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N-ary expansions (#113)

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Patrick Stevens
2020-04-12 12:16:20 +01:00
committed by GitHub
parent 380548134d
commit 269f2aa14f
11 changed files with 156 additions and 217 deletions
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1
Everything/Safe.agda
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@@ -95,6 +95,7 @@ open import Rings.UniqueFactorisationDomains.Definition
open import Rings.Examples.Examples
open import Rings.Orders.Total.Bounded
open import Rings.Orders.Partial.Bounded
open import Rings.Orders.Total.BaseExpansion
open import Setoids.Setoids
open import Setoids.DirectSum

9
Fields/CauchyCompletion/BaseExpansion.agda
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@@ -46,6 +46,7 @@ open import Fields.CauchyCompletion.Approximation order F
open import Rings.InitialRing R
open import Rings.Orders.Partial.Bounded pRing
open import Rings.Orders.Total.Bounded order
open import Rings.Orders.Total.Cauchy order
cauchyTimesBoundedIsCauchy : {s : Sequence A} → cauchy s → {t : Sequence A} → Bounded t → cauchy (apply _*_ s t)
cauchyTimesBoundedIsCauchy {s} cau {t} (K , bounded) e 0<e with allInvertible K (boundNonzero (K , bounded))
@@ -119,8 +120,10 @@ private
ofBaseExpansion : {n : ℕ} → .(1<n : 1 <N n) → (fromN n ∼ 0R → False) → Sequence (ℤn n (0<n 1<n)) → CauchyCompletion
ofBaseExpansion {succ n} 1<n charNotN seq = record { elts = ofBaseExpansionSeq (0<n 1<n) seq ; converges = boundedTimesCauchyIsCauchy (powerSeqConverges _ (1/nPositive n) (1/n<1 n (canRemoveSuccFrom<N (squash<N 1<n))) (1/nPositive n)) (digitExpansionBounded (0<n 1<n) seq)}
toBaseExpansion : {n : ℕ} → .(1<n : 1 <N n) → (fromN n ∼ 0R → False) → CauchyCompletion → Sequence (ℤn n (0<n 1<n))
toBaseExpansion {n} 1<n charNotN c = {!!}
toBaseExpansion : {n : ℕ} → .(1<n : 1 <N n) → (fromN n ∼ 0R → False) → (a : CauchyCompletion) → 0R r<C a → a <Cr 1R → Sequence (ℤn n (0<n 1<n))
Sequence.head (toBaseExpansion {n} 1<n charNotN c 0<c c<1) = {!!}
-- TOOD: approximate c to within 1/n^2, extract the first decimal of the result.
Sequence.tail (toBaseExpansion {n} 1<n charNotN c 0<c c<1) = toBaseExpansion 1<n charNotN {!!} {!!} {!!}
baseExpansionProof : {n : ℕ} → .{1<n : 1 <N n} → {charNotN : fromN n ∼ 0R → False} → (as : CauchyCompletion) → Setoid._∼_ cauchyCompletionSetoid (ofBaseExpansion 1<n charNotN (toBaseExpansion 1<n charNotN as)) as
baseExpansionProof : {n : ℕ} → .{1<n : 1 <N n} → {charNotN : fromN n ∼ 0R → False} → (as : CauchyCompletion) → (0<a : 0R r<C as) → (a<1 : as <Cr 1R) → Setoid._∼_ cauchyCompletionSetoid (ofBaseExpansion 1<n charNotN (toBaseExpansion 1<n charNotN as 0<a a<1)) as
baseExpansionProof = {!!}

1
Fields/FieldOfFractions/Lemmas.agda
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@@ -9,7 +9,6 @@ open import Rings.IntegralDomains.Definition
open import Setoids.Setoids
open import Sets.EquivalenceRelations
module Fields.FieldOfFractions.Lemmas {a b : _} {A : Set a} {S : Setoid {a} {b} A} {_+_ : A → A → A} {_*_ : A → A → A} {R : Ring S _+_ _*_} (I : IntegralDomain R) where
open import Fields.FieldOfFractions.Setoid I

238
Fields/FieldOfFractions/Order.agda
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@@ -17,7 +17,11 @@ module Fields.FieldOfFractions.Order {a b c : _} {A : Set a} {S : Setoid {a} {b}
open import Fields.FieldOfFractions.Setoid I
open import Fields.FieldOfFractions.Ring I
open import Fields.FieldOfFractions.Addition I
open import Fields.FieldOfFractions.Lemmas I
open Ring R
open Setoid S
open Equivalence eq
open SetoidTotalOrder (TotallyOrderedRing.total order)
open import Rings.Orders.Partial.Lemmas
open PartiallyOrderedRing pRing
@@ -28,17 +32,11 @@ fieldOfFractionsComparison (numA ,, (denomA , denomA!=0)) (numB ,, (denomB , den
fieldOfFractionsComparison (numA ,, (denomA , denomA!=0)) (numB ,, (denomB , denomB!=0)) | inl (inl 0<denomA) | inl (inl 0<denomB) = (numA * denomB) < (numB * denomA)
fieldOfFractionsComparison (numA ,, (denomA , denomA!=0)) (numB ,, (denomB , denomB!=0)) | inl (inl 0<denomA) | inl (inr denomB<0) = (numB * denomA) < (numA * denomB)
fieldOfFractionsComparison (numA ,, (denomA , denomA!=0)) (numB ,, (denomB , denomB!=0)) | inl (inl 0<denomA) | inr 0=denomB = exFalso (denomB!=0 (symmetric 0=denomB))
where
open Equivalence (Setoid.eq S)
fieldOfFractionsComparison (numA ,, (denomA , denomA!=0)) (numB ,, (denomB , denomB!=0)) | inl (inr denomA<0) with totality (Ring.0R R) denomB
fieldOfFractionsComparison (numA ,, (denomA , denomA!=0)) (numB ,, (denomB , denomB!=0)) | inl (inr denomA<0) | inl (inl 0<denomB) = (numB * denomA) < (numA * denomB)
fieldOfFractionsComparison (numA ,, (denomA , denomA!=0)) (numB ,, (denomB , denomB!=0)) | inl (inr denomA<0) | inl (inr denomB<0) = (numA * denomB) < (numB * denomA)
fieldOfFractionsComparison (numA ,, (denomA , denomA!=0)) (numB ,, (denomB , denomB!=0)) | inl (inr denomA<0) | inr 0=denomB = exFalso (denomB!=0 (symmetric 0=denomB))
where
open Equivalence (Setoid.eq S)
fieldOfFractionsComparison (numA ,, (denomA , denomA!=0)) (numB ,, (denomB , denomB!=0)) | inr 0=denomA = exFalso (denomA!=0 (symmetric 0=denomA))
where
open Equivalence (Setoid.eq S)
private
abstract
@@ -50,8 +48,6 @@ private
fieldOfFractionsOrderWellDefinedLeft {numX ,, (denomX , denomX!=0)} {numY ,, (denomY , denomY!=0)} {numZ ,, (denomZ , denomZ!=0)} x<y x=z | inl (inl 0<denomZ) | inl (inl 0<denomY) | inl (inl 0<denomX) with totality (Ring.0R R) denomY
fieldOfFractionsOrderWellDefinedLeft {numX ,, (denomX , denomX!=0)} {numY ,, (denomY , denomY!=0)} {numZ ,, (denomZ , denomZ!=0)} x<y x=z | inl (inl 0<denomZ) | inl (inl 0<denomY) | inl (inl 0<denomX) | inl (inl _) = s
where
open Ring R
open Equivalence (Setoid.eq S)
have : ((numX * denomY) * denomZ) < ((numY * denomX) * denomZ)
have = ringCanMultiplyByPositive pRing 0<denomZ x<y
