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<?xml version="1.0" encoding="UTF-8"?>
<!DOCTYPE html PUBLIC "-//W3C//DTD XHTML 1.0 Strict//EN" "http://www.w3.org/TR/xhtml1/DTD/xhtml1-strict.dtd">
<html>
<head>
<!-- Generated by HsColour, http://code.haskell.org/~malcolm/hscolour/ -->
<title>src/Control/Monad/Morph.hs</title>
<link type='text/css' rel='stylesheet' href='hscolour.css' />
</head>
<body>
<pre><a name="line-1"></a><span class='hs-comment'>{-# LANGUAGE CPP, RankNTypes #-}</span>
<a name="line-2"></a>
<a name="line-3"></a><span class='hs-comment'>{-| A monad morphism is a natural transformation:
<a name="line-4"></a>
<a name="line-5"></a>&gt; morph :: forall a . m a -&gt; n a
<a name="line-6"></a>
<a name="line-7"></a>    ... that obeys the following two laws:
<a name="line-8"></a>
<a name="line-9"></a>&gt; morph $ do x &lt;- m  =  do x &lt;- morph m
<a name="line-10"></a>&gt;            f x           morph (f x)
<a name="line-11"></a>&gt;
<a name="line-12"></a>&gt; morph (return x) = return x
<a name="line-13"></a>
<a name="line-14"></a>    ... which are equivalent to the following two functor laws:
<a name="line-15"></a>
<a name="line-16"></a>&gt; morph . (f &gt;=&gt; g) = morph . f &gt;=&gt; morph . g
<a name="line-17"></a>&gt;
<a name="line-18"></a>&gt; morph . return = return
<a name="line-19"></a>
<a name="line-20"></a>    Examples of monad morphisms include:
<a name="line-21"></a>
<a name="line-22"></a>    * 'lift' (from 'MonadTrans')
<a name="line-23"></a>
<a name="line-24"></a>    * 'squash' (See below)
<a name="line-25"></a>
<a name="line-26"></a>    * @'hoist' f@ (See below), if @f@ is a monad morphism
<a name="line-27"></a>
<a name="line-28"></a>    * @(f . g)@, if @f@ and @g@ are both monad morphisms
<a name="line-29"></a>
<a name="line-30"></a>    * 'id'
<a name="line-31"></a>
<a name="line-32"></a>    Monad morphisms commonly arise when manipulating existing monad transformer
<a name="line-33"></a>    code for compatibility purposes.  The 'MFunctor', 'MonadTrans', and
<a name="line-34"></a>    'MMonad' classes define standard ways to change monad transformer stacks:
<a name="line-35"></a>
<a name="line-36"></a>    * 'lift' introduces a new monad transformer layer of any type.
<a name="line-37"></a>
<a name="line-38"></a>    * 'squash' flattens two identical monad transformer layers into a single
<a name="line-39"></a>      layer of the same type.
<a name="line-40"></a>
<a name="line-41"></a>    * 'hoist' maps monad morphisms to modify deeper layers of the monad
<a name="line-42"></a>       transformer stack.
<a name="line-43"></a>
<a name="line-44"></a>-}</span>
<a name="line-45"></a>
<a name="line-46"></a><span class='hs-keyword'>module</span> <span class='hs-conid'>Control</span><span class='hs-varop'>.</span><span class='hs-conid'>Monad</span><span class='hs-varop'>.</span><span class='hs-conid'>Morph</span> <span class='hs-layout'>(</span>
<a name="line-47"></a>    <span class='hs-comment'>-- * Functors over Monads</span>
<a name="line-48"></a>    <span class='hs-conid'>MFunctor</span><span class='hs-layout'>(</span><span class='hs-keyglyph'>..</span><span class='hs-layout'>)</span><span class='hs-layout'>,</span>
<a name="line-49"></a>    <span class='hs-varid'>generalize</span><span class='hs-layout'>,</span>
<a name="line-50"></a>    <span class='hs-comment'>-- * Monads over Monads</span>
<a name="line-51"></a>    <span class='hs-conid'>MMonad</span><span class='hs-layout'>(</span><span class='hs-keyglyph'>..</span><span class='hs-layout'>)</span><span class='hs-layout'>,</span>
<a name="line-52"></a>    <span class='hs-conid'>MonadTrans</span><span class='hs-layout'>(</span><span class='hs-varid'>lift</span><span class='hs-layout'>)</span><span class='hs-layout'>,</span>
<a name="line-53"></a>    <span class='hs-varid'>squash</span><span class='hs-layout'>,</span>
<a name="line-54"></a>    <span class='hs-layout'>(</span><span class='hs-varop'>&gt;|&gt;</span><span class='hs-layout'>)</span><span class='hs-layout'>,</span>
<a name="line-55"></a>    <span class='hs-layout'>(</span><span class='hs-varop'>&lt;|&lt;</span><span class='hs-layout'>)</span><span class='hs-layout'>,</span>
<a name="line-56"></a>    <span class='hs-layout'>(</span><span class='hs-varop'>=&lt;|</span><span class='hs-layout'>)</span><span class='hs-layout'>,</span>
<a name="line-57"></a>    <span class='hs-layout'>(</span><span class='hs-varop'>|&gt;=</span><span class='hs-layout'>)</span>
<a name="line-58"></a>
<a name="line-59"></a>    <span class='hs-comment'>-- * Tutorial</span>
<a name="line-60"></a>    <span class='hs-comment'>-- $tutorial</span>
<a name="line-61"></a>
<a name="line-62"></a>    <span class='hs-comment'>-- ** Generalizing base monads</span>
<a name="line-63"></a>    <span class='hs-comment'>-- $generalize</span>
<a name="line-64"></a>
<a name="line-65"></a>    <span class='hs-comment'>-- ** Monad morphisms</span>
<a name="line-66"></a>    <span class='hs-comment'>-- $mmorph</span>
<a name="line-67"></a>
<a name="line-68"></a>    <span class='hs-comment'>-- ** Mixing diverse transformers</span>
<a name="line-69"></a>    <span class='hs-comment'>-- $interleave</span>
<a name="line-70"></a>
<a name="line-71"></a>    <span class='hs-comment'>-- ** Embedding transformers</span>
<a name="line-72"></a>    <span class='hs-comment'>-- $embed</span>
<a name="line-73"></a>    <span class='hs-layout'>)</span> <span class='hs-keyword'>where</span>
<a name="line-74"></a>
<a name="line-75"></a><span class='hs-keyword'>import</span> <span class='hs-conid'>Control</span><span class='hs-varop'>.</span><span class='hs-conid'>Monad</span><span class='hs-varop'>.</span><span class='hs-conid'>Trans</span><span class='hs-varop'>.</span><span class='hs-conid'>Class</span> <span class='hs-layout'>(</span><span class='hs-conid'>MonadTrans</span><span class='hs-layout'>(</span><span class='hs-varid'>lift</span><span class='hs-layout'>)</span><span class='hs-layout'>)</span>
<a name="line-76"></a><span class='hs-keyword'>import</span> <span class='hs-keyword'>qualified</span> <span class='hs-conid'>Control</span><span class='hs-varop'>.</span><span class='hs-conid'>Monad</span><span class='hs-varop'>.</span><span class='hs-conid'>Trans</span><span class='hs-varop'>.</span><span class='hs-conid'>Error</span>         <span class='hs-keyword'>as</span> <span class='hs-conid'>E</span>
<a name="line-77"></a><span class='hs-keyword'>import</span> <span class='hs-keyword'>qualified</span> <span class='hs-conid'>Control</span><span class='hs-varop'>.</span><span class='hs-conid'>Monad</span><span class='hs-varop'>.</span><span class='hs-conid'>Trans</span><span class='hs-varop'>.</span><span class='hs-conid'>Except</span>        <span class='hs-keyword'>as</span> <span class='hs-conid'>Ex</span>
<a name="line-78"></a><span class='hs-keyword'>import</span> <span class='hs-keyword'>qualified</span> <span class='hs-conid'>Control</span><span class='hs-varop'>.</span><span class='hs-conid'>Monad</span><span class='hs-varop'>.</span><span class='hs-conid'>Trans</span><span class='hs-varop'>.</span><span class='hs-conid'>Identity</span>      <span class='hs-keyword'>as</span> <span class='hs-conid'>I</span>
<a name="line-79"></a><span class='hs-keyword'>import</span> <span class='hs-keyword'>qualified</span> <span class='hs-conid'>Control</span><span class='hs-varop'>.</span><span class='hs-conid'>Monad</span><span class='hs-varop'>.</span><span class='hs-conid'>Trans</span><span class='hs-varop'>.</span><span class='hs-conid'>List</span>          <span class='hs-keyword'>as</span> <span class='hs-conid'>L</span>
<a name="line-80"></a><span class='hs-keyword'>import</span> <span class='hs-keyword'>qualified</span> <span class='hs-conid'>Control</span><span class='hs-varop'>.</span><span class='hs-conid'>Monad</span><span class='hs-varop'>.</span><span class='hs-conid'>Trans</span><span class='hs-varop'>.</span><span class='hs-conid'>Maybe</span>         <span class='hs-keyword'>as</span> <span class='hs-conid'>M</span>
<a name="line-81"></a><span class='hs-keyword'>import</span> <span class='hs-keyword'>qualified</span> <span class='hs-conid'>Control</span><span class='hs-varop'>.</span><span class='hs-conid'>Monad</span><span class='hs-varop'>.</span><span class='hs-conid'>Trans</span><span class='hs-varop'>.</span><span class='hs-conid'>Reader</span>        <span class='hs-keyword'>as</span> <span class='hs-conid'>R</span>
<a name="line-82"></a><span class='hs-keyword'>import</span> <span class='hs-keyword'>qualified</span> <span class='hs-conid'>Control</span><span class='hs-varop'>.</span><span class='hs-conid'>Monad</span><span class='hs-varop'>.</span><span class='hs-conid'>Trans</span><span class='hs-varop'>.</span><span class='hs-conid'>RWS</span><span class='hs-varop'>.</span><span class='hs-conid'>Lazy</span>      <span class='hs-keyword'>as</span> <span class='hs-conid'>RWS</span>
