X-Git-Url: http://matita.cs.unibo.it/gitweb/?a=blobdiff_plain;f=helm%2Fsoftware%2Fmatita%2Fcontribs%2Fdama%2Fdama%2Fmodels%2Fq_function.ma;h=b1ff639e44b959bd6a2a482c001df38d552316f9;hb=591ffe6f23ec9d2a4d368d2c1e7b213986189e44;hp=c458d418d3c46cf91efcd84b13b1ce2e3604c22e;hpb=5d5a1cc683e634f7b86cd1b8f36a52e204bcee32;p=helm.git

diff --git a/helm/software/matita/contribs/dama/dama/models/q_function.ma b/helm/software/matita/contribs/dama/dama/models/q_function.ma
index c458d418d..b1ff639e4 100644
--- a/helm/software/matita/contribs/dama/dama/models/q_function.ma
+++ b/helm/software/matita/contribs/dama/dama/models/q_function.ma
@@ -12,205 +12,46 @@
 (*                                                                        *)
 (**************************************************************************)
 
-include "Q/q/q.ma".
-include "list/list.ma".
-include "cprop_connectives.ma". 
-
-
-notation "\rationals" non associative with precedence 99 for @{'q}.
-interpretation "Q" 'q = Q. 
-
-definition bar ≝ ratio × ℚ. (* base (Qpos) , height *)
-record q_f : Type ≝ { start : ℚ; bars: list bar }.
-
-axiom qp : ℚ → ℚ → ℚ.
-axiom qm : ℚ → ℚ → ℚ.
-axiom qlt : ℚ → ℚ → CProp.
-
-interpretation "Q plus" 'plus x y = (qp x y).
-interpretation "Q minus" 'minus x y = (qm x y).
-interpretation "Q less than" 'lt x y = (qlt x y).
-
-inductive q_comparison (a,b:ℚ) : CProp ≝
- | q_eq : a = b → q_comparison a b 
- | q_lt : a < b → q_comparison a b
- | q_gt : b < a → q_comparison a b.
-
-axiom q_cmp:∀a,b:ℚ.q_comparison a b.
-
-definition qle ≝ λa,b:ℚ.a = b ∨ a < b.
-
-interpretation "Q less or equal than" 'leq x y = (qle x y).
-
-axiom q_le_minus: ∀a,b,c:ℚ. a ≤ c - b → a + b ≤ c.
-axiom q_lt_plus: ∀a,b,c:ℚ. a - b < c → a < c + b.
-
-axiom dist : ℚ → ℚ → ℚ.
-
-
-interpretation "list nth" 'nth = (nth _).
-interpretation "list nth" 'nth_appl l d i = (nth _ l d i).
-notation "'nth'" with precedence 90 for @{'nth}.
-notation < "'nth' \nbsp term 90 l \nbsp term 90 d \nbsp term 90 i" 
-with precedence 69 for @{'nth_appl $l $d $i}.
-
-notation < "\rationals \sup 2" non associative with precedence 90 for @{'q2}.
-interpretation "Q x Q" 'q2 = (Prod Q Q).
-
-definition make_list ≝
-  λA:Type.λdef:nat→A.
-    let rec make_list (n:nat) on n ≝
-      match n with [ O ⇒ nil ? | S m ⇒ def m :: make_list m]
-    in make_list.
-
-interpretation "'mk_list' appl" 'mk_list_appl f n = (make_list _ f n).
-interpretation "'mk_list'" 'mk_list = (make_list _).   
-notation "'mk_list'" with precedence 90 for @{'mk_list}.
-notation < "'mk_list' \nbsp term 90 f \nbsp term 90 n" 
-with precedence 69 for @{'mk_list_appl $f $n}.
-
-
-definition empty_bar : bar ≝ 〈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).
-
-notation "'len'" with precedence 90 for @{'len}.
-interpretation "len" 'len = (length _).
