.. _formal_verification:

##################################
SMTChecker and Formal Verification
##################################

Using formal verification it is possible to perform an automated mathematical
proof that your source code fulfills a certain formal specification.
The specification is still formal (just as the source code), but usually much
simpler.

Note that formal verification itself can only help you understand the
difference between what you did (the specification) and how you did it
(the actual implementation). You still need to check whether the specification
is what you wanted and that you did not miss any unintended effects of it.

Solidity implements a formal verification approach based on
`SMT (Satisfiability Modulo Theories) <https://en.wikipedia.org/wiki/Satisfiability_modulo_theories>`_ and
`Horn <https://en.wikipedia.org/wiki/Horn-satisfiability>`_ solving.
The SMTChecker module automatically tries to prove that the code satisfies the
specification given by ``require`` and ``assert`` statements. That is, it considers
``require`` statements as assumptions and tries to prove that the conditions
inside ``assert`` statements are always true.  If an assertion failure is
found, a counterexample may be given to the user showing how the assertion can
be violated. If no warning is given by the SMTChecker for a property,
it means that the property is safe.

The other verification targets that the SMTChecker checks at compile time are:

- Arithmetic underflow and overflow.
- Division by zero.
- Trivial conditions and unreachable code.
- Popping an empty array.
- Out of bounds index access.
- Insufficient funds for a transfer.

All the targets above are automatically checked by default if all engines are
enabled, except underflow and overflow for Solidity >=0.8.7.

The potential warnings that the SMTChecker reports are:

- ``<failing  property> happens here.``. This means that the SMTChecker proved that a certain property fails. A counterexample may be given, however in complex situations it may also not show a counterexample. This result may also be a false positive in certain cases, when the SMT encoding adds abstractions for Solidity code that is either hard or impossible to express.
- ``<failing property> might happen here``. This means that the solver could not prove either case within the given timeout. Since the result is unknown, the SMTChecker reports the potential failure for soundness. This may be solved by increasing the query timeout, but the problem might also simply be too hard for the engine to solve.

To enable the SMTChecker, you must select :ref:`which engine should run<smtchecker_engines>`,
where the default is no engine. Selecting the engine enables the SMTChecker on all files.

.. note::

    Prior to Solidity 0.8.4, the default way to enable the SMTChecker was via
    ``pragma experimental SMTChecker;`` and only the contracts containing the
    pragma would be analyzed. That pragma has been deprecated, and although it
    still enables the SMTChecker for backwards compatibility, it will be removed
    in Solidity 0.9.0. Note also that now using the pragma even in a single file
    enables the SMTChecker for all files.

.. note::

    The lack of warnings for a verification target represents an undisputed
    mathematical proof of correctness, assuming no bugs in the SMTChecker and
    the underlying solver. Keep in mind that these problems are
    *very hard* and sometimes *impossible* to solve automatically in the
    general case.  Therefore, several properties might not be solved or might
    lead to false positives for large contracts. Every proven property should
    be seen as an important achievement. For advanced users, see :ref:`SMTChecker Tuning <smtchecker_options>`
    to learn a few options that might help proving more complex
    properties.

********
Tutorial
********

Overflow
========

.. code-block:: Solidity

    // SPDX-License-Identifier: GPL-3.0
    pragma solidity >=0.8.0;

    contract Overflow {
        uint immutable x;
        uint immutable y;

        function add(uint _x, uint _y) internal pure returns (uint) {
            return _x + _y;
        }

        constructor(uint _x, uint _y) {
            (x, y) = (_x, _y);
        }

        function stateAdd() public view returns (uint) {
            return add(x, y);
        }
    }

The contract above shows an overflow check example.
The SMTChecker does not check underflow and overflow by default for Solidity >=0.8.7,
so we need to use the command line option ``--model-checker-targets "underflow,overflow"``
or the JSON option ``settings.modelChecker.targets = ["underflow", "overflow"]``.
See :ref:`this section for targets configuration<smtchecker_targets>`.
Here, it reports the following:

.. code-block:: text

    Warning: CHC: Overflow (resulting value larger than 2**256 - 1) happens here.
    Counterexample:
    x = 1, y = 115792089237316195423570985008687907853269984665640564039457584007913129639935
     = 0

