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Split partial and total order of rings (#61)

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Patrick Stevens
2019-11-02 18:42:37 +00:00
committed by GitHub
parent 55995ea801
commit 763ddb8dbb
26 changed files with 768 additions and 618 deletions
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12
Numbers/Integers/Order.agda
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@@ -7,7 +7,8 @@ open import Numbers.Integers.Addition
open import Numbers.Integers.Multiplication
open import Semirings.Definition
open import Rings.Definition
open import Rings.Orders.Definition
open import Rings.Orders.Partial.Definition
open import Rings.Orders.Total.Definition
open import Setoids.Setoids
open import Setoids.Orders
open import Orders
@@ -95,6 +96,9 @@ orderRespectsAddition (negSucc a) (negSucc b) (le x proof) (negSucc c) = le x (t
orderRespectsMultiplication : (a b : ℤ) → nonneg 0 <Z a → nonneg 0 <Z b → nonneg 0 <Z a *Z b
orderRespectsMultiplication (nonneg (succ a)) (nonneg (succ b)) 0<a 0<b = lessInherits (succIsPositive (b +N a *N succ b))
ℤOrderedRing : OrderedRing ℤRing (totalOrderToSetoidTotalOrder ℤOrder)
OrderedRing.orderRespectsAddition ℤOrderedRing {a} {b} = orderRespectsAddition a b
OrderedRing.orderRespectsMultiplication ℤOrderedRing {a} {b} = orderRespectsMultiplication a b
ℤPOrderedRing : PartiallyOrderedRing ℤRing (SetoidTotalOrder.partial (totalOrderToSetoidTotalOrder ℤOrder))
PartiallyOrderedRing.orderRespectsAddition ℤPOrderedRing {a} {b} = orderRespectsAddition a b
PartiallyOrderedRing.orderRespectsMultiplication ℤPOrderedRing {a} {b} = orderRespectsMultiplication a b
ℤOrderedRing : TotallyOrderedRing ℤPOrderedRing
TotallyOrderedRing.total ℤOrderedRing = totalOrderToSetoidTotalOrder ℤOrder

8
Numbers/Rationals.agda
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@@ -6,7 +6,8 @@ open import Numbers.Integers.Integers
open import Groups.Groups
open import Groups.Definition
open import Rings.Definition
open import Rings.Orders.Definition
open import Rings.Orders.Total.Definition
open import Rings.Orders.Partial.Definition
open import Fields.Fields
open import Numbers.Primes.PrimeNumbers
open import Setoids.Setoids
@@ -73,5 +74,8 @@ open Setoid (fieldOfFractionsSetoid ℤIntDom)
negateQWellDefined : (a b : ℚ) → (a =Q b) → (negateQ a) =Q (negateQ b)
negateQWellDefined a b a=b = inverseWellDefined (Ring.additiveGroup ℚRing) {a} {b} a=b
ℚOrdered : OrderedRing ℚRing (fieldOfFractionsTotalOrder ℤIntDom ℤOrderedRing)
ℚPOrdered : PartiallyOrderedRing ℚRing partial
ℚPOrdered = fieldOfFractionsPOrderedRing ℤIntDom ℤOrderedRing
ℚOrdered : TotallyOrderedRing ℚPOrdered
ℚOrdered = fieldOfFractionsOrderedRing ℤIntDom ℤOrderedRing

7
Numbers/RationalsLemmas.agda
View File

@@ -8,7 +8,8 @@ open import Numbers.Integers.Order
open import Groups.Groups
open import Groups.Definition
open import Rings.Definition
open import Rings.Orders.Definition
open import Rings.Orders.Partial.Definition
open import Rings.Orders.Total.Definition
open import Fields.Fields
open import Numbers.Primes.PrimeNumbers
open import Setoids.Setoids
@@ -23,8 +24,8 @@ open import Orders
module Numbers.RationalsLemmas where
open OrderedRing ℤOrderedRing
open import Rings.Orders.Lemmas ℤOrderedRing
open PartiallyOrderedRing ℤPOrderedRing
open import Rings.Orders.Total.Lemmas ℤOrderedRing
open SetoidTotalOrder (totalOrderToSetoidTotalOrder ℤOrder)
evenOrOdd : (a : ℕ) → (Sg ℕ (λ i → i *N 2 ≡ a)) || (Sg ℕ (λ i → succ (i *N 2) ≡ a))