agda-spa/Language.agda

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; +-assoc; +-comm)
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; _∷_; []; _++_; cast)
open import Data.Vec.Properties using (++-assoc; ++-identityʳ; lookup-++ˡ; lookup-cast₁; cast-sym)
open import Data.Vec.Relation.Binary.Equality.Cast using (cast-is-id)
open import Data.List using ([]; _∷_; List) renaming (foldr to foldrˡ; map to mapˡ; _++_ to _++ˡ_)
open import Data.List.Properties using () renaming (++-assoc to ++ˡ-assoc; map-++ to mapˡ-++ˡ; ++-identityʳ to ++ˡ-identityʳ)
open import Data.List.Membership.Propositional as MemProp using () renaming (_∈_ to _∈ˡ_)
open import Data.List.Membership.Propositional.Properties using () renaming (∈-++⁺ʳ to ∈ˡ-++⁺ʳ)
open import Data.List.Relation.Unary.All using (All; []; _∷_)
open import Data.List.Relation.Unary.Any as RelAny using ()
open import Data.List.Relation.Unary.Any.Properties using (++⁺ʳ)
open import Data.Fin using (Fin; suc; zero; fromℕ; inject₁; inject≤; _↑ʳ_; _↑ˡ_) renaming (_≟_ to _≟ᶠ_; cast to castᶠ)
open import Data.Fin.Properties using (suc-injective) renaming (cast-is-id to castᶠ-is-id)
open import Relation.Binary.PropositionalEquality as Eq using (subst; cong; _≡_; sym; trans; refl)
open import Relation.Nullary using (¬_)
open import Function using (_∘_)
open Eq.≡-Reasoning

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
        constructor MkGraph
        field
            size : ℕ

        Index : Set
        Index = Fin size

        Edge : Set
        Edge = Index × Index

        field
            nodes : Vec (List BasicStmt) size
            edges : List Edge

    ↑ˡ-Edge : ∀ {n} → (Fin n × Fin n) → ∀ m → (Fin (n +ⁿ m) × Fin (n +ⁿ m))
    ↑ˡ-Edge (idx₁ , idx₂) m = (idx₁ ↑ˡ m , idx₂ ↑ˡ m)

    _[_] : ∀ (g : Graph) → Graph.Index g → List BasicStmt
    _[_] g idx = lookup (Graph.nodes g) idx

    record _⊆_ (g₁ g₂ : Graph) : Set where
        constructor Mk-⊆
        field
            n : ℕ
            sg₂≡sg₁+n : Graph.size g₂ ≡ Graph.size g₁ +ⁿ n
            newNodes : Vec (List BasicStmt) n
            nsg₂≡nsg₁++newNodes : cast sg₂≡sg₁+n (Graph.nodes g₂) ≡ Graph.nodes g₁ ++ newNodes
            e∈g₁⇒e∈g₂ : ∀ {e : Graph.Edge g₁} →
                        e ∈ˡ (Graph.edges g₁) →
                        (↑ˡ-Edge e n) ∈ˡ (subst (λ m → List (Fin m × Fin m)) sg₂≡sg₁+n (Graph.edges g₂))

    castᵉ : ∀ {n m : ℕ} .(p : n ≡ m) → (Fin n × Fin n) → (Fin m × Fin m)
    castᵉ p (idx₁ , idx₂) = (castᶠ p idx₁ , castᶠ p idx₂)

    ↑ˡ-assoc : ∀ {s n₁ n₂} (f : Fin s) (p : s +ⁿ (n₁ +ⁿ n₂) ≡ s +ⁿ n₁ +ⁿ n₂) →
               f ↑ˡ n₁ ↑ˡ n₂ ≡ castᶠ p (f ↑ˡ (n₁ +ⁿ n₂))
    ↑ˡ-assoc zero p = refl
    ↑ˡ-assoc {suc s'} {n₁} {n₂} (suc f') p rewrite ↑ˡ-assoc f' (sym (+-assoc s' n₁ n₂)) = refl

