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Triangle inequality on the rationals (#20)

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Patrick Stevens
2019-01-18 13:00:15 +00:00
committed by GitHub
parent 20b31da82d
commit 9cccd51cbf
6 changed files with 356 additions and 22 deletions
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9
Groups/GroupsLemmas.agda
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@@ -24,3 +24,12 @@ module Groups.GroupsLemmas where
open Group G
open Equivalence eq
open Symmetric symmetricEq
swapInv : {a b : _} {A : Set a} {_+_ : A → A → A} {S : Setoid {a} {b} A} (G : Group S _+_) → {x y : A} → Setoid._∼_ S (Group.inverse G x) y → Setoid._∼_ S x (Group.inverse G y)
swapInv {S = S} G {x} {y} -x=y = transitive (symmetric (invInv G)) (inverseWellDefined G -x=y)
where
open Setoid S
open Group G
open Equivalence eq
open Symmetric symmetricEq
open Transitive transitiveEq

35
Numbers/Rationals.agda
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@@ -4,10 +4,12 @@ open import LogicalFormulae
open import Numbers.Naturals
open import Numbers.Integers
open import Groups.Groups
open import Groups.GroupDefinition
open import Rings.RingDefinition
open import Fields.Fields
open import PrimeNumbers
open import Setoids.Setoids
open import Setoids.Orders
open import Functions
open import Fields.FieldOfFractions
open import Fields.FieldOfFractionsOrder
@@ -26,8 +28,41 @@ module Numbers.Rationals where
ℚRing : Ring (fieldOfFractionsSetoid ℤIntDom) _+Q_ _*Q_
ℚRing = fieldOfFractionsRing ℤIntDom
0Q : ℚ
0Q = Ring.0R ℚRing
ℚField : Field ℚRing
ℚField = fieldOfFractions ℤIntDom
_<Q_ : ℚ → ℚ → Set
_<Q_ = fieldOfFractionsComparison ℤIntDom ℤOrderedRing
_=Q_ : ℚ → ℚ → Set
a =Q b = Setoid._∼_ (fieldOfFractionsSetoid ℤIntDom) a b
reflQ : {x : ℚ} → (x =Q x)
reflQ {x} = Reflexive.reflexive (Equivalence.reflexiveEq (Setoid.eq (fieldOfFractionsSetoid ℤIntDom))) {x}
_≤Q_ : ℚ → ℚ → Set
a ≤Q b = (a <Q b) || (a =Q b)
negateQ : ℚ → ℚ
negateQ a = Group.inverse (Ring.additiveGroup ℚRing) a
_-Q_ : ℚ → ℚ → ℚ
a -Q b = a +Q negateQ b
ℚPartialOrder : SetoidPartialOrder (fieldOfFractionsSetoid ℤIntDom) (fieldOfFractionsComparison ℤIntDom ℤOrderedRing)
ℚPartialOrder = fieldOfFractionsOrder ℤIntDom ℤOrderedRing
ℚTotalOrder : SetoidTotalOrder (fieldOfFractionsOrder ℤIntDom ℤOrderedRing)
ℚTotalOrder = fieldOfFractionsTotalOrder ℤIntDom ℤOrderedRing
absQ : ℚ → ℚ
absQ q with SetoidTotalOrder.totality (fieldOfFractionsTotalOrder ℤIntDom ℤOrderedRing) 0Q q
absQ q | inl (inl 0<q) = q
absQ q | inl (inr q<0) = Group.inverse (Ring.additiveGroup ℚRing) q
absQ q | inr x = 0Q
ℚOrdered : OrderedRing ℚRing (fieldOfFractionsTotalOrder ℤIntDom ℤOrderedRing)
ℚOrdered = fieldOfFractionsOrderedRing ℤIntDom ℤOrderedRing

271
Numbers/RationalsLemmas.agda Normal file
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@@ -0,0 +1,271 @@
{-# OPTIONS --safe --warning=error #-}
open import LogicalFormulae
open import Numbers.Naturals
open import Numbers.Integers
open import Numbers.Rationals
open import Groups.Groups
open import Groups.GroupsLemmas
open import Groups.GroupDefinition
open import Rings.RingDefinition
open import Fields.Fields
open import PrimeNumbers
open import Setoids.Setoids
open import Setoids.Orders
open import Functions
open import Fields.FieldOfFractions
open import Fields.FieldOfFractionsOrder
open import Rings.IntegralDomains
open import Rings.RingLemmas
module Numbers.RationalsLemmas where
triangleInequality : {a b : ℚ} → (absQ (a +Q b)) ≤Q ((absQ a) +Q (absQ b))
triangleInequality {a} {b} with SetoidTotalOrder.totality ℚTotalOrder 0Q a
triangleInequality {a} {b} | inl (inl 0<a) with SetoidTotalOrder.totality ℚTotalOrder 0Q b
triangleInequality {a} {b} | inl (inl 0<a) | inl (inl 0<b) with SetoidTotalOrder.totality ℚTotalOrder 0Q (a +Q b)
triangleInequality {a} {b} | inl (inl 0<a) | inl (inl 0<b) | inl (inl 0<a+b) = inr (reflexive {a +Q b})
where
open Reflexive (Equivalence.reflexiveEq (Setoid.eq (fieldOfFractionsSetoid ℤIntDom)))