    Transaction trace:
    Overflow.constructor(1, 115792089237316195423570985008687907853269984665640564039457584007913129639935)
    State: x = 1, y = 115792089237316195423570985008687907853269984665640564039457584007913129639935
    Overflow.stateAdd()
        Overflow.add(1, 115792089237316195423570985008687907853269984665640564039457584007913129639935) -- internal call
     --> o.sol:9:20:
      |
    9 |             return _x + _y;
      |                    ^^^^^^^

If we add ``require`` statements that filter out overflow cases,
the SMTChecker proves that no overflow is reachable (by not reporting warnings):

.. code-block:: Solidity

    // SPDX-License-Identifier: GPL-3.0
    pragma solidity >=0.8.0;

    contract Overflow {
        uint immutable x;
        uint immutable y;

        function add(uint _x, uint _y) internal pure returns (uint) {
            return _x + _y;
        }

        constructor(uint _x, uint _y) {
            (x, y) = (_x, _y);
        }

        function stateAdd() public view returns (uint) {
            require(x < type(uint128).max);
            require(y < type(uint128).max);
            return add(x, y);
        }
    }


Assert
======

An assertion represents an invariant in your code: a property that must be true
*for all transactions, including all input and storage values*, otherwise there is a bug.

The code below defines a function ``f`` that guarantees no overflow.
Function ``inv`` defines the specification that ``f`` is monotonically increasing:
for every possible pair ``(_a, _b)``, if ``_b > _a`` then ``f(_b) > f(_a)``.
Since ``f`` is indeed monotonically increasing, the SMTChecker proves that our
property is correct. You are encouraged to play with the property and the function
definition to see what results come out!

.. code-block:: Solidity

    // SPDX-License-Identifier: GPL-3.0
    pragma solidity >=0.8.0;

    contract Monotonic {
        function f(uint _x) internal pure returns (uint) {
            require(_x < type(uint128).max);
            return _x * 42;
        }

        function inv(uint _a, uint _b) public pure {
            require(_b > _a);
            assert(f(_b) > f(_a));
        }
    }

We can also add assertions inside loops to verify more complicated properties.
The following code searches for the maximum element of an unrestricted array of
numbers, and asserts the property that the found element must be greater or
equal every element in the array.

.. code-block:: Solidity

    // SPDX-License-Identifier: GPL-3.0
    pragma solidity >=0.8.0;

    contract Max {
        function max(uint[] memory _a) public pure returns (uint) {
            uint m = 0;
            for (uint i = 0; i < _a.length; ++i)
                if (_a[i] > m)
                    m = _a[i];

            for (uint i = 0; i < _a.length; ++i)
                assert(m >= _a[i]);

            return m;
        }
    }

Note that in this example the SMTChecker will automatically try to prove three properties:

1. ``++i`` in the first loop does not overflow.
2. ``++i`` in the second loop does not overflow.
3. The assertion is always true.

.. note::

    The properties involve loops, which makes it *much much* harder than the previous
    examples, so beware of loops!

All the properties are correctly proven safe. Feel free to change the
properties and/or add restrictions on the array to see different results.
For example, changing the code to

.. code-block:: Solidity

    // SPDX-License-Identifier: GPL-3.0
    pragma solidity >=0.8.0;

    contract Max {
        function max(uint[] memory _a) public pure returns (uint) {
            require(_a.length >= 5);
            uint m = 0;
            for (uint i = 0; i < _a.length; ++i)
                if (_a[i] > m)
                    m = _a[i];

            for (uint i = 0; i < _a.length; ++i)
                assert(m > _a[i]);

            return m;
        }
    }

gives us:

.. code-block:: text

    Warning: CHC: Assertion violation happens here.
    Counterexample:

    _a = [0, 0, 0, 0, 0]
     = 0

    Transaction trace:
    Test.constructor()
    Test.max([0, 0, 0, 0, 0])
      --> max.sol:14:4:
       |
    14 |            assert(m > _a[i]);


State Properties
================

So far the examples only demonstrated the use of the SMTChecker over pure code,
proving properties about specific operations or algorithms.
A common type of properties in smart contracts are properties that involve the
state of the contract. Multiple transactions might be needed to make an assertion
fail for such a property.

