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Add properties of end-to-end traces on loops

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Danila Fedorin 2024-04-28 12:53:25 -07:00
parent b4d395767d
commit aec15573fc
1 changed files with 31 additions and 3 deletions
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Language/Properties.agda
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@ -106,12 +106,40 @@ Trace-↦ʳ {g₁} {g₂} {idx₁} (Trace-edge ρ₁⇒ρ idx₁→idx tr')
(ListMemProp.∈-++⁺ˡ (x∈xs⇒fx∈fxs (Graph.size g₁ ↑ʳ_) idx₁→idx))) (ListMemProp.∈-++⁺ˡ (x∈xs⇒fx∈fxs (Graph.size g₁ ↑ʳ_) idx₁→idx)))
(Trace-↦ʳ {g₁} {g₂} tr') (Trace-↦ʳ {g₁} {g₂} tr')
Trace-loop : ∀ {g₁ : Graph} {idx₁ idx₂ : Graph.Index g₁} {ρ₁ ρ₂ : Env} → Trace-loop : ∀ {g : Graph} {idx₁ idx₂ : Graph.Index g} {ρ₁ ρ₂ : Env} →
Trace {g₁} idx₁ idx₂ ρ₁ ρ₂ → Trace {loop g₁} idx₁ idx₂ ρ₁ ρ₂ Trace {g} idx₁ idx₂ ρ₁ ρ₂ → Trace {loop g} idx₁ idx₂ ρ₁ ρ₂
Trace-loop {idx₁ = idx₁} {idx₁} (Trace-single ρ₁⇒ρ₂) = Trace-single ρ₁⇒ρ₂ Trace-loop {idx₁ = idx₁} {idx₁} (Trace-single ρ₁⇒ρ₂) = Trace-single ρ₁⇒ρ₂
Trace-loop {g₁} {idx₁} (Trace-edge ρ₁⇒ρ idx₁→idx tr') = Trace-loop {g} {idx₁} (Trace-edge ρ₁⇒ρ idx₁→idx tr') =
Trace-edge ρ₁⇒ρ (ListMemProp.∈-++⁺ˡ idx₁→idx) (Trace-loop tr') Trace-edge ρ₁⇒ρ (ListMemProp.∈-++⁺ˡ idx₁→idx) (Trace-loop tr')
EndToEndTrace-loop : ∀ {g : Graph} {ρ₁ ρ₂ : Env} →
EndToEndTrace {g} ρ₁ ρ₂ → EndToEndTrace {loop g} ρ₁ ρ₂
EndToEndTrace-loop etr = record
{ idx₁ = EndToEndTrace.idx₁ etr
; idx₁∈inputs = EndToEndTrace.idx₁∈inputs etr
; idx₂ = EndToEndTrace.idx₂ etr
; idx₂∈outputs = EndToEndTrace.idx₂∈outputs etr
; trace = Trace-loop (EndToEndTrace.trace etr)
}
EndToEndTrace-loop² : ∀ {g : Graph} {ρ₁ ρ₂ ρ₃ : Env} →
EndToEndTrace {loop g} ρ₁ ρ₂ →
EndToEndTrace {loop g} ρ₂ ρ₃ →
EndToEndTrace {loop g} ρ₁ ρ₃
EndToEndTrace-loop² {g} etr₁ etr₂ = record
{ idx₁ = EndToEndTrace.idx₁ etr₁
; idx₁∈inputs = EndToEndTrace.idx₁∈inputs etr₁
; idx₂ = EndToEndTrace.idx₂ etr₂
; idx₂∈outputs = EndToEndTrace.idx₂∈outputs etr₂
; trace =
let
o∈tr₁ = EndToEndTrace.idx₂∈outputs etr₁
i∈tr₂ = EndToEndTrace.idx₁∈inputs etr₂
oi∈es = ListMemProp.∈-++⁺ʳ (Graph.edges g) (∈-cartesianProduct o∈tr₁ i∈tr₂)
in
EndToEndTrace.trace etr₁ ++⟨ oi∈es ⟩ EndToEndTrace.trace etr₂
}
infixr 5 _++_ infixr 5 _++_
_++_ : ∀ {g₁ g₂ : Graph} {ρ₁ ρ₂ ρ₃ : Env} → _++_ : ∀ {g₁ g₂ : Graph} {ρ₁ ρ₂ ρ₃ : Env} →
EndToEndTrace {g₁} ρ₁ ρ₂ → EndToEndTrace {g₂} ρ₂ ρ₃ → EndToEndTrace {g₁} ρ₁ ρ₂ → EndToEndTrace {g₂} ρ₂ ρ₃ →