无法证明 Ada Spark 中看似微不足道的平等

Question

所以我有这两个文件。

测试广告

package Testing with
   SPARK_Mode
is

   function InefficientEuler1Sum2 (N: Natural) return Natural;

   procedure LemmaForTesting with
     Ghost,
     Post => (InefficientEuler1Sum2(0) = 0);

end Testing;

和测试.adb

package body Testing with
   SPARK_Mode
is

   function InefficientEuler1Sum2 (N: Natural) return Natural is
      Sum: Natural := 0;
   begin
      for I in 0..N loop
         if I mod 3 = 0 then
            Sum := Sum + I;
         end if;
         if I mod 5 = 0 then
            Sum := Sum + I;
         end if;
         if I mod 15 = 0 then
            Sum := Sum - I;
         end if;
      end loop;
      return Sum;
   end InefficientEuler1Sum2;

   procedure LemmaForTesting
   is
   begin
      null;
   end LemmaForTesting;

end Testing;

当我运行 SPARK -> Prove File 时，我收到这样的消息：

GNATprove
    E:\Ada\Testing SPARK\search\src\testing.ads
        10:14 medium: postcondition might fail
           cannot prove InefficientEuler1Sum2(0) = 0

为什么呢？我误解了什么或者做错了什么？提前致谢。

Answer 1

为了证明平凡的等式，您需要确保它被函数的后置条件所覆盖。如果是这样，您可以使用Assert如下示例所示的简单语句来证明相等性。此时不需要引理。

但是，后置条件不足以证明不存在运行时错误 (AoRTE)：给定函数的允许输入范围，对于 的某些值，求和可能会N溢出。为了缓解这个问题，您需要使用循环不变量限制 的输入值N并向证明者展示 的值Sum在循环期间保持有界（有关循环不变量的一些背景信息，请参见此处、此处和此处）。出于说明目的，我选择了 的保守界限(2 * I) * I，这将严格限制输入值的允许范围，但确实允许证明者证明示例中不存在运行时错误。

测试广告

package Testing with SPARK_Mode is

   --  Using the loop variant in the function body, one can guarantee that no
   --  overflow will occur for all values of N in the range 
   --
   --     0 .. Sqrt (Natural'Last / 2)   <=>   0 .. 32767
   --
   --  Of course, this bound is quite conservative, but it may be enough for a
   --  given application.
   --
   --  The post-condition can be used to prove the trivial equality as stated
   --  in your question.
   
   subtype Domain is Natural range 0 .. 32767;
   
   function Inefficient_Euler_1_Sum_2 (N : Domain) return Natural
     with Post => (if N = 0 then Inefficient_Euler_1_Sum_2'Result = 0);

end Testing;

测试.adb

package body Testing with SPARK_Mode  is

   -------------------------------
   -- Inefficient_Euler_1_Sum_2 --
   -------------------------------
   
   function Inefficient_Euler_1_Sum_2 (N : Domain) return Natural is
      Sum: Natural := 0;
   begin
      
      for I in 0 .. N loop
         
         if I mod 3 = 0 then
            Sum := Sum + I;
         end if;
         if I mod 5 = 0 then
            Sum := Sum + I;
         end if;
         if I mod 15 = 0 then
            Sum := Sum - I;
         end if;
         
         --  Changed slightly since initial post, no effect on Domain.
         pragma Loop_Invariant (Sum <= (2 * I) * I);
         
      end loop;
      
      return Sum;
      
   end Inefficient_Euler_1_Sum_2;

end Testing;

主文件

with Testing; use Testing;

procedure Main with SPARK_Mode is
begin
   pragma Assert (Inefficient_Euler_1_Sum_2 (0) = 0);   
end Main;

输出

$ gnatprove -Pdefault.gpr -j0 --level=1 --report=all
Phase 1 of 2: generation of Global contracts ...
Phase 2 of 2: flow analysis and proof ...
main.adb:5:19: info: assertion proved
testing.adb:13:15: info: division check proved
testing.adb:14:24: info: overflow check proved
testing.adb:16:15: info: division check proved
testing.adb:17:24: info: overflow check proved
testing.adb:19:15: info: division check proved
testing.adb:20:24: info: overflow check proved
testing.adb:20:24: info: range check proved
testing.adb:23:33: info: loop invariant preservation proved
testing.adb:23:33: info: loop invariant initialization proved
testing.adb:23:42: info: overflow check proved
testing.adb:23:46: info: overflow check proved
testing.ads:17:19: info: postcondition proved
Summary logged in /obj/gnatprove/gnatprove.out