diff --git a/DatapathVerification/BitHeap/BVComb.lean b/DatapathVerification/BitHeap/BVComb.lean
new file mode 100644
index 0000000..2f133ac
--- /dev/null
+++ b/DatapathVerification/BitHeap/BVComb.lean
@@ -0,0 +1,75 @@
+import DatapathVerification.BitHeap.BitHeap
+import DatapathVerification.BitHeap.Circuit
+import DatapathVerification.BitHeap.Compressors.DaddaTree
+import DatapathVerification.BitHeap.Compressors.NaiveCompression
+
+
+open BitHeap
+namespace Comb
+
+inductive ArithBinopKind
+| add
+| mul
+
+-- inductive ArithUnopKind
+-- | neg
+
+-- inductive BooleanBinopKind
+-- | and | or | xor
+
+inductive ArithCircuit : Nat → Type
+  | var (varIndex : Nat) : ArithCircuit w
+  | add (args : List (ArithCircuit w)) : ArithCircuit w
+  | mul (l r : ArithCircuit w) : ArithCircuit w
+  -- | arithunop (kind : ArithUnopKind) (width : Nat) (arg : ArithCircuit)
+  -- | bvbinop (kind : BooleanBinopKind) (width : Nat) (l r : ArithCircuit)
+
+/--
+Convert a bitheap into a new bitheap that has a single row,
+by using the naive compression algorithm.
+-/
+def BitHeap.toSingleRow (bh : BitHeap w) : CircuitVector :=
+    let (pp1, pp2) := NaiveCompression.naiveCompression bh
+    sorry
+
+namespace ArithCircuit
+/--
+Given a bitvector (x : BV 3), but a bitheap
+```
+*   *  *
+x2 x1 x0
+```
+-/
+def bitheapOfVar (varIndex : Nat) : BitHeap w :=
+  --  I want to create a bitheap that has one bit-variable per bit in the bitvector variable.
+  -- | We need to know that this index is unique which is a gigantic pain.
+  List.range w |>.foldl (fun bh i => bh.addBit i (BitHeap.Circuit.bit (varIndex * w + i))) (BitHeap.empty w)
+
+def toBitHeap : ArithCircuit w → BitHeap w
+  | .var varIndex => bitheapOfVar varIndex
+  | .add args => BitHeap.addBitHeap (args.map toBitHeap)
+  | .mul l r => BitHeap.truncate ((toBitHeap l).mulBitHeap (toBitHeap r)) w (by omega)
+
+-- def toBitHeap' (c : ArithCircuit) : BitHeap w :=
+--   match c with
+--   | .var width varIndex => bitheapOfVar width varIndex
+--   | .add width args => BitHeap.addBitHeap (args.map toBitHeap)
+--   | .mul width l r => BitHeap.truncate ((toBitHeap l).mulBitHeap (toBitHeap r)) width (by omega)
+  -- | .arithunop kind width arg =>`
+  --   match kind with
+  --   | .neg => (toBitHeap arg).negBitHeap
+  -- | .bvbinop kind width l r =>
+  --   match kind with
+  --   | .and =>
+  --     let lRow := (l.toBitHeap).toSingleRow
+  --     let rRow := (r.toBitHeap).toSingleRow
+  --     let newRow := Array.zipWith (fun lBit rBit => Circuit.and lBit rBit) lRow rRow
+  --     BitHeap.fromRow newRow
+
+def toCircuitVector (c : ArithCircuit w) : CircuitVector :=
+  let bh := c.toBitHeap
+  bh.toSingleRow
+
+end ArithCircuit
+
+end Comb
diff --git a/DatapathVerification/BitHeap/BitHeap.lean b/DatapathVerification/BitHeap/BitHeap.lean
index 251ba5f..c31dc91 100644
--- a/DatapathVerification/BitHeap/BitHeap.lean
+++ b/DatapathVerification/BitHeap/BitHeap.lean
@@ -29,6 +29,23 @@ def HornersMethod (env : BitEnv) : List Column → Nat
   | [] => 0
   | c :: rest => (c.eval env) + 2 * HornersMethod env rest
 
