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{-# OPTIONS --safe --without-K #-}
module Helium.Data.Pseudocode where
open import Algebra.Core
open import Data.Bool using (Bool; if_then_else_)
open import Data.Fin hiding (_+_; cast)
import Data.Fin.Properties as Finₚ
open import Data.Nat using (ℕ; zero; suc; _+_; _^_)
import Data.Vec as Vec
open import Level using (_⊔_) renaming (suc to ℓsuc)
open import Relation.Binary using (REL; Rel; Symmetric; Transitive; Decidable)
open import Relation.Binary.PropositionalEquality using (_≡_; refl; sym)
open import Relation.Nullary using (Dec; does)
open import Relation.Nullary.Decidable
private
map-False : ∀ {p q} {P : Set p} {Q : Set q} {P? : Dec P} {Q? : Dec Q} → (P → Q) → False Q? → False P?
map-False ⇒ f = fromWitnessFalse (λ x → toWitnessFalse f (⇒ x))
record RawPseudocode b₁ b₂ i₁ i₂ i₃ r₁ r₂ r₃ : Set (ℓsuc (b₁ ⊔ b₂ ⊔ i₁ ⊔ i₂ ⊔ i₃ ⊔ r₁ ⊔ r₂ ⊔ r₃)) where
infix 9 _^ᶻ_ _^ʳ_
infix 8 _⁻¹
infixr 7 _*ᶻ_ _*ʳ_
infix 6 -ᶻ_ -ʳ_
infixr 5 _+ᶻ_ _+ʳ_
infix 4 _≈ᵇ_ _≟ᵇ_ _≈ᶻ_ _≟ᶻ_ _<ᶻ_ _<?ᶻ_ _≈ʳ_ _≟ʳ_ _<ʳ_ _<?ʳ_
field
-- Types
Bits : ℕ → Set b₁
ℤ : Set i₁
ℝ : Set r₁
field
-- Relations
_≈ᵇ_ : ∀ {m n} → REL (Bits m) (Bits n) b₂
_≟ᵇ_ : ∀ {m n} → Decidable (_≈ᵇ_ {m} {n})
_≈ᶻ_ : Rel ℤ i₂
_≟ᶻ_ : Decidable _≈ᶻ_
_<ᶻ_ : Rel ℤ i₃
_<?ᶻ_ : Decidable _<ᶻ_
_≈ʳ_ : Rel ℝ r₂
_≟ʳ_ : Decidable _≈ʳ_
_<ʳ_ : Rel ℝ r₃
_<?ʳ_ : Decidable _<ʳ_
-- Constants
[] : Bits 0
0b : Bits 1
1b : Bits 1
0ℤ : ℤ
1ℤ : ℤ
0ℝ : ℝ
1ℝ : ℝ
field
-- Bitstring operations
ofFin : ∀ {n} → Fin (2 ^ n) → Bits n
cast : ∀ {m n} → .(eq : m ≡ n) → Bits m → Bits n
not : ∀ {n} → Op₁ (Bits n)
_and_ : ∀ {n} → Op₂ (Bits n)
_or_ : ∀ {n} → Op₂ (Bits n)
_∶_ : ∀ {m n} → Bits m → Bits n → Bits (m + n)
sliceᵇ : ∀ {n} (i : Fin (suc n)) j → Bits n → Bits (toℕ (i - j))
updateᵇ : ∀ {n} (i : Fin (suc n)) j → Bits (toℕ (i - j)) → Op₁ (Bits n)
field
-- Arithmetic operations
⟦_⟧ : ℤ → ℝ
round : ℝ → ℤ
_+ᶻ_ : Op₂ ℤ
_+ʳ_ : Op₂ ℝ
_*ᶻ_ : Op₂ ℤ
_*ʳ_ : Op₂ ℝ
-ᶻ_ : Op₁ ℤ
-ʳ_ : Op₁ ℝ
_⁻¹ : ∀ (y : ℝ) → .{False (y ≟ʳ 0ℝ)} → ℝ
-- Convenience operations
zeros : ∀ {n} → Bits n
zeros {zero} = []
zeros {suc n} = 0b ∶ zeros
_eor_ : ∀ {n} → Op₂ (Bits n)
x eor y = (x or y) and not (x and y)
getᵇ : ∀ {n} → Fin n → Bits n → Bits 1
getᵇ i x = cast (eq i) (sliceᵇ (suc i) (inject₁ (strengthen i)) x)
where
eq : ∀ {n} (i : Fin n) → toℕ (suc i - inject₁ (strengthen i)) ≡ 1
eq zero = refl
eq (suc i) = eq i
setᵇ : ∀ {n} → Fin n → Bits 1 → Op₁ (Bits n)
setᵇ i y = updateᵇ (suc i) (inject₁ (strengthen i)) (cast (sym (eq i)) y)
where
eq : ∀ {n} (i : Fin n) → toℕ (suc i - inject₁ (strengthen i)) ≡ 1
eq zero = refl
eq (suc i) = eq i
hasBit : ∀ {n} → Fin n → Bits n → Bool
hasBit i x = does (getᵇ i x ≟ᵇ 1b)
-- Stray constant cannot live with the others, because + is not defined at that point.