<a name="line-83"></a><span class='hs-keyword'>import</span> <span class='hs-keyword'>qualified</span> <span class='hs-conid'>Control</span><span class='hs-varop'>.</span><span class='hs-conid'>Monad</span><span class='hs-varop'>.</span><span class='hs-conid'>Trans</span><span class='hs-varop'>.</span><span class='hs-conid'>RWS</span><span class='hs-varop'>.</span><span class='hs-conid'>Strict</span>    <span class='hs-keyword'>as</span> <span class='hs-conid'>RWS'</span>
<a name="line-84"></a><span class='hs-keyword'>import</span> <span class='hs-keyword'>qualified</span> <span class='hs-conid'>Control</span><span class='hs-varop'>.</span><span class='hs-conid'>Monad</span><span class='hs-varop'>.</span><span class='hs-conid'>Trans</span><span class='hs-varop'>.</span><span class='hs-conid'>State</span><span class='hs-varop'>.</span><span class='hs-conid'>Lazy</span>    <span class='hs-keyword'>as</span> <span class='hs-conid'>S</span>
<a name="line-85"></a><span class='hs-keyword'>import</span> <span class='hs-keyword'>qualified</span> <span class='hs-conid'>Control</span><span class='hs-varop'>.</span><span class='hs-conid'>Monad</span><span class='hs-varop'>.</span><span class='hs-conid'>Trans</span><span class='hs-varop'>.</span><span class='hs-conid'>State</span><span class='hs-varop'>.</span><span class='hs-conid'>Strict</span>  <span class='hs-keyword'>as</span> <span class='hs-conid'>S'</span>
<a name="line-86"></a><span class='hs-keyword'>import</span> <span class='hs-keyword'>qualified</span> <span class='hs-conid'>Control</span><span class='hs-varop'>.</span><span class='hs-conid'>Monad</span><span class='hs-varop'>.</span><span class='hs-conid'>Trans</span><span class='hs-varop'>.</span><span class='hs-conid'>Writer</span><span class='hs-varop'>.</span><span class='hs-conid'>Lazy</span>   <span class='hs-keyword'>as</span> <span class='hs-conid'>W'</span>
<a name="line-87"></a><span class='hs-keyword'>import</span> <span class='hs-keyword'>qualified</span> <span class='hs-conid'>Control</span><span class='hs-varop'>.</span><span class='hs-conid'>Monad</span><span class='hs-varop'>.</span><span class='hs-conid'>Trans</span><span class='hs-varop'>.</span><span class='hs-conid'>Writer</span><span class='hs-varop'>.</span><span class='hs-conid'>Strict</span> <span class='hs-keyword'>as</span> <span class='hs-conid'>W</span>
<a name="line-88"></a><span class='hs-keyword'>import</span> <span class='hs-conid'>Data</span><span class='hs-varop'>.</span><span class='hs-conid'>Monoid</span> <span class='hs-layout'>(</span><span class='hs-conid'>Monoid</span><span class='hs-layout'>,</span> <span class='hs-varid'>mappend</span><span class='hs-layout'>)</span>
<a name="line-89"></a><span class='hs-keyword'>import</span> <span class='hs-conid'>Data</span><span class='hs-varop'>.</span><span class='hs-conid'>Functor</span><span class='hs-varop'>.</span><span class='hs-conid'>Compose</span> <span class='hs-layout'>(</span><span class='hs-conid'>Compose</span> <span class='hs-layout'>(</span><span class='hs-conid'>Compose</span><span class='hs-layout'>)</span><span class='hs-layout'>)</span>
<a name="line-90"></a><span class='hs-keyword'>import</span> <span class='hs-conid'>Data</span><span class='hs-varop'>.</span><span class='hs-conid'>Functor</span><span class='hs-varop'>.</span><span class='hs-conid'>Identity</span> <span class='hs-layout'>(</span><span class='hs-varid'>runIdentity</span><span class='hs-layout'>)</span>
<a name="line-91"></a><span class='hs-keyword'>import</span> <span class='hs-conid'>Data</span><span class='hs-varop'>.</span><span class='hs-conid'>Functor</span><span class='hs-varop'>.</span><span class='hs-conid'>Product</span> <span class='hs-layout'>(</span><span class='hs-conid'>Product</span> <span class='hs-layout'>(</span><span class='hs-conid'>Pair</span><span class='hs-layout'>)</span><span class='hs-layout'>)</span>
<a name="line-92"></a><span class='hs-keyword'>import</span> <span class='hs-conid'>Control</span><span class='hs-varop'>.</span><span class='hs-conid'>Applicative</span><span class='hs-varop'>.</span><span class='hs-conid'>Backwards</span> <span class='hs-layout'>(</span><span class='hs-conid'>Backwards</span> <span class='hs-layout'>(</span><span class='hs-conid'>Backwards</span><span class='hs-layout'>)</span><span class='hs-layout'>)</span>
<a name="line-93"></a><span class='hs-keyword'>import</span> <span class='hs-conid'>Control</span><span class='hs-varop'>.</span><span class='hs-conid'>Applicative</span><span class='hs-varop'>.</span><span class='hs-conid'>Lift</span> <span class='hs-layout'>(</span><span class='hs-conid'>Lift</span> <span class='hs-layout'>(</span><span class='hs-conid'>Pure</span><span class='hs-layout'>,</span> <span class='hs-conid'>Other</span><span class='hs-layout'>)</span><span class='hs-layout'>)</span>
<a name="line-94"></a>
<a name="line-95"></a><span class='hs-comment'>-- For documentation</span>
<a name="line-96"></a><span class='hs-keyword'>import</span> <span class='hs-conid'>Control</span><span class='hs-varop'>.</span><span class='hs-conid'>Exception</span> <span class='hs-layout'>(</span><span class='hs-varid'>try</span><span class='hs-layout'>,</span> <span class='hs-conid'>IOException</span><span class='hs-layout'>)</span>
<a name="line-97"></a><span class='hs-keyword'>import</span> <span class='hs-conid'>Control</span><span class='hs-varop'>.</span><span class='hs-conid'>Monad</span> <span class='hs-layout'>(</span><span class='hs-layout'>(</span><span class='hs-varop'>=&lt;&lt;</span><span class='hs-layout'>)</span><span class='hs-layout'>,</span> <span class='hs-layout'>(</span><span class='hs-varop'>&gt;=&gt;</span><span class='hs-layout'>)</span><span class='hs-layout'>,</span> <span class='hs-layout'>(</span><span class='hs-varop'>&lt;=&lt;</span><span class='hs-layout'>)</span><span class='hs-layout'>,</span> <span class='hs-varid'>join</span><span class='hs-layout'>)</span>
<a name="line-98"></a><span class='hs-keyword'>import</span> <span class='hs-conid'>Data</span><span class='hs-varop'>.</span><span class='hs-conid'>Functor</span><span class='hs-varop'>.</span><span class='hs-conid'>Identity</span> <span class='hs-layout'>(</span><span class='hs-conid'>Identity</span><span class='hs-layout'>)</span>
<a name="line-99"></a>
<a name="line-100"></a><span class='hs-comment'>{-| A functor in the category of monads, using 'hoist' as the analog of 'fmap':
<a name="line-101"></a>
<a name="line-102"></a>&gt; hoist (f . g) = hoist f . hoist g
<a name="line-103"></a>&gt;
<a name="line-104"></a>&gt; hoist id = id
<a name="line-105"></a>-}</span>
<a name="line-106"></a><span class='hs-keyword'>class</span> <span class='hs-conid'>MFunctor</span> <span class='hs-varid'>t</span> <span class='hs-keyword'>where</span>
<a name="line-107"></a>    <span class='hs-comment'>{-| Lift a monad morphism from @m@ to @n@ into a monad morphism from
<a name="line-108"></a>        @(t m)@ to @(t n)@
<a name="line-109"></a>    -}</span>
<a name="line-110"></a>    <span class='hs-varid'>hoist</span> <span class='hs-keyglyph'>::</span> <span class='hs-layout'>(</span><span class='hs-conid'>Monad</span> <span class='hs-varid'>m</span><span class='hs-layout'>)</span> <span class='hs-keyglyph'>=&gt;</span> <span class='hs-layout'>(</span><span class='hs-keyword'>forall</span> <span class='hs-varid'>a</span> <span class='hs-varop'>.</span> <span class='hs-varid'>m</span> <span class='hs-varid'>a</span> <span class='hs-keyglyph'>-&gt;</span> <span class='hs-varid'>n</span> <span class='hs-varid'>a</span><span class='hs-layout'>)</span> <span class='hs-keyglyph'>-&gt;</span> <span class='hs-varid'>t</span> <span class='hs-varid'>m</span> <span class='hs-varid'>b</span> <span class='hs-keyglyph'>-&gt;</span> <span class='hs-varid'>t</span> <span class='hs-varid'>n</span> <span class='hs-varid'>b</span>
<a name="line-111"></a>
<a name="line-112"></a><a name="instance%20MFunctor%20(E.ErrorT%20e)"></a><span class='hs-keyword'>instance</span> <span class='hs-conid'>MFunctor</span> <span class='hs-layout'>(</span><span class='hs-conid'>E</span><span class='hs-varop'>.</span><span class='hs-conid'>ErrorT</span> <span class='hs-varid'>e</span><span class='hs-layout'>)</span> <span class='hs-keyword'>where</span>
