Decimal: proofs that conversions from/to nat,N,Z are bijections

9 files changed, 1093 insertions, 69 deletions

diff --git a/theories/Arith/PeanoNat.v b/theories/Arith/PeanoNat.v
index bde6f1bb4..68060900c 100644
--- a/theories/Arith/PeanoNat.v
+++ b/theories/Arith/PeanoNat.v
@@ -724,6 +724,26 @@ Definition shiftr_spec a n m (_:0<=m) := shiftr_specif a n m.
 
 Include NExtraProp.
 
+(** Properties of tail-recursive addition and multiplication *)
+
+Lemma tail_add_spec n m : tail_add n m = n + m.
+Proof.
+ revert m. induction n as [|n IH]; simpl; trivial.
+ intros. now rewrite IH, add_succ_r.
+Qed.
+
+Lemma tail_addmul_spec r n m : tail_addmul r n m = r + n * m.
+Proof.
+ revert m r. induction n as [| n IH]; simpl; trivial.
+ intros. rewrite IH, tail_add_spec.
+ rewrite add_assoc. f_equal. apply add_comm.
+Qed.
+
+Lemma tail_mul_spec n m : tail_mul n m = n * m.
+Proof.
+ unfold tail_mul. now rewrite tail_addmul_spec.
+Qed.
+
 End Nat.
 
 (** Re-export notations that should be available even when
diff --git a/theories/Init/Decimal.v b/theories/Init/Decimal.v
index bf4764895..fa462f147 100644
--- a/theories/Init/Decimal.v
+++ b/theories/Init/Decimal.v
@@ -50,23 +50,29 @@ Bind Scope int_scope with int.
     and choose our canonical representation of 0 (here [D0 Nil]
     for unsigned numbers and [Pos (D0 Nil)] for signed numbers). *)
 
-Fixpoint nonzero_head d :=
+(** [nzhead] removes all head zero digits *)
+
+Fixpoint nzhead d :=
   match d with
-  | D0 d => nonzero_head d
+  | D0 d => nzhead d
   | _ => d
   end.
 
+(** [unorm] : normalization of unsigned integers *)
+
 Definition unorm d :=
-  match nonzero_head d with
+  match nzhead d with
   | Nil => zero
   | d => d
   end.
 
+(** [norm] : normalization of signed integers *)
+
 Definition norm d :=
   match d with
   | Pos d => Pos (unorm d)
   | Neg d =>
-    match nonzero_head d with
+    match nzhead d with
     | Nil => Pos zero
     | d => Neg d
     end
@@ -84,21 +90,23 @@ Definition opp (d:int) :=
 (** For conversions with binary numbers, it is easier to operate
     on little-endian numbers. *)
 
-Fixpoint rev (l l' : uint) :=
-  match l with
-  | Nil => l'
-  | D0 l => rev l (D0 l')
-  | D1 l => rev l (D1 l')
-  | D2 l => rev l (D2 l')
-  | D3 l => rev l (D3 l')
-  | D4 l => rev l (D4 l')
-  | D5 l => rev l (D5 l')
-  | D6 l => rev l (D6 l')
-  | D7 l => rev l (D7 l')
-  | D8 l => rev l (D8 l')
-  | D9 l => rev l (D9 l')
+Fixpoint revapp (d d' : uint) :=
+  match d with
+  | Nil => d'
+  | D0 d => revapp d (D0 d')
+  | D1 d => revapp d (D1 d')
+  | D2 d => revapp d (D2 d')
+  | D3 d => revapp d (D3 d')
+  | D4 d => revapp d (D4 d')
+  | D5 d => revapp d (D5 d')
+  | D6 d => revapp d (D6 d')
+  | D7 d => revapp d (D7 d')
+  | D8 d => revapp d (D8 d')
+  | D9 d => revapp d (D9 d')
   end.
 
+Definition rev d := revapp d Nil.
+
 Module Little.
 
 (** Successor of little-endian numbers *)
@@ -106,16 +114,16 @@ Module Little.
 Fixpoint succ d :=
   match d with
   | Nil => D1 Nil
-  | D0 l => D1 l
-  | D1 l => D2 l
-  | D2 l => D3 l
-  | D3 l => D4 l
-  | D4 l => D5 l
-  | D5 l => D6 l
-  | D6 l => D7 l
-  | D7 l => D8 l
-  | D8 l => D9 l
-  | D9 l => D0 (succ l)
+  | D0 d => D1 d
+  | D1 d => D2 d
+  | D2 d => D3 d
+  | D3 d => D4 d
+  | D4 d => D5 d
+  | D5 d => D6 d
+  | D6 d => D7 d
+  | D7 d => D8 d
+  | D8 d => D9 d
+  | D9 d => D0 (succ d)
   end.
 
 (** Doubling little-endian numbers *)
@@ -123,31 +131,31 @@ Fixpoint succ d :=
 Fixpoint double d :=
   match d with
   | Nil => Nil
-  | D0 l => D0 (double l)
-  | D1 l => D2 (double l)
-  | D2 l => D4 (double l)
-  | D3 l => D6 (double l)
-  | D4 l => D8 (double l)
-  | D5 l => D0 (succ_double l)
-  | D6 l => D2 (succ_double l)
-  | D7 l => D4 (succ_double l)
-  | D8 l => D6 (succ_double l)
-  | D9 l => D8 (succ_double l)
+  | D0 d => D0 (double d)
+  | D1 d => D2 (double d)
+  | D2 d => D4 (double d)
+  | D3 d => D6 (double d)
+  | D4 d => D8 (double d)
+  | D5 d => D0 (succ_double d)
+  | D6 d => D2 (succ_double d)
+  | D7 d => D4 (succ_double d)
+  | D8 d => D6 (succ_double d)
+  | D9 d => D8 (succ_double d)
   end
 
 with succ_double d :=
   match d with
   | Nil => D1 Nil
-  | D0 l => D1 (double l)
-  | D1 l => D3 (double l)
-  | D2 l => D5 (double l)
-  | D3 l => D7 (double l)
-  | D4 l => D9 (double l)
-  | D5 l => D1 (succ_double l)
-  | D6 l => D3 (succ_double l)
-  | D7 l => D5 (succ_double l)
-  | D8 l => D7 (succ_double l)
-  | D9 l => D9 (succ_double l)
+  | D0 d => D1 (double d)
+  | D1 d => D3 (double d)
+  | D2 d => D5 (double d)
+  | D3 d => D7 (double d)
+  | D4 d => D9 (double d)
+  | D5 d => D1 (succ_double d)
+  | D6 d => D3 (succ_double d)
+  | D7 d => D5 (succ_double d)
+  | D8 d => D7 (succ_double d)
+  | D9 d => D9 (succ_double d)
   end.
 
