0924 CEK machine.ipynb

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     "metadata": {},
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   ],
   "source": [
    ":!./install.sh"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# CEK 기계\n",
    "CEK 기계는 80년대에 Friedman과 Felleisen이 SECD 기계(Landin 1964)를 더 말끔하게 정리한 것이다.\n",
    "###### 참고자료\n",
    "* https://www.cs.bham.ac.uk/~hxt/2015/compilers/compiling-functional.pdf\n",
    "* https://en.wikipedia.org/wiki/SECD_machine\n",
    "* https://www.brics.dk/RS/03/33/\n",
    "* https://pages.cpsc.ucalgary.ca/~robin/FMCS/FMCS2014/Prashant_talk.pdf\n",
    "* https://github.com/kseo/secd"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "$ x \\in \\textit{Nm} $\n",
    "\n",
    "$ n \\in \\textit{Int} $\n",
    "\n",
    "$ e \\in \\textit{Exp} ~::=~ x ~\\mid~ (e_1~e_2) ~\\mid~ (\\lambda x.e) ~\\mid~ n $"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {},
   "outputs": [],
   "source": [
    "type Nm = String\n",
    "\n",
    "data Exp = Var Nm      -- x\n",
    "         | App Exp Exp -- e1 e2\n",
    "         | Lam Nm Exp  -- \\x.e\n",
    "         | Lit Int     -- n\n",
    "         deriving Show"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "$\\textit{CEK} ~=~ \\textit{Code} \\times \\textit{Env} \\times \\textit{Kont}$\n",
    "\n",
    "CEK 기계는 코드(Code), 환경(Env), 남은일(Kont) 세순서쌍으로 나타나며 그 세순서쌍의 각 요소는 아래와 같이 정의되는 집합의 원소이다.\n",
    "\n",
    " * 코드(Code)는 식(Exp)과 값(Value)의 합집합이다.\n",
    " \n",
    "    $ \\textit{Code} ~=~ \\textit{Exp} \\cup \\textit{Value} \\\\[]$\n",
    "\n",
    " * 환경(Env)은 이름(Nm)에서 값(Value)으로의 유한한 대응이다.\n",
    "\n",
    "    $ \\textit{Env} ~=~ \\textit{Nm} \\xrightarrow{\\text{fin}} \\textit{Value} \\\\[]$\n",
    "    \n",
    " * 남은일이란 프레임(Frame)의 목록이다.\n",
    " \n",
    "    $ \\textit{Kont} ~=~ \\textit{Frame}^{*} \\\\[]$\n",
    "\n",
    "\n",
    "값(Value)은 정수와 닫힘의 두 종류 값이 있다.\n",
    "\n",
    "$ \\textit{Value} ~=~ \\textit{Int} \\;\\cup\\; Closure $\n",
    "\n",
    "$ v \\in \\textit{Value} ~::=~ n ~\\mid~ \\langle\\lambda x.e,\\sigma\\rangle$\n",
    "\n",
    "이 때 닫힘(Closure)이란 어떤 식(Exp)을 그 자유변수를 모두 포함하는 환경(Env)과 함께 엮어 놓은 것이다.\n",
    "\n",
    "$ \\textit{Closure} ~=~ \\textit{Exp} \\times \\textit{Env}$\n",
    "\n",
    "\n",
    "프레임(Frame)이란 지금 하고 있는 계산의 값이 나오면 그것으로 다음에 할 일을 나타내는데, 아래와 같이 두 종류가 있다.\n",
    " * $(v~\\bigcirc) \\qquad$ \n",
    "   계산의 값이 나오면 인자로 $v$에 넘긴다. 즉, $v$를 함수로써 계산되어 나오는 값에 적용한다.\n",
    " * $(\\bigcirc~e~\\sigma) ~\\,\\quad$\n",
    "   계산의 값이 나오면 그 값을 함수로써 $e$에 적용한다. 물론 그러려면 $e$를 환경 $\\sigma$에서 값으로 계산하기도 해야 할 것이다.\n",
    "\n"
   ]
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   "cell_type": "code",
   "execution_count": 3,
   "metadata": {},
   "outputs": [],
   "source": [
    "type CEK = (Code,Env,Kont)\n",
    "\n",
    "data Code = OpE Exp\n",
    "          | OpV Value\n",
    "          deriving Show\n",
    "type Env  = [(Nm,Value)]\n",
    "type Kont = [Frame]\n",
    "\n",
    "data Value = Vint Int      -- n\n",
    "           | Clos Exp Env  -- < \\x.e, env >\n",
    "           deriving Show\n",
    "           \n",
    "data Frame = FrV Value   -- (v O)\n",
    "           | FrE Exp Env -- (O e env)\n",
    "           deriving Show"
   ]
  },
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   "cell_type": "markdown",
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   "source": [
    "$ \\big\\langle x ~\\big|~ \\sigma ~\\big|~ \\kappa \\big\\rangle\n",
    "  \\longrightarrow\n",
    "  \\big\\langle \\sigma(x) ~\\big|~ \\sigma ~\\big|~ \\kappa \\big\\rangle $\n",
    "\n",
    "$ \\big\\langle e_1~e_2 ~\\big|~ \\sigma ~\\big|~ \\kappa \\big\\rangle\n",
