X-Git-Url: http://matita.cs.unibo.it/gitweb/?a=blobdiff_plain;f=helm%2Fsoftware%2Fmatita%2Fcontribs%2Fdama%2Fdama%2Fmodels%2Fq_function.ma;h=a38cf6e4cda648377be8ea314956b673a2f2becd;hb=b12a46d53cf80d40b253ca5dd495397c5c0b4287;hp=48d3012ec66974d8c8993e5db02bd72e3b488229;hpb=59f65aaf6f8d23748e1294ecabffffaa903ae657;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 48d3012ec..a38cf6e4c 100644
--- a/helm/software/matita/contribs/dama/dama/models/q_function.ma
+++ b/helm/software/matita/contribs/dama/dama/models/q_function.ma
@@ -12,123 +12,316 @@
 (*                                                                        *)
 (**************************************************************************)
 
-include "Q/q/q.ma".
-include "list/list.ma".
-include "cprop_connectives.ma". 
+include "russell_support.ma".
+include "models/q_bars.ma".
 
+definition rebase_spec ≝ 
+ λl1,l2:q_f.λp:q_f × q_f. 
+   And3
+    (same_bases (bars (\fst p)) (bars (\snd p)))
+    (same_values l1 (\fst p)) 
+    (same_values l2 (\snd p)).
 
-notation "\rationals" non associative with precedence 99 for @{'q}.
-interpretation "Q" 'q = Q. 
+definition same_values_simpl ≝
+ λl1,l2.∀p1,p2,p3,p4,p5,p6.same_values (mk_q_f l1 p1 p2 p3) (mk_q_f l2 p4 p5 p6).
 
-record q_f : Type ≝ {
- start : ℚ;
- bars: list (ℚ × ℚ) (* base, height *)
-}.
-
-axiom qp : ℚ → ℚ → ℚ.
-
-interpretation "Q plus" 'plus x y = (qp x y).
-
-axiom qm : ℚ → ℚ → ℚ.
-
-interpretation "Q minus" 'minus x y = (qm x y).
-
-axiom qlt : ℚ → ℚ → CProp.
-
-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" 'le x y = (qle x y).
-
-notation "'nth'" with precedence 90 for @{'nth}.
-notation < "'nth' \nbsp l \nbsp d \nbsp i" with precedence 71 
-for @{'nth_appl $l $d $i}.
-interpretation "list nth" 'nth = (cic:/matita/list/list/nth.con _).
-interpretation "list nth" 'nth_appl l d i = (cic:/matita/list/list/nth.con _ l d i).
-
-notation < "\rationals \sup 2" non associative with precedence 90 for @{'q2}.
-interpretation "Q x Q" 'q2 = (Prod Q Q).
-
-let rec make_list (A:Type) (def:nat→A) (n:nat) on n ≝
-  match n with
-  [ O ⇒ nil ?
-  | S m ⇒ def m :: make_list A def m].
-
-notation "'mk_list'" with precedence 90 for @{'mk_list}.
-interpretation "'mk_list'" 'mk_list = (make_list _).   
-notation < "'mk_list' \nbsp f \nbsp n" 
-with precedence 71 for @{'mk_list_appl $f $n}.
-interpretation "'mk_list' appl" 'mk_list_appl f n = (make_list _ f n).
+alias symbol "pi2" = "pair pi2".
+alias symbol "pi1" = "pair pi1".
+definition rebase_spec_aux ≝ 
+ λl1,l2:list bar.λp:(list bar) × (list bar).
+   sorted q2_lt l1 → sorted q2_lt l2 →
+   (l1 ≠ [] → \snd (\nth l1 ▭ (pred (\len l1))) = 〈OQ,OQ〉) →
+   (l2 ≠ [] → \snd (\nth l2 ▭ (pred (\len l2))) = 〈OQ,OQ〉) →
+ And4
+   (nth_base l1 O = nth_base (\fst p) O ∨
+    nth_base l2 O = nth_base (\fst p) O) 
+   (sorted q2_lt (\fst p) ∧ sorted q2_lt (\snd p))
+   ((l1 ≠ [] → \snd (\nth (\fst p) ▭ (pred (\len (\fst p)))) = 〈OQ,OQ〉) ∧ 
+    (l2 ≠ [] → \snd (\nth (\snd p) ▭ (pred (\len (\snd p)))) = 〈OQ,OQ〉))  
+   (And3
+      (same_bases (\fst p) (\snd p))
+      (same_values_simpl l1 (\fst p)) 
+      (same_values_simpl l2 (\snd p))).   
 
