From dd97e0a58b377161fb8e9ab7b5524f63b875612c Mon Sep 17 00:00:00 2001
From: Greg Brown <greg.brown@cl.cam.ac.uk>
Date: Sat, 19 Mar 2022 17:26:39 +0000
Subject: Add definition of Hoare logic semantics.

---
 src/Helium/Semantics/Axiomatic/Term.agda | 385 +++++++++++++++++++++++++++++++
 1 file changed, 385 insertions(+)
 create mode 100644 src/Helium/Semantics/Axiomatic/Term.agda

(limited to 'src/Helium/Semantics/Axiomatic/Term.agda')

diff --git a/src/Helium/Semantics/Axiomatic/Term.agda b/src/Helium/Semantics/Axiomatic/Term.agda
new file mode 100644
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--- /dev/null
+++ b/src/Helium/Semantics/Axiomatic/Term.agda
@@ -0,0 +1,385 @@
+------------------------------------------------------------------------
+-- Agda Helium
+--
+-- Definition of terms for use in assertions
+------------------------------------------------------------------------
+
+{-# OPTIONS --safe --without-K #-}
+
+open import Helium.Data.Pseudocode.Types using (RawPseudocode)
+
+module Helium.Semantics.Axiomatic.Term
+  {b₁ b₂ i₁ i₂ i₃ r₁ r₂ r₃}
+  (rawPseudocode : RawPseudocode b₁ b₂ i₁ i₂ i₃ r₁ r₂ r₃)
+  where
+
+open RawPseudocode rawPseudocode
+
+import Data.Bool as Bool
+open import Data.Fin as Fin using (Fin; suc)
+import Data.Fin.Properties as Finₚ
+open import Data.Fin.Patterns
+open import Data.Nat as ℕ using (ℕ; suc)
+import Data.Nat.Properties as ℕₚ
+open import Data.Product using (∃; _×_; _,_; proj₁; proj₂; uncurry; dmap)
+open import Data.Vec as Vec using (Vec; []; _∷_; _++_; lookup)
+import Data.Vec.Functional as VecF
+import Data.Vec.Properties as Vecₚ
+open import Data.Vec.Relation.Unary.All as All using (All; []; _∷_)
+open import Function using (_∘_; _∘₂_; id; flip)
+open import Helium.Data.Pseudocode.Core
+import Helium.Data.Pseudocode.Manipulate as M
+open import Helium.Semantics.Axiomatic.Core rawPseudocode
+open import Level using (_⊔_; lift; lower)
+open import Relation.Binary.PropositionalEquality hiding ([_]) renaming (subst to ≡-subst)
+open import Relation.Nullary using (does; yes; no; ¬_)
+open import Relation.Nullary.Decidable.Core using (True; toWitness)
+open import Relation.Nullary.Negation using (contradiction)
+
+private
+  variable
+    t t′ t₁ t₂ : Type
+    m n o      : ℕ
+    Γ Δ Σ ts   : Vec Type m
+