    "  \\longrightarrow\n",
    "  \\big\\langle e_1 ~\\big|~ \\sigma ~\\big|~  (\\bigcirc\\,e_2\\;\\sigma),\\kappa \\big\\rangle $\n",
    "\n",
    "$ \\big\\langle \\lambda x.e ~\\big|~ \\sigma ~\\big|~ \\kappa \\big\\rangle\n",
    "  \\longrightarrow \n",
    "  \\big\\langle \\langle\\lambda x.e,\\sigma\\rangle ~\\big|~ \\sigma ~\\big|~ \\kappa \\big\\rangle $\n",
    "\n",
    "$ \\big\\langle v ~\\big|~ \\sigma_1 ~\\big|~ (\\bigcirc\\,e_2\\;\\sigma_2),\\kappa  \\big\\rangle\n",
    "  \\longrightarrow\n",
    "  \\big\\langle e_2 ~\\big|~ \\sigma_2 ~\\big|~ (v\\,\\bigcirc),\\kappa  \\big\\rangle $\n",
    "\n",
    "$ \\big\\langle v ~\\big|~ \\sigma_1 ~\\big|~ (\\langle\\lambda x.e,\\sigma_2\\rangle\\,\\bigcirc),\\kappa  \\big\\rangle\n",
    "  \\longrightarrow\n",
    "  \\big\\langle e ~\\big|~ \\sigma_2[x\\mapsto v] ~\\big|~ \\kappa \\big\\rangle $"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {},
   "outputs": [],
   "source": [
    "lookup' x env = case lookup x env of Just v  -> v\n",
    "                                     Nothing -> error(\"x is unknown\")\n",
    "step :: CEK -> Maybe CEK\n",
    "step (OpE(Var x),     env, k) = Just (OpV(lookup' x env), env, k)\n",
    "step (OpE(App e1 e2), env, k) = Just (OpE e1, env, FrE e2 env : k)\n",
    "step (OpE(e@Lam{}),   env, k) = Just (OpV(Clos e env), env, k)\n",
    "step (OpE(Lit n),     env, k) = Just (OpV(Vint n), env, k)\n",
    "step (OpV v, env1, FrE e2 env2 : k) = Just (OpE e2, env2, FrV v : k)\n",
    "step (OpV v, env1, FrV(Clos (Lam x e) env2) : k) = Just (OpE e, (x,v) : env2, k)\n",
    "step _ = Nothing"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "metadata": {},
   "outputs": [],
   "source": [
    "{-# LANGUAGE FlexibleInstances #-}\n",
    "import IHaskell.Display\n",
    "import Data.List (intersperse)\n",
    "\n",
    "class TeX a where\n",
    "  toTeX :: a -> String\n",
    "\n",
    "instance TeX Exp where\n",
    "  toTeX (Var x) = x\n",
    "  toTeX (App e1 e2) = \"(\"++toTeX e1++\"\\\\;\"++toTeX e2++\")\" \n",
    "  toTeX (Lam x e) = \"(\\\\lambda{}\"++x++\".\"++toTeX e++\")\"\n",
    "  toTeX (Lit n) = show n\n",
    "\n",
    "instance TeX Code where\n",
    "  toTeX (OpE e) = toTeX e \n",
    "  toTeX (OpV v) = toTeX v\n",
    "\n",
    "instance TeX Value where\n",
    "  toTeX (Vint n) = show n\n",
    "  toTeX (Clos e env) = \"\\\\langle{}\" ++ toTeX e ++ \",\" ++ revTeX env ++ \"\\\\rangle{}\"\n",
    "\n",
    "instance TeX Frame where\n",
    "  toTeX (FrV v) = \"(\"++toTeX v++\"\\\\;\\\\bigcirc{})\"\n",
    "  toTeX (FrE e env) = \"(\\\\bigcirc{}\\\\;\"++toTeX e++\"\\\\;\"++revTeX env++\")\"\n",
    "\n",
    "instance TeX (Nm,Value) where\n",
    "  toTeX (x,v) = x++\"\\\\mapsto{}\"++toTeX v\n",
    "\n",
    "instance TeX CEK where\n",
    "  toTeX (c,env,k) = \"\\\\big\\\\langle{}\"\n",
    "                 ++ toTeX c ++ \"\\\\;\\\\big|\\\\;\" ++ revTeX env ++ \"\\\\;\\\\big|\\\\;\" ++ toTeX k\n",
    "                 ++ \"\\\\big\\\\rangle{}\"\n",
    "\n",
    "instance TeX a => TeX (Maybe a) where\n",
    "  toTeX (Just x)= \"\\\\texttt{Just}(\"++toTeX x++\")\"\n",
    "  toTeX Nothing = \"\\\\texttt{Nothing}\"\n",
    "\n",
    "instance TeX a => TeX [a] where\n",
    "  toTeX xs = \"[\" ++ (concat . intersperse \",\" $ map toTeX xs) ++ \"]\"\n",
    "\n",
    "revTeX = toTeX . reverse\n",
    "\n",
    "newtype LaTeX a = LaTeX a\n",
    "\n",
    "htmlTeX a = html $ \"$\"++toTeX a++\"$\"\n",
    "\n",
    "instance TeX a => IHaskellDisplay (LaTeX a) where\n",
    "  display (LaTeX a) = return $ Display [htmlTeX a]"
   ]
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       "App (App (Lam \"x\" (Lam \"y\" (Var \"x\"))) (Lit 1)) (Lit 2)"