-definition q0 : ℚ × ℚ ≝ 〈OQ,OQ〉.
-notation < "0 \sub \rationals" with precedence 90 for @{'q0}.
-interpretation "q0" 'q0 = q0.
+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.
 
-notation < "[ \rationals \sup 2]" with precedence 90 for @{'lq2}.
-interpretation "lq2" 'lq2 = (list (Prod Q Q)).
-notation < "[ \rationals \sup 2] \sup 2" with precedence 90 for @{'lq22}.
-interpretation "lq22" 'lq22 = (Prod (list (Prod Q Q)) (list (Prod Q Q))).
+axiom devil : False.
+        
+definition copy ≝
+ λl:list bar.make_list ? (λn.〈nth_base l (\len l - S n),〈OQ,OQ〉〉) (\len l).
 
+lemma list_elim_with_len:
+  ∀T:Type.∀P: nat → list T → CProp.
+    P O [] → (∀l,a,n.P (\len l) l → P (S n) (a::l)) →
+     ∀l.P (\len l) l.
+intros;elim l; [assumption] simplify; apply H1; apply H2;
+qed.
+ 
+lemma sorted_near:
+ ∀r,l. sorted r l → ∀i,d. S i < \len l → r (\nth l d i) (\nth l d (S i)).
+ intros 3; elim H; 
+ [1: cases (not_le_Sn_O ? H1);
+ |2: simplify in H1; cases (not_le_Sn_O ? (le_S_S_to_le ?? H1));
+ |3: simplify; cases i in H4; intros; [apply H1]
+     apply H3; apply le_S_S_to_le; apply H4]
+ qed.  
+ 
+lemma sorted_copy:
+  ∀l:list bar.sorted q2_lt l → sorted q2_lt (copy l).
+intros 2; unfold copy; generalize in match (le_n (\len l)); 
+elim (\len l) in ⊢ (?%?→? ? (? ? ? %));
+simplify; [apply (sorted_nil q2_lt);] cases n in H1 H2;
+simplify; intros; [apply (sorted_one q2_lt);]
+apply (sorted_cons q2_lt);
+[2: apply H1; apply lt_to_le; apply H2;
+|1: elim l in H2 H; simplify; [simplify in H2; cases (not_le_Sn_O ? H2)]    
+    simplify in H3; unfold nth_base;
+    unfold canonical_q_lt; unfold q2_trel; unfold q2_lt; simplify;
+    change with (q2_lt (\nth (a::l1) ▭ (\len l1-S n1)) (\nth (a::l1) ▭ (\len l1-n1)));
+    cut (∃w.w = \len l1 - S n1); [2: exists[apply (\len l1 - S n1)] reflexivity]
+    cases Hcut; rewrite < H4; rewrite < (?:S w = \len l1 - n1);
+    [1: apply (sorted_near q2_lt (a::l1) H2); rewrite > H4;
+        simplify; apply le_S_S; elim (\len l1) in H3; simplify;
+        [ cases (not_le_Sn_O ? (le_S_S_to_le ?? H3));
+        | lapply le_S_S_to_le to H5 as H6;
+          lapply le_S_S_to_le to H6 as H7; clear H5 H6;
+          cases H7 in H3; intros; [rewrite < minus_n_n; apply le_S_S; apply le_O_n]
+          simplify in H5; apply le_S_S; apply (trans_le ???? (H5 ?));
+          [2: apply le_S_S; apply le_S_S; assumption;
+          |1: apply (lt_minus_S_n_to_le_minus_n n1 (S m) (S (minus m n1)) ?).
+              apply (not_le_to_lt (S (minus m n1)) (minus (S m) (S n1)) ?).
+              apply (not_le_Sn_n (minus (S m) (S n1))).]]
+    |2: rewrite > H4; lapply le_S_S_to_le to H3 as K;
+        clear H4 Hcut H3 H H1 H2; generalize in match K; clear K;
+        apply (nat_elim2 ???? n1 (\len l1)); simplify; intros;
+        [1: rewrite < minus_n_O; cases n2 in H; [intro; cases (not_le_Sn_O ? H)]
+            intros; cases n3; simplify; reflexivity;
+        |2: cases (not_le_Sn_O ? H);
+        |3: apply H; apply le_S_S_to_le; apply H1;]]]
+qed.
 
