diff --git a/lean/Spa/Language/Traces.lean b/lean/Spa/Language/Traces.lean
index 55478fb..550c6e0 100644
--- a/lean/Spa/Language/Traces.lean
+++ b/lean/Spa/Language/Traces.lean
@@ -57,4 +57,43 @@ inductive EndToEndTrace (g : Graph) (ρ₁ ρ₂ : Env) : Type
       (idx₂ : g.Index) (idx₂_mem : idx₂ ∈ g.outputs)
       (trace : Trace g idx₁ idx₂ ρ₁ ρ₂) : EndToEndTrace g ρ₁ ρ₂
 
+inductive MyReaches {prog : Program} : {s₁ s₂ : prog.State} → {ρ₁ ρ₂ : Env} →
+    Trace prog.cfg s₁ s₂ ρ₁ ρ₂ →
+    (s : prog.State) → (ρin ρout : Env) → Type
+  | single_here {s₁ : prog.State} {ρ₁ ρ₂ : Env}
+      (hnode : EvalBasicStmtOpt ρ₁ (prog.code s₁) ρ₂) :
+      MyReaches (.single hnode) s₁ ρ₁ ρ₂
+  | edge_here {s₁ s₂ s₃ : prog.State} {ρ₁ ρ₂ ρ₃ : Env}
+      (hnode : EvalBasicStmtOpt ρ₁ (prog.code s₁) ρ₂)
+      (hedge : (s₁, s₂) ∈ prog.cfg.edges) (rest : Trace prog.cfg s₂ s₃ ρ₂ ρ₃) :
+      MyReaches (.edge hnode hedge rest) s₁ ρ₁ ρ₂
+  | edge_there {s₁ s₂ s₃ : prog.State} {ρ₁ ρ₂ ρ₃ : Env}
+      (hnode : EvalBasicStmtOpt ρ₁ (prog.code s₁) ρ₂)
+      (hedge : (s₁, s₂) ∈ prog.cfg.edges) (rest : Trace prog.cfg s₂ s₃ ρ₂ ρ₃)
+      {s : prog.State} {ρin ρout : Env} :
+      MyReaches rest s ρin ρout →
+      MyReaches (.edge hnode hedge rest) s ρin ρout
+
+noncomputable def MyReaches.prefix {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
+
+noncomputable 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}
+    {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
+  | .edge_here hnode hedge hrest => hnode
+  | .edge_there hnode hedge hrest tmp' => tmp'.step
+
+
 end Spa