ps_set

add new element k to this set.

NYI: UNDER DEVELOPMENT: this should be integrated in the mutate effect!

add all elements of the given Sequence to this set

convert a Sequence's head into a 3-tuple.

ex.

[1,2,3,4,5].as_3tuple produces (1,2,3)
[1,2,3].as_3tuple produces (1,2,3)
[1,2].as_3tuple breaks the pre condition
[1].as_3tuple breaks the pre condition
[].as_3tuple breaks the pre condition

chop this Sequence into tuples of size 3. In case
the number of elements is not a multiple of 3,
the last count % 3 elements will be dropped silently.

ex.

[1,2,3,4,5,6,7].tuples3 produces [(1,2,3),(4,5,6)]

convert a Sequence's head into a 4-tuple.

ex.

[1,2,3,4,5].as_4tuple produces (1,2,3,4)
[1,2,3,4].as_4tuple produces (1,2,3,4)
[1,2,3].as_4tuple breaks the pre condition
[1,2].as_4tuple breaks the pre condition
[1].as_4tuple breaks the pre condition
[].as_4tuple breaks the pre condition

chop this Sequence into tuples of size 4. In case
the number of elements is not a multiple of 4,
the last count % 4 elements will be dropped silently.

ex.

[1,2,3,4,5].tuples4 produces [(1,2,3,4)]

convert a Sequence's head into a 5-tuple.

ex.

[1,2,3,4,5,6].as_5tuple produces (1,2,3,4,5)
[1,2,3,4,5].as_5tuple produces (1,2,3,4,5)
[1,2,3,4].as_5tuple breaks the pre condition
[1,2,3].as_5tuple breaks the pre condition
...
[].as_5tuple breaks the pre condition

chop this Sequence into tuples of size 5. In case
the number of elements is not a multiple of 5,
the last count % 5 elements will be dropped silently.

ex.

[1,2,3,4,5,6].tuples5 produces [(1,2,3,4,5)]

convert a Sequence's head into a 6-tuple.

ex.

[1,2,3,4,5,6,7].as_6tuple produces (1,2,3,4,5,6)
[1,2,3,4,5,6].as_6tuple produces (1,2,3,4,5,6)
[1,2,3,4,5].as_6tuple breaks the pre condition
[1,2,3,4].as_6tuple breaks the pre condition
...
[].as_6tuple breaks the pre condition

chop this Sequence into tuples of size 6. In case
the number of elements is not a multiple of 6,
the last count % 6 elements will be dropped silently.

ex.

[1,2,3,4,5,6,7].tuples6 produces [(1,2,3,4,5,6)]

create a sorted array from the elements of this set

create an array backed version of this sequence in case this is not array
backed. This will ensure that operations like index[] or drop perform
in constant time.

returns Sequence.this if is_array_backed.

wrap this sequence into an equatable type.

This requires the underlying element type to be equatable.

list representation of values in this set

wrap this sequence into an orderable type.

This requires the underlying element type to be orderable.

An empty `Sequence` is always less or equal to any other `Sequence`. Two non
empty `Sequence`s whose first elements are unequal are ordered by the
order of those two elements. If those two elements are equal, the tail
of both `Sequence`s will be used to find the order.

create a string representation of this Sequence including all the string
representations of its contents, separated by 'sep'.

NOTE: In case this Sequence is not finite, this will attempt to create an
infinitely long string resulting in failure due to resource exhaustion.

create a string representation of this Sequence including all the string
representations of its contents, separated by ", " and enclosed in '['
and ']'.

NOTE: In case this Sequence is not finite, this will attempt to create an
infinitely long string resulting in failure due to resource exhaustion.

call 'as_string' on the elements

convert a Sequence's head into a 2-tuple.

ex.

[1,2,3,4,5].as_tuple produces (1,2)
[1,2].as_tuple produces (1,2)
[1].as_tuple breaks the pre condition
[].as_tuple breaks the pre condition

chop this Sequence into tuples of size 2. In case
the number of elements is not a multiple of 2,
the last count % 2 elements will be dropped silently.

ex.

