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https://github.com/Smaug123/agdaproofs
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39 lines
1.3 KiB
Agda
39 lines
1.3 KiB
Agda
{-# OPTIONS --safe --warning=error --without-K #-}
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open import Functions.Definition
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open import Agda.Primitive using (Level; lzero; lsuc; _⊔_)
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open import LogicalFormulae
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open import Setoids.Subset
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open import Setoids.Setoids
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open import Setoids.Orders.Partial.Definition
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open import Fields.Fields
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open import Rings.Orders.Total.Definition
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open import Rings.Orders.Total.Lemmas
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open import Rings.Orders.Partial.Definition
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open import Rings.Definition
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open import Fields.Orders.LeastUpperBounds.Definition
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open import Fields.Orders.Total.Definition
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module Numbers.ClassicalReals.RealField where
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record RealField : Agda.Primitive.Setω where
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field
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a b c : _
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A : Set a
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S : Setoid {_} {b} A
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_+_ : A → A → A
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_*_ : A → A → A
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R : Ring S _+_ _*_
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F : Field R
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_<_ : Rel {_} {c} A
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pOrder : SetoidPartialOrder S _<_
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pOrderedRing : PartiallyOrderedRing R pOrder
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orderedRing : TotallyOrderedRing pOrderedRing
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lub : {d : _} → {pred : A → Set d} → (sub : subset S pred) → (nonempty : Sg A pred) → (boundedAbove : Sg A (UpperBound pOrder sub)) → Sg A (LeastUpperBound pOrder sub)
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open Setoid S
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open Field F
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charNot2 : (Ring.1R R + Ring.1R R) ∼ Ring.0R R → False
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charNot2 = orderedImpliesCharNot2 orderedRing nontrivial
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oField : TotallyOrderedField F pOrderedRing
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oField = record { oRing = orderedRing }
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