mirror of
https://github.com/ProgramSnail/pass_strategy_synthesis.git
synced 2026-04-26 16:24:50 +00:00
691 lines
18 KiB
Typst
691 lines
18 KiB
Typst
// #import "@preview/polylux:0.4.0": *
|
||
#import "@preview/simplebnf:0.1.1": *
|
||
// #import "@preview/zebraw:0.5.0": *
|
||
// #show: zebraw
|
||
#import "@preview/curryst:0.6.0": rule, prooftree, rule-set
|
||
#import "@preview/xarrow:0.4.0": xarrow, xarrowDashed, xarrowSquiggly
|
||
|
||
= Формальная модель используемого языка
|
||
|
||
#h(10pt)
|
||
|
||
// TODO: check correctnes for path, mem & type ??
|
||
|
||
== Syntax
|
||
|
||
#h(10pt)
|
||
|
||
#let isCorrect = `isCorrect`
|
||
|
||
#let isRead = `isRead`
|
||
#let isAlwaysWrite = `isAlwaysWrite`
|
||
#let isPossibleWrite = `isPossibleWrite`
|
||
#let isRef = `isRef`
|
||
#let isCopy = `isCopy`
|
||
#let isIn = `isIn`
|
||
#let isOut = `isOut`
|
||
|
||
#let mode = `mode`
|
||
#let expr = `expr`
|
||
#let stmt = `stmt`
|
||
#let decl = `decl`
|
||
#let prog = `prog`
|
||
#let path = `path`
|
||
#let type = `type`
|
||
#bnf(
|
||
Prod(`read`,
|
||
// NOTE: not three modalities for write, because read does not change value
|
||
// => it is not important to observe rsult, no differenc between always and maybe
|
||
{ Or[Read][read passed value]
|
||
Or[$not$ Read][] } ),
|
||
Prod(`write`,
|
||
{ Or[$square$ Write][in all cases there is a write to passed variable] // always write, requre at least one write in each flow variant
|
||
Or[$diamond$ Write][in some cases there is a write to passed variable] // possible write, no requirements (?)
|
||
Or[$not$ Write][] } ), // no write, require n owrites in all flow variants
|
||
Prod(`copy`,
|
||
{ Or[Ref][pass reference to the value]
|
||
Or[Value][pass copy of the value] } ),
|
||
Prod(`in`,
|
||
{ Or[In][parameter value used as input]
|
||
Or[$not$ In][] } ),
|
||
Prod(`out`,
|
||
{ Or[Out][parameter value returned]
|
||
Or[$not$ Out][] } ),
|
||
Prod(
|
||
`mode`,
|
||
{
|
||
Or[`read` #h(3pt) `write` #h(3pt) `copy` #h(3pt) `in` #h(3pt) `out`][]
|
||
}
|
||
),
|
||
Prod(
|
||
`path`,
|
||
{
|
||
// NOTE: global vars & local vars names could be used with one constructor
|
||
// Or[$\#x$][funciton or global variable itself]
|
||
Or[$@ X$][function argument or variable itself]
|
||
Or[$* path$][reference insede path]
|
||
Or[$path . n$][access $n$-th cell of the tuple]
|
||
// Or[$path : n$][access $n$-th cell of the union] // TODO: another notation ??
|
||
}
|
||
),
|
||
Prod(
|
||
`type`,
|
||
{
|
||
Or[$()$][simple type representing all primitive types] // `Unit`
|
||
Or[\& #h(3pt) `mode` #h(3pt) `type`][reference to structure, contains copy / ref choice] // `Ref`
|
||
Or[$[type+]$][tuple type] // `Prod`
|
||
// Or[`type` $times$ `type`][pair type, allows to make tuples] // `Prod`
|
||
// Or[`type` $+$ `type`][union type (important in some way ???)] // `Sum` // TODO ?
