Lang.agda

@@ -1,15 +1,5 @@
 {-# OPTIONS --guardedness #-}
-module Language.Flat where
+module Lang where
-
-open import Language.Values using
-    ( Value; int; bool
-    ; ValueType; tint; tbool
-    ; ⟦_⟧ˢ; ⟦⟧ˢ-empty; ⟦⟧ˢ-int; ⟦⟧ˢ-bool
-    ; s; s'; s''
-    ; st; st'; st''
-    ; b; b₁; b₂
-    ; z; z₁; z₂
-    )
 
 open import Data.Integer using (ℤ; _+_; +_)
 open import Data.Nat using (ℕ; suc; _<_)
@@ -20,6 +10,10 @@ open import Data.Unit using (⊤; tt)
 open import Data.Empty using (⊥; ⊥-elim)
 open import Data.Product using (Σ; _×_; _,_)
 
+data Value : Set where
+    int : ℤ → Value
+    bool : Bool → Value
+
 data Instruction : Set where
     pushint : ℤ → Instruction
     pushbool : Bool → Instruction
@@ -39,7 +33,10 @@ variable
     i : Instruction
     is is' is'' : Program
     S S' S'' : Env
+    s s' s'' : List Value
     n : ℕ
+    b : Bool
+    z z₁ z₂ : ℤ
 
 data _⇒ₓ_ : Program → Program → Set where
     pushint-skip : is ⇒ₓ is' → (pushint z ∷ is) ⇒ₓ is'
@@ -100,6 +97,9 @@ basic₂ = loop-evalⁿ thenᵉ pushbool-eval thenᵉ break-evalᵗ (pushint-ski
 basic₃ : ( [] , (loop 1 ∷ pushbool false ∷ break ∷ pushint (+ 1) ∷ end ∷ []) , [] ) ⇒* ( (int (+ 1) ∷ []) , [] , [] )
 basic₃ = loop-evalⁿ thenᵉ pushbool-eval thenᵉ break-evalᶠ thenᵉ pushint-eval thenᵉ end-eval thenᵉ (loop-eval⁰ (pushbool-skip (break-skip (pushint-skip end-skip)))) thenᵉ doneᵉ
 
+data ValueType : Set where
+    tint tbool : ValueType
+
 -- A stack type describes the current stack and the continuation types
 infix 35 _⟨_⟩
 record StackType : Set where
@@ -109,6 +109,7 @@ record StackType : Set where
         frames : List (List ValueType)
 
 variable
+    st st' st'' : List ValueType
     F F' F'' : List (List ValueType)
     σ σ' σ'' : StackType
 
@@ -145,9 +146,6 @@ _⊢_↦|_ σ is st = σ ⊢ is ↦* st ⟨ [] ⟩
 basict₁ : [] ⟨ [] ⟩ ⊢ (pushint (+ 1) ∷ pushint (+ 2) ∷ pushint (+ 3) ∷ add ∷ add ∷ []) ↦* (tint ∷ []) ⟨ [] ⟩
 basict₁ = pushint-t thenᵗ pushint-t thenᵗ pushint-t thenᵗ add-t thenᵗ add-t thenᵗ doneᵗ
 
-basict₄ : [] ⟨ [] ⟩ ⊢ (pushint (+ 1) ∷ loop 3 ∷ pushint (+ 2) ∷ add ∷ end ∷ []) ↦* (tint ∷ []) ⟨ [] ⟩
-basict₄ = pushint-t thenᵗ loop-t thenᵗ pushint-t thenᵗ add-t thenᵗ end-t thenᵗ doneᵗ
-
 -- doesn't typecheck: violates type discipline because on break, stack has different type than on end.
 --
 -- basict₂ : ([] ⟨ []  ⟩) ⊢ (loop 1 ∷ pushbool true ∷ break ∷ pushint (+ 1) ∷ end ∷ []) ↦* ([] ⟨ [] ⟩)
@@ -187,6 +185,12 @@ lemma₂ (loop-t thenᵗ ist⊢is↦|st'') =
     (_ , loop-skip is''⇒ₓis' is'₁⇒ₓis'₂  , is'₂↦|st'')
 lemma₂ (end-t thenᵗ rest) = (_ , end-skip , rest)
 