    ↑ˡ-Edge-assoc : ∀ {s n₁ n₂} (e : Fin s × Fin s) (p : s +ⁿ (n₁ +ⁿ n₂) ≡ s +ⁿ n₁ +ⁿ n₂) →
                    ↑ˡ-Edge (↑ˡ-Edge e n₁) n₂ ≡ castᵉ p (↑ˡ-Edge e (n₁ +ⁿ n₂))
    ↑ˡ-Edge-assoc (idx₁ , idx₂) p
        rewrite ↑ˡ-assoc idx₁ p
        rewrite ↑ˡ-assoc idx₂ p = refl

    ↑ˡ-identityʳ : ∀ {s} (f : Fin s) (p : s +ⁿ 0 ≡ s) →
               f ≡ castᶠ p (f ↑ˡ 0)
    ↑ˡ-identityʳ zero p = refl
    ↑ˡ-identityʳ {suc s'} (suc f') p rewrite sym (↑ˡ-identityʳ f' (+-comm s' 0)) = refl

    ↑ˡ-Edge-identityʳ : ∀ {s} (e : Fin s × Fin s) (p : s +ⁿ 0 ≡ s) →
                        e ≡ castᵉ p (↑ˡ-Edge e 0)
    ↑ˡ-Edge-identityʳ (idx₁ , idx₂) p
        rewrite sym (↑ˡ-identityʳ idx₁ p)
        rewrite sym (↑ˡ-identityʳ idx₂ p) = refl

    cast∈⇒∈subst : ∀ {n m : ℕ} (p : n ≡ m) (q : m ≡ n)
                   (e : Fin n × Fin n) (es : List (Fin m × Fin m)) →
                   castᵉ p e ∈ˡ es →
                   e ∈ˡ subst (λ m → List (Fin m × Fin m)) q es
    cast∈⇒∈subst refl refl (idx₁ , idx₂) es e∈es
        rewrite castᶠ-is-id refl idx₁
        rewrite castᶠ-is-id refl idx₂ = e∈es

    ⊆-trans : ∀ {g₁ g₂ g₃ : Graph} → g₁ ⊆ g₂ → g₂ ⊆ g₃ → g₁ ⊆ g₃
    ⊆-trans {MkGraph s₁ ns₁ es₁} {MkGraph s₂ ns₂ es₂} {MkGraph s₃ ns₃ es₃}
            (Mk-⊆ n₁ p₁@refl newNodes₁ nsg₂≡nsg₁++newNodes₁ e∈g₁⇒e∈g₂)
            (Mk-⊆ n₂ p₂@refl newNodes₂ nsg₃≡nsg₂++newNodes₂ e∈g₂⇒e∈g₃)
        rewrite cast-is-id refl ns₂
        rewrite cast-is-id refl ns₃
        with refl ← nsg₂≡nsg₁++newNodes₁
        with refl ← nsg₃≡nsg₂++newNodes₂ =
        record
            { n = n₁ +ⁿ n₂
            ; sg₂≡sg₁+n = +-assoc s₁ n₁ n₂
            ; newNodes = newNodes₁ ++ newNodes₂
            ; nsg₂≡nsg₁++newNodes = ++-assoc (+-assoc s₁ n₁ n₂) ns₁ newNodes₁ newNodes₂
            ; e∈g₁⇒e∈g₂ = λ {e} e∈g₁ →
                cast∈⇒∈subst (sym (+-assoc s₁ n₁ n₂)) (+-assoc s₁ n₁ n₂) _ _
                             (subst (λ e' → e' ∈ˡ es₃)
                                    (↑ˡ-Edge-assoc e (sym (+-assoc s₁ n₁ n₂)))
                                    (e∈g₂⇒e∈g₃ (e∈g₁⇒e∈g₂ e∈g₁)))
            }

    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 { (Mk-⊆ n refl _ _ _) idx → idx ↑ˡ n }
            }

        EdgeRelaxable : Relaxable Graph.Edge
        EdgeRelaxable = record
            { relax = λ g₁⊆g₂ (idx₁ , idx₂) →
                ( Relaxable.relax IndexRelaxable g₁⊆g₂ idx₁
                , Relaxable.relax IndexRelaxable 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

    relax-preserves-[]≡ : ∀ (g₁ g₂ : Graph) (g₁⊆g₂ : g₁ ⊆ g₂) (idx : Graph.Index g₁) →
                          g₁ [ idx ] ≡ g₂ [ relax g₁⊆g₂ idx ]
    relax-preserves-[]≡ g₁ g₂ (Mk-⊆ n refl newNodes nsg₂≡nsg₁++newNodes _) idx
        rewrite cast-is-id refl (Graph.nodes g₂)
        with refl ← nsg₂≡nsg₁++newNodes = sym (lookup-++ˡ (Graph.nodes g₁) _ _)

    -- Tools for graph construction. The most important is a 'monotonic function':
    -- one that takes a graph, and produces another graph, such that the
    -- new graph includes all the information from the old one.

