diff --git a/lean/Spa/Language/Traces.lean b/lean/Spa/Language/Traces.lean
index 550c6e0..a5afb6b 100644
--- a/lean/Spa/Language/Traces.lean
+++ b/lean/Spa/Language/Traces.lean
@@ -37,20 +37,102 @@ inductive Trace (g : Graph) : g.Index → g.Index → Env → Env → Type
       EvalBasicStmtOpt ρ₁ (g.nodes idx₁) ρ₂ → (idx₁, idx₂) ∈ g.edges →
       Trace g idx₂ idx₃ ρ₂ ρ₃ → Trace g idx₁ idx₃ ρ₁ ρ₃
 
+/-!
+
+## Open Traces
+
+A normal `Trace` starts right before one state, and ends right after another.
+This is convenient for inductively proving correctness / sufficience, but
+awkward because 1) no empty traces exist and 2) concatenation requires an extra
+edge.
+
+However, when attempting an "empty" trace, two types are equally possible:
+traces that end _right before_ executing a state (`Traceₗ`) and
+traces that begin _right after_ executing a state (`Traceᵣ`). They
+are symmetric and can be concatenated with full traces on the left
+and right, respectively. -/
+
+/-- Left-open trace, representing execution that ends right before `idx₂`. -/
+inductive Traceₗ (g : Graph) : g.Index → g.Index → Env → Env → Type where
+  | nil {idx : g.Index} {ρ : Env} : Traceₗ g idx idx ρ ρ
+  | cons {idx₁ idx₂ idx₃ : g.Index} {ρ₁ ρ₂ ρ₃ : Env} :
+      EvalBasicStmtOpt ρ₁ (g.nodes idx₁) ρ₂ →
+      (idx₁, idx₂) ∈ g.edges →
+      Traceₗ g idx₂ idx₃ ρ₂ ρ₃ → Traceₗ g idx₁ idx₃ ρ₁ ρ₃
+
+def Traceₗ.single (g : Graph) (idx : g.Index) (ρ : Env) : Traceₗ g idx idx ρ ρ := .nil
+
+/-- Right-open trace, representing execution that starts right after `idx₁`. -/
+inductive Traceᵣ (g : Graph) : g.Index → g.Index → Env → Env → Type where
+  | nil {idx : g.Index} {ρ : Env} : Traceᵣ g idx idx ρ ρ
+  | cons {idx₁ idx₂ idx₃ : g.Index} {ρ₁ ρ₂ ρ₃ : Env} :
+      Traceᵣ g idx₁ idx₂ ρ₁ ρ₂ →
+      (idx₂, idx₃) ∈ g.edges →
+      EvalBasicStmtOpt ρ₂ (g.nodes idx₃) ρ₃ → Traceᵣ g idx₁ idx₃ ρ₁ ρ₃
+
+def Traceᵣ.single (g : Graph) (idx : g.Index) (ρ : Env) : Traceᵣ g idx idx ρ ρ := .nil
+
 /-- Sequence two traces together. Since the endpoint of the first trace
     is _after_ its last basic block's execution, and the beginning of
     the next trace is _before_ its first basic block's execution,
     there must be an edge to connect the two. -/
-noncomputable def Trace.concat {g : Graph} {idx₁ idx₂ idx₃ idx₄ : g.Index}
+def Trace.concat {g : Graph} {idx₁ idx₂ idx₃ idx₄ : g.Index}
     {ρ₁ ρ₂ ρ₃ : Env} (tr₁ : Trace g idx₁ idx₂ ρ₁ ρ₂)
     (he : (idx₂, idx₃) ∈ g.edges) (tr₂ : Trace g idx₃ idx₄ ρ₂ ρ₃) :
-    Trace g idx₁ idx₄ ρ₁ ρ₃ := by
-  induction tr₁ with
-  | single hbs => exact Trace.edge hbs he tr₂
-  | edge hbs he' _ ih => exact Trace.edge hbs he' (ih he tr₂)
+    Trace g idx₁ idx₄ ρ₁ ρ₃ :=
+  match tr₁ with
+  | single hbs => edge hbs he tr₂
+  | edge hbs he' tr₁' => edge hbs he' (tr₁'.concat he tr₂)
 