p : ((numX * denomZ) * denomY) < ((numY * denomX) * denomZ)
@@ -64,13 +60,9 @@ private
s = ringCanCancelPositive order 0<denomX r
fieldOfFractionsOrderWellDefinedLeft {numX ,, (denomX , denomX!=0)} {numY ,, (denomY , denomY!=0)} {numZ ,, (denomZ , denomZ!=0)} x<y x=z | inl (inl 0<denomZ) | inl (inl 0<denomY) | inl (inl 0<denomX) | inl (inr x) = exFalso (SetoidPartialOrder.irreflexive pOrder (SetoidPartialOrder.<Transitive pOrder 0<denomY x))
fieldOfFractionsOrderWellDefinedLeft {numX ,, (denomX , denomX!=0)} {numY ,, (denomY , denomY!=0)} {numZ ,, (denomZ , denomZ!=0)} x<y x=z | inl (inl 0<denomZ) | inl (inl 0<denomY) | inl (inl 0<denomX) | inr x = exFalso (denomY!=0 (symmetric x))
where
open Equivalence (Setoid.eq S)
fieldOfFractionsOrderWellDefinedLeft {numX ,, (denomX , denomX!=0)} {numY ,, (denomY , denomY!=0)} {numZ ,, (denomZ , denomZ!=0)} x<y x=z | inl (inl 0<denomZ) | inl (inl 0<denomY) | inl (inr denomX<0) with totality (Ring.0R R) denomY
fieldOfFractionsOrderWellDefinedLeft {numX ,, (denomX , denomX!=0)} {numY ,, (denomY , denomY!=0)} {numZ ,, (denomZ , denomZ!=0)} x<y x=z | inl (inl 0<denomZ) | inl (inl 0<denomY) | inl (inr denomX<0) | inl (inl _) = ringCanCancelNegative order denomX<0 r
where
open Ring R
open Equivalence (Setoid.eq S)
p : ((numY * denomX) * denomZ) < ((numX * denomZ) * denomY)
p = SetoidPartialOrder.<WellDefined pOrder reflexive (transitive (symmetric *Associative) (transitive (*WellDefined reflexive *Commutative) *Associative)) (ringCanMultiplyByPositive pRing 0<denomZ x<y)
q : ((numY * denomX) * denomZ) < ((denomX * numZ) * denomY)
@@ -85,8 +77,6 @@ private
fieldOfFractionsOrderWellDefinedLeft {numX ,, (denomX , denomX!=0)} {numY ,, (denomY , denomY!=0)} {numZ ,, (denomZ , denomZ!=0)} x<y x=z | inl (inl 0<denomZ) | inl (inr denomY<0) | inl (inl 0<denomX) | inl (inl 0<denomY) = exFalso (SetoidPartialOrder.irreflexive pOrder (SetoidPartialOrder.<Transitive pOrder 0<denomY denomY<0))
fieldOfFractionsOrderWellDefinedLeft {numX ,, (denomX , denomX!=0)} {numY ,, (denomY , denomY!=0)} {numZ ,, (denomZ , denomZ!=0)} x<y x=z | inl (inl 0<denomZ) | inl (inr denomY<0) | inl (inl 0<denomX) | inl (inr _) = ringCanCancelPositive order 0<denomX r
where
open Ring R
open Equivalence (Setoid.eq S)
p : ((numY * denomX) * denomZ) < ((numX * denomY) * denomZ)
p = ringCanMultiplyByPositive pRing 0<denomZ x<y
q : ((numY * denomX) * denomZ) < ((denomX * numZ) * denomY)
@@ -98,8 +88,6 @@ private
fieldOfFractionsOrderWellDefinedLeft {numX ,, (denomX , denomX!=0)} {numY ,, (denomY , denomY!=0)} {numZ ,, (denomZ , denomZ!=0)} x<y x=z | inl (inl 0<denomZ) | inl (inr denomY<0) | inl (inr denomX<0) | inl (inl 0<denomY) = exFalso (SetoidPartialOrder.irreflexive pOrder (SetoidPartialOrder.<Transitive pOrder 0<denomY denomY<0))
fieldOfFractionsOrderWellDefinedLeft {numX ,, (denomX , denomX!=0)} {numY ,, (denomY , denomY!=0)} {numZ ,, (denomZ , denomZ!=0)} x<y x=z | inl (inl 0<denomZ) | inl (inr denomY<0) | inl (inr denomX<0) | inl (inr _) = ringCanCancelNegative order denomX<0 q
where
open Ring R
open Equivalence (Setoid.eq S)
p : ((numX * denomY) * denomZ) < ((numY * denomX) * denomZ)
p = ringCanMultiplyByPositive pRing 0<denomZ x<y
q : ((numZ * denomY) * denomX) < ((numY * denomZ) * denomX)
@@ -112,8 +100,6 @@ private
fieldOfFractionsOrderWellDefinedLeft {numX ,, (denomX , denomX!=0)} {numY ,, (denomY , denomY!=0)} {numZ ,, (denomZ , denomZ!=0)} x<y x=z | inl (inr denomZ<0) | inl (inl 0<denomY) | inl (inl 0<denomX) with totality (Ring.0R R) denomY
fieldOfFractionsOrderWellDefinedLeft {numX ,, (denomX , denomX!=0)} {numY ,, (denomY , denomY!=0)} {numZ ,, (denomZ , denomZ!=0)} x<y x=z | inl (inr denomZ<0) | inl (inl 0<denomY) | inl (inl 0<denomX) | inl (inl _) = ringCanCancelPositive order 0<denomX (SetoidPartialOrder.<WellDefined pOrder (transitive (symmetric *Associative) (transitive (*WellDefined reflexive *Commutative) *Associative)) (transitive (symmetric *Associative) (transitive (*WellDefined reflexive x=z) (transitive (*WellDefined reflexive (*Commutative)) (transitive *Associative (*WellDefined *Commutative reflexive))))) p)
where
open Ring R
open Equivalence (Setoid.eq S)
p : ((numY * denomX) * denomZ) < ((denomY * numX) * denomZ)
p = ringCanMultiplyByNegative pRing denomZ<0 (SetoidPartialOrder.<WellDefined pOrder *Commutative reflexive x<y)
fieldOfFractionsOrderWellDefinedLeft {numX ,, (denomX , denomX!=0)} {numY ,, (denomY , denomY!=0)} {numZ ,, (denomZ , denomZ!=0)} x<y x=z | inl (inr denomZ<0) | inl (inl 0<denomY) | inl (inl 0<denomX) | inl (inr denomY<0) = exFalso (SetoidPartialOrder.irreflexive pOrder (SetoidPartialOrder.<Transitive pOrder 0<denomY denomY<0))
@@ -121,29 +107,19 @@ private
fieldOfFractionsOrderWellDefinedLeft {numX ,, (denomX , denomX!=0)} {numY ,, (denomY , denomY!=0)} {numZ ,, (denomZ , denomZ!=0)} x<y x=z | inl (inr denomZ<0) | inl (inl 0<denomY) | inl (inr denomX<0) with totality (Ring.0R R) denomY
fieldOfFractionsOrderWellDefinedLeft {numX ,, (denomX , denomX!=0)} {numY ,, (denomY , denomY!=0)} {numZ ,, (denomZ , denomZ!=0)} x<y x=z | inl (inr denomZ<0) | inl (inl 0<denomY) | inl (inr denomX<0) | inl (inl _) = ringCanCancelNegative order denomX<0 (SetoidPartialOrder.<WellDefined pOrder (transitive (symmetric *Associative) (transitive (*WellDefined reflexive *Commutative) (transitive *Associative (transitive (*WellDefined x=z reflexive) (transitive (symmetric *Associative) *Commutative))))) (transitive (symmetric *Associative) (transitive (*WellDefined reflexive *Commutative) *Associative)) p)
where
open Ring R
open Equivalence (Setoid.eq S)
p : ((numX * denomY) * denomZ) < ((numY * denomX) * denomZ)
p = ringCanMultiplyByNegative pRing denomZ<0 x<y
fieldOfFractionsOrderWellDefinedLeft {numX ,, (denomX , denomX!=0)} {numY ,, (denomY , denomY!=0)} {numZ ,, (denomZ , denomZ!=0)} x<y x=z | inl (inr denomZ<0) | inl (inl 0<denomY) | inl (inr denomX<0) | inl (inr denomY<0) = exFalso (SetoidPartialOrder.irreflexive pOrder (SetoidPartialOrder.<Transitive pOrder denomY<0 0<denomY))
