84 lines
4.8 KiB
Agda
84 lines
4.8 KiB
Agda
module Language.Properties where
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open import Language.Base
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open import Language.Semantics
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open import Language.Graphs
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open import Language.Traces
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open import Data.Fin as Fin using (zero)
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open import Data.List using (_∷_; [])
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open import Data.List.Membership.Propositional.Properties as ListMemProp using ()
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open import Data.Product using (Σ; _,_)
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open import Data.Vec.Properties using (lookup-++ˡ; ++-identityʳ; lookup-++ʳ)
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open import Relation.Binary.PropositionalEquality as Eq using (_≡_; refl; sym)
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open import Utils using (x∈xs⇒fx∈fxs)
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buildCfg-input : ∀ (s : Stmt) → let g = buildCfg s in Σ (Graph.Index g) (λ idx → Graph.inputs g ≡ idx ∷ [])
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buildCfg-input ⟨ bs₁ ⟩ = (zero , refl)
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buildCfg-input (s₁ then s₂)
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with (idx , p) ← buildCfg-input s₁ rewrite p = (_ , refl)
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buildCfg-input (if _ then s₁ else s₂) = (zero , refl)
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buildCfg-input (while _ repeat s)
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with (idx , p) ← buildCfg-input s rewrite p = (_ , refl)
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buildCfg-output : ∀ (s : Stmt) → let g = buildCfg s in Σ (Graph.Index g) (λ idx → Graph.outputs g ≡ idx ∷ [])
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buildCfg-output ⟨ bs₁ ⟩ = (zero , refl)
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buildCfg-output (s₁ then s₂)
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with (idx , p) ← buildCfg-output s₂ rewrite p = (_ , refl)
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buildCfg-output (if _ then s₁ else s₂) = (_ , refl)
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buildCfg-output (while _ repeat s)
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with (idx , p) ← buildCfg-output s rewrite p = (_ , refl)
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Trace-∙ˡ : ∀ (g₁ g₂ : Graph) {idx₁ idx₂ : Graph.Index g₁} {ρ₁ ρ₂ : Env} →
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Trace {g₁} idx₁ idx₂ ρ₁ ρ₂ →
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Trace {g₁ ∙ g₂} (idx₁ Fin.↑ˡ Graph.size g₂) (idx₂ Fin.↑ˡ Graph.size g₂) ρ₁ ρ₂
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Trace-∙ˡ g₁ g₂ {idx₁} {idx₁} (Trace-single ρ₁⇒ρ₂)
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rewrite sym (lookup-++ˡ (Graph.nodes g₁) (Graph.nodes g₂) idx₁) =
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Trace-single ρ₁⇒ρ₂
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Trace-∙ˡ g₁ g₂ {idx₁} (Trace-edge ρ₁⇒ρ idx₁→idx tr')
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rewrite sym (lookup-++ˡ (Graph.nodes g₁) (Graph.nodes g₂) idx₁) =
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Trace-edge ρ₁⇒ρ (ListMemProp.∈-++⁺ˡ (x∈xs⇒fx∈fxs (_↑ˡ Graph.size g₂) idx₁→idx))
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(Trace-∙ˡ g₁ g₂ tr')
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Trace-∙ʳ : ∀ (g₁ g₂ : Graph) {idx₁ idx₂ : Graph.Index g₂} {ρ₁ ρ₂ : Env} →
