+++ /dev/null
-
-include "arithmetics/nat.ma".
-include "basics/sets.ma".
-
-(******************************** big O notation ******************************)
-
-(* O f g means g ∈ O(f) *)
-definition O: relation (nat→nat) ≝
- λf,g. ∃c.∃n0.∀n. n0 ≤ n → g n ≤ c* (f n).
-
-lemma O_refl: ∀s. O s s.
-#s %{1} %{0} #n #_ >commutative_times <times_n_1 @le_n qed.
-
-lemma O_trans: ∀s1,s2,s3. O s2 s1 → O s3 s2 → O s3 s1.
-#s1 #s2 #s3 * #c1 * #n1 #H1 * #c2 * # n2 #H2 %{(c1*c2)}
-%{(max n1 n2)} #n #Hmax
-@(transitive_le … (H1 ??)) [@(le_maxl … Hmax)]
->associative_times @le_times [//|@H2 @(le_maxr … Hmax)]
-qed.
-
-lemma sub_O_to_O: ∀s1,s2. O s1 ⊆ O s2 → O s2 s1.
-#s1 #s2 #H @H // qed.
-
-lemma O_to_sub_O: ∀s1,s2. O s2 s1 → O s1 ⊆ O s2.
-#s1 #s2 #H #g #Hg @(O_trans … H) // qed.
-
-lemma le_to_O: ∀s1,s2. (∀x.s1 x ≤ s2 x) → O s2 s1.
-#s1 #s2 #Hle %{1} %{0} #n #_ normalize <plus_n_O @Hle
-qed.
-
-definition sum_f ≝ λf,g:nat→nat.λn.f n + g n.
-interpretation "function sum" 'plus f g = (sum_f f g).
-
-lemma O_plus: ∀f,g,s. O s f → O s g → O s (f+g).
-#f #g #s * #cf * #nf #Hf * #cg * #ng #Hg
-%{(cf+cg)} %{(max nf ng)} #n #Hmax normalize
->distributive_times_plus_r @le_plus
- [@Hf @(le_maxl … Hmax) |@Hg @(le_maxr … Hmax) ]
-qed.
-
-lemma O_plus_l: ∀f,s1,s2. O s1 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_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.
-
-lemma O_absorbr: ∀f,g,s. O s f → O f g → O s (f+g).
-#f #g #s #Osf #Ofg @(O_plus … Osf) @(O_trans … Osf) //
-qed.
-
-lemma O_times_c: ∀f,c. O f (λx:ℕ.c*f x).
-#f #c %{c} %{0} //
-qed.
-
-lemma O_ext2: ∀f,g,s. O s f → (∀x.f x = g x) → O s g.
-#f #g #s * #c * #a #Osf #eqfg %{c} %{a} #n #lean <eqfg @Osf //
-qed.
-
-
-definition not_O ≝ λf,g.∀c,n0.∃n. n0 ≤ n ∧ c* (f n) < g n .
-
-(* 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.
+++ /dev/null
-
-include "arithmetics/minimization.ma".
-include "arithmetics/bigops.ma".
-include "arithmetics/pidgeon_hole.ma".
-include "arithmetics/iteration.ma".
-
-(************************** notation for miminimization ***********************)
-
-(* an alternative defintion of minimization
-definition Min ≝ λa,f.
- \big[min,a]_{i < a | f i} i. *)
-
-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 (S $b-$a) $a (λ${ident i}.$p)}.
-
-lemma f_min_true: ∀f,a,b.
- (∃i. a ≤ i ∧ i ≤ b ∧ f i = true) → f (μ_{i ∈[a,b]} (f i)) = true.
-#f #a #b * #i * * #Hil #Hir #Hfi @(f_min_true … (λx. f x)) <plus_minus_m_m
- [%{i} % // % [@Hil |@le_S_S @Hir]|@le_S @(transitive_le … Hil Hir)]
-qed.
-
-lemma min_up: ∀f,a,b.
- (∃i. a ≤ i ∧ i ≤ b ∧ f i = true) → μ_{i ∈[a,b]}(f i) ≤ b.
-#f #a #b * #i * * #Hil #Hir #Hfi @le_S_S_to_le
-cut ((S b) = S b - a + a) [@plus_minus_m_m @le_S @(transitive_le … Hil Hir)]
-#Hcut >Hcut in ⊢ (??%); @lt_min %{i} % // % [@Hil |<Hcut @le_S_S @Hir]
-qed.
-
-(*************************** Kleene's predicate *******************************)
-
-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 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.
-
-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.
-
-lemma decidable_test : ∀n,x,r,r1.
- (∀i. i < n → 〈i,x〉 ↓ r ∨ ¬ 〈i,x〉 ↓ r1) ∨
- (∃i. i < n ∧ (¬ 〈i,x〉 ↓ r ∧ 〈i,x〉 ↓ r1)).
-#n #x #r1 #r2
- cut (∀i0.decidable ((〈i0,x〉↓r1) ∨ ¬ 〈i0,x〉 ↓ r2))
- [#j @decidable_or [@terminate_dec |@decidable_not @terminate_dec ]] #Hdec
- cases(decidable_forall ? Hdec n)
- [#H %1 @H
- |#H %2 cases (not_forall_to_exists … Hdec H) #j * #leji #Hj
- %{j} % // %
- [@(not_to_not … Hj) #H %1 @H
- |cases (terminate_dec j x r2) // #H @False_ind cases Hj -Hj #Hj
- @Hj %2 @H
- ]
-qed.
-
-(**************************** the gap theorem *********************************)
-definition gapP ≝ λn,x,g,r. ∀i. i < n → 〈i,x〉 ↓ r ∨ ¬ 〈i,x〉 ↓ g r.
-
-lemma gapP_def : ∀n,x,g,r.
- gapP n x g r = ∀i. i < n → 〈i,x〉 ↓ r ∨ ¬ 〈i,x〉 ↓ g r.
-// qed.
-
-lemma upper_bound_aux: ∀g,b,n,x. (∀x. x ≤ g x) → ∀k.
- (∃j.j < k ∧
- (∀i. i < n → 〈i,x〉 ↓ g^j b ∨ ¬ 〈i,x〉 ↓ g^(S j) b)) ∨
- ∃l. |l| = k ∧ unique ? l ∧ ∀i. i ∈ l → i < n ∧ 〈i,x〉 ↓ g^k b .
-#g#b #n #x #Hg #k elim k
- [%2 %{([])} normalize % [% //|#x @False_ind]
- |#k0 *
- [* #j * #lej #H %1 %{j} % [@le_S // | @H ]
- |* #l * * #Hlen #Hunique #Hterm
- cases (decidable_test n x (g^k0 b) (g^(S k0) b))
- [#Hcase %1 %{k0} % [@le_n | @Hcase]
- |* #j * #ltjn * #H1 #H2 %2
- %{(j::l)} %
- [ % [normalize @eq_f @Hlen] whd % // % #H3
- @(absurd ?? H1) @(proj2 … (Hterm …)) @H3
- |#x *
- [#eqxj >eqxj % //
- |#Hmemx cases(Hterm … Hmemx) #lexn * #y #HU
- % [@lexn] %{y} @(monotonic_U ?????? HU) @Hg
- ]
- ]
- ]
- ]
- ]
-qed.
-
-lemma upper_bound: ∀g,b,n,x. (∀x. x ≤ g x) → ∃r.
- (* b ≤ r ∧ r ≤ g^n b ∧ ∀i. i < n → 〈i,x〉 ↓ r ∨ ¬ 〈i,x〉 ↓ g r. *)
- b ≤ r ∧ r ≤ g^n b ∧ gapP n x g r.
-#g #b #n #x #Hg
-cases (upper_bound_aux g b n x Hg n)
- [* #j * #Hj #H %{(g^j b)} % [2: @H] % [@le_iter //]
- @monotonic_iter2 // @lt_to_le //
- |* #l * * #Hlen #Hunique #Hterm %{(g^n b)} %
- [% [@le_iter // |@le_n]]
- #i #lein %1 @(proj2 … (Hterm ??))
- @(eq_length_to_mem_all … Hlen Hunique … lein)
- #x #memx @(proj1 … (Hterm ??)) //
- ]
-qed.
-
-definition gapb ≝ λn,x,g,r.
- \big[andb,true]_{i < n} ((termb i x r) ∨ ¬(termb i x (g r))).
-
-lemma gapb_def : ∀n,x,g,r. gapb n x g r =
- \big[andb,true]_{i < n} ((termb i x r) ∨ ¬(termb i x (g r))).
-// qed.
-
-lemma gapb_true_to_gapP : ∀n,x,g,r.
- gapb n x g r = true → ∀i. i < n → 〈i,x〉 ↓ r ∨ ¬(〈i,x〉 ↓ (g r)).
-#n #x #g #r elim n
- [>gapb_def >bigop_Strue //
- #H #i #lti0 @False_ind @(absurd … lti0) @le_to_not_lt //
- |#m #Hind >gapb_def >bigop_Strue //
- #H #i #leSm cases (le_to_or_lt_eq … leSm)
- [#lem @Hind [@(andb_true_r … H)|@le_S_S_to_le @lem]
- |#eqi >(injective_S … eqi) lapply (andb_true_l … H) -H #H cases (orb_true_l … H) -H
- [#H %1 @termb_true_to_term //
- |#H %2 % #H1 >(term_to_termb_true … H1) in H; normalize #H destruct
- ]
- ]
- ]
-qed.
-
-lemma gapP_to_gapb_true : ∀n,x,g,r.
- (∀i. i < n → 〈i,x〉 ↓ r ∨ ¬(〈i,x〉 ↓ (g r))) → gapb n x g r = true.
-#n #x #g #r elim n //
-#m #Hind #H >gapb_def >bigop_Strue // @true_to_andb_true
- [cases (H m (le_n …))
- [#H2 @orb_true_r1 @term_to_termb_true //
- |#H2 @orb_true_r2 @sym_eq @noteq_to_eqnot @sym_not_eq
- @(not_to_not … H2) @termb_true_to_term
- ]
- |@Hind #i0 #lei0 @H @le_S //
- ]
-qed.
-
-
-(* the gap function *)
-let rec gap g n on n ≝
- match n with
- [ O ⇒ 1
- | S m ⇒ let b ≝ gap g m in μ_{i ∈ [b,g^n b]} (gapb n n g i)
- ].
-
-lemma gapS: ∀g,m.
- gap g (S m) =
- (let b ≝ gap g m in
- μ_{i ∈ [b,g^(S m) b]} (gapb (S m) (S m) g i)).
-// qed.
-
-lemma upper_bound_gapb: ∀g,m. (∀x. x ≤ g x) →
- ∃r:ℕ.gap g m ≤ r ∧ r ≤ g^(S m) (gap g m) ∧ gapb (S m) (S m) g r = true.
-#g #m #leg
-lapply (upper_bound g (gap g m) (S m) (S m) leg) * #r * *
-#H1 #H2 #H3 %{r} %
- [% // |@gapP_to_gapb_true @H3]
-qed.
-
-lemma gapS_true: ∀g,m. (∀x. x ≤g x) → gapb (S m) (S m) g (gap g (S m)) = true.
-#g #m #leg @(f_min_true (gapb (S m) (S m) g)) @upper_bound_gapb //
-qed.
-
-theorem gap_theorem: ∀g,i.(∀x. x ≤ g x)→∃k.∀n.k < n →
- 〈i,n〉 ↓ (gap g n) ∨ ¬ 〈i,n〉 ↓ (g (gap g n)).
-#g #i #leg %{i} *
- [#lti0 @False_ind @(absurd ?? (not_le_Sn_O i) ) //
- |#m #leim lapply (gapS_true g m leg) #H
- @(gapb_true_to_gapP … H) //
- ]
-qed.
-
-(* an upper bound *)
-
-let rec sigma n ≝
- match n with
- [ O ⇒ 0 | S m ⇒ n + sigma m ].
-
-lemma gap_bound: ∀g. (∀x. x ≤ g x) → (monotonic ? le g) →
- ∀n.gap g n ≤ g^(sigma n) 1.
-#g #leg #gmono #n elim n
- [normalize //
- |#m #Hind >gapS @(transitive_le ? (g^(S m) (gap g m)))
- [@min_up @upper_bound_gapb //
- |@(transitive_le ? (g^(S m) (g^(sigma m) 1)))
- [@monotonic_iter // |>iter_iter >commutative_plus @le_n
- ]
- ]
-qed.
-
-lemma gap_bound2: ∀g. (∀x. x ≤ g x) → (monotonic ? le g) →
- ∀n.gap g n ≤ g^(n*n) 1.
-#g #leg #gmono #n elim n
- [normalize //
- |#m #Hind >gapS @(transitive_le ? (g^(S m) (gap g m)))
- [@min_up @upper_bound_gapb //
- |@(transitive_le ? (g^(S m) (g^(m*m) 1)))
- [@monotonic_iter //
- |>iter_iter @monotonic_iter2 [@leg | normalize <plus_n_Sm @le_S_S //
- ]
- ]
-qed.
-
-(*
-axiom universal: ∃u.∀i,x,y.
- ∃n. U u 〈i,x〉 n = Some y ↔ ∃m.U i x m = Some y. *)
-
-
-
-
-
-
-
-
-
-
-
-
+++ /dev/null
-
-include "arithmetics/nat.ma".
-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".
-
-(************************************ 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.
-
-alias id "max" = "cic:/matita/arithmetics/nat/max#def:2".
-alias id "mk_Aop" = "cic:/matita/arithmetics/bigops/Aop#con:0:1:2".
-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 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.
-
-(******************************** big O notation ******************************)
-
-(* O f g means g ∈ O(f) *)
-definition O: relation (nat→nat) ≝
- λf,g. ∃c.∃n0.∀n. n0 ≤ n → g n ≤ c* (f n).
-
-lemma O_refl: ∀s. O s s.
-#s %{1} %{0} #n #_ >commutative_times <times_n_1 @le_n qed.
-
-lemma O_trans: ∀s1,s2,s3. O s2 s1 → O s3 s2 → O s3 s1.
-#s1 #s2 #s3 * #c1 * #n1 #H1 * #c2 * # n2 #H2 %{(c1*c2)}
-%{(max n1 n2)} #n #Hmax
-@(transitive_le … (H1 ??)) [@(le_maxl … Hmax)]
->associative_times @le_times [//|@H2 @(le_maxr … Hmax)]
-qed.
-
-lemma sub_O_to_O: ∀s1,s2. O s1 ⊆ O s2 → O s2 s1.
-#s1 #s2 #H @H // qed.
-
-lemma O_to_sub_O: ∀s1,s2. O s2 s1 → O s1 ⊆ O s2.
-#s1 #s2 #H #g #Hg @(O_trans … H) // qed.
-
-definition sum_f ≝ λf,g:nat→nat.λn.f n + g n.
-interpretation "function sum" 'plus f g = (sum_f f g).
-
-lemma O_plus: ∀f,g,s. O s f → O s g → O s (f+g).
-#f #g #s * #cf * #nf #Hf * #cg * #ng #Hg
-%{(cf+cg)} %{(max nf ng)} #n #Hmax normalize
->distributive_times_plus_r @le_plus
- [@Hf @(le_maxl … Hmax) |@Hg @(le_maxr … Hmax) ]
-qed.
-
-lemma O_plus_l: ∀f,s1,s2. O s1 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_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.
-
-lemma O_absorbr: ∀f,g,s. O s f → O f g → O s (f+g).
-#f #g #s #Osf #Ofg @(O_plus … Osf) @(O_trans … Osf) //
-qed.
-
-(*
-lemma O_ff: ∀f,s. O s f → O s (f+f).
-#f #s #Osf /2/
-qed. *)
-
-lemma O_ext2: ∀f,g,s. O s f → (∀x.f x = g x) → O s g.
-#f #g #s * #c * #a #Osf #eqfg %{c} %{a} #n #lean <eqfg @Osf //
-qed.
-
-
-definition not_O ≝ λf,g.∀c,n0.∃n. n0 ≤ n ∧ c* (f n) < g n .
-
-(* 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.
-axiom fst : nat → nat.
-axiom snd : nat → nat.
-axiom fst_pair: ∀a,b. fst (pair a b) = a.
-axiom snd_pair: ∀a,b. snd (pair a b) = b.
-
-interpretation "abstract pair" 'pair f g = (pair f g).
-
-(************************ basic complexity notions ****************************)
-
-(* u is the deterministic configuration relation of the universal machine (one
- step)
-
-axiom u: nat → option nat.
-
-let rec U c n on n ≝
- match n with
- [ O ⇒ None ?
- | S m ⇒ match u c with
- [ None ⇒ Some ? c (* halting case *)
- | Some c1 ⇒ U c1 m
- ]
- ].
-
-lemma halt_U: ∀i,n,y. u i = None ? → U i n = Some ? y → y = i.
-#i #n #y #H cases n
- [normalize #H1 destruct |#m normalize >H normalize #H1 destruct //]
-qed.
-
-lemma Some_to_halt: ∀n,i,y. U i n = Some ? y → u y = None ? .
-#n elim n
- [#i #y normalize #H destruct (H)
- |#m #Hind #i #y normalize
- cut (u i = None ? ∨ ∃c. u i = Some ? c)
- [cases (u i) [/2/ | #c %2 /2/ ]]
- *[#H >H normalize #H1 destruct (H1) // |* #c #H >H normalize @Hind ]
- ]
-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
- [#i normalize #H destruct
- |#p #Hind #i <plus_n_Sm normalize
- cut (u i = None ? ∨ ∃c. u i = Some ? c)
- [cases (u i) [/2/ | #c %2 /2/ ]]
- *[#H1 >H1 normalize // |* #c #H >H normalize #H1 @Hind //]
- ]
-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,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}.
-
-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.
-#f #i #x normalize #H %
- [* #n * #y * #H1 #posy %{y} % //
- cases (H x) -H #m #H <(H (max n m)) [2:@(le_maxr … n) //]
- @(monotonic_U … H1) @(le_maxl … m) //
- |cases (H x) -H #m #Hm * #y #Hy %{m} %{y} >Hm //
- ]
-qed.
-
-(******************************* complexity classes ***************************)
-
-axiom size: nat → nat.
-axiom of_size: nat → nat.
-
-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|.
-
-definition size_f ≝ λf,n.Max_{i < S (of_size n) | eqb (|i|) n}|(f i)|.
-
-lemma size_f_def: ∀f,n. size_f f n =
- Max_{i < S (of_size n) | eqb (|i|) n}|(f i)|.
-// qed.
-
-(*
-definition Max_const : ∀f,p,n,a. a < n → p a →
- ∀n. f n = g n →
- Max_{i < n | p n}(f n) = *)
-
-lemma size_f_size : ∀f,n. size_f (f ∘ size) n = |(f n)|.
-#f #n @le_to_le_to_eq
- [@Max_le #a #lta #Ha normalize >(eqb_true_to_eq … Ha) //
- |<(size_of_size n) in ⊢ (?%?); >size_f_def
- @(le_Max (λi.|f (|i|)|) ? (S (of_size n)) (of_size n) ??)
- [@le_S_S // | @eq_to_eqb_true //]
- ]
-qed.
-
-lemma size_f_id : ∀n. size_f (λx.x) n = n.
-#n @le_to_le_to_eq
- [@Max_le #a #lta #Ha >(eqb_true_to_eq … Ha) //
- |<(size_of_size n) in ⊢ (?%?); >size_f_def
- @(le_Max (λi.|i|) ? (S (of_size n)) (of_size n) ??)
- [@le_S_S // | @eq_to_eqb_true //]
- ]
-qed.
-
-lemma size_f_fst : ∀n. size_f fst n ≤ n.
-#n @Max_le #a #lta #Ha <(eqb_true_to_eq … Ha) //
-qed.
-
-(* definition def ≝ λf:nat → option nat.λx.∃y. f x = Some ? y.*)
-
-(* 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.
-
-definition CF ≝ λs,f.∃i.code_for 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 * #i * #Hcode #HC %{i} %
- [#x cases (Hcode x) #a #H %{a} <Hext @H | //]
-qed.
-
-lemma monotonic_CF: ∀s1,s2,f. O s2 s1 → CF s1 f → CF s2 f.
-#s1 #s2 #f * #c1 * #a #H * #i * #Hcodef #HCs1 %{i} % //
-cases HCs1 #c2 * #b #H2 %{(c2*c1)} %{(max a b)}
-#x #Hmax cases (H2 x ?) [2:@(le_maxr … Hmax)] #y #Hy
-%{y} @(monotonic_U …Hy) >associative_times @le_times // @H @(le_maxl … Hmax)
-qed.
-
-(************************** The diagonal language *****************************)
-
-(* the diagonal language used for the hierarchy theorem *)
-
-definition diag ≝ λs,i.
- 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.
-#s #i %
- [whd in ⊢ (%→?); #H % [%{0} //] % * #x * #y *
- #H1 #Hy cut (0 = y) [@(unique_U … H H1)] #eqy /2/
- |* * #y cases y //
- #y0 #H * #H1 @False_ind @H1 -H1 whd %{(s (|i|))} %{(S y0)} % //
- ]
-qed.
-
-(* Let us define the characteristic function diag_cf for diag, and prove
-it correctness *)
-
-definition diag_cf ≝ λs,i.
- match U (fst i) i (s (|i|)) with
- [ None ⇒ None ?
- | Some y ⇒ if (eqb y 0) then (Some ? 1) else (Some ? 0)].
-
-lemma diag_cf_OK: ∀s,x. diag s x ↔ ∃y.diag_cf s x = Some ? y ∧ 0 < y.
-#s #x %
- [whd in ⊢ (%→?); #H %{1} % // whd in ⊢ (??%?); >H //
- |* #y * whd in ⊢ (??%?→?→%);
- 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) //
- |normalize #H destruct #H @False_ind @(absurd ? H) @lt_to_not_le //
- ]
- ]
- ]
-qed.
-
-lemma diag_spec: ∀s,i. code_for (diag_cf s) i → ∀x. lang i x ↔ diag s x.
-#s #i #Hcode #x @(iff_trans … (lang_cf … Hcode)) @iff_sym @diag_cf_OK
-qed.
-
-(******************************************************************************)
-
-lemma absurd1: ∀P. iff P (¬ P) →False.
-#P * #H1 #H2 cut (¬P) [% #H2 @(absurd … H2) @H1 //]
-#H3 @(absurd ?? H3) @H2 @H3
-qed.
-
-(* axiom weak_pad : ∀a,∃a0.∀n. a0 < 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 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) #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.
-
-lemma diag1_not_s1: ∀s1,s2. o s1 s2 → ¬ CF s2 (diag_cf s1).
-#s1 #s2 #H1 % * #i * #Hcode_i #Hs2_i
-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〉))
-@(iff_trans … (equiv_diag …)) >fst_pair
-%[* #_ // |#H6 % // ]
-qed.
-
-(******************************************************************************)
-
-definition to_Some ≝ λf.λx:nat. Some nat (f x).
-
-definition deopt ≝ λn. match n with
- [ None ⇒ 1
- | Some n ⇒ n].
-
-definition opt_comp ≝ λf,g:nat → option nat. λx.
- match g x with
- [ None ⇒ None ?
- | Some y ⇒ f y ].
-
-(* axiom CFU: ∀h,g,s. CF s (to_Some h) → CF s (to_Some (of_size ∘ g)) →
- CF (Slow s) (λx.U (h x) (g x)). *)
-
-axiom sU2: nat → nat → nat.
-axiom sU: nat → nat → nat → nat.
-
-(* axiom CFU: CF sU (λx.U (fst x) (snd x)). *)
-
-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|)).
-#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).
-
-(*
-axiom CF_comp: ∀f,g,s1, s2. CF s1 f → CF s2 g →
- CF (λx.s2 x + s1 (size (deopt (g x)))) (opt_comp f g).
-
-(* axiom CF_comp: ∀f,g,s1, s2. CF s1 f → CF s2 g →
- CF (s1 ∘ (λx. size (deopt (g x)))) (opt_comp f g). *)
-
-axiom CF_comp_strong: ∀f,g,s1,s2. CF s1 f → CF s2 g →
- CF (s1 ∘ s2) (opt_comp f g). *)
-
-definition IF ≝ λb,f,g:nat →option nat. λx.
- 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).
-
-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 (fst i) i (|of_size (s (|i|))|)) (λi.Some ? 1) (λi.Some ? 0) i.
-#s #i normalize >size_of_size // qed.
-
-(* and now ... *)
-axiom CF_pair: ∀f,g,s. CF s (λx.Some ? (f x)) → CF s (λx.Some ? (g x)) →
- CF s (λx.Some ? (pair (f x) (g x))).
-
-axiom CF_fst: ∀f,s. CF s (λx.Some ? (f x)) → CF s (λx.Some ? (fst (f x))).
-
-definition minimal ≝ λs. CF s (λn. Some ? n) ∧ ∀c. CF s (λn. Some ? c).
-
-
-(*
-axiom le_snd: ∀n. |snd n| ≤ |n|.
-axiom daemon: ∀P:Prop.P. *)
-
-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 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
-@ext_CF [2: #n @sym_eq @diag_cf_def | skip]
-@(monotonic_CF ???? (IF_CF (λi:ℕ.U (pair (fst i) i) (|of_size (s (|i|))|))
- … (λi.s i + s i + s i + (sU (size_f fst i) (size_f (λi.i) i) (size_f (λi.of_size (s (|i|))) i))) … (Hs_c 1) (Hs_c 0) … ))
- [2: @CFU [@CF_fst // | // |@Hs_constr]
- |@(O_ext2 (λn:ℕ.s n+sU (size_f fst n) n (s n) + (s n+s n+s n+s n)))
- [2: #i >size_f_size >size_of_size >size_f_id //]
- @O_absorbr
- [%{1} %{0} #n #_ >commutative_times <times_n_1 @le_plus //
- @monotonic_sU //
- |@O_plus_l @(O_plus … (O_refl s)) @(O_plus … (O_refl s))
- @(O_plus … (O_refl s)) //
- ]
-qed.
-*)
-
-(*************************** The hierachy theorem *****************************)
-
-(*
-theorem hierarchy_theorem_right: ∀s1,s2:nat→nat.
- O s1 idN → constructible s1 →
- not_O s2 s1 → ¬ CF s1 ⊆ CF s2.
-#s1 #s2 #Hs1 #monos1 #H % #H1
-@(absurd … (CF s2 (diag_cf s1)))
- [@H1 @diag_s // |@(diag1_not_s1 … H)]
-qed.
-*)
-
-theorem hierarchy_theorem_left: ∀s1,s2:nat→nat.
- O(s1) ⊆ O(s2) → CF s1 ⊆ CF s2.
-#s1 #s2 #HO #f * #i * #Hcode * #c * #a #Hs1_i %{i} % //
-cases (sub_O_to_O … HO) -HO #c1 * #b #Hs1s2
-%{(c*c1)} %{(max a b)} #x #lemax
-cases (Hs1_i x ?) [2: @(le_maxl …lemax)]
-#y #Hy %{y} @(monotonic_U … Hy) >associative_times
-@le_times // @Hs1s2 @(le_maxr … lemax)
-qed.
-
+++ /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.
-
-lemma fst_pU: ∀i,x,r. fst (pU i x r) = termb i x r.
-#i #x #r normalize cases (U i x r) normalize >fst_pair //
-qed.
-
-lemma snd_pU: ∀i,x,r. snd (pU i x r) = out i x r.
-#i #x #r normalize cases (U i x r) normalize >snd_pair //
-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: ∀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.
-axiom MSC_pair: ∀a,b. MSC 〈a,b〉 ≤ MSC a + MSC b.
-
-(* 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).
-
-lemma main_MSC: ∀h,f. CF h f → O h (λx.MSC (f x)).
-
-axiom CF_S: CF MSC S.
-axiom CF_fst: CF MSC fst.
-axiom CF_snd: CF MSC snd.
-
-lemma CF_compS: ∀h,f. CF h f → CF h (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. *)
-
-definition id ≝ λx:nat.x.
-
-axiom CF_id: CF MSC id.
-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).
-axiom CF_comp_pair: ∀h,f,g. CF h f → CF h g → CF h (λx. 〈f x,g x〉).
-
-lemma CF_fst: CF MSC fst.
-@(ext_CF (fst ∘ id)) [#n //] @(CF_comp_fst … CF_id)
-qed.
-
-lemma CF_snd: CF MSC snd.
-@(ext_CF (snd ∘ id)) [#n //] @(CF_comp_snd … CF_id)
-qed.
-
-(************************************** 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〉)).
-
-(****************************** constructibility ******************************)
-
-definition constructible ≝ λs. CF s s.
-
-lemma constr_comp : ∀s1,s2. constructible s1 → constructible s2 →
- (∀x. x ≤ s2 x) → constructible (s2 ∘ s1).
-#s1 #s2 #Hs1 #Hs2 #Hle @(CF_comp … Hs1 Hs2) @O_plus @le_to_O #x [@Hle | //]
-qed.
-
-lemma ext_constr: ∀s1,s2. (∀x.s1 x = s2 x) →
- constructible s1 → constructible s2.
-#s1 #s2 #Hext #Hs1 @(ext_CF … Hext) @(monotonic_CF … Hs1) #x >Hext //
-qed.
-
-(********************************* simulation *********************************)
-
-axiom sU : nat → nat.
-
-axiom monotonic_sU: ∀i1,i2,x1,x2,s1,s2. i1 ≤ i2 → x1 ≤ x2 → s1 ≤ s2 →
- sU 〈i1,〈x1,s1〉〉 ≤ sU 〈i2,〈x2,s2〉〉.
-
-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 sU_le: ∀i,x,s. s ≤ sU 〈i,〈x,s〉〉.
-axiom sU_le_i: ∀i,x,s. MSC i ≤ sU 〈i,〈x,s〉〉.
-axiom sU_le_x: ∀i,x,s. MSC x ≤ sU 〈i,〈x,s〉〉.
-
-definition pU_unary ≝ λp. pU (fst p) (fst (snd p)) (snd (snd p)).
-
-axiom CF_U : CF sU pU_unary.
-
-definition termb_unary ≝ λx:ℕ.termb (fst x) (fst (snd x)) (snd (snd x)).
-definition out_unary ≝ λx:ℕ.out (fst x) (fst (snd x)) (snd (snd x)).
-
-lemma CF_termb: CF sU termb_unary.
-@(ext_CF (fst ∘ pU_unary)) [2: @CF_comp_fst @CF_U]
-#n whd in ⊢ (??%?); whd in ⊢ (??(?%)?); >fst_pair %
-qed.
-
-lemma CF_out: CF sU out_unary.
-@(ext_CF (snd ∘ pU_unary)) [2: @CF_comp_snd @CF_U]
-#n whd in ⊢ (??%?); whd in ⊢ (??(?%)?); >snd_pair %
-qed.
-
-(*
-lemma CF_termb_comp: ∀f.CF (sU ∘ f) (termb_unary ∘ f).
-#f @(CF_comp … 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 (auxg h) → CF s (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
- (λp:ℕ.eqb (min_input h (fst p) (snd (snd p))) (snd (snd p))) →
- CF s2
- (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 (auxg h).
-#h #s1 #s2 #s #Hs1 #Hs2 #HO @(ext_CF (aux1g h))
- [#n whd in ⊢ (??%%); @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 (λp:ℕ.min_input h (fst p) (snd (snd p))) →
- CF s (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
- (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 (λp:ℕ.min_input h (fst p) (snd (snd p))).
-#h #s1 #s #Hs1 #HO @(ext_CF (min_input_aux h))
- [#n whd in ⊢ (??%%); @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
- (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〉)
- (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.
- (* constructible (λx. h (fst x) (snd x)) → *)
- (CF (λx. max (MSC x) (sU 〈fst (snd x),〈fst x,h (S (fst (snd x))) (fst x)〉〉))
- (λp.termb (fst (snd p)) (fst p) (h (S (fst (snd p))) (fst p)))).
-*)
-
-lemma compl_g6: ∀h.
- constructible (λx. h (fst x) (snd x)) →
- (CF (λx. sU 〈max (fst (snd x)) (snd (snd x)),〈fst x,h (S (fst (snd x))) (fst x)〉〉)
- (λp.termb (fst (snd p)) (fst p) (h (S (fst (snd p))) (fst p)))).
-#h #hconstr @(ext_CF (termb_aux h))
- [#n normalize >fst_pair >snd_pair >fst_pair >snd_pair // ]
-@(CF_comp … (λx.MSC x + h (S (fst (snd x))) (fst x)) … CF_termb)
- [@CF_comp_pair
- [@CF_comp_fst @(monotonic_CF … CF_snd) #x //
- |@CF_comp_pair
- [@(monotonic_CF … CF_fst) #x //
- |@(ext_CF ((λx.h (fst x) (snd x))∘(λx.〈S (fst (snd x)),fst x〉)))
- [#n normalize >fst_pair >snd_pair %]
- @(CF_comp … MSC …hconstr)
- [@CF_comp_pair [@CF_compS @CF_comp_fst // |//]
- |@O_plus @le_to_O [//|#n >fst_pair >snd_pair //]
- ]
- ]
- ]
- |@O_plus
- [@O_plus
- [@(O_trans … (λx.MSC (fst x) + MSC (max (fst (snd x)) (snd (snd x)))))
- [%{2} %{0} #x #_ cases (surj_pair x) #a * #b #eqx >eqx
- >fst_pair >snd_pair @(transitive_le … (MSC_pair …))
- >distributive_times_plus @le_plus [//]
- cases (surj_pair b) #c * #d #eqb >eqb
- >fst_pair >snd_pair @(transitive_le … (MSC_pair …))
- whd in ⊢ (??%); @le_plus
- [@monotonic_MSC @(le_maxl … (le_n …))
- |>commutative_times <times_n_1 @monotonic_MSC @(le_maxr … (le_n …))
- ]
- |@O_plus [@le_to_O #x @sU_le_x |@le_to_O #x @sU_le_i]
- ]
- |@le_to_O #n @sU_le
- ]
- |@le_to_O #x @monotonic_sU // @(le_maxl … (le_n …)) ]
- ]
-qed.
-
-(* 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.
- (∀x.MSC x≤h (S (fst (snd x))) (fst x)) →
- constructible (λx. h (fst x) (snd x)) →
- (∀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))〉〉)
- (λp:ℕ.min_input h (fst p) (snd (snd p))).
-#h #hle #hcostr #hmono @(monotonic_CF … (faux1 h))
- [#n normalize >fst_pair >snd_pair //]
-@compl_g5 [2:@(compl_g6 h hcostr)] #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.
-
-definition faux2 ≝ λh.
- (λx.MSC x + (snd (snd x)-fst x)*
- (λx.sU 〈max (fst(snd x)) (snd(snd x)),〈fst x,h (S (fst (snd x))) (fst x)〉〉) 〈snd (snd x),x〉).
-
-(* proviamo con x *)
-lemma compl_g7: ∀h.
- constructible (λx. h (fst x) (snd x)) →
- (∀n. monotonic ? le (h n)) →
- 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))〉〉)
- (λp:ℕ.min_input h (fst p) (snd (snd p))).
