Define end-to-end path concatenation and prove one more case

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

Language/Properties.agda

@@ -13,7 +13,7 @@ open import Data.Product using (Σ; _,_; _×_)
 open import Data.Vec.Properties using (lookup-++ˡ; ++-identityʳ; lookup-++ʳ)
 open import Relation.Binary.PropositionalEquality as Eq using (_≡_; refl; sym)
 
-open import Utils using (x∈xs⇒fx∈fxs)
+open import Utils using (x∈xs⇒fx∈fxs; ∈-cartesianProduct)
 
 
 buildCfg-input : ∀ (s : Stmt) → let g = buildCfg s in Σ (Graph.Index g) (λ idx → Graph.inputs g ≡ idx ∷ [])
@@ -83,6 +83,27 @@ Trace-loop g₁ {idx₁} {idx₁} (Trace-single ρ₁⇒ρ₂) = Trace-single ρ
 Trace-loop g₁ {idx₁} (Trace-edge ρ₁⇒ρ idx₁→idx tr') =
     Trace-edge ρ₁⇒ρ (ListMemProp.∈-++⁺ˡ idx₁→idx) (Trace-loop g₁ tr')
 
+_++_ : ∀ {g₁ g₂ : Graph} {ρ₁ ρ₂ ρ₃ : Env} →
+       EndToEndTrace {g₁} ρ₁ ρ₂ → EndToEndTrace {g₂} ρ₂ ρ₃ →
+       EndToEndTrace {g₁ ↦ g₂} ρ₁ ρ₃
+_++_ {g₁} {g₂} etr₁ etr₂
+    = record
+        { idx₁ = EndToEndTrace.idx₁ etr₁ Fin.↑ˡ Graph.size g₂
+        ; idx₁∈inputs = x∈xs⇒fx∈fxs (Fin._↑ˡ Graph.size g₂) (EndToEndTrace.idx₁∈inputs etr₁)
+        ; idx₂ = Graph.size g₁ Fin.↑ʳ EndToEndTrace.idx₂ etr₂
+        ; idx₂∈outputs = x∈xs⇒fx∈fxs (Graph.size g₁ Fin.↑ʳ_) (EndToEndTrace.idx₂∈outputs etr₂)
+        ; trace =
+            let
+                o∈tr₁ = x∈xs⇒fx∈fxs (Fin._↑ˡ Graph.size g₂) (EndToEndTrace.idx₂∈outputs etr₁)
+                i∈tr₂ = x∈xs⇒fx∈fxs (Graph.size g₁ Fin.↑ʳ_) (EndToEndTrace.idx₁∈inputs etr₂)
+                oi∈es = ListMemProp.∈-++⁺ʳ (Graph.edges g₁ ↑ˡᵉ Graph.size g₂)
+                                           (ListMemProp.∈-++⁺ʳ (Graph.size g₁ ↑ʳᵉ Graph.edges g₂)
+                                                               (∈-cartesianProduct o∈tr₁ i∈tr₂))
+            in
+                (Trace-↦ˡ g₁ g₂ (EndToEndTrace.trace etr₁)) ++⟨ oi∈es ⟩
+                (Trace-↦ʳ g₁ g₂ (EndToEndTrace.trace etr₂))
+        }
+
 buildCfg-sufficient : ∀ {s : Stmt} {ρ₁ ρ₂ : Env} → ρ₁ , s ⇒ˢ ρ₂ →
                       EndToEndTrace {buildCfg s} ρ₁ ρ₂
 buildCfg-sufficient (⇒ˢ-⟨⟩ ρ₁ ρ₂ bs ρ₁,bs⇒ρ₂) =
@@ -93,3 +114,5 @@ buildCfg-sufficient (⇒ˢ-⟨⟩ ρ₁ ρ₂ bs ρ₁,bs⇒ρ₂) =
         ; idx₂∈outputs = here refl
         ; trace = Trace-single (ρ₁,bs⇒ρ₂ ∷ [])
         }
+buildCfg-sufficient (⇒ˢ-then ρ₁ ρ₂ ρ₃ s₁ s₂ ρ₁,s₁⇒ρ₂ ρ₂,s₂⇒ρ₃) =
+    buildCfg-sufficient ρ₁,s₁⇒ρ₂ ++ buildCfg-sufficient ρ₂,s₂⇒ρ₃