mirror of
https://github.com/Smaug123/agdaproofs
synced 2025-10-11 06:38:39 +00:00
57 lines
2.2 KiB
Agda
57 lines
2.2 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 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 Sequences
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open import Numbers.Naturals.Semiring
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open import Numbers.Naturals.Order
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open import Semirings.Definition
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open import Functions
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open import Fields.Orders.Total.Definition
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open import Numbers.Primes.PrimeNumbers
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module Fields.Orders.Limits.Lemmas {a b c : _} {A : Set a} {S : Setoid {a} {b} A} {_+_ : A → A → A} {_*_ : A → A → A} {_<_ : Rel {_} {c} A} {R : Ring S _+_ _*_} {pOrder : SetoidPartialOrder S _<_} {F : Field R} {pRing : PartiallyOrderedRing R pOrder} (oF : TotallyOrderedField F pRing) where
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open Ring R
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open TotallyOrderedField oF
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open TotallyOrderedRing oRing
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open PartiallyOrderedRing pRing
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open import Rings.Orders.Total.Lemmas oRing
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open import Rings.Orders.Partial.Lemmas pRing
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open SetoidTotalOrder total
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open SetoidPartialOrder pOrder
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open Group additiveGroup
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open import Groups.Lemmas additiveGroup
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open Setoid S
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open Equivalence eq
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open Field F
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open import Fields.CauchyCompletion.Definition (TotallyOrderedField.oRing oF) F
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open import Fields.Orders.Limits.Definition oF
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open import Fields.Lemmas F
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open import Fields.Orders.Total.Lemmas oF
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open import Rings.Characteristic R
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open import Rings.InitialRing R
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open import Rings.Orders.Total.Cauchy oRing
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private
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2!=3 : 2 ≡ 3 → False
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2!=3 ()
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convergentSequenceCauchy : (nontrivial : 0R ∼ 1R → False) → {a : Sequence A} → {r : A} → a ~> r → cauchy a
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convergentSequenceCauchy _ {a} {r} a->r e 0<e with halve (λ i → charNotN 1 (transitive (transitive +Associative identRight) i)) e
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... | e/2 , prE/2 with a->r e/2 (halvePositive' prE/2 0<e)
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... | N , pr = N , λ {m} {n} → ans m n
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where
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ans : (m n : ℕ) → N <N m → N <N n → abs (index a m + inverse (index a n)) < e
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ans m n N<m N<n = <WellDefined reflexive prE/2 (bothNearImpliesNear {r} e/2 (halvePositive' prE/2 0<e) (pr m N<m) (pr n N<n))
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