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Trace {g₂} idx₁ idx₂ ρ₁ ρ₂ →
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Trace {g₁ ∙ g₂} (Graph.size g₁ Fin.↑ʳ idx₁) (Graph.size g₁ Fin.↑ʳ idx₂) ρ₁ ρ₂
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Trace-∙ʳ g₁ g₂ {idx₁} {idx₁} (Trace-single ρ₁⇒ρ₂)
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rewrite sym (lookup-++ʳ (Graph.nodes g₁) (Graph.nodes g₂) idx₁) =
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Trace-single ρ₁⇒ρ₂
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Trace-∙ʳ g₁ g₂ {idx₁} (Trace-edge ρ₁⇒ρ idx₁→idx tr')
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rewrite sym (lookup-++ʳ (Graph.nodes g₁) (Graph.nodes g₂) idx₁) =
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Trace-edge ρ₁⇒ρ (ListMemProp.∈-++⁺ʳ _ (x∈xs⇒fx∈fxs (Graph.size g₁ ↑ʳ_) idx₁→idx))
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(Trace-∙ʳ g₁ g₂ tr')
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Trace-↦ˡ : ∀ (g₁ g₂ : Graph) {idx₁ idx₂ : Graph.Index g₁} {ρ₁ ρ₂ : Env} →
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Trace {g₁} idx₁ idx₂ ρ₁ ρ₂ →
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Trace {g₁ ↦ g₂} (idx₁ Fin.↑ˡ Graph.size g₂) (idx₂ Fin.↑ˡ Graph.size g₂) ρ₁ ρ₂
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Trace-↦ˡ g₁ g₂ {idx₁} {idx₁} (Trace-single ρ₁⇒ρ₂)
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rewrite sym (lookup-++ˡ (Graph.nodes g₁) (Graph.nodes g₂) idx₁) =
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Trace-single ρ₁⇒ρ₂
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Trace-↦ˡ g₁ g₂ {idx₁} (Trace-edge ρ₁⇒ρ idx₁→idx tr')
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rewrite sym (lookup-++ˡ (Graph.nodes g₁) (Graph.nodes g₂) idx₁) =
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Trace-edge ρ₁⇒ρ (ListMemProp.∈-++⁺ˡ (x∈xs⇒fx∈fxs (_↑ˡ Graph.size g₂) idx₁→idx))
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(Trace-↦ˡ g₁ g₂ tr')
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Trace-↦ʳ : ∀ (g₁ g₂ : Graph) {idx₁ idx₂ : Graph.Index g₂} {ρ₁ ρ₂ : Env} →
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Trace {g₂} idx₁ idx₂ ρ₁ ρ₂ →
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Trace {g₁ ↦ g₂} (Graph.size g₁ Fin.↑ʳ idx₁) (Graph.size g₁ Fin.↑ʳ idx₂) ρ₁ ρ₂
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Trace-↦ʳ g₁ g₂ {idx₁} {idx₁} (Trace-single ρ₁⇒ρ₂)
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rewrite sym (lookup-++ʳ (Graph.nodes g₁) (Graph.nodes g₂) idx₁) =
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Trace-single ρ₁⇒ρ₂
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Trace-↦ʳ g₁ g₂ {idx₁} (Trace-edge ρ₁⇒ρ idx₁→idx tr')
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rewrite sym (lookup-++ʳ (Graph.nodes g₁) (Graph.nodes g₂) idx₁) =
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Trace-edge ρ₁⇒ρ (ListMemProp.∈-++⁺ʳ (Graph.edges g₁ ↑ˡᵉ Graph.size g₂)
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(ListMemProp.∈-++⁺ˡ (x∈xs⇒fx∈fxs (Graph.size g₁ ↑ʳ_) idx₁→idx)))
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(Trace-↦ʳ g₁ g₂ tr')
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Trace-loop : ∀ (g₁ : Graph) {idx₁ idx₂ : Graph.Index g₁} {ρ₁ ρ₂ : Env} →
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Trace {g₁} idx₁ idx₂ ρ₁ ρ₂ → Trace {loop g₁} idx₁ idx₂ ρ₁ ρ₂
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Trace-loop g₁ {idx₁} {idx₁} (Trace-single ρ₁⇒ρ₂) = Trace-single ρ₁⇒ρ₂
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Trace-loop g₁ {idx₁} (Trace-edge ρ₁⇒ρ idx₁→idx tr') =
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Trace-edge ρ₁⇒ρ (ListMemProp.∈-++⁺ˡ idx₁→idx) (Trace-loop g₁ tr')
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