fieldOfFractionsOrderWellDefinedLeft {numX ,, (denomX , denomX!=0)} {numY ,, (denomY , denomY!=0)} {numZ ,, (denomZ , denomZ!=0)} x<y x=z | inl (inr denomZ<0) | inl (inl 0<denomY) | inl (inr denomX<0) | inr 0=denomY = exFalso (denomY!=0 (Equivalence.symmetric (Setoid.eq S) 0=denomY))
fieldOfFractionsOrderWellDefinedLeft {numX ,, (denomX , denomX!=0)} {numY ,, (denomY , denomY!=0)} {numZ ,, (denomZ , denomZ!=0)} x<y x=z | inl (inr denomZ<0) | inl (inl 0<denomY) | inr x = exFalso (denomX!=0 (symmetric x))
where
open Equivalence (Setoid.eq S)
fieldOfFractionsOrderWellDefinedLeft {numX ,, (denomX , denomX!=0)} {numY ,, (denomY , denomY!=0)} {numZ ,, (denomZ , denomZ!=0)} x<y x=z | inl (inr denomZ<0) | inl (inr denomY<0) with totality (Ring.0R R) denomX
fieldOfFractionsOrderWellDefinedLeft {numX ,, (denomX , denomX!=0)} {numY ,, (denomY , denomY!=0)} {numZ ,, (denomZ , denomZ!=0)} x<y x=z | inl (inr denomZ<0) | inl (inr denomY<0) | inl (inl 0<denomX) with totality (Ring.0R R) denomY
fieldOfFractionsOrderWellDefinedLeft {numX ,, (denomX , denomX!=0)} {numY ,, (denomY , denomY!=0)} {numZ ,, (denomZ , denomZ!=0)} x<y x=z | inl (inr denomZ<0) | inl (inr denomY<0) | inl (inl 0<denomX) | inl (inl 0<denomY) = exFalso (SetoidPartialOrder.irreflexive pOrder (SetoidPartialOrder.<Transitive pOrder 0<denomY denomY<0))
fieldOfFractionsOrderWellDefinedLeft {numX ,, (denomX , denomX!=0)} {numY ,, (denomY , denomY!=0)} {numZ ,, (denomZ , denomZ!=0)} x<y x=z | inl (inr denomZ<0) | inl (inr denomY<0) | inl (inl 0<denomX) | inl (inr _) = ringCanCancelPositive order 0<denomX (SetoidPartialOrder.<WellDefined pOrder (transitive (*WellDefined *Commutative reflexive) (transitive (symmetric *Associative) (transitive (*WellDefined reflexive x=z) (transitive *Associative (transitive *Commutative *Associative))))) (transitive (symmetric *Associative) (transitive (*WellDefined reflexive *Commutative) *Associative)) (ringCanMultiplyByNegative pRing denomZ<0 x<y))
where
open Ring R
open Equivalence (Setoid.eq S)
fieldOfFractionsOrderWellDefinedLeft {numX ,, (denomX , denomX!=0)} {numY ,, (denomY , denomY!=0)} {numZ ,, (denomZ , denomZ!=0)} x<y x=z | inl (inr denomZ<0) | inl (inr denomY<0) | inl (inl 0<denomX) | inr x = exFalso (denomY!=0 (Equivalence.symmetric (Setoid.eq S) x))
fieldOfFractionsOrderWellDefinedLeft {numX ,, (denomX , denomX!=0)} {numY ,, (denomY , denomY!=0)} {numZ ,, (denomZ , denomZ!=0)} x<y x=z | inl (inr denomZ<0) | inl (inr denomY<0) | inl (inr denomX<0) with totality (Ring.0R R) denomY
fieldOfFractionsOrderWellDefinedLeft {numX ,, (denomX , denomX!=0)} {numY ,, (denomY , denomY!=0)} {numZ ,, (denomZ , denomZ!=0)} x<y x=z | inl (inr denomZ<0) | inl (inr denomY<0) | inl (inr denomX<0) | inl (inl 0<denomY) = exFalso (SetoidPartialOrder.irreflexive pOrder (SetoidPartialOrder.<Transitive pOrder 0<denomY denomY<0))
fieldOfFractionsOrderWellDefinedLeft {numX ,, (denomX , denomX!=0)} {numY ,, (denomY , denomY!=0)} {numZ ,, (denomZ , denomZ!=0)} x<y x=z | inl (inr denomZ<0) | inl (inr denomY<0) | inl (inr denomX<0) | inl (inr _) = ringCanCancelNegative order denomX<0 (SetoidPartialOrder.<WellDefined pOrder (transitive (symmetric *Associative) (transitive (*WellDefined reflexive *Commutative) *Associative)) (transitive (*WellDefined *Commutative reflexive) (transitive (symmetric *Associative) (transitive (*WellDefined reflexive x=z) (transitive (*WellDefined reflexive *Commutative) (transitive *Associative (*WellDefined *Commutative reflexive)))))) (ringCanMultiplyByNegative pRing denomZ<0 x<y))
where
open Ring R
open Equivalence (Setoid.eq S)
fieldOfFractionsOrderWellDefinedLeft {numX ,, (denomX , denomX!=0)} {numY ,, (denomY , denomY!=0)} {numZ ,, (denomZ , denomZ!=0)} x<y x=z | inl (inr denomZ<0) | inl (inr denomY<0) | inl (inr denomX<0) | inr x = exFalso (denomY!=0 (Equivalence.symmetric (Setoid.eq S) x))
fieldOfFractionsOrderWellDefinedLeft {numX ,, (denomX , denomX!=0)} {numY ,, (denomY , denomY!=0)} {numZ ,, (denomZ , denomZ!=0)} x<y x=z | inl (inr denomZ<0) | inl (inr denomY<0) | inr x = exFalso (denomX!=0 (Equivalence.symmetric (Setoid.eq S) x))
fieldOfFractionsOrderWellDefinedLeft {numX ,, (denomX , denomX!=0)} {numY ,, (denomY , denomY!=0)} {numZ ,, (denomZ , denomZ!=0)} x<y x=z | inl (inr denomZ<0) | inr x = exFalso (denomY!=0 (Equivalence.symmetric (Setoid.eq S) x))
@@ -154,55 +130,27 @@ private
fieldOfFractionsOrderWellDefinedRight {numX ,, (denomX , denomX!=0)} {numY ,, (denomY , denomY!=0)} {numZ ,, (denomZ , denomZ!=0)} x<y y=z | inl (inl 0<denomX) with totality (Ring.0R R) denomZ
fieldOfFractionsOrderWellDefinedRight {numX ,, (denomX , denomX!=0)} {numY ,, (denomY , denomY!=0)} {numZ ,, (denomZ , denomZ!=0)} x<y y=z | inl (inl 0<denomX) | inl (inl 0<denomZ) with totality (Ring.0R R) denomY
fieldOfFractionsOrderWellDefinedRight {numX ,, (denomX , denomX!=0)} {numY ,, (denomY , denomY!=0)} {numZ ,, (denomZ , denomZ!=0)} x<y y=z | inl (inl 0<denomX) | inl (inl 0<denomZ) | inl (inl 0<denomY) = ringCanCancelPositive order 0<denomY (SetoidPartialOrder.<WellDefined pOrder (transitive (symmetric *Associative) (transitive (*WellDefined reflexive *Commutative) *Associative)) (transitive (symmetric *Associative) (transitive (*WellDefined reflexive *Commutative) (transitive *Associative (transitive (*WellDefined y=z reflexive) (transitive (symmetric *Associative) *Commutative))))) (ringCanMultiplyByPositive pRing 0<denomZ x<y))
where
open Ring R
open Equivalence (Setoid.eq S)
fieldOfFractionsOrderWellDefinedRight {numX ,, (denomX , denomX!=0)} {numY ,, (denomY , denomY!=0)} {numZ ,, (denomZ , denomZ!=0)} x<y y=z | inl (inl 0<denomX) | inl (inl 0<denomZ) | inl (inr denomY<0) = ringCanCancelNegative order denomY<0 (SetoidPartialOrder.<WellDefined pOrder (transitive (symmetric *Associative) (transitive (*WellDefined reflexive *Commutative) (transitive *Associative (transitive (*WellDefined y=z reflexive) (transitive (symmetric *Associative) *Commutative))))) (transitive (symmetric *Associative) (transitive (*WellDefined reflexive *Commutative) *Associative)) (ringCanMultiplyByPositive pRing 0<denomZ x<y))