|
||
|
||
// NOTE: do not use names in type
|
||
// Or[$lambda_((x type)+)$][type of lambda or function pointer, defined by function declaration] // `Fun`
|
||
Or[$lambda_(type+)$][type of lambda or function pointer, defined by function declaration] // `Fun`
|
||
}
|
||
),
|
||
// FIXME: replace with expr
|
||
Prod(
|
||
`expr`,
|
||
{
|
||
Or[$()$][value of simple type] // `Unit`
|
||
Or[$path$][value from variable] // `Path`
|
||
Or[$\& #h(3pt) expr$][reference expr] // `Ref`
|
||
Or[$[expr+]$][tuple expr] // `Prod`
|
||
// NOTE: replaced with simple path value
|
||
// Or[$lambda_path$][function value from variable] // `Fun`
|
||
}
|
||
),
|
||
Prod(
|
||
`stmt`,
|
||
{
|
||
Or[`CALL` $f space expr+$][call function]
|
||
Or[`WRITE` $path$][write to variable]
|
||
Or[`READ` $path$][read from variable]
|
||
Or[$stmt ; stmt$][control flow operator, xecution ]
|
||
Or[$stmt | stmt$][control flow operator, excution of one statements]
|
||
}
|
||
),
|
||
Prod(
|
||
`decl`,
|
||
{
|
||
// TODO: path not allowed ??
|
||
Or[$"var" X : type = expr$][global variable declaration]
|
||
Or[$"fun" X ((X : type)+) = stmt$][function declaration]
|
||
}
|
||
),
|
||
Prod(
|
||
`prog`,
|
||
{
|
||
Or[$decl stmt$][declarations and executet statement]
|
||
}
|
||
),
|
||
)
|
||
|
||
== Value Model
|
||
|
||
// FIXME: check & add details
|
||
#let value = `value`
|
||
#bnf(
|
||
Prod(
|
||
`value`,
|
||
{
|
||
Or[$()$][value of simple type] // `Unit`
|
||
Or[$lambda_X$][function pointer value] // `Fun`
|
||
Or[$\& #h(3pt) value$][reference value] // `Ref`
|
||
Or[$[value+]$][tuple value] // `Prod`
|
||
}
|
||
),
|
||
)
|
||
|
||
$value$ - значения, которые могут лежать в переменных и передаваться как аргументы функций (то, во что вычисляется $expr$)
|
||
|
||
== Memory Model
|
||
|
||
// FIXME: check & add details
|
||
#let memelem = `memelem`
|
||
#let valuemem = `valuemem`
|
||
#bnf(
|
||
Prod(
|
||
`memelem`,
|
||
{
|
||
Or[$0$][cell with some value (always)]
|
||
Or[$\#$][cell with possible value or $bot$]
|
||
Or[$bot$][spoiled cell (always)]
|
||
}
|
||
),
|
||
Prod(
|
||
`valuemem`,
|
||
{
|
||
Or[$()$][one unit of memory, for simple vars] // `Unit`
|
||
Or[$lambda$][memory for lambda or function pointer, is not important in the memory model, 0 units] // `Fun`
|
||
Or[$\& #h(3pt) LL valuemem$][reference to structure memory start] // `Ref`
|
||
Or[$[valuemem+]$][memory specification for each tuple member] // `Prod`
|
||
}
|
||
),
|
||
)
|
||
|
||
#let mem = `mem`
|
||
Память преставляется в виде стека из значений `memelem`:
|
||
- $LL = NN$ - множество меток памяти
|
||
- $mu : mem = LL -> V$ - память, частично определённая функция
|
||
- $l in LL$ - длина используемого фрагмента памяти
|
||
|
||
Каждому значению $value$ в соответствие ставится $valuemem$ - расположние этого значения в пaмяти
|
||
|
||
Соответствие:
|
||
- $() -> ()$
|
||
- $lambda_X -> lambda$
|
||
- $\& #h(3pt) value -> \& #h(3pt) LL valuemem$
|
||
- $[value+] -> [valuemem+]$
|
||
|
||
// TODO: move allocation semantics there ??