<a name="line-113"></a>    <span class='hs-varid'>hoist</span> <span class='hs-varid'>nat</span> <span class='hs-varid'>m</span> <span class='hs-keyglyph'>=</span> <span class='hs-conid'>E</span><span class='hs-varop'>.</span><span class='hs-conid'>ErrorT</span> <span class='hs-layout'>(</span><span class='hs-varid'>nat</span> <span class='hs-layout'>(</span><span class='hs-conid'>E</span><span class='hs-varop'>.</span><span class='hs-varid'>runErrorT</span> <span class='hs-varid'>m</span><span class='hs-layout'>)</span><span class='hs-layout'>)</span>
<a name="line-114"></a>
<a name="line-115"></a><a name="instance%20MFunctor%20(Ex.ExceptT%20e)"></a><span class='hs-keyword'>instance</span> <span class='hs-conid'>MFunctor</span> <span class='hs-layout'>(</span><span class='hs-conid'>Ex</span><span class='hs-varop'>.</span><span class='hs-conid'>ExceptT</span> <span class='hs-varid'>e</span><span class='hs-layout'>)</span> <span class='hs-keyword'>where</span>
<a name="line-116"></a>    <span class='hs-varid'>hoist</span> <span class='hs-varid'>nat</span> <span class='hs-varid'>m</span> <span class='hs-keyglyph'>=</span> <span class='hs-conid'>Ex</span><span class='hs-varop'>.</span><span class='hs-conid'>ExceptT</span> <span class='hs-layout'>(</span><span class='hs-varid'>nat</span> <span class='hs-layout'>(</span><span class='hs-conid'>Ex</span><span class='hs-varop'>.</span><span class='hs-varid'>runExceptT</span> <span class='hs-varid'>m</span><span class='hs-layout'>)</span><span class='hs-layout'>)</span>
<a name="line-117"></a>
<a name="line-118"></a><a name="instance%20MFunctor%20I.IdentityT"></a><span class='hs-keyword'>instance</span> <span class='hs-conid'>MFunctor</span> <span class='hs-conid'>I</span><span class='hs-varop'>.</span><span class='hs-conid'>IdentityT</span> <span class='hs-keyword'>where</span>
<a name="line-119"></a>    <span class='hs-varid'>hoist</span> <span class='hs-varid'>nat</span> <span class='hs-varid'>m</span> <span class='hs-keyglyph'>=</span> <span class='hs-conid'>I</span><span class='hs-varop'>.</span><span class='hs-conid'>IdentityT</span> <span class='hs-layout'>(</span><span class='hs-varid'>nat</span> <span class='hs-layout'>(</span><span class='hs-conid'>I</span><span class='hs-varop'>.</span><span class='hs-varid'>runIdentityT</span> <span class='hs-varid'>m</span><span class='hs-layout'>)</span><span class='hs-layout'>)</span>
<a name="line-120"></a>
<a name="line-121"></a><a name="instance%20MFunctor%20L.ListT"></a><span class='hs-keyword'>instance</span> <span class='hs-conid'>MFunctor</span> <span class='hs-conid'>L</span><span class='hs-varop'>.</span><span class='hs-conid'>ListT</span> <span class='hs-keyword'>where</span>
<a name="line-122"></a>    <span class='hs-varid'>hoist</span> <span class='hs-varid'>nat</span> <span class='hs-varid'>m</span> <span class='hs-keyglyph'>=</span> <span class='hs-conid'>L</span><span class='hs-varop'>.</span><span class='hs-conid'>ListT</span> <span class='hs-layout'>(</span><span class='hs-varid'>nat</span> <span class='hs-layout'>(</span><span class='hs-conid'>L</span><span class='hs-varop'>.</span><span class='hs-varid'>runListT</span> <span class='hs-varid'>m</span><span class='hs-layout'>)</span><span class='hs-layout'>)</span>
<a name="line-123"></a>
<a name="line-124"></a><a name="instance%20MFunctor%20M.MaybeT"></a><span class='hs-keyword'>instance</span> <span class='hs-conid'>MFunctor</span> <span class='hs-conid'>M</span><span class='hs-varop'>.</span><span class='hs-conid'>MaybeT</span> <span class='hs-keyword'>where</span>
<a name="line-125"></a>    <span class='hs-varid'>hoist</span> <span class='hs-varid'>nat</span> <span class='hs-varid'>m</span> <span class='hs-keyglyph'>=</span> <span class='hs-conid'>M</span><span class='hs-varop'>.</span><span class='hs-conid'>MaybeT</span> <span class='hs-layout'>(</span><span class='hs-varid'>nat</span> <span class='hs-layout'>(</span><span class='hs-conid'>M</span><span class='hs-varop'>.</span><span class='hs-varid'>runMaybeT</span> <span class='hs-varid'>m</span><span class='hs-layout'>)</span><span class='hs-layout'>)</span>
<a name="line-126"></a>
<a name="line-127"></a><a name="instance%20MFunctor%20(R.ReaderT%20r)"></a><span class='hs-keyword'>instance</span> <span class='hs-conid'>MFunctor</span> <span class='hs-layout'>(</span><span class='hs-conid'>R</span><span class='hs-varop'>.</span><span class='hs-conid'>ReaderT</span> <span class='hs-varid'>r</span><span class='hs-layout'>)</span> <span class='hs-keyword'>where</span>
<a name="line-128"></a>    <span class='hs-varid'>hoist</span> <span class='hs-varid'>nat</span> <span class='hs-varid'>m</span> <span class='hs-keyglyph'>=</span> <span class='hs-conid'>R</span><span class='hs-varop'>.</span><span class='hs-conid'>ReaderT</span> <span class='hs-layout'>(</span><span class='hs-keyglyph'>\</span><span class='hs-varid'>i</span> <span class='hs-keyglyph'>-&gt;</span> <span class='hs-varid'>nat</span> <span class='hs-layout'>(</span><span class='hs-conid'>R</span><span class='hs-varop'>.</span><span class='hs-varid'>runReaderT</span> <span class='hs-varid'>m</span> <span class='hs-varid'>i</span><span class='hs-layout'>)</span><span class='hs-layout'>)</span>
<a name="line-129"></a>
<a name="line-130"></a><a name="instance%20MFunctor%20(RWS.RWST%20r%20w%20s)"></a><span class='hs-keyword'>instance</span> <span class='hs-conid'>MFunctor</span> <span class='hs-layout'>(</span><span class='hs-conid'>RWS</span><span class='hs-varop'>.</span><span class='hs-conid'>RWST</span> <span class='hs-varid'>r</span> <span class='hs-varid'>w</span> <span class='hs-varid'>s</span><span class='hs-layout'>)</span> <span class='hs-keyword'>where</span>
<a name="line-131"></a>    <span class='hs-varid'>hoist</span> <span class='hs-varid'>nat</span> <span class='hs-varid'>m</span> <span class='hs-keyglyph'>=</span> <span class='hs-conid'>RWS</span><span class='hs-varop'>.</span><span class='hs-conid'>RWST</span> <span class='hs-layout'>(</span><span class='hs-keyglyph'>\</span><span class='hs-varid'>r</span> <span class='hs-varid'>s</span> <span class='hs-keyglyph'>-&gt;</span> <span class='hs-varid'>nat</span> <span class='hs-layout'>(</span><span class='hs-conid'>RWS</span><span class='hs-varop'>.</span><span class='hs-varid'>runRWST</span> <span class='hs-varid'>m</span> <span class='hs-varid'>r</span> <span class='hs-varid'>s</span><span class='hs-layout'>)</span><span class='hs-layout'>)</span>
<a name="line-132"></a>
<a name="line-133"></a><a name="instance%20MFunctor%20(RWS'.RWST%20r%20w%20s)"></a><span class='hs-keyword'>instance</span> <span class='hs-conid'>MFunctor</span> <span class='hs-layout'>(</span><span class='hs-conid'>RWS'</span><span class='hs-varop'>.</span><span class='hs-conid'>RWST</span> <span class='hs-varid'>r</span> <span class='hs-varid'>w</span> <span class='hs-varid'>s</span><span class='hs-layout'>)</span> <span class='hs-keyword'>where</span>
<a name="line-134"></a>    <span class='hs-varid'>hoist</span> <span class='hs-varid'>nat</span> <span class='hs-varid'>m</span> <span class='hs-keyglyph'>=</span> <span class='hs-conid'>RWS'</span><span class='hs-varop'>.</span><span class='hs-conid'>RWST</span> <span class='hs-layout'>(</span><span class='hs-keyglyph'>\</span><span class='hs-varid'>r</span> <span class='hs-varid'>s</span> <span class='hs-keyglyph'>-&gt;</span> <span class='hs-varid'>nat</span> <span class='hs-layout'>(</span><span class='hs-conid'>RWS'</span><span class='hs-varop'>.</span><span class='hs-varid'>runRWST</span> <span class='hs-varid'>m</span> <span class='hs-varid'>r</span> <span class='hs-varid'>s</span><span class='hs-layout'>)</span><span class='hs-layout'>)</span>
<a name="line-135"></a>
<a name="line-136"></a><a name="instance%20MFunctor%20(S.StateT%20s)"></a><span class='hs-keyword'>instance</span> <span class='hs-conid'>MFunctor</span> <span class='hs-layout'>(</span><span class='hs-conid'>S</span><span class='hs-varop'>.</span><span class='hs-conid'>StateT</span> <span class='hs-varid'>s</span><span class='hs-layout'>)</span> <span class='hs-keyword'>where</span>
<a name="line-137"></a>    <span class='hs-varid'>hoist</span> <span class='hs-varid'>nat</span> <span class='hs-varid'>m</span> <span class='hs-keyglyph'>=</span> <span class='hs-conid'>S</span><span class='hs-varop'>.</span><span class='hs-conid'>StateT</span> <span class='hs-layout'>(</span><span class='hs-keyglyph'>\</span><span class='hs-varid'>s</span> <span class='hs-keyglyph'>-&gt;</span> <span class='hs-varid'>nat</span> <span class='hs-layout'>(</span><span class='hs-conid'>S</span><span class='hs-varop'>.</span><span class='hs-varid'>runStateT</span> <span class='hs-varid'>m</span> <span class='hs-varid'>s</span><span class='hs-layout'>)</span><span class='hs-layout'>)</span>
<a name="line-138"></a>
