X-Git-Url: http://matita.cs.unibo.it/gitweb/?a=blobdiff_plain;f=helm%2Fsoftware%2Fmatita%2Fcontribs%2Fdama%2Fdama%2Fmodels%2Fq_bars.ma;h=65066590f4baef3f754305ba709df1353d149bae;hb=02b4aca8654dd4b0c16cab14bf145bbc1ae963f8;hp=6be729db5492c6e55bbdeedfa97de1f011b21607;hpb=6b61a9e6698a7c1936adf217b599e34e65a5e4c9;p=helm.git

diff --git a/helm/software/matita/contribs/dama/dama/models/q_bars.ma b/helm/software/matita/contribs/dama/dama/models/q_bars.ma
index 6be729db5..65066590f 100644
--- a/helm/software/matita/contribs/dama/dama/models/q_bars.ma
+++ b/helm/software/matita/contribs/dama/dama/models/q_bars.ma
@@ -12,109 +12,212 @@
 (*                                                                        *)
 (**************************************************************************)
 
+include "nat_ordered_set.ma".
 include "models/q_support.ma".
-include "models/list_support.ma".
+include "models/list_support.ma". 
 include "cprop_connectives.ma". 
 
-definition bar ≝ ratio × ℚ. (* base (Qpos) , height *)
-record q_f : Type ≝ { start : ℚ; bars: list bar }.
+definition bar ≝ ℚ × ℚ.
 
 notation < "\rationals \sup 2" non associative with precedence 90 for @{'q2}.
 interpretation "Q x Q" 'q2 = (Prod Q Q).
 
-definition empty_bar : bar ≝ 〈one,OQ〉.
+definition empty_bar : bar ≝ 〈Qpos one,OQ〉.
 notation "\rect" with precedence 90 for @{'empty_bar}.
 interpretation "q0" 'empty_bar = empty_bar.
 
