Define end-to-end path concatenation and prove one more case
Signed-off-by: Danila Fedorin <danila.fedorin@gmail.com>
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@ -13,7 +13,7 @@ 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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open import Utils using (x∈xs⇒fx∈fxs; ∈-cartesianProduct)
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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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@ -83,6 +83,27 @@ Trace-loop g₁ {idx₁} {idx₁} (Trace-single ρ₁⇒ρ₂) = Trace-single ρ
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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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_++_ : ∀ {g₁ g₂ : Graph} {ρ₁ ρ₂ ρ₃ : Env} →
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EndToEndTrace {g₁} ρ₁ ρ₂ → EndToEndTrace {g₂} ρ₂ ρ₃ →
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EndToEndTrace {g₁ ↦ g₂} ρ₁ ρ₃
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_++_ {g₁} {g₂} etr₁ etr₂
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= record
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{ idx₁ = EndToEndTrace.idx₁ etr₁ Fin.↑ˡ Graph.size g₂
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; idx₁∈inputs = x∈xs⇒fx∈fxs (Fin._↑ˡ Graph.size g₂) (EndToEndTrace.idx₁∈inputs etr₁)
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; idx₂ = Graph.size g₁ Fin.↑ʳ EndToEndTrace.idx₂ etr₂
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; idx₂∈outputs = x∈xs⇒fx∈fxs (Graph.size g₁ Fin.↑ʳ_) (EndToEndTrace.idx₂∈outputs etr₂)
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; trace =
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let
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o∈tr₁ = x∈xs⇒fx∈fxs (Fin._↑ˡ Graph.size g₂) (EndToEndTrace.idx₂∈outputs etr₁)
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i∈tr₂ = x∈xs⇒fx∈fxs (Graph.size g₁ Fin.↑ʳ_) (EndToEndTrace.idx₁∈inputs etr₂)
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oi∈es = ListMemProp.∈-++⁺ʳ (Graph.edges g₁ ↑ˡᵉ Graph.size g₂)
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(ListMemProp.∈-++⁺ʳ (Graph.size g₁ ↑ʳᵉ Graph.edges g₂)
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(∈-cartesianProduct o∈tr₁ i∈tr₂))
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in
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(Trace-↦ˡ g₁ g₂ (EndToEndTrace.trace etr₁)) ++⟨ oi∈es ⟩
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(Trace-↦ʳ g₁ g₂ (EndToEndTrace.trace etr₂))
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}
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buildCfg-sufficient : ∀ {s : Stmt} {ρ₁ ρ₂ : Env} → ρ₁ , s ⇒ˢ ρ₂ →
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EndToEndTrace {buildCfg s} ρ₁ ρ₂
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buildCfg-sufficient (⇒ˢ-⟨⟩ ρ₁ ρ₂ bs ρ₁,bs⇒ρ₂) =
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@ -93,3 +114,5 @@ buildCfg-sufficient (⇒ˢ-⟨⟩ ρ₁ ρ₂ bs ρ₁,bs⇒ρ₂) =
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; idx₂∈outputs = here refl
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; trace = Trace-single (ρ₁,bs⇒ρ₂ ∷ [])
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}
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buildCfg-sufficient (⇒ˢ-then ρ₁ ρ₂ ρ₃ s₁ s₂ ρ₁,s₁⇒ρ₂ ρ₂,s₂⇒ρ₃) =
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buildCfg-sufficient ρ₁,s₁⇒ρ₂ ++ buildCfg-sufficient ρ₂,s₂⇒ρ₃
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