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Prolog でわかる TAPL 14日目 - 第Ⅵ部 高階の型システム (3)

サートゥルヌス
サートゥルヌス

サートゥルヌス (Sāturnus) は、ローマ神話に登場する農耕神。

英語ではサターン。ギリシア神話のクロノスと同一視され、土星の守護神ともされる。妻はオプス、あるいはレアーとされる。

彼の祝祭はサートゥルナーリア(Sāturnālia)と呼ばれ、毎年12月17日から7日間執り行われた。その間は、奴隷にも特別の自由が許され、楽しく陽気に祝われた。

サートゥルナーリアでは、人々はろうそくや小さな人形を贈物として交換した。この風習は、のちにキリスト教におけるクリスマスに受け継がれたという。

サートゥルヌスは四頭立て馬車(クアドリガ)に乗る姿でも表され、同様に表現されるヘーリオスと同一視される存在である。

最後に紹介するのはimmutableなオブジェクトを使った計算を行う純粋関数的オブジェクトを持った言語です。レコードの一部を書き換えたものを生成することでオブジェクトを書き換えれば参照を使わずとも計算できるという言語であり型システムです。

TAPL 32 多相更新

図32-1.多相更新

→ ∀ <: Top {} ← F<:(図26-1)とレコード(図11-7)に基づく

新しい構文形式

t ::= …             項:
  {}                レコード
  t<-l = t          フィールド更新
T ::= …             型:
 {ιi li:Ti i∈1..n} レコードの型
ι ::=
  #                 非変な(更新可能)フィールド
  omitted           共変な(更新不能)フィールド

新しい評価規則 t ==> t’

t1 ==> t1’
---------------------------------------- (E-Update1)
t1<-li=t2 ==> t1’<-li=t2

t2 ==> t2’
---------------------------------------- (E-Update2)
v1<-li=t2 ==> v1<-li=t2’

{ιj lj=vj j∈1..n}<-li=v
==> {ιj lj=vj j∈1..i-1, ιi li=v, ιk lk=vk k∈i+1..n}
                                        (E-UpdateV)

{ιi li=vi i∈1..n}.lj ==> vj             (E-ProjRcd)

tj ==> tj’
-------------------------------------------- (E-Rcd)
{ιi li=vi i∈1..j-1, ιj lj=tj, ιk lk=tk k∈j+1..n}
==> {ιi li=vi i∈1..j-1, ιj lj=tj’, ιk lk=tk k∈j+1..n}


新しい部分型付け S <: T

Γ ⊢ {ιi li:Ti i∈1..n+k}<:{ιi li:Ti i∈1..n}
                                          (S-RcdWidth)

各iに対して Γ ⊢ Si <: Ti
ιi = # ならば Γ ⊢ Ti <: Si
------------------------------------------ (S-RcdDepth)
Γ ⊢ {ιi li:Ti i∈1..n}<:{ιi li:Ti i∈1..n}

Γ ⊢ {…#li:Si…}<:{…li:Si…}               (S-RcdVariance)

新しい型付け規則 Γ ⊢ t : T

各iに対して Γ ⊢ ti : Ti
------------------------------------------ (T-Rcd)
Γ ⊢ {ιi li=ti i∈1..n}:{ιi li:Ti i∈1..n}

Γ ⊢ t1 : {ιi li:Ti i∈1..n}
------------------------------------------ (T-Proj)
Γ ⊢ t1.lj : Tj

Γ ⊢ r : R   Γ ⊢ R <: {#lj:Tj}
Γ ⊢ t : Tj
------------------------------------------ (T-Update)
Γ ⊢ r<-lj = t : R

実装

  • fullupdate フル純粋関数的オブジェクト 書き換え可能レコード bool+nat+unit+float+string+λ+let+letrec+fix+inert+as+top+<:+all+some+kind+variance(32章)

fullupdate.pl

プログラムの全体を見る
fullupdate.pl
:- discontiguous((\-)/2).
:- discontiguous((/-)/2).
:- op(1200, xfx, ['--', where]).
:- op(1100, xfy, [in]).
:- op(1050, xfy, ['=>']).
:- op(920, xfx, ['==>', '==>>', '<:']).
:- op(910, xfx, ['/-', '\\-']).
:- op(600, xfy, ['::', as]).
:- op(500, yfx, ['$', !, tsubst, tsubst2, subst, subst2, tmsubst, tmsubst2, '<-']).
:- op(400, yfx, ['#']).
term_expansion((A where B), (A :- B)).
:- op(920, xfx, ['<:']).
:- op(600, xfy, ['::']).
:- style_check(- singleton). 

% ------------------------   SYNTAX  ------------------------

:- use_module(rtg).

w ::= bool | nat | unit | float | string | top | true | false.  % キーワード:
syntax(x). x(X) :- \+ w(X), atom(X), sub_atom(X, 0, 1, _, P), (char_type(P, lower) ; P = '_'). % 識別子:
syntax(tx). tx(TX) :- atom(TX), sub_atom(TX, 0, 1, _, P), char_type(P, upper).  % 型変数:
syntax(floatl). floatl(F) :- float(F).     % 浮動小数点数
syntax(stringl). stringl(F) :- string(F).  % 文字列
syntax(l). l(L) :- atom(L) ; integer(L).   % ラベル
list(A) ::= [] | [A | list(A)].            % リスト

