Prove that graphs build by buildCfg are sufficient
That is, if we have a (semantic) trace, we can find a corresponding path through the CFG. Signed-off-by: Danila Fedorin <danila.fedorin@gmail.com>
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@ -79,19 +79,13 @@ _↦_ g₁ g₂ = record
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}
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loop : Graph → Graph
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loop g = record g
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{ edges = Graph.edges g List.++
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List.cartesianProduct (Graph.outputs g) (Graph.inputs g)
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}
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optional : Graph → Graph
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optional g = record
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loop g = record
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{ size = 2 Nat.+ Graph.size g
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; nodes = [] ∷ [] ∷ Graph.nodes g
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; edges = (2 ↑ʳᵉ Graph.edges g) List.++
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List.map (zero ,_) (2 ↑ʳⁱ Graph.inputs g) List.++
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List.map (_, suc zero) (2 ↑ʳⁱ Graph.outputs g) List.++
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((zero , suc zero) ∷ [])
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((suc zero , zero) ∷ (zero , suc zero) ∷ [])
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; inputs = zero ∷ []
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; outputs = (suc zero) ∷ []
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}
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@ -122,4 +116,4 @@ buildCfg : Stmt → Graph
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buildCfg ⟨ bs₁ ⟩ = singleton (bs₁ ∷ [])
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buildCfg (s₁ then s₂) = buildCfg s₁ ↦ buildCfg s₂
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buildCfg (if _ then s₁ else s₂) = singleton [] ↦ (buildCfg s₁ ∙ buildCfg s₂) ↦ singleton []
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buildCfg (while _ repeat s) = optional (loop (buildCfg s))
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buildCfg (while _ repeat s) = loop (buildCfg s)
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@ -5,12 +5,14 @@ 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 (List; _∷_; [])
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open import Data.List.Relation.Unary.Any using (here)
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open import Data.Fin as Fin using (suc; zero)
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open import Data.List as List using (List; _∷_; [])
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open import Data.List.Relation.Unary.Any using (here; there)
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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 as Vec using (_∷_)
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open import Data.Vec.Properties using (lookup-++ˡ; ++-identityʳ; lookup-++ʳ)
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open import Function using (_∘_)
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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; ∈-cartesianProduct)
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@ -107,49 +109,63 @@ Trace-↦ʳ {g₁} {g₂} {idx₁} (Trace-edge ρ₁⇒ρ idx₁→idx tr')
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(Trace-↦ʳ {g₁} {g₂} tr')
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Trace-loop : ∀ {g : Graph} {idx₁ idx₂ : Graph.Index g} {ρ₁ ρ₂ : Env} →
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Trace {g} idx₁ idx₂ ρ₁ ρ₂ → Trace {loop g} idx₁ idx₂ ρ₁ ρ₂
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Trace-loop {idx₁ = idx₁} {idx₁} (Trace-single ρ₁⇒ρ₂) = Trace-single ρ₁⇒ρ₂
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Trace-loop {g} {idx₁} (Trace-edge ρ₁⇒ρ idx₁→idx tr') =
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Trace-edge ρ₁⇒ρ (ListMemProp.∈-++⁺ˡ idx₁→idx) (Trace-loop tr')
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Trace {g} idx₁ idx₂ ρ₁ ρ₂ → Trace {loop g} (2 Fin.↑ʳ idx₁) (2 Fin.↑ʳ idx₂) ρ₁ ρ₂
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Trace-loop {g} {idx₁} {idx₁} (Trace-single ρ₁⇒ρ₂)
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rewrite sym (lookup-++ʳ (List.[] ∷ List.[] ∷ Vec.[]) (Graph.nodes g) idx₁) =
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Trace-single ρ₁⇒ρ₂
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Trace-loop {g} {idx₁} (Trace-edge ρ₁⇒ρ idx₁→idx tr')
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rewrite sym (lookup-++ʳ (List.[] ∷ List.[] ∷ Vec.[]) (Graph.nodes g) idx₁) =
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Trace-edge ρ₁⇒ρ (ListMemProp.∈-++⁺ˡ (x∈xs⇒fx∈fxs (2 ↑ʳ_) idx₁→idx))
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(Trace-loop {g} tr')
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EndToEndTrace-loop : ∀ {g : Graph} {ρ₁ ρ₂ : Env} →
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EndToEndTrace {g} ρ₁ ρ₂ → EndToEndTrace {loop g} ρ₁ ρ₂
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EndToEndTrace-loop etr = record
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{ idx₁ = EndToEndTrace.idx₁ etr
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; idx₁∈inputs = EndToEndTrace.idx₁∈inputs etr
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; idx₂ = EndToEndTrace.idx₂ etr
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; idx₂∈outputs = EndToEndTrace.idx₂∈outputs etr