As an example, consider a 2D grid where both axis have coordinates in the range (-2^128, 2^128 - 1).
Let us place a robot at position (0, 0). The robot can only move diagonally, one step at a time,
and cannot move outside the grid. The robot's state machine can be represented by the smart contract
below.

.. code-block:: Solidity

    // SPDX-License-Identifier: GPL-3.0
    pragma solidity >=0.8.0;

    contract Robot {
        int x = 0;
        int y = 0;

        modifier wall {
            require(x > type(int128).min && x < type(int128).max);
            require(y > type(int128).min && y < type(int128).max);
            _;
        }

        function moveLeftUp() wall public {
            --x;
            ++y;
        }

        function moveLeftDown() wall public {
            --x;
            --y;
        }

        function moveRightUp() wall public {
            ++x;
            ++y;
        }

        function moveRightDown() wall public {
            ++x;
            --y;
        }

        function inv() public view {
            assert((x + y) % 2 == 0);
        }
    }

Function ``inv`` represents an invariant of the state machine that ``x + y``
must be even.
The SMTChecker manages to prove that regardless how many commands we give the
robot, even if infinitely many, the invariant can *never* fail. The interested
reader may want to prove that fact manually as well.  Hint: this invariant is
inductive.

We can also trick the SMTChecker into giving us a path to a certain position we
think might be reachable.  We can add the property that (2, 4) is *not*
reachable, by adding the following function.

.. code-block:: Solidity

    function reach_2_4() public view {
        assert(!(x == 2 && y == 4));
    }

This property is false, and while proving that the property is false,
the SMTChecker tells us exactly *how* to reach (2, 4):

.. code-block:: text

    Warning: CHC: Assertion violation happens here.
    Counterexample:
    x = 2, y = 4

    Transaction trace:
    Robot.constructor()
    State: x = 0, y = 0
    Robot.moveLeftUp()
    State: x = (- 1), y = 1
    Robot.moveRightUp()
    State: x = 0, y = 2
    Robot.moveRightUp()
    State: x = 1, y = 3
    Robot.moveRightUp()
    State: x = 2, y = 4
    Robot.reach_2_4()
      --> r.sol:35:4:
       |
    35 |            assert(!(x == 2 && y == 4));
       |            ^^^^^^^^^^^^^^^^^^^^^^^^^^^

Note that the path above is not necessarily deterministic, as there are
other paths that could reach (2, 4). The choice of which path is shown
might change depending on the used solver, its version, or just randomly.

External Calls and Reentrancy
=============================

Every external call is treated as a call to unknown code by the SMTChecker.
The reasoning behind that is that even if the code of the called contract is
available at compile time, there is no guarantee that the deployed contract
will indeed be the same as the contract where the interface came from at
compile time.

In some cases, it is possible to automatically infer properties over state
variables that are still true even if the externally called code can do
anything, including reenter the caller contract.

.. code-block:: Solidity

    // SPDX-License-Identifier: GPL-3.0
    pragma solidity >=0.8.0;

    interface Unknown {
        function run() external;
    }

    contract Mutex {
        uint x;
        bool lock;

        Unknown immutable unknown;

        constructor(Unknown _u) {
            require(address(_u) != address(0));
            unknown = _u;
        }

        modifier mutex {
            require(!lock);
            lock = true;
            _;
            lock = false;
        }

        function set(uint _x) mutex public {
            x = _x;
        }

        function run() mutex public {
            uint xPre = x;
            unknown.run();
            assert(xPre == x);
        }
    }

The example above shows a contract that uses a mutex flag to forbid reentrancy.
The solver is able to infer that when ``unknown.run()`` is called, the contract
is already "locked", so it would not be possible to change the value of ``x``,
regardless of what the unknown called code does.

