Add formalization of 'traces through graph'

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

Language/Semantics.agda

@@ -5,7 +5,7 @@ open import Language.Base
 open import Data.Integer using (ℤ; +_) renaming (_+_ to _+ᶻ_; _-_ to _-ᶻ_)
 open import Data.Product using (_×_; _,_)
 open import Data.String using (String)
-open import Data.List using (List; _∷_)
+open import Data.List as List using (List)
 open import Data.Nat using (ℕ)
 open import Relation.Nullary using (¬_)
 open import Relation.Binary.PropositionalEquality using (_≡_)
@@ -17,8 +17,8 @@ Env : Set
 Env = List (String × Value)
 
 data _∈_ : (String × Value) → Env → Set where
-    here : ∀ (s : String) (v : Value) (ρ : Env) → (s , v) ∈ ((s , v) ∷ ρ)
-    there : ∀ (s s' : String) (v v' : Value) (ρ : Env) → ¬ (s ≡ s') → (s , v) ∈ ρ → (s , v) ∈ ((s' , v') ∷ ρ)
+    here : ∀ (s : String) (v : Value) (ρ : Env) → (s , v) ∈ ((s , v) List.∷ ρ)
+    there : ∀ (s s' : String) (v v' : Value) (ρ : Env) → ¬ (s ≡ s') → (s , v) ∈ ρ → (s , v) ∈ ((s' , v') List.∷ ρ)
 
 data _,_⇒ᵉ_ : Env → Expr → Value → Set where
     ⇒ᵉ-ℕ : ∀ (ρ : Env) (n : ℕ) → ρ , (# n) ⇒ᵉ (↑ᶻ (+ n))
@@ -33,7 +33,12 @@ data _,_⇒ᵉ_ : Env → Expr → Value → Set where
 data _,_⇒ᵇ_ : Env → BasicStmt → Env → Set where
     ⇒ᵇ-noop : ∀ (ρ : Env) → ρ , noop ⇒ᵇ ρ
     ⇒ᵇ-← : ∀ (ρ : Env) (x : String) (e : Expr) (v : Value) →
-           ρ , e ⇒ᵉ v → ρ , (x ← e) ⇒ᵇ ((x , v) ∷ ρ)
+           ρ , e ⇒ᵉ v → ρ , (x ← e) ⇒ᵇ ((x , v) List.∷ ρ)
+
+data _,_⇒ᵇˢ_ : Env → List BasicStmt → Env → Set where
+    [] : ∀ {ρ : Env} → ρ , List.[] ⇒ᵇˢ ρ
+    _∷_ : ∀ {ρ₁ ρ₂ ρ₃ : Env} {bs : BasicStmt} {bss : List BasicStmt} →
+          ρ₁ , bs ⇒ᵇ ρ₂ → ρ₂ , bss ⇒ᵇˢ ρ₃ → ρ₁ , (bs List.∷ bss) ⇒ᵇˢ ρ₃
 
 data _,_⇒ˢ_ : Env → Stmt → Env → Set where
     ⇒ˢ-⟨⟩ : ∀ (ρ₁ ρ₂ : Env) (bs : BasicStmt) →

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)