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; trace = Trace-loop (EndToEndTrace.trace etr)
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}
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EndToEndTrace-loop {g} etr =
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let
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zero→idx₁ = ListMemProp.∈-++⁺ʳ (2 ↑ʳᵉ Graph.edges g) (ListMemProp.∈-++⁺ˡ (x∈xs⇒fx∈fxs (zero ,_) (x∈xs⇒fx∈fxs (2 Fin.↑ʳ_) (EndToEndTrace.idx₁∈inputs etr))))
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idx₂→suc = ListMemProp.∈-++⁺ʳ (2 ↑ʳᵉ Graph.edges g) (ListMemProp.∈-++⁺ʳ (List.map (zero ,_) (2 ↑ʳⁱ Graph.inputs g)) (ListMemProp.∈-++⁺ˡ (x∈xs⇒fx∈fxs (_, suc zero) (x∈xs⇒fx∈fxs (2 Fin.↑ʳ_) (EndToEndTrace.idx₂∈outputs etr)))))
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in
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record
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{ idx₁ = zero
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; idx₁∈inputs = here refl
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; idx₂ = suc zero
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; idx₂∈outputs = here refl
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; trace =
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Trace-single [] ++⟨ zero→idx₁ ⟩
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Trace-loop {g} (EndToEndTrace.trace etr) ++⟨ idx₂→suc ⟩
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Trace-single []
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}
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EndToEndTrace-loop² : ∀ {g : Graph} {ρ₁ ρ₂ ρ₃ : Env} →
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EndToEndTrace {loop g} ρ₁ ρ₂ →
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EndToEndTrace {loop g} ρ₂ ρ₃ →
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EndToEndTrace {loop g} ρ₁ ρ₃
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EndToEndTrace-loop² {g} etr₁ etr₂ = record
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{ idx₁ = EndToEndTrace.idx₁ etr₁
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; idx₁∈inputs = EndToEndTrace.idx₁∈inputs etr₁
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; idx₂ = EndToEndTrace.idx₂ etr₂
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; idx₂∈outputs = EndToEndTrace.idx₂∈outputs etr₂
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; trace =
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let
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o∈tr₁ = EndToEndTrace.idx₂∈outputs etr₁
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i∈tr₂ = EndToEndTrace.idx₁∈inputs etr₂
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oi∈es = ListMemProp.∈-++⁺ʳ (Graph.edges g) (∈-cartesianProduct o∈tr₁ i∈tr₂)
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in
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EndToEndTrace.trace etr₁ ++⟨ oi∈es ⟩ EndToEndTrace.trace etr₂
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}
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EndToEndTrace-loop² {g} (MkEndToEndTrace zero (here refl) (suc zero) (here refl) tr₁)
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(MkEndToEndTrace zero (here refl) (suc zero) (here refl) tr₂) =
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let
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suc→zero = ListMemProp.∈-++⁺ʳ (2 ↑ʳᵉ Graph.edges g) (ListMemProp.∈-++⁺ʳ (List.map (zero ,_) (2 ↑ʳⁱ Graph.inputs g)) (ListMemProp.∈-++⁺ʳ (List.map (_, suc zero) (2 ↑ʳⁱ Graph.outputs g)) (here refl)))
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in
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record
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{ idx₁ = zero
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; idx₁∈inputs = here refl
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; idx₂ = suc zero
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; idx₂∈outputs = here refl
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; trace = tr₁ ++⟨ suc→zero ⟩ tr₂
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}
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Trace-optional : ∀ {g : Graph} {idx₁ idx₂ : Graph.Index g} {ρ₁ ρ₂ : Env} →
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Trace {g} idx₁ idx₂ ρ₁ ρ₂ → Trace {optional g} (2 Fin.↑ʳ idx₁) (2 Fin.↑ʳ idx₂) ρ₁ ρ₂
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Trace-optional = {!!}
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EndToEndTrace-optional : ∀ {g : Graph} {ρ₁ ρ₂ : Env} →
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EndToEndTrace {g} ρ₁ ρ₂ → EndToEndTrace {optional g} ρ₁ ρ₂
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EndToEndTrace-optional = {!!}
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EndToEndTrace-optional-ε : ∀ {g : Graph} {ρ : Env} → EndToEndTrace {optional g} ρ ρ
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EndToEndTrace-optional-ε = {!!}
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EndToEndTrace-loop⁰ : ∀ {g : Graph} {ρ : Env} →
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EndToEndTrace {loop g} ρ ρ
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EndToEndTrace-loop⁰ {g} {ρ} =
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let
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zero→suc = ListMemProp.∈-++⁺ʳ (2 ↑ʳᵉ Graph.edges g) (ListMemProp.∈-++⁺ʳ (List.map (zero ,_) (2 ↑ʳⁱ Graph.inputs g)) (ListMemProp.∈-++⁺ʳ (List.map (_, suc zero) (2 ↑ʳⁱ Graph.outputs g)) (there (here refl))))
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in
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record
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{ idx₁ = zero
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; idx₁∈inputs = here refl