triangleInequality {a} {b} | inl (inl 0<a) | inl (inl 0<b) | inl (inr a+b<0) = exFalso (exFalso (SetoidPartialOrder.irreflexive (fieldOfFractionsOrder ℤIntDom ℤOrderedRing) {a +Q b} (SetoidPartialOrder.transitive (fieldOfFractionsOrder ℤIntDom ℤOrderedRing) {a +Q b} {0Q} {a +Q b} a+b<0 (SetoidPartialOrder.wellDefined (fieldOfFractionsOrder ℤIntDom ℤOrderedRing) {0Q +Q 0Q} {0Q} {a +Q b} {a +Q b} (symmetric {0Q} {0Q +Q 0Q} (Group.multIdentRight (Ring.additiveGroup ℚRing) {0Q})) (reflexive {a +Q b}) (ringAddInequalities ℚOrdered {0Q} {a} {0Q} {b} 0<a 0<b)))))
where
open Group (Ring.additiveGroup ℚRing)
open Setoid (fieldOfFractionsSetoid ℤIntDom)
open Symmetric (Equivalence.symmetricEq eq)
open Reflexive (Equivalence.reflexiveEq eq)
open Transitive (Equivalence.transitiveEq eq)
triangleInequality {a} {b} | inl (inl 0<a) | inl (inl 0<b) | inr 0=a+b = exFalso (SetoidPartialOrder.irreflexive (fieldOfFractionsOrder ℤIntDom ℤOrderedRing) {0Q} (SetoidPartialOrder.wellDefined (fieldOfFractionsOrder ℤIntDom ℤOrderedRing) {0Q +Q 0Q} {0Q} {a +Q b} {0Q} (multIdentRight {0Q}) (symmetric {0Q} {a +Q b} 0=a+b) (ringAddInequalities ℚOrdered {0Q} {a} {0Q} {b} 0<a 0<b)))
where
open Group (Ring.additiveGroup ℚRing)
open Setoid (fieldOfFractionsSetoid ℤIntDom)
open Symmetric (Equivalence.symmetricEq eq)
open Reflexive (Equivalence.reflexiveEq eq)
open Transitive (Equivalence.transitiveEq eq)
triangleInequality {a} {b} | inl (inl 0<a) | inl (inr b<0) with SetoidTotalOrder.totality ℚTotalOrder 0Q (a +Q b)
triangleInequality {a} {b} | inl (inl 0<a) | inl (inr b<0) | inl (inl 0<a+b) = inl (SetoidPartialOrder.wellDefined (fieldOfFractionsOrder ℤIntDom ℤOrderedRing) {b +Q a} {a +Q b} {inverse b +Q a} {a +Q inverse b} (Ring.groupIsAbelian ℚRing {b} {a}) (Ring.groupIsAbelian ℚRing {inverse b} {a}) (OrderedRing.orderRespectsAddition ℚOrdered {b} {inverse b} (SetoidPartialOrder.transitive (fieldOfFractionsOrder ℤIntDom ℤOrderedRing) {b} {0Q} {inverse b} b<0 (ringMinusFlipsOrder'' ℚOrdered {b} b<0)) a))
where
open Group (Ring.additiveGroup ℚRing)
open Setoid (fieldOfFractionsSetoid ℤIntDom)
open Symmetric (Equivalence.symmetricEq eq)
open Reflexive (Equivalence.reflexiveEq eq)
open Transitive (Equivalence.transitiveEq eq)
triangleInequality {a} {b} | inl (inl 0<a) | inl (inr b<0) | inl (inr a+b<0) = inl (SetoidPartialOrder.wellDefined (fieldOfFractionsOrder ℤIntDom ℤOrderedRing) {inverse a +Q inverse b} {inverse (a +Q b)} {a +Q inverse b} {a +Q inverse b} (transitive {inverse a +Q inverse b} {inverse b +Q inverse a} {inverse (a +Q b)} (Ring.groupIsAbelian ℚRing {inverse a} {inverse b}) (symmetric {inverse (a +Q b)} {inverse b +Q inverse a} (invContravariant (Ring.additiveGroup ℚRing) {a} {b}))) (reflexive {a +Q inverse b}) (OrderedRing.orderRespectsAddition ℚOrdered {inverse a} {a} (SetoidPartialOrder.transitive (fieldOfFractionsOrder ℤIntDom ℤOrderedRing) {inverse a} {0Q} {a} (ringMinusFlipsOrder ℚOrdered {a} 0<a) 0<a) (inverse b)))
where
open Group (Ring.additiveGroup ℚRing)
open Setoid (fieldOfFractionsSetoid ℤIntDom)
open Symmetric (Equivalence.symmetricEq eq)
open Reflexive (Equivalence.reflexiveEq eq)
open Transitive (Equivalence.transitiveEq eq)
triangleInequality {a} {b} | inl (inl 0<a) | inl (inr b<0) | inr 0=a+b = inl (SetoidPartialOrder.transitive (fieldOfFractionsOrder ℤIntDom ℤOrderedRing) {0Q} {a} {a +Q (inverse b)} 0<a (SetoidPartialOrder.wellDefined (fieldOfFractionsOrder ℤIntDom ℤOrderedRing) {0Q +Q a} {a} {inverse b +Q a} {a +Q inverse b} (multIdentLeft {a}) (Ring.groupIsAbelian ℚRing {inverse b} {a}) (OrderedRing.orderRespectsAddition ℚOrdered {0Q} {inverse b} (ringMinusFlipsOrder'' ℚOrdered {b} b<0) a)))
where
open Group (Ring.additiveGroup ℚRing)
open Setoid (fieldOfFractionsSetoid ℤIntDom)
open Symmetric (Equivalence.symmetricEq eq)
open Reflexive (Equivalence.reflexiveEq eq)
open Transitive (Equivalence.transitiveEq eq)
triangleInequality {a} {b} | inl (inl 0<a) | inr 0=b with SetoidTotalOrder.totality ℚTotalOrder 0Q (a +Q b)
triangleInequality {a} {b} | inl (inl 0<a) | inr 0=b | inl (inl 0<a+b) = inr (wellDefined {a} {b} {a} {0Q} (reflexive {a}) (symmetric {0Q} {b} 0=b))
