From 535e4297a08c626d0e2e1923914727f914b8c9bd Mon Sep 17 00:00:00 2001
From: Greg Brown <greg.brown@cl.cam.ac.uk>
Date: Sat, 19 Mar 2022 13:28:42 +0000
Subject: Modify pseudocode definition.

This change makes the following changes to the definition of pseudocode:

- Add a separate type `bit` for single-bit values.
- Change `var` and `state` to take `Fin`s instead of bounded naturals.
- Make `[_]` and `unbox` work for any sliced type.
- Generalise `_:_` and `slice` into `splice` and `cut` respectively,
  making the two new operations inverses.
- Replace `tup` with `nil` and `cons` for building tuples.
- Add destructors `head` and `tail` for tuple types.
- Make function and procedure calls take a vector of arguments instead
  of a tuple.
- Introduce an `if_then_` expression for if statements with a trivial
  else branch.
---
 src/Helium/Data/Pseudocode/Core.agda | 261 ++++++++++++++++++++---------------
 1 file changed, 151 insertions(+), 110 deletions(-)

(limited to 'src/Helium/Data/Pseudocode/Core.agda')

diff --git a/src/Helium/Data/Pseudocode/Core.agda b/src/Helium/Data/Pseudocode/Core.agda
index 8a0c4e3..079e2ce 100644
--- a/src/Helium/Data/Pseudocode/Core.agda
+++ b/src/Helium/Data/Pseudocode/Core.agda
@@ -9,7 +9,8 @@
 module Helium.Data.Pseudocode.Core where
 
 open import Data.Bool using (Bool; true; false)
-open import Data.Fin using (Fin; #_)
+open import Data.Fin as Fin using (Fin)
+open import Data.Fin.Patterns
 open import Data.Nat as ℕ using (ℕ; zero; suc)
 open import Data.Nat.Properties using (+-comm)
 open import Data.Product using (∃; _,_; proj₂; uncurry)
@@ -19,7 +20,7 @@ open import Data.Vec.Relation.Unary.All using (All; []; _∷_; reduce)
 open import Data.Vec.Relation.Unary.Any using (Any; here; there)
 open import Function as F using (_∘_; _∘′_; _∋_)
 open import Relation.Binary.PropositionalEquality using (_≡_; refl)
-open import Relation.Nullary using (yes; no)
+open import Relation.Nullary using (Dec; yes; no)
 open import Relation.Nullary.Decidable.Core using (True; False; toWitness; fromWitness; map′)
 open import Relation.Nullary.Product using (_×-dec_)
 open import Relation.Nullary.Sum using (_⊎-dec_)
@@ -31,18 +32,17 @@ data Type : Set where
   int   : Type
   fin   : (n : ℕ) → Type
   real  : Type
+  bit   : Type
   bits  : (n : ℕ) → Type
   tuple : ∀ n → Vec Type n → Type
   array : Type → (n : ℕ) → Type
 
-bit : Type
-bit = bits 1
-
 data HasEquality : Type → Set where
   bool : HasEquality bool
   int  : HasEquality int
   fin  : ∀ {n} → HasEquality (fin n)
   real : HasEquality real
+  bit  : HasEquality bit
   bits : ∀ {n} → HasEquality (bits n)
 
 hasEquality? : Decidable HasEquality
@@ -50,6 +50,7 @@ hasEquality? bool        = yes bool
 hasEquality? int         = yes int
 hasEquality? (fin n)     = yes fin
 hasEquality? real        = yes real
+hasEquality? bit         = yes bit
 hasEquality? (bits n)    = yes bits
 hasEquality? (tuple n x) = no (λ ())
 hasEquality? (array t n) = no (λ ())
@@ -61,16 +62,17 @@ data IsNumeric : Type → Set where
 isNumeric? : Decidable IsNumeric
 isNumeric? bool        = no (λ ())
 isNumeric? int         = yes int
-isNumeric? real        = yes real
 isNumeric? (fin n)     = no (λ ())
+isNumeric? real        = yes real
+isNumeric? bit         = no (λ ())
 isNumeric? (bits n)    = no (λ ())
 isNumeric? (tuple n x) = no (λ ())
 isNumeric? (array t n) = no (λ ())
 
