more work to try to understand where the issue is

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 6be729db5492c6e55bbdeedfa97de1f011b21607..d0d043f47e053ae50df4e8aaba1ac9f510a83d72 100644 (file)
--- a/helm/software/matita/contribs/dama/dama/models/q_bars.ma
+++ b/helm/software/matita/contribs/dama/dama/models/q_bars.ma
@@ -29,7 +29,7 @@ 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 ≝
+let rec sum_bases (l:list bar) (i:nat) on i ≝
     match i with
     [ O ⇒ OQ
     | S m ⇒ 
@@ -148,3 +148,39 @@ try apply (sum_bases_ge_OQ []);
 apply (sum_bases_ge_OQ l1);
 qed.
 
+lemma sum_bases_increasing:
+  ∀l,x.sum_bases l x < sum_bases l (S x). 
+intro; elim l;
+[1: elim x;
+    [1: simplify; rewrite > q_plus_sym; rewrite > q_plus_OQ;
+        apply q_pos_lt_OQ;
+    |2: simplify in H ⊢ %; 
+        apply q_lt_plus; rewrite > q_elim_minus;
+        rewrite < q_plus_assoc; rewrite < q_elim_minus;
+        rewrite > q_plus_minus; rewrite > q_plus_OQ;
+        assumption;]
+|2: elim x;
+    [1: simplify; rewrite > q_plus_sym; rewrite > q_plus_OQ;
+        apply q_pos_lt_OQ;
+    |2: simplify; change in ⊢ (? ? (? % ?)) with (sum_bases l1 (S n)) ;  
+        apply q_lt_plus; rewrite > q_elim_minus;
+        rewrite < q_plus_assoc; rewrite < q_elim_minus;
+        rewrite > q_plus_minus; rewrite > q_plus_OQ; apply H]]
+qed.
+
+lemma sum_bases_lt_canc:
+  ∀l,x,y.sum_bases l (S x) < sum_bases l (S y) → sum_bases l x < sum_bases l y.
+intro; elim l; [apply (q_lt_canc_plus_r ?? (Qpos one));apply H]
+generalize in match H1;apply (nat_elim2 (?:? → ? → CProp) ??? x y);
+intros 2;
+[3: intros 2; simplify; apply q_lt_inj_plus_r; apply H;
+    apply (q_lt_canc_plus_r ?? (Qpos (\fst a))); apply H3;
+|2: cases (?:False); simplify in H2;
+    apply (q_lt_le_incompat (sum_bases l1 (S n)) OQ);[2: apply sum_bases_ge_OQ;]
+    apply (q_lt_canc_plus_r ?? (Qpos (\fst a))); apply H2;
+|1: cases n in H2; intro; 
+    [1: cases (?:False); apply (q_lt_corefl ? H2);
+    |2: simplify; apply  q_lt_plus_trans; [apply sum_bases_ge_OQ]
+        apply q_pos_lt_OQ;]]
+qed.
+

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 96cd96e49e644e08a40b84737b338f2a6aaa9e62..d3d63233c37520c7d982a414ed767a51f6eaa64f 100644 (file)
--- a/helm/software/matita/contribs/dama/dama/models/q_function.ma
+++ b/helm/software/matita/contribs/dama/dama/models/q_function.ma
@@ -15,62 +15,8 @@
 include "nat_ordered_set.ma".
 include "models/q_bars.ma".
 
-lemma sum_bars_increasing:
-  ∀l,x.sum_bases l x < sum_bases l (S x). 
-intro; elim l;
-[1: elim x;
-    [1: simplify; rewrite > q_plus_sym; rewrite > q_plus_OQ;
-        apply q_pos_lt_OQ;
-    |2: simplify in H ⊢ %; 
-        apply q_lt_plus; rewrite > q_elim_minus;
-        rewrite < q_plus_assoc; rewrite < q_elim_minus;
-        rewrite > q_plus_minus; rewrite > q_plus_OQ;
-        assumption;]
-|2: elim x;
-    [1: simplify; rewrite > q_plus_sym; rewrite > q_plus_OQ;
-        apply q_pos_lt_OQ;
-    |2: simplify; change in ⊢ (? ? (? % ?)) with (sum_bases l1 (S n)) ;  
-        apply q_lt_plus; rewrite > q_elim_minus;
-        rewrite < q_plus_assoc; rewrite < q_elim_minus;
-        rewrite > q_plus_minus; rewrite > q_plus_OQ; apply H]]
-qed.
-
-lemma q_lt_canc_plus_r:
-  ∀x,y,z:Q.x + z < y + z → x < y.
-intros; rewrite < (q_plus_OQ y); rewrite < (q_plus_minus z);
-rewrite > q_elim_minus; rewrite > q_plus_assoc;
-apply q_lt_plus; rewrite > q_elim_opp; assumption;
-qed.
-
-lemma q_lt_inj_plus_r:
-  ∀x,y,z:Q.x < y → x + z < y + z.
-intros; apply (q_lt_canc_plus_r ?? (Qopp z));
-do 2 (rewrite < q_plus_assoc;rewrite < q_elim_minus);
-rewrite > q_plus_minus;
-do 2 rewrite > q_plus_OQ; assumption;
-qed.
-
-lemma sum_bases_lt_canc:
-  ∀l,x,y.sum_bases l (S x) < sum_bases l (S y) → sum_bases l x < sum_bases l y.
-intro; elim l; [apply (q_lt_canc_plus_r ?? (Qpos one));apply H]
-generalize in match H1;apply (nat_elim2 (?:? → ? → CProp) ??? x y);
-intros 2;
-[3: intros 2; simplify; apply q_lt_inj_plus_r; apply H;
-    apply (q_lt_canc_plus_r ?? (Qpos (\fst a))); apply H3;
-|2: cases (?:False); simplify in H2;
-    apply (q_lt_le_incompat (sum_bases l1 (S n)) OQ);[2: apply sum_bases_ge_OQ;]
-    apply (q_lt_canc_plus_r ?? (Qpos (\fst a))); apply H2;
-|1: cases n in H2; intro; 
-    [1: cases (?:False); apply (q_lt_corefl ? H2);
-    |2: simplify; apply  q_lt_plus_trans; [apply sum_bases_ge_OQ]
-        apply q_pos_lt_OQ;]]
-qed.
-
-axiom q_minus_distrib:
-  ∀x,y,z:Q.x - (y + z) = x - y - z.
+axiom le_le_eq: ∀x,y:Q. x ≤ y → y ≤ x → x = y.
 