where
open Group (Ring.additiveGroup ℚRing)
open Setoid (fieldOfFractionsSetoid ℤIntDom)
open Symmetric (Equivalence.symmetricEq eq)
open Reflexive (Equivalence.reflexiveEq eq)
open Transitive (Equivalence.transitiveEq eq)
triangleInequality {a} {b} | inl (inl 0<a) | inr 0=b | inl (inr a+b<0) = exFalso (SetoidPartialOrder.irreflexive (fieldOfFractionsOrder ℤIntDom ℤOrderedRing) {a +Q b} (SetoidPartialOrder.transitive (fieldOfFractionsOrder ℤIntDom ℤOrderedRing) {a +Q b} {0Q} {a +Q b} a+b<0 (SetoidPartialOrder.wellDefined (fieldOfFractionsOrder ℤIntDom ℤOrderedRing) {0Q} {0Q} {a} {a +Q b} (reflexive {0Q}) (transitive {a} {a +Q 0Q} {a +Q b} (symmetric {a +Q 0Q} {a} (multIdentRight {a})) (wellDefined {a} {0Q} {a} {b} (reflexive {a}) 0=b)) 0<a)))
where
open Group (Ring.additiveGroup ℚRing)
open Setoid (fieldOfFractionsSetoid ℤIntDom)
open Symmetric (Equivalence.symmetricEq eq)
open Reflexive (Equivalence.reflexiveEq eq)
open Transitive (Equivalence.transitiveEq eq)
triangleInequality {a} {b} | inl (inl 0<a) | inr 0=b | inr 0=a+b = exFalso (SetoidPartialOrder.irreflexive (fieldOfFractionsOrder ℤIntDom ℤOrderedRing) {a +Q b} (SetoidPartialOrder.wellDefined (fieldOfFractionsOrder ℤIntDom ℤOrderedRing) {0Q} {a +Q b} {a} {a +Q b} 0=a+b (transitive {a} {a +Q 0Q} {a +Q b} (symmetric {a +Q 0Q} {a} (multIdentRight {a})) (wellDefined {a} {0Q} {a} {b} (reflexive {a}) 0=b)) 0<a))
where
open Group (Ring.additiveGroup ℚRing)
open Setoid (fieldOfFractionsSetoid ℤIntDom)
open Symmetric (Equivalence.symmetricEq eq)
open Reflexive (Equivalence.reflexiveEq eq)
open Transitive (Equivalence.transitiveEq eq)
triangleInequality {a} {b} | inl (inr a<0) with SetoidTotalOrder.totality ℚTotalOrder 0Q b
triangleInequality {a} {b} | inl (inr a<0) | inl (inl 0<b) with SetoidTotalOrder.totality ℚTotalOrder 0Q (a +Q b)
triangleInequality {a} {b} | inl (inr a<0) | inl (inl 0<b) | inl (inl 0<a+b) = inl (OrderedRing.orderRespectsAddition ℚOrdered {a} {inverse a} (SetoidPartialOrder.transitive (fieldOfFractionsOrder ℤIntDom ℤOrderedRing) {a} {0Q} {inverse a} a<0 (ringMinusFlipsOrder'' ℚOrdered {a} a<0)) b)
where
open Group (Ring.additiveGroup ℚRing)
open Setoid (fieldOfFractionsSetoid ℤIntDom)
open Symmetric (Equivalence.symmetricEq eq)
open Reflexive (Equivalence.reflexiveEq eq)
open Transitive (Equivalence.transitiveEq eq)
triangleInequality {a} {b} | inl (inr a<0) | inl (inl 0<b) | inl (inr a+b<0) = inl (SetoidPartialOrder.wellDefined (fieldOfFractionsOrder ℤIntDom ℤOrderedRing) {inverse a +Q inverse b} {inverse (a +Q b)} {inverse a +Q b} {inverse a +Q b} (transitive {inverse a +Q inverse b} {inverse b +Q inverse a} {inverse (a +Q b)} (Ring.groupIsAbelian ℚRing {inverse a} {inverse b}) (symmetric {inverse (a +Q b)} {inverse b +Q inverse a} (invContravariant (Ring.additiveGroup ℚRing) {a} {b}))) (reflexive {inverse a +Q b}) blah)
where
open Group (Ring.additiveGroup ℚRing)
open Setoid (fieldOfFractionsSetoid ℤIntDom)
open Symmetric (Equivalence.symmetricEq eq)
open Reflexive (Equivalence.reflexiveEq eq)
open Transitive (Equivalence.transitiveEq eq)
blah : (inverse a +Q inverse b) <Q (inverse a +Q b)
blah = SetoidPartialOrder.wellDefined (fieldOfFractionsOrder ℤIntDom ℤOrderedRing) {inverse b +Q inverse a} {inverse a +Q inverse b} {b +Q inverse a} {inverse a +Q b} (Ring.groupIsAbelian ℚRing {inverse b} {inverse a}) (Ring.groupIsAbelian ℚRing {b} {inverse a}) (OrderedRing.orderRespectsAddition ℚOrdered {inverse b} {b} (SetoidPartialOrder.transitive (fieldOfFractionsOrder ℤIntDom ℤOrderedRing) {inverse b} {0Q} {b} (ringMinusFlipsOrder ℚOrdered {b} 0<b) 0<b) (inverse a))
triangleInequality {a} {b} | inl (inr a<0) | inl (inl 0<b) | inr 0=a+b = inl (SetoidPartialOrder.transitive (fieldOfFractionsOrder ℤIntDom ℤOrderedRing) {0Q} {b} {inverse a +Q b} 0<b (SetoidPartialOrder.wellDefined (fieldOfFractionsOrder ℤIntDom ℤOrderedRing) {b +Q 0Q} {b} {b +Q inverse a} {inverse a +Q b} (multIdentRight {b}) (Ring.groupIsAbelian ℚRing {b} {inverse a}) (SetoidPartialOrder.wellDefined (fieldOfFractionsOrder ℤIntDom ℤOrderedRing) {0Q +Q b} {b +Q 0Q} {inverse a +Q b} {b +Q inverse a} (Ring.groupIsAbelian ℚRing {0Q} {b}) (Ring.groupIsAbelian ℚRing {inverse a} {b}) (OrderedRing.orderRespectsAddition ℚOrdered {0Q} {inverse a} (ringMinusFlipsOrder'' ℚOrdered {a} a<0) b))))
