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5IF(SFBU1PXFSPG newtypeT !NS@LPOO IUUQTLPOOTBODPN 4MJEFTBSFBWBJMBCMFBUhttp://bit.ly/derivia
 &YBNQMFDPEFTBSFPO(JU)VCkonn/newtype-talk-five

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4FMG*OUSPEVDUJPO w )JSPNJ*4)**ʢ!NS@LPOOʣ w %PDUPSBM$BOEJEBUFJO.BUIFNBUJDT w 3FTFBSDI"SFB.BUIFNBUJDBM-PHJD $PNQVUFS4DJFODF w 8SJUJOHBOEUFBDIJOH)BTLFMMGPSZFBST

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5IF(SFBU1PXFSPGnewtypeT 3PMFT 4BGF[FSPDPTUDPFSDJPOT BOE%FSJWJOH7JB ʙ.POPJE&'PMEBCMFJODMVEFEʙ (SFBU

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newtype

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newtype newtype Foo α = Bar α newtype Id = MkId Word

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newtype w "UZQFXJUIBTJOHMFDPOTUSVDUPSBOEpFME newtype Foo α = Bar α newtype Id = MkId Word

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newtype w "UZQFXJUIBTJOHMFDPOTUSVDUPSBOEpFME w )BTUIF4BNFSFQSFTFOUBUJPOBTJUTPOMZpFME newtype Foo α = Bar α newtype Id = MkId Word

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newtype w "UZQFXJUIBTJOHMFDPOTUSVDUPSBOEpFME w )BTUIF4BNFSFQSFTFOUBUJPOBTJUTPOMZpFME w %JTUJOHVJTIFEGSPNUIFPSJHJOBMUZQFBUUZQF MFWFM CVUIBTUIFTBNFNFNPSZSFQSFTFOUBUJPO BTUIFPSJHJOBM BOEFWBMVBUFETUSJDUMZ newtype Foo α = Bar α newtype Id = MkId Word

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5ZQJDBM/FXCJF2VFTUJPO

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5ZQJDBM/FXCJF2VFTUJPO 8IBUJTUIFEJ⒎FSFODFGSPNdata?"

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5ZQJDBM/FXCJF2VFTUJPO 8IBUJTUIFEJ⒎FSFODFGSPNdata?" *UTF⒏DJFOUUIBOLTUPJUTSFQSFTFOUBUJPO

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5ZQJDBM/FXCJF2VFTUJPO 8IBUJTUIFEJ⒎FSFODFGSPNdata?" *UTF⒏DJFOUUIBOLTUPJUTSFQSFTFOUBUJPO *UEPFTOUNBUUFSNVDIUPNF*ESBUIFS
 VTFdata

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5ZQJDBM/FXCJF2VFTUJPO 8IBUJTUIFEJ⒎FSFODFGSPNdata?" *UTF⒏DJFOUUIBOLTUPJUTSFQSFTFOUBUJPO *UEPFTOUNBUUFSNVDIUPNF*ESBUIFS
 VTFdata 8FMM XFIBWFfunpack-strict-fieldsBOZIPX

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3FBMMZ

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3PMFTPGnewtypeT

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3PMFTPGnewtypeT *NQMFNFOUBUJPO)JEJOH module Data.Id (Id ()) where newtype Id = MkId Word

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3PMFTPGnewtypeT *NQMFNFOUBUJPO)JEJOH module Data.Id (Id ()) where newtype Id = MkId Word %JTUJOHVJTIFEBUUZQFMFWFM CVUKVTUBWord internally

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3PMFTPGnewtypeT *NQMFNFOUBUJPO)JEJOH module Data.Id (Id ()) where newtype Id = MkId Word %JTUJOHVJTIFEBUUZQFMFWFM CVUKVTUBWord internally )JEFEBUBDPOTMkId
 outside the module

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3PMFTPGnewtypeT *NQMFNFOUBUJPO)JEJOH *NQMFNFOUBUJPO4IBSJOH module Data.Id (Id ()) where newtype Id = MkId Word %JTUJOHVJTIFEBUUZQFMFWFM CVUKVTUBWord internally {-# LANGUAGE GeneralizedNewtypeDeriving #-} newtype Id = MkId Word deriving (Num, Eq) )JEFEBUBDPOTMkId
 outside the module

