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Smaug123
2019-01-04 20:45:34 +00:00
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{-# OPTIONS --safe --warning=error #-}
open import LogicalFormulae
open import Groups
open import Rings
open import Setoids
open import Orders
open import IntegralDomains
open import Functions
open import Agda.Primitive using (Level; lzero; lsuc; _⊔_)
module Fields where
record Field {m n : _} {A : Set m} {S : Setoid {m} {n} A} {_+_ : A A A} {_*_ : A A A} (R : Ring S _+_ _*_) : Set (lsuc m n) where
open Ring R
open Group additiveGroup
open Setoid S
field
allInvertible : (a : A) ((a Group.identity (Ring.additiveGroup R)) False) Sg A (λ t t * a 1R)
nontrivial : (0R 1R) False
{-
record OrderedField {n} {A : Set n} {R : Ring A} (F : Field R) : Set (lsuc n) where
open Field F
field
ord : TotalOrder A
open TotalOrder ord
open Ring R
field
productPos : {a b : A} → (0R < a) → (0R < b) → (0R < (a * b))
orderRespectsAddition : {a b c : A} → (a < b) → (a + c) < (b + c)
-}