diff --git a/differential-dataflow/src/operators/join.rs b/differential-dataflow/src/operators/join.rs
index 477bc4b92..edf7120f6 100644
--- a/differential-dataflow/src/operators/join.rs
+++ b/differential-dataflow/src/operators/join.rs
@@ -400,19 +400,18 @@ where
     }
 }
 
-struct JoinThinker<'a, C1, C2>
-where
-    C1: Cursor,
-    C2: Cursor<Time = C1::Time>,
-{
-    pub history1: ValueHistory<'a, C1>,
-    pub history2: ValueHistory<'a, C2>,
+struct JoinThinker<V1, V2, T, D1, D2> {
+    pub history1: ValueHistory<V1, T, D1>,
+    pub history2: ValueHistory<V2, T, D2>,
 }
 
-impl<'a, C1, C2> JoinThinker<'a, C1, C2>
+impl<V1, V2, T, D1, D2> JoinThinker<V1, V2, T, D1, D2>
 where
-    C1: Cursor,
-    C2: Cursor<Time = C1::Time>,
+    V1: Copy + Ord,
+    V2: Copy + Ord,
+    T: Ord + Clone + Lattice,
+    D1: Clone + crate::difference::Semigroup,
+    D2: Clone + crate::difference::Semigroup,
 {
     fn new() -> Self {
         JoinThinker {
@@ -421,7 +420,7 @@ where
         }
     }
 
-    fn think<F: FnMut(C1::Val<'a>,C2::Val<'a>,C1::Time,&C1::Diff,&C2::Diff)>(&mut self, mut results: F) {
+    fn think<F: FnMut(V1, V2, T, &D1, &D2)>(&mut self, mut results: F) {
 
         // for reasonably sized edits, do the dead-simple thing.
         if self.history1.edits.len() < 10 || self.history2.edits.len() < 10 {
diff --git a/differential-dataflow/src/operators/mod.rs b/differential-dataflow/src/operators/mod.rs
index e511603e3..eb7880aab 100644
--- a/differential-dataflow/src/operators/mod.rs
+++ b/differential-dataflow/src/operators/mod.rs
@@ -19,12 +19,12 @@ use crate::lattice::Lattice;
 use crate::trace::Cursor;
 
 /// An accumulation of (value, time, diff) updates.
-struct EditList<'a, C: Cursor> {
-    values: Vec<(C::Val<'a>, usize)>,
-    edits: Vec<(C::Time, C::Diff)>,
+struct EditList<V, T, D> {
+    values: Vec<(V, usize)>,
+    edits: Vec<(T, D)>,
 }
 
-impl<'a, C: Cursor> EditList<'a, C> {
+impl<V: Copy, T: Ord, D: crate::difference::Semigroup> EditList<V, T, D> {
     /// Creates an empty list of edits.
     #[inline]
     fn new() -> Self {
@@ -34,9 +34,10 @@ impl<'a, C: Cursor> EditList<'a, C> {
         }
     }
     /// Loads the contents of a cursor.
-    fn load<L>(&mut self, cursor: &mut C, storage: &'a C::Storage, logic: L)
+    fn load<'a, C, L>(&mut self, cursor: &mut C, storage: &'a C::Storage, logic: L)
     where
-        L: Fn(C::TimeGat<'_>)->C::Time,
+        C: Cursor<Val<'a> = V, Time = T, Diff = D>,
+        L: Fn(C::TimeGat<'_>) -> T,
     {
         self.clear();
         while let Some(val) = cursor.get_val(storage) {
@@ -54,20 +55,20 @@ impl<'a, C: Cursor> EditList<'a, C> {
     fn len(&self) -> usize { self.edits.len() }
     /// Inserts a new edit for an as-yet undetermined value.
     #[inline]
-    fn push(&mut self, time: C::Time, diff: C::Diff) {
+    fn push(&mut self, time: T, diff: D) {
         // TODO: Could attempt "insertion-sort" like behavior here, where we collapse if possible.
         self.edits.push((time, diff));
     }
     /// Associates all edits pushed since the previous `seal_value` call with `value`.
     #[inline]
-    fn seal(&mut self, value: C::Val<'a>) {
+    fn seal(&mut self, value: V) {
         let prev = self.values.last().map(|x| x.1).unwrap_or(0);
         crate::consolidation::consolidate_from(&mut self.edits, prev);
         if self.edits.len() > prev {
             self.values.push((value, self.edits.len()));
         }
     }
-    fn map<F: FnMut(C::Val<'a>, &C::Time, &C::Diff)>(&self, mut logic: F) {
+    fn map<F: FnMut(V, &T, &D)>(&self, mut logic: F) {
         for index in 0 .. self.values.len() {
             let lower = if index == 0 { 0 } else { self.values[index-1].1 };
             let upper = self.values[index].1;
@@ -78,13 +79,13 @@ impl<'a, C: Cursor> EditList<'a, C> {
     }
 }
 