-notation < "'len' \nbsp term 90 l" with precedence 69 for @{'len_appl $l}.
-interpretation "len appl" 'len_appl l = (length _ l).
-
-(* a local letin makes russell fail *)
-definition cb0h ≝ (λl.mk_list (λi.〈\fst (nth l ▭ i),OQ〉) (len l)).
-
-lemma len_mk_list : ∀T:Type.∀f:nat→T.∀n.len (mk_list f n) = n.
-intros; elim n; [reflexivity] simplify; rewrite > H; reflexivity;
-qed.
-
-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)]].
-
-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) ≤ i - start f) 
-         (i - start f < sum_bases (bars f) (S (\fst p))) 
-         (\snd p = \snd (nth (bars f) ▭ (\fst p)))].
-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 ⇒ 〈O,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 (i - start f) (bars f)) (p Hp);
-        cases Hp; 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; clear H2;
-          simplify; apply q_le_minus; assumption;
-    |2,5: cases (value (q-Qpos (\fst b)) l1); cases H3; 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 H3; clear H3 value H2;
-        assumption;]
-|2: clear value H2; simplify; split;
-    [1: 
-    
-          
-definition same_shape ≝
-  λl1,l2:q_f.
-   ∀input.∃col.
-   
-    And3 
-      (sum_bases (bars l2) j ≤ offset - start l2)
-      (offset - start l2 ≤ sum_bases (bars l2) (S j))
-      (\snd (nth (bars l2)) q0 j) = \snd (nth (bars l1) q0 i).
-
-…┐─┌┐…
-\ldots\boxdl\boxh\boxdr\boxdl\ldots
+include "nat_ordered_set.ma".
+include "models/q_shift.ma".
 
 alias symbol "pi2" = "pair pi2".
 alias symbol "pi1" = "pair pi1".
 definition rebase_spec ≝ 
  ∀l1,l2:q_f.∃p:q_f × q_f.
    And4
-    (len (bars (\fst p)) = len (bars (\snd p)))
     (start (\fst p) = start (\snd p))
-    (∀i.\fst (nth (bars (\fst p)) q0 i) = \fst (nth (bars (\snd p)) q0 i))
-    (∀i,offset.
-      sum_bases (bars l1) i ≤ offset - start l1 →
-      offset - start l1 ≤ sum_bases (bars l1) (S i) →
-      ∃j.      
-        And3 
-         (sum_bases (bars (\fst p)) j ≤ offset - start (\fst p))
-         (offset - start (\fst p) ≤ sum_bases (bars (\fst p)) (S j))
-         (\snd (nth (bars (\fst p)) q0 j) = \snd (nth (bars l1) q0 i)) ∧
-        And3 
-         (sum_bases (bars (\snd p)) j ≤ offset - start (\snd p))
-         (offset - start (\snd p) ≤ sum_bases (bars (\snd p)) (S j))
-         (\snd (nth (bars (\snd p)) q0 j) = \snd (nth (bars l2) q0 i))).
+    (same_bases (bars (\fst p)) (bars (\snd p)))
+    (same_values l1 (\fst p)) 
+    (same_values l2 (\snd p)).
 
 definition rebase_spec_simpl ≝ 
- λl1,l2:list (ℚ × ℚ).λp:(list (ℚ × ℚ)) × (list (ℚ × ℚ)).
-    len ( (\fst p)) = len ( (\snd p)) ∧
-    (∀i.
-     \fst (nth ( (\fst p)) q0 i) = 
-     \fst (nth ( (\snd p)) q0 i)) ∧
-    ∀i,offset.
-      sum_bases ( l1) i ≤ offset ∧
-      offset ≤ sum_bases ( l1) (S i)
-       →
-      ∃j.  
-        sum_bases ( (\fst p)) j ≤ offset ∧
-        offset ≤ sum_bases ((\fst p)) (S j) ∧
-        \snd (nth ( (\fst p)) q0 j) = 
-        \snd (nth ( l1) q0 i).
