mirror of
https://github.com/ProgramSnail/pass_strategy_synthesis.git
synced 2026-04-26 16:24:50 +00:00
623 lines
15 KiB
Typst
623 lines
15 KiB
Typst
// #import "@preview/polylux:0.4.0": *
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#import "@preview/simplebnf:0.1.1": *
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// #import "@preview/zebraw:0.5.0": *
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// #show: zebraw
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#import "@preview/curryst:0.6.0": rule, prooftree, rule-set
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#import "@preview/xarrow:0.4.0": xarrow, xarrowDashed, xarrowSquiggly
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= Формальная модель используемого языка
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#h(10pt)
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// TODO: check correctnes for path, mem & type ??
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== Syntax
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#h(10pt)
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#let isCorrect = `isCorrect`
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#let isRead = `isRead`
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#let isAlwaysWrite = `isAlwaysWrite`
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#let isPossibleWrite = `isPossibleWrite`
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#let isRef = `isRef`
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#let isCopy = `isCopy`
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#let isIn = `isIn`
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#let isOut = `isOut`
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#let mode = `mode`
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#let expr = `expr`
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#let stmt = `stmt`
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#let decl = `decl`
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#let prog = `prog`
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#let path = `path`
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#let type = `type`
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#bnf(
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Prod(`read`,
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// NOTE: not three modalities for write, because read does not change value
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// => it is not important to observe rsult, no differenc between always and maybe
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{ Or[Read][read passed value]
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Or[$not$ Read][] } ),
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Prod(`write`,
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{ Or[$square$ Write][in all cases there is a write to passed variable] // always write, requre at least one write in each flow variant
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Or[$diamond$ Write][in some cases there is a write to passed variable] // possible write, no requirements (?)
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Or[$not$ Write][] } ), // no write, require n owrites in all flow variants
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Prod(`copy`,
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{ Or[Ref][pass reference to the value]
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Or[Value][pass copy of the value] } ),
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Prod(`in`,
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{ Or[In][parameter value used as input]
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Or[$not$ In][] } ),
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Prod(`out`,
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{ Or[Out][parameter value returned]
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Or[$not$ Out][] } ),
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Prod(
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`mode`,
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{
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Or[`read` #h(3pt) `write` #h(3pt) `copy` #h(3pt) `in` #h(3pt) `out`][]
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}
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),
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Prod(
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`path`,
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{
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// NOTE: global vars & local vars names could be used with one constructor
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// Or[$\#x$][funciton or global variable itself]
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Or[$@ X$][function argument or variable itself]
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Or[$* path$][reference insede path]
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Or[$path . n$][access $n$-th cell of the tuple]
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// Or[$path : n$][access $n$-th cell of the union] // TODO: another notation ??
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}
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),
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Prod(
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`type`,
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{
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Or[$()$][simple type representing all primitive types] // `Unit`
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Or[\& #h(3pt) `mode` #h(3pt) `type`][reference to structure, contains copy / ref choice] // `Ref`
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Or[$\[type+\]$][tuple type] // `Prod`
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// Or[`type` $times$ `type`][pair type, allows to make tuples] // `Prod`
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// Or[`type` $+$ `type`][union type (important in some way ???)] // `Sum` // TODO ?
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// NOTE: do not use names in type
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// Or[$lambda_((x type)+)$][type of lambda or function pointer, defined by function declaration] // `Fun`
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Or[$lambda_(type+)$][type of lambda or function pointer, defined by function declaration] // `Fun`
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}
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),
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// FIXME: replace with expr
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Prod(
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`expr`,
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{
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Or[$()$][value of simple type] // `Unit`
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Or[$path$][value from variable] // `Path`
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Or[$\& #h(3pt) expr$][reference expr] // `Ref`
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Or[\[$expr+$\]][tuple expr] // `Prod`
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// NOTE: replaced with simple path value
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// Or[$lambda_path$][function value from variable] // `Fun`
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}
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),
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Prod(
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`stmt`,
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{
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Or[`CALL` $f space expr+$][call function]
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Or[`WRITE` $path$][write to variable]
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Or[`READ` $path$][read from variable]
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Or[$stmt ; stmt$][control flow operator, xecution ]
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Or[$stmt | stmt$][control flow operator, excution of one statements]
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}
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),
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Prod(
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`decl`,
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{
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// TODO: path not allowed ??