+-- what it means for list of value types to describe a list of values
+data ⟦_⟧ˢ : List ValueType → List Value → Set where
+    ⟦⟧ˢ-empty : ⟦ [] ⟧ˢ []
+    ⟦⟧ˢ-int : ⟦ st ⟧ˢ s → ⟦ tint ∷ st ⟧ˢ (int z ∷ s)
+    ⟦⟧ˢ-bool : ⟦ st ⟧ˢ s → ⟦ tbool ∷ st ⟧ˢ (bool b ∷ s)
+
 -- Parameterize by final stack type. The reason to do this is that we want
 -- all continuations to end in the same type, so that control flow doesn't change
 -- what final type the program is evaluated to. This is an unfortunate consequence
@@ -273,4 +277,4 @@ stackToString (bool b ∷ s) = showBool b ++ " :: " ++ stackToString s
 open import IO
 open import Level using (0ℓ)
 
-main = run {0ℓ} (putStrLn (stackToString (proj₁ (static-dynamic _ (⟦⟧ˢ-empty , ⟦⟧ᶠ-empty , basict₄)))))
+main = run {0ℓ} (putStrLn (stackToString (proj₁ (static-dynamic _ (⟦⟧ˢ-empty , ⟦⟧ᶠ-empty , basict₁)))))

