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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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15
Rings/Orders/Total/Lemmas.agda
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@@ -10,7 +10,8 @@ open import Sets.EquivalenceRelations
open import Rings.Definition
open import Rings.Orders.Total.Definition
open import Rings.Orders.Partial.Definition
open import Numbers.Naturals.Semiring
open import Numbers.Naturals.Order
module Rings.Orders.Total.Lemmas {n m p : _} {A : Set n} {S : Setoid {n} {m} A} {_+_ : A → A → A} {_*_ : A → A → A} {R : Ring S _+_ _*_} {_<_ : Rel {_} {p} A} {pOrder : SetoidPartialOrder S _<_} {pOrderRing : PartiallyOrderedRing R pOrder} (order : TotallyOrderedRing pOrderRing) where
@@ -350,3 +351,15 @@ abstract
orderedImpliesCharNot2 : (0R ∼ 1R → False) → 1R + 1R ∼ 0R → False
orderedImpliesCharNot2 0!=1 x = irreflexive (<WellDefined (identRight {0R}) x (ringAddInequalities (0<1 0!=1) (0<1 0!=1)))
open import Rings.InitialRing R
open Equivalence eq
fromNIncreasing : (0R ∼ 1R → False) → (n : ℕ) → (fromN n) < (fromN (succ n))
fromNIncreasing 0!=1 zero = <WellDefined reflexive (symmetric identRight) (0<1 0!=1)
fromNIncreasing 0!=1 (succ n) = <WellDefined groupIsAbelian groupIsAbelian (orderRespectsAddition (fromNIncreasing 0!=1 n) 1R)
fromNPreservesOrder : (0R ∼ 1R → False) → {a b : ℕ} → (a <N b) → (fromN a) < (fromN b)
fromNPreservesOrder 0!=1 {zero} {succ zero} a<b = fromNIncreasing 0!=1 0
fromNPreservesOrder 0!=1 {zero} {succ (succ b)} a<b = <Transitive (fromNPreservesOrder 0!=1 (succIsPositive b)) (fromNIncreasing 0!=1 (succ b))
fromNPreservesOrder 0!=1 {succ a} {succ b} a<b = <WellDefined groupIsAbelian groupIsAbelian (orderRespectsAddition (fromNPreservesOrder 0!=1 (canRemoveSuccFrom<N a<b)) 1R)