<a name="line-139"></a><a name="instance%20MFunctor%20(S'.StateT%20s)"></a><span class='hs-keyword'>instance</span> <span class='hs-conid'>MFunctor</span> <span class='hs-layout'>(</span><span class='hs-conid'>S'</span><span class='hs-varop'>.</span><span class='hs-conid'>StateT</span> <span class='hs-varid'>s</span><span class='hs-layout'>)</span> <span class='hs-keyword'>where</span>
<a name="line-140"></a>    <span class='hs-varid'>hoist</span> <span class='hs-varid'>nat</span> <span class='hs-varid'>m</span> <span class='hs-keyglyph'>=</span> <span class='hs-conid'>S'</span><span class='hs-varop'>.</span><span class='hs-conid'>StateT</span> <span class='hs-layout'>(</span><span class='hs-keyglyph'>\</span><span class='hs-varid'>s</span> <span class='hs-keyglyph'>-&gt;</span> <span class='hs-varid'>nat</span> <span class='hs-layout'>(</span><span class='hs-conid'>S'</span><span class='hs-varop'>.</span><span class='hs-varid'>runStateT</span> <span class='hs-varid'>m</span> <span class='hs-varid'>s</span><span class='hs-layout'>)</span><span class='hs-layout'>)</span>
<a name="line-141"></a>
<a name="line-142"></a><a name="instance%20MFunctor%20(W.WriterT%20w)"></a><span class='hs-keyword'>instance</span> <span class='hs-conid'>MFunctor</span> <span class='hs-layout'>(</span><span class='hs-conid'>W</span><span class='hs-varop'>.</span><span class='hs-conid'>WriterT</span> <span class='hs-varid'>w</span><span class='hs-layout'>)</span> <span class='hs-keyword'>where</span>
<a name="line-143"></a>    <span class='hs-varid'>hoist</span> <span class='hs-varid'>nat</span> <span class='hs-varid'>m</span> <span class='hs-keyglyph'>=</span> <span class='hs-conid'>W</span><span class='hs-varop'>.</span><span class='hs-conid'>WriterT</span> <span class='hs-layout'>(</span><span class='hs-varid'>nat</span> <span class='hs-layout'>(</span><span class='hs-conid'>W</span><span class='hs-varop'>.</span><span class='hs-varid'>runWriterT</span> <span class='hs-varid'>m</span><span class='hs-layout'>)</span><span class='hs-layout'>)</span>
<a name="line-144"></a>
<a name="line-145"></a><a name="instance%20MFunctor%20(W'.WriterT%20w)"></a><span class='hs-keyword'>instance</span> <span class='hs-conid'>MFunctor</span> <span class='hs-layout'>(</span><span class='hs-conid'>W'</span><span class='hs-varop'>.</span><span class='hs-conid'>WriterT</span> <span class='hs-varid'>w</span><span class='hs-layout'>)</span> <span class='hs-keyword'>where</span>
<a name="line-146"></a>    <span class='hs-varid'>hoist</span> <span class='hs-varid'>nat</span> <span class='hs-varid'>m</span> <span class='hs-keyglyph'>=</span> <span class='hs-conid'>W'</span><span class='hs-varop'>.</span><span class='hs-conid'>WriterT</span> <span class='hs-layout'>(</span><span class='hs-varid'>nat</span> <span class='hs-layout'>(</span><span class='hs-conid'>W'</span><span class='hs-varop'>.</span><span class='hs-varid'>runWriterT</span> <span class='hs-varid'>m</span><span class='hs-layout'>)</span><span class='hs-layout'>)</span>
<a name="line-147"></a>
<a name="line-148"></a><a name="instance%20MFunctor%20(Compose%20f)"></a><span class='hs-keyword'>instance</span> <span class='hs-conid'>Functor</span> <span class='hs-varid'>f</span> <span class='hs-keyglyph'>=&gt;</span> <span class='hs-conid'>MFunctor</span> <span class='hs-layout'>(</span><span class='hs-conid'>Compose</span> <span class='hs-varid'>f</span><span class='hs-layout'>)</span> <span class='hs-keyword'>where</span>
<a name="line-149"></a>    <span class='hs-varid'>hoist</span> <span class='hs-varid'>nat</span> <span class='hs-layout'>(</span><span class='hs-conid'>Compose</span> <span class='hs-varid'>f</span><span class='hs-layout'>)</span> <span class='hs-keyglyph'>=</span> <span class='hs-conid'>Compose</span> <span class='hs-layout'>(</span><span class='hs-varid'>fmap</span> <span class='hs-varid'>nat</span> <span class='hs-varid'>f</span><span class='hs-layout'>)</span>
<a name="line-150"></a>
<a name="line-151"></a><a name="instance%20MFunctor%20(Product%20f)"></a><span class='hs-keyword'>instance</span> <span class='hs-conid'>MFunctor</span> <span class='hs-layout'>(</span><span class='hs-conid'>Product</span> <span class='hs-varid'>f</span><span class='hs-layout'>)</span> <span class='hs-keyword'>where</span>
<a name="line-152"></a>    <span class='hs-varid'>hoist</span> <span class='hs-varid'>nat</span> <span class='hs-layout'>(</span><span class='hs-conid'>Pair</span> <span class='hs-varid'>f</span> <span class='hs-varid'>g</span><span class='hs-layout'>)</span> <span class='hs-keyglyph'>=</span> <span class='hs-conid'>Pair</span> <span class='hs-varid'>f</span> <span class='hs-layout'>(</span><span class='hs-varid'>nat</span> <span class='hs-varid'>g</span><span class='hs-layout'>)</span>
<a name="line-153"></a>
<a name="line-154"></a><a name="instance%20MFunctor%20Backwards"></a><span class='hs-keyword'>instance</span> <span class='hs-conid'>MFunctor</span> <span class='hs-conid'>Backwards</span> <span class='hs-keyword'>where</span>
<a name="line-155"></a>    <span class='hs-varid'>hoist</span> <span class='hs-varid'>nat</span> <span class='hs-layout'>(</span><span class='hs-conid'>Backwards</span> <span class='hs-varid'>f</span><span class='hs-layout'>)</span> <span class='hs-keyglyph'>=</span> <span class='hs-conid'>Backwards</span> <span class='hs-layout'>(</span><span class='hs-varid'>nat</span> <span class='hs-varid'>f</span><span class='hs-layout'>)</span>
<a name="line-156"></a>
<a name="line-157"></a><a name="instance%20MFunctor%20Lift"></a><span class='hs-keyword'>instance</span> <span class='hs-conid'>MFunctor</span> <span class='hs-conid'>Lift</span> <span class='hs-keyword'>where</span>
<a name="line-158"></a>    <span class='hs-varid'>hoist</span> <span class='hs-keyword'>_</span>   <span class='hs-layout'>(</span><span class='hs-conid'>Pure</span> <span class='hs-varid'>a</span><span class='hs-layout'>)</span>  <span class='hs-keyglyph'>=</span> <span class='hs-conid'>Pure</span> <span class='hs-varid'>a</span>
<a name="line-159"></a>    <span class='hs-varid'>hoist</span> <span class='hs-varid'>nat</span> <span class='hs-layout'>(</span><span class='hs-conid'>Other</span> <span class='hs-varid'>f</span><span class='hs-layout'>)</span> <span class='hs-keyglyph'>=</span> <span class='hs-conid'>Other</span> <span class='hs-layout'>(</span><span class='hs-varid'>nat</span> <span class='hs-varid'>f</span><span class='hs-layout'>)</span>
<a name="line-160"></a>
<a name="line-161"></a><a name="generalize"></a><span class='hs-comment'>-- | A function that @generalize@s the 'Identity' base monad to be any monad.</span>
<a name="line-162"></a><span class='hs-definition'>generalize</span> <span class='hs-keyglyph'>::</span> <span class='hs-conid'>Monad</span> <span class='hs-varid'>m</span> <span class='hs-keyglyph'>=&gt;</span> <span class='hs-conid'>Identity</span> <span class='hs-varid'>a</span> <span class='hs-keyglyph'>-&gt;</span> <span class='hs-varid'>m</span> <span class='hs-varid'>a</span>
<a name="line-163"></a><span class='hs-definition'>generalize</span> <span class='hs-keyglyph'>=</span> <span class='hs-varid'>return</span> <span class='hs-varop'>.</span> <span class='hs-varid'>runIdentity</span>
<a name="line-164"></a><span class='hs-comment'>{-# INLINABLE generalize #-}</span>
<a name="line-165"></a>
<a name="line-166"></a><a name="MMonad"></a><span class='hs-comment'>{-| A monad in the category of monads, using 'lift' from 'MonadTrans' as the
<a name="line-167"></a>    analog of 'return' and 'embed' as the analog of ('=&lt;&lt;'):
<a name="line-168"></a>
<a name="line-169"></a>&gt; embed lift = id
<a name="line-170"></a>&gt;
<a name="line-171"></a>&gt; embed f (lift m) = f m
<a name="line-172"></a>&gt;
<a name="line-173"></a>&gt; embed g (embed f t) = embed (\m -&gt; embed g (f m)) t
<a name="line-174"></a>-}</span>
<a name="line-175"></a><a name="MMonad"></a><span class='hs-keyword'>class</span> <span class='hs-layout'>(</span><span class='hs-conid'>MFunctor</span> <span class='hs-varid'>t</span><span class='hs-layout'>,</span> <span class='hs-conid'>MonadTrans</span> <span class='hs-varid'>t</span><span class='hs-layout'>)</span> <span class='hs-keyglyph'>=&gt;</span> <span class='hs-conid'>MMonad</span> <span class='hs-varid'>t</span> <span class='hs-keyword'>where</span>
<a name="line-176"></a>    <span class='hs-comment'>{-| Embed a newly created 'MMonad' layer within an existing layer
<a name="line-177"></a>
<a name="line-178"></a>        'embed' is analogous to ('=&lt;&lt;')
<a name="line-179"></a>    -}</span>