 End Little.
diff --git a/theories/Init/Nat.v b/theories/Init/Nat.v
index 72d35827e..8f2648269 100644
--- a/theories/Init/Nat.v
+++ b/theories/Init/Nat.v
@@ -136,25 +136,21 @@ where "n ^ m" := (pow n m) : nat_scope.
 
 (** ** Tail-recursive versions of [add] and [mul] *)
 
-Module Tail.
-
-Fixpoint add n m :=
+Fixpoint tail_add n m :=
   match n with
     | O => m
-    | S n => add n (S m)
+    | S n => tail_add n (S m)
   end.
 
-(** [addmul r n m] is [r + n * m]. *)
+(** [tail_addmul r n m] is [r + n * m]. *)
 
-Fixpoint addmul r n m :=
+Fixpoint tail_addmul r n m :=
   match n with
     | O => r
-    | S n => addmul (add m r) n m
+    | S n => tail_addmul (tail_add m r) n m
   end.
 
-Definition mul n m := addmul 0 n m.
-
-End Tail.
+Definition tail_mul n m := tail_addmul 0 n m.
 
 (** ** Conversion with a decimal representation for printing/parsing *)
 
@@ -163,16 +159,16 @@ Local Notation ten := (S (S (S (S (S (S (S (S (S (S O)))))))))).
 Fixpoint of_uint_acc (d:Decimal.uint)(acc:nat) :=
   match d with
   | Decimal.Nil => acc
-  | Decimal.D0 d => of_uint_acc d (Tail.mul ten acc)
-  | Decimal.D1 d => of_uint_acc d (S (Tail.mul ten acc))
-  | Decimal.D2 d => of_uint_acc d (S (S (Tail.mul ten acc)))
-  | Decimal.D3 d => of_uint_acc d (S (S (S (Tail.mul ten acc))))
-  | Decimal.D4 d => of_uint_acc d (S (S (S (S (Tail.mul ten acc)))))
-  | Decimal.D5 d => of_uint_acc d (S (S (S (S (S (Tail.mul ten acc))))))
-  | Decimal.D6 d => of_uint_acc d (S (S (S (S (S (S (Tail.mul ten acc)))))))
-  | Decimal.D7 d => of_uint_acc d (S (S (S (S (S (S (S (Tail.mul ten acc))))))))
-  | Decimal.D8 d => of_uint_acc d (S (S (S (S (S (S (S (S (Tail.mul ten acc)))))))))
-  | Decimal.D9 d => of_uint_acc d (S (S (S (S (S (S (S (S (S (Tail.mul ten acc))))))))))
+  | Decimal.D0 d => of_uint_acc d (tail_mul ten acc)
+  | Decimal.D1 d => of_uint_acc d (S (tail_mul ten acc))
+  | Decimal.D2 d => of_uint_acc d (S (S (tail_mul ten acc)))
+  | Decimal.D3 d => of_uint_acc d (S (S (S (tail_mul ten acc))))
+  | Decimal.D4 d => of_uint_acc d (S (S (S (S (tail_mul ten acc)))))
+  | Decimal.D5 d => of_uint_acc d (S (S (S (S (S (tail_mul ten acc))))))
+  | Decimal.D6 d => of_uint_acc d (S (S (S (S (S (S (tail_mul ten acc)))))))
+  | Decimal.D7 d => of_uint_acc d (S (S (S (S (S (S (S (tail_mul ten acc))))))))
+  | Decimal.D8 d => of_uint_acc d (S (S (S (S (S (S (S (S (tail_mul ten acc)))))))))
+  | Decimal.D9 d => of_uint_acc d (S (S (S (S (S (S (S (S (S (tail_mul ten acc))))))))))
   end.
 
 Definition of_uint (d:Decimal.uint) := of_uint_acc d O.
@@ -184,7 +180,7 @@ Fixpoint to_little_uint n acc :=
   end.
 
 Definition to_uint n :=
-  Decimal.rev (to_little_uint n Decimal.zero) Decimal.Nil.
+  Decimal.rev (to_little_uint n Decimal.zero).
 
