### `++cap`

Tests whether the tree address `a` is in the head or the tail of a `noun`. Produces the constant `atom` `%2` if it is within the head (subtree `+2`), or the constant `atom` `%3` if it is within the tail (subtree `+3`).

#### Accepts

`a` is an `atom`.

#### Produces

A constant `atom`.

#### Source

``````        ++  cap
~/  %cap
|=  a=@
^-  ?(\$2 \$3)
?-  a
\$2        %2
\$3        %3
?(%0 %1)  !!
*         \$(a (div a 2))
==
``````

#### Examples

``````    > (cap 4)
%2

> (cap 6)
%3

%2

> (cap 1)                                    ::address '1' is in neither the head nor the tail
! exit

> (cap 0x40))
%2
> `@`0x40
64

> (cap 'a')
%3
> `@`'a'
97
``````

### `++mas`

Computes the tree address of `atom` `a` within either the head (`+2`) or tail (`+3`) of a `noun`.

#### Accepts

`a` is an `atom`.

#### Source

``````      ++  mas
~/  %mas
|=  a=@
^-  @
?-  a
1   !!
2   1
3   1
*   (add (mod a 2) (mul \$(a (div a 2)) 2))
==
``````

#### Examples

``````    > (mas 3)
1

> (mas 4)
2

> (mas 5)
3
> (cap 5)                                    ::`(cap a)` computes whether address `a` is in the head or the tail
%2

> (mas 7)
3
> (cap 7)
%3

> (mas 11)
7

> (mas (mas 11))
3

> (cap (mas 6))
%3

> (mas 0)
! exit                                       ::address `0` is in neither the head nor the tail

> (mas 1)
! exit                                       ::address `1` is in neither the head nor the tail
``````

#### Discussion

``````            1
/   \
/     \
2       3              <--here are the head (`+2`) and the tail (`+3`)
/ \      /\
4   5    6  7
/\   /\  /\  /\
(continues...)
``````

Running `(mas 7)` in the `Dojo` will return `3`, because address `+3` is what `+7` now occupies. The tree below helps illustrate the relationship. With parentheses are `a` values (if `a` is in subtree `+3`), and without parentheses are the values returned with `(mas a)`.

``````            1(3)                       ::new/(old) addresses
/    \
2       3
(6)     (7)
/ \       /\
/   \     /  \
4     5   6    7
(12) (13) (14) (15)
/ \    / \ / \   / \
(continues...)
``````

Notice how the old values in the head (subtree `+2`) were not illustrated in this case, because `+7` is within the tail (subtree `+3`).

### `++peg`

Computes the absolute address of `b`, a relative address within the subtree `a`.

#### Accepts

`a` is an `atom`.

`b` is an `atom`.

#### Source

``````    ++  peg
~/  %peg
|=  [a=@ b=@]
?<  =(0 a)
^-  @
?-  b
\$1  a
\$2  (mul a 2)
\$3  +((mul a 2))
*   (add (mod b 2) (mul \$(b (div b 2)) 2))
==
``````

#### Examples

``````    > (peg 4 1)
4

> (peg 1 4)
4

> (peg 4 2)
8

> (peg 4 8)
32

> (peg 4 (4 2))
32

> (peg 8 45)
269

> (cap (peg 4 2))                            ::`(cap a)` computes whether address `a` is in the head or the tail
%2

``````

#### Discussion

In other words, the subtree at address `a` is treated as a tree in its own right (starting with root `+1`, head `+2`, and tail `+3`). Relative address `b` is found with respect to `a`, and then its absolute address, within the greater tree, is returned.

Running `(peg 3 4)` in the `Dojo`, for example, will return `12`. Looking at a tree diagram makes it easy to see why.

``````                 1
/     \
/       \
/         \
2           3       <- here is the subtree `+3`. The subtree address is `a` in `(peg a b)`
/ \         / \
/   \       /   \
4     5     6     7
/ \   / \   / \   / \
8  9  10 11 12 13 14  15
/\  /\ /\ /\ /\ /\ /\  /\
(continues...)
``````

When we consider subtree at address `+3` by itself, it has relative addresses that are structured in the same way as its parent tree's absolute addresses. The absolute addresses are given in parentheses in the diagram below. Notice how relative address `+4` is at the same position as absolute address `+12`.

``````            1(3)                        ::new/(old) addresses
/    \
2       3
(6)     (7)
/ \       /\
/   \     /  \
4     5   6    7
(12) (13) (14)  (15)
/ \    / \ / \   / \
(continues...)
``````