where
open Group (Ring.additiveGroup ℚRing)
open Setoid (fieldOfFractionsSetoid ℤIntDom)
open Symmetric (Equivalence.symmetricEq eq)
open Reflexive (Equivalence.reflexiveEq eq)
open Transitive (Equivalence.transitiveEq eq)
triangleInequality {a} {b} | inl (inr a<0) | inl (inr b<0) with SetoidTotalOrder.totality ℚTotalOrder 0Q (a +Q b)
triangleInequality {a} {b} | inl (inr a<0) | inl (inr b<0) | inl (inl 0<a+b) = exFalso (SetoidPartialOrder.irreflexive (fieldOfFractionsOrder ℤIntDom ℤOrderedRing) {a +Q b} (SetoidPartialOrder.transitive (fieldOfFractionsOrder ℤIntDom ℤOrderedRing) {a +Q b} {0Q} {a +Q b} (SetoidPartialOrder.wellDefined (fieldOfFractionsOrder ℤIntDom ℤOrderedRing) {a +Q b} {a +Q b} {0Q +Q 0Q} {0Q} (reflexive {a +Q b}) (multIdentRight {0Q}) (ringAddInequalities ℚOrdered {a} {0Q} {b} {0Q} a<0 b<0)) 0<a+b))
where
open Group (Ring.additiveGroup ℚRing)
open Setoid (fieldOfFractionsSetoid ℤIntDom)
open Symmetric (Equivalence.symmetricEq eq)
open Reflexive (Equivalence.reflexiveEq eq)
open Transitive (Equivalence.transitiveEq eq)
triangleInequality {a} {b} | inl (inr a<0) | inl (inr b<0) | inl (inr a+b<0) = inr (transitive {inverse (fieldOfFractionsPlus ℤIntDom a b)} {fieldOfFractionsPlus ℤIntDom (inverse b) (inverse a)} {fieldOfFractionsPlus ℤIntDom (inverse a) (inverse b)} (invContravariant (Ring.additiveGroup ℚRing) {a} {b}) (Ring.groupIsAbelian ℚRing {inverse b} {inverse a}))
where
open Group (Ring.additiveGroup ℚRing)
open Setoid (fieldOfFractionsSetoid ℤIntDom)
open Symmetric (Equivalence.symmetricEq eq)
open Reflexive (Equivalence.reflexiveEq eq)
open Transitive (Equivalence.transitiveEq eq)
triangleInequality {a} {b} | inl (inr a<0) | inl (inr b<0) | inr 0=a+b = exFalso (SetoidPartialOrder.irreflexive (fieldOfFractionsOrder ℤIntDom ℤOrderedRing) {0Q} (SetoidPartialOrder.wellDefined (fieldOfFractionsOrder ℤIntDom ℤOrderedRing) {a +Q b} {0Q} {0Q} {0Q} (symmetric {0Q} {a +Q b} 0=a+b) (reflexive {0Q}) (SetoidPartialOrder.wellDefined (fieldOfFractionsOrder ℤIntDom ℤOrderedRing) {a +Q b} {a +Q b} {0Q +Q 0Q} {0Q} (reflexive {a +Q b}) (multIdentLeft {0Q}) (ringAddInequalities ℚOrdered {a} {0Q} {b} {0Q} a<0 b<0))))
where
open Group (Ring.additiveGroup ℚRing)
open Setoid (fieldOfFractionsSetoid ℤIntDom)
open Symmetric (Equivalence.symmetricEq eq)
open Reflexive (Equivalence.reflexiveEq eq)
open Transitive (Equivalence.transitiveEq eq)
triangleInequality {a} {b} | inl (inr a<0) | inr 0=b with SetoidTotalOrder.totality ℚTotalOrder 0Q (a +Q b)
triangleInequality {a} {b} | inl (inr a<0) | inr 0=b | inl (inl 0<a+b) = exFalso (SetoidPartialOrder.irreflexive (fieldOfFractionsOrder ℤIntDom ℤOrderedRing) {a} (SetoidPartialOrder.transitive (fieldOfFractionsOrder ℤIntDom ℤOrderedRing) {a} {0Q} {a} a<0 (SetoidPartialOrder.wellDefined (fieldOfFractionsOrder ℤIntDom ℤOrderedRing) {0Q} {0Q} {a +Q b} {a} (reflexive {0Q}) (transitive {a +Q b} {a +Q 0Q} {a} (wellDefined {a} {b} {a} {0Q} (reflexive {a}) (symmetric {0Q} {b} 0=b)) (multIdentRight {a})) 0<a+b)))
where
open Group (Ring.additiveGroup ℚRing)
open Setoid (fieldOfFractionsSetoid ℤIntDom)
open Symmetric (Equivalence.symmetricEq eq)
open Reflexive (Equivalence.reflexiveEq eq)
open Transitive (Equivalence.transitiveEq eq)
triangleInequality {a} {b} | inl (inr a<0) | inr 0=b | inl (inr a+b<0) = inr (transitive {inverse (a +Q b)} {(inverse b) +Q (inverse a)} {(inverse a) +Q 0Q} (invContravariant (Ring.additiveGroup ℚRing) {a} {b}) (transitive {inverse b +Q inverse a} {inverse a +Q inverse b} {inverse a +Q 0Q} (Ring.groupIsAbelian ℚRing {inverse b} {inverse a}) (wellDefined {inverse a} {inverse b} {inverse a} {0Q} (reflexive {inverse a}) (transitive {inverse b} {inverse 0Q} {0Q} (inverseWellDefined (Ring.additiveGroup ℚRing) {b} {0Q} (symmetric {0Q} {b} 0=b)) (symmetric {0Q} {inverse 0Q} (invIdentity (Ring.additiveGroup ℚRing)))))))
