Arith: full integration of the "Numbers" modular framework

 - The earlier proof-of-concept file NPeano (which instantiates
   the "Numbers" framework for nat) becomes now the entry point
   in the Arith lib, and gets renamed PeanoNat. It still provides
   an inner module "Nat" which sums up everything about type nat
   (functions, predicates and properties of them).
   This inner module Nat is usable as soon as you Require Import Arith,
   or just Arith_base, or simply PeanoNat.

 - Definitions of operations over type nat are now grouped in a new
   file Init/Nat.v. This file is meant to be used without "Import",
   hence providing for instance Nat.add or Nat.sqrt as soon as coqtop
   starts (but no proofs about them).

 - The definitions that used to be in Init/Peano.v (pred, plus, minus, mult)
   are now compatibility notations (for Nat.pred, Nat.add, Nat.sub, Nat.mul
   where here Nat is Init/Nat.v).

 - This Coq.Init.Nat module (with only pure definitions) is Include'd
   in the aforementioned Coq.Arith.PeanoNat.Nat. You might see Init.Nat
   sometimes instead of just Nat (for instance when doing "Print plus").
   Normally it should be ok to just ignore these "Init" since
   Init.Nat is included in the full PeanoNat.Nat. I'm investigating if
   it's possible to get rid of these "Init" prefixes.

 - Concerning predicates, orders le and lt are still defined in Init/Peano.v,
   with their notations "<=" and "<". Properties in PeanoNat.Nat directly
   refer to these predicates in Peano. For instantation reasons, PeanoNat.Nat
   also contains a Nat.le and Nat.lt (defined via "Definition le := Peano.le",
   we cannot yet include an Inductive to implement a Parameter), but these
   aliased predicates won't probably be very convenient to use.

 - Technical remark: I've split the previous property functor NProp in
   two parts (NBasicProp and NExtraProp), it helps a lot for building
   PeanoNat.Nat incrementally. Roughly speaking, we have the following schema:

   Module Nat.
     Include Coq.Init.Nat. (* definition of operations : add ... sqrt ... *)
     ... (** proofs of specifications for basic ops such as + * - *)
     Include NBasicProp. (** generic properties of these basic ops *)
     ... (** proofs of specifications for advanced ops (pow sqrt log2...)
             that may rely on proofs for + * - *)
     Include NExtraProp. (** all remaining properties *)
   End Nat.

- All other files in directory Arith are now taking advantage of PeanoNat :
  they are now filled with compatibility notations (when earlier lemmas
  have exact counterpart in the Nat module) or lemmas with one-line proofs
  based on the Nat module. All hints for database "arith" remain declared
  in these old-style file (such as Plus.v, Lt.v, etc). All the old-style
  files are still Require'd (or not) by Arith.v, just as before.

- Compatibility should be almost complete. For instance in the stdlib,
  the only adaptations were due to .ml code referring to some Coq constant
  name such as Coq.Init.Peano.pred, which doesn't live well with the
  new compatibility notations.

1 files changed, 40 insertions, 52 deletions

diff --git a/theories/Init/Peano.v b/theories/Init/Peano.v
index 6b0db724d..0de06c95d 100644
--- a/theories/Init/Peano.v
+++ b/theories/Init/Peano.v
@@ -26,6 +26,7 @@
 Require Import Notations.
 Require Import Datatypes.
 Require Import Logic.
+Require Coq.Init.Nat.
 
 Open Scope nat_scope.
 
@@ -37,10 +38,8 @@ Hint Resolve f_equal_nat: core.
 
 (** The predecessor function *)
 
-Definition pred (n:nat) : nat := match n with
-                                 | O => n
-                                 | S u => u
-                                 end.
+Notation pred := Nat.pred (compat "8.4").
+
 Definition f_equal_pred := f_equal pred.
 Hint Resolve f_equal_pred: v62.
 
@@ -82,13 +81,8 @@ Hint Resolve n_Sn: core.
 
 (** Addition *)
 
-Fixpoint plus (n m:nat) : nat :=
-  match n with
-  | O => m
-  | S p => S (p + m)
-  end
-
-where "n + m" := (plus n m) : nat_scope.
+Notation plus := Nat.add (compat "8.4").
+Infix "+" := Nat.add : nat_scope.
 
 Definition f_equal2_plus := f_equal2 plus.
 Hint Resolve f_equal2_plus: v62.
@@ -103,7 +97,7 @@ Hint Resolve plus_n_O: core.
 
 Lemma plus_O_n : forall n:nat, 0 + n = n.
 Proof.
-  auto.
+  reflexivity.
 Qed.
 
 Lemma plus_n_Sm : forall n m:nat, S (n + m) = n + S m.
@@ -114,7 +108,7 @@ Hint Resolve plus_n_Sm: core.
 