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3PMFTPGnewtypeT *NQMFNFOUBUJPO)JEJOH *NQMFNFOUBUJPO4IBSJOH module Data.Id (Id ()) where newtype Id = MkId Word %JTUJOHVJTIFEBUUZQFMFWFM CVUKVTUBWord internally {-# LANGUAGE GeneralizedNewtypeDeriving #-} newtype Id = MkId Word deriving (Num, Eq) 6OEFSJWBCMFJO)BTLFMM CVUXFDBOTIBSFWordTJNQM )JEFEBUBDPOTMkId
 outside the module

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3PMFTPGnewtypeT *NQMFNFOUBUJPO)JEJOH *NQMFNFOUBUJPO4IBSJOH module Data.Id (Id ()) where newtype Id = MkId Word %JTUJOHVJTIFEBUUZQFMFWFM CVUKVTUBWord internally {-# LANGUAGE GeneralizedNewtypeDeriving #-} newtype Id = MkId Word deriving (Num, Eq) 6OEFSJWBCMFJO)BTLFMM CVUXFDBOTIBSFWordTJNQM &WPMWFT NPSFJO %FSJWJOH7JB )JEFEBUBDPOTMkId
 outside the module

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3PMFTPGnewtypeT *NQMFNFOUBUJPO)JEJOH *NQMFNFOUBUJPO4IBSJOH *NQMFNFOUBUJPO4FMFDUJPO module Data.Id (Id ()) where newtype Id = MkId Word %JTUJOHVJTIFEBUUZQFMFWFM CVUKVTUBWord internally {-# LANGUAGE GeneralizedNewtypeDeriving #-} newtype Id = MkId Word deriving (Num, Eq) 6OEFSJWBCMFJO)BTLFMM CVUXFDBOTIBSFWordTJNQM &WPMWFT NPSFJO %FSJWJOH7JB )JEFEBUBDPOTMkId
 outside the module

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3PMFTPGnewtypeT *NQMFNFOUBUJPO)JEJOH *NQMFNFOUBUJPO4IBSJOH *NQMFNFOUBUJPO4FMFDUJPO {-# LANGUAGE GeneralizedNewtypeDeriving #-} newtype Id = MkId Word deriving (Num, Eq) 6OEFSJWBCMFJO)BTLFMM CVUXFDBOTIBSFWordTJNQM &WPMWFT NPSFJO %FSJWJOH7JB module Data.Id (Id ()) where newtype Id = MkId Word )JEFEBUBDPOTMkId
 outside the module

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3PMFTPGnewtypeT *NQMFNFOUBUJPO)JEJOH *NQMFNFOUBUJPO4IBSJOH *NQMFNFOUBUJPO4FMFDUJPO {-# LANGUAGE GeneralizedNewtypeDeriving #-} newtype Id = MkId Word deriving (Num, Eq) 6OEFSJWBCMFJO)BTLFMM CVUXFDBOTIBSFWordTJNQM &WPMWFT NPSFJO %FSJWJOH7JB module Data.Id (Id ()) where newtype Id = MkId Word )JEFEBUBDPOTMkId
 outside the module %JTUJOHVJTIFEBUUZQFMFWFM CVUKVTUBWord internally

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*NQMFNFOUBUJPO 4FMFDUJPO

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*NQMFNFOUBUJPO 4FMFDUJPO .POPJE'PMEBCMFBT&YBNQMFT

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&YFSDJTF-JTU4DBOOJOH 2 (JWFOBMJTUPGJOUFHFST DBMDVMBUFTJUTNBYJNVN BOEUPUBMTVNCZTDBOOJOHMJTUFYBDUMZPODF ˞%POPUVTFfoldlPSfoldsQBDLBHFT

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5ZQJDBM"OTXFS

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5ZQJDBM"OTXFS 'PMET

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5ZQJDBM"OTXFS 'PMET aggregate :: [ℕ] " (Maybe ℕ, ℕ) aggregate = foldr (λ a (m, s) " (Just a `max` m, a + s)) (Nothing, 0) ˞ℕJTTIPSUGPS*OUFHFS