|
||
|
||
== Semantics
|
||
|
||
// $V := memelem$ - значения памяти
|
||
|
||
$X$ - можество переменных
|
||
|
||
#let env = `env`
|
||
$sigma : env = X -> valuemem times type$ - #[ позиции памяти, соответстующие переменным контекста, частично определённая функция ]
|
||
|
||
$DD : X -> decl$ - глобальные определения, частично определённая функция
|
||
|
||
$d in decl, s in stmt, f in X, x in X, a in X$
|
||
|
||
$d space @ space overline(x)$ - запись применения функции (вида #decl) к аргументам
|
||
|
||
=== Path Accessors
|
||
|
||
Набор частично определённых фунций для работы с путями.
|
||
Для удобства значение, получаемое из текущего применением пути, будем называть полем.
|
||
// Все эти функции используются с префиксом `path.`.
|
||
|
||
// FIXME: types & description for functios
|
||
#let pathtype = `path.type`
|
||
#let pathmem = `path.mem`
|
||
#let pathmode = `path.mode`
|
||
#let pathvar = `path.var`
|
||
|
||
#let pathsize = `path.memsize`
|
||
#let pathbasepos = `path.membasepos`
|
||
#let pathoffset = `path.memoffset`
|
||
#let pathpos = `path.mempos`
|
||
|
||
#let pathenvtype = `path.envtype`
|
||
#let pathenvmem = `path.envmem`
|
||
#let pathenvmode = `path.envmode`
|
||
|
||
#let pathenvpos = `path.envpos`
|
||
#let pathenvval = `path.envval`
|
||
|
||
- #[ Получение типа поля
|
||
$ pathtype : type -> path -> type $
|
||
$ pathtype(t, @x) = t $
|
||
$ pathtype(\& #h(3pt) mode #h(3pt) t, *p) = pathtype(t, p) $
|
||
$ pathtype([t_1, t_2, ..., t_n], p.i) = pathtype(t_i, p) $
|
||
]
|
||
- #[ Получение начала памяти поля (предусловие: $pathtype(t, p) = \& #h(3pt) ...$)
|
||
$ pathmem : valuemem -> path -> LL $
|
||
$ pathmem(\& #h(3pt) y m, @x) = y $ // NOTE: only memory for refsi in midel ?? // TODO: decide
|
||
$ pathmem(\& #h(3pt) y m, *p) = pathmem(m, p) $
|
||
$ pathmem([m_1, m_2, ..., m_n], p.i) = pathmem(m_i, p) $
|
||
]
|
||
- #[ Получения тега поля (предусловие: $pathtype(t, p) = \& #h(3pt) ...$)
|
||
$ pathmode : type -> path -> mode $
|
||
$ pathmode(\& #h(3pt) mode #h(3pt) t, @x) = mode $
|
||
$ pathmode(\& #h(3pt) mode #h(3pt) t, *p) = pathmode(t, p) $
|
||
$ pathmode([t_1, t_2, ..., t_n], p.i) = pathmode(t_i, p) $
|
||
]
|
||
- #[ Получение \"корневой\" переменной пути
|
||
$ pathvar : path -> X $
|
||
$ pathvar(@x) = x $
|
||
$ pathvar(* p) = pathvar(p) $
|
||
$ pathvar(p.i) = pathvar(p) $
|
||
]
|
||
- #[ Получение размера поля в памяти
|
||
$ pathsize : valuemem -> path -> LL $
|
||
$ pathsize((), @x) = 1 $
|
||
$ pathsize(lambda, @x) = 0 $
|
||
$ pathsize(\& #h(3pt) y m, @x) = 0 $ // TODO: FIXME ?? always disconnected location ??
|
||
$ pathsize([m_1, m_2, ..., m_n], @x) = pathsize(m_1, @x) + ... + pathsize(m_n, @x) $
|
||
$ pathsize(\& #h(3pt) y m, *p) = pathsize(m, p) $
|
||
$ pathsize([m_1, m_2, ..., m_n], p.i) = pathsize(m_i, p) $
|
||
]
|
||
// TODO: FIXME ??
|
||
- #[ Получение позиции последного ссылочного предка поля в пaмяти
|
||
$ pathbasepos : value -> path -> X $
|
||
// $ pathbasepos(m, @x) = bot $ // NOTE: trivial if not present ??
