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+Require Import NPlusOrder.
+Require Export ZPlusOrder.
+Require Export ZPairsPlus.
+
+Module NatPairsOrder (Import NPlusMod : NPlusSig)
+                     (Import NOrderModule : NOrderSig
+                        with Module NBaseMod := NPlusMod.NBaseMod) <: ZOrderSignature.
+Module Import NPlusOrderPropertiesModule :=
+  NPlusOrderProperties NPlusMod NOrderModule.
+Module Export IntModule := NatPairsInt NPlusMod.
+Open Local Scope NatScope.
+
+Definition lt (p1 p2 : Z) := (fst p1) + (snd p2) < (fst p2) + (snd p1).
+Definition le (p1 p2 : Z) := (fst p1) + (snd p2) <= (fst p2) + (snd p1).
+
+Notation "x < y" := (lt x y) : IntScope.
+Notation "x <= y" := (le x y) : IntScope.
+
+Add Morphism lt with signature E ==> E ==> eq_bool as lt_wd.
+Proof.
+unfold lt, E; intros x1 y1 H1 x2 y2 H2; simpl.
+rewrite eq_true_iff; split; intro H.
+stepr (snd y1 + fst y2) by apply plus_comm.
+apply (plus_lt_repl_pair (fst x1) (snd x1)); [| assumption].
+stepl (snd y2 + fst x1) by apply plus_comm.
+stepr (fst y2 + snd x1) by apply plus_comm.
+apply (plus_lt_repl_pair (snd x2) (fst x2)).
+now stepl (fst x1 + snd x2) by apply plus_comm.
+stepl (fst y2 + snd x2) by apply plus_comm. now stepr (fst x2 + snd y2) by apply plus_comm.
+stepr (snd x1 + fst x2) by apply plus_comm.
+apply (plus_lt_repl_pair (fst y1) (snd y1)); [| now symmetry].
+stepl (snd x2 + fst y1) by apply plus_comm.
+stepr (fst x2 + snd y1) by apply plus_comm.
+apply (plus_lt_repl_pair (snd y2) (fst y2)).
+now stepl (fst y1 + snd y2) by apply plus_comm.
+stepl (fst x2 + snd y2) by apply plus_comm. now stepr (fst y2 + snd x2) by apply plus_comm.
+Qed.
+
+(* Below is a very long explanation why it would be useful to be
+able to use the fold tactic in hypotheses.
+We will prove the following statement not from scratch, like lt_wd,
+but expanding <= to < and == and then using lt_wd. The theorem we need
+to prove is (x1 <= x2) = (y1 <= y2) for all x1 == y1 and x2 == y2 : Z.
+To be able to express <= through < and ==, we need to expand <=%Int to
+<=%Nat, since we have not proved yet the properties of <=%Int. But
+then it would be convenient to fold back equalities from
+(fst x1 + snd x2 == fst x2 + snd x1)%Nat to (x1 == x2)%Int.
+The reason is that we will need to show that (x1 == x2)%Int <->
+(y1 == y2)%Int from (x1 == x2)%Int and (y1 == y2)%Int. If we fold
+equalities back to Int, then we could do simple rewriting, since we have
+already showed that ==%Int is an equivalence relation. On the other hand,
+if we leave equalities expanded to Nat, we will have to apply the
+transitivity of ==%Int by hand. *)
+
+Add Morphism le with signature E ==> E ==> eq_bool as le_wd.
+Proof.
+unfold le, E; intros x1 y1 H1 x2 y2 H2; simpl.
+rewrite eq_true_iff. do 2 rewrite le_lt.
+pose proof (lt_wd x1 y1 H1 x2 y2 H2) as H; unfold lt in H; rewrite H; clear H.
+(* This is a remark about an extra level of definitions created by
+"with Module NBaseMod := NPlusMod.NBaseMod" constraint in the beginning
+of this functor. We cannot just say "fold (x1 == x2)%Int" because it turns out
+that it expand to (NPlusMod.NBaseMod.NDomainModule.E ... ...), since
+NPlusMod was imported first. On the other hand, the goal uses
+NOrderModule.NBaseMod.NDomainModule.E, or just NDomainModule.E, since le_lt
+theorem was proved in NOrderDomain module. (E without qualifiers refers to
+ZDomainModule.E.) Therefore, we issue the "replace" command. It would be nicer,
+though, if the constraint "with Module NBaseMod := NPlusMod.NBaseMod" in the
+declaration of this functor would not create an extra level of definitions
+and there would be only one NDomainModule.E. *)
+replace NDomainModule.E with NPlusMod.NBaseMod.NDomainModule.E by reflexivity.
+fold (x1 == x2)%Int. fold (y1 == y2)%Int.
+assert (H1' : (x1 == y1)%Int); [exact H1 |].
+(* We do this instead of "fold (x1 == y1)%Int in H1" *)
+assert (H2' : (x2 == y2)%Int); [exact H2 |].
+rewrite H1'; rewrite H2'. reflexivity.
+Qed.
+
+Open Local Scope IntScope.
+
+Theorem le_lt : forall n m : Z, n <= m <-> n < m \/ n == m.
+Proof.
+intros n m; unfold lt, le, E; simpl. apply le_lt. (* refers to NOrderModule.le_lt *)
+Qed.
+
+Theorem lt_irr : forall n : Z, ~ (n < n).
+Proof.
+intros n; unfold lt, E; simpl. apply lt_irr.
+(* refers to NPlusOrderPropertiesModule.NOrderPropFunctModule.lt_irr *)
+Qed.
+
+Theorem lt_S : forall n m, n < (S m) <-> n <= m.
+Proof.
+intros n m; unfold lt, le, E; simpl. rewrite plus_S_l; apply lt_S.
+Qed.
+
+End NatPairsOrder.
+
+(* Since to define the order on integers we need both plus and order
+on natural numbers, we can export the properties of plus and order together *)
+Module NatPairsPlusOrderProperties (NPlusMod : NPlusSig)
+                                   (NOrderModule : NOrderSig
+                                     with Module NBaseMod := NPlusMod.NBaseMod).
+Module Export NatPairsPlusModule := NatPairsPlus NPlusMod.
+Module Export NatPairsOrderModule := NatPairsOrder NPlusMod NOrderModule.
+Module Export NatPairsPlusOrderPropertiesModule :=
+  ZPlusOrderProperties NatPairsPlusModule NatPairsOrderModule.
+End NatPairsPlusOrderProperties.
+
+(*
+ Local Variables:
+ tags-file-name: "~/coq/trunk/theories/Numbers/TAGS"
+ End:
+*)