Every exception that used to have type string is now a string Lazy.t.

[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 3acce20afe21277adfab0b15c6d023022afd5da5..0c8780f7e68a50e0b453a5fe2dff09707704c214 100644 (file)
--- a/helm/matita/library/nat/minus.ma
+++ b/helm/matita/library/nat/minus.ma
@@ -68,6 +68,18 @@ intros.simplify.reflexivity.
 intros.simplify.apply H.apply le_S_S_to_le.assumption.
 qed.
 
+theorem minus_plus_m_m: \forall n,m:nat.n = (n+m)-m.
+intros 2.
+generalize in match n.
+elim m.
+rewrite < minus_n_O.apply plus_n_O.
+elim n2.simplify.
+apply minus_n_n.
+rewrite < plus_n_Sm.
+change with S n3 = (S n3 + n1)-n1.
+apply H.
+qed.
+
 theorem plus_minus_m_m: \forall n,m:nat.
 m \leq n \to n = (n-m)+m.
 intros 2.
@@ -82,20 +94,22 @@ qed.
 
 theorem minus_to_plus :\forall n,m,p:nat.m \leq n \to n-m = p \to 
 n = m+p.
-intros.apply trans_eq ? ? ((n-m)+m) ?.
+intros.apply trans_eq ? ? ((n-m)+m).
 apply plus_minus_m_m.
 apply H.elim H1.
 apply sym_plus.
 qed.
 
-theorem plus_to_minus :\forall n,m,p:nat.m \leq n \to
+theorem plus_to_minus :\forall n,m,p:nat.
 n = m+p \to n-m = p.
 intros.
 apply inj_plus_r m.
-rewrite < H1.
+rewrite < H.
 rewrite < sym_plus.
 symmetry.
-apply plus_minus_m_m.assumption.
+apply plus_minus_m_m.rewrite > H.
+rewrite > sym_plus.
+apply le_plus_n.
 qed.
 
 theorem minus_S_S : \forall n,m:nat.
@@ -118,8 +132,8 @@ intros 2.
 apply nat_elim2 (\lambda n,m.n \leq m \to n-m = O).
 intros.simplify.reflexivity.
 intros.apply False_ind.
-(* ancora problemi con il not *)
-apply not_le_Sn_O n1 H.
+apply not_le_Sn_O.
+goal 13.apply H.
 intros.
 simplify.apply H.apply le_S_S_to_le. apply H1.
 qed.
@@ -220,7 +234,7 @@ apply (leb_elim z y).
     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 ? ? ?.
+      rewrite < plus_minus_m_m.
         reflexivity.
         apply le_times_r.assumption.
   intro.rewrite > eq_minus_n_m_O.
@@ -235,9 +249,9 @@ 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.
+cut m+p \le n \or m+p \nleq n.
   elim Hcut.
-    symmetry.apply plus_to_minus.assumption.
+    symmetry.apply plus_to_minus.
     rewrite > assoc_plus.rewrite > sym_plus p.rewrite < plus_minus_m_m.
       rewrite > sym_plus.rewrite < plus_minus_m_m.
         reflexivity.
@@ -259,8 +273,6 @@ p+(n-m) = n-(m-p).
 intros.
 apply sym_eq.
 apply plus_to_minus.
-apply le_plus_to_minus.
-apply trans_le ? n.assumption.rewrite < sym_plus.apply le_plus_n.
 rewrite < assoc_plus.
 rewrite < plus_minus_m_m.
 rewrite < sym_plus.