added a function to reorder the metasenv.

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

diff --git a/helm/matita/library/nat/minus.ma b/helm/matita/library/nat/minus.ma
index 6ff744dcb5a3aa5e8cc626b20ae2740bda00b49c..3acce20afe21277adfab0b15c6d023022afd5da5 100644 (file)
--- a/helm/matita/library/nat/minus.ma
+++ b/helm/matita/library/nat/minus.ma
@@ -215,24 +215,19 @@ qed.
 theorem distributive_times_minus: distributive nat times minus.
 simplify.
 intros.
-apply (leb_elim z y).intro.
-cut x*(y-z)+x*z = (x*y-x*z)+x*z.
-apply inj_plus_l (x*z).
-assumption.
-apply trans_eq nat ? (x*y).
-rewrite < distr_times_plus. 
-rewrite < plus_minus_m_m ? ? H.reflexivity.
-rewrite < plus_minus_m_m ? ? ?.reflexivity.
-apply le_times_r.
-assumption.
-intro.
-rewrite > eq_minus_n_m_O.
-rewrite > eq_minus_n_m_O (x*y).
-rewrite < sym_times.simplify.reflexivity.
-apply lt_to_le.
-apply not_le_to_lt.assumption.
-apply le_times_r.apply lt_to_le.
-apply not_le_to_lt.assumption.
+apply (leb_elim z y).
+  intro.cut x*(y-z)+x*z = (x*y-x*z)+x*z.
+    apply inj_plus_l (x*z).assumption.
+    apply trans_eq nat ? (x*y).
+      rewrite < distr_times_plus.rewrite < plus_minus_m_m ? ? H.reflexivity.
+      rewrite < plus_minus_m_m ? ? ?.
+        reflexivity.
+        apply le_times_r.assumption.
+  intro.rewrite > eq_minus_n_m_O.
+    rewrite > eq_minus_n_m_O (x*y).
+      rewrite < sym_times.simplify.reflexivity.
+        apply le_times_r.apply lt_to_le.apply not_le_to_lt.assumption.
+        apply lt_to_le.apply not_le_to_lt.assumption.
 qed.
 
 theorem distr_times_minus: \forall n,m,p:nat. n*(m-p) = n*m-n*p
@@ -241,26 +236,22 @@ theorem distr_times_minus: \forall n,m,p:nat. n*(m-p) = n*m-n*p
 theorem eq_minus_minus_minus_plus: \forall n,m,p:nat. (n-m)-p = n-(m+p).
 intros.
 cut m+p \le n \or \not m+p \le n.
-elim Hcut.
-apply sym_eq.
-apply plus_to_minus.assumption.
-rewrite > assoc_plus.
-rewrite > sym_plus p.
-rewrite < plus_minus_m_m.
-rewrite > sym_plus.
-rewrite < plus_minus_m_m.reflexivity.
-rewrite > eq_minus_n_m_O n (m+p).
-rewrite > eq_minus_n_m_O (n-m) p.reflexivity.
-apply decidable_le (m+p) n.
-apply le_plus_to_minus_r.
-rewrite > sym_plus.assumption.
-apply trans_le ? (m+p).
-rewrite < sym_plus.
-apply le_plus_n.assumption.
-apply lt_to_le.apply not_le_to_lt.assumption.
-apply le_plus_to_minus.
-apply lt_to_le.apply not_le_to_lt.
-rewrite < sym_plus.assumption.
+  elim Hcut.
+    symmetry.apply plus_to_minus.assumption.
+    rewrite > assoc_plus.rewrite > sym_plus p.rewrite < plus_minus_m_m.
+      rewrite > sym_plus.rewrite < plus_minus_m_m.
+        reflexivity.
+        apply trans_le ? (m+p).
+          rewrite < sym_plus.apply le_plus_n.
+          assumption.
+      apply le_plus_to_minus_r.rewrite > sym_plus.assumption.   
+    rewrite > eq_minus_n_m_O n (m+p).
+      rewrite > eq_minus_n_m_O (n-m) p.
+        reflexivity.
+      apply le_plus_to_minus.apply lt_to_le. rewrite < sym_plus.
+       apply not_le_to_lt. assumption.
+    apply lt_to_le.apply not_le_to_lt.assumption.          
+  apply decidable_le (m+p) n.
 qed.
 
 theorem eq_plus_minus_minus_minus: \forall n,m,p:nat. p \le m \to m \le n \to