-#h #hcostr #hmono @(monotonic_CF … (faux2 h))
- [#n normalize >fst_pair >snd_pair //]
-@compl_g5 [2:@(compl_g6 h hcostr)] #n #x #y #lexy >fst_pair >snd_pair
->fst_pair >snd_pair @monotonic_sU // @hmono @lexy
-qed.
-
-(* proviamo con x *)
-lemma compl_g71: ∀h.
- constructible (λx. h (fst x) (snd x)) →
- (∀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))〉〉)
- (λp:ℕ.min_input h (fst p) (snd (snd p))).
-#h #hcostr #hmono @(monotonic_CF … (compl_g7 h hcostr hmono)) #x
-@le_plus [@monotonic_MSC //]
-cases (decidable_le (fst x) (snd(snd x)))
- [#Hle @le_times // @monotonic_sU
- |#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))〉〉)
- (λp:ℕ.out (fst p) (snd (snd p)) (h (S (fst p)) (snd (snd p)))). *)
-
-definition out_aux ≝ λh.
- out_unary ∘ λp.〈fst p,〈snd(snd p),h (S (fst p)) (snd (snd p))〉〉.
-
-lemma compl_g8: ∀h.
- constructible (λx. h (fst x) (snd x)) →
- (CF (λx. sU 〈max (fst x) (snd x),〈snd(snd x),h (S (fst x)) (snd(snd x))〉〉)
- (λp.out (fst p) (snd (snd p)) (h (S (fst p)) (snd (snd p))))).
-#h #hconstr @(ext_CF (out_aux h))
- [#n normalize >fst_pair >snd_pair >fst_pair >snd_pair // ]
-@(CF_comp … (λx.h (S (fst x)) (snd(snd x)) + MSC x) … CF_out)
- [@CF_comp_pair
- [@(monotonic_CF … CF_fst) #x //
- |@CF_comp_pair
- [@CF_comp_snd @(monotonic_CF … CF_snd) #x //
- |@(ext_CF ((λx.h (fst x) (snd x))∘(λx.〈S (fst x),snd(snd x)〉)))
- [#n normalize >fst_pair >snd_pair %]
- @(CF_comp … MSC …hconstr)
- [@CF_comp_pair [@CF_compS // | @CF_comp_snd // ]
- |@O_plus @le_to_O [//|#n >fst_pair >snd_pair //]
- ]
- ]
- ]
- |@O_plus
- [@O_plus
- [@le_to_O #n @sU_le
- |@(O_trans … (λx.MSC (max (fst x) (snd x))))
- [%{2} %{0} #x #_ cases (surj_pair x) #a * #b #eqx >eqx
- >fst_pair >snd_pair @(transitive_le … (MSC_pair …))
- whd in ⊢ (??%); @le_plus
- [@monotonic_MSC @(le_maxl … (le_n …))
- |>commutative_times <times_n_1 @monotonic_MSC @(le_maxr … (le_n …))
- ]
- |@le_to_O #x @(transitive_le ???? (sU_le_i … )) //
- ]
- ]
- |@le_to_O #x @monotonic_sU [@(le_maxl … (le_n …))|//|//]
- ]
-qed.
-
-(*
-lemma compl_g81: ∀h.
- (∀x.MSC x≤h (S (fst x)) (snd(snd x))) →
- constructible (λx. h (fst x) (snd x)) →
- CF (λx. sU 〈max (fst x) (snd x),〈snd (snd x),h (S (fst x)) (snd (snd x))〉〉)
- (λp:ℕ.out (fst p) (snd (snd p)) (h (S (fst p)) (snd (snd p)))).
-#h #hle #hconstr @(monotonic_CF ???? (compl_g8 h hle hconstr)) #x @monotonic_sU // @(le_maxl … (le_n … ))
-qed. *)
-
-(* axiom daemon : False. *)
-
-lemma compl_g9 : ∀h.
- constructible (λx. h (fst x) (snd x)) →
- (∀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)〉〉)
- (auxg h).
-#h #hconstr #hmono #hantimono
-@(compl_g2 h ??? (compl_g3 … (compl_g71 h hconstr hmono)) (compl_g8 h hconstr))
-@O_plus
- [@O_plus_l @le_to_O #x >(times_n_1 (MSC x)) >commutative_times @le_times
- [// | @monotonic_MSC // ]]
-@(O_trans … (coroll2 ??))
- [#n #a #b #leab #ltb >fst_pair >fst_pair >snd_pair >snd_pair
- cut (b ≤ n) [@(transitive_le … (le_snd …)) @lt_to_le //] #lebn
- cut (max a n = n)
- [normalize >le_to_leb_true [//|@(transitive_le … leab lebn)]] #maxa
- cut (max b n = n) [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.
- constructible (λx. h (fst x) (snd x)) →
- (∀n. monotonic ? le (h n)) →
- (∀n,a,b. a ≤ b → b ≤ n → h b n ≤ h a n) →
- CF (sg h) (unary_g h).
-#h #hconstr #Hm #Ham @compl_g1 @(compl_g9 h hconstr 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.
-// 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.
-
-axiom h_of_constr : ∀r:nat →nat.
- (∀x. x ≤ r x) → monotonic ? le r → constructible r →
- constructible (h_of r).
-
-lemma speed_compl: ∀r:nat →nat.
- (∀x. x ≤ r x) → monotonic ? le r → constructible r →
- CF (h_of r) (unary_g (λi,x. r(h_of r 〈i,x〉))).
-#r #Hr #Hmono #Hconstr @(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)
- [%{(pred(b-a))} >S_pred [//] @lt_plus_to_minus_r @not_le_to_lt @ltab]
- * #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)
- |@(constr_comp … Hconstr Hr) @(ext_constr (h_of r))
- [#x cases (surj_pair x) #a * #b #eqx >eqx >fst_pair >snd_pair //]
- @(h_of_constr r Hr Hmono Hconstr)
- ]
-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 → constructible r →
- ∀i. CF (λx.h_of r 〈i,x〉) (λx.g (λi,x. r(h_of r 〈i,x〉)) i x).
-#r #Hr #Hmono #Hconstr #i
-@(ext_CF (λx.unary_g (λi,x. r(h_of r 〈i,x〉)) 〈i,x〉))
- [#n whd in ⊢ (??%%); @eq_f @sym_eq >fst_pair >snd_pair %]
-@smn @(ext_CF … (speed_compl r Hr Hmono Hconstr)) #n //
-qed.
-
-theorem pseudo_speedup:
- ∀r:nat →nat. (∀x. x ≤ r x) → monotonic ? le r → constructible r →
- ∃f.∀sf. CF sf f → ∃g,sg. f ≈ g ∧ CF sg g ∧ O sf (r ∘ sg).
-(* ∃m,a.∀n. a≤n → r(sg a) < m * sf n. *)
-#r #Hr #Hmono #Hconstr
-(* 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 Hconstr (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 → constructible r →
- ∃f.∀sf. CF sf f → ∃g,sg. f ≈ g ∧ CF sg g ∧
- (* ¬ O (r ∘ sg) sf. *)
- ∃m,a.∀n. a≤n → r(sg a) < m * sf n.
-#r #Hr #Hmono #Hconstr
-(* 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 Hconstr (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
+++ /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.
-
-lemma fst_pU: ∀i,x,r. fst (pU i x r) = termb i x r.
-#i #x #r normalize cases (U i x r) normalize >fst_pair //
-qed.
-
-lemma snd_pU: ∀i,x,r. snd (pU i x r) = out i x r.
-#i #x #r normalize cases (U i x r) normalize >snd_pair //
-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.
-
-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: ∀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.
-axiom MSC_pair: ∀a,b. MSC 〈a,b〉 ≤ MSC a + MSC b.
-
-(* 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 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.
-
-
-(**************************** primitive operations*****************************)
-
-definition id ≝ λx:nat.x.
-
-axiom CF_id: CF MSC id.
-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).
-axiom CF_comp_pair: ∀h,f,g. CF h f → CF h g → CF h (λx. 〈f x,g x〉).
-
-lemma CF_fst: CF MSC fst.
-@(ext_CF (fst ∘ id)) [#n //] @(CF_comp_fst … CF_id)
-qed.
-
-lemma CF_snd: CF MSC snd.
-@(ext_CF (snd ∘ id)) [#n //] @(CF_comp_snd … CF_id)
-qed.
-
-(************************************** eqb ***********************************)
-
-axiom CF_eqb: ∀h,f,g.
- CF h f → CF h g → CF h (λx.eqb (f x) (g 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〉)).
-
-(************************************* smn ************************************)
-axiom smn: ∀f,s. CF s f → ∀x. CF (λy.s 〈x,y〉) (λy.f 〈x,y〉).
-
-(****************************** constructibility ******************************)
-
-definition constructible ≝ λs. CF s s.
-
-lemma constr_comp : ∀s1,s2. constructible s1 → constructible s2 →
- (∀x. x ≤ s2 x) → constructible (s2 ∘ s1).
-#s1 #s2 #Hs1 #Hs2 #Hle @(CF_comp … Hs1 Hs2) @O_plus @le_to_O #x [@Hle | //]
-qed.
-
-lemma ext_constr: ∀s1,s2. (∀x.s1 x = s2 x) →
- constructible s1 → constructible s2.
-#s1 #s2 #Hext #Hs1 @(ext_CF … Hext) @(monotonic_CF … Hs1) #x >Hext //
-qed.
-
-(********************************* simulation *********************************)
-
-axiom sU : nat → nat.
-
-axiom monotonic_sU: ∀i1,i2,x1,x2,s1,s2. i1 ≤ i2 → x1 ≤ x2 → s1 ≤ s2 →
- sU 〈i1,〈x1,s1〉〉 ≤ sU 〈i2,〈x2,s2〉〉.
-
-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 sU_le: ∀i,x,s. s ≤ sU 〈i,〈x,s〉〉.
-axiom sU_le_i: ∀i,x,s. MSC i ≤ sU 〈i,〈x,s〉〉.
-axiom sU_le_x: ∀i,x,s. MSC x ≤ sU 〈i,〈x,s〉〉.
-
-definition pU_unary ≝ λp. pU (fst p) (fst (snd p)) (snd (snd p)).
-
-axiom CF_U : CF sU pU_unary.
-
-definition termb_unary ≝ λx:ℕ.termb (fst x) (fst (snd x)) (snd (snd x)).
-definition out_unary ≝ λx:ℕ.out (fst x) (fst (snd x)) (snd (snd x)).
-
-lemma CF_termb: CF sU termb_unary.
-@(ext_CF (fst ∘ pU_unary)) [2: @CF_comp_fst @CF_U]
-#n whd in ⊢ (??%?); whd in ⊢ (??(?%)?); >fst_pair %
-qed.
-
-lemma CF_out: CF sU out_unary.
-@(ext_CF (snd ∘ pU_unary)) [2: @CF_comp_snd @CF_U]
-#n whd in ⊢ (??%?); whd in ⊢ (??(?%)?); >snd_pair %
-qed.
-
-
-(******************** 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 (auxg h) → CF s (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
- (λp:ℕ.eqb (min_input h (fst p) (snd (snd p))) (snd (snd p))) →
- CF s2
- (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 (auxg h).
-#h #s1 #s2 #s #Hs1 #Hs2 #HO @(ext_CF (aux1g h))
- [#n whd in ⊢ (??%%); @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 (λp:ℕ.min_input h (fst p) (snd (snd p))) →
- CF s (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 compl_g4 : ∀h,s1,s.
- (CF s1
- (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 (λp:ℕ.min_input h (fst p) (snd (snd p))).
-#h #s1 #s #Hs1 #HO @(ext_CF (min_input_aux h))
- [#n whd in ⊢ (??%%); @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 //]
-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.
-
-(**************************** end of technical lemmas *************************)
-
-lemma compl_g5 : ∀h,s1.(∀n. monotonic ℕ le (λa:ℕ.s1 〈a,n〉)) →
- (CF s1
- (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〉)
- (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.
-
-lemma compl_g6: ∀h.
- constructible (λx. h (fst x) (snd x)) →
- (CF (λx. sU 〈max (fst (snd x)) (snd (snd x)),〈fst x,h (S (fst (snd x))) (fst x)〉〉)
- (λp.termb (fst (snd p)) (fst p) (h (S (fst (snd p))) (fst p)))).
-#h #hconstr @(ext_CF (termb_aux h))
- [#n normalize >fst_pair >snd_pair >fst_pair >snd_pair // ]
-@(CF_comp … (λx.MSC x + h (S (fst (snd x))) (fst x)) … CF_termb)
- [@CF_comp_pair
- [@CF_comp_fst @(monotonic_CF … CF_snd) #x //
- |@CF_comp_pair
- [@(monotonic_CF … CF_fst) #x //
- |@(ext_CF ((λx.h (fst x) (snd x))∘(λx.〈S (fst (snd x)),fst x〉)))
- [#n normalize >fst_pair >snd_pair %]
- @(CF_comp … MSC …hconstr)
- [@CF_comp_pair [@CF_compS @CF_comp_fst // |//]
- |@O_plus @le_to_O [//|#n >fst_pair >snd_pair //]
- ]
- ]
- ]
- |@O_plus
- [@O_plus
- [@(O_trans … (λx.MSC (fst x) + MSC (max (fst (snd x)) (snd (snd x)))))
- [%{2} %{0} #x #_ cases (surj_pair x) #a * #b #eqx >eqx
- >fst_pair >snd_pair @(transitive_le … (MSC_pair …))
- >distributive_times_plus @le_plus [//]
- cases (surj_pair b) #c * #d #eqb >eqb
- >fst_pair >snd_pair @(transitive_le … (MSC_pair …))
- whd in ⊢ (??%); @le_plus
- [@monotonic_MSC @(le_maxl … (le_n …))
- |>commutative_times <times_n_1 @monotonic_MSC @(le_maxr … (le_n …))
- ]
- |@O_plus [@le_to_O #x @sU_le_x |@le_to_O #x @sU_le_i]
- ]
- |@le_to_O #n @sU_le
- ]
- |@le_to_O #x @monotonic_sU // @(le_maxl … (le_n …)) ]
- ]
-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.
-
-definition faux2 ≝ λh.
- (λx.MSC x + (snd (snd x)-fst x)*
- (λx.sU 〈max (fst(snd x)) (snd(snd x)),〈fst x,h (S (fst (snd x))) (fst x)〉〉) 〈snd (snd x),x〉).
-
-lemma compl_g7: ∀h.
- constructible (λx. h (fst x) (snd x)) →
- (∀n. monotonic ? le (h n)) →
- 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))〉〉)
- (λp:ℕ.min_input h (fst p) (snd (snd p))).
-#h #hcostr #hmono @(monotonic_CF … (faux2 h))
- [#n normalize >fst_pair >snd_pair //]
-@compl_g5 [2:@(compl_g6 h hcostr)] #n #x #y #lexy >fst_pair >snd_pair
->fst_pair >snd_pair @monotonic_sU // @hmono @lexy
-qed.
-
-lemma compl_g71: ∀h.
- constructible (λx. h (fst x) (snd x)) →
- (∀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))〉〉)
- (λp:ℕ.min_input h (fst p) (snd (snd p))).
-#h #hcostr #hmono @(monotonic_CF … (compl_g7 h hcostr hmono)) #x
-@le_plus [@monotonic_MSC //]
-cases (decidable_le (fst x) (snd(snd x)))
- [#Hle @le_times // @monotonic_sU
- |#Hlt >(minus_to_0 … (lt_to_le … )) [// | @not_le_to_lt @Hlt]
- ]
-qed.
-
-definition out_aux ≝ λh.
- out_unary ∘ λp.〈fst p,〈snd(snd p),h (S (fst p)) (snd (snd p))〉〉.
-
-lemma compl_g8: ∀h.
- constructible (λx. h (fst x) (snd x)) →
- (CF (λx. sU 〈max (fst x) (snd x),〈snd(snd x),h (S (fst x)) (snd(snd x))〉〉)
- (λp.out (fst p) (snd (snd p)) (h (S (fst p)) (snd (snd p))))).
-#h #hconstr @(ext_CF (out_aux h))
- [#n normalize >fst_pair >snd_pair >fst_pair >snd_pair // ]
-@(CF_comp … (λx.h (S (fst x)) (snd(snd x)) + MSC x) … CF_out)
- [@CF_comp_pair
- [@(monotonic_CF … CF_fst) #x //
- |@CF_comp_pair
- [@CF_comp_snd @(monotonic_CF … CF_snd) #x //
- |@(ext_CF ((λx.h (fst x) (snd x))∘(λx.〈S (fst x),snd(snd x)〉)))
- [#n normalize >fst_pair >snd_pair %]
- @(CF_comp … MSC …hconstr)
- [@CF_comp_pair [@CF_compS // | @CF_comp_snd // ]
- |@O_plus @le_to_O [//|#n >fst_pair >snd_pair //]
- ]
- ]
- ]
- |@O_plus
- [@O_plus
- [@le_to_O #n @sU_le
- |@(O_trans … (λx.MSC (max (fst x) (snd x))))
- [%{2} %{0} #x #_ cases (surj_pair x) #a * #b #eqx >eqx
- >fst_pair >snd_pair @(transitive_le … (MSC_pair …))
- whd in ⊢ (??%); @le_plus
- [@monotonic_MSC @(le_maxl … (le_n …))
- |>commutative_times <times_n_1 @monotonic_MSC @(le_maxr … (le_n …))
- ]
- |@le_to_O #x @(transitive_le ???? (sU_le_i … )) //
- ]
- ]
- |@le_to_O #x @monotonic_sU [@(le_maxl … (le_n …))|//|//]
- ]
-qed.
-
-lemma compl_g9 : ∀h.
- constructible (λx. h (fst x) (snd x)) →
- (∀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)〉〉)
- (auxg h).
-#h #hconstr #hmono #hantimono
-@(compl_g2 h ??? (compl_g3 … (compl_g71 h hconstr hmono)) (compl_g8 h hconstr))
-@O_plus
- [@O_plus_l @le_to_O #x >(times_n_1 (MSC x)) >commutative_times @le_times
- [// | @monotonic_MSC // ]]
-@(O_trans … (coroll2 ??))
- [#n #a #b #leab #ltb >fst_pair >fst_pair >snd_pair >snd_pair
- cut (b ≤ n) [@(transitive_le … (le_snd …)) @lt_to_le //] #lebn
- cut (max a n = n)
- [normalize >le_to_leb_true [//|@(transitive_le … leab lebn)]] #maxa
- cut (max b n = n) [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.
- constructible (λx. h (fst x) (snd x)) →
- (∀n. monotonic ? le (h n)) →
- (∀n,a,b. a ≤ b → b ≤ n → h b n ≤ h a n) →
- CF (sg h) (unary_g h).
-#h #hconstr #Hm #Ham @compl_g1 @(compl_g9 h hconstr 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
- | 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.
-// 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.
-
-axiom h_of_constr : ∀r:nat →nat.
- (∀x. x ≤ r x) → monotonic ? le r → constructible r →
- constructible (h_of r).
-
-lemma speed_compl: ∀r:nat →nat.
- (∀x. x ≤ r x) → monotonic ? le r → constructible r →
- CF (h_of r) (unary_g (λi,x. r(h_of r 〈i,x〉))).
-#r #Hr #Hmono #Hconstr @(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)
- [%{(pred(b-a))} >S_pred [//] @lt_plus_to_minus_r @not_le_to_lt @ltab]
- * #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)
- |@(constr_comp … Hconstr Hr) @(ext_constr (h_of r))
- [#x cases (surj_pair x) #a * #b #eqx >eqx >fst_pair >snd_pair //]
- @(h_of_constr r Hr Hmono Hconstr)
- ]
-qed.
-
-lemma speed_compl_i: ∀r:nat →nat.
- (∀x. x ≤ r x) → monotonic ? le r → constructible r →
- ∀i. CF (λx.h_of r 〈i,x〉) (λx.g (λi,x. r(h_of r 〈i,x〉)) i x).
-#r #Hr #Hmono #Hconstr #i
-@(ext_CF (λx.unary_g (λi,x. r(h_of r 〈i,x〉)) 〈i,x〉))
- [#n whd in ⊢ (??%%); @eq_f @sym_eq >fst_pair >snd_pair %]
-@smn @(ext_CF … (speed_compl r Hr Hmono Hconstr)) #n //
-qed.
-
-(**************************** the speedup theorem *****************************)
-theorem pseudo_speedup:
- ∀r:nat →nat. (∀x. x ≤ r x) → monotonic ? le r → constructible r →
- ∃f.∀sf. CF sf f → ∃g,sg. f ≈ g ∧ CF sg g ∧ O sf (r ∘ sg).
-(* ∃m,a.∀n. a≤n → r(sg a) < m * sf n. *)
-#r #Hr #Hmono #Hconstr
-(* 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 Hconstr (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 → constructible r →
- ∃f.∀sf. CF sf f → ∃g,sg. f ≈ g ∧ CF sg g ∧
- (* ¬ O (r ∘ sg) sf. *)
- ∃m,a.∀n. a≤n → r(sg a) < m * sf n.
-#r #Hr #Hmono #Hconstr
-(* 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 Hconstr (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
+++ /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.
-
-lemma fst_pU: ∀i,x,r. fst (pU i x r) = termb i x r.
-#i #x #r normalize cases (U i x r) normalize >fst_pair //
-qed.
-
-lemma snd_pU: ∀i,x,r. snd (pU i x r) = out i x r.
-#i #x #r normalize cases (U i x r) normalize >snd_pair //
-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.
-
-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: ∀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.
-axiom MSC_pair: ∀a,b. MSC 〈a,b〉 ≤ MSC a + MSC b.
-
-(* 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 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.
-
-
-(**************************** primitive operations*****************************)
-
-definition id ≝ λx:nat.x.
-
-axiom CF_id: CF MSC id.
-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).
-axiom CF_comp_pair: ∀h,f,g. CF h f → CF h g → CF h (λx. 〈f x,g x〉).
-
-lemma CF_fst: CF MSC fst.
-@(ext_CF (fst ∘ id)) [#n //] @(CF_comp_fst … CF_id)
-qed.
-
-lemma CF_snd: CF MSC snd.
-@(ext_CF (snd ∘ id)) [#n //] @(CF_comp_snd … CF_id)
-qed.
-
-(************************************** eqb ***********************************)
-
-axiom CF_eqb: ∀h,f,g.
- CF h f → CF h g → CF h (λx.eqb (f x) (g 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〉)).
-
-(************************************* smn ************************************)
-axiom smn: ∀f,s. CF s f → ∀x. CF (λy.s 〈x,y〉) (λy.f 〈x,y〉).
-
-(****************************** constructibility ******************************)
-
-definition constructible ≝ λs. CF s s.
-
-lemma constr_comp : ∀s1,s2. constructible s1 → constructible s2 →
- (∀x. x ≤ s2 x) → constructible (s2 ∘ s1).
-#s1 #s2 #Hs1 #Hs2 #Hle @(CF_comp … Hs1 Hs2) @O_plus @le_to_O #x [@Hle | //]
-qed.
-
-lemma ext_constr: ∀s1,s2. (∀x.s1 x = s2 x) →
- constructible s1 → constructible s2.
-#s1 #s2 #Hext #Hs1 @(ext_CF … Hext) @(monotonic_CF … Hs1) #x >Hext //
-qed.
-
-(********************************* simulation *********************************)
-
-axiom sU : nat → nat.
-
-axiom monotonic_sU: ∀i1,i2,x1,x2,s1,s2. i1 ≤ i2 → x1 ≤ x2 → s1 ≤ s2 →
- sU 〈i1,〈x1,s1〉〉 ≤ sU 〈i2,〈x2,s2〉〉.
-
-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 sU_le: ∀i,x,s. s ≤ sU 〈i,〈x,s〉〉.
-axiom sU_le_i: ∀i,x,s. MSC i ≤ sU 〈i,〈x,s〉〉.
-axiom sU_le_x: ∀i,x,s. MSC x ≤ sU 〈i,〈x,s〉〉.
-
-definition pU_unary ≝ λp. pU (fst p) (fst (snd p)) (snd (snd p)).
-
-axiom CF_U : CF sU pU_unary.
-
-definition termb_unary ≝ λx:ℕ.termb (fst x) (fst (snd x)) (snd (snd x)).
-definition out_unary ≝ λx:ℕ.out (fst x) (fst (snd x)) (snd (snd x)).
-
-lemma CF_termb: CF sU termb_unary.
-@(ext_CF (fst ∘ pU_unary)) [2: @CF_comp_fst @CF_U]
-#n whd in ⊢ (??%?); whd in ⊢ (??(?%)?); >fst_pair %
-qed.
-
-lemma CF_out: CF sU out_unary.
-@(ext_CF (snd ∘ pU_unary)) [2: @CF_comp_snd @CF_U]
-#n whd in ⊢ (??%?); whd in ⊢ (??(?%)?); >snd_pair %
-qed.
-
-
-(******************** 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 (auxg h) → CF s (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
- (λp:ℕ.eqb (min_input h (fst p) (snd (snd p))) (snd (snd p))) →
- CF s2
- (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 (auxg h).
-#h #s1 #s2 #s #Hs1 #Hs2 #HO @(ext_CF (aux1g h))
- [#n whd in ⊢ (??%%); @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 (λp:ℕ.min_input h (fst p) (snd (snd p))) →
- CF s (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 compl_g4 : ∀h,s1,s.
- (CF s1
- (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 (λp:ℕ.min_input h (fst p) (snd (snd p))).
-#h #s1 #s #Hs1 #HO @(ext_CF (min_input_aux h))
- [#n whd in ⊢ (??%%); @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 //]
-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.
-
-(**************************** end of technical lemmas *************************)
-
-lemma compl_g5 : ∀h,s1.(∀n. monotonic ℕ le (λa:ℕ.s1 〈a,n〉)) →
- (CF s1
- (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〉)
- (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.
-
-lemma compl_g6: ∀h.
- constructible (λx. h (fst x) (snd x)) →
- (CF (λx. sU 〈max (fst (snd x)) (snd (snd x)),〈fst x,h (S (fst (snd x))) (fst x)〉〉)
- (λp.termb (fst (snd p)) (fst p) (h (S (fst (snd p))) (fst p)))).
-#h #hconstr @(ext_CF (termb_aux h))
- [#n normalize >fst_pair >snd_pair >fst_pair >snd_pair // ]
-@(CF_comp … (λx.MSC x + h (S (fst (snd x))) (fst x)) … CF_termb)
- [@CF_comp_pair
- [@CF_comp_fst @(monotonic_CF … CF_snd) #x //
- |@CF_comp_pair
- [@(monotonic_CF … CF_fst) #x //
- |@(ext_CF ((λx.h (fst x) (snd x))∘(λx.〈S (fst (snd x)),fst x〉)))
- [#n normalize >fst_pair >snd_pair %]
- @(CF_comp … MSC …hconstr)
- [@CF_comp_pair [@CF_compS @CF_comp_fst // |//]
- |@O_plus @le_to_O [//|#n >fst_pair >snd_pair //]
- ]
- ]
- ]
- |@O_plus
- [@O_plus
- [@(O_trans … (λx.MSC (fst x) + MSC (max (fst (snd x)) (snd (snd x)))))
- [%{2} %{0} #x #_ cases (surj_pair x) #a * #b #eqx >eqx
- >fst_pair >snd_pair @(transitive_le … (MSC_pair …))
- >distributive_times_plus @le_plus [//]
- cases (surj_pair b) #c * #d #eqb >eqb
- >fst_pair >snd_pair @(transitive_le … (MSC_pair …))
- whd in ⊢ (??%); @le_plus
- [@monotonic_MSC @(le_maxl … (le_n …))
- |>commutative_times <times_n_1 @monotonic_MSC @(le_maxr … (le_n …))
- ]
- |@O_plus [@le_to_O #x @sU_le_x |@le_to_O #x @sU_le_i]
- ]
- |@le_to_O #n @sU_le
- ]
- |@le_to_O #x @monotonic_sU // @(le_maxl … (le_n …)) ]
- ]
-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.
-
-definition faux2 ≝ λh.
- (λx.MSC x + (snd (snd x)-fst x)*
- (λx.sU 〈max (fst(snd x)) (snd(snd x)),〈fst x,h (S (fst (snd x))) (fst x)〉〉) 〈snd (snd x),x〉).
-
-lemma compl_g7: ∀h.
- constructible (λx. h (fst x) (snd x)) →
- (∀n. monotonic ? le (h n)) →
- 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))〉〉)
- (λp:ℕ.min_input h (fst p) (snd (snd p))).
-#h #hcostr #hmono @(monotonic_CF … (faux2 h))
- [#n normalize >fst_pair >snd_pair //]
-@compl_g5 [2:@(compl_g6 h hcostr)] #n #x #y #lexy >fst_pair >snd_pair
->fst_pair >snd_pair @monotonic_sU // @hmono @lexy
-qed.
-
-lemma compl_g71: ∀h.
- constructible (λx. h (fst x) (snd x)) →
- (∀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))〉〉)
- (λp:ℕ.min_input h (fst p) (snd (snd p))).
-#h #hcostr #hmono @(monotonic_CF … (compl_g7 h hcostr hmono)) #x
-@le_plus [@monotonic_MSC //]
-cases (decidable_le (fst x) (snd(snd x)))
- [#Hle @le_times // @monotonic_sU
- |#Hlt >(minus_to_0 … (lt_to_le … )) [// | @not_le_to_lt @Hlt]
- ]
-qed.
-
-definition out_aux ≝ λh.
- out_unary ∘ λp.〈fst p,〈snd(snd p),h (S (fst p)) (snd (snd p))〉〉.
-
-lemma compl_g8: ∀h.
- constructible (λx. h (fst x) (snd x)) →
- (CF (λx. sU 〈max (fst x) (snd x),〈snd(snd x),h (S (fst x)) (snd(snd x))〉〉)
- (λp.out (fst p) (snd (snd p)) (h (S (fst p)) (snd (snd p))))).
-#h #hconstr @(ext_CF (out_aux h))
- [#n normalize >fst_pair >snd_pair >fst_pair >snd_pair // ]
-@(CF_comp … (λx.h (S (fst x)) (snd(snd x)) + MSC x) … CF_out)
- [@CF_comp_pair
- [@(monotonic_CF … CF_fst) #x //
- |@CF_comp_pair
- [@CF_comp_snd @(monotonic_CF … CF_snd) #x //
- |@(ext_CF ((λx.h (fst x) (snd x))∘(λx.〈S (fst x),snd(snd x)〉)))
- [#n normalize >fst_pair >snd_pair %]
- @(CF_comp … MSC …hconstr)
- [@CF_comp_pair [@CF_compS // | @CF_comp_snd // ]
- |@O_plus @le_to_O [//|#n >fst_pair >snd_pair //]
- ]
- ]
- ]
- |@O_plus
- [@O_plus
- [@le_to_O #n @sU_le
- |@(O_trans … (λx.MSC (max (fst x) (snd x))))
- [%{2} %{0} #x #_ cases (surj_pair x) #a * #b #eqx >eqx
- >fst_pair >snd_pair @(transitive_le … (MSC_pair …))
- whd in ⊢ (??%); @le_plus
- [@monotonic_MSC @(le_maxl … (le_n …))
- |>commutative_times <times_n_1 @monotonic_MSC @(le_maxr … (le_n …))
- ]
- |@le_to_O #x @(transitive_le ???? (sU_le_i … )) //
- ]
- ]
- |@le_to_O #x @monotonic_sU [@(le_maxl … (le_n …))|//|//]
- ]
-qed.
-
-lemma compl_g9 : ∀h.
- constructible (λx. h (fst x) (snd x)) →
- (∀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)〉〉)
- (auxg h).
-#h #hconstr #hmono #hantimono
-@(compl_g2 h ??? (compl_g3 … (compl_g71 h hconstr hmono)) (compl_g8 h hconstr))
-@O_plus
- [@O_plus_l @le_to_O #x >(times_n_1 (MSC x)) >commutative_times @le_times
- [// | @monotonic_MSC // ]]
-@(O_trans … (coroll2 ??))
- [#n #a #b #leab #ltb >fst_pair >fst_pair >snd_pair >snd_pair
- cut (b ≤ n) [@(transitive_le … (le_snd …)) @lt_to_le //] #lebn
- cut (max a n = n)
- [normalize >le_to_leb_true [//|@(transitive_le … leab lebn)]] #maxa
- cut (max b n = n) [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.
- constructible (λx. h (fst x) (snd x)) →
- (∀n. monotonic ? le (h n)) →
- (∀n,a,b. a ≤ b → b ≤ n → h b n ≤ h a n) →
- CF (sg h) (unary_g h).
-#h #hconstr #Hm #Ham @compl_g1 @(compl_g9 h hconstr 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
- | 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.
-// 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.
-
-axiom h_of_constr : ∀r:nat →nat.
- (∀x. x ≤ r x) → monotonic ? le r → constructible r →
- constructible (h_of r).
-
-lemma speed_compl: ∀r:nat →nat.
- (∀x. x ≤ r x) → monotonic ? le r → constructible r →
- CF (h_of r) (unary_g (λi,x. r(h_of r 〈i,x〉))).
-#r #Hr #Hmono #Hconstr @(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)
- [%{(pred(b-a))} >S_pred [//] @lt_plus_to_minus_r @not_le_to_lt @ltab]
- * #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)
- |@(constr_comp … Hconstr Hr) @(ext_constr (h_of r))
- [#x cases (surj_pair x) #a * #b #eqx >eqx >fst_pair >snd_pair //]
- @(h_of_constr r Hr Hmono Hconstr)
- ]
-qed.
-
-lemma speed_compl_i: ∀r:nat →nat.
- (∀x. x ≤ r x) → monotonic ? le r → constructible r →
- ∀i. CF (λx.h_of r 〈i,x〉) (λx.g (λi,x. r(h_of r 〈i,x〉)) i x).
-#r #Hr #Hmono #Hconstr #i
-@(ext_CF (λx.unary_g (λi,x. r(h_of r 〈i,x〉)) 〈i,x〉))
- [#n whd in ⊢ (??%%); @eq_f @sym_eq >fst_pair >snd_pair %]
-@smn @(ext_CF … (speed_compl r Hr Hmono Hconstr)) #n //
-qed.
-
-(**************************** the speedup theorem *****************************)
-theorem pseudo_speedup:
- ∀r:nat →nat. (∀x. x ≤ r x) → monotonic ? le r → constructible r →
- ∃f.∀sf. CF sf f → ∃g,sg. f ≈ g ∧ CF sg g ∧ O sf (r ∘ sg).
-(* ∃m,a.∀n. a≤n → r(sg a) < m * sf n. *)
-#r #Hr #Hmono #Hconstr
-(* 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 Hconstr (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 → constructible r →
- ∃f.∀sf. CF sf f → ∃g,sg. f ≈ g ∧ CF sg g ∧
- (* ¬ O (r ∘ sg) sf. *)
- ∃m,a.∀n. a≤n → r(sg a) < m * sf n.
-#r #Hr #Hmono #Hconstr
-(* 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 Hconstr (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
+++ /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.
-
-lemma fst_pU: ∀i,x,r. fst (pU i x r) = termb i x r.
-#i #x #r normalize cases (U i x r) normalize >fst_pair //
-qed.
-
-lemma snd_pU: ∀i,x,r. snd (pU i x r) = out i x r.
-#i #x #r normalize cases (U i x r) normalize >snd_pair //
-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.
