∈[u,x[ | eqb (min_input h i x) x} (out i x (h (S i) x))).
+// qed.
+
+lemma le_u_to_g_1 : ∀h,u,x. x ≤ u → g h u x = 1.
+#h #u #x #lexu >g_def cut (x-u = 0) [/2 by minus_le_minus_minus_comm/]
+#eq0 >eq0 normalize // qed.
+
+lemma g_lt : ∀h,i,x. min_input h i x = x →
+ out i x (h (S i) x) < g h 0 x.
+#h #i #x #H @le_S_S @(le_MaxI … i) /2 by min_input_to_lt/
+qed.
+
+(*
+axiom ax1: ∀h,i.
+ (∃y.i < y ∧ (termb i y (h (S i) y)=true)) ∨
+ ∀y. i < y → (termb i y (h (S i) y)=false).
+
+lemma eventually_0: ∀h,u.∃nu.∀x. nu < x →
+ max_{i ∈ [0,u[ | eqb (min_input h i x) x} (out i x (h (S i) x)) = 0.
+#h #u elim u
+ [%{0} normalize //
+ |#u0 * #nu0 #Hind cases (ax1 h u0)
+ [* #x0 * #leu0x0 #Hx0 %{(max nu0 x0)}
+ #x #Hx >bigop_Sfalse
+ [>(minus_n_O u0) @Hind @(le_to_lt_to_lt … Hx) /2 by le_maxl/
+ |@not_eq_to_eqb_false % #Hf @(absurd (x ≤ x0))
+ [<Hf @true_to_le_min //
+ |@lt_to_not_le @(le_to_lt_to_lt … Hx) /2 by le_maxl/
+ ]
+ ]
+ |#H %{(max u0 nu0)} #x #Hx >bigop_Sfalse
+ [>(minus_n_O u0) @Hind @(le_to_lt_to_lt … Hx) @le_maxr //
+ |@not_eq_to_eqb_false >min_input_def
+ >(min_not_exists (λy.(termb (u0+0) y (h (S (u0+0)) y))))
+ [<plus_n_O <plus_n_Sm <plus_minus_m_m
+ [% #H1 /2/
+ |@lt_to_le @(le_to_lt_to_lt … Hx) @le_maxl //
+ ]
+ |/2 by /
+ ]
+ ]
+ ]
+ ]
+qed.
+
+definition almost_equal ≝ λf,g:nat → nat. ∃nu.∀x. nu < x → f x = g x.
+
+definition almost_equal1 ≝ λf,g:nat → nat. ¬ ∀nu.∃x. nu < x ∧ f x ≠ g x.
+
+interpretation "almost equal" 'napart f g = (almost_equal f g).
+
+lemma condition_1: ∀h,u.g h 0 ≈ g h u.
+#h #u cases (eventually_0 h u) #nu #H %{(max u nu)} #x #Hx @(eq_f ?? S)
+>(bigop_sumI 0 u x (λi:ℕ.eqb (min_input h i x) x) nat 0 MaxA)
+ [>H // @(le_to_lt_to_lt …Hx) /2 by le_maxl/
+ |@lt_to_le @(le_to_lt_to_lt …Hx) /2 by le_maxr/
+ |//
+ ]
+qed. *)
+
+lemma max_neq0 : ∀a,b. max a b ≠ 0 → a ≠ 0 ∨ b ≠ 0.
+#a #b whd in match (max a b); cases (true_or_false (leb a b)) #Hcase >Hcase
+ [#H %2 @H | #H %1 @H]
+qed.
+
+definition almost_equal ≝ λf,g:nat → nat. ¬ ∀nu.∃x. nu < x ∧ f x ≠ g x.
+interpretation "almost equal" 'napart f g = (almost_equal f g).
+
+lemma eventually_cancelled: ∀h,u.¬∀nu.∃x. nu < x ∧
+ max_{i ∈ [0,u[ | eqb (min_input h i x) x} (out i x (h (S i) x)) ≠ 0.
+#h #u elim u
+ [normalize % #H cases (H u) #x * #_ * #H1 @H1 //
+ |#u0 @not_to_not #Hind #nu cases (Hind nu) #x * #ltx
+ cases (true_or_false (eqb (min_input h (u0+O) x) x)) #Hcase
+ [>bigop_Strue [2:@Hcase] #Hmax cases (max_neq0 … Hmax) -Hmax
+ [2: #H %{x} % // <minus_n_O @H]
+ #Hneq0 (* if x is not enough we retry with nu=x *)
+ cases (Hind x) #x1 * #ltx1
+ >bigop_Sfalse
+ [#H %{x1} % [@transitive_lt //| <minus_n_O @H]
+ |@not_eq_to_eqb_false >(le_to_min_input … (eqb_true_to_eq … Hcase))
+ [@lt_to_not_eq @ltx1 | @lt_to_le @ltx1]
+ ]
+ |>bigop_Sfalse [2:@Hcase] #H %{x} % // <minus_n_O @H
+ ]
+ ]
+qed.
+
+
+lemma condition_1_weak: ∀h,u.g h 0 ≈ g h u.