<a name="line-180"></a>    <span class='hs-varid'>embed</span> <span class='hs-keyglyph'>::</span> <span class='hs-layout'>(</span><span class='hs-conid'>Monad</span> <span class='hs-varid'>n</span><span class='hs-layout'>)</span> <span class='hs-keyglyph'>=&gt;</span> <span class='hs-layout'>(</span><span class='hs-keyword'>forall</span> <span class='hs-varid'>a</span> <span class='hs-varop'>.</span> <span class='hs-varid'>m</span> <span class='hs-varid'>a</span> <span class='hs-keyglyph'>-&gt;</span> <span class='hs-varid'>t</span> <span class='hs-varid'>n</span> <span class='hs-varid'>a</span><span class='hs-layout'>)</span> <span class='hs-keyglyph'>-&gt;</span> <span class='hs-varid'>t</span> <span class='hs-varid'>m</span> <span class='hs-varid'>b</span> <span class='hs-keyglyph'>-&gt;</span> <span class='hs-varid'>t</span> <span class='hs-varid'>n</span> <span class='hs-varid'>b</span>
<a name="line-181"></a>
<a name="line-182"></a><a name="squash"></a><span class='hs-comment'>{-| Squash two 'MMonad' layers into a single layer
<a name="line-183"></a>
<a name="line-184"></a>    'squash' is analogous to 'join'
<a name="line-185"></a>-}</span>
<a name="line-186"></a><span class='hs-definition'>squash</span> <span class='hs-keyglyph'>::</span> <span class='hs-layout'>(</span><span class='hs-conid'>Monad</span> <span class='hs-varid'>m</span><span class='hs-layout'>,</span> <span class='hs-conid'>MMonad</span> <span class='hs-varid'>t</span><span class='hs-layout'>)</span> <span class='hs-keyglyph'>=&gt;</span> <span class='hs-varid'>t</span> <span class='hs-layout'>(</span><span class='hs-varid'>t</span> <span class='hs-varid'>m</span><span class='hs-layout'>)</span> <span class='hs-varid'>a</span> <span class='hs-keyglyph'>-&gt;</span> <span class='hs-varid'>t</span> <span class='hs-varid'>m</span> <span class='hs-varid'>a</span>
<a name="line-187"></a><span class='hs-definition'>squash</span> <span class='hs-keyglyph'>=</span> <span class='hs-varid'>embed</span> <span class='hs-varid'>id</span>
<a name="line-188"></a><span class='hs-comment'>{-# INLINABLE squash #-}</span>
<a name="line-189"></a>
<a name="line-190"></a><span class='hs-keyword'>infixr</span> <span class='hs-num'>2</span> <span class='hs-varop'>&gt;|&gt;</span><span class='hs-layout'>,</span> <span class='hs-varop'>=&lt;|</span>
<a name="line-191"></a><span class='hs-keyword'>infixl</span> <span class='hs-num'>2</span> <span class='hs-varop'>&lt;|&lt;</span><span class='hs-layout'>,</span> <span class='hs-varop'>|&gt;=</span>
<a name="line-192"></a>
<a name="line-193"></a><a name="%3e%7c%3e"></a><span class='hs-comment'>{-| Compose two 'MMonad' layer-building functions
<a name="line-194"></a>
<a name="line-195"></a>    ('&gt;|&gt;') is analogous to ('&gt;=&gt;')
<a name="line-196"></a>-}</span>
<a name="line-197"></a><span class='hs-layout'>(</span><span class='hs-varop'>&gt;|&gt;</span><span class='hs-layout'>)</span>
<a name="line-198"></a>    <span class='hs-keyglyph'>::</span> <span class='hs-layout'>(</span><span class='hs-conid'>Monad</span> <span class='hs-varid'>m3</span><span class='hs-layout'>,</span> <span class='hs-conid'>MMonad</span> <span class='hs-varid'>t</span><span class='hs-layout'>)</span>
<a name="line-199"></a>    <span class='hs-keyglyph'>=&gt;</span> <span class='hs-layout'>(</span><span class='hs-keyword'>forall</span> <span class='hs-varid'>a</span> <span class='hs-varop'>.</span> <span class='hs-varid'>m1</span> <span class='hs-varid'>a</span> <span class='hs-keyglyph'>-&gt;</span> <span class='hs-varid'>t</span> <span class='hs-varid'>m2</span> <span class='hs-varid'>a</span><span class='hs-layout'>)</span>
<a name="line-200"></a>    <span class='hs-keyglyph'>-&gt;</span> <span class='hs-layout'>(</span><span class='hs-keyword'>forall</span> <span class='hs-varid'>b</span> <span class='hs-varop'>.</span> <span class='hs-varid'>m2</span> <span class='hs-varid'>b</span> <span class='hs-keyglyph'>-&gt;</span> <span class='hs-varid'>t</span> <span class='hs-varid'>m3</span> <span class='hs-varid'>b</span><span class='hs-layout'>)</span>
<a name="line-201"></a>    <span class='hs-keyglyph'>-&gt;</span>             <span class='hs-varid'>m1</span> <span class='hs-varid'>c</span> <span class='hs-keyglyph'>-&gt;</span> <span class='hs-varid'>t</span> <span class='hs-varid'>m3</span> <span class='hs-varid'>c</span>
<a name="line-202"></a><span class='hs-layout'>(</span><span class='hs-varid'>f</span> <span class='hs-varop'>&gt;|&gt;</span> <span class='hs-varid'>g</span><span class='hs-layout'>)</span> <span class='hs-varid'>m</span> <span class='hs-keyglyph'>=</span> <span class='hs-varid'>embed</span> <span class='hs-varid'>g</span> <span class='hs-layout'>(</span><span class='hs-varid'>f</span> <span class='hs-varid'>m</span><span class='hs-layout'>)</span>
<a name="line-203"></a><span class='hs-comment'>{-# INLINABLE (&gt;|&gt;) #-}</span>
<a name="line-204"></a>
<a name="line-205"></a><a name="%3c%7c%3c"></a><span class='hs-comment'>{-| Equivalent to ('&gt;|&gt;') with the arguments flipped
<a name="line-206"></a>
<a name="line-207"></a>    ('&lt;|&lt;') is analogous to ('&lt;=&lt;')
<a name="line-208"></a>-}</span>
<a name="line-209"></a><span class='hs-layout'>(</span><span class='hs-varop'>&lt;|&lt;</span><span class='hs-layout'>)</span>
<a name="line-210"></a>    <span class='hs-keyglyph'>::</span> <span class='hs-layout'>(</span><span class='hs-conid'>Monad</span> <span class='hs-varid'>m3</span><span class='hs-layout'>,</span> <span class='hs-conid'>MMonad</span> <span class='hs-varid'>t</span><span class='hs-layout'>)</span>
<a name="line-211"></a>    <span class='hs-keyglyph'>=&gt;</span> <span class='hs-layout'>(</span><span class='hs-keyword'>forall</span> <span class='hs-varid'>b</span> <span class='hs-varop'>.</span> <span class='hs-varid'>m2</span> <span class='hs-varid'>b</span> <span class='hs-keyglyph'>-&gt;</span> <span class='hs-varid'>t</span> <span class='hs-varid'>m3</span> <span class='hs-varid'>b</span><span class='hs-layout'>)</span>
<a name="line-212"></a>    <span class='hs-keyglyph'>-&gt;</span> <span class='hs-layout'>(</span><span class='hs-keyword'>forall</span> <span class='hs-varid'>a</span> <span class='hs-varop'>.</span> <span class='hs-varid'>m1</span> <span class='hs-varid'>a</span> <span class='hs-keyglyph'>-&gt;</span> <span class='hs-varid'>t</span> <span class='hs-varid'>m2</span> <span class='hs-varid'>a</span><span class='hs-layout'>)</span>
<a name="line-213"></a>    <span class='hs-keyglyph'>-&gt;</span>             <span class='hs-varid'>m1</span> <span class='hs-varid'>c</span> <span class='hs-keyglyph'>-&gt;</span> <span class='hs-varid'>t</span> <span class='hs-varid'>m3</span> <span class='hs-varid'>c</span>
<a name="line-214"></a><span class='hs-layout'>(</span><span class='hs-varid'>g</span> <span class='hs-varop'>&lt;|&lt;</span> <span class='hs-varid'>f</span><span class='hs-layout'>)</span> <span class='hs-varid'>m</span> <span class='hs-keyglyph'>=</span> <span class='hs-varid'>embed</span> <span class='hs-varid'>g</span> <span class='hs-layout'>(</span><span class='hs-varid'>f</span> <span class='hs-varid'>m</span><span class='hs-layout'>)</span>
<a name="line-215"></a><span class='hs-comment'>{-# INLINABLE (&lt;|&lt;) #-}</span>
<a name="line-216"></a>
<a name="line-217"></a><a name="=%3c%7c"></a><span class='hs-comment'>{-| An infix operator equivalent to 'embed'
<a name="line-218"></a>
<a name="line-219"></a>    ('=&lt;|') is analogous to ('=&lt;&lt;')
<a name="line-220"></a>-}</span>
<a name="line-221"></a><span class='hs-layout'>(</span><span class='hs-varop'>=&lt;|</span><span class='hs-layout'>)</span> <span class='hs-keyglyph'>::</span> <span class='hs-layout'>(</span><span class='hs-conid'>Monad</span> <span class='hs-varid'>n</span><span class='hs-layout'>,</span> <span class='hs-conid'>MMonad</span> <span class='hs-varid'>t</span><span class='hs-layout'>)</span> <span class='hs-keyglyph'>=&gt;</span> <span class='hs-layout'>(</span><span class='hs-keyword'>forall</span> <span class='hs-varid'>a</span> <span class='hs-varop'>.</span> <span class='hs-varid'>m</span> <span class='hs-varid'>a</span> <span class='hs-keyglyph'>-&gt;</span> <span class='hs-varid'>t</span> <span class='hs-varid'>n</span> <span class='hs-varid'>a</span><span class='hs-layout'>)</span> <span class='hs-keyglyph'>-&gt;</span> <span class='hs-varid'>t</span> <span class='hs-varid'>m</span> <span class='hs-varid'>b</span> <span class='hs-keyglyph'>-&gt;</span> <span class='hs-varid'>t</span> <span class='hs-varid'>n</span> <span class='hs-varid'>b</span>
<a name="line-222"></a><span class='hs-layout'>(</span><span class='hs-varop'>=&lt;|</span><span class='hs-layout'>)</span> <span class='hs-keyglyph'>=</span> <span class='hs-varid'>embed</span>
<a name="line-223"></a><span class='hs-comment'>{-# INLINABLE (=&lt;|) #-}</span>
<a name="line-224"></a>
