Elevate the real numbers to actually existing (#65)

Numbers/Reals/Definition.agda

@@ -0,0 +1,39 @@
+{-# OPTIONS --warning=error --safe --guardedness #-}
+
+open import Setoids.Orders
+open import LogicalFormulae
+open import Rings.Definition
+open import Numbers.Rationals.Definition
+
+module Numbers.Reals.Definition where
+
+open import Fields.CauchyCompletion.Definition ℚOrdered ℚField
+open import Fields.CauchyCompletion.Setoid ℚOrdered ℚField ℚcharNot2
+open import Fields.CauchyCompletion.Addition ℚOrdered ℚField ℚcharNot2
+open import Fields.CauchyCompletion.Multiplication ℚOrdered ℚField ℚcharNot2
+open import Fields.CauchyCompletion.Ring ℚOrdered ℚField ℚcharNot2
+open import Fields.CauchyCompletion.Comparison ℚOrdered ℚField ℚcharNot2
+
+ℝ : Set
+ℝ = CauchyCompletion
+
+_+R_ : ℝ → ℝ → ℝ
+_+R_ = _+C_
+
+_*R_ : ℝ → ℝ → ℝ
+_*R_ = _*C_
+
+ℝRing : Ring cauchyCompletionSetoid _+R_ _*R_
+ℝRing = CRing
+
+injectionR : ℚ → ℝ
+injectionR = injection
+
+0R : ℝ
+0R = injection 0Q
+
+_<R_ : ℝ → ℝ → Set
+_<R_ = _<C_
+
+ℝPartialOrder : SetoidPartialOrder cauchyCompletionSetoid _<C_
+ℝPartialOrder = <COrder