Add formalization of 'traces through graph'

Signed-off-by: Danila Fedorin <danila.fedorin@gmail.com>

Language/Traces.agda

@@ -0,0 +1,24 @@
+module Language.Traces where
+
+open import Language.Base public
+open import Language.Semantics public
+open import Language.Graphs public
+
+open import Data.Product using (_,_)
+open import Data.List.Membership.Propositional as MemProp using ()
+
+module _ {g : Graph} where
+    open Graph g using (Index; edges)
+
+    data Trace : Index → Index → Env → Env → Set where
+        Trace-single : ∀ {ρ₁ ρ₂ : Env} {idx : Index} →
+                       ρ₁ , (g [ idx ]) ⇒ᵇˢ ρ₂ → Trace idx idx ρ₁ ρ₂
+        Trace-edge : ∀ {ρ₁ ρ₂ ρ₃ : Env} {idx₁ idx₂ idx₃ : Index} →
+                     ρ₁ , (g [ idx₁ ]) ⇒ᵇˢ ρ₂ → (idx₁ , idx₂) MemProp.∈ edges →
+                     Trace idx₂ idx₃ ρ₂ ρ₃ → Trace idx₁ idx₃ ρ₁ ρ₃
+
+    _++⟨_⟩_ : ∀ {idx₁ idx₂ idx₃ idx₄ : Index} {ρ₁ ρ₂ ρ₃ : Env} →
+           Trace idx₁ idx₂ ρ₁ ρ₂ → (idx₂ , idx₃) MemProp.∈ edges →
+           Trace idx₃ idx₄ ρ₂ ρ₃ → Trace idx₁ idx₄ ρ₁ ρ₃
+    _++⟨_⟩_ (Trace-single ρ₁⇒ρ₂) idx₂→idx₃ tr = Trace-edge ρ₁⇒ρ₂ idx₂→idx₃ tr
+    _++⟨_⟩_ (Trace-edge ρ₁⇒ρ₂ idx₁→idx' tr') idx₂→idx₃ tr = Trace-edge ρ₁⇒ρ₂ idx₁→idx' (tr' ++⟨ idx₂→idx₃ ⟩ tr)