Invariants

Invariants describe a property of the state of a contract that is always expected to hold.

Caution

Even if an invariant is verified, it may still be possible to violate it. This is a potential source of unsoundness. See Assumptions made while checking invariants for details.

Contents

• Invariants

  • Syntax

  • Overview

  • Assumptions made while checking invariants

  • Preserved blocks

  • Filters

  • Writing an invariant as a rule

  • Invariants and induction

Syntax

The syntax for invariants is given by the following EBNF grammar:

invariant ::= "invariant" id
              [ "(" params ")" ]
              expression
              [ "filtered" "{" id "->" expression "}" ]
              [ "{" { preserved_block } "}" ]

preserved_block ::= "preserved"
                    [ method_signature ]
                    [ "with" "(" params ")" ]
                    block

method_signature ::= id "(" [ evm_type [ id ] { "," evm_type [ id ] } ] ")"
                     | "fallback" "(" ")"

See Basic Syntax for the id production, Expressions for the expression production, and Statements for the block production.

Overview

In CVL, an invariant is a property of the contract state that is expected to be true whenever a contract method is not currently executing. This kind of invariant is sometimes called a “representation invariant”.

Each invariant has a name, possibly followed by a set of parameters, followed by a boolean expression. We say the invariant holds if the expression evaluates to true in every reachable state of the contract, and for all possible values of the parameters.

While verifying an invariant, the Prover checks two things. First, it checks that the invariant is established after calling any constructor. Second, it checks that the invariant holds after the execution of any contract method, assuming that it held before the method was executed (if it does hold, we say the method preserves the invariant).

If an invariant is proven, it is safe to assume that it holds in other rules and invariants. The requireInvariant command makes it easy to add this assumption to another rule, and is a quick way to rule out counterexamples that start in impossible states. See also Listing Safe Assumptions.

Note

Invariants are intended to describe the state of a contract at a particular point in time. Therefore, you should only use view functions inside of an invariant. Non-view functions are allowed, but the behavior is undefined.

Assumptions made while checking invariants

In Ethereum, the only way to change the storage state of a smart contract is using the smart contract’s methods. Therefore, if an invariant depends only on the storage of the contract, we can prove the invariant by checking it after calling each of the contract methods.

However, it is possible to write invariants whose value depends on things other than the contract’s storage. The truth of an expression may depend on the state of other contracts or on the environment. For these invariants, the expression can change from true to false without invoking a method on the main contract.

For example, consider the following contract:

contract Timestamp {
    uint256 public immutable timestamp;

    constructor() {
        timestamp = block.timestamp;
    }
}

The following invariant will be successfully verified, although it is clearly false:

invariant time_is_now(env e)
    timestamp() == e.block.timestamp;

The verification is successful because the action that falsifies the invariant is the passage of time, rather than the invocation of a contract method.

Similarly, an invariant that depends on an external contract can become false by calling a method on the external contract. For example:

contract SupplyTracker {
    address token;
    uint256 public supply;

    constructor(address _token) {
        token  = _token;
        supply = token.totalSupply();
    }
}

As above, an invariant stating that supply() == token.totalSupply() would be verified, but a method on token might change the total supply without updating the SupplyTracker contract. Since the Prover only checks the main contract’s methods for preservation, it will not report that the invariant can be falsified.

For this reason, invariants that depend on the environment or on the state of external contracts are a potential source of unsoundness, and should be used with care.

Todo

There is an additional source of unsoundness that occurs if the invariant expression reverts in the before state but not in the after state.

Preserved blocks

Often, the preservation of an invariant depends on another invariant, or on an external assumption about the system. These assumptions can be written in preserved blocks.

Caution

Adding require statements to preserved blocks can be a source of unsoundness, since the invariants are only guaranteed to hold if the requirements are true for every method invocation.

Recall that the Prover checks that a method preserves an invariant by first requiring the invariant (the prestate check), then executing the method, and then asserting the invariant (the poststate check). Preserved blocks are executed after the prestate check but before executing the method. They usually consist of require or requireInvariant statements, although other commands are also possible.

Preserved blocks are listed after the invariant expression (and after the filter block, if any), inside a set of curly braces ({ ... }). Each preserved block consists of the keyword preserved followed by an optional method signature, an optional with declaration, and finally the block of commands to execute.

If a preserved block specifies a method signature, the signature must either be fallback() or match one of the contract methods, and the preserved block only applies when checking preservation of that contract method. The fallback() preserved block applies only to the fallback() function that should be defined in the contract. The arguments of the method are in scope within the preserved block.
If there is no method signature, the preserved is a default block that applies to all methods that don’t have a specific preserved block, including the fallback() method.

