Version -10.0.0

Each function, combinator, or definition should be documented here.

## &

See and.

## &&

Combinator

Short-circuiting Boolean AND

Accept two quoted programs, run the first and expect a Boolean value, if it’s `true` pop it and run the second program (which should also return a Boolean value) otherwise pop the second program (leaving `false` on the stack.)

       [A] [B] &&
    ---------------- true
            B

       [A] [B] &&
    ---------------- false
         false

### Definition

    nulco [nullary [false]] dip branch

### Derivation

TODO: this is derived in one of the notebooks I think, look it up and link to it, or copy the content here.

### Discussion

This is seldom useful, I suspect, but this way you have it.

### Crosslinks

||

## *

See mul.

## •

See id.

## ^

See xor.

## =

See eq.

## !=

See ne.

## !-

Function

Not negative.

        n !-
    ----------- n < 0
       false

       n !-
    ---------- n >= 0
       true

### Definition

    0 \>=

### Discussion

Return a Boolean value indicating if a number is greater than or equal to zero.

## >

See gt.

## >=

See ge.

## >>

See rshift.

## -

See sub.

## –

See pred.

## <

See lt.

## <=

See le.

## <>

See ne.

## <{}

Function

       ... a <{}
    ----------------
       ... [] a

### Definition

    [] swap

### Discussion

Tuck an empty list just under the first item on the stack.

### Crosslinks

<<{}

## <<

See lshift.

## <<{}

Function

       ... b a <{}
    -----------------
       ... [] b a

### Definition

    [] rollup

### Discussion

Tuck an empty list just under the first two items on the stack.

### Crosslinks

<{}

## %

See mod.

## +

See add.

## ++

See succ.

## ?

Function

Is the item on the top of the stack “truthy”?

### Definition

> dup bool

### Discussion

You often want to test the truth value of an item on the stack without consuming the item.

### Crosslinks

bool

## /

See floordiv.

## //

See floordiv.

## /floor

See floordiv.

## ||

Combinator

Short-circuiting Boolean OR

### Definition

> nulco [nullary] dip [true] branch

### Discussion

Accept two quoted programs, run the first and expect a Boolean value, if it’s `false` pop it and run the second program (which should also return a Boolean value) otherwise pop the second program (leaving `true` on the stack.)

       [A] [B] ||
    ---------------- A -> false
            B

       [A] [B] ||
    ---------------- A -> true
          true

### Crosslinks

&&

## abs

Function

Return the absolute value of the argument.

### Definition

> dup 0 < [] [neg] branch

## add

Basis Function

Add two numbers together: a + b.

## anamorphism

Combinator

Build a list of values from a generator program `G` and a stopping predicate `P`.

               [P] [G] anamorphism
    -----------------------------------------
       [P] [pop []] [G] [dip swons] genrec

### Definition

> [pop []] swap [dip swons] genrec

### Example

The `range` function generates a list of the integers from 0 to n - 1:

> [0 <=] [-- dup] anamorphism

### Discussion

See the Recursion Combinators notebook.

## and

Basis Function

Logical bit-wise AND.

### Crosslinks

or xor

## app1

“apply one”

Combinator

Given a quoted program on TOS and anything as the second stack item run the program without disturbing the stack and replace the two args with the first result of the program.

             ... x [Q] app1
    ---------------------------------
       ... [x ...] [Q] infra first

This is the same effect as the unary combinator.

### Definition

> nullary popd

### Discussion

Just a specialization of `nullary` really. Its parallelizable cousins are more useful.

### Crosslinks

app2 app3 appN unary

## app2

Combinator

Like app1 with two items.

       ... y x [Q] . app2
    -----------------------------------
       ... [y ...] [Q] . infra first
           [x ...] [Q]   infra first

### Definition

> [grba swap grba swap] dip [infrst] cons ii

### Discussion

Unlike app1, which is essentially an alias for unary, this function is not the same as binary. Instead of running one program using exactly two items from the stack and pushing one result (as binary does) this function takes two items from the stack and runs the program twice, separately for each of the items, then puts both results onto the stack.

This is not currently implemented as parallel processes but it can (and should) be done.

### Crosslinks

app1 app3 appN unary

## app3

Combinator

Like app1 with three items.

         ... z y x [Q] . app3
    -----------------------------------
       ... [z ...] [Q] . infra first
           [y ...] [Q]   infra first
           [x ...] [Q]   infra first

### Definition

> 3 appN

### Discussion

See app2.

### Crosslinks

app1 app2 appN unary

## appN

Combinator

Like app1 with any number of items.

       ... xN ... x2 x1 x0 [Q] n . appN
    --------------------------------------
       ... [xN ...] [Q] . infra first
                       ...
           [x2 ...] [Q]   infra first
           [x1 ...] [Q]   infra first
           [x0 ...] [Q]   infra first

### Definition

> [grabN] codi map disenstacken

### Discussion

This function takes a quoted function `Q` and an integer and runs the function that many times on that many stack items. See also app2.

### Crosslinks

app1 app2 app3 unary

## at

See getitem.

## average

Function

Compute the average of a list of numbers. (Currently broken until I can figure out what to do about “numeric tower” in Thun.)

### Definition

> [sum] [size] cleave /

### Discussion

Theoretically this function would compute the sum and the size in two separate threads, then divide. This works but a compiled version would probably do better to sum and count the list once, in one thread, eh?

As an exercise in Functional Programming in Joy it would be fun to convert this into a catamorphism. See the Recursion Combinators notebook.

## b

Combinator

Run two quoted programs

       [P] [Q] b
    ---------------
          P Q

### Definition

> [i] dip i

### Discussion

This combinator may seem trivial but it comes in handy.

