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Lots of rings (#82)

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Patrick Stevens
2019-11-22 19:52:57 +00:00
committed by GitHub
parent b33baa5fb7
commit 660d7aa27c
40 changed files with 1246 additions and 881 deletions
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5
Fields/FieldOfFractions/Addition.agda
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@@ -6,7 +6,7 @@ open import Groups.Definition
open import Groups.Lemmas
open import Rings.Definition
open import Rings.Lemmas
open import Rings.IntegralDomains
open import Rings.IntegralDomains.Definition
open import Fields.Fields
open import Functions
open import Setoids.Setoids
@@ -25,8 +25,7 @@ fieldOfFractionsPlus (a ,, (b , b!=0)) (c ,, (d , d!=0)) = (((a * d) + (b * c))
open Ring R
ans : ((b * d) ∼ Ring.0R R) → False
ans pr with IntegralDomain.intDom I pr
ans pr | inl x = b!=0 x
ans pr | inr x = d!=0 x
ans pr | f = exFalso (d!=0 (f b!=0))
plusWellDefined : {a b c d : fieldOfFractionsSet} → (Setoid._∼_ fieldOfFractionsSetoid a c) → (Setoid._∼_ fieldOfFractionsSetoid b d) → Setoid._∼_ fieldOfFractionsSetoid (fieldOfFractionsPlus a b) (fieldOfFractionsPlus c d)
plusWellDefined {a ,, (b , b!=0)} {c ,, (d , d!=0)} {e ,, (f , f!=0)} {g ,, (h , h!=0)} af=be ch=dg = need

2
Fields/FieldOfFractions/Field.agda
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@@ -6,7 +6,7 @@ open import Groups.Definition
open import Groups.Lemmas
open import Rings.Definition
open import Rings.Lemmas
open import Rings.IntegralDomains
open import Rings.IntegralDomains.Definition
open import Fields.Fields
open import Functions
open import Setoids.Setoids

2
Fields/FieldOfFractions/Group.agda
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@@ -6,7 +6,7 @@ open import Groups.Definition
open import Groups.Lemmas
open import Rings.Definition
open import Rings.Lemmas
open import Rings.IntegralDomains
open import Rings.IntegralDomains.Definition
open import Fields.Fields
open import Functions
open import Setoids.Setoids

2
Fields/FieldOfFractions/Lemmas.agda
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@@ -8,7 +8,7 @@ open import Groups.Lemmas
open import Rings.Definition
open import Rings.Lemmas
open import Rings.Homomorphisms.Definition
open import Rings.IntegralDomains
open import Rings.IntegralDomains.Definition
open import Fields.Fields
open import Functions
open import Setoids.Setoids

5
Fields/FieldOfFractions/Multiplication.agda
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@@ -6,7 +6,7 @@ open import Groups.Definition
open import Groups.Lemmas
open import Rings.Definition
open import Rings.Lemmas
open import Rings.IntegralDomains
open import Rings.IntegralDomains.Definition
open import Fields.Fields
open import Functions
open import Setoids.Setoids
@@ -25,8 +25,7 @@ fieldOfFractionsTimes (a ,, (b , b!=0)) (c ,, (d , d!=0)) = (a * c) ,, ((b * d)
open Ring R
ans : ((b * d) ∼ Ring.0R R) → False
ans pr with IntegralDomain.intDom I pr
ans pr | inl x = b!=0 x
ans pr | inr x = d!=0 x
ans pr | f = exFalso (d!=0 (f b!=0))
fieldOfFractionsTimesWellDefined : {a b c d : fieldOfFractionsSet} → (Setoid._∼_ fieldOfFractionsSetoid a c) → (Setoid._∼_ fieldOfFractionsSetoid b d) → (Setoid._∼_ fieldOfFractionsSetoid (fieldOfFractionsTimes a b) (fieldOfFractionsTimes c d))
fieldOfFractionsTimesWellDefined {a ,, (b , _)} {c ,, (d , _)} {e ,, (f , _)} {g ,, (h , _)} af=be ch=dg = need

