mirror of
https://github.com/Smaug123/agdaproofs
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32 lines
1.2 KiB
Agda
32 lines
1.2 KiB
Agda
{-# OPTIONS --safe --warning=error --without-K --guardedness #-}
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open import Agda.Primitive using (Level; lzero; lsuc; _⊔_)
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open import Setoids.Setoids
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open import Rings.Definition
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open import Rings.Orders.Partial.Definition
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open import Rings.Orders.Total.Definition
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open import Sets.EquivalenceRelations
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open import Sequences
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open import Setoids.Orders
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open import Functions
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open import LogicalFormulae
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open import Numbers.Naturals.Semiring
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open import Groups.Definition
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module Rings.Orders.Total.Bounded {m n o : _} {A : Set m} {S : Setoid {m} {n} A} {_+_ : A → A → A} {_*_ : A → A → A} {_<_ : Rel {m} {o} A} {pOrder : SetoidPartialOrder S _<_} {R : Ring S _+_ _*_} {pRing : PartiallyOrderedRing R pOrder} (tRing : TotallyOrderedRing pRing) where
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open import Rings.Orders.Partial.Bounded pRing
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open Ring R
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open Group additiveGroup
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open import Groups.Lemmas (Ring.additiveGroup R)
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open Setoid S
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open Equivalence eq
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open SetoidPartialOrder pOrder
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open import Rings.Orders.Total.Lemmas tRing
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open PartiallyOrderedRing pRing
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boundGreaterThanZero : {s : Sequence A} → (b : Bounded s) → 0G < underlying b
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boundGreaterThanZero {s} (a , b) with b 0
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... | (l ,, r) = halvePositive a (<WellDefined invLeft reflexive (orderRespectsAddition (<Transitive l r) a))
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