-
-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: ∀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.
-axiom MSC_pair: ∀a,b. MSC 〈a,b〉 ≤ MSC a + MSC b.
-
-(* 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 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.
-
-(* primitve recursion *)
-
-let rec prim_rec (k,h:nat →nat) n m on n ≝
- match n with
- [ O ⇒ k m
- | S a ⇒ h 〈a,〈prim_rec k h a m, m〉〉].
-
-lemma prim_rec_S: ∀k,h,n,m.
- prim_rec k h (S n) m = h 〈n,〈prim_rec k h n m, m〉〉.
-// qed.
-
-definition unary_pr ≝ λk,h,x. prim_rec k h (fst x) (snd x).
-
-let rec prim_rec_compl (k,h,sk,sh:nat →nat) n m on n ≝
- match n with
- [ O ⇒ sk m
- | S a ⇒ prim_rec_compl k h sk sh a m + sh (prim_rec k h a m)].
-
-axiom CF_prim_rec: ∀k,h,sk,sh,sf. CF sk k → CF sh h →
- O sf (unary_pr sk (λx. fst (snd x) + sh 〈fst x,〈unary_pr k h 〈fst x,snd (snd x)〉,snd (snd x)〉〉))
- → CF sf (unary_pr k h).
-
-(* falso ????
-lemma prim_rec_O: ∀k1,h1,k2,h2. O k1 k2 → O h1 h2 →
- O (unary_pr k1 h1) (unary_pr k2 h2).
-#k1 #h1 #k2 #h2 #HO1 #HO2 whd *)
-
-
-(**************************** primitive operations*****************************)
-
-definition id ≝ λx:nat.x.
-
-axiom CF_id: CF MSC id.
-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).
-axiom CF_comp_pair: ∀h,f,g. CF h f → CF h g → CF h (λx. 〈f x,g x〉).
-
-lemma CF_fst: CF MSC fst.
-@(ext_CF (fst ∘ id)) [#n //] @(CF_comp_fst … CF_id)
-qed.
-
-lemma CF_snd: CF MSC snd.
-@(ext_CF (snd ∘ id)) [#n //] @(CF_comp_snd … CF_id)
-qed.
-
-(************************************** eqb ***********************************)
-
-axiom CF_eqb: ∀h,f,g.
- CF h f → CF h g → CF h (λx.eqb (f x) (g 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〉)).
-
-(************************************* smn ************************************)
-axiom smn: ∀f,s. CF s f → ∀x. CF (λy.s 〈x,y〉) (λy.f 〈x,y〉).
-
-(****************************** constructibility ******************************)
-
-definition constructible ≝ λs. CF s s.
-
-lemma constr_comp : ∀s1,s2. constructible s1 → constructible s2 →
- (∀x. x ≤ s2 x) → constructible (s2 ∘ s1).
-#s1 #s2 #Hs1 #Hs2 #Hle @(CF_comp … Hs1 Hs2) @O_plus @le_to_O #x [@Hle | //]
-qed.
-
-lemma ext_constr: ∀s1,s2. (∀x.s1 x = s2 x) →
- constructible s1 → constructible s2.
-#s1 #s2 #Hext #Hs1 @(ext_CF … Hext) @(monotonic_CF … Hs1) #x >Hext //
-qed.
-
-lemma constr_prim_rec: ∀s1,s2. constructible s1 → constructible s2 →
- (∀n,r,m. 2 * r ≤ s2 〈n,〈r,m〉〉) → constructible (unary_pr s1 s2).
-#s1 #s2 #Hs1 #Hs2 #Hincr @(CF_prim_rec … Hs1 Hs2) whd %{2} %{0}
-#x #_ lapply (surj_pair x) * #a * #b #eqx >eqx whd in match (unary_pr ???);
->fst_pair >snd_pair
-whd in match (unary_pr ???); >fst_pair >snd_pair elim a
- [normalize //
- |#n #Hind >prim_rec_S >fst_pair >snd_pair >fst_pair >snd_pair
- >prim_rec_S @transitive_le [| @(monotonic_le_plus_l … Hind)]
- @transitive_le [| @(monotonic_le_plus_l … (Hincr n ? b))]
- whd in match (unary_pr ???); >fst_pair >snd_pair //
- ]
-qed.
-
-(********************************* simulation *********************************)
-
-axiom sU : nat → nat.
-
-axiom monotonic_sU: ∀i1,i2,x1,x2,s1,s2. i1 ≤ i2 → x1 ≤ x2 → s1 ≤ s2 →
- sU 〈i1,〈x1,s1〉〉 ≤ sU 〈i2,〈x2,s2〉〉.
-
-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 sU_le: ∀i,x,s. s ≤ sU 〈i,〈x,s〉〉.
-axiom sU_le_i: ∀i,x,s. MSC i ≤ sU 〈i,〈x,s〉〉.
-axiom sU_le_x: ∀i,x,s. MSC x ≤ sU 〈i,〈x,s〉〉.
-
-definition pU_unary ≝ λp. pU (fst p) (fst (snd p)) (snd (snd p)).
-
-axiom CF_U : CF sU pU_unary.
-
-definition termb_unary ≝ λx:ℕ.termb (fst x) (fst (snd x)) (snd (snd x)).
-definition out_unary ≝ λx:ℕ.out (fst x) (fst (snd x)) (snd (snd x)).
-
-lemma CF_termb: CF sU termb_unary.
-@(ext_CF (fst ∘ pU_unary)) [2: @CF_comp_fst @CF_U]
-#n whd in ⊢ (??%?); whd in ⊢ (??(?%)?); >fst_pair %
-qed.
-
-lemma CF_out: CF sU out_unary.
-@(ext_CF (snd ∘ pU_unary)) [2: @CF_comp_snd @CF_U]
-#n whd in ⊢ (??%?); whd in ⊢ (??(?%)?); >snd_pair %
-qed.
-
-
-(******************** 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 (auxg h) → CF s (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
- (λp:ℕ.eqb (min_input h (fst p) (snd (snd p))) (snd (snd p))) →
- CF s2
- (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 (auxg h).
-#h #s1 #s2 #s #Hs1 #Hs2 #HO @(ext_CF (aux1g h))
- [#n whd in ⊢ (??%%); @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 (λp:ℕ.min_input h (fst p) (snd (snd p))) →
- CF s (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 compl_g4 : ∀h,s1,s.
- (CF s1
- (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 (λp:ℕ.min_input h (fst p) (snd (snd p))).
-#h #s1 #s #Hs1 #HO @(ext_CF (min_input_aux h))
- [#n whd in ⊢ (??%%); @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 //]
-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.
-
-(**************************** end of technical lemmas *************************)
-
-lemma compl_g5 : ∀h,s1.(∀n. monotonic ℕ le (λa:ℕ.s1 〈a,n〉)) →
- (CF s1
- (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〉)
- (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.
-
-lemma compl_g6: ∀h.
- constructible (λx. h (fst x) (snd x)) →
- (CF (λx. sU 〈max (fst (snd x)) (snd (snd x)),〈fst x,h (S (fst (snd x))) (fst x)〉〉)
- (λp.termb (fst (snd p)) (fst p) (h (S (fst (snd p))) (fst p)))).
-#h #hconstr @(ext_CF (termb_aux h))
- [#n normalize >fst_pair >snd_pair >fst_pair >snd_pair // ]
-@(CF_comp … (λx.MSC x + h (S (fst (snd x))) (fst x)) … CF_termb)
- [@CF_comp_pair
- [@CF_comp_fst @(monotonic_CF … CF_snd) #x //
- |@CF_comp_pair
- [@(monotonic_CF … CF_fst) #x //
- |@(ext_CF ((λx.h (fst x) (snd x))∘(λx.〈S (fst (snd x)),fst x〉)))
- [#n normalize >fst_pair >snd_pair %]
- @(CF_comp … MSC …hconstr)
- [@CF_comp_pair [@CF_compS @CF_comp_fst // |//]
- |@O_plus @le_to_O [//|#n >fst_pair >snd_pair //]
- ]
- ]
- ]
- |@O_plus
- [@O_plus
- [@(O_trans … (λx.MSC (fst x) + MSC (max (fst (snd x)) (snd (snd x)))))
- [%{2} %{0} #x #_ cases (surj_pair x) #a * #b #eqx >eqx
- >fst_pair >snd_pair @(transitive_le … (MSC_pair …))
- >distributive_times_plus @le_plus [//]
- cases (surj_pair b) #c * #d #eqb >eqb
- >fst_pair >snd_pair @(transitive_le … (MSC_pair …))
- whd in ⊢ (??%); @le_plus
- [@monotonic_MSC @(le_maxl … (le_n …))
- |>commutative_times <times_n_1 @monotonic_MSC @(le_maxr … (le_n …))
- ]
- |@O_plus [@le_to_O #x @sU_le_x |@le_to_O #x @sU_le_i]
- ]
- |@le_to_O #n @sU_le
- ]
- |@le_to_O #x @monotonic_sU // @(le_maxl … (le_n …)) ]
- ]
-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.
-
-definition faux2 ≝ λh.
- (λx.MSC x + (snd (snd x)-fst x)*
- (λx.sU 〈max (fst(snd x)) (snd(snd x)),〈fst x,h (S (fst (snd x))) (fst x)〉〉) 〈snd (snd x),x〉).
-
-lemma compl_g7: ∀h.
- constructible (λx. h (fst x) (snd x)) →
- (∀n. monotonic ? le (h n)) →
- 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))〉〉)
- (λp:ℕ.min_input h (fst p) (snd (snd p))).
-#h #hcostr #hmono @(monotonic_CF … (faux2 h))
- [#n normalize >fst_pair >snd_pair //]
-@compl_g5 [2:@(compl_g6 h hcostr)] #n #x #y #lexy >fst_pair >snd_pair
->fst_pair >snd_pair @monotonic_sU // @hmono @lexy
-qed.
-
-lemma compl_g71: ∀h.
- constructible (λx. h (fst x) (snd x)) →
- (∀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))〉〉)
- (λp:ℕ.min_input h (fst p) (snd (snd p))).
-#h #hcostr #hmono @(monotonic_CF … (compl_g7 h hcostr hmono)) #x
-@le_plus [@monotonic_MSC //]
-cases (decidable_le (fst x) (snd(snd x)))
- [#Hle @le_times // @monotonic_sU
- |#Hlt >(minus_to_0 … (lt_to_le … )) [// | @not_le_to_lt @Hlt]
- ]
-qed.
-
-definition out_aux ≝ λh.
- out_unary ∘ λp.〈fst p,〈snd(snd p),h (S (fst p)) (snd (snd p))〉〉.
-
-lemma compl_g8: ∀h.
- constructible (λx. h (fst x) (snd x)) →
- (CF (λx. sU 〈max (fst x) (snd x),〈snd(snd x),h (S (fst x)) (snd(snd x))〉〉)
- (λp.out (fst p) (snd (snd p)) (h (S (fst p)) (snd (snd p))))).
-#h #hconstr @(ext_CF (out_aux h))
- [#n normalize >fst_pair >snd_pair >fst_pair >snd_pair // ]
-@(CF_comp … (λx.h (S (fst x)) (snd(snd x)) + MSC x) … CF_out)
- [@CF_comp_pair
- [@(monotonic_CF … CF_fst) #x //
- |@CF_comp_pair
- [@CF_comp_snd @(monotonic_CF … CF_snd) #x //
- |@(ext_CF ((λx.h (fst x) (snd x))∘(λx.〈S (fst x),snd(snd x)〉)))
- [#n normalize >fst_pair >snd_pair %]
- @(CF_comp … MSC …hconstr)
- [@CF_comp_pair [@CF_compS // | @CF_comp_snd // ]
- |@O_plus @le_to_O [//|#n >fst_pair >snd_pair //]
- ]
- ]
- ]
- |@O_plus
- [@O_plus
- [@le_to_O #n @sU_le
- |@(O_trans … (λx.MSC (max (fst x) (snd x))))
- [%{2} %{0} #x #_ cases (surj_pair x) #a * #b #eqx >eqx
- >fst_pair >snd_pair @(transitive_le … (MSC_pair …))
- whd in ⊢ (??%); @le_plus
- [@monotonic_MSC @(le_maxl … (le_n …))
- |>commutative_times <times_n_1 @monotonic_MSC @(le_maxr … (le_n …))
- ]
- |@le_to_O #x @(transitive_le ???? (sU_le_i … )) //
- ]
- ]
- |@le_to_O #x @monotonic_sU [@(le_maxl … (le_n …))|//|//]
- ]
-qed.
-
-lemma compl_g9 : ∀h.
- constructible (λx. h (fst x) (snd x)) →
- (∀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)〉〉)
- (auxg h).
-#h #hconstr #hmono #hantimono
-@(compl_g2 h ??? (compl_g3 … (compl_g71 h hconstr hmono)) (compl_g8 h hconstr))
-@O_plus
- [@O_plus_l @le_to_O #x >(times_n_1 (MSC x)) >commutative_times @le_times
- [// | @monotonic_MSC // ]]
-@(O_trans … (coroll2 ??))
- [#n #a #b #leab #ltb >fst_pair >fst_pair >snd_pair >snd_pair
- cut (b ≤ n) [@(transitive_le … (le_snd …)) @lt_to_le //] #lebn
- cut (max a n = n)
- [normalize >le_to_leb_true [//|@(transitive_le … leab lebn)]] #maxa
- cut (max b n = n) [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.
- constructible (λx. h (fst x) (snd x)) →
- (∀n. monotonic ? le (h n)) →
- (∀n,a,b. a ≤ b → b ≤ n → h b n ≤ h a n) →
- CF (sg h) (unary_g h).
-#h #hconstr #Hm #Ham @compl_g1 @(compl_g9 h hconstr 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
- | 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.
-// 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.
-
-lemma h_of_aux_prim_rec : ∀r,c,n,b. h_of_aux r c n b =
- prim_rec (λx.c)
- (λx.let d ≝ S(fst x) in
- let b ≝ snd (snd x) in
- (S d)*(MSC 〈〈b-d,b〉,〈b-d,b〉〉) +
- d*(S d)*sU 〈〈b-d,b〉,〈b,r (fst (snd x))〉〉) n b.
-#r #c #n #b elim n
- [>h_of_aux_O normalize //
- |#n1 #Hind >h_of_aux_S >prim_rec_S >snd_pair >snd_pair >fst_pair
- >fst_pair <Hind //
- ]
-qed.
-
-lemma h_of_aux_constr :
-∀r,c. constructible (λx.h_of_aux r c (fst x) (snd x)).
-#r #c
- @(ext_constr …
- (unary_pr (λx.c)
- (λx.let d ≝ S(fst x) in
- let b ≝ snd (snd x) in
- (S d)*(MSC 〈〈b-d,b〉,〈b-d,b〉〉) +
- d*(S d)*sU 〈〈b-d,b〉,〈b,r (fst (snd x))〉〉)))
- [#n @sym_eq whd in match (unary_pr ???); @h_of_aux_prim_rec
- |@constr_prim_rec
-
-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.
-
-axiom h_of_constr : ∀r:nat →nat.
- (∀x. x ≤ r x) → monotonic ? le r → constructible r →
- constructible (h_of r).
-
-lemma speed_compl: ∀r:nat →nat.
- (∀x. x ≤ r x) → monotonic ? le r → constructible r →
- CF (h_of r) (unary_g (λi,x. r(h_of r 〈i,x〉))).
-#r #Hr #Hmono #Hconstr @(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)
- [%{(pred(b-a))} >S_pred [//] @lt_plus_to_minus_r @not_le_to_lt @ltab]
- * #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)
- |@(constr_comp … Hconstr Hr) @(ext_constr (h_of r))
- [#x cases (surj_pair x) #a * #b #eqx >eqx >fst_pair >snd_pair //]
- @(h_of_constr r Hr Hmono Hconstr)
- ]
-qed.
-
-lemma speed_compl_i: ∀r:nat →nat.
- (∀x. x ≤ r x) → monotonic ? le r → constructible r →
- ∀i. CF (λx.h_of r 〈i,x〉) (λx.g (λi,x. r(h_of r 〈i,x〉)) i x).
-#r #Hr #Hmono #Hconstr #i
-@(ext_CF (λx.unary_g (λi,x. r(h_of r 〈i,x〉)) 〈i,x〉))
- [#n whd in ⊢ (??%%); @eq_f @sym_eq >fst_pair >snd_pair %]
-@smn @(ext_CF … (speed_compl r Hr Hmono Hconstr)) #n //
-qed.
-
-(**************************** the speedup theorem *****************************)
-theorem pseudo_speedup:
- ∀r:nat →nat. (∀x. x ≤ r x) → monotonic ? le r → constructible r →
- ∃f.∀sf. CF sf f → ∃g,sg. f ≈ g ∧ CF sg g ∧ O sf (r ∘ sg).
-(* ∃m,a.∀n. a≤n → r(sg a) < m * sf n. *)
-#r #Hr #Hmono #Hconstr
-(* 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 Hconstr (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 → constructible r →
- ∃f.∀sf. CF sf f → ∃g,sg. f ≈ g ∧ CF sg g ∧
- (* ¬ O (r ∘ sg) sf. *)
- ∃m,a.∀n. a≤n → r(sg a) < m * sf n.
-#r #Hr #Hmono #Hconstr
-(* 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 Hconstr (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
--- /dev/null
+include "reverse_complexity/speedup.ma".
+
+definition aes : (nat → nat) → (nat → nat) → Prop ≝ λf,g.
+ ∃n.∀m. n ≤ m → f m = g m.
+
+lemma CF_almost: ∀f,g,s. CF s g → aes g f → CF s f.
+#f #g #s #CFg * #n lapply CFg -CFg lapply g -g
+elim n
+ [#g #CFg #H @(ext_CF … g) [#m @H // |//]
+ |#i #Hind #g #CFg #H
+ @(Hind (λx. if eqb i x then f i else g x))
+ [@CF_if
+ [@(O_to_CF … MSC) [@le_to_O @(MSC_in … CFg)] @CF_eqb //
+ |@(O_to_CF … MSC) [@le_to_O @(MSC_in … CFg)] @CF_const
+ |@CFg
+ ]
+ |#m #leim cases (le_to_or_lt_eq … leim)
+ [#ltim lapply (lt_to_not_eq … ltim) #noteqim
+ >(not_eq_to_eqb_false … noteqim) @H @ltim
+ |#eqim >eqim >eqb_n_n //
+ ]
+ ]
+ ]
+qed.
\ No newline at end of file
--- /dev/null
+include "basics/types.ma".
+include "arithmetics/nat.ma".
+
+(********************************** 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〉.
+
+lemma expand_pair: ∀x. x = 〈fst x, snd x〉.
+#x lapply (surj_pair x) * #a * #b #Hx >Hx >fst_pair >snd_pair //
+qed.
+
+(************************************* 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.
+
+lemma fst_pU: ∀i,x,r. fst (pU i x r) = termb i x r.
+#i #x #r normalize cases (U i x r) normalize >fst_pair //
+qed.
+
+lemma snd_pU: ∀i,x,r. snd (pU i x r) = out i x r.
+#i #x #r normalize cases (U i x r) normalize >snd_pair //
+qed.
+
+
+definition total ≝ λf.λx:nat. Some nat (f x).
+
+
--- /dev/null
+
+include "arithmetics/nat.ma".
+include "basics/sets.ma".
+
+(******************************** big O notation ******************************)
+
+(* O f g means g ∈ O(f) *)
+definition O: relation (nat→nat) ≝
+ λf,g. ∃c.∃n0.∀n. n0 ≤ n → g n ≤ c* (f n).
+
+lemma O_refl: ∀s. O s s.
+#s %{1} %{0} #n #_ >commutative_times <times_n_1 @le_n qed.
+
+lemma O_trans: ∀s1,s2,s3. O s2 s1 → O s3 s2 → O s3 s1.
+#s1 #s2 #s3 * #c1 * #n1 #H1 * #c2 * # n2 #H2 %{(c1*c2)}
+%{(max n1 n2)} #n #Hmax
+@(transitive_le … (H1 ??)) [@(le_maxl … Hmax)]
+>associative_times @le_times [//|@H2 @(le_maxr … Hmax)]
+qed.
+
+lemma sub_O_to_O: ∀s1,s2. O s1 ⊆ O s2 → O s2 s1.
+#s1 #s2 #H @H // qed.
+
+lemma O_to_sub_O: ∀s1,s2. O s2 s1 → O s1 ⊆ O s2.
+#s1 #s2 #H #g #Hg @(O_trans … H) // qed.
+
+lemma le_to_O: ∀s1,s2. (∀x.s1 x ≤ s2 x) → O s2 s1.
+#s1 #s2 #Hle %{1} %{0} #n #_ normalize <plus_n_O @Hle
+qed.
+
+definition sum_f ≝ λf,g:nat→nat.λn.f n + g n.
+interpretation "function sum" 'plus f g = (sum_f f g).
+
+lemma O_plus: ∀f,g,s. O s f → O s g → O s (f+g).
+#f #g #s * #cf * #nf #Hf * #cg * #ng #Hg
+%{(cf+cg)} %{(max nf ng)} #n #Hmax normalize
+>distributive_times_plus_r @le_plus
+ [@Hf @(le_maxl … Hmax) |@Hg @(le_maxr … Hmax) ]
+qed.
+
+lemma O_plus_l: ∀f,s1,s2. O s1 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_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.
+
+lemma O_absorbr: ∀f,g,s. O s f → O f g → O s (f+g).
+#f #g #s #Osf #Ofg @(O_plus … Osf) @(O_trans … Osf) //
+qed.
+
+lemma O_times_c: ∀f,c. O f (λx:ℕ.c*f x).
+#f #c %{c} %{0} //
+qed.
+
+lemma O_ext2: ∀f,g,s. O s f → (∀x.f x = g x) → O s g.
+#f #g #s * #c * #a #Osf #eqfg %{c} %{a} #n #lean <eqfg @Osf //
+qed.
+
+
+definition not_O ≝ λf,g.∀c,n0.∃n. n0 ≤ n ∧ c* (f n) < g n .
+
+(* 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.
--- /dev/null
+include "reverse_complexity/complexity.ma".
+include "reverse_complexity/sigma_diseq.ma".
+
+include alias "reverse_complexity/basics.ma".
+
+lemma bigop_prim_rec: ∀a,b,c,p,f,x.
+ bigop (b x-a x) (λi:ℕ.p 〈i+a x,x〉) ? (c 〈a x,x〉) plus (λi:ℕ.f 〈i+a x,x〉) =
+ prim_rec c
+ (λi.if p 〈fst i +fst (snd (snd i)),snd (snd (snd i))〉
+ then plus (f 〈fst i +fst (snd (snd i)),snd (snd (snd i))〉) (fst (snd i))
+ else fst (snd i)) (b x-a x) 〈a x ,x〉.
+#a #b #c #p #f #x normalize elim (b x-a x)
+ [normalize //
+ |#i #Hind >prim_rec_S
+ >fst_pair >snd_pair >snd_pair >fst_pair >snd_pair >fst_pair
+ cases (true_or_false (p 〈i+a x,x〉)) #Hcase >Hcase
+ [<Hind >bigop_Strue // |<Hind >bigop_Sfalse // ]
+ ]
+qed.
+
+lemma bigop_prim_rec_dec: ∀a,b,c,p,f,x.
+ bigop (b x-a x) (λi:ℕ.p 〈b x -i,x〉) ? (c 〈b x,x〉) plus (λi:ℕ.f 〈b x-i,x〉) =
+ prim_rec c
+ (λi.if p 〈fst (snd (snd i)) - fst i,snd (snd (snd i))〉
+ then plus (f 〈fst (snd (snd i)) - fst i,snd (snd (snd i))〉) (fst (snd i))
+ else fst (snd i)) (b x-a x) 〈b x ,x〉.
+#a #b #c #p #f #x normalize elim (b x-a x)
+ [normalize //
+ |#i #Hind >prim_rec_S
+ >fst_pair >snd_pair >snd_pair >fst_pair >snd_pair >fst_pair
+ cases (true_or_false (p 〈b x - i,x〉)) #Hcase >Hcase
+ [<Hind >bigop_Strue // |<Hind >bigop_Sfalse // ]
+ ]
+qed.
+
+lemma bigop_prim_rec_dec1: ∀a,b,c,p,f,x.
+ bigop (S(b x)-a x) (λi:ℕ.p 〈b x - i,x〉) ? (c 〈b x,x〉) plus (λi:ℕ.f 〈b x- i,x〉) =
+ prim_rec c
+ (λi.if p 〈fst (snd (snd i)) - (fst i),snd (snd (snd i))〉
+ then plus (f 〈fst (snd (snd i)) - (fst i),snd (snd (snd i))〉) (fst (snd i))
+ else fst (snd i)) (S(b x)-a x) 〈b x,x〉.
+#a #b #c #p #f #x elim (S(b x)-a x)
+ [normalize //
+ |#i #Hind >prim_rec_S
+ >fst_pair >snd_pair >snd_pair >fst_pair >snd_pair >fst_pair
+ cases (true_or_false (p 〈b x - i,x〉)) #Hcase >Hcase
+ [<Hind >bigop_Strue // |<Hind >bigop_Sfalse // ]
+ ]
+qed.
+
+lemma sum_prim_rec1: ∀a,b,p,f,x.
+ ∑_{i ∈[a x,b x[ | p 〈i,x〉 }(f 〈i,x〉) =
+ prim_rec (λi.0)
+ (λi.if p 〈fst i +fst (snd (snd i)),snd (snd (snd i))〉
+ then f 〈fst i +fst (snd (snd i)),snd (snd (snd i))〉 + fst (snd i)
+ else fst (snd i)) (b x-a x) 〈a x ,x〉.
+#a #b #p #f #x elim (b x-a x)
+ [normalize //
+ |#i #Hind >prim_rec_S
+ >fst_pair >snd_pair >snd_pair >fst_pair >snd_pair >fst_pair
+ cases (true_or_false (p 〈i+a x,x〉)) #Hcase >Hcase
+ [<Hind >bigop_Strue // |<Hind >bigop_Sfalse // ]
+ ]
+qed.
+
+lemma bigop_plus_c: ∀k,p,f,c.
+ c k + bigop k (λi.p i) ? 0 plus (λi.f i) =
+ bigop k (λi.p i) ? (c k) plus (λi.f i).
+#k #p #f elim k [normalize //]
+#i #Hind #c cases (true_or_false (p i)) #Hcase
+[>bigop_Strue // >bigop_Strue // <associative_plus >(commutative_plus ? (f i))
+ >associative_plus @eq_f @Hind
+|>bigop_Sfalse // >bigop_Sfalse //
+]
+qed.
+
+
+(*********************************** maximum **********************************)
+
+lemma max_gen: ∀a,b,p,f,x. a ≤b →
+ max_{i ∈[a,b[ | p 〈i,x〉 }(f 〈i,x〉) = max_{i < b | leb a i ∧ p 〈i,x〉 }(f 〈i,x〉).
+#a #b #p #f #x @(bigop_I_gen ????? MaxA)
+qed.
+
+lemma max_prim_rec_base: ∀a,b,p,f,x. a ≤b →
+ max_{i < b| p 〈i,x〉 }(f 〈i,x〉) =
+ prim_rec (λi.0)
+ (λi.if p 〈fst i,x〉 then max (f 〈fst i,snd (snd i)〉) (fst (snd i)) else fst (snd i)) b x.
+#a #b #p #f #x #leab >max_gen // elim b
+ [normalize //
+ |#i #Hind >prim_rec_S >fst_pair >snd_pair >snd_pair >fst_pair
+ cases (true_or_false (p 〈i,x〉)) #Hcase >Hcase
+ [<Hind >bigop_Strue // |<Hind >bigop_Sfalse // ]
+ ]
+qed.
+
+lemma max_prim_rec: ∀a,b,p,f,x. a ≤b →
+ max_{i ∈[a,b[ | p 〈i,x〉 }(f 〈i,x〉) =
+ prim_rec (λi.0)
+ (λi.if leb a (fst i) ∧ p 〈fst i,x〉 then max (f 〈fst i,snd (snd i)〉) (fst (snd i)) else fst (snd i)) b x.
+#a #b #p #f #x #leab >max_gen // elim b
+ [normalize //
+ |#i #Hind >prim_rec_S >fst_pair >snd_pair >snd_pair >fst_pair
+ cases (true_or_false (leb a i ∧ p 〈i,x〉)) #Hcase >Hcase
+ [<Hind >bigop_Strue // |<Hind >bigop_Sfalse // ]
+ ]
+qed.
+
+lemma max_prim_rec1: ∀a,b,p,f,x.
+ max_{i ∈[a x,b x[ | p 〈i,x〉 }(f 〈i,x〉) =
+ prim_rec (λi.0)
+ (λi.if p 〈fst i +fst (snd (snd i)),snd (snd (snd i))〉
+ then max (f 〈fst i +fst (snd (snd i)),snd (snd (snd i))〉) (fst (snd i))
+ else fst (snd i)) (b x-a x) 〈a x ,x〉.
+#a #b #p #f #x elim (b x-a x)
+ [normalize //
+ |#i #Hind >prim_rec_S
+ >fst_pair >snd_pair >snd_pair >fst_pair >snd_pair >fst_pair
+ cases (true_or_false (p 〈i+a x,x〉)) #Hcase >Hcase
+ [<Hind >bigop_Strue // |<Hind >bigop_Sfalse // ]
+ ]
+qed.
+
+(* the argument is 〈b-a,〈a,x〉〉 *)
+
+definition max_unary_pr ≝ λp,f.unary_pr (λx.0)
+ (λi.
+ let k ≝ fst i in
+ let r ≝ fst (snd i) in
+ let a ≝ fst (snd (snd i)) in
+ let x ≝ snd (snd (snd i)) in
+ if p 〈k + a,x〉 then max (f 〈k+a,x〉) r else r).
+
+lemma max_unary_pr1: ∀a,b,p,f,x.
+ max_{i ∈[a x,b x[ | p 〈i,x〉 }(f 〈i,x〉) =
+ ((max_unary_pr p f) ∘ (λx.〈b x - a x,〈a x,x〉〉)) x.
+#a #b #p #f #x normalize >fst_pair >snd_pair @max_prim_rec1
+qed.
+
+definition aux_compl ≝ λhp,hf.λi.
+ let k ≝ fst i in
+ let r ≝ fst (snd i) in
+ let a ≝ fst (snd (snd i)) in
+ let x ≝ snd (snd (snd i)) in
+ hp 〈k+a,x〉 + hf 〈k+a,x〉 + (* was MSC r*) MSC i .
+
+definition aux_compl1 ≝ λhp,hf.λi.
+ let k ≝ fst i in
+ let r ≝ fst (snd i) in
+ let a ≝ fst (snd (snd i)) in
+ let x ≝ snd (snd (snd i)) in
+ hp 〈k+a,x〉 + hf 〈k+a,x〉 + MSC r.
+
+lemma aux_compl1_def: ∀k,r,m,hp,hf.
+ aux_compl1 hp hf 〈k,〈r,m〉〉 =
+ let a ≝ fst m in
+ let x ≝ snd m in
+ hp 〈k+a,x〉 + hf 〈k+a,x〉 + MSC r.
+#k #r #m #hp #hf normalize >fst_pair >snd_pair >snd_pair >fst_pair //
+qed.
+
+lemma aux_compl1_def1: ∀k,r,a,x,hp,hf.
+ aux_compl1 hp hf 〈k,〈r,〈a,x〉〉〉 = hp 〈k+a,x〉 + hf 〈k+a,x〉 + MSC r.
+#k #r #a #x #hp #hf normalize >fst_pair >snd_pair >snd_pair >fst_pair
+>fst_pair >snd_pair //
+qed.
+
+axiom Oaux_compl: ∀hp,hf. O (aux_compl1 hp hf) (aux_compl hp hf).
+
+lemma MSC_max: ∀f,h,x. CF h f → MSC (max_{i < x}(f i)) ≤ max_{i < x}(h i).
+#f #h #x #hf elim x // #i #Hind >bigop_Strue [|//] >bigop_Strue [|//]
+whd in match (max ??);
+cases (true_or_false (leb (f i) (bigop i (λi0:ℕ.true) ? 0 max(λi0:ℕ.f i0))))
+#Hcase >Hcase
+ [@(transitive_le … Hind) @le_maxr //
+ |@(transitive_le … (MSC_out … hf i)) @le_maxl //
+ ]
+qed.
+
+lemma 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〉 + max_{i ∈ [a x, b x [ }(hf 〈i,x〉)))) →
+ CF s (λx.max_{i ∈[a x,b x[ | p 〈i,x〉 }(f 〈i,x〉)).
