// #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[$@ X$][function pointer value] // `Fun`
      Or[$\& #h(3pt) value$][reference value] // `Ref`
      Or[\[$value+$\]][tuple value] // `Prod`
    }
  ),
)

$value$ - значения, которые могут лежать в переменных на семантическом уровне (то, во что вычисляется $expr$)

== Memory Model

// FIXME: check & add details
#let memvalue = `memvalue`
#let argmem = `argmem`
#bnf(
  Prod(
    `memvalue`,
    {
      Or[$0$][cell with some value (always)]
      Or[$X$][cell with possible value or $bot$]
      Or[$bot$][spoiled cell (always)]
    }
  ),
  Prod(
    `argmem`,
    {
      Or[$@m$][memory id for simple type variable] // `Unit`
      Or[\& #h(3pt) `argmem`][reference to structure, contains copy / ref choice] // `Ref`
      // Or[\& #h(3pt) `mode` #h(3pt) `argmem`][reference to structure, contains copy / ref choice] // `Ref`
      Or[$\[argmem+\]$][pair type, allows specify memory for tuples] // `Prod`
      // Or[`argmem` $times$ `argmem`][pair type, allows specify memory for tuples] // `Prod`
      // Or[`argmem` $+$ `argmem`][union type (important in some way ???)] // `Sum` // TODO ?

      Or[$lambda$][memory for lambda or function pointer, defined by function declaration id] // `Fun` // why separated ??
      // Or[$F_m$][memory for lambda or function pointer, defined by function declaration id] // `Fun` // why separated ??
    }
  ),
)

== Semantics

// FIXME: make connected to syntax
*TODO*

#h(10pt)

$V := memvalue$ - значения памяти

// FIXME: not required, remove
// $L := NN$ - позиции в памяти

$X$ - можество переменных

$LL$ - множество меток памяти
_пока решил использовать всё-таки $NN$ для того, чтобы работать с размером памяти
и добавлением ячеек, может стоит поменять_

$FF$  -множество меток функций

$sigma : X -> argmem times type$ - #[ позиции памяти, соответстующие переменным контекста,
                           частично определённая функция ]

$mu : NN -> V$ - память, частично определённая функция

$l in NN$ - длина используемого фрагмента памяти

$DD : FF -> decl$ - определения функций, частично определённая функция

$d in decl, s in stmt, f in FF, x in X, a in X$

$d space @ space overline(x)$ - запись применения функции (вида #decl) к аргументам

#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)

=== Path

// FIXME: types & description for functios
#let pathtype = `pathtype`
$ 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) $

#let pathmem = `pathmem`
$ pathmem(@m, @x) = m $
$ pathmem(\& #h(3pt) m, *p) = pathmem(m, p) $
$ pathmem([m_1, m_2, ..., m_n], p.i) = pathmem(m_i, p) $

// NOTE: is replaced with pathtype
// #let pathfun = `pathfun`
// $ pathfun(F_m, @x) = m $
// $ pathfun(\& #h(3pt) m, *p) = pathfun(m, p) $
// $ pathfun([m_1, m_2, ..., m_n], p.i) = pathfun(m_i, p) $

#let pathtag = `pathtag`
$ pathtag(\& #h(3pt) mode #h(3pt) t, @x) = mode $
$ pathtag(\& #h(3pt) mode #h(3pt) t, *p) = pathtag(t, p) $
$ pathtag([t_1, t_2, ..., t_n], p.i) = pathtag(t_i, p) $

#let pathvar = `pathvar`
$ pathvar(@x) = x $
$ pathvar(* p) = pathvar(p) $
$ pathvar(p.i) = pathvar(p) $

#h(10pt)

// $ pathtype({t_1, t_2, ..., t_n}, x -> i) = t_i$

#let typeof = `typeof`
$ typeof(sigma, p) = pathtype(sigma[pathvar(p)].2, p) $

// TODO: two versions: write with change & read ??
#let accessmem = `accessmem`
$ accessmem(sigma, p) = pathmem(sigma[pathvar(p)].1, p) $

#let argtag = `argtag`
$ argtag(sigma, p) = pathtag(sigma[pathvar(p)].2, p) $

#let access = `access`
$ access(sigma, mu, p) = mu[accessmem(sigma, p)] $

#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
    $isRead mode -> isIn mode$,
    $isPossibleWrite mode and (isOut mode or not isCopy mode) -> isAlwaysWrite argtag(sigma, x)$, // NOTE: may mode => should sigma(x)
    $isRead mode -> access(mu, sigma, x) != bot and access(mu, sigma, x) != X$,

    $isCorrect_(cl sigma, mu cr) (mode, x)$,
  )
))

#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 ??
    $cl sigma, mu, l cr xarrowSquiggly(p : ())_copy cl accessmem(sigma, p) <- l, mu [l <- 0], l + 1 cr$,
  )
))

#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)
    $gamma stretch(=>)^((mode, x) : args)_sigma access(gamma, sigma, x) <- bot]$
  )
))

#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)
    $gamma stretch(=>)^((mode, x) : args)_sigma access(gamma, sigma, x) <- 0]$
  )
))

#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 $y space overline(x)$],

    $typeof(sigma, y) = F_f$,

    $cl sigma, mu, l cr
     xarrow("CALL" f space overline(x))
     cl sigma, gamma, l cr$,

    $cl sigma, mu, l cr
     xarrow("CALL_LAM" y space overline(x))
     cl sigma, gamma, l cr$,
  )
))

#h(10pt)

#align(center, prooftree(
  vertical-spacing: 4pt,
  rule(
    name: [ READ $x$],

    $access(mu, sigma, x) != bot$,
    $access(mu, sigma, x) != X$,

    $cl sigma, mu, l cr
     xarrow("READ" x)
     cl sigma, mu, l cr$,
  )
))

#h(10pt)

#align(center, prooftree(
  vertical-spacing: 4pt,
  rule(
    name: [ WRITE $x$],

    $isPossibleWrite sigma(x)$,

    $cl sigma, mu, l cr
     xarrow("WRITE" x)
     cl sigma, access(mu, sigma, x) <- 0, l cr$,
  )
))

#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 $

]