The document discusses records and row polymorphism in Haskell. It begins by explaining that Haskell records are nominally typed and cannot share field names. It then discusses structural typing and row polymorphism as ways to address these limitations. The document proposes modeling records as products in a universe, and corecords as sums. It shows how to implement records and row polymorphism in Haskell using type families, and how this recovers HLists. The document also discusses validating records, effects inside records, and modeling records as presheaves.

1. Programming in Vinyl
Jon Sterling
jonmsterling.com
May 18, 2014

2. Records in GHC 7.8

3. Records in GHC 7.8
Haskell records are nominally typed

4. Records in GHC 7.8
Haskell records are nominally typed
They may not share field names

5. Records in GHC 7.8
Haskell records are nominally typed
They may not share field names
data R = R { x :: X }

6. Records in GHC 7.8
Haskell records are nominally typed
They may not share field names
data R = R { x :: X }
data R’ = R’ { x :: X } −− ˆ Error

7. Structural Typing

8. Structural Typing
Sharing field names and accessors

9. Structural Typing
Sharing field names and accessors
Record types may be characterized structurally

10. Row Polymorphism

11. Row Polymorphism
How do we express the type of a function which adds a
field to a record?

12. Row Polymorphism
How do we express the type of a function which adds a
field to a record?
x : {foo : A}
f(x) : {foo : A, bar : B}

13. Row Polymorphism
How do we express the type of a function which adds a
field to a record?
x : {foo : A; rs}
f(x) : {foo : A, bar : B; rs}

14. Roll Your Own in Haskell

15. Roll Your Own in Haskell
data (s :: Symbol) ::: (t :: ∗) = Field

16. Roll Your Own in Haskell
data (s :: Symbol) ::: (t :: ∗) = Field
data Rec :: [∗] → ∗ where

17. Roll Your Own in Haskell
data (s :: Symbol) ::: (t :: ∗) = Field
data Rec :: [∗] → ∗ where
RNil :: Rec ’[]

18. Roll Your Own in Haskell
data (s :: Symbol) ::: (t :: ∗) = Field
data Rec :: [∗] → ∗ where
RNil :: Rec ’[]
(:&) :: !t → !(Rec rs) → Rec ((s ::: t) ’: rs)

19. Roll Your Own in Haskell
data (s :: Symbol) ::: (t :: ∗) = Field
data Rec :: [∗] → ∗ where
RNil :: Rec ’[]
(:&) :: !t → !(Rec rs) → Rec ((s ::: t) ’: rs)
class s ∈ (rs :: [∗])

20. Roll Your Own in Haskell
data (s :: Symbol) ::: (t :: ∗) = Field
data Rec :: [∗] → ∗ where
RNil :: Rec ’[]
(:&) :: !t → !(Rec rs) → Rec ((s ::: t) ’: rs)
class s ∈ (rs :: [∗])
class ss ⊆ (rs :: [∗]) where
cast :: Rec rs → Rec ss

21. Roll Your Own in Haskell
data (s :: Symbol) ::: (t :: ∗) = Field
data Rec :: [∗] → ∗ where
RNil :: Rec ’[]
(:&) :: !t → !(Rec rs) → Rec ((s ::: t) ’: rs)
class s ∈ (rs :: [∗])
class ss ⊆ (rs :: [∗]) where
cast :: Rec rs → Rec ss
(=:) : s ::: t → t → Rec ’[s ::: t]

22. Roll Your Own in Haskell
data (s :: Symbol) ::: (t :: ∗) = Field
data Rec :: [∗] → ∗ where
RNil :: Rec ’[]
(:&) :: !t → !(Rec rs) → Rec ((s ::: t) ’: rs)
class s ∈ (rs :: [∗])
class ss ⊆ (rs :: [∗]) where
cast :: Rec rs → Rec ss
(=:) : s ::: t → t → Rec ’[s ::: t]
(⊕) : Rec ss → Rec ts → Rec (ss ++ ts)

23. Roll Your Own in Haskell
data (s :: Symbol) ::: (t :: ∗) = Field
data Rec :: [∗] → ∗ where
RNil :: Rec ’[]
(:&) :: !t → !(Rec rs) → Rec ((s ::: t) ’: rs)
class s ∈ (rs :: [∗])
class ss ⊆ (rs :: [∗]) where
cast :: Rec rs → Rec ss
(=:) : s ::: t → t → Rec ’[s ::: t]
(⊕) : Rec ss → Rec ts → Rec (ss ++ ts)
lens : s ::: t ∈ rs ⇒ s ::: t → Lens’ (Rec rs) t