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Or[$"var" X : type = expr$][global variable declaration]
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Or[$"fun" X ((X : type)+) = stmt$][function declaration]
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}
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),
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Prod(
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`prog`,
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{
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Or[$decl stmt$][declarations and executet statement]
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}
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),
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)
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== Value Model
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// FIXME: check & add details
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#let value = `value`
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#bnf(
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Prod(
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`value`,
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{
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Or[$()$][value of simple type] // `Unit`
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Or[$@ X$][function pointer value] // `Fun`
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Or[$\& #h(3pt) value$][reference value] // `Ref`
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Or[\[$value+$\]][tuple value] // `Prod`
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}
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),
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)
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$value$ - значения, которые могут лежать в переменных на семантическом уровне (то, во что вычисляется $expr$)
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== Memory Model
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// FIXME: check & add details
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#let memvalue = `memvalue`
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#let argmem = `argmem`
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#bnf(
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Prod(
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`memvalue`,
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{
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Or[$0$][cell with some value (always)]
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Or[$X$][cell with possible value or $bot$]
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Or[$bot$][spoiled cell (always)]
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}
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),
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Prod(
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`argmem`,
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{
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Or[$@m$][memory id for simple type variable] // `Unit`
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Or[\& #h(3pt) `argmem`][reference to structure, contains copy / ref choice] // `Ref`
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// Or[\& #h(3pt) `mode` #h(3pt) `argmem`][reference to structure, contains copy / ref choice] // `Ref`
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Or[$\[argmem+\]$][pair type, allows specify memory for tuples] // `Prod`
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// Or[`argmem` $times$ `argmem`][pair type, allows specify memory for tuples] // `Prod`
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// Or[`argmem` $+$ `argmem`][union type (important in some way ???)] // `Sum` // TODO ?
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Or[$lambda$][memory for lambda or function pointer, defined by function declaration id] // `Fun` // why separated ??
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// Or[$F_m$][memory for lambda or function pointer, defined by function declaration id] // `Fun` // why separated ??
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}
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),
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)
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== Semantics
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// FIXME: make connected to syntax
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*TODO*
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#h(10pt)
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$V := memvalue$ - значения памяти
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// FIXME: not required, remove
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// $L := NN$ - позиции в памяти
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$X$ - можество переменных
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$LL$ - множество меток памяти
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_пока решил использовать всё-таки $NN$ для того, чтобы работать с размером памяти
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и добавлением ячеек, может стоит поменять_
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$FF$ -множество меток функций
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$sigma : X -> argmem times type$ - #[ позиции памяти, соответстующие переменным контекста,
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частично определённая функция ]
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$mu : NN -> V$ - память, частично определённая функция
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$l in NN$ - длина используемого фрагмента памяти
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$DD : FF -> decl$ - определения функций, частично определённая функция
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$d in decl, s in stmt, f in FF, x in X, a in X$
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$d space @ space overline(x)$ - запись применения функции (вида #decl) к аргументам
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#let args = `args`
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#[
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#let ref = `ref`
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#let copy = `copy`
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#let read = `read`
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#let cl = $chevron.l$
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#let cr = $chevron.r$
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// #align(center, grid(
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// columns: 3,
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// gutter: 5%,
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// align(bottom, prooftree(
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// ...
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// )),
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// align(bottom, prooftree(
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// ...
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// )),
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// align(bottom, prooftree(
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// ...
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// )),
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// ))
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// TODO: introduce spep env argument ??
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#h(10pt)
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=== Path
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// FIXME: types & description for functios
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#let pathtype = `pathtype`
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$ pathtype(t, @x) = t $
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$ pathtype(\& #h(3pt) mode #h(3pt) t, *p) = pathtype(t, p) $
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$ pathtype([t_1, t_2, ..., t_n], p.i) = pathtype(t_i, p) $
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#let pathmem = `pathmem`
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$ pathmem(@m, @x) = m $
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$ pathmem(\& #h(3pt) m, *p) = pathmem(m, p) $
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$ pathmem([m_1, m_2, ..., m_n], p.i) = pathmem(m_i, p) $
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// NOTE: is replaced with pathtype
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// #let pathfun = `pathfun`
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// $ pathfun(F_m, @x) = m $
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// $ pathfun(\& #h(3pt) m, *p) = pathfun(m, p) $
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// $ pathfun([m_1, m_2, ..., m_n], p.i) = pathfun(m_i, p) $
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#let pathtag = `pathtag`
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$ pathtag(\& #h(3pt) mode #h(3pt) t, @x) = mode $
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$ pathtag(\& #h(3pt) mode #h(3pt) t, *p) = pathtag(t, p) $
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$ pathtag([t_1, t_2, ..., t_n], p.i) = pathtag(t_i, p) $
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#let pathvar = `pathvar`
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$ pathvar(@x) = x $
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$ pathvar(* p) = pathvar(p) $
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$ pathvar(p.i) = pathvar(p) $
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#h(10pt)
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// $ pathtype({t_1, t_2, ..., t_n}, x -> i) = t_i$
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#let typeof = `typeof`
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$ typeof(sigma, p) = pathtype(sigma[pathvar(p)].2, p) $
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// TODO: two versions: write with change & read ??
