LamCalc.ipynb

{
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       "<style>/* Styles used for the Hoogle display in the pager */\n",
       ".hoogle-doc {\n",
       "display: block;\n",
       "padding-bottom: 1.3em;\n",
       "padding-left: 0.4em;\n",
       "}\n",
       ".hoogle-code {\n",
       "display: block;\n",
       "font-family: monospace;\n",
       "white-space: pre;\n",
       "}\n",
       ".hoogle-text {\n",
       "display: block;\n",
       "}\n",
       ".hoogle-name {\n",
       "color: green;\n",
       "font-weight: bold;\n",
       "}\n",
       ".hoogle-head {\n",
       "font-weight: bold;\n",
       "}\n",
       ".hoogle-sub {\n",
       "display: block;\n",
       "margin-left: 0.4em;\n",
       "}\n",
       ".hoogle-package {\n",
       "font-weight: bold;\n",
       "font-style: italic;\n",
       "}\n",
       ".hoogle-module {\n",
       "font-weight: bold;\n",
       "}\n",
       ".hoogle-class {\n",
       "font-weight: bold;\n",
       "}\n",
       ".get-type {\n",
       "color: green;\n",
       "font-weight: bold;\n",
       "font-family: monospace;\n",
       "display: block;\n",
       "white-space: pre-wrap;\n",
       "}\n",
       ".show-type {\n",
       "color: green;\n",
       "font-weight: bold;\n",
       "font-family: monospace;\n",
       "margin-left: 1em;\n",
       "}\n",
       ".mono {\n",
       "font-family: monospace;\n",
       "display: block;\n",
       "}\n",
       ".err-msg {\n",
       "color: red;\n",
       "font-style: italic;\n",
       "font-family: monospace;\n",
       "white-space: pre;\n",
       "display: block;\n",
       "}\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",
       "}\n",
       ".suggestion-warning { \n",
       "font-weight: bold;\n",
       "color: rgb(200, 130, 0);\n",
       "}\n",
       ".suggestion-error { \n",
       "font-weight: bold;\n",
       "color: red;\n",
       "}\n",
       ".suggestion-name {\n",
       "font-weight: bold;\n",
       "}\n",
       "</style><div class=\"suggestion-name\" style=\"clear:both;\">Unused LANGUAGE pragma</div><div class=\"suggestion-row\" style=\"float: left;\"><div class=\"suggestion-warning\">Found:</div><div class=\"highlight-code\" id=\"haskell\">{-# LANGUAGE DeriveDataTypeable, DeriveGeneric, FlexibleInstances,\n",
       "  FlexibleContexts, MultiParamTypeClasses #-}</div></div><div class=\"suggestion-row\" style=\"float: left;\"><div class=\"suggestion-warning\">Why Not:</div><div class=\"highlight-code\" id=\"haskell\">{-# LANGUAGE FlexibleInstances, FlexibleContexts,\n",
       "  MultiParamTypeClasses #-}</div></div>"
      ],
      "text/plain": [
       "Line 1: Unused LANGUAGE pragma\n",
       "Found:\n",
       "{-# LANGUAGE DeriveDataTypeable, DeriveGeneric, FlexibleInstances,\n",
       "  FlexibleContexts, MultiParamTypeClasses #-}\n",
       "Why not:\n",
       "{-# LANGUAGE FlexibleInstances, FlexibleContexts,\n",
       "  MultiParamTypeClasses #-}"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "{-# LANGUAGE DeriveDataTypeable\n",
    "           , DeriveGeneric\n",
    "           , FlexibleInstances\n",
    "           , FlexibleContexts\n",
    "           , MultiParamTypeClasses #-}\n",
    "\n",
    "import Unbound.Generics.LocallyNameless\n",
    "import Unbound.Generics.LocallyNameless.Internal.Fold\n",
    "\n",
    "import GHC.Generics (Generic)\n",
    "import Data.Typeable (Typeable)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {},
   "outputs": [],
   "source": [
    "-- | Names for expressions\n",
    "type Nm = Name Exp\n",
    "\n",
    "\n",
    "-- | Expressions\n",
    "data Exp = Var Nm            -- ^ variables\n",
    "         | Lam (Bind Nm Exp) -- ^ lambdas bind a variable within a body expression\n",
    "         | App Exp Exp       -- ^ application\n",
    "          deriving (Show, Generic, Typeable)\n",
    "\n",
    "-- Automatically construct alpha equiv, free vars, and binding operations.\n",
    "instance Alpha Exp\n",
    "\n",
    "-- semi-automatically implement capture avoiding substitution\n",
    "instance Subst Exp Exp where\n",
    "  -- `isvar` identifies the variable case in your AST.\n",
    "  isvar (Var x) = Just (SubstName x)\n",
    "  isvar _       = Nothing"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {},
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       "<style>/* Styles used for the Hoogle display in the pager */\n",
       ".hoogle-doc {\n",
       "display: block;\n",
       "padding-bottom: 1.3em;\n",
       "padding-left: 0.4em;\n",
       "}\n",
       ".hoogle-code {\n",
       "display: block;\n",
       "font-family: monospace;\n",
       "white-space: pre;\n",
       "}\n",
       ".hoogle-text {\n",
       "display: block;\n",
       "}\n",
       ".hoogle-name {\n",
       "color: green;\n",
       "font-weight: bold;\n",
       "}\n",
       ".hoogle-head {\n",
       "font-weight: bold;\n",
       "}\n",
       ".hoogle-sub {\n",
       "display: block;\n",
       "margin-left: 0.4em;\n",
       "}\n",
       ".hoogle-package {\n",
       "font-weight: bold;\n",
       "font-style: italic;\n",
       "}\n",
       ".hoogle-module {\n",
       "font-weight: bold;\n",
       "}\n",
       ".hoogle-class {\n",
       "font-weight: bold;\n",
       "}\n",
       ".get-type {\n",
       "color: green;\n",
       "font-weight: bold;\n",
       "font-family: monospace;\n",
       "display: block;\n",
       "white-space: pre-wrap;\n",
       "}\n",
       ".show-type {\n",
       "color: green;\n",