If we "forget" to use the ``mutex`` modifier on function ``set``, the
SMTChecker is able to synthesize the behavior of the externally called code so
that the assertion fails:

.. code-block:: text

    Warning: CHC: Assertion violation happens here.
    Counterexample:
    x = 1, lock = true, unknown = 1

    Transaction trace:
    Mutex.constructor(1)
    State: x = 0, lock = false, unknown = 1
    Mutex.run()
        unknown.run() -- untrusted external call, synthesized as:
            Mutex.set(1) -- reentrant call
      --> m.sol:32:3:
       |
    32 | 		assert(xPre == x);
       | 		^^^^^^^^^^^^^^^^^


.. _smtchecker_options:

*****************************
SMTChecker Options and Tuning
*****************************

Timeout
=======

The SMTChecker uses a hardcoded resource limit (``rlimit``) chosen per solver,
which is not precisely related to time. We chose the ``rlimit`` option as the default
because it gives more determinism guarantees than time inside the solver.

This options translates roughly to "a few seconds timeout" per query. Of course many properties
are very complex and need a lot of time to be solved, where determinism does not matter.
If the SMTChecker does not manage to solve the contract properties with the default ``rlimit``,
a timeout can be given in milliseconds via the CLI option ``--model-checker-timeout <time>`` or
the JSON option ``settings.modelChecker.timeout=<time>``, where 0 means no timeout.

.. _smtchecker_targets:

Verification Targets
====================

The types of verification targets created by the SMTChecker can also be
customized via the CLI option ``--model-checker-target <targets>`` or the JSON
option ``settings.modelChecker.targets=<targets>``.
In the CLI case, ``<targets>`` is a no-space-comma-separated list of one or
more verification targets, and an array of one or more targets as strings in
the JSON input.
The keywords that represent the targets are:

- Assertions: ``assert``.
- Arithmetic underflow: ``underflow``.
- Arithmetic overflow: ``overflow``.
- Division by zero: ``divByZero``.
- Trivial conditions and unreachable code: ``constantCondition``.
- Popping an empty array: ``popEmptyArray``.
- Out of bounds array/fixed bytes index access: ``outOfBounds``.
- Insufficient funds for a transfer: ``balance``.
- All of the above: ``default`` (CLI only).

A common subset of targets might be, for example:
``--model-checker-targets assert,overflow``.

All targets are checked by default, except underflow and overflow for Solidity >=0.8.7.

There is no precise heuristic on how and when to split verification targets,
but it can be useful especially when dealing with large contracts.

Unproved Targets
================

If there are any unproved targets, the SMTChecker issues one warning stating
how many unproved targets there are. If the user wishes to see all the specific
unproved targets, the CLI option ``--model-checker-show-unproved`` and
the JSON option ``settings.modelChecker.showUnproved = true`` can be used.

Verified Contracts
==================

By default all the deployable contracts in the given sources are analyzed separately as
the one that will be deployed. This means that if a contract has many direct
and indirect inheritance parents, all of them will be analyzed on their own,
even though only the most derived will be accessed directly on the blockchain.
This causes an unnecessary burden on the SMTChecker and the solver.  To aid
cases like this, users can specify which contracts should be analyzed as the
deployed one. The parent contracts are of course still analyzed, but only in
the context of the most derived contract, reducing the complexity of the
encoding and generated queries. Note that abstract contracts are by default
not analyzed as the most derived by the SMTChecker.

The chosen contracts can be given via a comma-separated list (whitespace is not
allowed) of <source>:<contract> pairs in the CLI:
``--model-checker-contracts "<source1.sol:contract1>,<source2.sol:contract2>,<source2.sol:contract3>"``,
and via the object ``settings.modelChecker.contracts`` in the :ref:`JSON input<compiler-api>`,
which has the following form:

.. code-block:: json

    "contracts": {
        "source1.sol": ["contract1"],
        "source2.sol": ["contract2", "contract3"]
    }

Division and Modulo With Slack Variables
========================================

Spacer, the default Horn solver used by the SMTChecker, often dislikes division
and modulo operations inside Horn rules. Because of that, by default the
Solidity division and modulo operations are encoded using the constraint
``a = b * d + m`` where ``d = a / b`` and ``m = a % b``.
However, other solvers, such as Eldarica, prefer the syntactically precise operations.
The command line flag ``--model-checker-div-mod-no-slacks`` and the JSON option
``settings.modelChecker.divModNoSlacks`` can be used to toggle the encoding
depending on the used solver preferences.