-axiom q_le_OQ_Qpos: ∀x.OQ ≤ Qpos x.
-       
 lemma initial_shift_same_values:
   ∀l1:q_f.∀init.init < start l1 →
    same_values l1 
@@ -137,25 +83,65 @@ whd in ⊢ (% → ?); simplify in H3;
                 rewrite < q_plus_assoc; rewrite < q_elim_minus;
                 rewrite > q_plus_minus;rewrite > q_plus_OQ; assumption;
             |2: rewrite < q_d_noabs; [2: apply q_lt_to_le; assumption]
-                rewrite > q_d_sym; apply (q_le_S ???? H5);apply sum_bases_ge_OQ;]]
+                rewrite > q_d_sym
+                
+                ; apply (q_le_S ???? H5);apply sum_bases_ge_OQ;]]
     |3: intro; cases H8; clear H8; rewrite > H11; rewrite > H7; clear H11 H7;
-        simplify in ⊢ (? ? ? (? ? ? (? ? % ? ?)));
+        simplify in H5 H6 ⊢ (? ? ? (? ? ? (? ? % ? ?)));
+
+axiom nth_nil: ∀T,n.∀d:T. nth [] d n = d.
+
+lemma key:
+  ∀init,input,l1,w1,w2,w.
+  Qpos w = start l1 - init →   
+  init < start l1 → 
+  start l1 < input →
+  sum_bases (〈w,OQ〉::bars l1) w1 ≤ ⅆ[input,init] →
+    ⅆ[input,init] < sum_bases (bars l1) w1 + (start l1-init) →
+  sum_bases (bars l1) w2 ≤ ⅆ[input,start l1] →
+    ⅆ[input,start l1] < sum_bases (bars l1) (S w2) →
+    \snd (nth (bars l1) ▭ w2) = \snd (nth (〈w,OQ〉::bars l1) ▭ w1).
+intros 4 (init input l); cases l (st l);
+change in match (start (mk_q_f st l)) with st;
+change in match (bars (mk_q_f st l)) with l;
+elim l;
+[1: rewrite > nth_nil; cases w1 in H4;
+    [1: rewrite > q_d_sym; rewrite > q_d_noabs; [2:
+          apply (q_le_trans ? st); apply q_lt_to_le; assumption]
+        do 2 rewrite > q_elim_minus; rewrite > q_plus_assoc;
+        intro X; lapply (q_lt_canc_plus_r ??? X) as Y;
+        simplify in Y; cases (?:False); 
+        apply (q_lt_corefl st); apply (q_lt_trans ??? H2);
+        apply (q_lt_le_trans ??? Y); rewrite > q_plus_sym; rewrite > q_plus_OQ;
+        apply q_eq_to_le; reflexivity;
+    |2: intros; simplify; rewrite > nth_nil; reflexivity;]       
+|2: FACTORIZE w1>0    
+    
+     (* interesting case: init < start < input *)
+        intro; cases H8; clear H8; rewrite > H11; rewrite > H7; clear H11 H7;
+        simplify in H5 H6 ⊢ (? ? ? (? ? ? (? ? % ? ?)));
+        elim (\fst w2) in H9 H10;
+        [1: elim (\fst w1) in H5 H6;
+            [1: cases (?:False); clear H5 H8 H7; 
+                apply (q_lt_antisym input (start l1)); [2: assumption]
+                rewrite > q_d_sym in H6; rewrite > q_d_noabs in H6; 
+                  [2: apply q_lt_to_le; assumption]
+                rewrite > q_plus_sym in H6; rewrite > q_plus_OQ in H6; 
+                rewrite > H2 in H6; apply (q_lt_canc_plus_r ?? (Qopp init)); 
+                do 2 rewrite < q_elim_minus; assumption;
+            |2: 
+                
         cut (\fst w1 = S (\fst w2)) as Key; [rewrite > Key; reflexivity;]
-        simplify in H5 H6;
-        cases (\fst w1) in H5 H6; intros;
-        [1: cases (?:False); clear H5 H9 H10; simplify in H6;
-            apply (q_lt_antisym input (start l1)); [2: assumption]
-            rewrite > q_d_sym in H6;
-            rewrite > q_d_noabs in H6; [2: apply q_lt_to_le; assumption]
-            rewrite > q_plus_sym in H6;
-            rewrite > q_plus_OQ in H6; rewrite > H2 in H6;
-            lapply (q_lt_plus ??? H6) as X; clear H6 H2 H3 H1 H H4 w1 w2 w;
-            rewrite > q_elim_minus in X; rewrite < q_plus_assoc in X;
-            rewrite > (q_plus_sym (Qopp init)) in X;
-            rewrite < q_elim_minus in X; rewrite > q_plus_minus in X;
-            rewrite > q_plus_OQ in X; assumption;
-        |2: simplify in H5; apply eq_f;
-            cut (sum_bases (bars l1) (\fst w2) < sum_bases (bars l1) (S n)+Qpos w);[2:
+        cases (\fst w1) in H5 H6; intros; [1: 
+          cases (?:False); clear H5 H9 H10; 
+          apply (q_lt_antisym input (start l1)); [2: assumption]
+          rewrite > q_d_sym in H6; rewrite > q_d_noabs in H6; 
+            [2: apply q_lt_to_le; assumption]
+          rewrite > q_plus_sym in H6; rewrite > q_plus_OQ in H6; 
+          rewrite > H2 in H6; apply (q_lt_canc_plus_r ?? (Qopp init)); 
+          do 2 rewrite < q_elim_minus; assumption;]
+       apply eq_f;
+            cut (sum_bases (bars l1) (\fst w2) < sum_bases (bars l1) (S n));[2:
               apply (q_le_lt_trans ??? H9);
               apply (q_lt_trans ??? ? H6);
               rewrite > q_d_sym; rewrite > q_d_noabs; [2:apply q_lt_to_le;assumption]