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5ZQJDBM"OTXFS 'PMET aggregate :: [ℕ] " (Maybe ℕ, ℕ) aggregate = foldr (λ a (m, s) " (Just a `max` m, a + s)) (Nothing, 0) 8FIBWFTJNJMBSPQFSBUJPOTPOCPUITJEFT ˞ℕJTTIPSUGPS*OUFHFS

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5ZQJDBM"OTXFS 'PMET aggregate :: [ℕ] " (Maybe ℕ, ℕ) aggregate = foldr (λ a (m, s) " (Just a `max` m, a + s)) (Nothing, 0) 8FIBWFTJNJMBSPQFSBUJPOTPOCPUITJEFT ˞ℕJTTIPSUGPS*OUFHFS

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5ZQJDBM"OTXFS 'PMET aggregate :: [ℕ] " (Maybe ℕ, ℕ) aggregate = foldr (λ a (m, s) " (Just a `max` m, a + s)) (Nothing, 0) 8FIBWFTJNJMBSPQFSBUJPOTPOCPUITJEFT .BQ#JOBSZ 0QFSBUJPO ˞ℕJTTIPSUGPS*OUFHFS

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5ZQJDBM"OTXFS 'PMET aggregate :: [ℕ] " (Maybe ℕ, ℕ) aggregate = foldr (λ a (m, s) " (Just a `max` m, a + s)) (Nothing, 0) 8FIBWFTJNJMBSPQFSBUJPOTPOCPUITJEFT .BQ#JOBSZ 0QFSBUJPO ˞ℕJTTIPSUGPS*OUFHFS

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5ZQJDBM"OTXFS 'PMET aggregate :: [ℕ] " (Maybe ℕ, ℕ) aggregate = foldr (λ a (m, s) " (Just a `max` m, a + s)) (Nothing, 0) 8FIBWFTJNJMBSPQFSBUJPOTPOCPUITJEFT .BQ#JOBSZ 0QFSBUJPO 6OJUT ˞ℕJTTIPSUGPS*OUFHFS

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5ZQJDBM"OTXFS 'PMET aggregate :: [ℕ] " (Maybe ℕ, ℕ) aggregate = foldr (λ a (m, s) " (Just a `max` m, a + s)) (Nothing, 0) 8FIBWFTJNJMBSPQFSBUJPOTPOCPUITJEFT .POPJET .BQ#JOBSZ 0QFSBUJPO 6OJUT ˞ℕJTTIPSUGPS*OUFHFS

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.POPJET w "OPQFSBUJPOXIJDIDBOCFDPNQVUFECPUI GSPNMFGUBOESJHIU XJUIVOJUFMFNFOU • Both `max` and (+) • … can be computed from either left or right, • … has Nothing (no max) and 0 as units. • Mapping to monoid + folding ⤳ Foldable! x ∙ (y ∙ z) = (x ∙ y) ∙ z x ∙ ε = x = ε ∙ x

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'PMEBCMFDMBTT class Foldable t where foldMap :: Monoid m 㱺 (a " m) " t a " m …

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'PMEBCMFDMBTT class Foldable t where foldMap :: Monoid m 㱺 (a " m) " t a " m … .BQUP.POPJE
 'PMEMFGUUPSJHIU

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'PMEBCMFDMBTT *UTV⒏DFTUPNBLFNBY pUJOUIJTGSBNFXPSL class Foldable t where foldMap :: Monoid m 㱺 (a " m) " t a " m … .BQUP.POPJE
 'PMEMFGUUPSJHIU

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'PMEBCMFDMBTT *UTV⒏DFTUPNBLFNBY pUJOUIJTGSBNFXPSL 8FDBOIBWFBUNPTUPOFJOTUBODFGPSMonoid Word class Foldable t where foldMap :: Monoid m 㱺 (a " m) " t a " m … .BQUP.POPJE
 'PMEMFGUUPSJHIU

