Lists · Integrity

This page documents library functions related to lists.

Tail

f_ ext{tail} left(, l , ight)

Get the tail of a list (all elements excluding the first).

Arguments

Argument	Description	Domain	Constraints	Notes
$l$	source list	$\forall T: [T, ...]$	Not empty.

Return

Value	Description	Codomain	Constraints	Notes
$\omega$	tail	$[T, ...]$		$\text{undefined}$ if $l$ is empty.

Instances

f_{tail}\left(\left[\right]\right)
=\frac{1}{0}

f_{tail}\left(\left[0\right]\right)-\left[\right]
f_{tail}\left(\left[0,\ 10\right]\right)-\left[10\right]
f_{tail}\left(\left[0,\ 10,\ 100\right]\right)-\left[10,\ 100\right]

Implementation

egin{align*} f_ ext{tail} left(, l , ight) = {& operatorname{length}left(l ight)<1: left[1/0 ight], \ & operatorname{length}left(l ight)=1: left[ ight], \ & left[lleft[i ight]operatorname{for}i=left[2...operatorname{length}left(l ight) ight] ight] \ }& end{align*}

Dependencies

None

Count Occurrences

f_ ext{count} left(, l, s , ight)

Count the number of occurrences of a value $s$ in a list $l$ .

Arguments

Argument	Description	Domain	Constraints	Notes
$l$	source list	$\forall T: [T, ...]$		The list can be empty.
$s$	target value	$T$

Return

Value	Description	Codomain	Constraints	Notes
$\omega$	occurrence count	$\mathbb{N}$		$\omega = 0$ for an empty list.

Instances

f_{count}\left(\left[\right],\ 1\right) - 0
f_{count}\left(\left[1\right],\ 1\right) - 1
f_{count}\left(\left[1,\ 1,\ 1\right],\ 1\right) - 3

Implementation

f_ ext{count} left(, l , ight) = operatorname{length}left(lleft[l=s ight] ight)

Dependencies

None

Count Occurrences (parallelised)

f_ ext{countl} left(, l, l_s , ight)

For each value $s$ in $l_s$ , count the number of times $s$ appears in $l$ .

Arguments

Argument	Description	Domain	Constraints	Notes
$l$	source list	$\forall T: [T, ...]$		The list can be empty.
$l_s$	target values	$[T, ...]$		The list can be empty.

Return

Value	Description	Codomain	Constraints	Notes
$\omega$	occurrence counts	$[\mathbb{N}, ...]$	Same lenth as $l_s$	Counts are in the same order as their corresponding values in $l_s$ .

Instances

f_{countl}\left(\left[\right],\ \left[\right]\right)-\left[\right]
f_{countl}\left(\left[0,\ 1\right],\ \left[\right]\right)-\left[\right]
f_{countl}\left(\left[\right],\ \left[1\right]\right)-0
f_{countl}\left(\left[0,\ 1,\ 1,\ 1\right],\ \left[1\right]\right)-3
f_{countl}\left(\left[0,\ 1,\ 1,\ 1,\ 10,\ 10\right],\ \left[1,\ 10\right]\right)-\left[3,\ 2\right]

Implementation

egin{align*} f_ ext{countl} left(, l, l_s , ight) =& { operatorname{length}left(l_{s} ight) = 0: left[ ight] \ & , left[operatorname{length}left(lleft[l=l_{s}left[i ight] ight] ight)operatorname{for}i=left[1...operatorname{length}left(l_{s} ight) ight] ight] \ & } end{align*}

Dependencies

None

Map Occurrences

f_ ext{counts} left(, l , ight)

Map each unique value $s$ in $l$ to a $(s, c)$ point, where $c$ is the number of times $s$ appears in $l$ .

Arguments

Argument	Description	Domain	Constraints	Notes
$l$	source list	$\forall T \subseteq \mathbb{R}: [T, ...]$		List can be empty.

Return

Value	Description	Codomain	Constraints	Notes
$\omega$	occurrence counts	$[(T, \mathbb{N}), ...]$	Same length as $l$ .	Points are in the same order as the corresponding values in $l$ .

Instances

f_{counts}\left(\left[\right]\right)-\left[\right]
f_{counts}\left(\left[0,\ 10,\ 10\right]\right)-\left[\left(0,\ 1\right),\ \left(10,\ 2\right)\right]
f_{counts}\left(\left[0,\ 10,\ 0,\ 100,\ 10\right]\right)-\left[\left(0,\ 2\right),\ \left(10,\ 2\right),\ \left(100,\ 1\right)\right]

Implementation

egin{align*} f_ ext{counts} left(, l , ight) =& { operatorname{length}left(l ight)=0: left[ ight] \ & quad, operatorname{unique}([ \ &quadquadleft(lleft[i ight], f_{count}left(l, lleft[i ight] ight) ight) \ &quadquadoperatorname{for}i=left[1...operatorname{length}left(l ight) ight] \ & quad]) \ & } end{align*}

Dependencies

None