mise sous CVS du repertoire theories/Arith

git-svn-id: svn+ssh://scm.gforge.inria.fr/svn/coq/trunk@311 85f007b7-540e-0410-9357-904b9bb8a0f7

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+
+(* $Id$ *)
+
+
+(**************************************************************************)
+(*          Subtraction (difference between two natural numbers           *)
+(**************************************************************************)
+
+
+Require Lt.
+Require Le.
+
+Fixpoint minus [n:nat] : nat -> nat := 
+  [m:nat]Cases n m of
+            O    _     =>  O
+         | (S k) O     => (S k)
+         | (S k) (S l) => (minus k l)
+        end. 
+
+Lemma minus_plus_simpl : 
+	(n,m,p:nat)((minus n m)=(minus (plus p n) (plus p m))).
+Proof.
+  Induction p; Simpl; Auto with arith.
+Qed.
+Hints Resolve minus_plus_simpl : arith v62.
+
+Lemma minus_n_O : (n:nat)(n=(minus n O)).
+Proof.
+Induction n; Simpl; Auto with arith.
+Qed.
+Hints Resolve minus_n_O : arith v62.
+
+Lemma minus_n_n : (n:nat)(O=(minus n n)).
+Proof.
+Induction n; Simpl; Auto with arith.
+Qed.
+Hints Resolve minus_n_n : arith v62.
+
+Lemma plus_minus : (n,m,p:nat)(n=(plus m p))->(p=(minus n m)).
+Proof.
+Intros n m p; Pattern m n; Apply nat_double_ind; Simpl; Intros.
+Replace (minus n0 O) with n0; Auto with arith.
+Absurd O=(S (plus n0 p)); Auto with arith.
+Auto with arith.
+Qed.
+Hints Immediate plus_minus : arith v62.
+
+Lemma minus_plus : (n,m:nat)(minus (plus n m) n)=m.
+Symmetry; Auto with arith.
+Save.
+Hints Resolve minus_plus : arith v62.
+
+Lemma le_plus_minus : (n,m:nat)(le n m)->(m=(plus n (minus m n))).
+Proof.
+Intros n m Le; Pattern n m; Apply le_elim_rel; Simpl; Auto with arith.
+Qed.
+Hints Resolve le_plus_minus : arith v62.
+
+Lemma le_plus_minus_r : (n,m:nat)(le n m)->(plus n (minus m n))=m.
+Proof.
+Symmetry; Auto with arith.
+Qed.
+Hints Resolve le_plus_minus_r : arith v62.
+
+
+Lemma minus_Sn_m : (n,m:nat)(le m n)->((S (minus n m))=(minus (S n) m)).
+Proof.
+Intros n m Le; Pattern m n; Apply le_elim_rel; Simpl; Auto with arith.
+Qed.
+Hints Resolve minus_Sn_m : arith v62.
+
+
+Lemma lt_minus : (n,m:nat)(le m n)->(lt O m)->(lt (minus n m) n).
+Proof.
+Intros n m Le; Pattern m n; Apply le_elim_rel; Simpl; Auto with arith.
+Intros; Absurd (lt O O); Auto with arith.
+Intros p q lepq Hp gtp.
+Elim (le_lt_or_eq O p); Auto with arith.
+Auto with arith.
+Induction 1; Elim minus_n_O; Auto with arith.
+Qed.
+Hints Resolve lt_minus : arith v62.
+
+Lemma lt_O_minus_lt : (n,m:nat)(lt O (minus n m))->(lt m n).
+Proof.
+Intros n m; Pattern n m; Apply nat_double_ind; Simpl; Auto with arith.
+Intros; Absurd (lt O O); Trivial with arith.
+Qed.
+Hints Immediate lt_O_minus_lt : arith v62.