-notation "'len'" with precedence 90 for @{'len}.
-interpretation "len" 'len = length.
-notation < "'len' \nbsp l" with precedence 70 for @{'len_appl $l}.
-interpretation "len appl" 'len_appl l = (length _ l).
+lemma make_list_ext: ∀T,f1,f2,n. (∀x.x<n → f1 x = f2 x) → make_list T f1 n = make_list T f2 n.
+intros 4;elim n; [reflexivity] simplify; rewrite > H1; [2: apply le_n]
+apply eq_f; apply H; intros; apply H1; apply (trans_le ??? H2); apply le_S; apply le_n;
+qed.
+    
+lemma len_copy: ∀l. \len l = \len (copy l).
+intro; elim l; [reflexivity] simplify; rewrite > H; clear H;
+apply eq_f; elim (\len (copy l1)) in ⊢ (??%(??(???%))); [reflexivity] simplify;
+rewrite > H in ⊢ (??%?); reflexivity;
+qed. 
 
-definition eject ≝
-  λP.λp:∃x:(list (ℚ × ℚ)) × (list (ℚ × ℚ)).P x.match p with [ex_introT p _ ⇒ p].
-coercion cic:/matita/dama/models/q_function/eject.con.
-definition inject ≝
-  λP.λp:(list (ℚ × ℚ)) × (list (ℚ × ℚ)).λh:P p. ex_introT ? P p h.
-(*coercion inject with 0 1 nocomposites.*)
-coercion cic:/matita/dama/models/q_function/inject.con 0 1 nocomposites.
+lemma same_bases_cons: ∀a,b,l1,l2.
+  same_bases l1 l2 → \fst a = \fst b → same_bases (a::l1) (b::l2).
+intros; intro; cases i; simplify; [assumption;] apply (H n);
+qed.
 
-definition cb0h ≝ (λl.mk_list (λi.〈\fst (nth l q0 i),OQ〉) (length ? l)).
+lemma minus_lt : ∀i,j. i < j → j - i = S (j - S i). 
+intros 2;
+apply (nat_elim2 ???? i j); simplify; intros;
+[1: cases n in H; intros; rewrite < minus_n_O; [cases (not_le_Sn_O ? H);]
+    simplify; rewrite < minus_n_O; reflexivity;
+|2: cases (not_le_Sn_O ? H);
+|3: apply H; apply le_S_S_to_le; assumption;]
+qed.
+    
+lemma copy_same_bases: ∀l. same_bases l (copy l).
+intro; unfold copy; elim l using list_elim_with_len; [1: intro;reflexivity]
+simplify; rewrite < minus_n_n;
+simplify in ⊢ (? ? (? ? (? ? ? % ?) ?));
+apply same_bases_cons; [2: reflexivity]
+cases l1 in H; [intros 2; reflexivity]
+simplify in ⊢ (? ? (? ? (λ_:?.? ? ? (? ? %) ?) ?)→?);
+simplify in ⊢ (?→? ? (? ? (λ_:?.? ? ? (? ? (? % ?)) ?) ?));
+intro; rewrite > (make_list_ext ?? (λn:nat.〈nth_base (b::l2) (\len l2-n),〈OQ,OQ〉〉));[assumption]
+intro; elim x; [simplify; rewrite < minus_n_O; reflexivity]
+simplify in ⊢ (? ? (? ? ? (? ? %) ?) ?);
+simplify in H2:(? ? %); rewrite > minus_lt; [reflexivity] apply le_S_S_to_le;
+assumption;
+qed.
 