-combineNumeric : ∀ t₁ t₂ → {isNumeric₁ : True (isNumeric? t₁)} → {isNumeric₂ : True (isNumeric? t₂)} → Type
-combineNumeric int  int  = int
-combineNumeric int  real = real
-combineNumeric real _    = real
+combineNumeric : ∀ t₁ t₂ → (isNumeric₁ : True (isNumeric? t₁)) → (isNumeric₂ : True (isNumeric? t₂)) → Type
+combineNumeric int  int  _ _ = int
+combineNumeric int  real _ _ = real
+combineNumeric real _    _ _ = real
 
 data Sliced : Set where
   bits  : Sliced
@@ -87,12 +89,13 @@ elemType (array t) = t
 --- Literals
 
 data Literal : Type → Set where
-  _′b : Bool → Literal bool
-  _′i : ℕ → Literal int
-  _′f : ∀ {n} → Fin n → Literal (fin n)
-  _′r : ℕ → Literal real
-  _′x : ∀ {n} → Vec Bool n → Literal (bits n)
-  _′a : ∀ {n t} → Literal t → Literal (array t n)
+  _′b  : Bool → Literal bool
+  _′i  : ℕ → Literal int
+  _′f  : ∀ {n} → Fin n → Literal (fin n)
+  _′r  : ℕ → Literal real
+  _′x  : Bool → Literal bit
+  _′xs : ∀ {n} → Vec Bool n → Literal (bits n)
+  _′a  : ∀ {n t} → Literal t → Literal (array t n)
 
 --- Expressions, references, statements, functions and procedures
 
@@ -100,10 +103,8 @@ module Code {o} (Σ : Vec Type o) where
   data Expression {n} (Γ : Vec Type n) : Type → Set
   data CanAssign {n Γ} : ∀ {t} → Expression {n} Γ t → Set
   canAssign? : ∀ {n Γ t} → Decidable (CanAssign {n} {Γ} {t})
-  canAssignAll? : ∀ {n Γ m ts} → Decidable {A = All (Expression {n} Γ) {m} ts} (All (CanAssign ∘ proj₂) ∘ (reduce (λ {x} e → x , e)))
-  data AssignsState {n Γ} : ∀ {t e} → CanAssign {n} {Γ} {t} e → Set
-  assignsState? : ∀ {n Γ t e} → Decidable (AssignsState {n} {Γ} {t} {e})
-  assignsStateAny? : ∀ {n Γ m ts es} → Decidable {A = All (CanAssign ∘ proj₂) (reduce {P = Expression {n} Γ} (λ {x} e → x , e) {m} {ts} es)} (Any (AssignsState ∘ proj₂) ∘ reduce (λ {x} a → x , a))
+  data ReferencesState {n Γ} : ∀ {t} → Expression {n} Γ t → Set
+  referencesState? : ∀ {n Γ t} → Decidable (ReferencesState {n} {Γ} {t})
   data Statement {n} (Γ : Vec Type n) : Set
   data ChangesState {n Γ} : Statement {n} Γ → Set
   changesState? : ∀ {n Γ} → Decidable (ChangesState {n} {Γ})
@@ -114,55 +115,61 @@ module Code {o} (Σ : Vec Type o) where
   infixr 7 _^_
   infixl 6 _*_ _and_ _>>_
   -- infixl 6 _/_
-  infixl 5 _+_ _or_ _&&_ _||_ _∶_
+  infixl 5 _+_ _or_ _&&_ _||_
   infix  4 _≟_ _<?_
 
