mirror of
https://github.com/Smaug123/agdaproofs
synced 2025-10-13 07:38:40 +00:00
46 lines
1.7 KiB
Agda
46 lines
1.7 KiB
Agda
{-# OPTIONS --safe --warning=error --without-K #-}
|
||
|
||
open import Rings.Definition
|
||
open import Rings.Orders.Partial.Definition
|
||
open import Rings.Orders.Total.Definition
|
||
open import Setoids.Setoids
|
||
open import Setoids.Orders.Partial.Definition
|
||
open import Setoids.Orders.Total.Definition
|
||
open import Functions.Definition
|
||
open import Fields.Fields
|
||
open import Fields.Orders.Total.Definition
|
||
open import Numbers.Naturals.Semiring
|
||
open import Numbers.Naturals.Order
|
||
open import Sets.EquivalenceRelations
|
||
open import LogicalFormulae
|
||
open import Groups.Definition
|
||
|
||
open import Agda.Primitive using (Level; lzero; lsuc; _⊔_)
|
||
|
||
module Fields.Orders.Total.Lemmas {m n : _} {A : Set m} {S : Setoid {m} {n} A} {_+_ : A → A → A} {_*_ : A → A → A} {R : Ring S _+_ _*_} {F : Field R} {p : _} {_<_ : Rel {_} {p} A} {pOrder : SetoidPartialOrder S _<_} {oR : PartiallyOrderedRing R pOrder} (oF : TotallyOrderedField F oR) where
|
||
|
||
open Ring R
|
||
open Group additiveGroup
|
||
open Setoid S
|
||
open Equivalence eq
|
||
open Field F
|
||
open TotallyOrderedField oF
|
||
open TotallyOrderedRing oRing
|
||
open PartiallyOrderedRing oR
|
||
open SetoidTotalOrder total
|
||
open SetoidPartialOrder pOrder
|
||
open import Rings.InitialRing R
|
||
open import Rings.Orders.Total.Lemmas oRing
|
||
open import Rings.Orders.Partial.Lemmas oR
|
||
open import Rings.Lemmas R
|
||
open import Groups.Lemmas additiveGroup
|
||
|
||
charNotN : (n : ℕ) → fromN (succ n) ∼ 0R → False
|
||
charNotN n pr = irreflexive (<WellDefined reflexive pr t)
|
||
where
|
||
t : 0R < fromN (succ n)
|
||
t = fromNPreservesOrder (0<1 (Field.nontrivial F)) (succIsPositive n)
|
||
|
||
charNot2 : Setoid._∼_ S ((Ring.1R R) + (Ring.1R R)) (Ring.0R R) → False
|
||
charNot2 pr = charNotN 1 (transitive (transitive +Associative identRight) pr)
|