 Definition of_int (d:Decimal.int) : option nat :=
   match Decimal.norm d with
diff --git a/theories/Numbers/DecimalFacts.v b/theories/Numbers/DecimalFacts.v
new file mode 100644
index 000000000..3eef63c7f
--- /dev/null
+++ b/theories/Numbers/DecimalFacts.v
@@ -0,0 +1,141 @@
+(************************************************************************)
+(*  v      *   The Coq Proof Assistant  /  The Coq Development Team     *)
+(* <O___,, *   INRIA - CNRS - LIX - LRI - PPS - Copyright 1999-2016     *)
+(*   \VV/  **************************************************************)
+(*    //   *      This file is distributed under the terms of the       *)
+(*         *       GNU Lesser General Public License Version 2.1        *)
+(************************************************************************)
+
+(** * DecimalFacts : some facts about Decimal numbers *)
+
+Require Import Decimal.
+
+Lemma uint_dec (d d' : uint) : { d = d' } + { d <> d' }.
+Proof.
+ decide equality.
+Defined.
+
+Lemma rev_revapp d d' :
+ rev (revapp d d') = revapp d' d.
+Proof.
+ revert d'. induction d; simpl; intros; now rewrite ?IHd.
+Qed.
+
+Lemma rev_rev d : rev (rev d) = d.
+Proof.
+ apply rev_revapp.
+Qed.
+
+(** Normalization on little-endian numbers *)
+
+Fixpoint nztail d :=
+ match d with
+ | Nil => Nil
+ | D0 d => match nztail d with Nil => Nil | d' => D0 d' end
+ | D1 d => D1 (nztail d)
+ | D2 d => D2 (nztail d)
+ | D3 d => D3 (nztail d)
+ | D4 d => D4 (nztail d)
+ | D5 d => D5 (nztail d)
+ | D6 d => D6 (nztail d)
+ | D7 d => D7 (nztail d)
+ | D8 d => D8 (nztail d)
+ | D9 d => D9 (nztail d)
+ end.
+
+Definition lnorm d :=
+ match nztail d with
+ | Nil => zero
+ | d => d
+ end.
+
+Lemma nzhead_revapp_0 d d' : nztail d = Nil ->
+ nzhead (revapp d d') = nzhead d'.
+Proof.
+ revert d'. induction d; intros d' [=]; simpl; trivial.
+ destruct (nztail d); now rewrite IHd.
+Qed.
+
+Lemma nzhead_revapp d d' : nztail d <> Nil ->
+ nzhead (revapp d d') = revapp (nztail d) d'.
+Proof.
+ revert d'.
+ induction d; intros d' H; simpl in *;
+ try destruct (nztail d) eqn:E;
+  (now rewrite ?nzhead_revapp_0) || (now rewrite IHd).
+Qed.
+
+Lemma nzhead_rev d : nztail d <> Nil ->
+ nzhead (rev d) = rev (nztail d).
+Proof.
+ apply nzhead_revapp.
+Qed.
+
+Lemma rev_nztail_rev d :
+  rev (nztail (rev d)) = nzhead d.
+Proof.
+ destruct (uint_dec (nztail (rev d)) Nil) as [H|H].
+ - rewrite H. unfold rev; simpl.
+   rewrite <- (rev_rev d). symmetry.
+   now apply nzhead_revapp_0.
+ - now rewrite <- nzhead_rev, rev_rev.
+Qed.
+
+Lemma revapp_nil_inv d d' : revapp d d' = Nil -> d = Nil /\ d' = Nil.
+Proof.
+ revert d'.
+ induction d; simpl; intros d' H; auto; now apply IHd in H.
+Qed.
+
+Lemma rev_nil_inv d : rev d = Nil -> d = Nil.
+Proof.
+ apply revapp_nil_inv.
+Qed.
+
+Lemma rev_lnorm_rev d :
+  rev (lnorm (rev d)) = unorm d.
+Proof.
+ unfold unorm, lnorm.
+ rewrite <- rev_nztail_rev.
+ destruct nztail; simpl; trivial;
+  destruct rev eqn:E; trivial; now apply rev_nil_inv in E.
+Qed.
+
+Lemma nzhead_nonzero d d' : nzhead d <> D0 d'.
+Proof.
+ induction d; easy.
+Qed.
+
+Lemma unorm_0 d : unorm d = zero <-> nzhead d = Nil.
+Proof.
+ unfold unorm. split.
+ - generalize (nzhead_nonzero d).
+   destruct nzhead; intros H [=]; trivial. now destruct (H u).
+ - now intros ->.
+Qed.
+
+Lemma unorm_nonnil d : unorm d <> Nil.
+Proof.
+ unfold unorm. now destruct nzhead.
+Qed.
+
+Lemma nzhead_invol d : nzhead (nzhead d) = nzhead d.
+Proof.
+ now induction d.
+Qed.
+
+Lemma unorm_invol d : unorm (unorm d) = unorm d.
+Proof.
+ unfold unorm.
+ destruct (nzhead d) eqn:E; trivial.
+ destruct (nzhead_nonzero _ _ E).
+Qed.
+
+Lemma norm_invol d : norm (norm d) = norm d.
+Proof.
+ unfold norm.
+ destruct d.
+ - f_equal. apply unorm_invol.
+ - destruct (nzhead d) eqn:E; auto.
+   destruct (nzhead_nonzero _ _ E).
+Qed.
diff --git a/theories/Numbers/DecimalN.v b/theories/Numbers/DecimalN.v
new file mode 100644
index 000000000..998f009a7
--- /dev/null
+++ b/theories/Numbers/DecimalN.v
@@ -0,0 +1,105 @@
+(************************************************************************)
+(*  v      *   The Coq Proof Assistant  /  The Coq Development Team     *)
+(* <O___,, *   INRIA - CNRS - LIX - LRI - PPS - Copyright 1999-2016     *)
+(*   \VV/  **************************************************************)
+(*    //   *      This file is distributed under the terms of the       *)
+(*         *       GNU Lesser General Public License Version 2.1        *)
+(************************************************************************)
+
+(** * DecimalN
+
+    Proofs that conversions between decimal numbers and [N]
+    are bijections *)
+
+Require Import Decimal DecimalFacts DecimalPos PArith NArith.