   data Expression {n} Γ where
-    lit   : ∀ {t} → Literal t → Expression Γ t
-    state : ∀ i {i<o : True (i ℕ.<? o)} → Expression Γ (lookup Σ (#_ i {m<n = i<o}))
-    var   : ∀ i {i<n : True (i ℕ.<? n)} → Expression Γ (lookup Γ (#_ i {m<n = i<n}))
-    abort : ∀ {t} → Expression Γ (fin 0) → Expression Γ t
-    _≟_   : ∀ {t} {hasEquality : True (hasEquality? t)} → Expression Γ t → Expression Γ t → Expression Γ bool
-    _<?_  : ∀ {t} {isNumeric : True (isNumeric? t)} → Expression Γ t → Expression Γ t → Expression Γ bool
-    inv   : Expression Γ bool → Expression Γ bool
-    _&&_  : Expression Γ bool → Expression Γ bool → Expression Γ bool
-    _||_  : Expression Γ bool → Expression Γ bool → Expression Γ bool
-    not   : ∀ {m} → Expression Γ (bits m) → Expression Γ (bits m)
-    _and_ : ∀ {m} → Expression Γ (bits m) → Expression Γ (bits m) → Expression Γ (bits m)
-    _or_  : ∀ {m} → Expression Γ (bits m) → Expression Γ (bits m) → Expression Γ (bits m)
-    [_]   : ∀ {t} → Expression Γ t → Expression Γ (array t 1)
-    unbox : ∀ {t} → Expression Γ (array t 1) → Expression Γ t
-    _∶_   : ∀ {m n t} → Expression Γ (asType t m) → Expression Γ (asType t n) → Expression Γ (asType t (n ℕ.+ m))
-    slice : ∀ {i j t} → Expression Γ (asType t (i ℕ.+ j)) → Expression Γ (fin (suc i)) → Expression Γ (asType t j)
-    cast  : ∀ {i j t} → .(eq : i ≡ j) → Expression Γ (asType t i) → Expression Γ (asType t j)
-    -_    : ∀ {t} {isNumeric : True (isNumeric? t)} → Expression Γ t → Expression Γ t
-    _+_   : ∀ {t₁ t₂} {isNumeric₁ : True (isNumeric? t₁)} {isNumeric₂ : True (isNumeric? t₂)} → Expression Γ t₁ → Expression Γ t₂ → Expression Γ (combineNumeric t₁ t₂ {isNumeric₁} {isNumeric₂})
-    _*_   : ∀ {t₁ t₂} {isNumeric₁ : True (isNumeric? t₁)} {isNumeric₂ : True (isNumeric? t₂)} → Expression Γ t₁ → Expression Γ t₂ → Expression Γ (combineNumeric t₁ t₂ {isNumeric₁} {isNumeric₂})
+    lit    : ∀ {t} → Literal t → Expression Γ t
+    state  : ∀ i → Expression Γ (lookup Σ i)
+    var    : ∀ i → Expression Γ (lookup Γ i)
+    abort  : ∀ {t} → Expression Γ (fin 0) → Expression Γ t
+    _≟_    : ∀ {t} {hasEquality : True (hasEquality? t)} → Expression Γ t → Expression Γ t → Expression Γ bool
+    _<?_   : ∀ {t} {isNumeric : True (isNumeric? t)} → Expression Γ t → Expression Γ t → Expression Γ bool
+    inv    : Expression Γ bool → Expression Γ bool
+    _&&_   : Expression Γ bool → Expression Γ bool → Expression Γ bool
+    _||_   : Expression Γ bool → Expression Γ bool → Expression Γ bool
+    not    : ∀ {m} → Expression Γ (bits m) → Expression Γ (bits m)
+    _and_  : ∀ {m} → Expression Γ (bits m) → Expression Γ (bits m) → Expression Γ (bits m)
+    _or_   : ∀ {m} → Expression Γ (bits m) → Expression Γ (bits m) → Expression Γ (bits m)
+    [_]    : ∀ {t} → Expression Γ (elemType t) → Expression Γ (asType t 1)
+    unbox  : ∀ {t} → Expression Γ (asType t 1) → Expression Γ (elemType t)
+    splice : ∀ {m n t} → Expression Γ (asType t m) → Expression Γ (asType t n) → Expression Γ (fin (suc n)) → Expression Γ (asType t (n ℕ.+ m))
+    cut    : ∀ {m n t} → Expression Γ (asType t (n ℕ.+ m)) → Expression Γ (fin (suc n)) → Expression Γ (tuple 2 (asType t m ∷ asType t n ∷ []))
+    cast   : ∀ {i j t} → .(eq : i ≡ j) → Expression Γ (asType t i) → Expression Γ (asType t j)
+    -_     : ∀ {t} {isNumeric : True (isNumeric? t)} → Expression Γ t → Expression Γ t
+    _+_    : ∀ {t₁ t₂} {isNumeric₁ : True (isNumeric? t₁)} {isNumeric₂ : True (isNumeric? t₂)} → Expression Γ t₁ → Expression Γ t₂ → Expression Γ (combineNumeric t₁ t₂ isNumeric₁ isNumeric₂)
+    _*_    : ∀ {t₁ t₂} {isNumeric₁ : True (isNumeric? t₁)} {isNumeric₂ : True (isNumeric? t₂)} → Expression Γ t₁ → Expression Γ t₂ → Expression Γ (combineNumeric t₁ t₂ isNumeric₁ isNumeric₂)
     -- _/_   : Expression Γ real → Expression Γ real → Expression Γ real
-    _^_   : ∀ {t} {isNumeric : True (isNumeric? t)} → Expression Γ t → ℕ → Expression Γ t
-    _>>_  : Expression Γ int → ℕ → Expression Γ int
-    rnd   : Expression Γ real → Expression Γ int
-    fin   : ∀ {k ms n} → (All (Fin) ms → Fin n) → Expression Γ (tuple k (map fin ms)) → Expression Γ (fin n)
-    asInt : ∀ {m} → Expression Γ (fin m) → Expression Γ int
-    tup   : ∀ {m ts} → All (Expression Γ) ts → Expression Γ (tuple m ts)
-    call  : ∀ {t m Δ} → Function Δ t → Expression Γ (tuple m Δ) → Expression Γ t
+    _^_    : ∀ {t} {isNumeric : True (isNumeric? t)} → Expression Γ t → ℕ → Expression Γ t
+    _>>_   : Expression Γ int → ℕ → Expression Γ int
+    rnd    : Expression Γ real → Expression Γ int
+    fin    : ∀ {k ms n} → (All (Fin) ms → Fin n) → Expression Γ (tuple k (map fin ms)) → Expression Γ (fin n)
+    asInt  : ∀ {m} → Expression Γ (fin m) → Expression Γ int
+    nil    : Expression Γ (tuple 0 [])
+    cons   : ∀ {m t ts} → Expression Γ t → Expression Γ (tuple m ts) → Expression Γ (tuple (suc m) (t ∷ ts))
+    head   : ∀ {m t ts} → Expression Γ (tuple (suc m) (t ∷ ts)) → Expression Γ t
+    tail   : ∀ {m t ts} → Expression Γ (tuple (suc m) (t ∷ ts)) → Expression Γ (tuple m ts)
+    call   : ∀ {t m Δ} → Function Δ t → All (Expression Γ) {m} Δ → Expression Γ t
     if_then_else_ : ∀ {t} → Expression Γ bool → Expression Γ t → Expression Γ t → Expression Γ t
 
