KEVM Lemmas

How to debug your KCFG and find KEVM reasoning gaps

In this section, we will verify Solady's mulWad function, demonstrating how to identify and write good lemmas.

The reasoning engine behind Kontrol is the K framework, which is accessed through the K definition of the EVM semantics, KEVM. Sometimes it is necessary to address reasoning gaps or suggest simplification strategies at the KEVM level. To gain a better understanding of the definitions and rules involved in expressions that are not being simplified as desired, you can explore the KEVM repository.

Verifying Solady's mulWad

Note: WAD = 10**18

For this example, we'll verify Solady's mulWadfunction, defined as:

function mulWad(uint256 x, uint256 y) internal pure returns (uint256 z) {
    assembly {
        if mul(y, gt(x, div(not(0), y))) {
            mstore(0x00, 0xbac65e5b)
            revert(0x1c, 0x04)
        z := div(mul(x, y), WAD)

Our goal is to demonstrate that this is equivalent to (x * y) / WAD for any two integers x and y. To achieve this, we define the following property test:

// Public wrapper for `FixedPointMathLib.mulWad` to ensure that `vm.expectRevert` works well
function mulWad(uint x, uint y) public pure returns (uint256) {
    return FixedPointMathLib.mulWad(x, y);

function testMulWad(uint256 x, uint256 y) public {
    if (y == 0 || x <= type(uint256).max / y) {
        uint256 zSpec = (x * y) / WAD;
        uint256 zImpl = FixedPointMathLib.mulWad(x, y);

        assert(zImpl == zSpec);
    } else {
        this.mulWad(x, y);

This property test asserts that if the product x*y does not result in an overflow, or if y is zero, then output of mulWad(x, y) should be equal to (x*y)/WAD. However, if an overflow occurs, the function mulWad should revert.

After symbolically executing this test, the generated message indicates that Kontrol was unable to prove that the property holds for all possible inputs. This output will look like the following:

1 Failure nodes. (0 pending and 1 failing)

Failing nodes:

  Node id: 23
  Failure reason:
    Implication check failed, the following is the remaining implication:
    ( ( #Not ( { VV1_y_114b9705:Int #Equals 0 } )
    #And ( { true #Equals 0 <=Int CALLER_ID:Int }
    #And ( { true #Equals 0 <=Int ORIGIN_ID:Int }
    #And ( { true #Equals 0 <=Int NUMBER_CELL:Int }
    #And ( { true #Equals 0 <=Int VV0_x_114b9705:Int }
    #And ( { true #Equals 0 <=Int VV1_y_114b9705:Int }
    #And ( { true #Equals CALLER_ID:Int <Int pow160 }
    #And ( { true #Equals ORIGIN_ID:Int <Int pow160 }
    #And ( { true #Equals NUMBER_CELL:Int <=Int maxSInt256 }
    #And ( { true #Equals VV0_x_114b9705:Int <Int pow256 }
    #And ( { true #Equals VV1_y_114b9705:Int <Int pow256 }
    #And ( { true #Equals VV0_x_114b9705:Int <=Int ( maxUInt256 /Int VV1_y_114b9705:Int ) }
    #And ( { false #Equals ( ( notBool VV0_x_114b9705:Int ==Int 0 ) andBool maxUInt256 /Word VV0_x_114b9705:Int <Int VV1_y_114b9705:Int ) }
    #And #Not ( { 0 #Equals chop ( ( VV1_y_114b9705:Int *Int bool2Word ( ( maxUInt256 /Int VV1_y_114b9705:Int ) <Int VV0_x_114b9705:Int ) ) ) } ) ) ) ) ) ) ) ) ) ) ) ) ) ) #Implies { true #Equals foundry_success ( ... statusCode: EVMC_REVERT , failed: #lookup ( .Map , 46308022326495007027972728677917914892729792999299745830475596687180801507328 ) , revertExpected: false , opcodeExpected: false , recordEventExpected: false , eventExpected: false ) } )
  Path condition:
    ( { true #Equals ( notBool VV1_y_114b9705:Int ==Int 0 ) }
#And ( { true #Equals ( notBool ( maxUInt256 /Int VV1_y_114b9705:Int ) <Int VV0_x_114b9705:Int ) }
#And { true #Equals ( notBool chop ( ( VV1_y_114b9705:Int *Int bool2Word ( ( maxUInt256 /Int VV1_y_114b9705:Int ) <Int VV0_x_114b9705:Int ) ) ) ==Int 0 ) } ) )

Our next step is to inspect the KCFG (kontrol view-kcfg) to understand which path condition is leading to node 23. To learn more about the KCFG: K Control Flow Graph (KCFG)

Identifying branching conditions

When we inspect the branching condition that leads to the failing node, we can see that it corresponds to the if statement of the mulWad function.

Let's unparse the branching condition. The condition is:

chop ( ( VV1_y_114b9705:Int *Int bool2Word ( ( maxUInt256 /Int VV1_y_114b9705:Int) <Int VV0_x_114b9705:Int ) ) ) ==Int 0

Taking into account that chop(x) is (x)mod[2**256], we can write the expression in a more palatable way:

(y *Int bool2Word((maxUInt256 /Int y) <Int x)) modInt 2**256 ==Int 0

There are three main things to know about this condition.

  • The operands *Int, <Int, modInt, /Int, and ==Int are K functions doing the obvious thing, with the suffix Int indicating the type of their arguments

  • The function bool2Word takes a boolean value and converts it into an EVM word. Particularly, it converts true to 1 and false to 0. You can find the definition here.