+#h #u @(not_to_not … (eventually_cancelled h u))
+#H #nu cases (H (max u nu)) #x * #ltx #Hdiff
+%{x} % [@(le_to_lt_to_lt … ltx) @(le_maxr … (le_n …))] @(not_to_not … Hdiff)
+#H @(eq_f ?? S) >(bigop_sumI 0 u x (λi:ℕ.eqb (min_input h i x) x) nat 0 MaxA)
+ [>H // |@lt_to_le @(le_to_lt_to_lt …ltx) /2 by le_maxr/ |//]
+qed.
+
+(******************************** Condition 2 *********************************)
+definition total ≝ λf.λx:nat. Some nat (f x).
+
+lemma exists_to_exists_min: ∀h,i. (∃x. i < x ∧ {i ⊙ x} ↓ h (S i) x) → ∃y. min_input h i y = y.
+#h #i * #x * #ltix #Hx %{(min_input h i x)} @min_spec_to_min @found //
+ [@(f_min_true (λy:ℕ.termb i y (h (S i) y))) %{x} % [% // | @term_to_termb_true //]
+ |#y #leiy #lty @(lt_min_to_false ????? lty) //
+ ]
+qed.
+
+lemma condition_2: ∀h,i. code_for (total (g h 0)) i → ¬∃x. i<x ∧ {i ⊙ x} ↓ h (S i) x.
+#h #i whd in ⊢(%→?); #H % #H1 cases (exists_to_exists_min … H1) #y #Hminy
+lapply (g_lt … Hminy)
+lapply (min_input_to_terminate … Hminy) * #r #termy
+cases (H y) -H #ny #Hy
+cut (r = g h 0 y) [@(unique_U … ny … termy) @Hy //] #Hr
+whd in match (out ???); >termy >Hr
+#H @(absurd ? H) @le_to_not_lt @le_n
+qed.
+
+
+(********************** complexity ***********************)
+
+(* We assume operations have a minimal structural complexity MSC.
+For instance, for time complexity, MSC is equal to the size of input.
+For space complexity, MSC is typically 0, since we only measure the
+space required in addition to dimension of the input. *)
+
+axiom MSC : nat → nat.
+axiom MSC_le: ∀n. MSC n ≤ n.
+axiom monotonic_MSC: monotonic ? le MSC.
+
+(* C s i means i is running in O(s) *)
+
+definition C ≝ λs,i.∃c.∃a.∀x.a ≤ x → ∃y.
+ U i x (c*(s x)) = Some ? y.
+
+(* C f s means f ∈ O(s) where MSC ∈O(s) *)
+definition CF ≝ λs,f.O s MSC ∧ ∃i.code_for (total f) i ∧ C s i.
+
+lemma ext_CF : ∀f,g,s. (∀n. f n = g n) → CF s f → CF s g.
+#f #g #s #Hext * #HO * #i * #Hcode #HC % // %{i} %
+ [#x cases (Hcode x) #a #H %{a} whd in match (total ??); <Hext @H | //]
+qed.
+
+(* lemma ext_CF_total : ∀f,g,s. (∀n. f n = g n) → CF s (total f) → CF s (total g).
+#f #g #s #Hext * #HO * #i * #Hcode #HC % // %{i} % [2:@HC]
+#x cases (Hcode x) #a #H %{a} #m #leam >(H m leam) normalize @eq_f @Hext
+qed. *)
+
+lemma monotonic_CF: ∀s1,s2,f.(∀x. s1 x ≤ s2 x) → CF s1 f → CF s2 f.
+#s1 #s2 #f #H * #HO * #i * #Hcode #Hs1 %
+ [cases HO #c * #a -HO #HO %{c} %{a} #n #lean @(transitive_le … (HO n lean))
+ @le_times //
+ |%{i} % [//] cases Hs1 #c * #a -Hs1 #Hs1 %{c} %{a} #n #lean
+ cases(Hs1 n lean) #y #Hy %{y} @(monotonic_U …Hy) @le_times //
+ ]
+qed.
+
+lemma O_to_CF: ∀s1,s2,f.O s2 s1 → CF s1 f → CF s2 f.
+#s1 #s2 #f #H * #HO * #i * #Hcode #Hs1 %
+ [@(O_trans … H) //
+ |%{i} % [//] cases Hs1 #c * #a -Hs1 #Hs1
+ cases H #c1 * #a1 #Ha1 %{(c*c1)} %{(a+a1)} #n #lean
+ cases(Hs1 n ?) [2:@(transitive_le … lean) //] #y #Hy %{y} @(monotonic_U …Hy)
+ >associative_times @le_times // @Ha1 @(transitive_le … lean) //
+ ]
+qed.
+
+lemma timesc_CF: ∀s,f,c.CF (λx.c*s x) f → CF s f.
+#s #f #c @O_to_CF @O_times_c
+qed.
+
+(********************************* composition ********************************)
+axiom CF_comp: ∀f,g,sf,sg,sh. CF sg g → CF sf f →
+ O sh (λx. sg x + sf (g x)) → CF sh (f ∘ g).
+
+lemma CF_comp_ext: ∀f,g,h,sh,sf,sg. CF sg g → CF sf f →
+ (∀x.f(g x) = h x) → O sh (λx. sg x + sf (g x)) → CF sh h.
