diff --git a/differential-dataflow/src/operators/arrange/arrangement.rs b/differential-dataflow/src/operators/arrange/arrangement.rs
index 868f69a93..05cab5328 100644
--- a/differential-dataflow/src/operators/arrange/arrangement.rs
+++ b/differential-dataflow/src/operators/arrange/arrangement.rs
@@ -283,22 +283,20 @@ where
     T1: TraceReader + Clone + 'static,
 {
     /// A direct implementation of `ReduceCore::reduce_abelian`.
-    pub fn reduce_abelian<L, K, V, Bu, T2>(&self, name: &str, mut logic: L) -> Arranged<G, TraceAgent<T2>>
+    pub fn reduce_abelian<L, Bu, T2>(&self, name: &str, mut logic: L) -> Arranged<G, TraceAgent<T2>>
     where
-        T1: TraceReader<KeyOwn = K>,
+        T1: TraceReader<KeyOwn: Ord>,
         T2: for<'a> Trace<
             Key<'a>= T1::Key<'a>,
-            KeyOwn = K,
-            ValOwn = V,
+            KeyOwn=T1::KeyOwn,
+            ValOwn: Data,
             Time=T1::Time,
             Diff: Abelian,
         >+'static,
-        K: Ord + 'static,
-        V: Data,
-        Bu: Builder<Time=G::Timestamp, Output = T2::Batch, Input: Container + PushInto<((K, V), T2::Time, T2::Diff)>>,
-        L: FnMut(T1::Key<'_>, &[(T1::Val<'_>, T1::Diff)], &mut Vec<(V, T2::Diff)>)+'static,
+        Bu: Builder<Time=G::Timestamp, Output = T2::Batch, Input: Container + PushInto<((T1::KeyOwn, T2::ValOwn), T2::Time, T2::Diff)>>,
+        L: FnMut(T1::Key<'_>, &[(T1::Val<'_>, T1::Diff)], &mut Vec<(T2::ValOwn, T2::Diff)>)+'static,
     {
-        self.reduce_core::<_,K,V,Bu,T2>(name, move |key, input, output, change| {
+        self.reduce_core::<_,Bu,T2>(name, move |key, input, output, change| {
             if !input.is_empty() {
                 logic(key, input, change);
             }
@@ -308,22 +306,20 @@ where
     }
 
     /// A direct implementation of `ReduceCore::reduce_core`.
-    pub fn reduce_core<L, K, V, Bu, T2>(&self, name: &str, logic: L) -> Arranged<G, TraceAgent<T2>>
+    pub fn reduce_core<L, Bu, T2>(&self, name: &str, logic: L) -> Arranged<G, TraceAgent<T2>>
     where
-        T1: TraceReader<KeyOwn = K>,
+        T1: TraceReader<KeyOwn: Ord>,
         T2: for<'a> Trace<
             Key<'a>=T1::Key<'a>,
-            KeyOwn = K,
-            ValOwn = V,
+            KeyOwn=T1::KeyOwn,
+            ValOwn: Data,
             Time=T1::Time,
         >+'static,
-        K: Ord + 'static,
-        V: Data,
-        Bu: Builder<Time=G::Timestamp, Output = T2::Batch, Input: Container + PushInto<((K, V), T2::Time, T2::Diff)>>,
-        L: FnMut(T1::Key<'_>, &[(T1::Val<'_>, T1::Diff)], &mut Vec<(V, T2::Diff)>, &mut Vec<(V, T2::Diff)>)+'static,
+        Bu: Builder<Time=G::Timestamp, Output = T2::Batch, Input: Container + PushInto<((T1::KeyOwn, T2::ValOwn), T2::Time, T2::Diff)>>,
+        L: FnMut(T1::Key<'_>, &[(T1::Val<'_>, T1::Diff)], &mut Vec<(T2::ValOwn, T2::Diff)>, &mut Vec<(T2::ValOwn, T2::Diff)>)+'static,
     {
         use crate::operators::reduce::reduce_trace;
-        reduce_trace::<_,_,Bu,_,_,V,_>(self, name, logic)
+        reduce_trace::<_,_,Bu,_,_>(self, name, logic)
     }
 }
 