### Crosslinks

dupdip ii

## binary

Combinator

Run a quoted program using exactly two stack values and leave the first item of the result on the stack.

       ... y x [P] binary
    -----------------------
            ... a

### Definition

> unary popd

### Discussion

Runs any other quoted function and returns its first result while consuming exactly two items from the stack.

### Crosslinks

nullary ternary unary

## bool

Basis Function

Convert the item on the top of the stack to a Boolean value.

### Discussion

For integers 0 is `false` and any other number is `true`; for lists the empty list is `false` and all other lists are `true`.

### Crosslinks

not

## branch

Basis Combinator

Use a Boolean value to select and run one of two quoted programs.

       false [F] [T] branch
    --------------------------
              F

       true [F] [T] branch
    -------------------------
                 T

### Definition

> rolldown choice i

### Discussion

This is one of the fundamental operations (although it can be defined in terms of choice as above). The more common “if..then..else” construct ifte adds a predicate function that is evaluated nullary.

### Crosslinks

choice ifte select

## ccccons

Function

       a b c d [...] ccccons
    ---------------------------
           [a b c d ...]

Do cons four times.

### Definition

> ccons ccons

### Crosslinks

ccons cons times

## ccons

Function

       a b [...] ccons
    ---------------------
          [a b ...]

Do cons two times.

### Definition

> cons cons

### Crosslinks

cons ccons

## choice

Basis Function

Use a Boolean value to select one of two items.

       a b false choice
    ----------------------
              a

       a b true choice
    ---------------------
              b

### Definition

> [pop] [popd] branch

### Discussion

It’s a matter of taste whether you implement this in terms of branch or the other way around.

### Crosslinks

branch select

## clear

Basis Function

Clear everything from the stack.

### Definition

> stack bool [pop stack bool] loop

### Crosslinks

stack swaack

## cleave

Combinator

Run two programs in parallel, consuming one additional item, and put their results on the stack.

       ... x [A] [B] cleave
    ------------------------
            ... a b

### Derivation

> fork popdd

### Example

       1 2 3 [+] [-] cleave
    --------------------------
             1 2 5 -1

### Discussion

One of a handful of useful parallel combinators.

### Crosslinks

clop fork map

## clop

Combinator

Run two programs in parallel, consuming two additional items, and put their results on the stack.

       ... x y [A] [B] clop
    --------------------------
            ... a b

### Definition

> cleave popdd

### Discussion

Like cleave but consumes an additional item from the stack.

       1 2 3 4 [+] [-] clop
    --------------------------
             1 2 7 -1

### Crosslinks

cleave fork map

## cmp

Combinator

Take two values and three quoted programs on the stack and run one of the three depending on the results of comparing the two values.

       a b [G] [E] [L] cmp
    ------------------------- a > b
            G

       a b [G] [E] [L] cmp
    ------------------------- a = b
                E

       a b [G] [E] [L] cmp
    ------------------------- a < b
                    L

### Discussion

This is useful sometimes, and you can dup or dupd with two quoted programs to handle the cases when you just want to deal with <= or >= and not all three possibilities, e.g.:

    [G] [EL] dup cmp

    [GE] [L] dupd cmp

Or even:

    [GL] [E] over cmp

### Crosslinks

TODO: link to tree notebooks where this was used.

## codi

Combinator

Take a quoted program from the stack, cons the next item onto it, then dip the whole thing under what was the third item on the stack.

       a b [F] . codi
    --------------------
             b . F a

### Definition

> cons dip

### Discussion

This is one of those weirdly specific functions that turns out to be useful in a few places.

### Crosslinks

appN codireco

## codireco

Combinator

This is part of the make_generator function. You would not use this combinator directly.

### Definition

> codi reco

### Discussion

See make_generator and the “Using `x` to Generate Values” notebook as well as Recursion Theory and Joy by Manfred von Thun.

### Crosslinks

make_generator

## concat

Function

Concatinate two lists.

       [a b c] [d e f] concat
    ----------------------------
           [a b c d e f]

### Crosslinks

first first_two flatten fourth getitem remove rest reverse rrest second shift shunt size sort split_at split_list swaack third zip

## cond

Combinator

This combinator works like a case statement. It expects a single quote on the stack that must contain zero or more condition quotes and a default quote. Each condition quote should contain a quoted predicate followed by the function expression to run if that predicate returns `true`. If no predicates return `true` the default function runs.

    [
        [ [Predicate0] Function0 ]
        [ [Predicate1] Function1 ]
        ...
        [ [PredicateN] FunctionN ]
        [Default]
    ]
    cond

### Discussion

It works by rewriting into a chain of nested ifte expressions, e.g.:

          [[[B0] T0] [[B1] T1] [D]] cond
    -----------------------------------------
       [B0] [T0] [[B1] [T1] [D] ifte] ifte

### Crosslinks

ifte

## cons

Basis Function

Given an item and a list, append the item to the list to make a new list.

       a [...] cons
    ------------------
         [a ...]

### Discussion

Cons is a venerable old function from Lisp. Its inverse operation is uncons.

### Crosslinks

uncons

## dinfrirst

Combinator

Specialist function (that means I forgot what it does and why.)

### Definition

> dip infrst

## dip

Basis Combinator

The `dip` combinator expects a quoted program on the stack and below it some item, it hoists the item into the expression and runs the program on the rest of the stack.

       ... x [Q] . dip
    ---------------------
             ... . Q x

### Discussion

This along with infra are enough to update any datastructure. See the “Traversing Datastructures with Zippers” notebook.

