NPeano improved, subsumes NatOrderedType

git-svn-id: svn+ssh://scm.gforge.inria.fr/svn/coq/trunk@12717 85f007b7-540e-0410-9357-904b9bb8a0f7

7 files changed, 141 insertions, 215 deletions

diff --git a/theories/Arith/Max.v b/theories/Arith/Max.v
index 3d7fe9fc2d..e49251a717 100644
--- a/theories/Arith/Max.v
+++ b/theories/Arith/Max.v
@@ -8,34 +8,35 @@
 
 (*i $Id$ i*)
 
-(** THIS FILE IS DEPRECATED. Use [MinMax] instead. *)
+(** THIS FILE IS DEPRECATED. Use [NPeano] and [MinMax] instead. *)
 
+Require Import NPeano.
 Require Export MinMax.
 
 Local Open Scope nat_scope.
 Implicit Types m n p : nat.
 
-Notation max := MinMax.max (only parsing).
+Notation max := NPeano.max (only parsing).
 
 Definition max_0_l := max_0_l.
 Definition max_0_r := max_0_r.
 Definition succ_max_distr := succ_max_distr.
 Definition plus_max_distr_l := plus_max_distr_l.
 Definition plus_max_distr_r := plus_max_distr_r.
-Definition max_case_strong := max_case_strong.
-Definition max_spec := max_spec.
-Definition max_dec := max_dec.
-Definition max_case := max_case.
-Definition max_idempotent := max_id.
-Definition max_assoc := max_assoc.
-Definition max_comm := max_comm.
-Definition max_l := max_l.
-Definition max_r := max_r.
-Definition le_max_l := le_max_l.
-Definition le_max_r := le_max_r.
-Definition max_lub_l := max_lub_l.
-Definition max_lub_r := max_lub_r.
-Definition max_lub := max_lub.
+Definition max_case_strong := Nat.max_case_strong.
+Definition max_spec := Nat.max_spec.
+Definition max_dec := Nat.max_dec.
+Definition max_case := Nat.max_case.
+Definition max_idempotent := Nat.max_id.
+Definition max_assoc := Nat.max_assoc.
+Definition max_comm := Nat.max_comm.
+Definition max_l := Nat.max_l.
+Definition max_r := Nat.max_r.
+Definition le_max_l := Nat.le_max_l.
+Definition le_max_r := Nat.le_max_r.
+Definition max_lub_l := Nat.max_lub_l.
+Definition max_lub_r := Nat.max_lub_r.
+Definition max_lub := Nat.max_lub.
 
 (* begin hide *)
 (* Compatibility *)
diff --git a/theories/Arith/Min.v b/theories/Arith/Min.v
index c52fc0dd08..44060b56c1 100644
--- a/theories/Arith/Min.v
+++ b/theories/Arith/Min.v
@@ -8,34 +8,35 @@
 
 (*i $Id$ i*)
 
-(** THIS FILE IS DEPRECATED. Use [MinMax] instead. *)
+(** THIS FILE IS DEPRECATED. Use [NPeano] and [MinMax] instead. *)
 
+Require Import NPeano.
 Require Export MinMax.
 
 Open Local Scope nat_scope.
 Implicit Types m n p : nat.
 
-Notation min := MinMax.min (only parsing).
+Notation min := NPeano.min (only parsing).
 
 Definition min_0_l := min_0_l.
 Definition min_0_r := min_0_r.
 Definition succ_min_distr := succ_min_distr.
 Definition plus_min_distr_l := plus_min_distr_l.
 Definition plus_min_distr_r := plus_min_distr_r.
-Definition min_case_strong := min_case_strong.
-Definition min_spec := min_spec.
-Definition min_dec := min_dec.
-Definition min_case := min_case.
-Definition min_idempotent := min_id.
-Definition min_assoc := min_assoc.
-Definition min_comm := min_comm.
-Definition min_l := min_l.
-Definition min_r := min_r.
-Definition le_min_l := le_min_l.
-Definition le_min_r := le_min_r.
-Definition min_glb_l := min_glb_l.
-Definition min_glb_r := min_glb_r.
-Definition min_glb := min_glb.
+Definition min_case_strong := Nat.min_case_strong.
+Definition min_spec := Nat.min_spec.
+Definition min_dec := Nat.min_dec.
+Definition min_case := Nat.min_case.
+Definition min_idempotent := Nat.min_id.
+Definition min_assoc := Nat.min_assoc.
+Definition min_comm := Nat.min_comm.
+Definition min_l := Nat.min_l.
+Definition min_r := Nat.min_r.
+Definition le_min_l := Nat.le_min_l.
+Definition le_min_r := Nat.le_min_r.
+Definition min_glb_l := Nat.min_glb_l.
+Definition min_glb_r := Nat.min_glb_r.
+Definition min_glb := Nat.min_glb.
 