diff --git a/differential-dataflow/src/operators/reduce.rs b/differential-dataflow/src/operators/reduce.rs
index 317b908f6..8d1880d87 100644
--- a/differential-dataflow/src/operators/reduce.rs
+++ b/differential-dataflow/src/operators/reduce.rs
@@ -89,7 +89,7 @@ where
 {
     fn reduce_named<L, V2: Data, R2: Ord+Abelian+'static>(&self, name: &str, logic: L) -> Collection<G, (K, V2), R2>
         where L: FnMut(&K, &[(&V, R)], &mut Vec<(V2, R2)>)+'static {
-        self.reduce_abelian::<_,K,V2,ValBuilder<_,_,_,_>,ValSpine<_,_,_,_>>(name, logic)
+        self.reduce_abelian::<_,ValBuilder<_,_,_,_>,ValSpine<K,V2,_,_>>(name, logic)
             .as_collection(|k,v| (k.clone(), v.clone()))
     }
 }
@@ -164,7 +164,7 @@ where
     T1: for<'a> TraceReader<Key<'a>=&'a K, KeyOwn = K, Val<'a>=&'a (), Diff=R1>+Clone+'static,
 {
     fn threshold_named<R2: Ord+Abelian+'static, F: FnMut(&K,&R1)->R2+'static>(&self, name: &str, mut thresh: F) -> Collection<G, K, R2> {
-        self.reduce_abelian::<_,K,(),KeyBuilder<K,G::Timestamp,R2>,KeySpine<K,G::Timestamp,R2>>(name, move |k,s,t| t.push(((), thresh(k, &s[0].1))))
+        self.reduce_abelian::<_,KeyBuilder<K,G::Timestamp,R2>,KeySpine<K,G::Timestamp,R2>>(name, move |k,s,t| t.push(((), thresh(k, &s[0].1))))
             .as_collection(|k,_| k.clone())
     }
 }
@@ -211,7 +211,7 @@ where
     T1: for<'a> TraceReader<Key<'a>=&'a K, KeyOwn = K, Val<'a>=&'a (), Diff=R>+Clone+'static,
 {
     fn count_core<R2: Ord + Abelian + From<i8> + 'static>(&self) -> Collection<G, (K, R), R2> {
-        self.reduce_abelian::<_,K,R,ValBuilder<K,R,G::Timestamp,R2>,ValSpine<K,R,G::Timestamp,R2>>("Count", |_k,s,t| t.push((s[0].1.clone(), R2::from(1i8))))
+        self.reduce_abelian::<_,ValBuilder<K,R,G::Timestamp,R2>,ValSpine<K,R,G::Timestamp,R2>>("Count", |_k,s,t| t.push((s[0].1.clone(), R2::from(1i8))))
             .as_collection(|k,c| (k.clone(), c.clone()))
     }
 }
@@ -303,22 +303,20 @@ where
             L: FnMut(&K, &[(&V, R)], &mut Vec<(V,T2::Diff)>, &mut Vec<(V, T2::Diff)>)+'static,
     {
         self.arrange_by_key_named(&format!("Arrange: {}", name))
-            .reduce_core::<_,_,_,Bu,_>(name, logic)
+            .reduce_core::<_,Bu,_>(name, logic)
     }
 }
 