 notation < "\ldots\rect\square\EmptySmallSquare\ldots" with precedence 90 for @{'lq2}.
-interpretation "lq2" 'lq2 = (list bar).
-
-let rec sum_bases (l:list bar) (i:nat)on i ≝
-    match i with
-    [ O ⇒ OQ
-    | S m ⇒ 
-         match l with
-         [ nil ⇒ sum_bases l m + Qpos one
-         | cons x tl ⇒ sum_bases tl m + Qpos (\fst x)]].
-         
-axiom sum_bases_empty_nat_of_q_ge_OQ:
-  ∀q:ℚ.OQ ≤ sum_bases [] (nat_of_q q). 
-axiom sum_bases_empty_nat_of_q_le_q:
-  ∀q:ℚ.sum_bases [] (nat_of_q q) ≤ q.
-axiom sum_bases_empty_nat_of_q_le_q_one:
-  ∀q:ℚ.q < sum_bases [] (nat_of_q q) + Qpos one.
-
-definition eject1 ≝
-  λP.λp:∃x:nat × ℚ.P x.match p with [ex_introT p _ ⇒ p].
-coercion eject1.
-definition inject1 ≝ λP.λp:nat × ℚ.λh:P p. ex_introT ? P p h.
-coercion inject1 with 0 1 nocomposites.
-
-definition value : 
-  ∀f:q_f.∀i:ℚ.∃p:nat × ℚ. 
-   match q_cmp i (start f) with
-   [ q_lt _ ⇒ \snd p = OQ
-   | _ ⇒ 
-        And3
-         (sum_bases (bars f) (\fst p) ≤ ⅆ[i,start f]) 
-         (ⅆ[i, start f] < sum_bases (bars f) (S (\fst p))) 
-         (\snd p = \snd (nth (bars f) ▭ (\fst p)))].
+interpretation "lq2" 'lq2 = (list bar). 
+
+definition q2_lt := mk_rel bar (λx,y:bar.\fst x < \fst y).
+
+interpretation "bar lt" 'lt x y = (rel_op _ q2_lt x y).
+
+lemma q2_trans : ∀a,b,c:bar. a < b → b < c → a < c.
+intros 3; cases a; cases b; cases c; unfold q2_lt; simplify; intros;
+apply (q_lt_trans ??? H H1);
+qed. 
+
+definition q2_trel := mk_trans_rel bar q2_lt q2_trans.
+
+interpretation "bar lt" 'lt x y = (FunClass_2_OF_trans_rel q2_trel x y).
+
+definition canonical_q_lt : rel bar → trans_rel ≝ λx:rel bar.q2_trel.
+
+coercion canonical_q_lt with nocomposites.
+
+interpretation "bar lt" 'lt x y = (FunClass_2_OF_trans_rel (canonical_q_lt _) x y).
+
+definition nth_base ≝ λf,n. \fst (\nth f ▭ n).
+definition nth_height ≝ λf,n. \snd (\nth f ▭ n).
+
+record q_f : Type ≝ {
+ bars: list bar; 
+ bars_sorted : sorted q2_lt bars;
+ bars_begin_OQ : nth_base bars O = OQ;
+ bars_end_OQ : nth_height bars (pred (\len bars)) = OQ
+}.
+
+lemma len_bases_gt_O: ∀f.O < \len (bars f).
+intros; generalize in match (bars_begin_OQ f); cases (bars f); intros;
+[2: simplify; apply le_S_S; apply le_O_n;
+|1: normalize in H; destruct H;]
+qed. 
+
+lemma all_bases_positive : ∀f:q_f.∀i. OQ < nth_base (bars f) (S i).
+intro f; generalize in match (bars_begin_OQ f); generalize in match (bars_sorted f);
+cases (len_gt_non_empty ?? (len_bases_gt_O f)); intros;
+cases (cmp_nat (\len l) i);
+[2: lapply (sorted_tail_bigger q2_lt ?? ▭ H ? H2) as K;  
+    simplify in H1; rewrite < H1; apply K;
+|1: simplify; elim l in i H2;[simplify; rewrite > nth_nil; apply (q_pos_OQ one)]
+    cases n in H3; intros; [simplify in H3; cases (not_le_Sn_O ? H3)] 
+    apply (H2 n1); simplify in H3; apply (le_S_S_to_le ?? H3);]
+qed.
+
+(*
+lemma lt_n_plus_n_Sm : ∀n,m:nat.n < n + S m.
+intros; rewrite > sym_plus; apply (le_S_S n (m+n)); apply (le_plus_n m n); qed.
+*)
+
+(*
+lemma all_bigger_can_concat_bigger:
+   ∀l1,l2,start,b,x,n.
+    (∀i.i< len l1 → nth_base l1 i < \fst b) →
+    (∀i.i< len l2 → \fst b ≤ nth_base l2 i) →
+    (∀i.i< len l1 → start ≤ i → x ≤ nth_base l1 i) →
+    start ≤ n → n < len (l1@b::l2) → x ≤ \fst b → x ≤ nth_base (l1@b::l2) n.
+intros; cases (cmp_nat n (len l1));
+[1: unfold nth_base;  rewrite > (nth_concat_lt_len ????? H6);
+    apply (H2 n); assumption;
+|2: rewrite > H6; unfold nth_base; rewrite > nth_len; assumption;
+|3: unfold nth_base; rewrite > nth_concat_ge_len; [2: apply lt_to_le; assumption]
+    rewrite > len_concat in H4; simplify in H4; rewrite < plus_n_Sm in H4;
+    lapply linear le_S_S_to_le to H4 as K; rewrite > sym_plus in K;
+    lapply linear le_plus_to_minus to K as X; 
+    generalize in match X; generalize in match (n - len l1); intro W; cases W; clear W X;
+    [intros; assumption] intros;
+    apply (q_le_trans ??? H5); apply (H1 n1); assumption;]
+qed. 
+*)
+
+
+inductive value_spec (f : q_f) (i : ℚ) : ℚ → nat → CProp ≝
+ value_of : ∀q,j. 
+   nth_height (bars f) j = q →  
+   nth_base (bars f) j < i →
+   (∀n.j < n → n < \len (bars f) → i ≤ nth_base (bars f) n) → value_spec f i q j. 
+     
+     
+inductive break_spec (T : Type) (n : nat) (l : list T) : list T → CProp ≝
+| break_to: ∀l1,x,l2. \len l1 = n → l = l1 @ [x] @ l2 → break_spec T n l l.     
+
+lemma list_break: ∀T,n,l. n < \len l → break_spec T n l l.
+intros 2; elim n;
+[1: elim l in H; [cases (not_le_Sn_O ? H)]
+    apply (break_to ?? ? [] a l1); reflexivity;
+|2: cases (H l); [2: apply lt_S_to_lt; assumption;] cases l2 in H3; intros;
+    [1: rewrite < H2 in H1; rewrite > H3 in H1; rewrite > append_nil in H1;
+        rewrite > len_append in H1; rewrite > plus_n_SO in H1;
+        cases (not_le_Sn_n ? H1);
+    |2: apply (break_to ?? ? (l1@[x]) t l3); 
+        [2: simplify; rewrite > associative_append; assumption;
+        |1: rewrite < H2; rewrite > len_append; rewrite > plus_n_SO; reflexivity]]]
+qed.
+
+definition value :  ∀f:q_f.∀i:ratio.∃p:ℚ.∃j.value_spec f (Qpos i) p j.
 intros;
-alias symbol "pi2" = "pair pi2".
-alias symbol "pi1" = "pair pi1".
-letin value ≝ (
-  let rec value (p: ℚ) (l : list bar) on l ≝