+#f #g #h #sh #sf #sg #Hg #Hf #Heq #H @(ext_CF (f ∘ g))
+ [#n normalize @Heq | @(CF_comp … H) //]
+qed.
+
+(*
+lemma CF_comp1: ∀f,g,s. CF s (total g) → CF s (total f) →
+ CF s (total (f ∘ g)).
+#f #g #s #Hg #Hf @(timesc_CF … 2) @(monotonic_CF … (CF_comp … Hg Hf))
+*)
+
+(*
+axiom CF_comp_ext2: ∀f,g,h,sf,sh. CF sh (total g) → CF sf (total f) →
+ (∀x.f(g x) = h x) →
+ (∀x. sf (g x) ≤ sh x) → CF sh (total h). *)
+
+(* axiom main_MSC: ∀h,f. CF h f → O h (λx.MSC (f x)).
+
+axiom CF_S: CF MSC (total S).
+axiom CF_fst: CF MSC (total fst).
+axiom CF_snd: CF MSC (total snd).
+
+lemma CF_compS: ∀h,f. CF h (total f) → CF h (total (S ∘ f)).
+#h #f #Hf @(CF_comp … Hf CF_S) @O_plus // @main_MSC //
+qed.
+
+lemma CF_comp_fst: ∀h,f. CF h (total f) → CF h (total (fst ∘ f)).
+#h #f #Hf @(CF_comp … Hf CF_fst) @O_plus // @main_MSC //
+qed.
+
+lemma CF_comp_snd: ∀h,f. CF h (total f) → CF h (total (snd ∘ f)).
+#h #f #Hf @(CF_comp … Hf CF_snd) @O_plus // @main_MSC //
+qed. *)
+
+axiom CF_compS: ∀h,f. CF h f → CF h (S ∘ f).
+axiom CF_comp_fst: ∀h,f. CF h f → CF h (fst ∘ f).
+axiom CF_comp_snd: ∀h,f. CF h f → CF h (snd ∘ f).
+
+(************************************** eqb ***********************************)
+(* definition btotal ≝
+ λf.λx:nat. match f x with [true ⇒ Some ? 0 |false ⇒ Some ? 1]. *)
+
+axiom CF_eqb: ∀h,f,g.
+ CF h f → CF h g → CF h (λx.eqb (f x) (g x)).
+
+(*
+axiom eqb_compl2: ∀h,f,g.
+ CF2 h (total2 f) → CF2 h (total2 g) →
+ CF2 h (btotal2 (λx1,x2.eqb (f x1 x2) (g x1 x2))).
+
+axiom eqb_min_input_compl:∀h,x.
+ CF (λi.∑_{y ∈ [S i,S x[ }(h i y))
+ (btotal (λi.eqb (min_input h i x) x)). *)
+(*********************************** maximum **********************************)
+
+axiom CF_max: ∀a,b.∀p:nat →bool.∀f,ha,hb,hp,hf,s.
+ CF ha a → CF hb b → CF hp p → CF hf f →
+ O s (λx.ha x + hb x + ∑_{i ∈[a x ,b x[ }(hp 〈i,x〉 + hf 〈i,x〉)) →
+ CF s (λx.max_{i ∈[a x,b x[ | p 〈i,x〉 }(f 〈i,x〉)).
+
+(******************************** minimization ********************************)
+
+axiom CF_mu: ∀a,b.∀f:nat →bool.∀sa,sb,sf,s.
+ CF sa a → CF sb b → CF sf f →
+ O s (λx.sa x + sb x + ∑_{i ∈[a x ,S(b x)[ }(sf 〈i,x〉)) →
+ CF s (λx.μ_{i ∈[a x,b x] }(f 〈i,x〉)).
+
+(********************************* simulation *********************************)
+
+axiom sU : nat → nat.
+definition termb_unary ≝ λx:ℕ.termb (fst x) (fst (snd x)) (snd (snd x)).
+
+axiom monotonic_sU: ∀i1,i2,x1,x2,s1,s2. i1 ≤ i2 → x1 ≤ x2 → s1 ≤ s2 →
+ sU 〈i1,〈x1,s1〉〉 ≤ sU 〈i2,〈x2,s2〉〉.
+
+axiom sU_le: ∀i,x,s. s ≤ sU 〈i,〈x,s〉〉.
+
+lemma monotonic_sU_aux : ∀x1,x2. fst x1 ≤ fst x2 → fst (snd x1) ≤ fst (snd x2) →
+snd (snd x1) ≤ snd (snd x2) → sU x1 ≤ sU x2.
+#x1 #x2 cases (surj_pair x1) #a1 * #y #eqx1 >eqx1 -eqx1 cases (surj_pair y)
+#b1 * #c1 #eqy >eqy -eqy
+cases (surj_pair x2) #a2 * #y2 #eqx2 >eqx2 -eqx2 cases (surj_pair y2)
+#b2 * #c2 #eqy2 >eqy2 -eqy2 >fst_pair >snd_pair >fst_pair >snd_pair
+>fst_pair >snd_pair >fst_pair >snd_pair @monotonic_sU
+qed.
+
+axiom CF_termb: CF sU (btotal (termb_unary)).