+#a #b #p #f #ha #hb #hp #hf #s #CFa #CFb #CFp #CFf #HO
+@ext_CF1 [|#x @max_unary_pr1]
+@(CF_comp ??? (λx.ha x + hb x))
+ [|@CF_comp_pair
+ [@CF_minus [@(O_to_CF … CFb) @O_plus_r // |@(O_to_CF … CFa) @O_plus_l //]
+ |@CF_comp_pair
+ [@(O_to_CF … CFa) @O_plus_l //
+ | @(O_to_CF … CF_id) @O_plus_r @le_to_O @(MSC_in … CFb)
+ ]
+ ]
+ |@(CF_prim_rec … MSC … (aux_compl1 hp hf))
+ [@CF_const
+ |@(O_to_CF … (Oaux_compl … ))
+ @CF_if
+ [@(CF_comp p … MSC … CFp)
+ [@CF_comp_pair
+ [@CF_plus [@CF_fst| @CF_comp_fst @CF_comp_snd @CF_snd]
+ |@CF_comp_snd @CF_comp_snd @CF_snd]
+ |@le_to_O #x normalize >commutative_plus >associative_plus @le_plus //
+ ]
+ |@CF_max1
+ [@(CF_comp f … MSC … CFf)
+ [@CF_comp_pair
+ [@CF_plus [@CF_fst| @CF_comp_fst @CF_comp_snd @CF_snd]
+ |@CF_comp_snd @CF_comp_snd @CF_snd]
+ |@le_to_O #x normalize >commutative_plus //
+ ]
+ |@CF_comp_fst @(monotonic_CF … CF_snd) normalize //
+ ]
+ |@CF_comp_fst @(monotonic_CF … CF_snd) normalize //
+ ]
+ |@O_refl
+ ]
+ |@(O_trans … HO) -HO
+ @(O_trans ? (λx:ℕ
+ .ha x+hb x
+ +bigop (b x-a x) (λi:ℕ.true) ? (MSC 〈a x,x〉) plus
+ (λi:ℕ
+ .(λi0:ℕ
+ .hp 〈i0,x〉+hf 〈i0,x〉
+ +bigop (b x-a x) (λi1:ℕ.true) ? 0 max
+ (λi1:ℕ.(λi2:ℕ.hf 〈i2,x〉) (i1+a x))) (i+a x))))
+ [
+ @le_to_O #n @le_plus // whd in match (unary_pr ???);
+ >fst_pair >snd_pair >bigop_prim_rec elim (b n - a n)
+ [normalize //
+ |#x #Hind >prim_rec_S >fst_pair >snd_pair >fst_pair >snd_pair >aux_compl1_def1
+ >prim_rec_S >fst_pair >snd_pair >fst_pair >snd_pair >fst_pair
+ >snd_pair normalize in ⊢ (??%); >commutative_plus @le_plus
+ [-Hind @le_plus // normalize >fst_pair >snd_pair
+ @(transitive_le ? (bigop x (λi1:ℕ.true) ? 0 (λn0:ℕ.λm:ℕ.if leb n0 m then m else n0 )
+ (λi1:ℕ.hf 〈i1+a n,n〉)))
+ [elim x [normalize @MSC_le]
+ #x0 #Hind >prim_rec_S >fst_pair >snd_pair >snd_pair >snd_pair
+ >fst_pair >fst_pair cases (p 〈x0+a n,n〉) normalize
+ [cases (true_or_false (leb (f 〈x0+a n,n〉)
+ (prim_rec (λx00:ℕ.O)
+ (λi:ℕ
+ .if p 〈fst i+fst (snd (snd i)),snd (snd (snd i))〉
+ then if leb (f 〈fst i+fst (snd (snd i)),snd (snd (snd i))〉)
+ (fst (snd i))
+ then fst (snd i)
+ else f 〈fst i+fst (snd (snd i)),snd (snd (snd i))〉
+ else fst (snd i) ) x0 〈a n,n〉))) #Hcase >Hcase normalize
+ [@(transitive_le … Hind) -Hind @(le_maxr … (le_n …))
+ |@(transitive_le … (MSC_out … CFf …)) @(le_maxl … (le_n …))
+ ]
+ |@(transitive_le … Hind) -Hind @(le_maxr … (le_n …))
+ ]
+ |@(le_maxr … (le_n …))
+ ]
+ |@(transitive_le … Hind) -Hind
+ generalize in match (bigop x (λi:ℕ.true) ? 0 max
+ (λi1:ℕ.(λi2:ℕ.hf 〈i2,n〉) (i1+a n))); #c
+ generalize in match (hf 〈x+a n,n〉); #c1
+ elim x [//] #x0 #Hind
+ >prim_rec_S >prim_rec_S normalize >fst_pair >fst_pair >snd_pair
+ >snd_pair >snd_pair >snd_pair >snd_pair >snd_pair >fst_pair >fst_pair
+ >fst_pair @le_plus
+ [@le_plus // cases (true_or_false (leb c1 c)) #Hcase
+ >Hcase normalize // @lt_to_le @not_le_to_lt @(leb_false_to_not_le ?? Hcase)
+ |@Hind
+ ]
+ ]
+ ]
+ |@O_plus [@O_plus_l //] @le_to_O #x
+ <bigop_plus_c @le_plus // @(transitive_le … (MSC_pair …)) @le_plus
+ [@MSC_out @CFa | @MSC_out @(O_to_CF MSC … (CF_const x)) @le_to_O @(MSC_in … CFb)]
+ ]
+qed.
+
+axiom daemon : ∀P:Prop.P.
+axiom O_extl: ∀s1,s2,f. (∀x.s1 x = s2 x) → O s1 f → O s2 f.
+
+lemma CF_max2: ∀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 +
+ (b x - a x)*max_{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〉)).
+#a #b #p #f #ha #hb #hp #hf #s #CFa #CFb #CFp #CFf #Os
+@(O_to_CF … Os (CF_max … CFa CFb CFp CFf ?)) @O_plus
+ [@O_plus_l @O_refl
+ |@O_plus_r @O_ext2 [|| #x @(bigop_op … plusAC)]
+ @O_plus
+ [@le_to_O normalize #x @sigma_to_Max
+ |@le_to_O #x @transitive_le [|@sigma_const] @le_times //
+ @Max_le #i #lti #_ @(transitive_le ???? (le_MaxI … ))
+ [@le_plus_n | // | @lt_minus_to_plus_r // | //]
+ ]
+ ]
+qed.
+
+(*
+lemma CF_max_monotonic: ∀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 +
+ (b x - a x)*max_{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〉)).
+#a #b #p #f #ha #hb #hp #hf #s #CFa #CFb #CFp #CFf #Os
+@(O_to_CF … Os (CF_max … CFa CFb CFp CFf ?)) @O_plus
+ [@O_plus_l @O_refl
+ |@O_plus_r @O_ext2 [|| #x @(bigop_op … plusAC)]
+ @O_plus
+ [@le_to_O normalize #x @sigma_to_Max
+ |@le_to_O #x @transitive_le [|@sigma_const] @le_times //
+ @Max_le #i #lti #_ @(transitive_le ???? (le_MaxI … ))
+ [@le_plus_n | // | @lt_minus_to_plus_r // | //]
+ ]
+ ]
+qed.
+*)
+
+(* old
+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 ********************************)
+
+let rec my_minim a f x k on k ≝
+ match k with
+ [O ⇒ a
+ |S p ⇒ if eqb (my_minim a f x p) (a+p)
+ then if f 〈a+p,x〉 then a+p else S(a+p)
+ else (my_minim a f x p) ].
+
+lemma my_minim_S : ∀a,f,x,k.
+ my_minim a f x (S k) =
+ if eqb (my_minim a f x k) (a+k)
+ then if f 〈a+k,x〉 then a+k else S(a+k)
+ else (my_minim a f x k) .
+// qed.
+
+lemma my_minim_fa : ∀a,f,x,k. f〈a,x〉 = true → my_minim a f x k = a.
+#a #f #x #k #H elim k // #i #Hind normalize >Hind
+cases (true_or_false (eqb a (a+i))) #Hcase >Hcase normalize //
+<(eqb_true_to_eq … Hcase) >H //
+qed.
+
+lemma my_minim_nfa : ∀a,f,x,k. f〈a,x〉 = false →
+my_minim a f x (S k) = my_minim (S a) f x k.
+#a #f #x #k #H elim k
+ [normalize <plus_n_O >H >eq_to_eqb_true //
+ |#i #Hind >my_minim_S >Hind >my_minim_S <plus_n_Sm //
+ ]
+qed.
+
+lemma my_min_eq : ∀a,f,x,k.
+ min k a (λi.f 〈i,x〉) = my_minim a f x k.
+#a #f #x #k lapply a -a elim k // #i #Hind #a normalize in ⊢ (??%?);
+cases (true_or_false (f 〈a,x〉)) #Hcase >Hcase
+ [>(my_minim_fa … Hcase) // | >Hind @sym_eq @(my_minim_nfa … Hcase) ]
+qed.
+
+(* cambiare qui: togliere S *)
+
+
+definition minim_unary_pr ≝ λf.unary_pr (λx.S(fst x))
+ (λi.
+ let k ≝ fst i in
+ let r ≝ fst (snd i) in
+ let b ≝ fst (snd (snd i)) in
+ let x ≝ snd (snd (snd i)) in
+ if f 〈b-k,x〉 then b-k else r).
+
+definition compl_minim_unary ≝ λhf:nat → nat.λi.
+ let k ≝ fst i in
+ let b ≝ fst (snd (snd i)) in
+ let x ≝ snd (snd (snd i)) in
+ hf 〈b-k,x〉 + MSC 〈k,〈S b,x〉〉.
+
+definition compl_minim_unary_aux ≝ λhf,i.
+ let k ≝ fst i in
+ let r ≝ fst (snd i) in
+ let b ≝ fst (snd (snd i)) in
+ let x ≝ snd (snd (snd i)) in
+ hf 〈b-k,x〉 + MSC i.
+
+lemma compl_minim_unary_aux_def : ∀hf,k,r,b,x.
+ compl_minim_unary_aux hf 〈k,〈r,〈b,x〉〉〉 = hf 〈b-k,x〉 + MSC 〈k,〈r,〈b,x〉〉〉.
+#hf #k #r #b #x normalize >snd_pair >snd_pair >snd_pair >fst_pair >fst_pair //
+qed.
+
+lemma compl_minim_unary_def : ∀hf,k,r,b,x.
+ compl_minim_unary hf 〈k,〈r,〈b,x〉〉〉 = hf 〈b-k,x〉 + MSC 〈k,〈S b,x〉〉.
+#hf #k #r #b #x normalize >snd_pair >snd_pair >snd_pair >fst_pair >fst_pair //
+qed.
+
+lemma compl_minim_unary_aux_def2 : ∀hf,k,r,x.
+ compl_minim_unary_aux hf 〈k,〈r,x〉〉 = hf 〈fst x-k,snd x〉 + MSC 〈k,〈r,x〉〉.
+#hf #k #r #x normalize >snd_pair >snd_pair >fst_pair //
+qed.
+
+lemma compl_minim_unary_def2 : ∀hf,k,r,x.
+ compl_minim_unary hf 〈k,〈r,x〉〉 = hf 〈fst x-k,snd x〉 + MSC 〈k,〈S(fst x),snd x〉〉.
+#hf #k #r #x normalize >snd_pair >snd_pair >fst_pair //
+qed.
+
+lemma min_aux: ∀a,f,x,k. min (S k) (a x) (λi:ℕ.f 〈i,x〉) =
+ ((minim_unary_pr f) ∘ (λx.〈S k,〈a x + k,x〉〉)) x.
+#a #f #x #k whd in ⊢ (???%); >fst_pair >snd_pair
+lapply a -a (* @max_prim_rec1 *)
+elim k
+ [normalize #a >fst_pair >snd_pair >fst_pair >snd_pair >snd_pair >fst_pair
+ <plus_n_O <minus_n_O >fst_pair //
+ |#i #Hind #a normalize in ⊢ (??%?); >prim_rec_S >fst_pair >snd_pair
+ >fst_pair >snd_pair >snd_pair >fst_pair <plus_n_Sm <(Hind (λx.S (a x)))
+ whd in ⊢ (???%); >minus_S_S <(minus_plus_m_m (a x) i) //
+qed.
+
+lemma minim_unary_pr1: ∀a,b,f,x.
+ μ_{i ∈[a x,b x]}(f 〈i,x〉) =
+ if leb (a x) (b x) then ((minim_unary_pr f) ∘ (λx.〈S (b x) - a x,〈b x,x〉〉)) x
+ else (a x).
+#a #b #f #x cases (true_or_false (leb (a x) (b x))) #Hcase >Hcase
+ [cut (b x = a x + (b x - a x))
+ [>commutative_plus <plus_minus_m_m // @leb_true_to_le // ]
+ #Hcut whd in ⊢ (???%); >minus_Sn_m [|@leb_true_to_le //]
+ >min_aux whd in ⊢ (??%?); <Hcut //
+ |>eq_minus_O // @not_le_to_lt @leb_false_to_not_le //
+ ]
+qed.
+
+axiom sum_inv: ∀a,b,f.∑_{i ∈ [a,S b[ }(f i) = ∑_{i ∈ [a,S b[ }(f (a + b - i)).
+
+(*
+#a #b #f @(bigop_iso … plusAC) whd %{(λi.b - a - i)} %{(λi.b - a -i)} %
+ [%[#i #lti #_ normalize @eq_f >commutative_plus <plus_minus_associative
+ [2: @le_minus_to_plus_r //
+ [// @eq_f @@sym_eq @plus_to_minus
+ |#i #Hi #_ % [% [@le_S_S
+*)
+
+example sum_inv_check : ∑_{i ∈ [3,6[ }(i*i) = ∑_{i ∈ [3,6[ }((8-i)*(8-i)).
+normalize // qed.
+
+(* rovesciamo la somma *)
+
+lemma 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〉 + MSC 〈b x - i,〈S(b x),x〉〉)) →
+ CF s (λx.μ_{i ∈[a x,b x] }(f 〈i,x〉)).
+#a #b #f #ha #hb #hf #s #CFa #CFb #CFf #HO
+@ext_CF1 [|#x @minim_unary_pr1]
+@CF_if
+ [@CF_le @(O_to_CF … HO)
+ [@(O_to_CF … CFa) @O_plus_l @O_plus_l @O_refl
+ |@(O_to_CF … CFb) @O_plus_l @O_plus_r @O_refl
+ ]
+ |@(CF_comp ??? (λx.ha x + hb x))
+ [|@CF_comp_pair
+ [@CF_minus [@CF_compS @(O_to_CF … CFb) @O_plus_r // |@(O_to_CF … CFa) @O_plus_l //]
+ |@CF_comp_pair
+ [@(O_to_CF … CFb) @O_plus_r //
+ |@(O_to_CF … CF_id) @O_plus_r @le_to_O @(MSC_in … CFb)
+ ]
+ ]
+ |@(CF_prim_rec_gen … MSC … (compl_minim_unary_aux hf))
+ [@((λx:ℕ.fst (snd x)
+ +compl_minim_unary hf
+ 〈fst x,
+ 〈unary_pr fst
+ (λi:ℕ
+ .(let (k:ℕ) ≝fst i in
+ let (r:ℕ) ≝fst (snd i) in
+ let (b:ℕ) ≝fst (snd (snd i)) in
+ let (x:ℕ) ≝snd (snd (snd i)) in if f 〈b-S k,x〉 then b-S k else r ))
+ 〈fst x,snd (snd x)〉,
+ snd (snd x)〉〉))
+ |@CF_compS @CF_fst
+ |@CF_if
+ [@(CF_comp f … MSC … CFf)
+ [@CF_comp_pair
+ [@CF_minus [@CF_comp_fst @CF_comp_snd @CF_snd|@CF_fst]
+ |@CF_comp_snd @CF_comp_snd @CF_snd]
+ |@le_to_O #x normalize >commutative_plus //
+ ]
+ |@(O_to_CF … MSC)
+ [@le_to_O #x normalize //
+ |@CF_minus
+ [@CF_comp_fst @CF_comp_snd @CF_snd |@CF_fst]
+ ]
+ |@CF_comp_fst @(monotonic_CF … CF_snd) normalize //
+ ]
+ |@O_plus
+ [@O_plus_l @O_refl
+ |@O_plus_r @O_ext2 [||#x >compl_minim_unary_aux_def2 @refl]
+ @O_trans [||@le_to_O #x >compl_minim_unary_def2 @le_n]
+ @O_plus [@O_plus_l //]
+ @O_plus_r
+ @O_trans [|@le_to_O #x @MSC_pair] @O_plus
+ [@le_to_O #x @monotonic_MSC @(transitive_le ???? (le_fst …))
+ >fst_pair @le_n]
+ @O_trans [|@le_to_O #x @MSC_pair] @O_plus
+ [@le_to_O #x @monotonic_MSC @(transitive_le ???? (le_snd …))
+ >snd_pair @(transitive_le ???? (le_fst …)) >fst_pair
+ normalize >snd_pair >fst_pair lapply (surj_pair x)
+ * #x1 * #x2 #Hx >Hx >fst_pair >snd_pair elim x1 //
+ #i #Hind >prim_rec_S >fst_pair >snd_pair >snd_pair >fst_pair
+ cases (f ?) [@le_S // | //]]
+ @le_to_O #x @monotonic_MSC @(transitive_le ???? (le_snd …)) >snd_pair
+ >(expand_pair (snd (snd x))) in ⊢ (?%?); @le_pair //
+ ]
+ ]
+ |cut (O s (λx.ha x + hb x +
+ ∑_{i ∈[a x ,S(b x)[ }(hf 〈a x+b x-i,x〉 + MSC 〈b x -(a x+b x-i),〈S(b x),x〉〉)))
+ [@(O_ext2 … HO) #x @eq_f @sum_inv] -HO #HO
+ @(O_trans … HO) -HO
+ @(O_trans ? (λx:ℕ.ha x+hb x
+ +bigop (S(b x)-a x) (λi:ℕ.true) ?
+ (MSC 〈b x,x〉) plus (λi:ℕ.(λj.hf j + MSC 〈b x - fst j,〈S(b (snd j)),snd j〉〉) 〈b x- i,x〉)))
+ [@le_to_O #n @le_plus // whd in match (unary_pr ???);
+ >fst_pair >snd_pair >(bigop_prim_rec_dec1 a b ? (λi.true))
+ (* it is crucial to recall that the index is bound by S(b x) *)
+ cut (S (b n) - a n ≤ S (b n)) [//]
+ elim (S(b n) - a n)
+ [normalize //
+ |#x #Hind #lex >prim_rec_S >fst_pair >snd_pair >fst_pair >snd_pair
+ >compl_minim_unary_def >prim_rec_S >fst_pair >snd_pair >fst_pair
+ >snd_pair >fst_pair >snd_pair >fst_pair whd in ⊢ (??%); >commutative_plus
+ @le_plus [2:@Hind @le_S @le_S_S_to_le @lex] -Hind >snd_pair
+ >minus_minus_associative // @le_S_S_to_le //
+ ]
+ |@O_plus [@O_plus_l //] @O_ext2 [||#x @bigop_plus_c]
+ @O_plus
+ [@O_plus_l @O_trans [|@le_to_O #x @MSC_pair]
+ @O_plus
+ [@O_plus_r @le_to_O @(MSC_out … CFb)
+ |@O_plus_r @le_to_O @(MSC_in … CFb)
+ ]
+ |@O_plus_r @(O_ext2 … (O_refl …)) #x @same_bigop
+ [//|#i #H #_ @eq_f2 [@eq_f @eq_f2 //|>fst_pair @eq_f @eq_f2 //]
+ ]
+ ]
+ ]
+ ]
+ |@(O_to_CF … CFa) @(O_trans … HO) @O_plus_l @O_plus_l @O_refl
+ ]
+
+qed.
+
+lemma CF_mu2: ∀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〉 + MSC〈S(b x),x〉)) →
+ CF s (λx.μ_{i ∈[a x,b x] }(f 〈i,x〉)).
+#a #b #f #sa #sb #sf #s #CFa #CFb #CFf #HO
+@(O_to_CF … HO (CF_mu … CFa CFb CFf ?)) @O_plus [@O_plus_l @O_refl]
+@O_plus_r @O_ext2 [|| #x @(bigop_op … plusAC)]
+@O_plus [@le_to_O #x @le_sigma //]
+@(O_trans ? (λx.∑_{i ∈[a x ,S(b x)[ }(MSC(b x -i)+MSC 〈S(b x),x〉)))
+ [@le_to_O #x @le_sigma //]
+@O_ext2 [|| #x @(bigop_op … plusAC)] @O_plus
+ [@le_to_O #x @le_sigma // #i #lti #_ @(transitive_le … (MSC 〈S (b x),x〉)) //
+ @monotonic_MSC @(transitive_le … (S(b x))) // @le_S //
+ |@le_to_O #x @le_sigma //
+ ]
+qed.
+
+(* uses MSC_S *)
+
+lemma CF_mu3: ∀a,b.∀f:nat →bool.∀sa,sb,sf,s. (∀x.sf x > 0) →
+ 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〉 + MSC〈b x,x〉)) →
+ CF s (λx.μ_{i ∈[a x,b x] }(f 〈i,x〉)).
+#a #b #f #sa #sb #sf #s #sfpos #CFa #CFb #CFf #HO
+@(O_to_CF … HO (CF_mu2 … CFa CFb CFf ?)) @O_plus [@O_plus_l @O_refl]
+@O_plus_r @O_ext2 [|| #x @(bigop_op … plusAC)]
+@O_plus [@le_to_O #x @le_sigma //]
+@le_to_O #x @le_sigma // #x #lti #_ >MSC_pair_eq >MSC_pair_eq <associative_plus
+@le_plus // @(transitive_le … (MSC_sublinear … )) /2 by monotonic_lt_plus_l/
+qed.
+
+lemma CF_mu4: ∀a,b.∀f:nat →bool.∀sa,sb,sf,s. (∀x.sf x > 0) →
+ CF sa a → CF sb b → CF sf f →
+ O s (λx.sa x + sb x + (S(b x) - a x)*Max_{i ∈[a x ,S(b x)[ }(sf 〈i,x〉)) →
+ CF s (λx.μ_{i ∈[a x,b x] }(f 〈i,x〉)).
+#a #b #f #sa #sb #sf #s #sfpos #CFa #CFb #CFf #HO
+@(O_to_CF … HO (CF_mu3 … sfpos CFa CFb CFf ?)) @O_plus [@O_plus_l @O_refl]
+@O_ext2 [|| #x @(bigop_op … plusAC)] @O_plus_r @O_plus
+ [@le_to_O #x @sigma_to_Max
+ |lapply (MSC_in … CFf) #Hle
+ %{1} %{0} #n #_ @(transitive_le … (sigma_const …))
+ >(commutative_times 1) <times_n_1
+ cases (decidable_le (S (b n)) (a n)) #H
+ [>(eq_minus_O … H) //
+ |lapply (le_S_S_to_le … (not_le_to_lt … H)) -H #H
+ @le_times // @(transitive_le … (Hle … ))
+ cut (b n = b n - a n + a n) [<plus_minus_m_m // ]
+ #Hcut >Hcut in ⊢ (?%?); @(le_Max (λi.sf 〈i+a n,n〉)) /2/
+ ]
+ ]
+qed.
+
+(*
+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〉)). *)
+
+
+
\ No newline at end of file
--- /dev/null
+include "reverse_complexity/basics.ma".
+include "reverse_complexity/big_O.ma".
+
+(********************************* 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_sublinear : ∀n. MSC (S n) ≤ S (MSC n).
+(* axiom MSC_S: O MSC (λx.MSC (S x)) . *)
+axiom MSC_le: ∀n. MSC n ≤ n.
+
+axiom monotonic_MSC: monotonic ? le MSC.
+axiom MSC_pair_eq: ∀a,b. MSC 〈a,b〉 = MSC a + MSC b.
+
+
+lemma MSC_pair: ∀a,b. MSC 〈a,b〉 ≤ MSC a + MSC b. // qed.
+
+(* 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.∃i.code_for (total f) i ∧ C s i.
+
+axiom MSC_in: ∀f,h. CF h f → ∀x. MSC x ≤ h x.
+axiom MSC_out: ∀f,h. CF h f → ∀x. MSC (f x) ≤ h x.
+
+
+lemma ext_CF : ∀f,g,s. (∀n. f n = g n) → CF s f → CF s g.
+#f #g #s #Hext * #i * #Hcode #HC %{i} %
+ [#x cases (Hcode x) #a #H %{a} whd in match (total ??); <Hext @H | //]
+qed.
+
+lemma ext_CF1 : ∀f,g,s. (∀n. f n = g n) → CF s g → CF s f.
+#f #g #s #Hext * #i * #Hcode #HC %{i} %
+ [#x cases (Hcode x) #a #H %{a} whd in match (total ??); >Hext @H | //]
+qed.
+
+lemma monotonic_CF: ∀s1,s2,f.(∀x. s1 x ≤ s2 x) → CF s1 f → CF s2 f.
+#s1 #s2 #f #H * #i * #Hcode #Hs1
+%{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 * #i * #Hcode #Hs1
+%{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.
+
+(************************************* smn ************************************)
+axiom smn: ∀f,s. CF s f → ∀x. CF (λy.s 〈x,y〉) (λy.f 〈x,y〉).
+
+(****************************** constructibility ******************************)
+
+definition constructible ≝ λs. CF s s.
+
+lemma constr_comp : ∀s1,s2. constructible s1 → constructible s2 →
+ (∀x. x ≤ s2 x) → constructible (s2 ∘ s1).
+#s1 #s2 #Hs1 #Hs2 #Hle @(CF_comp … Hs1 Hs2) @O_plus @le_to_O #x [@Hle | //]
+qed.
+
+lemma ext_constr: ∀s1,s2. (∀x.s1 x = s2 x) →
+ constructible s1 → constructible s2.
+#s1 #s2 #Hext #Hs1 @(ext_CF … Hext) @(monotonic_CF … Hs1) #x >Hext //
+qed.
+
+
+(***************************** primitive recursion ****************************)
+
+(* no arguments *)
+
+let rec prim_rec0 (k:nat) (h:nat →nat) n on n ≝
+ match n with
+ [ O ⇒ k
+ | S a ⇒ h 〈a, prim_rec0 k h a〉].
+
+lemma prim_rec0_S: ∀k,h,n.
+ prim_rec0 k h (S n) = h 〈n, prim_rec0 k h n〉.
+// qed.
+
+let rec prim_rec0_compl (k,sk:nat) (h,sh:nat →nat) n on n ≝
+ match n with
+ [ O ⇒ sk
+ | S a ⇒ prim_rec0_compl k sk h sh a + sh (prim_rec0 k h a)].
+
+axiom CF_prim_rec0_gen: ∀k,h,sk,sh,sh1,sf. CF sh h →
+ O sh1 (λx.snd x + sh 〈fst x,prim_rec0 k h (fst x)〉) →
+ O sf (prim_rec0 sk sh1) →
+ CF sf (prim_rec0 k h).
+
+lemma CF_prim_rec0: ∀k,h,sk,sh,sf. CF sh h →
+ O sf (prim_rec0 sk (λx. snd x + sh 〈fst x,prim_rec0 k h (fst x)〉))
+ → CF sf (prim_rec0 k h).
+#k #h #sk #sh #sf #Hh #HO @(CF_prim_rec0_gen … Hh … HO) @O_refl
+qed.
+
+(* with arguments. m is a vector of arguments *)
+let rec prim_rec (k,h:nat →nat) n m on n ≝
+ match n with
+ [ O ⇒ k m
+ | S a ⇒ h 〈a,〈prim_rec k h a m, m〉〉].
+
+lemma prim_rec_S: ∀k,h,n,m.
+ prim_rec k h (S n) m = h 〈n,〈prim_rec k h n m, m〉〉.
+// qed.
+
+lemma prim_rec_le: ∀k1,k2,h1,h2. (∀x.k1 x ≤ k2 x) →
+(∀x,y.x ≤y → h1 x ≤ h2 y) →
+ ∀x,m. prim_rec k1 h1 x m ≤ prim_rec k2 h2 x m.
+#k1 #k2 #h1 #h2 #lek #leh #x #m elim x //
+#n #Hind normalize @leh @le_pair // @le_pair //
+qed.
+
+definition unary_pr ≝ λk,h,x. prim_rec k h (fst x) (snd x).
+
+lemma prim_rec_unary: ∀k,h,a,b.
+ prim_rec k h a b = unary_pr k h 〈a,b〉.
+#k #h #a #b normalize >fst_pair >snd_pair //
+qed.
+
+let rec prim_rec_compl (k,h,sk,sh:nat →nat) n m on n ≝
+ match n with
+ [ O ⇒ sk m
+ | S a ⇒ prim_rec_compl k h sk sh a m + sh 〈a,〈prim_rec k h a m,m〉〉].
+
+axiom CF_prim_rec_gen: ∀k,h,sk,sh,sh1. CF sk k → CF sh h →
+ O sh1 (λx. fst (snd x) +
+ sh 〈fst x,〈unary_pr k h 〈fst x,snd (snd x)〉,snd (snd x)〉〉) →
+ CF (unary_pr sk sh1) (unary_pr k h).
+
+lemma CF_prim_rec: ∀k,h,sk,sh,sf. CF sk k → CF sh h →
+ O sf (unary_pr sk
+ (λx. fst (snd x) +
+ sh 〈fst x,〈unary_pr k h 〈fst x,snd (snd x)〉,snd (snd x)〉〉))
+ → CF sf (unary_pr k h).
+#k #h #sk #sh #sf #Hk #Hh #Osf @(O_to_CF … Osf) @(CF_prim_rec_gen … Hk Hh)
+@O_refl
+qed.
+
+lemma constr_prim_rec: ∀s1,s2. constructible s1 → constructible s2 →
+ (∀n,r,m. 2 * r ≤ s2 〈n,〈r,m〉〉) → constructible (unary_pr s1 s2).
+#s1 #s2 #Hs1 #Hs2 #Hincr @(CF_prim_rec … Hs1 Hs2) whd %{2} %{0}
+#x #_ lapply (surj_pair x) * #a * #b #eqx >eqx whd in match (unary_pr ???);
+>fst_pair >snd_pair
+whd in match (unary_pr ???); >fst_pair >snd_pair elim a
+ [normalize //
+ |#n #Hind >prim_rec_S >fst_pair >snd_pair >fst_pair >snd_pair
+ >prim_rec_S @transitive_le [| @(monotonic_le_plus_l … Hind)]
+ @transitive_le [| @(monotonic_le_plus_l … (Hincr n ? b))]
+ whd in match (unary_pr ???); >fst_pair >snd_pair //
+ ]
+qed.
+
+(**************************** primitive operations*****************************)
+
+definition id ≝ λx:nat.x.
+
+axiom CF_id: CF MSC id.
+axiom CF_const: ∀k.CF MSC (λx.k).
+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).
+axiom CF_comp_pair: ∀h,f,g. CF h f → CF h g → CF h (λx. 〈f x,g x〉).
+
+lemma CF_fst: CF MSC fst.
+@(ext_CF (fst ∘ id)) [#n //] @(CF_comp_fst … CF_id)
+qed.
+
+lemma CF_snd: CF MSC snd.
+@(ext_CF (snd ∘ id)) [#n //] @(CF_comp_snd … CF_id)
+qed.
+
+(**************************** arithmetic operations ***************************)
+
+axiom CF_compS: ∀h,f. CF h f → CF h (S ∘ f).
+axiom CF_le: ∀h,f,g. CF h f → CF h g → CF h (λx. leb (f x) (g x)).
+axiom CF_eqb: ∀h,f,g. CF h f → CF h g → CF h (λx.eqb (f x) (g x)).
+axiom CF_max1: ∀h,f,g. CF h f → CF h g → CF h (λx. max (f x) (g x)).
+axiom CF_plus: ∀h,f,g. CF h f → CF h g → CF h (λx. f x + g x).
+axiom CF_minus: ∀h,f,g. CF h f → CF h g → CF h (λx. f x - g x).
+
+(******************************** if then else ********************************)
+
+lemma if_prim_rec: ∀b:nat → bool. ∀f,g:nat → nat.∀x:nat.
+ if b x then f x else g x = prim_rec g (f ∘ snd ∘ snd) (b x) x.
+#b #f #g #x cases (b x) normalize //
+qed.
+
+lemma CF_if: ∀b:nat → bool. ∀f,g,s. CF s b → CF s f → CF s g →
+ CF s (λx. if b x then f x else g x).
+#b #f #g #s #CFb #CFf #CFg @(ext_CF (λx.unary_pr g (f ∘ snd ∘ snd) 〈b x,x〉))
+ [#n @sym_eq normalize >fst_pair >snd_pair @if_prim_rec
+ |@(CF_comp ??? s)
+ [|@CF_comp_pair // @(O_to_CF … CF_id) @le_to_O @MSC_in //
+ |@(CF_prim_rec_gen ??? (λx.MSC x + s(snd (snd x))) … CFg) [3:@O_refl|]
+ @(CF_comp … (λx.MSC x + s(snd x)) … CF_snd)
+ [@(CF_comp … s … CF_snd CFf) @O_refl
+ |@O_plus
+ [@O_plus_l @O_refl
+ |@O_plus
+ [@O_plus_l @le_to_O #x @monotonic_MSC //
+ |@O_plus_r @O_refl
+ ]
+ ]
+ ]
+ |%{6} %{0} #n #_ normalize in ⊢ (??%); <plus_n_O cases (b n) normalize
+ >snd_pair >fst_pair normalize
+ [@le_plus [//] >fst_pair >fst_pair >snd_pair >fst_pair
+ @le_plus [//] >snd_pair >snd_pair >snd_pair >snd_pair
+ <associative_plus <associative_plus @le_plus [|//]
+ @(transitive_le … (MSC_pair …)) >associative_plus @le_plus
+ [@(transitive_le ???? (MSC_in … CFf n)) @monotonic_MSC //
+ |@(transitive_le … (MSC_pair …)) @le_plus [|@(MSC_in … CFf)]
+ normalize @MSC_out @CFg
+ ]
+ |@le_plus //
+ ]
+ ]
+ ]
+qed.
+
+(********************************* simulation *********************************)
+
+definition pU_unary ≝ λp. pU (fst p) (fst (snd p)) (snd (snd p)).
+
+axiom sU : nat → nat.
+axiom CF_U : CF sU pU_unary.
+
+axiom monotonic_sU: ∀i1,i2,x1,x2,s1,s2. i1 ≤ i2 → x1 ≤ x2 → s1 ≤ s2 →
+ sU 〈i1,〈x1,s1〉〉 ≤ sU 〈i2,〈x2,s2〉〉.
+
+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 sU_pos: ∀x. 0 < sU x.
+
+axiom sU_le: ∀i,x,s. s ≤ sU 〈i,〈x,s〉〉.
+
+lemma sU_le_i: ∀i,x,s. MSC i ≤ sU 〈i,〈x,s〉〉.
+#i #x #s @(transitive_le ???? (MSC_in … CF_U 〈i,〈x,s〉〉)) @monotonic_MSC //
+qed.
+
+lemma sU_le_x: ∀i,x,s. MSC x ≤ sU 〈i,〈x,s〉〉.
+#i #x #s @(transitive_le ???? (MSC_in … CF_U 〈i,〈x,s〉〉)) @monotonic_MSC
+@(transitive_le … 〈x,s〉) //
+qed.
+
+
+
+
+definition termb_unary ≝ λx:ℕ.termb (fst x) (fst (snd x)) (snd (snd x)).
+definition out_unary ≝ λx:ℕ.out (fst x) (fst (snd x)) (snd (snd x)).
+
+lemma CF_termb: CF sU termb_unary.
+@(ext_CF (fst ∘ pU_unary)) [2: @CF_comp_fst @CF_U]
+#n whd in ⊢ (??%?); whd in ⊢ (??(?%)?); >fst_pair %
+qed.
+
+lemma CF_out: CF sU out_unary.
+@(ext_CF (snd ∘ pU_unary)) [2: @CF_comp_snd @CF_U]
+#n whd in ⊢ (??%?); whd in ⊢ (??(?%)?); >snd_pair %
+qed.
+
--- /dev/null
+
+include "arithmetics/minimization.ma".
+include "arithmetics/bigops.ma".
+include "arithmetics/pidgeon_hole.ma".
+include "arithmetics/iteration.ma".
+
+(************************** notation for miminimization ***********************)
+
+(* an alternative defintion of minimization
+definition Min ≝ λa,f.
+ \big[min,a]_{i < a | f i} i. *)
+
+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 (S $b-$a) $a (λ${ident i}.$p)}.
+
+lemma f_min_true: ∀f,a,b.
+ (∃i. a ≤ i ∧ i ≤ b ∧ f i = true) → f (μ_{i ∈[a,b]} (f i)) = true.
+#f #a #b * #i * * #Hil #Hir #Hfi @(f_min_true … (λx. f x)) <plus_minus_m_m
+ [%{i} % // % [@Hil |@le_S_S @Hir]|@le_S @(transitive_le … Hil Hir)]
+qed.
+
+lemma min_up: ∀f,a,b.
+ (∃i. a ≤ i ∧ i ≤ b ∧ f i = true) → μ_{i ∈[a,b]}(f i) ≤ b.
+#f #a #b * #i * * #Hil #Hir #Hfi @le_S_S_to_le
+cut ((S b) = S b - a + a) [@plus_minus_m_m @le_S @(transitive_le … Hil Hir)]
+#Hcut >Hcut in ⊢ (??%); @lt_min %{i} % // % [@Hil |<Hcut @le_S_S @Hir]
+qed.
+
+(*************************** Kleene's predicate *******************************)
+
+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 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.
+
+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.
+
+lemma decidable_test : ∀n,x,r,r1.