[1,2,3,4,5].tuples produces [(1,2),(3,4)]

the arithmetic mean of the sequence
https://en.wikipedia.org/wiki/Arithmetic_mean

create a new Sequence that contains the first elements of
this Sequence for which 'f e' is false

chop this Sequence into chunks of `chunk_size`.
the last chunk may be smaller than `chunk_size`.

create a new Sequence from the result of applying 'f' to the
elements all combinations of elements of this Sequence and
all elements of 'b' iterating of 'b' repeatedly as follows

Sequence.this[0] , b[0]
Sequence.this[0] , b[1]
Sequence.this[0] , b[2]
Sequence.this[0] , ...
Sequence.this[0] , b.last
Sequence.this[1] , b[0]
Sequence.this[1] , b[1]
Sequence.this[1] , ...
... , ...
Sequence.this.last, b.last

create a Sequence that consists of all the elements of this Sequence followed
by all the elements of s

General case: This will perform `as_list.concat_list` in case one of
`Sequence.this` or `s`is not known to be finite (`finite != trit.yet`) or
`s.count` is larger than `count`.

Otherwise, if `Sequence.this` or `s` is empty, the result will be `s` or
`Sequence.this`, respectively.

In all other cases, an instance of `container.expanding_array` will be
created and the elements of `this` and `s` will be added. Note that
`concat` is redefined for `container.expanding_array` to achieve
average `O(1)` performance for repeated concatenation with `s.count < c`
for some constant `c`.

Performance: O(count + s.count) worst case, O(s.count) average case

NOTE: For repeated concatenation `a.concat b` that fall into the general
case the resulting sequence will have iteration performance in `O(n²)`
for `n` concatenation. Explicitly use `container.expanding_array` to
avoid this. This implementation cannot do this automatically since
repeated wrapping of `a` into an `expanding_array` would result
in `O(n²)` performance for the concatenations.

does this set contain the given value?

count the number of elements in this Sequence. Note that this typically
runs forever if executed on an endless Sequence

For Sequences that are not array backed, this might require time in O(count).

count the number of elements in this Sequence that match the
given predicate. Note that this typically
runs forever if executed on an endless Sequence.

get the number of non-overlapping matches of l within this

get the number of matches of l

create a Sequence that repeats the current Sequence indefinitely. In case 'Sequence.this'
is empty, returns 'nil'

filter out consecutive duplicate elements.

Keep the order of elements unchanged.

ex.
[1,2,2,3,2,2,2,4].dedup = [1,2,3,2,4]

filter out consecutive duplicate elements using the
given relation.

Keep the order of elements unchanged.

ex.
[4,2,2,6,2,1,2,4].dedup (a,b -> a%2=b%2) = [4,1,2]
[4,2,2,6,2,1,2,4].dedup (<=) = [4,2,1]

this Set without all elements in other

create a Sequence that consists of the elements of this Sequence except the first
n elements

NOTE: this may have performance in O(n) unless it is backed by an immutable array.

Lazily drop the first elements of this Sequence for which predicate 'p' holds.

Get the dynamic type of this instance. For value instances `x`, this is
equal to `type_of x`, but for `x` with a `ref` type `x.dynamic_type` gives
the actual runtime type, while `type_of x` results in the static
compile-time type.

There is no dynamic type of a type instance since this would result in an
endless hierarchy of types. So for Type values, dynamic_type is redefined
to just return Type.type.

the euclidean norm of this sequence
i.e. the square of the sum of squares of this sequence

filter elements using predicate f
values for which f is false are dropped

get the index of pattern within this Sequence or nil if it does not exist

uses the Knuth-Morris-Pratt algorithm
port of racket code from this paper:
https://www.cambridge.org/core/services/aop-cambridge-core/content/view/8EFA77D663D585B68630E372BCE1EBA4/S0956796824000017a.pdf/knuth-morris-pratt-illustrated.pdf

worst-case performance: O( seq_length ) + O( pattern_length )
worst-case space complexity: O( pattern_length )

is this sequence known to be finite? For infinite sequences, features like
count diverge.

