From 53720903b96f9616f8bd53318575e8bcc13c6fce Mon Sep 17 00:00:00 2001
From: Pierre Roux
Date: Sat, 4 Apr 2020 22:36:02 +0200
Subject: Decimal: specify numeral notation for Q

---
 doc/stdlib/index-list.html.template |  1 +
 theories/Numbers/DecimalQ.v         | 95 +++++++++++++++++++++++++++++++++++++
 2 files changed, 96 insertions(+)
 create mode 100644 theories/Numbers/DecimalQ.v

diff --git a/doc/stdlib/index-list.html.template b/doc/stdlib/index-list.html.template
index 4a62888552..340e855a05 100644
--- a/doc/stdlib/index-list.html.template
+++ b/doc/stdlib/index-list.html.template
@@ -235,6 +235,7 @@ through the <tt>Require Import</tt> command.</p>
     theories/Numbers/DecimalPos.v
     theories/Numbers/DecimalN.v
     theories/Numbers/DecimalZ.v
+    theories/Numbers/DecimalQ.v
     theories/Numbers/DecimalString.v
     </dd>
 
diff --git a/theories/Numbers/DecimalQ.v b/theories/Numbers/DecimalQ.v
new file mode 100644
index 0000000000..d2cd061594
--- /dev/null
+++ b/theories/Numbers/DecimalQ.v
@@ -0,0 +1,95 @@
+(************************************************************************)
+(*         *   The Coq Proof Assistant / The Coq Development Team       *)
+(*  v      *         Copyright INRIA, CNRS and contributors             *)
+(* <O___,, * (see version control and CREDITS file for authors & dates) *)
+(*   \VV/  **************************************************************)
+(*    //   *    This file is distributed under the terms of the         *)
+(*         *     GNU Lesser General Public License Version 2.1          *)
+(*         *     (see LICENSE file for the text of the license)         *)
+(************************************************************************)
+
+(** * DecimalQ
+
+    Proofs that conversions between decimal numbers and [Q]
+    are bijections. *)
+
+Require Import Decimal DecimalFacts DecimalPos DecimalN DecimalZ QArith.
+
+Lemma of_to (q:Q) : forall d, to_decimal q = Some d -> of_decimal d = q.
+Admitted.
+
+(* normalize without fractional part, for instance norme 12.3e-1 is 123e-2 *)
+Definition dnorme (d:decimal) : decimal :=
+  let '(i, f, e) :=
+    match d with
+    | Decimal i f => (i, f, Pos Nil)
+    | DecimalExp i f e => (i, f, e)
+    end in
+  let i := norm (app_int i f) in
+  let e := norm (Z.to_int (Z.of_int e - Z.of_nat (nb_digits f))) in
+  match e with
+  | Pos zero => Decimal i Nil
+  | _ => DecimalExp i Nil e
+  end.
+
+(* normalize without exponent part, for instance norme 12.3e-1 is 1.23 *)
+Definition dnormf (d:decimal) : decimal :=
+  match dnorme d with
+  | Decimal i _ => Decimal i Nil
+  | DecimalExp i _ e =>
+    match Z.of_int e with
+    | Z0 => Decimal i Nil
+    | Zpos e => Decimal (norm (app_int i (Pos.iter D0 Nil e))) Nil
+    | Zneg e =>
+      let ne := Pos.to_nat e in
+      let ai := match i with Pos d | Neg d => d end in
+      let ni := nb_digits ai in
+      if ne <? ni then
+        Decimal (del_tail_int ne i) (del_head (ni - ne) ai)
+      else
+        let z := match i with Pos _ => Pos zero | Neg _ => Neg zero end in
+        Decimal z (Nat.iter (ne - ni) D0 ai)
+    end
+  end.
+
+Lemma dnorme_spec d :
+  match dnorme d with
+  | Decimal i Nil => i = norm i
+  | DecimalExp i Nil e => i = norm i /\ e = norm e /\ e <> Pos zero
+  | _ => False
+  end.
+Admitted.
+
+Lemma dnormf_spec d :
+  match dnormf d with
+  | Decimal i f => i = Neg zero \/ i = norm i
+  | _ => False
+  end.
+Admitted.
+
+Lemma dnorme_invol d : dnorme (dnorme d) = dnorme d.
+Admitted.
+
+Lemma dnormf_invol d : dnormf (dnormf d) = dnormf d.
+Admitted.
+
+Lemma to_of (d:decimal) :
+  to_decimal (of_decimal d) = Some (dnorme d)
+  \/ to_decimal (of_decimal d) = Some (dnormf d).
+Admitted.
+
+(** Some consequences *)
+
+Lemma to_decimal_inj q q' :
+  to_decimal q <> None -> to_decimal q = to_decimal q' -> q = q'.
+Admitted.
+
+Lemma to_decimal_surj d :
+  exists q, to_decimal q = Some (dnorme d) \/ to_decimal q = Some (dnormf d).
+Admitted.
+
+Lemma of_decimal_dnorme d : of_decimal (dnorme d) = of_decimal d.
+Admitted.
+
+Lemma of_decimal_dnormf d : of_decimal (dnormf d) = of_decimal d.
+Admitted.
-- 
cgit v1.2.3