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Tidy up groups (#64)

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This commit is contained in:
Patrick Stevens
2019-11-03 17:12:48 +00:00
committed by GitHub
parent e4daab7153
commit d95f510cdd
42 changed files with 1438 additions and 1038 deletions
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1
Fields/CauchyCompletion/Addition.agda
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@@ -7,6 +7,7 @@ open import Rings.Lemmas
open import Rings.Orders.Partial.Definition
open import Rings.Orders.Total.Definition
open import Groups.Definition
open import Groups.Lemmas
open import Groups.Groups
open import Fields.Fields
open import Sets.EquivalenceRelations

3
Fields/CauchyCompletion/Approximation.agda
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@@ -9,6 +9,7 @@ open import Rings.Orders.Partial.Definition
open import Rings.Orders.Total.Definition
open import Groups.Definition
open import Groups.Groups
open import Groups.Lemmas
open import Fields.Fields
open import Sets.EquivalenceRelations
open import Sequences
@@ -123,7 +124,7 @@ abstract
... | f with totality 0G (am + inverse aN)
r | am-aN<e/4 | inl (inl 0<am-aN) = SetoidPartialOrder.<Transitive pOrder (<WellDefined (Equivalence.transitive eq (invContravariant additiveGroup) (Equivalence.transitive eq groupIsAbelian (+WellDefined (Equivalence.reflexive eq) (invTwice additiveGroup _)))) (Equivalence.reflexive eq) (lemm2' _ 0<am-aN)) 0<e/4
r | f | inl (inr x) = <WellDefined (Equivalence.transitive eq (invContravariant additiveGroup) (Equivalence.transitive eq groupIsAbelian (+WellDefined (Equivalence.reflexive eq) (invTwice additiveGroup _)))) (Equivalence.reflexive eq) f
r | am-aN<e/4 | inr 0=am-aN = <WellDefined (Equivalence.transitive eq (Equivalence.symmetric eq 0=am-aN) (Equivalence.transitive eq (Equivalence.transitive eq (Equivalence.transitive eq (Equivalence.transitive eq (Equivalence.symmetric eq (invIdentity additiveGroup)) (inverseWellDefined additiveGroup 0=am-aN)) (inverseWellDefined additiveGroup groupIsAbelian)) (invContravariant additiveGroup)) (+WellDefined (Equivalence.reflexive eq) (invTwice additiveGroup _)))) (Equivalence.reflexive eq) am-aN<e/4
r | am-aN<e/4 | inr 0=am-aN = <WellDefined (Equivalence.transitive eq (Equivalence.symmetric eq 0=am-aN) (Equivalence.transitive eq (Equivalence.transitive eq (Equivalence.transitive eq (Equivalence.transitive eq (Equivalence.symmetric eq (invIdent additiveGroup)) (inverseWellDefined additiveGroup 0=am-aN)) (inverseWellDefined additiveGroup groupIsAbelian)) (invContravariant additiveGroup)) (+WellDefined (Equivalence.reflexive eq) (invTwice additiveGroup _)))) (Equivalence.reflexive eq) am-aN<e/4
q : ((inverse (index (CauchyCompletion.elts a) m)) + (index (CauchyCompletion.elts a) (succ N) + e/2)) < (e/4 + e/2)
q = <WellDefined (Equivalence.symmetric eq +Associative) (Equivalence.reflexive eq) (orderRespectsAddition r e/2)

1
Fields/CauchyCompletion/Comparison.agda
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@@ -8,6 +8,7 @@ open import Rings.Orders.Partial.Definition
open import Rings.Orders.Total.Definition
open import Groups.Definition
open import Groups.Groups
open import Groups.Lemmas
open import Fields.Fields
open import Fields.Orders.Total.Definition
open import Sets.EquivalenceRelations