+data Term (Σ : Vec Type o) (Γ : Vec Type n) (Δ : Vec Type m) : Type → Set (b₁ ⊔ i₁ ⊔ r₁) where
+  state : ∀ i → Term Σ Γ Δ (lookup Σ i)
+  var   : ∀ i → Term Σ Γ Δ (lookup Γ i)
+  meta  : ∀ i → Term Σ Γ Δ (lookup Δ i)
+  func  : Transform ts t → All (Term Σ Γ Δ) ts → Term Σ Γ Δ t
+
+castType : Term Σ Γ Δ t → t ≡ t′ → Term Σ Γ Δ t′
+castType (state i)   refl = state i
+castType (var i)     refl = var i
+castType (meta i)    refl = meta i
+castType (func f ts) eq   = func (≡-subst (Transform _) eq f) ts
+
+substState : Term Σ Γ Δ t → ∀ i → Term Σ Γ Δ (lookup Σ i) → Term Σ Γ Δ t
+substState (state i) j t′ with i Fin.≟ j
+... | yes refl = t′
+... | no _     = state i
+substState (var i) j t′ = var i
+substState (meta i) j t′ = meta i
+substState (func f ts) j t′ = func f (helper ts j t′)
+  where
+  helper : ∀ {n ts} → All (Term Σ Γ Δ) {n} ts → ∀ i → Term Σ Γ Δ (lookup Σ i) → All (Term Σ Γ Δ) ts
+  helper []       i t′ = []
+  helper (t ∷ ts) i t′ = substState t i t′ ∷ helper ts i t′
+
+substVar : Term Σ Γ Δ t → ∀ i → Term Σ Γ Δ (lookup Γ i) → Term Σ Γ Δ t
+substVar (state i) j t′ = state i
+substVar (var i) j t′ with i Fin.≟ j
+... | yes refl = t′
+... | no _     = var i
+substVar (meta i) j t′ = meta i
+substVar (func f ts) j t′ = func f (helper ts j t′)
+  where
+  helper : ∀ {n ts} → All (Term Σ Γ Δ) {n} ts → ∀ i → Term Σ Γ Δ (lookup Γ i) → All (Term Σ Γ Δ) ts
+  helper []       i t′ = []
+  helper (t ∷ ts) i t′ = substVar t i t′ ∷ helper ts i t′
+
+substVars : Term Σ Γ Δ t → All (Term Σ ts Δ) Γ → Term Σ ts Δ t
+substVars (state i)    ts = state i
+substVars (var i)      ts = All.lookup i ts
+substVars (meta i)     ts = meta i
+substVars (func f ts′) ts = func f (helper ts′ ts)
+  where
+  helper : ∀ {ts ts′} → All (Term Σ Γ Δ) {n} ts → All (Term {n = m} Σ ts′ Δ) Γ → All (Term Σ ts′ Δ) ts
+  helper []        ts = []
+  helper (t ∷ ts′) ts = substVars t ts ∷ helper ts′ ts
+
+elimVar : Term Σ (t′ ∷ Γ) Δ t → Term Σ Γ Δ t′ → Term Σ Γ Δ t
+elimVar (state i)     t′ = state i
+elimVar (var 0F)      t′ = t′
+elimVar (var (suc i)) t′ = var i
+elimVar (meta i)      t′ = meta i
+elimVar (func f ts)   t′ = func f (helper ts t′)
+  where
+  helper : ∀ {n ts} → All (Term Σ (_ ∷ Γ) Δ) {n} ts → Term Σ Γ Δ _ → All (Term Σ Γ Δ) ts
+  helper []       t′ = []
+  helper (t ∷ ts) t′ = elimVar t t′ ∷ helper ts t′
