(**************************************************************************)
-set "baseuri" "cic:/matita/nat/minus".
-
include "nat/le_arith.ma".
include "nat/compare.ma".
theorem minus_plus_m_m: \forall n,m:nat.n = (n+m)-m.
intros 2.
-generalize in match n.
-elim m.
+elim m in n ⊢ %.
rewrite < minus_n_O.apply plus_n_O.
-elim n2.simplify.
+elim n1.simplify.
apply minus_n_n.
rewrite < plus_n_Sm.
-change with (S n3 = (S n3 + n1)-n1).
+change with (S n2 = (S n2 + n)-n).
apply H.
qed.
assumption.
qed.
+theorem lt_minus_to_lt_plus:
+\forall n,m,p. n - m < p \to n < m + p.
+intros 2.
+apply (nat_elim2 ? ? ? ? n m)
+ [simplify.intros.autobatch.
+ |intros 2.rewrite < minus_n_O.
+ intro.assumption
+ |intros.
+ simplify.
+ cut (n1 < m1+p)
+ [autobatch
+ |apply H.
+ apply H1
+ ]
+ ]
+qed.
+
+theorem lt_plus_to_lt_minus:
+\forall n,m,p. m \le n \to n < m + p \to n - m < p.
+intros 2.
+apply (nat_elim2 ? ? ? ? n m)
+ [simplify.intros 3.
+ apply (le_n_O_elim ? H).
+ simplify.intros.assumption
+ |simplify.intros.assumption.
+ |intros.
+ simplify.
+ apply H
+ [apply le_S_S_to_le.assumption
+ |apply le_S_S_to_le.apply H2
+ ]
+ ]
+qed.
+
+theorem minus_m_minus_mn: \forall n,m. n\le m \to n=m-(m-n).
+intros.
+apply sym_eq.
+apply plus_to_minus.
+autobatch.
+qed.
+
theorem distributive_times_minus: distributive nat times minus.
unfold distributive.
intros.