Start working on proofs of Graph-related things
Signed-off-by: Danila Fedorin <danila.fedorin@gmail.com>
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@ -1,22 +1,23 @@
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module Language where
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open import Data.Nat using (ℕ; suc; pred)
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open import Data.Nat using (ℕ; suc; pred; _≤_) renaming (_+_ to _+ⁿ_)
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open import Data.Nat.Properties using (m≤n⇒m≤n+o; ≤-reflexive)
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open import Data.Integer using (ℤ; +_) renaming (_+_ to _+ᶻ_; _-_ to _-ᶻ_)
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open import Data.String using (String) renaming (_≟_ to _≟ˢ_)
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open import Data.Product using (_×_; Σ; _,_; proj₁; proj₂)
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open import Data.Vec using (Vec; foldr; lookup; _∷_)
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open import Data.Vec using (Vec; foldr; lookup; _∷_; []; _++_)
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open import Data.List using ([]; _∷_; List) renaming (foldr to foldrˡ; map to mapˡ)
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open import Data.List.Membership.Propositional as MemProp using () renaming (_∈_ to _∈ˡ_)
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open import Data.List.Relation.Unary.All using (All; []; _∷_)
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open import Data.List.Relation.Unary.Any as RelAny using ()
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open import Data.Fin using (Fin; suc; zero; fromℕ; inject₁) renaming (_≟_ to _≟ᶠ_)
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open import Data.Fin using (Fin; suc; zero; fromℕ; inject₁; inject≤; _↑ʳ_) renaming (_≟_ to _≟ᶠ_)
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open import Data.Fin.Properties using (suc-injective)
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open import Relation.Binary.PropositionalEquality using (cong; _≡_; refl)
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open import Relation.Binary.PropositionalEquality using (subst; cong; _≡_; refl)
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open import Relation.Nullary using (¬_)
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open import Function using (_∘_)
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open import Lattice
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open import Utils using (Unique; Unique-map; empty; push; x∈xs⇒fx∈fxs)
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open import Utils using (Unique; Unique-map; empty; push; x∈xs⇒fx∈fxs; _⊗_; _,_)
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data Expr : Set where
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_+_ : Expr → Expr → Expr
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@ -79,6 +80,94 @@ module Semantics where
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ρ , e ⇒ᵉ (↑ᶻ (+ 0)) →
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ρ , (while e repeat s) ⇒ˢ ρ
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module Graphs where
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open Semantics
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record Graph : Set where
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field
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size : ℕ
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Index : Set
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Index = Fin size
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Edge : Set
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Edge = Index × Index
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field
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nodes : Vec (List BasicStmt) size
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edges : List Edge
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_[_] : ∀ (g : Graph) → Graph.Index g → List BasicStmt
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_[_] g idx = lookup (Graph.nodes g) idx
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_⊆_ : Graph → Graph → Set
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_⊆_ g₁ g₂ =
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Σ (Graph.size g₁ ≤ Graph.size g₂) (λ n₁≤n₂ →
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( ∀ (idx : Graph.Index g₁) → g₁ [ idx ] ≡ g₂ [ inject≤ idx n₁≤n₂ ]
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× ∀ (idx₁ idx₂ : Graph.Index g₁) → (idx₁ , idx₂) ∈ˡ (Graph.edges g₁) →
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(inject≤ idx₁ n₁≤n₂ , inject≤ idx₂ n₁≤n₂) ∈ˡ (Graph.edges g₂)
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))
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-- Note: inject≤ doesn't seem to work as nicely with vector lookups.
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-- The ↑ˡ and ↑ʳ operators are way nicer. Can we reformulate the
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-- ⊆ property to use them?
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n≤n+m : ∀ (n m : ℕ) → n ≤ n +ⁿ m
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n≤n+m n m = m≤n⇒m≤n+o m (≤-reflexive (refl {x = n}))
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lookup-++ˡ : ∀ {a} {A : Set a} {n m : ℕ} (xs : Vec A n) (ys : Vec A m)
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(idx : Fin n) → lookup xs idx ≡ lookup (xs ++ ys) (inject≤ idx (n≤n+m n m))
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lookup-++ˡ = {!!}
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pushBasicBlock : List BasicStmt → (g₁ : Graph) → Σ Graph (λ g₂ → Graph.Index g₂ × g₁ ⊆ g₂)
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pushBasicBlock bss g₁ =
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let
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size' = Graph.size g₁ +ⁿ 1
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size≤size' = n≤n+m (Graph.size g₁) 1
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inject-Edge = λ (idx₁ , idx₂) → (inject≤ idx₁ size≤size' , inject≤ idx₂ size≤size')
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in
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( record
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{ size = size'
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; nodes = Graph.nodes g₁ ++ (bss ∷ [])
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; edges = mapˡ inject-Edge (Graph.edges g₁)
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}
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, ( (Graph.size g₁) ↑ʳ zero
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, ( size≤size'
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, λ idx → lookup-++ˡ (Graph.nodes g₁) (bss ∷ []) idx
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, λ idx₁ idx₂ e∈es → x∈xs⇒fx∈fxs inject-Edge e∈es
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)
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)
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)
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record Relaxable (T : Graph → Set) : Set where
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field relax : ∀ {g₁ g₂ : Graph} → g₁ ⊆ g₂ → T g₁ → T g₂
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instance
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IndexRelaxable : Relaxable Graph.Index
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IndexRelaxable = record
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{ relax = λ g₁⊆g₂ idx → inject≤ idx (proj₁ g₁⊆g₂)
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}
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EdgeRelaxable : Relaxable Graph.Edge
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EdgeRelaxable = record
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{ relax = λ {g₁} {g₂} g₁⊆g₂ (idx₁ , idx₂) →
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( Relaxable.relax IndexRelaxable {g₁} {g₂} g₁⊆g₂ idx₁
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, Relaxable.relax IndexRelaxable {g₁} {g₂} g₁⊆g₂ idx₂
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)
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}
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ProdRelaxable : ∀ {P : Graph → Set} {Q : Graph → Set} →
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{{ PRelaxable : Relaxable P }} → {{ QRelaxable : Relaxable Q }} →
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Relaxable (P ⊗ Q)
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ProdRelaxable {{pr}} {{qr}} = record
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{ relax = (λ { g₁⊆g₂ (p , q) →
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( Relaxable.relax pr g₁⊆g₂ p
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, Relaxable.relax qr g₁⊆g₂ q) }
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)
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}
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open Relaxable {{...}} public
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open import Lattice.MapSet _≟ˢ_
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renaming
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( MapSet to StringSet
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@ -68,3 +68,6 @@ data Pairwise {a} {b} {c} {A : Set a} {B : Set b} (P : A → B → Set c) : List
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_∷_ : ∀ {x : A} {y : B} {xs : List A} {ys : List B} →
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P x y → Pairwise P xs ys →
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Pairwise P (x ∷ xs) (y ∷ ys)
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data _⊗_ {a p q} {A : Set a} (P : A → Set p) (Q : A → Set q) : A → Set (a ⊔ℓ p ⊔ℓ q) where
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_,_ : ∀ {val : A} → P val → Q val → (P ⊗ Q) val
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