diff --git a/helm/software/matita/contribs/dama/dama/models/q_support.ma b/helm/software/matita/contribs/dama/dama/models/q_support.ma
index 82832c4bc64ee90cea02b82e867f4a321cc83328..2073c8a77afad47f5b6d6c64b56464a1db1ff0a0 100644 (file)
--- a/helm/software/matita/contribs/dama/dama/models/q_support.ma
+++ b/helm/software/matita/contribs/dama/dama/models/q_support.ma
@@ -33,6 +33,7 @@ axiom q_minus_r: ∀x,y. y + Qpos x = y - Qneg x.
 axiom q_plus_assoc: ∀x,y,z.x + (y + z) = x + y + z. 
 axiom q_elim_minus: ∀x,y.x - y = x + Qopp y.
 axiom q_elim_opp: ∀x,y.x - Qopp y = x + y.
+axiom q_minus_distrib:∀x,y,z:Q.x - (y + z) = x - y - z.
 
 (* order over Q *)
 axiom qlt : ℚ → ℚ → CProp.
@@ -68,6 +69,8 @@ axiom q_pos_lt_OQ: ∀x.OQ < Qpos x.
 axiom q_le_plus_trans: ∀x,y:Q. OQ ≤ x → OQ ≤ y → OQ ≤ x + y.
 axiom q_le_S: ∀x,y,z.OQ ≤ x → x + y ≤ z → y ≤ z.
 axiom q_eq_to_le: ∀x,y. x = y → x ≤ y.
+axiom q_le_OQ_Qpos: ∀x.OQ ≤ Qpos x.
+       
 
 inductive q_le_elimination (a,b:ℚ) : CProp ≝
 | q_le_from_eq : a = b → q_le_elimination a b
@@ -90,3 +93,19 @@ axiom q_d_sym:  ∀x,y. ⅆ[x,y] = ⅆ[y,x].
 (* integral part *)
 axiom nat_of_q: ℚ → nat.
 
+(* derived *)
+lemma q_lt_canc_plus_r:
+  ∀x,y,z:Q.x + z < y + z → x < y.
+intros; rewrite < (q_plus_OQ y); rewrite < (q_plus_minus z);
+rewrite > q_elim_minus; rewrite > q_plus_assoc;
+apply q_lt_plus; rewrite > q_elim_opp; assumption;
+qed.
+
+lemma q_lt_inj_plus_r:
+  ∀x,y,z:Q.x < y → x + z < y + z.
+intros; apply (q_lt_canc_plus_r ?? (Qopp z));
+do 2 (rewrite < q_plus_assoc;rewrite < q_elim_minus);
+rewrite > q_plus_minus;
+do 2 rewrite > q_plus_OQ; assumption;
+qed.
+