+ (∀i. i < n → 〈i,x〉 ↓ r ∨ ¬ 〈i,x〉 ↓ r1) ∨
+ (∃i. i < n ∧ (¬ 〈i,x〉 ↓ r ∧ 〈i,x〉 ↓ r1)).
+#n #x #r1 #r2
+ cut (∀i0.decidable ((〈i0,x〉↓r1) ∨ ¬ 〈i0,x〉 ↓ r2))
+ [#j @decidable_or [@terminate_dec |@decidable_not @terminate_dec ]] #Hdec
+ cases(decidable_forall ? Hdec n)
+ [#H %1 @H
+ |#H %2 cases (not_forall_to_exists … Hdec H) #j * #leji #Hj
+ %{j} % // %
+ [@(not_to_not … Hj) #H %1 @H
+ |cases (terminate_dec j x r2) // #H @False_ind cases Hj -Hj #Hj
+ @Hj %2 @H
+ ]
+qed.
+
+(**************************** the gap theorem *********************************)
+definition gapP ≝ λn,x,g,r. ∀i. i < n → 〈i,x〉 ↓ r ∨ ¬ 〈i,x〉 ↓ g r.
+
+lemma gapP_def : ∀n,x,g,r.
+ gapP n x g r = ∀i. i < n → 〈i,x〉 ↓ r ∨ ¬ 〈i,x〉 ↓ g r.
+// qed.
+
+lemma upper_bound_aux: ∀g,b,n,x. (∀x. x ≤ g x) → ∀k.
+ (∃j.j < k ∧
+ (∀i. i < n → 〈i,x〉 ↓ g^j b ∨ ¬ 〈i,x〉 ↓ g^(S j) b)) ∨
+ ∃l. |l| = k ∧ unique ? l ∧ ∀i. i ∈ l → i < n ∧ 〈i,x〉 ↓ g^k b .
+#g#b #n #x #Hg #k elim k
+ [%2 %{([])} normalize % [% //|#x @False_ind]
+ |#k0 *
+ [* #j * #lej #H %1 %{j} % [@le_S // | @H ]
+ |* #l * * #Hlen #Hunique #Hterm
+ cases (decidable_test n x (g^k0 b) (g^(S k0) b))
+ [#Hcase %1 %{k0} % [@le_n | @Hcase]
+ |* #j * #ltjn * #H1 #H2 %2
+ %{(j::l)} %
+ [ % [normalize @eq_f @Hlen] whd % // % #H3
+ @(absurd ?? H1) @(proj2 … (Hterm …)) @H3
+ |#x *
+ [#eqxj >eqxj % //
+ |#Hmemx cases(Hterm … Hmemx) #lexn * #y #HU
+ % [@lexn] %{y} @(monotonic_U ?????? HU) @Hg
+ ]
+ ]
+ ]
+ ]
+ ]
+qed.
+
+lemma upper_bound: ∀g,b,n,x. (∀x. x ≤ g x) → ∃r.
+ (* b ≤ r ∧ r ≤ g^n b ∧ ∀i. i < n → 〈i,x〉 ↓ r ∨ ¬ 〈i,x〉 ↓ g r. *)
+ b ≤ r ∧ r ≤ g^n b ∧ gapP n x g r.
+#g #b #n #x #Hg
+cases (upper_bound_aux g b n x Hg n)
+ [* #j * #Hj #H %{(g^j b)} % [2: @H] % [@le_iter //]
+ @monotonic_iter2 // @lt_to_le //
+ |* #l * * #Hlen #Hunique #Hterm %{(g^n b)} %
+ [% [@le_iter // |@le_n]]
+ #i #lein %1 @(proj2 … (Hterm ??))
+ @(eq_length_to_mem_all … Hlen Hunique … lein)
+ #x #memx @(proj1 … (Hterm ??)) //
+ ]
+qed.
+
+definition gapb ≝ λn,x,g,r.
+ \big[andb,true]_{i < n} ((termb i x r) ∨ ¬(termb i x (g r))).
+
+lemma gapb_def : ∀n,x,g,r. gapb n x g r =
+ \big[andb,true]_{i < n} ((termb i x r) ∨ ¬(termb i x (g r))).
+// qed.
+
+lemma gapb_true_to_gapP : ∀n,x,g,r.
+ gapb n x g r = true → ∀i. i < n → 〈i,x〉 ↓ r ∨ ¬(〈i,x〉 ↓ (g r)).
+#n #x #g #r elim n
+ [>gapb_def >bigop_Strue //
+ #H #i #lti0 @False_ind @(absurd … lti0) @le_to_not_lt //
+ |#m #Hind >gapb_def >bigop_Strue //
+ #H #i #leSm cases (le_to_or_lt_eq … leSm)
+ [#lem @Hind [@(andb_true_r … H)|@le_S_S_to_le @lem]
+ |#eqi >(injective_S … eqi) lapply (andb_true_l … H) -H #H cases (orb_true_l … H) -H
+ [#H %1 @termb_true_to_term //
+ |#H %2 % #H1 >(term_to_termb_true … H1) in H; normalize #H destruct
+ ]
+ ]
+ ]
+qed.
+
+lemma gapP_to_gapb_true : ∀n,x,g,r.
+ (∀i. i < n → 〈i,x〉 ↓ r ∨ ¬(〈i,x〉 ↓ (g r))) → gapb n x g r = true.
+#n #x #g #r elim n //
+#m #Hind #H >gapb_def >bigop_Strue // @true_to_andb_true
+ [cases (H m (le_n …))
+ [#H2 @orb_true_r1 @term_to_termb_true //
+ |#H2 @orb_true_r2 @sym_eq @noteq_to_eqnot @sym_not_eq
+ @(not_to_not … H2) @termb_true_to_term
+ ]
+ |@Hind #i0 #lei0 @H @le_S //
+ ]
+qed.
+
+
+(* the gap function *)
+let rec gap g n on n ≝
+ match n with
+ [ O ⇒ 1
+ | S m ⇒ let b ≝ gap g m in μ_{i ∈ [b,g^n b]} (gapb n n g i)
+ ].
+
+lemma gapS: ∀g,m.
+ gap g (S m) =
+ (let b ≝ gap g m in
+ μ_{i ∈ [b,g^(S m) b]} (gapb (S m) (S m) g i)).
+// qed.
+
+lemma upper_bound_gapb: ∀g,m. (∀x. x ≤ g x) →
+ ∃r:ℕ.gap g m ≤ r ∧ r ≤ g^(S m) (gap g m) ∧ gapb (S m) (S m) g r = true.
+#g #m #leg
+lapply (upper_bound g (gap g m) (S m) (S m) leg) * #r * *
+#H1 #H2 #H3 %{r} %
+ [% // |@gapP_to_gapb_true @H3]
+qed.
+
+lemma gapS_true: ∀g,m. (∀x. x ≤g x) → gapb (S m) (S m) g (gap g (S m)) = true.
+#g #m #leg @(f_min_true (gapb (S m) (S m) g)) @upper_bound_gapb //
+qed.
+
+theorem gap_theorem: ∀g,i.(∀x. x ≤ g x)→∃k.∀n.k < n →
+ 〈i,n〉 ↓ (gap g n) ∨ ¬ 〈i,n〉 ↓ (g (gap g n)).
+#g #i #leg %{i} *
+ [#lti0 @False_ind @(absurd ?? (not_le_Sn_O i) ) //
+ |#m #leim lapply (gapS_true g m leg) #H
+ @(gapb_true_to_gapP … H) //
+ ]
+qed.
+
+(* an upper bound *)
+
+let rec sigma n ≝
+ match n with
+ [ O ⇒ 0 | S m ⇒ n + sigma m ].
+
+lemma gap_bound: ∀g. (∀x. x ≤ g x) → (monotonic ? le g) →
+ ∀n.gap g n ≤ g^(sigma n) 1.
+#g #leg #gmono #n elim n
+ [normalize //
+ |#m #Hind >gapS @(transitive_le ? (g^(S m) (gap g m)))
+ [@min_up @upper_bound_gapb //
+ |@(transitive_le ? (g^(S m) (g^(sigma m) 1)))
+ [@monotonic_iter // |>iter_iter >commutative_plus @le_n
+ ]
+ ]
+qed.
+
+lemma gap_bound2: ∀g. (∀x. x ≤ g x) → (monotonic ? le g) →
+ ∀n.gap g n ≤ g^(n*n) 1.
+#g #leg #gmono #n elim n
+ [normalize //
+ |#m #Hind >gapS @(transitive_le ? (g^(S m) (gap g m)))
+ [@min_up @upper_bound_gapb //
+ |@(transitive_le ? (g^(S m) (g^(m*m) 1)))
+ [@monotonic_iter //
+ |>iter_iter @monotonic_iter2 [@leg | normalize <plus_n_Sm @le_S_S //
+ ]
+ ]
+qed.
+
+(*
+axiom universal: ∃u.∀i,x,y.
+ ∃n. U u 〈i,x〉 n = Some y ↔ ∃m.U i x m = Some y. *)
+
+
+
+
+
+
+
+
+
+
+
+
--- /dev/null
+
+include "arithmetics/nat.ma".
+include "arithmetics/log.ma".
+include "arithmetics/bigops.ma".
+include "arithmetics/bounded_quantifiers.ma".
+include "arithmetics/pidgeon_hole.ma".
+include "basics/sets.ma".
+include "basics/types.ma".
+
+(************************************ 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.
+
+alias id "max" = "cic:/matita/arithmetics/nat/max#def:2".
+alias id "mk_Aop" = "cic:/matita/arithmetics/bigops/Aop#con:0:1:2".
+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 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.
+
+(******************************** big O notation ******************************)
+
+(* O f g means g ∈ O(f) *)
+definition O: relation (nat→nat) ≝
+ λf,g. ∃c.∃n0.∀n. n0 ≤ n → g n ≤ c* (f n).
+
+lemma O_refl: ∀s. O s s.
+#s %{1} %{0} #n #_ >commutative_times <times_n_1 @le_n qed.
+
+lemma O_trans: ∀s1,s2,s3. O s2 s1 → O s3 s2 → O s3 s1.
+#s1 #s2 #s3 * #c1 * #n1 #H1 * #c2 * # n2 #H2 %{(c1*c2)}
+%{(max n1 n2)} #n #Hmax
+@(transitive_le … (H1 ??)) [@(le_maxl … Hmax)]
+>associative_times @le_times [//|@H2 @(le_maxr … Hmax)]
+qed.
+
+lemma sub_O_to_O: ∀s1,s2. O s1 ⊆ O s2 → O s2 s1.
+#s1 #s2 #H @H // qed.
+
+lemma O_to_sub_O: ∀s1,s2. O s2 s1 → O s1 ⊆ O s2.
+#s1 #s2 #H #g #Hg @(O_trans … H) // qed.
+
+definition sum_f ≝ λf,g:nat→nat.λn.f n + g n.
+interpretation "function sum" 'plus f g = (sum_f f g).
+
+lemma O_plus: ∀f,g,s. O s f → O s g → O s (f+g).
+#f #g #s * #cf * #nf #Hf * #cg * #ng #Hg
+%{(cf+cg)} %{(max nf ng)} #n #Hmax normalize
+>distributive_times_plus_r @le_plus
+ [@Hf @(le_maxl … Hmax) |@Hg @(le_maxr … Hmax) ]
+qed.
+
+lemma O_plus_l: ∀f,s1,s2. O s1 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_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.
+
+lemma O_absorbr: ∀f,g,s. O s f → O f g → O s (f+g).
+#f #g #s #Osf #Ofg @(O_plus … Osf) @(O_trans … Osf) //
+qed.
+
+lemma O_ext2: ∀f,g,s. O s f → (∀x.f x = g x) → O s g.
+#f #g #s * #c * #a #Osf #eqfg %{c} %{a} #n #lean <eqfg @Osf //
+qed.
+
+definition not_O ≝ λf,g.∀c,n0.∃n. n0 ≤ n ∧ c* (f n) < g n .
+
+(******************************* 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.
+axiom fst : nat → nat.
+axiom snd : nat → nat.
+axiom fst_pair: ∀a,b. fst (pair a b) = a.
+axiom snd_pair: ∀a,b. snd (pair a b) = b.
+
+interpretation "abstract pair" 'pair f g = (pair f g).
+
+(************************ basic complexity notions ****************************)
+
+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}.
+
+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.
+#f #i #x normalize #H %
+ [* #n * #y * #H1 #posy %{y} % //
+ cases (H x) -H #m #H <(H (max n m)) [2:@(le_maxr … n) //]
+ @(monotonic_U … H1) @(le_maxl … m) //
+ |cases (H x) -H #m #Hm * #y #Hy %{m} %{y} >Hm //
+ ]
+qed.
+
+(******************************* complexity classes ***************************)
+
+axiom size: nat → nat.
+axiom of_size: nat → nat.
+
+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|.
+
+definition size_f ≝ λf,n.Max_{i < S (of_size n) | eqb (|i|) n}|(f i)|.
+
+lemma size_f_def: ∀f,n. size_f f n =
+ Max_{i < S (of_size n) | eqb (|i|) n}|(f i)|.
+// qed.
+
+lemma size_f_size : ∀f,n. size_f (f ∘ size) n = |(f n)|.
+#f #n @le_to_le_to_eq
+ [@Max_le #a #lta #Ha normalize >(eqb_true_to_eq … Ha) //
+ |<(size_of_size n) in ⊢ (?%?); >size_f_def
+ @(le_Max (λi.|f (|i|)|) ? (S (of_size n)) (of_size n) ??)
+ [@le_S_S // | @eq_to_eqb_true //]
+ ]
+qed.
+
+lemma size_f_id : ∀n. size_f (λx.x) n = n.
+#n @le_to_le_to_eq
+ [@Max_le #a #lta #Ha >(eqb_true_to_eq … Ha) //
+ |<(size_of_size n) in ⊢ (?%?); >size_f_def
+ @(le_Max (λi.|i|) ? (S (of_size n)) (of_size n) ??)
+ [@le_S_S // | @eq_to_eqb_true //]
+ ]
+qed.
+
+lemma size_f_fst : ∀n. size_f fst n ≤ n.
+#n @Max_le #a #lta #Ha <(eqb_true_to_eq … Ha) //
+qed.
+
+(* C s i means that the complexity of i is O(s) *)
+
+definition C ≝ λs,i.∃c.∃a.∀x.a ≤ |x| → {i ⊙ x} ↓ (c*(s(|x|))).
+
+definition CF ≝ λs,f.∃i.code_for 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 * #i * #Hcode #HC %{i} %
+ [#x cases (Hcode x) #a #H %{a} <Hext @H | //]
+qed.
+
+lemma monotonic_CF: ∀s1,s2,f. O s2 s1 → CF s1 f → CF s2 f.
+#s1 #s2 #f * #c1 * #a #H * #i * #Hcodef #HCs1 %{i} % //
+cases HCs1 #c2 * #b #H2 %{(c2*c1)} %{(max a b)}
+#x #Hmax cases (H2 x ?) [2:@(le_maxr … Hmax)] #y #Hy
+%{y} @(monotonic_U …Hy) >associative_times @le_times // @H @(le_maxl … Hmax)
+qed.
+
+(*********************** The hierachy theorem (left) **************************)
+
+theorem hierarchy_theorem_left: ∀s1,s2:nat→nat.
+ O(s1) ⊆ O(s2) → CF s1 ⊆ CF s2.
+#s1 #s2 #HO #f * #i * #Hcode * #c * #a #Hs1_i %{i} % //
+cases (sub_O_to_O … HO) -HO #c1 * #b #Hs1s2
+%{(c*c1)} %{(max a b)} #x #lemax
+cases (Hs1_i x ?) [2: @(le_maxl …lemax)]
+#y #Hy %{y} @(monotonic_U … Hy) >associative_times
+@le_times // @Hs1s2 @(le_maxr … lemax)
+qed.
+
+(************************** The diagonal language *****************************)
+
+(* the diagonal language used for the hierarchy theorem *)
+
+definition diag ≝ λs,i.
+ 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.
+#s #i %
+ [whd in ⊢ (%→?); #H % [%{0} //] % * #x * #y *
+ #H1 #Hy cut (0 = y) [@(unique_U … H H1)] #eqy /2/
+ |* * #y cases y //
+ #y0 #H * #H1 @False_ind @H1 -H1 whd %{(s (|i|))} %{(S y0)} % //
+ ]
+qed.
+
+(* Let us define the characteristic function diag_cf for diag, and prove
+it correctness *)
+
+definition diag_cf ≝ λs,i.
+ match U (fst i) i (s (|i|)) with
+ [ None ⇒ None ?
+ | Some y ⇒ if (eqb y 0) then (Some ? 1) else (Some ? 0)].
+
+lemma diag_cf_OK: ∀s,x. diag s x ↔ ∃y.diag_cf s x = Some ? y ∧ 0 < y.
+#s #x %
+ [whd in ⊢ (%→?); #H %{1} % // whd in ⊢ (??%?); >H //
+ |* #y * whd in ⊢ (??%?→?→%);
+ 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) //
+ |normalize #H destruct #H @False_ind @(absurd ? H) @lt_to_not_le //
+ ]
+ ]
+ ]
+qed.
+
+lemma diag_spec: ∀s,i. code_for (diag_cf s) i → ∀x. lang i x ↔ diag s x.
+#s #i #Hcode #x @(iff_trans … (lang_cf … Hcode)) @iff_sym @diag_cf_OK
+qed.
+
+(******************************************************************************)
+
+lemma absurd1: ∀P. iff P (¬ P) →False.
+#P * #H1 #H2 cut (¬P) [% #H2 @(absurd … H2) @H1 //]
+#H3 @(absurd ?? H3) @H2 @H3
+qed.
+
+let rec f_img (f:nat →nat) n on n ≝
+ match n with
+ [O ⇒ [ ]
+ |S m ⇒ f m::f_img f m
+ ].
+
+(* a few lemma to prove injective_to_exists. This is a general result; a nice
+example of the pidgeon hole pricniple *)
+
+lemma f_img_to_exists:
+ ∀f.∀n,a. a ∈ f_img f n → ∃b. b < n ∧ f b = a.
+#f #n #a elim n normalize [@False_ind]
+#m #Hind *
+ [#Ha %{m} /2/ |#H cases(Hind H) #b * #Hb #Ha %{b} % // @le_S //]
+qed.
+
+lemma length_f_img: ∀f,n. |f_img f n| = n.
+#f #n elim n // normalize //
+qed.
+
+lemma unique_f_img: ∀f,n. injective … f → unique ? (f_img f n).
+#f #n #Hinj elim n normalize //
+#m #Hind % // % #H lapply(f_img_to_exists …H) * #b * #ltbm
+#eqbm @(absurd … ltbm) @le_to_not_lt >(Hinj… eqbm) //
+qed.
+
+lemma injective_to_exists: ∀f. injective nat nat f →
+ ∀n.(∀i.i < n → f i < n) → ∀a. a < n → ∃b. b<n ∧ f b = a.
+#f #finj #n #H1 #a #ltan @(f_img_to_exists f n a)
+@(eq_length_to_mem_all … (length_f_img …) (unique_f_img …finj …) …ltan)
+#x #Hx cases(f_img_to_exists … Hx) #b * #ltbn #eqx <eqx @H1 //
+qed.
+
+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 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) #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.
+
+lemma diag1_not_s1: ∀s1,s2. o s1 s2 → ¬ CF s2 (diag_cf s1).
+#s1 #s2 #H1 % * #i * #Hcode_i #Hs2_i
+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〉))
+@(iff_trans … (equiv_diag …)) >fst_pair
+%[* #_ // |#H6 % // ]
+qed.
+
+(******************************************************************************)
+
+definition to_Some ≝ λf.λx:nat. Some nat (f x).
+
+definition deopt ≝ λn. match n with
+ [ None ⇒ 1
+ | Some n ⇒ n].
+
+definition opt_comp ≝ λf,g:nat → option nat. λx.
+ match g x with
+ [ None ⇒ None ?
+ | Some y ⇒ f y ].
+
+axiom sU2: nat → nat → nat.
+axiom sU: nat → nat → nat → nat.
+
+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|)).
+#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.
+
+(* not used *)
+definition sU_space ≝ λi,x,r.i+x+r.
+definition sU_time ≝ λi,x,r.i+x+(i^2)*r*(log 2 r).
+
+definition IF ≝ λb,f,g:nat →option nat. λx.
+ 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).
+
+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 (fst i) i (|of_size (s (|i|))|)) (λi.Some ? 1) (λi.Some ? 0) i.
+#s #i normalize >size_of_size // qed.
+
+(* and now ... *)
+axiom CF_pair: ∀f,g,s. CF s (λx.Some ? (f x)) → CF s (λx.Some ? (g x)) →
+ CF s (λx.Some ? (pair (f x) (g x))).
+
+axiom CF_fst: ∀f,s. CF s (λx.Some ? (f x)) → CF s (λx.Some ? (fst (f x))).
+
+definition sufficiently_large ≝ λs. CF s (λn. Some ? n) ∧ ∀c. CF s (λn. Some ? c).
+
+definition constructible ≝ λs. CF s (λx.Some ? (of_size (s (|x|)))).
+
+lemma diag_s: ∀s. sufficiently_large 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.
\ No newline at end of file
--- /dev/null
+include "arithmetics/sigma_pi.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.
+
+
+(************************* 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_const: ∀c,a,b.
+ ∑_{i ∈ [a,b[ }c ≤ (b-a)*c.
+#c #a #b elim (b-a) // #n #Hind normalize @le_plus //
+qed.
+
+lemma sigma_to_Max: ∀h,a,b.
+ ∑_{i ∈ [a,b[ }(h i) ≤ (b-a)*Max_{i ∈ [a,b[ }(h i).
+#h #a #b elim (b-a)
+ [//
+ |#n #Hind >bigop_Strue [2://] whd in ⊢ (??%);
+ @le_plus
+ [>bigop_Strue // @(le_maxl … (le_n …))
+ |@(transitive_le … Hind) @le_times // >bigop_Strue //
+ @(le_maxr … (le_n …))
+ ]
+ ]
+qed.
+
+lemma sigma_bound1: ∀h,a,b. monotonic nat le h →
+ ∑_{i ∈ [a,b[ }(h i) ≤ (b-a)*h b.
+#h #a #b #H
+@(transitive_le … (sigma_to_Max …)) @le_times //
+@Max_le #i #lti #_ @H @lt_to_le @lt_minus_to_plus_r //
+qed.
+
+lemma sigma_bound_decr1: ∀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
+@(transitive_le … (sigma_to_Max …)) @le_times //
+@Max_le #i #lti #_ @H // @lt_minus_to_plus_r //
+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.
+
\ No newline at end of file
--- /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.
+
+lemma fst_pU: ∀i,x,r. fst (pU i x r) = termb i x r.
+#i #x #r normalize cases (U i x r) normalize >fst_pair //
+qed.
+
+lemma snd_pU: ∀i,x,r. snd (pU i x r) = out i x r.
+#i #x #r normalize cases (U i x r) normalize >snd_pair //
+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: ∀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.
+axiom MSC_pair: ∀a,b. MSC 〈a,b〉 ≤ MSC a + MSC b.
+
+(* 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).
+
+lemma main_MSC: ∀h,f. CF h f → O h (λx.MSC (f x)).
+
+axiom CF_S: CF MSC S.
+axiom CF_fst: CF MSC fst.
+axiom CF_snd: CF MSC snd.
+
+lemma CF_compS: ∀h,f. CF h f → CF h (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. *)
+
+definition id ≝ λx:nat.x.
+
+axiom CF_id: CF MSC id.
+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).
+axiom CF_comp_pair: ∀h,f,g. CF h f → CF h g → CF h (λx. 〈f x,g x〉).
+
+lemma CF_fst: CF MSC fst.
+@(ext_CF (fst ∘ id)) [#n //] @(CF_comp_fst … CF_id)
+qed.
+
+lemma CF_snd: CF MSC snd.
+@(ext_CF (snd ∘ id)) [#n //] @(CF_comp_snd … CF_id)
+qed.
+
+(************************************** 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〉)).
+
+(****************************** constructibility ******************************)
+
+definition constructible ≝ λs. CF s s.
+
+lemma constr_comp : ∀s1,s2. constructible s1 → constructible s2 →
+ (∀x. x ≤ s2 x) → constructible (s2 ∘ s1).
+#s1 #s2 #Hs1 #Hs2 #Hle @(CF_comp … Hs1 Hs2) @O_plus @le_to_O #x [@Hle | //]
+qed.
+
+lemma ext_constr: ∀s1,s2. (∀x.s1 x = s2 x) →
+ constructible s1 → constructible s2.
+#s1 #s2 #Hext #Hs1 @(ext_CF … Hext) @(monotonic_CF … Hs1) #x >Hext //
+qed.
+
+(********************************* simulation *********************************)
+
+axiom sU : nat → nat.
+
+axiom monotonic_sU: ∀i1,i2,x1,x2,s1,s2. i1 ≤ i2 → x1 ≤ x2 → s1 ≤ s2 →
+ sU 〈i1,〈x1,s1〉〉 ≤ sU 〈i2,〈x2,s2〉〉.
+
+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 sU_le: ∀i,x,s. s ≤ sU 〈i,〈x,s〉〉.
+axiom sU_le_i: ∀i,x,s. MSC i ≤ sU 〈i,〈x,s〉〉.
+axiom sU_le_x: ∀i,x,s. MSC x ≤ sU 〈i,〈x,s〉〉.
+
+definition pU_unary ≝ λp. pU (fst p) (fst (snd p)) (snd (snd p)).
+
+axiom CF_U : CF sU pU_unary.
+
+definition termb_unary ≝ λx:ℕ.termb (fst x) (fst (snd x)) (snd (snd x)).
+definition out_unary ≝ λx:ℕ.out (fst x) (fst (snd x)) (snd (snd x)).
+
+lemma CF_termb: CF sU termb_unary.
+@(ext_CF (fst ∘ pU_unary)) [2: @CF_comp_fst @CF_U]
+#n whd in ⊢ (??%?); whd in ⊢ (??(?%)?); >fst_pair %
+qed.
+
+lemma CF_out: CF sU out_unary.
+@(ext_CF (snd ∘ pU_unary)) [2: @CF_comp_snd @CF_U]
+#n whd in ⊢ (??%?); whd in ⊢ (??(?%)?); >snd_pair %
+qed.
+
+(*
+lemma CF_termb_comp: ∀f.CF (sU ∘ f) (termb_unary ∘ f).
+#f @(CF_comp … 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 (auxg h) → CF s (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
+ (λp:ℕ.eqb (min_input h (fst p) (snd (snd p))) (snd (snd p))) →
+ CF s2
+ (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 (auxg h).
+#h #s1 #s2 #s #Hs1 #Hs2 #HO @(ext_CF (aux1g h))
+ [#n whd in ⊢ (??%%); @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 (λp:ℕ.min_input h (fst p) (snd (snd p))) →
+ CF s (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
+ (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 (λp:ℕ.min_input h (fst p) (snd (snd p))).
+#h #s1 #s #Hs1 #HO @(ext_CF (min_input_aux h))
+ [#n whd in ⊢ (??%%); @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
+ (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〉)
+ (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.
+ (* constructible (λx. h (fst x) (snd x)) → *)
+ (CF (λx. max (MSC x) (sU 〈fst (snd x),〈fst x,h (S (fst (snd x))) (fst x)〉〉))
+ (λp.termb (fst (snd p)) (fst p) (h (S (fst (snd p))) (fst p)))).
+*)
+
+lemma compl_g6: ∀h.
+ constructible (λx. h (fst x) (snd x)) →
+ (CF (λx. sU 〈max (fst (snd x)) (snd (snd x)),〈fst x,h (S (fst (snd x))) (fst x)〉〉)
+ (λp.termb (fst (snd p)) (fst p) (h (S (fst (snd p))) (fst p)))).
+#h #hconstr @(ext_CF (termb_aux h))
+ [#n normalize >fst_pair >snd_pair >fst_pair >snd_pair // ]
+@(CF_comp … (λx.MSC x + h (S (fst (snd x))) (fst x)) … CF_termb)
+ [@CF_comp_pair
+ [@CF_comp_fst @(monotonic_CF … CF_snd) #x //
+ |@CF_comp_pair
+ [@(monotonic_CF … CF_fst) #x //
+ |@(ext_CF ((λx.h (fst x) (snd x))∘(λx.〈S (fst (snd x)),fst x〉)))
+ [#n normalize >fst_pair >snd_pair %]
+ @(CF_comp … MSC …hconstr)
+ [@CF_comp_pair [@CF_compS @CF_comp_fst // |//]
+ |@O_plus @le_to_O [//|#n >fst_pair >snd_pair //]
+ ]
+ ]
+ ]
+ |@O_plus
+ [@O_plus
+ [@(O_trans … (λx.MSC (fst x) + MSC (max (fst (snd x)) (snd (snd x)))))
+ [%{2} %{0} #x #_ cases (surj_pair x) #a * #b #eqx >eqx
+ >fst_pair >snd_pair @(transitive_le … (MSC_pair …))
+ >distributive_times_plus @le_plus [//]
+ cases (surj_pair b) #c * #d #eqb >eqb
+ >fst_pair >snd_pair @(transitive_le … (MSC_pair …))
+ whd in ⊢ (??%); @le_plus
+ [@monotonic_MSC @(le_maxl … (le_n …))
+ |>commutative_times <times_n_1 @monotonic_MSC @(le_maxr … (le_n …))
+ ]
+ |@O_plus [@le_to_O #x @sU_le_x |@le_to_O #x @sU_le_i]
+ ]
+ |@le_to_O #n @sU_le
+ ]
+ |@le_to_O #x @monotonic_sU // @(le_maxl … (le_n …)) ]
+ ]
+qed.
+
+(* 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.
+ (∀x.MSC x≤h (S (fst (snd x))) (fst x)) →
+ constructible (λx. h (fst x) (snd x)) →
+ (∀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))〉〉)
+ (λp:ℕ.min_input h (fst p) (snd (snd p))).
+#h #hle #hcostr #hmono @(monotonic_CF … (faux1 h))
+ [#n normalize >fst_pair >snd_pair //]
+@compl_g5 [2:@(compl_g6 h hcostr)] #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.
+
+definition faux2 ≝ λh.
+ (λx.MSC x + (snd (snd x)-fst x)*
+ (λx.sU 〈max (fst(snd x)) (snd(snd x)),〈fst x,h (S (fst (snd x))) (fst x)〉〉) 〈snd (snd x),x〉).
+
+(* proviamo con x *)
+lemma compl_g7: ∀h.
+ constructible (λx. h (fst x) (snd x)) →
+ (∀n. monotonic ? le (h n)) →
+ 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))〉〉)
+ (λp:ℕ.min_input h (fst p) (snd (snd p))).
+#h #hcostr #hmono @(monotonic_CF … (faux2 h))
+ [#n normalize >fst_pair >snd_pair //]
+@compl_g5 [2:@(compl_g6 h hcostr)] #n #x #y #lexy >fst_pair >snd_pair
+>fst_pair >snd_pair @monotonic_sU // @hmono @lexy
+qed.
+
+(* proviamo con x *)
+lemma compl_g71: ∀h.
+ constructible (λx. h (fst x) (snd x)) →
+ (∀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))〉〉)
+ (λp:ℕ.min_input h (fst p) (snd (snd p))).
+#h #hcostr #hmono @(monotonic_CF … (compl_g7 h hcostr hmono)) #x
+@le_plus [@monotonic_MSC //]
+cases (decidable_le (fst x) (snd(snd x)))
+ [#Hle @le_times // @monotonic_sU
+ |#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))〉〉)
+ (λp:ℕ.out (fst p) (snd (snd p)) (h (S (fst p)) (snd (snd p)))). *)
+
+definition out_aux ≝ λh.
+ out_unary ∘ λp.〈fst p,〈snd(snd p),h (S (fst p)) (snd (snd p))〉〉.
+
+lemma compl_g8: ∀h.
+ constructible (λx. h (fst x) (snd x)) →
+ (CF (λx. sU 〈max (fst x) (snd x),〈snd(snd x),h (S (fst x)) (snd(snd x))〉〉)
+ (λp.out (fst p) (snd (snd p)) (h (S (fst p)) (snd (snd p))))).
+#h #hconstr @(ext_CF (out_aux h))
+ [#n normalize >fst_pair >snd_pair >fst_pair >snd_pair // ]
+@(CF_comp … (λx.h (S (fst x)) (snd(snd x)) + MSC x) … CF_out)
+ [@CF_comp_pair
+ [@(monotonic_CF … CF_fst) #x //
+ |@CF_comp_pair
+ [@CF_comp_snd @(monotonic_CF … CF_snd) #x //
+ |@(ext_CF ((λx.h (fst x) (snd x))∘(λx.〈S (fst x),snd(snd x)〉)))
+ [#n normalize >fst_pair >snd_pair %]
+ @(CF_comp … MSC …hconstr)
+ [@CF_comp_pair [@CF_compS // | @CF_comp_snd // ]
+ |@O_plus @le_to_O [//|#n >fst_pair >snd_pair //]
+ ]
+ ]
+ ]
+ |@O_plus
+ [@O_plus
+ [@le_to_O #n @sU_le
+ |@(O_trans … (λx.MSC (max (fst x) (snd x))))
+ [%{2} %{0} #x #_ cases (surj_pair x) #a * #b #eqx >eqx
+ >fst_pair >snd_pair @(transitive_le … (MSC_pair …))
+ whd in ⊢ (??%); @le_plus
+ [@monotonic_MSC @(le_maxl … (le_n …))
+ |>commutative_times <times_n_1 @monotonic_MSC @(le_maxr … (le_n …))
+ ]
+ |@le_to_O #x @(transitive_le ???? (sU_le_i … )) //
+ ]
+ ]
+ |@le_to_O #x @monotonic_sU [@(le_maxl … (le_n …))|//|//]
+ ]
+qed.
+
+(*
+lemma compl_g81: ∀h.
+ (∀x.MSC x≤h (S (fst x)) (snd(snd x))) →
+ constructible (λx. h (fst x) (snd x)) →
+ CF (λx. sU 〈max (fst x) (snd x),〈snd (snd x),h (S (fst x)) (snd (snd x))〉〉)
+ (λp:ℕ.out (fst p) (snd (snd p)) (h (S (fst p)) (snd (snd p)))).
+#h #hle #hconstr @(monotonic_CF ???? (compl_g8 h hle hconstr)) #x @monotonic_sU // @(le_maxl … (le_n … ))
+qed. *)
+
+(* axiom daemon : False. *)
+
+lemma compl_g9 : ∀h.
+ constructible (λx. h (fst x) (snd x)) →
+ (∀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)〉〉)
+ (auxg h).
+#h #hconstr #hmono #hantimono
+@(compl_g2 h ??? (compl_g3 … (compl_g71 h hconstr hmono)) (compl_g8 h hconstr))
+@O_plus
+ [@O_plus_l @le_to_O #x >(times_n_1 (MSC x)) >commutative_times @le_times
+ [// | @monotonic_MSC // ]]
+@(O_trans … (coroll2 ??))
