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Dagstuhl18.agda
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Dagstuhl18.agda
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module Dagstuhl18 where
{-----------------------------------------------------------------------------
CHEESE TALK - Conor McBride in
888 888 888 d8b
888 888 888 88P
888 888 888 8P
8888888888 8888b. 88888b. .d8888b .d88b. .d8888b 888 888 " .d8888b
888 888 "88b 888 "88b d88P" d88""88b d88P" 888 .88P 88K
888 888 .d888888 888 888 888 888 888 888 888888K "Y8888b.
888 888 888 888 888 888 Y88b. Y88..88P Y88b. 888 "88b X88
888 888 "Y888888 888 888 "Y8888P "Y88P" "Y8888P 888 888 88888P'
888 888 888 .d888 888 888
888 888 888 d88P" 888 888
888 888 888 888 888 888
8888888888 8888b. 888 888888 8888888888 .d88b. 888 888 888d888
888 888 "88b 888 888 888 888 d88""88b 888 888 888P"
888 888 .d888888 888 888 888 888 888 888 888 888 888
888 888 888 888 888 888 888 888 Y88..88P Y88b 888 888
888 888 "Y888888 888 888 888 888 "Y88P" "Y88888 888
-----------------------------------------------------------------------------}
{----}
------------------------------------------------------------------------------
-- FUNCTIONAL PROGRAMMING
------------------------------------------------------------------------------
id : forall {l}{X : Set l} -> X -> X
id x = x
_-_ : forall {i j k}{A : Set i}{B : A -> Set j}{C : (a : A) -> B a -> Set k}
(f : {a : A}(b : B a) -> C a b)
(g : (a : A) -> B a) ->
(a : A) -> C a (g a)
-- "composition" is secretly the S combinator
(f - g) a = f (g a)
ko_ : forall {k l}{S : Set k}{T : Set l}(s : S)(t : T) -> S
ko_ s t = s
infixr 3 ko_
------------------------------------------------------------------------------
{----}
------------------------------------------------------------------------------
-- LITTLE THINGS
------------------------------------------------------------------------------
data Zero : Set where
record One : Set where constructor <>
data Two : Set where tt ff : Two
data _==_ {l}{X : Set l}(x : X) : X -> Set where
refl : x == x
{-# BUILTIN EQUALITY _==_ #-}
------------------------------------------------------------------------------
{----}
------------------------------------------------------------------------------
-- PAIRS OF THINGS
------------------------------------------------------------------------------
record Sg (S : Set)(T : S -> Set) : Set where
constructor _,_
field fst : S ; snd : T fst
open Sg
_*_ _+_ : Set -> Set -> Set
S * T = Sg S \ _ -> T
infixr 5 _*_ _,_
S + T = Sg Two \ { tt -> S ; ff -> T }
pattern ll_ s = tt , s ; pattern rr_ t = ff , t
infixr 4 _+_ ; infixr 3 ll_ rr_
-- Pi made to look like Sg
Pi : (S : Set)(T : S -> Set) -> Set
Pi S T = (s : S) -> T s
------------------------------------------------------------------------------
{----}
------------------------------------------------------------------------------
-- MORE FUNCTIONAL PROGRAMMING
------------------------------------------------------------------------------
\\_ : forall {S : Set}{T : S -> Set}{P : Sg S T -> Set} ->
((s : S) -> (t : T s) -> P (s , t)) ->
(st : Sg S T) -> P st -- uncurrying
(\\ f) (s , t) = f s t
infixr 3 \\_
tick_ : forall {S T : Set}{P : T -> Set} ->
((t : T) -> P t) ->
(st : S * T) -> P (snd st) -- disCARding
tick f = \\ ko f
infixr 3 tick_
boom : forall {T : Zero -> Set}(x : Zero) -> T x -- exploding
boom ()
------------------------------------------------------------------------------
{----}
------------------------------------------------------------------------------
