generalize no more required before elim

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

diff --git a/helm/software/matita/library/nat/minimization.ma b/helm/software/matita/library/nat/minimization.ma
index d2cde9b67c56d1b961cfc8832fadbc0e67eaf43a..5b1552dc6777de3c1a9ef298b846b62796af70ff 100644 (file)
--- a/helm/software/matita/library/nat/minimization.ma
+++ b/helm/software/matita/library/nat/minimization.ma
@@ -236,17 +236,16 @@ intros 2.
 elim n.absurd (le m O).assumption.
 cut (O < m).apply (lt_O_n_elim m Hcut).exact not_le_Sn_O.
 rewrite < (max_O_f f).assumption.
-generalize in match H1.
-elim (max_S_max f n1).
-elim H3.
+elim (max_S_max f n1) in H1 ⊢ %.
+elim H1.
 absurd (m \le S n1).assumption.
-apply lt_to_not_le.rewrite < H6.assumption.
-elim H3.
+apply lt_to_not_le.rewrite < H5.assumption.
+elim H1.
 apply (le_n_Sm_elim m n1 H2).
 intro.
-apply H.rewrite < H6.assumption.
+apply H.rewrite < H5.assumption.
 apply le_S_S_to_le.assumption.
-intro.rewrite > H7.assumption.
+intro.rewrite > H6.assumption.
 qed.
 
 theorem f_false_to_le_max: \forall f,n,p. (∃i:nat.i≤n∧f i=true) \to
@@ -366,10 +365,26 @@ apply not_eq_true_false.
 reflexivity.
 qed.
 
+theorem f_min_true: \forall f:nat \to bool. \forall m:nat.
+(\exists i. le i m \land f i = true) \to
+f (min m f) = true.
+intros.unfold min.
+apply f_min_aux_true.
+elim H.clear H.elim H1.clear H1.
+apply (ex_intro ? ? a).
+split
+  [split
+    [apply le_O_n
+    |rewrite < plus_n_O.assumption
+    ]
+  |assumption
+  ]
+qed.
+
 theorem lt_min_aux_to_false : \forall f:nat \to bool. 
 \forall n,off,m:nat. n \leq m \to m < (min_aux off n f) \to f m = false.
 intros 3.
-generalize in match n; clear n.
+generalize in match n; clear n;
 elim off.absurd (le n1 m).assumption.
 apply lt_to_not_le.rewrite < (min_aux_O_f f n1).assumption.
 elim (le_to_or_lt_eq ? ? H1);
@@ -406,8 +421,7 @@ qed.
 lemma le_min_aux : \forall f:nat \to bool. 
 \forall n,off:nat. n \leq (min_aux off n f).
 intros 3.
-generalize in match n. clear n.
-elim off.
+elim off in n ⊢ %.
 rewrite > (min_aux_O_f f n1).apply le_n.
 elim (min_aux_S f n n1).
 elim H1.rewrite > H3.apply le_n.
@@ -421,8 +435,7 @@ qed.
 theorem le_min_aux_r : \forall f:nat \to bool. 
 \forall n,off:nat. (min_aux off n f) \le n+off.
 intros.
-generalize in match n. clear n.
-elim off.simplify.
+elim off in n ⊢ %.simplify.
 elim (f n1).simplify.rewrite < plus_n_O.apply le_n.
 simplify.rewrite < plus_n_O.apply le_n.
 simplify.elim (f n1).