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'PMEBCMFDMBTT *UTV⒏DFTUPNBLFNBY pUJOUIJTGSBNFXPSL 8FDBOIBWFBUNPTUPOFJOTUBODFGPSMonoid Word *NQMFNFOUBUJPO4FMFDUJPOVTJOHnewtypeT class Foldable t where foldMap :: Monoid m 㱺 (a " m) " t a " m … .BQUP.POPJE
 'PMEMFGUUPSJHIU

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&YBNQMFJOTUBODFT newtype Sum a = Sum { getSum :: a} deriving (Num, Integral) instance Num a 㱺 Monoid (Sum a) where (<>) = (+); ε = 0 newtype Max a = Max { getMax :: a } deriving (Num, Integral, Ord) instance Ord a 㱺 Semigroup (Max a) where (<>) = max

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&YBNQMFJOTUBODFT newtype Sum a = Sum { getSum :: a} deriving (Num, Integral) instance Num a 㱺 Monoid (Sum a) where (<>) = (+); ε = 0 newtype Max a = Max { getMax :: a } deriving (Num, Integral, Ord) instance Ord a 㱺 Semigroup (Max a) where (<>) = max .POPJEPG OVNFSJDBEEJUJPO

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&YBNQMFJOTUBODFT newtype Sum a = Sum { getSum :: a} deriving (Num, Integral) instance Num a 㱺 Monoid (Sum a) where (<>) = (+); ε = 0 newtype Max a = Max { getMax :: a } deriving (Num, Integral, Ord) instance Ord a 㱺 Semigroup (Max a) where (<>) = max .POPJEPG OVNFSJDBEEJUJPO 4FNJHSPVQCZ UBLJOHNBY

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&YBNQMFJOTUBODFT newtype Sum a = Sum { getSum :: a} deriving (Num, Integral) instance Num a 㱺 Monoid (Sum a) where (<>) = (+); ε = 0 newtype Max a = Max { getMax :: a } deriving (Num, Integral, Ord) instance Ord a 㱺 Semigroup (Max a) where (<>) = max .POPJEPG OVNFSJDBEEJUJPO 4FNJHSPVQCZ UBLJOHNBY *NQMTIBSJOHCZ (/%

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1SBDUJDBM3FNBSL w 0OMZCPVOEFEUZQFTDBOCF\max min^NPOPJET 8F OFFENBYJNVNNJOJNVNFMFNFOUUPIBWFUIFVOJU w 8FTUJMMIBWFB4FNJSJOH XIJDIMBDLTVOJUT w 8FIBWFUPDPOWFSUJUUPNPOPJEUPVTFXJUI'PMEBCMF newtype Max a = Max a instance Ord a 㱺 Semigroup (Max a) instance Bounded a 㱺 Monoid (Max a)

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1SBDUJDBM3FNBSL newtype Option a = Option (Maybe a) instance Semigroup a 㱺 Monoid (Option a)

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1SBDUJDBM3FNBSL OptionBEKPJOTVOJU UVSOJOHTFNJHSPVQTJOUPNPOPJET newtype Option a = Option (Maybe a) instance Semigroup a 㱺 Monoid (Option a)

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1SBDUJDBM3FNBSL OptionBEKPJOTVOJU UVSOJOHTFNJHSPVQTJOUPNPOPJET ʙ()$ɿ.BZCFSFRVJSFT.POPJEBBTBDPOTUSBJOU newtype Option a = Option (Maybe a) instance Semigroup a 㱺 Monoid (Option a) instance Monoid a 㱺 Monoid (Maybe a)

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1SBDUJDBM3FNBSL OptionBEKPJOTVOJU UVSOJOHTFNJHSPVQTJOUPNPOPJET ʙ()$ɿ.BZCFSFRVJSFT.POPJEBBTBDPOTUSBJOU ()$ʙɿ3FRVJSFTPOMZSemigroup a, no need of 0QUJPO newtype Option a = Option (Maybe a) instance Semigroup a 㱺 Monoid (Option a) instance Monoid a 㱺 Monoid (Maybe a) instance Semigroup a 㱺 Monoid (Maybe a)