k ::=                      % カインド:
      '*'                  % 真の型のカインド
    | (k => k)             % 演算子のカインド
    .
i ::= #                    % 非変な(更新可能)フィールド
    | !                    % 共変な(更新不能)フィールド
.
t ::=                      % 型:
      bool                 % ブール値型
    | nat                  % 自然数型
    | unit                 % Unit型
    | float                % 浮動小数点数型
    | string               % 文字列型
    | top                  % 最大の型
    | tx                   % 型変数
    | (t -> t)             % 関数の型
    | {list(l : (i, t))}   % レコードの型
    | (all(tx :: t) => t)  % 全称型
    | (some(tx :: t) => t) % 存在型
    | abs(tx, k, t)        % 型抽象
    | t $ t                % 関数適用
    .
m ::=                      % 項:
      true                 % 真
    | false                % 偽
    | if(m, m, m)          % 条件式
    | 0                    % ゼロ
    | succ(m)              % 後者値
    | pred(m)              % 前者値
    | iszero(m)            % ゼロ判定
    | unit                 % 定数unit
    | floatl               % 浮動小数点数値
    | m * m                % 浮動小数点乗算
    | stringl              % 文字列定数
    | x                    % 変数
    | (fn(x : t) -> m)     % ラムダ抽象
    | m $ m                % 関数適用
    | (let(x) = m in m)    % let束縛
    | fix(m)               % mの不動点
    | inert(t)
    | m as t               % 型指定
    | {list(l = (i, m))}   % レコード
    | m # l <- m           % フィールド更新
    | m # l                % 射影
    | pack(t, m, t)        % パッケージ化
    | unpack(tx, x, m, m)  % アンパッケージ化
    | (fn(tx :: t) => m)   % 型抽象
    | m![t]                % 型適用
    .
n ::=                      % 数値:
      0                    % ゼロ
    | succ(n)              % 後者値
    .
v ::=                      % 値:
      true                 % 真
    | false                % 偽
    | n                    % 数値
    | unit                 % 定数unit
    | floatl               % 浮動小数点数値
    | stringl              % 文字列定数
    | (fn(x : t) -> m)     % ラムダ抽象
    | {list(l = (i, v))}   % レコード
    | pack(t, v, t)        % パッケージ化
    | (fn(tx :: t) => m)   % 型抽象
    . 

/*
k ::= * | (k => k).
i ::= # | ! .
t ::= bool | nat | unit | float | string | top | tx | (t -> t) | {list(l : (i, t))} | (all(tx :: t) => t) | (some(tx :: t) => t) | abs(tx, k, t) | t $ t.
m ::= true | false | if(m, m, m) | 0 | succ(m) | pred(m) | iszero(m) | unit | floatl | m * m | stringl | x | (fn(x : t) -> m) | m $ m | (let(x) = m in m) | fix(m) | inert(t) | m as t | {list(l = (i, m))} | m # l <- m | m # l | pack(t, m, t) | unpack(tx, x, m, m) | (fn(tx :: t) => m) | m![t] .
n ::= 0 | succ(n) .
v ::= true | false | n | unit | floatl | stringl | (fn(x : t) -> m) | {list(l = (i, v))} | pack(t, v, t) | (fn(tx :: t) => m). 
*/
% ------------------------   SUBSTITUTION  ------------------------

bool![(J -> S)] tsubst bool.
nat![(J -> S)] tsubst nat.
unit![(J -> S)] tsubst unit.
float![(J -> S)] tsubst float.
string![(J -> S)] tsubst string.
top![(J -> S)] tsubst top.
J![(J -> S)] tsubst S                            :- tx(J).
X![(J -> S)] tsubst X                            :- tx(X).
(T1 -> T2)![(J -> S)] tsubst (T1_ -> T2_)        :- T1![(J -> S)] tsubst T1_, T2![(J -> S)] tsubst T2_.
{Mf}![(J -> S)] tsubst {Mf_}                     :- maplist([L : (Vari, T), L : (Vari, T_)] >> (T![(J -> S)] tsubst T_), Mf, Mf_).
(all(TX :: T1) => T2)![(J -> S)] tsubst (all(TX :: T1_) => T2_)
                                                 :- T1![TX, (J -> S)] tsubst2 T1_, T2![TX, (J -> S)] tsubst2 T2_.
(some(TX :: T1) => T2)![(J -> S)] tsubst (some(TX :: T1_) => T2_)
                                                 :- T1![TX, (J -> S)] tsubst2 T1_, T2![TX, (J -> S)] tsubst2 T2_.
abs(TX, K, T2)![(J -> S)] tsubst abs(TX, K, T2_) :- T2![TX, (J -> S)] tsubst2 T2_.
T1 $ T2![(J -> S)] tsubst (T1_ $ T2_)            :- T1![(J -> S)] tsubst T1_, T2![(J -> S)] tsubst T2_.
T![X, (X -> S)] tsubst2 T.
T![X, (J -> S)] tsubst2 T_                       :- T![(J -> S)] tsubst T_.

true![(J -> M)] subst true.
false![(J -> M)] subst false.
if(M1, M2, M3)![(J -> M)] subst if(M1_, M2_, M3_)          :- M1![(J -> M)] subst M1_, M2![(J -> M)] subst M2_, M3![(J -> M)] subst M3_.
0![(J -> M)] subst 0.
succ(M1)![(J -> M)] subst succ(M1_)                        :- M1![(J -> M)] subst M1_.
pred(M1)![(J -> M)] subst pred(M1_)                        :- M1![(J -> M)] subst M1_.
iszero(M1)![(J -> M)] subst iszero(M1_)                    :- M1![(J -> M)] subst M1_.
unit![(J -> M)] subst unit.
F1![(J -> M)] subst F1                                     :- float(F1).
M1 * M2![(J -> M)] subst M1_ * M2_                         :- M1![(J -> M)] subst M1_, M2![(J -> M)] subst M2_.
X![(J -> M)] subst X                                       :- string(X).
J![(J -> M)] subst M                                       :- x(J).
X![(J -> M)] subst X                                       :- x(X).
(fn(X : T1) -> M2)![(J -> M)] subst (fn(X : T1) -> M2_)    :- M2![X, (J -> M)] subst2 M2_.
M1 $ M2![(J -> M)] subst (M1_ $ M2_)                       :- M1![(J -> M)] subst M1_, M2![(J -> M)] subst M2_.
(let(X) = M1 in M2)![(J -> M)] subst (let(X) = M1_ in M2_) :- M1![(J -> M)] subst M1_, M2![X, (J -> M)] subst2 M2_.
fix(M1)![(J -> M)] subst fix(M1_)                          :- M1![(J -> M)] subst M1_.
inert(T1)![(J -> M)] subst inert(T1).
(M1 as T1)![(J -> M)] subst (M1_ as T1)                    :- M1![(J -> M)] subst M1_.
{Mf}![(J -> M)] subst {Mf_}                                :- maplist([L = (Vari, Mi), L = (Vari, Mi_)] >> (Mi![(J -> M)] subst Mi_), Mf, Mf_).
M1 # L![(J -> M)] subst M1_ # L                            :- M1![(J -> M)] subst M1_.
(fn(TX :: T) => M2)![(J -> M)] subst (fn(TX :: T) => M2_)  :- M2![(J -> M)] subst M2_.
M1![T2]![(J -> M)] subst (M1_![T2])                        :- M1![(J -> M)] subst M1_.
pack(T1, M2, T3)![(J -> M)] subst pack(T1, M2_, T3)        :- M2![(J -> M)] subst M2_.
unpack(TX, X, M1, M2)![(J -> M)] subst unpack(TX, X, M1_, M2_)
                                                           :- M1![X, (J -> M)] subst2 M1_, M2![X, (J -> M)] subst2 M2_.
M1 # L <- M2![(J -> M)] subst (M1_ # L <- M2_)             :- M1![(J -> M)] subst M1_, M2![(J -> M)] subst M2_.
S![(J -> M)] subst _                                       :- writeln(error : subst(J, M, S)), fail.
S![J, (J -> M)] subst2 S.
S![X, (J -> M)] subst2 M_                                  :- S![(J -> M)] subst M_.

