diff --git a/exe/Main.idr b/exe/Main.idr
index 45a377d..177f438 100644
--- a/exe/Main.idr
+++ b/exe/Main.idr
@@ -164,7 +164,7 @@ private
 liftFromParser : Eff FromParserIO a -> Eff CompileStop a
 liftFromParser act =
   runEff act $ with Union.(::)
-    [handleExcept (\err => throw $ FromParserError err),
+    [handleExcept $ \err => throw $ FromParserError err,
      handleStateIORef !(asksAt STATE defs),
      handleStateIORef !(asksAt STATE ns),
      handleStateIORef !(asksAt STATE suf),
@@ -174,8 +174,7 @@ private
 liftErase : Q.Definitions -> Eff Erase a -> Eff CompileStop a
 liftErase defs act =
   runEff act
-    [\case Err e => throw $ EraseError e,
-     \case Ask   => pure defs,
+    [handleExcept $ \err => throw $ EraseError err,
      handleStateIORef !(asksAt STATE suf)]
 
 private
@@ -207,7 +206,7 @@ processFile file = withEarlyStop $ do
     traverse (uncurry Q.prettyDef) defList
   let defs = SortedMap.fromList defList
   erased <- liftErase defs $
-    traverse (\(x, d) => (x,) <$> eraseDef x d) defList
+    traverse (\(x, d) => (x,) <$> eraseDef defs x d) defList
   outputDocStopIf Erase $
     traverse (uncurry U.prettyDef) erased
   (scheme, mains) <- liftScheme $ map catMaybes $
diff --git a/lib/Quox/Equal.idr b/lib/Quox/Equal.idr
index 7055181..9a69ffb 100644
--- a/lib/Quox/Equal.idr
+++ b/lib/Quox/Equal.idr
@@ -170,12 +170,19 @@ compareType : Definitions -> EqContext n -> (s, t : Term 0 n) ->
               Eff EqualInner ()
 
 
+private
+0 NotRedexEq : {isRedex : RedexTest tm} -> CanWhnf tm isRedex =>
+               Definitions -> EqContext n -> SQty -> Pred (tm 0 n)
+NotRedexEq defs ctx sg t = NotRedex defs (toWhnfContext ctx) sg t
+
 namespace Term
   private covering
-  compare0' : (defs : Definitions) -> EqContext n -> (sg : SQty) ->
+  compare0' : (defs : Definitions) -> (ctx : EqContext n) -> (sg : SQty) ->
               (ty, s, t : Term 0 n) ->
-              (0 _ : NotRedex defs SZero ty) => (0 _ : So (isTyConE ty)) =>
-              (0 _ : NotRedex defs sg    s)  => (0 _ : NotRedex defs sg t) =>
+              (0 _ : NotRedexEq defs ctx SZero ty) =>
+              (0 _ : So (isTyConE ty)) =>
+              (0 _ : NotRedexEq defs ctx sg s) =>
+              (0 _ : NotRedexEq defs ctx sg t) =>
               Eff EqualInner ()
   compare0' defs ctx sg (TYPE {}) s t = compareType defs ctx s t
 
@@ -350,9 +357,10 @@ namespace Term
 
 
 private covering
-compareType' : (defs : Definitions) -> EqContext n -> (s, t : Term 0 n) ->
-               (0 _ : NotRedex defs SZero s) => (0 _ : So (isTyConE s)) =>
-               (0 _ : NotRedex defs SZero t) => (0 _ : So (isTyConE t)) =>
+compareType' : (defs : Definitions) -> (ctx : EqContext n) ->
+               (s, t : Term 0 n) ->
+               (0 _ : NotRedexEq defs ctx SZero s) => (0 _ : So (isTyConE s)) =>
+               (0 _ : NotRedexEq defs ctx SZero t) => (0 _ : So (isTyConE t)) =>
                (0 _ : So (sameTyCon s t)) =>
                Eff EqualInner ()
 -- equality is the same as subtyping, except with the
@@ -477,8 +485,9 @@ namespace Elim
   EqualElim = InnerErrEff :: EqualInner
 
   private covering
-  computeElimTypeE : (defs : Definitions) -> EqContext n -> (sg : SQty) ->
-                     (e : Elim 0 n) -> (0 ne : NotRedex defs sg e) =>
+  computeElimTypeE : (defs : Definitions) -> (ctx : EqContext n) ->
+                     (sg : SQty) ->
+                     (e : Elim 0 n) -> (0 ne : NotRedexEq defs ctx sg e) =>
                      Eff EqualElim (Term 0 n)
   computeElimTypeE defs ectx sg e = lift $
     computeElimType defs (toWhnfContext ectx) sg e
@@ -492,8 +501,8 @@ namespace Elim
   try act = lift $ catch putError $ lift act {fs' = EqualElim}
 
   private covering %inline
-  clashE : (defs : Definitions) -> EqContext n -> (sg : SQty) ->
-           (e, f : Elim 0 n) -> (0 nf : NotRedex defs sg f) =>
+  clashE : (defs : Definitions) -> (ctx : EqContext n) -> (sg : SQty) ->
+           (e, f : Elim 0 n) -> (0 nf : NotRedexEq defs ctx sg f) =>
            Eff EqualElim (Term 0 n)
   clashE defs ctx sg e f = do
     putError $ ClashE e.loc ctx !mode e f
@@ -522,9 +531,10 @@ namespace Elim
                   (e, f : Elim 0 n) -> Eff EqualElim (Term 0 n)
 
   private covering
-  compare0Inner' : (defs : Definitions) -> EqContext n -> (sg : SQty) ->
+  compare0Inner' : (defs : Definitions) -> (ctx : EqContext n) -> (sg : SQty) ->
                    (e, f : Elim 0 n) ->
-                   (0 ne : NotRedex defs sg e) -> (0 nf : NotRedex defs sg f) ->
+                   (0 ne : NotRedexEq defs ctx sg e) ->
+                   (0 nf : NotRedexEq defs ctx sg f) ->
                    Eff EqualElim (Term 0 n)
 
   compare0Inner' defs ctx sg e@(F {}) f _ _ = do
diff --git a/lib/Quox/Typechecker.idr b/lib/Quox/Typechecker.idr
index 13052b1..53677a5 100644
--- a/lib/Quox/Typechecker.idr
+++ b/lib/Quox/Typechecker.idr
@@ -215,8 +215,8 @@ mutual
 
   check' ctx sg (Let qty rhs body loc) ty = do
     eres <- inferC ctx (subjMult sg qty) rhs
-    qout <- checkC (extendTy (qty * sg.qty) body.name eres.type ctx) sg
-              body.term (weakT 1 ty)
+    qout <- checkC (extendTyLet (qty * sg.qty) body.name eres.type (E rhs) ctx)
+              sg body.term (weakT 1 ty)
         >>= popQ loc qty
     pure $ qty * eres.qout + qout
 
@@ -338,8 +338,8 @@ mutual
     pure $ lookupBound sg.qty i ctx.tctx
   where
     lookupBound : forall n. Qty -> Var n -> TContext d n -> InferResult' d n
-    lookupBound pi VZ (ctx :< type) =
-      InfRes {type = weakT 1 type, qout = zeroFor ctx :< pi}
+    lookupBound pi VZ (ctx :< var) =
+      InfRes {type = weakT 1 var.type, qout = zeroFor ctx :< pi}
     lookupBound pi (VS i) (ctx :< _) =
       let InfRes {type, qout} = lookupBound pi i ctx in
       InfRes {type = weakT 1 type, qout = qout :< Zero}
diff --git a/lib/Quox/Typing.idr b/lib/Quox/Typing.idr
index 31c059f..5d62e8a 100644
--- a/lib/Quox/Typing.idr
+++ b/lib/Quox/Typing.idr
@@ -70,7 +70,7 @@ parameters (defs : Definitions) {auto _ : (Has ErrorEff fs, Has NameGen fs)}
    parameters (ctx : TyContext d n) (sg : SQty) (loc : Loc)
     export covering
     whnf : {0 isRedex : RedexTest tm} -> CanWhnf tm isRedex =>
-           tm d n -> Eff fs (NonRedex tm d n defs sg)
+           tm d n -> Eff fs (NonRedex tm d n defs ? sg)
     whnf tm = do
       let Val n = ctx.termLen; Val d = ctx.dimLen
       res <- lift $ runExcept $ whnf defs (toWhnfContext ctx) sg tm
@@ -120,7 +120,7 @@ parameters (defs : Definitions) {auto _ : (Has ErrorEff fs, Has NameGen fs)}
    parameters (ctx : EqContext n) (sg : SQty) (loc : Loc)
     export covering
     whnf : {0 isRedex : RedexTest tm} -> CanWhnf tm isRedex =>
-           tm 0 n -> Eff fs (NonRedex tm 0 n defs sg)
+           tm 0 n -> Eff fs (NonRedex tm 0 n defs ? sg)
     whnf tm = do
       res <- lift $ runExcept $ whnf defs (toWhnfContext ctx) sg tm
       rethrow res
diff --git a/lib/Quox/Typing/Context.idr b/lib/Quox/Typing/Context.idr
index a1dd49a..f660a7c 100644
--- a/lib/Quox/Typing/Context.idr
+++ b/lib/Quox/Typing/Context.idr
@@ -14,9 +14,27 @@ public export
 QContext : Nat -> Type
 QContext = Context' Qty
 