where
open Ring R
open Equivalence (Setoid.eq S)
fieldOfFractionsOrderWellDefinedRight {numX ,, (denomX , denomX!=0)} {numY ,, (denomY , denomY!=0)} {numZ ,, (denomZ , denomZ!=0)} x<y y=z | inl (inl 0<denomX) | inl (inl 0<denomZ) | inr x = exFalso (denomY!=0 (Equivalence.symmetric (Setoid.eq S) x))
fieldOfFractionsOrderWellDefinedRight {numX ,, (denomX , denomX!=0)} {numY ,, (denomY , denomY!=0)} {numZ ,, (denomZ , denomZ!=0)} x<y y=z | inl (inl 0<denomX) | inl (inr denomZ<0) with totality (Ring.0R R) denomY
fieldOfFractionsOrderWellDefinedRight {numX ,, (denomX , denomX!=0)} {numY ,, (denomY , denomY!=0)} {numZ ,, (denomZ , denomZ!=0)} x<y y=z | inl (inl 0<denomX) | inl (inr denomZ<0) | inl (inl 0<denomY) = ringCanCancelPositive order 0<denomY (SetoidPartialOrder.<WellDefined pOrder (transitive (symmetric *Associative) (transitive (*WellDefined reflexive *Commutative) (transitive *Associative (transitive (*WellDefined y=z reflexive) (transitive (symmetric *Associative) *Commutative))))) (transitive (symmetric *Associative) (transitive (*WellDefined reflexive *Commutative) *Associative)) (ringCanMultiplyByNegative pRing denomZ<0 x<y))
where
open Ring R
open Equivalence (Setoid.eq S)
fieldOfFractionsOrderWellDefinedRight {numX ,, (denomX , denomX!=0)} {numY ,, (denomY , denomY!=0)} {numZ ,, (denomZ , denomZ!=0)} x<y y=z | inl (inl 0<denomX) | inl (inr denomZ<0) | inl (inr denomY<0) = ringCanCancelNegative order denomY<0 (SetoidPartialOrder.<WellDefined pOrder (transitive (symmetric *Associative) (transitive (*WellDefined reflexive *Commutative) *Associative)) (transitive (symmetric *Associative) (transitive (*WellDefined reflexive *Commutative) (transitive (*Associative) (transitive (*WellDefined y=z reflexive) (transitive (symmetric *Associative) *Commutative))))) (ringCanMultiplyByNegative pRing denomZ<0 x<y))
where
open Ring R
open Equivalence (Setoid.eq S)
fieldOfFractionsOrderWellDefinedRight {numX ,, (denomX , denomX!=0)} {numY ,, (denomY , denomY!=0)} {numZ ,, (denomZ , denomZ!=0)} x<y y=z | inl (inl 0<denomX) | inl (inr denomZ<0) | inr x = exFalso (denomY!=0 (Equivalence.symmetric (Setoid.eq S) x))
fieldOfFractionsOrderWellDefinedRight {numX ,, (denomX , denomX!=0)} {numY ,, (denomY , denomY!=0)} {numZ ,, (denomZ , denomZ!=0)} x<y y=z | inl (inl 0<denomX) | inr x = exFalso (denomZ!=0 (Equivalence.symmetric (Setoid.eq S) x))
fieldOfFractionsOrderWellDefinedRight {numX ,, (denomX , denomX!=0)} {numY ,, (denomY , denomY!=0)} {numZ ,, (denomZ , denomZ!=0)} x<y y=z | inl (inr denomX<0) with totality (Ring.0R R) denomZ
fieldOfFractionsOrderWellDefinedRight {numX ,, (denomX , denomX!=0)} {numY ,, (denomY , denomY!=0)} {numZ ,, (denomZ , denomZ!=0)} x<y y=z | inl (inr denomX<0) | inl (inl 0<denomZ) with totality (Ring.0R R) denomY
fieldOfFractionsOrderWellDefinedRight {numX ,, (denomX , denomX!=0)} {numY ,, (denomY , denomY!=0)} {numZ ,, (denomZ , denomZ!=0)} x<y y=z | inl (inr denomX<0) | inl (inl 0<denomZ) | inl (inl 0<denomY) = ringCanCancelPositive order 0<denomY (SetoidPartialOrder.<WellDefined pOrder (transitive (symmetric *Associative) (transitive (*WellDefined reflexive *Commutative) (transitive *Associative (transitive (*WellDefined y=z reflexive) (transitive (symmetric *Associative) *Commutative))))) (transitive (symmetric *Associative) (transitive (*WellDefined reflexive *Commutative) *Associative)) (ringCanMultiplyByPositive pRing 0<denomZ x<y))
where
open Ring R
open Equivalence (Setoid.eq S)
fieldOfFractionsOrderWellDefinedRight {numX ,, (denomX , denomX!=0)} {numY ,, (denomY , denomY!=0)} {numZ ,, (denomZ , denomZ!=0)} x<y y=z | inl (inr denomX<0) | inl (inl 0<denomZ) | inl (inr denomY<0) = ringCanCancelNegative order denomY<0 (SetoidPartialOrder.<WellDefined pOrder (transitive (symmetric *Associative) (transitive (*WellDefined reflexive *Commutative) *Associative)) (transitive (symmetric *Associative) (transitive (*WellDefined reflexive *Commutative) (transitive *Associative (transitive (*WellDefined y=z reflexive) (transitive (symmetric *Associative) *Commutative))))) (ringCanMultiplyByPositive pRing 0<denomZ x<y))
where
open Ring R
open Equivalence (Setoid.eq S)
fieldOfFractionsOrderWellDefinedRight {numX ,, (denomX , denomX!=0)} {numY ,, (denomY , denomY!=0)} {numZ ,, (denomZ , denomZ!=0)} x<y y=z | inl (inr denomX<0) | inl (inl 0<denomZ) | inr x = exFalso (denomY!=0 (Equivalence.symmetric (Setoid.eq S) x))
fieldOfFractionsOrderWellDefinedRight {numX ,, (denomX , denomX!=0)} {numY ,, (denomY , denomY!=0)} {numZ ,, (denomZ , denomZ!=0)} x<y y=z | inl (inr denomX<0) | inl (inr denomZ<0) with totality (Ring.0R R) denomY
fieldOfFractionsOrderWellDefinedRight {numX ,, (denomX , denomX!=0)} {numY ,, (denomY , denomY!=0)} {numZ ,, (denomZ , denomZ!=0)} x<y y=z | inl (inr denomX<0) | inl (inr denomZ<0) | inl (inl 0<denomY) = ringCanCancelPositive order 0<denomY (SetoidPartialOrder.<WellDefined pOrder (transitive (symmetric *Associative) (transitive (*WellDefined reflexive *Commutative) *Associative)) (transitive (symmetric *Associative) (transitive (*WellDefined reflexive *Commutative) (transitive *Associative (transitive (*WellDefined y=z reflexive) (transitive (symmetric *Associative) *Commutative))))) (ringCanMultiplyByNegative pRing denomZ<0 x<y))
where
open Ring R
open Equivalence (Setoid.eq S)
