Attempt at the INT construction

src/Data/Integer/IntConstruction.agda

@@ -0,0 +1,69 @@
+------------------------------------------------------------------------
+-- The Agda standard library
+--
+-- Construction of integers as a pair of naturals
+------------------------------------------------------------------------
+
+{-# OPTIONS --safe --cubical-compatible #-}
+
+module Data.Integer.IntConstruction where
+
+open import Data.Nat.Base as ℕ using (ℕ)
+open import Function.Base using (flip)
+open import Relation.Binary
+open import Relation.Binary.PropositionalEquality
+
+infixl 6 _⊖_
+
+record ℤ : Set where
+  constructor _⊖_
+  field
+    minuend : ℕ
+    subtrahend : ℕ
+
+infix 4 _≃_ _≤_ _≥_ _<_ _>_
+
+_≃_ : Rel ℤ _
+(a ⊖ b) ≃ (c ⊖ d) = a ℕ.+ d ≡ c ℕ.+ b
+
+_≤_ : Rel ℤ _
+(a ⊖ b) ≤ (c ⊖ d) = a ℕ.+ d ℕ.≤ c ℕ.+ b
+
+_≥_ : Rel ℤ _
+_≥_ = flip _≤_
+
+_<_ : Rel ℤ _
+(a ⊖ b) < (c ⊖ d) = a ℕ.+ d ℕ.< c ℕ.+ b
+
+_>_ : Rel ℤ _
+_>_ = flip _<_
+
+0ℤ : ℤ
+0ℤ = 0 ⊖ 0
+
+1ℤ : ℤ
+1ℤ = 1 ⊖ 0
+
+infixl 6 _+_
+_+_ : ℤ → ℤ → ℤ
+(a ⊖ b) + (c ⊖ d) = (a ℕ.+ c) ⊖ (b ℕ.+ d)
+
+infixl 7 _*_
+_*_ : ℤ → ℤ → ℤ
+(a ⊖ b) * (c ⊖ d) = (a ℕ.* c ℕ.+ b ℕ.* d) ⊖ (a ℕ.* d ℕ.+ b ℕ.* c)
+
+infix 8 -_
+-_ : ℤ → ℤ
+- (a ⊖ b) = b ⊖ a
+
+infix 8 ⁻_ ⁺_
+
+⁺_ : ℕ → ℤ
+⁺ n = n ⊖ 0
+
+⁻_ : ℕ → ℤ
+⁻ n = 0 ⊖ n
+
+∣_∣ : ℤ → ℕ
+∣ a ⊖ b ∣ = ℕ.∣ a - b ∣′
+