Natspec Function Abstraction
============================

Certain functions including common math methods such as ``pow``
and ``sqrt`` may be too complex to be analyzed in a fully automated way.
These functions can be annotated with Natspec tags that indicate to the
SMTChecker that these functions should be abstracted. This means that the
body of the function is not used, and when called, the function will:

- Return a nondeterministic value, and either keep the state variables unchanged if the abstracted function is view/pure, or also set the state variables to nondeterministic values otherwise. This can be used via the annotation ``/// @custom:smtchecker abstract-function-nondet``.
- Act as an uninterpreted function. This means that the semantics of the function (given by the body) are ignored, and the only property this function has is that given the same input it guarantees the same output. This is currently under development and will be available via the annotation ``/// @custom:smtchecker abstract-function-uf``.

.. _smtchecker_engines:

Model Checking Engines
======================

The SMTChecker module implements two different reasoning engines, a Bounded
Model Checker (BMC) and a system of Constrained Horn Clauses (CHC).  Both
engines are currently under development, and have different characteristics.
The engines are independent and every property warning states from which engine
it came. Note that all the examples above with counterexamples were
reported by CHC, the more powerful engine.

By default both engines are used, where CHC runs first, and every property that
was not proven is passed over to BMC. You can choose a specific engine via the CLI
option ``--model-checker-engine {all,bmc,chc,none}`` or the JSON option
``settings.modelChecker.engine={all,bmc,chc,none}``.

Bounded Model Checker (BMC)
---------------------------

The BMC engine analyzes functions in isolation, that is, it does not take the
overall behavior of the contract over multiple transactions into account when
analyzing each function.  Loops are also ignored in this engine at the moment.
Internal function calls are inlined as long as they are not recursive, directly
or indirectly. External function calls are inlined if possible. Knowledge
that is potentially affected by reentrancy is erased.

The characteristics above make BMC prone to reporting false positives,
but it is also lightweight and should be able to quickly find small local bugs.

Constrained Horn Clauses (CHC)
------------------------------

A contract's Control Flow Graph (CFG) is modelled as a system of
Horn clauses, where the life cycle of the contract is represented by a loop
that can visit every public/external function non-deterministically. This way,
the behavior of the entire contract over an unbounded number of transactions
is taken into account when analyzing any function. Loops are fully supported
by this engine. Internal function calls are supported, and external function
calls assume the called code is unknown and can do anything.

The CHC engine is much more powerful than BMC in terms of what it can prove,
and might require more computing resources.

SMT and Horn solvers
====================

The two engines detailed above use automated theorem provers as their logical
backends.  BMC uses an SMT solver, whereas CHC uses a Horn solver. Often the
same tool can act as both, as seen in `z3 <https://github.com/Z3Prover/z3>`_,
which is primarily an SMT solver and makes `Spacer
<https://spacer.bitbucket.io/>`_ available as a Horn solver, and `Eldarica
<https://github.com/uuverifiers/eldarica>`_ which does both.

The user can choose which solvers should be used, if available, via the CLI
option ``--model-checker-solvers {all,cvc4,smtlib2,z3}`` or the JSON option
``settings.modelChecker.solvers=[smtlib2,z3]``, where:

- ``cvc4`` is only available if the ``solc`` binary is compiled with it. Only BMC uses ``cvc4``.
- ``smtlib2`` outputs SMT/Horn queries in the `smtlib2 <http://smtlib.cs.uiowa.edu/>`_ format.
  These can be used together with the compiler's `callback mechanism <https://github.com/ethereum/solc-js>`_ so that
  any solver binary from the system can be employed to synchronously return the results of the queries to the compiler.
  This is currently the only way to use Eldarica, for example, since it does not have a C++ API.
  This can be used by both BMC and CHC depending on which solvers are called.
- ``z3`` is available

  - if ``solc`` is compiled with it;
  - if a dynamic ``z3`` library of version 4.8.x is installed in a Linux system (from Solidity 0.7.6);
  - statically in ``soljson.js`` (from Solidity 0.6.9), that is, the Javascript binary of the compiler.

Since both BMC and CHC use ``z3``, and ``z3`` is available in a greater variety
of environments, including in the browser, most users will almost never need to be
concerned about this option. More advanced users might apply this option to try
alternative solvers on more complex problems.