fieldOfFractionsOrderWellDefinedRight {numX ,, (denomX , denomX!=0)} {numY ,, (denomY , denomY!=0)} {numZ ,, (denomZ , denomZ!=0)} x<y y=z | inl (inr denomX<0) | inl (inr denomZ<0) | inl (inr denomY<0) = ringCanCancelNegative order denomY<0 (SetoidPartialOrder.<WellDefined pOrder (transitive (symmetric *Associative) (transitive (*WellDefined reflexive *Commutative) (transitive *Associative (transitive (*WellDefined y=z reflexive) (transitive (symmetric *Associative) *Commutative))))) (transitive (symmetric *Associative) (transitive (*WellDefined reflexive *Commutative) *Associative)) (ringCanMultiplyByNegative pRing denomZ<0 x<y))
where
open Ring R
open Equivalence (Setoid.eq S)
fieldOfFractionsOrderWellDefinedRight {numX ,, (denomX , denomX!=0)} {numY ,, (denomY , denomY!=0)} {numZ ,, (denomZ , denomZ!=0)} x<y y=z | inl (inr denomX<0) | inl (inr denomZ<0) | inr x = exFalso (denomY!=0 (Equivalence.symmetric (Setoid.eq S) x))
fieldOfFractionsOrderWellDefinedRight {numX ,, (denomX , denomX!=0)} {numY ,, (denomY , denomY!=0)} {numZ ,, (denomZ , denomZ!=0)} x<y y=z | inl (inr denomX<0) | inr x = exFalso (denomZ!=0 (Equivalence.symmetric (Setoid.eq S) x))
fieldOfFractionsOrderWellDefinedRight {numX ,, (denomX , denomX!=0)} {numY ,, (denomY , denomY!=0)} {numZ ,, (denomZ , denomZ!=0)} x<y y=z | inr x = exFalso (denomX!=0 (Equivalence.symmetric (Setoid.eq S) x))
swapLemma : {a b : _} {A : Set a} {S : Setoid {a} {b} A} {_+_ : A → A → A} {_*_ : A → A → A} (R : Ring S _+_ _*_) {x y z : A} → Setoid._∼_ S ((x * y) * z) ((x * z) * y)
swapLemma {S = S} R = transitive (symmetric *Associative) (transitive (*WellDefined reflexive *Commutative) *Associative)
where
open Setoid S
open Ring R
open Equivalence (Setoid.eq S)
swapLemma : {x y z : A} → Setoid._∼_ S ((x * y) * z) ((x * z) * y)
swapLemma = transitive (symmetric *Associative) (transitive (*WellDefined reflexive *Commutative) *Associative)
private
abstract
@@ -225,9 +173,6 @@ private
<transitive (numA ,, (denomA , denomA!=0)) (numB ,, (denomB , denomB!=0)) (numC ,, (denomC , denomC!=0)) a<b b<c | inl (inl 0<denomA) | inl (inl 0<denomC) | inl (inl x) with totality (Ring.0R R) denomC
<transitive (numA ,, (denomA , denomA!=0)) (numB ,, (denomB , denomB!=0)) (numC ,, (denomC , denomC!=0)) a<b b<c | inl (inl 0<denomA) | inl (inl 0<denomC) | inl (inl 0<denomB) | inl (inl _) = ringCanCancelPositive order 0<denomB p
where
open Setoid S
open Ring R
open Equivalence (Setoid.eq S)
inter : ((numA * denomB) * denomC) < ((numB * denomA) * denomC)
inter = ringCanMultiplyByPositive pRing 0<denomC a<b
p : ((numA * denomC) * denomB) < ((numC * denomA) * denomB)
@@ -236,79 +181,57 @@ private
<transitive (numA ,, (denomA , denomA!=0)) (numB ,, (denomB , denomB!=0)) (numC ,, (denomC , denomC!=0)) a<b b<c | inl (inl 0<denomA) | inl (inl 0<denomC) | inl (inl 0<denomB) | inr x = exFalso (denomC!=0 (Equivalence.symmetric (Setoid.eq S) x))
<transitive (numA ,, (denomA , denomA!=0)) (numB ,, (denomB , denomB!=0)) (numC ,, (denomC , denomC!=0)) a<b b<c | inl (inl 0<denomA) | inl (inl 0<denomC) | inl (inr denomB<0) with totality (Ring.0R R) denomC
<transitive (numA ,, (denomA , denomA!=0)) (numB ,, (denomB , denomB!=0)) (numC ,, (denomC , denomC!=0)) a<b b<c | inl (inl 0<denomA) | inl (inl 0<denomC) | inl (inr denomB<0) | inl (inl _) = ringCanCancelNegative order denomB<0 (SetoidPartialOrder.<Transitive pOrder (SetoidPartialOrder.<WellDefined pOrder (transitive (symmetric *Associative) (transitive (*WellDefined reflexive *Commutative) *Associative)) reflexive (ringCanMultiplyByPositive pRing 0<denomA b<c)) (SetoidPartialOrder.<WellDefined pOrder (transitive (symmetric *Associative) (transitive (*WellDefined reflexive *Commutative) *Associative)) (transitive (symmetric *Associative) (transitive (*WellDefined reflexive *Commutative) *Associative)) (ringCanMultiplyByPositive pRing 0<denomC a<b)))
where
open Setoid S
open Ring R
open Equivalence (Setoid.eq S)
<transitive (numA ,, (denomA , denomA!=0)) (numB ,, (denomB , denomB!=0)) (numC ,, (denomC , denomC!=0)) a<b b<c | inl (inl 0<denomA) | inl (inl 0<denomC) | inl (inr denomB<0) | inl (inr denomC<0) = exFalso (SetoidPartialOrder.irreflexive pOrder (SetoidPartialOrder.<Transitive pOrder 0<denomC denomC<0))
<transitive (numA ,, (denomA , denomA!=0)) (numB ,, (denomB , denomB!=0)) (numC ,, (denomC , denomC!=0)) a<b b<c | inl (inl 0<denomA) | inl (inl 0<denomC) | inl (inr denomB<0) | inr x = exFalso (denomC!=0 (Equivalence.symmetric (Setoid.eq S) x))
<transitive (numA ,, (denomA , denomA!=0)) (numB ,, (denomB , denomB!=0)) (numC ,, (denomC , denomC!=0)) a<b b<c | inl (inl 0<denomA) | inl (inl 0<denomC) | inr x = exFalso (denomB!=0 (Equivalence.symmetric (Setoid.eq S) x))
<transitive (numA ,, (denomA , denomA!=0)) (numB ,, (denomB , denomB!=0)) (numC ,, (denomC , denomC!=0)) a<b b<c | inl (inl 0<denomA) | inl (inr denomC<0) with totality (Ring.0R R) denomB
<transitive (numA ,, (denomA , denomA!=0)) (numB ,, (denomB , denomB!=0)) (numC ,, (denomC , denomC!=0)) a<b b<c | inl (inl 0<denomA) | inl (inr denomC<0) | inl (inl 0<denomB) with totality (Ring.0R R) denomC
<transitive (numA ,, (denomA , denomA!=0)) (numB ,, (denomB , denomB!=0)) (numC ,, (denomC , denomC!=0)) a<b b<c | inl (inl 0<denomA) | inl (inr denomC<0) | inl (inl 0<denomB) | inl (inl 0<denomC) = exFalso (SetoidPartialOrder.irreflexive pOrder (SetoidPartialOrder.<Transitive pOrder 0<denomC denomC<0))
<transitive (numA ,, (denomA , denomA!=0)) (numB ,, (denomB , denomB!=0)) (numC ,, (denomC , denomC!=0)) a<b b<c | inl (inl 0<denomA) | inl (inr denomC<0) | inl (inl 0<denomB) | inl (inr _) = ringCanCancelPositive order 0<denomB (SetoidPartialOrder.<Transitive pOrder have (SetoidPartialOrder.<WellDefined pOrder (swapLemma R) (swapLemma R) (ringCanMultiplyByNegative pRing denomC<0 a<b)))
<transitive (numA ,, (denomA , denomA!=0)) (numB ,, (denomB , denomB!=0)) (numC ,, (denomC , denomC!=0)) a<b b<c | inl (inl 0<denomA) | inl (inr denomC<0) | inl (inl 0<denomB) | inl (inr _) = ringCanCancelPositive order 0<denomB (SetoidPartialOrder.<Transitive pOrder have (SetoidPartialOrder.<WellDefined pOrder swapLemma swapLemma (ringCanMultiplyByNegative pRing denomC<0 a<b)))
where
open Setoid S