24. Roll Your Own in Haskell

25. Roll Your Own in Haskell
f :: Rec (”foo” ::: A ’: rs)
→ Rec (”bar” ::: B ’: ”foo” ::: A ’: rs)

26. Universes `a la Tarski

27. Universes `a la Tarski
A type U of codes for types.

28. Universes `a la Tarski
A type U of codes for types.
Function ElU : U → Type.

29. Universes `a la Tarski
A type U of codes for types.
Function ElU : U → Type.
Γ s : U
Γ ElU (s) : Type

30. Universes `a la Tarski
Type

31. Universes `a la Tarski

Type

32. Universes `a la Tarski

Type

33. Universes `a la Tarski

Type

34. A Closed Universe

35. A Closed Universe
Let A be a universe of address books:

36. A Closed Universe
Let A be a universe of address books:
Statics:

37. A Closed Universe
Let A be a universe of address books:
Statics:
A : Type

38. A Closed Universe
Let A be a universe of address books:
Statics:
A : Type Label : Type

39. A Closed Universe
Let A be a universe of address books:
Statics:
A : Type Label : Type Home, Office : Label

40. A Closed Universe
Let A be a universe of address books:
Statics:
A : Type Label : Type Home, Office : Label
Name : A

41. A Closed Universe
Let A be a universe of address books:
Statics:
A : Type Label : Type Home, Office : Label
Name : A
: Label
Phone[ ], Email[ ] : A

42. A Closed Universe
Let A be a universe of address books:
Statics:
A : Type Label : Type Home, Office : Label
Name : A
: Label
Phone[ ], Email[ ] : A
s : A
ElA(s) : Type

43. A Closed Universe
Let A be a universe of address books:
Statics:
A : Type Label : Type Home, Office : Label
Name : A
: Label
Phone[ ], Email[ ] : A
s : A
ElA(s) : Type
Dynamics:

44. A Closed Universe
Let A be a universe of address books:
Statics:
A : Type Label : Type Home, Office : Label
Name : A
: Label
Phone[ ], Email[ ] : A
s : A
ElA(s) : Type
Dynamics:
ElA(Name) string

45. A Closed Universe
Let A be a universe of address books:
Statics:
A : Type Label : Type Home, Office : Label
Name : A
: Label
Phone[ ], Email[ ] : A
s : A
ElA(s) : Type
Dynamics:
ElA(Name) string ElA(Email[ ]) string

46. A Closed Universe
Let A be a universe of address books:
Statics:
A : Type Label : Type Home, Office : Label
Name : A
: Label
Phone[ ], Email[ ] : A
s : A
ElA(s) : Type
Dynamics:
ElA(Name) string ElA(Email[ ]) string
ElA(Phone[ ]) list(N)

47. Records as Products

48. Records as Products
Records: the product of the image of ElU in Type
restricted to a subset of U.

49. Records as Products
Records: the product of the image of ElU in Type
restricted to a subset of U.
recordU
V⊆U V
ElU |V

50. Records as Products

Type

51. Records as Products

Type
×
×
×
×

52. Records as Products
recordU
V⊆U V
ElU |V

53. Example Record
recordU
V⊆U V
ElU |V

54. Example Record
recordU
V⊆U V
ElU |V
A {Name, Email Work}

55. Example Record
recordU
V⊆U V
ElU |V
A {Name, Email Work}
ex : recordU

56. Example Record
recordU
V⊆U V
ElU |V
A {Name, Email Work}
ex : recordU
ex A , λ.
{Name → ”Robert Harper”;
Email Work → ”rwh@cs.cmu.edu”}

57. Presheaves

58. Presheaves
A presheaf on some space X is a functor
O(X)op
→ Type, where O is the category of open sets of
X for whatever topology you have chosen.

