module Language.Properties where

open import Language.Base
open import Language.Semantics
open import Language.Graphs
open import Language.Traces

open import Data.Fin as Fin using (zero)
open import Data.List using (_∷_; [])
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)


buildCfg-input : ∀ (s : Stmt) → let g = buildCfg s in Σ (Graph.Index g) (λ idx → Graph.inputs g ≡ idx ∷ [])
buildCfg-input ⟨ bs₁ ⟩ = (zero , refl)
buildCfg-input (s₁ then s₂)
    with (idx , p) ← buildCfg-input s₁ rewrite p = (_ , refl)
buildCfg-input (if _ then s₁ else s₂) = (zero , refl)
buildCfg-input (while _ repeat s)
    with (idx , p) ← buildCfg-input s rewrite p = (_ , refl)

buildCfg-output : ∀ (s : Stmt) → let g = buildCfg s in Σ (Graph.Index g) (λ idx → Graph.outputs g ≡ idx ∷ [])
buildCfg-output ⟨ bs₁ ⟩ = (zero , refl)
buildCfg-output (s₁ then s₂)
    with (idx , p) ← buildCfg-output s₂ rewrite p = (_ , refl)
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')