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; idx₂ = suc zero
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; idx₂∈outputs = here refl
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; trace = Trace-single [] ++⟨ zero→suc ⟩ Trace-single []
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}
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infixr 5 _++_
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_++_ : ∀ {g₁ g₂ : Graph} {ρ₁ ρ₂ ρ₃ : Env} →
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@ -173,9 +189,9 @@ _++_ {g₁} {g₂} etr₁ etr₂
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(Trace-↦ʳ {g₁} {g₂} (EndToEndTrace.trace etr₂))
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}
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Trace-singleton : ∀ {bss : List BasicStmt} {ρ₁ ρ₂ : Env} →
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EndToEndTrace-singleton : ∀ {bss : List BasicStmt} {ρ₁ ρ₂ : Env} →
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ρ₁ , bss ⇒ᵇˢ ρ₂ → EndToEndTrace {singleton bss} ρ₁ ρ₂
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Trace-singleton ρ₁⇒ρ₂ = record
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EndToEndTrace-singleton ρ₁⇒ρ₂ = record
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{ idx₁ = zero
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; idx₁∈inputs = here refl
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; idx₂ = zero
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@ -183,23 +199,26 @@ Trace-singleton ρ₁⇒ρ₂ = record
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; trace = Trace-single ρ₁⇒ρ₂
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}
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Trace-singleton[] : ∀ (ρ : Env) → EndToEndTrace {singleton []} ρ ρ
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Trace-singleton[] env = Trace-singleton []
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EndToEndTrace-singleton[] : ∀ (ρ : Env) → EndToEndTrace {singleton []} ρ ρ
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EndToEndTrace-singleton[] env = EndToEndTrace-singleton []
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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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Trace-singleton (ρ₁,bs⇒ρ₂ ∷ [])
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EndToEndTrace-singleton (ρ₁,bs⇒ρ₂ ∷ [])
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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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buildCfg-sufficient (⇒ˢ-if-true ρ₁ ρ₂ _ _ s₁ s₂ _ _ ρ₁,s₁⇒ρ₂) =
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Trace-singleton[] ρ₁ ++
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EndToEndTrace-singleton[] ρ₁ ++
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(EndToEndTrace-∙ˡ (buildCfg-sufficient ρ₁,s₁⇒ρ₂)) ++
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Trace-singleton[] ρ₂
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EndToEndTrace-singleton[] ρ₂
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buildCfg-sufficient (⇒ˢ-if-false ρ₁ ρ₂ _ s₁ s₂ _ ρ₁,s₂⇒ρ₂) =
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Trace-singleton[] ρ₁ ++
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EndToEndTrace-singleton[] ρ₁ ++
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(EndToEndTrace-∙ʳ {buildCfg s₁} (buildCfg-sufficient ρ₁,s₂⇒ρ₂)) ++
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Trace-singleton[] ρ₂
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buildCfg-sufficient (⇒ˢ-while-true ρ₁ ρ₂ ρ₃ _ _ s _ _ ρ₁,s⇒ρ₂ ρ₂,ws⇒ρ₃) = {!!}
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EndToEndTrace-singleton[] ρ₂
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buildCfg-sufficient (⇒ˢ-while-true ρ₁ ρ₂ ρ₃ _ _ s _ _ ρ₁,s⇒ρ₂ ρ₂,ws⇒ρ₃) =
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EndToEndTrace-loop² {buildCfg s}
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(EndToEndTrace-loop {buildCfg s} (buildCfg-sufficient ρ₁,s⇒ρ₂))
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(buildCfg-sufficient ρ₂,ws⇒ρ₃)
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buildCfg-sufficient (⇒ˢ-while-false ρ _ s _) =
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EndToEndTrace-optional-ε {loop (buildCfg s)} {ρ}
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EndToEndTrace-loop⁰ {buildCfg s} {ρ}
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@ -17,6 +17,7 @@ module _ {g : Graph} where
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ρ₁ , (g [ idx₁ ]) ⇒ᵇˢ ρ₂ → (idx₁ , idx₂) ∈ edges →
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Trace idx₂ idx₃ ρ₂ ρ₃ → Trace idx₁ idx₃ ρ₁ ρ₃
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infixr 5 _++⟨_⟩_
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_++⟨_⟩_ : ∀ {idx₁ idx₂ idx₃ idx₄ : Index} {ρ₁ ρ₂ ρ₃ : Env} →
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Trace idx₁ idx₂ ρ₁ ρ₂ → (idx₂ , idx₃) ∈ edges →
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Trace idx₃ idx₄ ρ₂ ρ₃ → Trace idx₁ idx₄ ρ₁ ρ₃
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@ -24,6 +25,7 @@ module _ {g : Graph} where
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_++⟨_⟩_ (Trace-edge ρ₁⇒ρ₂ idx₁→idx' tr') idx₂→idx₃ tr = Trace-edge ρ₁⇒ρ₂ idx₁→idx' (tr' ++⟨ idx₂→idx₃ ⟩ tr)
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record EndToEndTrace (ρ₁ ρ₂ : Env) : Set where
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constructor MkEndToEndTrace
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field
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idx₁ : Index
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idx₁∈inputs : idx₁ ∈ inputs
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