diff --git a/Analysis/Forward.agda b/Analysis/Forward.agda
index aed8fa9..7f22991 100644
--- a/Analysis/Forward.agda
+++ b/Analysis/Forward.agda
@@ -80,6 +80,7 @@ module WithProg (prog : Program) where
         renaming
             ( FiniteMap to StateVariables
             ; isLattice to isLatticeᵐ
+            ; _≈_ to _≈ᵐ_
             ; _∈k_ to _∈kᵐ_
             ; locate to locateᵐ
             ; _≼_ to _≼ᵐ_
@@ -92,6 +93,11 @@ module WithProg (prog : Program) where
         renaming
             ( isFiniteHeightLattice to isFiniteHeightLatticeᵐ
             )
+    open IsFiniteHeightLattice isFiniteHeightLatticeᵐ
+        using ()
+        renaming
+            ( ≈-sym to ≈ᵐ-sym
+            )
 
     ≈ᵐ-dec = ≈ᵛ-dec⇒≈ᵐ-dec ≈ᵛ-dec
     fixedHeightᵐ = IsFiniteHeightLattice.fixedHeight isFiniteHeightLatticeᵐ
@@ -107,6 +113,9 @@ module WithProg (prog : Program) where
     variablesAt : State → StateVariables → VariableValues
     variablesAt s sv = proj₁ (locateᵐ {s} {sv} (states-in-Map s sv))
 
+    variablesAt-≈ : ∀ s sv₁ sv₂ → sv₁ ≈ᵐ sv₂ → variablesAt s sv₁ ≈ᵛ variablesAt s sv₂
+    variablesAt-≈ = {!!}
+
     -- build up the 'join' function, which follows from Exercise 4.26's
     --
     --    L₁ → (A → L₂)
@@ -270,3 +279,14 @@ module WithProg (prog : Program) where
                 updateAll-matches : ∀ {s sv ρ₁ ρ₂} → ρ₁ , (code s) ⇒ᵇˢ ρ₂ → ⟦ variablesAt s sv ⟧ᵛ ρ₁ → ⟦ variablesAt s (updateAll sv) ⟧ᵛ ρ₂
                 updateAll-matches {s} {sv} ρ₁,bss⇒ρ₂ ⟦vs⟧ρ₁ rewrite variablesAt-updateAll s sv =
                     updateVariablesForState-matches {s} {sv} ρ₁,bss⇒ρ₂ ⟦vs⟧ρ₁
+
+                walkTrace : ∀ {s₁ s₂ ρ₁ ρ₂} → ⟦ joinForKey s₁ result ⟧ᵛ ρ₁ → Trace {graph} s₁ s₂ ρ₁ ρ₂ → ⟦ variablesAt s₂ result ⟧ᵛ ρ₂
+                walkTrace {s₁} {s₁} {ρ₁} {ρ₂} ⟦joinForKey-s₁⟧ρ₁ (Trace-single ρ₁,bss⇒ρ₂) =
+                        let
+                            -- I'd use rewrite, but Agda gets a memory overflow (?!).
+                            ⟦joinAll-result⟧ρ₁ = subst (λ vs → ⟦ vs ⟧ᵛ ρ₁) (sym (variablesAt-joinAll s₁ result)) ⟦joinForKey-s₁⟧ρ₁ 
+                            ⟦analyze-result⟧ρ₂ = updateAll-matches {sv = joinAll result} ρ₁,bss⇒ρ₂ ⟦joinAll-result⟧ρ₁
+                            analyze-result≈result = ≈ᵐ-sym {result} {updateAll (joinAll result)} result≈analyze-result
+                            analyze-s₁≈s₁ = variablesAt-≈ s₁ (updateAll (joinAll result)) result (analyze-result≈result)
+                        in
+                            ⟦⟧ᵛ-respects-≈ᵛ {variablesAt s₁ (updateAll (joinAll result))} {variablesAt s₁ result} (analyze-s₁≈s₁) ρ₂ ⟦analyze-result⟧ρ₂
diff --git a/Language.agda b/Language.agda
index 971a2c5..56e3170 100644
--- a/Language.agda
+++ b/Language.agda
@@ -2,6 +2,7 @@ module Language where
 
 open import Language.Base public
 open import Language.Semantics public
+open import Language.Traces public
 open import Language.Graphs public
 open import Language.Properties public