true![(J -> S)] tmsubst true.
false![(J -> S)] tmsubst false.
if(M1, M2, M3)![(J -> S)] tmsubst if(M1_, M2_, M3_)          :- M1![(J -> S)] tmsubst M1_, M2![(J -> S)] tmsubst M2_, M3![(J -> S)] tmsubst M3_.
0![(J -> S)] tmsubst 0.
succ(M1)![(J -> S)] tmsubst succ(M1_)                        :- M1![(J -> S)] tmsubst M1_.
pred(M1)![(J -> S)] tmsubst pred(M1_)                        :- M1![(J -> S)] tmsubst M1_.
iszero(M1)![(J -> S)] tmsubst iszero(M1_)                    :- M1![(J -> S)] tmsubst M1_.
unit![(J -> M)] tmsubst unit.
F1![(J -> M)] tmsubst F1                                     :- float(F1).
M1 * M2![(J -> M)] tmsubst M1_ * M2_                         :- M1![(J -> M)] tmsubst M1_, M2![(J -> M)] tmsubst M2_.
X![(J -> M)] tmsubst X                                       :- string(X).
X![(J -> S)] tmsubst X                                       :- x(X).
(fn(X : T1) -> M2)![(J -> S)] tmsubst (fn(X : T1_) -> M2_)   :- T1![(J -> S)] tsubst T1_, M2![(J -> S)] tmsubst M2_.
M1 $ M2![(J -> S)] tmsubst (M1_ $ M2_)                       :- M1![(J -> S)] tmsubst M1_, M2![(J -> S)] tmsubst M2_.
(let(X) = M1 in M2)![(J -> S)] tmsubst (let(X) = M1_ in M2_) :- M1![(J -> S)] tmsubst M1_, M2![(J -> S)] tmsubst M2_.
fix(M1)![(J -> M)] tmsubst fix(M1_)                          :- M1![(J -> M)] tmsubst M1_.
inert(T1)![(J -> M)] tmsubst inert(T1).
(M1 as T1)![(J -> S)] tmsubst (M1_ as T1_)                   :- M1![(J -> S)] tmsubst M1_, T1![(J -> S)] tsubst T1_.
{Mf}![(J -> M)] tmsubst {Mf_}                                :- maplist([L = (Vari, Mi), L = (Vari, Mi_)] >> (Mi![(J -> M)] tmsubst Mi_), Mf, Mf_).
M1 # L![(J -> M)] tmsubst M1_ # L                            :- M1![(J -> M)] tmsubst M1_.
(fn(TX :: T1) => M2)![(J -> S)] tmsubst (fn(TX :: T1_) => M2_)
                                                             :- T1![TX, (J -> S)] tsubst2 T1_, M2![TX, (J -> S)] tmsubst2 M2_.
M1![T2]![(J -> S)] tmsubst (M1_![T2_])                       :- M1![(J -> S)] tmsubst M1_, T2![(J -> S)] tsubst T2_.
pack(T1, M2, T3)![(J -> S)] tmsubst pack(T1_, M2_, T3_)      :- T1![(J -> S)] tsubst T1_, M2![(J -> S)] tmsubst M2_, T3![(J -> S)] tsubst T3_.
unpack(TX, X, M1, M2)![(J -> S)] tmsubst unpack(TX, X, M1_, M2_)
                                                             :- M1![(J -> S)] tmsubst M1_, M2![(J -> S)] tmsubst M2_.
M1 # L <- M2![(J -> S)] tmsubst (M1_ # L <- M2_)             :- M1![(J -> S)] tmsubst M1_, M2![(J -> S)] tmsubst M2_.
T![X, (X -> S)] tmsubst2 T.
T![X, (J -> S)] tmsubst2 T_                                  :- T![(J -> S)] tmsubst T_.

getb(Γ, X, B) :- member(X - B, Γ).
gett(Γ, X, T) :- getb(Γ, X, bVar(T)).
gett(Γ, X, T) :- getb(Γ, X, bMAbb(_, T)). 
%gett(Γ,X,_) :- writeln(error:gett(Γ,X)),fail.

maketop('*', top).
maketop((K1 => K2), abs('_', K1, K2_)) :- maketop(K2, K2_). 

% ------------------------   EVALUATION  ------------------------

e([L = (Vari, M) | Mf], M, [L = (Vari, M_) | Mf], M_)  :- \+ v(M).
e([L = (Vari, M) | Mf], M1, [L = (Vari, M) | Mf_], M_) :- v(M), e(Mf, M1, Mf_, M_). 

%eval1(_,M,_) :- writeln(eval1:M),fail.