   data CanAssign {n} {Γ} where
-    state : ∀ i {i<o : True (i ℕ.<? o)} → CanAssign (state i {i<o})
-    var   : ∀ i {i<n : True (i ℕ.<? n)} → CanAssign (var i {i<n})
-    abort : ∀ {t} {e : Expression Γ (fin 0)} → CanAssign (abort {t = t} e)
-    _∶_   : ∀ {m n t} {e₁ : Expression Γ (asType t m)} {e₂ : Expression Γ (asType t n)} → CanAssign e₁ → CanAssign e₂ → CanAssign (e₁ ∶ e₂)
-    [_]   : ∀ {t} {e : Expression Γ t} → CanAssign e → CanAssign [ e ]
-    unbox : ∀ {t} {e : Expression Γ (array t 1)} → CanAssign e → CanAssign (unbox e)
-    slice : ∀ {i j t} {e₁ : Expression Γ (asType t (i ℕ.+ j))} → CanAssign e₁ → (e₂ : Expression Γ (fin (suc i))) → CanAssign (slice e₁ e₂)
-    cast  : ∀ {i j t} {e : Expression Γ (asType t i)} .(eq : i ≡ j) → CanAssign e → CanAssign (cast eq e)
-    tup   : ∀ {m ts} {es : All (Expression Γ) {m} ts} → All (CanAssign ∘ proj₂) (reduce (λ {x} e → x , e) es) → CanAssign (tup es)
+    state : ∀ i → CanAssign (state i)
+    var   : ∀ i → CanAssign (var i)
+    abort : ∀ {t} e → CanAssign (abort {t = t} e)
+    [_]   : ∀ {t e} → CanAssign e → CanAssign ([_] {t = t} e)
+    unbox : ∀ {t e} → CanAssign e → CanAssign (unbox {t = t} e)
+    splice : ∀ {m n t e₁ e₂} → CanAssign e₁ → CanAssign e₂ → ∀ e₃ → CanAssign (splice {m = m} {n} {t} e₁ e₂ e₃)
+    cut    : ∀ {m n t e₁} → CanAssign e₁ → ∀ e₂ → CanAssign (cut {m = m} {n} {t} e₁ e₂)
+    cast  : ∀ {i j t e} .(eq : i ≡ j) → CanAssign e → CanAssign (cast {t = t} eq e)
+    nil   : CanAssign nil
+    cons  : ∀ {m t ts e₁ e₂} → CanAssign e₁ → CanAssign e₂ → CanAssign (cons {m = m} {t} {ts} e₁ e₂)
+    head  : ∀ {m t ts e} → CanAssign e → CanAssign (head {m = m} {t} {ts} e)
+    tail  : ∀ {m t ts e} → CanAssign e → CanAssign (tail {m = m} {t} {ts} e)
 
