Lost of redundant typing hints removed from the functional induction term.

[helm.git] / helm / matita / library / list / sort.ma

diff --git a/helm/matita/library/list/sort.ma b/helm/matita/library/list/sort.ma
index 1cc127759f0b37f8778491389718224a0215a85f..455462d8d67f823e3712a4cc587cee284da76964 100644 (file)
--- a/helm/matita/library/list/sort.ma
+++ b/helm/matita/library/list/sort.ma
@@ -56,7 +56,7 @@ let rec insertionsort (A:Set) (le: A → A → bool) (l: list A) on l ≝
       let l'' ≝ insertionsort A le l' in
        insert A le he l''
   ].
-  
+
 lemma ordered_injective:
   ∀ A:Set. ∀ le:A → A → bool.
   ∀ l:list A.
@@ -90,64 +90,62 @@ lemma ordered_injective:
   ].
 qed.
 
+
 lemma insert_sorted:
   \forall A:Set. \forall le:A\to A\to bool.
   (\forall a,b:A. le a b = false \to le b a = true) \to
   \forall l:list A. \forall x:A.
     ordered A le l = true \to ordered A le (insert A le x l) = true.
-  intros 5 (A le H l x).
-  elim l;
-  [ 2: simplify;
-       apply (bool_elim ? (le x s));
-       [ intros;
-         simplify;
-         fold simplify (ordered ? le (s::l1));
-         apply andb_elim;
-         rewrite > H3;
-         assumption;
-       | generalize in match H2;
-         clear H1; clear H2;
-         generalize in match s;
-         clear s;
-         elim l1
-         [ simplify;
-           rewrite > (H x a H2);
-           reflexivity;
-         | simplify in \vdash (? ? (? ? ? %) ?);
-            apply (bool_elim ? (le x s));
-            [ intros;
-              simplify;
-              fold simplify (ordered A le (s::l2));
-              apply andb_elim;
-              rewrite > (H x a H3);
-              simplify;
-              fold simplify (ordered A le (s::l2));
-              apply andb_elim;
-              rewrite > H4;
-              apply (ordered_injective A le (a::s::l2));
-              assumption;
-            | intros;
-              simplify;
-              fold simplify (ordered A le (s::(insert A le x l2)));
-              apply andb_elim;
-              simplify in H2;
-              fold simplify (ordered A le (s::l2)) in H2;
-              generalize in match H2;
-              apply (andb_elim (le a s));
-              elim (le a s);
-              [ change with (ordered A le (s::l2) = true \to ordered A le (s::insert A le x l2) = true);
-                intros;
-                apply (H1 s);
-                assumption;
-              | simplify; intros; assumption
-              ]
-           ]
-         ]
+ intros 5 (A le H l x).
+ apply (
+  let rec insert_ind (l: list A) \def
+  match l in list
+  return
+    λli.
+     l = li → ordered ? le li = true →
+      ordered ? le (insert A le x li) = true
+  with
+   [ nil ⇒ ?
+   | (cons he l') ⇒
+       match le x he
+       return
+        λb. le x he = b → l = he::l' →
+         ordered ? le (he::l') = true → ordered ? le
+          (match b with 
+            [ true ⇒ x::he::l'
+            | false ⇒ he::(insert A le x l') ]) = true
+       with
+        [ true ⇒ ?
+        | false ⇒ let res ≝ insert_ind l' in ?         
+        ]
+       (refl_eq ? (le x he))
+   ] (refl_eq ? l) in insert_ind l);
+  
+  intros; simplify;
+  [ rewrite > insert_ind;
+    [ generalize in match (H ? ? H1); clear H1; intro;
+      generalize in match H3; clear H3;
+      elim l'; simplify;
+      [ rewrite > H4;
+        reflexivity
+      | elim (le x s); simplify;
+        [ rewrite > H4;
+          reflexivity
+        | simplify in H3;
+          rewrite > (andb_true_true ? ? H3);
+          reflexivity
+        ]
       ]
-   | simplify; reflexivity;
-   ]
+      
+    | apply (ordered_injective ? ? ? H3)
+    ]
+  | rewrite > H1;
+    rewrite > H3;
+    reflexivity
+  | reflexivity
+  ].
 qed.
-
+  
 theorem insertionsort_sorted:
   ∀A:Set.
   ∀le:A → A → bool.∀eq:A → A → bool.
@@ -161,22 +159,4 @@ theorem insertionsort_sorted:
   | apply (insert_sorted A le le_tot (insertionsort A le l1) s);
     assumption;
   ]
-qed.
-
-(*
-theorem insertionsort_correct:
-  ∀A:Set.
-  ∀le:A → A → bool.∀eq:A → A → bool.
-  (∀a,b:A. le a b = false → le b a = true) \to
-  ∀l,l':list A.
-  l' = insertionsort A le l
-  \to ordered A le l' = true
-    ∧ ((\forall x:A. mem A eq x l = true \to mem A eq x l' = true)
-        ∧ (\forall x:A. mem A eq x l = true \to mem A eq x l' = true)).
-  intros;
-  split;
-  [ rewrite > H1; 
-    apply (insertionsort_sorted A le eq H);
-  | split;
-    [ TO BE CONTINUED ...
-*)
+qed.
\ No newline at end of file