 scoped notation:65 tr₁:66 " ++< " he " >++ " tr₂:65 => Trace.concat tr₁ he tr₂
 
+def Trace.addEdge {g : Graph} {idx₁ idx₂ idx₃ : g.Index} {ρ₁ ρ₂ : Env} :
+    Trace g idx₁ idx₂ ρ₁ ρ₂ →
+    (idx₂, idx₃) ∈ g.edges →
+    Traceₗ g idx₁ idx₃ ρ₁ ρ₂
+  | .single hnode, hedge => .cons hnode hedge .nil
+  | .edge hnode hedge' rest, hedge => .cons hnode hedge' (rest.addEdge hedge)
+
+def Traceₗ.append {g : Graph} {idx₁ idx₂ idx₃ : g.Index} {ρ₁ ρ₂ ρ₃ : Env} :
+    Traceₗ g idx₁ idx₂ ρ₁ ρ₂ → Traceₗ g idx₂ idx₃ ρ₂ ρ₃ →
+    Traceₗ g idx₁ idx₃ ρ₁ ρ₃
+  | .nil, rhs => rhs
+  | .cons hnode hedge rest, rhs => .cons hnode hedge (rest.append rhs)
+
+def Traceₗ.appendTrace {g : Graph} {idx₁ idx₂ idx₃ : g.Index} {ρ₁ ρ₂ ρ₃ : Env} :
+    Traceₗ g idx₁ idx₂ ρ₁ ρ₂ → Trace g idx₂ idx₃ ρ₂ ρ₃ →
+    Trace g idx₁ idx₃ ρ₁ ρ₃
+  | .nil, rhs => rhs
+  | .cons hnode hedge rest, rhs => .edge hnode hedge (rest.appendTrace rhs)
+
+def Traceₗ.appendStep {g : Graph} {idx₁ idx₂ : g.Index} {ρ₁ ρ₂ ρ₃ : Env} :
+    Traceₗ g idx₁ idx₂ ρ₁ ρ₂ → EvalBasicStmtOpt ρ₂ (g.nodes idx₂) ρ₃ →
+    Trace g idx₁ idx₂ ρ₁ ρ₃
+  | .nil, rhs => .single rhs
+  | .cons hnode hedge rest, rhs => .edge hnode hedge (rest.appendStep rhs)
+
+def Trace.appendRight {g : Graph} {idx₁ idx₂ idx₃ : g.Index} {ρ₁ ρ₂ ρ₃ : Env} :
+    Trace g idx₁ idx₂ ρ₁ ρ₂ → Traceᵣ g idx₂ idx₃ ρ₂ ρ₃ →
+    Trace g idx₁ idx₃ ρ₁ ρ₃
+  | lhs, .nil => lhs
+  | lhs, .cons rest hedge hnode => Trace.concat (lhs.appendRight rest) hedge (.single hnode)
+
+instance instHAppendTraceLTraceL {g : Graph} {idx₁ idx₂ idx₃ : g.Index} {ρ₁ ρ₂ ρ₃ : Env} :
+    HAppend (Traceₗ g idx₁ idx₂ ρ₁ ρ₂) (Traceₗ g idx₂ idx₃ ρ₂ ρ₃) (Traceₗ g idx₁ idx₃ ρ₁ ρ₃) where
+  hAppend := Traceₗ.append
+
+instance instHAppendTraceLTrace {g : Graph} {idx₁ idx₂ idx₃ : g.Index} {ρ₁ ρ₂ ρ₃ : Env} :
+    HAppend (Traceₗ g idx₁ idx₂ ρ₁ ρ₂) (Trace g idx₂ idx₃ ρ₂ ρ₃) (Trace g idx₁ idx₃ ρ₁ ρ₃) where
+  hAppend := Traceₗ.appendTrace
+
+instance instHAppendTraceLStep {g : Graph} {idx₁ idx₂ : g.Index} {ρ₁ ρ₂ ρ₃ : Env} :
+    HAppend (Traceₗ g idx₁ idx₂ ρ₁ ρ₂) (EvalBasicStmtOpt ρ₂ (g.nodes idx₂) ρ₃) (Trace g idx₁ idx₂ ρ₁ ρ₃) where
+  hAppend := Traceₗ.appendStep
+
+instance instHAppendTraceTraceR {g : Graph} {idx₁ idx₂ idx₃ : g.Index} {ρ₁ ρ₂ ρ₃ : Env} :
+    HAppend (Trace g idx₁ idx₂ ρ₁ ρ₂) (Traceᵣ g idx₂ idx₃ ρ₂ ρ₃) (Trace g idx₁ idx₃ ρ₁ ρ₃) where
+  hAppend := Trace.appendRight
+
 /-- A beginning-to-end trace corresponding to the CFG `g`. -/
 inductive EndToEndTrace (g : Graph) (ρ₁ ρ₂ : Env) : Type
   | intro (idx₁ : g.Index) (idx₁_mem : idx₁ ∈ g.inputs)
@@ -74,21 +156,28 @@ inductive MyReaches {prog : Program} : {s₁ s₂ : prog.State} → {ρ₁ ρ₂
       MyReaches rest s ρin ρout →
       MyReaches (.edge hnode hedge rest) s ρin ρout
 