+public export
+record LocalVar d n where
+  constructor MkLocal
+  type : Term d n
+  term : Maybe (Term d n) -- if from a `let`
+%runElab deriveIndexed "LocalVar" [Show]
+
+export
+CanShift (LocalVar d) where
+  l // by = {type $= (// by), term $= map (// by)} l
+
+namespace LocalVar
+  subD : DSubst d1 d2 -> LocalVar d1 n -> LocalVar d2 n
+  subD th = {type $= (// th), term $= map (// th)}
+
+  weakD : LocalVar d n -> LocalVar (S d) n
+  weakD = subD $ shift 1
+
 public export
 TContext : TermLike
-TContext d = Context (Term d)
+TContext d = Context (\n => LocalVar d n)
 
 public export
 QOutput : Nat -> Type
@@ -67,10 +85,6 @@ record WhnfContext d n where
 %runElab deriveIndexed "WhnfContext" [Show]
 
 namespace TContext
-  export %inline
-  pushD : TContext d n -> TContext (S d) n
-  pushD tel = map (// shift 1) tel
-
   export %inline
   zeroFor : Context tm n -> QOutput n
   zeroFor ctx = Zero <$ ctx
@@ -89,6 +103,14 @@ public export
 CtxExtension0 : Nat -> Nat -> Nat -> Type
 CtxExtension0 d = Telescope ((BindName,) . Term d)
 
+public export
+CtxExtensionLet : Nat -> Nat -> Nat -> Type
+CtxExtensionLet d = Telescope ((Qty, BindName,) . LocalVar d)
+
+public export
+CtxExtensionLet0 : Nat -> Nat -> Nat -> Type
+CtxExtensionLet0 d = Telescope ((BindName,) . LocalVar d)
+
 namespace TyContext
   public export %inline
   empty : TyContext 0 0
@@ -100,21 +122,34 @@ namespace TyContext
   null ctx = null ctx.dnames && null ctx.tnames
 
   export %inline
-  extendTyN : CtxExtension d n1 n2 -> TyContext d n1 -> TyContext d n2
-  extendTyN xss (MkTyContext {termLen, dctx, dnames, tctx, tnames, qtys}) =
-    let (qs, xs, ss) = unzip3 xss in
+  extendTyLetN : CtxExtensionLet d n1 n2 -> TyContext d n1 -> TyContext d n2
+  extendTyLetN xss (MkTyContext {termLen, dctx, dnames, tctx, tnames, qtys}) =
+    let (qs, xs, ls) = unzip3 xss in
     MkTyContext {
       dctx, dnames,
       termLen = extendLen xss termLen,
-      tctx    = tctx . ss,
+      tctx    = tctx . ls,
       tnames  = tnames . xs,
       qtys    = qtys . qs
     }
 
+  export %inline
+  extendTyN : CtxExtension d n1 n2 -> TyContext d n1 -> TyContext d n2
+  extendTyN = extendTyLetN . map (\(q, x, s) => (q, x, MkLocal s Nothing))
+
+  export %inline
+  extendTyLetN0 : CtxExtensionLet0 d n1 n2 -> TyContext d n1 -> TyContext d n2
+  extendTyLetN0 xss = extendTyLetN (map (Zero,) xss)
+
   export %inline
   extendTyN0 : CtxExtension0 d n1 n2 -> TyContext d n1 -> TyContext d n2
   extendTyN0 xss = extendTyN (map (Zero,) xss)
 
+  export %inline
+  extendTyLet : Qty -> BindName -> Term d n -> Term d n ->
+                TyContext d n -> TyContext d (S n)
+  extendTyLet q x s e = extendTyLetN [< (q, x, MkLocal s (Just e))]
+
   export %inline
   extendTy : Qty -> BindName -> Term d n -> TyContext d n -> TyContext d (S n)
   extendTy q x s = extendTyN [< (q, x, s)]
@@ -130,7 +165,7 @@ namespace TyContext
       dctx   = dctx :<? Nothing,
       dnames = dnames :< x,
       dimLen = [|S dimLen|],
-      tctx   = pushD tctx,
+      tctx   = map weakD tctx,
       tnames, qtys
     }
 
@@ -169,7 +204,7 @@ makeEqContext' ctx th = MkEqContext {
   termLen = ctx.termLen,
   dassign = makeDAssign th,
   dnames  = ctx.dnames,
-  tctx    = map (// th) ctx.tctx,
+  tctx    = map (subD th) ctx.tctx,
   tnames  = ctx.tnames,
   qtys    = ctx.qtys
 }
@@ -191,21 +226,34 @@ namespace EqContext
   null ctx = null ctx.dnames && null ctx.tnames
 
   export %inline
-  extendTyN : CtxExtension 0 n1 n2 -> EqContext n1 -> EqContext n2
-  extendTyN xss (MkEqContext {termLen, dassign, dnames, tctx, tnames, qtys}) =
-    let (qs, xs, ss) = unzip3 xss in
+  extendTyLetN : CtxExtensionLet 0 n1 n2 -> EqContext n1 -> EqContext n2
+  extendTyLetN xss (MkEqContext {termLen, dassign, dnames, tctx, tnames, qtys}) =
+    let (qs, xs, ls) = unzip3 xss in
     MkEqContext {
       termLen = extendLen xss termLen,
-      tctx    = tctx . ss,
+      tctx    = tctx . ls,
       tnames  = tnames . xs,
       qtys    = qtys . qs,
       dassign, dnames
     }
 
+  export %inline
+  extendTyN : CtxExtension 0 n1 n2 -> EqContext n1 -> EqContext n2
+  extendTyN = extendTyLetN . map (\(q, x, s) => (q, x, MkLocal s Nothing))
+
+  export %inline
+  extendTyLetN0 : CtxExtensionLet0 0 n1 n2 -> EqContext n1 -> EqContext n2
+  extendTyLetN0 xss = extendTyLetN (map (Zero,) xss)
+
   export %inline
   extendTyN0 : CtxExtension0 0 n1 n2 -> EqContext n1 -> EqContext n2
   extendTyN0 xss = extendTyN (map (Zero,) xss)
 
+  export %inline
+  extendTyLet : Qty -> BindName -> Term 0 n -> Term 0 n ->
+                EqContext n -> EqContext (S n)
+  extendTyLet q x s e = extendTyLetN [< (q, x, MkLocal s (Just e))]
+
   export %inline
   extendTy : Qty -> BindName -> Term 0 n -> EqContext n -> EqContext (S n)
   extendTy q x s = extendTyN [< (q, x, s)]
@@ -225,7 +273,7 @@ namespace EqContext
   toTyContext (MkEqContext {dimLen, dassign, dnames, tctx, tnames, qtys}) =
     MkTyContext {
       dctx = fromGround dassign,
-      tctx = map (// shift0 dimLen) tctx,
+      tctx = map (subD $ shift0 dimLen) tctx,
       dnames, tnames, qtys
     }
 
@@ -252,7 +300,7 @@ namespace WhnfContext
     MkWhnfContext {
       dimLen = [|Val s + dimLen|],
       dnames = dnames ++ toSnocVect' ns,
-      tctx   = dweakT s <$> tctx,
+      tctx   = map (subD $ shift s) tctx,
       tnames
     }
 