+
+wknVar : Term Σ Γ Δ t → Term Σ (t′ ∷ Γ) Δ t
+wknVar (state i)   = state i
+wknVar (var i)     = var (suc i)
+wknVar (meta i)    = meta i
+wknVar (func f ts) = func f (helper ts)
+  where
+  helper : ∀ {n ts} → All (Term Σ Γ Δ) {n} ts → All (Term Σ (_ ∷ Γ) Δ) ts
+  helper []       = []
+  helper (t ∷ ts) = wknVar t ∷ helper ts
+
+wknVars : (ts : Vec Type n) → Term Σ Γ Δ t → Term Σ (ts ++ Γ) Δ t
+wknVars τs (state i)   = state i
+wknVars τs (var i)     = castType (var (Fin.raise (Vec.length τs) i)) (Vecₚ.lookup-++ʳ τs _ i)
+wknVars τs (meta i)    = meta i
+wknVars τs (func f ts) = func f (helper ts)
+  where
+  helper : ∀ {n ts} → All (Term Σ Γ Δ) {n} ts → All (Term Σ (τs ++ Γ) Δ) ts
+  helper []       = []
+  helper (t ∷ ts) = wknVars τs t ∷ helper ts
+
+addVars : Term Σ [] Δ t → Term Σ Γ Δ t
+addVars (state i)   = state i
+addVars (meta i)    = meta i
+addVars (func f ts) = func f (helper ts)
+  where
+  helper : ∀ {n ts} → All (Term Σ [] Δ) {n} ts → All (Term Σ Γ Δ) ts
+  helper []       = []
+  helper (t ∷ ts) = addVars t ∷ helper ts
+
+wknMetaAt : ∀ i → Term Σ Γ Δ t → Term Σ Γ (Vec.insert Δ i t′) t
+wknMetaAt′ : ∀ i → All (Term Σ Γ Δ) ts → All (Term Σ Γ (Vec.insert Δ i t′)) ts
+
+wknMetaAt i (state j)   = state j
+wknMetaAt i (var j)     = var j
+wknMetaAt i (meta j)    = castType (meta (Fin.punchIn i j)) (Vecₚ.insert-punchIn _ i _ j)
+wknMetaAt i (func f ts) = func f (wknMetaAt′ i ts)
+
+wknMetaAt′ i []       = []
+wknMetaAt′ i (t ∷ ts) = wknMetaAt i t ∷ wknMetaAt′ i ts
+
+wknMeta : Term Σ Γ Δ t → Term Σ Γ (t′ ∷ Δ) t
+wknMeta = wknMetaAt 0F
+
+wknMeta′ : All (Term Σ Γ Δ) ts → All (Term Σ Γ (t′ ∷ Δ)) ts
+wknMeta′ = wknMetaAt′ 0F
+
+wknMetas : (ts : Vec Type n) → Term Σ Γ Δ t → Term Σ Γ (ts ++ Δ) t
+wknMetas τs (state i)   = state i
+wknMetas τs (var i)     = var i
+wknMetas τs (meta i)    = castType (meta (Fin.raise (Vec.length τs) i)) (Vecₚ.lookup-++ʳ τs _ i)
+wknMetas τs (func f ts) = func f (helper ts)
+  where
+  helper : ∀ {n ts} → All (Term Σ Γ Δ) {n} ts → All (Term Σ Γ (τs ++ Δ)) ts
+  helper []       = []
+  helper (t ∷ ts) = wknMetas τs t ∷ helper ts
+
+func₀ : ⟦ t ⟧ₜ → Term Σ Γ Δ t
+func₀ f = func (λ _ → f) []
+
+func₁ : (⟦ t ⟧ₜ → ⟦ t′ ⟧ₜ) → Term Σ Γ Δ t → Term Σ Γ Δ t′
+func₁ f t = func (λ (x , _) → f x) (t ∷ [])
+