where
open Group (Ring.additiveGroup ℚRing)
open Setoid (fieldOfFractionsSetoid ℤIntDom)
open Symmetric (Equivalence.symmetricEq eq)
open Reflexive (Equivalence.reflexiveEq eq)
open Transitive (Equivalence.transitiveEq eq)
triangleInequality {a} {b} | inl (inr a<0) | inr 0=b | inr 0=a+b = exFalso (SetoidPartialOrder.irreflexive (fieldOfFractionsOrder ℤIntDom ℤOrderedRing) {0Q} (SetoidPartialOrder.wellDefined (fieldOfFractionsOrder ℤIntDom ℤOrderedRing) {a} {0Q} {0Q} {0Q} (transitive {a} {a +Q b} {0Q} (transitive {a} {a +Q 0Q} {a +Q b} (symmetric {a +Q 0Q} {a} (multIdentRight {a})) (wellDefined {a} {0Q} {a} {b} (reflexive {a}) 0=b)) (symmetric {0Q} {a +Q b} 0=a+b)) (reflexive {0Q}) a<0))
where
open Group (Ring.additiveGroup ℚRing)
open Setoid (fieldOfFractionsSetoid ℤIntDom)
open Symmetric (Equivalence.symmetricEq eq)
open Reflexive (Equivalence.reflexiveEq eq)
open Transitive (Equivalence.transitiveEq eq)
triangleInequality {a} {b} | inr 0=a with SetoidTotalOrder.totality ℚTotalOrder 0Q b
triangleInequality {a} {b} | inr 0=a | inl (inl 0<b) with SetoidTotalOrder.totality ℚTotalOrder 0Q (a +Q b)
triangleInequality {a} {b} | inr 0=a | inl (inl 0<b) | inl (inl 0<a+b) = inr (Group.wellDefined (Ring.additiveGroup ℚRing) {a} {b} {0Q} {b} (symmetric {0Q} {a} 0=a) (reflexive {b}))
where
open Setoid (fieldOfFractionsSetoid ℤIntDom)
open Symmetric (Equivalence.symmetricEq eq)
open Reflexive (Equivalence.reflexiveEq eq)
triangleInequality {a} {b} | inr 0=a | inl (inl 0<b) | inl (inr a+b<0) = exFalso (SetoidPartialOrder.irreflexive (fieldOfFractionsOrder ℤIntDom ℤOrderedRing) {a +Q b} (SetoidPartialOrder.transitive (fieldOfFractionsOrder ℤIntDom ℤOrderedRing) {a +Q b} {0Q} {a +Q b} a+b<0 (SetoidPartialOrder.wellDefined (fieldOfFractionsOrder ℤIntDom ℤOrderedRing) {0Q} {0Q} {b} {a +Q b} (reflexive {0Q}) (transitive {b} {0Q +Q b} {a +Q b} (symmetric {0Q +Q b} {b} (Group.multIdentLeft (Ring.additiveGroup ℚRing) {b})) (Group.wellDefined (Ring.additiveGroup ℚRing) {0Q} {b} {a} {b} 0=a (reflexive {b}))) 0<b)))
where
open Setoid (fieldOfFractionsSetoid ℤIntDom)
open Symmetric (Equivalence.symmetricEq eq)
open Reflexive (Equivalence.reflexiveEq eq)
open Transitive (Equivalence.transitiveEq eq)
triangleInequality {a} {b} | inr 0=a | inl (inl 0<b) | inr 0=a+b = exFalso (SetoidPartialOrder.irreflexive (fieldOfFractionsOrder ℤIntDom ℤOrderedRing) {b} (SetoidPartialOrder.wellDefined (fieldOfFractionsOrder ℤIntDom ℤOrderedRing) {0Q} {b} {b} {b} (transitive {0Q} {a +Q b} {b} 0=a+b (transitive {a +Q b} {0Q +Q b} {b} (wellDefined {a} {b} {0Q} {b} (symmetric {0Q} {a} 0=a) (reflexive {b})) (multIdentLeft {b}))) (reflexive {b}) 0<b))
where
open Group (Ring.additiveGroup ℚRing)
open Setoid (fieldOfFractionsSetoid ℤIntDom)
open Symmetric (Equivalence.symmetricEq eq)
open Reflexive (Equivalence.reflexiveEq eq)
open Transitive (Equivalence.transitiveEq eq)
triangleInequality {a} {b} | inr 0=a | inl (inr b<0) with SetoidTotalOrder.totality ℚTotalOrder 0Q (a +Q b)
triangleInequality {a} {b} | inr 0=a | inl (inr b<0) | inl (inl 0<a+b) = exFalso (SetoidPartialOrder.irreflexive (fieldOfFractionsOrder ℤIntDom ℤOrderedRing) {b} (SetoidPartialOrder.transitive (fieldOfFractionsOrder ℤIntDom ℤOrderedRing) {b} {0Q} {b} b<0 (SetoidPartialOrder.wellDefined (fieldOfFractionsOrder ℤIntDom ℤOrderedRing) {0Q} {0Q} {a +Q b} {b} (reflexive {0Q}) (transitive {a +Q b} {0Q +Q b} {b} (wellDefined {a} {b} {0Q} {b} (symmetric {0Q} {a} 0=a) (reflexive {b})) (multIdentLeft {b})) 0<a+b)))
where
open Group (Ring.additiveGroup ℚRing)
open Setoid (fieldOfFractionsSetoid ℤIntDom)
open Symmetric (Equivalence.symmetricEq eq)
open Reflexive (Equivalence.reflexiveEq eq)
open Transitive (Equivalence.transitiveEq eq)