Γ /- if(true, M2, _) ==> M2.
Γ /- if(false, _, M3) ==> M3.
Γ /- if(M1, M2, M3) ==> if(M1_, M2, M3)               where Γ /- M1 ==> M1_.
Γ /- succ(M1) ==> succ(M1_)                           where Γ /- M1 ==> M1_.
Γ /- pred(0) ==> 0.
Γ /- pred(succ(N1)) ==> N1                            where n(N1).
Γ /- pred(M1) ==> pred(M1_)                           where Γ /- M1 ==> M1_.
Γ /- iszero(0) ==> true.
Γ /- iszero(succ(N1)) ==> false                       where n(N1).
Γ /- iszero(M1) ==> iszero(M1_)                       where Γ /- M1 ==> M1_.
Γ /- F1 * F2 ==> F3                                   where float(F1), float(F2), F3 is F1 * F2.
Γ /- V1 * M2 ==> V1 * M2_                             where v(V1), Γ /- M2 ==> M2_.
Γ /- M1 * M2 ==> M1_ * M2                             where Γ /- M1 ==> M1_.
Γ /- X ==> M                                          where x(X), getb(Γ, X, bMAbb(M, _)).
Γ /- (fn(X : _) -> M12) $ V2 ==> R                    where v(V2), M12![(X -> V2)] subst R.
Γ /- V1 $ M2 ==> V1 $ M2_                             where v(V1), Γ /- M2 ==> M2_.
Γ /- M1 $ M2 ==> M1_ $ M2                             where Γ /- M1 ==> M1_.
Γ /- (let(X) = V1 in M2) ==> M2_                      where v(V1), M2![(X -> V1)] subst M2_.
Γ /- (let(X) = M1 in M2) ==> (let(X) = M1_ in M2)     where Γ /- M1 ==> M1_.
Γ /- fix((fn(X : T) -> M12)) ==> M12_                 where M12![(X -> fix((fn(X : T) -> M12)))] subst M12_.
Γ /- fix(M1) ==> fix(M1_)                             where Γ /- M1 ==> M1_.
Γ /- V1 as _ ==> V1                                   where v(V1).
Γ /- M1 as T ==> M1_ as T                             where Γ /- M1 ==> M1_.
Γ /- {Mf} ==> {Mf_}                                   where e(Mf, M, Mf_, M_), Γ /- M ==> M_.
Γ /- {Mf} # L ==> M                                   where v({Mf}), member(L = (_, M), Mf).
Γ /- M1 # L ==> M1_ # L                               where Γ /- M1 ==> M1_.
Γ /- pack(T1, M2, T3) ==> pack(T1, M2_, T3)           where Γ /- M2 ==> M2_.
Γ /- unpack(_, X, pack(T11, V12, _), M2) ==> M        where v(V12), M2![(X -> V12)] subst M2_, M2_![(X -> T11)] tmsubst M.
Γ /- unpack(TX, X, M1, M2) ==> unpack(TX, X, M1_, M2) where Γ /- M1 ==> M1_.
Γ /- {Mf} # L <- V2 ==> {Mf_}                         where v({Mf}), v(V2), maplist([ML = (Var, M), ML = (Var, R)] >> (ML = L, R = V2 ; R = M), Mf, Mf_).
Γ /- V1 # L <- M2 ==> V1 # L <- M2_                   where v(V1), Γ /- M2 ==> M2_.
Γ /- M1 # L <- M2 ==> M1_ # L <- M2                   where Γ /- M1 ==> M1_.
Γ /- (fn(X :: _) => M11)![T2] ==> M11_                where M11![(X -> T2)] tmsubst M11_.
Γ /- M1![T2] ==> M1_![T2]                             where Γ /- M1 ==> M1_.
% eval1(Γ,M,_) :- writeln(error:eval1(Γ,M)),fail.

Γ /- M ==>> M_ where Γ /- M ==> M1, Γ /- M1 ==>> M_.
Γ /- M ==>> M. 

% ------------------------   KINDING  ------------------------

gettabb(Γ, X, T)                   :- getb(Γ, X, bTAbb(T, _)).
compute(Γ, X, T)                   :- tx(X), gettabb(Γ, X, T).
compute(Γ, abs(X, _, T12) $ T2, T) :- T12![(X -> T2)] tsubst T.
simplify(Γ, T1 $ T2, T_)           :- simplify(Γ, T1, T1_), simplify2(Γ, T1_ $ T2, T_).
simplify(Γ, T, T_)                 :- simplify2(Γ, T, T_).
simplify2(Γ, T, T_)                :- compute(Γ, T, T1), simplify(Γ, T1, T_).
simplify2(Γ, T, T).

Γ /- S = T                         :- simplify(Γ, S, S_), simplify(Γ, T, T_), Γ /- S_ == T_.
Γ /- bool == bool.
Γ /- nat == nat.
Γ /- unit == unit.
Γ /- float == float.
Γ /- string == string.
Γ /- top == top.
Γ /- X == T                                          :- tx(X), gettabb(Γ, X, S), Γ /- S = T.
Γ /- S == X                                          :- tx(X), gettabb(Γ, X, T), Γ /- S = T.
Γ /- X == X                                          :- tx(X).
Γ /- (S1 -> S2) == (T1 -> T2)                        :- Γ /- S1 = T1, Γ /- S2 = T2.
Γ /- {Sf} == {Tf}                                    :- length(Sf, Len), length(Tf, Len),
                                                        maplist([L : (TVar, T)] >> (member(L : (TVar, S), Sf), Γ /- S = T), Tf).
Γ /- (all(TX :: S1) => S2) == (all(_ :: T1) => T2)   :- Γ /- S1 = T1, [TX - bName | Γ] /- S2 = T2.
Γ /- (some(TX :: S1) => S2) == (some(_ :: T1) => T2) :- Γ /- S1 = T1, [TX - bName | Γ] /- S2 = T2.
Γ /- abs(TX, K1, S2) == abs(_, K1, T2)               :- [TX - bName | g] /- S2 = T2.
Γ /- S1 $ S2 == T1 $ T2                              :- Γ /- S1 = T1, Γ /- S2 = T2.

