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Cauchy-completion, first parts (#52)
This commit is contained in:
Patrick Stevens
GitHub
@@ -7,6 +7,7 @@ open import Numbers.Integers.Addition
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open import Numbers.Integers.Multiplication
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open import Semirings.Definition
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open import Rings.Definition
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open import Rings.Order
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open import Setoids.Setoids
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open import Setoids.Orders
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open import Orders
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@@ -554,3 +554,9 @@ module Numbers.Naturals.Naturals where
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sumZeroImpliesOperandsZero : (a : ℕ) {b : ℕ} → a +N b ≡ 0 → (a ≡ 0) && (b ≡ 0)
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sumZeroImpliesOperandsZero zero {zero} pr = refl ,, refl
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inequalityShrinkRight : {a b c : ℕ} → a +N b <N c → b <N c
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inequalityShrinkRight {a} {b} {c} (le x proof) = le (x +N a) (transitivity (applyEquality succ (additionNIsAssociative x a b)) proof)
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inequalityShrinkLeft : {a b c : ℕ} → a +N b <N c → a <N c
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inequalityShrinkLeft {a} {b} {c} (le x proof) = le (x +N b) (transitivity (applyEquality succ (transitivity (additionNIsAssociative x b a) (applyEquality (x +N_) (additionNIsCommutative b a)))) proof)
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@@ -6,6 +6,7 @@ open import Numbers.Integers.Integers
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open import Groups.Groups
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open import Groups.Definition
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open import Rings.Definition
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open import Rings.Order
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open import Fields.Fields
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open import Numbers.Primes.PrimeNumbers
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open import Setoids.Setoids
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@@ -17,53 +18,57 @@ open import Sets.EquivalenceRelations
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module Numbers.Rationals where
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ℚ : Set
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ℚ = fieldOfFractionsSet ℤIntDom
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ℚ : Set
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ℚ = fieldOfFractionsSet ℤIntDom
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_+Q_ : ℚ → ℚ → ℚ
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a +Q b = fieldOfFractionsPlus ℤIntDom a b
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_+Q_ : ℚ → ℚ → ℚ
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a +Q b = fieldOfFractionsPlus ℤIntDom a b
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_*Q_ : ℚ → ℚ → ℚ
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a *Q b = fieldOfFractionsTimes ℤIntDom a b
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_*Q_ : ℚ → ℚ → ℚ
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a *Q b = fieldOfFractionsTimes ℤIntDom a b
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ℚRing : Ring (fieldOfFractionsSetoid ℤIntDom) _+Q_ _*Q_
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ℚRing = fieldOfFractionsRing ℤIntDom
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ℚRing : Ring (fieldOfFractionsSetoid ℤIntDom) _+Q_ _*Q_
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ℚRing = fieldOfFractionsRing ℤIntDom
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0Q : ℚ
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0Q = Ring.0R ℚRing
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0Q : ℚ
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0Q = Ring.0R ℚRing
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ℚField : Field ℚRing
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ℚField = fieldOfFractions ℤIntDom
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ℚField : Field ℚRing
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ℚField = fieldOfFractions ℤIntDom
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_<Q_ : ℚ → ℚ → Set
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_<Q_ = fieldOfFractionsComparison ℤIntDom ℤOrderedRing
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_<Q_ : ℚ → ℚ → Set
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_<Q_ = fieldOfFractionsComparison ℤIntDom ℤOrderedRing
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_=Q_ : ℚ → ℚ → Set
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a =Q b = Setoid._∼_ (fieldOfFractionsSetoid ℤIntDom) a b
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_=Q_ : ℚ → ℚ → Set
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a =Q b = Setoid._∼_ (fieldOfFractionsSetoid ℤIntDom) a b
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reflQ : {x : ℚ} → (x =Q x)
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reflQ {x} = Equivalence.reflexive (Setoid.eq (fieldOfFractionsSetoid ℤIntDom)) {x}
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reflQ : {x : ℚ} → (x =Q x)
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reflQ {x} = Equivalence.reflexive (Setoid.eq (fieldOfFractionsSetoid ℤIntDom)) {x}
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_≤Q_ : ℚ → ℚ → Set
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a ≤Q b = (a <Q b) || (a =Q b)
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_≤Q_ : ℚ → ℚ → Set
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a ≤Q b = (a <Q b) || (a =Q b)
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negateQ : ℚ → ℚ
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negateQ a = Group.inverse (Ring.additiveGroup ℚRing) a
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negateQ : ℚ → ℚ
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negateQ a = Group.inverse (Ring.additiveGroup ℚRing) a
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_-Q_ : ℚ → ℚ → ℚ
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a -Q b = a +Q negateQ b
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_-Q_ : ℚ → ℚ → ℚ
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a -Q b = a +Q negateQ b
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ℚPartialOrder : SetoidPartialOrder (fieldOfFractionsSetoid ℤIntDom) (fieldOfFractionsComparison ℤIntDom ℤOrderedRing)
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ℚPartialOrder = fieldOfFractionsOrder ℤIntDom ℤOrderedRing
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a-A : (a : ℚ) → (a -Q a) =Q 0Q
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a-A a = Group.invRight (Ring.additiveGroup ℚRing) {a}
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ℚTotalOrder : SetoidTotalOrder (fieldOfFractionsOrder ℤIntDom ℤOrderedRing)
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ℚTotalOrder = fieldOfFractionsTotalOrder ℤIntDom ℤOrderedRing