Note that the item that was on the top of the stack (`x` in the example above) will not be treated specially by the interpreter when it is reached again. This is something of a footgun. My advice is to avoid putting bare unquoted symbols onto the stack, but then you can’t use symbols as “atoms” and also use `dip` and `infra` to operate on compound datastructures with atoms in them. This is a kind of side-effect of the Continuation-Passing Style. The `dip` combinator could “set aside” the item and replace it after running `Q` but that means that there is an “extra space” where the item resides while `Q` runs. One of the nice things about CPS is that the whole state is recorded in the stack and pending expression (not counting modifications to the dictionary.)

### Crosslinks

dipd dipdd dupdip dupdipd infra

## dipd

Combinator

Like dip but expects two items.

       ... y x [Q] . dipd
    -------------------------
               ... . Q y x

### Discussion

See dip.

### Crosslinks

dip dipdd dupdip dupdipd infra

## dipdd

Combinator

Like dip but expects three items. :

       ... z y x [Q] . dip
    -----------------------------
                 ... . Q z y x

### Discussion

See dip.

### Crosslinks

dip dipd dupdip dupdipd infra

## disenstacken

Function

The `disenstacken` function expects a list on top of the stack and makes that the stack discarding the rest of the stack.

       1 2 3 [4 5 6] disenstacken
    --------------------------------
                6 5 4

### Definition

> [clear] dip reverse unstack

### Discussion

Note that the order of the list is not changed, it just looks that way because the stack is printed with the top on the right while lists are printed with the top or head on the left.

### Crosslinks

enstacken stack unstack

## div

See floordiv.

## divmod

Function

        x y divmod
    ------------------
         q      r
       (x/y)  (x%y)

Invariant: `qy + r = x`.

### Definition

> [floordiv] [mod] clop

## down_to_zero

Function

Given a number greater than zero put all the Natural numbers (including zero) less than that onto the stack.

### Example

       3 down_to_zero
    --------------------
          3 2 1 0

### Definition

> [0 >] [dup –] while

### Crosslinks

range

## drop

Function

Expects an integer and a quote on the stack and returns the quote with n items removed off the top.

### Example

       [a b c d] 2 drop
    ----------------------
           [c d]

### Definition

> [rest] times

### Crosslinks

take

## dup

Basis Function

“Dup”licate the top item on the stack.

       a dup
    -----------
        a a

### Crosslinks

dupd dupdd dupdip dupdipd

## dupd

Function

dup the second item down on the stack.

       a b dupd
    --------------
        a a b

### Definition

> [dup] dip

### Crosslinks

dup dupdd dupdip dupdipd

## dupdd

Function

dup the third item down on the stack.

       a b c dupdd
    -----------------
         a a b c

### Definition

> [dup] dipd

### Crosslinks

dup dupd dupdip dupdipd

## dupdip

Combinator

Apply a function `F` and dup the item under it on the stack.

       a [F] dupdip
    ------------------
          a F a

### Definition

> dupd dip

### Derivation

    a [F] dupdip
    a [F] dupd dip
    a [F] [dup] dip dip
    a dup [F] dip
    a a [F] dip
    a F a

### Discussion

A very common and useful combinator.

### Crosslinks

dupdipd

## dupdipd

Combinator

Run a copy of program `F` under the next item down on the stack.

       a [F] dupdipd
    -------------------
          F a [F]

### Definition

> dup dipd

### Crosslinks

dupdip

## enstacken

Function

Put the stack onto the stack replacing the contents of the stack.

       ... a b c enstacken
    -------------------------
           [c b a ...]

### Definition

> stack [clear] dip

### Discussion

This is a destructive version of stack. See the note under disenstacken about the apparent but illusory reversal of the stack.

### Crosslinks

stack [unstack] disenstacken

## eq

Basis Function

Compare the two items on the top of the stack for equality and replace them with a Boolean value.

       a b eq
    -------------
       Boolean
       (a = b)

### Crosslinks

cmp ge gt le lt ne

## first

Function

Replace a list with its first item.

       [a ...]
    --------------
          a

### Definition

> uncons pop

### Crosslinks

second third fourth rest

## first_two

Function

Replace a list with its first two items.

       [a b ...] first_two
    -------------------------
               a b

### Definition

> uncons first

### Crosslinks

first second third fourth rest

## flatten

Function

Given a list of lists, concatinate them.

### Example

       [[1 2] [3 [4] 5] [6 7]] flatten
    -------------------------------------
              [1 2 3 [4] 5 6 7]

### Definition

> <{} [concat] step

### Discussion

Note that only one “level” of lists is flattened. In the example above `[4]` is not unquoted.

### Crosslinks

concat first first_two fourth getitem remove rest reverse rrest second shift shunt size sort split_at split_list swaack third zip

## floor

Basis Function

Return the largest integer <= x.

### Discussion

This function doesn’t make sense (yet) to have because there are (as yet) only integers in the system.

## floordiv

Basis Function

I don’t know why this is called “floor” div, I think it rounds its result down (not towards zero or up.)

       a b floordiv
    ------------------
          (a/b)

### Discussion

All the division commands need to be revisited when the “numeric tower” for Thun gets nailed down.

### Crosslinks

divmod

## fork

Combinator

Run two quoted programs in parallel and replace them with their results.

       ... [F] [G] fork
    ----------------------
           ... f g

### Definition

> [i] app2

### Discussion

The basic parallelism combinator, the two programs are run independently.

### Crosslinks

cleave clop map

## fourth

Function

Replace a list with its fourth item.

       [a b c d ...] fourth
    --------------------------
              d

### Definition

> rest third

### Crosslinks

first second third rest

## gcd

Function

Take two integers from the stack and replace them with their Greatest Common Denominator.

