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https://github.com/Smaug123/agdaproofs
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40 lines
1.6 KiB
Agda
40 lines
1.6 KiB
Agda
{-# OPTIONS --safe --warning=error #-}
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open import Fields.Fields
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open import Functions
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open import Orders
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open import LogicalFormulae
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open import Numbers.Rationals
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open import Numbers.RationalsLemmas
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open import Numbers.Naturals
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open import Setoids.Setoids
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open import Setoids.Orders
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open import Agda.Primitive using (Level; lzero; lsuc; _⊔_)
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module Numbers.Reals where
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record ℝ : Set where
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field
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f : ℕ → ℚ
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converges : {ε : ℚ} → Sg ℕ (λ x → {y : ℕ} → x <N y → (absQ ((f x) -Q (f y))) <Q ε)
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_+R_ : ℝ → ℝ → ℝ
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ℝ.f (record { f = a ; converges = conA } +R record { f = b ; converges = conB }) n = (a n) +Q (b n)
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ℝ.converges (record { f = a ; converges = conA } +R record { f = b ; converges = conB }) {e} = {!absQ (a x +Q b x)!}
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negateR : ℝ → ℝ
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ℝ.f (negateR record { f = f ; converges = converges }) n = negateQ (f n)
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ℝ.converges (negateR record { f = f ; converges = converges }) {e} with converges {e}
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... | 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)
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_-R_ : ℝ → ℝ → ℝ
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a -R b = a +R (negateR b)
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_*R_ : ℝ → ℝ → ℝ
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ℝ.f (record { f = a ; converges = conA } *R record { f = b ; converges = conB }) n = (a n) *Q (b n)
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ℝ.converges (record { f = a ; converges = conA } *R record { f = b ; converges = conB}) {e} = {!!}
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realsSetoid : Setoid ℝ
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(realsSetoid Setoid.∼ record { f = a ; converges = convA }) record { f = b ; converges = convB } = {!!}
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Setoid.eq realsSetoid = {!!}
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