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|
module Encoded.Sum
import public Data.List
import public Data.List.Elem
import public Data.List.Quantifiers
import Control.Function.FunExt
import Encoded.Bool
import Encoded.Pair
import Encoded.Union
import Syntax.PreorderReasoning
import Term.Semantics
import Term.Syntax
%ambiguity_depth 4
-- Auxilliary Functions --------------------------------------------------------
public export
mapNonEmpty : NonEmpty xs -> NonEmpty (map f xs)
mapNonEmpty IsNonEmpty = IsNonEmpty
-- Types -----------------------------------------------------------------------
export
(+) : Ty -> Ty -> Ty
ty1 + ty2 = B * (ty1 <+> ty2)
export
Sum : (tys : List Ty) -> {auto 0 ok : NonEmpty tys} -> Ty
Sum = foldr1 (+)
-- Universal Properties --------------------------------------------------------
-- Binary
export
left : {ty1, ty2 : Ty} -> Term (ty1 ~> (ty1 + ty2)) ctx
left = Abs' (\t => App pair [<True, App inL [<t]])
export
right : {ty1, ty2 : Ty} -> Term (ty2 ~> (ty1 + ty2)) ctx
right = Abs' (\t => App pair [<False, App inR [<t]])
export
case' : {ty1, ty2, ty : Ty} -> Term ((ty1 + ty2) ~> (ty1 ~> ty) ~> (ty2 ~> ty) ~> ty) ctx
case' = Abs' (\t =>
App if'
[<App fst [<t]
, Abs $ Const $ App (Var Here . prL . snd) [<shift t]
, Const $ Abs $ App (Var Here . prR . snd) [<shift t]
])
-- Utilty
export
either : {ty1, ty2, ty : Ty} -> Term ((ty1 ~> ty) ~> (ty2 ~> ty) ~> (ty1 + ty2) ~> ty) ctx
either = Abs $ Abs $ Abs $
let f = Var $ There $ There Here in
let g = Var $ There Here in
let x = Var Here in
App case' [<x, f, g]
-- N-ary
export
any :
{tys : List Ty} ->
{ty : Ty} ->
{auto 0 ok : NonEmpty tys} ->
All (\ty' => Term (ty' ~> ty) ctx) tys ->
Term (Sum tys ~> ty) ctx
any [f] = f
any (f :: fs@(_ :: _)) = App either [<f, any fs]
export
tag :
{tys : List Ty} ->
{ty : Ty} ->
{auto 0 ok : NonEmpty tys} ->
Elem ty tys ->
Term (ty ~> Sum tys) ctx
tag {tys = [_]} Here = Id
tag {tys = _ :: _ :: _} Here = left
tag {tys = _ :: _ :: _} (There i) = right . tag i
-- Semantics -------------------------------------------------------------------
-- Conversion to Idris
export
toEither :
{ty1, ty2 : Ty} ->
TypeOf (ty1 + ty2) ->
Either (TypeOf ty1) (TypeOf ty2)
toEither x =
if' (Sem.fst x)
(Left (prL $ Sem.snd x))
(Right (prR $ Sem.snd x))
export
fromEither :
{ty1, ty2 : Ty} ->
Either (TypeOf ty1) (TypeOf ty2) ->
TypeOf (ty1 + ty2)
fromEither (Left x) = pair Sem.True (inL x)
fromEither (Right x) = pair Sem.False (inR x)
export
toFromId :
FunExt =>
{ty1, ty2 : Ty} ->
(x : Either (TypeOf ty1) (TypeOf ty2)) ->
toEither {ty1, ty2} (fromEither {ty1, ty2} x) = x
toFromId (Left x) =
rewrite retractL {ty1 = B, ty2 = ty1 <+> ty2} Sem.True in
cong Left $ Calc $
|~ (Sem.prL {ty1, ty2} . Sem.prR . Sem.inR . Sem.inL) x
~~ (prL {ty1, ty2} . inL) x ...(cong prL $ retractR (inL x))
~~ x ...(retractL {ty1, ty2} x)
toFromId (Right x) =
rewrite retractL {ty1 = B, ty2 = ty1 <+> ty2} Sem.False in
cong Right $ Calc $
|~ (Sem.prR {ty1, ty2} . Sem.prR . Sem.inR . Sem.inR) x
~~ (prR {ty1, ty2} . inR) x ...(cong prR $ retractR (inR x))
~~ x ...(retractR {ty1, ty2} x)
-- Redefinition in Idris
namespace Sem
public export
left : {ty1, ty2 : Ty} -> TypeOf ty1 -> TypeOf (ty1 + ty2)
left x = pair Sem.True (inL x)
public export
right : {ty1, ty2 : Ty} -> TypeOf ty2 -> TypeOf (ty1 + ty2)
right x = pair Sem.False (inR x)
public export
case' :
{ty1, ty2 : Ty} ->
TypeOf (ty1 + ty2) ->
(TypeOf ty1 -> a) ->
(TypeOf ty2 -> a) ->
a
case' x =
if' (Sem.fst x)
(\f, _ => f $ prL $ Sem.snd x)
(\_, g => g $ prR $ Sem.snd x)
-- Homomorphism
export
leftHomo :
FunExt =>
{ty1, ty2 : Ty} ->
(0 x : TypeOf ty1) ->
toEither {ty1, ty2} (left {ty1, ty2} x) = Left x
leftHomo x = irrelevantEq $ toFromId (Left x)
export
rightHomo :
FunExt =>
{ty1, ty2 : Ty} ->
(0 x : TypeOf ty2) ->
toEither {ty1, ty2} (right {ty1, ty2} x) = Right x
rightHomo x = irrelevantEq $ toFromId (Right x)
export
caseHomo :
FunExt =>
{ty1, ty2 : Ty} ->
(x : Either (TypeOf ty1) (TypeOf ty2)) ->
(f : TypeOf ty1 -> a) ->
(g : TypeOf ty2 -> a) ->
case' {ty1, ty2} (fromEither {ty1, ty2} x) f g = either f g x
caseHomo (Left x) f g =
rewrite retractL {ty1 = B, ty2 = ty1 <+> ty2} Sem.True in
cong f $ Calc $
|~ (Sem.prL {ty1, ty2} . Sem.prR . Sem.inR . Sem.inL) x
~~ (prL {ty1, ty2} . inL) x ...(cong prL $ retractR (inL x))
~~ x ...(retractL {ty1, ty2} x)
caseHomo (Right x) f g =
rewrite retractL {ty1 = B, ty2 = ty1 <+> ty2} Sem.False in
cong g $ Calc $
|~ (Sem.prR {ty1, ty2} . Sem.prR . Sem.inR . Sem.inR) x
~~ (prR {ty1, ty2} . inR) x ...(cong prR $ retractR (inR x))
~~ x ...(retractR {ty1, ty2} x)
|