       "font-weight: bold;\n",
       "font-family: monospace;\n",
       "margin-left: 1em;\n",
       "}\n",
       ".mono {\n",
       "font-family: monospace;\n",
       "display: block;\n",
       "}\n",
       ".err-msg {\n",
       "color: red;\n",
       "font-style: italic;\n",
       "font-family: monospace;\n",
       "white-space: pre;\n",
       "display: block;\n",
       "}\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",
       "}\n",
       ".suggestion-warning { \n",
       "font-weight: bold;\n",
       "color: rgb(200, 130, 0);\n",
       "}\n",
       ".suggestion-error { \n",
       "font-weight: bold;\n",
       "color: red;\n",
       "}\n",
       ".suggestion-name {\n",
       "font-weight: bold;\n",
       "}\n",
       "</style><span class='get-type'>unsafeUnbind :: forall p t. (Alpha p, Alpha t) => Bind p t -> (p, t)</span>"
      ],
      "text/plain": [
       "unsafeUnbind :: forall p t. (Alpha p, Alpha t) => Bind p t -> (p, t)"
      ]
     },
     "metadata": {},
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    }
   ],
   "source": [
    "import Unbound.Generics.LocallyNameless.Unsafe\n",
    ":type unsafeUnbind"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "<style>/* Styles used for the Hoogle display in the pager */\n",
       ".hoogle-doc {\n",
       "display: block;\n",
       "padding-bottom: 1.3em;\n",
       "padding-left: 0.4em;\n",
       "}\n",
       ".hoogle-code {\n",
       "display: block;\n",
       "font-family: monospace;\n",
       "white-space: pre;\n",
       "}\n",
       ".hoogle-text {\n",
       "display: block;\n",
       "}\n",
       ".hoogle-name {\n",
       "color: green;\n",
       "font-weight: bold;\n",
       "}\n",
       ".hoogle-head {\n",
       "font-weight: bold;\n",
       "}\n",
       ".hoogle-sub {\n",
       "display: block;\n",
       "margin-left: 0.4em;\n",
       "}\n",
       ".hoogle-package {\n",
       "font-weight: bold;\n",
       "font-style: italic;\n",
       "}\n",
       ".hoogle-module {\n",
       "font-weight: bold;\n",
       "}\n",
       ".hoogle-class {\n",
       "font-weight: bold;\n",
       "}\n",
       ".get-type {\n",
       "color: green;\n",
       "font-weight: bold;\n",
       "font-family: monospace;\n",
       "display: block;\n",
       "white-space: pre-wrap;\n",
       "}\n",
       ".show-type {\n",
       "color: green;\n",
       "font-weight: bold;\n",
       "font-family: monospace;\n",
       "margin-left: 1em;\n",
       "}\n",
       ".mono {\n",
       "font-family: monospace;\n",
       "display: block;\n",
       "}\n",
       ".err-msg {\n",
       "color: red;\n",
       "font-style: italic;\n",
       "font-family: monospace;\n",
       "white-space: pre;\n",
       "display: block;\n",
       "}\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",
       "}\n",
       ".suggestion-warning { \n",
       "font-weight: bold;\n",
       "color: rgb(200, 130, 0);\n",
       "}\n",
       ".suggestion-error { \n",
       "font-weight: bold;\n",
       "color: red;\n",
       "}\n",
       ".suggestion-name {\n",
       "font-weight: bold;\n",
       "}\n",
       "</style><div class=\"suggestion-name\" style=\"clear:both;\">Redundant bracket</div><div class=\"suggestion-row\" style=\"float: left;\"><div class=\"suggestion-warning\">Found:</div><div class=\"highlight-code\" id=\"haskell\">(Lam{})</div></div><div class=\"suggestion-row\" style=\"float: left;\"><div class=\"suggestion-warning\">Why Not:</div><div class=\"highlight-code\" id=\"haskell\">Lam{}</div></div><div class=\"suggestion-name\" style=\"clear:both;\">Redundant bracket</div><div class=\"suggestion-row\" style=\"float: left;\"><div class=\"suggestion-warning\">Found:</div><div class=\"highlight-code\" id=\"haskell\">(Var{})</div></div><div class=\"suggestion-row\" style=\"float: left;\"><div class=\"suggestion-warning\">Why Not:</div><div class=\"highlight-code\" id=\"haskell\">Var{}</div></div>"
      ],
      "text/plain": [
       "Line 13: Redundant bracket\n",
       "Found:\n",
       "(Lam{})\n",
       "Why not:\n",
       "Lam{}Line 15: Redundant bracket\n",
       "Found:\n",
       "(Var{})\n",
       "Why not:\n",
       "Var{}"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "import IHaskell.Display\n",
    "\n",
    "showExp = ppExp \"λ\" \".\"\n",
    "printExp = putStrLn . showExp\n",
    "dispExp e = html $ \"<code>\"++ppExp \"λ\" \".\" e++\"</code>\"\n",
    "displayExp = display . dispExp\n",
    "\n",
    "ppExp _ _ (Var x) = show x\n",
    "ppExp l d (Lam bnd) = l ++ show x ++ d ++ ppExp l d t\n",
    "  where (x, t) = unsafeUnbind bnd\n",
    "ppExp l d (App t s) = ppt t ++ \" \" ++ pps s\n",
    "  where\n",
    "  ppt t@(Lam{}) = paren (ppExp l d t)\n",
    "  ppt t         = ppExp l d t\n",
    "  pps s@(Var{}) = ppExp l d s\n",
    "  pps s         = paren (ppExp l d s)\n",
    "\n",
    "paren s = \"(\"++s++\")\""
   ]
  },
  {
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   "execution_count": 5,
   "metadata": {},
   "outputs": [
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       "<style>/* Styles used for the Hoogle display in the pager */\n",
       ".hoogle-doc {\n",
       "display: block;\n",
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       "padding-left: 0.4em;\n",
       "}\n",
       ".hoogle-code {\n",
       "display: block;\n",
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       "font-weight: bold;\n",
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       "}\n",