+-- Basically eval_insertColumn_eq_eval_add, but for HornersMethod. Adding a new column to a list is equivalent
+-- to adding the new column's evaluation to the old evaluation, and subtracting the old column's evaluation.
+theorem hornersMethod_set (env : BitEnv) (l : List Column) (k : Nat) (v : Column) (hk : k < l.length) :
+    (HornersMethod env (l.set k v) : Int)
+      = HornersMethod env l + 2^k * (v.eval env : Int) - 2^k * ((l[k]'hk).eval env : Int) := by
+  induction l generalizing k with
+  | nil =>
+    simp at hk
+  | cons c cs ih =>
+    cases k with
+    | zero =>
+      simp [HornersMethod, List.set]
+      grind
+    | succ j =>
+      simp [HornersMethod]
+      grind
+
 /--
 Evaluate a bit-heap, to compute the final sum of all the bits in the heap.
 -/
@@ -91,6 +108,29 @@ def addBit (column : Nat) (c : Circuit) (h : BitHeap w) : BitHeap w :=
       h.setColumn column (col.insert c) h1
       else addBit (column + 1) c (h.removeBit column c)
 
+def truncate (h : BitHeap w) (n : Nat) (hn : n ≤ w) : BitHeap n :=
+  ⟨Vector.ofFn (fun i => h.columns[i]'(by omega))⟩
+
+theorem evalMod_truncate (h : BitHeap w) (n : Nat) (hn : n ≤ w) (env : BitEnv) :
+    (h.truncate n hn).evalMod env = (h.eval env) % 2^n := by
+  simp [evalMod, eval, truncate]
+  sorry
+
+def addBitHeap (bhs : List (BitHeap w)) : BitHeap w:=
+  let h := BitHeap.empty w
+  let h := bhs.foldl (fun acc heap => heap.columns.zipIdx.foldl (fun acc' (column, index) =>
+             column.elems.toList.foldl (fun acc' c => acc'.addBit index c) acc') acc) h
+  h
+
+def mulBitHeap (h0 h1 : BitHeap w) : BitHeap (2 * w - 1) :=
+  let h := BitHeap.empty (2 * w - 1)
+  let h := h0.columns.zipIdx.foldl (fun acc (column0, i0) =>
+             h1.columns.zipIdx.foldl (fun acc' (column1, i2) =>
+               column0.elems.toList.foldl (fun acc'' c1 =>
+                 column1.elems.toList.foldl (fun acc''' c2 =>
+                   acc'''.addBit (i0 + i2) (Circuit.binop .and c1 c2)) acc'') acc') acc) h
+  h
+
 structure AdderResult (w : Nat) where
   heap : BitHeap w
   sum : Circuit
@@ -115,23 +155,6 @@ def fullAdder (column : Nat) (i j k : Circuit) (h : BitHeap w) : AdderResult w :
   let h := h.addBit (column + 1) carry
   ⟨h, sum, carry⟩
 
--- Basically eval_insertColumn_eq_eval_add, but for HornersMethod. Adding a new column to a list is equivalent
--- to adding the new column's evaluation to the old evaluation, and subtracting the old column's evaluation.
-theorem hornersMethod_set (env : BitEnv) (l : List Column) (k : Nat) (v : Column) (hk : k < l.length) :
-    (HornersMethod env (l.set k v) : Int)
-      = HornersMethod env l + 2^k * (v.eval env : Int) - 2^k * ((l[k]'hk).eval env : Int) := by
-  induction l generalizing k with
-  | nil =>
-    simp at hk
-  | cons c cs ih =>
-    cases k with
-    | zero =>
-      simp [HornersMethod, List.set]
-      grind
-    | succ j =>
-      simp [HornersMethod]
-      grind
-
 @[grind => ]
 theorem eval_insertColumn_eq_eval_add (h : BitHeap w) (k : Nat) (v : Column) (env : BitEnv) (h1 : k < w) :
     (h.setColumn k v h1).eval env
diff --git a/DatapathVerification/BitHeap/Circuit.lean b/DatapathVerification/BitHeap/Circuit.lean
index c34c06d..62cee0c 100644
--- a/DatapathVerification/BitHeap/Circuit.lean
+++ b/DatapathVerification/BitHeap/Circuit.lean
@@ -76,4 +76,16 @@ theorem eval_atLeastTwo (a b c : Circuit) (env : BitEnv) :
 