-- INDEXING THE THINGS
------------------------------------------------------------------------------
Ix : Set -> Set1
Ix I = I -> Set
_+:_ _*:_ _-:>_ : {I : Set} -> Ix I -> Ix I -> Ix I
(S +: T) i = S i + T i
(S *: T) i = S i * T i
(S -:> T) i = S i -> T i
infixr 3 _-:>_ ; infixr 4 _+:_ ; infixr 5 _*:_
[_] <_> : forall {I : Set} -> (I -> Set) -> Set
[ P ] = forall {i} -> P i -- P holds necessarily
< P > = Sg _ \ i -> P i -- P holds possibly
------------------------------------------------------------------------------
{----}
------------------------------------------------------------------------------
-- TONY HOARE
------------------------------------------------------------------------------
record Monad {I}(M : Ix I -> Ix I) : Set1 where
field
-----------------
skip : forall {Q} -> [ Q -:> M Q ]
_,,_ : forall {O P Q} -> [ O -:> M P ] -> [ P -:> M Q ]
----------------------------------
-> [ O -:> M Q ]
_=<<_ : forall {P Q} -> [ P -:> M Q ] -> [ M P -:> M Q ]
_=<<_ k = (id ,, k)
_>>=_ : forall {P Q} -> forall {i} -> M P i -- angel's i
-> (forall {j} -> P j -> M Q j) -- devil's j
-> M Q i
m >>= k = k =<< m
------------------------------------------------------------------------------
{----}
------------------------------------------------------------------------------
-- PETER HANCOCK
------------------------------------------------------------------------------
record _|>_ (I O : Set) : Set1 where
constructor Act
field
Comm : (o : O) -> Set
Resp : (o : O)(c : Comm o) -> Set
next : (o : O)(c : Comm o)(r : Resp o c) -> I
open _|>_
infixl 2 _|>_
_$_ : forall {I O} -> I |> O -> Ix I -> Ix O
{- computation -} {- environment -}
(F $ Q) o = Sg (Comm F o) \ c ->
Pi (Resp F o c) \ r ->
Q (next F o c r)
infixl 6 _$_
------------------------------------------------------------------------------
{----}
------------------------------------------------------------------------------
-- SOME INTERACTION STRUCTURES
------------------------------------------------------------------------------
Read : Set -> One |> One
Comm (Read X) o = One
Resp (Read X) <> <> = X
next (Read X) = _
Write : Set -> One |> One
Comm (Write X) o = X
Resp (Write X) <> x = One
next (Write X) = _
------------------------------------------------------------------------------
{----}
------------------------------------------------------------------------------
-- DISCRIMINATING EXCEPTIONS
------------------------------------------------------------------------------
Exn : forall {I} -> Ix I -> I |> I
Comm (Exn E) i = E i
Resp (Exn E) i e = Zero
next (Exn E) _ _ = boom
-- E gives "the set of exception packets appropriate to the state"
-- and that could be empty
------------------------------------------------------------------------------
{----}
------------------------------------------------------------------------------
-- INTERACTION STRUCTURES COMPOSE
------------------------------------------------------------------------------
Id : {I : Set} -> I |> I
Comm Id o = One
Resp Id o <> = One
next Id o <> <> = o
_&_ : forall {I J K} -> J |> I -> K |> J -> K |> I
Comm (C & D) i = Sg (Comm C i) \ c ->
Pi (Resp C i c) \ r ->
Comm D (next C i c r)
Resp (C & D) i (c , k) = Sg (Resp C i c) \ r ->
Resp D (next C i c r) (k r)
next (C & D) i (c , k) (r , s) = next D (next C i c r) (k r) s
------------------------------------------------------------------------------
{----}
------------------------------------------------------------------------------
-- CHOICE, SHARED STATE