  • The modulo operation appears because, inside assembly blocks, the Solidity compiler doesn't insert overflow checks

Thus, the condition is equivalent to y == 0 || types(uint256).max / y >= x. Now, since this is the condition to branch on the first if of the mulWad function, Kontrol will explore the two different branches, one asserting that (y *Int bool2Word((maxUInt256 /Int y) <Int x)) modInt 2**256 ==Int 0 holds, and another one asserting that the opposite holds (via the notBool operator).

The branch starting on node 21 explores the path of negating (y *Int bool2Word((maxUInt256 /Int y) <Int x)) modInt 2**256 ==Int 0. That is, entering the if statement, which is why we see that node 23 has statusCode: EVMC_REVERT.

Dealing with path constraints

Having a node with a status code EVMC_REVERT can mean that either there's a bug and that revert should not be happening, or that the branch is unfeasible but Kontrol did not realize it. Let's look at the path conditions of node 21 to see if a contradiction can be derived.

If we look a the first condition and the ones at the end, we can spot something suspicious. Let's inspect the following conditions:

  1. #Not ( { VV1_y_114b9705:Int #Equals 0 } )

  2. { VV0_x_114b9705:Int <=Int ( maxUInt256 /Int VV1_y_114b9705:Int ) #Equals true }

  3. ( notBool chop ( ( VV1_y_114b9705:Int *Int bool2Word ( ( maxUInt256 /Int VV1_y_114b9705:Int ) <Int VV0_x_114b9705:Int ) ) ) ==Int 0 )

After renaming the Matching Logic operators and other syntactic noise, the three conditions are

  1. y != 0

  2. x <= maxUInt256 / y

  3. y * bool2Word(maxUInt256 / y < x)) != 0

Given these constraints, we can see how 1 and 2 imply that condition 3 is false. Thus we're not facing a bug in mulWad, but rather a reasoning gap in Kontrol.

Bridging the (reasoning) gap

Kontrol is able to infer maxUInt256 / y < x = false from conditions 1 and 2, and a quick git grep -rin 'rule bool2Word' in the evm-semantics definition tells us that these are the defined rewrite rules for bool2Word:     rule bool2Word( true  ) => 1     rule bool2Word( false ) => 0

lemmas/lemmas.k:50:  rule bool2Word(A) |Int bool2Word(B) => bool2Word(A  orBool B) [simplification]
lemmas/lemmas.k:51:  rule bool2Word(A) &Int bool2Word(B) => bool2Word(A andBool B) [simplification]

lemmas/lemmas.k:56:  rule bool2Word(_B) |Int 1 => 1            [simplification]
lemmas/lemmas.k:57:  rule bool2Word( B) &Int 1 => bool2Word(B) [simplification]

lemmas/lemmas.k:174: rule bool2Word(X ==Int 1)        => X requires #rangeBool(X) [simplification]
lemmas/lemmas.k:178: rule bool2Word( B:Bool ) ==Int I => B ==K word2Bool(I)       [simplification, concrete(I)]

Knowing the existing rules for any function is important since this means knowing how the function is treated by KEVM and thus by Kontrol.

Note that none of those rules is telling Kontrol (actually, KEVM) to attempt to simplify the boolean arguments of bool2Word. Because none of these rules really apply to condition 3, no simplification is made. This is why Kontrol derives no contradiction from condition 3 above, it's treating that expression as purely symbolic. To tell Kontrol to attempt to simplify the boolean arguments of the bool2Word function, we need to add the following rules to the file described in the last section

rule bool2Word ( X ) => 1 requires X         [simplification]
rule bool2Word ( X ) => 0 requires notBool X [simplification]

Now Kontrol will attempt to simplify X to true or false in order to apply any of these two new rules. Indeed, if we run again our property test with these added rules, the proof will pass.

Finishing an existing proof

We've identified the missing reasoning link, great! To include our brand new lemmas in the KEVM definition (i.e., make them available for Kontrol to reason with) we need to rekompile the project, that is, building it again with the --rekompile flag (kontrol build --rekompile --require ${path_to_lemmas} --module-import ${module}). After this, re-running the proof with the --reinit flag will attempt (and succeed) to prove our property from scratch. But what if we don't want to spend more computational resources than necessary to finish the proof from where we left it? That is, what if don't want to explore those branches that we have already explored and know are not problematic? Enter KCFG manipulation.

Node pruning

From the screenshots above we know that the branch starting at node 21 should be pruned by Kontrol itself. However, due to the lack of reasoning arguments, it was not able to do so. Now that we've endowed Kontrol with the necessary weaponry, all we have to do is remove the reverting node, node 23, and restart the proof without the --reinit flag to let Kontrol resume the proof without the reverting node, and figure out that such an execution branch will never be traversed.

To remove the reverting node we can run the following command

kontrol remove-node FixedPointMathLibVerification.testMulWad 23

Note that FixedPointMathLibVerification is the contract where our property test lives.

With the node pruned, our configuration will look like this

Resuming execution

After this, we can resume the proof without the --reinit flag

kontrol prove --test FixedPointMathLibVerification.testMulWad

Indeed, after allowing Kontrol to reason again about the reverting branch, we're greeted with the following message

PROOF PASSED: FixedPointMathLibVerification.testMulWad(uint256,uint256):0

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