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       "}\n",
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       ".get-type {\n",
       "color: green;\n",
       "font-weight: bold;\n",
       "font-family: monospace;\n",
       "display: block;\n",
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       "}\n",
       ".show-type {\n",
       "color: green;\n",
       "font-weight: bold;\n",
       "font-family: monospace;\n",
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       ".mono {\n",
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       "color: red;\n",
       "font-style: italic;\n",
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       "</style><span class='get-type'>printExp :: Exp -> IO ()</span>"
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       "printExp :: Exp -> IO ()"
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       "display: block;\n",
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       "padding-left: 0.4em;\n",
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       ".hoogle-module {\n",
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       ".get-type {\n",
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       "margin-left: 1em;\n",
       "}\n",
       ".mono {\n",
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       ".err-msg {\n",
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       ".suggestion-warning { \n",
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       "}\n",
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       "}\n",
       "</style><span class='get-type'>dispExp :: Exp -> DisplayData</span>"
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       "}\n",
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       "display: block;\n",
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       "}\n",
       ".hoogle-package {\n",
       "font-weight: bold;\n",
       "font-style: italic;\n",
       "}\n",
       ".hoogle-module {\n",
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       ".hoogle-class {\n",
       "font-weight: bold;\n",
       "}\n",
       ".get-type {\n",
       "color: green;\n",
       "font-weight: bold;\n",
       "font-family: monospace;\n",
       "display: block;\n",
       "white-space: pre-wrap;\n",
       "}\n",
       ".show-type {\n",
       "color: green;\n",
       "font-weight: bold;\n",
       "font-family: monospace;\n",
       "margin-left: 1em;\n",
       "}\n",
       ".mono {\n",
       "font-family: monospace;\n",
       "display: block;\n",
       "}\n",
       ".err-msg {\n",
       "color: red;\n",
       "font-style: italic;\n",
       "font-family: monospace;\n",
       "white-space: pre;\n",
       "display: block;\n",
       "}\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",
       "}\n",
       ".suggestion-warning { \n",
       "font-weight: bold;\n",
       "color: rgb(200, 130, 0);\n",
       "}\n",
       ".suggestion-error { \n",
       "font-weight: bold;\n",
       "color: red;\n",
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       "}\n",
       "</style><span class='get-type'>displayExp :: Exp -> IO Display</span>"
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       "displayExp :: Exp -> IO Display"
      ]
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     "metadata": {},
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   "source": [
    ":type printExp\n",
    ":type dispExp\n",
    ":type displayExp"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "metadata": {},
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       ".hoogle-doc {\n",
       "display: block;\n",
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       "}\n",
       ".hoogle-code {\n",
       "display: block;\n",
       "font-family: monospace;\n",
       "white-space: pre;\n",
       "}\n",
       ".hoogle-text {\n",
       "display: block;\n",
       "}\n",
       ".hoogle-name {\n",
       "color: green;\n",
       "font-weight: bold;\n",
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       ".hoogle-head {\n",
       "font-weight: bold;\n",
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       ".hoogle-sub {\n",
       "display: block;\n",
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       ".hoogle-package {\n",
       "font-weight: bold;\n",
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       ".hoogle-module {\n",
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       ".hoogle-class {\n",
       "font-weight: bold;\n",
       "}\n",
       ".get-type {\n",
       "color: green;\n",
       "font-weight: bold;\n",
       "font-family: monospace;\n",
       "display: block;\n",
       "white-space: pre-wrap;\n",
       "}\n",
       ".show-type {\n",
       "color: green;\n",
       "font-weight: bold;\n",
       "font-family: monospace;\n",
       "margin-left: 1em;\n",
       "}\n",
       ".mono {\n",
       "font-family: monospace;\n",
       "display: block;\n",
       "}\n",
       ".err-msg {\n",
       "color: red;\n",
       "font-style: italic;\n",
       "font-family: monospace;\n",
       "white-space: pre;\n",
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       "#unshowable {\n",
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       ".highlight-code {\n",
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       "color: rgb(200, 130, 0);\n",
       "}\n",
       ".suggestion-error { \n",
       "font-weight: bold;\n",
       "color: red;\n",
       "}\n",
       ".suggestion-name {\n",
       "font-weight: bold;\n",