 /// A key-wise reduction of values in an input trace.
 ///
 /// This method exists to provide reduce functionality without opinions about qualifying trace types.
-pub fn reduce_trace<G, T1, Bu, T2, K, V, L>(trace: &Arranged<G, T1>, name: &str, mut logic: L) -> Arranged<G, TraceAgent<T2>>
+pub fn reduce_trace<G, T1, Bu, T2, L>(trace: &Arranged<G, T1>, name: &str, mut logic: L) -> Arranged<G, TraceAgent<T2>>
 where
     G: Scope<Timestamp=T1::Time>,
-    T1: for<'a> TraceReader<KeyOwn = K> + Clone + 'static,
-    T2: for<'a> Trace<Key<'a>=T1::Key<'a>, ValOwn = V, Time=T1::Time> + 'static,
-    K: Ord + 'static,
-    V: Data,
-    Bu: Builder<Time=T2::Time, Output = T2::Batch, Input: Container + PushInto<((K, V), T2::Time, T2::Diff)>>,
-    L: FnMut(T1::Key<'_>, &[(T1::Val<'_>, T1::Diff)], &mut Vec<(V,T2::Diff)>, &mut Vec<(V, T2::Diff)>)+'static,
+    T1: TraceReader<KeyOwn: Ord> + Clone + 'static,
+    T2: for<'a> Trace<Key<'a>=T1::Key<'a>, KeyOwn=T1::KeyOwn, ValOwn: Data, Time=T1::Time> + 'static,
+    Bu: Builder<Time=T2::Time, Output = T2::Batch, Input: Container + PushInto<((T1::KeyOwn, T2::ValOwn), T2::Time, T2::Diff)>>,
+    L: FnMut(T1::Key<'_>, &[(T1::Val<'_>, T1::Diff)], &mut Vec<(T2::ValOwn,T2::Diff)>, &mut Vec<(T2::ValOwn, T2::Diff)>)+'static,
 {
     let mut result_trace = None;
 
@@ -352,7 +350,7 @@ where
 
             // Our implementation maintains a list of outstanding `(key, time)` synthetic interesting times,
             // as well as capabilities for these times (or their lower envelope, at least).
-            let mut interesting = Vec::<(K, G::Timestamp)>::new();
+            let mut interesting = Vec::<(T1::KeyOwn, G::Timestamp)>::new();
             let mut capabilities = Vec::<Capability<G::Timestamp>>::new();
 
             // buffers and logic for computing per-key interesting times "efficiently".
@@ -451,7 +449,7 @@ where
                         //
                         // TODO: It would be better if all updates went into one batch, but timely dataflow prevents
                         //       this as long as it requires that there is only one capability for each message.
-                        let mut buffers = Vec::<(G::Timestamp, Vec<(V, G::Timestamp, T2::Diff)>)>::new();
+                        let mut buffers = Vec::<(G::Timestamp, Vec<(T2::ValOwn, G::Timestamp, T2::Diff)>)>::new();
                         let mut builders = Vec::new();
                         for cap in capabilities.iter() {
                             buffers.push((cap.time().clone(), Vec::new()));
diff --git a/differential-dataflow/src/trace/implementations/ord_neu.rs b/differential-dataflow/src/trace/implementations/ord_neu.rs
index 0c01c3378..218fc612a 100644
--- a/differential-dataflow/src/trace/implementations/ord_neu.rs
+++ b/differential-dataflow/src/trace/implementations/ord_neu.rs
@@ -135,11 +135,11 @@ pub mod layers {
     #[derive(Debug, Serialize, Deserialize)]
     pub struct Upds<O, T, D> {
         /// Offsets used to provide indexes from values to updates.
-        offs: O,
+        pub offs: O,
         /// Concatenated ordered lists of update times, bracketed by offsets in `offs`.
-        times: T,
+        pub times: T,
         /// Concatenated ordered lists of update diffs, bracketed by offsets in `offs`.
-        diffs: D,
+        pub diffs: D,
     }
 
     impl<O: for<'a> BatchContainer<ReadItem<'a> = usize>, T: BatchContainer, D: BatchContainer> Default for Upds<O, T, D> {
diff --git a/interactive/src/plan/mod.rs b/interactive/src/plan/mod.rs
index faad5e562..a8cbadf2e 100644
--- a/interactive/src/plan/mod.rs
+++ b/interactive/src/plan/mod.rs
@@ -170,7 +170,7 @@ impl<V: ExchangeData+Hash+Datum> Render for Plan<V> {
                         input_arrangement
                     };
 
-                    let output = input.reduce_abelian::<_,_,_,KeyBuilder<_,_,_>,KeySpine<_,_,_>>("Distinct", move |_,_,t| t.push(((), 1)));
+                    let output = input.reduce_abelian::<_,KeyBuilder<_,_,_>,KeySpine<_,_,_>>("Distinct", move |_,_,t| t.push(((), 1)));
 
                     arrangements.set_unkeyed(&self, &output.trace);
                     output.as_collection(|k,&()| k.clone())