# Spec.jl

Spec is a package for expressing specifications.

Spec.jl is small library for specfiying correctness properties of programs. These specifications can be used as documentation, for testing, and for debugging.

# Usage

Toadd:

- other meta dontcheck, e.g. existential propositions (example?)

### Pre Conditions

Preconditions are statements (also known as propositions or conditions) that must be true about the input before the function is executed. One purpose of using preconditions is for documentation -- by specifying precise constraints on the function's behaviour we are communicating more clearly what we expect it to do.

Preconditions are specified using the `@pre`

macro, for example:

```
f(x::Float64) = x * sqrt(x)
@pre f(::Float64) x >= 0
```

This should be interpreted as defining a precondition for the method `f(::Int)`

that specifies that the input `x`

should be non-negative.

If we are feeling lazy we can avoid writing the signature twice, and instead write:

```
f(x::Float64) = x * sqrt(x)
@pre f x > 0
```

This will define the precondition `x > 0`

for the most recently defined method for the generic function `f`

.

Multiple specifications can be expressed by simply adding additional lines. For example:

```
"Body mass index"
bmi(height, weight) = height / weight
@pre bmi(height, weight) = height > 0 "height must be positive"
@pre bmi(height, weight) = weight > 0 "weight must be positive"
```

The semantics of this is conjuctive: **all** the different specs should be true.
This example also shows us that we can attach a description to any specification.

There is a many-to-one relationship between methods and generic functions in Julia. One precondition can be applied to many methods of a generic function.

The following example adds specification for a `divide`

function, which works on Strings as well as numbers.

```
"is x a number?"
isnumstring(x::String) = occursin(r"^\d+$", x)
divide(x::String, y::String) = parse(Int64, x) / parse(Int64, y)
@pre divide(x::String, y::String) = isnumstring(x) && isnumstring(y) "String should be a number"
specapply(divide, "123", "114")
specapply(divide, "123", "aa4")
divide(x::Int, y::Int) = x / y
"Convert `x` to number if string"
asnum(x::String) = parse(Int64, x)
asnum(x) = x
# This precondition Applies to both methods
@pre divide(x, y) = asnum(y) != 0 "Denominator cannot be zero"
specapply(divide, 100, 0)
specapply(divide, "100", "0")
```

In this example the first precondition only applies to the string based method, whereas the second precondition applies to both.

## Post Conditions

The second kind of specification is a **post condition** which is a statement about the output and input of a proecdure after it has executed.
A canonical example of a post-condition is that of a sorted list, which states that a list is sorted if for every element `i`

and every element `j`

, if `i < j`

then the position of `i`

should be less than the position of `j`

```
"p implies q"
p → q = !(p && !q)
"is `xs` sorted"
issorted(xs) =
all((xs[i] < xs[j] → i < j for i = 1:length(xs) for j = 1:length(xs) if i != j))
"Is `y` sorted version of `x`"
isysortedx(xs, ys) =
length(ys) == length(xs) && all((y in xs for y in ys)) && issorted(y)
mysort(x) = sort(x)
@post mysort(x) = isysortedx(x, @ret) "Result is sorted version of input"
```

#### Capturing

As we have seen, a post-condition can also see values to a function. So why bother have two constructs at all? The main reason is that Julia procedures can have side-effects.

In the mutating sort function `sort!(xs)`

the input is mutated.
Therefore if we want to check that the result is a sorted version of the input we have a problem, because we have lost information about the input.
The solution is to use `@cap`

to capture variables in the input

`@post sort! isysortedx(@cap(x), @ret) "Result is sorted version of input"`

## Using Specifications

By default, pre and post conditions will not affect the behaviour of your program at all, and incur no runtime cost.
To actually check specificatiosn we use `specapply`

.

`specapply(f, args...)`

This will evaluate `f(args)`

, but for all function applications encountered in the execution of `f(args...)`

, each and every associated spec will be checked.

For example, given the specs above for `sort`

and `divide`

, the following snippeet will check the pre and post conditions:

```
function f(n)
x = rand(n)
y = sort(x)
z = map(divide, y)
end
specapply(f, n)
```

### For Debugging

One of the best uses of `specapply`

is debugging. Often a runtime error is the ressult of a pre-condition much further up the stack being violated.
For example:

```
function f()
x = rand(4, 4)
x[3,4]
end
```

Evaluating `f()`

gives us this horribly uninfomrative stack trace.

If instead we run this function with `specapply`

# Notes

- The concept of testing all the nested specs within a function call f(x) is orthogonal to pretty much everything else (that follows)
- For a given method, I may want to test all, some or none of the specs
- For a given method and given spec, I may want to test it on (i) one input, (ii) samples from a distribution of inputs, (iii) all inputs, finitely enumerable, (iv) all inputs abstractly.