       "}\n",
       "</style><span class='get-type'>eval :: forall (m :: * -> *). Fresh m => Exp -> m Exp</span>"
      ],
      "text/plain": [
       "eval :: forall (m :: * -> *). Fresh m => Exp -> m Exp"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "-- call-by-value evaluation\n",
    "-- evaluation takes an expression and returns a value while using a source of fresh names\n",
    "eval (Var x)     = fail $ \"unbound variable \" ++ show x\n",
    "eval e@(Lam _)  = return e\n",
    "eval (App e1 e2) = do\n",
    "  v1 <- eval e1\n",
    "  v2 <- eval e2\n",
    "  case v1 of\n",
    "   (Lam bnd) -> do\n",
    "     -- open the lambda by picking a fresh name for the bound variable x in body\n",
    "     (x, body) <- unbind bnd\n",
    "     eval (subst x v2 body)\n",
    "   _ -> fail \"application of non-lambda\"\n",
    "   \n",
    ":type eval"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "metadata": {},
   "outputs": [],
   "source": [
    "x = s2n \"x\"\n",
    "y = s2n \"y\"\n",
    "z = s2n \"z\"\n",
    "e = Lam $ bind x (Lam $ bind y (App (Var y) (Var x)))"
   ]
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  {
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   "execution_count": 8,
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    "lam x = Lam . bind x\n",
    ":type lam"
   ]
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       "</style><code>λx.x</code>"
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       "True"
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    "lam x (Var x)\n",
    "lam y (Var y)\n",
    "\n",
    "dispExp $ lam x (Var x)\n",
    "dispExp $ lam y (Var y)\n",
    "\n",
    "lam x (Var x) `aeq` lam y (Var y)"
   ]
  },
  {
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   "execution_count": 10,
   "metadata": {},
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    {
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   "source": [
    "e\n",
    "dispExp e"
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    "dispExp (App (App e e) e)\n",
    "dispExp $ runFreshM $ eval (App (App e e) e)"
   ]
  },
  {
   "cell_type": "code",
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   "metadata": {},
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   "source": [
    "tmId = lam x (Var x)\n",
    "tm1 = App tmId tmId `App` App tmId tmId\n",
    "tm2 = lam y $ App tmId tmId `App` App tmId tmId"
   ]
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       "display: block;\n",
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    "dispExp tmId\n",
    "dispExp $ runFreshM $ eval tmId\n",
    "dispExp tm1\n",
    "dispExp $ runFreshM $ eval tm1\n",
    "dispExp tm2\n",
    "dispExp $ runFreshM $ eval tm2"
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  {
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   "execution_count": 14,
   "metadata": {},
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     },
     "metadata": {},
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    }
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   "source": [
    ":type fv\n",
    ":type toListOf\n",
    "fv' e = toListOf fv e :: [Nm]\n",
    ":type fv'"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 15,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "[]"
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     },
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     "output_type": "display_data"
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    {
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    {
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       "[x,y]"
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     },
     "metadata": {},
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    {
     "data": {
      "text/plain": [
       "[z]"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "fv' tmId\n",
    "fv' (Var x)\n",
    "fv' (Var x `App` Var y)\n",
    "fv' (lam x . lam y $ (Var x `App` Var y) `App` Var z)"
   ]
  },
  {
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   "execution_count": 16,
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       "<style>/* Styles used for the Hoogle display in the pager */\n",
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       ".err-msg {\n",
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       ".suggestion-warning { \n",
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       "color: rgb(200, 130, 0);\n",
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       "color: red;\n",
       "}\n",
       ".suggestion-name {\n",
       "font-weight: bold;\n",
       "}\n",
       "</style><span class='get-type'>red :: forall (m :: * -> *). Fresh m => Exp -> m Exp</span>"
      ],
      "text/plain": [
       "red :: forall (m :: * -> *). Fresh m => Exp -> m Exp"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "-- reduce all the way (may diverge)\n",
    "red (App e1 e2) = do\n",
    "  e1' <- red e1\n",
    "  e2' <- red e2\n",
    "  case e1' of\n",
    "    Lam bnd -> do\n",
    "        (x, e1'') <- unbind bnd\n",
    "        return $ subst x e2' e1''\n",
    "    _ -> return $ App e1' e2'\n",
    "red (Lam bnd) = do\n",
    "   (x, e) <- unbind bnd\n",
    "   e' <- red e\n",
    "   case e of\n",
    "     App e1 (Var y) | y == x && x `notElem` fv' e1 -> return e1\n",
    "     _ -> return $ lam x e'\n",
    "red (Var x) = return $ Var x\n",
    "\n",