<a name="line-225"></a><a name="%7c%3e="></a><span class='hs-comment'>{-| Equivalent to ('=&lt;|') with the arguments flipped
<a name="line-226"></a>
<a name="line-227"></a>    ('|&gt;=') is analogous to ('&gt;&gt;=')
<a name="line-228"></a>-}</span>
<a name="line-229"></a><span class='hs-layout'>(</span><span class='hs-varop'>|&gt;=</span><span class='hs-layout'>)</span> <span class='hs-keyglyph'>::</span> <span class='hs-layout'>(</span><span class='hs-conid'>Monad</span> <span class='hs-varid'>n</span><span class='hs-layout'>,</span> <span class='hs-conid'>MMonad</span> <span class='hs-varid'>t</span><span class='hs-layout'>)</span> <span class='hs-keyglyph'>=&gt;</span> <span class='hs-varid'>t</span> <span class='hs-varid'>m</span> <span class='hs-varid'>b</span> <span class='hs-keyglyph'>-&gt;</span> <span class='hs-layout'>(</span><span class='hs-keyword'>forall</span> <span class='hs-varid'>a</span> <span class='hs-varop'>.</span> <span class='hs-varid'>m</span> <span class='hs-varid'>a</span> <span class='hs-keyglyph'>-&gt;</span> <span class='hs-varid'>t</span> <span class='hs-varid'>n</span> <span class='hs-varid'>a</span><span class='hs-layout'>)</span> <span class='hs-keyglyph'>-&gt;</span> <span class='hs-varid'>t</span> <span class='hs-varid'>n</span> <span class='hs-varid'>b</span>
<a name="line-230"></a><a name="t"></a><span class='hs-definition'>t</span> <span class='hs-varop'>|&gt;=</span> <span class='hs-varid'>f</span> <span class='hs-keyglyph'>=</span> <span class='hs-varid'>embed</span> <span class='hs-varid'>f</span> <span class='hs-varid'>t</span>
<a name="line-231"></a><span class='hs-comment'>{-# INLINABLE (|&gt;=) #-}</span>
<a name="line-232"></a>
<a name="line-233"></a><a name="instance%20MMonad%20(E.ErrorT%20e)"></a><span class='hs-keyword'>instance</span> <span class='hs-layout'>(</span><span class='hs-conid'>E</span><span class='hs-varop'>.</span><span class='hs-conid'>Error</span> <span class='hs-varid'>e</span><span class='hs-layout'>)</span> <span class='hs-keyglyph'>=&gt;</span> <span class='hs-conid'>MMonad</span> <span class='hs-layout'>(</span><span class='hs-conid'>E</span><span class='hs-varop'>.</span><span class='hs-conid'>ErrorT</span> <span class='hs-varid'>e</span><span class='hs-layout'>)</span> <span class='hs-keyword'>where</span>
<a name="line-234"></a>    <span class='hs-varid'>embed</span> <span class='hs-varid'>f</span> <span class='hs-varid'>m</span> <span class='hs-keyglyph'>=</span> <span class='hs-conid'>E</span><span class='hs-varop'>.</span><span class='hs-conid'>ErrorT</span> <span class='hs-layout'>(</span><span class='hs-keyword'>do</span>
<a name="line-235"></a>        <span class='hs-varid'>x</span> <span class='hs-keyglyph'>&lt;-</span> <span class='hs-conid'>E</span><span class='hs-varop'>.</span><span class='hs-varid'>runErrorT</span> <span class='hs-layout'>(</span><span class='hs-varid'>f</span> <span class='hs-layout'>(</span><span class='hs-conid'>E</span><span class='hs-varop'>.</span><span class='hs-varid'>runErrorT</span> <span class='hs-varid'>m</span><span class='hs-layout'>)</span><span class='hs-layout'>)</span>
<a name="line-236"></a>        <span class='hs-varid'>return</span> <span class='hs-layout'>(</span><span class='hs-keyword'>case</span> <span class='hs-varid'>x</span> <span class='hs-keyword'>of</span>
<a name="line-237"></a>            <span class='hs-conid'>Left</span>         <span class='hs-varid'>e</span>  <span class='hs-keyglyph'>-&gt;</span> <span class='hs-conid'>Left</span> <span class='hs-varid'>e</span>
<a name="line-238"></a>            <span class='hs-conid'>Right</span> <span class='hs-layout'>(</span><span class='hs-conid'>Left</span>  <span class='hs-varid'>e</span><span class='hs-layout'>)</span> <span class='hs-keyglyph'>-&gt;</span> <span class='hs-conid'>Left</span> <span class='hs-varid'>e</span>
<a name="line-239"></a>            <span class='hs-conid'>Right</span> <span class='hs-layout'>(</span><span class='hs-conid'>Right</span> <span class='hs-varid'>a</span><span class='hs-layout'>)</span> <span class='hs-keyglyph'>-&gt;</span> <span class='hs-conid'>Right</span> <span class='hs-varid'>a</span> <span class='hs-layout'>)</span> <span class='hs-layout'>)</span>
<a name="line-240"></a>
<a name="line-241"></a><a name="instance%20MMonad%20(Ex.ExceptT%20e)"></a><span class='hs-keyword'>instance</span> <span class='hs-conid'>MMonad</span> <span class='hs-layout'>(</span><span class='hs-conid'>Ex</span><span class='hs-varop'>.</span><span class='hs-conid'>ExceptT</span> <span class='hs-varid'>e</span><span class='hs-layout'>)</span> <span class='hs-keyword'>where</span>
<a name="line-242"></a>    <span class='hs-varid'>embed</span> <span class='hs-varid'>f</span> <span class='hs-varid'>m</span> <span class='hs-keyglyph'>=</span> <span class='hs-conid'>Ex</span><span class='hs-varop'>.</span><span class='hs-conid'>ExceptT</span> <span class='hs-layout'>(</span><span class='hs-keyword'>do</span>
<a name="line-243"></a>        <span class='hs-varid'>x</span> <span class='hs-keyglyph'>&lt;-</span> <span class='hs-conid'>Ex</span><span class='hs-varop'>.</span><span class='hs-varid'>runExceptT</span> <span class='hs-layout'>(</span><span class='hs-varid'>f</span> <span class='hs-layout'>(</span><span class='hs-conid'>Ex</span><span class='hs-varop'>.</span><span class='hs-varid'>runExceptT</span> <span class='hs-varid'>m</span><span class='hs-layout'>)</span><span class='hs-layout'>)</span>
<a name="line-244"></a>        <span class='hs-varid'>return</span> <span class='hs-layout'>(</span><span class='hs-keyword'>case</span> <span class='hs-varid'>x</span> <span class='hs-keyword'>of</span>
<a name="line-245"></a>            <span class='hs-conid'>Left</span>         <span class='hs-varid'>e</span>  <span class='hs-keyglyph'>-&gt;</span> <span class='hs-conid'>Left</span> <span class='hs-varid'>e</span>
<a name="line-246"></a>            <span class='hs-conid'>Right</span> <span class='hs-layout'>(</span><span class='hs-conid'>Left</span>  <span class='hs-varid'>e</span><span class='hs-layout'>)</span> <span class='hs-keyglyph'>-&gt;</span> <span class='hs-conid'>Left</span> <span class='hs-varid'>e</span>
<a name="line-247"></a>            <span class='hs-conid'>Right</span> <span class='hs-layout'>(</span><span class='hs-conid'>Right</span> <span class='hs-varid'>a</span><span class='hs-layout'>)</span> <span class='hs-keyglyph'>-&gt;</span> <span class='hs-conid'>Right</span> <span class='hs-varid'>a</span> <span class='hs-layout'>)</span> <span class='hs-layout'>)</span>
<a name="line-248"></a>
<a name="line-249"></a><a name="instance%20MMonad%20I.IdentityT"></a><span class='hs-keyword'>instance</span> <span class='hs-conid'>MMonad</span> <span class='hs-conid'>I</span><span class='hs-varop'>.</span><span class='hs-conid'>IdentityT</span> <span class='hs-keyword'>where</span>
<a name="line-250"></a>    <span class='hs-varid'>embed</span> <span class='hs-varid'>f</span> <span class='hs-varid'>m</span> <span class='hs-keyglyph'>=</span> <span class='hs-varid'>f</span> <span class='hs-layout'>(</span><span class='hs-conid'>I</span><span class='hs-varop'>.</span><span class='hs-varid'>runIdentityT</span> <span class='hs-varid'>m</span><span class='hs-layout'>)</span>
<a name="line-251"></a>
<a name="line-252"></a><a name="instance%20MMonad%20L.ListT"></a><span class='hs-keyword'>instance</span> <span class='hs-conid'>MMonad</span> <span class='hs-conid'>L</span><span class='hs-varop'>.</span><span class='hs-conid'>ListT</span> <span class='hs-keyword'>where</span>
<a name="line-253"></a>    <span class='hs-varid'>embed</span> <span class='hs-varid'>f</span> <span class='hs-varid'>m</span> <span class='hs-keyglyph'>=</span> <span class='hs-conid'>L</span><span class='hs-varop'>.</span><span class='hs-conid'>ListT</span> <span class='hs-layout'>(</span><span class='hs-keyword'>do</span>
<a name="line-254"></a>        <span class='hs-varid'>x</span> <span class='hs-keyglyph'>&lt;-</span> <span class='hs-conid'>L</span><span class='hs-varop'>.</span><span class='hs-varid'>runListT</span> <span class='hs-layout'>(</span><span class='hs-varid'>f</span> <span class='hs-layout'>(</span><span class='hs-conid'>L</span><span class='hs-varop'>.</span><span class='hs-varid'>runListT</span> <span class='hs-varid'>m</span><span class='hs-layout'>)</span><span class='hs-layout'>)</span>
<a name="line-255"></a>        <span class='hs-varid'>return</span> <span class='hs-layout'>(</span><span class='hs-varid'>concat</span> <span class='hs-varid'>x</span><span class='hs-layout'>)</span><span class='hs-layout'>)</span>
<a name="line-256"></a>