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    "fst_1_2 = App (App (Lam \"x\" $ Lam \"y\" $ Var \"x\") (Lit 1)) (Lit 2)\n",
    "fst_1_2\n",
    "LaTeX fst_1_2"
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      "text/plain": [
       "Just (OpE (App (Lam \"x\" (Lam \"y\" (Var \"x\"))) (Lit 1)),[],[FrE (Lit 2) []])"
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    "step (OpE fst_1_2, [], [])"
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     "data": {
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       "Just (OpE (App (Lam \"x\" (Lam \"y\" (Var \"x\"))) (Lit 1)),[],[FrE (Lit 2) []])"
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       "Just (OpE (Lam \"x\" (Lam \"y\" (Var \"x\"))),[],[FrE (Lit 1) [],FrE (Lit 2) []])"
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     "metadata": {},
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    {
     "data": {
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       "Just (OpV (Clos (Lam \"x\" (Lam \"y\" (Var \"x\"))) []),[],[FrE (Lit 1) [],FrE (Lit 2) []])"
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       "Just (OpE (Lit 1),[],[FrV (Clos (Lam \"x\" (Lam \"y\" (Var \"x\"))) []),FrE (Lit 2) []])"
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     "data": {
      "text/plain": [
       "Just (OpV (Vint 1),[],[FrV (Clos (Lam \"x\" (Lam \"y\" (Var \"x\"))) []),FrE (Lit 2) []])"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
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     "data": {
      "text/plain": [
       "Just (OpE (Lam \"y\" (Var \"x\")),[(\"x\",Vint 1)],[FrE (Lit 2) []])"
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     "data": {
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       "Just (OpV (Clos (Lam \"y\" (Var \"x\")) [(\"x\",Vint 1)]),[(\"x\",Vint 1)],[FrE (Lit 2) []])"
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     "data": {
      "text/plain": [
       "Just (OpE (Lit 2),[],[FrV (Clos (Lam \"y\" (Var \"x\")) [(\"x\",Vint 1)])])"
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     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "data": {
      "text/plain": [
       "Just (OpV (Vint 2),[],[FrV (Clos (Lam \"y\" (Var \"x\")) [(\"x\",Vint 1)])])"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "data": {
      "text/plain": [
       "Just (OpE (Var \"x\"),[(\"y\",Vint 2),(\"x\",Vint 1)],[])"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "data": {
      "text/plain": [
       "Just (OpV (Vint 1),[(\"y\",Vint 2),(\"x\",Vint 1)],[])"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "data": {
      "text/plain": [
       "Nothing"
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     "data": {
      "text/plain": [
       "Nothing"
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     "metadata": {},
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   ],
   "source": [
    "Just(OpE fst_1_2, [], []) :: Maybe CEK\n",
    "case it of Just cek -> step cek\n",
    "           Nothing  -> Nothing\n",
    "case it of Just cek -> step cek\n",
    "           Nothing  -> Nothing\n",
    "case it of Just cek -> step cek\n",
    "           Nothing  -> Nothing\n",
    "case it of Just cek -> step cek\n",
    "           Nothing  -> Nothing\n",
    "case it of Just cek -> step cek\n",
    "           Nothing  -> Nothing\n",
    "case it of Just cek -> step cek\n",
    "           Nothing  -> Nothing\n",
    "case it of Just cek -> step cek\n",
    "           Nothing  -> Nothing\n",
    "case it of Just cek -> step cek\n",
    "           Nothing  -> Nothing\n",
    "case it of Just cek -> step cek\n",