    ":type red"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 17,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "Lam (<y> App (App (Lam (<x> Var 0@0)) (Lam (<x> Var 0@0))) (App (Lam (<x> Var 0@0)) (Lam (<x> Var 0@0))))"
      ]
     },
     "metadata": {},
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       "<style>/* Styles used for the Hoogle display in the pager */\n",
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       "}\n",
       ".hoogle-code {\n",
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       ".hoogle-name {\n",
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       ".hoogle-head {\n",
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       "color: red;\n",
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       ".suggestion-name {\n",
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       "</style><code>λy.(λx.x) (λx.x) ((λx.x) (λx.x))</code>"
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       "<style>/* Styles used for the Hoogle display in the pager */\n",
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       "font-weight: bold;\n",
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       ".show-type {\n",
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       "}\n",
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       "font-family: monospace;\n",
       "display: block;\n",
       "}\n",
       ".err-msg {\n",
       "color: red;\n",
       "font-style: italic;\n",
       "font-family: monospace;\n",
       "white-space: pre;\n",
       "display: block;\n",
       "}\n",
       "#unshowable {\n",
       "color: red;\n",
       "font-weight: bold;\n",
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       "padding-top: 0.7em;\n",
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       "color: red;\n",
       "}\n",
       ".suggestion-name {\n",
       "font-weight: bold;\n",
       "}\n",
       "</style><code>λy.λx5.x5</code>"
      ]
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     "metadata": {},
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    }
   ],
   "source": [
    "tm2\n",
    "dispExp $ runFreshM $ eval tm2\n",
    "dispExp $ runFreshM $ red tm2"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 18,
   "metadata": {},
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    {
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       "}\n",
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       "</style><span class='get-type'>step :: forall (f :: * -> *). (Alternative f, Fresh f) => Exp -> f Exp</span>"
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       ".hoogle-sub {\n",
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       "}\n",
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       "font-family: monospace;\n",
       "display: block;\n",
       "}\n",
       ".err-msg {\n",
       "color: red;\n",
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       "color: red;\n",
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       "font-weight: bold;\n",
       "color: rgb(200, 130, 0);\n",
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       "color: red;\n",
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       "</style><span class='get-type'>beta :: forall (m :: * -> *). (Fresh m, Alternative m) => Exp -> m Exp</span>"
      ],
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       "beta :: forall (m :: * -> *). (Fresh m, Alternative m) => Exp -> m Exp"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "import Control.Applicative \n",
    "\n",
    "-- single step reduction\n",
    "step e@(App e1 e2) = beta e\n",
    "                 <|> (App <$> step e1 <*> pure e2)\n",
    "                 <|> (App <$> pure e1 <*> step e2)\n",
    "step (Lam bnd) = do (x, e) <- unbind bnd\n",
    "                    lam x <$> step e\n",
    "step (Var x) = empty\n",
    "\n",
    "\n",
    "beta (App (Lam bnd) e2) = do (x, e1) <- unbind bnd\n",
    "                             return $ subst x e2 e1\n",
    "beta _                  = empty\n",
    "\n",
    "\n",
    ":type step\n",
    ":type beta"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 19,
   "metadata": {},
   "outputs": [
    {
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       "white-space: pre-wrap;\n",
       "}\n",
       ".show-type {\n",
       "color: green;\n",
       "font-weight: bold;\n",
       "font-family: monospace;\n",
       "margin-left: 1em;\n",
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       "font-family: monospace;\n",
       "display: block;\n",
       "}\n",
       ".err-msg {\n",
       "color: red;\n",
       "font-style: italic;\n",
       "font-family: monospace;\n",
       "white-space: pre;\n",
       "display: block;\n",
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       "#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",
       "}\n",
       ".suggestion-warning { \n",
       "font-weight: bold;\n",
       "color: rgb(200, 130, 0);\n",
       "}\n",
       ".suggestion-error { \n",
       "font-weight: bold;\n",
       "color: red;\n",
       "}\n",
       ".suggestion-name {\n",
       "font-weight: bold;\n",
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       "<style>/* Styles used for the Hoogle display in the pager */\n",
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       ".err-msg {\n",
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      ]
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     "metadata": {},
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    }
   ],
   "source": [
    "omega = lam x $ Var x `App` Var x \n",