 Lemma plus_Sn_m : forall n m:nat, S n + m = S (n + m).
 Proof.
-  auto.
+  reflexivity.
 Qed.
 
 (** Standard associated names *)
@@ -124,13 +118,9 @@ Notation plus_succ_r_reverse := plus_n_Sm (compat "8.2").
 
 (** Multiplication *)
 
-Fixpoint mult (n m:nat) : nat :=
-  match n with
-  | O => 0
-  | S p => m + p * m
-  end
+Notation mult := Nat.mul (compat "8.4").
+Infix "*" := Nat.mul : nat_scope.
 
-where "n * m" := (mult n m) : nat_scope.
 Definition f_equal2_mult := f_equal2 mult.
 Hint Resolve f_equal2_mult: core.
 
@@ -155,14 +145,8 @@ Notation mult_succ_r_reverse := mult_n_Sm (compat "8.2").
 
 (** Truncated subtraction: [m-n] is [0] if [n>=m] *)
 
-Fixpoint minus (n m:nat) : nat :=
-  match n, m with
-  | O, _ => n
-  | S k, O => n
-  | S k, S l => k - l
-  end
-
-where "n - m" := (minus n m) : nat_scope.
+Notation minus := Nat.sub (compat "8.4").
+Infix "-" := Nat.sub : nat_scope.
 
 (** Definition of the usual orders, the basic properties of [le] and [lt]
     can be found in files Le and Lt *)
@@ -206,6 +190,16 @@ Proof.
 intros n m. exact (le_pred (S n) (S m)).
 Qed.
 
+Theorem le_0_n : forall n, 0 <= n.
+Proof.
+ induction n; constructor; trivial.
+Qed.
+
+Theorem le_n_S : forall n m, n <= m -> S n <= S m.
+Proof.
+ induction 1; constructor; trivial.
+Qed.
+
 (** Case analysis *)
 
 Theorem nat_case :
@@ -228,44 +222,38 @@ Qed.
 
 (** Maximum and minimum : definitions and specifications *)
 
-Fixpoint max n m : nat :=
-  match n, m with
-    | O, _ => m
-    | S n', O => n
-    | S n', S m' => S (max n' m')
-  end.
+Notation max := Nat.max (compat "8.4").
+Notation min := Nat.min (compat "8.4").
 
-Fixpoint min n m : nat :=
-  match n, m with
-    | O, _ => 0
-    | S n', O => 0
-    | S n', S m' => S (min n' m')
-  end.
-
-Theorem max_l : forall n m : nat, m <= n -> max n m = n.
+Lemma max_l n m : m <= n -> Nat.max n m = n.
 Proof.
-induction n; destruct m; simpl; auto. inversion 1.
-intros. apply f_equal. apply IHn. apply le_S_n. trivial.
+ revert m; induction n; destruct m; simpl; trivial.
+ - inversion 1.
+ - intros. apply f_equal, IHn, le_S_n; trivial.
 Qed.
 
-Theorem max_r : forall n m : nat, n <= m -> max n m = m.
+Lemma max_r n m : n <= m -> Nat.max n m = m.
 Proof.
-induction n; destruct m; simpl; auto. inversion 1.
-intros. apply f_equal. apply IHn. apply le_S_n. trivial.
+ revert m; induction n; destruct m; simpl; trivial.
+ - inversion 1.
+ - intros. apply f_equal, IHn, le_S_n; trivial.
 Qed.
 
-Theorem min_l : forall n m : nat, n <= m -> min n m = n.
+Lemma min_l n m : n <= m -> Nat.min n m = n.
 Proof.
-induction n; destruct m; simpl; auto. inversion 1.
-intros. apply f_equal. apply IHn. apply le_S_n. trivial.
+ revert m; induction n; destruct m; simpl; trivial.
+ - inversion 1.
+ - intros. apply f_equal, IHn, le_S_n; trivial.
 Qed.
 
-Theorem min_r : forall n m : nat, m <= n -> min n m = m.
+Lemma min_r n m : m <= n -> Nat.min n m = m.
 Proof.
-induction n; destruct m; simpl; auto. inversion 1.
-intros. apply f_equal. apply IHn. apply le_S_n. trivial.
+ revert m; induction n; destruct m; simpl; trivial.
+ - inversion 1.
+ - intros. apply f_equal, IHn, le_S_n; trivial.
 Qed.
 
+
 Lemma nat_rect_succ_r {A} (f: A -> A) (x:A) n :
   nat_rect (fun _ => A) x (fun _ => f) (S n) = nat_rect (fun _ => A) (f x) (fun _ => f) n.
 Proof.