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ℚPartialOrder : SetoidPartialOrder (fieldOfFractionsSetoid ℤIntDom) (fieldOfFractionsComparison ℤIntDom ℤOrderedRing)
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ℚPartialOrder = fieldOfFractionsOrder ℤIntDom ℤOrderedRing
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absQ : ℚ → ℚ
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absQ q with SetoidTotalOrder.totality (fieldOfFractionsTotalOrder ℤIntDom ℤOrderedRing) 0Q q
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absQ q | inl (inl 0<q) = q
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absQ q | inl (inr q<0) = Group.inverse (Ring.additiveGroup ℚRing) q
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absQ q | inr x = 0Q
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ℚTotalOrder : SetoidTotalOrder (fieldOfFractionsOrder ℤIntDom ℤOrderedRing)
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ℚTotalOrder = fieldOfFractionsTotalOrder ℤIntDom ℤOrderedRing
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ℚOrdered : OrderedRing ℚRing (fieldOfFractionsTotalOrder ℤIntDom ℤOrderedRing)
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ℚOrdered = fieldOfFractionsOrderedRing ℤIntDom ℤOrderedRing
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open SetoidTotalOrder (fieldOfFractionsTotalOrder ℤIntDom ℤOrderedRing)
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open SetoidPartialOrder partial
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open Setoid (fieldOfFractionsSetoid ℤIntDom)
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negateQWellDefined : (a b : ℚ) → (a =Q b) → (negateQ a) =Q (negateQ b)
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negateQWellDefined a b a=b = inverseWellDefined (Ring.additiveGroup ℚRing) {a} {b} a=b
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ℚOrdered : OrderedRing ℚRing (fieldOfFractionsTotalOrder ℤIntDom ℤOrderedRing)
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ℚOrdered = fieldOfFractionsOrderedRing ℤIntDom ℤOrderedRing
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@@ -1,209 +0,0 @@
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{-# OPTIONS --safe --warning=error #-}
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open import LogicalFormulae
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open import Numbers.Naturals.Naturals
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open import Numbers.Integers.Integers
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open import Numbers.Rationals
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open import Groups.Groups
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open import Groups.Lemmas
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open import Groups.Definition
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open import Rings.Definition
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open import Fields.Fields
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open import Numbers.Primes.PrimeNumbers
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open import Setoids.Setoids
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open import Setoids.Orders
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open import Functions
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open import Fields.FieldOfFractions
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open import Fields.FieldOfFractionsOrder
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open import Rings.IntegralDomains
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open import Rings.Lemmas
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open import Sets.EquivalenceRelations
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module Numbers.RationalsLemmas where
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triangleInequality : {a b : ℚ} → (absQ (a +Q b)) ≤Q ((absQ a) +Q (absQ b))
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triangleInequality {a} {b} with SetoidTotalOrder.totality ℚTotalOrder 0Q a
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triangleInequality {a} {b} | inl (inl 0<a) with SetoidTotalOrder.totality ℚTotalOrder 0Q b
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triangleInequality {a} {b} | inl (inl 0<a) | inl (inl 0<b) with SetoidTotalOrder.totality ℚTotalOrder 0Q (a +Q b)
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triangleInequality {a} {b} | inl (inl 0<a) | inl (inl 0<b) | inl (inl 0<a+b) = inr (reflexive {a +Q b})
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where
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open Equivalence (Setoid.eq (fieldOfFractionsSetoid ℤIntDom))
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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.identRight (Ring.additiveGroup ℚRing) {0Q})) (reflexive {a +Q b}) (ringAddInequalities ℚOrdered {0Q} {a} {0Q} {b} 0<a 0<b)))))
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where
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open Group (Ring.additiveGroup ℚRing)
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open Setoid (fieldOfFractionsSetoid ℤIntDom)
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open Equivalence eq
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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} (identRight {0Q}) (symmetric {0Q} {a +Q b} 0=a+b) (ringAddInequalities ℚOrdered {0Q} {a} {0Q} {b} 0<a 0<b)))
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where
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open Group (Ring.additiveGroup ℚRing)
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open Setoid (fieldOfFractionsSetoid ℤIntDom)
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open Equivalence eq
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triangleInequality {a} {b} | inl (inl 0<a) | inl (inr b<0) with SetoidTotalOrder.totality ℚTotalOrder 0Q (a +Q b)
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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))
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where
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open Group (Ring.additiveGroup ℚRing)
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open Setoid (fieldOfFractionsSetoid ℤIntDom)
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open Equivalence eq
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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)))
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where
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open Group (Ring.additiveGroup ℚRing)
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open Setoid (fieldOfFractionsSetoid ℤIntDom)
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open Equivalence eq
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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} (identLeft {a}) (Ring.groupIsAbelian ℚRing {inverse b} {a}) (OrderedRing.orderRespectsAddition ℚOrdered {0Q} {inverse b} (ringMinusFlipsOrder'' ℚOrdered {b} b<0) a)))
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where
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open Group (Ring.additiveGroup ℚRing)
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open Setoid (fieldOfFractionsSetoid ℤIntDom)
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open Equivalence eq
