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Mostly show sqrt 2 is irrational (#53)

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Patrick Stevens
2019-10-22 20:56:58 +01:00
committed by GitHub
parent 6eaa181104
commit f5f4cf1376
8 changed files with 244 additions and 37 deletions
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Fields/CauchyCompletion/Definition.agda Normal file
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{-# 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.Lemmas
open import Rings.Order
open import Groups.Definition
open import Groups.Groups
open import Fields.Fields
open import Sets.EquivalenceRelations
open import Sequences
open import Setoids.Orders
open import Functions
open import LogicalFormulae
open import Numbers.Naturals.Naturals
module Fields.CauchyCompletion.Definition {m n o : _} {A : Set m} {S : Setoid {m} {n} A} {_+_ : A → A → A} {_*_ : A → A → A} {_<_ : Rel {m} {o} A} {pOrder : SetoidPartialOrder S _<_} {tOrder : SetoidTotalOrder {_<_ = _<_} pOrder} {R : Ring S _+_ _*_} (order : OrderedRing R tOrder) (F : Field R) where
open Setoid S
open SetoidTotalOrder tOrder
open SetoidPartialOrder pOrder
open Equivalence eq
open OrderedRing order
open Ring R
open Group additiveGroup
open Field F
open import Rings.Orders.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)
record CauchyCompletion : Set (m ⊔ o) where
field
elts : Sequence A
converges : cauchy elts
injection : A → CauchyCompletion
CauchyCompletion.elts (injection a) = constSequence a
CauchyCompletion.converges (injection a) = λ ε 0<e → 0 , λ {m} {n} _ _ → <WellDefined (symmetric (identityOfIndiscernablesRight _∼_ (absWellDefined (index (constSequence a) m + inverse (index (constSequence a) n)) 0R (t m n)) (absZero order))) reflexive 0<e
where
t : (m n : ℕ) → index (constSequence a) m + inverse (index (constSequence a) n) ∼ 0R
t m n = identityOfIndiscernablesLeft _∼_ (identityOfIndiscernablesLeft _∼_ invRight (equalityCommutative (applyEquality (λ i → a + inverse i) (indexAndConst a n)))) (applyEquality (_+ inverse (index (constSequence a) n)) (equalityCommutative (indexAndConst a m)))