|
||
$ pathbasepos(\& #h(3pt) y m, *p) = pathbasepos(m, p) | y $
|
||
$ pathbasepos([m_1, m_2, ..., m_n], p.i) = pathbasepos(m_i, p) $
|
||
]
|
||
- #[ Получение смещения поля в последнем монолитном отрезке памяти
|
||
$ pathoffset : value -> path -> X $
|
||
$ pathoffset(m, @x) = 0 $
|
||
$ pathoffset(\& #h(3pt) y m, *p) = pathoffset(m, p) $
|
||
$ pathoffset([m_1, m_2, ..., m_n], p.i) = (i - 1) + pathoffset(m_i, p) $
|
||
]
|
||
- #[ Получение позиции поля в пaмяти
|
||
$ pathpos : value -> path -> X $
|
||
$ pathpos(m, p) = pathbasepos(m, p) + pathoffset(m, p)$
|
||
]
|
||
- #[ Получение типа поля по окружению
|
||
$ pathenvtype : env -> path -> type $
|
||
$ pathenvtype(sigma, p) = pathtype(sigma[pathvar(p)].2, p) $
|
||
]
|
||
- #[ Получение начала памяти поля по окружению (предусловие: $pathtype(t, p) = \& #h(3pt) ...$)
|
||
$ pathenvmem : env -> path -> LL $
|
||
$ pathenvmem(sigma, p) = pathmem(sigma[pathvar(p)].1, p) $
|
||
]
|
||
- #[ Получения тега поля по окружению (предусловие: $pathtype(t, p) = \& #h(3pt) ...$)
|
||
$ pathenvmode : env -> path -> mode $
|
||
$ pathenvmode(sigma, p) = pathmode(sigma[pathvar(p)].2, p) $
|
||
]
|
||
- #[ Получение позиции поля в пaмяти по окружению
|
||
$ pathenvpos : env -> path -> X $
|
||
$ pathenvpos(m, p) = pathpos(sigma[pathvar(p)].1, p)$
|
||
]
|
||
- #[ Получение поля в пaмяти
|
||
$ pathenvval : env -> mem -> path -> memelem $
|
||
$ pathenvval(sigma, mu, p) = mu[pathenvpos(sigma, p)]$
|
||
]
|
||
|
||
=== Mode Correctness
|
||
|
||
Функции проверки тегов, имеют тип $mode -> #[bool]$:
|
||
|
||
#let modevar = $(r space w space c space i space o)$
|
||
|
||
$ isRead modevar = r == "Read" $
|
||
$ isAlwaysWrite modevar = w == square "Write" $
|
||
$ isPossibleWrite modevar = w == diamond "Write" || w == square "Write" $
|
||
$ isRef modevar = c == "Ref" $
|
||
$ isCopy modevar = c == "Copy" $
|
||
$ isIn modevar = i == "In" $
|
||
$ isOut modevar = o == "Out" $
|
||
|
||
Требования к тегам:
|
||
|
||
$ isOut mode -> isAlwaysWrite mode $
|
||
$ isRead mode -> isIn mode $
|
||
|
||
// TODO: rest conditions ??
|
||
|
||
=== Eval Rules
|
||
|
||
// FIXME: make connected to syntax
|
||
*TODO*
|
||
|
||
#h(10pt)
|
||
|
||
#let args = `args`
|
||
|
||
#[
|
||
|
||
#let ref = `ref`
|
||
#let copy = `copy`
|
||
#let read = `read`
|
||
|
||
#let cl = $chevron.l$
|
||
#let cr = $chevron.r$
|
||
|
||
// #align(center, grid(
|
||
// columns: 3,
|
||
// gutter: 5%,
|
||
// align(bottom, prooftree(
|
||
// ...
|
||
// )),
|
||
// align(bottom, prooftree(
|
||
// ...
|
||
// )),
|
||
// align(bottom, prooftree(
|
||
// ...
|
||
// )),
|
||
// ))
|
||
|
||
// TODO: introduce spep env argument ??
|
||
|
||
#h(10pt)
|
||
|
||
=== Correctness
|
||
|
||
// TODO: FIXME: well formatness for mode, extract
|
||
// TODO: FIXME: check for mode, is recursion required ??