4
Fields/CauchyCompletion/Group.agda
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@@ -7,6 +7,8 @@ open import Rings.Lemmas
open import Rings.Orders.Partial.Definition
open import Rings.Orders.Total.Definition
open import Groups.Definition
open import Groups.Lemmas
open import Groups.Homomorphisms.Definition
open import Groups.Groups
open import Fields.Fields
open import Sets.EquivalenceRelations
@@ -50,7 +52,7 @@ CinvRight : {a : CauchyCompletion} → Setoid._∼_ cauchyCompletionSetoid (a +C
CinvRight {a} ε 0<e = 0 , ans
where
ans : {m : ℕ} → (0 <N m) → abs (index (apply _+_ (CauchyCompletion.elts (a +C (-C a))) (map inverse (CauchyCompletion.elts (injection 0G)))) m) < ε
ans {m} 0<m rewrite indexAndApply (CauchyCompletion.elts (a +C (-C a))) (map inverse (CauchyCompletion.elts (injection 0G))) _+_ {m} | indexAndApply (CauchyCompletion.elts a) (map inverse (CauchyCompletion.elts a)) _+_ {m} | equalityCommutative (mapAndIndex (CauchyCompletion.elts a) inverse m) | equalityCommutative (mapAndIndex (constSequence 0G) inverse m) | indexAndConst 0G m = <WellDefined (Equivalence.symmetric eq (Equivalence.transitive eq (absWellDefined _ _ (Equivalence.transitive eq (+WellDefined invRight (invIdentity (Ring.additiveGroup R))) identRight)) (identityOfIndiscernablesRight _∼_ (Equivalence.reflexive eq) absZero))) (Equivalence.reflexive eq) 0<e
ans {m} 0<m rewrite indexAndApply (CauchyCompletion.elts (a +C (-C a))) (map inverse (CauchyCompletion.elts (injection 0G))) _+_ {m} | indexAndApply (CauchyCompletion.elts a) (map inverse (CauchyCompletion.elts a)) _+_ {m} | equalityCommutative (mapAndIndex (CauchyCompletion.elts a) inverse m) | equalityCommutative (mapAndIndex (constSequence 0G) inverse m) | indexAndConst 0G m = <WellDefined (Equivalence.symmetric eq (Equivalence.transitive eq (absWellDefined _ _ (Equivalence.transitive eq (+WellDefined invRight (invIdent (Ring.additiveGroup R))) identRight)) (identityOfIndiscernablesRight _∼_ (Equivalence.reflexive eq) absZero))) (Equivalence.reflexive eq) 0<e
CGroup : Group cauchyCompletionSetoid _+C_
Group.+WellDefined CGroup {a} {b} {c} {d} x y = additionWellDefined {a} {c} {b} {d} x y

1
Fields/CauchyCompletion/Multiplication.agda
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@@ -8,6 +8,7 @@ open import Rings.Orders.Partial.Definition
open import Rings.Orders.Total.Definition
open import Groups.Definition
open import Groups.Groups
open import Groups.Lemmas
open import Fields.Fields
open import Sets.EquivalenceRelations
open import Sequences

1
Fields/CauchyCompletion/Ring.agda
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@@ -8,6 +8,7 @@ open import Rings.Orders.Partial.Definition
open import Rings.Orders.Total.Definition
open import Groups.Definition
open import Groups.Groups
open import Groups.Lemmas
open import Fields.Fields
open import Sets.EquivalenceRelations
open import Sequences

1
Fields/CauchyCompletion/Setoid.agda
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@@ -8,6 +8,7 @@ open import Rings.Orders.Partial.Definition
open import Rings.Orders.Total.Definition
open import Groups.Definition
open import Groups.Groups
open import Groups.Lemmas
open import Fields.Fields
open import Sets.EquivalenceRelations
open import Sequences

1
Fields/FieldOfFractions/Lemmas.agda
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@@ -2,6 +2,7 @@
open import LogicalFormulae
open import Groups.Groups
open import Groups.Homomorphisms.Definition
open import Groups.Definition
open import Groups.Lemmas
open import Rings.Definition

1
Fields/Orders/Lemmas.agda
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@@ -2,6 +2,7 @@
open import LogicalFormulae
open import Groups.Groups
open import Groups.Lemmas
open import Groups.Definition
open import Rings.Definition
open import Rings.Orders.Partial.Definition