Prove 'sufficiency' for if-else.

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

Language/Graphs.agda

@@ -52,7 +52,7 @@ _↑ˡᵉ_ l m = List.map (_↑ˡ m) l
 _↑ʳᵉ_ : ∀ {m} n → List (Fin m × Fin m) → List (Fin (n Nat.+ m) × Fin (n Nat.+ m))
 _↑ʳᵉ_ n l = List.map (n ↑ʳ_) l
 
-infixl 5 _∙_
+infixr 5 _∙_
 _∙_ : Graph → Graph → Graph
 _∙_ g₁ g₂ = record
     { size = Graph.size g₁ Nat.+ Graph.size g₂
@@ -65,7 +65,7 @@ _∙_ g₁ g₂ = record
                 (Graph.size g₁ ↑ʳⁱ Graph.outputs g₂)
     }
 
-infixl 5 _↦_
+infixr 5 _↦_
 _↦_ : Graph → Graph → Graph
 _↦_ g₁ g₂ = record
     { size = Graph.size g₁ Nat.+ Graph.size g₂
@@ -84,7 +84,7 @@ loop g = record g
               List.cartesianProduct (Graph.outputs g) (Graph.inputs g)
     }
 
-infixl 5 _skipto_
+infixr 5 _skipto_
 _skipto_ : Graph → Graph → Graph
 _skipto_ g₁ g₂ = record (g₁ ∙ g₂)
     { edges = Graph.edges (g₁ ∙ g₂) List.++

Language/Properties.agda

@@ -112,6 +112,7 @@ Trace-loop {idx₁ = idx₁} {idx₁} (Trace-single ρ₁⇒ρ₂) = Trace-singl
 Trace-loop {g₁} {idx₁} (Trace-edge ρ₁⇒ρ idx₁→idx tr') =
     Trace-edge ρ₁⇒ρ (ListMemProp.∈-++⁺ˡ idx₁→idx) (Trace-loop tr')
 
+infixr 5 _++_
 _++_ : ∀ {g₁ g₂ : Graph} {ρ₁ ρ₂ ρ₃ : Env} →
        EndToEndTrace {g₁} ρ₁ ρ₂ → EndToEndTrace {g₂} ρ₂ ρ₃ →
        EndToEndTrace {g₁ ↦ g₂} ρ₁ ρ₃
@@ -152,3 +153,11 @@ buildCfg-sufficient (⇒ˢ-⟨⟩ ρ₁ ρ₂ bs ρ₁,bs⇒ρ₂) =
     Trace-singleton (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
+buildCfg-sufficient (⇒ˢ-if-true ρ₁ ρ₂ _ _ s₁ s₂ _ _ ρ₁,s₁⇒ρ₂) =
+    Trace-singleton[] ρ₁ ++
+    (EndToEndTrace-∙ˡ (buildCfg-sufficient ρ₁,s₁⇒ρ₂)) ++
+    Trace-singleton[] ρ₂
+buildCfg-sufficient (⇒ˢ-if-false ρ₁ ρ₂ _ s₁ s₂ _ ρ₁,s₂⇒ρ₂) =
+    Trace-singleton[] ρ₁ ++
+    (EndToEndTrace-∙ʳ {buildCfg s₁} (buildCfg-sufficient ρ₁,s₂⇒ρ₂)) ++
+    Trace-singleton[] ρ₂