+func₂ : (⟦ t₁ ⟧ₜ → ⟦ t₂ ⟧ₜ → ⟦ t′ ⟧ₜ) → Term Σ Γ Δ t₁ → Term Σ Γ Δ t₂ → Term Σ Γ Δ t′
+func₂ f t₁ t₂ = func (λ (x , y , _) → f x y) (t₁ ∷ t₂ ∷ [])
+
+[_][_≟_] : HasEquality t → Term Σ Γ Δ t → Term Σ Γ Δ t → Term Σ Γ Δ bool
+[ bool ][ t ≟ t′ ] = func₂ (λ (lift x) (lift y) → lift (does (x Bool.≟ y))) t t′
+[ int ][ t ≟ t′ ] = func₂ (λ (lift x) (lift y) → lift (does (x ≟ᶻ y))) t t′
+[ fin ][ t ≟ t′ ] = func₂ (λ (lift x) (lift y) → lift (does (x Fin.≟ y))) t t′
+[ real ][ t ≟ t′ ] = func₂ (λ (lift x) (lift y) → lift (does (x ≟ʳ y))) t t′
+[ bit ][ t ≟ t′ ] = func₂ (λ (lift x) (lift y) → lift (does (x ≟ᵇ₁ y))) t t′
+[ bits ][ t ≟ t′ ] = func₂ (λ xs ys → lift (does (VecF.fromVec (Vec.map lower xs) ≟ᵇ VecF.fromVec (Vec.map lower ys)))) t t′
+
+[_][_<?_] : IsNumeric t → Term Σ Γ Δ t → Term Σ Γ Δ t → Term Σ Γ Δ bool
+[ int ][ t <? t′ ] = func₂ (λ (lift x) (lift y) → lift (does (x <ᶻ? y))) t t′
+[ real ][ t <? t′ ] = func₂ (λ (lift x) (lift y) → lift (does (x <ʳ? y))) t t′
+
+[_][_] : ∀ t → Term Σ Γ Δ (elemType t) → Term Σ Γ Δ (asType t 1)
+[ bits ][ t ] = func₁ (_∷ []) t
+[ array τ ][ t ] = func₁ (_∷ []) t
+
+unbox : ∀ t → Term Σ Γ Δ (asType t 1) → Term Σ Γ Δ (elemType t)
+unbox bits = func₁ Vec.head
+unbox (array t) = func₁ Vec.head
+
+castV : ∀ {a} {A : Set a} {m n} → .(eq : m ≡ n) → Vec A m → Vec A n
+castV {n = 0}     eq []       = []
+castV {n = suc n} eq (x ∷ xs) = x ∷ castV (ℕₚ.suc-injective eq) xs
+
+cast′ : ∀ t → .(eq : m ≡ n) → ⟦ asType t m ⟧ₜ → ⟦ asType t n ⟧ₜ
+cast′ bits      eq = castV eq
+cast′ (array τ) eq = castV eq
+
+cast : ∀ t → .(eq : m ≡ n) → Term Σ Γ Δ (asType t m) → Term Σ Γ Δ (asType t n)
+cast τ eq = func₁ (cast′ τ eq)
+
+[_][-_] : IsNumeric t → Term Σ Γ Δ t → Term Σ Γ Δ t
+[ int ][- t ] = func₁ (lift ∘ ℤ.-_ ∘ lower) t
+[ real ][- t ] = func₁ (lift ∘ ℝ.-_ ∘ lower) t
+
+[_,_,_,_][_+_] : ∀ t₁ t₂ → (isNum₁ : True (isNumeric? t₁)) → (isNum₂ : True (isNumeric? t₂)) → Term Σ Γ Δ t₁ → Term Σ Γ Δ t₂ → Term Σ Γ Δ (combineNumeric t₁ t₂ isNum₁ isNum₂)
+[ int , int , _ , _ ][ t + t′ ] = func₂ (λ (lift x) (lift y) → lift (x ℤ.+ y)) t t′
+[ int , real , _ , _ ][ t + t′ ] = func₂ (λ (lift x) (lift y) → lift (x /1 ℝ.+ y)) t t′
+[ real , int , _ , _ ][ t + t′ ] = func₂ (λ (lift x) (lift y) → lift (x ℝ.+ y /1)) t t′
+[ real , real , _ , _ ][ t + t′ ] = func₂ (λ (lift x) (lift y) → lift (x ℝ.+ y)) t t′
