module Language where open import Data.Nat using (ℕ; suc; pred; _≤_) renaming (_+_ to _+ⁿ_) open import Data.Nat.Properties using (m≤n⇒m≤n+o; ≤-reflexive) open import Data.Integer using (ℤ; +_) renaming (_+_ to _+ᶻ_; _-_ to _-ᶻ_) open import Data.String using (String) renaming (_≟_ to _≟ˢ_) open import Data.Product using (_×_; Σ; _,_; proj₁; proj₂) open import Data.Vec using (Vec; foldr; lookup; _∷_; []; _++_) open import Data.List using ([]; _∷_; List) renaming (foldr to foldrˡ; map to mapˡ) open import Data.List.Membership.Propositional as MemProp using () renaming (_∈_ to _∈ˡ_) open import Data.List.Relation.Unary.All using (All; []; _∷_) open import Data.List.Relation.Unary.Any as RelAny using () open import Data.Fin using (Fin; suc; zero; fromℕ; inject₁; inject≤; _↑ʳ_) renaming (_≟_ to _≟ᶠ_) open import Data.Fin.Properties using (suc-injective) open import Relation.Binary.PropositionalEquality using (subst; cong; _≡_; refl) open import Relation.Nullary using (¬_) open import Function using (_∘_) open import Lattice open import Utils using (Unique; Unique-map; empty; push; x∈xs⇒fx∈fxs; _⊗_; _,_) data Expr : Set where _+_ : Expr → Expr → Expr _-_ : Expr → Expr → Expr `_ : String → Expr #_ : ℕ → Expr data BasicStmt : Set where _←_ : String → Expr → BasicStmt noop : BasicStmt data Stmt : Set where ⟨_⟩ : BasicStmt → Stmt _then_ : Stmt → Stmt → Stmt if_then_else_ : Expr → Stmt → Stmt → Stmt while_repeat_ : Expr → Stmt → Stmt module Semantics where data Value : Set where ↑ᶻ : ℤ → Value Env : Set Env = List (String × Value) data _∈_ : (String × Value) → Env → Set where here : ∀ (s : String) (v : Value) (ρ : Env) → (s , v) ∈ ((s , v) ∷ ρ) there : ∀ (s s' : String) (v v' : Value) (ρ : Env) → ¬ (s ≡ s') → (s , v) ∈ ρ → (s , v) ∈ ((s' , v') ∷ ρ) data _,_⇒ᵉ_ : Env → Expr → Value → Set where ⇒ᵉ-ℕ : ∀ (ρ : Env) (n : ℕ) → ρ , (# n) ⇒ᵉ (↑ᶻ (+ n)) ⇒ᵉ-Var : ∀ (ρ : Env) (x : String) (v : Value) → (x , v) ∈ ρ → ρ , (` x) ⇒ᵉ v ⇒ᵉ-+ : ∀ (ρ : Env) (e₁ e₂ : Expr) (z₁ z₂ : ℤ) → ρ , e₁ ⇒ᵉ (↑ᶻ z₁) → ρ , e₂ ⇒ᵉ (↑ᶻ z₂) → ρ , (e₁ + e₂) ⇒ᵉ (↑ᶻ (z₁ +ᶻ z₂)) ⇒ᵉ-- : ∀ (ρ : Env) (e₁ e₂ : Expr) (z₁ z₂ : ℤ) → ρ , e₁ ⇒ᵉ (↑ᶻ z₁) → ρ , e₂ ⇒ᵉ (↑ᶻ z₂) → ρ , (e₁ - e₂) ⇒ᵉ (↑ᶻ (z₁ -ᶻ z₂)) data _,_⇒ᵇ_ : Env → BasicStmt → Env → Set where ⇒ᵇ-noop : ∀ (ρ : Env) → ρ , noop ⇒ᵇ ρ ⇒ᵇ-← : ∀ (ρ : Env) (x : String) (e : Expr) (v : Value) → ρ , e ⇒ᵉ v → ρ , (x ← e) ⇒ᵇ ((x , v) ∷ ρ) data _,_⇒ˢ_ : Env → Stmt → Env → Set where ⇒ˢ-⟨⟩ : ∀ (ρ₁ ρ₂ : Env) (bs : BasicStmt) → ρ₁ , bs ⇒ᵇ ρ₂ → ρ₁ , ⟨ bs ⟩ ⇒ˢ ρ₂ ⇒ˢ-then : ∀ (ρ₁ ρ₂ ρ₃ : Env) (s₁ s₂ : Stmt) → ρ₁ , s₁ ⇒ˢ ρ₂ → ρ₂ , s₂ ⇒ˢ ρ₃ → ρ₁ , (s₁ then s₂) ⇒ˢ ρ₃ ⇒ˢ-if-true : ∀ (ρ₁ ρ₂ : Env) (e : Expr) (z : ℤ) (s₁ s₂ : Stmt) → ρ₁ , e ⇒ᵉ (↑ᶻ z) → ¬ z ≡ (+ 0) → ρ₁ , s₁ ⇒ˢ ρ₂ → ρ₁ , (if e then s₁ else s₂) ⇒ˢ ρ₂ ⇒ˢ-if-false : ∀ (ρ₁ ρ₂ : Env) (e : Expr) (s₁ s₂ : Stmt) → ρ₁ , e ⇒ᵉ (↑ᶻ (+ 0)) → ρ₁ , s₂ ⇒ˢ ρ₂ → ρ₁ , (if e then s₁ else s₂) ⇒ˢ ρ₂ ⇒ˢ-while-true : ∀ (ρ₁ ρ₂ ρ₃ : Env) (e : Expr) (z : ℤ) (s : Stmt) → ρ₁ , e ⇒ᵉ (↑ᶻ z) → ¬ z ≡ (+ 0) → ρ₁ , s ⇒ˢ ρ₂ → ρ₂ , (while e repeat s) ⇒ˢ ρ₃ → ρ₁ , (while e repeat s) ⇒ˢ ρ₃ ⇒ˢ-while-false : ∀ (ρ : Env) (e : Expr) (s : Stmt) → ρ , e ⇒ᵉ (↑ᶻ (+ 0)) → ρ , (while e repeat s) ⇒ˢ ρ module Graphs where open Semantics record Graph : Set where field size : ℕ Index : Set Index = Fin size Edge : Set Edge = Index × Index field nodes : Vec (List BasicStmt) size edges : List Edge _[_] : ∀ (g : Graph) → Graph.Index g → List BasicStmt _[_] g idx = lookup (Graph.nodes g) idx _⊆_ : Graph → Graph → Set _⊆_ g₁ g₂ = Σ (Graph.size g₁ ≤ Graph.size g₂) (λ n₁≤n₂ → ( ∀ (idx : Graph.Index g₁) → g₁ [ idx ] ≡ g₂ [ inject≤ idx n₁≤n₂ ] × ∀ (idx₁ idx₂ : Graph.Index g₁) → (idx₁ , idx₂) ∈ˡ (Graph.edges g₁) → (inject≤ idx₁ n₁≤n₂ , inject≤ idx₂ n₁≤n₂) ∈ˡ (Graph.edges g₂) )) -- Note: inject≤ doesn't seem to work as nicely with vector lookups. -- The ↑ˡ and ↑ʳ operators are way nicer. Can we reformulate the -- ⊆ property to use them? n≤n+m : ∀ (n m : ℕ) → n ≤ n +ⁿ m n≤n+m n m = m≤n⇒m≤n+o m (≤-reflexive (refl {x = n})) lookup-++ˡ : ∀ {a} {A : Set a} {n m : ℕ} (xs : Vec A n) (ys : Vec A m) (idx : Fin n) → lookup xs idx ≡ lookup (xs ++ ys) (inject≤ idx (n≤n+m n m)) lookup-++ˡ = {!!} pushBasicBlock : List BasicStmt → (g₁ : Graph) → Σ Graph (λ g₂ → Graph.Index g₂ × g₁ ⊆ g₂) pushBasicBlock bss g₁ = let size' = Graph.size g₁ +ⁿ 1 size≤size' = n≤n+m (Graph.size g₁) 1 inject-Edge = λ (idx₁ , idx₂) → (inject≤ idx₁ size≤size' , inject≤ idx₂ size≤size') in ( record { size = size' ; nodes = Graph.nodes g₁ ++ (bss ∷ []) ; edges = mapˡ inject-Edge (Graph.edges g₁) } , ( (Graph.size g₁) ↑ʳ zero , ( size≤size' , λ idx → lookup-++ˡ (Graph.nodes g₁) (bss ∷ []) idx , λ idx₁ idx₂ e∈es → x∈xs⇒fx∈fxs inject-Edge e∈es ) ) ) record Relaxable (T : Graph → Set) : Set where field relax : ∀ {g₁ g₂ : Graph} → g₁ ⊆ g₂ → T g₁ → T g₂ instance IndexRelaxable : Relaxable Graph.Index IndexRelaxable = record { relax = λ g₁⊆g₂ idx → inject≤ idx (proj₁ g₁⊆g₂) } EdgeRelaxable : Relaxable Graph.Edge EdgeRelaxable = record { relax = λ {g₁} {g₂} g₁⊆g₂ (idx₁ , idx₂) → ( Relaxable.relax IndexRelaxable {g₁} {g₂} g₁⊆g₂ idx₁ , Relaxable.relax IndexRelaxable {g₁} {g₂} g₁⊆g₂ idx₂ ) } ProdRelaxable : ∀ {P : Graph → Set} {Q : Graph → Set} → {{ PRelaxable : Relaxable P }} → {{ QRelaxable : Relaxable Q }} → Relaxable (P ⊗ Q) ProdRelaxable {{pr}} {{qr}} = record { relax = (λ { g₁⊆g₂ (p , q) → ( Relaxable.relax pr g₁⊆g₂ p , Relaxable.relax qr g₁⊆g₂ q) } ) } open Relaxable {{...