   canAssign? (lit x)      = no λ ()
   canAssign? (state i)    = yes (state i)
   canAssign? (var i)      = yes (var i)
-  canAssign? (abort e)    = yes abort
+  canAssign? (abort e)    = yes (abort e)
   canAssign? (e ≟ e₁)     = no λ ()
   canAssign? (e <? e₁)    = no λ ()
   canAssign? (inv e)      = no λ ()
@@ -173,8 +180,8 @@ module Code {o} (Σ : Vec Type o) where
   canAssign? (e or e₁)    = no λ ()
   canAssign? [ e ]        = map′ [_] (λ { [ e ] → e }) (canAssign? e)
   canAssign? (unbox e)    = map′ unbox (λ { (unbox e) → e }) (canAssign? e)
-  canAssign? (e ∶ e₁)     = map′ (uncurry _∶_) (λ { (e ∶ e₁) → e , e₁ }) (canAssign? e ×-dec canAssign? e₁)
-  canAssign? (slice e e₁) = map′ (λ e → slice e e₁) (λ { (slice e e₁) → e }) (canAssign? e)
+  canAssign? (splice e e₁ e₂) = map′ (λ (e , e₁) → splice e e₁ e₂) (λ { (splice e e₁ _) → e , e₁ }) (canAssign? e ×-dec canAssign? e₁)
+  canAssign? (cut e e₁)       = map′ (λ e → cut e e₁) (λ { (cut e e₁) → e }) (canAssign? e)
   canAssign? (cast eq e)  = map′ (cast eq) (λ { (cast eq e) → e }) (canAssign? e)
   canAssign? (- e)        = no λ ()
   canAssign? (e + e₁)     = no λ ()
@@ -185,38 +192,60 @@ module Code {o} (Σ : Vec Type o) where
   canAssign? (rnd e)      = no λ ()
   canAssign? (fin x e)    = no λ ()
   canAssign? (asInt e)    = no λ ()
-  canAssign? (tup es)     = map′ tup (λ { (tup es) → es }) (canAssignAll? es)
+  canAssign? nil          = yes nil
+  canAssign? (cons e e₁)  = map′ (uncurry cons) (λ { (cons e e₁) → e , e₁ }) (canAssign? e ×-dec canAssign? e₁)
+  canAssign? (head e)     = map′ head (λ { (head e) → e }) (canAssign? e)
+  canAssign? (tail e)     = map′ tail (λ { (tail e) → e }) (canAssign? e)
   canAssign? (call x e)   = no λ ()
   canAssign? (if e then e₁ else e₂) = no λ ()
 