    "tmOmega = App omega omega -- well known diverging term\n",
    "\n",
    "dispExp omega\n",
    "dispExp tmOmega"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 20,
   "metadata": {},
   "outputs": [
    {
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       "}\n",
       ".hoogle-code {\n",
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       ".hoogle-text {\n",
       "display: block;\n",
       "}\n",
       ".hoogle-name {\n",
       "color: green;\n",
       "font-weight: bold;\n",
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       ".hoogle-head {\n",
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       ".hoogle-sub {\n",
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       ".hoogle-package {\n",
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       "margin-left: 1em;\n",
       "}\n",
       ".mono {\n",
       "font-family: monospace;\n",
       "display: block;\n",
       "}\n",
       ".err-msg {\n",
       "color: red;\n",
       "font-style: italic;\n",
       "font-family: monospace;\n",
       "white-space: pre;\n",
       "display: block;\n",
       "}\n",
       "#unshowable {\n",
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       "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",
       "}\n",
       ".suggestion-warning { \n",
       "font-weight: bold;\n",
       "color: rgb(200, 130, 0);\n",
       "}\n",
       ".suggestion-error { \n",
       "font-weight: bold;\n",
       "color: red;\n",
       "}\n",
       ".suggestion-name {\n",
       "font-weight: bold;\n",
       "}\n",
       "</style><span class='get-type'>runFreshMT . step $ tmOmega :: forall (m :: * -> *). MonadPlus m => m Exp</span>"
      ],
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       "runFreshMT . step $ tmOmega :: forall (m :: * -> *). MonadPlus m => m Exp"
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       ".mono {\n",
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       "display: block;\n",
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       ".err-msg {\n",
       "color: red;\n",
       "font-style: italic;\n",
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       "#unshowable {\n",
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       "font-weight: bold;\n",
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       ".err-msg.in.collapse {\n",
       "padding-top: 0.7em;\n",
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       "color: rgb(200, 130, 0);\n",
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       "</style><code>(λx.x x) (λx.x x)</code>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    ":type runFreshMT . step $ tmOmega\n",
    "map dispExp . runFreshMT . step $ tmOmega"
   ]
  },
  {
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       ".hoogle-sub {\n",
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       "font-family: monospace;\n",
       "margin-left: 1em;\n",
       "}\n",
       ".mono {\n",
       "font-family: monospace;\n",
       "display: block;\n",
       "}\n",
       ".err-msg {\n",
       "color: red;\n",
       "font-style: italic;\n",
       "font-family: monospace;\n",
       "white-space: pre;\n",
       "display: block;\n",
       "}\n",
       "#unshowable {\n",
       "color: red;\n",
       "font-weight: bold;\n",
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       ".err-msg.in.collapse {\n",
       "padding-top: 0.7em;\n",
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       ".highlight-code {\n",
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       "font-family: monospace;\n",
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       ".suggestion-warning { \n",
       "font-weight: bold;\n",
       "color: rgb(200, 130, 0);\n",
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       ".suggestion-error { \n",
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       "color: red;\n",
       "}\n",
       ".suggestion-name {\n",
       "font-weight: bold;\n",
       "}\n",
       "</style><span class='get-type'>runFreshMT . step $ tm1 :: forall (m :: * -> *). MonadPlus m => m Exp</span>"
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       "runFreshMT . step $ tm1 :: forall (m :: * -> *). MonadPlus m => m Exp"
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       "}\n",
       ".hoogle-code {\n",
       "display: block;\n",
       "font-family: monospace;\n",
       "white-space: pre;\n",
       "}\n",
       ".hoogle-text {\n",
       "display: block;\n",
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       ".hoogle-name {\n",
       "color: green;\n",
       "font-weight: bold;\n",
       "}\n",
       ".hoogle-head {\n",
       "font-weight: bold;\n",
       "}\n",
       ".hoogle-sub {\n",
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       ".hoogle-package {\n",
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       ".hoogle-module {\n",
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       "}\n",
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       "font-family: monospace;\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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       "font-family: monospace;\n",
       "white-space: pre;\n",
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       "#unshowable {\n",
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       "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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       ".suggestion-warning { \n",
