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Prove that graphs build by buildCfg are sufficient

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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>
This commit is contained in:
Danila Fedorin 2024-04-29 20:57:43 -07:00
parent bbfba34e05
commit c574ca9c56
3 changed files with 72 additions and 57 deletions
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12
Language/Graphs.agda
View File

@ -79,19 +79,13 @@ _↦_ g₁ g₂ = record
} }
loop : Graph → Graph loop : Graph → Graph
loop g = record g loop g = record
{ edges = Graph.edges g List.++
List.cartesianProduct (Graph.outputs g) (Graph.inputs g)
}
optional : Graph → Graph
optional g = record
{ size = 2 Nat.+ Graph.size g { size = 2 Nat.+ Graph.size g
; nodes = [] ∷ [] ∷ Graph.nodes g ; nodes = [] ∷ [] ∷ Graph.nodes g
; edges = (2 ↑ʳᵉ Graph.edges g) List.++ ; edges = (2 ↑ʳᵉ Graph.edges g) List.++
List.map (zero ,_) (2 ↑ʳⁱ Graph.inputs g) List.++ List.map (zero ,_) (2 ↑ʳⁱ Graph.inputs g) List.++
List.map (_, suc zero) (2 ↑ʳⁱ Graph.outputs g) List.++ List.map (_, suc zero) (2 ↑ʳⁱ Graph.outputs g) List.++
((zero , suc zero) ∷ []) ((suc zero , zero) ∷ (zero , suc zero) ∷ [])
; inputs = zero ∷ [] ; inputs = zero ∷ []
; outputs = (suc zero) ∷ [] ; outputs = (suc zero) ∷ []
} }
@ -122,4 +116,4 @@ buildCfg : Stmt → Graph
buildCfg ⟨ bs₁ ⟩ = singleton (bs₁ ∷ []) buildCfg ⟨ bs₁ ⟩ = singleton (bs₁ ∷ [])
buildCfg (s₁ then s₂) = buildCfg s₁ ↦ buildCfg s₂ buildCfg (s₁ then s₂) = buildCfg s₁ ↦ buildCfg s₂
buildCfg (if _ then s₁ else s₂) = singleton [] ↦ (buildCfg s₁ ∙ buildCfg s₂) ↦ singleton [] buildCfg (if _ then s₁ else s₂) = singleton [] ↦ (buildCfg s₁ ∙ buildCfg s₂) ↦ singleton []
buildCfg (while _ repeat s) = optional (loop (buildCfg s)) buildCfg (while _ repeat s) = loop (buildCfg s)

107
Language/Properties.agda
View File

@ -5,12 +5,14 @@ open import Language.Semantics
open import Language.Graphs open import Language.Graphs
open import Language.Traces open import Language.Traces
open import Data.Fin as Fin using (zero) open import Data.Fin as Fin using (suc; zero)
open import Data.List using (List; _∷_; []) open import Data.List as List using (List; _∷_; [])
open import Data.List.Relation.Unary.Any using (here) open import Data.List.Relation.Unary.Any using (here; there)
open import Data.List.Membership.Propositional.Properties as ListMemProp using () open import Data.List.Membership.Propositional.Properties as ListMemProp using ()
open import Data.Product using (Σ; _,_; _×_) open import Data.Product using (Σ; _,_; _×_)