    MonotonicGraphFunction : (Graph → Set) → Set
    MonotonicGraphFunction T = (g₁ : Graph) → Σ Graph (λ g₂ → T g₂ × g₁ ⊆ g₂)

    -- Now, define some operations on monotonic functions; these are useful
    -- to save the work of threading intermediate graphs in and out of operations.

    infixr 2 _⟨⊗⟩_
    _⟨⊗⟩_ : ∀ {T₁ T₂ : Graph → Set} {{ T₁Relaxable : Relaxable T₁ }} →
          MonotonicGraphFunction T₁ → MonotonicGraphFunction T₂ →
          MonotonicGraphFunction (T₁ ⊗ T₂)
    _⟨⊗⟩_ {{r}} f₁ f₂ g
        with (g' , (t₁ , g⊆g')) ← f₁ g
        with (g'' , (t₂ , g'⊆g'')) ← f₂ g' =
        (g'' , ((Relaxable.relax r g'⊆g'' t₁ , t₂) , ⊆-trans g⊆g' g'⊆g''))

    infixl 2 _update_
    _update_ : ∀ {T : Graph → Set} {{ TRelaxable : Relaxable T }} →
             MonotonicGraphFunction T → (∀ (g : Graph) → T g → Σ Graph (λ g' → g ⊆ g')) →
             MonotonicGraphFunction T
    _update_ {{r}} f mod g
        with (g' , (t , g⊆g')) ← f g
        with (g'' , g'⊆g'') ← mod g' t =
        (g'' , ((Relaxable.relax r g'⊆g'' t , ⊆-trans g⊆g' g'⊆g'')))

    infixl 2 _map_
    _map_ : ∀ {T₁ T₂ : Graph → Set} →
          MonotonicGraphFunction T₁ → (∀ (g : Graph) → T₁ g → T₂ g) →
          MonotonicGraphFunction T₂
    _map_ f fn g = let (g' , (t₁ , g⊆g')) = f g in (g' , (fn g' t₁ , g⊆g'))


    -- To reason about monotonic functions and what we do, we need a way
    -- to describe values they produce. A 'graph-value predicate' is
    -- just a predicate for some (dependent) value.

    GraphValuePredicate : (Graph → Set) → Set₁
    GraphValuePredicate T = ∀ (g : Graph) → T g → Set

    Both : {T₁ T₂ : Graph → Set} → GraphValuePredicate T₁ → GraphValuePredicate T₂ →
           GraphValuePredicate (T₁ ⊗ T₂)
    Both P Q = (λ { g (t₁ , t₂) → (P g t₁ × Q g t₂) })

    -- Since monotnic functions keep adding on to a function, proofs of
    -- graph-value predicates go stale fast (they describe old values of
    -- the graph). To keep propagating them through, we need them to still
    -- on 'bigger graphs'. We call such predicates monotonic as well, since
    -- they respect the ordering of graphs.

    MonotonicPredicate : ∀ {T : Graph → Set} {{ TRelaxable : Relaxable T }} →
                         GraphValuePredicate T → Set
    MonotonicPredicate {T} P = ∀ (g₁ g₂ : Graph) (t₁ : T g₁) (g₁⊆g₂ : g₁ ⊆ g₂) →
                               P g₁ t₁ → P g₂ (relax g₁⊆g₂ t₁)


    -- A 'map' has a certain property if its ouputs satisfy that property
    -- for all inputs.

    always : ∀ {T : Graph → Set} → GraphValuePredicate T → MonotonicGraphFunction T → Set
    always P m = ∀ g₁ → let (g₂ , t , _) = m g₁ in P g₂ t