2ℤ : ℤ
2ℤ = 1ℤ +ᶻ 1ℤ
_^ᶻ_ : ℤ → ℕ → ℤ
x ^ᶻ zero = 1ℤ
x ^ᶻ suc y = x *ᶻ x ^ᶻ y
_^ʳ_ : ℝ → ℕ → ℝ
x ^ʳ zero = 1ℝ
x ^ʳ suc y = x *ʳ x ^ʳ y
_<<_ : ℤ → ℕ → ℤ
x << n = x *ᶻ 2ℤ ^ᶻ n
uint : ∀ {n} → Bits n → ℤ
uint x = Vec.foldr (λ _ → ℤ) _+ᶻ_ 0ℤ (Vec.tabulate (λ i → if hasBit i x then 1ℤ << toℕ i else 0ℤ))
sint : ∀ {n} → Bits n → ℤ
sint {zero} x = 0ℤ
sint {suc n} x = uint x +ᶻ (if hasBit (fromℕ n) x then -ᶻ 1ℤ << suc n else 0ℤ)
module divmod
(≈ᶻ-trans : Transitive _≈ᶻ_)
(round∘⟦⟧ : ∀ x → x ≈ᶻ round ⟦ x ⟧)
(round-cong : ∀ {x y} → x ≈ʳ y → round x ≈ᶻ round y)
(0#-homo-round : round 0ℝ ≈ᶻ 0ℤ)
where
infix 7 _div_ _mod_
_div_ : ∀ (x y : ℤ) → .{≢0 : False (y ≟ᶻ 0ℤ)} → ℤ
(x div y) {≢0} =
let f = (λ y≈0 → ≈ᶻ-trans (round∘⟦⟧ y) (≈ᶻ-trans (round-cong y≈0) 0#-homo-round)) in
round (⟦ x ⟧ *ʳ (⟦ y ⟧ ⁻¹) {map-False f ≢0})
_mod_ : ∀ (x y : ℤ) → .{≢0 : False (y ≟ᶻ 0ℤ)} → ℤ
(x mod y) {≢0} = x +ᶻ -ᶻ y *ᶻ (x div y) {≢0}
module 2^n≢0
(≈ᶻ-trans : Transitive _≈ᶻ_)
(round∘⟦⟧ : ∀ x → x ≈ᶻ round ⟦ x ⟧)
(round-cong : ∀ {x y} → x ≈ʳ y → round x ≈ᶻ round y)
(0#-homo-round : round 0ℝ ≈ᶻ 0ℤ)
(2^n≢0 : ∀ n → False (2ℤ ^ᶻ n ≟ᶻ 0ℤ))
where
open divmod ≈ᶻ-trans round∘⟦⟧ round-cong 0#-homo-round public
_>>_ : ℤ → ℕ → ℤ
x >> n = (x div (2ℤ ^ᶻ n)) {2^n≢0 n}
getᶻ : ℕ → ℤ → Bits 1
getᶻ i x = if (does ((x mod (2ℤ ^ᶻ suc i)) {2^n≢0 (suc i)} <?ᶻ 2ℤ ^ᶻ i)) then 0b else 1b
module sliceᶻ
(≈ᶻ-trans : Transitive _≈ᶻ_)
(round∘⟦⟧ : ∀ x → x ≈ᶻ round ⟦ x ⟧)
(round-cong : ∀ {x y} → x ≈ʳ y → round x ≈ᶻ round y)
(0#-homo-round : round 0ℝ ≈ᶻ 0ℤ)
(2^n≢0 : ∀ n → False (2ℤ ^ᶻ n ≟ᶻ 0ℤ))
(*ᶻ-identityʳ : ∀ x → x *ᶻ 1ℤ ≈ᶻ x)
where
open 2^n≢0 ≈ᶻ-trans round∘⟦⟧ round-cong 0#-homo-round 2^n≢0 public
sliceᶻ : ∀ n i → ℤ → Bits (n ℕ-ℕ i)
sliceᶻ zero zero z = []
sliceᶻ (suc n) zero z = getᶻ n z ∶ sliceᶻ n zero z
sliceᶻ (suc n) (suc i) z = sliceᶻ n i ((z div 2ℤ) {2≢0})
where
2≢0 = map-False (≈ᶻ-trans (*ᶻ-identityʳ 2ℤ)) (2^n≢0 1)
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