open import Data.Vec as Vec using (_∷_)
open import Data.Vec.Properties using (lookup-++ˡ; ++-identityʳ; lookup-++ʳ) open import Data.Vec.Properties using (lookup-++ˡ; ++-identityʳ; lookup-++ʳ)
open import Function using (_∘_)
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; ∈-cartesianProduct) open import Utils using (x∈xs⇒fx∈fxs; ∈-cartesianProduct)
@ -107,49 +109,63 @@ Trace-↦ʳ {g₁} {g₂} {idx₁} (Trace-edge ρ₁⇒ρ idx₁→idx tr')
(Trace-↦ʳ {g₁} {g₂} tr') (Trace-↦ʳ {g₁} {g₂} tr')
Trace-loop : ∀ {g : Graph} {idx₁ idx₂ : Graph.Index g} {ρ₁ ρ₂ : Env} → Trace-loop : ∀ {g : Graph} {idx₁ idx₂ : Graph.Index g} {ρ₁ ρ₂ : Env} →
Trace {g} idx₁ idx₂ ρ₁ ρ₂ → Trace {loop g} idx₁ idx₂ ρ₁ ρ₂ Trace {g} idx₁ idx₂ ρ₁ ρ₂ → Trace {loop g} (2 Fin.↑ʳ idx₁) (2 Fin.↑ʳ idx₂) ρ₁ ρ₂
Trace-loop {idx₁ = idx₁} {idx₁} (Trace-single ρ₁⇒ρ₂) = Trace-single ρ₁⇒ρ₂ Trace-loop {g} {idx₁} {idx₁} (Trace-single ρ₁⇒ρ₂)
Trace-loop {g} {idx₁} (Trace-edge ρ₁⇒ρ idx₁→idx tr') = rewrite sym (lookup-++ʳ (List.[] ∷ List.[] ∷ Vec.[]) (Graph.nodes g) idx₁) =
Trace-edge ρ₁⇒ρ (ListMemProp.∈-++⁺ˡ idx₁→idx) (Trace-loop tr') Trace-single ρ₁⇒ρ₂
Trace-loop {g} {idx₁} (Trace-edge ρ₁⇒ρ idx₁→idx tr')
rewrite sym (lookup-++ʳ (List.[] ∷ List.[] ∷ Vec.[]) (Graph.nodes g) idx₁) =
Trace-edge ρ₁⇒ρ (ListMemProp.∈-++⁺ˡ (x∈xs⇒fx∈fxs (2 ↑ʳ_) idx₁→idx))
(Trace-loop {g} tr')
EndToEndTrace-loop : ∀ {g : Graph} {ρ₁ ρ₂ : Env} → EndToEndTrace-loop : ∀ {g : Graph} {ρ₁ ρ₂ : Env} →
EndToEndTrace {g} ρ₁ ρ₂ → EndToEndTrace {loop g} ρ₁ ρ₂ EndToEndTrace {g} ρ₁ ρ₂ → EndToEndTrace {loop g} ρ₁ ρ₂
EndToEndTrace-loop etr = record EndToEndTrace-loop {g} etr =
{ idx₁ = EndToEndTrace.idx₁ etr let
; idx₁∈inputs = EndToEndTrace.idx₁∈inputs etr 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))))
; idx₂ = EndToEndTrace.idx₂ etr 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)))))
; idx₂∈outputs = EndToEndTrace.idx₂∈outputs etr in
; trace = Trace-loop (EndToEndTrace.trace etr) record
{ idx₁ = zero
; idx₁∈inputs = here refl
; idx₂ = suc zero
; idx₂∈outputs = here refl
; trace =
Trace-single [] ++⟨ zero→idx₁ ⟩
Trace-loop {g} (EndToEndTrace.trace etr) ++⟨ idx₂→suc ⟩
Trace-single []
} }
EndToEndTrace-loop² : ∀ {g : Graph} {ρ₁ ρ₂ ρ₃ : Env} → EndToEndTrace-loop² : ∀ {g : Graph} {ρ₁ ρ₂ ρ₃ : Env} →
EndToEndTrace {loop g} ρ₁ ρ₂ → EndToEndTrace {loop g} ρ₁ ρ₂ →
EndToEndTrace {loop g} ρ₂ ρ₃ → EndToEndTrace {loop g} ρ₂ ρ₃ →
EndToEndTrace {loop g} ρ₁ ρ₃ EndToEndTrace {loop g} ρ₁ ρ₃