+
+[_,_,_,_][_*_] : ∀ t₁ t₂ → (isNum₁ : True (isNumeric? t₁)) → (isNum₂ : True (isNumeric? t₂)) → Term Σ Γ Δ t₁ → Term Σ Γ Δ t₂ → Term Σ Γ Δ (combineNumeric t₁ t₂ isNum₁ isNum₂)
+[ int , int , _ , _ ][ t * t′ ] = func₂ (λ (lift x) (lift y) → lift (x ℤ.* y)) t t′
+[ int , real , _ , _ ][ t * t′ ] = func₂ (λ (lift x) (lift y) → lift (x /1 ℝ.* y)) t t′
+[ real , int , _ , _ ][ t * t′ ] = func₂ (λ (lift x) (lift y) → lift (x ℝ.* y /1)) t t′
+[ real , real , _ , _ ][ t * t′ ] = func₂ (λ (lift x) (lift y) → lift (x ℝ.* y)) t t′
+
+[_][_^_] : IsNumeric t → Term Σ Γ Δ t → ℕ → Term Σ Γ Δ t
+[ int ][ t ^ n ] = func₁ (lift ∘ (ℤ′._^′ n) ∘ lower) t
+[ real ][ t ^ n ] = func₁ (lift ∘ (ℝ′._^′ n) ∘ lower) t
+
+2≉0 : Set _
+2≉0 = ¬ 2 ℝ′.×′ 1ℝ ℝ.≈ 0ℝ
+
+[_][_>>_] : 2≉0 → Term Σ Γ Δ int → ℕ → Term Σ Γ Δ int
+[ 2≉0 ][ t >> n ] = func₁ (lift ∘ ⌊_⌋ ∘ (ℝ._* 2≉0 ℝ.⁻¹ ℝ′.^′ n) ∘ _/1 ∘ lower) t
+
+-- 0 of y is 0 of result
+join′ : ∀ t → ⟦ asType t m ⟧ₜ → ⟦ asType t n ⟧ₜ → ⟦ asType t (n ℕ.+ m) ⟧ₜ
+join′ bits      = flip _++_
+join′ (array t) = flip _++_
+
+take′ : ∀ t m {n} → ⟦ asType t (m ℕ.+ n) ⟧ₜ → ⟦ asType t m ⟧ₜ
+take′ bits      m = Vec.take m
+take′ (array t) m = Vec.take m
+
+drop′ : ∀ t m {n} → ⟦ asType t (m ℕ.+ n) ⟧ₜ → ⟦ asType t n ⟧ₜ
+drop′ bits      m = Vec.drop m
+drop′ (array t) m = Vec.drop m
+
+private
+  m≤n⇒m+k≡n : ∀ {m n} → m ℕ.≤ n → ∃ λ k → m ℕ.+ k ≡ n
+  m≤n⇒m+k≡n ℕ.z≤n       = _ , refl
+  m≤n⇒m+k≡n (ℕ.s≤s m≤n) = dmap id (cong suc) (m≤n⇒m+k≡n m≤n)
+
+  slicedSize : ∀ n m (i : Fin (suc n)) → ∃ λ k → n ℕ.+ m ≡ Fin.toℕ i ℕ.+ (m ℕ.+ k) × Fin.toℕ i ℕ.+ k ≡ n
+  slicedSize n m i = k , (begin
+    n ℕ.+ m                 ≡˘⟨ cong (ℕ._+ m) (proj₂ i+k≡n) ⟩
+    (Fin.toℕ i ℕ.+ k) ℕ.+ m ≡⟨  ℕₚ.+-assoc (Fin.toℕ i) k m ⟩
+    Fin.toℕ i ℕ.+ (k ℕ.+ m) ≡⟨  cong (Fin.toℕ i ℕ.+_) (ℕₚ.+-comm k m) ⟩
+    Fin.toℕ i ℕ.+ (m ℕ.+ k) ∎) ,
+    proj₂ i+k≡n
+    where
+    open ≡-Reasoning
+    i+k≡n = m≤n⇒m+k≡n (ℕₚ.≤-pred (Finₚ.toℕ<n i))
+    k = proj₁ i+k≡n
+
+-- 0 of x is i of result
+splice′ : ∀ t → ⟦ asType t m ⟧ₜ → ⟦ asType t n ⟧ₜ → ⟦ fin (suc n) ⟧ₜ → ⟦ asType t (n ℕ.+ m) ⟧ₜ
+splice′ {m = m} t x y (lift i) = cast′ t eq (join′ t (join′ t high x) low)
+  where
+  reasoning = slicedSize _ m i
+  k = proj₁ reasoning
+  n≡i+k = sym (proj₂ (proj₂ reasoning))
+  low = take′ t (Fin.toℕ i) (cast′ t n≡i+k y)
