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https://github.com/Smaug123/agdaproofs
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26 lines
1.1 KiB
Agda
26 lines
1.1 KiB
Agda
{-# OPTIONS --safe --warning=error --without-K #-}
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open import Groups.Definition
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open import Setoids.Setoids
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open import Setoids.Orders.Partial.Definition
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open import Functions
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open import Rings.Definition
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open import Agda.Primitive using (Level; lzero; lsuc; _⊔_)
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module Rings.Orders.Partial.Definition {n m : _} {A : Set n} {S : Setoid {n} {m} A} {_+_ : A → A → A} {_*_ : A → A → A} (R : Ring S _+_ _*_) where
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open Ring R
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open Group additiveGroup
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open Setoid S
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open import Groups.Orders.Partial.Definition
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record PartiallyOrderedRing {p : _} {_<_ : Rel {_} {p} A} (pOrder : SetoidPartialOrder S _<_) : Set (lsuc n ⊔ m ⊔ p) where
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field
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orderRespectsAddition : {a b : A} → (a < b) → (c : A) → (a + c) < (b + c)
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orderRespectsMultiplication : {a b : A} → (0R < a) → (0R < b) → (0R < (a * b))
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open SetoidPartialOrder pOrder
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toGroup : {p : _} {_<_ : Rel {_} {p} A} {pOrder : SetoidPartialOrder S _<_} → PartiallyOrderedRing pOrder → PartiallyOrderedGroup additiveGroup pOrder
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PartiallyOrderedGroup.orderRespectsAddition (toGroup p) = PartiallyOrderedRing.orderRespectsAddition p
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