+++ /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.
-
-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.
-
projects / helm.git / blobdiff