+  high = drop′ t (Fin.toℕ i) (cast′ t n≡i+k y)
+  eq = sym (proj₁ (proj₂ reasoning))
+
+splice : ∀ t → Term Σ Γ Δ (asType t m) → Term Σ Γ Δ (asType t n) → Term Σ Γ Δ (fin (suc n)) → Term Σ Γ Δ (asType t (n ℕ.+ m))
+splice τ t₁ t₂ t′ = func (λ (x , y , i , _) → splice′ τ x y i) (t₁ ∷ t₂ ∷ t′ ∷ [])
+
+-- i of x is 0 of first
+cut′ : ∀ t → ⟦ asType t (n ℕ.+ m) ⟧ₜ → ⟦ fin (suc n) ⟧ₜ → ⟦ asType t m ∷ asType t n ∷ [] ⟧ₜ′
+cut′ {m = m} t x (lift i) = middle , cast′ t i+k≡n (join′ t high low) , _
+  where
+  reasoning = slicedSize _ m i
+  k = proj₁ reasoning
+  i+k≡n = proj₂ (proj₂ reasoning)
+  eq = proj₁ (proj₂ reasoning)
+  low = take′ t (Fin.toℕ i) (cast′ t eq x)
+  middle = take′ t m (drop′ t (Fin.toℕ i) (cast′ t eq x))
+  high = drop′ t m (drop′ t (Fin.toℕ i) (cast′ t eq x))
+
+cut : ∀ t → Term Σ Γ Δ (asType t (n ℕ.+ m)) → Term Σ Γ Δ (fin (suc n)) → Term Σ Γ Δ (tuple _ (asType t m ∷ asType t n ∷ []))
+cut τ t t′ = func₂ (cut′ τ) t t′
+
+flatten : ∀ {ms : Vec ℕ n} → ⟦ Vec.map fin ms ⟧ₜ′ → All Fin ms
+flatten {ms = []}     _             = []
+flatten {ms = _ ∷ ms} (lift x , xs) = x ∷ flatten xs
+
+𝒦 : Literal t → Term Σ Γ Δ t
+𝒦 (x ′b) = func₀ (lift x)
+𝒦 (x ′i) = func₀ (lift (x ℤ′.×′ 1ℤ))
+𝒦 (x ′f) = func₀ (lift x)
+𝒦 (x ′r) = func₀ (lift (x ℝ′.×′ 1ℝ))
+𝒦 (x ′x) = func₀ (lift (Bool.if x then 1𝔹 else 0𝔹))
+𝒦 ([] ′xs) = func₀ []
+𝒦 ((x ∷ xs) ′xs) = func₂ (flip Vec._∷ʳ_) (𝒦 (x ′x)) (𝒦 (xs ′xs))
+𝒦 (x ′a) = func₁ Vec.replicate (𝒦 x)
+
+module _ (2≉0 : 2≉0) where
+  ℰ : Code.Expression Σ Γ t → Term Σ Γ Δ t
+  ℰ e = (uncurry helper) (M.elimFunctions e)
+    where
+    1+m⊔n≡1+k : ∀ m n → ∃ λ k → suc m ℕ.⊔ n ≡ suc k
+    1+m⊔n≡1+k m 0       = m , refl
+    1+m⊔n≡1+k m (suc n) = m ℕ.⊔ n , refl
+
+    pull-0₂ : ∀ {x y} → x ℕ.⊔ y ≡ 0 → x ≡ 0
+    pull-0₂ {0} {0}     refl = refl
+    pull-0₂ {0} {suc y} ()
+
+    pull-0₃ : ∀ {x y z} → x ℕ.⊔ y ℕ.⊔ z ≡ 0 → x ≡ 0
+    pull-0₃ {0}     {0}     {0} refl = refl
+    pull-0₃ {0}     {suc y} {0}     ()
+    pull-0₃ {suc x} {0}     {0}     ()
+    pull-0₃ {suc x} {0}     {suc z} ()
+
+    pull-1₃ : ∀ x {y z} → x ℕ.⊔ y ℕ.⊔ z ≡ 0 → y ≡ 0
+    pull-1₃ 0       {0}     {0}     refl = refl
+    pull-1₃ 0       {suc y} {0}     ()
+    pull-1₃ (suc x) {0}     {0}     ()
+    pull-1₃ (suc x) {0}     {suc z} ()
+
+    pull-last : ∀ {x y} → x ℕ.⊔ y ≡ 0 → y ≡ 0