-  canAssignAll? []       = yes []
-  canAssignAll? (e ∷ es) = map′ (uncurry _∷_) (λ { (e ∷ es) → e , es }) (canAssign? e ×-dec canAssignAll? es)
-
-  data AssignsState where
-    state : ∀ i {i<o} → AssignsState (state i {i<o})
-    _∶ˡ_ : ∀ {m n t e₁ e₂ a₁} → AssignsState a₁ → ∀ a₂ → AssignsState (_∶_ {m = m} {n} {t} {e₁} {e₂} a₁ a₂)
-    _∶ʳ_ : ∀ {m n t e₁ e₂} a₁ {a₂} → AssignsState a₂ → AssignsState (_∶_ {m = m} {n} {t} {e₁} {e₂} a₁ a₂)
-    [_] : ∀ {t e a} → AssignsState a → AssignsState ([_] {t = t} {e} a)
-    unbox : ∀ {t e a} → AssignsState a → AssignsState (unbox {t = t} {e} a)
-    slice : ∀ {i j t e₁ a} → AssignsState a → ∀ e₂ → AssignsState (slice {i = i} {j} {t} {e₁} a e₂)
-    cast : ∀ {i j t e} .(eq : i ≡ j) {a} → AssignsState a → AssignsState (cast {t = t} {e} eq a)
-    tup : ∀ {m ts es as} → Any (AssignsState ∘ proj₂) (reduce (λ {x} a → x , a) as) → AssignsState (tup {m = m} {ts} {es} as)
-
-  assignsState? (state i)    = yes (state i)
-  assignsState? (var i)      = no λ ()
-  assignsState? abort        = no λ ()
-  assignsState? (a ∶ a₁)     = map′ [ (_∶ˡ a₁) , (a ∶ʳ_) ]′ (λ { (s ∶ˡ _) → inj₁ s ; (_ ∶ʳ s) → inj₂ s }) (assignsState? a ⊎-dec assignsState? a₁)
-  assignsState? [ a ]        = map′ [_] (λ { [ s ] → s }) (assignsState? a)
-  assignsState? (unbox a)    = map′ unbox (λ { (unbox s) → s }) (assignsState? a)
-  assignsState? (slice a e₂) = map′ (λ s → slice s e₂ ) (λ { (slice s _) → s }) (assignsState? a)
-  assignsState? (cast eq a)  = map′ (cast eq) (λ { (cast _ s) → s }) (assignsState? a)
-  assignsState? (tup as)     = map′ tup (λ { (tup ss) → ss }) (assignsStateAny? as)
-
-  assignsStateAny? {es = []}     []       = no λ ()
-  assignsStateAny? {es = e ∷ es} (a ∷ as) = map′ [ here , there ]′ (λ { (here s) → inj₁ s ; (there ss) → inj₂ ss }) (assignsState? a ⊎-dec assignsStateAny? as)
+  data ReferencesState where
+    state : ∀ i → ReferencesState (state i)
+    [_] : ∀ {t e} → ReferencesState e → ReferencesState ([_] {t = t} e)
+    unbox : ∀ {t e} → ReferencesState e → ReferencesState (unbox {t = t} e)
+    spliceˡ : ∀ {m n t e} → ReferencesState e → ∀ e₁ e₂ → ReferencesState (splice {m = m} {n} {t} e e₁ e₂)
+    spliceʳ : ∀ {m n t} e {e₁} → ReferencesState e₁ → ∀ e₂ → ReferencesState (splice {m = m} {n} {t} e e₁ e₂)
+    cut : ∀ {m n t e} → ReferencesState e → ∀ e₁ → ReferencesState (cut {m = m} {n} {t} e e₁)
+    cast : ∀ {i j t} .(eq : i ≡ j) {e} → ReferencesState e → ReferencesState (cast {t = t} eq e)
+    consˡ : ∀ {m t ts e} → ReferencesState e → ∀ e₁ → ReferencesState (cons {m = m} {t} {ts} e e₁)
+    consʳ : ∀ {m t ts} e {e₁} → ReferencesState e₁ → ReferencesState (cons {m = m} {t} {ts} e e₁)
+    head : ∀ {m t ts e} → ReferencesState e → ReferencesState (head {m = m} {t} {ts} e)
+    tail : ∀ {m t ts e} → ReferencesState e → ReferencesState (tail {m = m} {t} {ts} e)
+
+  referencesState? (lit x) = no λ ()
+  referencesState? (state i) = yes (state i)
+  referencesState? (var i) = no λ ()
+  referencesState? (abort e) = no λ ()
+  referencesState? (e ≟ e₁) = no λ ()
+  referencesState? (e <? e₁) = no λ ()
+  referencesState? (inv e) = no λ ()
+  referencesState? (e && e₁) = no λ ()
+  referencesState? (e || e₁) = no λ ()
+  referencesState? (not e) = no λ ()
+  referencesState? (e and e₁) = no λ ()
+  referencesState? (e or e₁) = no λ ()
+  referencesState? [ e ] = map′ [_] (λ { [ e ] → e }) (referencesState? e)
+  referencesState? (unbox e) = map′ unbox (λ { (unbox e) → e }) (referencesState? e)
+  referencesState? (splice e e₁ e₂) = map′ [ (λ e → spliceˡ e e₁ e₂) , (λ e₁ → spliceʳ e e₁ e₂) ]′ (λ { (spliceˡ e e₁ e₂) → inj₁ e ; (spliceʳ e e₁ e₂) → inj₂ e₁ }) (referencesState? e ⊎-dec referencesState? e₁)
+  referencesState? (cut e e₁) = map′ (λ e → cut e e₁) (λ { (cut e e₁) → e }) (referencesState? e)
+  referencesState? (cast eq e) = map′ (cast eq) (λ { (cast eq e) → e }) (referencesState? e)
+  referencesState? (- e) = no λ ()
+  referencesState? (e + e₁) = no λ ()
+  referencesState? (e * e₁) = no λ ()
+  referencesState? (e ^ x) = no λ ()
+  referencesState? (e >> x) = no λ ()
+  referencesState? (rnd e) = no λ ()
+  referencesState? (fin x e) = no λ ()
+  referencesState? (asInt e) = no λ ()
+  referencesState? nil = no λ ()
+  referencesState? (cons e e₁) = map′ [ (λ e → consˡ e e₁) , (λ e₁ → consʳ e e₁) ]′ (λ { (consˡ e e₁) → inj₁ e ; (consʳ e e₁) → inj₂ e₁ }) (referencesState? e ⊎-dec referencesState? e₁)
+  referencesState? (head e) = map′ head (λ { (head e) → e }) (referencesState? e)
+  referencesState? (tail e) = map′ tail (λ { (tail e) → e }) (referencesState? e)
+  referencesState? (call f es) = no λ ()
+  referencesState? (if e then e₁ else e₂) = no λ ()
 
   infix  4 _≔_
-  infixl 2 if_then_else_
+  infixl 2 if_then_else_ if_then_
   infixl 1 _∙_ _∙return_
   infix  1 _∙end
 