-    match l with
-    [ nil ⇒ 〈nat_of_q p,OQ〉
-    | cons x tl ⇒
-        match q_cmp p (Qpos (\fst x)) with
-        [ q_lt _ ⇒ 〈O, \snd x〉
-        | _ ⇒
-           let rc ≝ value (p - Qpos (\fst x)) tl in
-           〈S (\fst rc),\snd rc〉]]
-  in value :
-  ∀acc,l.∃p:nat × ℚ. OQ ≤ acc →
-     And3
-       (sum_bases l (\fst p) ≤ acc) 
-       (acc < sum_bases l (S (\fst p))) 
-       (\snd p = \snd (nth l ▭ (\fst p))));
-[5: clearbody value; 
-    cases (q_cmp i (start f));
-    [2: exists [apply 〈O,OQ〉] simplify; reflexivity;
-    |*: cases (value ⅆ[i,start f] (bars f)) (p Hp);
-        cases (Hp (q_dist_ge_OQ ? ?)); clear Hp value;
-        exists[1,3:apply p]; simplify; split; assumption;]
-|1,3: intros; split;
-    [1,4: clear H2; cases (value (q-Qpos (\fst b)) l1);
-           cases (H2 (q_le_to_diff_ge_OQ ?? (? H1)));
-          [1,3: intros; [apply q_lt_to_le|apply q_eq_to_le;symmetry] assumption]
-          simplify; apply q_le_minus; assumption;
-    |2,5: cases (value (q-Qpos (\fst b)) l1); 
-          cases (H4 (q_le_to_diff_ge_OQ ?? (? H1)));
-          [1,3: intros; [apply q_lt_to_le|apply q_eq_to_le;symmetry] assumption]
-          clear H3 H2 value;
-          change with (q < sum_bases l1 (S (\fst w)) + Qpos (\fst b));
-          apply q_lt_plus; assumption;
-    |*: cases (value (q-Qpos (\fst b)) l1); simplify; 
-        cases (H4 (q_le_to_diff_ge_OQ ?? (? H1))); 
-        [1,3: intros; [apply q_lt_to_le|apply q_eq_to_le;symmetry] assumption]
-        assumption;]
-|2: clear value H2; simplify; intros; split; [assumption|3:reflexivity]
-    rewrite > q_plus_sym; rewrite > q_plus_OQ; assumption;
-|4: simplify; intros; split; 
-    [1: apply sum_bases_empty_nat_of_q_le_q;
-    |2: apply sum_bases_empty_nat_of_q_le_q_one;
-    |3: elim (nat_of_q q); [reflexivity] simplify; assumption]] 
+letin P ≝ 
+  (λx:bar.match q_cmp (Qpos i) (\fst x) with[ q_leq _ ⇒ true| q_gt _ ⇒ false]);
+exists [apply (nth_height (bars f) (pred (find ? P (bars f) ▭)));]
+exists [apply (pred (find ? P (bars f) ▭))] apply value_of;
+[1: reflexivity
+|2: cases (cases_find bar P (bars f) ▭);
+    [1: cases i1 in H H1 H2 H3; simplify; intros;
+        [1: generalize in match (bars_begin_OQ f); 
+            cases (len_gt_non_empty ?? (len_bases_gt_O f)); simplify; intros;
+            rewrite > H4; apply q_pos_OQ;
+        |2: cases (len_gt_non_empty ?? (len_bases_gt_O f)) in H3;
+            intros; lapply (H3 n (le_n ?)) as K; unfold P in K;
+            cases (q_cmp (Qpos i) (\fst (\nth (x::l) ▭ n))) in K;
+            simplify; intros; [destruct H5] assumption] 
+    |2: destruct H; cases (len_gt_non_empty ?? (len_bases_gt_O f)) in H2;
+        simplify; intros; lapply (H (\len l) (le_n ?)) as K; clear H;
+        unfold P in K; cases (q_cmp (Qpos i) (\fst (\nth (x::l) ▭ (\len l)))) in K;
+        simplify; intros; [destruct H2] assumption;]     
+|3: intro; cases (cases_find bar P (bars f) ▭); intros;
+    [1: generalize in match (bars_sorted f); 
+        cases (list_break ??? H) in H1; rewrite > H6;
+        rewrite < H1; simplify; rewrite > nth_len; unfold P; 
+        cases (q_cmp (Qpos i) (\fst x)); simplify; 
+        intros (X Hs); [2: destruct X] clear X;
+        cases (sorted_pivot q2_lt ??? ▭ Hs);
+        cut (\len l1 ≤ n) as Hn; [2:
+          rewrite > H1;  cases i1 in H4; simplify; intro X; [2: assumption]
+          apply lt_to_le; assumption;]
+        unfold nth_base; rewrite > (nth_append_ge_len ????? Hn);
+        cut (n - \len l1 < \len (x::l2)) as K; [2:
+          simplify; rewrite > H1; rewrite > (?:\len l2 = \len (bars f) - \len (l1 @ [x]));[2:
+            rewrite > H6; repeat rewrite > len_append; simplify;
+            repeat rewrite < plus_n_Sm; rewrite < plus_n_O; simplify;
+            rewrite > sym_plus; rewrite < minus_plus_m_m; reflexivity;]
+          rewrite > len_append; rewrite > H1; simplify; rewrite < plus_n_SO;
+          apply le_S_S; clear H1 H6 H7 Hs H8 H9 Hn x l2 l1 H4 H3 H2 H P i;
+          elim (\len (bars f)) in i1 n H5; [cases (not_le_Sn_O ? H);]
+          simplify; cases n2; [ repeat rewrite < minus_n_O; apply le_S_S_to_le; assumption]
+          cases n1 in H1; [intros; rewrite > eq_minus_n_m_O; apply le_O_n]
+          intros; simplify; apply H; apply le_S_S_to_le; assumption;]
+        cases (n - \len l1) in K; simplify; intros; [ assumption]
+        lapply (H9 ? (le_S_S_to_le ?? H10)) as W; apply (q_le_trans ??? H7);
+        apply q_lt_to_le; apply W;
+    |2: cases (not_le_Sn_n i1); rewrite > H in ⊢ (??%);
+        apply (trans_le ??? ? H4); cases i1 in H3; intros; apply le_S_S; 
+        [ apply le_O_n; | assumption]]]
+qed.    
+
+lemma value_OQ_l:
+  ∀l,i.i < start l → \snd (\fst (value l i)) = OQ.
+intros; cases (value l i) (q Hq); cases Hq; clear Hq; simplify; cases H1; clear H1;
+try assumption; cases H2; cases (?:False); apply (q_lt_le_incompat ?? H H6);
+qed.
+    
+lemma value_OQ_r:
+  ∀l,i.start l + sum_bases (bars l) (len (bars l)) ≤ i → \snd (\fst (value l i)) = OQ.
+intros; cases (value l i) (q Hq); cases Hq; clear Hq; simplify; cases H1; clear H1;
+try assumption; cases H2; cases (?:False); apply (q_lt_le_incompat ?? H7 H);
 qed.
     
