{-# OPTIONS --safe --warning=error --without-K --guardedness #-} open import Agda.Primitive using (Level; lzero; lsuc; _⊔_) open import Setoids.Setoids open import Rings.Definition open import Rings.Orders.Partial.Definition open import Rings.Orders.Total.Definition open import Groups.Definition open import Sets.EquivalenceRelations open import Sequences open import Setoids.Orders open import Functions open import LogicalFormulae open import Numbers.Naturals.Semiring open import Numbers.Naturals.Order 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 open Setoid S open SetoidTotalOrder (TotallyOrderedRing.total order) open SetoidPartialOrder pOrder open Equivalence eq open TotallyOrderedRing order open Ring R open Group additiveGroup open import Rings.Orders.Total.Lemmas order cauchy : Sequence A → Set (m ⊔ o) cauchy s = ∀ (ε : A) → (0R < ε) → Sg ℕ (λ N → ∀ {m n : ℕ} → (N