Add some helpers and rewrite code

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

Language/Graphs.agda

@@ -79,13 +79,19 @@ _↦_ g₁ g₂ = record
     }
 
 loop : Graph → Graph
-loop g = record
-    { size = Graph.size g
-    ; nodes = Graph.nodes g
-    ; edges = Graph.edges g List.++
+loop g = record g
+    { edges = Graph.edges g List.++
               List.cartesianProduct (Graph.outputs g) (Graph.inputs g)
-    ; inputs = Graph.inputs g
-    ; outputs = Graph.outputs g
+    }
+
+infixl 5 _skipto_
+_skipto_ : Graph → Graph → Graph
+_skipto_ g₁ g₂ = record (g₁ ∙ g₂)
+    { edges = Graph.edges (g₁ ∙ g₂) List.++
+              (List.cartesianProduct (Graph.inputs g₁ ↑ˡⁱ Graph.size g₂)
+                                     (Graph.size g₁ ↑ʳⁱ Graph.inputs g₂))
+    ; inputs = Graph.inputs g₁ ↑ˡⁱ Graph.size g₂
+    ; outputs = Graph.size g₁ ↑ʳⁱ Graph.inputs g₂
     }
 
 _[_] : ∀ (g : Graph) → Graph.Index g → List BasicStmt
@@ -104,4 +110,4 @@ 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 (buildCfg s ↦ singleton [])
+buildCfg (while _ repeat s) = (loop (singleton [] ↦ buildCfg s)) skipto (singleton [])

Language/Properties.agda

@@ -6,10 +6,15 @@ open import Language.Graphs
 open import Language.Traces
 
 open import Data.Fin as Fin using (zero)
-open import Data.List using (_∷_; [])
-open import Data.Product using (Σ; _,_)
+open import Data.List using (List; _∷_; [])
+open import Data.List.Relation.Unary.Any using (here)
+open import Data.List.Membership.Propositional.Properties as ListMemProp using ()
+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; ∈-cartesianProduct)
 
-open import Relation.Binary.PropositionalEquality as Eq using (_≡_; refl)
 