+ λstart.λl1,l2:list bar.λp:(list bar) × (list bar).
+   And3
+    (same_bases (\fst p) (\snd p))
+    (same_values (mk_q_f start l1) (mk_q_f start (\fst p))) 
+    (same_values (mk_q_f start l2) (mk_q_f start (\snd p))).
+
+(* a local letin makes russell fail *)
+definition cb0h : list bar → list bar ≝ 
+  λl.mk_list (λi.〈\fst (nth l ▭ i),OQ〉) (len l).
 
 definition eject ≝
   λP.λp:∃x:(list bar) × (list bar).P x.match p with [ex_introT p _ ⇒ p].
 coercion eject.
 definition inject ≝ λP.λp:(list bar) × (list bar).λh:P p. ex_introT ? P p h.
 coercion inject with 0 1 nocomposites.
+
+axiom devil : False.
         
 definition rebase: rebase_spec.
 intros 2 (f1 f2); cases f1 (s1 l1); cases f2 (s2 l2); clear f1 f2;
 letin spec ≝ (
-  λl1,l2:list (ℚ × ℚ).λm:nat.λz:(list (ℚ × ℚ)) × (list (ℚ × ℚ)).
-    len l1 + len l2 < m → rebase_spec_simpl l1 l2 z);
+  λs.λl1,l2.λm.λz.len l1 + len l2 < m → rebase_spec_simpl s l1 l2 z);
+alias symbol "pi1" (instance 34) = "exT \fst".
+alias symbol "pi1" (instance 21) = "exT \fst".
 letin aux ≝ ( 
-let rec aux (l1,l2:list (ℚ × ℚ)) (n:nat) on n : (list (ℚ × ℚ)) × (list (ℚ × ℚ)) ≝
+let rec aux (l1,l2:list bar) (n:nat) on n : (list bar) × (list bar) ≝
 match n with
 [ O ⇒ 〈 nil ? , nil ? 〉
 | S m ⇒ 
@@ -220,43 +61,133 @@ match n with
      match l2 with
      [ nil ⇒ 〈l1, cb0h l1〉
      | cons he2 tl2 ⇒  
-         let base1 ≝ (\fst he1) in
-         let base2 ≝ (\fst he2) in
+         let base1 ≝ Qpos (\fst he1) in
+         let base2 ≝ Qpos (\fst he2) in
          let height1 ≝ (\snd he1) in
          let height2 ≝ (\snd he2) in
          match q_cmp base1 base2 with
          [ q_eq _ ⇒ 
              let rc ≝ aux tl1 tl2 m in 
              〈he1 :: \fst rc,he2 :: \snd rc〉 
-         | q_lt _ ⇒
+         | q_lt Hp ⇒
              let rest ≝ base2 - base1 in
-             let rc ≝ aux tl1 (〈rest,height2〉 :: tl2) m in
-             〈〈base1,height1〉 :: \fst rc,〈base1,height2〉 :: \snd rc〉 
-         | q_gt _ ⇒ 
+             let rc ≝ aux tl1 (〈\fst (unpos rest ?),height2〉 :: tl2) m in
+             〈〈\fst he1,height1〉 :: \fst rc,〈\fst he1,height2〉 :: \snd rc〉 
+         | q_gt Hp ⇒ 
              let rest ≝ base1 - base2 in
-             let rc ≝ aux (〈rest,height1〉 :: tl1) tl2 m in
-             〈〈base2,height1〉 :: \fst rc,〈base2,height2〉 :: \snd rc〉
+             let rc ≝ aux (〈\fst (unpos rest ?),height1〉 :: tl1) tl2 m in
+             〈〈\fst he2,height1〉 :: \fst rc,〈\fst he2,height2〉 :: \snd rc〉
 ]]]]
-in aux : ∀l1,l2,m.∃z.spec l1 l2 m z); unfold spec;
-[7: clearbody aux; unfold spec in aux; clear spec;
+in aux : ∀l1,l2,m.∃z.∀s.spec s l1 l2 m z); unfold spec;