-struct ValueHistory<'storage, C: Cursor> {
-    edits: EditList<'storage, C>,
-    history: Vec<(C::Time, C::Time, usize, usize)>,     // (time, meet, value_index, edit_offset)
-    buffer: Vec<((C::Val<'storage>, C::Time), C::Diff)>,   // where we accumulate / collapse updates.
+struct ValueHistory<V, T, D> {
+    edits: EditList<V, T, D>,
+    history: Vec<(T, T, usize, usize)>,     // (time, meet, value_index, edit_offset)
+    buffer: Vec<((V, T), D)>,               // where we accumulate / collapse updates.
 }
 
-impl<'storage, C: Cursor> ValueHistory<'storage, C> {
+impl<V: Copy + Ord, T: Ord + Clone + Lattice, D: crate::difference::Semigroup> ValueHistory<V, T, D> {
     fn new() -> Self {
         ValueHistory {
             edits: EditList::new(),
@@ -97,9 +98,10 @@ impl<'storage, C: Cursor> ValueHistory<'storage, C> {
         self.history.clear();
         self.buffer.clear();
     }
-    fn load<L>(&mut self, cursor: &mut C, storage: &'storage C::Storage, logic: L)
+    fn load<'a, C, L>(&mut self, cursor: &mut C, storage: &'a C::Storage, logic: L)
     where
-        L: Fn(C::TimeGat<'_>)->C::Time,
+        C: Cursor<Val<'a> = V, Time = T, Diff = D>,
+        L: Fn(C::TimeGat<'_>) -> T,
     {
         self.edits.load(cursor, storage, logic);
     }
@@ -107,15 +109,16 @@ impl<'storage, C: Cursor> ValueHistory<'storage, C> {
     /// Loads and replays a specified key.
     ///
     /// If the key is absent, the replayed history will be empty.
-    fn replay_key<'history, L>(
+    fn replay_key<'a, 'history, C, L>(
         &'history mut self,
         cursor: &mut C,
-        storage: &'storage C::Storage,
-        key: C::Key<'storage>,
-        logic: L
-    ) -> HistoryReplay<'storage, 'history, C>
+        storage: &'a C::Storage,
+        key: C::Key<'a>,
+        logic: L,
+    ) -> HistoryReplay<'history, V, T, D>
     where
-        L: Fn(C::TimeGat<'_>)->C::Time,
+        C: Cursor<Val<'a> = V, Time = T, Diff = D>,
+        L: Fn(C::TimeGat<'_>) -> T,
     {
         self.clear();
         cursor.seek_key(storage, key);
@@ -126,7 +129,7 @@ impl<'storage, C: Cursor> ValueHistory<'storage, C> {
     }
 
     /// Organizes history based on current contents of edits.
-    fn replay<'history>(&'history mut self) -> HistoryReplay<'storage, 'history, C> {
+    fn replay<'history>(&'history mut self) -> HistoryReplay<'history, V, T, D> {
 
         self.buffer.clear();
         self.history.clear();
@@ -150,18 +153,18 @@ impl<'storage, C: Cursor> ValueHistory<'storage, C> {
     }
 }
 
