diff --git a/lib/Quox/Lexer.idr b/lib/Quox/Lexer.idr
index cd1bfd5..22c1a4b 100644
--- a/lib/Quox/Lexer.idr
+++ b/lib/Quox/Lexer.idr
@@ -215,7 +215,11 @@ reserved =
    W "λ" `Or` W "fun",
    W "δ" `Or` W "dfun",
    W "ω" `Or` S "#",
-   S "★" `Or` W "Type"]
+   S "★" `Or` W "Type",
+   W1 "def",
+   W1 "def0",
+   W "defω" `Or` W "def#",
+   S "≔" `Or` S ":="]
 
 ||| `IsReserved str` is true if `R str` might actually show up in
 ||| the token stream
diff --git a/lib/Quox/Parser.idr b/lib/Quox/Parser.idr
index 703e31b..ae2f5ec 100644
--- a/lib/Quox/Parser.idr
+++ b/lib/Quox/Parser.idr
@@ -21,18 +21,6 @@ public export
 Grammar = Core.Grammar () Token
 %hide Core.Grammar
 
-private
-data PArg = T PTerm | D PDim
-
-
-private
-apply : PTerm -> List PArg -> PTerm
-apply = foldl $ \f, x => case x of T x => f :@ x; D p => f :% p
-
-private
-annotate : PTerm -> Maybe PTerm -> PTerm
-annotate s a = maybe s (s :#) a
-
 
 export
 res : (str : String) -> (0 _ : IsReserved str) => Grammar True ()
@@ -167,19 +155,15 @@ private
 makeSig : MakeBinder 0
 makeSig = ("×", \([], x, s) => Sig x s)
 
-private
-plural : Nat -> a -> a -> a
-plural 1 s p = s
-plural _ s p = p
-
 private
 makeBinder : {m, n : Nat} -> MakeBinder m -> PBinderHead n -> PTerm ->
              Grammar False PTerm
 makeBinder (str, f) h t =
   case decEq m n of
     Yes Refl => pure $ f h t
-    No  _    => fatalError
-      "'\{str}' expects \{show m} quantit\{plural m "y" "ies"}, got \{show n}"
+    No  _    =>
+      let q = if m == 1 then "quantity" else "quantities" in
+      fatalError "'\{str}' expects \{show m} \{q}, got \{show n}"
 
 private
 binderInfix : Grammar True (n ** MakeBinder n)
@@ -208,6 +192,9 @@ mutual
      <|> caseTerm
      <|> bindTerm
      <|> [|annotate infixEqTerm (optional $ resC "∷" *> term)|]
+  where
+    annotate : PTerm -> Maybe PTerm -> PTerm
+    annotate s a = maybe s (s :#) a
 
   private covering
   lamTerm : Grammar True PTerm
@@ -248,11 +235,15 @@ mutual
   appTerm = resC "★"  *> [|TYPE nat|]
         <|> resC "Eq" *> [|Eq (bracks optBinderTerm) aTerm aTerm|]
         <|> [|apply aTerm (many appArg)|]
+  where
+    data PArg = T PTerm | D PDim
 
-  private covering
-  appArg : Grammar True PArg
-  appArg = [|D $ resC "@" *> dim|]
-       <|> [|T aTerm|]
+    appArg : Grammar True PArg
+    appArg = [|D $ resC "@" *> dim|]
+         <|> [|T aTerm|]
+
+    apply : PTerm -> List PArg -> PTerm
+    apply = foldl $ \f, x => case x of T x => f :@ x; D p => f :% p
 
   private covering
   aTerm : Grammar True PTerm
@@ -269,8 +260,24 @@ mutual
     name <- bname
     resC ":"
     ty <- term
-    pure (_ ** (qs.snd, name, ty))
+    pure (qs.fst ** (qs.snd, name, ty))
 
