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Lots of without-K (#110)

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Patrick Stevens
2020-04-11 12:14:03 +01:00
committed by GitHub
parent 412edaf4c7
commit e9aa1bcc05
30 changed files with 424 additions and 98 deletions
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12
Rings/Orders/Partial/Lemmas.agda
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@@ -9,7 +9,8 @@ open import Functions
open import Sets.EquivalenceRelations
open import Rings.Definition
open import Rings.Orders.Partial.Definition
open import Numbers.Naturals.Semiring
open import Numbers.Naturals.Order
module Rings.Orders.Partial.Lemmas {n m p : _} {A : Set n} {S : Setoid {n} {m} A} {_+_ : A → A → A} {_*_ : A → A → A} {_<_ : Rel {_} {p} A} {R : Ring S _+_ _*_} {pOrder : SetoidPartialOrder S _<_} (pRing : PartiallyOrderedRing R pOrder) where
@@ -78,3 +79,12 @@ abstract
moveInequality' : {a b : A} → a < b → (a + inverse b) < 0R
moveInequality' {a} {b} a<b = <WellDefined reflexive invRight (orderRespectsAddition a<b (inverse b))
greaterImpliesNotEqual : {a b : A} → a < b → (a ∼ b → False)
greaterImpliesNotEqual {a} {b} a<b a=b = irreflexive (<WellDefined a=b reflexive a<b)
greaterImpliesNotEqual' : {a b : A} → a < b → (b ∼ a → False)
greaterImpliesNotEqual' {a} {b} a<b a=b = irreflexive (<WellDefined reflexive a=b a<b)
negativeInequality : {a : A} → a < 0G → 0G < inverse a
negativeInequality {a} a<0 = <WellDefined invRight identLeft (orderRespectsAddition a<0 (inverse a))