Define semantics and encode types up to pairs.

2 files changed, 85 insertions, 0 deletions

diff --git a/src/Term/Semantics.idr b/src/Term/Semantics.idr
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+++ b/src/Term/Semantics.idr
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+module Term.Semantics
+
+import Term
+import public Data.SnocList.Quantifiers
+
+public export
+TypeOf : Ty -> Type
+TypeOf N = Nat
+TypeOf (ty ~> ty') = TypeOf ty -> TypeOf ty'
+
+rec : Nat -> a -> (a -> a) -> a
+rec 0 x f = x
+rec (S k) x f = f (rec k x f)
+
+fullSem : FullTerm ty ctx -> ctx `Thins` ctx' -> All TypeOf ctx' -> TypeOf ty
+fullSem Var thin ctx = indexAll (index thin Here) ctx
+fullSem (Const t) thin ctx = const (fullSem t thin ctx)
+fullSem (Abs t) thin ctx = \v => fullSem t (Keep thin) (ctx :< v)
+fullSem (App (MakePair (t `Over` thin1) (u `Over` thin2) _)) thin ctx =
+  fullSem t (thin . thin1) ctx (fullSem u (thin . thin2) ctx)
+fullSem Zero thin ctx = 0
+fullSem (Suc t) thin ctx = S (fullSem t thin ctx)
+fullSem
+  (Rec (MakePair
+    (t `Over` thin1)
+    (MakePair (u `Over` thin2) (v `Over` thin3) _ `Over` thin')
+    _))
+  thin
+  ctx =
+  let thin' = thin . thin' in
+  rec
+    (fullSem t (thin . thin1) ctx)
+    (fullSem u (thin' . thin2) ctx)
+    (fullSem v (thin' . thin3) ctx)
+
+export
+sem : Term ty ctx -> All TypeOf ctx -> TypeOf ty
+sem (t `Over` thin) ctx = fullSem t thin ctx
diff --git a/src/Term/Syntax.idr b/src/Term/Syntax.idr
new file mode 100644
index 0000000..a990600
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+++ b/src/Term/Syntax.idr
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+module Term.Syntax
+
+import public Data.SnocList
+import public Term
+
+-- Combinators
+
+export
+Id : Term (ty ~> ty) ctx
+Id = Abs (Var Here)
+
+export
+Arb : {ty : Ty} -> Term ty ctx
+Arb {ty = N} = Zero
+Arb {ty = ty ~> ty'} = Const Arb
+
+export
+Lit : Nat -> Term N ctx
+Lit 0 = Zero
+Lit (S k) = Suc (Lit k)
+
+-- HOAS
+
+infix 4 ~>*
+
+public export
+(~>*) : SnocList Ty -> Ty -> Ty
+sty ~>* ty = foldr (~>) ty sty
+
+export
+Abs' : (Term ty (ctx :< ty) -> Term ty' (ctx :< ty)) -> Term (ty ~> ty') ctx
+Abs' f = Abs (f $ Var Here)
+
+export
+App' : {ty : Ty} -> Term (ty ~> ty') ctx -> Term ty ctx -> Term ty' ctx
+App' (Const t `Over` thin) u = t `Over` thin
+App' (Abs t `Over` thin) u = subst (t `Over` Keep thin) (Base Id :< u)
+App' t u = App t u
+
+export
+App : {sty : SnocList Ty} -> Term (sty ~>* ty) ctx -> All (flip Term ctx) sty -> Term ty ctx
+App t [<] = t
+App t (us :< u) = App' (App t us) u
+
+export
+(.) : {ty, ty' : Ty} -> Term (ty' ~> ty'') ctx -> Term (ty ~> ty') ctx -> Term (ty ~> ty'') ctx
+t . u = Abs (App (shift t) [<App (shift u) [<Var Here]])