agda-spa/Language/Properties.agda

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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')
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')