open Ring R
open Equivalence (Setoid.eq S)
have : ((numC * denomA) * denomB) < ((numB * denomC) * denomA)
have = SetoidPartialOrder.<WellDefined pOrder (swapLemma R) reflexive (ringCanMultiplyByPositive pRing 0<denomA b<c)
have = SetoidPartialOrder.<WellDefined pOrder swapLemma reflexive (ringCanMultiplyByPositive pRing 0<denomA b<c)
<transitive (numA ,, (denomA , denomA!=0)) (numB ,, (denomB , denomB!=0)) (numC ,, (denomC , denomC!=0)) a<b b<c | inl (inl 0<denomA) | inl (inr denomC<0) | inl (inl 0<denomB) | inr x = exFalso (denomC!=0 (Equivalence.symmetric (Setoid.eq S) x))
<transitive (numA ,, (denomA , denomA!=0)) (numB ,, (denomB , denomB!=0)) (numC ,, (denomC , denomC!=0)) a<b b<c | inl (inl 0<denomA) | inl (inr denomC<0) | inl (inr denomB<0) with totality (Ring.0R R) denomC
... | (inl (inl 0<denomC)) = exFalso (SetoidPartialOrder.irreflexive pOrder (SetoidPartialOrder.<Transitive pOrder 0<denomC denomC<0))
... | (inl (inr _)) = ringCanCancelNegative order denomB<0 (SetoidPartialOrder.<Transitive pOrder have (SetoidPartialOrder.<WellDefined pOrder (swapLemma R) (swapLemma R) (ringCanMultiplyByPositive pRing 0<denomA b<c)))
... | (inl (inr _)) = ringCanCancelNegative order denomB<0 (SetoidPartialOrder.<Transitive pOrder have (SetoidPartialOrder.<WellDefined pOrder (swapLemma) (swapLemma) (ringCanMultiplyByPositive pRing 0<denomA b<c)))
where
open Setoid S
open Ring R
open Equivalence (Setoid.eq S)
have : ((numA * denomC) * denomB) < ((numB * denomA) * denomC)
have = SetoidPartialOrder.<WellDefined pOrder (swapLemma R) reflexive (ringCanMultiplyByNegative pRing denomC<0 a<b)
have = SetoidPartialOrder.<WellDefined pOrder (swapLemma) reflexive (ringCanMultiplyByNegative pRing denomC<0 a<b)
... | inr x = exFalso (denomC!=0 (Equivalence.symmetric (Setoid.eq S) x))
<transitive (numA ,, (denomA , denomA!=0)) (numB ,, (denomB , denomB!=0)) (numC ,, (denomC , denomC!=0)) a<b b<c | inl (inl 0<denomA) | inl (inr denomC<0) | inr x = exFalso (denomB!=0 (Equivalence.symmetric (Setoid.eq S) x))
<transitive (numA ,, (denomA , denomA!=0)) (numB ,, (denomB , denomB!=0)) (numC ,, (denomC , denomC!=0)) a<b b<c | inl (inl 0<denomA) | inr x = exFalso (denomC!=0 (Equivalence.symmetric (Setoid.eq S) x))
<transitive (numA ,, (denomA , denomA!=0)) (numB ,, (denomB , denomB!=0)) (numC ,, (denomC , denomC!=0)) a<b b<c | inl (inr denomA<0) with totality (Ring.0R R) denomC
<transitive (numA ,, (denomA , denomA!=0)) (numB ,, (denomB , denomB!=0)) (numC ,, (denomC , denomC!=0)) a<b b<c | inl (inr denomA<0) | inl (inl 0<denomC) with totality (Ring.0R R) denomB
<transitive (numA ,, (denomA , denomA!=0)) (numB ,, (denomB , denomB!=0)) (numC ,, (denomC , denomC!=0)) a<b b<c | inl (inr denomA<0) | inl (inl 0<denomC) | inl (inl 0<denomB) with totality (Ring.0R R) denomC
<transitive (numA ,, (denomA , denomA!=0)) (numB ,, (denomB , denomB!=0)) (numC ,, (denomC , denomC!=0)) a<b b<c | inl (inr denomA<0) | inl (inl 0<denomC) | inl (inl 0<denomB) | inl (inl _) = ringCanCancelPositive order 0<denomB (SetoidPartialOrder.<Transitive pOrder (SetoidPartialOrder.<WellDefined pOrder (swapLemma R) (swapLemma R) (ringCanMultiplyByNegative pRing denomA<0 b<c)) have)
<transitive (numA ,, (denomA , denomA!=0)) (numB ,, (denomB , denomB!=0)) (numC ,, (denomC , denomC!=0)) a<b b<c | inl (inr denomA<0) | inl (inl 0<denomC) | inl (inl 0<denomB) | inl (inl _) = ringCanCancelPositive order 0<denomB (SetoidPartialOrder.<Transitive pOrder (SetoidPartialOrder.<WellDefined pOrder (swapLemma) (swapLemma) (ringCanMultiplyByNegative pRing denomA<0 b<c)) have)
where
open Setoid S
open Ring R
open Equivalence (Setoid.eq S)
have : ((numB * denomA) * denomC) < ((numA * denomC) * denomB)
have = SetoidPartialOrder.<WellDefined pOrder reflexive (swapLemma R) (ringCanMultiplyByPositive pRing 0<denomC a<b)
have = SetoidPartialOrder.<WellDefined pOrder reflexive (swapLemma) (ringCanMultiplyByPositive pRing 0<denomC a<b)
<transitive (numA ,, (denomA , denomA!=0)) (numB ,, (denomB , denomB!=0)) (numC ,, (denomC , denomC!=0)) a<b b<c | inl (inr denomA<0) | inl (inl 0<denomC) | inl (inl 0<denomB) | inl (inr denomC<0) = exFalso (SetoidPartialOrder.irreflexive pOrder (SetoidPartialOrder.<Transitive pOrder 0<denomC denomC<0))
<transitive (numA ,, (denomA , denomA!=0)) (numB ,, (denomB , denomB!=0)) (numC ,, (denomC , denomC!=0)) a<b b<c | inl (inr denomA<0) | inl (inl 0<denomC) | inl (inl 0<denomB) | inr x = exFalso (denomC!=0 (Equivalence.symmetric (Setoid.eq S) x))
<transitive (numA ,, (denomA , denomA!=0)) (numB ,, (denomB , denomB!=0)) (numC ,, (denomC , denomC!=0)) a<b b<c | inl (inr denomA<0) | inl (inl 0<denomC) | inl (inr denomB<0) with totality (Ring.0R R) denomC
<transitive (numA ,, (denomA , denomA!=0)) (numB ,, (denomB , denomB!=0)) (numC ,, (denomC , denomC!=0)) a<b b<c | inl (inr denomA<0) | inl (inl 0<denomC) | inl (inr denomB<0) | inl (inl _) = ringCanCancelNegative order denomB<0 (SetoidPartialOrder.<Transitive pOrder have (SetoidPartialOrder.<WellDefined pOrder (swapLemma R) (swapLemma R) (ringCanMultiplyByNegative pRing denomA<0 b<c)))
<transitive (numA ,, (denomA , denomA!=0)) (numB ,, (denomB , denomB!=0)) (numC ,, (denomC , denomC!=0)) a<b b<c | inl (inr denomA<0) | inl (inl 0<denomC) | inl (inr denomB<0) | inl (inl _) = ringCanCancelNegative order denomB<0 (SetoidPartialOrder.<Transitive pOrder have (SetoidPartialOrder.<WellDefined pOrder (swapLemma) (swapLemma) (ringCanMultiplyByNegative pRing denomA<0 b<c)))
where
open Setoid S
open Ring R
open Equivalence (Setoid.eq S)
have : ((numA * denomC) * denomB) < ((numB * denomA) * denomC)
have = SetoidPartialOrder.<WellDefined pOrder (swapLemma R) reflexive (ringCanMultiplyByPositive pRing 0<denomC a<b)
have = SetoidPartialOrder.<WellDefined pOrder (swapLemma) reflexive (ringCanMultiplyByPositive pRing 0<denomC a<b)