@@ -264,10 +312,20 @@ namespace WhnfContext
 private
 prettyTContextElt : {opts : _} ->
                     BContext d -> BContext n ->
-                    Qty -> BindName -> Term d n -> Eff Pretty (Doc opts)
-prettyTContextElt dnames tnames q x s =
-  pure $ hsep [hcat [!(prettyQty q), !dotD, !(prettyTBind x)], !colonD,
-               !(withPrec Outer $ prettyTerm dnames tnames s)]
+                    Qty -> BindName -> LocalVar d n -> Eff Pretty (Doc opts)
+prettyTContextElt dnames tnames q x s = do
+  q  <- prettyQty q;   dot <- dotD
+  x  <- prettyTBind x; colon <- colonD
+  ty <- withPrec Outer $ prettyTerm dnames tnames s.type; eq <- cstD
+  tm <- traverse (withPrec Outer . prettyTerm dnames tnames) s.term
+  d  <- askAt INDENT
+  let qx = hcat [q, dot, x]
+  pure $ case tm of
+    Nothing =>
+      ifMultiline (hsep [qx, colon, ty]) (vsep [qx, indent d $ colon <++> ty])
+    Just tm =>
+      ifMultiline (hsep [qx, colon, ty, eq, tm])
+                  (vsep [qx, indent d $ colon <++> ty, indent d $ eq <++> tm])
 
 private
 prettyTContext' : {opts : _} ->
diff --git a/lib/Quox/Untyped/Erase.idr b/lib/Quox/Untyped/Erase.idr
index 701eaad..c51c5f5 100644
--- a/lib/Quox/Untyped/Erase.idr
+++ b/lib/Quox/Untyped/Erase.idr
@@ -88,16 +88,16 @@ parameters {opts : LayoutOpts} (showContext : Bool)
 
 public export
 Erase : List (Type -> Type)
-Erase = [Except Error, DefsReader, NameGen]
+Erase = [Except Error, NameGen]
 
 export
 liftWhnf : Eff Whnf a -> Eff Erase a
 liftWhnf act = lift $ wrapErr WrapTypeError act
 
 export covering
-computeElimType : ErasureContext d n -> SQty -> Elim d n -> Eff Erase (Term d n)
-computeElimType ctx sg e = do
-  defs <- askAt DEFS
+computeElimType : Q.Definitions -> ErasureContext d n -> SQty ->
+                  Elim d n -> Eff Erase (Term d n)
+computeElimType defs ctx sg e = do
   let ctx = toWhnfContext ctx
   liftWhnf $ do
     Element e _ <- whnf defs ctx sg e
@@ -106,10 +106,11 @@ computeElimType ctx sg e = do
 
 private %macro
 wrapExpect : TTImp ->
-             Elab (TyContext d n -> Loc -> Term d n -> Eff Erase a)
+             Elab (Q.Definitions -> TyContext d n -> Loc ->
+                   Term d n -> Eff Erase a)
 wrapExpect f_ = do
   f <- check `(\x => ~(f_) x)
-  pure $ \ctx, loc, s => liftWhnf $ f !(askAt DEFS) ctx SZero loc s
+  pure $ \defs, ctx, loc, s => liftWhnf $ f defs ctx SZero loc s
 
 
 public export
@@ -119,27 +120,36 @@ record EraseElimResult d n where
   term : U.Term n
 
 
+export covering
+eraseTerm' : (defs : Q.Definitions) -> (ctx : ErasureContext d n) ->
+             (ty, tm : Q.Term d n) ->
+             (0 _ : NotRedex defs (toWhnfContext ctx) SZero ty) =>
+             Eff Erase (U.Term n)
+
 -- "Ψ | Γ | Σ ⊢ s ⤋ s' ⇐ A" for `s' <- eraseTerm (Ψ,Γ,Σ) A s`
 --
 -- in the below comments, Ψ, Γ, Σ are implicit and
 -- only their extensions are written
 export covering
-eraseTerm : ErasureContext d n ->
+eraseTerm : Q.Definitions -> ErasureContext d n ->
             (ty, tm : Q.Term d n) -> Eff Erase (U.Term n)
+eraseTerm defs ctx ty tm = do
+  Element ty _ <- liftWhnf $ Interface.whnf defs (toWhnfContext ctx) SZero ty
+  eraseTerm' defs ctx ty tm
 
 
 -- "Ψ | Γ | Σ ⊢ e ⤋ e' ⇒ A" for `EraRes A e' <- eraseElim (Ψ,Γ,Σ) e`
 export covering
-eraseElim : ErasureContext d n -> (tm : Q.Elim d n) ->
+eraseElim : Q.Definitions -> ErasureContext d n -> (tm : Q.Elim d n) ->
             Eff Erase (EraseElimResult d n)
 
-eraseTerm ctx _ s@(TYPE {}) =
+eraseTerm' defs ctx _ s@(TYPE {}) =
   throw $ CompileTimeOnly ctx s
 
-eraseTerm ctx _ s@(IOState {}) =
+eraseTerm' defs ctx _ s@(IOState {}) =
   throw $ CompileTimeOnly ctx s
 
-eraseTerm ctx _ s@(Pi {}) =
+eraseTerm' defs ctx _ s@(Pi {}) =
   throw $ CompileTimeOnly ctx s
 
 --      x : A | 0.x ⊢ s ⤋ s' ⇐ B
@@ -149,66 +159,66 @@ eraseTerm ctx _ s@(Pi {}) =
 --    x : A | π.x ⊢ s ⤋ s' ⇐ B   π ≠ 0
 -- ----------------------------------------
 --  (λ x ⇒ s) ⤋ (λ x ⇒ s') ⇐ π.(x : A) → B
-eraseTerm ctx ty (Lam body loc) = do
+eraseTerm' defs ctx ty (Lam body loc) = do
   let x = body.name
-  (qty, arg, res) <- wrapExpect `(expectPi) ctx loc ty
-  body            <- eraseTerm (extendTy qty x arg ctx) res.term body.term
+  (qty, arg, res) <- wrapExpect `(expectPi) defs ctx loc ty
+  body            <- eraseTerm defs (extendTy qty x arg ctx) res.term body.term
   pure $ case isErased qty of
     Kept   => U.Lam x body loc
     Erased => sub1 (Erased loc) body
 
-eraseTerm ctx _ s@(Sig {}) =
+eraseTerm' defs ctx _ s@(Sig {}) =
   throw $ CompileTimeOnly ctx s
 
 --   s ⤋ s' ⇐ A    t ⤋ t' ⇐ B[s/x]
 -- ---------------------------------
 --  (s, t) ⤋ (s', t') ⇐ (x : A) × B
-eraseTerm ctx ty (Pair fst snd loc) = do
-  (a, b) <- wrapExpect `(expectSig) ctx loc ty
+eraseTerm' defs ctx ty (Pair fst snd loc) = do
+  (a, b) <- wrapExpect `(expectSig) defs ctx loc ty
   let b = sub1 b (Ann fst a a.loc)
-  fst <- eraseTerm ctx a fst
-  snd <- eraseTerm ctx b snd
+  fst <- eraseTerm defs ctx a fst
+  snd <- eraseTerm defs ctx b snd
   pure $ Pair fst snd loc
 
-eraseTerm ctx _ s@(Enum {}) =
+eraseTerm' defs ctx _ s@(Enum {}) =
   throw $ CompileTimeOnly ctx s
 
 -- '𝐚 ⤋ '𝐚 ⇐ {⋯}
-eraseTerm ctx _ (Tag tag loc) =
+eraseTerm' defs ctx _ (Tag tag loc) =
   pure $ Tag tag loc
 
-eraseTerm ctx ty s@(Eq {}) =
+eraseTerm' defs ctx ty s@(Eq {}) =
   throw $ CompileTimeOnly ctx s
 
 --        𝑖 ⊢ s ⤋ s' ⇐ A
 -- ---------------------------------
 --  (δ 𝑖 ⇒ s) ⤋ s' ⇐ Eq (𝑖 ⇒ A) l r
-eraseTerm ctx ty (DLam body loc) = do
-  a <- fst <$> wrapExpect `(expectEq) ctx loc ty
-  eraseTerm (extendDim body.name ctx) a.term body.term
+eraseTerm' defs ctx ty (DLam body loc) = do
+  a <- fst <$> wrapExpect `(expectEq) defs ctx loc ty
+  eraseTerm defs (extendDim body.name ctx) a.term body.term
 