1362
Fields/FieldOfFractions/Order.agda
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File diff suppressed because it is too large Load Diff

2
Fields/FieldOfFractions/Ring.agda
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@@ -6,7 +6,7 @@ open import Groups.Definition
open import Groups.Lemmas
open import Rings.Definition
open import Rings.Lemmas
open import Rings.IntegralDomains
open import Rings.IntegralDomains.Definition
open import Fields.Fields
open import Functions
open import Setoids.Setoids

11
Fields/FieldOfFractions/Setoid.agda
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@@ -6,16 +6,17 @@ open import Groups.Definition
open import Groups.Lemmas
open import Rings.Definition
open import Rings.Lemmas
open import Rings.IntegralDomains
open import Fields.Fields
open import Functions
open import Setoids.Setoids
open import Sets.EquivalenceRelations
open import Rings.IntegralDomains.Definition
open import Agda.Primitive using (Level; lzero; lsuc; _⊔_)
module Fields.FieldOfFractions.Setoid {a b : _} {A : Set a} {S : Setoid {a} {b} A} {_+_ : A → A → A} {_*_ : A → A → A} {R : Ring S _+_ _*_} (I : IntegralDomain R) where
fieldOfFractionsSet : Set (a ⊔ b)
fieldOfFractionsSet = (A && (Sg A (λ a → (Setoid._∼_ S a (Ring.0R R) → False))))
@@ -36,9 +37,7 @@ Equivalence.transitive (Setoid.eq fieldOfFractionsSetoid) {a ,, (b , b!=0)} {c ,
p2 = transitive (transitive (symmetric *Associative) (transitive (*WellDefined reflexive *Commutative) *Associative)) (transitive p (transitive (symmetric *Associative) (*WellDefined reflexive cf=de)))
p3 : (a * f) * d ∼ (b * e) * d
p3 = transitive p2 (transitive (*WellDefined reflexive *Commutative) *Associative)
p4 : (d ∼ 0R) || ((a * f) ∼ (b * e))
p4 = cancelIntDom I (transitive *Commutative (transitive p3 *Commutative))
p4 : ((d ∼ 0R) → False) → ((a * f) ∼ (b * e))
p4 = cancelIntDom R I (transitive *Commutative (transitive p3 *Commutative))
p5 : (a * f) ∼ (b * e)
p5 with p4
p5 | inl d=0 = exFalso (d!=0 d=0)
p5 | inr x = x
p5 = p4 d!=0

4
Fields/Fields.agda
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@@ -8,7 +8,7 @@ open import Rings.Lemmas
open import Setoids.Setoids
open import Setoids.Orders
open import Orders
open import Rings.IntegralDomains
open import Rings.IntegralDomains.Definition
open import Functions
open import Sets.EquivalenceRelations
@@ -23,6 +23,8 @@ record Field {m n : _} {A : Set m} {S : Setoid {m} {n} A} {_+_ : A → A → A}
field
allInvertible : (a : A) → ((a ∼ Group.0G (Ring.additiveGroup R)) → False) → Sg A (λ t → t * a ∼ 1R)
nontrivial : (0R ∼ 1R) → False
0F : A
0F = Ring.0R R
record Field' {m n : _} : Set (lsuc m ⊔ lsuc n) where
field

