Move towards base-n expansions (#112)

Rings/Orders/Partial/Bounded.agda

@@ -0,0 +1,35 @@
+{-# 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 Sets.EquivalenceRelations
+open import Sequences
+open import Setoids.Orders
+open import Functions
+open import LogicalFormulae
+open import Numbers.Naturals.Semiring
+open import Groups.Definition
+
+module Rings.Orders.Partial.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) where
+
+open Group (Ring.additiveGroup R)
+open import Groups.Lemmas (Ring.additiveGroup R)
+open Setoid S
+open Equivalence eq
+open SetoidPartialOrder pOrder
+
+BoundedAbove : Sequence A → Set (m ⊔ o)
+BoundedAbove x = Sg A (λ K → (n : ℕ) → index x n < K)
+
+BoundedBelow : Sequence A → Set (m ⊔ o)
+BoundedBelow x = Sg A (λ K → (n : ℕ) → K < index x n)
+
+Bounded : Sequence A → Set (m ⊔ o)
+Bounded x = Sg A (λ K → (n : ℕ) → ((Group.inverse (Ring.additiveGroup R) K) < index x n) && (index x n < K))
+
+boundNonzero : {s : Sequence A} → (b : Bounded s) → underlying b ∼ 0G → False
+boundNonzero {s} (a , b) isEq with b 0
+... | bad1 ,, bad2 = irreflexive (<Transitive bad1 (<WellDefined reflexive (transitive isEq (symmetric (transitive (inverseWellDefined isEq) invIdent))) bad2))