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Define end-to-end path concatenation and prove one more case

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Signed-off-by: Danila Fedorin <danila.fedorin@gmail.com>
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Danila Fedorin 2024-04-27 17:34:50 -07:00
parent 316e56f2bc
commit 4fee16413a
1 changed files with 24 additions and 1 deletions
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25
Language/Properties.agda
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@ -13,7 +13,7 @@ open import Data.Product using (Σ; _,_; _×_)
open import Data.Vec.Properties using (lookup-++ˡ; ++-identityʳ; lookup-++ʳ) open import Data.Vec.Properties using (lookup-++ˡ; ++-identityʳ; lookup-++ʳ)
open import Relation.Binary.PropositionalEquality as Eq using (_≡_; refl; sym) open import Relation.Binary.PropositionalEquality as Eq using (_≡_; refl; sym)
open import Utils using (x∈xs⇒fx∈fxs) open import Utils using (x∈xs⇒fx∈fxs; ∈-cartesianProduct)
buildCfg-input : ∀ (s : Stmt) → let g = buildCfg s in Σ (Graph.Index g) (λ idx → Graph.inputs g ≡ idx ∷ []) buildCfg-input : ∀ (s : Stmt) → let g = buildCfg s in Σ (Graph.Index g) (λ idx → Graph.inputs g ≡ idx ∷ [])
@ -83,6 +83,27 @@ Trace-loop g₁ {idx₁} {idx₁} (Trace-single ρ₁⇒ρ₂) = Trace-single ρ
Trace-loop g₁ {idx₁} (Trace-edge ρ₁⇒ρ idx₁→idx tr') = Trace-loop g₁ {idx₁} (Trace-edge ρ₁⇒ρ idx₁→idx tr') =
Trace-edge ρ₁⇒ρ (ListMemProp.∈-++⁺ˡ idx₁→idx) (Trace-loop g₁ tr') Trace-edge ρ₁⇒ρ (ListMemProp.∈-++⁺ˡ idx₁→idx) (Trace-loop g₁ tr')
_++_ : ∀ {g₁ g₂ : Graph} {ρ₁ ρ₂ ρ₃ : Env} →
EndToEndTrace {g₁} ρ₁ ρ₂ → EndToEndTrace {g₂} ρ₂ ρ₃ →
EndToEndTrace {g₁ ↦ g₂} ρ₁ ρ₃
_++_ {g₁} {g₂} etr₁ etr₂
= record
{ idx₁ = EndToEndTrace.idx₁ etr₁ Fin.↑ˡ Graph.size g₂
; idx₁∈inputs = x∈xs⇒fx∈fxs (Fin._↑ˡ Graph.size g₂) (EndToEndTrace.idx₁∈inputs etr₁)
; idx₂ = Graph.size g₁ Fin.↑ʳ EndToEndTrace.idx₂ etr₂
; idx₂∈outputs = x∈xs⇒fx∈fxs (Graph.size g₁ Fin.↑ʳ_) (EndToEndTrace.idx₂∈outputs etr₂)
; trace =
let
o∈tr₁ = x∈xs⇒fx∈fxs (Fin._↑ˡ Graph.size g₂) (EndToEndTrace.idx₂∈outputs etr₁)
i∈tr₂ = x∈xs⇒fx∈fxs (Graph.size g₁ Fin.↑ʳ_) (EndToEndTrace.idx₁∈inputs etr₂)
oi∈es = ListMemProp.∈-++⁺ʳ (Graph.edges g₁ ↑ˡᵉ Graph.size g₂)
(ListMemProp.∈-++⁺ʳ (Graph.size g₁ ↑ʳᵉ Graph.edges g₂)
(∈-cartesianProduct o∈tr₁ i∈tr₂))
in
(Trace-↦ˡ g₁ g₂ (EndToEndTrace.trace etr₁)) ++⟨ oi∈es ⟩
(Trace-↦ʳ g₁ g₂ (EndToEndTrace.trace etr₂))
}
buildCfg-sufficient : ∀ {s : Stmt} {ρ₁ ρ₂ : Env} → ρ₁ , s ⇒ˢ ρ₂ → buildCfg-sufficient : ∀ {s : Stmt} {ρ₁ ρ₂ : Env} → ρ₁ , s ⇒ˢ ρ₂ →
EndToEndTrace {buildCfg s} ρ₁ ρ₂ EndToEndTrace {buildCfg s} ρ₁ ρ₂
buildCfg-sufficient (⇒ˢ-⟨⟩ ρ₁ ρ₂ bs ρ₁,bs⇒ρ₂) = buildCfg-sufficient (⇒ˢ-⟨⟩ ρ₁ ρ₂ bs ρ₁,bs⇒ρ₂) =
@ -93,3 +114,5 @@ buildCfg-sufficient (⇒ˢ-⟨⟩ ρ₁ ρ₂ bs ρ₁,bs⇒ρ₂) =
; idx₂∈outputs = here refl ; idx₂∈outputs = here refl
; trace = Trace-single (ρ₁,bs⇒ρ₂ ∷ []) ; trace = Trace-single (ρ₁,bs⇒ρ₂ ∷ [])
} }
buildCfg-sufficient (⇒ˢ-then ρ₁ ρ₂ ρ₃ s₁ s₂ ρ₁,s₁⇒ρ₂ ρ₂,s₂⇒ρ₃) =
buildCfg-sufficient ρ₁,s₁⇒ρ₂ ++ buildCfg-sufficient ρ₂,s₂⇒ρ₃