diff --git a/src/hst/interleave.cc b/src/hst/interleave.cc
index 533238b..2f31fd3 100644
--- a/src/hst/interleave.cc
+++ b/src/hst/interleave.cc
@@ -89,7 +89,8 @@ Interleave::initials(std::function<void(Event)> op) const
     //                ∪ (Ps = {STOP}? {✔}: {})                          [rule 4]
 
     bool any_events = false;
-    for (const Process* p : ps_) {
+    for (const auto& process_and_count : ps_) {
+        const Process* p = process_and_count.first;
         p->initials([&any_events, &op](Event initial) {
             any_events = true;
             if (initial == Event::tick()) {
@@ -119,9 +120,10 @@ Interleave::normal_afters(Event initial,
     // basic structure: Ps' = Ps ∖ {P} ∪ {P'}.  Each Ps' starts with Ps, so go
     // ahead and add that to our Ps' set once.
     Process::Bag ps_prime(ps_);
-    for (const Process* p : ps_) {
+    for (const auto& process_and_count : ps_) {
+        const Process* p = process_and_count.first;
         // Set Ps' to Ps ∖ {P}
-        ps_prime.erase(ps_prime.find(p));
+        ps_prime.erase(p);
         // Grab afters(P, a)
         p->afters(initial, [this, &op, &ps_prime](const Process& p_prime) {
             // ps_prime currently contains Ps.  Add P' and remove P to produce
@@ -130,7 +132,7 @@ Interleave::normal_afters(Event initial,
             // Create ⫴ (Ps ∖ {P} ∪ {P'}) as a result.
             op(*env_->interleave(ps_prime));
             // Reset Ps' back to Ps ∖ {P}.
-            ps_prime.erase(ps_prime.find(&p_prime));
+            ps_prime.erase(&p_prime);
         });
         // Reset Ps' back to Ps.
         ps_prime.insert(p);
@@ -153,7 +155,8 @@ Interleave::tau_afters(Event initial,
     // ahead and add that to our Ps' set once.
     Process::Bag ps_prime(ps_);
     // Find each P ∈ Ps where ✔ ∈ initials(P).
-    for (const Process* p : ps_) {
+    for (const auto& process_and_count : ps_) {
+        const Process* p = process_and_count.first;
         bool any_tick = false;
         p->initials([&any_tick](Event initial) {
             if (initial == Event::tick()) {
@@ -162,11 +165,11 @@ Interleave::tau_afters(Event initial,
         });
         if (any_tick) {
             // Create Ps ∖ {P} ∪ {STOP}) as a result.
-            ps_prime.erase(ps_prime.find(p));
+            ps_prime.erase(p);
             ps_prime.insert(env_->stop());
             op(*env_->interleave(ps_prime));
             // Reset Ps' back to Ps.
-            ps_prime.erase(ps_prime.find(env_->stop()));
+            ps_prime.erase(env_->stop());
             ps_prime.insert(p);
         }
     }
@@ -177,7 +180,8 @@ Interleave::tick_afters(Event initial,
                         std::function<void(const Process&)> op) const
 {
     // afters(⫴ {STOP}, ✔) = {STOP}                                     [rule 4]
-    for (const Process* p : ps_) {
+    for (const auto& process_and_count : ps_) {
+        const Process* p = process_and_count.first;
         bool any_events = false;
         p->initials([&any_events](Event _) { any_events = true; });
         if (any_events) {
@@ -204,8 +208,9 @@ Interleave::afters(Event initial, std::function<void(const Process&)> op) const
 void
 Interleave::subprocesses(std::function<void(const Process&)> op) const
 {
-    for (const Process* process : ps_) {
-        op(*process);
+    for (const auto& process_and_count : ps_) {
+        const Process* p = process_and_count.first;
+        op(*p);
     }
 }
 
diff --git a/src/hst/process.cc b/src/hst/process.cc
index 3a2fb65..f6ea406 100644
--- a/src/hst/process.cc
+++ b/src/hst/process.cc
@@ -8,6 +8,7 @@
 #include "hst/process.h"
 
 #include <algorithm>
+#include <assert.h>
 #include <string>
 #include <utility>
 #include <vector>
@@ -44,34 +45,57 @@ NormalizedProcess::afters(Event initial,
     }
 }
 
+void
+Process::Bag::insert(const Process* process)
+{
+    ++counts_[process];
+}
+
+void
+Process::Bag::erase(const Process* process)
+{
+    auto it = counts_.find(process);
+    assert(it != counts_.end() && it->second != 0);
+    if (--it->second == 0) {
+        counts_.erase(it);
+    }
+}
+
 std::size_t
 Process::Bag::hash() const
 {
     static hash_scope scope;
     hst::hasher hash(scope);
-    std::vector<const Process*> sorted(begin(), end());
-    std::sort(sorted.begin(), sorted.end());
-    for (const Process* process : sorted) {
+    for (const Process* process : sorted()) {
         hash.add(*process);
     }
     return hash.value();
 }
 
-std::ostream& operator<<(std::ostream& out, const Process::Bag& processes)
+std::vector<const Process*>
+Process::Bag::sorted() const
 {
-    // We want reproducible output, so we sort the processes in the set before
-    // rendering them into the stream.  We the process's index to print out the
-    // processes in the order that they were defined.
-    std::vector<const Process*> sorted_processes(processes.begin(),
-                                                 processes.end());
-    std::sort(sorted_processes.begin(), sorted_processes.end(),
+    std::vector<const Process*> sorted;
+    for (const auto& process_and_count : *this) {
+        const Process* process = process_and_count.first;
+        size_type count = process_and_count.second;
+        sorted.insert(sorted.end(), count, process);
+    }
+    std::sort(sorted.begin(), sorted.end(),
               [](const Process* p1, const Process* p2) {
                   return p1->index() < p2->index();
               });
+    return sorted;
+}
 
