= x == y Z == (S y) = False (S x) == Z = False x /= y = not (x == y) 23 I t i s h ar d t o i m agi n e m an y c as e s w h e r e t h e /= m e t h od w i l l b e an y t h i n g ot h e r t h an t h e h av ai l ab l e , or r e t u r n Nothing i f on e or b ot h ar e n ot ( “f ai l f as t ”) . M an agi n g t h e Nothing c as e s i s ac h i e v e d b y t h e >>= op e r at or , h i d d e n b y t h e do i n g d e p e n d e n t t y p e s , t h e t y p e i t s e l f ge t s n o r m a l i s ed . S o i m agi n e w e w an t t o p r ov e t h e f ol l ow i n g t h e or e m ab ou t t h e r e d u c t i on b e h

= x == y Z == (S y) = False (S x) == Z = False x /= y = not (x == y) 23 I t i s h ar d t o i m agi n e m an y c as e s w h e r e t h e /= m e t h od w i l l b e an y t h i n g ot h e r t h an t h e h av ai l ab l e , or r e t u r n Nothing i f on e or b ot h ar e n ot ( “f ai l f as t ”) . M an agi n g t h e Nothing c as e s i s ac h i e v e d b y t h e >>= op e r at or , h i d d e n b y t h e do i n g d e p e n d e n t t y p e s , t h e t y p e i t s e l f ge t s n o r m a l i s ed . S o i m agi n e w e w an t t o p r ov e t h e f ol l ow i n g t h e or e m ab ou t t h e r e d u c t i on b e h

p r i n t x b e c au s e head [] d oe s n ’ t t y p e c h e c k . How e v e r , w e m i gh t i m agi n e a f u n c t i on , unsafeHead, t h at i s i d e n t i c al t o I d r i s ’ head f u n c t i on y) = x == y Z == (S y) = False (S x) == Z = False x /= y = not (x == y) I t i s h ar d t o i m agi n e m an y c as e s w h e r e t h e /= m e t h od w i l l b e an y t h i n g ot h e r t h an t h e h av ai l ab l e , or r e t u r n Nothing i f on e or b ot h ar e n ot ( “f ai l f as t ”) . M an agi n g t h e Nothing c as e s i s ac h i e v e d b y t h e >>= op e r at or , h i d d e n b y t h e do

p r i n t x b e c au s e head [] d oe s n ’ t t y p e c h e c k . How e v e r , w e m i gh t i m agi n e a f u n c t i on , unsafeHead, t h at i s i d e n t i c al t o I d r i s ’ head f u n c t i on y) = x == y Z == (S y) = False (S x) == Z = False x /= y = not (x == y) I t i s h ar d t o i m agi n e m an y c as e s w h e r e t h e /= m e t h od w i l l b e an y t h i n g ot h e r t h an t h e h av ai l ab l e , or r e t u r n Nothing i f on e or b ot h ar e n ot ( “f ai l f as t ”) . M an agi n g t h e Nothing c as e s i s ac h i e v e d b y t h e >>= op e r at or , h i d d e n b y t h e do

y) = x == y Z == (S y) = False (S x) == Z = False x /= y = not (x == y) I t i s h ar d t o i m agi n e m an y c as e s w h e r e t h e /= m e t h od w i l l b e an y t h i n g ot h e r t h an t h e h av ai l ab l e , or r e t u r n Nothing i f on e or b ot h ar e n ot ( “f ai l f as t ”) . M an agi n g t h e Nothing c as e s i s ac h i e v e d b y t h e >>= op e r at or , h i d d e n b y t h e do i n g d e p e n d e n t t y p e s , t h e t y p e i t s e l f ge t s n o r m a l i s ed . S o i m agi n e w e w an t t o p r ov e t h e f ol l ow i n g t h e or e m ab ou t t h e r e d u c t i on b e h

y) = x == y Z == (S y) = False (S x) == Z = False x /= y = not (x == y) I t i s h ar d t o i m agi n e m an y c as e s w h e r e t h e /= m e t h od w i l l b e an y t h i n g ot h e r t h an t h e h av ai l ab l e , or r e t u r n Nothing i f on e or b ot h ar e n ot ( “f ai l f as t ”) . M an agi n g t h e Nothing c as e s i s ac h i e v e d b y t h e >>= op e r at or , h i d d e n b y t h e do i n g d e p e n d e n t t y p e s , t h e t y p e i t s e l f ge t s n o r m a l i s ed . S o i m agi n e w e w an t t o p r ov e t h e f ol l ow i n g t h e or e m ab ou t t h e r e d u c t i on b e h

y) = x == y Z == (S y) = False (S x) == Z = False x /= y = not (x == y) I t i s h ar d t o i m agi n e m an y c as e s w h e r e t h e /= m e t h od w i l l b e an y t h i n g ot h e r t h an t h e h av ai l ab l e , or r e t u r n Nothing i f on e or b ot h ar e n ot ( “f ai l f as t ”) . M an agi n g t h e Nothing c as e s i s ac h i e v e d b y t h e >>= op e r at or , h i d d e n b y t h e do i n g d e p e n d e n t t y p e s , t h e t y p e i t s e l f ge t s n o r m a l i s ed . S o i m agi n e w e w an t t o p r ov e t h e f ol l ow i n g t h e or e m ab ou t t h e r e d u c t i on b e h

y) = x == y Z == (S y) = False (S x) == Z = False x /= y = not (x == y) I t i s h ar d t o i m agi n e m an y c as e s w h e r e t h e /= m e t h od w i l l b e an y t h i n g ot h e r t h an t h e h av ai l ab l e , or r e t u r n Nothing i f on e or b ot h ar e n ot ( “f ai l f as t ”) . M an agi n g t h e Nothing c as e s i s ac h i e v e d b y t h e >>= op e r at or , h i d d e n b y t h e do i n g d e p e n d e n t t y p e s , t h e t y p e i t s e l f ge t s n o r m a l i s ed . S o i m agi n e w e w an t t o p r ov e t h e f ol l ow i n g t h e or e m ab ou t t h e r e d u c t i on b e h

y) = x == y Z == (S y) = False (S x) == Z = False x /= y = not (x == y) I t i s h ar d t o i m agi n e m an y c as e s w h e r e t h e /= m e t h od w i l l b e an y t h i n g ot h e r t h an t h e h av ai l ab l e , or r e t u r n Nothing i f on e or b ot h ar e n ot ( “f ai l f as t ”) . M an agi n g t h e Nothing c as e s i s ac h i e v e d b y t h e >>= op e r at or , h i d d e n b y t h e do i n g d e p e n d e n t t y p e s , t h e t y p e i t s e l f ge t s n o r m a l i s ed . S o i m agi n e w e w an t t o p r ov e t h e f ol l ow i n g t h e or e m ab ou t t h e r e d u c t i on b e h

y) = x == y Z == (S y) = False (S x) == Z = False x /= y = not (x == y) I t i s h ar d t o i m agi n e m an y c as e s w h e r e t h e /= m e t h od w i l l b e an y t h i n g ot h e r t h an t h e h av ai l ab l e , or r e t u r n Nothing i f on e or b ot h ar e n ot ( “f ai l f as t ”) . M an agi n g t h e Nothing c as e s i s ac h i e v e d b y t h e >>= op e r at or , h i d d e n b y t h e do i n g d e p e n d e n t t y p e s , t h e t y p e i t s e l f ge t s n o r m a l i s ed . S o i m agi n e w e w an t t o p r ov e t h e f ol l ow i n g t h e or e m ab ou t t h e r e d u c t i on b e h