Build rational arithmetic.

Yggdrasil/Rational.agda

@@ -1,11 +1,11 @@
 module Yggdrasil.Rational where
 
 open import Data.Bool using (true; false; T)
-open import Data.Integer as ℤ using (ℤ; +_)
+open import Data.Integer as ℤ using (ℤ; +_; -[1+_])
 open import Data.Nat as ℕ using (ℕ; suc; zero)
 open import Data.Nat.GCD as GCD using (GCD; gcd)
 open import Data.Nat.Divisibility using (_∣_; divides)
-open import Data.Nat.Coprimality using (coprime?; gcd-coprime; Bézout-coprime)
+open import Data.Nat.Coprimality using (Coprime; coprime?; gcd-coprime; Bézout-coprime)
 open import Data.Nat.Properties using (*-comm; *-assoc)
 open import Data.Product renaming (_,_ to ⟨_,_⟩)
 open import Data.Rational using (ℚ) renaming (_÷_ to _÷†_)
@@ -13,9 +13,9 @@ open import Data.Unit using (⊤; tt)
 open import Data.Empty using (⊥; ⊥-elim)
 open import Data.Sign using (Sign) renaming (+ to s+; - to s-)
 open import Relation.Nullary using (yes; no; ¬_)
-open import Relation.Nullary.Decidable using (True; False; ⌊_⌋; fromWitness)
+open import Relation.Nullary.Decidable using (True; False; ⌊_⌋; fromWitness; fromWitnessFalse)
 import Relation.Binary.PropositionalEquality as Eq
-open Eq using (_≡_; _≢_; refl; subst₂; sym; cong)
+open Eq using (_≡_; _≢_; refl; subst₂; sym; cong; trans)
 open Eq.≡-Reasoning using (_≡⟨_⟩_; _∎; begin_)
 
 open ℚ
@@ -23,14 +23,22 @@ open ℚ
 infixl 6 _+_ _-_
 infixl 7 _*_ _÷_
 
-1+≢*0 : ∀ x y → suc x ≢ y ℕ.* 0
-1+≢*0 x zero ()
-1+≢*0 x (suc y) = 1+≢*0 x y
+private
+  1+≢*0 : ∀ x y → suc x ≢ y ℕ.* 0
+  1+≢*0 x zero ()
+  1+≢*0 x (suc y) = 1+≢*0 x y
+
+  1≢0 : ∀ {n} → suc n ≢ zero
+  1≢0 ()
 
 simp : ∀ x y-1 → ℚ
 simp x y-1 with GCD.Bézout.lemma x (suc y-1)
-... | GCD.Bézout.result 0 (GCD.is ⟨ _ , divides y′ y-eq ⟩ _) _ = ⊥-elim (1+≢*0 y-1 y′ y-eq)
-... | GCD.Bézout.result (suc d-1) (GCD.is ⟨ divides x′ x-eq , divides y′ y-eq ⟩ _) bézout = _÷†_ (+ x′) y′ {fromWitness {!(Bézout-coprime bézout′)!}}
+... | GCD.Bézout.result 0 (GCD.is ⟨ _ , divides y′ y-eq ⟩ _) _ =
+  ⊥-elim (1+≢*0 y-1 y′ y-eq)
+... | GCD.Bézout.result (suc d-1)
+      (GCD.is ⟨ divides x′ x-eq , divides y′ y-eq ⟩ _) bézout =
+        _÷†_ (+ x′) y′ {fromWitness (λ {i} → Bézout-coprime bézout′)}
+          {fromWitnessFalse y′≢0}
   where
     y = suc y-1
     d = suc d-1
@@ -38,23 +46,32 @@ simp x y-1 with GCD.Bézout.lemma x (suc y-1)
     bézout′ : GCD.Bézout.Identity d (x′ ℕ.* d) (y′ ℕ.* d)
     bézout′ = subst₂ (GCD.Bézout.Identity d) x-eq y-eq bézout
 
