未能断言 libsparkcrypto SHA256 结果相等

Question

我的问题总结

我将libsparkcrypto library 用于我的 SHA256 函数。我发现我不能 Assert x = y 暗示 Sha256(x) = Sha256(y)。任何帮助将不胜感激。

代码

testpackage.adb

package body TestPackage with
 SPARK_Mode
is
   
   function Is_Equal (X, Y : LSC.Types.Bytes) return Boolean is
   begin
      if X = Y then
         pragma Assert (LSC.SHA2.Hash_SHA256 (X) = LSC.SHA2.Hash_SHA256 (Y));
         return True;
      end if;
      return (LSC.SHA2.Hash_SHA256 (X) = LSC.SHA2.Hash_SHA256 (Y));
   end Is_Equal;

end TestPackage;

testpackage.ads

with LSC.Types; use LSC.Types;
with LSC.SHA2;

package TestPackage with
 SPARK_Mode
is
   
   function Is_Equal (X, Y : LSC.Types.Bytes) return Boolean with
     Post => Is_Equal'Result = (LSC.SHA2.Hash_SHA256 (X) = LSC.SHA2.Hash_SHA256 (Y));
   
end TestPackage;

输出

我收到的错误是：

testpackage.adb:8:25: medium：断言可能失败，无法证明 LSC.SHA2.Hash_SHA256 (X) = LSC.SHA2.Hash_SHA256 (Y) [可能 解释： testpackage.ads:8 的子程序应该在 前置条件][#0]

我的gnatprove.out：

Summary of SPARK analysis
=========================

-------------------------------------------------------------------------------------------
SPARK Analysis results        Total      Flow   CodePeer     Provers   Justified   Unproved
-------------------------------------------------------------------------------------------
Data Dependencies                 .         .          .           .           .          .
Flow Dependencies                 .         .          .           .           .          .
Initialization                    .         .          .           .           .          .
Non-Aliasing                      .         .          .           .           .          .
Run-time Checks                   6         .          .    6 (CVC4)           .          .
Assertions                        1         .          .           .           .          1
Functional Contracts              1         .          .    1 (CVC4)           .          .
LSP Verification                  .         .          .           .           .          .
Termination                       .         .          .           .           .          .
Concurrency                       .         .          .           .           .          .
-------------------------------------------------------------------------------------------
Total                             8         .          .     7 (88%)           .    1 (13%)


max steps used for successful proof: 1

Analyzed 2 units
in unit main, 0 subprograms and packages out of 1 analyzed
  Main at main.adb:8 skipped
in unit testpackage, 2 subprograms and packages out of 2 analyzed
  TestPackage at testpackage.ads:4 flow analyzed (0 errors, 0 checks and 0 warnings) and proved (0 checks)
  TestPackage.Is_Equal at testpackage.ads:8 flow analyzed (0 errors, 0 checks and 0 warnings) and not proved, 7 checks out of 8 proved

Answer 1

虽然这不是问题的答案，但对较小示例的一些调查表明该问题并非特定于 libsparkcrypto 库中的 LSC.SHA2.Hash_SHA256 函数。看起来在证明具有数组类型参数的函数的“纯度”方面存在普遍的困难。另一方面，具有标量类型参数的函数按预期证明。

所以问题可能是数组上缺少一些条件，求解器超时时间太短，或者只是 SPARK 目前无法证明这样的事情（例如，参见 SPARK UG 中的 7.8.3 部分）。关于缺少的条件：我不确定（还）这些缺少的条件是什么，我已经添加了很多，但似乎没有任何帮助。

如果您是证明专家，那么您可以通过检查在手动证明环境中失败的“目标”来进一步调查问题（有关详细信息，另请参阅 SPARK UG 中的 7.1.8 部分）。不幸的是，我在这里缺少合适的博士来了解 SPARK 工具的这一部分并对此提供任何帮助;-)。

pkg.ads

package Pkg with SPARK_Mode, Pure is 
   
   --------------------------------------------
   -- Functions with a scalar type parameter --
   --------------------------------------------
   
   function Fcn_Scalar_1 (X : Integer) return Integer;
   
   function Fcn_Scalar_2 (X : Integer) return Integer
     with Pure_Function;

   function Fcn_Scalar_3 (X : Integer) return Integer
     with 
       Global  => null, 
       Depends => (Fcn_Scalar_3'Result => X);

   function Fcn_Scalar_4 (X : Integer) return Integer
     with Post => Fcn_Scalar_4'Result = X; 
   