+lemma copy_rebases: 
+  ∀l1.rebase_spec_aux l1 [] 〈l1, copy l1〉.
+intros; elim l1; intros 4;
+[1: split; [left; reflexivity]; split; try assumption; unfold; intros;
+    unfold same_values; intros; reflexivity;
+|2: rewrite > H3; [2: intro X; destruct X]
+    split; [left; reflexivity] split; 
+    unfold same_values_simpl; unfold same_values; intros; try reflexivity;
+    try assumption; [4: normalize in p2; destruct p2|2: cases H5; reflexivity;]
+    [1: apply (sorted_copy ? H1);
+    |2: apply (copy_same_bases (a::l));]]
+qed.
+               
+lemma copy_rebases_r: 
+  ∀l1.rebase_spec_aux [] l1 〈copy l1, l1〉.
+intros; elim l1; intros 4;
+[1: split; [left; reflexivity]; split; try assumption; unfold; intros;
+    unfold same_values; intros; reflexivity;
+|2: rewrite > H4; [2: intro X; destruct X]
+    split; [right; simplify; rewrite < minus_n_n; reflexivity] split; 
+    unfold same_values_simpl; unfold same_values; intros; try reflexivity;
+    try assumption; [4: normalize in p2; destruct p2|2: cases H5; reflexivity;]
+    [1: apply (sorted_copy ? H2);
+    |2: intro; symmetry; apply (copy_same_bases (a::l));]]
+qed.
+               
+definition rebase: ∀l1,l2:q_f.∃p:q_f × q_f.rebase_spec l1 l2 p.
+intros 2 (f1 f2); cases f1 (b1 Hs1 Hb1 He1); cases f2 (b2 Hs2 Hb2 He2); clear f1 f2;
+alias symbol "plus" = "natural plus".
 alias symbol "pi2" = "pair pi2".
 alias symbol "pi1" = "pair pi1".
-definition rebase: 
-  q_f → q_f → 
-    ∃p:q_f × q_f.∀i.
-     \fst (nth (bars (\fst p)) q0 i) = 
-     \fst (nth (bars (\snd p)) q0 i).
-intros (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)).True);
+alias symbol "minus" = "Q minus".
 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 ⇒ 
+[ O ⇒ 〈[], []〉
+| S m ⇒
   match l1 with
-  [ nil ⇒ 〈cb0h l2, l2〉
+  [ nil ⇒ 〈copy l2, l2〉
   | cons he1 tl1 ⇒
      match l2 with
-     [ nil ⇒ 〈l1, cb0h l1〉
+     [ nil ⇒ 〈l1, copy l1〉
      | cons he2 tl2 ⇒  
-         let base1 ≝ (\fst he1) in
-         let base2 ≝ (\fst he2) in
-         let height1 ≝ (\snd he1) in
-         let height2 ≝ (\snd he2) in
+         let base1 ≝ \fst he1 in
+         let base2 ≝ \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 _ ⇒ 
-             let rest ≝ base2 - base1 in
-             let rc ≝ aux tl1 (〈rest,height2〉 :: tl2) m in
-             〈〈base1,height1〉 :: \fst rc,〈base1,height2〉 :: \snd rc〉
-         | q_gt _ ⇒ 
+         [ q_leq Hp1 ⇒ 
+             match q_cmp base2 base1 with
+             [ q_leq Hp2 ⇒
+                 let rc ≝ aux tl1 tl2 m in 
+                 〈he1 :: \fst rc,he2 :: \snd rc〉
+             | q_gt 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 Hp ⇒ 
              let rest ≝ base1 - base2 in