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triangleInequality {a} {b} | inl (inl 0<a) | inr 0=b with SetoidTotalOrder.totality ℚTotalOrder 0Q (a +Q b)
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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))
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where
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open Group (Ring.additiveGroup ℚRing)
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open Setoid (fieldOfFractionsSetoid ℤIntDom)
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open Equivalence eq
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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} (identRight {a})) (+WellDefined {a} {0Q} {a} {b} (reflexive {a}) 0=b)) 0<a)))
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where
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open Group (Ring.additiveGroup ℚRing)
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open Setoid (fieldOfFractionsSetoid ℤIntDom)
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open Equivalence eq
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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} (identRight {a})) (+WellDefined {a} {0Q} {a} {b} (reflexive {a}) 0=b)) 0<a))
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where
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open Group (Ring.additiveGroup ℚRing)
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open Setoid (fieldOfFractionsSetoid ℤIntDom)
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open Equivalence eq
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triangleInequality {a} {b} | inl (inr a<0) with SetoidTotalOrder.totality ℚTotalOrder 0Q b
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triangleInequality {a} {b} | inl (inr a<0) | inl (inl 0<b) with SetoidTotalOrder.totality ℚTotalOrder 0Q (a +Q b)
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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)
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where
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open Group (Ring.additiveGroup ℚRing)
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open Setoid (fieldOfFractionsSetoid ℤIntDom)
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open Equivalence eq
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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)
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where
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open Group (Ring.additiveGroup ℚRing)
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open Setoid (fieldOfFractionsSetoid ℤIntDom)
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open Equivalence eq
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blah : (inverse a +Q inverse b) <Q (inverse a +Q b)
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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))
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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} (identRight {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))))
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where
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open Group (Ring.additiveGroup ℚRing)
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open Setoid (fieldOfFractionsSetoid ℤIntDom)
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open Equivalence eq
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triangleInequality {a} {b} | inl (inr a<0) | inl (inr b<0) with SetoidTotalOrder.totality ℚTotalOrder 0Q (a +Q b)
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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}) (identRight {0Q}) (ringAddInequalities ℚOrdered {a} {0Q} {b} {0Q} a<0 b<0)) 0<a+b))
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where
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open Group (Ring.additiveGroup ℚRing)
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open Setoid (fieldOfFractionsSetoid ℤIntDom)
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open Equivalence eq
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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}))
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where
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open Group (Ring.additiveGroup ℚRing)
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open Setoid (fieldOfFractionsSetoid ℤIntDom)
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open Equivalence eq
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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}) (identLeft {0Q}) (ringAddInequalities ℚOrdered {a} {0Q} {b} {0Q} a<0 b<0))))
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where
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open Group (Ring.additiveGroup ℚRing)
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open Setoid (fieldOfFractionsSetoid ℤIntDom)
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open Equivalence eq
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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)) (identRight {a})) 0<a+b)))
|
||||
where
|
||||
open Group (Ring.additiveGroup ℚRing)
|
||||
open Setoid (fieldOfFractionsSetoid ℤIntDom)
|
||||
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 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} (identRight {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 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 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.identLeft (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 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})) (identLeft {b}))) (reflexive {b}) 0<b))
|
||||
where
|
||||
open Group (Ring.additiveGroup ℚRing)
|
||||
open Setoid (fieldOfFractionsSetoid ℤIntDom)
|
||||
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})) (identLeft {b})) 0<a+b)))
|
||||
where
|
||||
open Group (Ring.additiveGroup ℚRing)
|
||||
open Setoid (fieldOfFractionsSetoid ℤIntDom)
|
||||
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 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})) (identLeft {b}))) b<0))
|
||||
where
|
||||
open Group (Ring.additiveGroup ℚRing)
|
||||
open Setoid (fieldOfFractionsSetoid ℤIntDom)
|
||||
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)) (identRight {0Q})) 0<a+b))
|
||||
where
|
||||
open Group (Ring.additiveGroup ℚRing)
|
||||
open Setoid (fieldOfFractionsSetoid ℤIntDom)
|
||||
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)) (identRight {0Q})) (reflexive {0Q}) a+b<0))
|
||||
where
|
||||
open Group (Ring.additiveGroup ℚRing)
|
||||
open Setoid (fieldOfFractionsSetoid ℤIntDom)
|
||||
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)))
|
||||
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 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 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 Equivalence eq
|
||||
absNegation a | inl (inr a<0) with SetoidTotalOrder.totality ℚTotalOrder 0Q (negateQ 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 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 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 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 Equivalence eq
|
||||
absNegation a | inr 0=a | inr _ = refl
|
||||
Reference in New Issue
Block a user