Language/Nested.agda

@@ -1,189 +0,0 @@
-{-# OPTIONS --guardedness #-}
-module Language.Nested where
-
-open import Language.Values using
-    ( Value; int; bool
-    ; ValueType; tint; tbool
-    ; ⟦_⟧ˢ; ⟦⟧ˢ-empty; ⟦⟧ˢ-int; ⟦⟧ˢ-bool
-    ; s; s'; s''
-    ; st; st'; st''
-    ; b; b₁; b₂
-    ; z; z₁; z₂
-    )
-
-open import Data.Integer using (ℤ; _+_; +_)
-open import Data.Nat using (ℕ; suc)
-open import Data.Bool using (Bool; true; false)
-open import Data.List using (List; _∷_; []; length)
-open import Data.Unit using (⊤; tt)
-open import Data.Empty using (⊥; ⊥-elim)
-open import Data.Product using (Σ; _×_; _,_)
-open import Data.Maybe using (Maybe; just; nothing)
-open import Data.Sum using (_⊎_; inj₁; inj₂)
-open import Relation.Binary.PropositionalEquality using (_≡_; refl)
-
-data Instruction : Set where
-    pushint : ℤ → Instruction
-    pushbool : Bool → Instruction
-    add : Instruction
-
-    loop : ℕ → List Instruction → Instruction
-    break : Instruction
-
-Program : Set
-Program = List Instruction
-
-Env : Set
-Env = List (List Instruction)
-
-variable
-    i : Instruction
-    is is' is'' : Program
-    S S' S'' : Env
-    n : ℕ
-
-Triple : Set
-Triple = List Value × Program × Env
-
-data _⇒_ : Triple → Triple → Set where
-    pushint-eval : ( s , (pushint z ∷ is) , S ) ⇒ ( ((int z) ∷ s) , is , S )
-    pushbool-eval : ( s , (pushbool b ∷ is) , S ) ⇒ ( ((bool b) ∷ s) , is , S )
-    add-eval : ( (int z₁ ∷ int z₂ ∷ s) , (add ∷ is), S ) ⇒ ( ((int (z₁ + z₂)) ∷ s) , is , S )
-
-    loop-eval⁰ : (s , loop 0 is' ∷ is , S) ⇒ (s , is , S)
-    loop-evalⁿ : (s , loop (suc n) is' ∷ is , S) ⇒ (s , is' , (loop n is' ∷ is) ∷ S)
-    break-evalᶠ : (bool false ∷ s , break ∷ is , is' ∷ S) ⇒ (s , is , is' ∷ S)
-    break-evalᵗ : (bool true ∷ s , break ∷ is , (loop n is'' ∷ is') ∷ S) ⇒ (s , is' , S)
-
-    empty-eval : (s , [] , is' ∷ S) ⇒ (s , is' , S)
-
-infixr 30 _thenᵉ_
-data _⇒*_ : Triple → Triple → Set where
-    doneᵉ : ( s , [] , []  ) ⇒* ( s , [] , [] )
-    _thenᵉ_ : ( s , is , S ) ⇒ ( s' , is' , S' ) →
-             ( s' , is' , S' ) ⇒* ( s'' , is'' , S'' ) →
-             ( s , is , S ) ⇒* ( s'' , is'' , S'' )
-
-StackType : Set
-StackType = List ValueType
-
-variable
-    F F' F'' : List (List ValueType)
-    σ σ' σ'' : StackType
-    mst : Maybe StackType
-
-interleaved mutual
-    data _⊢⟨_⟩_↦_ : StackType → Maybe StackType → Instruction → StackType → Set
-    data _⊢⟨_⟩_↦*_ : StackType → Maybe StackType → Program → StackType → Set
-
-    data _⊢⟨_⟩_↦_ where
-        pushint-t : st ⊢⟨ mst ⟩ (pushint z) ↦ (tint ∷ st)
-        pushbool-t : st ⊢⟨ mst ⟩ (pushbool b) ↦ (tbool ∷ st)
-        add-t : (tint ∷ tint ∷ st) ⊢⟨ mst ⟩ add ↦ (tint ∷ st)
-
-        loop-t : st ⊢⟨ just st ⟩ is ↦* st →
-                 st ⊢⟨ mst ⟩ loop n is ↦ st
-        break-t : (tbool ∷ st) ⊢⟨ just st ⟩ break ↦ st
-
-    infixr 30 _thenᵗ_
-    data _⊢⟨_⟩_↦*_ where
-        doneᵗ-just : st ⊢⟨ just st ⟩ [] ↦* st
-        doneᵗ-nothing : st ⊢⟨ nothing ⟩ [] ↦* st
-        _thenᵗ_ : st ⊢⟨ mst ⟩ i ↦ st' →
-                  st' ⊢⟨ mst ⟩ is ↦* st'' →
-                  st ⊢⟨ mst ⟩ (i ∷ is) ↦* st''
-
-_⊢_↦|_ : StackType → Program → StackType → Set
-_⊢_↦|_ st is st' = st ⊢⟨ nothing ⟩ is ↦* st'
-
-basict₄ : [] ⊢ (pushint (+ 1) ∷ loop 3 (pushint (+ 2) ∷ add ∷ []) ∷ pushint (+ 1) ∷ add ∷ []) ↦| (tint ∷ [])