              let rc ≝ aux (〈rest,height1〉 :: tl1) tl2 m in
-             〈〈base2,height1〉 :: \fst rc,〈base2,height2〉 :: \snd rc〉
-]]]]
-in aux : ∀l1,l2,m.∃z.spec l1 l2 m z); 
-qed.
\ No newline at end of file
+             〈〈base2,height1〉 :: \fst rc,〈base2,height2〉 :: \snd rc〉]]]]
+in aux : ∀l1,l2,m.∃z.\len l1 + \len l2 ≤ m → rebase_spec_aux l1 l2 z);
+[7: clearbody aux; cases (aux b1 b2 (\len b1 + \len b2)) (w Hw); clear aux;
+    cases (Hw (le_n ?) Hs1 Hs2 (λ_.He1) (λ_.He2)); clear Hw; cases H1; cases H2; cases H3; clear H3 H1 H2;
+    exists [constructor 1;constructor 1;[apply (\fst w)|5:apply (\snd w)]] try assumption;
+    [1,3: apply hide; cases H (X X); try rewrite < (H8 O); try rewrite < X; assumption
+    |2,4: apply hide;[apply H6|apply H7]intro X;[rewrite > X in Hb1|rewrite > X in Hb2]
+         normalize in Hb1 Hb2; [destruct Hb1|destruct Hb2]]
+    unfold; unfold same_values; simplify in ⊢ (? (? % %) ? ?); 
+    simplify in match (\snd 〈?,?〉); simplify in match (\fst 〈?,?〉);
+    split; [assumption; |apply H9;|apply H10]
+|6: intro ABS; unfold; intros 4; clear H1 H2;
+    cases l in ABS H3; intros 1; [2: simplify in H1; cases (not_le_Sn_O ? H1)]
+    cases l1 in H4 H1; intros; [2: simplify in H2; cases (not_le_Sn_O ? H2)]
+    split; [ left; reflexivity|split; apply (sorted_nil q2_lt);|split; assumption;]
+    split; unfold; intros; unfold same_values; intros; reflexivity;
+|5: intros; apply copy_rebases_r;
+|4: intros; rewrite < H1; apply copy_rebases;
+|3: cut (\fst b = \fst b3) as K; [2: apply q_le_to_le_to_eq; assumption] clear H6 H5 H4 H3;
+    intros; cases (aux l2 l3 n1); intros 4; simplify in match (\fst ≪w,H≫);
+    simplify in match (\fst 〈?,?〉); simplify in match (\snd 〈?,?〉);
+    cases H4; 
+      [2: apply le_S_S_to_le; apply (trans_le ???? H3); simplify;
+          rewrite < plus_n_Sm; apply le_S; apply le_n;
+      |3,4: apply (sorted_tail q2_lt); [2: apply H5|4:apply H6]
+      |5: intro; cases l2 in H7 H9; intros; [cases H9; reflexivity]
+          simplify in H7 ⊢ %; apply H7; intro; destruct H10;
+      |6: intro; cases l3 in H8 H9; intros; [cases H9; reflexivity]
+          simplify in H8 ⊢ %; apply H8; intro; destruct H10;]
+    clear aux; split; 
+    [1: left; reflexivity;
+    |2: cases H10;  
+
+    
+     
+ unfold rebase_spec_aux; intros; cases l1 in H2 H4 H6; intros; [ simplify in H2; destruct H2;]
+    lapply H6 as H7; [2: intro X; destruct X] clear H6 H5;
+    rewrite > H7; split; [right; simplify;
+    
+     split; [left;reflexivity]
+    split; 
+
+,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)));
+    
+
+
+definition excess ≝ 
+  λf,g.∃i.\snd (\fst (value f i)) < \snd (\fst (value g i)).
+