diff --git a/lib/Quox/Syntax/Qty.idr b/lib/Quox/Syntax/Qty.idr
index e5ed436..2318649 100644
--- a/lib/Quox/Syntax/Qty.idr
+++ b/lib/Quox/Syntax/Qty.idr
@@ -48,7 +48,7 @@ interface Eq q => IsQty q where
   ||| subject of a typing judgement [@qtt, §2.3].
   IsSubj : Pred q
   isSubj : Dec1 IsSubj
-  zeroIsSubj : IsSubj zero
+  zeroIsSubj : forall pi. IsZero pi -> IsSubj pi
   oneIsSubj  : IsSubj one
   timesSubj  : forall pi, rh. IsSubj pi -> IsSubj rh -> IsSubj (pi * rh)
 
@@ -56,7 +56,7 @@ interface Eq q => IsQty q where
   ||| quantity. so not exact usage counts, maybe.
   IsGlobal : Pred q
   isGlobal : Dec1 IsGlobal
-  zeroIsGlobal : IsGlobal zero
+  zeroIsGlobal : forall pi. IsZero pi -> IsGlobal zero
 
 public export
 0 SQty : (q : Type) -> IsQty q => Type
@@ -64,16 +64,27 @@ SQty q = Subset q IsSubj
 
 public export %inline
 szero : IsQty q => SQty q
-szero = Element zero zeroIsSubj
+szero = Element zero $ zeroIsSubj zeroIsZero
 
 public export %inline
 sone : IsQty q => SQty q
 sone = Element one oneIsSubj
 
+||| "σ ⨴ π"
+|||
+||| ``sg `subjMult` pi`` is equal to `pi` if it is zero, otherwise it
+||| is equal to `sg`.
+public export %inline
+subjMult : IsQty q => SQty q -> q -> SQty q
+subjMult sg pi = case isZero pi of
+  Yes y => Element pi $ zeroIsSubj y
+  No  _ => sg
+
+
 public export
 0 GQty : (q : Type) -> IsQty q => Type
 GQty q = Subset q IsGlobal
 
 public export %inline
 gzero : IsQty q => GQty q
-gzero = Element zero zeroIsGlobal
+gzero = Element zero $ zeroIsGlobal zeroIsZero
diff --git a/lib/Quox/Syntax/Qty/Three.idr b/lib/Quox/Syntax/Qty/Three.idr
index 702cafd..2bc6585 100644
--- a/lib/Quox/Syntax/Qty/Three.idr
+++ b/lib/Quox/Syntax/Qty/Three.idr
@@ -99,13 +99,13 @@ IsQty Three where
 
   IsSubj     = IsSubj3
   isSubj     = isSubj3
-  zeroIsSubj = SZero
+  zeroIsSubj = \Refl => SZero
   oneIsSubj  = SOne
   timesSubj  = timesSubj3
 
   IsGlobal     = IsGlobal3
   isGlobal     = isGlobal3
-  zeroIsGlobal = GZero
+  zeroIsGlobal = \Refl => GZero
 
 
 export Uninhabited (IsGlobal3 One) where uninhabited _ impossible
diff --git a/lib/Quox/Typechecker.idr b/lib/Quox/Typechecker.idr
index c875078..436023e 100644
--- a/lib/Quox/Typechecker.idr
+++ b/lib/Quox/Typechecker.idr
@@ -52,15 +52,6 @@ private %inline
 lookupFree : CanTC' q g m => Name -> m (Definition' q g)
 lookupFree x = lookupFree' !ask x
 
-||| ``sg `subjMult` pi`` is zero if either argument is, otherwise it
-||| is equal to `sg`. (written as "σ ⨴ π" to emphasise its left bias.)
-|||
-||| only certain quantities can occur for a typing judgement to avoid losing
-||| subject reduction [@qtt, §2.3], which is why this is not just `sg*pi`.
-private %inline
-subjMult : IsQty q => SQty q -> q -> SQty q
-subjMult sg pi = if isYes $ isZero pi then szero else sg
-
 
 parameters {auto _ : IsQty q} {auto _ : CanTC q m}
  mutual