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1SBDUJDBM3FNBSL OptionBEKPJOTVOJU UVSOJOHTFNJHSPVQTJOUPNPOPJET ʙ()$ɿ.BZCFSFRVJSFT.POPJEBBTBDPOTUSBJOU ()$ʙɿ3FRVJSFTPOMZSemigroup a, no need of 0QUJPO 8FTUJMMIBWFUPVTF0QUJPOUPXSJUFBQPSUBCMFDPEFTUIPVHI newtype Option a = Option (Maybe a) instance Semigroup a 㱺 Monoid (Option a) instance Monoid a 㱺 Monoid (Maybe a) instance Semigroup a 㱺 Monoid (Maybe a)

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'PMEBCMFOFXUZQFWFSTJPO import Control.Arrow aggregate :: [ℕ] " (Maybe ℕ, ℕ) aggregate = fmap getMax *** getSum ○ foldMap (Just . Max &&& Sum)

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'PMEBCMFOFXUZQFWFSTJPO 4PDPODJTF import Control.Arrow aggregate :: [ℕ] " (Maybe ℕ, ℕ) aggregate = fmap getMax *** getSum ○ foldMap (Just . Max &&& Sum)

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'PMEBCMFOFXUZQFWFSTJPO 4PDPODJTF *UTTUJMMUFEJPVTUPVOXSBQ.BYBOE4VNUIFZ TJUTJOOFTUFEUZQFT import Control.Arrow aggregate :: [ℕ] " (Maybe ℕ, ℕ) aggregate = fmap getMax *** getSum ○ foldMap (Just . Max &&& Sum)

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'PMEBCMFOFXUZQFWFSTJPO 4PDPODJTF *UTTUJMMUFEJPVTUPVOXSBQ.BYBOE4VNUIFZ TJUTJOOFTUFEUZQFT import Control.Arrow aggregate :: [ℕ] " (Maybe ℕ, ℕ) aggregate = fmap getMax *** getSum ○ foldMap (Just . Max &&& Sum) ‎;FSP$PTU$PFSDJPOT

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4BGF;FSP$PTU $PFSDJPOTBOE3PMFT

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4BGF;FSP$PTU $PFSDJPOTBOE3PMFT [1] Breitner, Eisenberg, Peyton Jones and Weirich, 2014 "HSFBUJOWFOUJPO
 PQFOJOHVQUIFOFXFSBPGOFXUZQFT

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#FGPSF )BTLFMMFSTDPNQMBJOU

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*OEFFE OFXUZQFTBSF DPOWJOFOUGPSJNQM TFMFDUJPO

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#VUXFIBWFUPVOXSBQ UIFNPOFCZPOF

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%PJOHTPJTOPUTP F⒏DJFOU

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4JODFXFLOPXJUTTBGF XFDBOVTF VOTBGF$PFSDF

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*UTOPURVJUFTNBSU

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#VU

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*O 
 5IF3FWPMVUJPOUPPLQMBDF UPOFXUZQFT

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;FSP$PTU$PFSDJPO

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;FSP$PTU$PFSDJPO import Data.Coerce (coerce) coerce :: Coercible a b 㱺 a " b

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;FSP$PTU$PFSDJPO w $PFSDJCMFSFMBUFTUXPUZQFTXJUIUIFTBNFNFNPSZSFQS import Data.Coerce (coerce) coerce :: Coercible a b 㱺 a " b

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;FSP$PTU$PFSDJPO w $PFSDJCMFSFMBUFTUXPUZQFTXJUIUIFTBNFNFNPSZSFQS w *UTFFNTMJLFBUZQFDMBTT CVU()$HFOFSBUFTBOJOGPSNBUJPO BUDPNQJMFUJNF BOEVTFSDBOOPUBEEDVTUPNJOTUBODF import Data.Coerce (coerce) coerce :: Coercible a b 㱺 a " b

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;FSP$PTU$PFSDJPO w $PFSDJCMFSFMBUFTUXPUZQFTXJUIUIFTBNFNFNPSZSFQS w *UTFFNTMJLFBUZQFDMBTT CVU()$HFOFSBUFTBOJOGPSNBUJPO BUDPNQJMFUJNF BOEVTFSDBOOPUBEEDVTUPNJOTUBODF w 8JUIcoerceGSPNData.Coerce XFDBOEP[FSPDPTUDBTUT import Data.Coerce (coerce) coerce :: Coercible a b 㱺 a " b