### Definition

> true [tuck mod dup 0 >] loop pop

### Discussion

Euclid’s Algorithm

## gcd2

Function

Compiled GCD function.

### Discussion

See gcd.

### Crosslinks

gcd

## ge

Basis Function

Greater-than-or-equal-to comparison of two numbers.

       a b ge
    --------------
       Boolean
       (a >= b)

### Crosslinks

cmp eq gt le lt ne

## genrec

Combinator

General Recursion Combinator.

                          [if] [then] [rec1] [rec2] genrec
    ---------------------------------------------------------------------
       [if] [then] [rec1 [[if] [then] [rec1] [rec2] genrec] rec2] ifte

### Definition

> [[genrec] ccccons] nullary swons concat ifte

(Note that this definition includes the `genrec` symbol itself, it is self-referential. This is possible because the definition machinery does not check that symbols in defs are in the dictionary. `genrec` is the only self-referential definition.)

### Discussion

See the Recursion Combinators notebook.

From “Recursion Theory and Joy” by Manfred von Thun:

> “The genrec combinator takes four program parameters in addition to whatever data parameters it needs. Fourth from the top is an if-part, followed by a then-part. If the if-part yields true, then the then-part is executed and the combinator terminates. The other two parameters are the rec1-part and the rec2-part. If the if-part yields false, the rec1-part is executed. Following that the four program parameters and the combinator are again pushed onto the stack bundled up in a quoted form. Then the rec2-part is executed, where it will find the bundled form. Typically it will then execute the bundled form, either with i or with app2, or some other combinator.”

The way to design one of these is to fix your base case `[then]` and the test `[if]`, and then treat `rec1` and `rec2` as an else-part “sandwiching” a quotation of the whole function.

For example, given a (general recursive) function `F`:

    F == [I] [T] [R1] [R2] genrec

If the `[I]` if-part fails you must derive `R1` and `R2` from: :

    ... R1 [F] R2

Just set the stack arguments in front, and figure out what `R1` and `R2` have to do to apply the quoted `[F]` in the proper way. In effect, the `genrec` combinator turns into an ifte combinator with a quoted copy of the original definition in the else-part:

    F == [I] [T] [R1]   [R2] genrec
      == [I] [T] [R1 [F] R2] ifte

Tail recursive functions are those where `R2` is the `i` combinator:

    P == [I] [T] [R] tailrec
      == [I] [T] [R [P] i] ifte
      == [I] [T] [R P] ifte

### Crosslinks

anamorphism tailrec x

## getitem

Function

Expects an integer and a quote on the stack and returns the item at the nth position in the quote counting from 0.

### Example

       [a b c d] 2 getitem
    -------------------------
            c

### Definition

> drop first

### Discussion

If the number isn’t a valid index into the quote `getitem` will cause some sort of problem (the exact nature of which is implementation-dependant.)

### Crosslinks

concat first first_two flatten fourth remove rest reverse rrest second shift shunt size sort split_at split_list swaack third zip

## grabN

Function

Expect a number on the top of the satck and put that many items from uner it onto a new list.

### Example

       a b c d e 3 grabN
    -----------------------
          a b [c d e]

### Definition

> <{} [cons] times

## grba

Function

A weird function used in app2 that does this:

          ... 1 2 3 4 5 grba
    -------------------------------
       ... 1 2 3 [4 3 2 1 ...] 5

It grabs the stack under the top item, and substitutes it for the second item down on the stack.

### Definition

> [stack popd] dip

### Discussion

This function “grabs” an item from the stack along with a copy of the stack. It’s part of the app2 definition.

### Crosslinks

app2

## gt

Basis Function

Greater-than comparison of two numbers.

       a b gt
    --------------
       Boolean
       (a > b)

### Crosslinks

cmp eq ge le lt ne

## help

Function

Accepts a quoted symbol on the top of the stack and prints its documentation.

       [foo] help
    ----------------

### Discussion

Technically this is equivalent to `pop`, but it will only work if the item on the top of the stack is a quoted symbol.

## hypot

Function

             x y hypot
    ---------------------------
       sqrt(sqr(x) + sqr(y))

### Definition

> [sqr] ii + sqrt

### Discussion

This is another function that has to wait on the numeric tower.

### Crosslinks

sqrt

## i

Basis Combinator

Append a quoted expression onto the pending expression.

       [Q] . i
    -------------
           . Q

### Discussion

This is a fundamental combinator. It is used in all kinds of places. For example, the x combinator can be defined as `dup i`.

## id

Basis Function

The identity function.

### Discussion

Does nothing. It’s kind of a mathematical thing, but it occasionally comes in handy.

## ifte

Combinator

If-Then-Else combinator, a common and convenient specialization of branch.

            [if] [then] [else] ifte
    ---------------------------------------
       [if] nullary [else] [then] branch

### Definition

> [nullary] dipd swap branch

### Crosslinks

branch loop while

## ii

Combinator

Take a quoted program from the stack and run it twice, first under the top item, then again with the top item.

    ... a [Q] ii
    ------------------
     ... Q a Q

### Definition

> [dip] dupdip i

### Example

It’s a little tricky to understand how this works so here’s an example trace:

          1 2 3 4 [++] • [dip] dupdip i
    1 2 3 4 [++] [dip] • dupdip i
          1 2 3 4 [++] • dip [++] i
                 1 2 3 • ++ 4 [++] i
                 1 2 4 • 4 [++] i
               1 2 4 4 • [++] i
          1 2 4 4 [++] • i
               1 2 4 4 • ++
               1 2 4 5 •

### Discussion

In some cases (like the example above) this is the same effect as using app2 but most of the time it’s not:

       1 2 3 4 [+] ii
    --------------------
            1 9

       1 2 3 4 [+] app2
    ----------------------
           1 2 5 6

### Crosslinks

app2 b

## infra

Combinator

Accept a quoted program and a list on the stack and run the program with the list as its stack. Does not affect the stack (below the list.)