 (* begin hide *)
 (* Compatibility *)
diff --git a/theories/Arith/MinMax.v b/theories/Arith/MinMax.v
index 6e86a88c59..9ce43ab8ce 100644
--- a/theories/Arith/MinMax.v
+++ b/theories/Arith/MinMax.v
@@ -6,61 +6,10 @@
 (*         *       GNU Lesser General Public License Version 2.1        *)
 (************************************************************************)
 
-Require Import Orders NatOrderedType GenericMinMax.
-
-(** * Maximum and Minimum of two natural numbers *)
-
-Fixpoint max n m : nat :=
-  match n, m with
-    | O, _ => m
-    | S n', O => n
-    | S n', S m' => S (max n' m')
-  end.
-
-Fixpoint min n m : nat :=
-  match n, m with
-    | O, _ => 0
-    | S n', O => 0
-    | S n', S m' => S (min n' m')
-  end.
-
-(** These functions implement indeed a maximum and a minimum *)
-
-Lemma max_l : forall x y, y<=x -> max x y = x.
-Proof.
- induction x; destruct y; simpl; auto with arith.
-Qed.
-
-Lemma max_r : forall x y, x<=y -> max x y = y.
-Proof.
- induction x; destruct y; simpl; auto with arith.
-Qed.
-
-Lemma min_l : forall x y, x<=y -> min x y = x.
-Proof.
- induction x; destruct y; simpl; auto with arith.
-Qed.
-
-Lemma min_r : forall x y, y<=x -> min x y = y.
-Proof.
- induction x; destruct y; simpl; auto with arith.
-Qed.
-
-
-Module NatHasMinMax <: HasMinMax Nat_as_OT.
- Definition max := max.
- Definition min := min.
- Definition max_l := max_l.
- Definition max_r := max_r.
- Definition min_l := min_l.
- Definition min_r := min_r.
-End NatHasMinMax.
-
-(** We obtain hence all the generic properties of [max] and [min],
-    see file [GenericMinMax] or use SearchAbout. *)
-
-Module Export MMP := UsualMinMaxProperties Nat_as_OT NatHasMinMax.
+Require Import Orders NPeano.
 
+(** The Definition of maximum and minimum of two natural numbers
+    is now in NPeano, as well as generic properties. *)
 
 (** * Properties specific to the [nat] domain *)
 
@@ -88,26 +37,26 @@ Proof. auto. Qed.
 
 Lemma plus_max_distr_l : forall n m p, max (p + n) (p + m) = p + max n m.
 Proof.
-intros. apply max_monotone. repeat red; auto with arith.
+intros. apply Nat.max_monotone. repeat red; auto with arith.
 Qed.
 
 Lemma plus_max_distr_r : forall n m p, max (n + p) (m + p) = max n m + p.
 Proof.
-intros. apply max_monotone with (f:=fun x => x + p).
+intros. apply Nat.max_monotone with (f:=fun x => x + p).
 repeat red; auto with arith.
 Qed.
 
 Lemma plus_min_distr_l : forall n m p, min (p + n) (p + m) = p + min n m.
 Proof.
-intros. apply min_monotone. repeat red; auto with arith.
+intros. apply Nat.min_monotone. repeat red; auto with arith.
 Qed.
 
 Lemma plus_min_distr_r : forall n m p, min (n + p) (m + p) = min n m + p.
 Proof.
-intros. apply min_monotone with (f:=fun x => x + p).
+intros. apply Nat.min_monotone with (f:=fun x => x + p).
 repeat red; auto with arith.
 Qed.
 
