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Move equiv rels (#46)

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Patrick Stevens
2019-09-28 22:24:41 +01:00
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parent b92e6b2dd8
commit 00ce1dfdf8
20 changed files with 455 additions and 769 deletions
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3
Numbers/Rationals.agda
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@@ -13,6 +13,7 @@ open import Setoids.Orders
open import Functions
open import Fields.FieldOfFractions
open import Fields.FieldOfFractionsOrder
open import Sets.EquivalenceRelations
module Numbers.Rationals where
@@ -41,7 +42,7 @@ module Numbers.Rationals where
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}
reflQ {x} = Equivalence.reflexive (Setoid.eq (fieldOfFractionsSetoid ℤIntDom)) {x}
_≤Q_ : ℚ → ℚ → Set
a ≤Q b = (a <Q b) || (a =Q b)

132
Numbers/RationalsLemmas.agda
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@@ -17,6 +17,7 @@ open import Fields.FieldOfFractions
open import Fields.FieldOfFractionsOrder
open import Rings.IntegralDomains
open import Rings.Lemmas
open import Sets.EquivalenceRelations
module Numbers.RationalsLemmas where
@@ -26,191 +27,142 @@ module Numbers.RationalsLemmas where
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)))
open Equivalence (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)
open Equivalence 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)
open Equivalence 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)
open Equivalence 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)
open Equivalence 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)
open Equivalence 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)
open Equivalence 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)
open Equivalence 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)
open Equivalence 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)
open Equivalence 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)
open Equivalence 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)
open Equivalence 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)
open Equivalence 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)
open Equivalence 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)
open Equivalence 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)
open Equivalence 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)
open Equivalence 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)
open Equivalence 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)
open Equivalence 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)
open Equivalence 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)
open Equivalence 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)
open Equivalence 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)
open Equivalence 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)
open Equivalence 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)
open Equivalence 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)
open Equivalence 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)))
@@ -220,52 +172,38 @@ module Numbers.RationalsLemmas where
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)
open Equivalence 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)
open Equivalence 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)
open Equivalence 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 (inl _) = Equivalence.reflexive (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)
open Equivalence 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)
open Equivalence 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)
open Equivalence 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)
open Equivalence eq
absNegation a | inr 0=a | inr _ = refl