------------------------------------------------------------------------------
_|+|_ : forall {I O} -> I |> O -> I |> O -> I |> O
Comm (A |+| B) = Comm A +: Comm B
Resp (A |+| B) o (ll ac) = Resp A o ac
next (A |+| B) o (ll ac) ar = next A o ac ar
Resp (A |+| B) o (rr bc) = Resp B o bc
next (A |+| B) o (rr bc) br = next B o bc br
State : Set -> One |> One
State S = Read S |+| Write S
------------------------------------------------------------------------------
{----}
------------------------------------------------------------------------------
-- FRAMING
------------------------------------------------------------------------------
_>|_ : forall {I} -> (I |> I) -> forall J -> (I * J |> I * J)
Comm (C >| J) (i , j) = Comm C i
Resp (C >| J) (i , j) c = Resp C i c
next (C >| J) (i , j) c r = next C i c r , j
_|<_ : forall I {J} -> (J |> J) -> (I * J |> I * J)
Comm (I |< D) (i , j) = Comm D j
Resp (I |< D) (i , j) d = Resp D j d
next (I |< D) (i , j) d r = i , next D j d r
_>|<_ : forall {I J} -> (I |> I) -> (J |> J) -> (I * J |> I * J)
C >|< D = (C >| _) |+| (_ |< D)
------------------------------------------------------------------------------
{----}
------------------------------------------------------------------------------
-- FILE HANDLE JIGGERY POKERY (I)
------------------------------------------------------------------------------
postulate Path : Set
postulate Char : Set
data Status : Set where opened closed : Status
data FileComm (Mode : One |> One) : Status -> Set where
fopen : Path -> FileComm Mode closed
fact : Comm Mode <> -> FileComm Mode opened
fclose : FileComm Mode opened
-- A gives the actions available while the file is open.
------------------------------------------------------------------------------
{----}
------------------------------------------------------------------------------
-- FILE HANDLE JIGGERY POKERY (II)
------------------------------------------------------------------------------
FH : (Mode : One |> One) -> Status |> Status
Comm (FH Mode) o = FileComm Mode o
Resp (FH Mode) .closed (fopen x) = Two -- did the file actually open?
next (FH Mode) .closed (fopen x) tt = opened
next (FH Mode) .closed (fopen x) ff = closed
Resp (FH Mode) .opened (fact a) = Resp Mode <> a
next (FH Mode) .opened (fact a) r = opened
Resp (FH Mode) .opened fclose = One
next (FH Mode) .opened fclose r = closed
-- consider Mode as Read or Write...
------------------------------------------------------------------------------
{----}
------------------------------------------------------------------------------
-- FREE MONADS ON INDEXED SETS
------------------------------------------------------------------------------
data FIx {I}(F : I |> I)(Q : Ix I) : Ix I where
return : [ Q -:> FIx F Q ]
!_ : [ F $ FIx F Q -:> FIx F Q ]
infixr 3 !_
_=<<_ : forall {I}{F : I |> I}{P Q} -> [ P -:> FIx F Q ]
-> [ FIx F P -:> FIx F Q ]
k =<< return p = k p
k =<< (! c , j) = ! c , \ r -> k =<< j r
Free : forall {I}(F : I |> I) -> Monad (FIx F)
Monad.skip (Free F) = return
Monad._,,_ (Free F) = \ op pq o -> pq =<< op o
------------------------------------------------------------------------------
{----}
------------------------------------------------------------------------------
-- HANDLING
------------------------------------------------------------------------------
handle : forall {I}(F : I |> I){X Q : Ix I} ->
[ X -:> Q ] -> -- how to handle return
[ F $ Q -:> Q ] -> -- how to handle commands
[ FIx F X -:> Q ] -- the handler
handle F ret op (return x) = ret x
