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# Quantification

Quantification allows us to talk about all values in a set

**Syntax:**

`forall (<type> <id> in <expression>...<expression>) <expression>`

`forall (<type> <id> in <expression>) <expression>`

Universal quantifiers are powerful expressions that allow users to talk about all values that fit a criteria. Scribble supports 2 types of universal quantifiers - quantifiers over numeric ranges and quantifiers over arrays and maps.

Quantifiers can talk about properties that hold over all numbers in a given range. The syntax for such quantifiers is

`forall(<type> <id> in <expression>...<expression>) <expression>`

. The start expression is

**inclusive**, while the end range expression is**exclusive.**The most common use-case of quantification over numeric ranges is describing all indices of an array (see below example).

Example 1: Forall over arrays

In the below example we check that the array

`arr`

is sorted./// #invariant forall(uint i in 0...arr.length-1) arr[i] < arr[i+1];

contract Token {

uint[] arr;

...

}

Note that the type of the quantifier variable (

`uint i`

in the above example) must always be a numeric type, and it must be able to fit both the start and end expressions. So for example you can't write:
`forall (int8 i in 0...arr.length-1) ...`

in the above example instead, since `int8`

is too small of a type to store the array length (which is `uint256`

).The second flavor of universal quantifiers talks about all indices in an array or keys in a map. The syntax is

`forall (<type> <id> in <expression>) <expression>`

.Example 1: Forall over arrays

Example 2: Forall over maps

In the below example, we check that each stakeholder has a positive stake

/// #invariant

/// forall(uint i in stakeHolders) stakes[stakeHolders[i]] > 0;

contract Token {

address[] stakeHolders;

mapping(uint=>uint) stakes;

}

In the below example, the mapping

`authorized`

is a mapping from account addresses to operators addresses, authorized to act on behalf of the given account. The property below states that for no account is the `0x0`

address authorized to act on their behalf./// #invariant

/// forall(address acct in authorized) authorized[acct] != address(0);

contract Token {

mapping(address=>address) authorized;

}

Any

`forall`

over arrays can be equivalently expressed as a `forall`

over numeric ranges.
I.e.`forall(<type> i in arr) <expr>`

is equivalent to `forall(uint i in 0...arr.length) <expr>`

Note that in Solidity maps are technically defined over their complete input range (i.e. a

`mapping(uint=>uint)`

is defined for all `uint256`

s, just its 0 for most of them). Due to this, when we say that we "quantify over all keys in a map", we mean all keys that have:
1. Been set explicitly at least once
2. Have not been deleted with the `delete`

keyword.To support this, under the hood we re-write all annotated maps with our custom data type that track key insertion and deletions.

The

`delete`

operation is equivalent to zeroing out a value. However, we do not treat them equivalently. Given a map `m`

, when you do `delete m[x]`

, we will remove `x`

from the keys that we store in `m`

. However, if you do `m[x] = 0`

we will **not**remove x from the keys we store in`m`

.You can nest

`forall`

statements arbitrarily. For example, we can extend the example above to store a list of authorized users for a given address and require that none of the authorized users is `0x0`

:Authorized Users 2

In the below example, the mapping

`authorizedUsers`

stores a list of addresses of operators authorized to act on behalf of a given address. The property below states that the `0x0`

address is not authorized to act on anyone's behalf./// #invariant

/// forall(address acct in authorized)

/// forall(uint i in authorized[acct])

/// authorized[acct][i] != address(0);

contract Token {

mapping(address=>address[]) authorized;

}