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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+
+(* $Id$ *)
+
+(**************************************************************************)
+(*                         Properties of addition                         *)
+(**************************************************************************)
+
+Require Le.
+Require Lt.
+
+Lemma plus_sym : (n,m:nat)((plus n m)=(plus m n)).
+Proof.
+Intros n m ; Elim n ; Simpl ; Auto with arith.
+Intros y H ; Elim (plus_n_Sm m y) ; Auto with arith.
+Qed.
+Hints Immediate plus_sym : arith v62.
+
+Lemma plus_Snm_nSm : 
+  (n,m:nat)(plus (S n) m)=(plus n (S m)).
+Intros.
+Simpl.
+Rewrite -> (plus_sym n m).
+Rewrite -> (plus_sym n (S m)).
+Trivial with arith.
+Qed.
+
+Lemma simpl_plus_l : (n,m,p:nat)((plus n m)=(plus n p))->(m=p).
+Proof.
+Induction n ; Simpl ; Auto with arith.
+Qed.
+
+Lemma plus_assoc_l : (n,m,p:nat)((plus n (plus m p))=(plus (plus n m) p)).
+Proof.
+Intros n m p; Elim n; Simpl; Auto with arith.
+Qed.
+Hints Resolve plus_assoc_l : arith v62.
+
+Lemma plus_permute : (n,m,p:nat) ((plus n (plus m p))=(plus m (plus n p))).
+Proof. 
+Intros; Rewrite (plus_assoc_l m n p); Rewrite (plus_sym m n); Auto with arith.
+Qed.
+
+Lemma plus_assoc_r : (n,m,p:nat)((plus (plus n m) p)=(plus n (plus m p))).
+Proof.
+Auto with arith.
+Qed.
+Hints Resolve plus_assoc_r : arith v62.
+
+Lemma simpl_le_plus_l : (p,n,m:nat)(le (plus p n) (plus p m))->(le n m).
+Proof.
+Induction p; Simpl; Auto with arith.
+Qed.
+
+Lemma le_reg_l : (n,m,p:nat)(le n m)->(le (plus p n) (plus p m)).
+Proof.
+Induction p; Simpl; Auto with arith.
+Qed.
+Hints Resolve le_reg_l : arith v62.
+
+Lemma le_reg_r : (a,b,c:nat) (le a b)->(le (plus a c) (plus b c)).
+Proof.
+Induction 1 ; Simpl; Auto with arith.
+Qed.
+Hints Resolve le_reg_r : arith v62.
+
+Lemma le_plus_plus : 
+	(n,m,p,q:nat) (le n m)->(le p q)->(le (plus n p) (plus m q)).
+Proof.
+Intros n m p q H H0.
+Elim H; Simpl; Auto with arith.
+Qed.
+
+Lemma le_plus_l : (n,m:nat)(le n (plus n m)).
+Proof.
+Induction n; Simpl; Auto with arith.
+Qed.
+Hints Resolve le_plus_l : arith v62.
+
+Lemma le_plus_r : (n,m:nat)(le m (plus n m)).
+Proof.
+Intros n m; Elim n; Simpl; Auto with arith.
+Qed.
+Hints Resolve le_plus_r : arith v62.
+
+Theorem le_plus_trans : (n,m,p:nat)(le n m)->(le n (plus m p)).
+Proof.
+Intros; Apply le_trans with m; Auto with arith.
+Qed.
+Hints Resolve le_plus_trans : arith v62.
+
+Lemma simpl_lt_plus_l : (n,m,p:nat)(lt (plus p n) (plus p m))->(lt n m).
+Proof.
+Induction p; Simpl; Auto with arith.
+Qed.
+
+Lemma lt_reg_l : (n,m,p:nat)(lt n m)->(lt (plus p n) (plus p m)).
+Proof.
+Induction p; Simpl; Auto with arith.
+Qed.
+Hints Resolve lt_reg_l : arith v62.
+
+Lemma lt_reg_r : (n,m,p:nat)(lt n m) -> (lt (plus n p) (plus m p)).
+Proof.
+Intros n m p H ; Rewrite (plus_sym n p) ; Rewrite (plus_sym m p).
+Elim p; Auto with arith.
+Qed.
+Hints Resolve lt_reg_r : arith v62.
+
+Theorem lt_plus_trans : (n,m,p:nat)(lt n m)->(lt n (plus m p)).
+Proof.
+Intros; Apply lt_le_trans with m; Auto with arith.
+Qed.
+Hints Immediate lt_plus_trans : arith v62.