       ... x y z [a b c] [Q] infra
    ---------------------------------
        c b a Q [z y x ...] swaack

### Definition

> swons swaack [i] dip swaack

### Discussion

This is one of the more useful combinators. It allows a quoted expression to serve as a stack for a program, effectively running it in a kind of “pocket universe”. If the list represents a datastructure then `infra` lets you work on its internal structure.

### Crosslinks

swaack

## infrst

Combinator

Does infra and then extracts the first item from the resulting list.

### Definition

> infra first

## inscribe

Create a new Joy function definition in the Joy dictionary. A definition is given as a quote with a name followed by a Joy expression.

### Example

    [sqr dup mul] inscribe

### Discussion

This is the only function that modifies the dictionary. It’s provided as a convenience, for tinkering with new definitions before entering them into the `defs.txt` file. It can be abused, which you should avoid unless you know what you’re doing.

## le

Basis Function

Less-Than-or-Equal-to comparison of the two items on the top of the stack, replacing them with a Boolean value.

       a b le
    -------------
       Boolean
       (a <= b)

### Crosslinks

cmp eq ge gt lt ne

## loop

Basis Combinator

Expect a quoted program `Q` and a Boolean value on the stack. If the value is false discard the quoted program, otherwise run a copy of `Q` and `loop` again.

       false [Q] loop
    --------------------

       true [Q] . loop
    --------------------------
                . Q [Q] loop

### Discussion

This, along with branch and fork, is one of the four main combinators of all programming. The fourth, sequence, is implied by juxtaposition. That is to say, in Joy `F G` is like `G(F(...))` in a language bassed on function application. Or again, to quote the Joy Wikipedia entry,

> In Joy, the meaning function is a homomorphism from the syntactic monoid onto the semantic monoid. That is, the syntactic relation of concatenation of symbols maps directly onto the semantic relation of composition of functions.

Anyway, branch, fork, amd loop are the fundamental combinators in Joy. Just as branch has it’s more common and convenient form ifte, loop has while.

### Crosslinks

branch fork while

## lshift

Basis Function

Logical Left-Shift

       a n lshift
    ----------------
         (a×2ⁿ)

### Crosslinks

rshift

## lt

Basis Function

Less-Than comparison of the two items on the top of the stack, replacing them with a Boolean value.

       a b lt
    -------------
       Boolean
       (a < b)

### Crosslinks

cmp eq ge gt le ne

## make_generator

Function

Given an initial state value and a quoted generator function build a generator quote.

       state [generator function] make_generator
    -----------------------------------------------
         [state [generator function] codireco]

### Example

       230 [dup ++] make_generator
    ---------------------------------
         [230 [dup ++] codireco]

And then:

       [230 [dup ++] codireco] 5 [x] times pop
    ---------------------------------------------
                 230 231 232 233 234

### Definition

> [codireco] ccons

### Discussion

See the “Using `x` to Generate Values” notebook.

### Crosslinks

codireco

## map

Combinator

Given a list of items and a quoted program run the program for each item in the list (with the rest of the stack) and replace the old list and the program with a list of the results.

### Example

       5 [1 2 3] [++ *] map
    --------------------------
           5 [10 15 20]

### Discussion

This is a common operation in many languages. In Joy it can be a parallelism combinator due to the “pure” nature of the language.

### Crosslinks

app1 app2 app3 appN fork

## max

Basis Function

Given a list find the maximum.

### Example

       [1 2 3 4] max
    -------------------
             4

### Crosslinks

min size sum

## min

Basis Function

Given a list find the minimum.

### Example

       [1 2 3 4] min
    -------------------
             1

### Crosslinks

max size sum

## mod

Basis Function

Return the remainder of `a` divided by `b`.

       a b mod
    -------------
        (a%b)

### Crosslinks

divmod mul

## modulus

See mod.

## mul

Basis Function

Multiply two numbers.

       a b mul
    -------------
        (a×b)

### Crosslinks

div product

## ne

Basis Function

Not-Equal comparison of the two items on the top of the stack, replacing them with a Boolean value.

       a b ne
    -------------
       Boolean
       (a = b)

### Crosslinks

cmp eq ge gt le lt

## neg

Function

Invert the sign of a number.

       a neg
    -----------
        -a

### Definition

> 0 swap -

## not

Function

Like bool but convert the item on the top of the stack to the inverse Boolean value.

       true not
    --------------
        false

       false not
    ---------------
         true

### Definition

> bool [true] [false] branch

### Crosslinks

bool

## nulco

Function

Take the item on the top of the stack and cons it onto `[nullary]`.

         [F] nulco
    -------------------
       [[F] nullary]

### Definition

> [nullary] cons

### Discussion

Helper function for || and &&.

### Crosslinks

&& ||

## nullary

Combinator

Run a quoted program without using any stack values and leave the first item of the result on the stack.

       ... [P] nullary
    ---------------------
            ... a

### Definition

> [stack] dip infra first

### Example

    ... [P] nullary
    ... [P] [stack] dip infra first
    ... stack [P] infra first
    ... [...] [P] infra first
    ... [a ...] first
    ...  a

### Discussion

A very useful function that runs any other quoted function and returns it’s first result without disturbing the stack (under the quoted program.)

### Crosslinks

unary binary ternary

## of

Function

Like getitem but swaps the order of arguments.

### Example

       2 [a b c d] of
    --------------------
             c

### Definition

> swap getitem

### Crosslinks

getitem

## or

Basis Function

Logical bit-wise OR.

