Prove another (simple) case

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

Language/Graphs.agda

@@ -84,6 +84,18 @@ loop g = record g
               List.cartesianProduct (Graph.outputs g) (Graph.inputs g)
     }
 
+optional : Graph → Graph
+optional g = record
+    { size = 2 Nat.+ Graph.size g
+    ; nodes = [] ∷ [] ∷ Graph.nodes g
+    ; edges = (2 ↑ʳᵉ Graph.edges g) List.++
+              List.map (zero ,_) (2 ↑ʳⁱ Graph.inputs g) List.++
+              List.map (_, suc zero) (2 ↑ʳⁱ Graph.outputs g) List.++
+              ((zero , suc zero) ∷ [])
+    ; inputs = zero ∷ []
+    ; outputs = (suc zero) ∷ []
+    }
+
 infixr 5 _skipto_
 _skipto_ : Graph → Graph → Graph
 _skipto_ g₁ g₂ = record (g₁ ∙ g₂)
@@ -110,4 +122,4 @@ buildCfg : Stmt → Graph
 buildCfg ⟨ bs₁ ⟩ = singleton (bs₁ ∷ [])
 buildCfg (s₁ then s₂) = buildCfg s₁ ↦ buildCfg s₂
 buildCfg (if _ then s₁ else s₂) = singleton [] ↦ (buildCfg s₁ ∙ buildCfg s₂) ↦ singleton []
-buildCfg (while _ repeat s) = (loop (singleton [] ↦ buildCfg s)) skipto (singleton [])
+buildCfg (while _ repeat s) = optional (loop (buildCfg s))

Language/Properties.agda

@@ -106,12 +106,51 @@ Trace-↦ʳ {g₁} {g₂} {idx₁} (Trace-edge ρ₁⇒ρ idx₁→idx tr')
                                         (ListMemProp.∈-++⁺ˡ (x∈xs⇒fx∈fxs (Graph.size g₁ ↑ʳ_) idx₁→idx)))
                     (Trace-↦ʳ {g₁} {g₂} tr')
 
-Trace-loop : ∀ {g₁ : Graph} {idx₁ idx₂ : Graph.Index g₁} {ρ₁ ρ₂ : Env} →
+Trace-loop : ∀ {g : Graph} {idx₁ idx₂ : Graph.Index g} {ρ₁ ρ₂ : Env} →
-             Trace {g₁} idx₁ idx₂ ρ₁ ρ₂ → Trace {loop g₁} idx₁ idx₂ ρ₁ ρ₂
+             Trace {g} idx₁ idx₂ ρ₁ ρ₂ → Trace {loop g} idx₁ idx₂ ρ₁ ρ₂
 Trace-loop {idx₁ = idx₁} {idx₁} (Trace-single ρ₁⇒ρ₂) = Trace-single ρ₁⇒ρ₂
-Trace-loop {g₁} {idx₁} (Trace-edge ρ₁⇒ρ idx₁→idx tr') =
+Trace-loop {g} {idx₁} (Trace-edge ρ₁⇒ρ idx₁→idx tr') =
     Trace-edge ρ₁⇒ρ (ListMemProp.∈-++⁺ˡ idx₁→idx) (Trace-loop tr')
 
+EndToEndTrace-loop : ∀ {g : Graph} {ρ₁ ρ₂ : Env} →
+                     EndToEndTrace {g} ρ₁ ρ₂ → EndToEndTrace {loop g} ρ₁ ρ₂
+EndToEndTrace-loop etr = record
+    { idx₁ = EndToEndTrace.idx₁ etr
+    ; idx₁∈inputs = EndToEndTrace.idx₁∈inputs etr
+    ; idx₂ = EndToEndTrace.idx₂ etr
+    ; idx₂∈outputs = EndToEndTrace.idx₂∈outputs etr
+    ; trace = Trace-loop (EndToEndTrace.trace etr)
+    }
+
+EndToEndTrace-loop² : ∀ {g : Graph} {ρ₁ ρ₂ ρ₃ : Env} →
+                      EndToEndTrace {loop g} ρ₁ ρ₂ →
+                      EndToEndTrace {loop g} ρ₂ ρ₃ →
+                      EndToEndTrace {loop g} ρ₁ ρ₃
+EndToEndTrace-loop² {g} etr₁ etr₂ = record
+    { idx₁ = EndToEndTrace.idx₁ etr₁
+    ; idx₁∈inputs = EndToEndTrace.idx₁∈inputs etr₁
+    ; idx₂ = EndToEndTrace.idx₂ etr₂
+    ; idx₂∈outputs = EndToEndTrace.idx₂∈outputs etr₂
+    ; trace =
+        let
+            o∈tr₁ = EndToEndTrace.idx₂∈outputs etr₁
+            i∈tr₂ = EndToEndTrace.idx₁∈inputs etr₂
+            oi∈es = ListMemProp.∈-++⁺ʳ (Graph.edges g) (∈-cartesianProduct o∈tr₁ i∈tr₂)
+        in
+            EndToEndTrace.trace etr₁ ++⟨ oi∈es ⟩ EndToEndTrace.trace etr₂
+    }
+
+Trace-optional : ∀ {g : Graph} {idx₁ idx₂ : Graph.Index g} {ρ₁ ρ₂ : Env} →
+                 Trace {g} idx₁ idx₂ ρ₁ ρ₂ → Trace {optional g} (2 Fin.↑ʳ idx₁) (2 Fin.↑ʳ idx₂) ρ₁ ρ₂
+Trace-optional = {!!}
+
+EndToEndTrace-optional : ∀ {g : Graph} {ρ₁ ρ₂ : Env} →
+                         EndToEndTrace {g} ρ₁ ρ₂ → EndToEndTrace {optional g} ρ₁ ρ₂
+EndToEndTrace-optional = {!!}
+
+EndToEndTrace-optional-ε : ∀ {g : Graph} {ρ : Env} → EndToEndTrace {optional g} ρ ρ
+EndToEndTrace-optional-ε = {!!}
+
 infixr 5 _++_
 _++_ : ∀ {g₁ g₂ : Graph} {ρ₁ ρ₂ ρ₃ : Env} →
        EndToEndTrace {g₁} ρ₁ ρ₂ → EndToEndTrace {g₂} ρ₂ ρ₃ →
@@ -161,3 +200,6 @@ 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
+buildCfg-sufficient (⇒ˢ-while-true ρ₁ ρ₂ ρ₃ _ _ s _ _ ρ₁,s⇒ρ₂ ρ₂,ws⇒ρ₃) = {!!}
+buildCfg-sufficient (⇒ˢ-while-false ρ _ s _) =
+    EndToEndTrace-optional-ε {loop (buildCfg s)} {ρ}