handle F ret op (! c , k) = op (c , \ r -> handle F ret op (k r))
------------------------------------------------------------------------------
{----}
------------------------------------------------------------------------------
-- COMPLETELY ITERATIVE MONADS (THE COINDUCTIVE STORY)
------------------------------------------------------------------------------
record CoFIx {I}(F : I |> I)(Q : Ix I)(i : I) : Set where
coinductive
constructor node
field
force : (Q +: F $ CoFIx F Q) i
open CoFIx
retCo : forall {I}{F : I |> I}{Q : Ix I} -> [ Q -:> CoFIx F Q ]
retCo q = node (ll q)
-- !!_ : forall {I}{F : I |> I}{Q : Ix I} -> [ F $ CoFIx F Q -:> CoFIx F Q ]
pattern !!_ f = node (rr f)
infixl 3 !!_
_=<<Co_ : forall {I}{F : I |> I}{P Q} -> [ P -:> CoFIx F Q ]
-> [ CoFIx F P -:> CoFIx F Q ]
force (k =<<Co b) with force b
force (k =<<Co b) | ll q = force (k q)
force (k =<<Co b) | rr c , j = rr c , \ r -> k =<<Co j r
------------------------------------------------------------------------------
{----}
------------------------------------------------------------------------------
-- FOOTLING
------------------------------------------------------------------------------
footle : forall {I}(F : I |> I){P Q : Ix I} -> [ P -:> Q +: F $ P ]
-> [ P -:> CoFIx F Q ]
force (footle F d p) with d p
force (footle F d p) | ll q = ll q
force (footle F d p) | rr c , k = rr c , \ r -> footle F d (k r)
------------------------------------------------------------------------------
{----}
------------------------------------------------------------------------------
-- CONTAINER DRIVERS (IN MEMORIAM MARK E SMITH)
------------------------------------------------------------------------------
Driver : forall {I J} (_~_ : I -> J -> Set) (Hi : I |> I) (Lo : J |> J) -> Set
Driver _~_ Hi Lo = forall {i j} ->
{- Hi -} i ~ j -> {- Lo -}
Pi (Comm Hi i) \ c ->
Sg (Comm Lo j) \ c' ->
Pi (Resp Lo j c') \ r' ->
Sg (Resp Hi i c) \ r ->
next Hi i c r ~ next Lo j c' r'
------------------------------------------------------------------------------
{----}
------------------------------------------------------------------------------
-- THE SYNC ADJUNCTION
------------------------------------------------------------------------------
module SYNC {I J} (_~_ : I -> J -> Set) where
SgSyncR : Ix I -> Ix J ; PiSyncL : Ix J -> Ix I
SgSyncR P j = < P *: (_~ j) > ; PiSyncL Q i = [ (i ~_) -:> Q ]
module SYNCADJUNCTION {P : Ix I}{Q : Ix J} where
{- [ P -:> PiSyncL Q ]
===========================
[ SgSyncR P -:> Q ] -}
mkSyncR-:> : [ P -:> PiSyncL Q ] -> [ SgSyncR P -:> Q ]
mk-:>SyncL : [ SgSyncR P -:> Q ] -> [ P -:> PiSyncL Q ]
mkSyncR-:> f (_ , p , s) = f p s
mk-:>SyncL f p s = f (_ , p , s)
open SYNC ; open SYNCADJUNCTION
------------------------------------------------------------------------------
{----}
------------------------------------------------------------------------------
-- CONTAINER DRIVING
------------------------------------------------------------------------------
drive : forall {I J} (_~_ : I -> J -> Set)(Hi : I |> I)(Lo : J |> J)
(D : Driver _~_ Hi Lo){Q : Ix I} ->
[ SgSyncR _~_ (FIx Hi Q) -:> FIx Lo (SgSyncR _~_ Q) ]
drive _~_ Hi Lo D = mkSyncR-:> _~_ {Q = FIx Lo _}
(handle Hi {Q = PiSyncL _~_ _}
(\ { q s -> return (_ , q , s) })
(\ { (c , k) s -> let c' , D' = D s c in ! c' , \ r' ->
let r , s' = D' r' in k r s' })
)
------------------------------------------------------------------------------
{----}
------------------------------------------------------------------------------
-- HENRY FORD