}} public open import Lattice.MapSet _≟ˢ_ renaming ( MapSet to StringSet ; insert to insertˢ ; to-List to to-Listˢ ; empty to emptyˢ ; singleton to singletonˢ ; _⊔_ to _⊔ˢ_ ; `_ to `ˢ_ ; _∈_ to _∈ˢ_ ; ⊔-preserves-∈k₁ to ⊔ˢ-preserves-∈k₁ ; ⊔-preserves-∈k₂ to ⊔ˢ-preserves-∈k₂ ) data _∈ᵉ_ : String → Expr → Set where in⁺₁ : ∀ {e₁ e₂ : Expr} {k : String} → k ∈ᵉ e₁ → k ∈ᵉ (e₁ + e₂) in⁺₂ : ∀ {e₁ e₂ : Expr} {k : String} → k ∈ᵉ e₂ → k ∈ᵉ (e₁ + e₂) in⁻₁ : ∀ {e₁ e₂ : Expr} {k : String} → k ∈ᵉ e₁ → k ∈ᵉ (e₁ - e₂) in⁻₂ : ∀ {e₁ e₂ : Expr} {k : String} → k ∈ᵉ e₂ → k ∈ᵉ (e₁ - e₂) here : ∀ {k : String} → k ∈ᵉ (` k) data _∈ᵇ_ : String → BasicStmt → Set where in←₁ : ∀ {k : String} {e : Expr} → k ∈ᵇ (k ← e) in←₂ : ∀ {k k' : String} {e : Expr} → k ∈ᵉ e → k ∈ᵇ (k' ← e) private Expr-vars : Expr → StringSet Expr-vars (l + r) = Expr-vars l ⊔ˢ Expr-vars r Expr-vars (l - r) = Expr-vars l ⊔ˢ Expr-vars r Expr-vars (` s) = singletonˢ s Expr-vars (# _) = emptyˢ -- ∈-Expr-vars⇒∈ : ∀ {k : String} (e : Expr) → k ∈ˢ (Expr-vars e) → k ∈ᵉ e -- ∈-Expr-vars⇒∈ {k} (e₁ + e₂) k∈vs -- with Expr-Provenance k ((`ˢ (Expr-vars e₁)) ∪ (`ˢ (Expr-vars e₂))) k∈vs -- ... | in₁ (single k,tt∈vs₁) _ = (in⁺₁ (∈-Expr-vars⇒∈ e₁ (forget k,tt∈vs₁))) -- ... | in₂ _ (single k,tt∈vs₂) = (in⁺₂ (∈-Expr-vars⇒∈ e₂ (forget k,tt∈vs₂))) -- ... | bothᵘ (single k,tt∈vs₁) _ = (in⁺₁ (∈-Expr-vars⇒∈ e₁ (forget k,tt∈vs₁))) -- ∈-Expr-vars⇒∈ {k} (e₁ - e₂) k∈vs -- with Expr-Provenance k ((`ˢ (Expr-vars e₁)) ∪ (`ˢ (Expr-vars e₂))) k∈vs -- ... | in₁ (single k,tt∈vs₁) _ = (in⁻₁ (∈-Expr-vars⇒∈ e₁ (forget k,tt∈vs₁))) -- ... | in₂ _ (single k,tt∈vs₂) = (in⁻₂ (∈-Expr-vars⇒∈ e₂ (forget k,tt∈vs₂))) -- ... | bothᵘ (single k,tt∈vs₁) _ = (in⁻₁ (∈-Expr-vars⇒∈ e₁ (forget k,tt∈vs₁))) -- ∈-Expr-vars⇒∈ {k} (` k) (RelAny.here refl) = here -- ∈⇒∈-Expr-vars : ∀ {k : String} {e : Expr} → k ∈ᵉ e → k ∈ˢ (Expr-vars e) -- ∈⇒∈-Expr-vars {k} {e₁ + e₂} (in⁺₁ k∈e₁) = -- ⊔ˢ-preserves-∈k₁ {m₁ = Expr-vars e₁} -- {m₂ = Expr-vars e₂} -- (∈⇒∈-Expr-vars k∈e₁) -- ∈⇒∈-Expr-vars {k} {e₁ + e₂} (in⁺₂ k∈e₂) = -- ⊔ˢ-preserves-∈k₂ {m₁ = Expr-vars e₁} -- {m₂ = Expr-vars e₂} -- (∈⇒∈-Expr-vars k∈e₂) -- ∈⇒∈-Expr-vars {k} {e₁ - e₂} (in⁻₁ k∈e₁) = -- ⊔ˢ-preserves-∈k₁ {m₁ = Expr-vars e₁} -- {m₂ = Expr-vars e₂} -- (∈⇒∈-Expr-vars