-eraseTerm ctx _ s@(NAT {}) =
+eraseTerm' defs ctx _ s@(NAT {}) =
   throw $ CompileTimeOnly ctx s
 
 -- n ⤋ n ⇐ ℕ
-eraseTerm _ _ (Nat n loc) =
+eraseTerm' _ _ _ (Nat n loc) =
   pure $ Nat n loc
 
 --       s ⤋ s' ⇐ ℕ
 -- -----------------------
 --  succ s ⤋ succ s' ⇐ ℕ
-eraseTerm ctx ty (Succ p loc) = do
-  p <- eraseTerm ctx ty p
+eraseTerm' defs ctx ty (Succ p loc) = do
+  p <- eraseTerm defs ctx ty p
   pure $ Succ p loc
 
-eraseTerm ctx ty s@(STRING {}) =
+eraseTerm' defs ctx ty s@(STRING {}) =
   throw $ CompileTimeOnly ctx s
 
 -- s ⤋ s ⇐ String
-eraseTerm _ _ (Str s loc) =
+eraseTerm' _ _ _ (Str s loc) =
   pure $ Str s loc
 
-eraseTerm ctx ty s@(BOX {}) =
+eraseTerm' defs ctx ty s@(BOX {}) =
   throw $ CompileTimeOnly ctx s
 
 -- [s] ⤋ ⌷ ⇐ [0.A]
@@ -216,48 +226,49 @@ eraseTerm ctx ty s@(BOX {}) =
 --  π ≠ 0   s ⤋ s' ⇐ A
 -- --------------------
 --   [s] ⤋ s' ⇐ [π.A]
-eraseTerm ctx ty (Box val loc) = do
-  (qty, a) <- wrapExpect `(expectBOX) ctx loc ty
+eraseTerm' defs ctx ty (Box val loc) = do
+  (qty, a) <- wrapExpect `(expectBOX) defs ctx loc ty
   case isErased qty of
     Erased => pure $ Erased loc
-    Kept   => eraseTerm ctx a val
+    Kept   => eraseTerm defs ctx a val
 
 --            s ⤋ s' ⇐ A
 -- ---------------------------------
 --   let0 x = e in s ⤋ s'[⌷/x] ⇐ A
 --
---     e ⤋ e'  s ⤋ s' ⇐ A   π ≠ 0
+--        e ⤋ e' ⇒ E   π ≠ 0
+--      x : E ≔ e ⊢ s ⤋ s' ⇐ A
 -- -------------------------------------
 --  letπ x = e in s ⤋ let x = e' in s'
-eraseTerm ctx ty (Let pi e s loc) = do
+eraseTerm' defs ctx ty (Let pi e s loc) = do
   let x = s.name
   case isErased pi of
     Erased => do
-      ety <- computeElimType ctx SZero e
-      s'  <- eraseTerm (extendTy pi x ety ctx) (weakT 1 ty) s.term
+      ety <- computeElimType defs ctx SZero e
+      s'  <- eraseTerm defs (extendTyLet pi x ety (E e) ctx) (weakT 1 ty) s.term
       pure $ sub1 (Erased e.loc) s'
     Kept => do
-      EraRes ety e' <- eraseElim ctx e
-      s' <- eraseTerm (extendTy pi x ety ctx) (weakT 1 ty) s.term
+      EraRes ety e' <- eraseElim defs ctx e
+      s' <- eraseTerm defs (extendTyLet pi x ety (E e) ctx) (weakT 1 ty) s.term
       pure $ Let x e' s' loc
 
 --  e ⤋ e' ⇒ B
 -- ------------
 --  e ⤋ e' ⇐ A
-eraseTerm ctx ty (E e) =
-  term <$> eraseElim ctx e
+eraseTerm' defs ctx ty (E e) =
+  term <$> eraseElim defs ctx e
 
-eraseTerm ctx ty (CloT (Sub term th)) =
-  eraseTerm ctx ty $ pushSubstsWith' id th term
+eraseTerm' defs ctx ty (CloT (Sub term th)) =
+  eraseTerm defs ctx ty $ pushSubstsWith' id th term
 
-eraseTerm ctx ty (DCloT (Sub term th)) =
-  eraseTerm ctx ty $ pushSubstsWith' th id term
+eraseTerm' defs ctx ty (DCloT (Sub term th)) =
+  eraseTerm defs ctx ty $ pushSubstsWith' th id term
 
 --  defω x : A = s
 -- ----------------
 --    x ⤋ x ⇒ A
-eraseElim ctx e@(F x u loc) = do
-  Just def <- asksAt DEFS $ lookup x
+eraseElim defs ctx e@(F x u loc) = do
+  let Just def = lookup x defs
     | Nothing => throw $ notInScope loc x
   case isErased def.qty.qty of
     Erased => throw $ CompileTimeOnly ctx $ E e
@@ -266,10 +277,10 @@ eraseElim ctx e@(F x u loc) = do
 --  π.x ∈ Σ   π ≠ 0
 -- -----------------
 --     x ⤋ x ⇒ A
-eraseElim ctx e@(B i loc) = do
+eraseElim defs ctx e@(B i loc) = do
   case isErased $ ctx.qtys !!! i of
     Erased => throw $ CompileTimeOnly ctx $ E e
-    Kept   => pure $ EraRes (ctx.tctx !! i) $ B i loc
+    Kept   => pure $ EraRes (ctx.tctx !! i).type $ B i loc
 
 --  f ⤋ f' ⇒ π.(x : A) → B   s ⤋ s' ⇒ A   π ≠ 0
 -- ---------------------------------------------
@@ -278,13 +289,13 @@ eraseElim ctx e@(B i loc) = do
 --  f ⤋ f' ⇒ 0.(x : A) → B
 -- -------------------------
 --    f s ⤋ f' ⇒ B[s/x]
-eraseElim ctx (App fun arg loc) = do
-  efun              <- eraseElim ctx fun
-  (qty, targ, tres) <- wrapExpect `(expectPi) ctx loc efun.type
+eraseElim defs ctx (App fun arg loc) = do
+  efun              <- eraseElim defs ctx fun
+  (qty, targ, tres) <- wrapExpect `(expectPi) defs ctx loc efun.type
   let ty = sub1 tres (Ann arg targ arg.loc)
   case isErased qty of
     Erased => pure $ EraRes ty efun.term
-    Kept   => do arg <- eraseTerm ctx targ arg
+    Kept   => do arg <- eraseTerm defs ctx targ arg
                  pure $ EraRes ty $ App efun.term arg loc
 
 --                        e ⇒ (x : A) × B
@@ -298,16 +309,16 @@ eraseElim ctx (App fun arg loc) = do
 --   (caseρ e return z ⇒ R of {(x, y) ⇒ s}) ⤋
 --     ⤋
 --   let xy = e' in let x = fst xy in let y = snd xy in s' ⇒ R[e/z]
-eraseElim ctx (CasePair qty pair ret body loc) = do
+eraseElim defs ctx (CasePair qty pair ret body loc) = do
   let [< x, y] = body.names
   case isErased qty of
     Kept => do
-      EraRes ety eterm <- eraseElim ctx pair
+      EraRes ety eterm <- eraseElim defs ctx pair
       let ty = sub1 (ret // shift 2) $
                Ann (Pair (BVT 0 loc) (BVT 1 loc) loc) (weakT 2 ety) loc
-      (tfst, tsnd) <- wrapExpect `(expectSig) ctx loc ety
+      (tfst, tsnd) <- wrapExpect `(expectSig) defs ctx loc ety
       let ctx' = extendTyN [< (qty, x, tfst), (qty, y, tsnd.term)] ctx
-      body' <- eraseTerm ctx' ty body.term
+      body' <- eraseTerm defs ctx' ty body.term
       p <- mnb "p" loc
       pure $ EraRes (sub1 ret pair) $
         Let p eterm
@@ -316,28 +327,28 @@ eraseElim ctx (CasePair qty pair ret body loc) = do
               (body' // (B VZ loc ::: B (VS VZ) loc ::: shift 3))
               loc) loc) loc
     Erased => do
-      ety <- computeElimType ctx SOne pair
+      ety <- computeElimType defs ctx SOne pair
       let ty = sub1 (ret // shift 2) $
                Ann (Pair (BVT 0 loc) (BVT 1 loc) loc) (weakT 2 ety) loc
-      (tfst, tsnd) <- wrapExpect `(expectSig) ctx loc ety
+      (tfst, tsnd) <- wrapExpect `(expectSig) defs ctx loc ety
       let ctx' = extendTyN0 [< (x, tfst), (y, tsnd.term)] ctx
-      body' <- eraseTerm ctx' ty body.term
+      body' <- eraseTerm defs ctx' ty body.term
       pure $ EraRes (sub1 ret pair) $ subN [< Erased loc, Erased loc] body'
 