 end Circuit
 
+
+/-- A vector of circuits, used to represent symbolic BitVectors. -/
+def CircuitVector := Array Circuit
+
+namespace CircuitVector
+
+def eval (vec : CircuitVector) (env : Circuit.BitEnv) : Int :=
+  (vec.mapIdx (fun i c => 2^i * (if c.eval env then 1 else 0))).sum
+
+end CircuitVector
+
+
 end BitHeap
diff --git a/DatapathVerification/BitHeap/Examples/Examples.lean b/DatapathVerification/BitHeap/Examples/Examples.lean
index 4702bae..f5146e6 100644
--- a/DatapathVerification/BitHeap/Examples/Examples.lean
+++ b/DatapathVerification/BitHeap/Examples/Examples.lean
@@ -99,6 +99,49 @@ info: 6
 #eval (applyChain compressionChain fourBitsInCol1).eval
         (show BitEnv from fun n => n = 1 || n = 2 || n = 3)
 
+----------------------------
+
+def exampleHeap1 : BitHeap 4 :=
+  let h := BitHeap.empty 4
+  let h := h.addBit 0 (Circuit.bit 0)
+  let h := h.addBit 1 (Circuit.bit 1)
+  let h := h.addBit 2 (Circuit.bit 2)
+  let h := h.addBit 3 (Circuit.bit 3)
+  h
+
+def exampleHeap2 : BitHeap 4 :=
+  let h := BitHeap.empty 4
+  let h := h.addBit 0 (Circuit.bit 4)
+  let h := h.addBit 1 (Circuit.bit 5)
+  let h := h.addBit 2 (Circuit.bit 6)
+  let h := h.addBit 3 (Circuit.bit 7)
+  h
+
+def exampleHeap3 : BitHeap 4 :=
+  let h := BitHeap.empty 4
+  let h := h.addBit 0 (Circuit.bit 8)
+  let h := h.addBit 1 (Circuit.bit 9)
+  let h := h.addBit 2 (Circuit.bit 10)
+  let h := h.addBit 3 (Circuit.bit 11)
+  h
+/--
+info: {0 ↦ [b4, b8, b0], 1 ↦ [b1, b5, b9], 2 ↦ [b2, b10, b6], 3 ↦ [b3, b11, b7]}
+-/
+#guard_msgs in
+#eval addBitHeap [exampleHeap1, exampleHeap2, exampleHeap3]
+
+/--
+info: {0 ↦ [(b0 ∧ b4)], 1 ↦ [(b0 ∧ b5), (b1 ∧ b4)], 2 ↦ [(b1 ∧ b5), (b2 ∧ b4), (b0 ∧ b6)], 3 ↦ [(b3 ∧ b4), (b0 ∧ b7), (b2 ∧ b5), (b1 ∧ b6)], 4 ↦ [(b2 ∧ b6), (b3 ∧ b5), (b1 ∧ b7)], 5 ↦ [(b3 ∧ b6), (b2 ∧ b7)], 6 ↦ [(b3 ∧ b7)]}
+-/
+#guard_msgs in
+#eval mulBitHeap exampleHeap1 exampleHeap2
+
+/--
+info: {0 ↦ [b8], 1 ↦ [b9]}
+-/
+#guard_msgs in
+#eval truncate exampleHeap3 2 (by omega)
+
 end Examples
 
 end BitHeap