   --------------------------------------------
   -- Functions with an array type parameter --
   --------------------------------------------
   
   type Arr is array (Natural range <>) of Integer;
   
   function Fcn_Array_1 (X : Arr) return Integer;
   
   function Fcn_Array_2 (X : Arr) return Integer
     with Pure_Function;

   function Fcn_Array_3 (X : Arr) return Integer
     with       
       Global  => null, 
       Depends => (Fcn_Array_3'Result => X);

   function Fcn_Array_4 (X : Arr) return Arr
     with Post => Fcn_Array_4'Result = X; 

end Pkg;

test.ads

with Pkg; use Pkg;

package Test with SPARK_Mode is
   
   --  Is_Equal_Scalar_1 : Postcondition proved.
   --  Is_Equal_Scalar_2 : Postcondition proved.
   --  Is_Equal_Scalar_3 : Postcondition proved. 
   --  Is_Equal_Scalar_4 : Postcondition proved.  
   
   function Is_Equal_Scalar_1 (X, Y : Integer) return Boolean is
     (if X = Y then True else Fcn_Scalar_1 (X) = Fcn_Scalar_1 (Y))
       with Post => Is_Equal_Scalar_1'Result = (Fcn_Scalar_1 (X) = Fcn_Scalar_1 (Y));
   
   function Is_Equal_Scalar_2 (X, Y : Integer) return Boolean is
     (if X = Y then True else Fcn_Scalar_2 (X) = Fcn_Scalar_2 (Y))
       with Post => Is_Equal_Scalar_2'Result = (Fcn_Scalar_2 (X) = Fcn_Scalar_2 (Y));
   
   function Is_Equal_Scalar_3 (X, Y : Integer) return Boolean is
     (if X = Y then True else Fcn_Scalar_3 (X) = Fcn_Scalar_3(Y))
       with Post => Is_Equal_Scalar_3'Result = (Fcn_Scalar_3 (X) = Fcn_Scalar_3 (Y));
   
   function Is_Equal_Scalar_4 (X, Y : Integer) return Boolean is
     (if X = Y then True else Fcn_Scalar_4 (X) = Fcn_Scalar_4(Y))
       with Post => Is_Equal_Scalar_4'Result = (Fcn_Scalar_4 (X) = Fcn_Scalar_4 (Y));
  
   --  Is_Equal_Array_1 : Postcondition NOT proved.
   --  Is_Equal_Array_2 : Postcondition NOT proved.
   --  Is_Equal_Array_3 : Postcondition NOT proved.
   --  Is_Equal_Array_4 : Postcondition proved, but only because of the postcondition on Fcn_Array_4.
   
   function Is_Equal_Array_1 (X, Y : Arr) return Boolean is
     (if X = Y then True else Fcn_Array_1 (X) = Fcn_Array_1 (Y))
         Pre  => X'First = 0 and Y'First = 0 and X'Length = Y'Length and X'Length > 0 and Y'Length > 0,
         Post => Is_Equal_Array_1'Result = (Fcn_Array_1 (X) = Fcn_Array_1 (Y));
   
   function Is_Equal_Array_2 (X, Y : Arr) return Boolean is
     (if X = Y then True else Fcn_Array_2 (X) = Fcn_Array_2 (Y))
       with 
         Pre  => X'First = 0 and Y'First = 0 and X'Length = Y'Length and X'Length > 0 and Y'Length > 0,
         Post => Is_Equal_Array_2'Result = (Fcn_Array_2 (X) = Fcn_Array_2 (Y));
   
   function Is_Equal_Array_3 (X, Y : Arr) return Boolean is
     (if X = Y then True else Fcn_Array_3 (X) = Fcn_Array_3 (Y))
       with 
         Pre  => X'First = 0 and Y'First = 0 and X'Length = Y'Length and X'Length > 0 and Y'Length > 0,
         Post => Is_Equal_Array_3'Result = (Fcn_Array_3 (X) = Fcn_Array_3 (Y));
   
   function Is_Equal_Array_4 (X, Y : Arr) return Boolean is
     (if X = Y then True else Fcn_Array_4 (X) = Fcn_Array_4 (Y))
       with Post => Is_Equal_Array_4'Result = (Fcn_Array_4 (X) = Fcn_Array_4 (Y));

end Test;