+[9: clearbody aux; unfold spec in aux; clear spec;
     cases (q_cmp s1 s2);
     [1: cases (aux l1 l2 (S (len l1 + len l2)));
-        cases (H1 (le_n ?)); clear H1;
-        exists [apply 〈mk_q_f s1 (\fst w), mk_q_f s2 (\snd w)〉] repeat split;
-        [1: cases H2; assumption;
+        cases (H1 s1 (le_n ?)); clear H1;
+        exists [apply 〈mk_q_f s1 (\fst w), mk_q_f s2 (\snd w)〉] split;
+        [1,2: assumption;
+        |3: intro; apply (H3 input);
+        |4: intro; rewrite > H in H4; 
+            rewrite > (H4 input) in ⊢ (? ? % ?); reflexivity;]
+    |2: letin l2' ≝ (〈\fst (unpos (s2-s1) ?),OQ〉::l2);[
+          apply q_lt_minus; rewrite > q_plus_sym; rewrite > q_plus_OQ;
+          assumption]
+        cases (aux l1 l2' (S (len l1 + len l2')));
+        cases (H1 s1 (le_n ?)); clear H1 aux;
+        exists [apply 〈mk_q_f s1 (\fst w), mk_q_f s1 (\snd w)〉] split;
+        [1: reflexivity
         |2: assumption;
-        |3: cases H2; assumption;
-        |4: intros; cases (H3 i (offset - s1)); 
-            [2: 
-        
-        
-|1,2: simplify; generalize in ⊢ (? ? (? (? ? (? ? ? (? ? %)))) (? (? ? (? ? ? (? ? %))))); intro X; 
-      cases X (rc OK); clear X; simplify; apply eq_f; assumption;
-|3: cases (aux l4 l5 n1) (rc OK); simplify; apply eq_f; assumption;
-|4,5: simplify; unfold cb0h; rewrite > len_mk_list; reflexivity;
-|6: reflexivity]
-clearbody aux; unfold spec in aux; clear spec; 
+        |3: assumption;
+        |4: intro; 
+            rewrite > (initial_shift_same_values (mk_q_f s2 l2) s1 H input) in ⊢ (? ? % ?);
+            rewrite < (H4 input)in ⊢ (? ? ? %); reflexivity;]
+    |3: letin l1' ≝ (〈\fst (unpos (s1-s2) ?),OQ〉::l1);[
+          apply q_lt_minus; rewrite > q_plus_sym; rewrite > q_plus_OQ;
+          assumption]
+        cases (aux l1' l2 (S (len l1' + len l2)));
+        cases (H1 s2 (le_n ?)); clear H1 aux;
+        exists [apply 〈mk_q_f s2 (\fst w), mk_q_f s2 (\snd w)〉] split;
+        [1: reflexivity
+        |2: assumption;
+        |4: assumption;
+        |3: intro; simplify in ⊢ (? ? ? (? ? ? (? ? ? (? % ?))));
+            rewrite > (initial_shift_same_values (mk_q_f s1 l1) s2 H input) in ⊢ (? ? % ?);
+            rewrite < (H3 input) in ⊢ (? ? ? %); reflexivity;]]
+|1,2: unfold rest; apply q_lt_minus; rewrite > q_plus_sym; rewrite > q_plus_OQ;
+      assumption;        
+|8: intros; cases (?:False); apply (not_le_Sn_O ? H1);
+|3: intros; generalize in match (unpos ??); intro X; cases X; clear X;
+    simplify in ⊢ (???? (??? (??? (??? (?? (? (?? (??? % ?) ?) ??)))) ?));
+    simplify in ⊢ (???? (???? (??? (??? (?? (? (?? (??? % ?) ?) ??)))))); 
