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------------------------------------------------------------------------
-- Agda Helium
--
-- Definition of types and operations used by the Armv8-M pseudocode.
------------------------------------------------------------------------
{-# OPTIONS --safe --without-K #-}
module Helium.Data.Pseudocode.Algebra where
open import Algebra.Core
import Algebra.Definitions.RawSemiring as RS
open import Data.Bool.Base using (Bool; if_then_else_)
open import Data.Empty using (⊥-elim)
open import Data.Fin.Base as Fin hiding (cast)
import Data.Fin.Properties as Fₚ
import Data.Fin.Induction as Induction
open import Data.Nat.Base using (ℕ; zero; suc)
open import Data.Sum using (_⊎_; inj₁; inj₂; [_,_]′)
open import Data.Vec.Functional
open import Data.Vec.Functional.Relation.Binary.Pointwise using (Pointwise)
import Data.Vec.Functional.Relation.Binary.Pointwise.Properties as Pwₚ
open import Function using (_$_; _∘′_; id)
open import Helium.Algebra.Ordered.StrictTotal.Bundles
open import Helium.Algebra.Decidable.Bundles
using (BooleanAlgebra; RawBooleanAlgebra)
import Helium.Algebra.Decidable.Construct.Pointwise as Pw
open import Helium.Algebra.Morphism.Structures
open import Level using (_⊔_) renaming (suc to ℓsuc)
open import Relation.Binary.Core using (Rel)
open import Relation.Binary.Definitions
open import Relation.Binary.PropositionalEquality as P using (_≡_)
import Relation.Binary.Reasoning.StrictPartialOrder as Reasoning
open import Relation.Binary.Structures using (IsStrictTotalOrder)
open import Relation.Nullary using (does; yes; no)
open import Relation.Nullary.Decidable.Core
using (False; toWitnessFalse; fromWitnessFalse)
record RawPseudocode b₁ b₂ i₁ i₂ i₃ r₁ r₂ r₃ : Set (ℓsuc (b₁ ⊔ b₂ ⊔ i₁ ⊔ i₂ ⊔ i₃ ⊔ r₁ ⊔ r₂ ⊔ r₃)) where
field
bitRawBooleanAlgebra : RawBooleanAlgebra b₁ b₂
integerRawRing : RawRing i₁ i₂ i₃
realRawField : RawField r₁ r₂ r₃
bitsRawBooleanAlgebra : ℕ → RawBooleanAlgebra b₁ b₂
bitsRawBooleanAlgebra = Pw.rawBooleanAlgebra bitRawBooleanAlgebra
module 𝔹 = RawBooleanAlgebra bitRawBooleanAlgebra
renaming (Carrier to Bit; ⊤ to 1𝔹; ⊥ to 0𝔹)
module Bits {n} = RawBooleanAlgebra (bitsRawBooleanAlgebra n)
renaming (⊤ to ones; ⊥ to zeros)
module ℤ = RawRing integerRawRing renaming (Carrier to ℤ; 1# to 1ℤ; 0# to 0ℤ)
module ℝ = RawField realRawField renaming (Carrier to ℝ; 1# to 1ℝ; 0# to 0ℝ)
module ℤ′ = RS ℤ.Unordered.rawSemiring
module ℝ′ = RS ℝ.Unordered.rawSemiring
Bits : ℕ → Set b₁