 Hint Resolve
- max_l max_r le_max_l le_max_r
- min_l min_r le_min_l le_min_r : arith v62.
+ Nat.max_l Nat.max_r Nat.le_max_l Nat.le_max_r
+ Nat.min_l Nat.min_r Nat.le_min_l Nat.le_min_r : arith v62.
diff --git a/theories/Arith/NatOrderedType.v b/theories/Arith/NatOrderedType.v
deleted file mode 100644
index df5b37e019..0000000000
--- a/theories/Arith/NatOrderedType.v
+++ /dev/null
@@ -1,64 +0,0 @@
-(************************************************************************)
-(*  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        *)
-(************************************************************************)
-
-Require Import Lt Peano_dec Compare_dec EqNat
- Equalities Orders OrdersTac.
-
-
-(** * DecidableType structure for Peano numbers *)
-
-Module Nat_as_UBE <: UsualBoolEq.
- Definition t := nat.
- Definition eq := @eq nat.
- Definition eqb := beq_nat.
- Definition eqb_eq := beq_nat_true_iff.
-End Nat_as_UBE.
-
-Module Nat_as_DT <: UsualDecidableTypeFull := Make_UDTF Nat_as_UBE.
-
-(** Note that the last module fulfills by subtyping many other
-    interfaces, such as [DecidableType] or [EqualityType]. *)
-
-
-
-(** * OrderedType structure for Peano numbers *)
-
-Module Nat_as_OT <: OrderedTypeFull.
- Include Nat_as_DT.
- Definition lt := lt.
- Definition le := le.
- Definition compare := nat_compare.
-
- Instance lt_strorder : StrictOrder lt.
- Proof. split; [ exact lt_irrefl | exact lt_trans ]. Qed.
-
- Instance lt_compat : Proper (Logic.eq==>Logic.eq==>iff) lt.
- Proof. repeat red; intros; subst; auto. Qed.
-
- Definition le_lteq := le_lt_or_eq_iff.
- Definition compare_spec := nat_compare_spec.
-
-End Nat_as_OT.
-
-(** Note that [Nat_as_OT] can also be seen as a [UsualOrderedType]
-   and a [OrderedType] (and also as a [DecidableType]). *)
-
-
-
-(** * An [order] tactic for Peano numbers *)
-
-Module NatOrder := OTF_to_OrderTac Nat_as_OT.
-Ltac nat_order := NatOrder.order.
-
-(** Note that [nat_order] is domain-agnostic: it will not prove
-    [1<=2] or [x<=x+x], but rather things like [x<=y -> y<=x -> x=y]. *)
-
-Section Test.
-Let test : forall x y : nat, x<=y -> y<=x -> x=y.
-Proof. nat_order. Qed.
-End Test.
diff --git a/theories/Arith/vo.itarget b/theories/Arith/vo.itarget
index c3f29d2143..d8bff0ab91 100644
--- a/theories/Arith/vo.itarget
+++ b/theories/Arith/vo.itarget
@@ -19,5 +19,4 @@ Mult.vo
 Peano_dec.vo
 Plus.vo
 Wf_nat.vo
-NatOrderedType.vo
 MinMax.vo
diff --git a/theories/Numbers/Natural/Peano/NPeano.v b/theories/Numbers/Natural/Peano/NPeano.v
index d42bb810d2..e5c6824193 100644
--- a/theories/Numbers/Natural/Peano/NPeano.v
+++ b/theories/Numbers/Natural/Peano/NPeano.v
@@ -10,11 +10,44 @@
 
 (*i $Id$ i*)
 
-Require Import Arith MinMax NAxioms NProperties.
+Require Import Peano Peano_dec Compare_dec EqNat NAxioms NProperties NDiv.
+
+(** Some functions not already encountered: min max div mod *)
+
+Fixpoint max n m : nat :=
+  match n, m with
+    | O, _ => m
+    | S n', O => n
+    | S n', S m' => S (max n' m')
+  end.
+
+Fixpoint min n m : nat :=
+  match n, m with
+    | O, _ => 0
+    | S n', O => 0
+    | S n', S m' => S (min n' m')
+  end.
+
+Definition divF div x y := if leb y x then S (div (x-y) y) else 0.
+Definition modF mod x y := if leb y x then mod (x-y) y else x.
+Definition initF (_ _ : nat) := 0.
+
+Fixpoint loop {A} (F:A->A)(i:A) (n:nat) : A :=
+ match n with
+  | 0 => i
+  | S n => F (loop F i n)
+ end.
+
+Definition div x y := loop divF initF x x y.
+Definition modulo x y := loop modF initF x x y.
+Infix "/" := div : nat_scope.
+Infix "mod" := modulo (at level 40, no associativity) : nat_scope.
+
 
 (** * Implementation of [NAxiomsSig] by [nat] *)
 
-Module NPeanoAxiomsMod <: NAxiomsSig.
+Module Nat
+ <: NAxiomsSig <: UsualDecidableTypeFull <: OrderedTypeFull <: TotalOrder.
 
 (** Bi-directional induction. *)
 
@@ -57,7 +90,7 @@ Qed.
 