+    clear H4; cases (aux (〈w,\snd b〉::l4) l5 n1); clear aux;
+    cut (len (〈w,\snd b〉::l4) + len l5 < n1) as K;[2:
+      simplify in H5; simplify; rewrite > sym_plus in H5; simplify in H5;
+      rewrite > sym_plus in H5; apply le_S_S_to_le; apply H5;] 
+    split;
+    [1: simplify in ⊢ (? % ?); simplify in ⊢ (? ? %); 
+        cases (H4 s K); clear K H4; intro input; cases input; [reflexivity]
+        simplify; apply H7; 
+    |2: simplify in ⊢ (? ? %); cases (H4 s K); clear H4 K H5 spec;
+        intro;
+        (* input < s + b1 || input >= s + b1 *)
+    |3: simplify in ⊢ (? ? %);]   
+|4: intros; generalize in match (unpos ??); intro X; cases X; clear X;
+    (* duale del 3 *)
+|5: intros; (* triviale, caso in cui non fa nulla *)
+|6,7: (* casi base in cui allunga la lista più corta *) 
+]
+elim devil;
+qed.
+
+include "Q/q/qtimes.ma".
+
+let rec area (l:list bar) on l ≝
+  match l with 
+  [ nil ⇒ OQ
+  | cons he tl ⇒ area tl + Qpos (\fst he) * ⅆ[OQ,\snd he]].
+
+alias symbol "pi1" = "exT \fst".
+alias symbol "minus" = "Q minus".
+alias symbol "exists" = "CProp exists".
+definition minus_spec_bar ≝
+ λf,g,h:list bar.
+   same_bases f g → len f = len g →
+     ∀s,i:ℚ. \snd (\fst (value (mk_q_f s h) i)) = 
+       \snd (\fst (value (mk_q_f s f) i)) - \snd (\fst (value (mk_q_f s g) i)). 
 
+definition minus_spec ≝
+ λf,g:q_f.
+   ∃h:q_f. 
+     ∀i:ℚ. \snd (\fst (value h i)) = 
+       \snd (\fst (value f i)) - \snd (\fst (value g i)). 
+
+definition eject_bar : ∀P:list bar → CProp.(∃l:list bar.P l) → list bar ≝
+ λP.λp.match p with [ex_introT x _ ⇒ x].
+definition inject_bar ≝ ex_introT (list bar).
+
+coercion inject_bar with 0 1 nocomposites.
+coercion eject_bar with 0 0 nocomposites.
+
+lemma minus_q_f : ∀f,g. minus_spec f g.
+intros;
+letin aux ≝ (
+  let rec aux (l1, l2 : list bar) on l1 ≝
+    match l1 with
+    [ nil ⇒ []
+    | cons he1 tl1 ⇒
+        match l2 with
+        [ nil ⇒ []
+        | cons he2 tl2 ⇒ 〈\fst he1, \snd he1 - \snd he2〉 :: aux tl1 tl2]]
+  in aux : ∀l1,l2 : list bar.∃h.minus_spec_bar l1 l2 h);
+[2: intros 4; simplify in H3; destruct H3;
+|3: intros 4; simplify in H3; cases l1 in H2; [2: intro X; simplify in X; destruct X]    
+    intros; rewrite > (value_OQ_e (mk_q_f s []) i); [2: reflexivity]
+    rewrite > q_elim_minus; rewrite > q_plus_OQ; reflexivity;
+|1: cases (aux l2 l3); unfold in H2; intros 4;
+    simplify in ⊢ (? ? (? ? ? (? ? ? (? % ?))) ?);
+    cases (q_cmp i (s + Qpos (\fst b)));
+    
 
 
-qed.
\ No newline at end of file
+definition excess ≝ 
+  λf,g.∃i.\snd (\fst (value f i)) < \snd (\fst (value g i)).
+