   private covering
   optBinderTerm : Grammar True (BName, PTerm)
   optBinderTerm = [|MkPair optNameBinder term|]
+
+
+export covering
+defIntro : Grammar True PQty
+defIntro = Zero <$ resC "def0"
+       <|> Any  <$ resC "defω"
+       <|> resC "def" *> option Any (qty <* resC "·")
+
+export covering
+definition : Grammar True PDefinition
+definition =
+  [|MkPDef defIntro name (resC ":" *> term) (resC "≔" *> term)|] <* resC ";"
+
+export covering
+input : Grammar False (List PDefinition)
+input = many definition
diff --git a/lib/Quox/Parser/Syntax.idr b/lib/Quox/Parser/Syntax.idr
index edd13fc..126c962 100644
--- a/lib/Quox/Parser/Syntax.idr
+++ b/lib/Quox/Parser/Syntax.idr
@@ -2,6 +2,7 @@ module Quox.Parser.Syntax
 
 import public Quox.Syntax
 import public Quox.Syntax.Qty.Three
+import public Quox.Definition
 
 import public Control.Monad.Either
 
@@ -65,6 +66,17 @@ namespace PTerm
 %runElab deriveMutual ["PTerm", "PCaseBody"] [Eq, Ord, Show]
 
 
+public export
+record PDefinition where
+  constructor MkPDef
+  qty  : PQty
+  name : Name
+  type : PTerm
+  term : PTerm
+%name PDefinition def
+%runElab derive "PDefinition" [Eq, Ord, Show]
+
+
 export
 toPDimWith : Context' BaseName d -> Dim d -> PDim
 toPDimWith ds (K e) = K e
@@ -163,6 +175,7 @@ data FromPTermError =
     AnnotationNeeded PTerm
   | DuplicatesInEnum (List TagVal)
   | DimNotInScope Name
+  | QtyNotGlobal PQty
 
 public export
 FromPTerm : (Type -> Type) -> Type
@@ -189,8 +202,8 @@ export
 fromPDimWith : FromPTerm m =>
                Context' BName d -> PDim -> m (Dim d)
 fromPDimWith ds (K e) = pure $ K e
-fromPDimWith ds (V i) =
-  fromBaseName (pure . B) (throwError . DimNotInScope) ds i
+fromPDimWith ds (V i) = fromBaseName (pure . B) (throwError . DimNotInScope) ds i
+
 
 mutual
   export
@@ -298,3 +311,22 @@ mutual
 export %inline
 fromPTerm : FromPTerm m => PTerm -> m (Term Three 0 0)
 fromPTerm = fromPTermWith [<] [<]
+
+
+export
+globalPQty : FromPTerm m => (q : PQty) -> m (IsGlobal q)
+globalPQty Zero = pure GZero
+globalPQty One  = throwError $ QtyNotGlobal One
+globalPQty Any  = pure GAny
+
+
+-- [todo] extend substitutions so they can do this injection. that's the sort of
+-- thing they are for.
+export
+fromPDefinition : FromPTerm m => PDefinition -> m (Name, Definition Three)
+fromPDefinition (MkPDef {name, qty, type, term}) =
+  pure (name, MkDef' {
+    qty, qtyGlobal = !(globalPQty qty),
+    type = T $ inject !(fromPTerm type),
+    term = Just $ T $ inject !(fromPTerm term)
+  })
diff --git a/lib/Quox/Syntax/Dim.idr b/lib/Quox/Syntax/Dim.idr
index 54faaa6..620903d 100644
--- a/lib/Quox/Syntax/Dim.idr
+++ b/lib/Quox/Syntax/Dim.idr
@@ -127,3 +127,9 @@ DecEq (Dim d) where
 public export %inline
 BV : (i : Nat) -> (0 _ : LT i d) => Dim d
 BV i = B $ V i
+
+
+export
+inject : {0 d' : Nat} -> Dim d -> Dim (d + d')
+inject (K e) = K e
+inject (B i) = B $ inject i
diff --git a/lib/Quox/Syntax/Shift.idr b/lib/Quox/Syntax/Shift.idr
index b03830e..cdd32e3 100644
--- a/lib/Quox/Syntax/Shift.idr
+++ b/lib/Quox/Syntax/Shift.idr
@@ -120,6 +120,21 @@ ssDownViaNat by =
 %transform "Shift.ssDown" ssDown = ssDownViaNat
 
