agdaproofs/Fields/Fields.agda

{-# OPTIONS --safe --warning=error --without-K #-}

open import LogicalFormulae
open import Groups.Definition
open import Rings.Definition
open import Setoids.Setoids

open import Agda.Primitive using (Level; lzero; lsuc; _⊔_)

module Fields.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.0G (Ring.additiveGroup R)) → False) → Sg A (λ t → t * a ∼ 1R)
    nontrivial : (0R ∼ 1R) → False
  0F : A
  0F = Ring.0R R

record Field' {m n : _} : Set (lsuc m ⊔ lsuc n) where
  field
    A : Set m
    S : Setoid {m} {n} A
    _+_ : A → A → A
    _*_ : A → A → A
    R : Ring S _+_ _*_
    isField : Field R

encapsulateField : {m n : _} {A : Set m} {S : Setoid {m} {n} A} {_+_ : A → A → A} {_*_ : A → A → A} {R : Ring S _+_ _*_} (F : Field R) → Field'
encapsulateField {A = A} {S = S} {_+_} {_*_} {R} F = record { A = A ; S = S ; _+_ = _+_ ; _*_ = _*_ ; R = R ; isField = F }


{-
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)

-}