k∈e₁) -- ∈⇒∈-Expr-vars {k} {e₁ - e₂} (in⁻₂ k∈e₂) = -- ⊔ˢ-preserves-∈k₂ {m₁ = Expr-vars e₁} -- {m₂ = Expr-vars e₂} -- (∈⇒∈-Expr-vars k∈e₂) -- ∈⇒∈-Expr-vars here = RelAny.here refl BasicStmt-vars : BasicStmt → StringSet BasicStmt-vars (x ← e) = (singletonˢ x) ⊔ˢ (Expr-vars e) BasicStmt-vars noop = emptyˢ Stmt-vars : Stmt → StringSet Stmt-vars ⟨ bs ⟩ = BasicStmt-vars bs Stmt-vars (s₁ then s₂) = (Stmt-vars s₁) ⊔ˢ (Stmt-vars s₂) Stmt-vars (if e then s₁ else s₂) = ((Expr-vars e) ⊔ˢ (Stmt-vars s₁)) ⊔ˢ (Stmt-vars s₂) Stmt-vars (while e repeat s) = (Expr-vars e) ⊔ˢ (Stmt-vars s) -- ∈-Stmt-vars⇒∈ : ∀ {k : String} (s : Stmt) → k ∈ˢ (Stmt-vars s) → k ∈ᵇ s -- ∈-Stmt-vars⇒∈ {k} (k' ← e) k∈vs -- with Expr-Provenance k ((`ˢ (singletonˢ k')) ∪ (`ˢ (Expr-vars e))) k∈vs -- ... | in₁ (single (RelAny.here refl)) _ = in←₁ -- ... | in₂ _ (single k,tt∈vs') = in←₂ (∈-Expr-vars⇒∈ e (forget k,tt∈vs')) -- ... | bothᵘ (single (RelAny.here refl)) _ = in←₁ -- ∈⇒∈-Stmt-vars : ∀ {k : String} {s : Stmt} → k ∈ᵇ s → k ∈ˢ (Stmt-vars s) -- ∈⇒∈-Stmt-vars {k} {k ← e} in←₁ = -- ⊔ˢ-preserves-∈k₁ {m₁ = singletonˢ k} -- {m₂ = Expr-vars e} -- (RelAny.here refl) -- ∈⇒∈-Stmt-vars {k} {k' ← e} (in←₂ k∈e) = -- ⊔ˢ-preserves-∈k₂ {m₁ = singletonˢ k'} -- {m₂ = Expr-vars e} -- (∈⇒∈-Expr-vars k∈e) Stmts-vars : ∀ {n : ℕ} → Vec Stmt n → StringSet Stmts-vars = foldr (λ n → StringSet) (λ {k} stmt acc → (Stmt-vars stmt) ⊔ˢ acc) emptyˢ -- ∈-Stmts-vars⇒∈ : ∀ {n : ℕ} {k : String} (ss : Vec Stmt n) → -- k ∈ˢ (Stmts-vars ss) → Σ (Fin n) (λ f → k ∈ᵇ lookup ss f) -- ∈-Stmts-vars⇒∈ {suc n'} {k} (s ∷ ss') k∈vss -- with Expr-Provenance k ((`ˢ (Stmt-vars s)) ∪ (`ˢ (Stmts-vars ss'))) k∈vss -- ... | in₁ (single k,tt∈vs) _ = (zero , ∈-Stmt-vars⇒∈ s (forget k,tt∈vs)) -- ... | in₂ _ (single k,tt∈vss') = -- let -- (f' , k∈s') = ∈-Stmts-vars⇒∈ ss' (forget k,tt∈vss') -- in -- (suc f' , k∈s') -- ... | bothᵘ (single k,tt∈vs) _ = (zero , ∈-Stmt-vars⇒∈ s (forget k,tt∈vs)) -- ∈⇒∈-Stmts-vars : ∀ {n : ℕ} {k : String} {ss : Vec Stmt n} {f : Fin n} → -- k ∈ᵇ lookup ss f → k ∈ˢ (Stmts-vars ss) -- ∈⇒∈-Stmts-vars {suc n} {k} {s ∷ ss'} {zero} k∈s = -- ⊔ˢ-preserves-∈k₁ {m₁ = Stmt-vars s} -- {m₂ = Stmts-vars ss'} -- (∈⇒∈-Stmt-vars k∈s) -- ∈⇒∈-Stmts-vars {suc n} {k} {s ∷ ss'} {suc f'} k∈ss' = -- ⊔ˢ-preserves-∈k₂ {m₁ = Stmt-vars s} -- {m₂ = Stmts-vars ss'} -- (∈⇒∈-Stmts-vars {n} {k} {ss'} {f'} k∈ss') -- Creating a new number from a natural number can never create one -- equal to one you get from weakening the bounds on another number. z≢sf : ∀ {n : ℕ} (f : Fin n) → ¬ (zero ≡ suc f) z≢sf f () z≢mapsfs : ∀ {n : ℕ} (fs : List (Fin n)) → All (λ sf → ¬ zero ≡ sf) (mapˡ suc fs) z≢mapsfs [] = [] z≢mapsfs (f ∷ fs') = z≢sf f ∷ z≢mapsfs fs' indices : ∀ (n : ℕ) → Σ (List (Fin n)) Unique indices 0 = ([] , empty) indices (suc n') = let (inds' , unids') = indices n' in ( zero ∷ mapˡ suc inds' , push (z≢mapsfs inds') (Unique-map suc suc-injective unids') ) indices-complete : ∀ (n : ℕ) (f : Fin n) → f ∈ˡ (proj₁ (indices n)) indices-complete (suc n') zero = RelAny.here refl indices-complete (suc n') (suc f') = RelAny.there (x∈xs⇒fx∈fxs suc (indices-complete n' f')) -- Sketch, 'build control flow graph' -- -- Create new block, mark it as the current insertion point. -- emptyBlock : m Id -- currentBlock : m Id -- -- Create a new block, and insert the statement into it. Shold restore insertion pont. -- createBlock : Stmt → m (Id × Id) -- -- Note that the given ID is a successor / predecessor of the given -- -- insertion point. -- noteSuccessor : Id → m () -- notePredecessor : Id → m () -- noteEdge : Id → Id → m () -- -- Insert the given statment into the current insertion point. -- buildCfg : Stmt → m Cfg -- buildCfg { bs₁ } = push bs₁ -- buildCfg (s₁ ; s₂ ) = buildCfg s₁ >> buildCfg s₂ -- buildCfg (if _ then s₁ else s₂) = do -- (b₁ , b₁') ← createBlock s₁ -- noteSuccessor b₁ -- (b₂ , b₂') ← createBlock s₂ -- noteSuccessor b₂ -- b ← emptyBlock -- notePredecessor b₁' -- notePredecessor b₂' -- buildCfg (while e repeat s) = do -- (b₁, b₁') ← createBlock s -- noteSuccessor b₁ -- noteEdge b₁' b₁ -- b ← emptyBlock -- notePredecessor b₁' -- For now, just represent the program and CFG as one type, without branching. record Program : Set where field length : ℕ stmts : Vec Stmt length private vars-Set : StringSet vars-Set = Stmts-vars stmts vars : List String vars = to-Listˢ vars-Set vars-Unique : Unique vars vars-Unique = proj₂ vars-Set State : Set State = Fin length states : List State states = proj₁ (indices length) states-complete : ∀ (s : State) → s ∈ˡ states states-complete = indices-complete length states-Unique : Unique states states-Unique = proj₂ (indices length) code : State → Stmt code = lookup stmts -- vars-complete : ∀ {k : String} (s : State) → k ∈ᵇ (code s) → k ∈ˡ vars -- vars-complete {k} s = ∈⇒∈-Stmts-vars {length} {k} {stmts} {s} _≟_ : IsDecidable (_≡_ {_} {State}) _≟_ = _≟ᶠ_ -- Computations for incoming and outgoing edges will have to change too -- when we support branching etc. incoming : State → List State incoming with length ... | 0 = (λ ()) ... | suc n' = (λ { zero → [] ; (suc f') → (inject₁ f') ∷ [] })