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Simplify proofs about 'loop' using concatenation lemma

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Signed-off-by: Danila Fedorin <danila.fedorin@gmail.com>
This commit is contained in:
Danila Fedorin 2024-04-29 21:28:21 -07:00
parent c574ca9c56
commit 91b5d108f6
2 changed files with 33 additions and 6 deletions
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32
Language/Properties.agda
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@ -8,6 +8,7 @@ open import Language.Traces
open import Data.Fin as Fin using (suc; zero)
open import Data.List as List using (List; _∷_; [])
open import Data.List.Relation.Unary.Any using (here; there)
open import Data.List.Membership.Propositional as ListMem using ()
open import Data.List.Membership.Propositional.Properties as ListMemProp using ()
open import Data.Product using (Σ; _,_; _×_)
open import Data.Vec as Vec using (_∷_)
@ -15,7 +16,7 @@ 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 Utils using (x∈xs⇒fx∈fxs; ∈-cartesianProduct)
open import Utils using (x∈xs⇒fx∈fxs; ∈-cartesianProduct; concat-∈)
buildCfg-input : ∀ (s : Stmt) → let g = buildCfg s in Σ (Graph.Index g) (λ idx → Graph.inputs g ≡ idx ∷ [])
@ -108,6 +109,19 @@ Trace-↦ʳ {g₁} {g₂} {idx₁} (Trace-edge ρ₁⇒ρ idx₁→idx tr')
(ListMemProp.∈-++⁺ˡ (x∈xs⇒fx∈fxs (Graph.size g₁ ↑ʳ_) idx₁→idx)))
(Trace-↦ʳ {g₁} {g₂} tr')
loop-edge-groups : ∀ (g : Graph) → List (List (Graph.Edge (loop g)))
loop-edge-groups g =
(2 ↑ʳᵉ Graph.edges g) ∷
(List.map (zero ,_) (2 ↑ʳⁱ Graph.inputs g)) ∷
(List.map (_, suc zero) (2 ↑ʳⁱ Graph.outputs g)) ∷
((suc zero , zero) ∷ (zero , suc zero) ∷ []) ∷
[]
loop-edge-help : ∀ (g : Graph) {l : List (Graph.Edge (loop g))} {e : Graph.Edge (loop g)} →
e ListMem.∈ l → l ListMem.∈ loop-edge-groups g →
e ListMem.∈ Graph.edges (loop g)
loop-edge-help g e∈l l∈ess = concat-∈ e∈l l∈ess
Trace-loop : ∀ {g : Graph} {idx₁ idx₂ : Graph.Index g} {ρ₁ ρ₂ : Env} →
Trace {g} idx₁ idx₂ ρ₁ ρ₂ → Trace {loop g} (2 Fin.↑ʳ idx₁) (2 Fin.↑ʳ idx₂) ρ₁ ρ₂
Trace-loop {g} {idx₁} {idx₁} (Trace-single ρ₁⇒ρ₂)
@ -122,8 +136,14 @@ EndToEndTrace-loop : ∀ {g : Graph} {ρ₁ ρ₂ : Env} →
EndToEndTrace {g} ρ₁ ρ₂ → EndToEndTrace {loop g} ρ₁ ρ₂
EndToEndTrace-loop {g} etr =
let
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→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)))))
zero→idx₁' = x∈xs⇒fx∈fxs (zero ,_)
(x∈xs⇒fx∈fxs (2 Fin.↑ʳ_)
(EndToEndTrace.idx₁∈inputs etr))
zero→idx₁ = loop-edge-help g zero→idx₁' (there (here refl))
idx₂→suc' = x∈xs⇒fx∈fxs (_, suc zero)
(x∈xs⇒fx∈fxs (2 Fin.↑ʳ_)
(EndToEndTrace.idx₂∈outputs etr))
idx₂→suc = loop-edge-help g idx₂→suc' (there (there (here refl)))
in
record
{ idx₁ = zero
@ -143,7 +163,8 @@ EndToEndTrace-loop² : ∀ {g : Graph} {ρ₁ ρ₂ ρ₃ : Env} →
EndToEndTrace-loop² {g} (MkEndToEndTrace zero (here refl) (suc zero) (here refl) tr₁)
(MkEndToEndTrace zero (here refl) (suc zero) (here refl) tr₂) =
let
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)))
suc→zero = loop-edge-help g (here refl)
(there (there (there (here refl))))
in
record
{ idx₁ = zero
@ -157,7 +178,8 @@ EndToEndTrace-loop⁰ : ∀ {g : Graph} {ρ : Env} →
EndToEndTrace {loop g} ρ ρ
EndToEndTrace-loop⁰ {g} {ρ} =
let
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))))
zero→suc = loop-edge-help g (there (here refl))
(there (there (there (here refl))))
in
record
{ idx₁ = zero

7
Utils.agda
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@ -3,7 +3,7 @@ module Utils where
open import Agda.Primitive using () renaming (_⊔_ to _⊔ℓ_)
open import Data.Product as Prod using ()
open import Data.Nat using (ℕ; suc)
open import Data.List using (List; cartesianProduct; []; _∷_; _++_) renaming (map to mapˡ)
open import Data.List using (List; cartesianProduct; []; _∷_; _++_; foldr) renaming (map to mapˡ)
open import Data.List.Membership.Propositional using (_∈_)
open import Data.List.Membership.Propositional.Properties as ListMemProp using ()
open import Data.List.Relation.Unary.All using (All; []; _∷_; map)
@ -86,3 +86,8 @@ proj₂ (_ , v) = v
x ∈ xs → y ∈ ys → (x Prod., y) ∈ cartesianProduct xs ys
∈-cartesianProduct {x = x} (here refl) y∈ys = ListMemProp.∈-++⁺ˡ (x∈xs⇒fx∈fxs (x Prod.,_) y∈ys)
∈-cartesianProduct {x = x} {xs = x' ∷ _} {ys = ys} (there x∈rest) y∈ys = ListMemProp.∈-++⁺ʳ (mapˡ (x' Prod.,_) ys) (∈-cartesianProduct x∈rest y∈ys)
concat-∈ : ∀ {a} {A : Set a} {x : A} {l : List A} {ls : List (List A)} →
x ∈ l → l ∈ ls → x ∈ foldr _++_ [] ls
concat-∈ x∈l (here refl) = ListMemProp.∈-++⁺ˡ x∈l
concat-∈ {ls = l' ∷ ls'} x∈l (there l∈ls') = ListMemProp.∈-++⁺ʳ l' (concat-∈ x∈l l∈ls')