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       "color: rgb(200, 130, 0);\n",
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       "}\n",
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       "display: block;\n",
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       "font-weight: bold;\n",
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       ".hoogle-head {\n",
       "font-weight: bold;\n",
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       ".hoogle-sub {\n",
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       ".hoogle-package {\n",
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       "}\n",
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       "color: green;\n",
       "font-weight: bold;\n",
       "font-family: monospace;\n",
       "margin-left: 1em;\n",
       "}\n",
       ".mono {\n",
       "font-family: monospace;\n",
       "display: block;\n",
       "}\n",
       ".err-msg {\n",
       "color: red;\n",
       "font-style: italic;\n",
       "font-family: monospace;\n",
       "white-space: pre;\n",
       "display: block;\n",
       "}\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",
       "}\n",
       ".suggestion-warning { \n",
       "font-weight: bold;\n",
       "color: rgb(200, 130, 0);\n",
       "}\n",
       ".suggestion-error { \n",
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       "color: red;\n",
       "}\n",
       ".suggestion-name {\n",
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       "</style><code>(λx.x) (λx.x) (λx.x)</code>"
      ]
     },
     "metadata": {},
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    }
   ],
   "source": [
    ":type runFreshMT . step $ tm1\n",
    "map dispExp . runFreshMT . step $ tm1"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 22,
   "metadata": {},
   "outputs": [
    {
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       "}\n",
       ".hoogle-code {\n",
       "display: block;\n",
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       "color: green;\n",
       "font-weight: bold;\n",
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       ".hoogle-head {\n",
       "font-weight: bold;\n",
       "}\n",
       ".hoogle-sub {\n",
       "display: block;\n",
       "margin-left: 0.4em;\n",
       "}\n",
       ".hoogle-package {\n",
       "font-weight: bold;\n",
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       ".hoogle-class {\n",
       "font-weight: bold;\n",
       "}\n",
       ".get-type {\n",
       "color: green;\n",
       "font-weight: bold;\n",
       "font-family: monospace;\n",
       "display: block;\n",
       "white-space: pre-wrap;\n",
       "}\n",
       ".show-type {\n",
       "color: green;\n",
       "font-weight: bold;\n",
       "font-family: monospace;\n",
       "margin-left: 1em;\n",
       "}\n",
       ".mono {\n",
       "font-family: monospace;\n",
       "display: block;\n",
       "}\n",
       ".err-msg {\n",
       "color: red;\n",
       "font-style: italic;\n",
       "font-family: monospace;\n",
       "white-space: pre;\n",
       "display: block;\n",
       "}\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",
       "}\n",
       ".suggestion-warning { \n",
       "font-weight: bold;\n",
       "color: rgb(200, 130, 0);\n",
       "}\n",
       ".suggestion-error { \n",
       "font-weight: bold;\n",
       "color: red;\n",
       "}\n",
       ".suggestion-name {\n",
       "font-weight: bold;\n",
       "}\n",
       "</style><span class='get-type'>runFreshMT . step $ tm2 :: forall (m :: * -> *). MonadPlus m => m Exp</span>"
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      "text/plain": [
       "runFreshMT . step $ tm2 :: forall (m :: * -> *). MonadPlus m => m Exp"
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       "<style>/* Styles used for the Hoogle display in the pager */\n",
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       "}\n",
       ".hoogle-code {\n",
       "display: block;\n",
       "font-family: monospace;\n",
       "white-space: pre;\n",
       "}\n",
       ".hoogle-text {\n",
       "display: block;\n",
       "}\n",
       ".hoogle-name {\n",
       "color: green;\n",
       "font-weight: bold;\n",
       "}\n",
       ".hoogle-head {\n",
       "font-weight: bold;\n",
       "}\n",
       ".hoogle-sub {\n",
       "display: block;\n",
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       ".hoogle-package {\n",
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       "}\n",
       ".show-type {\n",
       "color: green;\n",
       "font-weight: bold;\n",
       "font-family: monospace;\n",
       "margin-left: 1em;\n",
       "}\n",
       ".mono {\n",
       "font-family: monospace;\n",
       "display: block;\n",
       "}\n",
       ".err-msg {\n",
       "color: red;\n",
       "font-style: italic;\n",
       "font-family: monospace;\n",
       "white-space: pre;\n",
       "display: block;\n",
       "}\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",
       "}\n",
       ".suggestion-warning { \n",
       "font-weight: bold;\n",
       "color: rgb(200, 130, 0);\n",
       "}\n",
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       "color: red;\n",
       "}\n",
       ".suggestion-name {\n",
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       "}\n",
       "</style><code>λy.(λx.x) ((λx.x) (λx.x))</code>"
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    {
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       "<style>/* Styles used for the Hoogle display in the pager */\n",
