X-Git-Url: http://matita.cs.unibo.it/gitweb/?a=blobdiff_plain;f=helm%2Fsoftware%2Fmatita%2Fcontribs%2Fdama%2Fdama%2Fmodels%2Fq_bars.ma;h=de39589073d51967d81528130c9afa59c4e858c4;hb=b5564e329d48efa6c2ca01da18203def26a70294;hp=65066590f4baef3f754305ba709df1353d149bae;hpb=02b4aca8654dd4b0c16cab14bf145bbc1ae963f8;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 65066590f..de3958907 100644
--- a/helm/software/matita/contribs/dama/dama/models/q_bars.ma
+++ b/helm/software/matita/contribs/dama/dama/models/q_bars.ma
@@ -15,14 +15,14 @@
 include "nat_ordered_set.ma".
 include "models/q_support.ma".
 include "models/list_support.ma". 
-include "cprop_connectives.ma". 
+include "logic/cprop_connectives.ma". 
 
-definition bar ≝ ℚ × ℚ.
+definition bar ≝ ℚ × (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 ≝ 〈Qpos one,OQ〉.
+definition empty_bar : bar ≝ 〈Qpos one,〈OQ,OQ〉〉.
 notation "\rect" with precedence 90 for @{'empty_bar}.
 interpretation "q0" 'empty_bar = empty_bar.
 
@@ -55,7 +55,7 @@ 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
+ bars_end_OQ : nth_height bars (pred (\len bars)) = 〈OQ,OQ〉
 }.
 
 lemma len_bases_gt_O: ∀f.O < \len (bars f).
@@ -75,62 +75,17 @@ cases (cmp_nat (\len l) i);
     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.
+coinductive value_spec (f : q_f) (i : ℚ) : ℚ × ℚ → CProp ≝
+| value_of : ∀j,q. 
+    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. 
+         
+definition value_lemma :  ∀f:q_f.∀i:ratio.∃p:ℚ×ℚ.value_spec f (Qpos i) p.
 intros;
 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;
+apply (value_of ?? (pred (find ? P (bars f) ▭)));
 [1: reflexivity
 |2: cases (cases_find bar P (bars f) ▭);
     [1: cases i1 in H H1 H2 H3; simplify; intros;
@@ -175,55 +130,21 @@ exists [apply (pred (find ? P (bars f) ▭))] apply value_of;
         [ 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.
+lemma value : q_f → ratio → ℚ × ℚ.
+intros; cases (value_lemma q r); apply w; 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.
+lemma cases_value : ∀f,i. value_spec f (Qpos i) (value f i).
+intros; unfold value; cases (value_lemma f i); assumption; qed.
 
-definition same_values ≝
-  λl1,l2:q_f.
-   ∀input.\snd (\fst (value l1 input)) = \snd (\fst (value l2 input)). 
+definition same_values ≝ λl1,l2:q_f.∀input. value l1 input = value l2 input. 
 
-definition same_bases ≝ 
-  λl1,l2:list bar. (∀i.\fst (nth l1 ▭ i) = \fst (nth l2 ▭ i)).
+definition same_bases ≝ 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.
 intro; cases x; intros; [2:exists [apply r] reflexivity]
 cases (?:False);
-[ apply (q_lt_corefl ? H)|apply (q_neg_gt ? H)]
+[ apply (q_lt_corefl ? H)| cases (q_lt_le_incompat ?? (q_neg_gt ?) (q_lt_to_le ?? H))]
 qed.
 
 notation < "x \blacksquare" non associative with precedence 50 for @{'unpos $x}.