### Crosslinks

and xor

## over

Function

dup the second item on the stack `over` the first.

       a b over
    --------------
        a b a

### Definition

There are many many ways to define this function.

> swap tuck

> [pop] nullary

> [dup] dip swap

> unit dupdip

> unit dupdipd first

And so on…

### Discussion

A fine old word from Forth.

### Crosslinks

tuck

## pam

Combinator

Take a list of quoted functions from the stack and replace it with a list of the first results from running those functions (on copies of the rest of the stack.)

### Example

       5 7 [[+][-][*][/][%]] pam
    -------------------------------
          5 7 [12 -2 35 0 5]

### Definition

> [i] map

### Discussion

A specialization of map that runs a list of functions in parallel (if the underlying map function is so implemented, of course.)

### Crosslinks

map

## pick

See getitem.

## pm

Function

Plus or minus. Replace two numbers with their sum and difference.

          a b pm
    -----------------
       (a+b) (a-b)

### Definition

> [+] [-] clop

## pop

Basis Function

Pop the top item from the stack and discard it.

       a pop
    -----------

### Crosslinks

popd popdd popop popopd popopdd popopop

## popd

Function

pop the second item down on the stack.

       a b popd
    --------------
          b

### Definition

> swap pop

### Crosslinks

pop popdd popop popopd popopdd popopop

## popdd

Function

pop the third item on the stack.

       a b c popdd
    -----------------
           b c

### Definition

> rolldown pop

### Crosslinks

pop popd popop popopd popopdd popopop

## popop

Function

pop two items from the stack.

       a b popop
    ---------------

### Definition

> pop pop

### Crosslinks

pop popd popdd popopd popopdd popopop

## popopd

Function

pop the second and third items from the stack.

       a b c popopd
    ------------------
            c

### Definition

> rollup popop

### Crosslinks

pop popd popdd popop popopdd popopop

## popopdd

Function

       a b c d popopdd
    ---------------------
            c d

### Definition

> [popop] dipd

### Crosslinks

pop popd popdd popop popopd popopop

## popopop

Function

pop three items from the stack.

       a b c popopop
    -------------------

### Definition

> pop popop

### Crosslinks

pop popd popdd popop popopd popopdd

## pow

Basis Function

Take two numbers `a` and `b` from the stack and raise `a` to the `n`th power. (`b` is on the top of the stack.)

       a n pow
    -------------
        (aⁿ)

### Example

       2 [2 3 4 5 6 7 8 9] [pow] map
    -----------------------------------
        2 [4 8 16 32 64 128 256 512]

## pred

Function

Predecessor. Decrement TOS.

### Definition

> 1 -

### Crosslinks

succ

## primrec

Combinator

From the “Overview of the language JOY”

> The primrec combinator expects two quoted programs in addition to a data parameter. For an integer data parameter it works like this: If the data parameter is zero, then the first quotation has to produce the value to be returned. If the data parameter is positive then the second has to combine the data parameter with the result of applying the function to its predecessor.

> 5 [1] [*] primrec

> Then primrec tests whether the top element on the stack (initially the 5) is equal to zero. If it is, it pops it off and executes one of the quotations, the [1] which leaves 1 on the stack as the result. Otherwise it pushes a decremented copy of the top element and recurses. On the way back from the recursion it uses the other quotation, [*], to multiply what is now a factorial on top of the stack by the second element on the stack.

       0 [Base] [Recur] primrec
    ------------------------------
          Base

             n [Base] [Recur] primrec
    ------------------------------------------ n > 0
       n (n-1) [Base] [Recur] primrec Recur

### Discussion

Simple and useful specialization of the genrec combinator from the original Joy system.

### Crosslinks

genrec tailrec

## product

Function

Just as sum sums a list of numbers, this function multiplies them together.

### Definition

> 1 swap [mul] step

Or,

> [1] [mul] primrec

## quoted

Function

“Quote D” Wrap the second item on the stack in a list.

       a b quoted
    ----------------
         [a] b

### Definition

> [unit] dip

### Discussion

This comes from the original Joy stuff.

### Crosslinks

unit

## range

Function

Expect a number `n` on the stack and replace it with a list: `[(n-1)...0]`.

### Example

         5 range
    -----------------
       [4 3 2 1 0]

       -5 range
    --------------
          []

### Definition

> [0 <=] [1 - dup] anamorphism

### Discussion

If `n` is less than 1 the resulting list is empty.

### Crosslinks

range_to_zero

## range_to_zero

Function

Take a number `n` from the stack and replace it with a list `[0...n]`.

### Example

       5 range_to_zero
    ---------------------
        [0 1 2 3 4 5]

### Definition

> unit [down_to_zero] infra

### Discussion

Note that the order is reversed compared to range.

### Crosslinks

down_to_zero range

## reco

Function

Replace the first item in a list with the item under it.

       a [b ...] reco
    --------------------
         [a ...]

### Definition

> rest cons

### Crosslinks

codireco make_generator

## rem

See mod.

## remainder

See mod.

## remove

Function

Expects an item on the stack and a quote under it and removes that item from the the quote. The item is only removed once. If the list is empty or the item isn’t in the list then the list is unchanged.

       [1 2 3 1] 1 remove
    ------------------------
            [2 3 1]

### Definition

See the “Remove Function” notebook.

## rest

Basis Function

       [a ...] rest
    ------------------
          [...]

### Crosslinks

first uncons

## reverse

Function

Reverse the list on the top of the stack.