+
+Module Unsigned.
+
+Lemma of_to (n:N) : N.of_uint (N.to_uint n) = n.
+Proof.
+ destruct n.
+ - reflexivity.
+ - apply DecimalPos.Unsigned.of_to.
+Qed.
+
+Lemma to_of (d:uint) : N.to_uint (N.of_uint d) = unorm d.
+Proof.
+ exact (DecimalPos.Unsigned.to_of d).
+Qed.
+
+Lemma to_uint_inj n n' : N.to_uint n = N.to_uint n' -> n = n'.
+Proof.
+ intros E. now rewrite <- (of_to n), <- (of_to n'), E.
+Qed.
+
+Lemma to_uint_surj d : exists p, N.to_uint p = unorm d.
+Proof.
+ exists (N.of_uint d). apply to_of.
+Qed.
+
+Lemma of_uint_norm d : N.of_uint (unorm d) = N.of_uint d.
+Proof.
+ now induction d.
+Qed.
+
+Lemma of_inj d d' :
+ N.of_uint d = N.of_uint d' -> unorm d = unorm d'.
+Proof.
+ intros. rewrite <- !to_of. now f_equal.
+Qed.
+
+Lemma of_iff d d' : N.of_uint d = N.of_uint d' <-> unorm d = unorm d'.
+Proof.
+ split. apply of_inj. intros E. rewrite <- of_uint_norm, E.
+ apply of_uint_norm.
+Qed.
+
+End Unsigned.
+
+(** Conversion from/to signed decimal numbers *)
+
+Module Signed.
+
+Lemma of_to (n:N) : N.of_int (N.to_int n) = Some n.
+Proof.
+ unfold N.to_int, N.of_int, norm. f_equal.
+ rewrite Unsigned.of_uint_norm. apply Unsigned.of_to.
+Qed.
+
+Lemma to_of (d:int)(n:N) : N.of_int d = Some n -> N.to_int n = norm d.
+Proof.
+ unfold N.of_int.
+ destruct (norm d) eqn:Hd; intros [= <-].
+ unfold N.to_int. rewrite Unsigned.to_of. f_equal.
+ revert Hd; destruct d; simpl.
+ - intros [= <-]. apply unorm_invol.
+ - destruct (nzhead d); now intros [= <-].
+Qed.
+
+Lemma to_int_inj n n' : N.to_int n = N.to_int n' -> n = n'.
+Proof.
+ intro E.
+ assert (E' : Some n = Some n').
+ { now rewrite <- (of_to n), <- (of_to n'), E. }
+ now injection E'.
+Qed.
+
+Lemma to_int_pos_surj d : exists n, N.to_int n = norm (Pos d).
+Proof.
+ exists (N.of_uint d). unfold N.to_int. now rewrite Unsigned.to_of.
+Qed.
+
+Lemma of_int_norm d : N.of_int (norm d) = N.of_int d.
+Proof.
+ unfold N.of_int. now rewrite norm_invol.
+Qed.
+
+Lemma of_inj_pos d d' :
+ N.of_int (Pos d) = N.of_int (Pos d') -> unorm d = unorm d'.
+Proof.
+ unfold N.of_int. simpl. intros [= H]. apply Unsigned.of_inj.
+ change Pos.of_uint with N.of_uint in H.
+ now rewrite <- Unsigned.of_uint_norm, H, Unsigned.of_uint_norm.
+Qed.
+
+End Signed.
diff --git a/theories/Numbers/DecimalNat.v b/theories/Numbers/DecimalNat.v
new file mode 100644
index 000000000..4aa189e24
--- /dev/null
+++ b/theories/Numbers/DecimalNat.v
@@ -0,0 +1,300 @@
+(************************************************************************)
+(*  v      *   The Coq Proof Assistant  /  The Coq Development Team     *)
+(* <O___,, *   INRIA - CNRS - LIX - LRI - PPS - Copyright 1999-2016     *)
+(*   \VV/  **************************************************************)
+(*    //   *      This file is distributed under the terms of the       *)
+(*         *       GNU Lesser General Public License Version 2.1        *)
+(************************************************************************)
+
+(** * DecimalNat
+
+    Proofs that conversions between decimal numbers and [nat]
+    are bijections. *)
+
+Require Import Decimal DecimalFacts Arith.
+
+Module Unsigned.
+
+(** A few helper functions used during proofs *)
+
+Definition hd d :=
+ match d with
+ | Nil => 0
+ | D0 _ => 0
+ | D1 _ => 1
+ | D2 _ => 2
+ | D3 _ => 3
+ | D4 _ => 4
+ | D5 _ => 5
+ | D6 _ => 6
+ | D7 _ => 7
+ | D8 _ => 8
+ | D9 _ => 9
+end.
+
+Definition tl d :=
+ match d with
+ | Nil => d
+ | D0 d | D1 d | D2 d | D3 d | D4 d | D5 d | D6 d | D7 d | D8 d | D9 d => d
+end.
+
+Fixpoint usize (d:uint) : nat :=
+  match d with
+  | Nil => 0
+  | D0 d => S (usize d)
+  | D1 d => S (usize d)
+  | D2 d => S (usize d)
+  | D3 d => S (usize d)
+  | D4 d => S (usize d)
+  | D5 d => S (usize d)
+  | D6 d => S (usize d)
+  | D7 d => S (usize d)
+  | D8 d => S (usize d)
+  | D9 d => S (usize d)
+  end.
+
+(** A direct version of [to_little_uint], not tail-recursive *)
+Fixpoint to_lu n :=
+ match n with
+ | 0 => Decimal.zero
+ | S n => Little.succ (to_lu n)
+ end.
+
+(** A direct version of [of_little_uint] *)
+Fixpoint of_lu (d:uint) : nat :=
+  match d with
+  | Nil => 0
+  | D0 d => 10 * of_lu d
+  | D1 d => 1 + 10 * of_lu d
+  | D2 d => 2 + 10 * of_lu d
+  | D3 d => 3 + 10 * of_lu d
+  | D4 d => 4 + 10 * of_lu d
+  | D5 d => 5 + 10 * of_lu d
+  | D6 d => 6 + 10 * of_lu d
+  | D7 d => 7 + 10 * of_lu d
+  | D8 d => 8 + 10 * of_lu d
+  | D9 d => 9 + 10 * of_lu d
+  end.
+
+(** Properties of [to_lu] *)
+
+Lemma to_lu_succ n : to_lu (S n) = Little.succ (to_lu n).
+Proof.
+ reflexivity.
+Qed.
+
+Lemma to_little_uint_succ n d :
+ Nat.to_little_uint n (Little.succ d) =
+  Little.succ (Nat.to_little_uint n d).
