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Total order on the rationals (#17)

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Patrick Stevens
2019-01-13 12:05:30 +00:00
committed by GitHub
parent 9e22ba78f5
commit 2866b11be6
9 changed files with 624 additions and 52 deletions
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44
Fields/Fields.agda
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@@ -2,8 +2,11 @@
open import LogicalFormulae
open import Groups.Groups
open import Groups.GroupDefinition
open import Rings.RingDefinition
open import Rings.RingLemmas
open import Setoids.Setoids
open import Setoids.Orders
open import Orders
open import Rings.IntegralDomains
open import Functions
@@ -19,6 +22,47 @@ module Fields.Fields where
allInvertible : (a : A) → ((a ∼ Group.identity (Ring.additiveGroup R)) → False) → Sg A (λ t → t * a ∼ 1R)
nontrivial : (0R ∼ 1R) → False
orderedFieldIsIntDom : {a b c : _} {A : Set a} {S : Setoid {a} {b} A} {_+_ : A → A → A} {_*_ : A → A → A} {_<_ : Rel {_} {c} A} {R : Ring S _+_ _*_} {pOrder : SetoidPartialOrder S _<_} {tOrder : SetoidTotalOrder pOrder} (O : OrderedRing R tOrder) (F : Field R) → IntegralDomain R
IntegralDomain.intDom (orderedFieldIsIntDom {S = S} {_*_ = _*_} {R = R} {tOrder = tOrder} O F) {a} {b} ab=0 with SetoidTotalOrder.totality tOrder (Ring.0R R) a
IntegralDomain.intDom (orderedFieldIsIntDom {A = A} {S = S} {_*_ = _*_} {R = R} {pOrder = pOrder} {tOrder = tOrder} O F) {a} {b} ab=0 | inl (inl x) = inr (transitive (transitive (symmetric multIdentIsIdent) (multWellDefined q reflexive)) p')
where
open Setoid S
open Symmetric (Equivalence.symmetricEq (Setoid.eq S))
open Reflexive (Equivalence.reflexiveEq (Setoid.eq S))
open Transitive (Equivalence.transitiveEq (Setoid.eq S))
open Ring R
a!=0 : (a ∼ Group.identity additiveGroup) → False
a!=0 pr = SetoidPartialOrder.irreflexive pOrder (SetoidPartialOrder.wellDefined pOrder (symmetric pr) reflexive x)
invA : A
invA = underlying (Field.allInvertible F a a!=0)
q : 1R ∼ (invA * a)
q with Field.allInvertible F a a!=0
... | invA , pr = symmetric pr
p : invA * (a * b) ∼ invA * Group.identity additiveGroup
p = multWellDefined reflexive ab=0
p' : (invA * a) * b ∼ Group.identity additiveGroup
p' = transitive (symmetric multAssoc) (transitive p (ringTimesZero R))
IntegralDomain.intDom (orderedFieldIsIntDom {A = A} {S = S} {_*_ = _*_} {R = R} {pOrder = pOrder} {tOrder = tOrder} O F) {a} {b} ab=0 | inl (inr x) = inr (transitive (transitive (symmetric multIdentIsIdent) (multWellDefined q reflexive)) p')
where
open Setoid S
open Symmetric (Equivalence.symmetricEq (Setoid.eq S))
open Reflexive (Equivalence.reflexiveEq (Setoid.eq S))
open Transitive (Equivalence.transitiveEq (Setoid.eq S))
open Ring R
a!=0 : (a ∼ Group.identity additiveGroup) → False
a!=0 pr = SetoidPartialOrder.irreflexive pOrder (SetoidPartialOrder.wellDefined pOrder reflexive (symmetric pr) x)
invA : A
invA = underlying (Field.allInvertible F a a!=0)
q : 1R ∼ (invA * a)
q with Field.allInvertible F a a!=0
... | invA , pr = symmetric pr
p : invA * (a * b) ∼ invA * Group.identity additiveGroup
p = multWellDefined reflexive ab=0
p' : (invA * a) * b ∼ Group.identity additiveGroup
p' = transitive (symmetric multAssoc) (transitive p (ringTimesZero R))
IntegralDomain.intDom (orderedFieldIsIntDom {S = S} {_*_ = _*_} {R = R} {tOrder = tOrder} O F) {a} {b} ab=0 | inr x = inl (Symmetric.symmetric (Equivalence.symmetricEq (Setoid.eq S)) x)
IntegralDomain.nontrivial (orderedFieldIsIntDom {S = S} O F) pr = Field.nontrivial F (Symmetric.symmetric (Equivalence.symmetricEq (Setoid.eq S)) pr)
record Field' {m n : _} : Set (lsuc m ⊔ lsuc n) where
field
A : Set m