<a name="line-257"></a><a name="instance%20MMonad%20M.MaybeT"></a><span class='hs-keyword'>instance</span> <span class='hs-conid'>MMonad</span> <span class='hs-conid'>M</span><span class='hs-varop'>.</span><span class='hs-conid'>MaybeT</span> <span class='hs-keyword'>where</span>
<a name="line-258"></a>    <span class='hs-varid'>embed</span> <span class='hs-varid'>f</span> <span class='hs-varid'>m</span> <span class='hs-keyglyph'>=</span> <span class='hs-conid'>M</span><span class='hs-varop'>.</span><span class='hs-conid'>MaybeT</span> <span class='hs-layout'>(</span><span class='hs-keyword'>do</span>
<a name="line-259"></a>        <span class='hs-varid'>x</span> <span class='hs-keyglyph'>&lt;-</span> <span class='hs-conid'>M</span><span class='hs-varop'>.</span><span class='hs-varid'>runMaybeT</span> <span class='hs-layout'>(</span><span class='hs-varid'>f</span> <span class='hs-layout'>(</span><span class='hs-conid'>M</span><span class='hs-varop'>.</span><span class='hs-varid'>runMaybeT</span> <span class='hs-varid'>m</span><span class='hs-layout'>)</span><span class='hs-layout'>)</span>
<a name="line-260"></a>        <span class='hs-varid'>return</span> <span class='hs-layout'>(</span><span class='hs-keyword'>case</span> <span class='hs-varid'>x</span> <span class='hs-keyword'>of</span>
<a name="line-261"></a>            <span class='hs-conid'>Nothing</span>       <span class='hs-keyglyph'>-&gt;</span> <span class='hs-conid'>Nothing</span>
<a name="line-262"></a>            <span class='hs-conid'>Just</span> <span class='hs-conid'>Nothing</span>  <span class='hs-keyglyph'>-&gt;</span> <span class='hs-conid'>Nothing</span>
<a name="line-263"></a>            <span class='hs-conid'>Just</span> <span class='hs-layout'>(</span><span class='hs-conid'>Just</span> <span class='hs-varid'>a</span><span class='hs-layout'>)</span> <span class='hs-keyglyph'>-&gt;</span> <span class='hs-conid'>Just</span> <span class='hs-varid'>a</span> <span class='hs-layout'>)</span> <span class='hs-layout'>)</span>
<a name="line-264"></a>
<a name="line-265"></a><a name="instance%20MMonad%20(R.ReaderT%20r)"></a><span class='hs-keyword'>instance</span> <span class='hs-conid'>MMonad</span> <span class='hs-layout'>(</span><span class='hs-conid'>R</span><span class='hs-varop'>.</span><span class='hs-conid'>ReaderT</span> <span class='hs-varid'>r</span><span class='hs-layout'>)</span> <span class='hs-keyword'>where</span>
<a name="line-266"></a>    <span class='hs-varid'>embed</span> <span class='hs-varid'>f</span> <span class='hs-varid'>m</span> <span class='hs-keyglyph'>=</span> <span class='hs-conid'>R</span><span class='hs-varop'>.</span><span class='hs-conid'>ReaderT</span> <span class='hs-layout'>(</span><span class='hs-keyglyph'>\</span><span class='hs-varid'>i</span> <span class='hs-keyglyph'>-&gt;</span> <span class='hs-conid'>R</span><span class='hs-varop'>.</span><span class='hs-varid'>runReaderT</span> <span class='hs-layout'>(</span><span class='hs-varid'>f</span> <span class='hs-layout'>(</span><span class='hs-conid'>R</span><span class='hs-varop'>.</span><span class='hs-varid'>runReaderT</span> <span class='hs-varid'>m</span> <span class='hs-varid'>i</span><span class='hs-layout'>)</span><span class='hs-layout'>)</span> <span class='hs-varid'>i</span><span class='hs-layout'>)</span>
<a name="line-267"></a>
<a name="line-268"></a><a name="instance%20MMonad%20(W.WriterT%20w)"></a><span class='hs-keyword'>instance</span> <span class='hs-layout'>(</span><span class='hs-conid'>Monoid</span> <span class='hs-varid'>w</span><span class='hs-layout'>)</span> <span class='hs-keyglyph'>=&gt;</span> <span class='hs-conid'>MMonad</span> <span class='hs-layout'>(</span><span class='hs-conid'>W</span><span class='hs-varop'>.</span><span class='hs-conid'>WriterT</span> <span class='hs-varid'>w</span><span class='hs-layout'>)</span> <span class='hs-keyword'>where</span>
<a name="line-269"></a>    <span class='hs-varid'>embed</span> <span class='hs-varid'>f</span> <span class='hs-varid'>m</span> <span class='hs-keyglyph'>=</span> <span class='hs-conid'>W</span><span class='hs-varop'>.</span><span class='hs-conid'>WriterT</span> <span class='hs-layout'>(</span><span class='hs-keyword'>do</span>
<a name="line-270"></a>        <span class='hs-keyglyph'>~</span><span class='hs-layout'>(</span><span class='hs-layout'>(</span><span class='hs-varid'>a</span><span class='hs-layout'>,</span> <span class='hs-varid'>w1</span><span class='hs-layout'>)</span><span class='hs-layout'>,</span> <span class='hs-varid'>w2</span><span class='hs-layout'>)</span> <span class='hs-keyglyph'>&lt;-</span> <span class='hs-conid'>W</span><span class='hs-varop'>.</span><span class='hs-varid'>runWriterT</span> <span class='hs-layout'>(</span><span class='hs-varid'>f</span> <span class='hs-layout'>(</span><span class='hs-conid'>W</span><span class='hs-varop'>.</span><span class='hs-varid'>runWriterT</span> <span class='hs-varid'>m</span><span class='hs-layout'>)</span><span class='hs-layout'>)</span>
<a name="line-271"></a>        <span class='hs-varid'>return</span> <span class='hs-layout'>(</span><span class='hs-varid'>a</span><span class='hs-layout'>,</span> <span class='hs-varid'>mappend</span> <span class='hs-varid'>w1</span> <span class='hs-varid'>w2</span><span class='hs-layout'>)</span> <span class='hs-layout'>)</span>
<a name="line-272"></a>
<a name="line-273"></a><a name="instance%20MMonad%20(W'.WriterT%20w)"></a><span class='hs-keyword'>instance</span> <span class='hs-layout'>(</span><span class='hs-conid'>Monoid</span> <span class='hs-varid'>w</span><span class='hs-layout'>)</span> <span class='hs-keyglyph'>=&gt;</span> <span class='hs-conid'>MMonad</span> <span class='hs-layout'>(</span><span class='hs-conid'>W'</span><span class='hs-varop'>.</span><span class='hs-conid'>WriterT</span> <span class='hs-varid'>w</span><span class='hs-layout'>)</span> <span class='hs-keyword'>where</span>
<a name="line-274"></a>    <span class='hs-varid'>embed</span> <span class='hs-varid'>f</span> <span class='hs-varid'>m</span> <span class='hs-keyglyph'>=</span> <span class='hs-conid'>W'</span><span class='hs-varop'>.</span><span class='hs-conid'>WriterT</span> <span class='hs-layout'>(</span><span class='hs-keyword'>do</span>
<a name="line-275"></a>        <span class='hs-layout'>(</span><span class='hs-layout'>(</span><span class='hs-varid'>a</span><span class='hs-layout'>,</span> <span class='hs-varid'>w1</span><span class='hs-layout'>)</span><span class='hs-layout'>,</span> <span class='hs-varid'>w2</span><span class='hs-layout'>)</span> <span class='hs-keyglyph'>&lt;-</span> <span class='hs-conid'>W'</span><span class='hs-varop'>.</span><span class='hs-varid'>runWriterT</span> <span class='hs-layout'>(</span><span class='hs-varid'>f</span> <span class='hs-layout'>(</span><span class='hs-conid'>W'</span><span class='hs-varop'>.</span><span class='hs-varid'>runWriterT</span> <span class='hs-varid'>m</span><span class='hs-layout'>)</span><span class='hs-layout'>)</span>
<a name="line-276"></a>        <span class='hs-varid'>return</span> <span class='hs-layout'>(</span><span class='hs-varid'>a</span><span class='hs-layout'>,</span> <span class='hs-varid'>mappend</span> <span class='hs-varid'>w1</span> <span class='hs-varid'>w2</span><span class='hs-layout'>)</span> <span class='hs-layout'>)</span>
<a name="line-277"></a>
<a name="line-278"></a><span class='hs-comment'>{- $tutorial
<a name="line-279"></a>    Monad morphisms solve the common problem of fixing monadic code after the
<a name="line-280"></a>    fact without modifying the original source code or type signatures.  The
<a name="line-281"></a>    following sections illustrate various examples of transparently modifying
<a name="line-282"></a>    existing functions.
<a name="line-283"></a>-}</span>
<a name="line-284"></a>
<a name="line-285"></a><span class='hs-comment'>{- $generalize
<a name="line-286"></a>    Imagine that some library provided the following 'S.State' code:
<a name="line-287"></a>
<a name="line-288"></a>&gt; import Control.Monad.Trans.State
<a name="line-289"></a>&gt;
<a name="line-290"></a>&gt; tick :: State Int ()
<a name="line-291"></a>&gt; tick = modify (+1)
<a name="line-292"></a>
<a name="line-293"></a>    ... but we would prefer to reuse @tick@ within a larger
<a name="line-294"></a>    @('S.StateT' Int 'IO')@ block in order to mix in 'IO' actions.
<a name="line-295"></a>
<a name="line-296"></a>    We could patch the original library to generalize @tick@'s type signature:
<a name="line-297"></a>
<a name="line-298"></a>&gt; tick :: (Monad m) =&gt; StateT Int m ()
<a name="line-299"></a>
<a name="line-300"></a>    ... but we would prefer not to fork upstream code if possible.  How could
<a name="line-301"></a>    we generalize @tick@'s type without modifying the original code?