|
||
// TODO: FIXME: check mode & access corectness in os correct
|
||
|
||
// TODO: check all requirements
|
||
#align(center, prooftree(
|
||
vertical-spacing: 4pt,
|
||
rule(
|
||
name: [ is correct],
|
||
$isOut mode -> isAlwaysWrite mode$, // NOTE; strong requirment should write
|
||
$isRead mode -> isIn mode$,
|
||
$isPossibleWrite mode and (isOut mode or not isCopy mode) -> isAlwaysWrite pathenvmode(sigma, x)$, // NOTE: may mode => should sigma(x)
|
||
$isRead mode -> pathenvval(mu, sigma, x) != bot and pathenvval(mu, sigma, x) != X$,
|
||
|
||
$isCorrect_(cl sigma, mu cr) (mode, x)$,
|
||
)
|
||
))
|
||
|
||
#h(10pt)
|
||
|
||
=== Call Initialization
|
||
|
||
Отсутствующий нижний индекс ($ref$, $copy$) означает произвольный индекс.
|
||
Считается, что выбранный индекс одинаков в рамках одного правила.
|
||
|
||
// NOTE: no empty type
|
||
// #align(center, prooftree(
|
||
// vertical-spacing: 4pt,
|
||
// rule(
|
||
// name: [ add paths init],
|
||
|
||
// $cl sigma, mu, l cr stretch(~>)^nothing cl sigma, mu, l cr$,
|
||
// )
|
||
// ))
|
||
|
||
// #h(10pt)
|
||
|
||
#align(center, prooftree(
|
||
vertical-spacing: 4pt,
|
||
rule(
|
||
name: [ add paths field by copy],
|
||
|
||
// TODO: check that access is what required ??
|
||
$cl sigma, mu, l cr xarrowSquiggly(p : ())_copy cl pathenvmem(sigma, p) <- l, mu [l <- 0], l + 1 cr$,
|
||
)
|
||
))
|
||
|
||
#h(10pt)
|
||
|
||
// NOTE: do nothing, ref init by default
|
||
#align(center, prooftree(
|
||
vertical-spacing: 4pt,
|
||
rule(
|
||
name: [ add paths field by reference],
|
||
|
||
$cl sigma, mu, l cr xarrowSquiggly(p : ())_ref cl sigma, mu, l cr$,
|
||
)
|
||
))
|
||
|
||
#h(10pt)
|
||
|
||
#align(center, prooftree(
|
||
vertical-spacing: 4pt,
|
||
rule(
|
||
name: [ add paths ref],
|
||
$cl sigma, mu, l cr xarrowSquiggly(*p : t)_ref cl sigma', mu', l' cr$,
|
||
$isRef mode$,
|
||
|
||
$cl sigma, mu, l cr xarrowSquiggly(p : \& mode t) cl sigma', mu', l' cr$,
|
||
)
|
||
))
|
||
|
||
#h(10pt)
|
||
|
||
#align(center, prooftree(
|
||
vertical-spacing: 4pt,
|
||
rule(
|
||
name: [ add paths ref],
|
||
$cl sigma, mu, l cr xarrowSquiggly(*p : t)_copy cl sigma, mu, l cr$,
|
||
$isCopy mode$,
|
||
|
||
$cl sigma, mu, l cr xarrowSquiggly(p : \& mode t) cl sigma', mu', l' cr$,
|
||
)
|
||
))
|
||
|
||
#h(10pt)
|
||
|
||