59. Topologies on some space X

60. Topologies on some space X
What are the open sets on X?

61. Topologies on some space X
What are the open sets on X?
The empty set and X are open sets
The union of open sets is open
Finite intersections of open sets are open

62. Records are Presheaves

63. Records are Presheaves
Let O = P, the discrete topology

64. Records are Presheaves
Let O = P, the discrete topology
Then records on a universe X give rise to a presheaf
R: subset inclusions are taken to casts from larger to
smaller records

65. Records are Presheaves
Let O = P, the discrete topology
Then records on a universe X give rise to a presheaf
R: subset inclusions are taken to casts from larger to
smaller records
for V ⊆ X R(V ) :≡
V
ElX|V : Type

66. Records are Presheaves
Let O = P, the discrete topology
Then records on a universe X give rise to a presheaf
R: subset inclusions are taken to casts from larger to
smaller records
for V ⊆ X R(V ) :≡
V
ElX|V : Type
for i : V → U R(i) :≡ cast : R(U) → R(V )

67. Records are Sheaves

68. Records are Sheaves
For a cover U = i Ui on X, then:
R(U) i R(Ui) i,j R(Ui ∩ Uj)
e p
q
is an equalizer, where
e = λr.λi. castUi
(r)
p = λf.λi.λj. castUi∩Uj
(f(i))
q = λf.λi.λj. castUi∩Uj
(f(j))

69. Records are Sheaves
For a cover U = i Ui on X, then:
R(U) i R(Ui) i,j R(Ui ∩ Uj)
Γ
e
m
!u
p
q
where
e = λr.λi. castUi
(r)
p = λf.λi.λj. castUi∩Uj
(f(i))
q = λf.λi.λj. castUi∩Uj
(f(j))

70. Corecords as Sums

71. Corecords as Sums
Corecords (extensible variants): the sum of the image of
ElU in Type restricted to a subset of U.

72. Corecords as Sums
Corecords (extensible variants): the sum of the image of
ElU in Type restricted to a subset of U.
corecordU
V⊆U V
ElU |V

73. Corecords as Sums

Type

74. Corecords as Sums

Type
+
+
+
+

75. Corecords as Sums
corecordU
V⊆U V
ElU |V

76. Doing it in Haskell

77. Doing it in Haskell
Create a universe U at the type-level

78. Doing it in Haskell
Create a universe U at the type-level
Use type families to approximate ElU

79. Doing it in Haskell
Create a universe U at the type-level
Use type families to approximate ElU
Parameterize Rec by U, ElU ?

80. Records in Haskell

81. Records in Haskell
data Rec :: (U → ∗) → [ U ] → ∗ where

82. Records in Haskell
data Rec :: (U → ∗) → [ U ] → ∗ where
RNil :: Rec elU ’[]

83. Records in Haskell
data Rec :: (U → ∗) → [ U ] → ∗ where
RNil :: Rec elU ’[]
(:&) :: !(elU r) → !(Rec elU rs) → Rec elU (r ’: rs)

84. Records in Haskell (Actually)

85. Records in Haskell (Actually)
data TyFun :: ∗ → ∗ → ∗

86. Records in Haskell (Actually)
data TyFun :: ∗ → ∗ → ∗
type family (f :: TyFun k l → ∗) $ (x :: k) :: l

87. Records in Haskell (Actually)
data TyFun :: ∗ → ∗ → ∗
type family (f :: TyFun k l → ∗) $ (x :: k) :: l
data Rec :: (TyFun U ∗ → ∗) → [ U ] → ∗ where
RNil :: Rec elU ’[]
(:&) :: !(elU $ r) → !(Rec elU rs) → Rec elU (r ’: rs)

88. Recovering HList

89. Recovering HList
data Id :: (TyFun k k) → ∗ where
type instance Id $ x = x

90. Recovering HList
data Id :: (TyFun k k) → ∗ where
type instance Id $ x = x
type HList rs = Rec Id rs

91. Recovering HList
data Id :: (TyFun k k) → ∗ where
type instance Id $ x = x
type HList rs = Rec Id rs
ex :: HList [Z, Bool, String]

92. Recovering HList
data Id :: (TyFun k k) → ∗ where
type instance Id $ x = x
type HList rs = Rec Id rs
ex :: HList [Z, Bool, String]
ex = 34 :& True :& ”vinyl” :& RNil

93. Validating Records
bob :: Rec ElA [Name, Email Work]

94. Validating Records
bob :: Rec ElA [Name, Email Work]
bob = Name =: ”Robert Harper”
⊕ Email Work =: ”rwh@cs.cmu.edu”

95. Validating Records
bob :: Rec ElA [Name, Email Work]
bob = Name =: ”Robert Harper”
⊕ Email Work =: ”rwh@cs.cmu.edu”
validateName :: String → Either Error String
validateEmail :: String → Either Error String
validatePhone :: [N] → Either Error [N]