-          
+lemma value_OQ_e:
+  ∀l,i.bars l = [] → \snd (\fst (value l i)) = OQ.
+intros; cases (value l i) (q Hq); cases Hq; clear Hq; simplify; cases H1; clear H1;
+try assumption; cases H2; cases (?:False); apply (H1 H);
+qed.
+
+inductive value_ok_spec (f : q_f) (i : ℚ) : nat × ℚ → Type ≝
+ | value_ok : ∀n,q. n ≤ (len (bars f)) → 
+      q = \snd (nth (bars f) ▭ n) →
+      sum_bases (bars f) n ≤ ⅆ[i,start f] →
+           ⅆ[i, start f] < sum_bases (bars f) (S n) → value_ok_spec f i 〈n,q〉.
+  
+lemma value_ok:
+  ∀f,i.bars f ≠ [] → start f ≤ i → i < start f + sum_bases (bars f) (len (bars f)) → 
+    value_ok_spec f i (\fst (value f i)). 
+intros; cases (value f i); simplify;   
+cases H3; simplify; clear H3; cases H4; clear H4;
+[1,2,3: cases (?:False); 
+  [1: apply (q_lt_le_incompat ?? H3 H1);
+  |2: apply (q_lt_le_incompat ?? H2 H3);
+  |3: apply (H H3);]
+|4: cases H7; clear H7; cases w in H3 H4 H5 H6 H8; simplify; intros;
+    constructor 1; assumption;]
+qed.
+
 definition same_values ≝
   λl1,l2:q_f.
    ∀input.\snd (\fst (value l1 input)) = \snd (\fst (value l2 input)). 
 
