generalize no more required before elim

[helm.git] / helm / software / matita / library / nat / iteration2.ma

diff --git a/helm/software/matita/library/nat/iteration2.ma b/helm/software/matita/library/nat/iteration2.ma
index 6dbbfd572a2ac824de1d69bff781d06eb18009cc..3a4128e70d4727d81aad445bf59036a715b6511d 100644 (file)
--- a/helm/software/matita/library/nat/iteration2.ma
+++ b/helm/software/matita/library/nat/iteration2.ma
@@ -12,14 +12,11 @@
 (*                                                                        *)
 (**************************************************************************)
 
-set "baseuri" "cic:/matita/nat/iteration2".
-
 include "nat/primes.ma".
 include "nat/ord.ma".
 include "nat/generic_iter_p.ma".
 include "nat/count.ma".(*necessary just to use bool_to_nat and bool_to_nat_andb*)
 
-
 (* sigma_p on nautral numbers is a specialization of sigma_p_gen *)
 definition sigma_p: nat \to (nat \to bool) \to (nat \to nat) \to nat \def
 \lambda n, p, g. (iter_p_gen n p nat g O plus).
@@ -289,17 +286,16 @@ theorem le_sigma_p:
 (\forall i. i < n \to p i = true \to g1 i \le g2 i ) \to 
 sigma_p n p g1 \le sigma_p n p g2.
 intros.
-generalize in match H.
-elim n
+elim n in H ⊢ %
   [apply le_n.
   |apply (bool_elim ? (p n1));intros
     [rewrite > true_to_sigma_p_Sn
       [rewrite > true_to_sigma_p_Sn in ⊢ (? ? %)
         [apply le_plus
-          [apply H2[apply le_n|assumption]
-          |apply H1.
+          [apply H1[apply le_n|assumption]
+          |apply H.
            intros.
-           apply H2[apply le_S.assumption|assumption]
+           apply H1[apply le_S.assumption|assumption]
           ]
         |assumption
         ]
@@ -307,9 +303,9 @@ elim n
       ]
     |rewrite > false_to_sigma_p_Sn
       [rewrite > false_to_sigma_p_Sn in ⊢ (? ? %)
-        [apply H1.
+        [apply H.
          intros.
-         apply H2[apply le_S.assumption|assumption]
+         apply H1[apply le_S.assumption|assumption]
         |assumption
         ]
       |assumption
@@ -325,25 +321,24 @@ theorem le_sigma_p1:
 bool_to_nat (p1 i)*(g1 i) \le bool_to_nat (p2 i)*g2 i) \to 
 sigma_p n p1 g1 \le sigma_p n p2 g2.
 intros.
-generalize in match H.
-elim n
+elim n in H ⊢ %
   [apply le_n.
   |apply (bool_elim ? (p1 n1));intros
     [apply (bool_elim ? (p2 n1));intros
       [rewrite > true_to_sigma_p_Sn
         [rewrite > true_to_sigma_p_Sn in ⊢ (? ? %)
           [apply le_plus
-            [lapply (H2 n1) as H5
-              [rewrite > H3 in H5.
-               rewrite > H4 in H5.
+            [lapply (H1 n1) as H5
+              [rewrite > H2 in H5.
+               rewrite > H3 in H5.
                simplify in H5.
                rewrite < plus_n_O in H5.
                rewrite < plus_n_O in H5.
                assumption
               |apply le_S_S.apply le_n
               ]
-            |apply H1.intros.
-             apply H2.apply le_S.assumption
+            |apply H.intros.
+             apply H1.apply le_S.assumption
             ]
           |assumption
           ]
@@ -353,16 +348,16 @@ elim n
         [rewrite > false_to_sigma_p_Sn in ⊢ (? ? %)
           [change in ⊢ (? ? %) with (O + sigma_p n1 p2 g2).
            apply le_plus
-            [lapply (H2 n1) as H5
-              [rewrite > H3 in H5.
-               rewrite > H4 in H5.
+            [lapply (H1 n1) as H5
+              [rewrite > H2 in H5.
+               rewrite > H3 in H5.
                simplify in H5.
                rewrite < plus_n_O in H5.
                assumption
               |apply le_S_S.apply le_n
               ]
-            |apply H1.intros.
-             apply H2.apply le_S.assumption
+            |apply H.intros.
+             apply H1.apply le_S.assumption
             ]
           |assumption
           ]
@@ -374,16 +369,16 @@ elim n
         [rewrite > true_to_sigma_p_Sn in ⊢ (? ? %)
           [change in ⊢ (? % ?) with (O + sigma_p n1 p1 g1).
            apply le_plus
-            [lapply (H2 n1) as H5
-              [rewrite > H3 in H5.
-               rewrite > H4 in H5.
+            [lapply (H1 n1) as H5
+              [rewrite > H2 in H5.
+               rewrite > H3 in H5.
                simplify in H5.
                rewrite < plus_n_O in H5.
                assumption
               |apply le_S_S.apply le_n
               ]
-            |apply H1.intros.
-             apply H2.apply le_S.assumption
+            |apply H.intros.
+             apply H1.apply le_S.assumption
             ]
           |assumption
           ]
@@ -391,8 +386,8 @@ elim n
         ]
       |rewrite > false_to_sigma_p_Sn
         [rewrite > false_to_sigma_p_Sn in ⊢ (? ? %)
-          [apply H1.intros.
-           apply H2.apply le_S.assumption
+          [apply H.intros.
+           apply H1.apply le_S.assumption
           |assumption
           ]
         |assumption
@@ -400,7 +395,7 @@ elim n
       ]
     ]
   ]
-qed.    
+qed.
 