    "           Nothing  -> Nothing\n",
    "case it of Just cek -> step cek\n",
    "           Nothing  -> Nothing\n",
    "case it of Just cek -> step cek\n",
    "           Nothing  -> Nothing\n",
    "case it of Just cek -> step cek\n",
    "           Nothing  -> Nothing\n",
    "case it of Just cek -> step cek\n",
    "           Nothing  -> Nothing"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 10,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "Just (OpE (App (App (Lam \"x\" (Lam \"y\" (Var \"x\"))) (Lit 1)) (Lit 2)),[],[])"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "data": {
      "text/plain": [
       "Just (OpE (App (Lam \"x\" (Lam \"y\" (Var \"x\"))) (Lit 1)),[],[FrE (Lit 2) []])"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "data": {
      "text/plain": [
       "Just (OpE (Lam \"x\" (Lam \"y\" (Var \"x\"))),[],[FrE (Lit 1) [],FrE (Lit 2) []])"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "data": {
      "text/plain": [
       "Just (OpV (Clos (Lam \"x\" (Lam \"y\" (Var \"x\"))) []),[],[FrE (Lit 1) [],FrE (Lit 2) []])"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "data": {
      "text/plain": [
       "Just (OpE (Lit 1),[],[FrV (Clos (Lam \"x\" (Lam \"y\" (Var \"x\"))) []),FrE (Lit 2) []])"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "data": {
      "text/plain": [
       "Just (OpV (Vint 1),[],[FrV (Clos (Lam \"x\" (Lam \"y\" (Var \"x\"))) []),FrE (Lit 2) []])"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "data": {
      "text/plain": [
       "Just (OpE (Lam \"y\" (Var \"x\")),[(\"x\",Vint 1)],[FrE (Lit 2) []])"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "data": {
      "text/plain": [
       "Just (OpV (Clos (Lam \"y\" (Var \"x\")) [(\"x\",Vint 1)]),[(\"x\",Vint 1)],[FrE (Lit 2) []])"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "data": {
      "text/plain": [
       "Just (OpE (Lit 2),[],[FrV (Clos (Lam \"y\" (Var \"x\")) [(\"x\",Vint 1)])])"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "data": {
      "text/plain": [
       "Just (OpV (Vint 2),[],[FrV (Clos (Lam \"y\" (Var \"x\")) [(\"x\",Vint 1)])])"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "data": {
      "text/plain": [
       "Just (OpE (Var \"x\"),[(\"y\",Vint 2),(\"x\",Vint 1)],[])"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "data": {
      "text/plain": [
       "Just (OpV (Vint 1),[(\"y\",Vint 2),(\"x\",Vint 1)],[])"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "data": {
      "text/plain": [
       "Nothing"
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     "data": {
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       "Nothing"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "Just(OpE fst_1_2, [], []) :: Maybe CEK\n",
    "step =<< it\n",
    "step =<< it\n",
    "step =<< it\n",
    "step =<< it\n",
    "step =<< it\n",
    "step =<< it\n",
    "step =<< it\n",
    "step =<< it\n",
    "step =<< it\n",
    "step =<< it\n",
    "step =<< it\n",
    "step =<< it\n",
    "step =<< it"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 11,
   "metadata": {},
   "outputs": [],
   "source": [
    "m0  =  Just(OpE fst_1_2, [], []) :: Maybe CEK\n",
    "m1  =  step =<< m0\n",
    "m2  =  step =<< m1\n",
    "m3  =  step =<< m2\n",
    "m4  =  step =<< m3\n",
    "m5  =  step =<< m4\n",
    "m6  =  step =<< m5\n",
    "m7  =  step =<< m6\n",
    "m8  =  step =<< m7\n",
    "m9  =  step =<< m8\n",
    "m10 =  step =<< m9\n",
    "m11 =  step =<< m10\n",
    "m12 =  step =<< m11\n",
    "m13 =  step =<< m12"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 12,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "Just (OpE (App (App (Lam \"x\" (Lam \"y\" (Var \"x\"))) (Lit 1)) (Lit 2)),[],[])"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "data": {