@@ -225,27 +254,30 @@ module Code {o} (Σ : Vec Type o) where
     skip          : Statement Γ
     _≔_           : ∀ {t} → (ref : Expression Γ t) → {canAssign : True (canAssign? ref)} → Expression Γ t → Statement Γ
     declare       : ∀ {t} → Expression Γ t → Statement (t ∷ Γ) → Statement Γ
-    invoke        : ∀ {m Δ} → Procedure Δ → Expression Γ (tuple m Δ) → Statement Γ
+    invoke        : ∀ {m Δ} → Procedure Δ → All (Expression Γ) {m} Δ → Statement Γ
+    if_then_      : Expression Γ bool → Statement Γ → Statement Γ
     if_then_else_ : Expression Γ bool → Statement Γ → Statement Γ → Statement Γ
     for           : ∀ m → Statement (fin m ∷ Γ) → Statement Γ
 
   data ChangesState where
-    _∙ˡ_           : ∀ {s₁} → ChangesState s₁ → ∀ s₂ → ChangesState (s₁ ∙ s₂)
-    _∙ʳ_           : ∀ s₁ {s₂} → ChangesState s₂ → ChangesState (s₁ ∙ s₂)
-    _≔_            : ∀ {t ref} canAssign {assignsState : True (assignsState? (toWitness canAssign))} e₂ → ChangesState (_≔_ {t = t} ref {canAssign} e₂)
+    _∙ˡ_           : ∀ {s} → ChangesState s → ∀ s₁ → ChangesState (s ∙ s₁)
+    _∙ʳ_           : ∀ s {s₁} → ChangesState s₁ → ChangesState (s ∙ s₁)
+    _≔_            : ∀ {t} ref {canAssign : True (canAssign? ref)} {refsState : True (referencesState? ref)} e₂ → ChangesState (_≔_ {t = t} ref {canAssign} e₂)
     declare        : ∀ {t} e {s} → ChangesState s → ChangesState (declare {t = t} e s)
-    invoke         : ∀ {m Δ} p e → ChangesState (invoke {m = m} {Δ} p e)
-    if_then′_else_ : ∀ e {s₁} → ChangesState s₁ → ∀ s₂ → ChangesState (if e then s₁ else s₂)
-    if_then_else′_ : ∀ e s₁ {s₂} → ChangesState s₂ → ChangesState (if e then s₁ else s₂)
+    invoke         : ∀ {m Δ} p es → ChangesState (invoke {m = m} {Δ} p es)
+    if_then_       : ∀ e {s} → ChangesState s → ChangesState (if e then s)
+    if_then′_else_ : ∀ e {s} → ChangesState s → ∀ s₁ → ChangesState (if e then s else s₁)
+    if_then_else′_ : ∀ e s {s₁} → ChangesState s₁ → ChangesState (if e then s else s₁)
     for            : ∀ m {s} → ChangesState s → ChangesState (for m s)
 
-  changesState? (s₁ ∙ s₂)              = map′ [ _∙ˡ s₂ , s₁ ∙ʳ_ ]′ (λ { (s ∙ˡ _) → inj₁ s ; (_ ∙ʳ s) → inj₂ s }) (changesState? s₁ ⊎-dec changesState? s₂)
-  changesState? skip                   = no λ ()
-  changesState? (_≔_ ref {a} e)        = map′ (λ s → _≔_ a {fromWitness s} e) (λ { (_≔_ _ {s} _) → toWitness s }) (assignsState? (toWitness a))
-  changesState? (declare e s)          = map′ (declare e) (λ { (declare e s) → s }) (changesState? s)
-  changesState? (invoke p e)           = yes (invoke p e)
-  changesState? (if e then s₁ else s₂) = map′ [ if e then′_else s₂ , if e then s₁ else′_ ]′ (λ { (if _ then′ s else _) → inj₁ s ; (if _ then _ else′ s) → inj₂ s }) (changesState? s₁ ⊎-dec changesState? s₂)
-  changesState? (for m s)              = map′ (for m) (λ { (for m s) → s }) (changesState? s)
+  changesState? (s ∙ s₁)              = map′ [ _∙ˡ s₁ , s ∙ʳ_ ]′ (λ { (s ∙ˡ s₁) → inj₁ s ; (s ∙ʳ s₁) → inj₂ s₁ }) (changesState? s ⊎-dec changesState? s₁)
+  changesState? skip                  = no λ ()
+  changesState? (_≔_ ref e)           = map′ (λ refsState → _≔_ ref {refsState = fromWitness refsState} e) (λ { (_≔_ ref {refsState = refsState} e) → toWitness refsState }) (referencesState? ref)
+  changesState? (declare e s)         = map′ (declare e) (λ { (declare e s) → s }) (changesState? s)
+  changesState? (invoke p e)          = yes (invoke p e)
+  changesState? (if e then s)         = map′ (if e then_) (λ { (if e then s) → s }) (changesState? s)
+  changesState? (if e then s else s₁) = map′ [ if e then′_else s₁ , if e then s else′_ ]′ (λ { (if e then′ s else s₁) → inj₁ s ; (if e then s else′ s₁) → inj₂ s₁ }) (changesState? s ⊎-dec changesState? s₁)
+  changesState? (for m s)             = map′ (for m) (λ { (for m s) → s }) (changesState? s)
 