+ [#n #a #b #leab #ltb >fst_pair >fst_pair >snd_pair >snd_pair
+ cut (b ≤ n) [@(transitive_le … (le_snd …)) @lt_to_le //] #lebn
+ cut (max a n = n)
+ [normalize >le_to_leb_true [//|@(transitive_le … leab lebn)]] #maxa
+ cut (max b n = n) [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.
+ constructible (λx. h (fst x) (snd x)) →
+ (∀n. monotonic ? le (h n)) →
+ (∀n,a,b. a ≤ b → b ≤ n → h b n ≤ h a n) →
+ CF (sg h) (unary_g h).
+#h #hconstr #Hm #Ham @compl_g1 @(compl_g9 h hconstr 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.
+// 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.
+
+axiom h_of_constr : ∀r:nat →nat.
+ (∀x. x ≤ r x) → monotonic ? le r → constructible r →
+ constructible (h_of r).
+
+lemma speed_compl: ∀r:nat →nat.
+ (∀x. x ≤ r x) → monotonic ? le r → constructible r →
+ CF (h_of r) (unary_g (λi,x. r(h_of r 〈i,x〉))).
+#r #Hr #Hmono #Hconstr @(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)
+ [%{(pred(b-a))} >S_pred [//] @lt_plus_to_minus_r @not_le_to_lt @ltab]
+ * #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)
+ |@(constr_comp … Hconstr Hr) @(ext_constr (h_of r))
+ [#x cases (surj_pair x) #a * #b #eqx >eqx >fst_pair >snd_pair //]
+ @(h_of_constr r Hr Hmono Hconstr)
+ ]
+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 → constructible r →
+ ∀i. CF (λx.h_of r 〈i,x〉) (λx.g (λi,x. r(h_of r 〈i,x〉)) i x).
+#r #Hr #Hmono #Hconstr #i
+@(ext_CF (λx.unary_g (λi,x. r(h_of r 〈i,x〉)) 〈i,x〉))
+ [#n whd in ⊢ (??%%); @eq_f @sym_eq >fst_pair >snd_pair %]
+@smn @(ext_CF … (speed_compl r Hr Hmono Hconstr)) #n //
+qed.
+
+theorem pseudo_speedup:
+ ∀r:nat →nat. (∀x. x ≤ r x) → monotonic ? le r → constructible r →
+ ∃f.∀sf. CF sf f → ∃g,sg. f ≈ g ∧ CF sg g ∧ O sf (r ∘ sg).
+(* ∃m,a.∀n. a≤n → r(sg a) < m * sf n. *)
+#r #Hr #Hmono #Hconstr
+(* 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 Hconstr (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 → constructible r →
+ ∃f.∀sf. CF sf f → ∃g,sg. f ≈ g ∧ CF sg g ∧
+ (* ¬ O (r ∘ sg) sf. *)
+ ∃m,a.∀n. a≤n → r(sg a) < m * sf n.
+#r #Hr #Hmono #Hconstr
+(* 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 Hconstr (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
--- /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.
+
+lemma fst_pU: ∀i,x,r. fst (pU i x r) = termb i x r.
+#i #x #r normalize cases (U i x r) normalize >fst_pair //
+qed.
+
+lemma snd_pU: ∀i,x,r. snd (pU i x r) = out i x r.
+#i #x #r normalize cases (U i x r) normalize >snd_pair //
+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.
+
+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: ∀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.
+axiom MSC_pair: ∀a,b. MSC 〈a,b〉 ≤ MSC a + MSC b.
+
+(* 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 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.
+
+
+(**************************** primitive operations*****************************)
+
+definition id ≝ λx:nat.x.
+
+axiom CF_id: CF MSC id.
+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).
+axiom CF_comp_pair: ∀h,f,g. CF h f → CF h g → CF h (λx. 〈f x,g x〉).
+
+lemma CF_fst: CF MSC fst.
+@(ext_CF (fst ∘ id)) [#n //] @(CF_comp_fst … CF_id)
+qed.
+
+lemma CF_snd: CF MSC snd.
+@(ext_CF (snd ∘ id)) [#n //] @(CF_comp_snd … CF_id)
+qed.
+
+(************************************** eqb ***********************************)
+
+axiom CF_eqb: ∀h,f,g.
+ CF h f → CF h g → CF h (λx.eqb (f x) (g 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〉)).
+
+(************************************* smn ************************************)
+axiom smn: ∀f,s. CF s f → ∀x. CF (λy.s 〈x,y〉) (λy.f 〈x,y〉).
+
+(****************************** constructibility ******************************)
+
+definition constructible ≝ λs. CF s s.
+
+lemma constr_comp : ∀s1,s2. constructible s1 → constructible s2 →
+ (∀x. x ≤ s2 x) → constructible (s2 ∘ s1).
+#s1 #s2 #Hs1 #Hs2 #Hle @(CF_comp … Hs1 Hs2) @O_plus @le_to_O #x [@Hle | //]
+qed.
+
+lemma ext_constr: ∀s1,s2. (∀x.s1 x = s2 x) →
+ constructible s1 → constructible s2.
+#s1 #s2 #Hext #Hs1 @(ext_CF … Hext) @(monotonic_CF … Hs1) #x >Hext //
+qed.
+
+(********************************* simulation *********************************)
+
+axiom sU : nat → nat.
+
+axiom monotonic_sU: ∀i1,i2,x1,x2,s1,s2. i1 ≤ i2 → x1 ≤ x2 → s1 ≤ s2 →
+ sU 〈i1,〈x1,s1〉〉 ≤ sU 〈i2,〈x2,s2〉〉.
+
+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 sU_le: ∀i,x,s. s ≤ sU 〈i,〈x,s〉〉.
+axiom sU_le_i: ∀i,x,s. MSC i ≤ sU 〈i,〈x,s〉〉.
+axiom sU_le_x: ∀i,x,s. MSC x ≤ sU 〈i,〈x,s〉〉.
+
+definition pU_unary ≝ λp. pU (fst p) (fst (snd p)) (snd (snd p)).
+
+axiom CF_U : CF sU pU_unary.
+
+definition termb_unary ≝ λx:ℕ.termb (fst x) (fst (snd x)) (snd (snd x)).
+definition out_unary ≝ λx:ℕ.out (fst x) (fst (snd x)) (snd (snd x)).
+
+lemma CF_termb: CF sU termb_unary.
+@(ext_CF (fst ∘ pU_unary)) [2: @CF_comp_fst @CF_U]
+#n whd in ⊢ (??%?); whd in ⊢ (??(?%)?); >fst_pair %
+qed.
+
+lemma CF_out: CF sU out_unary.
+@(ext_CF (snd ∘ pU_unary)) [2: @CF_comp_snd @CF_U]
+#n whd in ⊢ (??%?); whd in ⊢ (??(?%)?); >snd_pair %
+qed.
+
+
+(******************** 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 (auxg h) → CF s (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
+ (λp:ℕ.eqb (min_input h (fst p) (snd (snd p))) (snd (snd p))) →
+ CF s2
+ (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 (auxg h).
+#h #s1 #s2 #s #Hs1 #Hs2 #HO @(ext_CF (aux1g h))
+ [#n whd in ⊢ (??%%); @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 (λp:ℕ.min_input h (fst p) (snd (snd p))) →
+ CF s (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 compl_g4 : ∀h,s1,s.
+ (CF s1
+ (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 (λp:ℕ.min_input h (fst p) (snd (snd p))).
+#h #s1 #s #Hs1 #HO @(ext_CF (min_input_aux h))
+ [#n whd in ⊢ (??%%); @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 //]
+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.
+
+(**************************** end of technical lemmas *************************)
+
+lemma compl_g5 : ∀h,s1.(∀n. monotonic ℕ le (λa:ℕ.s1 〈a,n〉)) →
+ (CF s1
+ (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〉)
+ (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.
+
+lemma compl_g6: ∀h.
+ constructible (λx. h (fst x) (snd x)) →
+ (CF (λx. sU 〈max (fst (snd x)) (snd (snd x)),〈fst x,h (S (fst (snd x))) (fst x)〉〉)
+ (λp.termb (fst (snd p)) (fst p) (h (S (fst (snd p))) (fst p)))).
+#h #hconstr @(ext_CF (termb_aux h))
+ [#n normalize >fst_pair >snd_pair >fst_pair >snd_pair // ]
+@(CF_comp … (λx.MSC x + h (S (fst (snd x))) (fst x)) … CF_termb)
+ [@CF_comp_pair
+ [@CF_comp_fst @(monotonic_CF … CF_snd) #x //
+ |@CF_comp_pair
+ [@(monotonic_CF … CF_fst) #x //
+ |@(ext_CF ((λx.h (fst x) (snd x))∘(λx.〈S (fst (snd x)),fst x〉)))
+ [#n normalize >fst_pair >snd_pair %]
+ @(CF_comp … MSC …hconstr)
+ [@CF_comp_pair [@CF_compS @CF_comp_fst // |//]
+ |@O_plus @le_to_O [//|#n >fst_pair >snd_pair //]
+ ]
+ ]
+ ]
+ |@O_plus
+ [@O_plus
+ [@(O_trans … (λx.MSC (fst x) + MSC (max (fst (snd x)) (snd (snd x)))))
+ [%{2} %{0} #x #_ cases (surj_pair x) #a * #b #eqx >eqx
+ >fst_pair >snd_pair @(transitive_le … (MSC_pair …))
+ >distributive_times_plus @le_plus [//]
+ cases (surj_pair b) #c * #d #eqb >eqb
+ >fst_pair >snd_pair @(transitive_le … (MSC_pair …))
+ whd in ⊢ (??%); @le_plus
+ [@monotonic_MSC @(le_maxl … (le_n …))
+ |>commutative_times <times_n_1 @monotonic_MSC @(le_maxr … (le_n …))
+ ]
+ |@O_plus [@le_to_O #x @sU_le_x |@le_to_O #x @sU_le_i]
+ ]
+ |@le_to_O #n @sU_le
+ ]
+ |@le_to_O #x @monotonic_sU // @(le_maxl … (le_n …)) ]
+ ]
+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.
+
+definition faux2 ≝ λh.
+ (λx.MSC x + (snd (snd x)-fst x)*
+ (λx.sU 〈max (fst(snd x)) (snd(snd x)),〈fst x,h (S (fst (snd x))) (fst x)〉〉) 〈snd (snd x),x〉).
+
+lemma compl_g7: ∀h.
+ constructible (λx. h (fst x) (snd x)) →
+ (∀n. monotonic ? le (h n)) →
+ 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))〉〉)
+ (λp:ℕ.min_input h (fst p) (snd (snd p))).
+#h #hcostr #hmono @(monotonic_CF … (faux2 h))
+ [#n normalize >fst_pair >snd_pair //]
+@compl_g5 [2:@(compl_g6 h hcostr)] #n #x #y #lexy >fst_pair >snd_pair
+>fst_pair >snd_pair @monotonic_sU // @hmono @lexy
+qed.
+
+lemma compl_g71: ∀h.
+ constructible (λx. h (fst x) (snd x)) →
+ (∀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))〉〉)
+ (λp:ℕ.min_input h (fst p) (snd (snd p))).
+#h #hcostr #hmono @(monotonic_CF … (compl_g7 h hcostr hmono)) #x
+@le_plus [@monotonic_MSC //]
+cases (decidable_le (fst x) (snd(snd x)))
+ [#Hle @le_times // @monotonic_sU
+ |#Hlt >(minus_to_0 … (lt_to_le … )) [// | @not_le_to_lt @Hlt]
+ ]
+qed.
+
+definition out_aux ≝ λh.
+ out_unary ∘ λp.〈fst p,〈snd(snd p),h (S (fst p)) (snd (snd p))〉〉.
+
+lemma compl_g8: ∀h.
+ constructible (λx. h (fst x) (snd x)) →
+ (CF (λx. sU 〈max (fst x) (snd x),〈snd(snd x),h (S (fst x)) (snd(snd x))〉〉)
+ (λp.out (fst p) (snd (snd p)) (h (S (fst p)) (snd (snd p))))).
+#h #hconstr @(ext_CF (out_aux h))
+ [#n normalize >fst_pair >snd_pair >fst_pair >snd_pair // ]
+@(CF_comp … (λx.h (S (fst x)) (snd(snd x)) + MSC x) … CF_out)
+ [@CF_comp_pair
+ [@(monotonic_CF … CF_fst) #x //
+ |@CF_comp_pair
+ [@CF_comp_snd @(monotonic_CF … CF_snd) #x //
+ |@(ext_CF ((λx.h (fst x) (snd x))∘(λx.〈S (fst x),snd(snd x)〉)))
+ [#n normalize >fst_pair >snd_pair %]
+ @(CF_comp … MSC …hconstr)
+ [@CF_comp_pair [@CF_compS // | @CF_comp_snd // ]
+ |@O_plus @le_to_O [//|#n >fst_pair >snd_pair //]
+ ]
+ ]
+ ]
+ |@O_plus
+ [@O_plus
+ [@le_to_O #n @sU_le
+ |@(O_trans … (λx.MSC (max (fst x) (snd x))))
+ [%{2} %{0} #x #_ cases (surj_pair x) #a * #b #eqx >eqx
+ >fst_pair >snd_pair @(transitive_le … (MSC_pair …))
+ whd in ⊢ (??%); @le_plus
+ [@monotonic_MSC @(le_maxl … (le_n …))
+ |>commutative_times <times_n_1 @monotonic_MSC @(le_maxr … (le_n …))
+ ]
+ |@le_to_O #x @(transitive_le ???? (sU_le_i … )) //
+ ]
+ ]
+ |@le_to_O #x @monotonic_sU [@(le_maxl … (le_n …))|//|//]
+ ]
+qed.
+
+lemma compl_g9 : ∀h.
+ constructible (λx. h (fst x) (snd x)) →
+ (∀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)〉〉)
+ (auxg h).
+#h #hconstr #hmono #hantimono
+@(compl_g2 h ??? (compl_g3 … (compl_g71 h hconstr hmono)) (compl_g8 h hconstr))
+@O_plus
+ [@O_plus_l @le_to_O #x >(times_n_1 (MSC x)) >commutative_times @le_times
+ [// | @monotonic_MSC // ]]
+@(O_trans … (coroll2 ??))
+ [#n #a #b #leab #ltb >fst_pair >fst_pair >snd_pair >snd_pair
+ cut (b ≤ n) [@(transitive_le … (le_snd …)) @lt_to_le //] #lebn
+ cut (max a n = n)
+ [normalize >le_to_leb_true [//|@(transitive_le … leab lebn)]] #maxa
+ cut (max b n = n) [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.
+ constructible (λx. h (fst x) (snd x)) →
+ (∀n. monotonic ? le (h n)) →
+ (∀n,a,b. a ≤ b → b ≤ n → h b n ≤ h a n) →
+ CF (sg h) (unary_g h).
+#h #hconstr #Hm #Ham @compl_g1 @(compl_g9 h hconstr 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
+ | 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.
+// 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.
+
+axiom h_of_constr : ∀r:nat →nat.
+ (∀x. x ≤ r x) → monotonic ? le r → constructible r →
+ constructible (h_of r).
+
+lemma speed_compl: ∀r:nat →nat.
+ (∀x. x ≤ r x) → monotonic ? le r → constructible r →
+ CF (h_of r) (unary_g (λi,x. r(h_of r 〈i,x〉))).
+#r #Hr #Hmono #Hconstr @(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)
+ [%{(pred(b-a))} >S_pred [//] @lt_plus_to_minus_r @not_le_to_lt @ltab]
+ * #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)
+ |@(constr_comp … Hconstr Hr) @(ext_constr (h_of r))
+ [#x cases (surj_pair x) #a * #b #eqx >eqx >fst_pair >snd_pair //]
+ @(h_of_constr r Hr Hmono Hconstr)
+ ]
+qed.
+
+lemma speed_compl_i: ∀r:nat →nat.
+ (∀x. x ≤ r x) → monotonic ? le r → constructible r →
+ ∀i. CF (λx.h_of r 〈i,x〉) (λx.g (λi,x. r(h_of r 〈i,x〉)) i x).
+#r #Hr #Hmono #Hconstr #i
+@(ext_CF (λx.unary_g (λi,x. r(h_of r 〈i,x〉)) 〈i,x〉))
+ [#n whd in ⊢ (??%%); @eq_f @sym_eq >fst_pair >snd_pair %]
+@smn @(ext_CF … (speed_compl r Hr Hmono Hconstr)) #n //
+qed.
+
+(**************************** the speedup theorem *****************************)
+theorem pseudo_speedup:
+ ∀r:nat →nat. (∀x. x ≤ r x) → monotonic ? le r → constructible r →
+ ∃f.∀sf. CF sf f → ∃g,sg. f ≈ g ∧ CF sg g ∧ O sf (r ∘ sg).
+(* ∃m,a.∀n. a≤n → r(sg a) < m * sf n. *)
+#r #Hr #Hmono #Hconstr
+(* 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 Hconstr (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 → constructible r →
+ ∃f.∀sf. CF sf f → ∃g,sg. f ≈ g ∧ CF sg g ∧
+ (* ¬ O (r ∘ sg) sf. *)
+ ∃m,a.∀n. a≤n → r(sg a) < m * sf n.
+#r #Hr #Hmono #Hconstr
+(* 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 Hconstr (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
--- /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.
+
+lemma fst_pU: ∀i,x,r. fst (pU i x r) = termb i x r.
+#i #x #r normalize cases (U i x r) normalize >fst_pair //
+qed.
+
+lemma snd_pU: ∀i,x,r. snd (pU i x r) = out i x r.
+#i #x #r normalize cases (U i x r) normalize >snd_pair //
+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.
+
+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: ∀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.
+axiom MSC_pair: ∀a,b. MSC 〈a,b〉 ≤ MSC a + MSC b.
+
+(* 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 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.
+
+
+(**************************** primitive operations*****************************)
+
+definition id ≝ λx:nat.x.
+
+axiom CF_id: CF MSC id.
+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).
+axiom CF_comp_pair: ∀h,f,g. CF h f → CF h g → CF h (λx. 〈f x,g x〉).
+
+lemma CF_fst: CF MSC fst.
+@(ext_CF (fst ∘ id)) [#n //] @(CF_comp_fst … CF_id)
+qed.
+
+lemma CF_snd: CF MSC snd.
+@(ext_CF (snd ∘ id)) [#n //] @(CF_comp_snd … CF_id)
+qed.
+
+(************************************** eqb ***********************************)
+
+axiom CF_eqb: ∀h,f,g.
+ CF h f → CF h g → CF h (λx.eqb (f x) (g 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〉)).
+
+(************************************* smn ************************************)
+axiom smn: ∀f,s. CF s f → ∀x. CF (λy.s 〈x,y〉) (λy.f 〈x,y〉).
+
+(****************************** constructibility ******************************)
+
+definition constructible ≝ λs. CF s s.
+
+lemma constr_comp : ∀s1,s2. constructible s1 → constructible s2 →
+ (∀x. x ≤ s2 x) → constructible (s2 ∘ s1).
+#s1 #s2 #Hs1 #Hs2 #Hle @(CF_comp … Hs1 Hs2) @O_plus @le_to_O #x [@Hle | //]
+qed.
+
+lemma ext_constr: ∀s1,s2. (∀x.s1 x = s2 x) →
+ constructible s1 → constructible s2.
+#s1 #s2 #Hext #Hs1 @(ext_CF … Hext) @(monotonic_CF … Hs1) #x >Hext //
+qed.
+
+(********************************* simulation *********************************)
+
+axiom sU : nat → nat.
+
+axiom monotonic_sU: ∀i1,i2,x1,x2,s1,s2. i1 ≤ i2 → x1 ≤ x2 → s1 ≤ s2 →
+ sU 〈i1,〈x1,s1〉〉 ≤ sU 〈i2,〈x2,s2〉〉.
+
+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 sU_le: ∀i,x,s. s ≤ sU 〈i,〈x,s〉〉.
+axiom sU_le_i: ∀i,x,s. MSC i ≤ sU 〈i,〈x,s〉〉.
+axiom sU_le_x: ∀i,x,s. MSC x ≤ sU 〈i,〈x,s〉〉.
+
+definition pU_unary ≝ λp. pU (fst p) (fst (snd p)) (snd (snd p)).
+
+axiom CF_U : CF sU pU_unary.
+
+definition termb_unary ≝ λx:ℕ.termb (fst x) (fst (snd x)) (snd (snd x)).
+definition out_unary ≝ λx:ℕ.out (fst x) (fst (snd x)) (snd (snd x)).
+
+lemma CF_termb: CF sU termb_unary.
+@(ext_CF (fst ∘ pU_unary)) [2: @CF_comp_fst @CF_U]
+#n whd in ⊢ (??%?); whd in ⊢ (??(?%)?); >fst_pair %
+qed.
+
+lemma CF_out: CF sU out_unary.
+@(ext_CF (snd ∘ pU_unary)) [2: @CF_comp_snd @CF_U]
+#n whd in ⊢ (??%?); whd in ⊢ (??(?%)?); >snd_pair %
+qed.
+
+
+(******************** 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 (auxg h) → CF s (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
+ (λp:ℕ.eqb (min_input h (fst p) (snd (snd p))) (snd (snd p))) →
+ CF s2
+ (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 (auxg h).
+#h #s1 #s2 #s #Hs1 #Hs2 #HO @(ext_CF (aux1g h))
+ [#n whd in ⊢ (??%%); @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 (λp:ℕ.min_input h (fst p) (snd (snd p))) →
+ CF s (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 compl_g4 : ∀h,s1,s.
+ (CF s1
+ (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 (λp:ℕ.min_input h (fst p) (snd (snd p))).
+#h #s1 #s #Hs1 #HO @(ext_CF (min_input_aux h))
+ [#n whd in ⊢ (??%%); @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 //]
+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.
+
+(**************************** end of technical lemmas *************************)
+
+lemma compl_g5 : ∀h,s1.(∀n. monotonic ℕ le (λa:ℕ.s1 〈a,n〉)) →
+ (CF s1
+ (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〉)
+ (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.
+
+lemma compl_g6: ∀h.
+ constructible (λx. h (fst x) (snd x)) →
+ (CF (λx. sU 〈max (fst (snd x)) (snd (snd x)),〈fst x,h (S (fst (snd x))) (fst x)〉〉)
+ (λp.termb (fst (snd p)) (fst p) (h (S (fst (snd p))) (fst p)))).
+#h #hconstr @(ext_CF (termb_aux h))
+ [#n normalize >fst_pair >snd_pair >fst_pair >snd_pair // ]
+@(CF_comp … (λx.MSC x + h (S (fst (snd x))) (fst x)) … CF_termb)
+ [@CF_comp_pair
+ [@CF_comp_fst @(monotonic_CF … CF_snd) #x //
+ |@CF_comp_pair
+ [@(monotonic_CF … CF_fst) #x //
+ |@(ext_CF ((λx.h (fst x) (snd x))∘(λx.〈S (fst (snd x)),fst x〉)))
+ [#n normalize >fst_pair >snd_pair %]
+ @(CF_comp … MSC …hconstr)
+ [@CF_comp_pair [@CF_compS @CF_comp_fst // |//]
+ |@O_plus @le_to_O [//|#n >fst_pair >snd_pair //]
+ ]
+ ]
+ ]
+ |@O_plus
+ [@O_plus
+ [@(O_trans … (λx.MSC (fst x) + MSC (max (fst (snd x)) (snd (snd x)))))
+ [%{2} %{0} #x #_ cases (surj_pair x) #a * #b #eqx >eqx
+ >fst_pair >snd_pair @(transitive_le … (MSC_pair …))
+ >distributive_times_plus @le_plus [//]
+ cases (surj_pair b) #c * #d #eqb >eqb
+ >fst_pair >snd_pair @(transitive_le … (MSC_pair …))
+ whd in ⊢ (??%); @le_plus
+ [@monotonic_MSC @(le_maxl … (le_n …))
+ |>commutative_times <times_n_1 @monotonic_MSC @(le_maxr … (le_n …))
+ ]
+ |@O_plus [@le_to_O #x @sU_le_x |@le_to_O #x @sU_le_i]
+ ]
+ |@le_to_O #n @sU_le
+ ]
+ |@le_to_O #x @monotonic_sU // @(le_maxl … (le_n …)) ]
+ ]
+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.
+
+definition faux2 ≝ λh.
+ (λx.MSC x + (snd (snd x)-fst x)*
+ (λx.sU 〈max (fst(snd x)) (snd(snd x)),〈fst x,h (S (fst (snd x))) (fst x)〉〉) 〈snd (snd x),x〉).
+
+lemma compl_g7: ∀h.
+ constructible (λx. h (fst x) (snd x)) →
+ (∀n. monotonic ? le (h n)) →
+ 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))〉〉)
+ (λp:ℕ.min_input h (fst p) (snd (snd p))).
+#h #hcostr #hmono @(monotonic_CF … (faux2 h))
+ [#n normalize >fst_pair >snd_pair //]
+@compl_g5 [2:@(compl_g6 h hcostr)] #n #x #y #lexy >fst_pair >snd_pair
+>fst_pair >snd_pair @monotonic_sU // @hmono @lexy
+qed.
+
+lemma compl_g71: ∀h.
+ constructible (λx. h (fst x) (snd x)) →
+ (∀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))〉〉)
+ (λp:ℕ.min_input h (fst p) (snd (snd p))).
+#h #hcostr #hmono @(monotonic_CF … (compl_g7 h hcostr hmono)) #x
+@le_plus [@monotonic_MSC //]
+cases (decidable_le (fst x) (snd(snd x)))
+ [#Hle @le_times // @monotonic_sU
+ |#Hlt >(minus_to_0 … (lt_to_le … )) [// | @not_le_to_lt @Hlt]
+ ]
+qed.
+
+definition out_aux ≝ λh.
+ out_unary ∘ λp.〈fst p,〈snd(snd p),h (S (fst p)) (snd (snd p))〉〉.
+
+lemma compl_g8: ∀h.
+ constructible (λx. h (fst x) (snd x)) →
+ (CF (λx. sU 〈max (fst x) (snd x),〈snd(snd x),h (S (fst x)) (snd(snd x))〉〉)
+ (λp.out (fst p) (snd (snd p)) (h (S (fst p)) (snd (snd p))))).
+#h #hconstr @(ext_CF (out_aux h))
+ [#n normalize >fst_pair >snd_pair >fst_pair >snd_pair // ]
+@(CF_comp … (λx.h (S (fst x)) (snd(snd x)) + MSC x) … CF_out)
+ [@CF_comp_pair
+ [@(monotonic_CF … CF_fst) #x //
+ |@CF_comp_pair
+ [@CF_comp_snd @(monotonic_CF … CF_snd) #x //
+ |@(ext_CF ((λx.h (fst x) (snd x))∘(λx.〈S (fst x),snd(snd x)〉)))
+ [#n normalize >fst_pair >snd_pair %]
+ @(CF_comp … MSC …hconstr)
+ [@CF_comp_pair [@CF_compS // | @CF_comp_snd // ]
+ |@O_plus @le_to_O [//|#n >fst_pair >snd_pair //]
+ ]
+ ]
+ ]
+ |@O_plus
+ [@O_plus
+ [@le_to_O #n @sU_le
+ |@(O_trans … (λx.MSC (max (fst x) (snd x))))
+ [%{2} %{0} #x #_ cases (surj_pair x) #a * #b #eqx >eqx
+ >fst_pair >snd_pair @(transitive_le … (MSC_pair …))
+ whd in ⊢ (??%); @le_plus
+ [@monotonic_MSC @(le_maxl … (le_n …))
+ |>commutative_times <times_n_1 @monotonic_MSC @(le_maxr … (le_n …))
+ ]
+ |@le_to_O #x @(transitive_le ???? (sU_le_i … )) //
+ ]
+ ]
+ |@le_to_O #x @monotonic_sU [@(le_maxl … (le_n …))|//|//]
+ ]
+qed.
+
+lemma compl_g9 : ∀h.
+ constructible (λx. h (fst x) (snd x)) →
+ (∀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)〉〉)
+ (auxg h).
+#h #hconstr #hmono #hantimono
+@(compl_g2 h ??? (compl_g3 … (compl_g71 h hconstr hmono)) (compl_g8 h hconstr))
+@O_plus
+ [@O_plus_l @le_to_O #x >(times_n_1 (MSC x)) >commutative_times @le_times
+ [// | @monotonic_MSC // ]]
+@(O_trans … (coroll2 ??))
+ [#n #a #b #leab #ltb >fst_pair >fst_pair >snd_pair >snd_pair
+ cut (b ≤ n) [@(transitive_le … (le_snd …)) @lt_to_le //] #lebn
+ cut (max a n = n)
+ [normalize >le_to_leb_true [//|@(transitive_le … leab lebn)]] #maxa
+ cut (max b n = n) [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.
+ constructible (λx. h (fst x) (snd x)) →
+ (∀n. monotonic ? le (h n)) →
+ (∀n,a,b. a ≤ b → b ≤ n → h b n ≤ h a n) →
+ CF (sg h) (unary_g h).
+#h #hconstr #Hm #Ham @compl_g1 @(compl_g9 h hconstr 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
+ | 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.
+// 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.
+
+axiom h_of_constr : ∀r:nat →nat.
+ (∀x. x ≤ r x) → monotonic ? le r → constructible r →
+ constructible (h_of r).
+
+lemma speed_compl: ∀r:nat →nat.
+ (∀x. x ≤ r x) → monotonic ? le r → constructible r →
+ CF (h_of r) (unary_g (λi,x. r(h_of r 〈i,x〉))).
+#r #Hr #Hmono #Hconstr @(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)
+ [%{(pred(b-a))} >S_pred [//] @lt_plus_to_minus_r @not_le_to_lt @ltab]
+ * #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)
+ |@(constr_comp … Hconstr Hr) @(ext_constr (h_of r))
+ [#x cases (surj_pair x) #a * #b #eqx >eqx >fst_pair >snd_pair //]
+ @(h_of_constr r Hr Hmono Hconstr)
+ ]
+qed.
+
+lemma speed_compl_i: ∀r:nat →nat.
+ (∀x. x ≤ r x) → monotonic ? le r → constructible r →
+ ∀i. CF (λx.h_of r 〈i,x〉) (λx.g (λi,x. r(h_of r 〈i,x〉)) i x).
+#r #Hr #Hmono #Hconstr #i
+@(ext_CF (λx.unary_g (λi,x. r(h_of r 〈i,x〉)) 〈i,x〉))
+ [#n whd in ⊢ (??%%); @eq_f @sym_eq >fst_pair >snd_pair %]
+@smn @(ext_CF … (speed_compl r Hr Hmono Hconstr)) #n //
+qed.
+
+(**************************** the speedup theorem *****************************)
+theorem pseudo_speedup:
+ ∀r:nat →nat. (∀x. x ≤ r x) → monotonic ? le r → constructible r →
+ ∃f.∀sf. CF sf f → ∃g,sg. f ≈ g ∧ CF sg g ∧ O sf (r ∘ sg).
+(* ∃m,a.∀n. a≤n → r(sg a) < m * sf n. *)
+#r #Hr #Hmono #Hconstr
+(* 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 Hconstr (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 → constructible r →
+ ∃f.∀sf. CF sf f → ∃g,sg. f ≈ g ∧ CF sg g ∧
+ (* ¬ O (r ∘ sg) sf. *)
+ ∃m,a.∀n. a≤n → r(sg a) < m * sf n.
+#r #Hr #Hmono #Hconstr
+(* 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 Hconstr (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
--- /dev/null
+include "reverse_complexity/bigops_compl.ma".
+
+(********************************* 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.
+
+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: ∀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 *********************************)
+
+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 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 (auxg h) → CF s (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
+ (λp:ℕ.eqb (min_input h (fst p) (snd (snd p))) (snd (snd p))) →
+ CF s2
+ (λp:ℕ.out (fst p) (snd (snd p)) (h (S (fst p)) (snd (snd p)))) →
+ O s (λx.MSC x + (snd x - fst x)*Max_{i ∈[fst x ,snd x[ }(s1 〈i,x〉+s2 〈i,x〉)) →
+ CF s (auxg h).
+#h #s1 #s2 #s #Hs1 #Hs2 #HO @(ext_CF (aux1g h))
+ [#n whd in ⊢ (??%%); @eq_aux]
+@(CF_max2 … 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 (λp:ℕ.min_input h (fst p) (snd (snd p))) →
+ CF s (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 … (le_to_O … (MSC_in … 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 compl_g4 : ∀h,s1,s. (∀x. 0 < s1 x) →
+ (CF s1
+ (λp.termb (fst (snd p)) (fst p) (h (S (fst (snd p))) (fst p)))) →
+ (O s (λx.MSC x + ((snd(snd x) - (fst x)))*Max_{i ∈[S(fst x) ,S(snd (snd x))[ }(s1 〈i,x〉))) →
+ CF s (λp:ℕ.min_input h (fst p) (snd (snd p))).
+#h #s1 #s #pos_s1 #Hs1 #HO @(ext_CF (min_input_aux h))
+ [#n whd in ⊢ (??%%); @min_input_eq]
+@(CF_mu4 … MSC MSC … pos_s1 … Hs1)
+ [@CF_compS @CF_fst
+ |@CF_comp_snd @CF_snd
+ |@(O_trans … HO) @O_plus [@O_plus @O_plus_l // | @O_plus_r //]
+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 coroll3: ∀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.(snd x - fst x)*Max_{i ∈[fst x,snd x[ }(s1 〈i,x〉)).
+#s1 #Hs1 @le_to_O #i @le_times // @Max_le #a #lta #_ @Hs1 //
+@lt_minus_to_plus_r //
+qed.
+
+(**************************** end of technical lemmas *************************)
+
+lemma compl_g5 : ∀h,s1.
+ (∀n. 0 < s1 n) →
+ (∀n. monotonic ℕ le (λa:ℕ.s1 〈a,n〉)) →
+ (CF s1
+ (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〉)
+ (λp:ℕ.min_input h (fst p) (snd (snd p))).
+#h #s1 #Hpos #Hmono #Hs1 @(compl_g4 … Hpos Hs1) @O_plus
+[@O_plus_l // |@O_plus_r @le_to_O #n @le_times //
+@Max_le #i #lti #_ @Hmono @le_S_S_to_le @lt_minus_to_plus_r @lti
+qed.
+
+lemma compl_g6: ∀h.
+ constructible (λx. h (fst x) (snd x)) →
+ (CF (λx. sU 〈max (fst (snd x)) (snd (snd x)),〈fst x,h (S (fst (snd x))) (fst x)〉〉)
+ (λp.termb (fst (snd p)) (fst p) (h (S (fst (snd p))) (fst p)))).