+
+axiom CF_compb: ∀f,g,sf,sg,sh. CF sg (total g) → CF sf (btotal f) →
+ O sh (λx. sg x + sf (g x)) → CF sh (btotal (f ∘ g)).
+
+(*
+lemma CF_termb_comp: ∀f.CF (sU ∘ f) (btotal (termb_unary ∘ f)).
+#f @(CF_compb … CF_termb) *)
+
+(******************** complexity of g ********************)
+
+definition unary_g ≝ λh.λux. g h (fst ux) (snd ux).
+definition auxg ≝
+ λh,ux. max_{i ∈[fst ux,snd ux[ | eqb (min_input h i (snd ux)) (snd ux)}
+ (out i (snd ux) (h (S i) (snd ux))).
+
+lemma compl_g1 : ∀h,s.
+ CF s (total (auxg h)) → CF s (total (unary_g h)).
+#h #s #H1 @(CF_compS ? (auxg h) H1)
+qed.
+
+definition aux1g ≝
+ λh,ux. max_{i ∈[fst ux,snd ux[ | (λp. eqb (min_input h (fst p) (snd (snd p))) (snd (snd p))) 〈i,ux〉}
+ ((λp.out (fst p) (snd (snd p)) (h (S (fst p)) (snd (snd p)))) 〈i,ux〉).
+
+lemma eq_aux : ∀h,x.aux1g h x = auxg h x.
+#h #x @same_bigop
+ [#n #_ >fst_pair >snd_pair // |#n #_ #_ >fst_pair >snd_pair //]
+qed.
+
+lemma compl_g2 : ∀h,s1,s2,s.
+ CF s1
+ (btotal (λp:ℕ.eqb (min_input h (fst p) (snd (snd p))) (snd (snd p)))) →
+ CF s2
+ (total (λp:ℕ.out (fst p) (snd (snd p)) (h (S (fst p)) (snd (snd p))))) →
+ O s (λx.MSC x + ∑_{i ∈[fst x ,snd x[ }(s1 〈i,x〉+s2 〈i,x〉)) →
+ CF s (total (auxg h)).
+#h #s1 #s2 #s #Hs1 #Hs2 #HO @(ext_CF (total (aux1g h)))
+ [#n whd in ⊢ (??%%); @eq_f @eq_aux]
+@(CF_max … CF_fst CF_snd Hs1 Hs2 …) @(O_trans … HO)
+@O_plus [@O_plus @O_plus_l // | @O_plus_r //]
+qed.
+
+lemma compl_g3 : ∀h,s.
+ CF s (total (λp:ℕ.min_input h (fst p) (snd (snd p)))) →
+ CF s (btotal (λp:ℕ.eqb (min_input h (fst p) (snd (snd p))) (snd (snd p)))).
+#h #s #H @(CF_eqb … H) @(CF_comp … CF_snd CF_snd) @(O_trans … (proj1 … H))
+@O_plus // %{1} %{0} #n #_ >commutative_times <times_n_1 @monotonic_MSC //
+qed.
+
+definition min_input_aux ≝ λh,p.
+ μ_{y ∈ [S (fst p),snd (snd p)] }
+ ((λx.termb (fst (snd x)) (fst x) (h (S (fst (snd x))) (fst x))) 〈y,p〉).
+
+lemma min_input_eq : ∀h,p.
+ min_input_aux h p =
+ min_input h (fst p) (snd (snd p)).
+#h #p >min_input_def whd in ⊢ (??%?); >minus_S_S @min_f_g #i #_ #_
+whd in ⊢ (??%%); >fst_pair >snd_pair //
+qed.
+
+definition termb_aux ≝ λh.
+ termb_unary ∘ λp.〈fst (snd p),〈fst p,h (S (fst (snd p))) (fst p)〉〉.
+
+(*
+lemma termb_aux : ∀h,p.
+ (λx:ℕ.termb (fst x) (fst (snd x)) (snd (snd x)))
+ 〈fst (snd p),〈fst p,h (S (fst (snd p))) (fst p)〉〉 =
+ termb (fst (snd p)) (fst p) (h (S (fst (snd p))) (fst p)) .
+#h #p normalize >fst_pair >snd_pair >fst_pair >snd_pair //
+qed. *)
+
+lemma compl_g4 : ∀h,s1,s.
+ (CF s1
+ (btotal
+ (λp.termb (fst (snd p)) (fst p) (h (S (fst (snd p))) (fst p))))) →
+ (O s (λx.MSC x + ∑_{i ∈[S(fst x) ,S(snd (snd x))[ }(s1 〈i,x〉))) →
+ CF s (total (λp:ℕ.min_input h (fst p) (snd (snd p)))).
+#h #s1 #s #Hs1 #HO @(ext_CF (total (min_input_aux h)))
+ [#n whd in ⊢ (??%%); @eq_f @min_input_eq]
+@(CF_mu … MSC MSC … Hs1)
+ [@CF_compS @CF_fst
+ |@CF_comp_snd @CF_snd
+ |@(O_trans … HO) @O_plus [@O_plus @O_plus_l // | @O_plus_r //]
+(* @(ext_CF (btotal (termb_aux h)))
+ [#n normalize >fst_pair >snd_pair >fst_pair >snd_pair // ]
+@(CF_compb … CF_termb) *)
+qed.
