Move towards base-n expansions (#112)

Rings/Orders/Total/Cauchy.agda

@@ -0,0 +1,30 @@
+{-# 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 <N m) → (N <N n) → abs ((index s m) -R (index s n)) < ε)