Alternative LUB property (#97)

Numbers/ClassicalReals/Sequences.agda

@@ -0,0 +1,72 @@
+{-# OPTIONS --safe --warning=error --without-K --guardedness #-}
+
+open import Agda.Primitive using (Level; lzero; lsuc; _⊔_)
+
+open import Numbers.ClassicalReals.RealField
+open import LogicalFormulae
+open import Setoids.Subset
+open import Setoids.Setoids
+open import Setoids.Orders
+open import Sets.EquivalenceRelations
+open import Rings.Orders.Total.Definition
+open import Rings.Orders.Partial.Definition
+open import Rings.Definition
+open import Fields.Fields
+open import Groups.Definition
+open import Numbers.Naturals.Semiring
+open import Numbers.Naturals.Order
+open import Numbers.Naturals.Order.Lemmas
+open import Sequences
+
+module Numbers.ClassicalReals.Sequences (ℝ : RealField) where
+
+open RealField ℝ
+open Setoid S
+open Equivalence eq
+open Ring R
+open Field F
+open SetoidPartialOrder pOrder
+open import Fields.Orders.LeastUpperBounds.Definition pOrder
+open import Rings.Orders.Total.Lemmas orderedRing
+open import Rings.Orders.Partial.Lemmas pOrderedRing
+open Group additiveGroup
+open PartiallyOrderedRing pOrderedRing
+open SetoidTotalOrder (TotallyOrderedRing.total orderedRing)
+open import Rings.InitialRing R
+open import Fields.Orders.Lemmas oField
+open import Rings.Lemmas R
+open import Groups.Lemmas additiveGroup
+open import Numbers.Intervals.Definition pOrderedRing
+open import Numbers.Intervals.Arithmetic pOrderedRing
+open import Fields.Lemmas F
+open import Fields.Orders.Total.Definition F
+
+orderedField : TotallyOrderedField pOrderedRing
+orderedField = record { oRing = orderedRing }
+
+open import Fields.Orders.Limits.Definition orderedField
+
+StrictlyIncreasing : Sequence A → Set c
+StrictlyIncreasing x = (n : ℕ) → (index x n) < (index x (succ n))
+
+Increasing : Sequence A → Set (b ⊔ c)
+Increasing x = (n : ℕ) → ((index x n) < (index x (succ n))) || ((index x n) ∼ (index x (succ n)))
+
+Bounded : Sequence A → Set (a ⊔ c)
+Bounded x = Sg A (λ K → (n : ℕ) → index x n < K)
+
+sequencePredicate : (x : Sequence A) → A → Set b
+sequencePredicate x a = Sg ℕ (λ n → index x n ∼ a)
+
+sequenceSubset : (x : Sequence A) → subset S (sequencePredicate x)
+sequenceSubset sequence {x} {y} x=y (n , sn=x) = n , transitive sn=x x=y
+
+boundedSequenceBounds : (K : A) (x : Sequence A) → ((n : ℕ) → index x n < K) → UpperBound (sequenceSubset x) K
+boundedSequenceBounds K x pr y (n , y=xn) = inl (<WellDefined y=xn reflexive (pr n))
+
+increasingBoundedLimit : (x : Sequence A) → Increasing x → Bounded x → A
+increasingBoundedLimit x increasing (K , kIsBound) with lub (sequenceSubset x) ((index x 0) , (0 , reflexive)) (K , boundedSequenceBounds K x kIsBound)
+... | a , _ = a
+
+increasingBoundedConverges : (x : Sequence A) → (increasing : Increasing x) → (bounded : Bounded x) → x ~> (increasingBoundedLimit x increasing bounded)
+increasingBoundedConverges x increasing bounded = {!!}