+
+(************************* a couple of technical lemmas ***********************)
+lemma minus_to_0: ∀a,b. a ≤ b → minus a b = 0.
+#a elim a // #n #Hind *
+ [#H @False_ind /2 by absurd/ | #b normalize #H @Hind @le_S_S_to_le /2/]
+qed.
+
+lemma sigma_bound: ∀h,a,b. monotonic nat le h →
+ ∑_{i ∈ [a,S b[ }(h i) ≤ (S b-a)*h b.
+#h #a #b #H cases (decidable_le a b)
+ [#leab cut (b = pred (S b - a + a))
+ [<plus_minus_m_m // @le_S //] #Hb >Hb in match (h b);
+ generalize in match (S b -a);
+ #n elim n
+ [//
+ |#m #Hind >bigop_Strue [2://] @le_plus
+ [@H @le_n |@(transitive_le … Hind) @le_times [//] @H //]
+ ]
+ |#ltba lapply (not_le_to_lt … ltba) -ltba #ltba
+ cut (S b -a = 0) [@minus_to_0 //] #Hcut >Hcut //
+ ]
+qed.
+
+lemma sigma_bound_decr: ∀h,a,b. (∀a1,a2. a1 ≤ a2 → a2 < b → h a2 ≤ h a1) →
+ ∑_{i ∈ [a,b[ }(h i) ≤ (b-a)*h a.
+#h #a #b #H cases (decidable_le a b)
+ [#leab cut ((b -a) +a ≤ b) [/2 by le_minus_to_plus_r/] generalize in match (b -a);
+ #n elim n
+ [//
+ |#m #Hind >bigop_Strue [2://] #Hm
+ cut (m+a ≤ b) [@(transitive_le … Hm) //] #Hm1
+ @le_plus [@H // |@(transitive_le … (Hind Hm1)) //]
+ ]
+ |#ltba lapply (not_le_to_lt … ltba) -ltba #ltba
+ cut (b -a = 0) [@minus_to_0 @lt_to_le @ltba] #Hcut >Hcut //
+ ]
+qed.
+
+
+lemma coroll: ∀s1:nat→nat. (∀n. monotonic ℕ le (λa:ℕ.s1 〈a,n〉)) →
+O (λx.(snd (snd x)-fst x)*(s1 〈snd (snd x),x〉))
+ (λx.∑_{i ∈[S(fst x) ,S(snd (snd x))[ }(s1 〈i,x〉)).
+#s1 #Hs1 %{1} %{0} #n #_ >commutative_times <times_n_1
+@(transitive_le … (sigma_bound …)) [@Hs1|>minus_S_S //]
+qed.
+
+lemma coroll2: ∀s1:nat→nat. (∀n,a,b. a ≤ b → b < snd n → s1 〈b,n〉 ≤ s1 〈a,n〉) →
+O (λx.(snd x - fst x)*s1 〈fst x,x〉) (λx.∑_{i ∈[fst x,snd x[ }(s1 〈i,x〉)).
+#s1 #Hs1 %{1} %{0} #n #_ >commutative_times <times_n_1
+@(transitive_le … (sigma_bound_decr …)) [2://] @Hs1
+qed.
+
+lemma compl_g5 : ∀h,s1.(∀n. monotonic ℕ le (λa:ℕ.s1 〈a,n〉)) →
+ (CF s1
+ (btotal
+ (λp.termb (fst (snd p)) (fst p) (h (S (fst (snd p))) (fst p))))) →
+ CF (λx.MSC x + (snd (snd x)-fst x)*s1 〈snd (snd x),x〉)
+ (total (λp:ℕ.min_input h (fst p) (snd (snd p)))).
+#h #s1 #Hmono #Hs1 @(compl_g4 … Hs1) @O_plus
+[@O_plus_l // |@O_plus_r @coroll @Hmono]
+qed.
+
+axiom compl_g6: ∀h.
+ (CF (λx. sU 〈fst (snd x),〈fst x,h (S (fst (snd x))) (fst x)〉〉)
+ (btotal
+ (λp.termb (fst (snd p)) (fst p) (h (S (fst (snd p))) (fst p))))).
+(* #h #s1 @(ext_CF (btotal (termb_aux h)))
+ [#n normalize >fst_pair >snd_pair >fst_pair >snd_pair // ]
+@(CF_compb … CF_termb) *)
+
+definition faux1 ≝ λh.
+ (λx.MSC x + (snd (snd x)-fst x)*(λx.sU 〈fst (snd x),〈fst x,h (S (fst (snd x))) (fst x)〉〉) 〈snd (snd x),x〉).
+
+(* complexity of min_input *)
+lemma compl_g7: ∀h. (∀n. monotonic ? le (h n)) →
+ CF (λx.MSC x + (snd (snd x)-fst x)*sU 〈fst x,〈snd (snd x),h (S (fst x)) (snd (snd x))〉〉)
+ (total (λp:ℕ.min_input h (fst p) (snd (snd p)))).
+#h #hmono @(monotonic_CF … (faux1 h))
+ [#n normalize >fst_pair >snd_pair //]
+@compl_g5 [2:@compl_g6] #n #x #y #lexy >fst_pair >snd_pair
+>fst_pair >snd_pair @monotonic_sU // @hmono @lexy
+qed.