### Example

       [1 2 3] reverse
    ---------------------
           [3 2 1]

### Definition

> <{} shunt

## rolldown

Function

       a b c rolldown
    --------------------
           b c a

### Definition

> swapd swap

### Crosslinks

rollup

## rollup

Function

       a b c rollup
    ------------------
          c a b

### Definition

> swap swapd

### Crosslinks

rolldown

## roll>

See rollup.

## roll<

See rolldown.

## round

Function

Round a number to a given precision in decimal digits.

### Discussion

Another one that won’t make sense until the “numeric tower” is nailed down.

## rrest

Function

       [a b ...] rrest
    ---------------------
            [...]

### Definition

> rest rest

### Crosslinks

rest

## rshift

Basis Function

Logical Right-Shift

       a n rshift
    ----------------
         (a∕2ⁿ)

### Crosslinks

lshift

## run

Function

Run a quoted program in a list.

### Example

       [1 2 +] run
    -----------------
           [3]

### Definition

> <{} infra

## second

Function

       [a b ...] second
    ----------------------
              b

### Definition

> rest first

### Crosslinks

first third fourth

## select

Basis Function

Use a Boolean value to select one of two items from a sequence. :

       [a b] false select
    ------------------------
               a

       [a b] true select
    -----------------------
               b

### Discussion

The sequence can contain more than two items but not fewer.

### Crosslinks

choice

## sharing

Function

Print redistribution information.

### Discussion

Mathematically this is a form of id, but it has the side-effect of printing out the GPL notice.

### Crosslinks

warranty

## shift

Function

Move the top item from one list to another.

### Example

       [x y z] [a b c] shift
    ---------------------------
          [a x y z] [b c]

### Definition

> uncons [swons] dip

### Crosslinks

shunt

## shunt

Function

Like concat but reverse the top list into the second.

### Example

       [a b c] [d e f] shunt
    ---------------------------
           [f e d a b c]

### Definition

> [swons] step

### Discussion

This is more efficient than concat so prefer it if you don’t need to preserve order.

### Crosslinks

concat reverse shift

## size

Function

Replace a list with its size.

### Example

       [23 [cats] 4] size
    ------------------------
               3

### Definition

> [pop ++] step_zero

## sort

Function

Given a list return it sorted.

### Example

       [4 2 5 7 1] sort
    ----------------------
          [1 2 4 5 7]

## spiral_next

Function

Example code.

### Discussion

See the “Square Spiral Example Joy Code” notebook.

## split_at

Function

Split a list (second on the stack) at the position given by the number on the top of the stack.

### Example

       [1 2 3 4 5 6 7] 4 split_at
    --------------------------------
           [5 6 7] [4 3 2 1]

### Definition

> [drop] [take] clop

### Discussion

Take a list and a number `n` from the stack, take `n` items from the top of the list and shunt them onto a new list that replaces the number `n` on the top of the stack.

### Crosslinks

split_list

## split_list

Function

Split a list (second on the stack) at the position given by the number on the top of the stack such that concat would reconstruct the original list.

       [1 2 3 4 5 6 7] 4 split_list
    ----------------------------------
            [1 2 3 4] [5 6 7]

### Definition

> [take reverse] [drop] clop

### Discussion

Compare with split_at. This function does extra work to ensure that concat would reconstruct the original list.

### Crosslinks

split_at

## sqr

Function

Square the number on the top of the stack.

       n  sqr
    ------------
         n²

### Definition

> dup mul

## sqrt

Basis Function Combinator

Return the square root of the number a. Negative numbers return complex roots.

### Discussion

Another “numeric tower” hatch…

## stack

Function

Put the stack onto the stack.

          ... c b a stack
    ---------------------------
       ... c b a [a b c ...]

### Definition

> [] swaack dup swaack first

### Discussion

This function forms a pair with [unstack], and together they form the complement to the “destructive” pair enstacken and disenstacken.

### Crosslinks

[unstack] enstacken disenstacken

## stackd

Function

Grab the stack under the top item and put it onto the stack.

### Example

       ... 1 2 3 stackd
    ------------------------
      ... 1 2 [2 1 ...] 3

### Definition

> [stack] dip

## step

Combinator

Run a quoted program on each item in a sequence.

       ... [] [Q] step
    ---------------------
             ...

       ... [a] [Q] step
    ----------------------
          ... a Q

       ... [a b c] [Q] . step
    ----------------------------------------
                 ... a . Q [b c] [Q] step

### Discussion

See the Recursion Combinators notebook.

### Crosslinks

step_zero

## step_zero

Combinator

Like step but with 0 as the initial value.

       [...] [F] step_zero
    -------------------------
         0 [...] [F] step

### Definition

> 0 roll> step

### Discussion

size and sum can both be defined in terms of this specialization of step.

### Crosslinks

step

## stuncons

Function

Take the stack and uncons the top item.

### Example

       1 2 3 stuncons
    --------------------
       1 2 3 3 [2 1]

### Definition

> stack uncons

## stununcons

Function

Take the stack and uncons the top two items.

### Example

       1 2 3 stununcons
    ----------------------
        1 2 3 3 2 [1]

### Definition

> stack uncons uncons

### Crosslinks

stuncons

## sub

Basis Function

Subtract the number on the top of the stack from the number below it.

       a b sub
    -------------
        (a-b)

### Crosslinks

add

## succ

Function

Successor. Increment TOS.

### Definition

> 1 +

### Crosslinks

pred

## sum

Combinator

Given a quoted sequence of numbers return the sum.

### Example

       [1 2 3 4 5] sum
    ---------------------
             15

### Definition

> [+] step_zero

### Crosslinks

size

## swaack

Basis Function

Swap stack. Take a list from the top of the stack, replace the stack with the list, and put the old stack onto it.

### Example

       1 2 3 [4 5 6] swaack
    --------------------------
       6 5 4 [3 2 1]

### Discussion

This function works as a kind of “context switch”. It’s used in the definition of infra.