-    eq-prf : x ℕ.* y′ ≡ x′ ℕ.* y
-    eq-prf = begin
-      x ℕ.* y′           ≡⟨ cong (λ z → z ℕ.* y′) x-eq ⟩
-      x′ ℕ.* d ℕ.* y′    ≡⟨ *-assoc x′ d y′ ⟩
-      x′ ℕ.* (d ℕ.* y′)  ≡⟨ sym (cong (ℕ._*_ x′) (*-comm y′ d)) ⟩
-      x′ ℕ.* (y′ ℕ.* d)  ≡⟨ sym (cong (ℕ._*_ x′) y-eq)  ⟩
-      x′ ℕ.* y           ∎
+    y′≢0 : y′ ≢ 0
+    y′≢0 y′≡0 = ⊥-elim (1≢0 (trans y-eq (cong (ℕ._* d) y′≡0)))
 
-postulate
-  _÷_ : ℤ → (d : ℕ) → {d≢0 : False (d ℕ.≟ 0)} → ℚ
+-_ : ℚ → ℚ
+- q = _÷†_ (ℤ.- numerator q) (suc (denominator-1 q))
+  {fromWitness (λ{ {i} ⟨ i∣n , i∣d ⟩ → coprime q ⟨ ∣-abs⇒∣abs i (numerator q) i∣n , i∣d ⟩})}
+  where
+    -abs≡abs : ∀ i → ℤ.∣ ℤ.- i ∣ ≡ ℤ.∣ i ∣
+    -abs≡abs (+ zero) = refl
+    -abs≡abs (+ (suc n)) = refl
+    -abs≡abs -[1+ n ] = refl
+
+    ∣-abs⇒∣abs : ∀ i j → i ∣ ℤ.∣ ℤ.- j ∣ → i ∣ ℤ.∣ j ∣
+    ∣-abs⇒∣abs i j (divides k j=k*i) = divides k (sym (begin
+      k ℕ.* i      ≡⟨ sym j=k*i ⟩
+      ℤ.∣ ℤ.- j ∣  ≡⟨ -abs≡abs j ⟩
+      ℤ.∣ j ∣      ∎))
+    
+_÷_ : ℤ → (d : ℕ) → {d≢0 : False (d ℕ.≟ 0)} → ℚ
+_÷_ n zero {}
+(+ n) ÷ (suc d-1) = simp n d-1
+(-[1+ n-1 ]) ÷ (suc d-1) = - simp (suc n-1) d-1
 
 ∣_∣ : ℚ → ℚ
 ∣ q ∣ = _÷†_ (+ ℤ.∣ numerator q ∣) (suc (denominator-1 q)) {isCoprime q}
 
--_ : ℚ → ℚ
-- q = _÷_ (ℤ.- numerator q) (suc (denominator-1 q))
-
 _+_ : ℚ → ℚ → ℚ
 a + b = let
     n-a = numerator a
@@ -77,46 +94,3 @@ a * b = let
 
 _-_ : ℚ → ℚ → ℚ
 a - b = a + (- b)
-
-
---gcd (suc (denominator-1 a)) (suc (denominator-1 b))
---... | ⟨ c , denom-gcd ⟩ with GCD.commonDivisor denom-gcd
---...   | ⟨ divides d₁ d₁*c≡da , divides d₂ d₂*c≡db ⟩ = let
-
-
-
---data ℚ′ : Set where
---
---∣◃∣-≡ : (n : ℕ) → (s : Sign) → ∣ s ◃ n ∣ ≡ n
---∣◃∣-≡ = ?
---
---normalise : ℚ′ → ℚ
---normalise (_÷′_ n zero {()})
---normalise (n ÷′ suc d) with gcd ∣ n ∣ (suc d)
-----... | ⟨ 1 , gcd ⟩ = record
-----  { numerator = n
-----  ; denominator-1 = d
-----  ; isCoprime = fromWitness {Q = coprime? ∣ n ∣ (suc d)} (gcd-coprime gcd) }
---... | ⟨ _ , gcd₁ ⟩ with GCD.commonDivisor gcd₁
---...   | ⟨ divides n′ _ , divides d′ _ ⟩ with d′ | gcd n′ d′ | sign n
---...     | suc d′ | ⟨ suc (suc m) , gcd₂ ⟩ | _ = ?
---...     | suc d′ | ⟨ 1 , gcd₂ ⟩ | s+ = record
---with coprime? ∣ n ∣ (suc d)
---... | yes cp = record
---... | no ¬cp = ?