Triangle inequality on the rationals (#20)

Numbers/Reals.agda

@@ -1,32 +1,39 @@
 {-# OPTIONS --safe --warning=error #-}
 
-open import Fields
-open import Rings
+open import Fields.Fields
 open import Functions
 open import Orders
 open import LogicalFormulae
-open import Rationals
-open import Naturals
+open import Numbers.Rationals
+open import Numbers.RationalsLemmas
+open import Numbers.Naturals
+open import Setoids.Setoids
+open import Setoids.Orders
 
 open import Agda.Primitive using (Level; lzero; lsuc; _⊔_)
 
-module Reals where
-  record Subset {m n : _} (A : Set m) (B : Set n) : Set (m ⊔ n) where
-    field
-      inj : A → B
-      injInj : Injection inj
-
-  --record RealField {n : _} {A : Set n} {R : Ring A} {F : Field R} : Set _ where
-  --  field
-  --    orderedField : OrderedField F
-  --  open OrderedField orderedField
-  --  open TotalOrder ord
-  --  open Ring R
-  --  field
-  --    complete : {c : _} {C : Set c} → (S : Subset C A) → (upperBound : A) → ({s : C} → (Subset.inj S s) < upperBound) → Sg A (λ lub → ({s : C} → (Subset.inj S s) < lub) && ({s : C} → (Subset.inj S s) < upperBound → (Subset.inj S s) < lub))
-
-
-  record Real : Set where
+module Numbers.Reals where
+  record ℝ : Set where
     field
       f : ℕ → ℚ
-      converges : {ε : ℚ} → Sg ℕ (λ x → {y : ℕ} → x <N y → (f x) +Q (f y) <Q ε)
+      converges : {ε : ℚ} → Sg ℕ (λ x → {y : ℕ} → x <N y → (absQ ((f x) -Q (f y))) <Q ε)
+
+  _+R_ : ℝ → ℝ → ℝ
+  ℝ.f (record { f = a ; converges = conA } +R record { f = b ; converges = conB }) n = (a n) +Q (b n)
+  ℝ.converges (record { f = a ; converges = conA } +R record { f = b ; converges = conB }) {e} = {!absQ (a x +Q b x)!}
+
+  negateR : ℝ → ℝ
+  ℝ.f (negateR record { f = f ; converges = converges }) n = negateQ (f n)
+  ℝ.converges (negateR record { f = f ; converges = converges }) {e} with converges {e}
+  ... | n , pr = n , λ {y} x<y → SetoidPartialOrder.wellDefined ℚPartialOrder {absQ ((f n) -Q (f y))} {absQ ((negateQ (f n)) -Q (negateQ (f y)))} {e} {e} {!!} {!reflQ e!} (pr {y} x<y)
+
+  _-R_ : ℝ → ℝ → ℝ
+  a -R b = a +R (negateR b)
+
+  _*R_ : ℝ → ℝ → ℝ
+  ℝ.f (record { f = a ; converges = conA } *R record { f = b ; converges = conB }) n = (a n) *Q (b n)
+  ℝ.converges (record { f = a ; converges = conA } *R record { f = b ; converges = conB}) {e} = {!!}
+
+  realsSetoid : Setoid ℝ
+  (realsSetoid Setoid.∼ record { f = a ; converges = convA }) record { f = b ; converges = convB } = {!!}
+  Setoid.eq realsSetoid = {!!}