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;FSP$PTU$PFSDJPO w $PFSDJCMFSFMBUFTUXPUZQFTXJUIUIFTBNFNFNPSZSFQS w *UTFFNTMJLFBUZQFDMBTT CVU()$HFOFSBUFTBOJOGPSNBUJPO BUDPNQJMFUJNF BOEVTFSDBOOPUBEEDVTUPNJOTUBODF w 8JUIcoerceGSPNData.Coerce XFDBOEP[FSPDPTUDBTUT w *OGFSSFEQFSNPEVMF XFOFFEUIFJOGPPGEBUBDPOTUSVDUPS UPDBMMcoerce import Data.Coerce (coerce) coerce :: Coercible a b 㱺 a " b

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8JUIDPFSDF import Control.Arrow aggregate :: [ℕ] " (Maybe ℕ, ℕ) aggregate = fmap getMax *** getSum ○ foldMap (Just . Max &&& Sum)

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8FHFUJU import Control.Arrow import Data.Coerce aggregate :: [ℕ] " (Maybe ℕ, ℕ) aggregate = coerce ○ foldMap (Just . Max &&& Sum)

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8FHFUJU import Control.Arrow import Data.Coerce aggregate :: [ℕ] " (Maybe ℕ, ℕ) aggregate = coerce ○ foldMap (Just . Max &&& Sum) $BTUJOH OFTUFE UZQFTXJUI[FSP DPTU

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8FHFUJU import Control.Arrow import Data.Coerce aggregate :: [ℕ] " (Maybe ℕ, ℕ) aggregate = coerce ○ foldMap (Just . Max &&& Sum) coerce :: (Maybe (Max ), Sum ) → (Maybe , ) $BTUJOH OFTUFE UZQFTXJUI[FSP DPTU

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8FHFUJU import Control.Arrow import Data.Coerce aggregate :: [ℕ] " (Maybe ℕ, ℕ) aggregate = coerce ○ foldMap (Just . Max &&& Sum) w /PF⒎FDUPOUIFPGTDBOOJOHTJODFJUT[FSPDPTU coerce :: (Maybe (Max ), Sum ) → (Maybe , ) $BTUJOH OFTUFE UZQFTXJUI[FSP DPTU

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8FHFUJU import Control.Arrow import Data.Coerce aggregate :: [ℕ] " (Maybe ℕ, ℕ) aggregate = coerce ○ foldMap (Just . Max &&& Sum) w /PF⒎FDUPOUIFPGTDBOOJOHTJODFJUT[FSPDPTU w +VTUPOFDBMMGPSDPFSDFUPNBLFJUEPOF coerce :: (Maybe (Max ), Sum ) → (Maybe , ) $BTUJOH OFTUFE UZQFTXJUI[FSP DPTU

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$BTUJOHCXOFTUFEUZQFT

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$BTUJOHCXOFTUFEUZQFT *UTDPOWFOJFOUUIBUXFDBODBTUBOZOFTUFEUZQFT

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$BTUJOHCXOFTUFEUZQFT *UTDPOWFOJFOUUIBUXFDBODBTUBOZOFTUFEUZQFT 3FBMMZ

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$BTUJOHCXOFTUFEUZQFT *UTDPOWFOJFOUUIBUXFDBODBTUBOZOFTUFEUZQFT 3FBMMZ newtype Down a = Down a instance Ord a 㱺 Ord (Down a) where a ≤ b = b ≤ a data Heap a minView :: Heap a " Maybe a

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$BTUJOHCXOFTUFEUZQFT *UTDPOWFOJFOUUIBUXFDBODBTUBOZOFTUFEUZQFT 3FBMMZ newtype Down a = Down a instance Ord a 㱺 Ord (Down a) where a ≤ b = b ≤ a data Heap a minView :: Heap a " Maybe a 3FWFSTFE0SEFS