get the first element of this Sequence

get the first element of this Sequence or default if sequence is empty

map each element of this Sequence to Sequence
Then flatten the result by one level,
essentially combining all the sequences.

fold the elements of this Sequence using the given monoid.

e.g., to sum the elements of a Sequence of i32, use s.fold i32.sum

fold the elements of this non-empty Sequence using the given function

e.g., to find the minimum of a Sequence of i32, use `s.fold1 (<=)`

fold the elements of this Sequence using the given function and initial
value.

In case this Sequence is empty, the result is `e`.

e.g., to find the product of a Sequence of i32, use `s.foldf (*) 1`

fold the elements of this Sequence using the given monoid right-to-left.

e.g., to concat the elements of a Sequence `s` of Strings, use `s.foldr String.concat`

fold the elements of this Sequence using the given monoid and initial value right-to-left.

Used to fold a Sequence tail-recursively

fold the elements of this non-empty Sequence using the given function right-to-left

e.g., to concat the elements of a non-empty Sequence of Sequeces, use `foldr1 (++)`

fold the elements of this Sequence using the given function right-to-left.

e.g., to concat the elements of a Sequence of Sequences, use `foldrf [] (++)`

run `f` on each element in this Sequence in order.

Note that this makes sense only if `f` has some side-effect, e.g., to print
each element, use `s.for_each say`

apply 'f' to each element 'e' as long as 'f e'

group the elements of this sequence by a key of type K
using a mutable_tree_map

f determines the key of an element

group the elements of this sequence by a key of type K
using a custom Mutable_Map

f determines the key of an element

group the elements of this sequence by a key of type and applies function f to all elements

example: group characters by category and add underscores around each character

values array codepoint := ["A", "1", "b", "?", "X", "4"]

classify (c codepoint) String => c.is_ascii_letter ? "letter" : c.is_ascii_digit ? "digit" : "other"

values.group_map String String classify (x->"_{x}_")

=> {(digit => [_1_, _4_]), (letter => [_A_, _b_, _X_]), (other => [_?_])}

group elements using key_f and
reduce elements within a group by first applying f and then using reduce_f to reduce

example: count occurrences of letters, numbers and other characters

values array codepoint := ["A", "1", "b", "?", "X", "4"]

classify (c codepoint) String => c.is_ascii_letter ? "letter" : c.is_ascii_digit ? "digit" : "other"

values.group_map_reduce String i32 classify (_->1) (+)

=> {(digit => 2), (letter => 3), (other => 1)}

group elements using key_f and reduce elements within a group using reduce_f
in contrast to the more general version, values in the resulting map must have the same type as in the input

example: sum even and odd numbers individually

(0..10).group_reduce String (x -> x%%2 ? "even" : "odd") (+)

=> {(even => 30), (odd => 25)}

check if an element equal to given element k is part of this set

get a function that, given an index, returns the element at that index

the nth element in the sequence, must exist

determine the index of element x within this list. 0 if x is at the
head of the list, 1 if it comes directly after head, etc. nil if x is
not in the list.

adds the corresponding index to
every element in the sequence

adds an index to every element
in the sequence starting at start_idx

create a Sequence that repeats the current Sequence `n` times.

infix operand synonym for concat

map the Sequence to a new Sequence applying function f to all elements

This is an infix operator alias of map enabling piping code like

l := 1..10 | *10 | 300-

to obtain 290,280,270,...200

Note that map and therefore also this operator is lazy, so

_ := (1..10 | say)

will not print anything while

(1..10 | say).for_each _->unit

will print the elements since `for_each` is not lazy.