+
+definition big : nat →nat ≝ λx.
+ let m ≝ max (fst x) (snd x) in 〈m,m〉.
+
+lemma big_def : ∀a,b. big 〈a,b〉 = 〈max a b,max a b〉.
+#a #b normalize >fst_pair >snd_pair // qed.
+
+lemma le_big : ∀x. x ≤ big x.
+#x cases (surj_pair x) #a * #b #eqx >eqx @le_pair >fst_pair >snd_pair
+[@(le_maxl … (le_n …)) | @(le_maxr … (le_n …))]
+qed.
+
+(* proviamo con x *)
+lemma compl_g71: ∀h. (∀n. monotonic ? le (h n)) →
+ 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))〉〉)
+ (total (λp:ℕ.min_input h (fst p) (snd (snd p)))).
+#h #hmono @(monotonic_CF … (compl_g7 h hmono)) #x
+@le_plus [@monotonic_MSC //]
+cases (decidable_le (fst x) (snd(snd x)))
+ [#Hle @le_times // @monotonic_sU // @(le_maxl … (le_n … ))
+ |#Hlt >(minus_to_0 … (lt_to_le … )) [// | @not_le_to_lt @Hlt]
+ ]
+qed.
+
+axiom compl_g8: ∀h.
+ CF (λx. sU 〈fst x,〈snd (snd x),h (S (fst x)) (snd (snd x))〉〉)
+ (total (λp:ℕ.out (fst p) (snd (snd p)) (h (S (fst p)) (snd (snd p))))).
+
+lemma compl_g81: ∀h.
+ CF (λx. sU 〈max (fst x) (snd x),〈snd (snd x),h (S (fst x)) (snd (snd x))〉〉)
+ (total (λp:ℕ.out (fst p) (snd (snd p)) (h (S (fst p)) (snd (snd p))))).
+#h @(monotonic_CF … (compl_g8 h)) #x @monotonic_sU // @(le_maxl … (le_n … ))
+qed.
+
+axiom daemon : False.
+
+lemma compl_g9 : ∀h.
+ (∀n. monotonic ? le (h n)) →
+ (∀n,a,b. a ≤ b → b ≤ n → h b n ≤ h a n) →
+ CF (λx. (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)〉〉)
+ (total (auxg h)).
+#h #hmono #hantimono @(compl_g2 h ??? (compl_g3 … (compl_g71 h hmono)) (compl_g81 h))
+@O_plus [@O_plus_l @le_to_O #x elim daemon]
+@(O_trans … (coroll2 ??))
+ [#n #a #b #leab #ltb >fst_pair >fst_pair >snd_pair >snd_pair
+ cut (max a n = n)
+ [normalize >le_to_leb_true [//|elim daemon (*@(transitive_le … leab lebn)*)]] #maxa
+ cut (max b n = n) [elim daemon (*normalize >le_to_leb_true //*)] #maxb
+ @le_plus
+ [@le_plus [>big_def >big_def >maxa >maxb //]
+ @le_times
+ [/2 by monotonic_le_minus_r/
+ |@monotonic_sU // @hantimono [@le_S_S // |@ltb]
+ ]
+ |@monotonic_sU // @hantimono [@le_S_S // |@ltb]
+ ]
+ |@le_to_O #n >fst_pair >snd_pair
+ cut (max (fst n) n = n) [normalize >le_to_leb_true //] #Hmax >Hmax
+ >associative_plus >distributive_times_plus
+ @le_plus [@le_times [@le_S // |>big_def >Hmax //] |//]
+ ]
+qed.
+
+definition sg ≝ λh,x.
+ (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)〉〉.
+
+lemma sg_def : ∀h,a,b.
+ sg h 〈a,b〉 = (S (b-a))*MSC 〈〈a,b〉,〈a,b〉〉 +
+ (b-a)*(S(b-a))*sU 〈〈a,b〉,〈b,h (S a) b〉〉.
+#h #a #b whd in ⊢ (??%?); >fst_pair >snd_pair //
+qed.
+
+lemma compl_g11 : ∀h.
+ (∀n. monotonic ? le (h n)) →
+ (∀n,a,b. a ≤ b → b ≤ n → h b n ≤ h a n) →
+ CF (sg h) (total (unary_g h)).
+#h #Hm #Ham @compl_g1 @(compl_g9 h Hm Ham)
+qed.
+
+(**************************** closing the argument ****************************)
+
+let rec h_of_aux (r:nat →nat) (c,d,b:nat) on d : nat ≝
+ match d with
+ [ O ⇒ c (* MSC 〈〈b,b〉,〈b,b〉〉 *)
+ | S d1 ⇒ (S d)*(MSC 〈〈b-d,b〉,〈b-d,b〉〉) +
+ d*(S d)*sU 〈〈b-d,b〉,〈b,r (h_of_aux r c d1 b)〉〉].
+
+lemma h_of_aux_O: ∀r,c,b.
+ h_of_aux r c O b = c (* MSC 〈〈b,b〉,〈b,b〉〉*) .
+// qed.
+
+lemma h_of_aux_S : ∀r,c,d,b.
