mirror of
https://github.com/Smaug123/agdaproofs
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71 lines
2.8 KiB
Agda
71 lines
2.8 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 Numbers.ClassicalReals.RealField
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open import LogicalFormulae
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open import Setoids.Subset
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open import Setoids.Setoids
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open import Setoids.Orders
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open import Sets.EquivalenceRelations
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open import Rings.Orders.Total.Definition
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open import Rings.Orders.Partial.Definition
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open import Rings.Definition
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open import Fields.Fields
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open import Groups.Definition
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open import Numbers.Naturals.Semiring
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open import Numbers.Naturals.Order
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open import Numbers.Naturals.Order.Lemmas
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open import Sequences
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module Numbers.ClassicalReals.Sequences (ℝ : RealField) where
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open RealField ℝ
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open Setoid S
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open Equivalence eq
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open Ring R
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open Field F
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open SetoidPartialOrder pOrder
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open import Fields.Orders.LeastUpperBounds.Definition pOrder
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open import Rings.Orders.Total.Lemmas orderedRing
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open import Rings.Orders.Partial.Lemmas pOrderedRing
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open Group additiveGroup
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open PartiallyOrderedRing pOrderedRing
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open SetoidTotalOrder (TotallyOrderedRing.total orderedRing)
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open import Rings.InitialRing R
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open import Fields.Orders.Lemmas oField
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open import Rings.Lemmas R
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open import Groups.Lemmas additiveGroup
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open import Numbers.Intervals.Definition pOrderedRing
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open import Numbers.Intervals.Arithmetic pOrderedRing
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open import Fields.Lemmas F
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open import Fields.Orders.Total.Definition F
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orderedField : TotallyOrderedField pOrderedRing
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orderedField = record { oRing = orderedRing }
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open import Fields.Orders.Limits.Definition orderedField
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StrictlyIncreasing : Sequence A → Set c
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StrictlyIncreasing x = (n : ℕ) → (index x n) < (index x (succ n))
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Increasing : Sequence A → Set (b ⊔ c)
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Increasing x = (n : ℕ) → ((index x n) < (index x (succ n))) || ((index x n) ∼ (index x (succ n)))
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sequencePredicate : (x : Sequence A) → A → Set b
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sequencePredicate x a = Sg ℕ (λ n → index x n ∼ a)
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sequenceSubset : (x : Sequence A) → subset S (sequencePredicate x)
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sequenceSubset sequence {x} {y} x=y (n , sn=x) = n , transitive sn=x x=y
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boundedSequenceBounds : (K : A) (x : Sequence A) → ((n : ℕ) → index x n < K) → UpperBound (sequenceSubset x) K
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boundedSequenceBounds K x pr y (n , y=xn) = inl (<WellDefined y=xn reflexive (pr n))
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increasingBoundedLimit : (x : Sequence A) → Increasing x → Bounded x → A
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increasingBoundedLimit x increasing (K , kIsBound) with lub (sequenceSubset x) ((index x 0) , (0 , reflexive)) (K , boundedSequenceBounds K x kIsBound)
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... | a , _ = a
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increasingBoundedConverges : (x : Sequence A) → (increasing : Increasing x) → (bounded : Bounded x) → x ~> (increasingBoundedLimit x increasing bounded)
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increasingBoundedConverges x increasing (K , prBound) with lub (sequenceSubset x) (Sequence.head x , (0 , reflexive)) (K , boundedSequenceBounds K x prBound)
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... | lub , isLub = {!!}
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