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More lecture notes (#126)

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Patrick Stevens
2020-04-19 20:15:02 +01:00
committed by GitHub
parent 485b27e009
commit cab004f6d8
11 changed files with 163 additions and 36 deletions
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Fields/Lemmas.agda
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@@ -8,6 +8,7 @@ open import Sets.EquivalenceRelations
open import LogicalFormulae
open import Rings.IntegralDomains.Definition
open import Rings.IntegralDomains.Lemmas
open import Setoids.Subset
module Fields.Lemmas {m n : _} {A : Set m} {S : Setoid {m} {n} A} {_+_ : A → A → A} {_*_ : A → A → A} {R : Ring S _+_ _*_} (F : Field R) where
@@ -30,13 +31,30 @@ abstract
halfHalves : {x : A} (1/2 : A) (pr : 1/2 + 1/2 ∼ 1R) → (x + x) * 1/2 ∼ x
halfHalves {x} 1/2 pr = transitive (transitive (transitive *Commutative (transitive (transitive *DistributesOver+ (transitive (+WellDefined *Commutative *Commutative) (symmetric *DistributesOver+))) *Commutative)) (*WellDefined pr (reflexive))) identIsIdent
fieldIsIntDom : (Setoid._∼_ S (Ring.1R R) (Ring.0R R) → False) → IntegralDomain R
IntegralDomain.intDom (fieldIsIntDom 1!=0) {a} {b} ab=0 a!=0 with Field.allInvertible F a a!=0
IntegralDomain.intDom (fieldIsIntDom _) {a} {b} ab=0 a!=0 | 1/a , prA = transitive (symmetric identIsIdent) (transitive (*WellDefined (symmetric prA) reflexive) (transitive (symmetric *Associative) (transitive (*WellDefined reflexive ab=0) (Ring.timesZero R))))
IntegralDomain.nontrivial (fieldIsIntDom 1!=0) = 1!=0
fieldIsIntDom : IntegralDomain R
IntegralDomain.intDom fieldIsIntDom {a} {b} ab=0 a!=0 with Field.allInvertible F a a!=0
IntegralDomain.intDom fieldIsIntDom {a} {b} ab=0 a!=0 | 1/a , prA = transitive (symmetric identIsIdent) (transitive (*WellDefined (symmetric prA) reflexive) (transitive (symmetric *Associative) (transitive (*WellDefined reflexive ab=0) (Ring.timesZero R))))
IntegralDomain.nontrivial fieldIsIntDom 1=0 = Field.nontrivial F (symmetric 1=0)
allInvertibleWellDefined : {a b : A} {a!=0 : (a ∼ 0F) → False} {b!=0 : (b ∼ 0F) → False} → (a ∼ b) → underlying (allInvertible a a!=0) ∼ underlying (allInvertible b b!=0)
allInvertibleWellDefined {a} {b} {a!=0} {b!=0} a=b with allInvertible a a!=0
... | x , prX with allInvertible b b!=0
... | y , prY with transitive (transitive prX (symmetric prY)) (*WellDefined reflexive (symmetric a=b))
... | xa=ya = cancelIntDom (fieldIsIntDom λ p → a!=0 (oneZeroImpliesAllZero (symmetric p))) (transitive *Commutative (transitive xa=ya *Commutative)) a!=0
... | xa=ya = cancelIntDom fieldIsIntDom (transitive *Commutative (transitive xa=ya *Commutative)) a!=0
private
mulNonzeros : Sg A (λ m → (Setoid._∼_ S m (Ring.0R R)) → False) → Sg A (λ m → (Setoid._∼_ S m (Ring.0R R)) → False) → Sg A (λ m → (Setoid._∼_ S m (Ring.0R R)) → False)
mulNonzeros (a , a!=0) (b , b!=0) = (a * b) , λ ab=0 → b!=0 (IntegralDomain.intDom (fieldIsIntDom) ab=0 a!=0)
fieldMultiplicativeGroup : Group (subsetSetoid S {pred = λ m → ((Setoid._∼_ S m (Ring.0R R)) → False)}(λ {x} {y} x=y x!=0 → λ y=0 → x!=0 (Equivalence.transitive (Setoid.eq S) x=y y=0))) (mulNonzeros)
Group.+WellDefined (fieldMultiplicativeGroup) {x , prX} {y , prY} {z , prZ} {w , prW} = Ring.*WellDefined R
Group.0G (fieldMultiplicativeGroup) = Ring.1R R , λ 1=0 → Field.nontrivial F (Equivalence.symmetric (Setoid.eq S) 1=0)
Group.inverse (fieldMultiplicativeGroup) (x , pr) with Field.allInvertible F x pr
... | 1/x , pr1/x = 1/x , λ 1/x=0 → Field.nontrivial F (Equivalence.transitive (Setoid.eq S) (Equivalence.symmetric (Setoid.eq S) (Equivalence.transitive (Setoid.eq S) (Ring.*WellDefined R 1/x=0 (Equivalence.reflexive (Setoid.eq S))) (Ring.timesZero' R))) pr1/x)
Group.+Associative (fieldMultiplicativeGroup) {x , prX} {y , prY} {z , prZ} = Ring.*Associative R
Group.identRight (fieldMultiplicativeGroup) {x , prX} = Ring.identIsIdent' R
Group.identLeft (fieldMultiplicativeGroup) {x , prX} = Ring.identIsIdent R
Group.invLeft (fieldMultiplicativeGroup) {x , prX} with Field.allInvertible F x prX
... | 1/x , pr1/x = pr1/x
Group.invRight (fieldMultiplicativeGroup) {x , prX} with Field.allInvertible F x prX
... | 1/x , pr1/x = Equivalence.transitive (Setoid.eq S) (Ring.*Commutative R) pr1/x