added a function to reorder the metasenv.

[helm.git] / helm / matita / library / nat / minimization.ma

diff --git a/helm/matita/library/nat/minimization.ma b/helm/matita/library/nat/minimization.ma
index da76606b4233393971bd70580184a34740676700..748399fbcc699f20c308228bc8e600e64fc2a4d5 100644 (file)
--- a/helm/matita/library/nat/minimization.ma
+++ b/helm/matita/library/nat/minimization.ma
@@ -15,7 +15,6 @@
 set "baseuri" "cic:/matita/nat/minimization".
 
 include "nat/minus.ma".
-include "datatypes/bool.ma".
 
 let rec max i f \def
   match (f i) with 
@@ -40,6 +39,39 @@ simplify.left.split.reflexivity.reflexivity.
 simplify.right.split.reflexivity.reflexivity.
 qed.
 
+theorem le_max_n : \forall f: nat \to bool. \forall n:nat.
+max n f \le n.
+intros.elim n.rewrite > max_O_f.apply le_n.
+simplify.elim (f (S n1)).simplify.apply le_n.
+simplify.apply le_S.assumption.
+qed.
+
+theorem le_to_le_max : \forall f: nat \to bool. \forall n,m:nat.
+n\le m  \to max n f \le max m f.
+intros.elim H.
+apply le_n.
+apply trans_le ? (max n1 f).apply H2.
+cut (f (S n1) = true \land max (S n1) f = (S n1)) \lor 
+(f (S n1) = false \land max (S n1) f = max n1 f).
+elim Hcut.elim H3.
+rewrite > H5.
+apply le_S.apply le_max_n.
+elim H3.rewrite > H5.apply le_n.
+apply max_S_max.
+qed.
+
+theorem f_m_to_le_max: \forall f: nat \to bool. \forall n,m:nat.
+m\le n \to f m = true \to m \le max n f.
+intros 3.elim n.apply le_n_O_elim m H.
+apply le_O_n.
+apply le_n_Sm_elim m n1 H1.
+intro.apply trans_le ? (max n1 f).
+apply H.apply le_S_S_to_le.assumption.assumption.
+apply le_to_le_max.apply le_n_Sn.
+intro.simplify.rewrite < H3. 
+rewrite > H2.simplify.apply le_n.
+qed.
+
 definition max_spec \def \lambda f:nat \to bool.\lambda n: nat.
 ex nat (\lambda i:nat. (le i n) \land (f i = true)) \to
 (f n) = true \land (\forall i. i < n \to (f i = false)).
@@ -55,7 +87,6 @@ apply bool_ind (\lambda b:bool.
 (f (S n1) = b) \to (f ([\lambda b:bool.nat] match b in bool with
 [ true \Rightarrow (S n1)
 | false  \Rightarrow (max n1 f)])) = true) ? ? ?.
-reflexivity.
 simplify.intro.assumption.
 simplify.intro.apply H.
 elim H1.elim H3.generalize in match H5.
@@ -65,6 +96,7 @@ apply ex_intro nat ? a.
 split.apply le_S_S_to_le.assumption.assumption.
 intros.apply False_ind.apply not_eq_true_false ?.
 rewrite < H2.rewrite < H7.rewrite > H6. reflexivity.
+reflexivity.
 qed.
 
 theorem lt_max_to_false : \forall f:nat \to bool. 
@@ -132,7 +164,6 @@ apply bool_ind (\lambda b:bool.
 (f (m-(S n)) = b) \to (f ([\lambda b:bool.nat] match b in bool with
 [ true \Rightarrow m-(S n)
 | false  \Rightarrow (min_aux n m f)])) = true) ? ? ?.
-reflexivity.
 simplify.intro.assumption.
 simplify.intro.apply H.
 elim H1.elim H3.elim H4.
@@ -144,6 +175,7 @@ assumption.assumption.
 absurd f a = false.rewrite < H8.assumption.
 rewrite > H5.
 apply not_eq_true_false.
+reflexivity.
 qed.
 
 theorem lt_min_aux_to_false : \forall f:nat \to bool. 
@@ -159,8 +191,9 @@ apply lt_to_not_le.rewrite < H6.assumption.
 elim H3.
 elim le_to_or_lt_eq (n -(S n1)) m.
 apply H.apply lt_minus_S_n_to_le_minus_n.assumption.
-rewrite < H6.assumption.assumption.
+rewrite < H6.assumption.
 rewrite < H7.assumption.
+assumption.
 qed.
 
 theorem le_min_aux : \forall f:nat \to bool.