[Ada] Proof of the runtime support for attribute 'Width

diff --git a/gcc/ada/libgnat/s-widlllu.ads b/gcc/ada/libgnat/s-widlllu.ads
--- a/gcc/ada/libgnat/s-widlllu.ads
+++ b/gcc/ada/libgnat/s-widlllu.ads
@@ -34,8 +34,9 @@ 
 with System.Width_U;
 with System.Unsigned_Types;
 
-package System.Wid_LLLU is
-
+package System.Wid_LLLU
+  with SPARK_Mode
+is
    subtype Long_Long_Long_Unsigned is Unsigned_Types.Long_Long_Long_Unsigned;
 
    function Width_Long_Long_Long_Unsigned is


diff --git a/gcc/ada/libgnat/s-widllu.ads b/gcc/ada/libgnat/s-widllu.ads
--- a/gcc/ada/libgnat/s-widllu.ads
+++ b/gcc/ada/libgnat/s-widllu.ads
@@ -34,8 +34,9 @@ 
 with System.Width_U;
 with System.Unsigned_Types;
 
-package System.Wid_LLU is
-
+package System.Wid_LLU
+  with SPARK_Mode
+is
    subtype Long_Long_Unsigned is Unsigned_Types.Long_Long_Unsigned;
 
    function Width_Long_Long_Unsigned is new Width_U (Long_Long_Unsigned);


diff --git a/gcc/ada/libgnat/s-widthu.adb b/gcc/ada/libgnat/s-widthu.adb
--- a/gcc/ada/libgnat/s-widthu.adb
+++ b/gcc/ada/libgnat/s-widthu.adb
@@ -29,10 +29,87 @@ 
 --                                                                          --
 ------------------------------------------------------------------------------
 
+with Ada.Numerics.Big_Numbers.Big_Integers;
+use Ada.Numerics.Big_Numbers.Big_Integers;
+
 function System.Width_U (Lo, Hi : Uns) return Natural is
+
+   --  Ghost code, loop invariants and assertions in this unit are meant for
+   --  analysis only, not for run-time checking, as it would be too costly
+   --  otherwise. This is enforced by setting the assertion policy to Ignore.
+
+   pragma Assertion_Policy (Ghost          => Ignore,
+                            Loop_Invariant => Ignore,
+                            Assert         => Ignore);
+
    W : Natural;
    T : Uns;
 
+   package Unsigned_Conversion is new Unsigned_Conversions (Int => Uns);
+
+   function Big (Arg : Uns) return Big_Integer is
+     (Unsigned_Conversion.To_Big_Integer (Arg))
+   with Ghost;
+
+   --  Maximum value of exponent for 10 that fits in Uns'Base
+   function Max_Log10 return Natural is
+     (case Uns'Base'Size is
+        when 8   => 2,
+        when 16  => 4,
+        when 32  => 9,
+        when 64  => 19,
+        when 128 => 38,
+        when others => raise Program_Error)
+   with Ghost;
+
+   Max_W  : constant Natural := Max_Log10 with Ghost;
+   Big_10 : constant Big_Integer := Big (10) with Ghost;
+
+   procedure Lemma_Lower_Mult (A, B, C : Big_Natural)
+   with
+     Ghost,
+     Pre  => A <= B,
+     Post => A * C <= B * C;
+
+   procedure Lemma_Div_Commutation (X, Y : Uns)
+   with
+     Ghost,
+     Pre  => Y /= 0,
+     Post => Big (X) / Big (Y) = Big (X / Y);
+
+   procedure Lemma_Div_Twice (X : Big_Natural; Y, Z : Big_Positive)
+   with
+     Ghost,
+     Post => X / Y / Z = X / (Y * Z);
+
+   procedure Lemma_Lower_Mult (A, B, C : Big_Natural) is
+   begin
+      null;
+   end Lemma_Lower_Mult;
+
+   procedure Lemma_Div_Commutation (X, Y : Uns) is
+   begin
+      null;
+   end Lemma_Div_Commutation;
+
+   procedure Lemma_Div_Twice (X : Big_Natural; Y, Z : Big_Positive) is
+      XY  : constant Big_Natural := X / Y;
+      YZ  : constant Big_Natural := Y * Z;
+      XYZ : constant Big_Natural := X / Y / Z;
+      R   : constant Big_Natural := (XY rem Z) * Y + (X rem Y);
+   begin
+      pragma Assert (X = XY * Y + (X rem Y));
+      pragma Assert (XY = XY / Z * Z + (XY rem Z));
+      pragma Assert (X = XYZ * YZ + R);
+      pragma Assert ((XY rem Z) * Y <= (Z - 1) * Y);
+      pragma Assert (R <= YZ - 1);
+      pragma Assert (X / YZ = (XYZ * YZ + R) / YZ);
+      pragma Assert (X / YZ = XYZ + R / YZ);
+   end Lemma_Div_Twice;
+
+   Pow    : Big_Integer := 1 with Ghost;
+   T_Init : constant Uns := Uns'Max (Lo, Hi) with Ghost;
+
 begin
    if Lo > Hi then
       return 0;
@@ -50,10 +127,40 @@  begin
       --  Increase value if more digits required
 
       while T >= 10 loop
+         Lemma_Div_Commutation (T, 10);
+         Lemma_Div_Twice (Big (T_Init), Big_10 ** (W - 2), Big_10);
+
          T := T / 10;
          W := W + 1;
+         Pow := Pow * 10;
+
+         pragma Loop_Variant (Decreases => T);
+         pragma Loop_Invariant (W in 3 .. Max_W + 3);
+         pragma Loop_Invariant (Pow = Big_10 ** (W - 2));
+         pragma Loop_Invariant (Big (T) = Big (T_Init) / Pow);
       end loop;
 
+      declare
+         F : constant Big_Integer := Big_10 ** (W - 2) with Ghost;
+         Q : constant Big_Integer := Big (T_Init) / F with Ghost;
+         R : constant Big_Integer := Big (T_Init) rem F with Ghost;
+      begin
+         pragma Assert (Q < Big_10);
+         pragma Assert (Big (T_Init) = Q * F + R);
+         Lemma_Lower_Mult (Q, Big (9), F);
+         pragma Assert (Big (T_Init) <= Big (9) * F + F - 1);
+         pragma Assert (Big (T_Init) < Big_10 * F);
+         pragma Assert (Big_10 * F = Big_10 ** (W - 1));
+      end;
+
+      --  This is an expression of the functional postcondition for Width_U,
+      --  which cannot be expressed readily as a postcondition as this would
+      --  require making the instantiation Unsigned_Conversion and function
+      --  Big available from the spec.
+
+      pragma Assert (Big (Lo) < Big_10 ** (W - 1));
+      pragma Assert (Big (Hi) < Big_10 ** (W - 1));
+
       return W;
    end if;
 


diff --git a/gcc/ada/libgnat/s-widuns.ads b/gcc/ada/libgnat/s-widuns.ads
--- a/gcc/ada/libgnat/s-widuns.ads
+++ b/gcc/ada/libgnat/s-widuns.ads
@@ -34,8 +34,9 @@ 
 with System.Width_U;
 with System.Unsigned_Types;
 
-package System.Wid_Uns is
-
+package System.Wid_Uns
+  with SPARK_Mode
+is
    subtype Unsigned is Unsigned_Types.Unsigned;
 
    function Width_Unsigned is new Width_U (Unsigned);