 
+public export
+inject : Shift from to -> Shift (from + n') (to + n')
+inject SZ      = SZ
+inject (SS by) = SS (inject by)
+
+public export
+injectViaNat : Shift from to -> Shift (from + n') (to + n')
+injectViaNat by =
+  rewrite shiftDiff by in
+  rewrite sym $ plusAssociative by.nat from n' in
+  fromNat by.nat
+
+%transform "Shift.inject" Shift.inject = Shift.injectViaNat
+
+
 public export
 shift : Shift from to -> Var from -> Var to
 shift SZ      i = i
diff --git a/lib/Quox/Syntax/Term/Subst.idr b/lib/Quox/Syntax/Term/Subst.idr
index c62de69..36b309f 100644
--- a/lib/Quox/Syntax/Term/Subst.idr
+++ b/lib/Quox/Syntax/Term/Subst.idr
@@ -299,3 +299,77 @@ mutual
       pushSubstsWith th (comp th ps ph) e
     pushSubstsWith th ph (DCloE e ps) =
       pushSubstsWith (ps . th) ph e
+
+
+parameters {0 d', n' : Nat}
+ mutual
+  namespace Term
+    export
+    inject : Term q d n -> Term q (d + d') (n + n')
+    inject (TYPE l) = TYPE l
+    inject (Pi qty arg res) = Pi qty (inject arg) (inject res)
+    inject (Lam body) = Lam (inject body)
+    inject (Sig fst snd) = Sig (inject fst) (inject snd)
+    inject (Pair fst snd) = Pair (inject fst) (inject snd)
+    inject (Enum cases) = Enum cases
+    inject (Tag tag) = Tag tag
+    inject (Eq ty l r) = Eq (inject ty) (inject l) (inject r)
+    inject (DLam body) = DLam (inject body)
+    inject (E e) = E (inject e)
+    inject (CloT tm th) = CloT (inject tm) (inject th)
+    inject (DCloT tm th) = DCloT (inject tm) (inject th)
+
+  namespace Elim
+    export
+    inject : Elim q d n -> Elim q (d + d') (n + n')
+    inject (F x) = F x
+    inject (B i) = B $ inject i
+    inject (fun :@ arg) = (inject fun) :@ (inject arg)
+    inject (CasePair qty pair ret body) =
+      CasePair qty (inject pair) (inject ret) (inject body)
+    inject (CaseEnum qty tag ret arms) =
+      CaseEnum qty (inject tag) (inject ret)
+        (assert_total $ map inject arms)
+    inject (fun :% arg) = (inject fun) :% (inject arg)
+    inject (tm :# ty) = (inject tm) :# (inject ty)
+    inject (CloE el th) = CloE (inject el) (inject th)
+    inject (DCloE el th) = DCloE (inject el) (inject th)
+
+  namespace ScopeTerm
+    export
+    inject : ScopeTermN s q d n -> ScopeTermN s q (d + d') (n + n')
+    inject (S names (N body)) = S names (N (inject body))
+    inject (S names (Y body)) {s, n} =
+      S names $ Y $ rewrite plusAssociative s n n' in inject body
+
+  namespace DScopeTerm
+    export
+    inject : DScopeTermN s q d n -> DScopeTermN s q (d + d') (n + n')
+    inject (S names (N body)) = S names (N (inject body))
+    inject (S names (Y body)) {s, d} =
+      S names $ Y $ rewrite plusAssociative s d d' in inject body
+
+  namespace TSubst
+    export
+    inject : TSubst q d from to -> TSubst q (d + d') (from + n') (to + n')
+    inject (Shift by) = Shift $ inject by
+    inject (t ::: th) = inject t ::: inject th
+
+  namespace DSubst
+    export
+    inject : DSubst from to -> DSubst (from + d') (to + d')
+    inject (Shift by) = Shift $ inject by
+    inject (p ::: th) = inject p ::: inject th
+
+
+-- [fixme]
+-- Error: Linearity checking failed on metavar Quox.Syntax.Term.Subst.{b:7362}
+-- (Int not a function type)
+{-
+%transform "Term.inject" Term.inject t = believe_me t
+%transform "Elim.inject" Elim.inject e = believe_me e
+%transform "ScopeTerm.inject" ScopeTerm.inject t = believe_me t
+%transform "DScopeTerm.inject" DScopeTerm.inject t = believe_me t
+%transform "TSubst.inject" TSubst.inject th = believe_me th
+%transform "DSubst.inject" DSubst.inject th = believe_me th
+-}
diff --git a/lib/Quox/Syntax/Var.idr b/lib/Quox/Syntax/Var.idr
index 2813cdc..95959fd 100644
--- a/lib/Quox/Syntax/Var.idr
+++ b/lib/Quox/Syntax/Var.idr
@@ -111,6 +111,18 @@ toFromNat 0     (LTESucc x) = Refl
 toFromNat (S k) (LTESucc x) = cong S $ toFromNat k x
 
