include "arithmetics/nat.ma".
-include "arithmetics/log.ma".
-(* include "arithmetics/ord.ma". *)
+include "arithmetics/log.ma".
include "arithmetics/bigops.ma".
include "arithmetics/bounded_quantifiers.ma".
-include "arithmetics/pidgeon_hole.ma".
+include "arithmetics/pidgeon_hole.ma".
include "basics/sets.ma".
include "basics/types.ma".
#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)).
#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) *)
(************************ 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.
∃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}.
+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.
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) //
#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 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.
%{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 *)
U (fst i) i (s (|i|)) = Some ? 0.
lemma equiv_diag: ∀s,i.
- diag s i ↔ [fst i,i] ↓ s (|i|) ∧ ¬lang (fst i) i.
+ diag s i ↔ {fst i ⊙ i} ↓ s(|i|) ∧ ¬lang (fst i) i.
#s #i %
[whd in ⊢ (%→?); #H % [%{0} //] % * #x * #y *
#H1 #Hy cut (0 = y) [@(unique_U … H H1)] #eqy /2/
#H3 @(absurd ?? H3) @H2 @H3
qed.
-(* axiom weak_pad : ∀a,∃a0.∀n. a0 < n → ∃b. |〈a,b〉| = n. *)
+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))
|#H lapply(not_forall_to_exists … Hdec H)
* #b * #H1 #H2 %{b} @not_lt_to_le @H2
]
-qed.
+qed.
lemma pad : ∀n,a. ∃b. n ≤ |〈a,b〉|.
#n #a cases (weak_pad1 (of_size n) a) #b #Hb
qed.
lemma o_to_ex: ∀s1,s2. o s1 s2 → ∀i. C s2 i →
- ∃b.[i, 〈i,b〉] ↓ s1 (|〈i,b〉|).
+ ∃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〉 ?)
[ 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)) →
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).
-
-(*
-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 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
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 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. minimal s → constructible s →
+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 //]
|%{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.
-
+qed.
\ No newline at end of file
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