      "text/plain": [
       "Just (OpE (App (Lam \"x\" (Lam \"y\" (Var \"x\"))) (Lit 1)),[],[FrE (Lit 2) []])"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "data": {
      "text/plain": [
       "Just (OpE (Lam \"x\" (Lam \"y\" (Var \"x\"))),[],[FrE (Lit 1) [],FrE (Lit 2) []])"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "data": {
      "text/plain": [
       "Just (OpV (Clos (Lam \"x\" (Lam \"y\" (Var \"x\"))) []),[],[FrE (Lit 1) [],FrE (Lit 2) []])"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "data": {
      "text/plain": [
       "Just (OpE (Lit 1),[],[FrV (Clos (Lam \"x\" (Lam \"y\" (Var \"x\"))) []),FrE (Lit 2) []])"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "data": {
      "text/plain": [
       "Just (OpV (Vint 1),[],[FrV (Clos (Lam \"x\" (Lam \"y\" (Var \"x\"))) []),FrE (Lit 2) []])"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "data": {
      "text/plain": [
       "Just (OpE (Lam \"y\" (Var \"x\")),[(\"x\",Vint 1)],[FrE (Lit 2) []])"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "data": {
      "text/plain": [
       "Just (OpV (Clos (Lam \"y\" (Var \"x\")) [(\"x\",Vint 1)]),[(\"x\",Vint 1)],[FrE (Lit 2) []])"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "data": {
      "text/plain": [
       "Just (OpE (Lit 2),[],[FrV (Clos (Lam \"y\" (Var \"x\")) [(\"x\",Vint 1)])])"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "data": {
      "text/plain": [
       "Just (OpV (Vint 2),[],[FrV (Clos (Lam \"y\" (Var \"x\")) [(\"x\",Vint 1)])])"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "data": {
      "text/plain": [
       "Just (OpE (Var \"x\"),[(\"y\",Vint 2),(\"x\",Vint 1)],[])"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "data": {
      "text/plain": [
       "Just (OpV (Vint 1),[(\"y\",Vint 2),(\"x\",Vint 1)],[])"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "data": {
      "text/plain": [
       "Nothing"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "data": {
      "text/plain": [
       "Nothing"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "m0\n",
    "m1\n",
    "m2\n",
    "m3\n",
    "m4\n",
    "m5\n",
    "m6\n",
    "m7\n",
    "m8\n",
    "m9\n",
    "m10\n",
    "m11\n",
    "m12\n",
    "m13"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 13,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "Just (OpE (App (App (Lam \"x\" (Lam \"y\" (Var \"x\"))) (Lit 1)) (Lit 2)),[],[])"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "data": {
      "text/plain": [
       "Just (OpE (App (Lam \"x\" (Lam \"y\" (Var \"x\"))) (Lit 1)),[],[FrE (Lit 2) []])"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "data": {
      "text/plain": [
       "Just (OpE (Lam \"x\" (Lam \"y\" (Var \"x\"))),[],[FrE (Lit 1) [],FrE (Lit 2) []])"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "data": {
      "text/plain": [
       "Just (OpV (Clos (Lam \"x\" (Lam \"y\" (Var \"x\"))) []),[],[FrE (Lit 1) [],FrE (Lit 2) []])"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "data": {
      "text/plain": [
       "Just (OpE (Lit 1),[],[FrV (Clos (Lam \"x\" (Lam \"y\" (Var \"x\"))) []),FrE (Lit 2) []])"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "data": {
      "text/plain": [
       "Just (OpV (Vint 1),[],[FrV (Clos (Lam \"x\" (Lam \"y\" (Var \"x\"))) []),FrE (Lit 2) []])"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "data": {
      "text/plain": [
       "Just (OpE (Lam \"y\" (Var \"x\")),[(\"x\",Vint 1)],[FrE (Lit 2) []])"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "data": {
