lost in the wood

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 32122371004982cad24e4deccf33ec5f71dad0e1..96cd96e49e644e08a40b84737b338f2a6aaa9e62 100644 (file)
--- a/helm/software/matita/contribs/dama/dama/models/q_function.ma
+++ b/helm/software/matita/contribs/dama/dama/models/q_function.ma
@@ -12,8 +12,65 @@
 (*                                                                        *)
 (**************************************************************************)
 
+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 q_le_OQ_Qpos: ∀x.OQ ≤ Qpos x.
+       
 lemma initial_shift_same_values:
   ∀l1:q_f.∀init.init < start l1 →
    same_values l1 
@@ -78,15 +135,92 @@ whd in ⊢ (% → ?); simplify in H3;
             [1: rewrite > H2; do 2 rewrite > q_elim_minus;
                 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;
+                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;]]
-    |3:
-    
-        
-STOP            
+                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 ⊢ (? ? ? (? ? ? (? ? % ? ?)));
+        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:
+              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]
+              rewrite > q_d_sym; rewrite > q_d_noabs; [2:apply q_lt_to_le;assumption]
+              do 2 rewrite > q_elim_minus; rewrite > (q_plus_sym ? (Qopp init));
+              apply q_lt_plus; rewrite > q_plus_sym;
+              rewrite > q_elim_minus; rewrite < q_plus_assoc;
+              rewrite < q_elim_minus; rewrite > q_plus_minus;
+              rewrite > q_plus_OQ; apply q_lt_opp_opp; assumption]
+            clear H9 H6;
+            cut (ⅆ[input,init] - Qpos w = ⅆ[input,start l1]);[2:
+              rewrite > q_d_sym; rewrite > q_d_noabs; [2:apply q_lt_to_le;assumption]
+              rewrite > q_d_sym; rewrite > q_d_noabs; [2:apply q_lt_to_le;assumption]
+              rewrite > H2; rewrite > (q_elim_minus (start ?));
+              rewrite > q_minus_distrib; rewrite > q_elim_opp;
+              do 2 rewrite > q_elim_minus;
+              do 2 rewrite < q_plus_assoc;
+              rewrite > (q_plus_sym ? init);
+              rewrite > (q_plus_assoc ? init);
+              rewrite > (q_plus_sym ? init);
+              rewrite < (q_elim_minus init); rewrite > q_plus_minus;
+              rewrite > (q_plus_sym OQ); rewrite > q_plus_OQ;
+              rewrite < q_elim_minus; reflexivity;]
+            cut (sum_bases (bars l1) n < sum_bases (bars l1) (S (\fst w2)));[2:
+              apply (q_le_lt_trans ???? H10); rewrite < Hcut1;
+              rewrite > q_elim_minus; apply q_le_minus_r; rewrite > q_elim_opp;
+              assumption;] clear Hcut1 H5 H10;
+            generalize in match Hcut;generalize in match Hcut2;clear Hcut Hcut2; 
+            apply (nat_elim2 ???? n (\fst w2)); 
+            [3: intros (x y); apply eq_f; apply H5; clear H5;
+                [1: clear H7; apply sum_bases_lt_canc; assumption;
+                |2: clear H6; ]
+            |2: intros; cases (?:False); clear H6;
+                cases n1 in H5; intro;
+                [1: apply (q_lt_corefl ? H5);
+                |2: cases (bars l1) in H5; intro;
+                    [1: simplify in H5; 
+                        apply (q_lt_le_incompat ?? (q_lt_canc_plus_r ??? H5));
+                        apply q_le_plus_trans; [apply sum_bases_ge_OQ]
+                        apply q_le_OQ_Qpos;
+                    |2: simplify in H5:(??%);
+                        lapply (q_lt_canc_plus_r (sum_bases l (S n2)) ?? H5) as X;
+                        apply (q_lt_le_incompat ?? X); apply sum_bases_ge_OQ]]
+            |1: intro; cases n1 [intros; reflexivity] intros; cases (?:False);
+                elim n2 in H5 H6;
+            
+            
+             elim (bars l1) 0; 
+                [1: intro; elim n1; [reflexivity] cases (?:False);
+                 
+            
+                intros; clear H5;
+                elim n1 in H6; [reflexivity] cases (?:False);
+                [1: apply (q_lt_corefl ? H5);
+                |2: cases (bars l1) in H5; intro;
+                    [1: simplify in H5; 
+                        apply (q_lt_le_incompat ?? (q_lt_canc_plus_r ??? H5));
+                        apply q_le_plus_trans; [apply sum_bases_ge_OQ]
+                        apply q_le_OQ_Qpos;
+                    |2: simplify in H5:(??%);
+                        lapply (q_lt_canc_plus_r (sum_bases l (S n2)) ?? H5) as X;
+                        apply (q_lt_le_incompat ?? X); apply sum_bases_ge_OQ]]
+qed.
+
+            
         
 alias symbol "pi2" = "pair pi2".
 alias symbol "pi1" = "pair pi1".

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 f10113d501b9c640fa705f1e5db6d275f0a44cc4..82832c4bc64ee90cea02b82e867f4a321cc83328 100644 (file)
--- a/helm/software/matita/contribs/dama/dama/models/q_support.ma
+++ b/helm/software/matita/contribs/dama/dama/models/q_support.ma
@@ -51,7 +51,7 @@ axiom q_le_minus: ∀a,b,c:ℚ. a ≤ c - b → a + b ≤ c.
 axiom q_le_minus_r: ∀a,b,c:ℚ. a - b ≤ c → a ≤ c + b.
 axiom q_lt_plus: ∀a,b,c:ℚ. a - b < c → a < c + b.
 axiom q_lt_minus: ∀a,b,c:ℚ. a + b < c → a < c - b.
-
+axiom q_lt_opp_opp: ∀a,b.b < a → Qopp a < Qopp b.
 axiom q_lt_to_le: ∀a,b:ℚ.a < b → a ≤ b.
 axiom q_le_to_diff_ge_OQ : ∀a,b.a ≤ b → OQ ≤ b-a.
 axiom q_lt_corefl: ∀x:Q.x < x → False.
@@ -59,6 +59,8 @@ axiom q_lt_antisym: ∀x,y:Q.x < y → y < x → False.
 axiom q_lt_le_incompat: ∀x,y:Q.x < y → y ≤ x → False.
 axiom q_neg_gt: ∀r:ratio.OQ < Qneg r → False.
 axiom q_lt_trans: ∀x,y,z:Q. x < y → y < z → x < z.
+axiom q_lt_le_trans: ∀x,y,z:Q. x < y → y ≤ z → x < z.
+axiom q_le_lt_trans: ∀x,y,z:Q. x ≤ y → y < z → x < z.
 axiom q_le_trans: ∀x,y,z:Q. x ≤ y → y ≤ z → x ≤ z.
 axiom q_pos_OQ: ∀x.Qpos x ≤ OQ → False.
 axiom q_lt_plus_trans: ∀x,y:Q.OQ ≤ x → OQ < y → OQ < x + y.