+Proof.
+ revert d; induction n; simpl; trivial.
+Qed.
+
+Lemma to_lu_equiv n :
+  to_lu n = Nat.to_little_uint n zero.
+Proof.
+ induction n; simpl; trivial.
+ now rewrite IHn, <- to_little_uint_succ.
+Qed.
+
+Lemma to_uint_alt n :
+ Nat.to_uint n = rev (to_lu n).
+Proof.
+ unfold Nat.to_uint. f_equal. symmetry. apply to_lu_equiv.
+Qed.
+
+(** Properties of [of_lu] *)
+
+Lemma of_lu_eqn d :
+ of_lu d = hd d + 10 * of_lu (tl d).
+Proof.
+ induction d; simpl; trivial.
+Qed.
+
+Ltac simpl_of_lu :=
+ match goal with
+ | |- context [ of_lu (?f ?x) ] =>
+   rewrite (of_lu_eqn (f x)); simpl hd; simpl tl
+ end.
+
+Lemma of_lu_succ d :
+ of_lu (Little.succ d) = S (of_lu d).
+Proof.
+ induction d; trivial.
+ simpl_of_lu. rewrite IHd. simpl_of_lu.
+ now rewrite Nat.mul_succ_r, <- (Nat.add_comm 10).
+Qed.
+
+Lemma of_to_lu n :
+ of_lu (to_lu n) = n.
+Proof.
+ induction n; simpl; trivial. rewrite of_lu_succ. now f_equal.
+Qed.
+
+Lemma of_lu_revapp d d' :
+of_lu (revapp d d') =
+ of_lu (rev d) + of_lu d' * 10^usize d.
+Proof.
+ revert d'.
+ induction d; intro d'; simpl usize;
+ [ simpl; now rewrite Nat.mul_1_r | .. ];
+ unfold rev; simpl revapp; rewrite 2 IHd;
+ rewrite <- Nat.add_assoc; f_equal; simpl_of_lu; simpl of_lu;
+ rewrite Nat.pow_succ_r'; ring.
+Qed.
+
+Lemma of_uint_acc_spec n d :
+ Nat.of_uint_acc d n = of_lu (rev d) + n * 10^usize d.
+Proof.
+ revert n. induction d; intros;
+ simpl Nat.of_uint_acc; rewrite ?Nat.tail_mul_spec, ?IHd;
+ simpl rev; simpl usize; rewrite ?Nat.pow_succ_r';
+ [ simpl; now rewrite Nat.mul_1_r | .. ];
+ unfold rev at 2; simpl revapp; rewrite of_lu_revapp;
+ simpl of_lu; ring.
+Qed.
+
+Lemma of_uint_alt d : Nat.of_uint d = of_lu (rev d).
+Proof.
+ unfold Nat.of_uint. now rewrite of_uint_acc_spec.
+Qed.
+
+(** First main bijection result *)
+
+Lemma of_to (n:nat) : Nat.of_uint (Nat.to_uint n) = n.
+Proof.
+ rewrite to_uint_alt, of_uint_alt, rev_rev. apply of_to_lu.
+Qed.
+
+(** The other direction *)
+
+Lemma to_lu_tenfold n : n<>0 ->
+ to_lu (10 * n) = D0 (to_lu n).
+Proof.
+ induction n.
+ - simpl. now destruct 1.
+ - intros _.
+   destruct (Nat.eq_dec n 0) as [->|H]; simpl; trivial.
+   rewrite !Nat.add_succ_r.
+   simpl in *. rewrite (IHn H). now destruct (to_lu n).
+Qed.
+
+Lemma of_lu_0 d : of_lu d = 0 <-> nztail d = Nil.
+Proof.
+ induction d; try simpl_of_lu; try easy.
+ rewrite Nat.add_0_l.
+ split; intros H.
+ - apply Nat.eq_mul_0_r in H; auto.
+   rewrite IHd in H. simpl. now rewrite H.
+ - simpl in H. destruct (nztail d); try discriminate.
+   now destruct IHd as [_ ->].
+Qed.
+
+Lemma to_of_lu_tenfold d :
+ to_lu (of_lu d) = lnorm d ->
+ to_lu (10 * of_lu d) = lnorm (D0 d).
+Proof.
+ intro IH.
+ destruct (Nat.eq_dec (of_lu d) 0) as [H|H].
+ - rewrite H. simpl. rewrite of_lu_0 in H.
+   unfold lnorm. simpl. now rewrite H.
+ - rewrite (to_lu_tenfold _ H), IH.
+   rewrite of_lu_0 in H.
+   unfold lnorm. simpl. now destruct (nztail d).
+Qed.
+
+Lemma to_of_lu d : to_lu (of_lu d) = lnorm d.
+Proof.
+ induction d; [ reflexivity | .. ];
+ simpl_of_lu;
+ rewrite ?Nat.add_succ_l, Nat.add_0_l, ?to_lu_succ, to_of_lu_tenfold
+  by assumption;
+ unfold lnorm; simpl; now destruct nztail.
+Qed.
+
+(** Second bijection result *)
+
+Lemma to_of (d:uint) : Nat.to_uint (Nat.of_uint d) = unorm d.
+Proof.
+ rewrite to_uint_alt, of_uint_alt, to_of_lu.
+ apply rev_lnorm_rev.
+Qed.
+
+(** Some consequences *)
+
+Lemma to_uint_inj n n' : Nat.to_uint n = Nat.to_uint n' -> n = n'.
+Proof.
+ intro EQ.
+ now rewrite <- (of_to n), <- (of_to n'), EQ.
+Qed.
+
+Lemma to_uint_surj d : exists n, Nat.to_uint n = unorm d.
+Proof.
+ exists (Nat.of_uint d). apply to_of.
+Qed.
+
+Lemma of_uint_norm d : Nat.of_uint (unorm d) = Nat.of_uint d.
+Proof.
+ unfold Nat.of_uint. now induction d.
+Qed.
+
+Lemma of_inj d d' :
+ Nat.of_uint d = Nat.of_uint d' -> unorm d = unorm d'.
+Proof.
+ intros. rewrite <- !to_of. now f_equal.
+Qed.
+
+Lemma of_iff d d' : Nat.of_uint d = Nat.of_uint d' <-> unorm d = unorm d'.
+Proof.
+ split. apply of_inj. intros E. rewrite <- of_uint_norm, E.
+ apply of_uint_norm.
+Qed.
+
+End Unsigned.
+
+(** Conversion from/to signed decimal numbers *)
+
+Module Signed.
+
+Lemma of_to (n:nat) : Nat.of_int (Nat.to_int n) = Some n.
+Proof.
+ unfold Nat.to_int, Nat.of_int, norm. f_equal.
+ rewrite Unsigned.of_uint_norm. apply Unsigned.of_to.
+Qed.
+
+Lemma to_of (d:int)(n:nat) : Nat.of_int d = Some n -> Nat.to_int n = norm d.
+Proof.
+ unfold Nat.of_int.
+ destruct (norm d) eqn:Hd; intros [= <-].
+ unfold Nat.to_int. rewrite Unsigned.to_of. f_equal.
+ revert Hd; destruct d; simpl.
+ - intros [= <-]. apply unorm_invol.
+ - destruct (nzhead d); now intros [= <-].
+Qed.
+