      "text/plain": [
       "Just (OpV (Clos (Lam \"y\" (Var \"x\")) [(\"x\",Vint 1)]),[(\"x\",Vint 1)],[FrE (Lit 2) []])"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "data": {
      "text/plain": [
       "Just (OpE (Lit 2),[],[FrV (Clos (Lam \"y\" (Var \"x\")) [(\"x\",Vint 1)])])"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "data": {
      "text/plain": [
       "Just (OpV (Vint 2),[],[FrV (Clos (Lam \"y\" (Var \"x\")) [(\"x\",Vint 1)])])"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "data": {
      "text/plain": [
       "Just (OpE (Var \"x\"),[(\"y\",Vint 2),(\"x\",Vint 1)],[])"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "data": {
      "text/plain": [
       "Just (OpV (Vint 1),[(\"y\",Vint 2),(\"x\",Vint 1)],[])"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "data": {
      "text/plain": [
       "Nothing"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "data": {
      "text/plain": [
       "Nothing"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "print m0\n",
    "print m1\n",
    "print m2\n",
    "print m3\n",
    "print m4\n",
    "print m5\n",
    "print m6\n",
    "print m7\n",
    "print m8\n",
    "print m9\n",
    "print m10\n",
    "print m11\n",
    "print m12\n",
    "print m13"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 14,
   "metadata": {},
   "outputs": [
    {
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       ".suggestion-name {\n",
       "font-weight: bold;\n",
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       "</style>$\\texttt{Just}(\\big\\langle{}(((\\lambda{}x.(\\lambda{}y.x))\\;1)\\;2)\\;\\big|\\;[]\\;\\big|\\;[]\\big\\rangle{})$"
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       ".err-msg.in.collapse {\n",
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       "</style>$\\texttt{Just}(\\big\\langle{}(\\lambda{}x.(\\lambda{}y.x))\\;\\big|\\;[]\\;\\big|\\;[(\\bigcirc{}\\;1\\;[]),(\\bigcirc{}\\;2\\;[])]\\big\\rangle{})$"
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       "}\n",
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       "}\n",
       ".mono {\n",
       "font-family: monospace;\n",
       "display: block;\n",
       "}\n",
       ".err-msg {\n",
       "color: red;\n",
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       "}\n",
       "#unshowable {\n",
       "color: red;\n",
       "font-weight: bold;\n",
       "}\n",
       ".err-msg.in.collapse {\n",
       "padding-top: 0.7em;\n",
       "}\n",
       ".highlight-code {\n",
       "white-space: pre;\n",
       "font-family: monospace;\n",
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       "Just (OpE (App (App (Lam \"x\" (Lam \"y\" (Var \"x\"))) (Lit 1)) (Lit 2)),[],[])\n",
       "Just (OpE (App (Lam \"x\" (Lam \"y\" (Var \"x\"))) (Lit 1)),[],[FrE (Lit 2) []])\n",
       "Just (OpE (Lam \"x\" (Lam \"y\" (Var \"x\"))),[],[FrE (Lit 1) [],FrE (Lit 2) []])\n",
       "Just (OpV (Clos (Lam \"x\" (Lam \"y\" (Var \"x\"))) []),[],[FrE (Lit 1) [],FrE (Lit 2) []])\n",
       "Just (OpE (Lit 1),[],[FrV (Clos (Lam \"x\" (Lam \"y\" (Var \"x\"))) []),FrE (Lit 2) []])\n",
       "Just (OpV (Vint 1),[],[FrV (Clos (Lam \"x\" (Lam \"y\" (Var \"x\"))) []),FrE (Lit 2) []])\n",
       "Just (OpE (Lam \"y\" (Var \"x\")),[(\"x\",Vint 1)],[FrE (Lit 2) []])\n",
       "Just (OpV (Clos (Lam \"y\" (Var \"x\")) [(\"x\",Vint 1)]),[(\"x\",Vint 1)],[FrE (Lit 2) []])\n",
       "Just (OpE (Lit 2),[],[FrV (Clos (Lam \"y\" (Var \"x\")) [(\"x\",Vint 1)])])\n",
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       "Just (OpE (Var \"x\"),[(\"y\",Vint 2),(\"x\",Vint 1)],[])\n",
       "Just (OpV (Vint 1),[(\"y\",Vint 2),(\"x\",Vint 1)],[])\n",
