module Language.Properties where

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

open import MonotonicState _⊆_ ⊆-trans renaming (MonotonicState to MonotonicGraphFunction)
open import Utils using (_⊗_; _,_)
open Relaxable {{...}}

open import Data.Fin using (zero)
open import Data.List using (List; _∷_; [])
open import Data.Vec using (_∷_; [])
open import Data.Vec.Properties using (cast-is-id; lookup-++ˡ; lookup-++ʳ)
open import Relation.Binary.PropositionalEquality as Eq using (_≡_; refl; sym; trans; subst)

relax-preserves-[]≡ : ∀ (g₁ g₂ : Graph) (g₁⊆g₂ : g₁ ⊆ g₂) (idx : Graph.Index g₁) →
                      g₁ [ idx ] ≡ g₂ [ relax g₁⊆g₂ idx ]
relax-preserves-[]≡ g₁ g₂ (Mk-⊆ n refl newNodes nsg₂≡nsg₁++newNodes _) idx
    rewrite cast-is-id refl (Graph.nodes g₂)
    with refl ← nsg₂≡nsg₁++newNodes = sym (lookup-++ˡ (Graph.nodes g₁) _ _)

instance
    NodeEqualsMonotonic : ∀ {bss : List BasicStmt} →
                          MonotonicPredicate (λ g n → g [ n ] ≡ bss)
    NodeEqualsMonotonic = record
        { relaxPredicate = λ g₁ g₂ idx g₁⊆g₂ g₁[idx]≡bss →
            trans (sym (relax-preserves-[]≡ g₁ g₂ g₁⊆g₂ idx)) g₁[idx]≡bss
        }

pushBasicBlock-works : ∀ (bss : List BasicStmt) → Always (λ g idx → g [ idx ] ≡ bss) (pushBasicBlock bss)
pushBasicBlock-works bss = MkAlways (λ g → lookup-++ʳ (Graph.nodes g) (bss ∷ []) zero)

TransformsEnv : ∀ (ρ₁ ρ₂ : Env) → DependentPredicate (Graph.Index ⊗ Graph.Index)
TransformsEnv ρ₁ ρ₂ g (idx₁ , idx₂) = Trace {g} idx₁ idx₂ ρ₁ ρ₂

instance
    TransformsEnvMonotonic : ∀ {ρ₁ ρ₂ : Env} → MonotonicPredicate (TransformsEnv ρ₁ ρ₂)
    TransformsEnvMonotonic = {!!}

buildCfg-sufficient : ∀ {ρ₁ ρ₂ : Env} {s : Stmt} → ρ₁ , s ⇒ˢ ρ₂ → Always (TransformsEnv ρ₁ ρ₂) (buildCfg s)
buildCfg-sufficient {ρ₁} {ρ₂} {⟨ bs ⟩} (⇒ˢ-⟨⟩ ρ₁ ρ₂ bs ρ₁,bs⇒ρ₂) =
    pushBasicBlock-works (bs ∷ [])
    map-reason
        (λ g idx g[idx]≡[bs] → Trace-single (subst (ρ₁ ,_⇒ᵇˢ ρ₂)
                                                   (sym g[idx]≡[bs])
                                                   (ρ₁,bs⇒ρ₂ ∷ [])))
buildCfg-sufficient {ρ₁} {ρ₂} {s₁ then s₂} (⇒ˢ-then ρ₁ ρ ρ₂ s₁ s₂ ρ₁,s₁⇒ρ₂ ρ₂,s₂⇒ρ₃) =
    (buildCfg-sufficient ρ₁,s₁⇒ρ₂ ⟨⊗⟩-reason buildCfg-sufficient ρ₂,s₂⇒ρ₃)
    update-reason {!!}
    map-reason {!!}