+Lemma to_int_inj n n' : Nat.to_int n = Nat.to_int n' -> n = n'.
+Proof.
+ intro E.
+ assert (E' : Some n = Some n').
+ { now rewrite <- (of_to n), <- (of_to n'), E. }
+ now injection E'.
+Qed.
+
+Lemma to_int_pos_surj d : exists n, Nat.to_int n = norm (Pos d).
+Proof.
+ exists (Nat.of_uint d). unfold Nat.to_int. now rewrite Unsigned.to_of.
+Qed.
+
+Lemma of_int_norm d : Nat.of_int (norm d) = Nat.of_int d.
+Proof.
+ unfold Nat.of_int. now rewrite norm_invol.
+Qed.
+
+Lemma of_inj_pos d d' :
+ Nat.of_int (Pos d) = Nat.of_int (Pos d') -> unorm d = unorm d'.
+Proof.
+ unfold Nat.of_int. simpl. intros [= H]. apply Unsigned.of_inj.
+ now rewrite <- Unsigned.of_uint_norm, H, Unsigned.of_uint_norm.
+Qed.
+
+End Signed.
diff --git a/theories/Numbers/DecimalPos.v b/theories/Numbers/DecimalPos.v
new file mode 100644
index 000000000..40c8f5a5a
--- /dev/null
+++ b/theories/Numbers/DecimalPos.v
@@ -0,0 +1,381 @@
+(************************************************************************)
+(*  v      *   The Coq Proof Assistant  /  The Coq Development Team     *)
+(* <O___,, *   INRIA - CNRS - LIX - LRI - PPS - Copyright 1999-2016     *)
+(*   \VV/  **************************************************************)
+(*    //   *      This file is distributed under the terms of the       *)
+(*         *       GNU Lesser General Public License Version 2.1        *)
+(************************************************************************)
+
+(** * DecimalPos
+
+    Proofs that conversions between decimal numbers and [positive]
+    are bijections. *)
+
+Require Import Decimal DecimalFacts PArith NArith.
+
+Module Unsigned.
+
+Local Open Scope N.
+
+(** A direct version of [of_little_uint] *)
+Fixpoint of_lu (d:uint) : N :=
+  match d with
+  | Nil => 0
+  | D0 d => 10 * of_lu d
+  | D1 d => 1 + 10 * of_lu d
+  | D2 d => 2 + 10 * of_lu d
+  | D3 d => 3 + 10 * of_lu d
+  | D4 d => 4 + 10 * of_lu d
+  | D5 d => 5 + 10 * of_lu d
+  | D6 d => 6 + 10 * of_lu d
+  | D7 d => 7 + 10 * of_lu d
+  | D8 d => 8 + 10 * of_lu d
+  | D9 d => 9 + 10 * of_lu d
+  end.
+
+Definition hd d :=
+match d with
+ | Nil => 0
+ | D0 _ => 0
+ | D1 _ => 1
+ | D2 _ => 2
+ | D3 _ => 3
+ | D4 _ => 4
+ | D5 _ => 5
+ | D6 _ => 6
+ | D7 _ => 7
+ | D8 _ => 8
+ | D9 _ => 9
+end.
+
+Definition tl d :=
+ match d with
+ | Nil => d
+ | D0 d | D1 d | D2 d | D3 d | D4 d | D5 d | D6 d | D7 d | D8 d | D9 d => d
+end.
+
+Lemma of_lu_eqn d :
+ of_lu d = hd d + 10 * (of_lu (tl d)).
+Proof.
+ induction d; simpl; trivial.
+Qed.
+
+Ltac simpl_of_lu :=
+ match goal with
+ | |- context [ of_lu (?f ?x) ] =>
+   rewrite (of_lu_eqn (f x)); simpl hd; simpl tl
+ end.
+
+Fixpoint usize (d:uint) : N :=
+  match d with
+  | Nil => 0
+  | D0 d => N.succ (usize d)
+  | D1 d => N.succ (usize d)
+  | D2 d => N.succ (usize d)
+  | D3 d => N.succ (usize d)
+  | D4 d => N.succ (usize d)
+  | D5 d => N.succ (usize d)
+  | D6 d => N.succ (usize d)
+  | D7 d => N.succ (usize d)
+  | D8 d => N.succ (usize d)
+  | D9 d => N.succ (usize d)
+  end.
+
+Lemma of_lu_revapp d d' :
+ of_lu (revapp d d') =
+  of_lu (rev d) + of_lu d' * 10^usize d.
+Proof.
+ revert d'.
+ induction d; simpl; intro d'; [ now rewrite N.mul_1_r | .. ];
+ unfold rev; simpl revapp; rewrite 2 IHd;
+ rewrite <- N.add_assoc; f_equal; simpl_of_lu; simpl of_lu;
+ rewrite N.pow_succ_r'; ring.
+Qed.
+
+Definition Nadd n p :=
+ match n with
+ | N0 => p
+ | Npos p0 => (p0+p)%positive
+ end.
+
+Lemma Nadd_simpl n p q : Npos (Nadd n (p * q)) = n + Npos p * Npos q.
+Proof.
+ now destruct n.
+Qed.
+
+Lemma of_uint_acc_eqn d acc : d<>Nil ->
+ Pos.of_uint_acc d acc = Pos.of_uint_acc (tl d) (Nadd (hd d) (10*acc)).
+Proof.
+ destruct d; simpl; trivial. now destruct 1.
+Qed.
+
+Lemma of_uint_acc_rev d acc :
+ Npos (Pos.of_uint_acc d acc) =
+  of_lu (rev d) + (Npos acc) * 10^usize d.
+Proof.
+ revert acc.
+ induction d; intros; simpl usize;
+ [ simpl; now rewrite Pos.mul_1_r | .. ];
+ rewrite N.pow_succ_r';
+ unfold rev; simpl revapp; try rewrite of_lu_revapp; simpl of_lu;
+ rewrite of_uint_acc_eqn by easy; simpl tl; simpl hd;
+ rewrite IHd, Nadd_simpl; ring.
+Qed.
+
+Lemma of_uint_alt d : Pos.of_uint d = of_lu (rev d).
+Proof.
+ induction d; simpl; trivial; unfold rev; simpl revapp;
+ rewrite of_lu_revapp; simpl of_lu; try apply of_uint_acc_rev.
+ rewrite IHd. ring.
+Qed.
+
+Lemma of_lu_rev d : Pos.of_uint (rev d) = of_lu d.
+Proof.
+ rewrite of_uint_alt. now rewrite rev_rev.
+Qed.
+
+Lemma of_lu_double_gen d :
+  of_lu (Little.double d) = N.double (of_lu d) /\
+  of_lu (Little.succ_double d) = N.succ_double (of_lu d).
+Proof.
+ rewrite N.double_spec, N.succ_double_spec.
+ induction d; try destruct IHd as (IH1,IH2);
+ simpl Little.double; simpl Little.succ_double;