+    pull-last {0}     {0} refl = refl
+    pull-last {suc x} {0} ()
+
+    helper : ∀ (e : Code.Expression Σ Γ t) → M.callDepth e ≡ 0 → Term Σ Γ Δ t
+    helper (Code.lit x) eq = 𝒦 x
+    helper (Code.state i) eq = state i
+    helper (Code.var i) eq = var i
+    helper (Code.abort e) eq = func₁ (λ ()) (helper e eq)
+    helper (Code._≟_ {hasEquality = hasEq} e e₁) eq = [ toWitness hasEq ][ helper e (pull-0₂ eq) ≟ helper e₁ (pull-last eq) ]
+    helper (Code._<?_ {isNumeric = isNum} e e₁) eq = [ toWitness isNum ][ helper e (pull-0₂ eq) <? helper e₁ (pull-last eq) ]
+    helper (Code.inv e) eq = func₁ (lift ∘ Bool.not ∘ lower) (helper e eq)
+    helper (e Code.&& e₁) eq = func₂ (λ (lift x) (lift y) → lift (x Bool.∧ y)) (helper e (pull-0₂ eq)) (helper e₁ (pull-last eq))
+    helper (e Code.|| e₁) eq = func₂ (λ (lift x) (lift y) → lift (x Bool.∨ y)) (helper e (pull-0₂ eq)) (helper e₁ (pull-last eq))
+    helper (Code.not e) eq = func₁ (Vec.map (lift ∘ 𝔹.¬_ ∘ lower)) (helper e eq)
+    helper (e Code.and e₁) eq = func₂ (λ xs ys → Vec.zipWith (λ (lift x) (lift y) → lift (x 𝔹.∧ y)) xs ys) (helper e (pull-0₂ eq)) (helper e₁ (pull-last eq))
+    helper (e Code.or e₁) eq = func₂ (λ xs ys → Vec.zipWith (λ (lift x) (lift y) → lift (x 𝔹.∨ y)) xs ys) (helper e (pull-0₂ eq)) (helper e₁ (pull-last eq))
+    helper (Code.[_] {t = t} e) eq = [ t ][ helper e eq ]
+    helper (Code.unbox {t = t} e) eq = unbox t (helper e eq)
+    helper (Code.splice {t = t} e e₁ e₂) eq = splice t (helper e (pull-0₃ eq)) (helper e₁ (pull-1₃ (M.callDepth e) eq)) (helper e₂ (pull-last eq))
+    helper (Code.cut {t = t} e e₁) eq = cut t (helper e (pull-0₂ eq)) (helper e₁ (pull-last eq))
+    helper (Code.cast {t = t} i≡j e) eq = cast t i≡j (helper e eq)
+    helper (Code.-_ {isNumeric = isNum} e) eq = [ toWitness isNum ][- helper e eq ]
+    helper (Code._+_ {isNumeric₁ = isNum₁} {isNumeric₂ = isNum₂} e e₁) eq = [ _ , _ , isNum₁ , isNum₂ ][ helper e (pull-0₂ eq) + helper e₁ (pull-last eq) ]
+    helper (Code._*_ {isNumeric₁ = isNum₁} {isNumeric₂ = isNum₂} e e₁) eq = [ _ , _ , isNum₁ , isNum₂ ][ helper e (pull-0₂ eq) * helper e₁ (pull-last eq) ]
+    helper (Code._^_ {isNumeric = isNum} e y) eq = [ toWitness isNum ][ helper e eq ^ y ]
+    helper (e Code.>> n) eq = [ 2≉0 ][ helper e eq >> n ]