check if predicate f holds for all elements

check if predicate f holds for at least one element

infix variant for Set.difference
Note: the operator is NOT a backslash, but the unicode character U+2216 (SET MINUS)

infix variant for Set.intersection

infix variant for Set.union

insert element v at position at

create an intersection of these two Sets

apply transducer to sequence, returning a sequence of results

example usage:
human(age i32) is
ages := map (Sequence i32) human i32 (x -> x.age)
gt_ten := filter (Sequence i32) i32 (x -> x > 10)
xf := ages ∘ gt_ten
say ([human 4, human 12, human 30].into xf) # [12,30]

is this Sequence known to be array backed? If so, this means that operations
like index[] are fast.

is this Sequence empty?

is this Sequence sorted?

is this Set a subset of other?
true, iff other contains all
elements that this Set contains.

is this Set a superset of other?
true, iff this Set contains all
elements that other contains.

check if argument is a valid index in this sequence.

Note that this may have a performance in O(i) unless this
Sequence is_array_backed.

get the last element of this Sequence

This may take time in O(count), in particular, it may not terminate
for an infinite Sequence.

get the last element of this Sequence or default if sequence is empty

map the Sequence to a new Sequence applying function f to all elements

This performs a lazy mapping, f is called only when the elements
in the resulting list are accessed.

Map this Sequence to f applied to neighboring pairs of values
in this Sequence.

In case this Sequence has less than two elements, the result will
be the empty list.

ex. to obtain a Sequence of differences, you may use `map_pairs (-)`:

[2,3,5,7,11,13,17,19,23,29].map_pairs a,b->b-a

results in `[1,2,2,4,2,4,2,4,6]`

get the highest element in this set

the median of the sequence
https://en.wikipedia.org/wiki/Median

get the lowest element in this set

the nth element in the sequence if it exists, wrapped in an option,
nil otherwise.

Complexity: if Sequence is array backed O(1) otherwise O(n)

create a new Sequence from tuples of all combinations of elements
of this Sequence and all elements of 'b' iterating of 'b' repeatedly
as follows

(Sequence.this[0] , b[0] )
(Sequence.this[0] , b[1] )
(Sequence.this[0] , b[2] )
(Sequence.this[0] , ... )
(Sequence.this[0] , b.last)
(Sequence.this[1] , b[0] )
(Sequence.this[1] , b[1] )
(Sequence.this[1] , ... )
(... , ... )
(Sequence.this.last, b.last)

calls `f` for each element in the Sequence.

Unlike `for_each` this returns itself
allowing easier composition with
other Sequence features.

example:

[1,2,3,4,5]
.filter is_prime
.peek say
.drop_while <10

convenience prefix operator to create a string from a value.

This permits usage of `$` as a prefix operator in a similar way both
inside and outside of constant strings: $x and "$x" will produce the
same string.

generic product of the elements of a Sequence of numeric.

This allows multiplying the elements of a list, as in

l := [1,2,3]
say <| l.product # '6'

reduce this Sequence to R with an initial value init
and a reducing function f.
the reduction is finished once f yields abort or
if the end of the sequence is reached.

reduce this Sequence to `outcome R`
with an initial value `init` and a reducing function `f`.
the reduction is finished once `f` yields `abort` or
if the end of the sequence is reached.

replace all occurrences of old by new

replace the first n occurrences of old by new

reverse the order of the elements in this Sequence

map this Sequence to a list that contains the result of folding
all prefixes using the given monoid.

e.g., for a Sequence of i32 s, s.scan i32.sum creates a list of
partial sums (0..).map x->(s.take x).fold i32.sum

map this Sequence to a Sequence that contains the result of folding
all prefixes using the given function and initial value.

e.g., for a Sequence s `[3,8,10]`, `s.scan 0 (+)` creates a Sequence of
partial sums `[0, 3, 11, 21]`

scan1 works like its counterpart with an initial value, except
that the initial value is taken to be the first element of the given
Sequence.

for example, (1::id).scan (+) would create [1, 2, 3, 4, ...], while
(1..).scan (+) would create [1, 3, 6, 10, ...].

number of entries in this set. May be undefined, i.e., a range of
floating point numbers or an infinite set.

