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|
module SOAS
import Data.List.Quantifiers
parameters {Ty : Type}
data Ctx : Type where
Lin : Ctx
(:<) : (ctx : Ctx) -> (ty : Ty) -> Ctx
data (.var) : Ctx -> Ty -> Type where
Here : (ctx :< ty).var ty
There : ctx.var ty -> (ctx :< sy).var ty
Family, SortedFamily : Type
Family = Ctx -> Type
SortedFamily = Ty -> Family
Var : SortedFamily
Var = flip (.var)
infixr 3 -|> , ~> , -<>, ^
infixl 3 <<<
(-|>) : (src,tgt : SortedFamily) -> Type
(src -|> tgt) = {ty : Ty} -> {ctx : Ctx} -> src ty ctx -> tgt ty ctx
(++) : (ctx1,ctx2 : Ctx) -> Ctx
ctx1 ++ [<] = ctx1
ctx1 ++ (ctx2 :< ty) = (ctx1 ++ ctx2) :< ty
(<<<) : SortedFamily -> Ctx -> SortedFamily
(f <<< ctx0) ty ctx = f ty (ctx ++ ctx0)
(.subst) : SortedFamily -> Ctx -> Ctx -> Type
f.subst ctx1 ctx2 = {ty : Ty} -> ctx1.var ty -> f ty ctx2
(~>) : (src, tgt : Ctx) -> Type
(~>) = (flip (.var)).subst
weakl : (ctx1, ctx2 : Ctx) -> ctx1 ~> (ctx1 ++ ctx2)
weakl ctx1 [<] x = x
weakl ctx1 (ctx2 :< z) x = There $ weakl ctx1 ctx2 x
weakr : (ctx1, ctx2 : Ctx) -> ctx2 ~> (ctx1 ++ ctx2)
weakr ctx1 (ctx2 :< _) Here = Here
weakr ctx1 (ctx2 :< sy) (There x) = There (weakr ctx1 ctx2 x)
(.copair) : (f : SortedFamily) -> {ctx2 : Ctx} -> f.subst ctx1 ctx -> f.subst ctx2 ctx -> f.subst (ctx1 ++ ctx2) ctx
f.copair {ctx2 = [<] } g1 g2 x = g1 x
f.copair {ctx2 = ctx2 :< ty} g1 g2 Here = g2 Here
f.copair {ctx2 = ctx2 :< ty} g1 g2 (There x) = f.copair g1 (g2 . There) x
extend : (f : SortedFamily) -> {ctx1 : Ctx} -> {ty : Ty} -> f ty ctx2 -> f.subst ctx1 ctx2 -> f.subst (ctx1 :< ty) ctx2
extend f {ctx2, ty} x theta = f.copair {ctx2 = [< ty]} theta workaround -- (\case {Here => x})
where
workaround : f.subst [< ty] ctx2
workaround Here = x
(-<>) : (src, tgt : SortedFamily) -> SortedFamily
(src -<> tgt) ty ctx = {ctx' : Ctx} -> src ty ctx' -> tgt ty (ctx ++ ctx')
Nil : SortedFamily -> SortedFamily
Nil f ty ctx = {ctx' : Ctx} -> ctx ~> ctx' -> f ty ctx'
-- TODO: (Setoid) coalgebras
(^) : (tgt, src : SortedFamily) -> SortedFamily
(tgt ^ src) ty ctx = {ctx' : Ctx} -> src.subst ctx ctx' -> tgt ty ctx'
hideCtx : {0 a : Ctx -> Type} ->
((ctx : Ctx) -> a ctx) -> {ctx : Ctx} -> a ctx
hideCtx f {ctx} = f ctx
(*) : (derivative, tangent : SortedFamily) -> SortedFamily
(derivative * tangent) ty ctx = (ctx' : Ctx ** (derivative ty ctx' , tangent.subst ctx' ctx))
record MonStruct {Ty : Type} (m : SortedFamily {Ty}) where
constructor MkSubstMonoidStruct
var : Var -|> m
mult : m -|> (m ^ m)
(.sub) : {Ty : Type} -> {m : SortedFamily {Ty}} -> {ty,sy : Ty} -> {ctx : Ctx {Ty}} ->
(mon : MonStruct m) => m sy (ctx :< ty) -> m ty ctx -> m sy ctx
t.sub s = mon.mult t (extend m s mon.var)
(.sub2) : {Ty : Type} -> {m : SortedFamily {Ty}} -> {ty1,ty2,sy : Ty} -> {ctx : Ctx {Ty}} ->
(mon : MonStruct m) => m sy (ctx :< ty1 :< ty2) -> m ty1 ctx -> m ty2 ctx -> m sy ctx
t.sub2 s1 s2 = mon.mult t (extend m s2 (extend m s1 mon.var))
record PointedCoalgStruct (x : SortedFamily) where
constructor MkPointedCoalgStruct
ren : x -|> [] x
var : Var -|> x
lift : (ctx : Ctx) -> (mon : PointedCoalgStruct p) => {ctx2 : Ctx} ->
(p.subst ctx1 ctx2) -> p.subst (ctx1 ++ ctx) (ctx2 ++ ctx)
lift [<] f x = f x
lift (ctx :< ty) f Here = mon.var Here
lift (ctx :< ty) f (There x) = mon.ren (lift ctx f x) There
Strength : {Ty : Type} -> (f : SortedFamily {Ty} -> SortedFamily {Ty}) -> Type
Strength f = {p,x : SortedFamily {Ty}} -> (mon : PointedCoalgStruct p) =>
(f (x ^ p)) -|> ((f x) ^ p)
SortedFunctor : {type : Type} -> Type
SortedFunctor {type} = SortedFamily {Ty=type} ->
SortedFamily {Ty=type}
SortedFunctorMap : {type : Type} -> SortedFunctor {type} -> Type
SortedFunctorMap {type} signature
= {a,b : SortedFamily {Ty = type}} -> (a -|> b) ->
signature a -|> signature b
parameters {Ty : Type} (signature : SortedFunctor {type = Ty})
(str : Strength {Ty} signature)
record (.MonoidStruct) (x : SortedFamily {Ty}) where
constructor MkSignatureMonoid
mon : MonStruct x
alg : signature x -|> x
parameters (meta : SortedFamily {Ty})
record MetaAlg (a : SortedFamily {Ty}) where
constructor MkMetaAlg
alg : signature a -|> a
var : Var -|> a
mvar : meta -|> (a ^ a)
traverse : {p,a : SortedFamily {Ty}} ->
(coalg : PointedCoalgStruct p) =>
(alg : signature a -|> a) ->
(phi : p -|> a) ->
(chi : meta -|> (a ^ a)) -> MetaAlg (a ^ p)
traverse {p,a} alg phi chi = MkMetaAlg
{ alg = \h,s => alg $ str h s
, var = \v,s => phi (s v)
, mvar = \m,e,s => chi m (\v => e v s)
}
namespace TermDef
mutual
{- alas, non obviously strictly positive because we can't tell
Idris that signature must be strictly positive.
