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|
--
-- Prelude for the transfer language.
--
--
-- Basic functions
--
const : (A:Type) -> (B:Type) -> A -> B -> A
const _ _ x _ = x
id : (A:Type) -> A -> A
id _ x = x
--
-- The Bool type
--
data Bool : Type where
True : Bool
False : Bool
not : Bool -> Bool
not b = if b then False else True
--
-- The Add class
--
Add : Type -> Type
Add = sig zero : A
plus : A -> A -> A
zero : (A : Type) -> Add A -> A
zero _ d = d.zero
plus : (A : Type) -> Add A -> A -> A -> A
plus _ d = d.plus
sum : (A:Type) -> Add A -> List A -> A
sum _ d (Nil _) = d.zero
sum A d (Cons _ x xs) = d.plus x (sum A d xs)
-- Operators:
{-
(x + y) => (plus ? ? x y)
-}
-- Instances:
add_Integer : Add Integer
add_Integer = rec zero = 0
plus = prim_add_Int
add_String : Add String
add_String = rec zero = ""
plus = prim_add_Str
--
-- The Prod class
--
Prod : Type -> Type
Prod = sig one : A
times : A -> A -> A
one : (A : Type) -> Prod A -> A
one _ d = d.one
times : (A : Type) -> Prod A -> A -> A -> A
times _ d = d.times
product : (A:Type) -> Prod A -> List A -> A
product _ d (Nil _) = d.one
product A d (Cons _ x xs) = d.times x (product A d xs)
-- Operators:
{-
(x * y) => (times ? ? x y)
-}
-- Instances:
prod_Integer : Add Integer
prod_Integer = rec one = 1
times = prim_mul_Int
--
-- The Neg class
--
Neg : Type -> Type
Neg = sig negate : A -> A
negate : (A : Type) -> Neg A -> A -> A
negate _ d = d.neg
-- Operators:
{-
(-x) => negate ? ? x
-}
-- Instances:
neg_Integer : Neg Integer
neg_Integer = rec negate = prim_neg_Int
neg_Bool : Neg Bool
neg_Bool = rec negate = not
--
-- The Eq class
--
Eq : Type -> Type
Eq A = sig eq : A -> A -> Bool
eq : (A : Type) -> Eq A -> A -> A -> Bool
eq _ d = d.eq
neq : (A : Type) -> Eq A -> A -> A -> Bool
neq A d x y = not (eq A d x y)
-- Operators:
{-
(x == y) => (eq ? ? x y)
(x /= y) => (neq ? ? x y)
-}
-- Instances:
eq_Integer : Eq Integer
eq_Integer = rec eq = prim_eq_Int
eq_String : Eq String
eq_String = rec eq = prim_eq_Str
--
-- The Ord class
--
data Ordering : Type where
LT : Ordering
EQ : Ordering
GT : Ordering
Ord : Type -> Type
Ord A = sig eq : A -> A -> Bool
compare : A -> A -> Ordering
compare : (A : Type) -> Ord A -> A -> A -> Ordering
compare _ d = d.compare
ordOp : (Ordering -> Bool) -> (A : Type) -> Ord A -> A -> A -> Bool
ordOp f A d x y = f (compare A d x y)
lt : (A : Type) -> Ord A -> A -> A -> Bool
lt = ordOp (\o -> case o of { LT -> True; _ -> False })
le : (A : Type) -> Ord A -> A -> A -> Bool
le = ordOp (\o -> case o of { GT -> False; _ -> True })
ge : (A : Type) -> Ord A -> A -> A -> Bool
ge = ordOp (\o -> case o of { LT -> False; _ -> True })
gt : (A : Type) -> Ord A -> A -> A -> Bool
gt = ordOp (\o -> case o of { GT -> True; _ -> False })
-- Operators:
{-
(x < y) => (lt ? ? x y)
(x <= y) => (le ? ? x y)
(x >= y) => (ge ? ? x y)
(x > y) => (gt ? ? x y)
-}
-- Instances:
ord_Integer : Ord Integer
ord_Integer = rec eq = prim_eq_Int
compare = prim_cmp_Int
ord_String : Ord String
ord_String = rec eq = prim_eq_Str
compare = prim_cmp_Str
--
-- The Show class
--
Show : Type -> Type
Show A = sig show : A -> String
show : (A : Type) -> Show A -> A -> String
show _ d = d.show
-- Instances:
show_Integer : Show Integer
show_Integer = rec show = prim_show_Int
show_String : Show String
show_String = rec show = prim_show_Str
--
-- The Monoid class
--
Monoid : Type -> Type
Monoid = sig mzero : A
mplus : A -> A -> A
--
-- The Compos class
--
Compos : Type -> Type
Compos T = sig
C : Type
composOp : (c : C) -> ((a : C) -> T a -> T a) -> T c -> T c
composFold : (B : Type) -> Monoid B -> (c : C) -> ((a : C) -> T a -> b) -> T c -> b
composOp : (T : Type) -> (d : Compos T)
-> (c : d.C) -> ((a : d.C) -> T a -> T a) -> T c -> T c
composOp _ d = d.composOp
composFold : (T : Type) -> (d : Compos T) -> (B : Type) -> Monoid B
-> (c : d.C) -> ((a : d.C) -> T a -> b) -> T c -> b
composFold _ _ d = d.composFold
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