96. Validating Records
bob :: Rec ElA [Name, Email Work]
bob = Name =: ”Robert Harper”
⊕ Email Work =: ”rwh@cs.cmu.edu”
validateName :: String → Either Error String
validateEmail :: String → Either Error String
validatePhone :: [N] → Either Error [N]
*unnnnnhhh...*

97. Validating Records
bob :: Rec ElA [Name, Email Work]
bob = Name =: ”Robert Harper”
⊕ Email Work =: ”rwh@cs.cmu.edu”
validateName :: String → Either Error String
validateEmail :: String → Either Error String
validatePhone :: [N] → Either Error [N]
*unnnnnhhh...*
validateContact
:: Rec ElA [Name, Email Work]
→ Either Error (Rec ElA [Name, Email Work])

98. Welp.

99. Effects inside records
data Rec :: (TyFun U ∗ → ∗) → [ U ] → ∗ where
RNil :: Rec elU ’[]
(:&) :: !(elU $ r) → !(Rec elU rs) → Rec elU (r ’: rs)

100. Effects inside records
data Rec :: (TyFun U ∗ → ∗) → (∗ → ∗) → [ U ] → ∗ where
RNil :: Rec elU f ’[]
(:&) :: !(f (elU $ r)) → !(Rec elU f rs) → Rec elU f (r ’: rs)

101. Effects inside records
data Rec :: (TyFun U ∗ → ∗) → (∗ → ∗) → [ U ] → ∗ where
RNil :: Rec elU f ’[]
(:&) :: !(f (elU $ r)) → !(Rec elU f rs) → Rec elU f (r ’: rs)
(=:) : Applicative f ⇒ sing r → elU $ r → Rec elU f ’[r]

102. Effects inside records
data Rec :: (TyFun U ∗ → ∗) → (∗ → ∗) → [ U ] → ∗ where
RNil :: Rec elU f ’[]
(:&) :: !(f (elU $ r)) → !(Rec elU f rs) → Rec elU f (r ’: rs)
(=:) : Applicative f ⇒ sing r → elU $ r → Rec elU f ’[r]
k =: x = pure x :& RNil

103. Effects inside records
data Rec :: (TyFun U ∗ → ∗) → (∗ → ∗) → [ U ] → ∗ where
RNil :: Rec elU f ’[]
(:&) :: !(f (elU $ r)) → !(Rec elU f rs) → Rec elU f (r ’: rs)
(=:) : Applicative f ⇒ sing r → elU $ r → Rec elU f ’[r]
k =: x = pure x :& RNil
(⇐): sing r → f (elU $ r) → Rec elU f ’[r]

104. Effects inside records
data Rec :: (TyFun U ∗ → ∗) → (∗ → ∗) → [ U ] → ∗ where
RNil :: Rec elU f ’[]
(:&) :: !(f (elU $ r)) → !(Rec elU f rs) → Rec elU f (r ’: rs)
(=:) : Applicative f ⇒ sing r → elU $ r → Rec elU f ’[r]
k =: x = pure x :& RNil
(⇐): sing r → f (elU $ r) → Rec elU f ’[r]
k ⇐ x = x :& RNil

105. Compositional Validation
type RecA = Rec ElA

106. Compositional Validation
type RecA = Rec ElA
bob :: RecA Identity [Name, Email Work]

107. Compositional Validation
type RecA = Rec ElA
bob :: RecA Identity [Name, Email Work]
bob = Name =: ”Robert Harper”
⊕ Email Work =: ”rwh@cs.cmu.edu”

108. Compositional Validation
type RecA = Rec ElA
bob :: RecA Identity [Name, Email Work]
bob = Name =: ”Robert Harper”
⊕ Email Work =: ”rwh@cs.cmu.edu”

109. Compositional Validation
type Validator a = a → Either Error a

110. Compositional Validation
type Validator a = a → Either Error a
validateName :: RecA Validator ’[Name]
validatePhone :: ∀ . RecA Validator ’[Phone ]
validateEmail :: ∀ . RecA Validator ’[Email ]