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$BTUJOHCXOFTUFEUZQFT *UTDPOWFOJFOUUIBUXFDBODBTUBOZOFTUFEUZQFT 3FBMMZ newtype Down a = Down a instance Ord a 㱺 Ord (Down a) where a ≤ b = b ≤ a data Heap a minView :: Heap a " Maybe a 3FWFSTFE0SEFS )FBQ

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$BTUJOHCXOFTUFEUZQFT *UTDPOWFOJFOUUIBUXFDBODBTUBOZOFTUFEUZQFT 3FBMMZ newtype Down a = Down a instance Ord a 㱺 Ord (Down a) where a ≤ b = b ≤ a data Heap a minView :: Heap a " Maybe a 3FWFSTFE0SEFS )FBQ .JOJNVN 0

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$BTUJOHCXOFTUFEUZQFT *UTDPOWFOJFOUUIBUXFDBODBTUBOZOFTUFEUZQFT 3FBMMZ newtype Down a = Down a instance Ord a 㱺 Ord (Down a) where a ≤ b = b ≤ a data Heap a minView :: Heap a " Maybe a 4FNBOUJDBMMZ Heap a.645/05CFDBTUFEUPHeap (Down a) 3FWFSTFE0SEFS )FBQ .JOJNVN 0

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3PMFT

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3PMFT w 8FDBODBTUUIFNXJUIDPFSDF

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3PMFT w 8FDBODBTUUIFNXJUIDPFSDF ghci> h = fromList [1,2,3] :: Heap Int ghci> minView (coerce h :: Heap (Down Int)) Just 1

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3PMFT w 8FDBODBTUUIFNXJUIDPFSDF ghci> h = fromList [1,2,3] :: Heap Int ghci> minView (coerce h :: Heap (Down Int)) Just 1 .VTUCFJust 3!

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3PMFT w 8FDBODBTUUIFNXJUIDPFSDF ghci> h = fromList [1,2,3] :: Heap Int ghci> minView (coerce h :: Heap (Down Int)) Just 1 w 5IFOXFTQFDJGZUIF3PMF .VTUCFJust 3!

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3PMFT w 8FDBODBTUUIFNXJUIDPFSDF type role Heap nominal ghci> h = fromList [1,2,3] :: Heap Int ghci> minView (coerce h :: Heap (Down Int)) Just 1 w 5IFOXFTQFDJGZUIF3PMF .VTUCFJust 3!

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3PMFT w 8FDBODBTUUIFNXJUIDPFSDF type role Heap nominal ghci> h = fromList [1,2,3] :: Heap Int ghci> minView (coerce h :: Heap (Down Int)) Just 1 w 5IFOXFTQFDJGZUIF3PMF ghci> minView (coerce h :: Heap (Down Int)) error: Couldn't match type ‘Int’ with ‘Down Int’ .VTUCFJust 3!

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&YBNQMFPG*TP data Blog = Blog { authors :: [Author] , articles :: [Article] } deriving (Generic) deriving (Semigroup, Monoid) via Blog `SameRepAs` ([Author], Dual [Article]) {- ghci> mconcat [Blog ["1"] ["1"], Blog ["2"] ["3","4"]] Blog {authors = ["1","2"], articles = ["3","4","1"]} -}

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&YBNQMFPG*TP DPOU newtype SameRepAs a b = SameRepAs { runSameRepAs :: a } type Iso a b = (Generic a, Generic b, Coercible (Rep a ()) (Rep b ())) instance (Semigroup b, Iso a b) 㱺 Semigroup (SameRepAs a b) where SameRepAs a <> SameRepAs b = ... instance (Monoid b, Iso a b) 㱺 Monoid (SameRepAs a b) where mempty = SameRepAs $ toA mempty where toA :: b -> a toA = to . (coerce :: Rep b () -> Rep a ()) . from

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3FGFSFODFT 1. J. Breitner, R. A. Eisenberg, S. P. Jones and S. Weirich, Safe Zero-cost Coercions for Haskell, ICFP 2014. 2. Baldur Blöndal, Andres Löh and Ryan Scott, Deriving Via: How to Turn Hand-Written Instances into an Anti-Pattern, ICFP18.