EndToEndTrace-loop² {g} etr₁ etr₂ = record EndToEndTrace-loop² {g} (MkEndToEndTrace zero (here refl) (suc zero) (here refl) tr₁)
{ idx₁ = EndToEndTrace.idx₁ etr₁ (MkEndToEndTrace zero (here refl) (suc zero) (here refl) tr₂) =
; idx₁∈inputs = EndToEndTrace.idx₁∈inputs etr₁
; idx₂ = EndToEndTrace.idx₂ etr₂
; idx₂∈outputs = EndToEndTrace.idx₂∈outputs etr₂
; trace =
let let
o∈tr₁ = EndToEndTrace.idx₂∈outputs etr₁ 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)))
i∈tr₂ = EndToEndTrace.idx₁∈inputs etr₂
oi∈es = ListMemProp.∈-++⁺ʳ (Graph.edges g) (∈-cartesianProduct o∈tr₁ i∈tr₂)
in in
EndToEndTrace.trace etr₁ ++⟨ oi∈es ⟩ EndToEndTrace.trace etr₂ record
{ idx₁ = zero
; idx₁∈inputs = here refl
; idx₂ = suc zero
; idx₂∈outputs = here refl
; trace = tr₁ ++⟨ suc→zero ⟩ tr₂
} }
Trace-optional : ∀ {g : Graph} {idx₁ idx₂ : Graph.Index g} {ρ₁ ρ₂ : Env} → EndToEndTrace-loop⁰ : ∀ {g : Graph} {ρ : Env} →
Trace {g} idx₁ idx₂ ρ₁ ρ₂ → Trace {optional g} (2 Fin.↑ʳ idx₁) (2 Fin.↑ʳ idx₂) ρ₁ ρ₂ EndToEndTrace {loop g} ρ ρ
Trace-optional = {!!} EndToEndTrace-loop⁰ {g} {ρ} =
let
EndToEndTrace-optional : ∀ {g : Graph} {ρ₁ ρ₂ : Env} → 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))))
EndToEndTrace {g} ρ₁ ρ₂ → EndToEndTrace {optional g} ρ₁ ρ₂ in
EndToEndTrace-optional = {!!} record
{ idx₁ = zero
EndToEndTrace-optional-ε : ∀ {g : Graph} {ρ : Env} → EndToEndTrace {optional g} ρ ρ ; idx₁∈inputs = here refl
EndToEndTrace-optional-ε = {!!} ; idx₂ = suc zero
; idx₂∈outputs = here refl
; trace = Trace-single [] ++⟨ zero→suc ⟩ Trace-single []
}
infixr 5 _++_ infixr 5 _++_
_++_ : ∀ {g₁ g₂ : Graph} {ρ₁ ρ₂ ρ₃ : Env} → _++_ : ∀ {g₁ g₂ : Graph} {ρ₁ ρ₂ ρ₃ : Env} →
@ -173,9 +189,9 @@ _++_ {g₁} {g₂} etr₁ etr₂
(Trace-↦ʳ {g₁} {g₂} (EndToEndTrace.trace etr₂)) (Trace-↦ʳ {g₁} {g₂} (EndToEndTrace.trace etr₂))
} }
Trace-singleton : ∀ {bss : List BasicStmt} {ρ₁ ρ₂ : Env} → EndToEndTrace-singleton : ∀ {bss : List BasicStmt} {ρ₁ ρ₂ : Env} →
ρ₁ , bss ⇒ᵇˢ ρ₂ → EndToEndTrace {singleton bss} ρ₁ ρ₂ ρ₁ , bss ⇒ᵇˢ ρ₂ → EndToEndTrace {singleton bss} ρ₁ ρ₂
Trace-singleton ρ₁⇒ρ₂ = record EndToEndTrace-singleton ρ₁⇒ρ₂ = record
{ idx₁ = zero { idx₁ = zero
; idx₁∈inputs = here refl ; idx₁∈inputs = here refl
; idx₂ = zero ; idx₂ = zero
@ -183,23 +199,26 @@ Trace-singleton ρ₁⇒ρ₂ = record
; trace = Trace-single ρ₁⇒ρ₂ ; trace = Trace-single ρ₁⇒ρ₂
} }
Trace-singleton[] : ∀ (ρ : Env) → EndToEndTrace {singleton []} ρ ρ EndToEndTrace-singleton[] : ∀ (ρ : Env) → EndToEndTrace {singleton []} ρ ρ
Trace-singleton[] env = Trace-singleton [] EndToEndTrace-singleton[] env = EndToEndTrace-singleton []