The with declaration is used to give a name to the environment used while invoking the method. It can be used to restrict the transactions that are considered. For example, the following preserved block rules out counterexamples where the msg.sender is the 0 address:

invariant zero_address_has_no_balance()
    balanceOf(0) == 0
    { preserved with (env e) { require e.msg.sender != 0; } }

The variables defined as parameters to the invariant are also available in preserved blocks, which allows restricting the arguments that are considered when checking that a method preserves an invariant. As always, you should use caution when adding additional require statements, as they can rule out important cases.

Caution

A common source of confusion is the difference between env parameters to an invariant and the env variables defined by the with declaration. Compare the following to the previous example:

invariant zero_address_has_no_balance_v2(env e)
    balanceOf(e, 0) == 0
    { preserved { require e.msg.sender != 0; } }

In this example, we require the msg.sender argument to balanceOf to be nonzero, but makes no restrictions on the environment for the call to the method we are checking for preservation.

To see why this is not the desired behavior, consider a deposit method that increases the message sender’s balance. When the zero_address_has_no_balance_v2 invariant is checked on deposit, the Prover will effectively check the following (see Writing an invariant as a rule):

env e;
require balanceOf(e,0) == 0;

env calledEnv;
require e.msg.sender != 0; // from the `preserved` block
deposit(calledEnv, ...);

assert balanceOf(e,0) == 0;

Notice that the calledEnv is not restricted by the preserved block.

The Prover will report a violation with the msg.sender set to 0 in the call to deposit and set to a nonzero value in the calls to balanceOf. This counterexample is not ruled out by the preserved block because the preserved block only places restrictions on the environment passed to balanceOf.

In general, you should be cautious of invariants that depend on an environment; see Assumptions made while checking invariants for more details.

Filters

For performance reasons, you may want to avoid checking that an invariant is preserved by a particular method or set of methods. Invariant filters provide a method for skipping verification on a method-by-method basis.

Caution

Filtering out methods while checking invariants is unsound. If you are filtering out a method because the invariant doesn’t pass, consider using a preserved block instead; this allows you to add assumptions in a fine-grained way.

To filter out methods from an invariant, add a filtered block after the expression defining the invariant. The body of the filtered block must contain a single filter of the form var -> expr, where var is a variable name, and expr is a boolean expression that may depend on var.

Before verifying that a method preserves an invariant, the expr is evaluated with var bound to a method object. This allows expr to refer to the checked method using var’s fields, such as var.selector and var.isView. See The method and calldataarg types for a list of the fields available on method objects.

If the expression evaluates to false with var replaced by a given method, the Prover will not check that the method preserves the invariant. For example, the following invariant will not be checked on the deposit(uint) method:

invariant balance_is_0(address a)
    balanceOf(a) == 0
    filtered {
        f -> f.selector != sig:deposit(uint).selector
    }

In this example, when the variable f is bound to deposit(uint), the expression f.selector != sig:deposit(uint).selector evaluates to false, so the method will be skipped.

Note

If there is a preserved block for a method, the method will be verified even if the filter would normally exclude it.

Writing an invariant as a rule

Above we explained that verifying an invariant requires two checks: an initial-state check that the constructor establishes the invariant, and a preservation check that each method preserves the invariant.

Invariants are the only mechanism in CVL for specifying properties of constructors, but parametric rules can be used to write the preservation check in a different way. This is useful for two reasons: First, it can help you understand what the preservation check is doing. Second, it can help break down a complicated invariant by defining new intermediate variables.

The following example demonstrates all of the features of invariants:

invariant complex_example(env e1, uint arg)
    property_of(e1, arg)
    filtered {
        m -> m.selector != sig:ignored(uint, address).selector
    }
    {
        preserved with (env e2) {
            require e2.msg.sender != 0;
        }
        preserved special_method(address a) with (env e3) {
            require a != 0;
            require e3.block.timestamp > 0;
        }
    }

The preservation check for this invariant could be written as a parametric rule as follows:

rule complex_example_as_rule(env e1, uint arg, method f)
filtered {
    f -> f.selector != sig:ignored(uint, address).selector
}
{
    // pre-state check
    require property_of(e1, arg);

    if (f.selector == sig:special_method(address).selector) {
        // special_method preserved block
        address a;
        env e3;
        require a != 0;
        require e3.block.timestamp > 0;

        // method execution
        special_method(e3, a);
    } else {
        // general preserved block
        calldataarg args;
        env e2;
        require e2.msg.sender != 0;

        // method execution
        f(e2, args);
    }

    // post-state check
    assert property_of(e1, arg);
}

Invariants and induction

This section describes the logical justification for invariant checks. You do not need to understand this section to use the Prover correctly, but it helps explain the connection between the invariant checks and mathematical proofs for those who are familiar with writing proofs. This section also justifies the safety of arbitrary requireInvariant statements in preserved blocks.

This section assumes familiarity with basic proofs by induction. We use the symbols \(∀\), \(⇒\), and \(∧\) to stand for “for all”, “implies”, and “and” respectively.