-noncomputable def MyReaches.prefix {prog : Program} {s₁ s₂ s: prog.State}
+def MyReaches.pre {prog : Program} {s₁ s₂ s: prog.State}
+    {ρ₁ ρ₂ ρin ρout : Env} {tr : Trace prog.cfg s₁ s₂ ρ₁ ρ₂} :
+    (r : MyReaches tr s ρin ρout) → Traceₗ prog.cfg s₁ s ρ₁ ρin
+  | .single_here _ => .nil
+  | .edge_here _ _ _ => .nil
+  | .edge_there hnode hedge _ r => .cons hnode hedge r.pre
+
+def MyReaches.post {prog : Program} {s₁ s₂ s: prog.State}
     {ρ₁ ρ₂ ρin ρout : Env} {tr : Trace prog.cfg s₁ s₂ ρ₁ ρ₂} :
     (r : MyReaches tr s ρin ρout) → Trace prog.cfg s₁ s ρ₁ ρout
-  | .single_here hnode => Spa.Trace.single hnode
-  | .edge_here hnode hedge hrest => Spa.Trace.single hnode
-  | .edge_there hnode hedge hrest tmp' => Spa.Trace.edge hnode hedge tmp'.prefix
+  | .single_here hnode => .single hnode
+  | .edge_here hnode _ _ => .single hnode
+  | .edge_there hnode hedge _ r => .edge hnode hedge r.post
 
-noncomputable def MyReaches.first {prog : Program} {s₁ s₂ s: prog.State}
+def MyReaches.first {prog : Program} {s₁ s₂ s: prog.State}
     {ρ₁ ρ₂ ρin ρout : Env} {tr : Trace prog.cfg s₁ s₂ ρ₁ ρ₂} :
     (r : MyReaches tr s ρin ρout) → Σ ρ₁', MyReaches tr s₁ ρ₁ ρ₁'
   | .single_here hnode => ⟨_, .single_here hnode⟩
   | .edge_here hnode hedge hrest => ⟨_, .edge_here hnode hedge hrest⟩
   | .edge_there hnode hedge hrest tmp' => ⟨_, .edge_here hnode hedge hrest⟩
 
-noncomputable def MyReaches.step {prog : Program} {s₁ s₂ s: prog.State}
+def MyReaches.step {prog : Program} {s₁ s₂ s: prog.State}
     {ρ₁ ρ₂ ρin ρout : Env} {tr : Trace prog.cfg s₁ s₂ ρ₁ ρ₂} :
     (r : MyReaches tr s ρin ρout) → EvalBasicStmtOpt ρin (prog.code s) ρout
   | .single_here hnode => hnode