Γ /- T :: K                        where Γ \- T :: K, !.
Γ /- T :: K                        where writeln(error : kindof(T, K)), fail.
Γ \- X :: '*'                      where tx(X), \+ member(X - _, Γ).
Γ \- X :: K                        where tx(X), getb(Γ, X, bTVar(T)), Γ /- T :: K, !.
Γ \- X :: K                        where tx(X), !, getb(Γ, X, bTAbb(_, K)).
Γ \- (T1 -> T2) :: '*'             where !, Γ /- T1 :: '*', Γ /- T2 :: '*'.
Γ \- {Tf} :: '*'                   where maplist([L : (_, S)] >> (Γ /- S :: '*'), Tf).
Γ \- (all(TX :: T1) => T2) :: '*'  where !, [TX - bTVar(T1) | Γ] /- T2 :: '*'.
Γ \- abs(TX, K1, T2) :: (K1 => K)  where !, maketop(K1, T1), [TX - bTVar(T1) | Γ] /- T2 :: K.
Γ \- T1 $ T2 :: K12                where !, Γ /- T1 :: (K11 => K12), Γ /- T2 :: K11.
Γ \- (some(TX :: T1) => T2) :: '*' where !, [TX - bTVar(T1) | Γ] /- T2 :: '*'.
Γ \- T :: '*'. 

% ------------------------   SUBTYPING  ------------------------

promote(Γ, X, T)          :- tx(X), getb(Γ, X, bTVar(T)).
promote(Γ, S $ T, S_ $ T) :- promote(Γ, S, S_).

Γ /- S <: T                                          where Γ /- S = T.
Γ /- S <: T                                          where simplify(Γ, S, S_), simplify(Γ, T, T_), Γ \- S_ <: T_.
Γ \- _ <: top.
Γ \- X <: T                                          where tx(X), promote(Γ, X, S), Γ /- S <: T.
Γ \- (S1 -> S2) <: (T1 -> T2)                        where Γ /- T1 <: S1, Γ /- S2 <: T2.
Γ \- {SF} <: {TF}                                    where maplist([L : (Vart, T)] >> (
                                                             member(L : (Vars, S), SF),
                                                             (Vars = # ; Vart = !), Γ /- S <: T
                                                           ), TF).
Γ \- T1 $ T2 <: T                                    where promote(Γ, T1 $ T2, S), Γ /- S <: T.
Γ \- abs(TX, K1, S2) <: abs(_, K1, T2)               where maketop(K1, T1), [TX - bTVar(T1) | Γ] /- S2 <: T2.
Γ \- (all(TX :: S1) => S2) <: (all(_ :: T1) => T2)   where Γ /- S1 <: T1, Γ /- T1 <: S1, [TX - bTVar(T1) | Γ] /- S2 <: T2.
Γ \- (some(TX :: S1) => S2) <: (some(_ :: T1) => T2) where Γ /- S1 <: T1, Γ /- T1 <: S1, [TX - bTVar(T1) | Γ] /- S2 <: T2.

Γ /- S /\ T : T                                        :- Γ /- S <: T.
Γ /- S /\ T : S                                        :- Γ /- T <: S.
Γ /- S /\ T : R                                        :- simplify(Γ, S, S_), simplify(Γ, T, T_), Γ \- S_ /\ T_ : R.
Γ \- {SF} /\ {TF} : {UF_}                              :- include([L : _] >> member(L : _, TF), SF, UF),
                                                          maplist([L : (SVar, S), L : (Var, T_)] >> (
                                                            member(L : (TVar, T), TF),
                                                            (SVar = TVar, Var = SVar ; Var = #),
                                                            Γ /- S /\ T : T_
                                                          ), UF, UF_).
Γ \- (all(TX :: S1) => S2) /\ (all(_ :: T1) => T2)
                              : (all(TX :: S1) => T2_) :- Γ /- S1 <: T1, Γ /- T1 <: S1, [TX - bTVar(T1) | Γ] /- T1 /\ T2 : T2_.
Γ \- (all(TX :: S1) => S2) /\ (all(_ :: T1) => T2) : top.
Γ \- (S1 -> S2) /\ (T1 -> T2) : (S_ -> T_)             :- Γ /- S1 \/ T1 : S_, Γ /- S2 /\ T2 : T_.

Γ \- _ /\ _ : top.
Γ /- S \/ T : S                                        :- Γ /- S <: T.
Γ /- S \/ T : T                                        :- Γ /- T <: S.
Γ /- S \/ T : R                                        :- simplify(Γ, S, S_), simplify(Γ, T, T_), Γ \- S_ \/ T_ : R.
Γ \- {SF} \/ {TF} : {UF_}                              :- maplist([L : (SVar, S), L : (Var, T_)] >> (
                                                          member(L : (TVar, T), TF),
                                                          (SVar = TVar, Var = SVar ; Var = !),
                                                          Γ /- S \/ T : T_ ; T_ = S
                                                        ), SF, SF_),
                                                        include([L : _] >> (\+ member(L : _, SF)), TF, TF_),
                                                        append(SF_, TF_, UF_).
Γ \- (all(TX :: S1) => S2) \/ (all(_ :: T1) => T2)
                              : (all(TX :: S1) => T2_) :- Γ /- S1 <: T1, Γ /- T1 <: S1, [TX - bTVar(T1) | Γ] /- T1 \/ T2 : T2_.
Γ \- (S1 -> S2) \/ (T1 -> T2) : (S_ -> T_)             :- Γ /- S1 /\ T1 : S_, Γ /- S2 \/ T2 : T_. 

% ------------------------   TYPING  ------------------------

lcst(Γ, S, T) :- simplify(Γ, S, S_), lcst2(Γ, S_, T).
lcst2(Γ, S, T) :- promote(Γ, S, S_), lcst(Γ, S_, T).
lcst2(Γ, T, T). 

%typeof(Γ,M,_) :- writeln(typeof(Γ,M)),fail.