-basict₄ = pushint-t thenᵗ loop-t (pushint-t thenᵗ add-t thenᵗ doneᵗ-just) thenᵗ pushint-t thenᵗ add-t thenᵗ doneᵗ-nothing
-
-data _⊢_↦*_ : StackType → (Program × Env) → StackType → Set where
-    program-typed-flat : st ⊢⟨ nothing ⟩ is ↦* st' →
-                         st ⊢ (is , []) ↦* st'
-    program-typed-rec : ∀ {stˡ} →
-                        st ⊢⟨ just st' ⟩ is ↦* st' →
-                        st' ⊢ (loop n is'' ∷ is' , S) ↦* stˡ →
-                        st ⊢ (is , (loop n is'' ∷ is') ∷ S) ↦* stˡ
-
-skip-loop-continuation : st ⊢ (loop n is ∷ is' , S) ↦* st' →
-                         st ⊢ (is' , S) ↦* st'
-skip-loop-continuation (program-typed-flat (loop-t _ thenᵗ rest)) = (program-typed-flat rest)
-skip-loop-continuation (program-typed-rec (loop-t _ thenᵗ rest) frames) = (program-typed-rec rest frames)
-
-preservation : ∀ {stˡ} →
-               (s , is , S) ⇒ (s' , is' , S') →
-               ⟦ st ⟧ˢ s →
-               st ⊢ (is , S) ↦* stˡ →
-               Σ StackType (λ st' → ⟦ st' ⟧ˢ s' × st' ⊢ (is' , S') ↦* stˡ)
-preservation pushint-eval ⟦st⟧ (program-typed-flat (pushint-t thenᵗ rest)) = (_ , ⟦⟧ˢ-int ⟦st⟧ , (program-typed-flat rest))
-preservation pushint-eval ⟦st⟧ (program-typed-rec (pushint-t thenᵗ rest) frames) = (_ , ⟦⟧ˢ-int ⟦st⟧ , (program-typed-rec rest frames))
-preservation pushbool-eval ⟦st⟧ (program-typed-flat (pushbool-t thenᵗ rest)) = (_ , ⟦⟧ˢ-bool ⟦st⟧ , (program-typed-flat rest))
-preservation pushbool-eval ⟦st⟧ (program-typed-rec (pushbool-t thenᵗ rest) frames) = (_ , ⟦⟧ˢ-bool ⟦st⟧ , (program-typed-rec rest frames))
-preservation add-eval (⟦⟧ˢ-int (⟦⟧ˢ-int ⟦st⟧)) (program-typed-flat (add-t thenᵗ rest)) = (_ , ⟦⟧ˢ-int ⟦st⟧ , (program-typed-flat rest))
-preservation add-eval (⟦⟧ˢ-int (⟦⟧ˢ-int ⟦st⟧)) (program-typed-rec (add-t thenᵗ rest) frames) = (_ , ⟦⟧ˢ-int ⟦st⟧ , (program-typed-rec rest frames))
-preservation loop-eval⁰ ⟦st⟧ (program-typed-flat ((loop-t _) thenᵗ rest)) = (_ , ⟦st⟧ , (program-typed-flat rest))
-preservation loop-eval⁰ ⟦st⟧ (program-typed-rec ((loop-t _) thenᵗ rest) frames) = (_ , ⟦st⟧ , (program-typed-rec rest) frames)
-preservation loop-evalⁿ ⟦st⟧ (program-typed-flat ((loop-t loopt) thenᵗ rest)) = (_ , ⟦st⟧ , (program-typed-rec loopt (program-typed-flat (loop-t loopt thenᵗ rest))))
-preservation loop-evalⁿ ⟦st⟧ (program-typed-rec ((loop-t loopt) thenᵗ rest) frames) = (_ , ⟦st⟧ , (program-typed-rec loopt (program-typed-rec (loop-t loopt thenᵗ rest) frames)))
-preservation break-evalᵗ (⟦⟧ˢ-bool ⟦st⟧) (program-typed-rec (break-t thenᵗ rest) frames) = (_ , ⟦st⟧ , skip-loop-continuation frames)
-preservation break-evalᶠ (⟦⟧ˢ-bool ⟦st⟧) (program-typed-rec (break-t thenᵗ rest) frames) = (_ , ⟦st⟧ , (program-typed-rec rest frames))
-preservation empty-eval ⟦st⟧ (program-typed-rec (doneᵗ-just) frames) = (_ , ⟦st⟧ , frames)
-
-progress : ∀ {stˡ} →
-           st ⊢ (is , S) ↦* stˡ →
-           ⟦ st ⟧ˢ s →
-           (Σ Triple ((s , is , S) ⇒_)) ⊎ (is ≡ [] × S ≡ [] × ⟦ stˡ ⟧ˢ s)
-progress (program-typed-flat (pushint-t thenᵗ rest)) ⟦st⟧ = inj₁ (_ , pushint-eval)
-progress (program-typed-rec (pushint-t thenᵗ rest) frames) ⟦st⟧ = inj₁ (_ , pushint-eval)
-progress (program-typed-flat (pushbool-t thenᵗ rest)) ⟦st⟧ = inj₁ (_ , pushbool-eval)
-progress (program-typed-rec (pushbool-t thenᵗ rest) frames) ⟦st⟧ = inj₁ (_ , pushbool-eval)