+#h #hconstr @(ext_CF (termb_aux h))
+ [#n normalize >fst_pair >snd_pair >fst_pair >snd_pair // ]
+@(CF_comp … (λx.MSC x + h (S (fst (snd x))) (fst x)) … CF_termb)
+ [@CF_comp_pair
+ [@CF_comp_fst @(monotonic_CF … CF_snd) #x //
+ |@CF_comp_pair
+ [@(monotonic_CF … CF_fst) #x //
+ |@(ext_CF ((λx.h (fst x) (snd x))∘(λx.〈S (fst (snd x)),fst x〉)))
+ [#n normalize >fst_pair >snd_pair %]
+ @(CF_comp … MSC …hconstr)
+ [@CF_comp_pair [@CF_compS @CF_comp_fst // |//]
+ |@O_plus @le_to_O [//|#n >fst_pair >snd_pair //]
+ ]
+ ]
+ ]
+ |@O_plus
+ [@O_plus
+ [@(O_trans … (λx.MSC (fst x) + MSC (max (fst (snd x)) (snd (snd x)))))
+ [%{2} %{0} #x #_ cases (surj_pair x) #a * #b #eqx >eqx
+ >fst_pair >snd_pair @(transitive_le … (MSC_pair …))
+ >distributive_times_plus @le_plus [//]
+ cases (surj_pair b) #c * #d #eqb >eqb
+ >fst_pair >snd_pair @(transitive_le … (MSC_pair …))
+ whd in ⊢ (??%); @le_plus
+ [@monotonic_MSC @(le_maxl … (le_n …))
+ |>commutative_times <times_n_1 @monotonic_MSC @(le_maxr … (le_n …))
+ ]
+ |@O_plus [@le_to_O #x @sU_le_x |@le_to_O #x @sU_le_i]
+ ]
+ |@le_to_O #n @sU_le
+ ]
+ |@le_to_O #x @monotonic_sU // @(le_maxl … (le_n …)) ]
+ ]
+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.
+
+definition faux2 ≝ λh.
+ (λx.MSC x + (snd (snd x)-fst x)*
+ (λx.sU 〈max (fst(snd x)) (snd(snd x)),〈fst x,h (S (fst (snd x))) (fst x)〉〉) 〈snd (snd x),x〉).
+
+lemma compl_g7: ∀h.
+ constructible (λx. h (fst x) (snd x)) →
+ (∀n. monotonic ? le (h n)) →
+ 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))〉〉)
+ (λp:ℕ.min_input h (fst p) (snd (snd p))).
+#h #hcostr #hmono @(monotonic_CF … (faux2 h))
+ [#n normalize >fst_pair >snd_pair //]
+@compl_g5 [#n @sU_pos | 3:@(compl_g6 h hcostr)] #n #x #y #lexy >fst_pair >snd_pair
+>fst_pair >snd_pair @monotonic_sU // @hmono @lexy
+qed.
+
+lemma compl_g71: ∀h.
+ constructible (λx. h (fst x) (snd x)) →
+ (∀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))〉〉)
+ (λp:ℕ.min_input h (fst p) (snd (snd p))).
+#h #hcostr #hmono @(monotonic_CF … (compl_g7 h hcostr hmono)) #x
+@le_plus [@monotonic_MSC //]
+cases (decidable_le (fst x) (snd(snd x)))
+ [#Hle @le_times // @monotonic_sU
+ |#Hlt >(minus_to_0 … (lt_to_le … )) [// | @not_le_to_lt @Hlt]
+ ]
+qed.
+
+definition out_aux ≝ λh.
+ out_unary ∘ λp.〈fst p,〈snd(snd p),h (S (fst p)) (snd (snd p))〉〉.
+
+lemma compl_g8: ∀h.
+ constructible (λx. h (fst x) (snd x)) →
+ (CF (λx. sU 〈max (fst x) (snd x),〈snd(snd x),h (S (fst x)) (snd(snd x))〉〉)
+ (λp.out (fst p) (snd (snd p)) (h (S (fst p)) (snd (snd p))))).
+#h #hconstr @(ext_CF (out_aux h))
+ [#n normalize >fst_pair >snd_pair >fst_pair >snd_pair // ]
+@(CF_comp … (λx.h (S (fst x)) (snd(snd x)) + MSC x) … CF_out)
+ [@CF_comp_pair
+ [@(monotonic_CF … CF_fst) #x //
+ |@CF_comp_pair
+ [@CF_comp_snd @(monotonic_CF … CF_snd) #x //
+ |@(ext_CF ((λx.h (fst x) (snd x))∘(λx.〈S (fst x),snd(snd x)〉)))
+ [#n normalize >fst_pair >snd_pair %]
+ @(CF_comp … MSC …hconstr)
+ [@CF_comp_pair [@CF_compS // | @CF_comp_snd // ]
+ |@O_plus @le_to_O [//|#n >fst_pair >snd_pair //]
+ ]
+ ]
+ ]
+ |@O_plus
+ [@O_plus
+ [@le_to_O #n @sU_le
+ |@(O_trans … (λx.MSC (max (fst x) (snd x))))
+ [%{2} %{0} #x #_ cases (surj_pair x) #a * #b #eqx >eqx
+ >fst_pair >snd_pair @(transitive_le … (MSC_pair …))
+ whd in ⊢ (??%); @le_plus
+ [@monotonic_MSC @(le_maxl … (le_n …))
+ |>commutative_times <times_n_1 @monotonic_MSC @(le_maxr … (le_n …))
+ ]
+ |@le_to_O #x @(transitive_le ???? (sU_le_i … )) //
+ ]
+ ]
+ |@le_to_O #x @monotonic_sU [@(le_maxl … (le_n …))|//|//]
+ ]
+qed.
+
+lemma compl_g9 : ∀h.
+ constructible (λx. h (fst x) (snd x)) →
+ (∀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)〉〉)
+ (auxg h).
+#h #hconstr #hmono #hantimono
+@(compl_g2 h ??? (compl_g3 … (compl_g71 h hconstr hmono)) (compl_g8 h hconstr))
+@O_plus
+ [@O_plus_l @le_to_O #x >(times_n_1 (MSC x)) >commutative_times @le_times
+ [// | @monotonic_MSC // ]]
+@(O_trans … (coroll3 ??))
+ [#n #a #b #leab #ltb >fst_pair >fst_pair >snd_pair >snd_pair
+ cut (b ≤ n) [@(transitive_le … (le_snd …)) @lt_to_le //] #lebn
+ cut (max a n = n)
+ [normalize >le_to_leb_true [//|@(transitive_le … leab lebn)]] #maxa
+ cut (max b n = n) [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 c ≝ λx.(S (snd x-fst x))*MSC 〈x,x〉.
+
+definition sg ≝ λh,x.
+ let a ≝ fst x in
+ let b ≝ snd x in
+ c x + (b-a)*(S(b-a))*sU 〈x,〈snd x,h (S a) b〉〉.
+
+lemma sg_def1 : ∀h,a,b.
+ sg h 〈a,b〉 = c 〈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 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 normalize >fst_pair >snd_pair //
+qed.
+
+lemma compl_g11 : ∀h.
+ constructible (λx. h (fst x) (snd x)) →
+ (∀n. monotonic ? le (h n)) →
+ (∀n,a,b. a ≤ b → b ≤ n → h b n ≤ h a n) →
+ CF (sg h) (unary_g h).
+#h #hconstr #Hm #Ham @compl_g1 @(compl_g9 h hconstr 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
+ | 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.
+// 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.
+
+axiom h_of_constr : ∀r:nat →nat.
+ (∀x. x ≤ r x) → monotonic ? le r → constructible r →
+ constructible (h_of r).
+
+lemma speed_compl: ∀r:nat →nat.
+ (∀x. x ≤ r x) → monotonic ? le r → constructible r →
+ CF (h_of r) (unary_g (λi,x. r(h_of r 〈i,x〉))).
+#r #Hr #Hmono #Hconstr @(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)
+ [%{(pred(b-a))} >S_pred [//] @lt_plus_to_minus_r @not_le_to_lt @ltab]
+ * #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)
+ |@(constr_comp … Hconstr Hr) @(ext_constr (h_of r))
+ [#x cases (surj_pair x) #a * #b #eqx >eqx >fst_pair >snd_pair //]
+ @(h_of_constr r Hr Hmono Hconstr)
+ ]
+qed.
+
+lemma speed_compl_i: ∀r:nat →nat.
+ (∀x. x ≤ r x) → monotonic ? le r → constructible r →
+ ∀i. CF (λx.h_of r 〈i,x〉) (λx.g (λi,x. r(h_of r 〈i,x〉)) i x).
+#r #Hr #Hmono #Hconstr #i
+@(ext_CF (λx.unary_g (λi,x. r(h_of r 〈i,x〉)) 〈i,x〉))
+ [#n whd in ⊢ (??%%); @eq_f @sym_eq >fst_pair >snd_pair %]
+@smn @(ext_CF … (speed_compl r Hr Hmono Hconstr)) #n //
+qed.
+
+(**************************** the speedup theorem *****************************)
+theorem pseudo_speedup:
+ ∀r:nat →nat. (∀x. x ≤ r x) → monotonic ? le r → constructible r →
+ ∃f.∀sf. CF sf f → ∃g,sg. f ≈ g ∧ CF sg g ∧ O sf (r ∘ sg).
+(* ∃m,a.∀n. a≤n → r(sg a) < m * sf n. *)
+#r #Hr #Hmono #Hconstr
+(* f is (g (λi,x. r(h_of r 〈i,x〉)) 0) *)
+%{(g (λi,x. r(h_of r 〈i,x〉)) 0)} #sf * #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 Hconstr (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 → constructible r →
+ ∃f.∀sf. CF sf f → ∃g,sg. f ≈ g ∧ CF sg g ∧
+ (* ¬ O (r ∘ sg) sf. *)
+ ∃m,a.∀n. a≤n → r(sg a) < m * sf n.
+#r #Hr #Hmono #Hconstr
+(* f is (g (λi,x. r(h_of r 〈i,x〉)) 0) *)
+%{(g (λi,x. r(h_of r 〈i,x〉)) 0)} #sf * #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 〈S i,x〉) *)
+%{(λx. h_of r 〈S i,x〉)}
+lapply (speed_compl_i … Hr Hmono Hconstr (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
--- /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〉.
+
+lemma expand_pair: ∀x. x = 〈fst x, snd x〉.
+#x lapply (surj_pair x) * #a * #b #Hx >Hx >fst_pair >snd_pair //
+qed.
+
+(************************************* 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.
+
+lemma fst_pU: ∀i,x,r. fst (pU i x r) = termb i x r.
+#i #x #r normalize cases (U i x r) normalize >fst_pair //
+qed.
+
+lemma snd_pU: ∀i,x,r. snd (pU i x r) = out i x r.
+#i #x #r normalize cases (U i x r) normalize >snd_pair //
+qed.
+
+
+definition total ≝ λf.λx:nat. Some nat (f x).
+
+
+(********************************* 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.
+axiom MSC_pair: ∀a,b. MSC 〈a,b〉 ≤ MSC a + MSC b.
+
+(* 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_CF1 : ∀f,g,s. (∀n. f n = g n) → CF s g → CF s f.
+#f #g #s #Hext * #HO * #i * #Hcode #HC % // %{i} %
+ [#x cases (Hcode x) #a #H %{a} whd in match (total ??); >Hext @H | //]
+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.
+
+(* primitve recursion *)
+
+(* no arguments *)
+
+let rec prim_rec0 (k:nat) (h:nat →nat) n on n ≝
+ match n with
+ [ O ⇒ k
+ | S a ⇒ h 〈a, prim_rec0 k h a〉].
+
+lemma prim_rec0_S: ∀k,h,n.
+ prim_rec0 k h (S n) = h 〈n, prim_rec0 k h n〉.
+// qed.
+
+let rec prim_rec0_compl (k,sk:nat) (h,sh:nat →nat) n on n ≝
+ match n with
+ [ O ⇒ sk
+ | S a ⇒ prim_rec0_compl k sk h sh a + sh (prim_rec0 k h a)].
+
+axiom CF_prim_rec0_gen: ∀k,h,sk,sh,sh1,sf. CF sh h →
+ O sh1 (λx.snd x + sh 〈fst x,prim_rec0 k h (fst x)〉) →
+ O sf (prim_rec0 sk sh1) →
+ CF sf (prim_rec0 k h).
+
+lemma CF_prim_rec0: ∀k,h,sk,sh,sf. CF sh h →
+ O sf (prim_rec0 sk (λx. snd x + sh 〈fst x,prim_rec0 k h (fst x)〉))
+ → CF sf (prim_rec0 k h).
+#k #h #sk #sh #sf #Hh #HO @(CF_prim_rec0_gen … Hh … HO) @O_refl
+qed.
+
+(* with argument(s) m *)
+let rec prim_rec (k,h:nat →nat) n m on n ≝
+ match n with
+ [ O ⇒ k m
+ | S a ⇒ h 〈a,〈prim_rec k h a m, m〉〉].
+
+lemma prim_rec_S: ∀k,h,n,m.
+ prim_rec k h (S n) m = h 〈n,〈prim_rec k h n m, m〉〉.
+// qed.
+
+lemma prim_rec_le: ∀k1,k2,h1,h2. (∀x.k1 x ≤ k2 x) →
+(∀x,y.x ≤y → h1 x ≤ h2 y) →
+ ∀x,m. prim_rec k1 h1 x m ≤ prim_rec k2 h2 x m.
+#k1 #k2 #h1 #h2 #lek #leh #x #m elim x //
+#n #Hind normalize @leh @le_pair // @le_pair //
+qed.
+
+definition unary_pr ≝ λk,h,x. prim_rec k h (fst x) (snd x).
+
+lemma prim_rec_unary: ∀k,h,a,b.
+ prim_rec k h a b = unary_pr k h 〈a,b〉.
+#k #h #a #b normalize >fst_pair >snd_pair //
+qed.
+
+
+let rec prim_rec_compl (k,h,sk,sh:nat →nat) n m on n ≝
+ match n with
+ [ O ⇒ sk m
+ | S a ⇒ prim_rec_compl k h sk sh a m + sh (prim_rec k h a m)].
+
+axiom CF_prim_rec_gen: ∀k,h,sk,sh,sh1. CF sk k → CF sh h →
+ O sh1 (λx. fst (snd x) + sh 〈fst x,〈unary_pr k h 〈fst x,snd (snd x)〉,snd (snd x)〉〉) →
+ CF (unary_pr sk sh1) (unary_pr k h).
+
+lemma CF_prim_rec: ∀k,h,sk,sh,sf. CF sk k → CF sh h →
+ O sf (unary_pr sk (λx. fst (snd x) + sh 〈fst x,〈unary_pr k h 〈fst x,snd (snd x)〉,snd (snd x)〉〉))
+ → CF sf (unary_pr k h).
+#k #h #sk #sh #sf #Hk #Hh #Osf @(O_to_CF … Osf) @(CF_prim_rec_gen … Hk Hh) @O_refl
+qed.
+
+(**************************** primitive operations*****************************)
+
+definition id ≝ λx:nat.x.
+
+axiom CF_id: CF MSC id.
+axiom CF_const: ∀k.CF MSC (λx.k).
+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).
+axiom CF_comp_pair: ∀h,f,g. CF h f → CF h g → CF h (λx. 〈f x,g x〉).
+
+lemma CF_fst: CF MSC fst.
+@(ext_CF (fst ∘ id)) [#n //] @(CF_comp_fst … CF_id)
+qed.
+
+lemma CF_snd: CF MSC snd.
+@(ext_CF (snd ∘ id)) [#n //] @(CF_comp_snd … CF_id)
+qed.
+
+(************************************** eqb ***********************************)
+
+axiom CF_eqb: ∀h,f,g.
+ CF h f → CF h g → CF h (λx.eqb (f x) (g x)).
+
+(*********************************** maximum **********************************)
+
+alias symbol "pair" (instance 13) = "abstract pair".
+alias symbol "pair" (instance 12) = "abstract pair".
+alias symbol "and" (instance 11) = "boolean and".
+alias symbol "plus" (instance 8) = "natural plus".
+alias symbol "pair" (instance 7) = "abstract pair".
+alias symbol "plus" (instance 6) = "natural plus".
+alias symbol "pair" (instance 5) = "abstract pair".
+alias id "max" = "cic:/matita/arithmetics/nat/max#def:2".
+alias symbol "minus" (instance 3) = "natural minus".
+lemma max_gen: ∀a,b,p,f,x. a ≤b →
+ max_{i ∈[a,b[ | p 〈i,x〉 }(f 〈i,x〉) = max_{i < b | leb a i ∧ p 〈i,x〉 }(f 〈i,x〉).
+#a #b #p #f #x @(bigop_I_gen ????? MaxA)
+qed.
+
+lemma max_prim_rec_base: ∀a,b,p,f,x. a ≤b →
+ max_{i < b| p 〈i,x〉 }(f 〈i,x〉) =
+ prim_rec (λi.0)
+ (λi.if p 〈fst i,x〉 then max (f 〈fst i,snd (snd i)〉) (fst (snd i)) else fst (snd i)) b x.
+#a #b #p #f #x #leab >max_gen // elim b
+ [normalize //
+ |#i #Hind >prim_rec_S >fst_pair >snd_pair >snd_pair >fst_pair
+ cases (true_or_false (p 〈i,x〉)) #Hcase >Hcase
+ [<Hind >bigop_Strue // |<Hind >bigop_Sfalse // ]
+ ]
+qed.
+
+lemma max_prim_rec: ∀a,b,p,f,x. a ≤b →
+ max_{i ∈[a,b[ | p 〈i,x〉 }(f 〈i,x〉) =
+ prim_rec (λi.0)
+ (λi.if leb a (fst i) ∧ p 〈fst i,x〉 then max (f 〈fst i,snd (snd i)〉) (fst (snd i)) else fst (snd i)) b x.
+#a #b #p #f #x #leab >max_gen // elim b
+ [normalize //
+ |#i #Hind >prim_rec_S >fst_pair >snd_pair >snd_pair >fst_pair
+ cases (true_or_false (leb a i ∧ p 〈i,x〉)) #Hcase >Hcase
+ [<Hind >bigop_Strue // |<Hind >bigop_Sfalse // ]
+ ]
+qed.
+
+lemma max_prim_rec1: ∀a,b,p,f,x.
+ max_{i ∈[a x,b x[ | p 〈i,x〉 }(f 〈i,x〉) =
+ prim_rec (λi.0)
+ (λi.if p 〈fst i +fst (snd (snd i)),snd (snd (snd i))〉
+ then max (f 〈fst i +fst (snd (snd i)),snd (snd (snd i))〉) (fst (snd i))
+ else fst (snd i)) (b x-a x) 〈a x ,x〉.
+#a #b #p #f #x elim (b x-a x)
+ [normalize //
+ |#i #Hind >prim_rec_S
+ >fst_pair >snd_pair >snd_pair >fst_pair >snd_pair >fst_pair
+ cases (true_or_false (p 〈i+a x,x〉)) #Hcase >Hcase
+ [<Hind >bigop_Strue // |<Hind >bigop_Sfalse // ]
+ ]
+qed.
+
+lemma sum_prim_rec1: ∀a,b,p,f,x.
+ ∑_{i ∈[a x,b x[ | p 〈i,x〉 }(f 〈i,x〉) =
+ prim_rec (λi.0)
+ (λi.if p 〈fst i +fst (snd (snd i)),snd (snd (snd i))〉
+ then plus (f 〈fst i +fst (snd (snd i)),snd (snd (snd i))〉) (fst (snd i))
+ else fst (snd i)) (b x-a x) 〈a x ,x〉.
+#a #b #p #f #x elim (b x-a x)
+ [normalize //
+ |#i #Hind >prim_rec_S
+ >fst_pair >snd_pair >snd_pair >fst_pair >snd_pair >fst_pair
+ cases (true_or_false (p 〈i+a x,x〉)) #Hcase >Hcase
+ [<Hind >bigop_Strue // |<Hind >bigop_Sfalse // ]
+ ]
+qed.
+
+lemma bigop_prim_rec: ∀a,b,c,p,f,x.
+ bigop (b x-a x) (λi:ℕ.p 〈i+a x,x〉) ? (c 〈a x,x〉) plus (λi:ℕ.f 〈i+a x,x〉) =
+ prim_rec c
+ (λi.if p 〈fst i +fst (snd (snd i)),snd (snd (snd i))〉
+ then plus (f 〈fst i +fst (snd (snd i)),snd (snd (snd i))〉) (fst (snd i))
+ else fst (snd i)) (b x-a x) 〈a x ,x〉.
+#a #b #c #p #f #x normalize elim (b x-a x)
+ [normalize //
+ |#i #Hind >prim_rec_S
+ >fst_pair >snd_pair >snd_pair >fst_pair >snd_pair >fst_pair
+ cases (true_or_false (p 〈i+a x,x〉)) #Hcase >Hcase
+ [<Hind >bigop_Strue // |<Hind >bigop_Sfalse // ]
+ ]
+qed.
+
+lemma bigop_prim_rec_dec: ∀a,b,c,p,f,x.
+ bigop (b x-a x) (λi:ℕ.p 〈b x -i,x〉) ? (c 〈b x,x〉) plus (λi:ℕ.f 〈b x-i,x〉) =
+ prim_rec c
+ (λi.if p 〈fst (snd (snd i)) - fst i,snd (snd (snd i))〉
+ then plus (f 〈fst (snd (snd i)) - fst i,snd (snd (snd i))〉) (fst (snd i))
+ else fst (snd i)) (b x-a x) 〈b x ,x〉.
+#a #b #c #p #f #x normalize elim (b x-a x)
+ [normalize //
+ |#i #Hind >prim_rec_S
+ >fst_pair >snd_pair >snd_pair >fst_pair >snd_pair >fst_pair
+ cases (true_or_false (p 〈b x - i,x〉)) #Hcase >Hcase
+ [<Hind >bigop_Strue // |<Hind >bigop_Sfalse // ]
+ ]
+qed.
+
+lemma bigop_prim_rec_dec1: ∀a,b,c,p,f,x.
+ bigop (S(b x)-a x) (λi:ℕ.p 〈b x - i,x〉) ? (c 〈b x,x〉) plus (λi:ℕ.f 〈b x- i,x〉) =
+ prim_rec c
+ (λi.if p 〈fst (snd (snd i)) - (fst i),snd (snd (snd i))〉
+ then plus (f 〈fst (snd (snd i)) - (fst i),snd (snd (snd i))〉) (fst (snd i))
+ else fst (snd i)) (S(b x)-a x) 〈b x,x〉.
+#a #b #c #p #f #x elim (S(b x)-a x)
+ [normalize //
+ |#i #Hind >prim_rec_S
+ >fst_pair >snd_pair >snd_pair >fst_pair >snd_pair >fst_pair
+ cases (true_or_false (p 〈b x - i,x〉)) #Hcase >Hcase
+ [<Hind >bigop_Strue // |<Hind >bigop_Sfalse // ]
+ ]
+qed.
+
+(*
+lemma bigop_aux_1: ∀k,c,f.
+ bigop (S k) (λi:ℕ.true) ? c plus (λi:ℕ.f i) =
+ f 0 + bigop k (λi:ℕ.true) ? c plus (λi:ℕ.f (S i)).
+#k #c #f elim k [normalize //] #i #Hind >bigop_Strue //
+
+lemma bigop_prim_rec_dec: ∀a,b,c,f,x.
+ bigop (b x-a x) (λi:ℕ.true) ? (c 〈b x,x〉) plus (λi:ℕ.f 〈i+a x,x〉) =
+ prim_rec c
+ (λi. plus (f 〈fst (snd (snd i)) - fst i,snd (snd (snd i))〉) (fst (snd i)))
+ (b x-a x) 〈b x ,x〉.
+#a #b #c #f #x normalize elim (b x-a x)
+ [normalize //
+ |#i #Hind >prim_rec_S
+ >fst_pair >snd_pair >snd_pair >fst_pair >snd_pair >fst_pair
+ <Hind >bigop_Strue // |<Hind >bigop_Sfalse // ]
+ ]
+qed. *)
+
+lemma bigop_plus_c: ∀k,p,f,c.
+ c k + bigop k (λi.p i) ? 0 plus (λi.f i) =
+ bigop k (λi.p i) ? (c k) plus (λi.f i).
+#k #p #f elim k [normalize //]
+#i #Hind #c cases (true_or_false (p i)) #Hcase
+[>bigop_Strue // >bigop_Strue // <associative_plus >(commutative_plus ? (f i))
+ >associative_plus @eq_f @Hind
+|>bigop_Sfalse // >bigop_Sfalse //
+]
+qed.
+
+(* the argument is 〈b-a,〈a,x〉〉 *)
+
+
+definition max_unary_pr ≝ λp,f.unary_pr (λx.0)
+ (λi.
+ let k ≝ fst i in
+ let r ≝ fst (snd i) in
+ let a ≝ fst (snd (snd i)) in
+ let x ≝ snd (snd (snd i)) in
+ if p 〈k + a,x〉 then max (f 〈k+a,x〉) r else r).
+
+lemma max_unary_pr1: ∀a,b,p,f,x.
+ max_{i ∈[a x,b x[ | p 〈i,x〉 }(f 〈i,x〉) =
+ ((max_unary_pr p f) ∘ (λx.〈b x - a x,〈a x,x〉〉)) x.
+#a #b #p #f #x normalize >fst_pair >snd_pair @max_prim_rec1
+qed.
+
+(*
+lemma max_unary_pr: ∀a,b,p,f,x.
+ max_{i ∈[a,b[ | p 〈i,x〉 }(f 〈i,x〉) =
+ prim_rec (λi.0)
+ (λi.if p 〈fst i +a,x〉 then max (f 〈fst i +a ,snd (snd i)〉) (fst (snd i)) else fst (snd i)) (b-a) x.
+*)
+
+(*
+definition unary_compl ≝ λp,f,hf.
+ unary_pr MSC
+ (λx:ℕ
+ .fst (snd x)
+ +hf
+ 〈fst x,
+ 〈unary_pr (λx0:ℕ.O)
+ (λi:ℕ
+ .(let (k:ℕ) ≝fst i in
+ let (r:ℕ) ≝fst (snd i) in
+ let (a:ℕ) ≝fst (snd (snd i)) in
+ let (x:ℕ) ≝snd (snd (snd i)) in
+ if p 〈k+a,x〉 then max (f 〈k+a,x〉) r else r )) 〈fst x,snd (snd x)〉,
+ snd (snd x)〉〉). *)
+
+definition aux_compl ≝ λhp,hf.λi.
+ let k ≝ fst i in
+ let r ≝ fst (snd i) in
+ let a ≝ fst (snd (snd i)) in
+ let x ≝ snd (snd (snd i)) in
+ hp 〈k+a,x〉 + hf 〈k+a,x〉 + (* was MSC r*) MSC i .
+
+definition aux_compl1 ≝ λhp,hf.λi.
+ let k ≝ fst i in
+ let r ≝ fst (snd i) in
+ let a ≝ fst (snd (snd i)) in
+ let x ≝ snd (snd (snd i)) in
+ hp 〈k+a,x〉 + hf 〈k+a,x〉 + MSC r.
+
+lemma aux_compl1_def: ∀k,r,m,hp,hf.
+ aux_compl1 hp hf 〈k,〈r,m〉〉 =
+ let a ≝ fst m in
+ let x ≝ snd m in
+ hp 〈k+a,x〉 + hf 〈k+a,x〉 + MSC r.
+#k #r #m #hp #hf normalize >fst_pair >snd_pair >snd_pair >fst_pair //
+qed.
+
+lemma aux_compl1_def1: ∀k,r,a,x,hp,hf.
+ aux_compl1 hp hf 〈k,〈r,〈a,x〉〉〉 = hp 〈k+a,x〉 + hf 〈k+a,x〉 + MSC r.
+#k #r #a #x #hp #hf normalize >fst_pair >snd_pair >snd_pair >fst_pair
+>fst_pair >snd_pair //
+qed.
+
+
+axiom Oaux_compl: ∀hp,hf. O (aux_compl1 hp hf) (aux_compl hp hf).
+
+(*
+definition IF ≝ λb,f,g:nat →option nat. λx.
+ match b x with
+ [None ⇒ None ?
+ |Some n ⇒ if (eqb n 0) then f x else g x].
+*)
+
+axiom CF_if: ∀b:nat → bool. ∀f,g,s. CF s b → CF s f → CF s g →
+ CF s (λx. if b x then f x else g x).
+
+(*
+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.
+*)
+
+axiom CF_le: ∀h,f,g. CF h f → CF h g → CF h (λx. leb (f x) (g x)).
+axiom CF_max1: ∀h,f,g. CF h f → CF h g → CF h (λx. max (f x) (g x)).
+axiom CF_plus: ∀h,f,g. CF h f → CF h g → CF h (λx. f x + g x).
+axiom CF_minus: ∀h,f,g. CF h f → CF h g → CF h (λx. f x - g x).
+
+axiom MSC_prop: ∀f,h. CF h f → ∀x. MSC (f x) ≤ h x.
+
+lemma MSC_max: ∀f,h,x. CF h f → MSC (max_{i < x}(f i)) ≤ max_{i < x}(h i).
+#f #h #x #hf elim x // #i #Hind >bigop_Strue [|//] >bigop_Strue [|//]
+whd in match (max ??);
+cases (true_or_false (leb (f i) (bigop i (λi0:ℕ.true) ? 0 max(λi0:ℕ.f i0))))
+#Hcase >Hcase
+ [@(transitive_le … Hind) @le_maxr //
+ |@(transitive_le … (MSC_prop … hf i)) @le_maxl //
+ ]
+qed.
+
+lemma 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〉 + max_{i ∈ [a x, b x [ }(hf 〈i,x〉)))) →
+ CF s (λx.max_{i ∈[a x,b x[ | p 〈i,x〉 }(f 〈i,x〉)).
+#a #b #p #f #ha #hb #hp #hf #s #CFa #CFb #CFp #CFf #HO
+@ext_CF1 [|#x @max_unary_pr1]
+@(CF_comp ??? (λx.ha x + hb x))
+ [|@CF_comp_pair
+ [@CF_minus [@(O_to_CF … CFb) @O_plus_r // |@(O_to_CF … CFa) @O_plus_l //]
+ |@CF_comp_pair
+ [@(O_to_CF … CFa) @O_plus_l //
+ | @(O_to_CF … CF_id) @O_plus_r @(proj1 … CFb)
+ ]
+ ]
+ |@(CF_prim_rec … MSC … (aux_compl1 hp hf))
+ [@CF_const
+ |@(O_to_CF … (Oaux_compl … ))
+ @CF_if
+ [@(CF_comp p … MSC … CFp)
+ [@CF_comp_pair
+ [@CF_plus [@CF_fst| @CF_comp_fst @CF_comp_snd @CF_snd]
+ |@CF_comp_snd @CF_comp_snd @CF_snd]
+ |@le_to_O #x normalize >commutative_plus >associative_plus @le_plus //
+ ]
+ |@CF_max1
+ [@(CF_comp f … MSC … CFf)
+ [@CF_comp_pair
+ [@CF_plus [@CF_fst| @CF_comp_fst @CF_comp_snd @CF_snd]
+ |@CF_comp_snd @CF_comp_snd @CF_snd]
+ |@le_to_O #x normalize >commutative_plus //
+ ]
+ |@CF_comp_fst @(monotonic_CF … CF_snd) normalize //
+ ]
+ |@CF_comp_fst @(monotonic_CF … CF_snd) normalize //
+ ]
+ |@O_refl
+ ]
+ |@(O_trans … HO) -HO
+ @(O_trans ? (λx:ℕ
+ .ha x+hb x
+ +bigop (b x-a x) (λi:ℕ.true) ? (MSC 〈a x,x〉) plus
+ (λi:ℕ
+ .(λi0:ℕ
+ .hp 〈i0,x〉+hf 〈i0,x〉
+ +bigop (b x-a x) (λi1:ℕ.true) ? 0 max
+ (λi1:ℕ.(λi2:ℕ.hf 〈i2,x〉) (i1+a x))) (i+a x))))
+ [
+ @le_to_O #n @le_plus // whd in match (unary_pr ???);
+ >fst_pair >snd_pair >bigop_prim_rec elim (b n - a n)
+ [normalize //
+ |#x #Hind >prim_rec_S >fst_pair >snd_pair >fst_pair >snd_pair >aux_compl1_def1
+ >prim_rec_S >fst_pair >snd_pair >fst_pair >snd_pair >fst_pair
+ >snd_pair normalize in ⊢ (??%); >commutative_plus @le_plus
+ [-Hind @le_plus // normalize >fst_pair >snd_pair
+ @(transitive_le ? (bigop x (λi1:ℕ.true) ? 0 (λn0:ℕ.λm:ℕ.if leb n0 m then m else n0 )
+ (λi1:ℕ.hf 〈i1+a n,n〉)))
+ [elim x [normalize @MSC_le]
+ #x0 #Hind >prim_rec_S >fst_pair >snd_pair >snd_pair >snd_pair
+ >fst_pair >fst_pair cases (p 〈x0+a n,n〉) normalize
+ [cases (true_or_false (leb (f 〈x0+a n,n〉)
+ (prim_rec (λx00:ℕ.O)
+ (λi:ℕ
+ .if p 〈fst i+fst (snd (snd i)),snd (snd (snd i))〉
+ then if leb (f 〈fst i+fst (snd (snd i)),snd (snd (snd i))〉)
+ (fst (snd i))
+ then fst (snd i)
+ else f 〈fst i+fst (snd (snd i)),snd (snd (snd i))〉
+ else fst (snd i) ) x0 〈a n,n〉))) #Hcase >Hcase normalize
+ [@(transitive_le … Hind) -Hind @(le_maxr … (le_n …))
+ |@(transitive_le … (MSC_prop … CFf …)) @(le_maxl … (le_n …))
+ ]
+ |@(transitive_le … Hind) -Hind @(le_maxr … (le_n …))
+ ]
+ |@(le_maxr … (le_n …))
+ ]
+ |@(transitive_le … Hind) -Hind
+ generalize in match (bigop x (λi:ℕ.true) ? 0 max
+ (λi1:ℕ.(λi2:ℕ.hf 〈i2,n〉) (i1+a n))); #c
+ generalize in match (hf 〈x+a n,n〉); #c1
+ elim x [//] #x0 #Hind
+ >prim_rec_S >prim_rec_S normalize >fst_pair >fst_pair >snd_pair
+ >snd_pair >snd_pair >snd_pair >snd_pair >snd_pair >fst_pair >fst_pair
+ >fst_pair @le_plus
+ [@le_plus // cases (true_or_false (leb c1 c)) #Hcase
+ >Hcase normalize // @lt_to_le @not_le_to_lt @(leb_false_to_not_le ?? Hcase)
+ |@Hind
+ ]
+ ]
+ ]
+ |@O_plus [@O_plus_l //] @le_to_O #x
+ <bigop_plus_c @le_plus // @(transitive_le … (MSC_pair …)) @le_plus
+ [@MSC_prop @CFa | @MSC_prop @(O_to_CF MSC … (CF_const x)) @(proj1 … CFb)]
+ ]
+qed.
+
+(* old
+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 ********************************)
+
+alias symbol "plus" (instance 2) = "natural plus".
+alias symbol "plus" (instance 5) = "natural plus".
+alias symbol "plus" (instance 1) = "natural plus".
+alias symbol "plus" (instance 4) = "natural plus".
+alias symbol "pair" (instance 3) = "abstract pair".
+alias id "O" = "cic:/matita/arithmetics/nat/nat#con:0:1:0".
+let rec my_minim a f x k on k ≝
+ match k with
+ [O ⇒ a
+ |S p ⇒ if eqb (my_minim a f x p) (a+p)
+ then if f 〈a+p,x〉 then a+p else S(a+p)
+ else (my_minim a f x p) ].
