added override for \circ (notation for function composition)

[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 748399fbcc699f20c308228bc8e600e64fc2a4d5..72812cb651ecf447983be299aa9e94d00794af64 100644 (file)
--- a/helm/matita/library/nat/minimization.ma
+++ b/helm/matita/library/nat/minimization.ma
@@ -72,12 +72,13 @@ 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
+\exists i. (le i n) \land (f i = true) \to
 (f n) = true \land (\forall i. i < n \to (f i = false)).
 
 theorem f_max_true : \forall f:nat \to bool. \forall n:nat.
-ex nat (\lambda i:nat. (le i n) \land (f i = true)) \to f (max n f) = true. 
+(\exists i:nat. le i n \land f i = true) \to f (max n f) = true. 
 intros 2.
 elim n.elim H.elim H1.generalize in match H3.
 apply le_n_O_elim a H2.intro.simplify.rewrite > H4.
@@ -86,7 +87,7 @@ simplify.
 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) ? ? ?.
+| false  \Rightarrow (max n1 f)])) = true).
 simplify.intro.assumption.
 simplify.intro.apply H.
 elim H1.elim H3.generalize in match H5.
@@ -94,7 +95,7 @@ apply le_n_Sm_elim a n1 H4.
 intros.
 apply ex_intro nat ? a.
 split.apply le_S_S_to_le.assumption.assumption.
-intros.apply False_ind.apply not_eq_true_false ?.
+intros.apply False_ind.apply not_eq_true_false.
 rewrite < H2.rewrite < H7.rewrite > H6. reflexivity.
 reflexivity.
 qed.
@@ -106,7 +107,6 @@ 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.
-(* ?? non posso generalizzare su un goal implicativo ?? *)
 elim max_S_max f n1.
 elim H3.
 absurd m \le S n1.assumption.
@@ -152,7 +152,7 @@ simplify.right.split.reflexivity.reflexivity.
 qed.
 
 theorem f_min_aux_true: \forall f:nat \to bool. \forall off,m:nat.
-ex nat (\lambda i:nat. (le (m-off) i) \land (le i m) \land (f i = true)) \to
+(\exists i. le (m-off) i \land le i m \land f i = true) \to
 f (min_aux off m f) = true. 
 intros 2.
 elim off.elim H.elim H1.elim H2.
@@ -163,7 +163,7 @@ simplify.
 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) ? ? ?.
+| false  \Rightarrow (min_aux n m f)])) = true).
 simplify.intro.assumption.
 simplify.intro.apply H.
 elim H1.elim H3.elim H4.
@@ -204,7 +204,7 @@ rewrite > min_aux_O_f f n.apply le_n.
 elim min_aux_S f n1 n.
 elim H1.rewrite > H3.apply le_n.
 elim H1.rewrite > H3.
-apply trans_le (n-(S n1)) (n-n1) ?.
+apply trans_le (n-(S n1)) (n-n1).
 apply monotonic_le_minus_r.
 apply le_n_Sn.
 assumption.