create a slice from this Sequence that consists of the elements starting at index
from (including) up to index to (excluding).

sliding window with step size one
blocks of size elements, each is offset by one element to the previous one

example:
`(0..5).sliding 3`
=> `[[0, 1, 2], [1, 2, 3], [2, 3, 4], [3, 4, 5]]`

sliding window
blocks of size elements, each is offset by step elements to the previous one

examples:
`(0..5).sliding 3 1`
=> `[[0, 1, 2], [1, 2, 3], [2, 3, 4], [3, 4, 5]]`

`(0..9).sliding 3 2`
=> `[[0, 1, 2], [2, 3, 4], [4, 5, 6], [6, 7, 8]]`

sort this Sequence

sort this Sequence using the order defined by less_or_equal

create a tuple of two Sequences by splitting this at the given index, i.e.,
a Sequence of length 'at' and one of length 'count-at'.

at may be <= 0 or >= count, in which case the resulting tuple will be the
(empty list, Sequence.this.as_list) or (Sequence.this.as_list, empty list), resp.

does this sequence start with l?

the standard deviation of the sequence
https://en.wikipedia.org/wiki/Standard_deviation

generic sum of the elements of a Sequence of numeric.

This allows summing the elements of a list, as in

l := [1,2,3]
say <| l.sum # '6'

create a lazy list of all the tails of this Sequence, including the complete Sequence
as a list and the empty list 'nil'.

create a Sequence that consists only of the first n elements of this
Sequence, fewer if this Sequence has fewer elements

Lazily take the first elements of this Sequence for which predicate 'p' holds.

takes a transducer xf, a reducer f and an initial value
returns the result of applying the reducer xf f to the Sequence

create a union of these two Sets

filter out duplicate elements.

Keep the order of elements unchanged.

Other languages call this 'distinct' (eg., Java, C# or Kotlin)
or `nub` (Haskell).

ex.
[4,1,2,2,3,2,2,2,4].unique = [4, 1, 2, 3]

the variance of the sequence
https://en.wikipedia.org/wiki/Variance

create a new Sequence from the result of applying 'f' to the
elements of this Sequence and 'b' in order.

string representation of this type to be used for debugging.

result has the form "Type of '<name>'", but this might change in the future

Maximum number of elements shown for on a call to `as_string` for a non-finite
Sequence.

monoid of Sequences with infix concatenation operation.

There is no dynamic type of a type instance since this would result in an
endless hierarchy of types, so dynamic_type is redefined to just return
Type.type here.

an empty ps_set

equality implements the default equality relation for values of this type.

This relation must be

- reflexive (equality a b),
- symmetric (equality a b = equality b a), and
- transitive ((equality a b && equality b c) : equality a c).

result is true iff 'a' is considered to represent the same abstract value
as 'b'.

Is this type assignable to a type parameter with constraint `T`?

The result of this is a compile-time constant that can be used to specialize
code for a particular type.

is_of_integer_type(n T : numeric) => T : integer
say (is_of_integer_type 1234) # true
say (is_of_integer_type 3.14) # false

it is most useful in conjunction preconditions or `if` statements as in

pair(a,b T) is
same
pre T : property.equatable
=>
a = b

or

val(n T) is

name of this type, including type parameters, e.g. 'option (list i32)'.

initialize a partially sorted set from one Sequence

This feature creates a pre-initialized instance of ps_set.

convenience prefix operator to create a string from a value.

This permits usage of `$` as a prefix operator in a similar way both
inside and outside of constant strings: $x and "$x" will produce the
same string.

NYI: Redefinition allows the type feature to be distinguished from its normal counterpart, see #3913

Get a type as a value.

This is a feature with the effect equivalent to Fuzion's `expr.type` call tail.
It is recommended to use `expr.type` and not `expr.type_value`.

`type_value` is here to show how this can be implemented and to illustrate the
difference to `dynamic_type`.

monoid of Set with infix ∪ operation.