-struct HistoryReplay<'storage, 'history, C: Cursor> {
-    replay: &'history mut ValueHistory<'storage, C>
+struct HistoryReplay<'history, V, T, D> {
+    replay: &'history mut ValueHistory<V, T, D>,
 }
 
-impl<'storage, 'history, C: Cursor> HistoryReplay<'storage, 'history, C> {
-    fn time(&self) -> Option<&C::Time> { self.replay.history.last().map(|x| &x.0) }
-    fn meet(&self) -> Option<&C::Time> { self.replay.history.last().map(|x| &x.1) }
-    fn edit(&self) -> Option<(C::Val<'storage>, &C::Time, &C::Diff)> {
+impl<'history, V: Copy + Ord, T: Ord + Clone + Lattice, D: Clone + crate::difference::Semigroup> HistoryReplay<'history, V, T, D> {
+    fn time(&self) -> Option<&T> { self.replay.history.last().map(|x| &x.0) }
+    fn meet(&self) -> Option<&T> { self.replay.history.last().map(|x| &x.1) }
+    fn edit(&self) -> Option<(V, &T, &D)> {
         self.replay.history.last().map(|&(ref t, _, v, e)| (self.replay.edits.values[v].0, t, &self.replay.edits.edits[e].1))
     }
 
-    fn buffer(&self) -> &[((C::Val<'storage>, C::Time), C::Diff)] {
+    fn buffer(&self) -> &[((V, T), D)] {
         &self.replay.buffer[..]
     }
 
@@ -169,7 +172,7 @@ impl<'storage, 'history, C: Cursor> HistoryReplay<'storage, 'history, C> {
         let (time, _, value_index, edit_offset) = self.replay.history.pop().unwrap();
         self.replay.buffer.push(((self.replay.edits.values[value_index].0, time), self.replay.edits.edits[edit_offset].1.clone()));
     }
-    fn step_while_time_is(&mut self, time: &C::Time) -> bool {
+    fn step_while_time_is(&mut self, time: &T) -> bool {
         let mut found = false;
         while self.time() == Some(time) {
             found = true;
@@ -177,7 +180,7 @@ impl<'storage, 'history, C: Cursor> HistoryReplay<'storage, 'history, C> {
         }
         found
     }
-    fn advance_buffer_by(&mut self, meet: &C::Time) {
+    fn advance_buffer_by(&mut self, meet: &T) {
         for element in self.replay.buffer.iter_mut() {
             (element.0).1 = (element.0).1.join(meet);
         }
diff --git a/differential-dataflow/src/operators/reduce.rs b/differential-dataflow/src/operators/reduce.rs
index 046a5e32c..b6ebb4bb2 100644
--- a/differential-dataflow/src/operators/reduce.rs
+++ b/differential-dataflow/src/operators/reduce.rs
@@ -312,7 +312,6 @@ fn sort_dedup<T: Ord>(list: &mut Vec<T>) {
 mod history_replay {
 
     use timely::progress::Antichain;
-    use timely::PartialOrder;
 
     use crate::lattice::Lattice;
     use crate::trace::Cursor;
@@ -322,32 +321,28 @@ mod history_replay {
 
