mirror of
https://github.com/Smaug123/agdaproofs
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50 lines
1.3 KiB
Agda
50 lines
1.3 KiB
Agda
{-# OPTIONS --warning=error --safe --guardedness --without-K #-}
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open import Setoids.Orders
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open import Setoids.Setoids
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open import LogicalFormulae
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open import Rings.Definition
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open import Numbers.Rationals.Definition
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open import Functions
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module Numbers.Reals.Definition where
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open import Fields.CauchyCompletion.Definition ℚOrdered ℚField
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open import Fields.CauchyCompletion.Setoid ℚOrdered ℚField ℚcharNot2
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open import Fields.CauchyCompletion.Addition ℚOrdered ℚField ℚcharNot2
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open import Fields.CauchyCompletion.Multiplication ℚOrdered ℚField ℚcharNot2
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open import Fields.CauchyCompletion.Ring ℚOrdered ℚField ℚcharNot2
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open import Fields.CauchyCompletion.Comparison ℚOrdered ℚField ℚcharNot2
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ℝ : Set
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ℝ = CauchyCompletion
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_+R_ : ℝ → ℝ → ℝ
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_+R_ = _+C_
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_*R_ : ℝ → ℝ → ℝ
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_*R_ = _*C_
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ℝSetoid = cauchyCompletionSetoid
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_=R_ : ℝ → ℝ → Set
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a =R b = Setoid._∼_ cauchyCompletionSetoid a b
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ℝRing : Ring cauchyCompletionSetoid _+R_ _*R_
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ℝRing = CRing
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injectionR : ℚ → ℝ
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injectionR = injection
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injectionRInjective : Injection injectionR
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injectionRInjective = CInjection'
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0R : ℝ
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0R = injection 0Q
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_<R_ : ℝ → ℝ → Set
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_<R_ = _<C_
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ℝPartialOrder : SetoidPartialOrder cauchyCompletionSetoid _<C_
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ℝPartialOrder = <COrder
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