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       "}\n",
       ".hoogle-code {\n",
       "display: block;\n",
       "font-family: monospace;\n",
       "white-space: pre;\n",
       "}\n",
       ".hoogle-text {\n",
       "display: block;\n",
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       ".hoogle-name {\n",
       "color: green;\n",
       "font-weight: bold;\n",
       "}\n",
       ".hoogle-head {\n",
       "font-weight: bold;\n",
       "}\n",
       ".hoogle-sub {\n",
       "display: block;\n",
       "margin-left: 0.4em;\n",
       "}\n",
       ".hoogle-package {\n",
       "font-weight: bold;\n",
       "font-style: italic;\n",
       "}\n",
       ".hoogle-module {\n",
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       ".get-type {\n",
       "color: green;\n",
       "font-weight: bold;\n",
       "font-family: monospace;\n",
       "display: block;\n",
       "white-space: pre-wrap;\n",
       "}\n",
       ".show-type {\n",
       "color: green;\n",
       "font-weight: bold;\n",
       "font-family: monospace;\n",
       "margin-left: 1em;\n",
       "}\n",
       ".mono {\n",
       "font-family: monospace;\n",
       "display: block;\n",
       "}\n",
       ".err-msg {\n",
       "color: red;\n",
       "font-style: italic;\n",
       "font-family: monospace;\n",
       "white-space: pre;\n",
       "display: block;\n",
       "}\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",
       "}\n",
       ".suggestion-warning { \n",
       "font-weight: bold;\n",
       "color: rgb(200, 130, 0);\n",
       "}\n",
       ".suggestion-error { \n",
       "font-weight: bold;\n",
       "color: red;\n",
       "}\n",
       ".suggestion-name {\n",
       "font-weight: bold;\n",
       "}\n",
       "</style><code>λy.(λx.x) (λx.x) (λx.x)</code>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    ":type runFreshMT . step $ tm2\n",
    "map dispExp . runFreshMT . step $ tm2"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 23,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "<style>/* Styles used for the Hoogle display in the pager */\n",
       ".hoogle-doc {\n",
       "display: block;\n",
       "padding-bottom: 1.3em;\n",
       "padding-left: 0.4em;\n",
       "}\n",
       ".hoogle-code {\n",
       "display: block;\n",
       "font-family: monospace;\n",
       "white-space: pre;\n",
       "}\n",
       ".hoogle-text {\n",
       "display: block;\n",
       "}\n",
       ".hoogle-name {\n",
       "color: green;\n",
       "font-weight: bold;\n",
       "}\n",
       ".hoogle-head {\n",
       "font-weight: bold;\n",
       "}\n",
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       "}\n",
       "</style><div class=\"suggestion-name\" style=\"clear:both;\">Redundant bracket</div><div class=\"suggestion-row\" style=\"float: left;\"><div class=\"suggestion-warning\">Found:</div><div class=\"highlight-code\" id=\"haskell\">(App (lam x \\$ Var x) (lam y \\$ Var y) `App`\n",
       "   App (lam x \\$ Var x) (lam z \\$ Var z))</div></div><div class=\"suggestion-row\" style=\"float: left;\"><div class=\"suggestion-warning\">Why Not:</div><div class=\"highlight-code\" id=\"haskell\">App (lam x \\$ Var x) (lam y \\$ Var y) `App`\n",
       "  App (lam x \\$ Var x) (lam z \\$ Var z)</div></div>"
      ],
      "text/plain": [
       "Line 1: Redundant bracket\n",
       "Found:\n",
       "(App (lam x $ Var x) (lam y $ Var y) `App`\n",
       "   App (lam x $ Var x) (lam z $ Var z))\n",
       "Why not:\n",
       "App (lam x $ Var x) (lam y $ Var y) `App`\n",
       "  App (lam x $ Var x) (lam z $ Var z)"
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   ],
   "source": [
    "tm10 = (App (lam x $ Var x) (lam y $ Var y) `App` App (lam x $ Var x) (lam z $ Var z))\n",
    "\n",
    "list1 :: [Exp]\n",
    "list1 = runFreshMT . step $ tm10\n",
    "list2 :: [[Exp]]\n",
    "list2 = map (runFreshMT . step) list1\n",
    "\n",
    "dispExp tm10\n",
    "putStrLn \"list1:\"\n",
    "map dispExp list1\n",
    "putStrLn \"list2:\"\n",
    "map (map dispExp) list2"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 24,
   "metadata": {},
   "outputs": [],
   "source": [
    "import Data.Tree hiding (drawTree)\n",
    "import Data.Tree.View\n",
    "\n",
    "redTree e = Node e (map redTree . runFreshMT . step $ e)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 25,
   "metadata": {},
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       "showTree . fmap showExp :: Tree Exp -> String"
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   ],
   "source": [
    ":type showTree . fmap showExp\n",
    "drawTreeExp = showTree . fmap showExp"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 26,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "(λx.x) (λy.y) ((λx.x) (λz.z))\n",
       " ├╴(λy.y) ((λx.x) (λz.z))\n",
       " │  ├╴(λx.x) (λz.z)\n",
       " │  │  └╴λz.z\n",
       " │  └╴(λy.y) (λz.z)\n",
       " │     └╴λz.z\n",
       " └╴(λx.x) (λy.y) (λz.z)\n",
       "    └╴(λy.y) (λz.z)\n",
       "       └╴λz.z"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "putStrLn $ drawTreeExp $ redTree (App (lam x $ Var x) (lam y $ Var y) `App` App (lam x $ Var x) (lam z $ Var z))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": []
  }
 ],
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   "display_name": "Haskell",
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   "file_extension": ".hs",
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