     /// The `HistoryReplayer` is a compute strategy based on moving through existing inputs, interesting times, etc in
     /// time order, maintaining consolidated representations of updates with respect to future interesting times.
-    pub struct HistoryReplayer<'a, C1, C2, C3, V>
-    where
-        C1: Cursor,
-        C2: Cursor<Key<'a> = C1::Key<'a>, Time = C1::Time>,
-        C3: Cursor<Key<'a> = C1::Key<'a>, Val<'a> = C1::Val<'a>, Time = C1::Time, Diff = C1::Diff>,
-        V: Clone + Ord,
-    {
-        input_history: ValueHistory<'a, C1>,
-        output_history: ValueHistory<'a, C2>,
-        batch_history: ValueHistory<'a, C3>,
-        input_buffer: Vec<(C1::Val<'a>, C1::Diff)>,
-        output_buffer: Vec<(V, C2::Diff)>,
-        update_buffer: Vec<(V, C2::Diff)>,
-        output_produced: Vec<((V, C2::Time), C2::Diff)>,
-        synth_times: Vec<C1::Time>,
-        meets: Vec<C1::Time>,
-        times_current: Vec<C1::Time>,
-        temporary: Vec<C1::Time>,
+    pub struct HistoryReplayer<V1, V2, V, T, D1, D2> {
+        input_history: ValueHistory<V1, T, D1>,
+        output_history: ValueHistory<V2, T, D2>,
+        batch_history: ValueHistory<V1, T, D1>,
+        input_buffer: Vec<(V1, D1)>,
+        output_buffer: Vec<(V, D2)>,
+        update_buffer: Vec<(V, D2)>,
+        output_produced: Vec<((V, T), D2)>,
+        synth_times: Vec<T>,
+        meets: Vec<T>,
+        times_current: Vec<T>,
+        temporary: Vec<T>,
     }
 
-    impl<'a, C1, C2, C3, V> HistoryReplayer<'a, C1, C2, C3, V>
+    impl<V1, V2, V, T, D1, D2> HistoryReplayer<V1, V2, V, T, D1, D2>
     where
-        C1: Cursor,
-        C2: for<'b> Cursor<Key<'a> = C1::Key<'a>, ValOwn = V, Time = C1::Time>,
-        C3: Cursor<Key<'a> = C1::Key<'a>, Val<'a> = C1::Val<'a>, Time = C1::Time, Diff = C1::Diff>,
+        V1: Copy + Ord,
+        V2: Copy + Ord,
         V: Clone + Ord,
+        T: Ord + Clone + Lattice,
+        D1: Clone + crate::difference::Semigroup,
+        D2: Clone + crate::difference::Semigroup,
     {
         pub fn new() -> Self {
             HistoryReplayer {
@@ -365,24 +360,23 @@ mod history_replay {
             }
         }
         #[inline(never)]
-        pub fn compute<L>(
+        pub fn compute<'a, K, C1, C2, C3, L>(
             &mut self,
-            key: C1::Key<'a>,
+            key: K,
             (source_cursor, source_storage): (&mut C1, &'a C1::Storage),
             (output_cursor, output_storage): (&mut C2, &'a C2::Storage),
             (batch_cursor, batch_storage): (&mut C3, &'a C3::Storage),
-            times: &Vec<C1::Time>,
+            times: &Vec<T>,
             logic: &mut L,
-            upper_limit: &Antichain<C1::Time>,
-            outputs: &mut [(C2::Time, Vec<(V, C2::Time, C2::Diff)>)],
-            new_interesting: &mut Vec<C1::Time>)
+            upper_limit: &Antichain<T>,
+            outputs: &mut [(T, Vec<(V, T, D2)>)],
+            new_interesting: &mut Vec<T>)
         where
-            L: FnMut(
-                C1::Key<'a>,
-                &[(C1::Val<'a>, C1::Diff)],
-                &mut Vec<(V, C2::Diff)>,
-                &mut Vec<(V, C2::Diff)>,
-            )
+            C1: Cursor<Key<'a> = K, Val<'a> = V1, Time = T, Diff = D1>,
+            C2: Cursor<Key<'a> = K, Val<'a> = V2, ValOwn = V, Time = T, Diff = D2>,
+            C3: Cursor<Key<'a> = K, Val<'a> = V1, Time = T, Diff = D1>,
+            K: Copy + Ord,
+            L: FnMut(K, &[(V1, D1)], &mut Vec<(V, D2)>, &mut Vec<(V, D2)>),
         {
 
             // The work we need to perform is at times defined principally by the contents of `batch_cursor`