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) =
Trace-singleton (ρ₁,bs⇒ρ₂ ∷ []) EndToEndTrace-singleton (ρ₁,bs⇒ρ₂ ∷ [])
buildCfg-sufficient (⇒ˢ-then ρ₁ ρ₂ ρ₃ s₁ s₂ ρ₁,s₁⇒ρ₂ ρ₂,s₂⇒ρ₃) = 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 buildCfg-sufficient ρ₁,s₁⇒ρ₂ ++ buildCfg-sufficient ρ₂,s₂⇒ρ₃
buildCfg-sufficient (⇒ˢ-if-true ρ₁ ρ₂ _ _ s₁ s₂ _ _ ρ₁,s₁⇒ρ₂) = buildCfg-sufficient (⇒ˢ-if-true ρ₁ ρ₂ _ _ s₁ s₂ _ _ ρ₁,s₁⇒ρ₂) =
Trace-singleton[] ρ₁ ++ EndToEndTrace-singleton[] ρ₁ ++
(EndToEndTrace-∙ˡ (buildCfg-sufficient ρ₁,s₁⇒ρ₂)) ++ (EndToEndTrace-∙ˡ (buildCfg-sufficient ρ₁,s₁⇒ρ₂)) ++
Trace-singleton[] ρ₂ EndToEndTrace-singleton[] ρ₂
buildCfg-sufficient (⇒ˢ-if-false ρ₁ ρ₂ _ s₁ s₂ _ ρ₁,s₂⇒ρ₂) = buildCfg-sufficient (⇒ˢ-if-false ρ₁ ρ₂ _ s₁ s₂ _ ρ₁,s₂⇒ρ₂) =
Trace-singleton[] ρ₁ ++ EndToEndTrace-singleton[] ρ₁ ++
(EndToEndTrace-∙ʳ {buildCfg s₁} (buildCfg-sufficient ρ₁,s₂⇒ρ₂)) ++ (EndToEndTrace-∙ʳ {buildCfg s₁} (buildCfg-sufficient ρ₁,s₂⇒ρ₂)) ++
Trace-singleton[] ρ₂ EndToEndTrace-singleton[] ρ₂
buildCfg-sufficient (⇒ˢ-while-true ρ₁ ρ₂ ρ₃ _ _ s _ _ ρ₁,s⇒ρ₂ ρ₂,ws⇒ρ₃) = {!!} buildCfg-sufficient (⇒ˢ-while-true ρ₁ ρ₂ ρ₃ _ _ s _ _ ρ₁,s⇒ρ₂ ρ₂,ws⇒ρ₃) =
EndToEndTrace-loop² {buildCfg s}
(EndToEndTrace-loop {buildCfg s} (buildCfg-sufficient ρ₁,s⇒ρ₂))
(buildCfg-sufficient ρ₂,ws⇒ρ₃)
buildCfg-sufficient (⇒ˢ-while-false ρ _ s _) = buildCfg-sufficient (⇒ˢ-while-false ρ _ s _) =
EndToEndTrace-optional-ε {loop (buildCfg s)} {ρ} EndToEndTrace-loop⁰ {buildCfg s} {ρ}

2
Language/Traces.agda
View File

@ -17,6 +17,7 @@ module _ {g : Graph} where
ρ₁ , (g [ idx₁ ]) ⇒ᵇˢ ρ₂ → (idx₁ , idx₂) ∈ edges → ρ₁ , (g [ idx₁ ]) ⇒ᵇˢ ρ₂ → (idx₁ , idx₂) ∈ edges →
Trace idx₂ idx₃ ρ₂ ρ₃ → Trace idx₁ idx₃ ρ₁ ρ₃ Trace idx₂ idx₃ ρ₂ ρ₃ → Trace idx₁ idx₃ ρ₁ ρ₃
infixr 5 _++⟨_⟩_
_++⟨_⟩_ : ∀ {idx₁ idx₂ idx₃ idx₄ : Index} {ρ₁ ρ₂ ρ₃ : Env} → _++⟨_⟩_ : ∀ {idx₁ idx₂ idx₃ idx₄ : Index} {ρ₁ ρ₂ ρ₃ : Env} →
Trace idx₁ idx₂ ρ₁ ρ₂ → (idx₂ , idx₃) ∈ edges → Trace idx₁ idx₂ ρ₁ ρ₂ → (idx₂ , idx₃) ∈ edges →
Trace idx₃ idx₄ ρ₂ ρ₃ → Trace idx₁ idx₄ ρ₁ ρ₃ Trace idx₃ idx₄ ρ₂ ρ₃ → Trace idx₁ idx₄ ρ₁ ρ₃
@ -24,6 +25,7 @@ module _ {g : Graph} where
_++⟨_⟩_ (Trace-edge ρ₁⇒ρ₂ idx₁→idx' tr') idx₂→idx₃ tr = Trace-edge ρ₁⇒ρ₂ idx₁→idx' (tr' ++⟨ idx₂→idx₃ ⟩ tr) _++⟨_⟩_ (Trace-edge ρ₁⇒ρ₂ idx₁→idx' tr') idx₂→idx₃ tr = Trace-edge ρ₁⇒ρ₂ idx₁→idx' (tr' ++⟨ idx₂→idx₃ ⟩ tr)
record EndToEndTrace (ρ₁ ρ₂ : Env) : Set where record EndToEndTrace (ρ₁ ρ₂ : Env) : Set where
constructor MkEndToEndTrace
field field
idx₁ : Index idx₁ : Index
idx₁∈inputs : idx₁ ∈ inputs idx₁∈inputs : idx₁ ∈ inputs