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#let accessmem = `accessmem`
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$ accessmem(sigma, p) = pathmem(sigma[pathvar(p)].1, p) $
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#let argtag = `argtag`
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$ argtag(sigma, p) = pathtag(sigma[pathvar(p)].2, p) $
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#let access = `access`
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$ access(sigma, mu, p) = mu[accessmem(sigma, p)] $
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#h(10pt)
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=== Correctness
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// TODO: FIXME: well formatness for mode, extract
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// TODO: FIXME: check for mode, is recursion required ??
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// TODO: FIXME: check mode & access corectness in os correct
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// TODO: check all requirements
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#align(center, prooftree(
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vertical-spacing: 4pt,
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rule(
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name: [ is correct],
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$isOut mode -> isAlwaysWrite mode$, // NOTE; strong requirment should write
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$isRead mode -> isIn mode$,
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$isPossibleWrite mode and (isOut mode or not isCopy mode) -> isAlwaysWrite argtag(sigma, x)$, // NOTE: may mode => should sigma(x)
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$isRead mode -> access(mu, sigma, x) != bot and access(mu, sigma, x) != X$,
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$isCorrect_(cl sigma, mu cr) (mode, x)$,
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)
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))
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#h(10pt)
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=== Call Initialization
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Отсутствующий нижний индекс ($ref$, $copy$) означает произвольный индекс.
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Считается, что выбранный индекс одинаков в рамках одного правила.
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// NOTE: no empty type
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// #align(center, prooftree(
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// vertical-spacing: 4pt,
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// rule(
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// name: [ add paths init],
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// $cl sigma, mu, l cr stretch(~>)^nothing cl sigma, mu, l cr$,
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// )
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// ))
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// #h(10pt)
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#align(center, prooftree(
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vertical-spacing: 4pt,
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rule(
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name: [ add paths field by copy],
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// TODO: check that access is what required ??
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$cl sigma, mu, l cr xarrowSquiggly(p : ())_copy cl accessmem(sigma, p) <- l, mu [l <- 0], l + 1 cr$,
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)
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))
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#h(10pt)
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// NOTE: do nothing, ref init by default
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#align(center, prooftree(
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vertical-spacing: 4pt,
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rule(
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name: [ add paths field by reference],
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$cl sigma, mu, l cr xarrowSquiggly(p : ())_ref cl sigma, mu, l cr$,
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)
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))
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#h(10pt)
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#align(center, prooftree(
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vertical-spacing: 4pt,
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rule(
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name: [ add paths ref],
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$cl sigma, mu, l cr xarrowSquiggly(*p : t)_ref cl sigma', mu', l' cr$,
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$isRef mode$,
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$cl sigma, mu, l cr xarrowSquiggly(p : \& mode t) cl sigma', mu', l' cr$,
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)
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))
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#h(10pt)
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#align(center, prooftree(
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vertical-spacing: 4pt,
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rule(
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name: [ add paths ref],
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$cl sigma, mu, l cr xarrowSquiggly(*p : t)_copy cl sigma, mu, l cr$,
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$isCopy mode$,
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$cl sigma, mu, l cr xarrowSquiggly(p : \& mode t) cl sigma', mu', l' cr$,
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)
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))
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#h(10pt)
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#align(center, prooftree(
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vertical-spacing: 4pt,
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rule(
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name: [ add paths tuple],
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$cl sigma, mu, l cr xarrowSquiggly(p.1 : t_1) cl sigma_1, mu_1, l_1 cr$,
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$...$,
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$cl sigma_(n - 1), mu_(n - 1), l_(n - 1) cr xarrowSquiggly(p.n : t_n) cl sigma_n, mu_n, l_n cr$,
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$cl sigma, mu, l cr xarrowSquiggly(p : [t_1, ... t_n]) cl sigma_n, mu_n, l_n cr$,
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)
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))
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#h(10pt)
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#align(center, prooftree(
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vertical-spacing: 4pt,
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rule(
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name: [ add paths funciton pointer],
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$cl sigma, mu, l cr xarrowSquiggly(F_x) cl sigma, mu, l cr$,
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)
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))
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|
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#h(10pt)
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=== Call Finalization
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#align(center, prooftree(
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vertical-spacing: 4pt,
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rule(
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name: [ spoil init],
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$mu stretch(=>)^nothing_(cl sigma, mu cr) mu$,
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||
)
|
||
))
|
||
|
||
#h(10pt)
|
||
|
||
#align(center, prooftree(
|
||
vertical-spacing: 4pt,
|
||
rule(
|
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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)
|
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$gamma stretch(=>)^((mode, x) : args)_sigma access(gamma, sigma, x) <- bot]$
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||
)
|
||
))
|
||
|
||
#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 $
|
||
|
||
]
|