 --  e ⤋ e' ⇒ (x : A) × B
 -- ----------------------
 --   fst e ⤋ fst e' ⇒ A
-eraseElim ctx (Fst pair loc) = do
-  epair      <- eraseElim ctx pair
-  a          <- fst <$> wrapExpect `(expectSig) ctx loc epair.type
+eraseElim defs ctx (Fst pair loc) = do
+  epair      <- eraseElim defs ctx pair
+  a          <- fst <$> wrapExpect `(expectSig) defs ctx loc epair.type
   pure $ EraRes a $ Fst epair.term loc
 
 --     e ⤋ e' ⇒ (x : A) × B
 -- -----------------------------
 --  snd e ⤋ snd e' ⇒ B[fst e/x]
-eraseElim ctx (Snd pair loc) = do
-  epair      <- eraseElim ctx pair
-  b          <- snd <$> wrapExpect `(expectSig) ctx loc epair.type
+eraseElim defs ctx (Snd pair loc) = do
+  epair      <- eraseElim defs ctx pair
+  b          <- snd <$> wrapExpect `(expectSig) defs ctx loc epair.type
   pure $ EraRes (sub1 b (Fst pair loc)) $ Snd epair.term loc
 
 -- caseρ e return z ⇒ R of {} ⤋ absurd ⇒ R[e/z]
@@ -349,23 +360,23 @@ eraseElim ctx (Snd pair loc) = do
 --         e ⤋ e' ⇒ A   sᵢ ⤋ s'ᵢ ⇐ R[𝐚ᵢ/z]   ρ ≠ 0   i ≠ 0
 -- -------------------------------------------------------------------
 --  caseρ e return z ⇒ R of {𝐚ᵢ ⇒ sᵢ} ⤋ case e of {𝐚ᵢ ⇒ s'ᵢ} ⇒ R[e/z]
-eraseElim ctx e@(CaseEnum qty tag ret arms loc) = do
+eraseElim defs ctx e@(CaseEnum qty tag ret arms loc) = do
   let ty = sub1 ret tag
   case isErased qty of
     Erased => case SortedMap.toList arms of
       []         => pure $ EraRes ty $ Absurd loc
       [(t, rhs)] => do
         let ty' = sub1 ret (Ann (Tag t loc) (enum [t] loc) loc)
-        rhs' <- eraseTerm ctx ty' rhs
+        rhs' <- eraseTerm defs ctx ty' rhs
         pure $ EraRes ty rhs'
       _ => throw $ CompileTimeOnly ctx $ E e
     Kept => case List1.fromList $ SortedMap.toList arms of
       Nothing   => pure $ EraRes ty $ Absurd loc
       Just arms => do
-        etag <- eraseElim ctx tag
+        etag <- eraseElim defs ctx tag
         arms <- for arms $ \(t, rhs) => do
           let ty' = sub1 ret (Ann (Tag t loc) etag.type loc)
-          rhs' <- eraseTerm ctx ty' rhs
+          rhs' <- eraseTerm defs ctx ty' rhs
           pure (t, rhs')
         pure $ EraRes ty $ CaseEnum etag.term arms loc
 
@@ -382,12 +393,12 @@ eraseElim ctx e@(CaseEnum qty tag ret arms loc) = do
 --      caseρ n return z ⇒ R of {0 ⇒ z; succ m, 0.ih ⇒ s}
 --        ⤋
 --      case n' of {0 ⇒ z'; succ m ⇒ s'[⌷/ih]} ⇒ R[n/z]
-eraseElim ctx (CaseNat qty qtyIH nat ret zero succ loc) = do
+eraseElim defs ctx (CaseNat qty qtyIH nat ret zero succ loc) = do
   let ty = sub1 ret nat
-  enat <- eraseElim ctx nat
-  zero <- eraseTerm ctx (sub1 ret (Ann (Zero loc) (NAT loc) loc)) zero
+  enat <- eraseElim defs ctx nat
+  zero <- eraseTerm defs ctx (sub1 ret (Ann (Zero loc) (NAT loc) loc)) zero
   let [< p, ih] = succ.names
-  succ' <- eraseTerm
+  succ' <- eraseTerm defs
     (extendTyN [< (qty, p, NAT loc),
                   (qtyIH, ih, sub1 (ret // shift 1) (BV 0 loc))] ctx)
     (sub1 (ret // shift 2) (Ann (Succ (BVT 1 loc) loc) (NAT loc) loc))
@@ -404,56 +415,56 @@ eraseElim ctx (CaseNat qty qtyIH nat ret zero succ loc) = do
 --   b ⇒ [π.A]   x : A | 0.x ⊢ s ⤋ s' ⇐ R[[x]∷[0.A]/z]   πρ = 0
 -- -------------------------------------------------------------
 --  caseρ b return z ⇒ R of {[x] ⇒ s} ⤋ s'[⌷/x] ⇒ R[b/z]
-eraseElim ctx (CaseBox qty box ret body loc) = do
-  tbox <- computeElimType ctx SOne box -- [fixme] is there any way to avoid this?
-  (pi, tinner) <- wrapExpect `(expectBOX) ctx loc tbox
+eraseElim defs ctx (CaseBox qty box ret body loc) = do
+  tbox <- computeElimType defs ctx SOne box
+  (pi, tinner) <- wrapExpect `(expectBOX) defs ctx loc tbox
   let ctx' = extendTy (pi * qty) body.name tinner ctx
       bty  = sub1 (ret // shift 1) $
              Ann (Box (BVT 0 loc) loc) (weakT 1 tbox) loc
   case isErased $ qty * pi of
     Kept => do
-      ebox  <- eraseElim ctx box
-      ebody <- eraseTerm ctx' bty body.term
+      ebox  <- eraseElim defs ctx box
+      ebody <- eraseTerm defs ctx' bty body.term
       pure $ EraRes (sub1 ret box) $ Let body.name ebox.term ebody loc
     Erased => do
-      body' <- eraseTerm ctx' bty body.term
+      body' <- eraseTerm defs ctx' bty body.term
       pure $ EraRes (sub1 ret box) $ body' // one (Erased loc)
 
 --    f ⤋ f' ⇒ Eq (𝑖 ⇒ A) l r
 -- ------------------------------
 --      f @r ⤋ f' ⇒ A‹r/𝑖›
-eraseElim ctx (DApp fun arg loc) = do
-  efun <- eraseElim ctx fun
-  a    <- fst <$> wrapExpect `(expectEq) ctx loc efun.type
+eraseElim defs ctx (DApp fun arg loc) = do
+  efun <- eraseElim defs ctx fun
+  a    <- fst <$> wrapExpect `(expectEq) defs ctx loc efun.type
   pure $ EraRes (dsub1 a arg) efun.term
 
 --    s ⤋ s' ⇐ A
 -- ----------------
 --  s ∷ A ⤋ s' ⇒ A
-eraseElim ctx (Ann tm ty loc) =
-  EraRes ty <$> eraseTerm ctx ty tm
+eraseElim defs ctx (Ann tm ty loc) =
+  EraRes ty <$> eraseTerm defs ctx ty tm
 
 --        s ⤋ s' ⇐ A‹p/𝑖›
 -- -----------------------------------
 --  coe (𝑖 ⇒ A) @p @q s ⤋ s' ⇒ A‹q/𝑖›
-eraseElim ctx (Coe ty p q val loc) = do
-  val <- eraseTerm ctx (dsub1 ty p) val
+eraseElim defs ctx (Coe ty p q val loc) = do
+  val <- eraseTerm defs ctx (dsub1 ty p) val
   pure $ EraRes (dsub1 ty q) val
 