+ h_of_aux r c (S d) b =
+ (S (S d))*(MSC 〈〈b-(S d),b〉,〈b-(S d),b〉〉) +
+ (S d)*(S (S d))*sU 〈〈b-(S d),b〉,〈b,r(h_of_aux r c d b)〉〉.
+// qed.
+
+definition h_of ≝ λr,p.
+ let m ≝ max (fst p) (snd p) in
+ h_of_aux r (MSC 〈〈m,m〉,〈m,m〉〉) (snd p - fst p) (snd p).
+
+lemma h_of_O: ∀r,a,b. b ≤ a →
+ h_of r 〈a,b〉 = let m ≝ max a b in MSC 〈〈m,m〉,〈m,m〉〉.
+#r #a #b #Hle normalize >fst_pair >snd_pair >(minus_to_0 … Hle) //
+qed.
+
+lemma h_of_def: ∀r,a,b.h_of r 〈a,b〉 =
+ let m ≝ max a b in
+ h_of_aux r (MSC 〈〈m,m〉,〈m,m〉〉) (b - a) b.
+#r #a #b normalize >fst_pair >snd_pair //
+qed.
+
+lemma mono_h_of_aux: ∀r.(∀x. x ≤ r x) → monotonic ? le r →
+ ∀d,d1,c,c1,b,b1.c ≤ c1 → d ≤ d1 → b ≤ b1 →
+ h_of_aux r c d b ≤ h_of_aux r c1 d1 b1.
+#r #Hr #monor #d #d1 lapply d -d elim d1
+ [#d #c #c1 #b #b1 #Hc #Hd @(le_n_O_elim ? Hd) #leb
+ >h_of_aux_O >h_of_aux_O //
+ |#m #Hind #d #c #c1 #b #b1 #lec #led #leb cases (le_to_or_lt_eq … led)
+ [#ltd @(transitive_le … (Hind … lec ? leb)) [@le_S_S_to_le @ltd]
+ >h_of_aux_S @(transitive_le ???? (le_plus_n …))
+ >(times_n_1 (h_of_aux r c1 m b1)) in ⊢ (?%?);
+ >commutative_times @le_times [//|@(transitive_le … (Hr ?)) @sU_le]
+ |#Hd >Hd >h_of_aux_S >h_of_aux_S
+ cut (b-S m ≤ b1 - S m) [/2 by monotonic_le_minus_l/] #Hb1
+ @le_plus [@le_times //]
+ [@monotonic_MSC @le_pair @le_pair //
+ |@le_times [//] @monotonic_sU
+ [@le_pair // |// |@monor @Hind //]
+ ]
+ ]
+ ]
+qed.
+
+lemma mono_h_of2: ∀r.(∀x. x ≤ r x) → monotonic ? le r →
+ ∀i,b,b1. b ≤ b1 → h_of r 〈i,b〉 ≤ h_of r 〈i,b1〉.
+#r #Hr #Hmono #i #a #b #leab >h_of_def >h_of_def
+cut (max i a ≤ max i b)
+ [@to_max
+ [@(le_maxl … (le_n …))|@(transitive_le … leab) @(le_maxr … (le_n …))]]
+#Hmax @(mono_h_of_aux r Hr Hmono)
+ [@monotonic_MSC @le_pair @le_pair @Hmax |/2 by monotonic_le_minus_l/ |@leab]
+qed.
+
+lemma speed_compl: ∀r:nat →nat.
+ (∀x. x ≤ r x) → monotonic ? le r →
+ CF (h_of r) (total (unary_g (λi,x. r(h_of r 〈i,x〉)))).
+#r #Hr #Hmono @(monotonic_CF … (compl_g11 …))
+ [#x cases (surj_pair x) #a * #b #eqx >eqx
+ >sg_def cases (decidable_le b a)
+ [#leba >(minus_to_0 … leba) normalize in ⊢ (?%?);
+ <plus_n_O <plus_n_O >h_of_def
+ cut (max a b = a)
+ [normalize cases (le_to_or_lt_eq … leba)
+ [#ltba >(lt_to_leb_false … ltba) %
+ |#eqba <eqba >(le_to_leb_true … (le_n ?)) % ]]
+ #Hmax >Hmax normalize >(minus_to_0 … leba) normalize
+ @monotonic_MSC @le_pair @le_pair //
+ |#ltab >h_of_def >h_of_def
+ cut (max a b = b)
+ [normalize >(le_to_leb_true … ) [%] @lt_to_le @not_le_to_lt @ltab]
+ #Hmax >Hmax
+ cut (max (S a) b = b)
+ [whd in ⊢ (??%?); >(le_to_leb_true … ) [%] @not_le_to_lt @ltab]
+ #Hmax1 >Hmax1
+ cut (∃d.b - a = S d) [elim daemon] * #d #eqd >eqd
+ cut (b-S a = d) [//] #eqd1 >eqd1 >h_of_aux_S >eqd1
+ cut (b - S d = a)
+ [@plus_to_minus >commutative_plus @minus_to_plus
+ [@lt_to_le @not_le_to_lt // | //]] #eqd2 >eqd2
+ normalize //
+ ]
+ |#n #a #b #leab #lebn >h_of_def >h_of_def
+ cut (max a n = n)
+ [normalize >le_to_leb_true [%|@(transitive_le … leab lebn)]] #Hmaxa
+ cut (max b n = n)
+ [normalize >(le_to_leb_true … lebn) %] #Hmaxb
+ >Hmaxa >Hmaxb @Hmono @(mono_h_of_aux r … Hr Hmono) // /2 by monotonic_le_minus_r/
+ |#n #a #b #leab @Hmono @(mono_h_of2 … Hr Hmono … leab)
+ ]
+qed.