输出（gnatprove）

$ gnatprove -Pdefault.gpr --level=2 -j0 -u test.ads --report=statistics
Phase 1 of 2: generation of Global contracts ...
Phase 2 of 2: flow analysis and proof ...
test.ads:12:21: info: postcondition proved (CVC4: 2 VC in max 0.0 seconds and 1 step)
test.ads:16:21: info: postcondition proved (CVC4: 2 VC in max 0.0 seconds and 1 step)
test.ads:20:21: info: postcondition proved (CVC4: 2 VC in max 0.0 seconds and 1 step)
test.ads:24:21: info: postcondition proved (CVC4: 2 VC in max 0.0 seconds and 1 step)
test.ads:35:18: medium: postcondition might fail, cannot prove Is_Equal_Array_1'Result = (Fcn_Array_1 (X) = Fcn_Array_1 (Y)) (e.g. when X = (others => 0) and X'First = 0 and X'Last = 0 and Y = (others => 0) and Y'First = 0 and Y'Last = 0)
test.ads:41:18: medium: postcondition might fail, cannot prove Is_Equal_Array_2'Result = (Fcn_Array_2 (X) = Fcn_Array_2 (Y)) (e.g. when X = (others => 0) and X'First = 0 and X'Last = 0 and Y = (others => 0) and Y'First = 0 and Y'Last = 0)
test.ads:47:18: medium: postcondition might fail, cannot prove Is_Equal_Array_3'Result = (Fcn_Array_3 (X) = Fcn_Array_3 (Y)) (e.g. when X = (others => 0) and X'First = 0 and X'Last = 0 and Y = (others => 0) and Y'First = 0 and Y'Last = 0)
test.ads:51:21: info: postcondition proved (CVC4: 2 VC in max 0.0 seconds and 1 step)
Summary logged in /obj/gnatprove/gnatprove.out

---

更新

转念一想，还有更多。虽然仍然无法解决问题，但我意识到 Is_Equal 函数的先决条件在数组类型的情况下是强制性的。这是因为数组的相等运算符在 Ada 中的行为方式。数组上的相等运算符不考虑索引边界 (RM 4.5.2 (18))，它只检查数组长度及其组件值。因此，以下数组 A1 和 A2 被认为是相等的：

type Arr is array (Natural range <>) of Integer;

A1 : constant Arr (0 .. 3) := (1, 2, 3, 4);
A2 : constant Arr (1 .. 4) := (1, 2, 3, 4);   --  Bounds on index differ.

现在将简单函数First_Index定义为：

function First_Index (A : Arr) return Integer is (A'First);

此函数返回数组的索引下限。不幸的是，gnatprove 将无法证明这个 First_Index 函数的 Is_Equal 函数只有一个后置条件，原因很明显。

function Is_Equal (X, Y : Arr) return Boolean is
     (if X = Y then True else First_Index (X) = First_Index (Y))
       with Post => Is_Equal'Result = (First_Index (X) = First_Index (Y));

因此，前提条件是强制性的，因为“纯”函数的结果可能取决于数组的边界。有了这个前提，就可以证明这个功能（见下文）。对于前面示例中的情况，这是行不通的。

ma​​in.adb

with Ada.Text_IO; use Ada.Text_IO;

procedure Main with SPARK_Mode is
   
   type Arr is array (Natural range <>) of Integer;   
   
   function First_Index (A : Arr) return Integer is (A'First);
   
   function Is_Equal (X, Y : Arr) return Boolean is
     (if X = Y then True else First_Index (X) = First_Index (Y))
       with  
         Pre  => X'First = 0 and Y'First = 0,
         Post => Is_Equal'Result = (First_Index (X) = First_Index (Y));   
   
   A1 : constant Arr (0 .. 3) := (1, 2, 3, 4);
   A2 : constant Arr (1 .. 4) := (1, 2, 3, 4);   --  Bounds on index differ.
   
begin
   if (A1 = A2) then
      Put_Line ("Equal");
   else
      Put_Line ("Not Equal");
   end if;
end Main;

输出（主要）

Equal

输出（gnatprove）

$ gnatprove -Pdefault.gpr --level=1 -j0 -u main.adb --report=statistics
Phase 1 of 2: generation of Global contracts ...
Phase 2 of 2: flow analysis and proof ...
main.adb:16:18: info: postcondition proved (CVC4: 2 VC in max 0.0 seconds and 1 step)
Summary logged in obj/gnatprove/gnatprove.out

Answer 2

可能的解释（是的，我知道，爸爸开玩笑）。您没有为 X 和 Y 设置任何前置条件检查，因此 SPARK 无法验证它们。即使他们是同一类型。尝试设置任何检查，看看会发生什么。一般来说，SPARK 喜欢一切都在合同中，越多越好。