 Theorem sub_succ_r : forall n m : nat, n - (S m) = pred (n - m).
 Proof.
-intros n m; induction n m using nat_double_ind; simpl; auto. apply sub_0_r.
+induction n; destruct m; simpl; auto. apply sub_0_r.
 Qed.
 
 Theorem mul_0_l : forall n : nat, 0 * n = 0.
@@ -67,49 +100,61 @@ Qed.
 
 Theorem mul_succ_l : forall n m : nat, S n * m = n * m + m.
 Proof.
-intros n m; now rewrite plus_comm.
+assert (add_S_r : forall n m, n+S m = S(n+m)) by (induction n; auto).
+assert (add_comm : forall n m, n+m = m+n).
+ induction n; simpl; auto. intros; rewrite add_S_r; auto.
+intros n m; now rewrite add_comm.
 Qed.
 
 (** Order on natural numbers *)
 
 Program Instance lt_wd : Proper (eq==>eq==>iff) lt.
 
-Theorem lt_eq_cases : forall n m : nat, n <= m <-> n < m \/ n = m.
+Theorem lt_succ_r : forall n m : nat, n < S m <-> n <= m.
 Proof.
-intros n m; split.
-apply le_lt_or_eq.
-intro H; destruct H as [H | H].
-now apply lt_le_weak. rewrite H; apply le_refl.
+unfold lt.
+split; [|induction 1; auto].
+assert (le_pred : forall n m, n <= m -> pred n <= pred m).
+ induction 1 as [|m' H IH]; auto.
+ destruct m'. inversion H; subst; auto. simpl; auto.
+exact (le_pred (S n) (S m)).
 Qed.
 
-Theorem lt_irrefl : forall n : nat, ~ (n < n).
+
+Theorem lt_eq_cases : forall n m : nat, n <= m <-> n < m \/ n = m.
 Proof.
-exact lt_irrefl.
+split.
+inversion 1; auto. rewrite lt_succ_r; auto.
+destruct 1; [|subst; auto]. rewrite <- lt_succ_r; auto.
 Qed.
 
-Theorem lt_succ_r : forall n m : nat, n < S m <-> n <= m.
+Theorem lt_irrefl : forall n : nat, ~ (n < n).
 Proof.
-intros n m; split; [apply lt_n_Sm_le | apply le_lt_n_Sm].
+induction n. intro H; inversion H. rewrite lt_succ_r; auto.
 Qed.
 
 Theorem min_l : forall n m : nat, n <= m -> min n m = n.
 Proof.
-exact min_l.
+induction n; destruct m; simpl; auto. inversion 1.
+rewrite (lt_succ_r n m). auto.
 Qed.
 
 Theorem min_r : forall n m : nat, m <= n -> min n m = m.
 Proof.
-exact min_r.
+induction n; destruct m; simpl; auto. inversion 1.
+rewrite (lt_succ_r m n). auto.
 Qed.
 
 Theorem max_l : forall n m : nat, m <= n -> max n m = n.
 Proof.
-exact max_l.
+induction n; destruct m; simpl; auto. inversion 1.
+rewrite (lt_succ_r m n). auto.
 Qed.
 
 Theorem max_r : forall n m : nat, n <= m -> max n m = m.
 Proof.
-exact max_r.
+induction n; destruct m; simpl; auto. inversion 1.
+rewrite (lt_succ_r n m). auto.
 Qed.
 
 (** Facts specific to natural numbers, not integers. *)
@@ -149,6 +194,8 @@ Qed.
 
 Definition t := nat.
 Definition eq := @eq nat.
+Definition eqb := beq_nat.
+Definition compare := nat_compare.
 Definition zero := 0.
 Definition succ := S.
 Definition pred := pred.
@@ -160,30 +207,16 @@ Definition le := le.
 Definition min := min.
 Definition max := max.
 
-End NPeanoAxiomsMod.
-
-(** Now we apply the largest property functor *)
+Definition eqb_eq := beq_nat_true_iff.
+Definition compare_spec := nat_compare_spec.
+Definition eq_dec := eq_nat_dec.
 
-Module Export NPeanoPropMod := NPropFunct NPeanoAxiomsMod.
+(** Generic Properties *)
 
+Include NPropFunct
+ <+ UsualMinMaxLogicalProperties <+ UsualMinMaxDecProperties.
 