111. Compositional Validation
type Validator a = a → Either Error a
validateName :: RecA Validator ’[Name]
validatePhone :: ∀ . RecA Validator ’[Phone ]
validateEmail :: ∀ . RecA Validator ’[Email ]
type TotalContact =
[ Name, Email Home, Email Work
, Phone Home, Phone Work ]

112. Compositional Validation
type Validator a = a → Either Error a
validateName :: RecA Validator ’[Name]
validatePhone :: ∀ . RecA Validator ’[Phone ]
validateEmail :: ∀ . RecA Validator ’[Email ]
type TotalContact =
[ Name, Email Home, Email Work
, Phone Home, Phone Work ]
validateContact :: RecA Validator TotalContact
validateContact = validateName
⊕ validateEmail
⊕ validateEmail
⊕ validatePhone
⊕ validatePhone

113. Record Operators

114. Record Operators
newtype Lift o f g x = Lift { runLift :: f x ‘o‘ g x }

115. Record Operators
newtype Lift o f g x = Lift { runLift :: f x ‘o‘ g x }
type Validator = Lift (→) Identity (Either Error)

116. Record Operators
newtype Lift o f g x = Lift { runLift :: f x ‘o‘ g x }
type Validator = Lift (→) Identity (Either Error)
( ) :: RecU (Lift (→) f g) rs → RecU f rs → RecU g rs

117. Record Operators
newtype Lift o f g x = Lift { runLift :: f x ‘o‘ g x }
type Validator = Lift (→) Identity (Either Error)
( ) :: RecU (Lift (→) f g) rs → RecU f rs → RecU g rs
rdist :: Applicative f ⇒ RecU f rs → f (RecU Identity rs)

118. Compositional Validation
newtype Lift o f g x = Lift { runLift :: f x ‘o‘ g x }
type Validator = Lift (→) Identity (Either Error)
( ) :: RecU (Lift (→) f g) rs → RecU f rs → RecU g rs
rdist :: Applicative f ⇒ RecU f rs → f (RecU Identity rs)

119. Compositional Validation
newtype Lift o f g x = Lift { runLift :: f x ‘o‘ g x }
type Validator = Lift (→) Identity (Either Error)
( ) :: RecU (Lift (→) f g) rs → RecU f rs → RecU g rs
rdist :: Applicative f ⇒ RecU f rs → f (RecU Identity rs)
validateContact :: RecA Validator TotalContact

120. Compositional Validation
newtype Lift o f g x = Lift { runLift :: f x ‘o‘ g x }
type Validator = Lift (→) Identity (Either Error)
( ) :: RecU (Lift (→) f g) rs → RecU f rs → RecU g rs
rdist :: Applicative f ⇒ RecU f rs → f (RecU Identity rs)
validateContact :: RecA Validator TotalContact
bobValid :: RecA (Either Error) [Name, Email Work]

121. Compositional Validation
newtype Lift o f g x = Lift { runLift :: f x ‘o‘ g x }
type Validator = Lift (→) Identity (Either Error)
( ) :: RecU (Lift (→) f g) rs → RecU f rs → RecU g rs
rdist :: Applicative f ⇒ RecU f rs → f (RecU Identity rs)
validateContact :: RecA Validator TotalContact
bobValid :: RecA (Either Error) [Name, Email Work]
bobValid = cast validateContact bob

122. Compositional Validation
newtype Lift o f g x = Lift { runLift :: f x ‘o‘ g x }
type Validator = Lift (→) Identity (Either Error)
( ) :: RecU (Lift (→) f g) rs → RecU f rs → RecU g rs
rdist :: Applicative f ⇒ RecU f rs → f (RecU Identity rs)
validateContact :: RecA Validator TotalContact
bobValid :: RecA (Either Error) [Name, Email Work]
bobValid = cast validateContact bob
validBob :: Either Error (RecA Identity [Name, Email Work])

123. Compositional Validation
newtype Lift o f g x = Lift { runLift :: f x ‘o‘ g x }
type Validator = Lift (→) Identity (Either Error)
( ) :: RecU (Lift (→) f g) rs → RecU f rs → RecU g rs
rdist :: Applicative f ⇒ RecU f rs → f (RecU Identity rs)
validateContact :: RecA Validator TotalContact
bobValid :: RecA (Either Error) [Name, Email Work]
bobValid = cast validateContact bob
validBob :: Either Error (RecA Identity [Name, Email Work])
validBob = rdist bobValid