Please note that certain combinations of chosen engine and solver will lead to
the SMTChecker doing nothing, for example choosing CHC and ``cvc4``.

*******************************
Abstraction and False Positives
*******************************

The SMTChecker implements abstractions in an incomplete and sound way: If a bug
is reported, it might be a false positive introduced by abstractions (due to
erasing knowledge or using a non-precise type). If it determines that a
verification target is safe, it is indeed safe, that is, there are no false
negatives (unless there is a bug in the SMTChecker).

If a target cannot be proven you can try to help the solver by using the tuning
options in the previous section.
If you are sure of a false positive, adding ``require`` statements in the code
with more information may also give some more power to the solver.

SMT Encoding and Types
======================

The SMTChecker encoding tries to be as precise as possible, mapping Solidity types
and expressions to their closest `SMT-LIB <http://smtlib.cs.uiowa.edu/>`_
representation, as shown in the table below.

+-----------------------+--------------------------------+-----------------------------+
|Solidity type          |SMT sort                        |Theories                     |
+=======================+================================+=============================+
|Boolean                |Bool                            |Bool                         |
+-----------------------+--------------------------------+-----------------------------+
|intN, uintN, address,  |Integer                         |LIA, NIA                     |
|bytesN, enum, contract |                                |                             |
+-----------------------+--------------------------------+-----------------------------+
|array, mapping, bytes, |Tuple                           |Datatypes, Arrays, LIA       |
|string                 |(Array elements, Integer length)|                             |
+-----------------------+--------------------------------+-----------------------------+
|struct                 |Tuple                           |Datatypes                    |
+-----------------------+--------------------------------+-----------------------------+
|other types            |Integer                         |LIA                          |
+-----------------------+--------------------------------+-----------------------------+

Types that are not yet supported are abstracted by a single 256-bit unsigned
integer, where their unsupported operations are ignored.

For more details on how the SMT encoding works internally, see the paper
`SMT-based Verification of Solidity Smart Contracts <https://github.com/leonardoalt/text/blob/master/solidity_isola_2018/main.pdf>`_.

Function Calls
==============

In the BMC engine, function calls to the same contract (or base contracts) are
inlined when possible, that is, when their implementation is available.  Calls
to functions in other contracts are not inlined even if their code is
available, since we cannot guarantee that the actual deployed code is the same.

The CHC engine creates nonlinear Horn clauses that use summaries of the called
functions to support internal function calls. External function calls are treated
as calls to unknown code, including potential reentrant calls.

Complex pure functions are abstracted by an uninterpreted function (UF) over
the arguments.

+-----------------------------------+--------------------------------------+
|Functions                          |BMC/CHC behavior                      |
+===================================+======================================+
|``assert``                         |Verification target.                  |
+-----------------------------------+--------------------------------------+
|``require``                        |Assumption.                           |
+-----------------------------------+--------------------------------------+
|internal call                      |BMC: Inline function call.            |
|                                   |CHC: Function summaries.              |
+-----------------------------------+--------------------------------------+
|external call to known code        |BMC: Inline function call or          |
|                                   |erase knowledge about state variables |
|                                   |and local storage references.         |
|                                   |CHC: Assume called code is unknown.   |
|                                   |Try to infer invariants that hold     |
|                                   |after the call returns.               |
+-----------------------------------+--------------------------------------+
|Storage array push/pop             |Supported precisely.                  |
|                                   |Checks whether it is popping an       |
|                                   |empty array.                          |
+-----------------------------------+--------------------------------------+
|ABI functions                      |Abstracted with UF.                   |
+-----------------------------------+--------------------------------------+
|``addmod``, ``mulmod``             |Supported precisely.                  |
+-----------------------------------+--------------------------------------+
|``gasleft``, ``blockhash``,        |Abstracted with UF.                   |
|``keccak256``, ``ecrecover``       |                                      |
|``ripemd160``                      |                                      |
+-----------------------------------+--------------------------------------+
|pure functions without             |Abstracted with UF                    |
|implementation (external or        |                                      |
|complex)                           |                                      |
+-----------------------------------+--------------------------------------+
|external functions without         |BMC: Erase state knowledge and assume |
|implementation                     |result is nondeterminisc.             |
|                                   |CHC: Nondeterministic summary.        |
|                                   |Try to infer invariants that hold     |
|                                   |after the call returns.               |
+-----------------------------------+--------------------------------------+
|transfer                           |BMC: Checks whether the contract's    |
|                                   |balance is sufficient.                |
|                                   |CHC: does not yet perform the check.  |
+-----------------------------------+--------------------------------------+
|others                             |Currently unsupported                 |
+-----------------------------------+--------------------------------------+