Γ /- true : bool.
Γ /- false : bool.
Γ /- if(M1, M2, M3) : T                           where Γ /- M1 : T1, Γ /- T1 <: bool,
                                                        Γ /- M2 : T2, Γ /- M3 : T3, Γ /- T2 /\ T3 : T.
Γ /- 0 : nat.
Γ /- succ(M1) : nat                               where Γ /- M1 : T1, Γ /- T1 <: nat.
Γ /- pred(M1) : nat                               where Γ /- M1 : T1, Γ /- T1 <: nat.
Γ /- iszero(M1) : bool                            where Γ /- M1 : T1, Γ /- T1 <: nat.
Γ /- unit : unit.
Γ /- F1 : float                                   where float(F1).
Γ /- M1 * M2 : float                              where Γ /- M1 : T1, Γ /- T1 <: float, Γ /- M2 : T2, Γ /- T2 <: float.
Γ /- X : string                                   where string(X).
Γ /- X : T                                        where x(X), !, gett(Γ, X, T).
Γ /- (fn(X : T1) -> M2) : (T1 -> T2_)             where Γ /- T1 :: '*', [X - bVar(T1) | Γ] /- M2 : T2_, !.
Γ /- M1 $ M2 : T12                                where Γ /- M1 : T1, lcst(Γ, T1, (T11 -> T12)), Γ /- M2 : T2, Γ /- T2 <: T11.
Γ /- (let(X) = M1 in M2) : T                      where Γ /- M1 : T1, [X - bVar(T1) | Γ] /- M2 : T.
Γ /- fix(M1) : T12                                where Γ /- M1 : T1, lcst(Γ, T1, (T11 -> T12)), Γ /- T12 <: T11.
Γ /- inert(T) : T.
Γ /- (M1 as T) : T                                where Γ /- T :: '*', Γ /- M1 : T1, Γ /- T1 <: T.
Γ /- {Mf} : {Tf}                                  where maplist([L = (Var, M), L : (Var, T)] >> (Γ /- M : T), Mf, Tf), !.
Γ /- M1 # L : T                                   where Γ /- M1 : T1, lcst(Γ, T1, {Tf}), member(L : (_, T), Tf).
Γ /- M1 # L <- M2 : T1                            where Γ /- M1 : T1, Γ /- M2 : T2, lcst(Γ, T1, {Tf}),
                                                        member(L : (#, T), Tf), Γ /- T2 <: T.
Γ /- pack(T1, M2, T) : T                          where Γ /- T :: '*', simplify(Γ, T, (some(Y :: TBound) => T2)),
                                                        Γ /- T1 <: TBound, Γ /- M2 : S2, T2![(Y -> T1)] tsubst T2_, Γ /- S2 <: T2_.
Γ /- unpack(TX, X, M1, M2) : T2                   where Γ /- M1 : T1, lcst(Γ, T1, (some(_ :: TBound) => T11)),
                                                        [X - bVar(T11), TX - bTVar(TBound) | Γ] /- M2 : T2.
Γ /- (fn(TX :: T1) => M2) : (all(TX :: T1) => T2) where [TX - bTVar(T1) | Γ] /- M2 : T2, !.
Γ /- M1![T2] : T12_                               where Γ /- M1 : T1, lcst(Γ, T1, (all(X :: T11) => T12)),
                                                        Γ /- T2 <: T11, T12![(X -> T2)] tsubst T12_.
Γ /- M : _                                        where writeln(error : typeof(Γ, M)), fail. 

% ------------------------   MAIN  ------------------------

show(Γ, X, bName)       :- format('~w\n', [X]).
show(Γ, X, bVar(T))     :- format('~w : ~w\n', [X, T]).
show(Γ, X, bTVar(T))    :- format('~w <: ~w\n', [X, T]).
show(Γ, X, bMAbb(M, T)) :- format('~w : ~w\n', [X, T]).
show(Γ, X, bTAbb(T, K)) :- format('~w :: ~w\n', [X, K]).

check_someBind(TBody, pack(_, T12, _), bMAbb(T12, some(TBody))).
check_someBind(TBody, _, bVar(TBody)).

run({(TX, X)} = M, Γ, [X - B, TX - bTVar(TBound) | Γ])
                                           :- tx(TX), x(X), m(M), !, Γ /- M : T, lcst(Γ, T, (some(_ :: TBound) => TBody)),
                                              Γ /- M ==>> M_, check_someBind(TBody, M_, B), format('~w\n~w : ~w\n', [TX, X, TBody]).
run(type(X) = T, Γ, [X - bTAbb(T, K) | Γ]) :- tx(X), t(T), Γ /- T :: K, show(Γ, X, bTAbb(T, K)).
run(X <: T, Γ, [X - bTVar(T) | Γ])         :- tx(X), t(T), Γ /- T :: _, write(X), show(Γ, X, bTVar(T), R), writeln(R).
run(X : T, Γ, [X - bVar(T) | Γ])           :- x(X), t(T), show(Γ, X, bVar(T)).
run(X : T = M, Γ, [X - bMAbb(M_, T) | Γ])  :- x(X), t(T), m(M), Γ /- M : T1, Γ /- T1 <: T, Γ /- M ==>> M_, show(Γ, X, bMAbb(M_, T)).
run(X = M, Γ, [X - bMAbb(M_, T) | Γ])      :- !, x(X), m(M), !, Γ /- M : T, Γ /- M ==>> M_, show(Γ, X, bMAbb(M_, T)), !.
run(M, Γ, Γ)                               :- !, m(M), !, Γ /- M : T, !, Γ /- M ==>> M_, !, writeln(M_ : T).

run(Ls)                                    :- foldl(run, Ls, [], _). 