-
-(** Euclidean Division *)
-
-Definition divF div x y := if leb y x then S (div (x-y) y) else 0.
-Definition modF mod x y := if leb y x then mod (x-y) y else x.
-Definition initF (_ _ : nat) := 0.
-
-Fixpoint loop {A} (F:A->A)(i:A) (n:nat) : A :=
- match n with
-  | 0 => i
-  | S n => F (loop F i n)
- end.
-
-Definition div x y := loop divF initF x x y.
-Definition modulo x y := loop modF initF x x y.
-Infix "/" := div : nat_scope.
-Infix "mod" := modulo (at level 40, no associativity) : nat_scope.
+(** Proofs of specification for [div] and [mod]. *)
 
 Lemma div_mod : forall x y, y<>0 -> x = y*(x/y) + x mod y.
 Proof.
@@ -195,14 +228,13 @@ Proof.
  simpl; unfold divF at 1, modF at 1.
  intros.
  destruct (leb y x) as [ ]_eqn:L;
-  [apply leb_complete in L | apply leb_complete_conv in L].
+  [apply leb_complete in L | apply leb_complete_conv in L; now nzsimpl].
  rewrite mul_succ_r, <- add_assoc, (add_comm y), add_assoc.
  rewrite <- IHn; auto.
  symmetry; apply sub_add; auto.
- rewrite <- NPeanoAxiomsMod.lt_succ_r.
+ rewrite <- lt_succ_r.
  apply lt_le_trans with x; auto.
- apply lt_minus; auto. rewrite <- neq_0_lt_0; auto.
- nzsimpl; auto.
+ apply sub_lt; auto. rewrite <- neq_0_lt_0; auto.
 Qed.
 
 Lemma mod_upper_bound : forall x y, y<>0 -> x mod y < y.
@@ -216,22 +248,30 @@ Proof.
  destruct (leb y x) as [ ]_eqn:L;
   [apply leb_complete in L | apply leb_complete_conv in L]; auto.
  apply IHn; auto.
- rewrite <- NPeanoAxiomsMod.lt_succ_r.
+ rewrite <- lt_succ_r.
  apply lt_le_trans with x; auto.
- apply lt_minus; auto. rewrite <- neq_0_lt_0; auto.
+ apply sub_lt; auto. rewrite <- neq_0_lt_0; auto.
 Qed.
 
-Require Import NDiv.
+Definition div := div.
+Definition modulo := modulo.
+Program Instance div_wd : Proper (eq==>eq==>eq) div.
+Program Instance mod_wd : Proper (eq==>eq==>eq) modulo.
+
+(** Generic properties of [div] and [mod] *)
+
+Include NDivPropFunct.
+
+End Nat.
+
+(** [Nat] contains an [order] tactic for natural numbers *)
 
-Module NDivMod <: NDivSig.
- Include NPeanoAxiomsMod.
- Definition div := div.
- Definition modulo := modulo.
- Definition div_mod := div_mod.
- Definition mod_upper_bound := mod_upper_bound.
- Local Obligation Tactic := simpl_relation.
- Program Instance div_wd : Proper (eq==>eq==>eq) div.
- Program Instance mod_wd : Proper (eq==>eq==>eq) modulo.
-End NDivMod.
+(** Note that [Nat.order] is domain-agnostic: it will not prove
+    [1<=2] or [x<=x+x], but rather things like [x<=y -> y<=x -> x=y]. *)
 
-Module Export NDivPropMod := NDivPropFunct NDivMod NPeanoPropMod.
+Section TestOrder.
+ Let test : forall x y, x<=y -> y<=x -> x=y.
+ Proof.
+ Nat.order.
+ Qed.
+End TestOrder.
diff --git a/theories/Structures/OrdersEx.v b/theories/Structures/OrdersEx.v
index 90840ce1f6..b0702f6818 100644
--- a/theories/Structures/OrdersEx.v
+++ b/theories/Structures/OrdersEx.v
@@ -13,7 +13,7 @@
 
 (* $Id$ *)
 
-Require Import Orders NatOrderedType POrderedType NArith
+Require Import Orders NPeano POrderedType NArith
  ZArith RelationPairs EqualitiesFacts.
 
 (** * Examples of Ordered Type structures. *)
@@ -21,7 +21,7 @@ Require Import Orders NatOrderedType POrderedType NArith
 
 (** Ordered Type for [nat], [Positive], [N], [Z] with the usual order. *)
 
-Module Nat_as_OT := NatOrderedType.Nat_as_OT.
+Module Nat_as_OT := NPeano.Nat.
 Module Positive_as_OT := POrderedType.Positive_as_OT.
 Module N_as_OT := NArith.N.
 Module Z_as_OT := ZArith.Z.