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-(************************************************************************)
-(*  v      *   The Coq Proof Assistant  /  The Coq Development Team     *)
-(* <O___,, * CNRS-Ecole Polytechnique-INRIA Futurs-Universite Paris Sud *)
-(*   \VV/  **************************************************************)
-(*    //   *      This file is distributed under the terms of the       *)
-(*         *       GNU Lesser General Public License Version 2.1        *)
-(************************************************************************)
-
-(*i $Id: Minus.v,v 1.1.2.1 2004/07/16 19:31:24 herbelin Exp $ i*)
-
-(** Subtraction (difference between two natural numbers) *)
-
-Require Lt.
-Require Le.
-
-V7only [Import nat_scope.].
-Open Local Scope nat_scope.
-
-Implicit Variables Type m,n,p:nat.
-
-(** 0 is right neutral *)
-
-Lemma minus_n_O : (n:nat)(n=(minus n O)).
-Proof.
-NewInduction n; Simpl; Auto with arith.
-Qed.
-Hints Resolve minus_n_O : arith v62.
-
-(** Permutation with successor *)
-
-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.
-
-Theorem pred_of_minus : (x:nat)(pred x)=(minus x (S O)).
-Intro x; NewInduction x; Simpl; Auto with arith.
-Qed.
-
-(** Diagonal *)
-
-Lemma minus_n_n : (n:nat)(O=(minus n n)).
-Proof.
-NewInduction n; Simpl; Auto with arith.
-Qed.
-Hints Resolve minus_n_n : arith v62.
-
-(** Simplification *)
-
-Lemma minus_plus_simpl : 
-	(n,m,p:nat)((minus n m)=(minus (plus p n) (plus p m))).
-Proof.
-  NewInduction p; Simpl; Auto with arith.
-Qed.
-Hints Resolve minus_plus_simpl : arith v62.
-
-(** Relation with plus *)
-
-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.
-Qed.
-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.
-
-(** Relation with order *)
-
-Theorem le_minus: (i,h:nat) (le (minus i h) i).
-Proof.
-Intros i h;Pattern i h; Apply nat_double_ind; [
-  Auto
-| Auto
-| Intros m n H; Simpl; Apply le_trans with m:=m; Auto ].
-Qed.
-
-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.
-NewInduction 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.
-
-Theorem inj_minus_aux: (x,y:nat) ~(le y x) -> (minus x y) = O.
-Intros y x; Pattern y x ; Apply nat_double_ind; [
-  Simpl; Trivial with arith
-| Intros n H; Absurd (le O (S n)); [ Assumption | Apply le_O_n]
-| Simpl; Intros n m H1 H2; Apply H1;
-  Unfold not ; Intros H3; Apply H2; Apply le_n_S; Assumption].
-Qed.