<transitive (numA ,, (denomA , denomA!=0)) (numB ,, (denomB , denomB!=0)) (numC ,, (denomC , denomC!=0)) a<b b<c | inl (inr denomA<0) | inl (inl 0<denomC) | inl (inr denomB<0) | inl (inr denomC<0) = exFalso (SetoidPartialOrder.irreflexive pOrder (SetoidPartialOrder.<Transitive pOrder 0<denomC denomC<0))
<transitive (numA ,, (denomA , denomA!=0)) (numB ,, (denomB , denomB!=0)) (numC ,, (denomC , denomC!=0)) a<b b<c | inl (inr denomA<0) | inl (inl 0<denomC) | inl (inr denomB<0) | inr x = exFalso (denomC!=0 (Equivalence.symmetric (Setoid.eq S) x))
<transitive (numA ,, (denomA , denomA!=0)) (numB ,, (denomB , denomB!=0)) (numC ,, (denomC , denomC!=0)) a<b b<c | inl (inr denomA<0) | inl (inl 0<denomC) | inr x = exFalso (denomB!=0 (Equivalence.symmetric (Setoid.eq S) x))
<transitive (numA ,, (denomA , denomA!=0)) (numB ,, (denomB , denomB!=0)) (numC ,, (denomC , denomC!=0)) a<b b<c | inl (inr denomA<0) | inl (inr denomC<0) with totality (Ring.0R R) denomB
<transitive (numA ,, (denomA , denomA!=0)) (numB ,, (denomB , denomB!=0)) (numC ,, (denomC , denomC!=0)) a<b b<c | inl (inr denomA<0) | inl (inr denomC<0) | inl (inl 0<denomB) with totality (Ring.0R R) denomC
<transitive (numA ,, (denomA , denomA!=0)) (numB ,, (denomB , denomB!=0)) (numC ,, (denomC , denomC!=0)) a<b b<c | inl (inr denomA<0) | inl (inr denomC<0) | inl (inl 0<denomB) | inl (inl 0<denomC) = exFalso (SetoidPartialOrder.irreflexive pOrder (SetoidPartialOrder.<Transitive pOrder 0<denomC denomC<0))
<transitive (numA ,, (denomA , denomA!=0)) (numB ,, (denomB , denomB!=0)) (numC ,, (denomC , denomC!=0)) a<b b<c | inl (inr denomA<0) | inl (inr denomC<0) | inl (inl 0<denomB) | inl (inr _) = ringCanCancelPositive order 0<denomB (SetoidPartialOrder.<Transitive pOrder have (SetoidPartialOrder.<WellDefined pOrder (swapLemma R) (swapLemma R) (ringCanMultiplyByNegative pRing denomA<0 b<c)))
<transitive (numA ,, (denomA , denomA!=0)) (numB ,, (denomB , denomB!=0)) (numC ,, (denomC , denomC!=0)) a<b b<c | inl (inr denomA<0) | inl (inr denomC<0) | inl (inl 0<denomB) | inl (inr _) = ringCanCancelPositive order 0<denomB (SetoidPartialOrder.<Transitive pOrder have (SetoidPartialOrder.<WellDefined pOrder (swapLemma) (swapLemma) (ringCanMultiplyByNegative pRing denomA<0 b<c)))
where
open Setoid S
open Ring R
open Equivalence (Setoid.eq S)
have : ((numA * denomC) * denomB) < ((numB * denomA) * denomC)
have = SetoidPartialOrder.<WellDefined pOrder (swapLemma R) reflexive (ringCanMultiplyByNegative pRing denomC<0 a<b)
have = SetoidPartialOrder.<WellDefined pOrder (swapLemma) reflexive (ringCanMultiplyByNegative pRing denomC<0 a<b)
<transitive (numA ,, (denomA , denomA!=0)) (numB ,, (denomB , denomB!=0)) (numC ,, (denomC , denomC!=0)) a<b b<c | inl (inr denomA<0) | inl (inr denomC<0) | inl (inl 0<denomB) | inr x = exFalso (denomC!=0 (Equivalence.symmetric (Setoid.eq S) x))
<transitive (numA ,, (denomA , denomA!=0)) (numB ,, (denomB , denomB!=0)) (numC ,, (denomC , denomC!=0)) a<b b<c | inl (inr denomA<0) | inl (inr denomC<0) | inl (inr denomB<0) with totality (Ring.0R R) denomC
<transitive (numA ,, (denomA , denomA!=0)) (numB ,, (denomB , denomB!=0)) (numC ,, (denomC , denomC!=0)) a<b b<c | inl (inr denomA<0) | inl (inr denomC<0) | inl (inr denomB<0) | inl (inl 0<denomC) = exFalso (SetoidPartialOrder.irreflexive pOrder (SetoidPartialOrder.<Transitive pOrder 0<denomC denomC<0))
<transitive (numA ,, (denomA , denomA!=0)) (numB ,, (denomB , denomB!=0)) (numC ,, (denomC , denomC!=0)) a<b b<c | inl (inr denomA<0) | inl (inr denomC<0) | inl (inr denomB<0) | inl (inr _) = ringCanCancelNegative order denomB<0 (SetoidPartialOrder.<Transitive pOrder (SetoidPartialOrder.<WellDefined pOrder (swapLemma R) (swapLemma R) (ringCanMultiplyByNegative pRing denomA<0 b<c)) have)
<transitive (numA ,, (denomA , denomA!=0)) (numB ,, (denomB , denomB!=0)) (numC ,, (denomC , denomC!=0)) a<b b<c | inl (inr denomA<0) | inl (inr denomC<0) | inl (inr denomB<0) | inl (inr _) = ringCanCancelNegative order denomB<0 (SetoidPartialOrder.<Transitive pOrder (SetoidPartialOrder.<WellDefined pOrder (swapLemma) (swapLemma) (ringCanMultiplyByNegative pRing denomA<0 b<c)) have)
where
open Setoid S
open Ring R
open Equivalence (Setoid.eq S)
have : ((numB * denomA) * denomC) < ((numA * denomC) * denomB)
have = SetoidPartialOrder.<WellDefined pOrder reflexive (swapLemma R) (ringCanMultiplyByNegative pRing denomC<0 a<b)
have = SetoidPartialOrder.<WellDefined pOrder reflexive (swapLemma) (ringCanMultiplyByNegative pRing denomC<0 a<b)
<transitive (numA ,, (denomA , denomA!=0)) (numB ,, (denomB , denomB!=0)) (numC ,, (denomC , denomC!=0)) a<b b<c | inl (inr denomA<0) | inl (inr denomC<0) | inl (inr denomB<0) | inr x = exFalso (denomC!=0 (Equivalence.symmetric (Setoid.eq S) x))
<transitive (numA ,, (denomA , denomA!=0)) (numB ,, (denomB , denomB!=0)) (numC ,, (denomC , denomC!=0)) a<b b<c | inl (inr denomA<0) | inl (inr denomC<0) | inr x = exFalso (denomB!=0 (Equivalence.symmetric (Setoid.eq S) x))
<transitive (numA ,, (denomA , denomA!=0)) (numB ,, (denomB , denomB!=0)) (numC ,, (denomC , denomC!=0)) a<b b<c | inl (inr denomA<0) | inr x = exFalso (denomC!=0 (Equivalence.symmetric (Setoid.eq S) x))
@@ -366,57 +289,25 @@ private
ineqLemma {x} {y} 0<xy 0<x with totality (Ring.0R R) y
ineqLemma {x} {y} 0<xy 0<x | inl (inl 0<y) = 0<y
ineqLemma {x} {y} 0<xy 0<x | inl (inr y<0) = exFalso (SetoidPartialOrder.irreflexive pOrder (SetoidPartialOrder.<Transitive pOrder 0<xy (SetoidPartialOrder.<WellDefined pOrder reflexive (transitive *Commutative (Ring.timesZero R)) (ringCanMultiplyByNegative pRing y<0 0<x))))
where
open Setoid S
open Ring R
open Equivalence (Setoid.eq S)
ineqLemma {x} {y} 0<xy 0<x | inr 0=y = exFalso (SetoidPartialOrder.irreflexive pOrder (SetoidPartialOrder.<WellDefined pOrder reflexive (transitive (*WellDefined reflexive (symmetric 0=y)) (Ring.timesZero R)) 0<xy))
where
open Setoid S
open Ring R
open Equivalence (Setoid.eq S)
ineqLemma' : {x y : A} → (Ring.0R R) < (x * y) → x < (Ring.0R R) → y < (Ring.0R R)
