ocaml 3.09 transition

[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 4910f3003541fb0a94f99b1072012ee0ea21d6d0..710418d72644022cb4d799fb95f67354da4af89b 100644 (file)
--- a/helm/matita/library/nat/minus.ma
+++ b/helm/matita/library/nat/minus.ma
@@ -170,7 +170,7 @@ qed.
 theorem lt_minus_m: \forall n,m:nat. O < n \to O < m \to n-m \lt n.
 intros.apply (lt_O_n_elim n H).intro.
 apply (lt_O_n_elim m H1).intro.
-simplify.apply le_S_S.apply le_minus_m.
+simplify.unfold lt.apply le_S_S.apply le_minus_m.
 qed.
 
 theorem minus_le_O_to_le: \forall n,m:nat. n-m \leq O \to n \leq m.
@@ -225,9 +225,19 @@ apply H.apply le_S_S_to_le.
 rewrite > plus_n_Sm.assumption.
 qed.
 
+(* minus and lt - to be completed *)
+theorem lt_minus_to_plus: \forall n,m,p. (lt n (p-m)) \to (lt (n+m) p).
+intros 3.apply (nat_elim2 (\lambda m,p.(lt n (p-m)) \to (lt (n+m) p))).
+intro.rewrite < plus_n_O.rewrite < minus_n_O.intro.assumption.
+simplify.intros.apply False_ind.apply (not_le_Sn_O n H).
+simplify.intros.unfold lt.
+apply le_S_S.
+rewrite < plus_n_Sm.
+apply H.apply H1.
+qed.
 
 theorem distributive_times_minus: distributive nat times minus.
-simplify.
+unfold distributive.
 intros.
 apply ((leb_elim z y)).
   intro.cut (x*(y-z)+x*z = (x*y-x*z)+x*z).
@@ -247,6 +257,15 @@ qed.
 theorem distr_times_minus: \forall n,m,p:nat. n*(m-p) = n*m-n*p
 \def distributive_times_minus.
 
+theorem eq_minus_plus_plus_minus: \forall n,m,p:nat. p \le m \to (n+m)-p = n+(m-p).
+intros.
+apply plus_to_minus.
+rewrite > sym_plus in \vdash (? ? ? %).
+rewrite > assoc_plus.
+rewrite < plus_minus_m_m.
+reflexivity.assumption.
+qed.
+
 theorem eq_minus_minus_minus_plus: \forall n,m,p:nat. (n-m)-p = n-(m+p).
 intros.
 cut (m+p \le n \or m+p \nleq n).