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    ":type iterate\n",
    ":type \\m -> step =<< m\n",
    ":type (step =<<)\n",
    ":type iterate (step =<<)\n",
    ":type iterate (step =<<) m0"
   ]
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   "metadata": {},
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    {
     "data": {
      "text/plain": [
       "[Just (OpE (App (App (Lam \"x\" (Lam \"y\" (Var \"x\"))) (Lit 1)) (Lit 2)),[],[]),Just (OpE (App (Lam \"x\" (Lam \"y\" (Var \"x\"))) (Lit 1)),[],[FrE (Lit 2) []]),Just (OpE (Lam \"x\" (Lam \"y\" (Var \"x\"))),[],[FrE (Lit 1) [],FrE (Lit 2) []]),Just (OpV (Clos (Lam \"x\" (Lam \"y\" (Var \"x\"))) []),[],[FrE (Lit 1) [],FrE (Lit 2) []]),Just (OpE (Lit 1),[],[FrV (Clos (Lam \"x\" (Lam \"y\" (Var \"x\"))) []),FrE (Lit 2) []]),Just (OpV (Vint 1),[],[FrV (Clos (Lam \"x\" (Lam \"y\" (Var \"x\"))) []),FrE (Lit 2) []]),Just (OpE (Lam \"y\" (Var \"x\")),[(\"x\",Vint 1)],[FrE (Lit 2) []]),Just (OpV (Clos (Lam \"y\" (Var \"x\")) [(\"x\",Vint 1)]),[(\"x\",Vint 1)],[FrE (Lit 2) []]),Just (OpE (Lit 2),[],[FrV (Clos (Lam \"y\" (Var \"x\")) [(\"x\",Vint 1)])]),Just (OpV (Vint 2),[],[FrV (Clos (Lam \"y\" (Var \"x\")) [(\"x\",Vint 1)])]),Just (OpE (Var \"x\"),[(\"y\",Vint 2),(\"x\",Vint 1)],[]),Just (OpV (Vint 1),[(\"y\",Vint 2),(\"x\",Vint 1)],[]),Nothing,Nothing]"
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   ],
   "source": [
    "take 14 $ iterate (step =<<) m0"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 19,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "Just (OpE (App (App (Lam \"x\" (Lam \"y\" (Var \"x\"))) (Lit 1)) (Lit 2)),[],[])\n",
       "Just (OpE (App (Lam \"x\" (Lam \"y\" (Var \"x\"))) (Lit 1)),[],[FrE (Lit 2) []])\n",
       "Just (OpE (Lam \"x\" (Lam \"y\" (Var \"x\"))),[],[FrE (Lit 1) [],FrE (Lit 2) []])\n",
       "Just (OpV (Clos (Lam \"x\" (Lam \"y\" (Var \"x\"))) []),[],[FrE (Lit 1) [],FrE (Lit 2) []])\n",
       "Just (OpE (Lit 1),[],[FrV (Clos (Lam \"x\" (Lam \"y\" (Var \"x\"))) []),FrE (Lit 2) []])\n",
       "Just (OpV (Vint 1),[],[FrV (Clos (Lam \"x\" (Lam \"y\" (Var \"x\"))) []),FrE (Lit 2) []])\n",
       "Just (OpE (Lam \"y\" (Var \"x\")),[(\"x\",Vint 1)],[FrE (Lit 2) []])\n",
       "Just (OpV (Clos (Lam \"y\" (Var \"x\")) [(\"x\",Vint 1)]),[(\"x\",Vint 1)],[FrE (Lit 2) []])\n",
       "Just (OpE (Lit 2),[],[FrV (Clos (Lam \"y\" (Var \"x\")) [(\"x\",Vint 1)])])\n",
       "Just (OpV (Vint 2),[],[FrV (Clos (Lam \"y\" (Var \"x\")) [(\"x\",Vint 1)])])\n",
       "Just (OpE (Var \"x\"),[(\"y\",Vint 2),(\"x\",Vint 1)],[])\n",
       "Just (OpV (Vint 1),[(\"y\",Vint 2),(\"x\",Vint 1)],[])\n",
       "Nothing\n",
       "Nothing"
      ]
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   "source": [
    "mapM_ print . take 14 $ iterate (step =<<) m0"
   ]
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  {
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   "execution_count": 20,
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   "source": [
    "map LaTeX . take 14 $ iterate (step =<<) m0"
   ]
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  {
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   "source": [
    "isJust (Just _) = True\n",
    "isJust Nothing  = False"
   ]
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    {
     "data": {
      "text/plain": [