% ------------------------   TEST  ------------------------

% lambda x:Top. x;
:- run([(fn(x : top) -> x)]). 
% (lambda x:Top. x) (lambda x:Top. x);
:- run([(fn(x : top) -> x) $ (fn(x : top) -> x)]). 
% (lambda x:Top->Top. x) (lambda x:Top. x);
:- run([(fn(x : (top -> top)) -> x) $ (fn(x : top) -> x)]). 
% (lambda r:{x:Top->Top}. r.x r.x) 
%   {x=lambda z:Top.z, y=lambda z:Top.z}; 
:- run([(fn(r : {[x : (!, (top -> top))]}) -> r # x $ r # x) $ {[x = (!, (fn(z : top) -> z)), y = (!, (fn(z : top) -> z))]}]). 
% "hello";
:- run(["hello"]). 
% unit;
:- run([unit]). 
% lambda x:A. x;
:- run([(fn(x : 'A') -> x)]). 
% let x=true in x;
:- run([(let(x) = true in x)]). 
% {x=true, y=false};
:- run([{[x = (!, true), y = (!, false)]}]). 
% {x=true, y=false}.x;
:- run([{[x = (!, true), y = (!, false)]} # x]). 
% {true, false};
:- run([{[1 = (!, true), 2 = (!, false)]}]). 
% {true, false}.1;
:- run([{[1 = (!, true), 2 = (!, false)]} # 1]). 
% if true then {x=true,y=false,a=false} else {y=false,x={},b=false};
:- run([if(true, {[x = (!, true), y = (!, false), a = (!, false)]}, {[y = (!, false), x = (!, {[]}), b = (!, false)]})]). 
% timesfloat 2.0 3.14159;
:- run([2.0 * 3.14159]). 
% lambda X. lambda x:X. x;
:- run([(fn('X' :: top) => fn(x : 'X') -> x)]). 
% (lambda X. lambda x:X. x) [All X.X->X];
:- run([(fn('X' :: top) => fn(x : 'X') -> x)![(all('X' :: top) => 'X' -> 'X')]]). 
% lambda X<:Top->Top. lambda x:X. x x; 
:- run([(fn('X' :: (top -> top)) => fn(x : 'X') -> x $ x)]). 
% lambda x:Bool. x;
:- run([(fn(x : bool) -> x)]). 
% (lambda x:Bool->Bool. if x false then true else false) 
%   (lambda x:Bool. if x then false else true); 
:- run([(fn(x : (bool -> bool)) -> if(x $ false, true, false)) $ (fn(x : bool) -> if(x, false, true))]). 
% lambda x:Nat. succ x;
:- run([(fn(x : nat) -> succ(x))]).  
% (lambda x:Nat. succ (succ x)) (succ 0); 
:- run([(fn(x : nat) -> succ(succ(x))) $ succ(0)]).  
% T = Nat->Nat;
% lambda f:T. lambda x:Nat. f (f x);
:- run([type('T') = (nat -> nat), (fn(f : 'T') -> fn(x : nat) -> f $ (f $ x))]). 
% {*All Y.Y, lambda x:(All Y.Y). x} as {Some X,X->X};
:- run([pack((all('Y' :: top) => 'Y'), (fn(x : (all('Y' :: top) => 'Y')) -> x), (some('X' :: top) => 'X' -> 'X'))]). 


% {*Nat, {c=0, f=lambda x:Nat. succ x}}
%   as {Some X, {c:X, f:X->Nat}};
:- run([pack(nat, {[c = (!, 0), f = (!, (fn(x : nat) -> succ(x)))]}, (some('X' :: top) => {[c : (!, 'X'), f : (!, ('X' -> nat))]}))]). 
% let {X,ops} = {*Nat, {c=0, f=lambda x:Nat. succ x}}
%               as {Some X, {c:X, f:X->Nat}}
% in (ops.f ops.c);
:- run([unpack('X', ops, pack(nat, {[c = (!, 0), f = (!, (fn(x : nat) -> succ(x)))]}, (some('X' :: top) => {[c : (!, 'X'), f : (!, ('X' -> nat))]})),
      ops # f $ ops # c)]).
:- run([ 
% Pair = lambda X. lambda Y. All R. (X->Y->R) -> R;
type('Pair') = abs('X', '*', abs('Y', '*', (all('R' :: top) => ('X' -> 'Y' -> 'R') -> 'R'))),  
% pair = lambda X.lambda Y.lambda x:X.lambda y:Y.lambda R.lambda p:X->Y->R.p x y;
pair = (fn('X' :: top) => fn('Y' :: top) => fn(x : 'X') -> fn(y : 'Y') -> fn('R' :: top) => fn(p : ('X' -> 'Y' -> 'R')) -> p $ x $ y),  
% fst = lambda X.lambda Y.lambda p:Pair X Y.p [X] (lambda x:X.lambda y:Y.x);
fst = (fn('X' :: top) => fn('Y' :: top) => fn(p : 'Pair' $ 'X' $ 'Y') -> p!['X'] $ (fn(x : 'X') -> fn(y : 'Y') -> x)),  
% snd = lambda X.lambda Y.lambda p:Pair X Y.p [Y] (lambda x:X.lambda y:Y.y);
snd = (fn('X' :: top) => fn('Y' :: top) => fn(p : 'Pair' $ 'X' $ 'Y') -> p!['Y'] $ (fn(x : 'X') -> fn(y : 'Y') -> y)),  
% List = lambda X. All R. (X->R->R) -> R -> R; 
type('List') = abs('X', '*', (all('R' :: top) => ('X' -> 'R' -> 'R') -> 'R' -> 'R')),  
% diverge =
% lambda X.
%   lambda _:Unit.
%   fix (lambda x:X. x);
diverge = (fn('X' :: top) =>
  fn('_' : unit) ->
  fix((fn(x : 'X') -> x))),  
% nil = lambda X.
%       (lambda R. lambda c:X->R->R. lambda n:R. n)
%       as List X; 
nil = (fn('X' :: top) =>
  (fn('R' :: top) => fn(c : ('X' -> 'R' -> 'R')) -> fn(n : 'R') -> n)
  as 'List' $ 'X'),  
% cons = 
% lambda X.
%   lambda hd:X. lambda tl: List X.
%      (lambda R. lambda c:X->R->R. lambda n:R. c hd (tl [R] c n))
%      as List X; 
cons = (fn('X' :: top) =>
  fn(hd : 'X') -> fn(tl : 'List' $ 'X') ->
    (fn('R' :: top) => fn(c : ('X' -> 'R' -> 'R')) -> fn(n : 'R') -> c $ hd $ (tl!['R'] $ c $ n))
    as 'List' $ 'X'),  
% isnil =  
% lambda X. 
%   lambda l: List X. 
%     l [Bool] (lambda hd:X. lambda tl:Bool. false) true; 
isnil = (fn('X' :: top) =>
  fn(l : 'List' $ 'X') ->
    l![bool] $ (fn(hd : 'X') -> fn(tl : bool) -> false) $ true),  
% head = 
% lambda X. 
%   lambda l: List X. 
%     (l [Unit->X] (lambda hd:X. lambda tl:Unit->X. lambda _:Unit. hd) (diverge [X]))
%     unit; 
head = (fn('X' :: top) =>
  fn(l : 'List' $ 'X') ->
    l![(unit -> 'X')] $ (fn(hd : 'X') -> fn(tl : (unit -> 'X')) -> fn('_' : unit) -> hd) $ (diverge!['X']) $ unit),  
% tail =  
% lambda X.  
%   lambda l: List X. 
%     (fst [List X] [List X] ( 
%       l [Pair (List X) (List X)]
%         (lambda hd: X. lambda tl: Pair (List X) (List X). 
%           pair [List X] [List X] 
%             (snd [List X] [List X] tl)  
%             (cons [X] hd (snd [List X] [List X] tl))) 
%         (pair [List X] [List X] (nil [X]) (nil [X]))))
%     as List X; 
tail = (fn('X' :: top) =>
  fn(l : 'List' $ 'X') ->
    fst!['List' $ 'X']!['List' $ 'X'] $ (
      l!['Pair' $ ('List' $ 'X') $ ('List' $ 'X')] $
        (fn(hd : 'X') -> fn(tl : 'Pair' $ ('List' $ 'X') $ ('List' $ 'X')) ->
          pair!['List' $ 'X']!['List' $ 'X'] $
            (snd!['List' $ 'X']!['List' $ 'X'] $ tl) $
            (cons!['X'] $ hd $ (snd!['List' $ 'X']!['List' $ 'X'] $ tl))) $
        (pair!['List' $ 'X']!['List' $ 'X'] $ (nil!['X']) $ (nil!['X'])))
    as 'List' $ 'X')
]).
:- halt.

