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https://github.com/Smaug123/agdaproofs
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31 lines
1.2 KiB
Agda
31 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 Groups.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 Numbers.Naturals.Order
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module Rings.Orders.Total.Cauchy {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} (order : TotallyOrderedRing pRing) where
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open Setoid S
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open SetoidTotalOrder (TotallyOrderedRing.total order)
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open SetoidPartialOrder pOrder
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open Equivalence eq
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open TotallyOrderedRing order
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open Ring R
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open Group additiveGroup
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open import Rings.Orders.Total.Lemmas order
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cauchy : Sequence A → Set (m ⊔ o)
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cauchy s = ∀ (ε : A) → (0R < ε) → Sg ℕ (λ N → ∀ {m n : ℕ} → (N <N m) → (N <N n) → abs ((index s m) -R (index s n)) < ε)
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