patrick/agdaproofs
1
0
Fork 0
mirror of https://github.com/Smaug123/agdaproofs synced 2025-10-20 18:48:39 +00:00
Code Issues Packages Projects Releases Wiki Activity

Split partial and total order of rings (#61)

Browse Source
This commit is contained in:
Patrick Stevens
2019-11-02 18:42:37 +00:00
committed by GitHub
parent 55995ea801
commit 763ddb8dbb
26 changed files with 768 additions and 618 deletions
Download Patch File Download Diff File Expand all files Collapse all files

13
Fields/CauchyCompletion/Definition.agda
View File

@@ -4,7 +4,8 @@ open import Agda.Primitive using (Level; lzero; lsuc; _⊔_)
open import Setoids.Setoids
open import Rings.Definition
open import Rings.Lemmas
open import Rings.Orders.Definition
open import Rings.Orders.Partial.Definition
open import Rings.Orders.Total.Definition
open import Groups.Definition
open import Groups.Groups
open import Fields.Fields
@@ -15,18 +16,18 @@ 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
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 _<_} {R : Ring S _+_ _*_} {pRing : PartiallyOrderedRing R pOrder} (order : TotallyOrderedRing pRing) (F : Field R) where
open Setoid S
open SetoidTotalOrder tOrder
open SetoidTotalOrder (TotallyOrderedRing.total order)
open SetoidPartialOrder pOrder
open Equivalence eq
open OrderedRing order
open TotallyOrderedRing order
open Ring R
open Group additiveGroup
open Field F
open import Rings.Orders.Lemmas(order)
open import Rings.Orders.Total.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)
@@ -38,7 +39,7 @@ record CauchyCompletion : Set (m ⊔ o) where
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
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)) 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)))