 definition same_bases ≝ 
-  λl1,l2:q_f.
-    (∀i.\fst (nth (bars l1) ▭ i) = \fst (nth (bars l2) ▭ i)).
+  λl1,l2:list bar. (∀i.\fst (nth l1 ▭ i) = \fst (nth l2 ▭ i)).
 
 alias symbol "lt" = "Q less than".
 lemma unpos: ∀x:ℚ.OQ < x → ∃r:ratio.Qpos r = x.
@@ -123,28 +226,6 @@ cases (?:False);
 [ apply (q_lt_corefl ? H)|apply (q_neg_gt ? H)]
 qed.
 
-notation < "\blacksquare" non associative with precedence 90 for @{'hide}.
-definition hide ≝ λT:Type.λx:T.x.
-interpretation "hide" 'hide = (hide _ _).
-
-lemma sum_bases_ge_OQ:
-  ∀l,n. OQ ≤ sum_bases l n.
-intro; elim l; simplify; intros;
-[1: elim n; [apply q_eq_to_le;reflexivity] simplify;
-    apply q_le_plus_trans; try assumption; apply q_lt_to_le; apply q_pos_lt_OQ;
-|2: cases n; [apply q_eq_to_le;reflexivity] simplify; 
-    apply q_le_plus_trans; [apply H| apply q_lt_to_le; apply q_pos_lt_OQ;]]
-qed.
-
-lemma sum_bases_O:
-  ∀l:q_f.∀x.sum_bases (bars l) x ≤ OQ → x = O.
-intros; cases x in H; [intros; reflexivity] intro; cases (?:False);
-cases (q_le_cases ?? H); 
-[1: apply (q_lt_corefl OQ); rewrite < H1 in ⊢ (?? %); 
-|2: apply (q_lt_antisym ??? H1);] clear H H1; cases (bars l);
-simplify; apply q_lt_plus_trans;
-try apply q_pos_lt_OQ; 
-try apply (sum_bases_ge_OQ []);
-apply (sum_bases_ge_OQ l1);
-qed.
+notation < "x \blacksquare" non associative with precedence 50 for @{'unpos $x}.
+interpretation "hide unpos proof" 'unpos x = (unpos x _).