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)