+
+lemma my_minim_S : ∀a,f,x,k.
+ my_minim a f x (S k) =
+ if eqb (my_minim a f x k) (a+k)
+ then if f 〈a+k,x〉 then a+k else S(a+k)
+ else (my_minim a f x k) .
+// qed.
+
+lemma my_minim_fa : ∀a,f,x,k. f〈a,x〉 = true → my_minim a f x k = a.
+#a #f #x #k #H elim k // #i #Hind normalize >Hind
+cases (true_or_false (eqb a (a+i))) #Hcase >Hcase normalize //
+<(eqb_true_to_eq … Hcase) >H //
+qed.
+
+lemma my_minim_nfa : ∀a,f,x,k. f〈a,x〉 = false →
+my_minim a f x (S k) = my_minim (S a) f x k.
+#a #f #x #k #H elim k
+ [normalize <plus_n_O >H >eq_to_eqb_true //
+ |#i #Hind >my_minim_S >Hind >my_minim_S <plus_n_Sm //
+ ]
+qed.
+
+lemma my_min_eq : ∀a,f,x,k.
+ min k a (λi.f 〈i,x〉) = my_minim a f x k.
+#a #f #x #k lapply a -a elim k // #i #Hind #a normalize in ⊢ (??%?);
+cases (true_or_false (f 〈a,x〉)) #Hcase >Hcase
+ [>(my_minim_fa … Hcase) // | >Hind @sym_eq @(my_minim_nfa … Hcase) ]
+qed.
+
+(* cambiare qui: togliere S *)
+
+
+definition minim_unary_pr ≝ λf.unary_pr (λx.S(fst x))
+ (λi.
+ let k ≝ fst i in
+ let r ≝ fst (snd i) in
+ let b ≝ fst (snd (snd i)) in
+ let x ≝ snd (snd (snd i)) in
+ if f 〈b-k,x〉 then b-k else r).
+
+definition compl_minim_unary ≝ λhf:nat → nat.λi.
+ let k ≝ fst i in
+ let b ≝ fst (snd (snd i)) in
+ let x ≝ snd (snd (snd i)) in
+ hf 〈b-k,x〉 + MSC 〈k,〈S b,x〉〉.
+
+definition compl_minim_unary_aux ≝ λhf,i.
+ let k ≝ fst i in
+ let r ≝ fst (snd i) in
+ let b ≝ fst (snd (snd i)) in
+ let x ≝ snd (snd (snd i)) in
+ hf 〈b-k,x〉 + MSC i.
+
+lemma compl_minim_unary_aux_def : ∀hf,k,r,b,x.
+ compl_minim_unary_aux hf 〈k,〈r,〈b,x〉〉〉 = hf 〈b-k,x〉 + MSC 〈k,〈r,〈b,x〉〉〉.
+#hf #k #r #b #x normalize >snd_pair >snd_pair >snd_pair >fst_pair >fst_pair //
+qed.
+
+lemma compl_minim_unary_def : ∀hf,k,r,b,x.
+ compl_minim_unary hf 〈k,〈r,〈b,x〉〉〉 = hf 〈b-k,x〉 + MSC 〈k,〈S b,x〉〉.
+#hf #k #r #b #x normalize >snd_pair >snd_pair >snd_pair >fst_pair >fst_pair //
+qed.
+
+lemma compl_minim_unary_aux_def2 : ∀hf,k,r,x.
+ compl_minim_unary_aux hf 〈k,〈r,x〉〉 = hf 〈fst x-k,snd x〉 + MSC 〈k,〈r,x〉〉.
+#hf #k #r #x normalize >snd_pair >snd_pair >fst_pair //
+qed.
+
+lemma compl_minim_unary_def2 : ∀hf,k,r,x.
+ compl_minim_unary hf 〈k,〈r,x〉〉 = hf 〈fst x-k,snd x〉 + MSC 〈k,〈S(fst x),snd x〉〉.
+#hf #k #r #x normalize >snd_pair >snd_pair >fst_pair //
+qed.
+
+lemma min_aux: ∀a,f,x,k. min (S k) (a x) (λi:ℕ.f 〈i,x〉) =
+ ((minim_unary_pr f) ∘ (λx.〈S k,〈a x + k,x〉〉)) x.
+#a #f #x #k whd in ⊢ (???%); >fst_pair >snd_pair
+lapply a -a (* @max_prim_rec1 *)
+elim k
+ [normalize #a >fst_pair >snd_pair >fst_pair >snd_pair >snd_pair >fst_pair
+ <plus_n_O <minus_n_O >fst_pair //
+ |#i #Hind #a normalize in ⊢ (??%?); >prim_rec_S >fst_pair >snd_pair
+ >fst_pair >snd_pair >snd_pair >fst_pair <plus_n_Sm <(Hind (λx.S (a x)))
+ whd in ⊢ (???%); >minus_S_S <(minus_plus_m_m (a x) i) //
+qed.
+
+lemma minim_unary_pr1: ∀a,b,f,x.
+ μ_{i ∈[a x,b x]}(f 〈i,x〉) =
+ if leb (a x) (b x) then ((minim_unary_pr f) ∘ (λx.〈S (b x) - a x,〈b x,x〉〉)) x
+ else (a x).
+#a #b #f #x cases (true_or_false (leb (a x) (b x))) #Hcase >Hcase
+ [cut (b x = a x + (b x - a x))
+ [>commutative_plus <plus_minus_m_m // @leb_true_to_le // ]
+ #Hcut whd in ⊢ (???%); >minus_Sn_m [|@leb_true_to_le //]
+ >min_aux whd in ⊢ (??%?); <Hcut //
+ |>eq_minus_O // @not_le_to_lt @leb_false_to_not_le //
+ ]
+qed.
+
+axiom sum_inv: ∀a,b,f.∑_{i ∈ [a,S b[ }(f i) = ∑_{i ∈ [a,S b[ }(f (a + b - i)).
+
+(*
+#a #b #f @(bigop_iso … plusAC) whd %{(λi.b - a - i)} %{(λi.b - a -i)} %
+ [%[#i #lti #_ normalize @eq_f >commutative_plus <plus_minus_associative
+ [2: @le_minus_to_plus_r //
+ [// @eq_f @@sym_eq @plus_to_minus
+ |#i #Hi #_ % [% [@le_S_S
+*)
+
+example sum_inv_check : ∑_{i ∈ [3,6[ }(i*i) = ∑_{i ∈ [3,6[ }((8-i)*(8-i)).
+normalize // qed.
+
+(* provo rovesciando la somma *)
+
+axiom daemon: ∀P:Prop.P.
+
+lemma 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〉 + MSC 〈b x - i,〈S(b x),x〉〉)) →
+ CF s (λx.μ_{i ∈[a x,b x] }(f 〈i,x〉)).
+#a #b #f #ha #hb #hf #s #CFa #CFb #CFf #HO
+@ext_CF1 [|#x @minim_unary_pr1]
+@CF_if
+ [@CF_le @(O_to_CF … HO)
+ [@(O_to_CF … CFa) @O_plus_l @O_plus_l @O_refl
+ |@(O_to_CF … CFb) @O_plus_l @O_plus_r @O_refl
+ ]
+ |@(CF_comp ??? (λx.ha x + hb x))
+ [|@CF_comp_pair
+ [@CF_minus [@CF_compS @(O_to_CF … CFb) @O_plus_r // |@(O_to_CF … CFa) @O_plus_l //]
+ |@CF_comp_pair
+ [@(O_to_CF … CFb) @O_plus_r //
+ |@(O_to_CF … CF_id) @O_plus_r @(proj1 … CFb)
+ ]
+ ]
+ |@(CF_prim_rec_gen … MSC … (compl_minim_unary_aux hf))
+ [@((λx:ℕ.fst (snd x)
+ +compl_minim_unary hf
+ 〈fst x,
+ 〈unary_pr fst
+ (λi:ℕ
+ .(let (k:ℕ) ≝fst i in
+ let (r:ℕ) ≝fst (snd i) in
+ let (b:ℕ) ≝fst (snd (snd i)) in
+ let (x:ℕ) ≝snd (snd (snd i)) in if f 〈b-S k,x〉 then b-S k else r ))
+ 〈fst x,snd (snd x)〉,
+ snd (snd x)〉〉))
+ |@CF_compS @CF_fst
+ |@CF_if
+ [@(CF_comp f … MSC … CFf)
+ [@CF_comp_pair
+ [@CF_minus [@CF_comp_fst @CF_comp_snd @CF_snd|@CF_fst]
+ |@CF_comp_snd @CF_comp_snd @CF_snd]
+ |@le_to_O #x normalize >commutative_plus //
+ ]
+ |@(O_to_CF … MSC)
+ [@le_to_O #x normalize //
+ |@CF_minus
+ [@CF_comp_fst @CF_comp_snd @CF_snd |@CF_fst]
+ ]
+ |@CF_comp_fst @(monotonic_CF … CF_snd) normalize //
+ ]
+ |@O_plus
+ [@O_plus_l @O_refl
+ |@O_plus_r @O_ext2 [||#x >compl_minim_unary_aux_def2 @refl]
+ @O_trans [||@le_to_O #x >compl_minim_unary_def2 @le_n]
+ @O_plus [@O_plus_l //]
+ @O_plus_r
+ @O_trans [|@le_to_O #x @MSC_pair] @O_plus
+ [@le_to_O #x @monotonic_MSC @(transitive_le ???? (le_fst …))
+ >fst_pair @le_n]
+ @O_trans [|@le_to_O #x @MSC_pair] @O_plus
+ [@le_to_O #x @monotonic_MSC @(transitive_le ???? (le_snd …))
+ >snd_pair @(transitive_le ???? (le_fst …)) >fst_pair
+ normalize >snd_pair >fst_pair lapply (surj_pair x)
+ * #x1 * #x2 #Hx >Hx >fst_pair >snd_pair elim x1 //
+ #i #Hind >prim_rec_S >fst_pair >snd_pair >snd_pair >fst_pair
+ cases (f ?) [@le_S // | //]]
+ @le_to_O #x @monotonic_MSC @(transitive_le ???? (le_snd …)) >snd_pair
+ >(expand_pair (snd (snd x))) in ⊢ (?%?); @le_pair //
+ ]
+ ]
+ |cut (O s (λx.ha x + hb x +
+ ∑_{i ∈[a x ,S(b x)[ }(hf 〈a x+b x-i,x〉 + MSC 〈b x -(a x+b x-i),〈S(b x),x〉〉)))
+ [@(O_ext2 … HO) #x @eq_f @sum_inv] -HO #HO
+ @(O_trans … HO) -HO
+ @(O_trans ? (λx:ℕ.ha x+hb x
+ +bigop (S(b x)-a x) (λi:ℕ.true) ?
+ (MSC 〈b x,x〉) plus (λi:ℕ.(λj.hf j + MSC 〈b x - fst j,〈S(b (snd j)),snd j〉〉) 〈b x- i,x〉)))
+ [@le_to_O #n @le_plus // whd in match (unary_pr ???);
+ >fst_pair >snd_pair >(bigop_prim_rec_dec1 a b ? (λi.true))
+ (* it is crucial to recall that the index is bound by S(b x) *)
+ cut (S (b n) - a n ≤ S (b n)) [//]
+ elim (S(b n) - a n)
+ [normalize //
+ |#x #Hind #lex >prim_rec_S >fst_pair >snd_pair >fst_pair >snd_pair
+ >compl_minim_unary_def >prim_rec_S >fst_pair >snd_pair >fst_pair
+ >snd_pair >fst_pair >snd_pair >fst_pair whd in ⊢ (??%); >commutative_plus
+ @le_plus [2:@Hind @le_S @le_S_S_to_le @lex] -Hind >snd_pair
+ >minus_minus_associative // @le_S_S_to_le //
+ ]
+ |@O_plus [@O_plus_l //] @O_ext2 [||#x @bigop_plus_c]
+ @O_plus
+ [@O_plus_l @O_trans [|@le_to_O #x @MSC_pair]
+ @O_plus [@O_plus_r @le_to_O @(MSC_prop … CFb)|@O_plus_r @(proj1 … CFb)]
+ |@O_plus_r @(O_ext2 … (O_refl …)) #x @same_bigop
+ [//|#i #H #_ @eq_f2 [@eq_f @eq_f2 //|>fst_pair @eq_f @eq_f2 //]
+ ]
+ ]
+ ]
+ ]
+ |@(O_to_CF … CFa) @(O_trans … HO) @O_plus_l @O_plus_l @O_refl
+ ]
+qed.
+
+(*
+lemma 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〉)).
+#a #b #f #ha #hb #hf #s #CFa #CFb #CFf #HO
+@ext_CF1 [|#x @minim_unary_pr1]
+@(CF_comp ??? (λx.ha x + hb x))
+ [|@CF_comp_pair
+ [@CF_minus [@CF_compS @(O_to_CF … CFb) @O_plus_r // |@(O_to_CF … CFa) @O_plus_l //]
+ |@CF_comp_pair
+ [@CF_max1 [@(O_to_CF … CFa) @O_plus_l // | @CF_compS @(O_to_CF … CFb) @O_plus_r //]
+ | @(O_to_CF … CF_id) @O_plus_r @(proj1 … CFb)
+ ]
+ ]
+ |@(CF_prim_rec … MSC … (compl_minim_unary_aux hf))
+ [@CF_fst
+ |@CF_if
+ [@(CF_comp f … MSC … CFf)
+ [@CF_comp_pair
+ [@CF_minus [@CF_comp_fst @CF_comp_snd @CF_snd|@CF_compS @CF_fst]
+ |@CF_comp_snd @CF_comp_snd @CF_snd]
+ |@le_to_O #x normalize >commutative_plus //
+ ]
+ |@(O_to_CF … MSC)
+ [@le_to_O #x normalize //
+ |@CF_minus
+ [@CF_comp_fst @CF_comp_snd @CF_snd |@CF_compS @CF_fst]
+ ]
+ |@CF_comp_fst @(monotonic_CF … CF_snd) normalize //
+ ]
+ |@O_refl
+ ]
+ |@(O_trans … HO) -HO
+ @(O_trans ? (λx:ℕ
+ .ha x+hb x
+ +bigop (S(b x)-a x) (λi:ℕ.true) ? (MSC 〈a x,x〉) plus (λi:ℕ.hf 〈i+a x,x〉)))
+ [@le_to_O #n @le_plus // whd in match (unary_pr ???);
+ >fst_pair >snd_pair >(bigop_prim_rec ? (λn.S(b n)) ? (λi.true)) elim (S(b n) - a n)
+ [normalize @monotonic_MSC @le_pair
+ |#x #Hind >prim_rec_S >fst_pair >snd_pair >fst_pair >snd_pair >compl_minim_unary_def
+ >prim_rec_S >fst_pair >snd_pair >fst_pair >snd_pair >fst_pair
+ >snd_pair normalize in ⊢ (??%); >commutative_plus @le_plus
+ [-Hind @le_plus // normalize >fst_pair >snd_pair
+ @(transitive_le ? (bigop x (λi1:ℕ.true) ? 0 (λn0:ℕ.λm:ℕ.if leb n0 m then m else n0 )
+ (λi1:ℕ.hf 〈i1+a n,n〉)))
+ [elim x [normalize @MSC_le]
+ #x0 #Hind >prim_rec_S >fst_pair >snd_pair >snd_pair >snd_pair
+ >fst_pair >fst_pair cases (p 〈x0+a n,n〉) normalize
+ [cases (true_or_false (leb (f 〈x0+a n,n〉)
+ (prim_rec (λx00:ℕ.O)
+ (λi:ℕ
+ .if p 〈fst i+fst (snd (snd i)),snd (snd (snd i))〉
+ then if leb (f 〈fst i+fst (snd (snd i)),snd (snd (snd i))〉)
+ (fst (snd i))
+ then fst (snd i)
+ else f 〈fst i+fst (snd (snd i)),snd (snd (snd i))〉
+ else fst (snd i) ) x0 〈a n,n〉))) #Hcase >Hcase normalize
+ [@(transitive_le … Hind) -Hind @(le_maxr … (le_n …))
+ |@(transitive_le … (MSC_prop … CFf …)) @(le_maxl … (le_n …))
+ ]
+ |@(transitive_le … Hind) -Hind @(le_maxr … (le_n …))
+ ]
+ |@(le_maxr … (le_n …))
+ ]
+
+
+ @(O_trans ? (λx:ℕ
+ .ha x+hb x
+ +bigop (b x-a x) (λi:ℕ.true) ? (MSC 〈a x,x〉) plus
+ (λi:ℕ
+ .(λi0:ℕ
+ .hp 〈i0,x〉+hf 〈i0,x〉
+ +bigop (b x-a x) (λi1:ℕ.true) ? 0 max
+ (λi1:ℕ.(λi2:ℕ.hf 〈i2,x〉) (i1+a x))) (i+a x))))
+ [
+ @le_to_O #n @le_plus // whd in match (unary_pr ???);
+ >fst_pair >snd_pair >bigop_prim_rec elim (b n - a n)
+ [normalize //
+ |#x #Hind >prim_rec_S >fst_pair >snd_pair >fst_pair >snd_pair >aux_compl1_def1
+ >prim_rec_S >fst_pair >snd_pair >fst_pair >snd_pair >fst_pair
+ >snd_pair normalize in ⊢ (??%); >commutative_plus @le_plus
+ [-Hind @le_plus // normalize >fst_pair >snd_pair
+ @(transitive_le ? (bigop x (λi1:ℕ.true) ? 0 (λn0:ℕ.λm:ℕ.if leb n0 m then m else n0 )
+ (λi1:ℕ.hf 〈i1+a n,n〉)))
+ [elim x [normalize @MSC_le]
+ #x0 #Hind >prim_rec_S >fst_pair >snd_pair >snd_pair >snd_pair
+ >fst_pair >fst_pair cases (p 〈x0+a n,n〉) normalize
+ [cases (true_or_false (leb (f 〈x0+a n,n〉)
+ (prim_rec (λx00:ℕ.O)
+ (λi:ℕ
+ .if p 〈fst i+fst (snd (snd i)),snd (snd (snd i))〉
+ then if leb (f 〈fst i+fst (snd (snd i)),snd (snd (snd i))〉)
+ (fst (snd i))
+ then fst (snd i)
+ else f 〈fst i+fst (snd (snd i)),snd (snd (snd i))〉
+ else fst (snd i) ) x0 〈a n,n〉))) #Hcase >Hcase normalize
+ [@(transitive_le … Hind) -Hind @(le_maxr … (le_n …))
+ |@(transitive_le … (MSC_prop … CFf …)) @(le_maxl … (le_n …))
+ ]
+ |@(transitive_le … Hind) -Hind @(le_maxr … (le_n …))
+ ]
+ |@(le_maxr … (le_n …))
+ ]
+ |@(transitive_le … Hind) -Hind
+ generalize in match (bigop x (λi:ℕ.true) ? 0 max
+ (λi1:ℕ.(λi2:ℕ.hf 〈i2,n〉) (i1+a n))); #c
+ generalize in match (hf 〈x+a n,n〉); #c1
+ elim x [//] #x0 #Hind
+ >prim_rec_S >prim_rec_S normalize >fst_pair >fst_pair >snd_pair
+ >snd_pair >snd_pair >snd_pair >snd_pair >snd_pair >fst_pair >fst_pair
+ >fst_pair @le_plus
+ [@le_plus // cases (true_or_false (leb c1 c)) #Hcase
+ >Hcase normalize // @lt_to_le @not_le_to_lt @(leb_false_to_not_le ?? Hcase)
+ |@Hind
+ ]
+ ]
+ ]
+ |@O_plus [@O_plus_l //] @le_to_O #x
+ <bigop_plus_c @le_plus // @(transitive_le … (MSC_pair …)) @le_plus
+ [@MSC_prop @CFa | @MSC_prop @(O_to_CF MSC … (CF_const x)) @(proj1 … CFb)]
+ ]
+qed.
+
+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〉)).*)
+
+(************************************* smn ************************************)
+axiom smn: ∀f,s. CF s f → ∀x. CF (λy.s 〈x,y〉) (λy.f 〈x,y〉).
+
+(****************************** constructibility ******************************)
+
+definition constructible ≝ λs. CF s s.
+
+lemma constr_comp : ∀s1,s2. constructible s1 → constructible s2 →
+ (∀x. x ≤ s2 x) → constructible (s2 ∘ s1).
+#s1 #s2 #Hs1 #Hs2 #Hle @(CF_comp … Hs1 Hs2) @O_plus @le_to_O #x [@Hle | //]
+qed.
+
+lemma ext_constr: ∀s1,s2. (∀x.s1 x = s2 x) →
+ constructible s1 → constructible s2.
+#s1 #s2 #Hext #Hs1 @(ext_CF … Hext) @(monotonic_CF … Hs1) #x >Hext //
+qed.
+
+lemma constr_prim_rec: ∀s1,s2. constructible s1 → constructible s2 →
+ (∀n,r,m. 2 * r ≤ s2 〈n,〈r,m〉〉) → constructible (unary_pr s1 s2).
+#s1 #s2 #Hs1 #Hs2 #Hincr @(CF_prim_rec … Hs1 Hs2) whd %{2} %{0}
+#x #_ lapply (surj_pair x) * #a * #b #eqx >eqx whd in match (unary_pr ???);
+>fst_pair >snd_pair
+whd in match (unary_pr ???); >fst_pair >snd_pair elim a
+ [normalize //
+ |#n #Hind >prim_rec_S >fst_pair >snd_pair >fst_pair >snd_pair
+ >prim_rec_S @transitive_le [| @(monotonic_le_plus_l … Hind)]
+ @transitive_le [| @(monotonic_le_plus_l … (Hincr n ? b))]
+ whd in match (unary_pr ???); >fst_pair >snd_pair //
+ ]
+qed.
+
+(********************************* simulation *********************************)
+
+axiom sU : nat → nat.
+
+axiom monotonic_sU: ∀i1,i2,x1,x2,s1,s2. i1 ≤ i2 → x1 ≤ x2 → s1 ≤ s2 →
+ sU 〈i1,〈x1,s1〉〉 ≤ sU 〈i2,〈x2,s2〉〉.
+
+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 sU_le: ∀i,x,s. s ≤ sU 〈i,〈x,s〉〉.
+axiom sU_le_i: ∀i,x,s. MSC i ≤ sU 〈i,〈x,s〉〉.
+axiom sU_le_x: ∀i,x,s. MSC x ≤ sU 〈i,〈x,s〉〉.
+
+definition pU_unary ≝ λp. pU (fst p) (fst (snd p)) (snd (snd p)).
+
+axiom CF_U : CF sU pU_unary.
+
+definition termb_unary ≝ λx:ℕ.termb (fst x) (fst (snd x)) (snd (snd x)).
+definition out_unary ≝ λx:ℕ.out (fst x) (fst (snd x)) (snd (snd x)).
+
+lemma CF_termb: CF sU termb_unary.
+@(ext_CF (fst ∘ pU_unary)) [2: @CF_comp_fst @CF_U]
+#n whd in ⊢ (??%?); whd in ⊢ (??(?%)?); >fst_pair %
+qed.
+
+lemma CF_out: CF sU out_unary.
+@(ext_CF (snd ∘ pU_unary)) [2: @CF_comp_snd @CF_U]
+#n whd in ⊢ (??%?); whd in ⊢ (??(?%)?); >snd_pair %
+qed.
+
+
+(******************** 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 (auxg h) → CF s (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
+ (λp:ℕ.eqb (min_input h (fst p) (snd (snd p))) (snd (snd p))) →
+ CF s2
+ (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 (auxg h).
+#h #s1 #s2 #s #Hs1 #Hs2 #HO @(ext_CF (aux1g h))
+ [#n whd in ⊢ (??%%); @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 (λp:ℕ.min_input h (fst p) (snd (snd p))) →
+ CF s (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 compl_g4 : ∀h,s1,s.
+ (CF s1
+ (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 (λp:ℕ.min_input h (fst p) (snd (snd p))).
+#h #s1 #s #Hs1 #HO @(ext_CF (min_input_aux h))
+ [#n whd in ⊢ (??%%); @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 //]
+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.
+
+(**************************** end of technical lemmas *************************)
+
+(*lemma compl_g5 : ∀h,s1.(∀n. monotonic ℕ le (λa:ℕ.s1 〈a,n〉)) →
+ (CF s1
+ (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〉)
+ (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.
+
+lemma compl_g6: ∀h.
+ constructible (λx. h (fst x) (snd x)) →
+ (CF (λx. sU 〈max (fst (snd x)) (snd (snd x)),〈fst x,h (S (fst (snd x))) (fst x)〉〉)
+ (λp.termb (fst (snd p)) (fst p) (h (S (fst (snd p))) (fst p)))).
+#h #hconstr @(ext_CF (termb_aux h))
+ [#n normalize >fst_pair >snd_pair >fst_pair >snd_pair // ]
+@(CF_comp … (λx.MSC x + h (S (fst (snd x))) (fst x)) … CF_termb)
+ [@CF_comp_pair
+ [@CF_comp_fst @(monotonic_CF … CF_snd) #x //
+ |@CF_comp_pair
+ [@(monotonic_CF … CF_fst) #x //
+ |@(ext_CF ((λx.h (fst x) (snd x))∘(λx.〈S (fst (snd x)),fst x〉)))
+ [#n normalize >fst_pair >snd_pair %]
+ @(CF_comp … MSC …hconstr)
+ [@CF_comp_pair [@CF_compS @CF_comp_fst // |//]
+ |@O_plus @le_to_O [//|#n >fst_pair >snd_pair //]
+ ]
+ ]
+ ]
+ |@O_plus
+ [@O_plus
+ [@(O_trans … (λx.MSC (fst x) + MSC (max (fst (snd x)) (snd (snd x)))))
+ [%{2} %{0} #x #_ cases (surj_pair x) #a * #b #eqx >eqx
+ >fst_pair >snd_pair @(transitive_le … (MSC_pair …))
+ >distributive_times_plus @le_plus [//]
+ cases (surj_pair b) #c * #d #eqb >eqb
+ >fst_pair >snd_pair @(transitive_le … (MSC_pair …))
+ whd in ⊢ (??%); @le_plus
+ [@monotonic_MSC @(le_maxl … (le_n …))
+ |>commutative_times <times_n_1 @monotonic_MSC @(le_maxr … (le_n …))
+ ]
+ |@O_plus [@le_to_O #x @sU_le_x |@le_to_O #x @sU_le_i]
+ ]
+ |@le_to_O #n @sU_le
+ ]
+ |@le_to_O #x @monotonic_sU // @(le_maxl … (le_n …)) ]
+ ]
+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.
+
+definition faux2 ≝ λh.
+ (λx.MSC x + (snd (snd x)-fst x)*
+ (λx.sU 〈max (fst(snd x)) (snd(snd x)),〈fst x,h (S (fst (snd x))) (fst x)〉〉) 〈snd (snd x),x〉).
+
+(*lemma compl_g7: ∀h.
+ constructible (λx. h (fst x) (snd x)) →
+ (∀n. monotonic ? le (h n)) →
+ 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))〉〉)
+ (λp:ℕ.min_input h (fst p) (snd (snd p))).
+#h #hcostr #hmono @(monotonic_CF … (faux2 h))
+ [#n normalize >fst_pair >snd_pair //]
+@compl_g5 [2:@(compl_g6 h hcostr)] #n #x #y #lexy >fst_pair >snd_pair
+>fst_pair >snd_pair @monotonic_sU // @hmono @lexy
+qed.
+
+lemma compl_g71: ∀h.
+ constructible (λx. h (fst x) (snd x)) →
+ (∀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))〉〉)
+ (λp:ℕ.min_input h (fst p) (snd (snd p))).
+#h #hcostr #hmono @(monotonic_CF … (compl_g7 h hcostr hmono)) #x
+@le_plus [@monotonic_MSC //]
+cases (decidable_le (fst x) (snd(snd x)))
+ [#Hle @le_times // @monotonic_sU
+ |#Hlt >(minus_to_0 … (lt_to_le … )) [// | @not_le_to_lt @Hlt]
+ ]
+qed.*)
+
+definition out_aux ≝ λh.
+ out_unary ∘ λp.〈fst p,〈snd(snd p),h (S (fst p)) (snd (snd p))〉〉.
+
+lemma compl_g8: ∀h.
+ constructible (λx. h (fst x) (snd x)) →
+ (CF (λx. sU 〈max (fst x) (snd x),〈snd(snd x),h (S (fst x)) (snd(snd x))〉〉)
+ (λp.out (fst p) (snd (snd p)) (h (S (fst p)) (snd (snd p))))).
+#h #hconstr @(ext_CF (out_aux h))
+ [#n normalize >fst_pair >snd_pair >fst_pair >snd_pair // ]
+@(CF_comp … (λx.h (S (fst x)) (snd(snd x)) + MSC x) … CF_out)
+ [@CF_comp_pair
+ [@(monotonic_CF … CF_fst) #x //
+ |@CF_comp_pair
+ [@CF_comp_snd @(monotonic_CF … CF_snd) #x //
+ |@(ext_CF ((λx.h (fst x) (snd x))∘(λx.〈S (fst x),snd(snd x)〉)))
+ [#n normalize >fst_pair >snd_pair %]
+ @(CF_comp … MSC …hconstr)
+ [@CF_comp_pair [@CF_compS // | @CF_comp_snd // ]
+ |@O_plus @le_to_O [//|#n >fst_pair >snd_pair //]
+ ]
+ ]
+ ]
+ |@O_plus
+ [@O_plus
+ [@le_to_O #n @sU_le
+ |@(O_trans … (λx.MSC (max (fst x) (snd x))))
+ [%{2} %{0} #x #_ cases (surj_pair x) #a * #b #eqx >eqx
+ >fst_pair >snd_pair @(transitive_le … (MSC_pair …))
+ whd in ⊢ (??%); @le_plus
+ [@monotonic_MSC @(le_maxl … (le_n …))
+ |>commutative_times <times_n_1 @monotonic_MSC @(le_maxr … (le_n …))
+ ]
+ |@le_to_O #x @(transitive_le ???? (sU_le_i … )) //
+ ]
+ ]
+ |@le_to_O #x @monotonic_sU [@(le_maxl … (le_n …))|//|//]
+ ]
+qed.
+
+(*lemma compl_g9 : ∀h.
+ constructible (λx. h (fst x) (snd x)) →
+ (∀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)〉〉)
+ (auxg h).
+#h #hconstr #hmono #hantimono
+@(compl_g2 h ??? (compl_g3 … (compl_g71 h hconstr hmono)) (compl_g8 h hconstr))
+@O_plus
+ [@O_plus_l @le_to_O #x >(times_n_1 (MSC x)) >commutative_times @le_times
+ [// | @monotonic_MSC // ]]
+@(O_trans … (coroll2 ??))
+ [#n #a #b #leab #ltb >fst_pair >fst_pair >snd_pair >snd_pair
+ cut (b ≤ n) [@(transitive_le … (le_snd …)) @lt_to_le //] #lebn
+ cut (max a n = n)
+ [normalize >le_to_leb_true [//|@(transitive_le … leab lebn)]] #maxa
+ cut (max b n = n) [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.
+ constructible (λx. h (fst x) (snd x)) →
+ (∀n. monotonic ? le (h n)) →
+ (∀n,a,b. a ≤ b → b ≤ n → h b n ≤ h a n) →
+ CF (sg h) (unary_g h).
+#h #hconstr #Hm #Ham @compl_g1 @(compl_g9 h hconstr 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
+ | 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.
+// 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.
+
+lemma h_of_aux_prim_rec : ∀r,c,n,b. h_of_aux r c n b =
+ prim_rec (λx.c)
+ (λx.let d ≝ S(fst x) in
+ let b ≝ snd (snd x) in
+ (S d)*(MSC 〈〈b-d,b〉,〈b-d,b〉〉) +
+ d*(S d)*sU 〈〈b-d,b〉,〈b,r (fst (snd x))〉〉) n b.
+#r #c #n #b elim n
+ [>h_of_aux_O normalize //
+ |#n1 #Hind >h_of_aux_S >prim_rec_S >snd_pair >snd_pair >fst_pair
+ >fst_pair <Hind //
+ ]
+qed.
+
+(*
+lemma h_of_aux_constr :
+∀r,c. constructible (λx.h_of_aux r c (fst x) (snd x)).
+#r #c
+ @(ext_constr …
+ (unary_pr (λx.c)
+ (λx.let d ≝ S(fst x) in
+ let b ≝ snd (snd x) in
+ (S d)*(MSC 〈〈b-d,b〉,〈b-d,b〉〉) +
+ d*(S d)*sU 〈〈b-d,b〉,〈b,r (fst (snd x))〉〉)))
+ [#n @sym_eq whd in match (unary_pr ???); @h_of_aux_prim_rec
+ |@constr_prim_rec*)
+
+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.
+
+axiom h_of_constr : ∀r:nat →nat.
+ (∀x. x ≤ r x) → monotonic ? le r → constructible r →
+ constructible (h_of r).
+
+(*lemma speed_compl: ∀r:nat →nat.
+ (∀x. x ≤ r x) → monotonic ? le r → constructible r →
+ CF (h_of r) (unary_g (λi,x. r(h_of r 〈i,x〉))).
+#r #Hr #Hmono #Hconstr @(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)
+ [%{(pred(b-a))} >S_pred [//] @lt_plus_to_minus_r @not_le_to_lt @ltab]
+ * #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)
+ |@(constr_comp … Hconstr Hr) @(ext_constr (h_of r))
+ [#x cases (surj_pair x) #a * #b #eqx >eqx >fst_pair >snd_pair //]
+ @(h_of_constr r Hr Hmono Hconstr)
+ ]
+qed.
+
+lemma speed_compl_i: ∀r:nat →nat.
+ (∀x. x ≤ r x) → monotonic ? le r → constructible r →
+ ∀i. CF (λx.h_of r 〈i,x〉) (λx.g (λi,x. r(h_of r 〈i,x〉)) i x).
+#r #Hr #Hmono #Hconstr #i
+@(ext_CF (λx.unary_g (λi,x. r(h_of r 〈i,x〉)) 〈i,x〉))
+ [#n whd in ⊢ (??%%); @eq_f @sym_eq >fst_pair >snd_pair %]
+@smn @(ext_CF … (speed_compl r Hr Hmono Hconstr)) #n //
+qed.*)
+
+(**************************** the speedup theorem *****************************)
+(*theorem pseudo_speedup:
+ ∀r:nat →nat. (∀x. x ≤ r x) → monotonic ? le r → constructible r →
+ ∃f.∀sf. CF sf f → ∃g,sg. f ≈ g ∧ CF sg g ∧ O sf (r ∘ sg).
+(* ∃m,a.∀n. a≤n → r(sg a) < m * sf n. *)
+#r #Hr #Hmono #Hconstr
+(* 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 Hconstr (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 → constructible r →
+ ∃f.∀sf. CF sf f → ∃g,sg. f ≈ g ∧ CF sg g ∧
+ (* ¬ O (r ∘ sg) sf. *)
+ ∃m,a.∀n. a≤n → r(sg a) < m * sf n.
+#r #Hr #Hmono #Hconstr
+(* 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 Hconstr (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.*)
+
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projects / helm.git / commitdiff