It will be, though, if we complete the termination project
-}
public export
data Sub : {type : Type} -> {signature : SortedFunctor {type}} ->
SortedFamily {Ty = type} -> Ctx {Ty = type} ->
Ctx {Ty = type} -> Type where
Lin : Sub {type, signature} x [<] ctx
(:<) : Sub {type, signature} x shape ctx ->
signature.Term x ty ctx ->
Sub {type,signature} x (shape :< ty) ctx
public export
data (.Term) : {type : Type} ->
(signature : SortedFunctor {type}) ->
SortedFamily {Ty = type} ->
SortedFamily {Ty = type}
where
Op : {signature : SortedFunctor {type}} ->
signature (signature.Term x) ty ctx ->
signature.Term x ty ctx
Va : Var ty ctx -> signature.Term x ty ctx
Me : {ctx1, ctx2 : Ctx {Ty = type}} ->
x ty ctx2 ->
Sub (signature.Term {type} x)
ctx2 ctx1 ->
signature.Term {type} x ty ctx1
parameters {type : Type} {signature : SortedFunctor {type}}
MetaContext : Type
MetaContext = SnocList (Ctx {Ty=type}, type)
(.metaEnv) : SortedFamily {Ty=type} -> MetaContext ->
Family {Ty=type}
x.metaEnv [<] ctx = ()
x.metaEnv (mctx :< meta) ctx =
(x.metaEnv mctx ctx, (x <<< (fst meta)) (snd meta) ctx)
parameters {type : Type} {signature : SortedFunctor {type}}
{auto signatureMap : SortedFunctorMap signature}
{auto str : Strength {Ty = type} signature}
parameters {x : SortedFamily {Ty = type}}
(a : SortedFamily {Ty = type})
{auto metalg : MetaAlg {Ty = type} signature str x a}
(.envSem) : {ctx : Ctx} -> {mctx : MetaContext {signature}} ->
(signature.Term x).metaEnv mctx ctx ->
a.metaEnv mctx ctx
(.subSem) : Sub (signature.Term x) ctx1 ctx2 ->
a.subst ctx1 ctx2
(.sem) : signature.Term x -|> a
(.envSem) {mctx = [<] } menv = ()
(.envSem) {mctx = mctx :< meta} (menv,v) =
( (.envSem) menv
, (.sem) v
)
(.sem) (Op args) = MetaAlg.alg _ _ _ metalg
$ signatureMap {b = a} (.sem) args
(.sem) (Va x ) = MetaAlg.var _ _ _ metalg x
(.sem) {ctx = ctx1''} (Me m env) = MetaAlg.mvar _ _ _ metalg m $ ((.subSem) env)
data (+) : {type : Type} -> (signature1, signature2 : SortedFunctor {type}) ->
SortedFunctor {type} where
Lft : {signature1, signature2 : SortedFunctor {type}} ->
(op : sig1 x ty ctx) -> (sig1 + sig2) x ty ctx
Rgt : {signature1, signature2 : SortedFunctor {type}} ->
(op : sig2 x ty ctx) -> (sig1 + sig2) x ty ctx
sum : {type : Type} -> (signatures : List $ (String, SortedFunctor {type})) ->
SortedFunctor {type}
(sum signatures) x ty ctx = Any (\(name,sig) => sig x ty ctx) signatures
prod : {type : Type} -> (signatures : List $ SortedFunctor {type}) ->
SortedFunctor {type}
(prod signatures) x ty ctx = All (\sig => sig x ty ctx) signatures
bind : {type : Type} -> (tys : Ctx {Ty = type}) -> SortedFunctor {type}
bind tys = (<<< tys)
infixr 3 -:>
data TypeSTLC = B | (-:>) TypeSTLC TypeSTLC
data STLC : SortedFunctor {type = TypeSTLC} where
App : (f : a (ty1 -:> ty2) ctx) -> (x : a ty1 ctx) -> STLC a ty2 ctx
Lam : (body : a ty2 (ctx :< ty1)) ->
STLC a (ty1 -:> ty2) ctx
foo : STLC .Term Var (B -:> (B -:> B) -:> B) [<]
foo = Op (Lam (Op (Lam (Op (App (Va Here) (Op (App (Va Here) (Va (There Here)))))))))
|