   data Function Γ ret where
     _∙return_ : (s : Statement Γ) → {False (changesState? s)} → Expression Γ ret → Function Γ ret
@@ -253,36 +285,45 @@ module Code {o} (Σ : Vec Type o) where
 
   data Procedure Γ where
     _∙end   : Statement Γ → Procedure Γ
-    declare : ∀ {t} → Expression Γ t → Procedure (t ∷ Γ) → Procedure Γ
 
   infixl  6 _<<_
-  infixl 5 _-_
+  infixl 5 _-_ _∶_
+
+  tup : ∀ {n Γ m ts} → All (Expression {n} Γ) ts → Expression Γ (tuple m ts)
+  tup []       = nil
+  tup (e ∷ es) = cons e (tup es)
+
+  _∶_ : ∀ {n Γ i j t} → Expression {n} Γ (asType t j) → Expression Γ (asType t i) → Expression Γ (asType t (i ℕ.+ j))
+  e₁ ∶ e₂ = splice e₁ e₂ (lit (Fin.fromℕ _ ′f))
+
+  slice : ∀ {n Γ i j t} → Expression {n} Γ (asType t (i ℕ.+ j)) → Expression Γ (fin (suc i)) → Expression Γ (asType t j)
+  slice e₁ e₂ = head (cut e₁ e₂)
 
   slice′ : ∀ {n Γ i j t} → Expression {n} Γ (asType t (i ℕ.+ j)) → Expression Γ (fin (suc j)) → Expression Γ (asType t i)
   slice′ {i = i} e₁ e₂ = slice (cast (+-comm i _) e₁) e₂
 
-  _-_   : ∀ {n Γ t₁ t₂} {isNumeric₁ : True (isNumeric? t₁)} {isNumeric₂ : True (isNumeric? t₂)} → Expression {n} Γ t₁ → Expression Γ t₂ → Expression Γ (combineNumeric t₁ t₂ {isNumeric₁} {isNumeric₂})
+  _-_   : ∀ {n Γ t₁ t₂} {isNumeric₁ : True (isNumeric? t₁)} {isNumeric₂ : True (isNumeric? t₂)} → Expression {n} Γ t₁ → Expression Γ t₂ → Expression Γ (combineNumeric t₁ t₂ isNumeric₁ isNumeric₂)
   _-_ {isNumeric₂ = isNumeric₂} x y = x + (-_ {isNumeric = isNumeric₂} y)
 
   _<<_ : ∀ {n Γ} → Expression {n} Γ int → ℕ → Expression Γ int
   x << n = rnd (x * lit (2 ′r) ^ n)
 
   get : ∀ {n Γ} → ℕ → Expression {n} Γ int → Expression Γ bit
-  get i x = if x - x >> suc i << suc i <? lit (2 ′i) ^ i then lit ((false ∷ []) ′x) else lit ((true ∷ []) ′x)
+  get i x = if x - x >> suc i << suc i <? lit (2 ′i) ^ i then lit (false ′x) else lit (true ′x)
 
   uint : ∀ {n Γ m} → Expression {n} Γ (bits m) → Expression Γ int
   uint {m = zero}  x = lit (0 ′i)
   uint {m = suc m} x =
-    lit (2 ′i) * uint {m = m} (slice x (lit ((# 1) ′f))) +
-    ( if slice′ x (lit ((# 0) ′f)) ≟ lit ((true ∷ []) ′x)
+    lit (2 ′i) * uint {m = m} (slice x (lit (1F ′f))) +
+    ( if slice′ x (lit (0F ′f)) ≟ lit ((true ∷ []) ′xs)
       then lit (1 ′i)
       else lit (0 ′i))
 
   sint : ∀ {n Γ m} → Expression {n} Γ (bits m) → Expression Γ int
   sint {m = zero}        x = lit (0 ′i)
-  sint {m = suc zero}    x = if x ≟ lit ((true ∷ []) ′x) then - lit (1 ′i) else lit (0 ′i)
+  sint {m = suc zero}    x = if x ≟ lit ((true ∷ []) ′xs) then - lit (1 ′i) else lit (0 ′i)
   sint {m = suc (suc m)} x =
-    lit (2 ′i) * sint (slice {i = 1} x (lit ((# 1) ′f))) +
-    ( if slice′ x (lit ((# 0) ′f)) ≟ lit ((true ∷ []) ′x)
+    lit (2 ′i) * sint (slice {i = 1} x (lit (1F ′f))) +
+    ( if slice′ x (lit (0F ′f)) ≟ lit ((true ∷ []) ′xs)
       then lit (1 ′i)
       else lit (0 ′i))
-- 
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