#align(center, prooftree(
|
||
vertical-spacing: 4pt,
|
||
rule(
|
||
name: [ add paths tuple],
|
||
$cl sigma, mu, l cr xarrowSquiggly(p.1 : t_1) cl sigma_1, mu_1, l_1 cr$,
|
||
$...$,
|
||
$cl sigma_(n - 1), mu_(n - 1), l_(n - 1) cr xarrowSquiggly(p.n : t_n) cl sigma_n, mu_n, l_n cr$,
|
||
|
||
$cl sigma, mu, l cr xarrowSquiggly(p : [t_1, ... t_n]) cl sigma_n, mu_n, l_n cr$,
|
||
)
|
||
))
|
||
|
||
#h(10pt)
|
||
|
||
#align(center, prooftree(
|
||
vertical-spacing: 4pt,
|
||
rule(
|
||
name: [ add paths funciton pointer],
|
||
|
||
$cl sigma, mu, l cr xarrowSquiggly(F_x) cl sigma, mu, l cr$,
|
||
)
|
||
))
|
||
|
||
#h(10pt)
|
||
|
||
=== Call Finalization
|
||
|
||
#align(center, prooftree(
|
||
vertical-spacing: 4pt,
|
||
rule(
|
||
name: [ spoil init],
|
||
$mu stretch(=>)^nothing_(cl sigma, mu cr) mu$,
|
||
)
|
||
))
|
||
|
||
#h(10pt)
|
||
|
||
#align(center, prooftree(
|
||
vertical-spacing: 4pt,
|
||
rule(
|
||
name: [ spoil step],
|
||
|
||
$mu stretch(=>)^args_sigma gamma$,
|
||
|
||
$isPossibleWrite mode$, // NOTE: weak requirement: may write
|
||
$not isCopy mode$,
|
||
$not isOut mode$,
|
||
|
||
$isCorrect_(cl sigma, mu cr) (mode, x)$,
|
||
|
||
// gamma - memory (as mu)
|
||
$gamma stretch(=>)^((mode, x) : args)_sigma pathenvval(gamma, sigma, x) <- bot]$
|
||
)
|
||
))
|
||
|
||
#h(10pt)
|
||
|
||
#align(center, prooftree(
|
||
vertical-spacing: 4pt,
|
||
rule(
|
||
name: [ fix step],
|
||
|
||
$mu stretch(=>)^args_sigma gamma$,
|
||
|
||
$isAlwaysWrite mode$, // NOTE: strong requirement: should write
|
||
$isOut mode$,
|
||
|
||
$isCorrect_(cl sigma, mu cr) (mode, x)$,
|
||
|
||
// gamma - memory (as mu)
|
||
$gamma stretch(=>)^((mode, x) : args)_sigma pathenvval(gamma, sigma, x) <- 0]$
|
||
)
|
||
))
|
||
|
||
#h(10pt)
|
||
|
||
#align(center, prooftree(
|
||
vertical-spacing: 4pt,
|
||
rule(
|
||
name: [ skip step],
|
||
|
||
$mu stretch(=>)^args_sigma gamma$,
|
||
|
||
$not "spoil step"$,
|
||
$not "fix step"$,
|
||
|
||
$isCorrect_(cl sigma, mu cr) (mode, x)$,
|
||
|
||
// mu
|
||
$gamma stretch(=>)^((mode, x) : args)_sigma gamma$
|
||
)
|
||
))
|
||
|
||
|
||
#h(10pt)
|
||
|
||
=== Function Evaluation
|
||
|
||
#align(center, prooftree(
|
||
vertical-spacing: 4pt,
|
||
rule(
|
||
name: [ $(lambda a : t. d) m$],
|
||
|
||
// TODO: verify that type of m is t ??