------------------------------------------------------------------------------
fordOut : forall {I}{Q : Ix I} -> [ SgSyncR _==_ Q -:> Q ]
fordOut (_ , q , refl) = q
fordIn : forall {I}{Q : Ix I} -> [ Q -:> SgSyncR _==_ Q ]
fordIn q = (_ , q , refl)
fordDrive : forall {I}(Hi Lo : I |> I)(D : Driver _==_ Hi Lo){Q : Ix I}
->
[ FIx Hi Q -:> FIx Lo Q ]
fordDrive Hi Lo D t =
(return - fordOut) =<< drive _==_ Hi Lo D (fordIn t)
------------------------------------------------------------------------------
{----}
------------------------------------------------------------------------------
-- COINDUCTIVE CONTAINER DRIVING
------------------------------------------------------------------------------
driveCo : forall {I J} (_~_ : I -> J -> Set)(Hi : I |> I)(Lo : J |> J)
(D : Driver _~_ Hi Lo){Q : Ix I} ->
[ SgSyncR _~_ (CoFIx Hi Q) -:> CoFIx Lo (SgSyncR _~_ Q) ]
driveCo _~_ Hi Lo D {Q} = footle Lo {P = SgSyncR _~_ _} help where
help : [ SgSyncR _~_ (CoFIx Hi Q) -:>
SgSyncR _~_ Q +: Lo $ SgSyncR _~_ (CoFIx Hi Q) ]
help (_ , t , s) with force t
... | (ll q) = ll _ , q , s
... | (rr c , k) = rr let c' , D' = D s c in c' , \ r' ->
let r , s' = D' r' in _ , k r , s'
fordDriveCo : forall {I}(Hi Lo : I |> I)(D : Driver _==_ Hi Lo){Q : Ix I} ->
[ CoFIx Hi Q -:> CoFIx Lo Q ]
fordDriveCo Hi Lo D t =
(retCo - fordOut) =<<Co driveCo _==_ Hi Lo D (fordIn t)
------------------------------------------------------------------------------
{----}
------------------------------------------------------------------------------
-- "SHELL SCRIPTING"
------------------------------------------------------------------------------
Responses : forall {I}(C : I |> I) {o} -> FIx C (ko One) o -> Set
Responses C (return <>) = One
Responses C (! c , k) = Sg (Resp C _ c) \ r -> Responses C (k r)
peterHan : forall {I} (C : I |> I) (o : I) (t : FIx C (ko One) o) ->
Responses C t -> I
peterHan C o (return x) <> = o
peterHan C o (! c , k) (r , rs) = peterHan C (next C o c r) (k r) rs
Star : forall {I} -> I |> I -> I |> I
Comm (Star C) = FIx C (ko One)
Resp (Star C) o = Responses C
next (Star C) = peterHan C
------------------------------------------------------------------------------
{----}
------------------------------------------------------------------------------
-- "BASH"
------------------------------------------------------------------------------
run : forall {I}(C : I |> I){Q : I -> Set}
->
[ FIx (Star C) Q -:> FIx C Q ]
run C {Q} = handle (Star C) {Q} {FIx C Q} return (\\ help) where
help : forall {o}
(t : FIx C (ko One) o)
(k : (s : Responses C t) -> FIx C Q (peterHan C o t s))
-> FIx C Q o
help (return <>) k = k <>
help (! c , j) k = ! c , \ r -> help (j r) \ rs -> k (r , rs)
------------------------------------------------------------------------------
{----}
------------------------------------------------------------------------------
-- "CoBASH"
------------------------------------------------------------------------------
runCo : forall {I}(C : I |> I){Q : I -> Set}
->
[ CoFIx (C & Star C) Q -:> CoFIx C Q ]
runCo C {Q} = footle C {CoFIx (C & Star C) Q}{Q} help where
help : [ CoFIx (C & Star C) Q -:> Q +: C $ CoFIx (C & Star C) Q ]
help t with force t
... | ll q = ll q
... | rr (c , j) , k = rr c , \ r -> buff (j r) \ rs -> k (r , rs) where
buff : forall {o}(t : FIx C (ko One) o) ->
((s : Responses C t) -> CoFIx (C & Star C) Q (peterHan C o t s)) ->
CoFIx (C & Star C) Q o
buff (return <>) k = k <>
buff (! c , j) k = !! (c , j) , k
------------------------------------------------------------------------------