<a name="line-302"></a>
<a name="line-303"></a>    We can solve this if we realize that 'S.State' is a type synonym for
<a name="line-304"></a>    'S.StateT' with an 'Identity' base monad:
<a name="line-305"></a>
<a name="line-306"></a>&gt; type State s = StateT s Identity
<a name="line-307"></a>
<a name="line-308"></a>    ... which means that @tick@'s true type is actually:
<a name="line-309"></a>
<a name="line-310"></a>&gt; tick :: StateT Int Identity ()
<a name="line-311"></a>
<a name="line-312"></a>    Now all we need is a function that @generalize@s the 'Identity' base monad
<a name="line-313"></a>    to be any monad:
<a name="line-314"></a>
<a name="line-315"></a>&gt; import Data.Functor.Identity
<a name="line-316"></a>&gt;
<a name="line-317"></a>&gt; generalize :: (Monad m) =&gt; Identity a -&gt; m a
<a name="line-318"></a>&gt; generalize m = return (runIdentity m)
<a name="line-319"></a>
<a name="line-320"></a>    ... which we can 'hoist' to change @tick@'s base monad:
<a name="line-321"></a>
<a name="line-322"></a>&gt; hoist :: (Monad m, MFunctor t) =&gt; (forall a . m a -&gt; n a) -&gt; t m b -&gt; t n b
<a name="line-323"></a>&gt;
<a name="line-324"></a>&gt; hoist generalize :: (Monad m, MFunctor t) =&gt; t Identity b -&gt; t m b
<a name="line-325"></a>&gt;
<a name="line-326"></a>&gt; hoist generalize tick :: (Monad m) =&gt; StateT Int m ()
<a name="line-327"></a>
<a name="line-328"></a>    This lets us mix @tick@ alongside 'IO' using 'lift':
<a name="line-329"></a>
<a name="line-330"></a>&gt; import Control.Monad.Morph
<a name="line-331"></a>&gt; import Control.Monad.Trans.Class
<a name="line-332"></a>&gt;
<a name="line-333"></a>&gt; tock                        ::                   StateT Int IO ()
<a name="line-334"></a>&gt; tock = do
<a name="line-335"></a>&gt;     hoist generalize tick   :: (Monad      m) =&gt; StateT Int m  ()
<a name="line-336"></a>&gt;     lift $ putStrLn "Tock!" :: (MonadTrans t) =&gt; t          IO ()
<a name="line-337"></a>
<a name="line-338"></a>&gt;&gt;&gt; runStateT tock 0
<a name="line-339"></a>Tock!
<a name="line-340"></a>((), 1)
<a name="line-341"></a>
<a name="line-342"></a>-}</span>
<a name="line-343"></a>
<a name="line-344"></a><span class='hs-comment'>{- $mmorph
<a name="line-345"></a>    Notice that @generalize@ is a monad morphism, and the following two proofs
<a name="line-346"></a>    show how @generalize@ satisfies the monad morphism laws.  You can refer to
<a name="line-347"></a>    these proofs as an example for how to prove a function obeys the monad
<a name="line-348"></a>    morphism laws:
<a name="line-349"></a>
<a name="line-350"></a>&gt; generalize (return x)
<a name="line-351"></a>&gt;
<a name="line-352"></a>&gt; -- Definition of 'return' for the Identity monad
<a name="line-353"></a>&gt; = generalize (Identity x)
<a name="line-354"></a>&gt;
<a name="line-355"></a>&gt; -- Definition of 'generalize'
<a name="line-356"></a>&gt; = return (runIdentity (Identity x))
<a name="line-357"></a>&gt;
<a name="line-358"></a>&gt; -- runIdentity (Identity x) = x
<a name="line-359"></a>&gt; = return x
<a name="line-360"></a>
<a name="line-361"></a>&gt; generalize $ do x &lt;- m
<a name="line-362"></a>&gt;                 f x
<a name="line-363"></a>&gt;
<a name="line-364"></a>&gt; -- Definition of (&gt;&gt;=) for the Identity monad
<a name="line-365"></a>&gt; = generalize (f (runIdentity m))
<a name="line-366"></a>&gt;
<a name="line-367"></a>&gt; -- Definition of 'generalize'
<a name="line-368"></a>&gt; = return (runIdentity (f (runIdentity m)))
<a name="line-369"></a>&gt;
<a name="line-370"></a>&gt; -- Monad law: Left identity
<a name="line-371"></a>&gt; = do x &lt;- return (runIdentity m)
<a name="line-372"></a>&gt;      return (runIdentity (f x))
<a name="line-373"></a>&gt;
<a name="line-374"></a>&gt; -- Definition of 'generalize' in reverse
<a name="line-375"></a>&gt; = do x &lt;- generalize m
<a name="line-376"></a>&gt;      generalize (f x)
<a name="line-377"></a>-}</span>
<a name="line-378"></a>
<a name="line-379"></a><span class='hs-comment'>{- $interleave
<a name="line-380"></a>    You can combine 'hoist' and 'lift' to insert arbitrary layers anywhere
<a name="line-381"></a>    within a monad transformer stack.  This comes in handy when interleaving two
<a name="line-382"></a>    diverse stacks.
<a name="line-383"></a>
<a name="line-384"></a>    For example, we might want to combine the following @save@ function:
<a name="line-385"></a>
<a name="line-386"></a>&gt; import Control.Monad.Trans.Writer
<a name="line-387"></a>&gt;
<a name="line-388"></a>&gt; -- i.e. :: StateT Int (WriterT [Int] Identity) ()
<a name="line-389"></a>&gt; save    :: StateT Int (Writer  [Int]) ()
<a name="line-390"></a>&gt; save = do
<a name="line-391"></a>&gt;     n &lt;- get
<a name="line-392"></a>&gt;     lift $ tell [n]
<a name="line-393"></a>
<a name="line-394"></a>    ... with our previous @tock@ function:
<a name="line-395"></a>
<a name="line-396"></a>&gt; tock :: StateT Int IO ()
<a name="line-397"></a>
<a name="line-398"></a>    However, @save@ and @tock@ differ in two ways:
<a name="line-399"></a>
<a name="line-400"></a>    * @tock@ lacks a 'W.WriterT' layer
<a name="line-401"></a>
<a name="line-402"></a>    * @save@ has an 'Identity' base monad
<a name="line-403"></a>
<a name="line-404"></a>    We can mix the two by inserting a 'W.WriterT' layer for @tock@ and
<a name="line-405"></a>    generalizing @save@'s base monad:
<a name="line-406"></a>
<a name="line-407"></a>&gt; import Control.Monad
<a name="line-408"></a>&gt;
<a name="line-409"></a>&gt; program ::                   StateT Int (WriterT [Int] IO) ()
<a name="line-410"></a>&gt; program = replicateM_ 4 $ do
<a name="line-411"></a>&gt;     hoist lift tock
<a name="line-412"></a>&gt;         :: (MonadTrans t) =&gt; StateT Int (t             IO) ()
<a name="line-413"></a>&gt;     hoist (hoist generalize) save
<a name="line-414"></a>&gt;         :: (Monad      m) =&gt; StateT Int (WriterT [Int] m ) ()
<a name="line-415"></a>
<a name="line-416"></a>&gt;&gt;&gt; execWriterT (runStateT program 0)
<a name="line-417"></a>Tock!
<a name="line-418"></a>Tock!
<a name="line-419"></a>Tock!
<a name="line-420"></a>Tock!
<a name="line-421"></a>[1,2,3,4]
<a name="line-422"></a>
<a name="line-423"></a>-}</span>
<a name="line-424"></a>
<a name="line-425"></a><span class='hs-comment'>{- $embed
<a name="line-426"></a>    Suppose we decided to @check@ all 'IOException's using a combination of
<a name="line-427"></a>    'try' and 'ErrorT':
<a name="line-428"></a>
<a name="line-429"></a>&gt; import Control.Exception
<a name="line-430"></a>&gt; import Control.Monad.Trans.Class
<a name="line-431"></a>&gt; import Control.Monad.Trans.Error
<a name="line-432"></a>&gt;
<a name="line-433"></a>&gt; check :: IO a -&gt; ErrorT IOException IO a
<a name="line-434"></a>&gt; check io = ErrorT (try io)
<a name="line-435"></a>
<a name="line-436"></a>    ... but then we forget to use @check@ in one spot, mistakenly using 'lift'
<a name="line-437"></a>    instead:
<a name="line-438"></a>
<a name="line-439"></a>&gt; program :: ErrorT IOException IO ()
<a name="line-440"></a>&gt; program = do
<a name="line-441"></a>&gt;     str &lt;- lift $ readFile "test.txt"
<a name="line-442"></a>&gt;     check $ putStr str
<a name="line-443"></a>
<a name="line-444"></a>&gt;&gt;&gt; runErrorT program
<a name="line-445"></a>*** Exception: test.txt: openFile: does not exist (No such file or directory)
<a name="line-446"></a>
<a name="line-447"></a>    How could we go back and fix 'program' without modifying its source code?
<a name="line-448"></a>
<a name="line-449"></a>    Well, @check@ is a monad morphism, but we can't 'hoist' it to modify the
<a name="line-450"></a>    base monad because then we get two 'E.ErrorT' layers instead of one:
<a name="line-451"></a>
<a name="line-452"></a>&gt; hoist check :: (MFunctor t) =&gt; t IO a -&gt; t (ErrorT IOException IO) a
<a name="line-453"></a>&gt;
<a name="line-454"></a>&gt; hoist check program :: ErrorT IOException (ErrorT IOException IO) ()
<a name="line-455"></a>
<a name="line-456"></a>    We'd prefer to 'embed' all newly generated exceptions in the existing
<a name="line-457"></a>    'E.ErrorT' layer:
<a name="line-458"></a>
<a name="line-459"></a>&gt; embed check :: ErrorT IOException IO a -&gt; ErrorT IOException IO a
<a name="line-460"></a>&gt;
<a name="line-461"></a>&gt; embed check program :: ErrorT IOException IO ()
<a name="line-462"></a>
<a name="line-463"></a>    This correctly checks the exceptions that slipped through the cracks:
<a name="line-464"></a>
<a name="line-465"></a>&gt;&gt;&gt; import Control.Monad.Morph
<a name="line-466"></a>&gt;&gt;&gt; runErrorT (embed check program)
<a name="line-467"></a>Left test.txt: openFile: does not exist (No such file or directory)
<a name="line-468"></a>
<a name="line-469"></a>-}</span>
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