 buildCfg-input : ∀ (s : Stmt) → let g = buildCfg s in Σ (Graph.Index g) (λ idx → Graph.inputs g ≡ idx ∷ [])
 buildCfg-input ⟨ bs₁ ⟩ = (zero , refl)
@@ -26,3 +31,95 @@ buildCfg-output (s₁ then s₂)
 buildCfg-output (if _ then s₁ else s₂) = (_ , refl)
 buildCfg-output (while _ repeat s)
     with (idx , p) ← buildCfg-output s rewrite p = (_ , refl)
+
+Trace-∙ˡ : ∀ (g₁ g₂ : Graph) {idx₁ idx₂ : Graph.Index g₁} {ρ₁ ρ₂ : Env} →
+           Trace {g₁} idx₁ idx₂ ρ₁ ρ₂ →
+           Trace {g₁ ∙ g₂} (idx₁ Fin.↑ˡ Graph.size g₂) (idx₂ Fin.↑ˡ Graph.size g₂) ρ₁ ρ₂
+Trace-∙ˡ g₁ g₂ {idx₁} {idx₁} (Trace-single ρ₁⇒ρ₂)
+    rewrite sym (lookup-++ˡ (Graph.nodes g₁) (Graph.nodes g₂) idx₁) =
+    Trace-single ρ₁⇒ρ₂
+Trace-∙ˡ g₁ g₂ {idx₁} (Trace-edge ρ₁⇒ρ idx₁→idx tr')
+    rewrite sym (lookup-++ˡ (Graph.nodes g₁) (Graph.nodes g₂) idx₁) =
+    Trace-edge ρ₁⇒ρ (ListMemProp.∈-++⁺ˡ (x∈xs⇒fx∈fxs (_↑ˡ Graph.size g₂) idx₁→idx))
+                    (Trace-∙ˡ g₁ g₂ tr')
+
+Trace-∙ʳ : ∀ (g₁ g₂ : Graph) {idx₁ idx₂ : Graph.Index g₂} {ρ₁ ρ₂ : Env} →
+           Trace {g₂} idx₁ idx₂ ρ₁ ρ₂ →
+           Trace {g₁ ∙ g₂} (Graph.size g₁ Fin.↑ʳ idx₁) (Graph.size g₁ Fin.↑ʳ idx₂) ρ₁ ρ₂
+Trace-∙ʳ g₁ g₂ {idx₁} {idx₁} (Trace-single ρ₁⇒ρ₂)
+    rewrite sym (lookup-++ʳ (Graph.nodes g₁) (Graph.nodes g₂) idx₁) =
+    Trace-single ρ₁⇒ρ₂
+Trace-∙ʳ g₁ g₂ {idx₁} (Trace-edge ρ₁⇒ρ idx₁→idx tr')
+    rewrite sym (lookup-++ʳ (Graph.nodes g₁) (Graph.nodes g₂) idx₁) =
+    Trace-edge ρ₁⇒ρ (ListMemProp.∈-++⁺ʳ _ (x∈xs⇒fx∈fxs (Graph.size g₁ ↑ʳ_) idx₁→idx))
+                    (Trace-∙ʳ g₁ g₂ tr')
+
+Trace-↦ˡ : ∀ (g₁ g₂ : Graph) {idx₁ idx₂ : Graph.Index g₁} {ρ₁ ρ₂ : Env} →
+           Trace {g₁} idx₁ idx₂ ρ₁ ρ₂ →
+           Trace {g₁ ↦ g₂} (idx₁ Fin.↑ˡ Graph.size g₂) (idx₂ Fin.↑ˡ Graph.size g₂) ρ₁ ρ₂
+Trace-↦ˡ g₁ g₂ {idx₁} {idx₁} (Trace-single ρ₁⇒ρ₂)
+    rewrite sym (lookup-++ˡ (Graph.nodes g₁) (Graph.nodes g₂) idx₁) =
+    Trace-single ρ₁⇒ρ₂
+Trace-↦ˡ g₁ g₂ {idx₁} (Trace-edge ρ₁⇒ρ idx₁→idx tr')
+    rewrite sym (lookup-++ˡ (Graph.nodes g₁) (Graph.nodes g₂) idx₁) =
+    Trace-edge ρ₁⇒ρ (ListMemProp.∈-++⁺ˡ (x∈xs⇒fx∈fxs (_↑ˡ Graph.size g₂) idx₁→idx))
+                    (Trace-↦ˡ g₁ g₂ tr')
+
+Trace-↦ʳ : ∀ (g₁ g₂ : Graph) {idx₁ idx₂ : Graph.Index g₂} {ρ₁ ρ₂ : Env} →
+           Trace {g₂} idx₁ idx₂ ρ₁ ρ₂ →
+           Trace {g₁ ↦ g₂} (Graph.size g₁ Fin.↑ʳ idx₁) (Graph.size g₁ Fin.↑ʳ idx₂) ρ₁ ρ₂
+Trace-↦ʳ g₁ g₂ {idx₁} {idx₁} (Trace-single ρ₁⇒ρ₂)
+    rewrite sym (lookup-++ʳ (Graph.nodes g₁) (Graph.nodes g₂) idx₁) =
+    Trace-single ρ₁⇒ρ₂
+Trace-↦ʳ g₁ g₂ {idx₁} (Trace-edge ρ₁⇒ρ idx₁→idx tr')
+    rewrite sym (lookup-++ʳ (Graph.nodes g₁) (Graph.nodes g₂) idx₁) =
+    Trace-edge ρ₁⇒ρ (ListMemProp.∈-++⁺ʳ (Graph.edges g₁ ↑ˡᵉ Graph.size g₂)
+                                        (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 {g₁} idx₁ idx₂ ρ₁ ρ₂ → Trace {loop g₁} idx₁ idx₂ ρ₁ ρ₂
+Trace-loop g₁ {idx₁} {idx₁} (Trace-single ρ₁⇒ρ₂) = Trace-single ρ₁⇒ρ₂
+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))
+        }
+
+Trace-singleton : ∀ {bss : List BasicStmt} {ρ₁ ρ₂ : Env} →
+                  ρ₁ , bss ⇒ᵇˢ ρ₂ → EndToEndTrace {singleton bss} ρ₁ ρ₂
+Trace-singleton ρ₁⇒ρ₂ = record
+    { idx₁ = zero
+    ; idx₁∈inputs = here refl
+    ; idx₂ = zero
+    ; idx₂∈outputs = here refl
+    ; trace = Trace-single ρ₁⇒ρ₂
+    }
+
+Trace-singleton[] : ∀ (ρ : Env) → EndToEndTrace {singleton []} ρ ρ
+Trace-singleton[] env = Trace-singleton []
+
+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) =
+    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

Language/Traces.agda

@@ -1,24 +1,34 @@
 module Language.Traces where
 
-open import Language.Base public
-open import Language.Semantics public
-open import Language.Graphs public
+open import Language.Base
+open import Language.Semantics using (Env; _,_⇒ᵇˢ_)
+open import Language.Graphs
 
 open import Data.Product using (_,_)
-open import Data.List.Membership.Propositional as MemProp using ()
+open import Data.List.Membership.Propositional using (_∈_)
 
 module _ {g : Graph} where
-    open Graph g using (Index; edges)
+    open Graph g using (Index; edges; inputs; outputs)
 
     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 →
+                     ρ₁ , (g [ idx₁ ]) ⇒ᵇˢ ρ₂ → (idx₁ , idx₂) ∈ 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 idx₁ idx₂ ρ₁ ρ₂ → (idx₂ , idx₃) ∈ 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)
+
+    record EndToEndTrace (ρ₁ ρ₂ : Env) : Set where
+        field
+            idx₁ : Index
+            idx₁∈inputs : idx₁ ∈ inputs
+
+            idx₂ : Index
+            idx₂∈outputs : idx₂ ∈ outputs
+
+            trace : Trace idx₁ idx₂ ρ₁ ρ₂