### Crosslinks

infra

## swap

Basis Function

Swap the top two items on the stack.

       a b swap
    --------------
         b a

### Crosslinks

swapd

## swapd

Function

Swap the second and third items on the stack.

       a b c swapd
    -----------------
          b a c

### Definition

> [swap] dip

### Crosslinks

over tuck

## swoncat

Function

concat two lists, but swap the lists first.

### Definition

> swap concat

### Crosslinks

concat

## swons

Function

Like cons but swap the item and list.

       [...] a swons
    -------------------
          [a ...]

### Definition

> swap cons

## tailrec

Combinator

A specialization of the genrec combinator.

### Definition

> [i] genrec

### Discussion

Some recursive functions do not need to store additional data or pending actions per-call. These are called “tail recursive” functions. In Joy, they appear as genrec definitions that have i for the second half of their recursive branch.

See the Recursion Combinators notebook.

### Crosslinks

genrec

## take

Function

Expects an integer `n` and a list on the stack and replace them with a list with just the top `n` items in reverse order.

       [a b c d] 2 take
    ----------------------
            [b a]

### Definition

> <<{} [shift] times pop

## ternary

Combinator

Run a quoted program using exactly three stack values and leave the first item of the result on the stack.

       ... z y x [P] ternary
    -------------------------
             ... a

### Definition

> binary popd

### Discussion

Runs any other quoted function and returns its first result while consuming exactly three items from the stack.

### Crosslinks

binary nullary unary

## third

Function

       [a b c ...] third
    -----------------------
               c

### Definition

> rest second

### Crosslinks

first second fourth rest

## times

Combinator

Expect a quoted program and an integer `n` on the stack and do the program `n` times.

       ... n [Q] . times
    -----------------------  w/ n <= 0
             ... .

       ... 1 [Q] . times
    -----------------------
             ... . Q

       ... n [Q] . times
    -------------------------------------  w/ n > 1
             ... . Q (n-1) [Q] times

### Definition

> [-- dip] cons [swap] infra [0 >] swap while pop :

### Discussion

This works by building a little while program and running it:

                     1 3 [++] • [-- dip] cons [swap] infra [0 >] swap while pop
            1 3 [++] [-- dip] • cons [swap] infra [0 >] swap while pop
            1 3 [[++] -- dip] • [swap] infra [0 >] swap while pop
     1 3 [[++] -- dip] [swap] • infra [0 >] swap while pop
                  dip -- [++] • swap [3 1] swaack [0 >] swap while pop
                  dip [++] -- • [3 1] swaack [0 >] swap while pop
            dip [++] -- [3 1] • swaack [0 >] swap while pop
            1 3 [-- [++] dip] • [0 >] swap while pop
      1 3 [-- [++] dip] [0 >] • swap while pop
      1 3 [0 >] [-- [++] dip] • while pop

This is a common pattern in Joy. You accept some parameters from the stack which typically include qouted programs and use them to build another program which does the actual work. This is kind of like macros in Lisp, or preprocessor directives in C.

## truthy

See bool.

## tuck

Function

dup the item on the top of the stack under the second item on the stack.

       a b tuck
    --------------
        b a b

### Definition

> dup [swap] dip

### Crosslinks

over

## unary

(Combinator)

Run a quoted program using exactly one stack value and leave the first item of the result on the stack.

       ... x [P] unary
    ---------------------
           ... a

### Definition

> nullary popd

### Discussion

Runs any other quoted function and returns its first result while consuming exactly one item from the stack.

### Crosslinks

binary nullary ternary

## uncons

Basis Function

Removes an item from a list and leaves it on the stack under the rest of the list. You cannot `uncons` an item from an empty list.

       [a ...] uncons
    --------------------
          a [...]

### Discussion

This is the inverse of cons.

### Crosslinks

cons

## unique

Function

Given a list remove duplicate items.

## unit

Function

       a unit
    ------------
        [a]

### Definition

> [] cons

## unquoted

Combinator

Unquote (using i) the list that is second on the stack.

### Example

       1 2 [3 4] 5 unquoted
    --------------------------
             1 2 3 4 5

### Definition

> [i] dip

### Crosslinks

unit

## unswons

Function

       [a ...] unswons
    ---------------------
           [...] a

### Definition

> uncons swap

## void

Basis Function

True if the form on TOS is void otherwise False.

### Discussion

A form is any Joy expression composed solely of lists. This represents a binary Boolean logical formula in the arithmetic of the “Laws of Form”, see The Markable Mark

## warranty

Basis Function

Print warranty information.

## while

Combinator

A specialization of loop that accepts a quoted predicate program `P` and runs it nullary.

       [P] [F] while
    ------------------- P -> false

        [P] [F] while
    --------------------- P -> true
       F [P] [F] while

### Definition

> swap nulco dupdipd concat loop

### Crosslinks

loop

## words

Basis Function

Print all the words in alphabetical order.

### Discussion

Mathematically this is a form of id.

### Crosslinks

help

## x

Combinator

Take a quoted function `F` and run it with itself as the first item on the stack.

       [F] x
    -----------
       [F] F

### Definition

    dup i

### Discussion

The simplest recursive pattern.

See the Recursion Combinators notebook. as well as Recursion Theory and Joy by Manfred von

## xor

Basis Function

Logical bit-wise eXclusive OR.

### Crosslinks

and or

## zip

Function

Replace the two lists on the top of the stack with a list of the pairs from each list. The smallest list sets the length of the result list.

### Example

       [1 2 3] [4 5 6] zip
    -------------------------
       [[4 1] [5 2] [6 3]]