124. Laziness as an effect

125. Laziness as an effect
newtype Identity a = Identity { runIdentity :: a }

126. Laziness as an effect
newtype Identity a = Identity { runIdentity :: a }
data Thunk a = Thunk { unThunk :: a }

127. Laziness as an effect
newtype Identity a = Identity { runIdentity :: a }
data Thunk a = Thunk { unThunk :: a }
type PlainRecU rs = RecU Identity rs

128. Laziness as an effect
newtype Identity a = Identity { runIdentity :: a }
data Thunk a = Thunk { unThunk :: a }
type PlainRecU rs = RecU Identity rs
type LazyRecU rs = RecU Thunk rs

129. Concurrent Records with Async

130. Concurrent Records with Async
fetchName :: RecA IO ’[Name]

131. Concurrent Records with Async
fetchName :: RecA IO ’[Name]
fetchName = Name ⇐ someOperation

132. Concurrent Records with Async
fetchName :: RecA IO ’[Name]
fetchName = Name ⇐ someOperation
fetchWorkEmail :: RecA IO ’[Email Work]

133. Concurrent Records with Async
fetchName :: RecA IO ’[Name]
fetchName = Name ⇐ someOperation
fetchWorkEmail :: RecA IO ’[Email Work]
fetchWorkEmail = Email Work ⇐ anotherOperation

134. Concurrent Records with Async
fetchName :: RecA IO ’[Name]
fetchName = Name ⇐ someOperation
fetchWorkEmail :: RecA IO ’[Email Work]
fetchWorkEmail = Email Work ⇐ anotherOperation
fetchBob :: RecA IO [Name, Email Work]

135. Concurrent Records with Async
fetchName :: RecA IO ’[Name]
fetchName = Name ⇐ someOperation
fetchWorkEmail :: RecA IO ’[Email Work]
fetchWorkEmail = Email Work ⇐ anotherOperation
fetchBob :: RecA IO [Name, Email Work]
fetchBob = fetchName ⊕ fetchWorkEmail

136. Concurrent Records with Async

137. Concurrent Records with Async
newtype Concurrently a
= Concurrently { runConcurrently :: IO a }

138. Concurrent Records with Async
newtype Concurrently a
= Concurrently { runConcurrently :: IO a }
( $ ) :: (∀ a. f a → g a) → RecU f rs → RecU g rs

139. Concurrent Records with Async
newtype Concurrently a
= Concurrently { runConcurrently :: IO a }
( $ ) :: (∀ a. f a → g a) → RecU f rs → RecU g rs
bobConcurrently :: RecA Concurrently [Name, Email Work]

140. Concurrent Records with Async
newtype Concurrently a
= Concurrently { runConcurrently :: IO a }
( $ ) :: (∀ a. f a → g a) → RecU f rs → RecU g rs
bobConcurrently :: RecA Concurrently [Name, Email Work]
bobConcurrently = Concurrently $ fetchBob

141. Concurrent Records with Async
newtype Concurrently a
= Concurrently { runConcurrently :: IO a }
( $ ) :: (∀ a. f a → g a) → RecU f rs → RecU g rs
bobConcurrently :: RecA Concurrently [Name, Email Work]
bobConcurrently = Concurrently $ fetchBob
concurrentBob :: Concurrently (RecA Identity [...])

142. Concurrent Records with Async
newtype Concurrently a
= Concurrently { runConcurrently :: IO a }
( $ ) :: (∀ a. f a → g a) → RecU f rs → RecU g rs
bobConcurrently :: RecA Concurrently [Name, Email Work]
bobConcurrently = Concurrently $ fetchBob
concurrentBob :: Concurrently (RecA Identity [...])
concurrentBob = rdist bobConcurrently

143. Concurrent Records with Async

144. Concurrent Records with Async
fetchBob :: RecA IO [Name, Email Work]
bobConcurrently :: RecA Concurrently [Name, Email Work]
concurrentBob :: Concurrently (RecA Identity [...])