-progress (program-typed-flat (add-t thenᵗ rest)) (⟦⟧ˢ-int (⟦⟧ˢ-int ⟦st⟧)) = inj₁ (_ , add-eval)
-progress (program-typed-rec (add-t thenᵗ rest) frames) (⟦⟧ˢ-int (⟦⟧ˢ-int ⟦st⟧)) = inj₁ (_ , add-eval)
-progress {is = loop 0 is' ∷ is} (program-typed-flat (loop-t _ thenᵗ rest)) ⟦st⟧ = inj₁ (_ , loop-eval⁰)
-progress {is = loop 0 is' ∷ is} (program-typed-rec (loop-t _ thenᵗ rest) frames) ⟦st⟧ = inj₁ (_ , loop-eval⁰)
-progress {is = loop (suc n) is' ∷ is} (program-typed-flat (loop-t _ thenᵗ rest)) ⟦st⟧ = inj₁ (_ , loop-evalⁿ)
-progress {is = loop (suc n) is' ∷ is} (program-typed-rec (loop-t _ thenᵗ rest) frames) ⟦st⟧ = inj₁ (_ , loop-evalⁿ)
-progress {s = bool true ∷ s} (program-typed-rec (break-t thenᵗ rest) frames) (⟦⟧ˢ-bool ⟦st⟧) = inj₁ (_ , break-evalᵗ)
-progress {s = bool false ∷ s} (program-typed-rec (break-t thenᵗ rest) frames) (⟦⟧ˢ-bool ⟦st⟧) = inj₁ (_ , break-evalᶠ)
-progress {is = []} (program-typed-rec (doneᵗ-just) frames) ⟦st⟧ = inj₁ (_ , empty-eval)
-progress {is = []} (program-typed-flat (doneᵗ-nothing)) ⟦st⟧ = inj₂ (refl , refl , ⟦st⟧)
-
-{-# TERMINATING #-} -- evaluation always terminates, but proving it is tedious
-static-dynamic-help : ∀ {stˡ} →
-                      st ⊢ (is , S) ↦* stˡ →
-                      ⟦ st ⟧ˢ s →
-                      Σ (List Value) (λ sˡ → ⟦ stˡ ⟧ˢ sˡ × (s , is , S) ⇒* (sˡ , [] , []))
-static-dynamic-help {s = s} frames ⟦st⟧
-    with progress frames ⟦st⟧
-...   | inj₁ ((s' , is' , S') , is⇒is') =
-        let (st' , (⟦st'⟧ , frames')) = preservation is⇒is' ⟦st⟧ frames in
-        let (sˡ , (⟦stˡ⟧ , is'⇒*[])) = static-dynamic-help frames' ⟦st'⟧ in
-        (sˡ , (⟦stˡ⟧ , is⇒is' thenᵉ is'⇒*[]))
-...   | inj₂ (refl , refl , ⟦stˡ⟧) = (s , (⟦stˡ⟧ , doneᵉ))
-
-
-static-dynamic : ∀ {stˡ} →
-                 st ⊢ is ↦| stˡ →
-                 ⟦ st ⟧ˢ s →
-                 Σ (List Value) (λ sˡ → ⟦ stˡ ⟧ˢ sˡ × (s , is , []) ⇒* (sˡ , [] , []))
-static-dynamic st⊢is↦|stˡ = static-dynamic-help (program-typed-flat st⊢is↦|stˡ)
-
--- Copied from Flat.agda
-
-open import Data.Nat.Show using () renaming (show to showℕ)
-open import Data.Integer.Show using () renaming (show to showℤ)
-open import Data.Bool.Show using () renaming (show to showBool)
-open import Data.String using (String; _++_)
-open import Data.Product using (proj₁)
-
-stackToString : List Value → String
-stackToString [] = "[]"
-stackToString (int i ∷ s) = showℤ i ++ " :: " ++ stackToString s
-stackToString (bool b ∷ s) = showBool b ++ " :: " ++ stackToString s
-
-open import IO
-open import Level using (0ℓ)
-
-main = run {0ℓ} (putStrLn (stackToString (proj₁ (static-dynamic basict₄ ⟦⟧ˢ-empty))))

Language/Values.agda

@@ -1,24 +0,0 @@
-module Language.Values where
-
-open import Data.Integer using (ℤ)
-open import Data.Bool using (Bool)
-open import Data.List using (List; []; _∷_)
-
-data Value : Set where
-    int : ℤ → Value
-    bool : Bool → Value
-
-data ValueType : Set where
-    tint tbool : ValueType
-
-variable
-    s s' s'' : List Value
-    st st' st'' : List ValueType
-    b b₁ b₂ : Bool
-    z z₁ z₂ : ℤ
-
--- what it means for list of value types to describe a list of values
-data ⟦_⟧ˢ : List ValueType → List Value → Set where
-    ⟦⟧ˢ-empty : ⟦ [] ⟧ˢ []
-    ⟦⟧ˢ-int : ⟦ st ⟧ˢ s → ⟦ tint ∷ st ⟧ˢ (int z ∷ s)
-    ⟦⟧ˢ-bool : ⟦ st ⟧ˢ s → ⟦ tbool ∷ st ⟧ˢ (bool b ∷ s)