 theorem lt_sigma_p: 
 \forall n:nat. \forall p:nat \to bool. \forall g1,g2:nat \to nat.
@@ -629,32 +624,33 @@ elim n
 qed.
 
 theorem sigma_p_plus_1: \forall n:nat. \forall f,g:nat \to nat.
-sigma_p n (\lambda b:nat. true) (\lambda a:nat.(f a) + (g a)) = 
-sigma_p n (\lambda b:nat. true) f + sigma_p n (\lambda b:nat. true) g.
+\forall p.
+sigma_p n p (\lambda a:nat.(f a) + (g a)) = 
+sigma_p n p f + sigma_p n p g.
 intros.
 elim n
 [ simplify.
   reflexivity
-| rewrite > true_to_sigma_p_Sn
-  [ rewrite > (true_to_sigma_p_Sn n1 (\lambda c:nat.true) f)
-    [ rewrite > (true_to_sigma_p_Sn n1 (\lambda c:nat.true) g)
-      [ rewrite > assoc_plus in \vdash (? ? ? %).
-        rewrite < assoc_plus in \vdash (? ? ? (? ? %)).
-        rewrite < sym_plus in \vdash (? ? ? (? ? (? % ?))).
-        rewrite > assoc_plus in \vdash (? ? ? (? ? %)).
-        rewrite < assoc_plus in \vdash (? ? ? %).
-        apply eq_f.
-        assumption
-      | reflexivity
-      ]
-    | reflexivity
-    ]
-  | reflexivity
-  ]
-]
+| apply (bool_elim ? (p n1)); intro;
+  [ rewrite > true_to_sigma_p_Sn
+    [ rewrite > (true_to_sigma_p_Sn n1 p f)
+      [ rewrite > (true_to_sigma_p_Sn n1 p g)
+        [ rewrite > assoc_plus in \vdash (? ? ? %).
+          rewrite < assoc_plus in \vdash (? ? ? (? ? %)).
+          rewrite < sym_plus in \vdash (? ? ? (? ? (? % ?))).
+          rewrite > assoc_plus in \vdash (? ? ? (? ? %)).
+          rewrite < assoc_plus in \vdash (? ? ? %).
+          apply eq_f.
+          assumption]]]
+   assumption
+ | rewrite > false_to_sigma_p_Sn
+    [ rewrite > (false_to_sigma_p_Sn n1 p f)
+      [ rewrite > (false_to_sigma_p_Sn n1 p g)
+        [assumption]]]
+   assumption
+]]
 qed.
 
-
 theorem eq_sigma_p_sigma_p_times1 : \forall n,m:nat.\forall f:nat \to nat.
 sigma_p (n*m) (\lambda x:nat.true) f =
 sigma_p m (\lambda x:nat.true)