 --          s ⤋ s' ⇐ A
 -- --------------------------------
 --  comp A @p @q s @r {⋯} ⤋ s' ⇒ A
-eraseElim ctx (Comp ty p q val r zero one loc) =
-  EraRes ty <$> eraseTerm ctx ty val
+eraseElim defs ctx (Comp ty p q val r zero one loc) =
+  EraRes ty <$> eraseTerm defs ctx ty val
 
-eraseElim ctx t@(TypeCase ty ret arms def loc) =
+eraseElim defs ctx t@(TypeCase ty ret arms def loc) =
   throw $ CompileTimeOnly ctx $ E t
 
-eraseElim ctx (CloE (Sub term th)) =
-  eraseElim ctx $ pushSubstsWith' id th term
+eraseElim defs ctx (CloE (Sub term th)) =
+  eraseElim defs ctx $ pushSubstsWith' id th term
 
-eraseElim ctx (DCloE (Sub term th)) =
-  eraseElim ctx $ pushSubstsWith' th id term
+eraseElim defs ctx (DCloE (Sub term th)) =
+  eraseElim defs ctx $ pushSubstsWith' th id term
 
 
 export
@@ -539,8 +550,8 @@ trimLets (Erased loc) = Erased loc
 
 
 export covering
-eraseDef : Name -> Q.Definition -> Eff Erase U.Definition
-eraseDef name def@(MkDef qty type body scheme isMain loc) =
+eraseDef : Q.Definitions -> Name -> Q.Definition -> Eff Erase U.Definition
+eraseDef defs name def@(MkDef qty type body scheme isMain loc) =
   wrapErr (WhileErasing name def) $
   case isErased qty.qty of
     Erased => do
@@ -552,4 +563,4 @@ eraseDef name def@(MkDef qty type body scheme isMain loc) =
         Nothing  => case body of
           Postulate     => throw $ Postulate loc name
           Concrete body => KeptDef isMain . trimLets <$>
-                           eraseTerm empty type body
+                           eraseTerm defs empty type body
diff --git a/lib/Quox/Whnf/Coercion.idr b/lib/Quox/Whnf/Coercion.idr
index 0e35665..4b900bf 100644
--- a/lib/Quox/Whnf/Coercion.idr
+++ b/lib/Quox/Whnf/Coercion.idr
@@ -28,7 +28,7 @@ parameters {auto _ : CanWhnf Term Interface.isRedexT}
   export covering
   piCoe : (ty : DScopeTerm d n) -> (p, q : Dim d) ->
           (val, s : Term d n) -> Loc ->
-          Eff Whnf (Subset (Elim d n) (No . isRedexE defs sg))
+          Eff Whnf (Subset (Elim d n) (No . isRedexE defs ctx sg))
   piCoe sty@(S [< i] ty) p q val s loc = do
     -- (coe [i ⇒ π.(x : A) → B] @p @q t) s ⇝
     -- coe [i ⇒ B[𝒔‹i›/x] @p @q ((t ∷ (π.(x : A) → B)‹p/i›) 𝒔‹p›)
@@ -49,7 +49,7 @@ parameters {auto _ : CanWhnf Term Interface.isRedexT}
   sigCoe : (qty : Qty) ->
            (ty : DScopeTerm d n) -> (p, q : Dim d) -> (val : Term d n) ->
            (ret : ScopeTerm d n) -> (body : ScopeTermN 2 d n) -> Loc ->
-           Eff Whnf (Subset (Elim d n) (No . isRedexE defs sg))
+           Eff Whnf (Subset (Elim d n) (No . isRedexE defs ctx sg))
   sigCoe qty sty@(S [< i] ty) p q val ret body loc = do
     -- caseπ (coe [i ⇒ (x : A) × B] @p @q s) return z ⇒ C of { (a, b) ⇒ e }
     --   ⇝
@@ -74,7 +74,7 @@ parameters {auto _ : CanWhnf Term Interface.isRedexT}
   ||| reduce a pair projection `Fst (Coe ty p q val) loc`
   export covering
   fstCoe : (ty : DScopeTerm d n) -> (p, q : Dim d) -> (val : Term d n) ->
-           Loc -> Eff Whnf (Subset (Elim d n) (No . isRedexE defs sg))
+           Loc -> Eff Whnf (Subset (Elim d n) (No . isRedexE defs ctx sg))
   fstCoe sty@(S [< i] ty) p q val loc = do
     -- fst (coe (𝑖 ⇒ (x : A) × B) @p @q s)
     --   ⇝
@@ -91,7 +91,7 @@ parameters {auto _ : CanWhnf Term Interface.isRedexT}
   ||| reduce a pair projection `Snd (Coe ty p q val) loc`
   export covering
   sndCoe : (ty : DScopeTerm d n) -> (p, q : Dim d) -> (val : Term d n) ->
-           Loc -> Eff Whnf (Subset (Elim d n) (No . isRedexE defs sg))
+           Loc -> Eff Whnf (Subset (Elim d n) (No . isRedexE defs ctx sg))
   sndCoe sty@(S [< i] ty) p q val loc = do
     -- snd (coe (𝑖 ⇒ (x : A) × B) @p @q s)
     --   ⇝
@@ -115,7 +115,7 @@ parameters {auto _ : CanWhnf Term Interface.isRedexT}
   export covering
   eqCoe : (ty : DScopeTerm d n) -> (p, q : Dim d) -> (val : Term d n) ->
           (r : Dim d) -> Loc ->
-          Eff Whnf (Subset (Elim d n) (No . isRedexE defs sg))
+          Eff Whnf (Subset (Elim d n) (No . isRedexE defs ctx sg))
   eqCoe sty@(S [< j] ty) p q val r loc = do
     -- (coe [j ⇒ Eq [i ⇒ A] L R] @p @q eq) @r
     --   ⇝
@@ -133,7 +133,7 @@ parameters {auto _ : CanWhnf Term Interface.isRedexT}
   boxCoe : (qty : Qty) ->
            (ty : DScopeTerm d n) -> (p, q : Dim d) -> (val : Term d n) ->
            (ret : ScopeTerm d n) -> (body : ScopeTerm d n) -> Loc ->
-           Eff Whnf (Subset (Elim d n) (No . isRedexE defs sg))
+           Eff Whnf (Subset (Elim d n) (No . isRedexE defs ctx sg))
   boxCoe qty sty@(S [< i] ty) p q val ret body loc = do
     -- caseπ (coe [i ⇒ [ρ. A]] @p @q s) return z ⇒ C of { [a] ⇒ e }
     --   ⇝
@@ -151,7 +151,7 @@ parameters {auto _ : CanWhnf Term Interface.isRedexT}
   pushCoe : BindName ->
             (ty : Term (S d) n) -> (p, q : Dim d) -> (s : Term d n) -> Loc ->
             (0 pc : So (canPushCoe sg ty s)) =>
-            Eff Whnf (NonRedex Elim d n defs sg)
+            Eff Whnf (NonRedex Elim d n defs ctx sg)
   pushCoe i ty p q s loc =
     case ty of
       -- (coe ★ᵢ @_ @_ s) ⇝ (s ∷ ★ᵢ)
diff --git a/lib/Quox/Whnf/ComputeElimType.idr b/lib/Quox/Whnf/ComputeElimType.idr
index 441ba1e..87891ff 100644
--- a/lib/Quox/Whnf/ComputeElimType.idr
+++ b/lib/Quox/Whnf/ComputeElimType.idr
@@ -14,8 +14,8 @@ export covering
 computeElimType :
   CanWhnf Term Interface.isRedexT =>
   CanWhnf Elim Interface.isRedexE =>
-  (defs : Definitions) -> WhnfContext d n -> (0 sg : SQty) ->
-  (e : Elim d n) -> (0 ne : No (isRedexE defs sg e)) =>
+  (defs : Definitions) -> (ctx : WhnfContext d n) -> (0 sg : SQty) ->
+  (e : Elim d n) -> (0 ne : No (isRedexE defs ctx sg e)) =>
   Eff Whnf (Term d n)
 