triangleInequality {a} {b} | inr 0=a | inl (inr b<0) | inl (inr a+b<0) = inr (transitive {inverse (a +Q b)} {inverse b +Q inverse a} {0Q +Q inverse b} (invContravariant (Ring.additiveGroup ℚRing) {a} {b}) (transitive {inverse b +Q inverse a} {inverse a +Q inverse b} {0Q +Q inverse b} (Ring.groupIsAbelian ℚRing {inverse b} {inverse a}) (wellDefined {inverse a} {inverse b} {0Q} {inverse b} (transitive {inverse a} {inverse 0Q} {0Q} (symmetric {inverse 0Q} {inverse a} (inverseWellDefined (Ring.additiveGroup ℚRing) {0Q} {a} 0=a)) (invIdentity (Ring.additiveGroup ℚRing))) (reflexive {inverse b}))))
where
open Group (Ring.additiveGroup ℚRing)
open Setoid (fieldOfFractionsSetoid ℤIntDom)
open Symmetric (Equivalence.symmetricEq eq)
open Reflexive (Equivalence.reflexiveEq eq)
open Transitive (Equivalence.transitiveEq eq)
triangleInequality {a} {b} | inr 0=a | inl (inr b<0) | inr 0=a+b = exFalso (SetoidPartialOrder.irreflexive (fieldOfFractionsOrder ℤIntDom ℤOrderedRing) {b} (SetoidPartialOrder.wellDefined (fieldOfFractionsOrder ℤIntDom ℤOrderedRing) {b} {b} {0Q} {b} (reflexive {b}) (transitive {0Q} {a +Q b} {b} 0=a+b (transitive {a +Q b} {0Q +Q b} {b} (wellDefined {a} {b} {0Q} {b} (symmetric {0Q} {a} 0=a) (reflexive {b})) (multIdentLeft {b}))) b<0))
where
open Group (Ring.additiveGroup ℚRing)
open Setoid (fieldOfFractionsSetoid ℤIntDom)
open Symmetric (Equivalence.symmetricEq eq)
open Reflexive (Equivalence.reflexiveEq eq)
open Transitive (Equivalence.transitiveEq eq)
triangleInequality {a} {b} | inr 0=a | inr 0=b with SetoidTotalOrder.totality ℚTotalOrder 0Q (a +Q b)
triangleInequality {a} {b} | inr 0=a | inr 0=b | inl (inl 0<a+b) = exFalso (SetoidPartialOrder.irreflexive (fieldOfFractionsOrder ℤIntDom ℤOrderedRing) {0Q} (SetoidPartialOrder.wellDefined (fieldOfFractionsOrder ℤIntDom ℤOrderedRing) {0Q} {0Q} {a +Q b} {0Q} (reflexive {0Q}) (transitive {a +Q b} {0Q +Q 0Q} {0Q} (wellDefined {a} {b} {0Q} {0Q} (symmetric {0Q} {a} 0=a) (symmetric {0Q} {b} 0=b)) (multIdentRight {0Q})) 0<a+b))
where
open Group (Ring.additiveGroup ℚRing)
open Setoid (fieldOfFractionsSetoid ℤIntDom)
open Symmetric (Equivalence.symmetricEq eq)
open Reflexive (Equivalence.reflexiveEq eq)
open Transitive (Equivalence.transitiveEq eq)
triangleInequality {a} {b} | inr 0=a | inr 0=b | inl (inr a+b<0) = exFalso (SetoidPartialOrder.irreflexive (fieldOfFractionsOrder ℤIntDom ℤOrderedRing) {0Q} (SetoidPartialOrder.wellDefined (fieldOfFractionsOrder ℤIntDom ℤOrderedRing) {a +Q b} {0Q} {0Q} {0Q} (transitive {a +Q b} {0Q +Q 0Q} {0Q} (wellDefined {a} {b} {0Q} {0Q} (symmetric {0Q} {a} 0=a) (symmetric {0Q} {b} 0=b)) (multIdentRight {0Q})) (reflexive {0Q}) a+b<0))
where
open Group (Ring.additiveGroup ℚRing)
open Setoid (fieldOfFractionsSetoid ℤIntDom)
open Symmetric (Equivalence.symmetricEq eq)
open Reflexive (Equivalence.reflexiveEq eq)
open Transitive (Equivalence.transitiveEq eq)
triangleInequality {a} {b} | inr 0=a | inr 0=b | inr 0=a+b = inr refl
absNegation : (a : ℚ) → ((absQ a) =Q (absQ (negateQ a)))
absNegation a with SetoidTotalOrder.totality ℚTotalOrder 0Q a
absNegation a | inl (inl 0<a) with SetoidTotalOrder.totality ℚTotalOrder 0Q (negateQ a)
absNegation a | inl (inl 0<a) | inl (inl 0<-a) = exFalso (SetoidPartialOrder.irreflexive (fieldOfFractionsOrder ℤIntDom ℤOrderedRing) {a} (SetoidPartialOrder.transitive (fieldOfFractionsOrder ℤIntDom ℤOrderedRing) {a} {0Q} {a} (ringMinusFlipsOrder''' ℚOrdered {a} 0<-a) 0<a))
where
open Group (Ring.additiveGroup ℚRing)
open Setoid (fieldOfFractionsSetoid ℤIntDom)
open Symmetric (Equivalence.symmetricEq eq)
open Reflexive (Equivalence.reflexiveEq eq)
open Transitive (Equivalence.transitiveEq eq)
absNegation a | inl (inl 0<a) | inl (inr _) = symmetric {inverse (inverse a)} {a} (invTwice (Ring.additiveGroup ℚRing) a)
where
open Group (Ring.additiveGroup ℚRing)
open Setoid (fieldOfFractionsSetoid ℤIntDom)
open Symmetric (Equivalence.symmetricEq eq)
open Reflexive (Equivalence.reflexiveEq eq)
open Transitive (Equivalence.transitiveEq eq)
absNegation a | inl (inl 0<a) | inr 0=-a = exFalso (SetoidPartialOrder.irreflexive (fieldOfFractionsOrder ℤIntDom ℤOrderedRing) {0Q} (SetoidPartialOrder.wellDefined (fieldOfFractionsOrder ℤIntDom ℤOrderedRing) {0Q} {0Q} {a} {0Q} (reflexive {0Q}) (transitive {a} {inverse 0Q} {0Q} (swapInv (Ring.additiveGroup ℚRing) {a} {0Q} (symmetric {0Q} {inverse a} 0=-a)) (invIdentity (Ring.additiveGroup ℚRing))) 0<a))
