Editing Monad (functional programming) (section)

=== Verifying the monad laws ===
Returning to the <code>Maybe</code> example, its components were declared to make up a monad, but no proof was given that it satisfies the monad laws.

This can be rectified by plugging the specifics of <code>Maybe</code> into one side of the general laws, then algebraically building a chain of equalities to reach the other side:
 '''Law 1:'''  eta(a) >>= f(x)  ⇔  (Just a) >>= f(x)  ⇔  f(a)

 '''Law 2:''' <!--@ col 8 --> ma >>= eta(x)          <!--@ col 32      --> ⇔  ma
 
        <!--@ col 8    --> '''if''' ma '''is''' (Just a) '''then'''
            <!--@ col 12      --> eta(a)             <!--@ col 32      --> ⇔ Just a
        <!--@ col 8    --> '''else'''                    <!--@ col 36-->    '''or'''
            <!--@ col 12      --> Nothing            <!--@ col 32      --> ⇔ Nothing
        <!--@ col 8    --> '''end if'''

 '''Law 3:''' <!--@ col 8 --> '''('''ma >>= f(x)''')''' >>= g(y)           <!--@ col 52          -->            ⇔  ma >>= '''('''f(x) >>= g(y)''')'''
 
        <!--@ col 8       --> '''if''' (ma >>= f(x)) '''is''' (Just b) '''then'''  <!--@ col 52 -->             '''if''' ma '''is''' (Just a) '''then'''
            <!--@ col 12         --> g(ma >>= f(x))                          <!--@ col 56                   -->      (f(x) >>= g(y)) a
        <!--@ col 8       --> '''else'''                                   <!--@ col 52             -->         '''else'''
            <!--@ col 12         --> Nothing                                   <!--@ col 56                   -->      Nothing
        <!--@ col 8       --> '''end if'''                                 <!--@ col 52             -->         '''end if'''
 
                <!--@ col 16            --> ⇔  '''if''' ma '''is''' (Just a) '''and''' f(a) '''is''' (Just b) '''then'''      
                    <!--@ col 20               -->    (g ∘ f) a
                <!--@ col 16            -->    '''else if''' ma '''is''' (Just a) '''and''' f(a) '''is Nothing then'''
                    <!--@ col 20               -->    Nothing
                <!--@ col 16            -->    '''else'''
                    <!--@ col 20               -->    Nothing
                <!--@ col 16            -->    '''end if'''