 
@@ -24,8 +24,8 @@ export covering
 computeWhnfElimType0 :
   CanWhnf Term Interface.isRedexT =>
   CanWhnf Elim Interface.isRedexE =>
-  (defs : Definitions) -> WhnfContext d n -> (0 sg : SQty) ->
-  (e : Elim d n) -> (0 ne : No (isRedexE defs sg e)) =>
+  (defs : Definitions) -> (ctx : WhnfContext d n) -> (0 sg : SQty) ->
+  (e : Elim d n) -> (0 ne : No (isRedexE defs ctx sg e)) =>
   Eff Whnf (Term d n)
 
 computeElimType defs ctx sg e =
@@ -36,7 +36,7 @@ computeElimType defs ctx sg e =
       pure $ def.typeWithAt ctx.dimLen ctx.termLen u
 
     B i _ =>
-      pure $ ctx.tctx !! i
+      pure (ctx.tctx !! i).type
 
     App f s loc =>
       case !(computeWhnfElimType0 defs ctx sg f {ne = noOr1 ne}) of
diff --git a/lib/Quox/Whnf/Interface.idr b/lib/Quox/Whnf/Interface.idr
index 15a8978..d2b11da 100644
--- a/lib/Quox/Whnf/Interface.idr
+++ b/lib/Quox/Whnf/Interface.idr
@@ -18,13 +18,14 @@ Whnf = [Except Error, NameGen]
 
 public export
 0 RedexTest : TermLike -> Type
-RedexTest tm = {0 d, n : Nat} -> Definitions -> SQty -> tm d n -> Bool
+RedexTest tm =
+  {0 d, n : Nat} -> Definitions -> WhnfContext d n -> SQty -> tm d n -> Bool
 
 public export
 interface CanWhnf (0 tm : TermLike) (0 isRedex : RedexTest tm) | tm
 where
   whnf : (defs : Definitions) -> (ctx : WhnfContext d n) -> (q : SQty) ->
-         tm d n -> Eff Whnf (Subset (tm d n) (No . isRedex defs q))
+         tm d n -> Eff Whnf (Subset (tm d n) (No . isRedex defs ctx q))
   -- having isRedex be part of the class header, and needing to be explicitly
   -- quantified on every use since idris can't infer its type, is a little ugly.
   -- but none of the alternatives i've thought of so far work. e.g. in some
@@ -37,19 +38,20 @@ whnf0 defs ctx q t = fst <$> whnf defs ctx q t
 
 public export
 0 IsRedex, NotRedex : {isRedex : RedexTest tm} -> CanWhnf tm isRedex =>
-                      Definitions -> SQty -> Pred (tm d n)
-IsRedex  defs q = So . isRedex defs q
-NotRedex defs q = No . isRedex defs q
+                      Definitions -> WhnfContext d n -> SQty -> Pred (tm d n)
+IsRedex  defs ctx q = So . isRedex defs ctx q
+NotRedex defs ctx q = No . isRedex defs ctx q
 
 public export
 0 NonRedex : (tm : TermLike) -> {isRedex : RedexTest tm} ->
              CanWhnf tm isRedex => (d, n : Nat) ->
-             (defs : Definitions) -> SQty -> Type
-NonRedex tm d n defs q = Subset (tm d n) (NotRedex defs q)
+             Definitions -> WhnfContext d n -> SQty -> Type
+NonRedex tm d n defs ctx q = Subset (tm d n) (NotRedex defs ctx q)
 
 public export %inline
 nred : {0 isRedex : RedexTest tm} -> (0 _ : CanWhnf tm isRedex) =>
-       (t : tm d n) -> (0 nr : NotRedex defs q t) => NonRedex tm d n defs q
+       (t : tm d n) -> (0 nr : NotRedex defs ctx q t) =>
+       NonRedex tm d n defs ctx q
 nred t = Element t nr
 
 
@@ -193,50 +195,52 @@ mutual
   ||| a reducible elimination
   |||
   ||| 1. a free variable, if its definition is known
-  ||| 2. an elimination whose head is reducible
-  ||| 3. an "active" elimination:
+  ||| 2. a bound variable pointing to a `let`
+  ||| 3. an elimination whose head is reducible
+  ||| 4. an "active" elimination:
   |||    an application whose head is an annotated lambda,
   |||    a case expression whose head is an annotated constructor form, etc
-  ||| 4. a redundant annotation, or one whose term or type is reducible
-  ||| 5. a coercion `coe (𝑖 ⇒ A) @p @q s` where:
+  ||| 5. a redundant annotation, or one whose term or type is reducible
+  ||| 6. a coercion `coe (𝑖 ⇒ A) @p @q s` where:
   |||      a. `A` is reducible or a type constructor, or
   |||      b. `𝑖` is not mentioned in `A`
   |||         ([fixme] should be A‹0/𝑖› = A‹1/𝑖›), or
   |||      c. `p = q`
-  ||| 6. a composition `comp A @p @q s @r {⋯}`
+  ||| 7. a composition `comp A @p @q s @r {⋯}`
   |||    where `p = q`, `r = 0`, or `r = 1`
-  ||| 7. a closure
+  ||| 8. a closure
   public export
   isRedexE : RedexTest Elim
-  isRedexE defs sg (F {x, u, _}) = isJust $ lookupElim0 x u defs
-  isRedexE _ sg (B {}) = False
-  isRedexE defs sg (App {fun, _}) =
-    isRedexE defs sg fun || isLamHead fun
-  isRedexE defs sg (CasePair {pair, _}) =
-    isRedexE defs sg pair || isPairHead pair || isYes (sg `decEq` SZero)
-  isRedexE defs sg (Fst pair _) =
-    isRedexE defs sg pair || isPairHead pair
-  isRedexE defs sg (Snd pair _) =
-    isRedexE defs sg pair || isPairHead pair
-  isRedexE defs sg (CaseEnum {tag, _}) =
-    isRedexE defs sg tag || isTagHead tag
-  isRedexE defs sg (CaseNat {nat, _}) =
-    isRedexE defs sg nat || isNatHead nat
-  isRedexE defs sg (CaseBox {box, _}) =
-    isRedexE defs sg box || isBoxHead box
-  isRedexE defs sg (DApp {fun, arg, _}) =
-    isRedexE defs sg fun || isDLamHead fun || isK arg
-  isRedexE defs sg (Ann {tm, ty, _}) =
-    isE tm || isRedexT defs sg tm || isRedexT defs SZero ty
-  isRedexE defs sg (Coe {ty = S _ (N _), _}) = True
-  isRedexE defs sg (Coe {ty = S _ (Y ty), p, q, val, _}) =
-    isRedexT defs SZero ty || canPushCoe sg ty val || isYes (p `decEqv` q)
-  isRedexE defs sg (Comp {ty, p, q, r, _}) =
+  isRedexE defs ctx sg (F {x, u, _}) = isJust $ lookupElim0 x u defs
+  isRedexE _    ctx sg (B {i, _}) = isJust (ctx.tctx !! i).term
+  isRedexE defs ctx sg (App {fun, _}) =
+    isRedexE defs ctx sg fun || isLamHead fun
+  isRedexE defs ctx sg (CasePair {pair, _}) =
+    isRedexE defs ctx sg pair || isPairHead pair || isYes (sg `decEq` SZero)
+  isRedexE defs ctx sg (Fst pair _) =
+    isRedexE defs ctx sg pair || isPairHead pair
+  isRedexE defs ctx sg (Snd pair _) =
+    isRedexE defs ctx sg pair || isPairHead pair
+  isRedexE defs ctx sg (CaseEnum {tag, _}) =
+    isRedexE defs ctx sg tag || isTagHead tag
+  isRedexE defs ctx sg (CaseNat {nat, _}) =
+    isRedexE defs ctx sg nat || isNatHead nat
+  isRedexE defs ctx sg (CaseBox {box, _}) =
+    isRedexE defs ctx sg box || isBoxHead box
+  isRedexE defs ctx sg (DApp {fun, arg, _}) =
+    isRedexE defs ctx sg fun || isDLamHead fun || isK arg
+  isRedexE defs ctx sg (Ann {tm, ty, _}) =
+    isE tm || isRedexT defs ctx sg tm || isRedexT defs ctx SZero ty
+  isRedexE defs ctx sg (Coe {ty = S _ (N _), _}) = True
+  isRedexE defs ctx sg (Coe {ty = S [< i] (Y ty), p, q, val, _}) =
+    isRedexT defs (extendDim i ctx) SZero ty ||
+    canPushCoe sg ty val || isYes (p `decEqv` q)
+  isRedexE defs ctx sg (Comp {ty, p, q, r, _}) =
     isYes (p `decEqv` q) || isK r
-  isRedexE defs sg (TypeCase {ty, ret, _}) =
-    isRedexE defs sg ty || isRedexT defs sg ret || isAnnTyCon ty
-  isRedexE _ _ (CloE  {}) = True
-  isRedexE _ _ (DCloE {}) = True
+  isRedexE defs ctx sg (TypeCase {ty, ret, _}) =
+    isRedexE defs ctx sg ty || isRedexT defs ctx sg ret || isAnnTyCon ty
+  isRedexE _ _ _ (CloE  {}) = True
+  isRedexE _ _ _ (DCloE {}) = True
 