+
+lemma unary_g_def : ∀h,i,x. g h i x = unary_g h 〈i,x〉.
+#h #i #x whd in ⊢ (???%); >fst_pair >snd_pair %
+qed.
+
+(* smn *)
+axiom smn: ∀f,s. CF s f → ∀x. CF (λy.s 〈x,y〉) (λy.f 〈x,y〉).
+
+lemma speed_compl_i: ∀r:nat →nat.
+ (∀x. x ≤ r x) → monotonic ? le r →
+ ∀i. CF (λx.h_of r 〈i,x〉) (total (λx.g (λi,x. r(h_of r 〈i,x〉)) i x)).
+#r #Hr #Hmono #i
+@(ext_CF (total (λx.unary_g (λi,x. r(h_of r 〈i,x〉)) 〈i,x〉)))
+ [#n whd in ⊢ (??%%); @eq_f @sym_eq @unary_g_def]
+@smn @(ext_CF … (speed_compl r Hr Hmono)) #n //
+qed.
+
+theorem pseudo_speedup:
+ ∀r:nat →nat. (∀x. x ≤ r x) → monotonic ? le r →
+ ∃f.∀sf. CF sf (total f) → ∃g,sg. f ≈ g ∧ CF sg (total g) ∧ O sf (r ∘ sg).
+(* ∃m,a.∀n. a≤n → r(sg a) < m * sf n. *)
+#r #Hr #Hmono
+(* f is (g (λi,x. r(h_of r 〈i,x〉)) 0) *)
+%{(g (λi,x. r(h_of r 〈i,x〉)) 0)} #sf * #H * #i *
+#Hcodei #HCi
+(* g is (g (λi,x. r(h_of r 〈i,x〉)) (S i)) *)
+%{(g (λi,x. r(h_of r 〈i,x〉)) (S i))}
+(* sg is (λx.h_of r 〈i,x〉) *)
+%{(λx. h_of r 〈S i,x〉)}
+lapply (speed_compl_i … Hr Hmono (S i)) #Hg
+%[%[@condition_1 |@Hg]
+ |cases Hg #H1 * #j * #Hcodej #HCj
+ lapply (condition_2 … Hcodei) #Hcond2 (* @not_to_not *)
+ cases HCi #m * #a #Ha %{m} %{(max (S i) a)} #n #ltin @lt_to_le @not_le_to_lt
+ @(not_to_not … Hcond2) -Hcond2 #Hlesf %{n} %
+ [@(transitive_le … ltin) @(le_maxl … (le_n …))]
+ cases (Ha n ?) [2: @(transitive_le … ltin) @(le_maxr … (le_n …))]
+ #y #Uy %{y} @(monotonic_U … Uy) @(transitive_le … Hlesf) //
+ ]
+qed.
+
+theorem pseudo_speedup':
+ ∀r:nat →nat. (∀x. x ≤ r x) → monotonic ? le r →
+ ∃f.∀sf. CF sf (total f) → ∃g,sg. f ≈ g ∧ CF sg (total g) ∧
+ (* ¬ O (r ∘ sg) sf. *)
+ ∃m,a.∀n. a≤n → r(sg a) < m * sf n.
+#r #Hr #Hmono
+(* f is (g (λi,x. r(h_of r 〈i,x〉)) 0) *)
+%{(g (λi,x. r(h_of r 〈i,x〉)) 0)} #sf * #H * #i *
+#Hcodei #HCi
+(* g is (g (λi,x. r(h_of r 〈i,x〉)) (S i)) *)
+%{(g (λi,x. r(h_of r 〈i,x〉)) (S i))}
+(* sg is (λx.h_of r 〈i,x〉) *)
+%{(λx. h_of r 〈S i,x〉)}
+lapply (speed_compl_i … Hr Hmono (S i)) #Hg
+%[%[@condition_1 |@Hg]
+ |cases Hg #H1 * #j * #Hcodej #HCj
+ lapply (condition_2 … Hcodei) #Hcond2 (* @not_to_not *)
+ cases HCi #m * #a #Ha
+ %{m} %{(max (S i) a)} #n #ltin @not_le_to_lt @(not_to_not … Hcond2) -Hcond2 #Hlesf
+ %{n} % [@(transitive_le … ltin) @(le_maxl … (le_n …))]
+ cases (Ha n ?) [2: @(transitive_le … ltin) @(le_maxr … (le_n …))]
+ #y #Uy %{y} @(monotonic_U … Uy) @(transitive_le … Hlesf)
+ @Hmono @(mono_h_of2 … Hr Hmono … ltin)
+ ]
+qed.
+
\ No newline at end of file
projects / helm.git / commitdiff