66
Fields/Orders/Lemmas.agda
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@@ -11,7 +11,7 @@ open import Rings.Lemmas
open import Setoids.Setoids
open import Setoids.Orders
open import Orders
open import Rings.IntegralDomains
open import Rings.IntegralDomains.Definition
open import Functions
open import Sets.EquivalenceRelations
open import Fields.Fields
@@ -84,35 +84,39 @@ abstract
r : (1/2 * ((1R + 1R) * x)) ∼ (1/2' * ((1R + 1R) * x))
r = Equivalence.transitive eq *Associative (Equivalence.transitive eq q (Equivalence.symmetric eq *Associative))
private
orderedFieldIntDom : {a b : A} → (a * b ∼ 0R) → (a ∼ 0R) || (b ∼ 0R)
orderedFieldIntDom {a} {b} ab=0 with totality 0R a
... | inl (inl x) = inr (Equivalence.transitive eq (Equivalence.transitive eq (symmetric identIsIdent) (*WellDefined q reflexive)) p')
where
open Equivalence eq
a!=0 : (a ∼ Group.0G 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.0G additiveGroup
p = *WellDefined reflexive ab=0
p' : (invA * a) * b ∼ Group.0G additiveGroup
p' = Equivalence.transitive eq (symmetric *Associative) (Equivalence.transitive eq p (Ring.timesZero R))
orderedFieldIntDom {a} {b} ab=0 | inl (inr x) = inr (Equivalence.transitive eq (Equivalence.transitive eq (symmetric identIsIdent) (*WellDefined q reflexive)) p')
where
open Equivalence eq
a!=0 : (a ∼ Group.0G 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.0G additiveGroup
p = *WellDefined reflexive ab=0
p' : (invA * a) * b ∼ Group.0G additiveGroup
p' = Equivalence.transitive eq (symmetric *Associative) (Equivalence.transitive eq p (Ring.timesZero R))
orderedFieldIntDom {a} {b} ab=0 | inr x = inl (Equivalence.symmetric (Setoid.eq S) x)
orderedFieldIsIntDom : IntegralDomain R
IntegralDomain.intDom orderedFieldIsIntDom {a} {b} ab=0 with totality (Ring.0R R) a
IntegralDomain.intDom orderedFieldIsIntDom {a} {b} ab=0 | inl (inl x) = inr (Equivalence.transitive eq (Equivalence.transitive eq (symmetric identIsIdent) (*WellDefined q reflexive)) p')
where
open Equivalence eq
a!=0 : (a ∼ Group.0G 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.0G additiveGroup
p = *WellDefined reflexive ab=0
p' : (invA * a) * b ∼ Group.0G additiveGroup
p' = Equivalence.transitive eq (symmetric *Associative) (Equivalence.transitive eq p (Ring.timesZero R))
IntegralDomain.intDom orderedFieldIsIntDom {a} {b} ab=0 | inl (inr x) = inr (Equivalence.transitive eq (Equivalence.transitive eq (symmetric identIsIdent) (*WellDefined q reflexive)) p')
where
open Equivalence eq
a!=0 : (a ∼ Group.0G 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.0G additiveGroup
p = *WellDefined reflexive ab=0
p' : (invA * a) * b ∼ Group.0G additiveGroup
p' = Equivalence.transitive eq (symmetric *Associative) (Equivalence.transitive eq p (Ring.timesZero R))
IntegralDomain.intDom orderedFieldIsIntDom {a} {b} ab=0 | inr x = inl (Equivalence.symmetric (Setoid.eq S) x)
IntegralDomain.intDom orderedFieldIsIntDom = decidedIntDom R orderedFieldIntDom
IntegralDomain.nontrivial orderedFieldIsIntDom pr = Field.nontrivial F (Equivalence.symmetric (Setoid.eq S) pr)

1
Fields/Orders/Partial/Definition.agda
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@@ -9,7 +9,6 @@ open import Rings.Lemmas
open import Setoids.Setoids
open import Setoids.Orders
open import Orders
open import Rings.IntegralDomains
open import Functions
open import Sets.EquivalenceRelations
open import Fields.Fields

1
Fields/Orders/Total/Definition.agda
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@@ -10,7 +10,6 @@ open import Rings.Lemmas
open import Setoids.Setoids
open import Setoids.Orders
open import Orders
open import Rings.IntegralDomains
open import Functions
open import Sets.EquivalenceRelations
open import Fields.Fields