|
||
|
||
$cl sigma [a <- (m, t)], mu, l cr
|
||
xarrowSquiggly(t)
|
||
cl sigma', mu', l' cr$,
|
||
|
||
$cl sigma', mu', l' cr
|
||
xarrowDashed(d space @ space overline(y))
|
||
cl sigma'', mu'', l'' cr$,
|
||
|
||
$isRead mode$,
|
||
$not isCopy mode$,
|
||
|
||
// NOTE: correctness checked in CALL f
|
||
|
||
$cl sigma, mu, l cr
|
||
xarrowDashed((lambda a. d) space @ space x space overline(y))
|
||
cl sigma'', mu'', l'' cr$,
|
||
)
|
||
))
|
||
|
||
#h(10pt)
|
||
|
||
#align(center, prooftree(
|
||
vertical-spacing: 4pt,
|
||
rule(
|
||
name: [decl body],
|
||
|
||
$cl sigma, mu, l cr
|
||
attach(stretch(->)^overline(s), tr: *)
|
||
cl sigma', mu', l' cr$,
|
||
|
||
$d = overline(s)$,
|
||
|
||
$cl sigma, mu, l cr
|
||
xarrowDashed(d space @)
|
||
cl sigma', mu', l' cr$,
|
||
)
|
||
))
|
||
|
||
#h(10pt)
|
||
|
||
=== Statement Evaluation
|
||
|
||
#align(center, prooftree(
|
||
vertical-spacing: 4pt,
|
||
rule(
|
||
name: [ CALL $f space overline(x)$],
|
||
|
||
$cl [], mu, l cr
|
||
xarrowDashed(d space @ space overline(x))
|
||
cl sigma', mu', l' cr$,
|
||
|
||
// TODO: FIXME define args in some way
|
||
$mu attach(stretch(=>)^args_sigma, tr: *) gamma$,
|
||
|
||
$DD(f) := d$,
|
||
|
||
$cl sigma, mu, l cr
|
||
xarrow("CALL" f space overline(x))
|
||
cl sigma, gamma, l cr$,
|
||
)
|
||
))
|
||
|
||
#h(10pt)
|
||
|
||
#align(center, prooftree(
|
||
vertical-spacing: 4pt,
|
||
rule(
|
||
name: [ CALL_LAM $y space overline(x)$],
|
||
|
||
$pathenvtype(sigma, y) = F_f$,
|
||
|
||
$cl sigma, mu, l cr
|
||
xarrow("CALL" f space overline(x))
|
||
cl sigma, gamma, l cr$,
|
||
|
||
$cl sigma, mu, l cr
|
||
xarrow("CALL_LAM" y space overline(x))
|
||
cl sigma, gamma, l cr$,
|
||
)
|
||
))
|
||
|
||
#h(10pt)
|
||
|
||
#align(center, prooftree(
|
||
vertical-spacing: 4pt,
|
||
rule(
|
||
name: [ READ $x$],
|
||
|
||
$pathenvval(mu, sigma, x) != bot$,
|
||
$pathenvval(mu, sigma, x) != X$,
|
||
|
||
$cl sigma, mu, l cr
|
||
xarrow("READ" x)
|
||
cl sigma, mu, l cr$,
|
||
)
|
||
))
|
||
|
||
#h(10pt)
|
||
|
||
#align(center, prooftree(
|
||
vertical-spacing: 4pt,
|
||
rule(
|
||
name: [ WRITE $x$],
|
||
|
||
$isPossibleWrite sigma(x)$,
|
||
|
||
$cl sigma, mu, l cr
|
||
xarrow("WRITE" x)
|
||
cl sigma, pathenvval(mu, sigma, x) <- 0, l cr$,
|
||
)
|
||
))
|
||
|
||
#h(10pt)
|
||
|
||
#let combine = `combine`
|
||
|
||
#align(center, prooftree(
|
||
vertical-spacing: 4pt,
|
||
rule(
|
||
name: [ CHOICE $overline(s)$ $overline(t)$],
|
||
|
||
$cl sigma, mu, l cr
|
||
attach(stretch(->)^overline(s), tr: *)
|
||
cl sigma_s, mu_s, l_s cr$,
|
||
|
||
$cl sigma, mu, l cr
|
||
attach(stretch(->)^overline(t), tr: *)
|
||
cl sigma_t, mu_t, l_t cr$,
|
||
|
||
$l_t = l_s$,
|
||
$sigma_s = sigma_t$,
|
||
|
||
// TODO changes ?? two ways ??
|
||
$cl sigma, mu, l cr
|
||
xarrow("CHOICE" overline(s) space overline(t))
|
||
cl sigma, combine(mu_s, mu_t), l cr$,
|
||
)
|
||
))
|
||
|
||
#h(10pt)
|
||
|
||
=== Combination
|
||
|
||
$ combine(mu_1, mu_2)[i] = combine_e (mu_1[i], mu_2[i]) $
|
||
$ combine_e (bot, bot) = bot $
|
||
$ combine_e (0, 0) = 0 $
|
||
$ combine_e (\_, \_) = X $
|
||
|
||
]
|