145. Concurrent Records with Async
fetchBob :: RecA IO [Name, Email Work]
bobConcurrently :: RecA Concurrently [Name, Email Work]
concurrentBob :: Concurrently (RecA Identity [...])
bob :: IO (RecA Identity [Name, Email Work])

146. Concurrent Records with Async
fetchBob :: RecA IO [Name, Email Work]
bobConcurrently :: RecA Concurrently [Name, Email Work]
concurrentBob :: Concurrently (RecA Identity [...])
bob :: IO (RecA Identity [Name, Email Work])
bob = runConcurrently concurrentBob

147. Containers: The Syntax for Data Types
container : Type

148. Containers: The Syntax for Data Types
container : Type
U : Type ElU : U → Type
U ElU : container

149. Containers: The Syntax for Data Types
container : Type
U : Type ElU : U → Type
U ElU : container
C : container
C.Sh : Type

150. Containers: The Syntax for Data Types
container : Type
U : Type ElU : U → Type
U ElU : container
C : container
C.Sh : Type
C U ElU
C.Sh U

151. Containers: The Syntax for Data Types
container : Type
U : Type ElU : U → Type
U ElU : container
C : container
C.Sh : Type
C U ElU
C.Sh U
C : container
C.Po : C.Sh → Type

152. Containers: The Syntax for Data Types
container : Type
U : Type ElU : U → Type
U ElU : container
C : container
C.Sh : Type
C U ElU
C.Sh U
C : container
C.Po : C.Sh → Type
C U ElU
C.Po ElU

153. Restricting Containers
C : container V ⊆ C.Sh
C|V : container

154. Restricting Containers
C : container V ⊆ C.Sh
C|V : container
C U ElU
C|V V ElU |V

155. Container Lifting
C : container F : Type → Type
C ↑ F : container

156. Container Lifting
C : container F : Type → Type
C ↑ F : container
C U ElU
C ↑ F U F ◦ ElU

157. A Menagerie of Quantifiers

158. A Menagerie of Quantifiers
Dependent Products:
Γ A : Type Γ, x : A B : Type
Γ A B : Type
Γ, x : A e : B[x]
Γ λx.e : A B

159. A Menagerie of Quantifiers
Dependent Sums:
Γ A : Type Γ, x : A B : Type
Γ A B : Type
Γ a : A Γ b : B[a]
Γ a, b : A B

160. A Menagerie of Quantifiers
Inductive Types:
Γ A : Type Γ, x : A B : Type
Γ WA B : Type
Γ a : A Γ, v : B[a] b : WA B
Γ sup(a; v. b) : WA B

161. A Menagerie of Quantifiers
Coinductive Types:
Γ A : Type Γ, x : A B : Type
Γ MA B : Type
Γ a : A Γ, v : B[a] b : ∞ (MA B)
Γ inf(a; v. b) : MA B

162. A Scheme for Quantifiers

163. A Scheme for Quantifiers
Γ, A : Type, (x : A B : Type) QAB : Type
Γ Q quantifier

164. Quantifiers Give Containers Semantics
Γ, A : Type, (x : A B : Type) QAB : Type
Γ Q quantifier

165. Quantifiers Give Containers Semantics
Γ, A : Type, (x : A B : Type) QAB : Type
Γ Q quantifier
C : container Q quantifier
C Q : Type

166. Quantifiers Give Containers Semantics
Γ, A : Type, (x : A B : Type) QAB : Type
Γ Q quantifier
C : container Q quantifier
C Q : Type
C U ElU
C Q QU ElU

167. Vinyl Records as Containers

168. Vinyl Records as Containers
Records and corecords are finite products and sums
respectively.

169. Vinyl Records as Containers
Records and corecords are finite products and sums
respectively.
Rec ElU F rs ∼= (U ElU )|rs − ↑ F Π

170. Vinyl Records as Containers
Records and corecords are finite products and sums
respectively.
Rec ElU F rs ∼= (U ElU )|rs − ↑ F Π
CoRec ElU F rs ∼= (U ElU )|rs − ↑ F Σ

171. Vinyl Records as Containers
Records and corecords are finite products and sums
respectively.
Rec ElU F rs ∼= (U ElU )|rs − ↑ F Π
CoRec ElU F rs ∼= (U ElU )|rs − ↑ F Σ
??? ElU F rs ∼= (U ElU )|rs − ↑ F W

172. Vinyl Records as Containers
Records and corecords are finite products and sums
respectively.
Rec ElU F rs ∼= (U ElU )|rs − ↑ F Π
CoRec ElU F rs ∼= (U ElU )|rs − ↑ F Σ
??? ElU F rs ∼= (U ElU )|rs − ↑ F W
??? ElU F rs ∼= (U ElU )|rs − ↑ F M

173. Questions