+std::ostream& operator<<(std::ostream& out, const Process::Bag& processes)
+{
+    // We want reproducible output, so we sort the processes in the set before
+    // rendering them into the stream.  We use the process's index as the sort
+    // key to print out the processes in the order that they were defined.
     bool first = true;
     out << "{";
-    for (const Process* process : sorted_processes) {
+    for (const Process* process : processes.sorted()) {
         if (first) {
             first = false;
         } else {
@@ -87,14 +111,23 @@ Process::Set::hash() const
 {
     static hash_scope scope;
     hst::hasher hash(scope);
-    std::vector<const Process*> sorted(begin(), end());
-    std::sort(sorted.begin(), sorted.end());
-    for (const Process* process : sorted) {
+    for (const Process* process : sorted()) {
         hash.add(*process);
     }
     return hash.value();
 }
 
+std::vector<const Process*>
+Process::Set::sorted() const
+{
+    std::vector<const Process*> sorted(begin(), end());
+    std::sort(sorted.begin(), sorted.end(),
+              [](const Process* p1, const Process* p2) {
+                  return p1->index() < p2->index();
+              });
+    return sorted;
+}
+
 void
 Process::Set::tau_close()
 {
@@ -118,16 +151,9 @@ std::ostream& operator<<(std::ostream& out, const Process::Set& processes)
     // We want reproducible output, so we sort the processes in the set before
     // rendering them into the stream.  We the process's index to print out the
     // processes in the order that they were defined.
-    std::vector<const Process*> sorted_processes(processes.begin(),
-                                                 processes.end());
-    std::sort(sorted_processes.begin(), sorted_processes.end(),
-              [](const Process* p1, const Process* p2) {
-                  return p1->index() < p2->index();
-              });
-
     bool first = true;
     out << "{";
-    for (const Process* process : sorted_processes) {
+    for (const Process* process : processes.sorted()) {
         if (first) {
             first = false;
         } else {
diff --git a/src/hst/process.h b/src/hst/process.h
index 3ebd6c5..65a34c5 100644
--- a/src/hst/process.h
+++ b/src/hst/process.h
@@ -11,9 +11,11 @@
 #include <algorithm>
 #include <assert.h>
 #include <functional>
+#include <initializer_list>
 #include <memory>
 #include <ostream>
 #include <string>
+#include <unordered_map>
 #include <unordered_set>
 #include <vector>
 
@@ -123,14 +125,48 @@ class NormalizedProcess : public Process {
     void bfs(std::function<void(const NormalizedProcess&)> op) const;
 };
 
-class Process::Bag : public std::unordered_multiset<const Process*> {
+class Process::Bag {
   private:
-    using Parent = std::unordered_multiset<const Process*>;
+    using size_type = unsigned int;
+    using CountMap = std::unordered_map<const Process*, size_type>;
 
   public:
-    using Parent::unordered_multiset;
+    using const_iterator = CountMap::const_iterator;
+    using iterator = CountMap::iterator;
+
+    Bag() = default;
+    Bag(const Bag& other) = default;
+    explicit Bag(std::initializer_list<const Process*> elements)
+    {
+        for (const Process* process : elements) {
+            insert(process);
+        }
+    }
+
+    void insert(const Process* process);
+
+    // Unlike the STL multimap classes, this only removes ONE copy of process.
+    void erase(const Process* process);
 
     std::size_t hash() const;
+
+    // Returns a vector containing a sorted copy of the bag.  Duplicate elements
+    // will be added to the vector multiple times.
+    std::vector<const Process*> sorted() const;
+
+    bool operator==(const Bag& other) const { return counts_ == other.counts_; }
+    bool operator!=(const Bag& other) const { return counts_ != other.counts_; }
+
+    // Returns an iterator of (Process, count) pairs
+    iterator begin() { return counts_.begin(); }
+    iterator end() { return counts_.end(); }
+    const_iterator begin() const { return counts_.begin(); }
+    const_iterator end() const { return counts_.end(); }
+    const_iterator cbegin() const { return counts_.begin(); }
+    const_iterator cend() const { return counts_.end(); }
+
+  private:
+    CountMap counts_;
 };
 
 std::ostream& operator<<(std::ostream& out, const Process::Bag& processes);
@@ -144,6 +180,9 @@ class Process::Set : public std::unordered_set<const Process*> {
 
     std::size_t hash() const;
 
+    // Returns a vector containing a sorted copy of the set.
+    std::vector<const Process*> sorted() const;
+
     // Updates this set of processes to be τ-closed.  (That is, we add any
     // additional processes you can reach by following τ one or more times.)
     void tau_close();
@@ -271,12 +310,7 @@ Process::print_subprocesses(std::ostream& out, const T& processes,
     // We want reproducible output, so we sort the processes in the set before
     // rendering them into the stream.  We the process's index to print out the
     // processes in the order that they were defined.
-    std::vector<const Process*> sorted_processes(processes.begin(),
-                                                 processes.end());
-    std::sort(sorted_processes.begin(), sorted_processes.end(),
-              [](const Process* p1, const Process* p2) {
-                  return p1->index() < p2->index();
-              });
+    std::vector<const Process*> sorted_processes = processes.sorted();
 
     if (sorted_processes.size() == 2) {
         const Process* lhs = sorted_processes[0];