    ⟨⊗⟩-reason : ∀ {T₁ T₂ : Graph → Set} {{ T₁Relaxable : Relaxable T₁ }}
                 {P : GraphValuePredicate T₁} {Q : GraphValuePredicate T₂}
                 {P-Mono : MonotonicPredicate P}
                 {m₁ : MonotonicGraphFunction T₁} {m₂ : MonotonicGraphFunction T₂} →
                 always P m₁ → always Q m₂ → always (Both P Q) (m₁ ⟨⊗⟩ m₂)
    ⟨⊗⟩-reason {P-Mono = P-Mono} {m₁ = m₁} {m₂ = m₂} aP aQ g
        with p ← aP g
        with (g' , (t₁ , g⊆g')) ← m₁ g
        with q ← aQ g'
        with (g'' , (t₂ , g'⊆g'')) ← m₂ g' = (P-Mono _ _ _ g'⊆g'' p , q)

    module Construction where
        pushBasicBlock : List BasicStmt → MonotonicGraphFunction Graph.Index
        pushBasicBlock bss g =
            ( record
                { size = Graph.size g +ⁿ 1
                ; nodes = Graph.nodes g ++ (bss ∷ [])
                ; edges = mapˡ (λ e → ↑ˡ-Edge e 1) (Graph.edges g)
                }
            , ( Graph.size g ↑ʳ zero
              , record
                  { n = 1
                  ; sg₂≡sg₁+n = refl
                  ; newNodes = (bss ∷ [])
                  ; nsg₂≡nsg₁++newNodes = cast-is-id refl _
                  ; e∈g₁⇒e∈g₂ = λ e∈g₁ → x∈xs⇒fx∈fxs (λ e → ↑ˡ-Edge e 1) e∈g₁
                  }
              )
            )

        addEdges : ∀ (g : Graph) → List (Graph.Edge g) → Σ Graph (λ g' → g ⊆ g')
        addEdges (MkGraph s ns es) es' =
            ( record
                { size = s
                ; nodes = ns
                ; edges = es' ++ˡ es
                }
            , record
                { n = 0
                ; sg₂≡sg₁+n = +-comm 0 s
                ; newNodes = []
                ; nsg₂≡nsg₁++newNodes = cast-sym _ (++-identityʳ (+-comm s 0) ns)
                ; e∈g₁⇒e∈g₂ = λ {e} e∈es →
                    cast∈⇒∈subst (+-comm s 0) (+-comm 0 s) _ _
                                 (subst (λ e' → e' ∈ˡ _)
                                        (↑ˡ-Edge-identityʳ e (+-comm s 0))
                                        (∈ˡ-++⁺ʳ es' e∈es))
                }
            )

        pushEmptyBlock : MonotonicGraphFunction Graph.Index
        pushEmptyBlock = pushBasicBlock []

        buildCfg : Stmt → MonotonicGraphFunction (Graph.Index ⊗ Graph.Index)
        buildCfg ⟨ bs₁ ⟩ = pushBasicBlock (bs₁ ∷ []) map (λ g idx → (idx , idx))
        buildCfg (s₁ then s₂) =
            (buildCfg s₁ ⟨⊗⟩ buildCfg s₂)
            update (λ { g ((idx₁ , idx₂) , (idx₃ , idx₄)) → addEdges g ((idx₂ , idx₃) ∷ []) })
            map (λ { g ((idx₁ , idx₂) , (idx₃ , idx₄)) → (idx₁ , idx₄) })
        buildCfg (if _ then s₁ else s₂) =
            (buildCfg s₁ ⟨⊗⟩ buildCfg s₂ ⟨⊗⟩ pushEmptyBlock ⟨⊗⟩ pushEmptyBlock)
            update (λ { g ((idx₁ , idx₂) , (idx₃ , idx₄) , idx , idx') →
                      addEdges g ((idx , idx₁) ∷ (idx , idx₃) ∷ (idx₂ , idx') ∷ (idx₄ , idx') ∷ []) })
            map (λ { g ((idx₁ , idx₂) , (idx₃ , idx₄) , idx , idx') → (idx , idx') })
        buildCfg (while _ repeat s) =
            (buildCfg s ⟨⊗⟩ pushEmptyBlock ⟨⊗⟩ pushEmptyBlock)
            update (λ { g ((idx₁ , idx₂) , idx , idx') →
                      addEdges g ((idx , idx') ∷ (idx₂ , idx) ∷ []) })
            map (λ { g ((idx₁ , idx₂) , idx , idx') → (idx , idx') })

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'))


-- 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') ∷ []
            })