src/Data/Integer/IntConstruction/IntProperties.agda

@@ -0,0 +1,72 @@
+------------------------------------------------------------------------
+-- The Agda standard library
+--
+-- To be merged into Data.Integer.Properties before merging!
+------------------------------------------------------------------------
+
+{-# OPTIONS --safe --cubical-compatible #-}
+
+module Data.Integer.IntConstruction.IntProperties where
+
+open import Data.Integer.Base
+open import Data.Integer.IntConstruction as INT using (_≃_)
+open import Data.Integer.IntConstruction.Tmp
+open import Data.Integer.Properties
+import Data.Nat.Base as ℕ
+open import Data.Product.Base
+open import Function.Base
+open import Relation.Binary.PropositionalEquality
+
+fromINT-cong : ∀ {i j} → i ≃ j → fromINT i ≡ fromINT j
+fromINT-cong {a INT.⊖ b} {c INT.⊖ d} a+d≡c+b = begin
+  a ⊖ b                       ≡⟨ m-n≡m⊖n a b ⟨
+  + a - + b                   ≡⟨ cong (_- + b) (+-identityʳ (+ a)) ⟨
+  (+ a + 0ℤ) - + b            ≡⟨ cong (λ z → (+ a + z) - + b) (+-inverseʳ (+ d)) ⟨
+  (+ a + (+ d - + d)) - + b   ≡⟨ cong (_- + b) (+-assoc (+ a) (+ d) (- + d)) ⟨
+  (+ (a ℕ.+ d) - + d) - + b   ≡⟨ cong (λ z → (+ z - + d) - + b) a+d≡c+b ⟩
+  (+ (c ℕ.+ b) - + d) - + b   ≡⟨ cong (_- + b) (+-assoc (+ c) (+ b) (- + d)) ⟩
+  (+ c + (+ b - + d)) - + b   ≡⟨ cong (λ z → (+ c + z) - + b) (+-comm (+ b) (- + d)) ⟩
+  (+ c + (- + d + + b)) - + b ≡⟨ cong (_- + b) (+-assoc (+ c) (- + d) (+ b)) ⟨
+  ((+ c - + d) + + b) - + b   ≡⟨ +-assoc (+ c - + d) (+ b) (- + b) ⟩
+  (+ c - + d) + (+ b - + b)   ≡⟨ cong₂ _+_ (m-n≡m⊖n c d) (+-inverseʳ (+ b)) ⟩
+  c ⊖ d + 0ℤ                  ≡⟨ +-identityʳ (c ⊖ d) ⟩
+  c ⊖ d                       ∎
+  where open ≡-Reasoning
+
+fromINT-injective : ∀ {i j} → fromINT i ≡ fromINT j → i ≃ j
+fromINT-injective {a INT.⊖ b} {c INT.⊖ d} a⊖b≡c⊖d = +-injective $ begin
+  + a + + d                   ≡⟨ cong (_+ + d) (+-identityʳ (+ a)) ⟨
+  (+ a + 0ℤ) + + d            ≡⟨ cong (λ z → (+ a + z) + + d) (+-inverseˡ (+ b)) ⟨
+  (+ a + (- + b + + b)) + + d ≡⟨ cong (_+ + d) (+-assoc (+ a) (- + b) (+ b)) ⟨
+  ((+ a - + b) + + b) + + d   ≡⟨ cong (λ z → (z + + b) + + d) (m-n≡m⊖n a b) ⟩
+  (a ⊖ b + + b) + + d         ≡⟨ cong (λ z → (z + + b) + + d) a⊖b≡c⊖d ⟩
+  (c ⊖ d + + b) + + d         ≡⟨ cong (λ z → (z + + b) + + d) (m-n≡m⊖n c d) ⟨
+  ((+ c - + d) + + b) + + d   ≡⟨ cong (_+ + d) (+-assoc (+ c) (- + d) (+ b)) ⟩
+  (+ c + (- + d + + b)) + + d ≡⟨ cong (λ z → (+ c + z) + + d) (+-comm (- + d) (+ b)) ⟩
+  (+ c + (+ b - + d)) + + d   ≡⟨ cong (_+ + d) (+-assoc (+ c) (+ b) (- + d)) ⟨
+  ((+ c + + b) - + d) + + d   ≡⟨ +-assoc (+ c + + b) (- + d) (+ d) ⟩
+  (+ c + + b) + (- + d + + d) ≡⟨ cong (_+_ (+ c + + b)) (+-inverseˡ (+ d)) ⟩
+  (+ c + + b) + 0ℤ            ≡⟨ +-identityʳ (+ c + + b) ⟩
+  + c + + b                   ∎
+  where open ≡-Reasoning
+
+fromINT-surjective : ∀ j → ∃[ i ] ∀ {z} → z ≃ i → fromINT z ≡ j
+fromINT-surjective j .proj₁ = toINT j
+fromINT-surjective (+ n) .proj₂ {a INT.⊖ b} a+0≡n+b = begin
+  a ⊖ b             ≡⟨ m-n≡m⊖n a b ⟨
+  + a - + b         ≡⟨ cong (_- + b) (+-identityʳ (+ a)) ⟨
+  (+ a + 0ℤ) - + b  ≡⟨ cong (λ z → + z - + b) a+0≡n+b ⟩
+  (+ n + + b) - + b ≡⟨ +-assoc (+ n) (+ b) (- + b) ⟩
+  + n + (+ b - + b) ≡⟨ cong (_+_ (+ n)) (+-inverseʳ (+ b)) ⟩
+  + n + 0ℤ          ≡⟨ +-identityʳ (+ n) ⟩
+  + n               ∎
+  where open ≡-Reasoning
+fromINT-surjective (-[1+ n ]) .proj₂ {a INT.⊖ b} a+sn≡b = begin
+  a ⊖ b                     ≡⟨ m-n≡m⊖n a b ⟨
+  + a - + b                 ≡⟨ cong (λ z → + a - + z) a+sn≡b ⟨
+  + a - (+ a + + ℕ.suc n)   ≡⟨ cong (_+_ (+ a)) (neg-distrib-+ (+ a) (+ ℕ.suc n)) ⟩
+  + a + (- + a - + ℕ.suc n) ≡⟨ +-assoc (+ a) (- + a) (- + ℕ.suc n) ⟨
+  (+ a - + a) - + ℕ.suc n   ≡⟨ cong (_- + ℕ.suc n) (+-inverseʳ (+ a)) ⟩
+  0ℤ - + ℕ.suc n            ≡⟨ +-identityˡ (- + ℕ.suc n) ⟩
+  -[1+ n ]                  ∎
+  where open ≡-Reasoning