Bits n = Bits.Carrier {n}
open 𝔹 public using (Bit; 1𝔹; 0𝔹)
open Bits public using (ones; zeros)
open ℤ public using (ℤ; 1ℤ; 0ℤ)
open ℝ public using (ℝ; 1ℝ; 0ℝ)
infix 4 _≟ᶻ_ _<ᶻ?_ _≟ʳ_ _<ʳ?_ _≟ᵇ₁_ _≟ᵇ_
field
_≟ᶻ_ : Decidable ℤ._≈_
_<ᶻ?_ : Decidable ℤ._<_
_≟ʳ_ : Decidable ℝ._≈_
_<ʳ?_ : Decidable ℝ._<_
_≟ᵇ₁_ : Decidable 𝔹._≈_
_≟ᵇ_ : ∀ {n} → Decidable (Bits._≈_ {n})
_≟ᵇ_ = Pwₚ.decidable _≟ᵇ₁_
field
_/1 : ℤ → ℝ
⌊_⌋ : ℝ → ℤ
record Pseudocode b₁ b₂ i₁ i₂ i₃ r₁ r₂ r₃ :
Set (ℓsuc (b₁ ⊔ b₂ ⊔ i₁ ⊔ i₂ ⊔ i₃ ⊔ r₁ ⊔ r₂ ⊔ r₃)) where
field
bitBooleanAlgebra : BooleanAlgebra b₁ b₂
integerRing : CommutativeRing i₁ i₂ i₃
realField : Field r₁ r₂ r₃
bitsBooleanAlgebra : ℕ → BooleanAlgebra b₁ b₂
bitsBooleanAlgebra = Pw.booleanAlgebra bitBooleanAlgebra
module 𝔹 = BooleanAlgebra bitBooleanAlgebra
renaming (Carrier to Bit; ⊤ to 1𝔹; ⊥ to 0𝔹)
module Bits {n} = BooleanAlgebra (bitsBooleanAlgebra n)
renaming (⊤ to ones; ⊥ to zeros)
module ℤ = CommutativeRing integerRing
renaming (Carrier to ℤ; 1# to 1ℤ; 0# to 0ℤ)
module ℝ = Field realField
renaming (Carrier to ℝ; 1# to 1ℝ; 0# to 0ℝ)
Bits : ℕ → Set b₁
Bits n = Bits.Carrier {n}
open 𝔹 public using (Bit; 1𝔹; 0𝔹)
open Bits public using (ones; zeros)
open ℤ public using (ℤ; 1ℤ; 0ℤ)
open ℝ public using (ℝ; 1ℝ; 0ℝ)
module ℤ-Reasoning = Reasoning ℤ.strictPartialOrder
module ℝ-Reasoning = Reasoning ℝ.strictPartialOrder
field
integerDiscrete : ∀ x y → y ℤ.≤ x ⊎ x ℤ.+ 1ℤ ℤ.≤ y
1≉0 : 1ℤ ℤ.≉ 0ℤ
_/1 : ℤ → ℝ
⌊_⌋ : ℝ → ℤ
/1-isHomo : IsRingHomomorphism ℤ.Unordered.rawRing ℝ.Unordered.rawRing _/1
⌊⌋-isHomo : IsRingHomomorphism ℝ.Unordered.rawRing ℤ.Unordered.rawRing ⌊_⌋
/1-mono-< : ∀ x y → x ℤ.< y → x /1 ℝ.< y /1
⌊⌋-mono-≤ : ∀ x y → x ℝ.≤ y → ⌊ x ⌋ ℤ.≤ ⌊ y ⌋
⌊⌋-floor : ∀ x y → x ℝ.< y /1 → ⌊ x ⌋ ℤ.< y
⌊⌋∘/1≗id : ∀ x → ⌊ x /1 ⌋ ℤ.≈ x
module /1 = IsRingHomomorphism /1-isHomo renaming (⟦⟧-cong to cong)
module ⌊⌋ = IsRingHomomorphism ⌊⌋-isHomo renaming (⟦⟧-cong to cong)
bitRawBooleanAlgebra : RawBooleanAlgebra b₁ b₂
bitRawBooleanAlgebra = record
{ _≈_ = _≈_
; _∨_ = _∨_
; _∧_ = _∧_
; ¬_ = ¬_
; ⊤ = ⊤
; ⊥ = ⊥
}
where open BooleanAlgebra bitBooleanAlgebra
rawPseudocode : RawPseudocode b₁ b₂ i₁ i₂ i₃ r₁ r₂ r₃
rawPseudocode = record
{ bitRawBooleanAlgebra = bitRawBooleanAlgebra
; integerRawRing = ℤ.rawRing
; realRawField = ℝ.rawField
; _≟ᶻ_ = ℤ._≟_
; _<ᶻ?_ = ℤ._<?_
; _≟ʳ_ = ℝ._≟_
; _<ʳ?_ = ℝ._<?_
; _≟ᵇ₁_ = 𝔹._≟_
; _/1 = _/1
; ⌊_⌋ = ⌊_⌋
}
open RawPseudocode rawPseudocode using (module ℤ′; module ℝ′) public
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