where
open Group (Ring.additiveGroup ℚRing)
open Setoid (fieldOfFractionsSetoid ℤIntDom)
open Symmetric (Equivalence.symmetricEq eq)
open Reflexive (Equivalence.reflexiveEq eq)
open Transitive (Equivalence.transitiveEq eq)
absNegation a | inl (inr a<0) with SetoidTotalOrder.totality ℚTotalOrder 0Q (negateQ a)
absNegation a | inl (inr a<0) | inl (inl _) = Reflexive.reflexive (Equivalence.reflexiveEq (Setoid.eq (fieldOfFractionsSetoid ℤIntDom))) {Group.inverse (Ring.additiveGroup ℚRing) a}
absNegation a | inl (inr a<0) | inl (inr -a<0) = exFalso (SetoidPartialOrder.irreflexive (fieldOfFractionsOrder ℤIntDom ℤOrderedRing) {0Q} (SetoidPartialOrder.transitive (fieldOfFractionsOrder ℤIntDom ℤOrderedRing) {0Q} {a} {0Q} (ringMinusFlipsOrder' ℚOrdered {a} -a<0) a<0))
where
open Group (Ring.additiveGroup ℚRing)
open Setoid (fieldOfFractionsSetoid ℤIntDom)
open Symmetric (Equivalence.symmetricEq eq)
open Reflexive (Equivalence.reflexiveEq eq)
open Transitive (Equivalence.transitiveEq eq)
absNegation a | inl (inr a<0) | inr -a=0 = exFalso (SetoidPartialOrder.irreflexive (fieldOfFractionsOrder ℤIntDom ℤOrderedRing) {0Q} (SetoidPartialOrder.wellDefined (fieldOfFractionsOrder ℤIntDom ℤOrderedRing) {a} {0Q} {0Q} {0Q} (transitive {a} {inverse 0Q} {0Q} (swapInv (Ring.additiveGroup ℚRing) {a} {0Q} (symmetric {0Q} {inverse a} -a=0)) (invIdentity (Ring.additiveGroup ℚRing))) (reflexive {0Q}) a<0))
where
open Group (Ring.additiveGroup ℚRing)
open Setoid (fieldOfFractionsSetoid ℤIntDom)
open Symmetric (Equivalence.symmetricEq eq)
open Reflexive (Equivalence.reflexiveEq eq)
open Transitive (Equivalence.transitiveEq eq)
absNegation a | inr 0=a with SetoidTotalOrder.totality ℚTotalOrder 0Q (negateQ a)
absNegation a | inr 0=a | inl (inl 0<-a) = exFalso (SetoidPartialOrder.irreflexive (fieldOfFractionsOrder ℤIntDom ℤOrderedRing) {0Q} (SetoidPartialOrder.wellDefined (fieldOfFractionsOrder ℤIntDom ℤOrderedRing) {0Q} {0Q} {inverse a} {0Q} (reflexive {0Q}) (transitive {inverse a} {inverse 0Q} {0Q} (symmetric {inverse 0Q} {inverse a} (inverseWellDefined (Ring.additiveGroup ℚRing) {0Q} {a} 0=a)) (invIdentity (Ring.additiveGroup ℚRing))) 0<-a))
where
open Group (Ring.additiveGroup ℚRing)
open Setoid (fieldOfFractionsSetoid ℤIntDom)
open Symmetric (Equivalence.symmetricEq eq)
open Reflexive (Equivalence.reflexiveEq eq)
open Transitive (Equivalence.transitiveEq eq)
absNegation a | inr 0=a | inl (inr -a<0) = exFalso (SetoidPartialOrder.irreflexive (fieldOfFractionsOrder ℤIntDom ℤOrderedRing) {0Q} (SetoidPartialOrder.wellDefined (fieldOfFractionsOrder ℤIntDom ℤOrderedRing) {inverse a} {0Q} {0Q} {0Q} (transitive {inverse a} {inverse 0Q} {0Q} (symmetric {inverse 0Q} {inverse a} (inverseWellDefined (Ring.additiveGroup ℚRing) {0Q} {a} 0=a)) (invIdentity (Ring.additiveGroup ℚRing))) (reflexive {0Q}) -a<0))
where
open Group (Ring.additiveGroup ℚRing)
open Setoid (fieldOfFractionsSetoid ℤIntDom)
open Symmetric (Equivalence.symmetricEq eq)
open Reflexive (Equivalence.reflexiveEq eq)
open Transitive (Equivalence.transitiveEq eq)
absNegation a | inr 0=a | inr _ = refl

51
Numbers/Reals.agda
View File

@@ -1,32 +1,39 @@
{-# OPTIONS --safe --warning=error #-}
open import Fields
open import Rings
open import Fields.Fields
open import Functions
open import Orders
open import LogicalFormulae
open import Rationals
open import Naturals
open import Numbers.Rationals
open import Numbers.RationalsLemmas
open import Numbers.Naturals
open import Setoids.Setoids
open import Setoids.Orders
open import Agda.Primitive using (Level; lzero; lsuc; _⊔_)
module Reals where
record Subset {m n : _} (A : Set m) (B : Set n) : Set (m ⊔ n) where
field
inj : A → B
injInj : Injection inj
--record RealField {n : _} {A : Set n} {R : Ring A} {F : Field R} : Set _ where
-- field
-- orderedField : OrderedField F
-- open OrderedField orderedField
-- open TotalOrder ord
-- open Ring R
-- field