+    helper (Code.rnd e) eq = func₁ (lift ∘ ⌊_⌋ ∘ lower) (helper e eq)
+    helper (Code.fin f e) eq = func₁ (lift ∘ f ∘ flatten) (helper e eq)
+    helper (Code.asInt e) eq = func₁ (lift ∘ (ℤ′._×′ 1ℤ) ∘ Fin.toℕ ∘ lower) (helper e eq)
+    helper Code.nil eq = func₀ _
+    helper (Code.cons e e₁) eq = func₂ _,_ (helper e (pull-0₂ eq)) (helper e₁ (pull-last eq))
+    helper (Code.head e) eq = func₁ proj₁ (helper e eq)
+    helper (Code.tail e) eq = func₁ proj₂ (helper e eq)
+    helper (Code.call f es) eq = contradiction (trans (sym eq) (proj₂ (1+m⊔n≡1+k (M.funCallDepth f) (M.callDepth′ es)))) ℕₚ.0≢1+n
+    helper (Code.if e then e₁ else e₂) eq = func (λ (lift b , x , x₁ , _) → Bool.if b then x else x₁) (helper e (pull-0₃ eq) ∷ helper e₁ (pull-1₃ (M.callDepth e) eq) ∷ helper e₂ (pull-last eq) ∷ [])
+
+  ℰ′ : All (Code.Expression Σ Γ) ts → All (Term Σ Γ Δ) ts
+  ℰ′ []       = []
+  ℰ′ (e ∷ es) = ℰ e ∷ ℰ′ es
+
+  subst : Term Σ Γ Δ t → {e : Code.Expression Σ Γ t′} → Code.CanAssign Σ e → Term Σ Γ Δ t′ → Term Σ Γ Δ t
+  subst t (Code.state i) t′ = substState t i t′
+  subst t (Code.var i) t′ = substVar t i t′
+  subst t (Code.abort e) t′ = func₁ (λ ()) (ℰ e)
+  subst t (Code.[_] {t = τ} ref) t′ = subst t ref (unbox τ t′)
+  subst t (Code.unbox {t = τ} ref) t′ = subst t ref [ τ ][ t′ ]
+  subst t (Code.splice {t = τ} ref ref₁ e₃) t′ = subst (subst t ref (func₁ proj₁ (cut τ t′ (ℰ e₃)))) ref₁ (func₁ (proj₁ ∘ proj₂) (cut τ t′ (ℰ e₃)))
+  subst t (Code.cut {t = τ} ref e₂) t′ = subst t ref (splice τ (func₁ proj₁ t′) (func₁ (proj₁ ∘ proj₂) t′) (ℰ e₂))
+  subst t (Code.cast {t = τ} eq ref) t′ = subst t ref (cast τ (sym eq) t′)
+  subst t Code.nil t′ = t
+  subst t (Code.cons ref ref₁) t′ = subst (subst t ref (func₁ proj₁ t′)) ref₁ (func₁ proj₂ t′)
+  subst t (Code.head {e = e} ref) t′ = subst t ref (func₂ _,_ t′ (func₁ proj₂ (ℰ e)))
+  subst t (Code.tail {t = τ} {e = e} ref) t′ = subst t ref (func₂ {t₁ = τ} _,_ (func₁ proj₁ (ℰ e)) t′)
+
+⟦_⟧ : Term Σ Γ Δ t → ⟦ Σ ⟧ₜ′ → ⟦ Γ ⟧ₜ′ → ⟦ Δ ⟧ₜ′ → ⟦ t ⟧ₜ
+⟦_⟧′ : ∀ {ts} → All (Term Σ Γ Δ) {n} ts → ⟦ Σ ⟧ₜ′ → ⟦ Γ ⟧ₜ′ → ⟦ Δ ⟧ₜ′ → ⟦ ts ⟧ₜ′
+
+⟦ state i ⟧ σ γ δ = fetch i σ
+⟦ var i ⟧ σ γ δ = fetch i γ
+⟦ meta i ⟧ σ γ δ = fetch i δ
+⟦ func f ts ⟧ σ γ δ = f (⟦ ts ⟧′ σ γ δ)
+
+⟦ [] ⟧′     σ γ δ = _
+⟦ t ∷ ts ⟧′ σ γ δ = ⟦ t ⟧ σ γ δ , ⟦ ts ⟧′ σ γ δ
-- 
cgit v1.2.3