まとめ

最後のプログラムはレコードの一部を書き換えることが出来る高階多相部分型付けのある言語でした。

2週間に渡って、TAPLの図とサンプルプログラムをPrologに移植したものを見てきました。
詳細については触れていませんが、かなりの量の図とプログラムがありました。
型理論の論文を概観するには図や推論規則を見ることが大事です。
そして、推論規則用いられている一階述語論理に近い形で Prolog で書くことに慣れれば、一階述語論理はそう難しくないものになります。
しかし、大きなプログラムを書くためには型システムの補助があると便利です。Prologは動的な型システムであり、静的な型システムがないので、大きなプログラムを書くと些細なバグに悩まされることも多くなります。
できれば、より柔軟な型システムがPrologに欲しいところです。
TAPLでは型再構築を含みますが、System F 以降の型システムは必ず型を書かなくてはなりませんでした。
しかし、世の中にはもっと様々な型システムがあります。let多相な型システムであれば型推論は可能です。
TypeScript の集合論的な型システムは、漸進的な型システムと組み合わせることで JavaScript の動的な型システムの世界に、より堅牢な静的型システムを導入出来ます。同様の型システムを開発することで型理論の図やPrologのプログラムは型システムにより型安全性を保証することができるようになるでしょう。

TAPL の本は分厚く、多くの演習問題や証明などがあり、その1つ1つは多くの研究者によって構築されてきました。
図だけを見れば意外と入門的な内容しか触れていないわけですが、詳細に踏み込めばより多くの努力が必要であることがわかります。

しかし人生はそう長くありません。すべての問題を解き、理解できるのが理想でしょうが、まずは全体像を把握することから始めても悪くないのではないでしょうか?

Saturnalia From Wikipedia, the free encyclopedia

Saturnalia was an ancient Roman festival in honour of the god Saturn, held on 17 December of the Julian calendar and later expanded with festivities through to 23 December. The holiday was celebrated with a sacrifice at the Temple of Saturn, in the Roman Forum, and a public banquet, followed by private gift-giving, continual partying, and a carnival atmosphere that overturned Roman social norms: gambling was permitted, and masters provided table service for their slaves as it was seen as a time of liberty for both slaves and freedmen alike.[1] A common custom was the election of a "King of the Saturnalia", who would give orders to people, which were to be followed and preside over the merrymaking. The gifts exchanged were usually gag gifts or small figurines made of wax or pottery known as sigillaria. The poet Catullus called it "the best of days".[2]

Saturnalia was the Roman equivalent to the earlier Greek holiday of Kronia, which was celebrated during the Attic month of Hekatombaion in late midsummer. It held theological importance for some Romans, who saw it as a restoration of the ancient Golden Age, when the world was ruled by Saturn. The Neoplatonist philosopher Porphyry interpreted the freedom associated with Saturnalia as symbolizing the "freeing of souls into immortality". Saturnalia may have influenced some of the customs associated with later celebrations in western Europe occurring in midwinter, particularly traditions associated with Christmas, the Feast of the Holy Innocents, and Epiphany. In particular, the historical western European Christmas custom of electing a "Lord of Misrule" may have its roots in Saturnalia celebrations.

Saturnalia by Antoine Callet

なんだが、サートルヌスだの、サツマイモみてぇな名前だな。
サターンがサタンだと悪魔みたいだからサンタさんみてーになったんかもな?サトル君が悟ったんだべw
孫悟空さは空を悟った孫だべw
アンダーソンさんみてーなもんが?
サタンは悪魔だーというユダヤ人の考えがあって騙されんなーみたいな気持ちもあったのかもなぁ
サートルヌスがヘーリオスと同一視されるとかも、盆暮れ正月の暮れと正月はおんなじ休みだがらだっつーごともあるんだべなぁ。始まりと終わりだから繋がってんだっペw

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