-- complete : {c : _} {C : Set c} → (S : Subset C A) → (upperBound : A) → ({s : C} → (Subset.inj S s) < upperBound) → Sg A (λ lub → ({s : C} → (Subset.inj S s) < lub) && ({s : C} → (Subset.inj S s) < upperBound → (Subset.inj S s) < lub))
record Real : Set where
module Numbers.Reals where
record ℝ : Set where
field
f : ℕ → ℚ
converges : {ε : ℚ} → Sg ℕ (λ x → {y : ℕ} → x <N y → (f x) +Q (f y) <Q ε)
converges : {ε : ℚ} → Sg ℕ (λ x → {y : ℕ} → x <N y → (absQ ((f x) -Q (f y))) <Q ε)
_+R_ : ℝ → ℝ → ℝ
ℝ.f (record { f = a ; converges = conA } +R record { f = b ; converges = conB }) n = (a n) +Q (b n)
ℝ.converges (record { f = a ; converges = conA } +R record { f = b ; converges = conB }) {e} = {!absQ (a x +Q b x)!}
negateR : ℝ → ℝ
ℝ.f (negateR record { f = f ; converges = converges }) n = negateQ (f n)
ℝ.converges (negateR record { f = f ; converges = converges }) {e} with converges {e}
... | n , pr = n , λ {y} x<y → SetoidPartialOrder.wellDefined ℚPartialOrder {absQ ((f n) -Q (f y))} {absQ ((negateQ (f n)) -Q (negateQ (f y)))} {e} {e} {!!} {!reflQ e!} (pr {y} x<y)
_-R_ : ℝ → ℝ → ℝ
a -R b = a +R (negateR b)
_*R_ : ℝ → ℝ → ℝ
ℝ.f (record { f = a ; converges = conA } *R record { f = b ; converges = conB }) n = (a n) *Q (b n)
ℝ.converges (record { f = a ; converges = conA } *R record { f = b ; converges = conB}) {e} = {!!}
realsSetoid : Setoid ℝ
(realsSetoid Setoid.∼ record { f = a ; converges = convA }) record { f = b ; converges = convB } = {!!}
Setoid.eq realsSetoid = {!!}

6
Rings/RingDefinition.agda
View File

@@ -61,3 +61,9 @@ module Rings.RingDefinition where
-- ringHom : RingHom R S f
-- bijective : Bijection f
abs : {a b c : _} {A : Set a} {S : Setoid {a} {b} A} {_+_ : A → A → A} {_*_ : A → A → A} {_<_ : Rel {_} {c} A} {pOrder : SetoidPartialOrder S _<_} (R : Ring S _+_ _*_) (order : SetoidTotalOrder pOrder) (x : A) → A
abs R order x with SetoidTotalOrder.totality order (Ring.0R R) x
... | inl (inl 0<x) = x
... | inl (inr x<0) = Group.inverse (Ring.additiveGroup R) x
... | inr 0=x = Ring.0R R

6
Rings/RingLemmas.agda
View File

@@ -88,6 +88,12 @@ module Rings.RingLemmas where
open Reflexive (Equivalence.reflexiveEq (Setoid.eq S))
open Symmetric (Equivalence.symmetricEq (Setoid.eq S))
ringMinusFlipsOrder'' : {a b c : _} {A : Set a} {S : Setoid {a} {b} A} {_+_ : A → A → A} {_*_ : A → A → A} {R : Ring S _+_ _*_} {_<_ : Rel {_} {c} A} {pOrder : SetoidPartialOrder S _<_} {tOrder : SetoidTotalOrder pOrder} (oRing : OrderedRing R tOrder) → {x : A} → x < (Ring.0R R) → (Ring.0R R) < Group.inverse (Ring.additiveGroup R) x
ringMinusFlipsOrder'' {S = S} {R = R} {pOrder = pOrder} oRing {x} x<0 = ringMinusFlipsOrder' oRing (SetoidPartialOrder.wellDefined pOrder {x} {Group.inverse (Ring.additiveGroup R) (Group.inverse (Ring.additiveGroup R) x)} {Ring.0R R} {Ring.0R R} (Symmetric.symmetric (Equivalence.symmetricEq (Setoid.eq S)) (invInv (Ring.additiveGroup R))) (Reflexive.reflexive (Equivalence.reflexiveEq (Setoid.eq S))) x<0)
ringMinusFlipsOrder''' : {a b c : _} {A : Set a} {S : Setoid {a} {b} A} {_+_ : A → A → A} {_*_ : A → A → A} {R : Ring S _+_ _*_} {_<_ : Rel {_} {c} A} {pOrder : SetoidPartialOrder S _<_} {tOrder : SetoidTotalOrder pOrder} (oRing : OrderedRing R tOrder) → {x : A} → (Ring.0R R) < (Group.inverse (Ring.additiveGroup R) x) → x < (Ring.0R R)
ringMinusFlipsOrder''' {S = S} {R = R} {pOrder = pOrder} oRing {x} 0<-x = SetoidPartialOrder.wellDefined pOrder (invInv (Ring.additiveGroup R)) (Reflexive.reflexive (Equivalence.reflexiveEq (Setoid.eq S))) (ringMinusFlipsOrder oRing 0<-x)
ringCanMultiplyByPositive : {a b c : _} {A : Set a} {S : Setoid {a} {b} A} {_+_ : A → A → A} {_*_ : A → A → A} {R : Ring S _+_ _*_} {_<_ : Rel {_} {c} A} {pOrder : SetoidPartialOrder S _<_} {tOrder : SetoidTotalOrder pOrder} (oRing : OrderedRing R tOrder) → {x y c : A} → (Ring.0R R) < c → x < y → (x * c) < (y * c)
ringCanMultiplyByPositive {S = S} {_+_ = _+_} {_*_ = _*_} {R = R} {_<_ = _<_} {pOrder = pOrder} {tOrder = tOrder} oRing {x} {y} {c} 0<c x<y = SetoidPartialOrder.wellDefined pOrder reflexive (Group.multIdentRight additiveGroup) q'
where