Consider an invariant i(x) that is verified by the Prover. For the moment, let’s assume that i(x) has no preserved blocks. We will prove that for all reachable states of the contract, i(x) is true.

A state s is reachable if we can start with an newly created state (that is, where all storage variables are 0), apply any constructor, and then call any number of contract methods to produce s.

Let \(P_i(x,n)\) be the statement “if we start from the newly created state, apply any constructor, and then call \(n\) contract methods, then the resulting state satisfies i(x).” Our goal is then to prove \(∀ n, ∀ x, P_i(x,n)\). We will prove this by induction on \(n\).

In the base case we want to show that for any \(x\), if we apply any constructor to the newly created contract, that the resulting state satisfies i(x). This is exactly what the Prover verifies in the initial state check. In other words, the initial state check proves that \(∀ x, P_i(x,0)\).

For the inductive step, we assume that any \(n\) contract calls produce a state that satisfies i(x), and we want to show that a state produced after \(n+1\) calls also satisfies i(x). This is exactly what the Prover verifies in the preservation check: that if the state before the last method call satisfies i(x) then after the last method call it still satisfies i(x). In other words, the preservation check proves that \(∀ n, ∀ x, P_i(x,n) ⇒ P_i(x, n+1)\).

This completes the proof that together, the initial state check and the preservation check ensure that the invariant i holds on all reachable states.

Now, let us consider preserved blocks. Adding require statements to a preserved block for invariant i adds an additional assumption Q to the preservation check. Now, instead of

\[∀ n, ∀ x, P_i(x,n) ⇒ P_i(x, n+1),\]

the preservation check only proves

\[∀ n, ∀ x, P_i(x,n) {\bf ∧ Q} ⇒ P_i(x, n+1).\]

The addition of the assumption \(Q\) invalidates the above proof if we don’t have reason to believe that \(Q\) actually holds, which is why we caution against adding require statements to preserved blocks.

However, it is important to note that adding requireInvariant j(y) to a preserved block is safe (assuming that j is verified), even if the preserved block for j requires the invariant i. To demonstrate this, we consider three examples.

For the first example, consider the spec

invariant i(uint x) ... { preserved { requireInvariant i(x); } }

Although this may seem like circular logic (we require i in the proof of i), it is not. The verification of the preservation check for i proves the statement

\[∀ n, ∀ x, P_i(x, n) ∧ P_i(x, n) ⇒ P_i(x, n+1),\]

which is logically equivalent to the preservation check without the preserved block (since \(P_i(x,n) ∧ P_i(x,n)\) is equivalent to just \(P_i(x,n)\)).

For the second example, consider the following spec:

invariant i(uint x) ...  { preserved { requireInvariant j(x); } }

invariant j(uint x) ...  { preserved { requireInvariant i(x); } }

Verifying these invariants gives us the preservation check for i:

\[∀ n, ∀ x, P_i(x, n) ∧ P_j(x, n) ⇒ P_i(x, n + 1)\]

and for j:

\[∀ n, ∀ x, P_j(x, n) ∧ P_i(x, n) ⇒ P_j(x, n + 1)\]

Putting these together allows us to conclude

\[∀ n, ∀ x, P_i(x,n) ∧ P_j(x,n) ⇒ P_i(x,n+1) ∧ P_j(x,n+1)\]

which is exactly what we need for an inductive proof of the statement \(∀ n, ∀ x, P_i(x,n) ∧ P_j(x,n)\). This statement then shows that both i(x) and j(x) are true in all reachable states.

For the third example, consider the following spec:

invariant i(uint x) ... { preserved { requireInvariant i(f(x)); } }

The preservation check now proves

\[∀ n, ∀ x, P_i(x,n) ∧ P_i(f(x), n) ⇒ P_i(x, n+1).\]

Seeing that this gives us enough to write an inductive proof that \(∀ n, ∀ x, P_i(x,n)\) takes a little more effort, but it only requires a simple trick. Let \(Q(n)\) be the statement \(∀ x, P_i(x,n)\). We prove \(∀ n, Q(n)\) by induction.

The base case comes directly from the initial state check for i.

For the inductive step, choose an arbitrary \(n\) and assume \(Q(n)\). We want to show \(Q(n+1)\), i.e. that \(∀ x, P_i(x, n+1)\). Fix an arbitrary \(x\). We can apply \(Q(n)\) to \(x\) to conclude \(P_i(x,n)\). We can also apply \(Q(n)\) to \(f(x)\) to conclude \(P_i(f(x), n)\). These facts together with the preservation check show \(P_i(x, n+1)\). Since \(x\) was arbitrary, we can conclude \(∀ x, P(x, n+1)\), which is \(Q(n+1)\). This completes the inductive step, and thus the proof.

The techniques used in these three examples can be used to demonstrate that it is always logically sound to add a requireInvariant to a preserved block, even for complicated interdependent invariants (as long as the required invariants have been verified).