+ repeat (simpl_of_lu; rewrite ?IH1, ?IH2); split; reflexivity || ring.
+Qed.
+
+Lemma of_lu_double d :
+  of_lu (Little.double d) = N.double (of_lu d).
+Proof.
+ apply of_lu_double_gen.
+Qed.
+
+Lemma of_lu_succ_double d :
+  of_lu (Little.succ_double d) = N.succ_double (of_lu d).
+Proof.
+ apply of_lu_double_gen.
+Qed.
+
+(** First bijection result *)
+
+Lemma of_to (p:positive) : Pos.of_uint (Pos.to_uint p) = Npos p.
+Proof.
+ unfold Pos.to_uint.
+ rewrite of_lu_rev.
+ induction p; simpl; trivial.
+ - now rewrite of_lu_succ_double, IHp.
+ - now rewrite of_lu_double, IHp.
+Qed.
+
+(** The other direction *)
+
+Definition to_lu n :=
+  match n with
+  | N0 => Decimal.zero
+  | Npos p => Pos.to_little_uint p
+  end.
+
+Lemma succ_double_alt d :
+  Little.succ_double d = Little.succ (Little.double d).
+Proof.
+ now induction d.
+Qed.
+
+Lemma double_succ d :
+  Little.double (Little.succ d) =
+  Little.succ (Little.succ_double d).
+Proof.
+ induction d; simpl; f_equal; auto using succ_double_alt.
+Qed.
+
+Lemma to_lu_succ n :
+ to_lu (N.succ n) = Little.succ (to_lu n).
+Proof.
+ destruct n; simpl; trivial.
+ induction p; simpl; rewrite ?IHp;
+  auto using succ_double_alt, double_succ.
+Qed.
+
+Lemma nat_iter_S n {A} (f:A->A) i :
+ Nat.iter (S n) f i = f (Nat.iter n f i).
+Proof.
+ reflexivity.
+Qed.
+
+Lemma nat_iter_0 {A} (f:A->A) i : Nat.iter 0 f i = i.
+Proof.
+ reflexivity.
+Qed.
+
+Lemma to_ldec_tenfold p :
+ to_lu (10 * Npos p) = D0 (to_lu (Npos p)).
+Proof.
+ induction p using Pos.peano_rect.
+ - trivial.
+ - change (N.pos (Pos.succ p)) with (N.succ (N.pos p)).
+   rewrite N.mul_succ_r.
+   change 10 at 2 with (Nat.iter 10%nat N.succ 0).
+   rewrite ?nat_iter_S, nat_iter_0.
+   rewrite !N.add_succ_r, N.add_0_r, !to_lu_succ, IHp.
+   destruct (to_lu (N.pos p)); simpl; auto.
+Qed.
+
+Lemma of_lu_0 d : of_lu d = 0 <-> nztail d = Nil.
+Proof.
+ induction d; try simpl_of_lu; split; trivial; try discriminate;
+ try (intros H; now apply N.eq_add_0 in H).
+ - rewrite N.add_0_l. intros H.
+   apply N.eq_mul_0_r in H; [|easy]. rewrite IHd in H.
+   simpl. now rewrite H.
+ - simpl. destruct (nztail d); try discriminate.
+   now destruct IHd as [_ ->].
+Qed.
+
+Lemma to_of_lu_tenfold d :
+ to_lu (of_lu d) = lnorm d ->
+ to_lu (10 * of_lu d) = lnorm (D0 d).
+Proof.
+ intro IH.
+ destruct (N.eq_dec (of_lu d) 0) as [H|H].
+ - rewrite H. simpl. rewrite of_lu_0 in H.
+   unfold lnorm. simpl. now rewrite H.
+ - destruct (of_lu d) eqn:Eq; [easy| ].
+   rewrite to_ldec_tenfold; auto. rewrite IH.
+   rewrite <- Eq in H. rewrite of_lu_0 in H.
+   unfold lnorm. simpl. now destruct (nztail d).
+Qed.
+
+Lemma Nadd_alt n m : n + m = Nat.iter (N.to_nat n) N.succ m.
+Proof.
+ destruct n. trivial.
+ induction p using Pos.peano_rect.
+ - now rewrite N.add_1_l.
+ - change (N.pos (Pos.succ p)) with (N.succ (N.pos p)).
+   now rewrite N.add_succ_l, IHp, N2Nat.inj_succ.
+Qed.
+
+Ltac simpl_to_nat := simpl N.to_nat; unfold Pos.to_nat; simpl Pos.iter_op.
+
+Lemma to_of_lu d : to_lu (of_lu d) = lnorm d.
+Proof.
+ induction d; [reflexivity|..];
+ simpl_of_lu; rewrite Nadd_alt; simpl_to_nat;
+ rewrite ?nat_iter_S, nat_iter_0, ?to_lu_succ, to_of_lu_tenfold by assumption;
+ unfold lnorm; simpl; destruct nztail; auto.
+Qed.
+
+(** Second bijection result *)
+
+Lemma to_of (d:uint) : N.to_uint (Pos.of_uint d) = unorm d.
+Proof.
+ rewrite of_uint_alt.
+ unfold N.to_uint, Pos.to_uint.
+ destruct (of_lu (rev d)) eqn:H.
+ - rewrite of_lu_0 in H. rewrite <- rev_lnorm_rev.
+   unfold lnorm. now rewrite H.
+ - change (Pos.to_little_uint p) with (to_lu (N.pos p)).
+   rewrite <- H. rewrite to_of_lu. apply rev_lnorm_rev.
+Qed.
+
+(** Some consequences *)
+
+Lemma to_uint_nonzero p : Pos.to_uint p <> zero.
+Proof.
+ intro E. generalize (of_to p). now rewrite E.
+Qed.
+
+Lemma to_uint_nonnil p : Pos.to_uint p <> Nil.
+Proof.
+ intros E. generalize (of_to p). now rewrite E.
+Qed.
+
+Lemma to_uint_inj p p' : Pos.to_uint p = Pos.to_uint p' -> p = p'.
+Proof.
+ intro E.
+ assert (E' : N.pos p = N.pos p').
+ { now rewrite <- (of_to p), <- (of_to p'), E. }
+ now injection E'.
+Qed.
+
+Lemma to_uint_pos_surj d :
+ unorm d<>zero -> exists p, Pos.to_uint p = unorm d.
+Proof.
+ intros.
+ destruct (Pos.of_uint d) eqn:E.
+ - destruct H. generalize (to_of d). now rewrite E.
+ - exists p. generalize (to_of d). now rewrite E.
+Qed.
+
+Lemma of_uint_norm d : Pos.of_uint (unorm d) = Pos.of_uint d.
+Proof.
+ now induction d.
+Qed.
+
+Lemma of_inj d d' :
+ Pos.of_uint d = Pos.of_uint d' -> unorm d = unorm d'.
+Proof.
+ intros. rewrite <- !to_of. now f_equal.
+Qed.
+
+Lemma of_iff d d' : Pos.of_uint d = Pos.of_uint d' <-> unorm d = unorm d'.