       "[Just (OpE (App (App (Lam \"x\" (Lam \"y\" (Var \"x\"))) (Lit 1)) (Lit 2)),[],[]),Just (OpE (App (Lam \"x\" (Lam \"y\" (Var \"x\"))) (Lit 1)),[],[FrE (Lit 2) []]),Just (OpE (Lam \"x\" (Lam \"y\" (Var \"x\"))),[],[FrE (Lit 1) [],FrE (Lit 2) []]),Just (OpV (Clos (Lam \"x\" (Lam \"y\" (Var \"x\"))) []),[],[FrE (Lit 1) [],FrE (Lit 2) []]),Just (OpE (Lit 1),[],[FrV (Clos (Lam \"x\" (Lam \"y\" (Var \"x\"))) []),FrE (Lit 2) []]),Just (OpV (Vint 1),[],[FrV (Clos (Lam \"x\" (Lam \"y\" (Var \"x\"))) []),FrE (Lit 2) []]),Just (OpE (Lam \"y\" (Var \"x\")),[(\"x\",Vint 1)],[FrE (Lit 2) []]),Just (OpV (Clos (Lam \"y\" (Var \"x\")) [(\"x\",Vint 1)]),[(\"x\",Vint 1)],[FrE (Lit 2) []]),Just (OpE (Lit 2),[],[FrV (Clos (Lam \"y\" (Var \"x\")) [(\"x\",Vint 1)])]),Just (OpV (Vint 2),[],[FrV (Clos (Lam \"y\" (Var \"x\")) [(\"x\",Vint 1)])]),Just (OpE (Var \"x\"),[(\"y\",Vint 2),(\"x\",Vint 1)],[]),Just (OpV (Vint 1),[(\"y\",Vint 2),(\"x\",Vint 1)],[])]"
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   "source": [
    "takeWhile isJust $ iterate (step =<<) m0"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 23,
   "metadata": {},
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    {
     "data": {
      "text/plain": [
       "Just (OpE (App (App (Lam \"x\" (Lam \"y\" (Var \"x\"))) (Lit 1)) (Lit 2)),[],[])\n",
       "Just (OpE (App (Lam \"x\" (Lam \"y\" (Var \"x\"))) (Lit 1)),[],[FrE (Lit 2) []])\n",
       "Just (OpE (Lam \"x\" (Lam \"y\" (Var \"x\"))),[],[FrE (Lit 1) [],FrE (Lit 2) []])\n",
       "Just (OpV (Clos (Lam \"x\" (Lam \"y\" (Var \"x\"))) []),[],[FrE (Lit 1) [],FrE (Lit 2) []])\n",
       "Just (OpE (Lit 1),[],[FrV (Clos (Lam \"x\" (Lam \"y\" (Var \"x\"))) []),FrE (Lit 2) []])\n",
       "Just (OpV (Vint 1),[],[FrV (Clos (Lam \"x\" (Lam \"y\" (Var \"x\"))) []),FrE (Lit 2) []])\n",
       "Just (OpE (Lam \"y\" (Var \"x\")),[(\"x\",Vint 1)],[FrE (Lit 2) []])\n",
       "Just (OpV (Clos (Lam \"y\" (Var \"x\")) [(\"x\",Vint 1)]),[(\"x\",Vint 1)],[FrE (Lit 2) []])\n",
       "Just (OpE (Lit 2),[],[FrV (Clos (Lam \"y\" (Var \"x\")) [(\"x\",Vint 1)])])\n",
       "Just (OpV (Vint 2),[],[FrV (Clos (Lam \"y\" (Var \"x\")) [(\"x\",Vint 1)])])\n",
       "Just (OpE (Var \"x\"),[(\"y\",Vint 2),(\"x\",Vint 1)],[])\n",
       "Just (OpV (Vint 1),[(\"y\",Vint 2),(\"x\",Vint 1)],[])"
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   "source": [
    "mapM_ print $ takeWhile isJust $ iterate (step =<<) m0"
   ]
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   "source": [
    "map LaTeX . takeWhile isJust $ iterate (step =<<) m0"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 25,
   "metadata": {},
   "outputs": [],
   "source": [
    "run cek = case step cek of Just cek' -> run cek'\n",
    "                           Nothing   -> cek\n",
    "\n",
    "runExp e = run (OpE e, [], [])"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 26,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "(OpV (Vint 1),[(\"y\",Vint 2),(\"x\",Vint 1)],[])"
      ]
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     "metadata": {},
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       ".mono {\n",
       "font-family: monospace;\n",
       "display: block;\n",
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       ".err-msg {\n",
       "color: red;\n",
       "font-style: italic;\n",
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       ".highlight-code {\n",
       "white-space: pre;\n",
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       ".suggestion-warning { \n",
       "font-weight: bold;\n",
       "color: rgb(200, 130, 0);\n",
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       "</style>$\\big\\langle{}1\\;\\big|\\;[x\\mapsto{}1,y\\mapsto{}2]\\;\\big|\\;[]\\big\\rangle{}$"
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    }
   ],
   "source": [
    "runExp fst_1_2\n",
    "display (LaTeX it)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": []
  }
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