Using abstraction means loss of precise knowledge, but in many cases it does
not mean loss of proving power.

.. code-block:: solidity

    // SPDX-License-Identifier: GPL-3.0
    pragma solidity >=0.8.0;

    contract Recover
    {
        function f(
            bytes32 hash,
            uint8 _v1, uint8 _v2,
            bytes32 _r1, bytes32 _r2,
            bytes32 _s1, bytes32 _s2
        ) public pure returns (address) {
            address a1 = ecrecover(hash, _v1, _r1, _s1);
            require(_v1 == _v2);
            require(_r1 == _r2);
            require(_s1 == _s2);
            address a2 = ecrecover(hash, _v2, _r2, _s2);
            assert(a1 == a2);
            return a1;
        }
    }

In the example above, the SMTChecker is not expressive enough to actually
compute ``ecrecover``, but by modelling the function calls as uninterpreted
functions we know that the return value is the same when called on equivalent
parameters. This is enough to prove that the assertion above is always true.

Abstracting a function call with an UF can be done for functions known to be
deterministic, and can be easily done for pure functions.  It is however
difficult to do this with general external functions, since they might depend
on state variables.

Reference Types and Aliasing
============================

Solidity implements aliasing for reference types with the same :ref:`data
location<data-location>`.
That means one variable may be modified through a reference to the same data
area.
The SMTChecker does not keep track of which references refer to the same data.
This implies that whenever a local reference or state variable of reference
type is assigned, all knowledge regarding variables of the same type and data
location is erased.
If the type is nested, the knowledge removal also includes all the prefix base
types.

.. code-block:: solidity

    // SPDX-License-Identifier: GPL-3.0
    pragma solidity >=0.8.0;

    contract Aliasing
    {
        uint[] array1;
        uint[][] array2;
        function f(
            uint[] memory a,
            uint[] memory b,
            uint[][] memory c,
            uint[] storage d
        ) internal {
            array1[0] = 42;
            a[0] = 2;
            c[0][0] = 2;
            b[0] = 1;
            // Erasing knowledge about memory references should not
            // erase knowledge about state variables.
            assert(array1[0] == 42);
            // However, an assignment to a storage reference will erase
            // storage knowledge accordingly.
            d[0] = 2;
            // Fails as false positive because of the assignment above.
            assert(array1[0] == 42);
            // Fails because `a == b` is possible.
            assert(a[0] == 2);
            // Fails because `c[i] == b` is possible.
            assert(c[0][0] == 2);
            assert(d[0] == 2);
            assert(b[0] == 1);
        }
        function g(
            uint[] memory a,
            uint[] memory b,
            uint[][] memory c,
            uint x
        ) public {
            f(a, b, c, array2[x]);
        }
    }

After the assignment to ``b[0]``, we need to clear knowledge about ``a`` since
it has the same type (``uint[]``) and data location (memory).  We also need to
clear knowledge about ``c``, since its base type is also a ``uint[]`` located
in memory. This implies that some ``c[i]`` could refer to the same data as
``b`` or ``a``.

Notice that we do not clear knowledge about ``array`` and ``d`` because they
are located in storage, even though they also have type ``uint[]``.  However,
if ``d`` was assigned, we would need to clear knowledge about ``array`` and
vice-versa.

**********************
Real World Assumptions
**********************

Some scenarios can be expressed in Solidity and the EVM, but are expected to
never occur in practice.
One of such cases is the length of a dynamic storage array overflowing during a
push: If the ``push`` operation is applied to an array of length 2^256 - 1, its
length silently overflows.
However, this is unlikely to happen in practice, since the operations required
to grow the array to that point would take billions of years to execute.