+Proof.
+ split. apply of_inj. intros E. rewrite <- of_uint_norm, E.
+ apply of_uint_norm.
+Qed.
+
+End Unsigned.
+
+(** Conversion from/to signed decimal numbers *)
+
+Module Signed.
+
+Lemma of_to (p:positive) : Pos.of_int (Pos.to_int p) = Some p.
+Proof.
+ unfold Pos.to_int, Pos.of_int, norm.
+ now rewrite Unsigned.of_to.
+Qed.
+
+Lemma to_of (d:int)(p:positive) :
+ Pos.of_int d = Some p -> Pos.to_int p = norm d.
+Proof.
+ unfold Pos.of_int.
+ destruct d; [ | intros [=]].
+ simpl norm. rewrite <- Unsigned.to_of.
+ destruct (Pos.of_uint d); now intros [= <-].
+Qed.
+
+Lemma to_int_inj p p' : Pos.to_int p = Pos.to_int p' -> p = p'.
+Proof.
+ intro E.
+ assert (E' : Some p = Some p').
+ { now rewrite <- (of_to p), <- (of_to p'), E. }
+ now injection E'.
+Qed.
+
+Lemma to_int_pos_surj d :
+ unorm d <> zero -> exists p, Pos.to_int p = norm (Pos d).
+Proof.
+ simpl. unfold Pos.to_int. intros H.
+ destruct (Unsigned.to_uint_pos_surj d H) as (p,Hp).
+ exists p. now f_equal.
+Qed.
+
+Lemma of_int_norm d : Pos.of_int (norm d) = Pos.of_int d.
+Proof.
+ unfold Pos.of_int.
+ destruct d.
+ - simpl. now rewrite Unsigned.of_uint_norm.
+ - simpl. now destruct (nzhead d) eqn:H.
+Qed.
+
+Lemma of_inj_pos d d' :
+ Pos.of_int (Pos d) = Pos.of_int (Pos d') -> unorm d = unorm d'.
+Proof.
+ unfold Pos.of_int.
+ destruct (Pos.of_uint d) eqn:Hd, (Pos.of_uint d') eqn:Hd';
+  intros [=].
+ - apply Unsigned.of_inj; now rewrite Hd, Hd'.
+ - apply Unsigned.of_inj; rewrite Hd, Hd'; now f_equal.
+Qed.
+
+End Signed.
diff --git a/theories/Numbers/DecimalZ.v b/theories/Numbers/DecimalZ.v
new file mode 100644
index 000000000..92d66ecfb
--- /dev/null
+++ b/theories/Numbers/DecimalZ.v
@@ -0,0 +1,73 @@
+(************************************************************************)
+(*  v      *   The Coq Proof Assistant  /  The Coq Development Team     *)
+(* <O___,, *   INRIA - CNRS - LIX - LRI - PPS - Copyright 1999-2016     *)
+(*   \VV/  **************************************************************)
+(*    //   *      This file is distributed under the terms of the       *)
+(*         *       GNU Lesser General Public License Version 2.1        *)
+(************************************************************************)
+
+(** * DecimalZ
+
+    Proofs that conversions between decimal numbers and [Z]
+    are bijections. *)
+
+Require Import Decimal DecimalFacts DecimalPos DecimalN ZArith.
+
+Lemma of_to (z:Z) : Z.of_int (Z.to_int z) = z.
+Proof.
+ destruct z; simpl.
+ - trivial.
+ - unfold Z.of_uint. rewrite DecimalPos.Unsigned.of_to. now destruct p.
+ - unfold Z.of_uint. rewrite DecimalPos.Unsigned.of_to. destruct p; auto.
+Qed.
+
+Lemma to_of (d:int) : Z.to_int (Z.of_int d) = norm d.
+Proof.
+ destruct d; simpl; unfold Z.to_int, Z.of_uint.
+ - rewrite <- (DecimalN.Unsigned.to_of d). unfold N.of_uint.
+   now destruct (Pos.of_uint d).
+ - destruct (Pos.of_uint d) eqn:Hd; simpl; f_equal.
+   + generalize (DecimalPos.Unsigned.to_of d). rewrite Hd. simpl.
+     intros H. symmetry in H. apply unorm_0 in H. now rewrite H.
+   + assert (Hp := DecimalPos.Unsigned.to_of d). rewrite Hd in Hp. simpl in *.
+     rewrite Hp. unfold unorm in *.
+     destruct (nzhead d); trivial.
+     generalize (DecimalPos.Unsigned.of_to p). now rewrite Hp.
+Qed.
+
+(** Some consequences *)
+
+Lemma to_int_inj n n' : Z.to_int n = Z.to_int n' -> n = n'.
+Proof.
+ intro EQ.
+ now rewrite <- (of_to n), <- (of_to n'), EQ.
+Qed.
+
+Lemma to_int_surj d : exists n, Z.to_int n = norm d.
+Proof.
+ exists (Z.of_int d). apply to_of.
+Qed.
+
+Lemma of_int_norm d : Z.of_int (norm d) = Z.of_int d.
+Proof.
+ unfold Z.of_int, Z.of_uint.
+ destruct d.
+ - simpl. now rewrite DecimalPos.Unsigned.of_uint_norm.
+ - simpl. destruct (nzhead d) eqn:H;
+   [ induction d; simpl; auto; discriminate |
+     destruct (nzhead_nonzero _ _ H) | .. ];
+   f_equal; f_equal; apply DecimalPos.Unsigned.of_iff;
+   unfold unorm; now rewrite H.
+Qed.
+
+Lemma of_inj d d' :
+ Z.of_int d = Z.of_int d' -> norm d = norm d'.
+Proof.
+ intros. rewrite <- !to_of. now f_equal.
+Qed.
+
+Lemma of_iff d d' : Z.of_int d = Z.of_int d' <-> norm d = norm d'.
+Proof.
+ split. apply of_inj. intros E. rewrite <- of_int_norm, E.
+ apply of_int_norm.
+Qed.
diff --git a/theories/PArith/BinPosDef.v b/theories/PArith/BinPosDef.v
index 85c401cf4..a77c26e5a 100644
--- a/theories/PArith/BinPosDef.v
+++ b/theories/PArith/BinPosDef.v
@@ -608,7 +608,7 @@ Fixpoint to_little_uint p :=
   | p~0 => Decimal.Little.double (to_little_uint p)
   end.
 
-Definition to_uint p := Decimal.rev (to_little_uint p) Decimal.Nil.
+Definition to_uint p := Decimal.rev (to_little_uint p).
 
 Definition to_int n := Decimal.Pos (to_uint n).