   ||| a reducible term
   |||
@@ -248,9 +252,9 @@ mutual
   ||| 5. a `let` expression
   public export
   isRedexT : RedexTest Term
-  isRedexT _    _  (CloT  {})  = True
-  isRedexT _    _  (DCloT {})  = True
-  isRedexT _    _  (Let   {})  = True
-  isRedexT defs sg (E {e, _})  = isAnn e || isRedexE defs sg e
-  isRedexT _    _  (Succ p {}) = isNatConst p
-  isRedexT _    _  _           = False
+  isRedexT _    _   _  (CloT  {})  = True
+  isRedexT _    _   _  (DCloT {})  = True
+  isRedexT _    _   _  (Let   {})  = True
+  isRedexT defs ctx sg (E {e, _})  = isAnn e || isRedexE defs ctx sg e
+  isRedexT _    _   _  (Succ p {}) = isNatConst p
+  isRedexT _    _   _  _           = False
diff --git a/lib/Quox/Whnf/Main.idr b/lib/Quox/Whnf/Main.idr
index 9248edf..2c0a67a 100644
--- a/lib/Quox/Whnf/Main.idr
+++ b/lib/Quox/Whnf/Main.idr
@@ -20,7 +20,10 @@ CanWhnf Elim Interface.isRedexE where
     _ | Just y  = whnf defs ctx sg $ setLoc loc $ injElim ctx y
     _ | Nothing = pure $ Element (F x u loc) $ rewrite eq in Ah
 
-  whnf _ _ _ (B i loc) = pure $ nred $ B i loc
+  whnf defs ctx sg (B i loc) with (ctx.tctx !! i) proof eq1
+    _ | l with (l.term) proof eq2
+      _ | Just y  = whnf defs ctx sg $ Ann y l.type loc
+      _ | Nothing = pure $ Element (B i loc) $ rewrite eq1 in rewrite eq2 in Ah
 
   -- ((λ x ⇒ t) ∷ (π.x : A) → B) s ⇝ t[s∷A/x] ∷ B[s∷A/x]
   whnf defs ctx sg (App f s appLoc) = do
diff --git a/lib/Quox/Whnf/TypeCase.idr b/lib/Quox/Whnf/TypeCase.idr
index a42c0d9..2fc34ba 100644
--- a/lib/Quox/Whnf/TypeCase.idr
+++ b/lib/Quox/Whnf/TypeCase.idr
@@ -35,7 +35,7 @@ parameters {auto _ : CanWhnf Term Interface.isRedexT}
   ||| for an elim returns a pair of type-cases that will reduce to that;
   ||| for other intro forms error
   export covering
-  tycasePi : (t : Term d n) -> (0 tnf : No (isRedexT defs SZero t)) =>
+  tycasePi : (t : Term d n) -> (0 tnf : No (isRedexT defs ctx SZero t)) =>
              Eff Whnf (Term d n, ScopeTerm d n)
   tycasePi (Pi {arg, res, _}) = pure (arg, res)
   tycasePi (E e) {tnf} = do
@@ -53,7 +53,7 @@ parameters {auto _ : CanWhnf Term Interface.isRedexT}
   ||| for an elim returns a pair of type-cases that will reduce to that;
   ||| for other intro forms error
   export covering
-  tycaseSig : (t : Term d n) -> (0 tnf : No (isRedexT defs SZero t)) =>
+  tycaseSig : (t : Term d n) -> (0 tnf : No (isRedexT defs ctx SZero t)) =>
               Eff Whnf (Term d n, ScopeTerm d n)
   tycaseSig (Sig {fst, snd, _}) = pure (fst, snd)
   tycaseSig (E e) {tnf} = do
@@ -71,7 +71,7 @@ parameters {auto _ : CanWhnf Term Interface.isRedexT}
   ||| for an elim returns a type-case that will reduce to that;
   ||| for other intro forms error
   export covering
-  tycaseBOX : (t : Term d n) -> (0 tnf : No (isRedexT defs SZero t)) =>
+  tycaseBOX : (t : Term d n) -> (0 tnf : No (isRedexT defs ctx SZero t)) =>
               Eff Whnf (Term d n)
   tycaseBOX (BOX {ty, _}) = pure ty
   tycaseBOX (E e) {tnf} = do
@@ -83,7 +83,7 @@ parameters {auto _ : CanWhnf Term Interface.isRedexT}
   ||| for an elim returns five type-cases that will reduce to that;
   ||| for other intro forms error
   export covering
-  tycaseEq : (t : Term d n) -> (0 tnf : No (isRedexT defs SZero t)) =>
+  tycaseEq : (t : Term d n) -> (0 tnf : No (isRedexT defs ctx SZero t)) =>
              Eff Whnf (Term d n, Term d n, DScopeTerm d n, Term d n, Term d n)
   tycaseEq (Eq {ty, l, r, _}) = pure (ty.zero, ty.one, ty, l, r)
   tycaseEq (E e) {tnf} = do
@@ -107,7 +107,7 @@ parameters {auto _ : CanWhnf Term Interface.isRedexT}
   reduceTypeCase : (ty : Term d n) -> (u : Universe) -> (ret : Term d n) ->
                    (arms : TypeCaseArms d n) -> (def : Term d n) ->
                    (0 _ : So (isTyCon ty)) => Loc ->
-                   Eff Whnf (Subset (Elim d n) (No . isRedexE defs SZero))
+                   Eff Whnf (Subset (Elim d n) (No . isRedexE defs ctx SZero))
   reduceTypeCase ty u ret arms def loc = case ty of
     -- (type-case ★ᵢ ∷ _ return Q of { ★ ⇒ s; ⋯ }) ⇝ s ∷ Q
     TYPE {} =>
diff --git a/tests/AstExtra.idr b/tests/AstExtra.idr
index f9cc1fd..76257b7 100644
--- a/tests/AstExtra.idr
+++ b/tests/AstExtra.idr
@@ -24,13 +24,17 @@ anys : {n : Nat} -> QContext n
 anys {n = 0}   = [<]
 anys {n = S n} = anys :< Any
 
+public export
+locals : Context (Term d) n -> Context (LocalVar d) n
+locals = map $ \t => MkLocal t Nothing
+
 public export
 ctx, ctx01 : {n : Nat} -> Context (\n => (BindName, Term 0 n)) n ->
              TyContext 0 n
 ctx tel = let (ns, ts) = unzip tel in
-          MkTyContext new [<] ts ns anys
+          MkTyContext new [<] (locals ts) ns anys
 ctx01 tel = let (ns, ts) = unzip tel in
-            MkTyContext ZeroIsOne [<] ts ns anys
+            MkTyContext ZeroIsOne [<] (locals ts) ns anys
 
 export
 mkDef : GQty -> Term 0 0 -> Term 0 0 -> Definition
diff --git a/tests/Tests/Reduce.idr b/tests/Tests/Reduce.idr
index 66595cb..40c071e 100644
--- a/tests/Tests/Reduce.idr
+++ b/tests/Tests/Reduce.idr
@@ -33,7 +33,7 @@ parameters {0 isRedex : RedexTest tm} {auto _ : CanWhnf tm isRedex} {d, n : Nat}
 
 private
 ctx : {n : Nat} -> Context (\n => (BindName, Term 0 n)) n -> WhnfContext 0 n
-ctx xs = let (ns, ts) = unzip xs in MkWhnfContext [<] ns ts
+ctx xs = let (ns, ts) = unzip xs in MkWhnfContext [<] ns (locals ts)
 
 
 export