 
+export
+inject : Var m -> Var (m + n)
+inject VZ     = VZ
+inject (VS i) = VS $ inject i
+
+private
+injectViaNat : Var m -> Var (m + n)
+injectViaNat i = fromNatWith i.nat (toNatLT i `transitive` lteAddRight m)
+
+%transform "Var.inject" Var.inject = injectViaNat
+
+
 -- not using %transform like other things because weakSpec requires the proof
 -- to be relevant. but since only `LTESucc` is ever possible that seems
 -- to be an instance of <https://github.com/idris-lang/Idris2/issues/1259>?
diff --git a/tests/Tests/Lexer.idr b/tests/Tests/Lexer.idr
index 537e2af..0a1a105 100644
--- a/tests/Tests/Lexer.idr
+++ b/tests/Tests/Lexer.idr
@@ -54,10 +54,10 @@ tests = "lexer" :- [
 
   "identifiers & keywords" :- [
     lexes "abc" [I "abc"],
-    lexes "abc def" [I "abc", I "def"],
+    lexes "abc def" [I "abc", R "def"],
     lexes "abc_def" [I "abc_def"],
     lexes "abc-def" [I "abc-def"],
-    lexes "abc{-comment-}def" [I "abc", I "def"],
+    lexes "abc{-comment-}def" [I "abc", R "def"],
     lexes "λ" [R "λ"],
     lexes "fun" [R "λ"],
     lexes "δ" [R "δ"],
diff --git a/tests/Tests/Parser.idr b/tests/Tests/Parser.idr
index f1b4c1c..f1c71d3 100644
--- a/tests/Tests/Parser.idr
+++ b/tests/Tests/Parser.idr
@@ -237,5 +237,25 @@ tests = "parser" :- [
       Case Any (V "t") (Nothing, V "A") (CaseEnum []),
     parsesAs term "case# t return A of {}" $
       Case Any (V "t") (Nothing, V "A") (CaseEnum [])
+  ],
+
+  "definitions" :- [
+    parsesAs definition "defω x : (_: {a}) × {b} ≔ ('a, 'b);" $
+      MkPDef Any "x" (Sig Nothing (Enum ["a"]) (Enum ["b"]))
+        (Pair (Tag "a") (Tag "b")),
+    parsesAs definition "def# x : (_: {a}) ** {b} ≔ ('a, 'b);" $
+      MkPDef Any "x" (Sig Nothing (Enum ["a"]) (Enum ["b"]))
+        (Pair (Tag "a") (Tag "b")),
+    parsesAs definition "def ω·x : (_: {a}) × {b} ≔ ('a, 'b);" $
+      MkPDef Any "x" (Sig Nothing (Enum ["a"]) (Enum ["b"]))
+        (Pair (Tag "a") (Tag "b")),
+    parsesAs definition "def x : (_: {a}) × {b} ≔ ('a, 'b);" $
+      MkPDef Any "x" (Sig Nothing (Enum ["a"]) (Enum ["b"]))
+        (Pair (Tag "a") (Tag "b")),
+    parsesAs definition "def0 A : ★₀ ≔ {a, b, c};" $
+      MkPDef Zero "A" (TYPE 0) (Enum ["a", "b", "c"]),
+    parsesAs input "def0 A : ★₀ ≔ {}; def0 B : ★₁ ≔ A;" $
+      [MkPDef Zero "A" (TYPE 0) (Enum []),
+       MkPDef Zero "B" (TYPE 1) (V "A")]
   ]
 ]