(**************************************************************************)
-set "baseuri" "cic:/matita/nat/minus".
-
include "nat/le_arith.ma".
include "nat/compare.ma".
theorem eq_minus_S_pred: \forall n,m. n - (S m) = pred(n -m).
apply nat_elim2
[intro.reflexivity
- |intro.simplify.auto
+ |intro.simplify.autobatch
|intros.simplify.assumption
]
qed.
apply nat_elim2
[intros.apply False_ind.apply (not_le_Sn_O ? H)
|intros.rewrite < minus_n_O.
- auto
+ autobatch
|intros.
generalize in match H2.
apply (nat_case n1)
rewrite > eq_minus_S_pred.
apply lt_pred
[unfold lt.apply le_plus_to_minus_r.applyS H1
- |apply H[auto|assumption]
+ |apply H[autobatch|assumption]
]
]
qed.
apply H.apply H1.
qed.
+theorem lt_O_minus_to_lt: \forall a,b:nat.
+O \lt b-a \to a \lt b.
+intros.
+rewrite > (plus_n_O a).
+rewrite > (sym_plus a O).
+apply (lt_minus_to_plus O a b).
+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.