ineqLemma' {x} {y} 0<xy x<0 with totality (Ring.0R R) y
... | inl (inl 0<y) = exFalso (SetoidPartialOrder.irreflexive pOrder (SetoidPartialOrder.<Transitive pOrder 0<xy (SetoidPartialOrder.<WellDefined pOrder *Commutative (transitive *Commutative (Ring.timesZero R)) (ringCanMultiplyByNegative pRing x<0 0<y))))
where
open Setoid S
open Ring R
open Equivalence (Setoid.eq S)
... | inl (inr y<0) = y<0
... | (inr 0=y) = exFalso (SetoidPartialOrder.irreflexive pOrder (SetoidPartialOrder.<WellDefined pOrder reflexive (transitive (*WellDefined reflexive (symmetric 0=y)) (Ring.timesZero R)) 0<xy))
where
open Setoid S
open Ring R
open Equivalence (Setoid.eq S)
ineqLemma'' : {x y : A} → (x * y) < (Ring.0R R) → (Ring.0R R) < x → y < (Ring.0R R)
ineqLemma'' {x} {y} xy<0 0<x with totality (Ring.0R R) y
... | inl (inl 0<y) = exFalso (SetoidPartialOrder.irreflexive pOrder (SetoidPartialOrder.<Transitive pOrder xy<0 (orderRespectsMultiplication 0<x 0<y)))
where
open Setoid S
open Ring R
open Equivalence (Setoid.eq S)
... | inl (inr y<0) = y<0
... | (inr 0=y) = exFalso (SetoidPartialOrder.irreflexive pOrder (SetoidPartialOrder.<WellDefined pOrder (transitive (*WellDefined reflexive (symmetric 0=y)) (Ring.timesZero R)) reflexive xy<0))
where
open Setoid S
open Ring R
open Equivalence (Setoid.eq S)
ineqLemma''' : {x y : A} → (x * y) < (Ring.0R R) → x < (Ring.0R R) → (Ring.0R R) < y
ineqLemma''' {x} {y} xy<0 x<0 with totality (Ring.0R R) y
... | inl (inl 0<y) = 0<y
... | inl (inr y<0) = exFalso (SetoidPartialOrder.irreflexive pOrder (SetoidPartialOrder.<Transitive pOrder xy<0 (SetoidPartialOrder.<WellDefined pOrder (transitive *Commutative (Ring.timesZero R)) reflexive (ringCanMultiplyByNegative pRing y<0 x<0))))
where
open Setoid S
open Ring R
open Equivalence (Setoid.eq S)
... | inr 0=y = exFalso (SetoidPartialOrder.irreflexive pOrder (SetoidPartialOrder.<WellDefined pOrder (transitive (*WellDefined reflexive (symmetric 0=y)) (Ring.timesZero R)) reflexive xy<0))
where
open Setoid S
open Ring R
open Equivalence (Setoid.eq S)
private
<orderRespectsAddition : (a b : fieldOfFractionsSet) (a<b : fieldOfFractionsComparison a b) (c : fieldOfFractionsSet) → fieldOfFractionsComparison (fieldOfFractionsPlus a c) (fieldOfFractionsPlus b c)
@@ -426,23 +317,17 @@ private
<orderRespectsAddition (numA ,, (denomA , denomA!=0)) (numB ,, (denomB , denomB!=0)) a<b (numC ,, (denomC , denomC!=0)) | inl (inl 0<dAdC) | inl (inl 0<dBdC) | inl (inl 0<dA) with totality (Ring.0R R) denomB
<orderRespectsAddition (numA ,, (denomA , denomA!=0)) (numB ,, (denomB , denomB!=0)) a<b (numC ,, (denomC , denomC!=0)) | inl (inl 0<dAdC) | inl (inl 0<dBdC) | inl (inl 0<dA) | inl (inl 0<dB) = SetoidPartialOrder.<WellDefined pOrder (symmetric *Associative) (symmetric *Associative) (ringCanMultiplyByPositive pRing 0<dC (SetoidPartialOrder.<WellDefined pOrder (transitive (Group.+WellDefined additiveGroup *Commutative *Commutative) (transitive (symmetric *DistributesOver+) *Commutative)) (transitive (Group.+WellDefined additiveGroup *Commutative *Commutative) (transitive (symmetric *DistributesOver+) *Commutative)) ans))
where
open Setoid S
open Equivalence (Setoid.eq S)
open Ring R
0<dC : 0R < denomC
0<dC with totality 0R denomC
0<dC | inl (inl x) = x
0<dC | inl (inr dC<0) = exFalso (SetoidPartialOrder.irreflexive pOrder (SetoidPartialOrder.<Transitive pOrder 0<dBdC (SetoidPartialOrder.<WellDefined pOrder reflexive (transitive *Commutative (Ring.timesZero R)) (ringCanMultiplyByNegative pRing dC<0 0<dB))))
0<dC | inr x = exFalso (denomC!=0 (Equivalence.symmetric (Setoid.eq S) x))
p : ((numA * denomC) * denomB) < ((numB * denomC) * denomA)
p = SetoidPartialOrder.<WellDefined pOrder (swapLemma R) (swapLemma R) (ringCanMultiplyByPositive pRing 0<dC a<b)
p = SetoidPartialOrder.<WellDefined pOrder (swapLemma) (swapLemma) (ringCanMultiplyByPositive pRing 0<dC a<b)
ans : ((((numA * denomC) * denomB) + ((denomA * numC) * denomB))) < ((((numB * denomC) * denomA) + ((denomB * numC) * denomA)))
ans = SetoidPartialOrder.<WellDefined pOrder reflexive (Group.+WellDefined additiveGroup reflexive (transitive (*WellDefined *Commutative reflexive) (transitive (transitive (symmetric *Associative) (transitive (*WellDefined reflexive *Commutative) *Associative)) (*WellDefined *Commutative reflexive)))) (PartiallyOrderedRing.orderRespectsAddition pRing p ((denomA * numC) * denomB))
<orderRespectsAddition (numA ,, (denomA , denomA!=0)) (numB ,, (denomB , denomB!=0)) a<b (numC ,, (denomC , denomC!=0)) | inl (inl 0<dAdC) | inl (inl 0<dBdC) | inl (inl 0<dA) | inl (inr dB<0) = exFalso bad
where
open Setoid S
open Equivalence (Setoid.eq S)
open Ring R
dC<0 : denomC < 0R
dC<0 with totality 0R denomC
... | inl (inl x) = exFalso (SetoidPartialOrder.irreflexive pOrder (SetoidPartialOrder.<Transitive pOrder 0<dBdC (SetoidPartialOrder.<WellDefined pOrder reflexive (transitive *Commutative (Ring.timesZero R)) (ringCanMultiplyByPositive pRing x dB<0))))
@@ -454,9 +339,6 @@ private
<orderRespectsAddition (numA ,, (denomA , denomA!=0)) (numB ,, (denomB , denomB!=0)) a<b (numC ,, (denomC , denomC!=0)) | inl (inl 0<dAdC) | inl (inl 0<dBdC) | inl (inr dA<0) with totality (Ring.0R R) denomB
<orderRespectsAddition (numA ,, (denomA , denomA!=0)) (numB ,, (denomB , denomB!=0)) a<b (numC ,, (denomC , denomC!=0)) | inl (inl 0<dAdC) | inl (inl 0<dBdC) | inl (inr dA<0) | inl (inl 0<dB) = exFalso bad
where
open Setoid S
open Equivalence (Setoid.eq S)
open Ring R
0<dC : 0R < denomC
0<dC with totality 0R denomC
0<dC | inl (inl x) = x
@@ -471,9 +353,6 @@ private
bad = SetoidPartialOrder.irreflexive pOrder (SetoidPartialOrder.<Transitive pOrder 0<dC dC<0)
<orderRespectsAddition (numA ,, (denomA , denomA!=0)) (numB ,, (denomB , denomB!=0)) a<b (numC ,, (denomC , denomC!=0)) | inl (inl 0<dAdC) | inl (inl 0<dBdC) | inl (inr dA<0) | inl (inr dB<0) = SetoidPartialOrder.<WellDefined pOrder (symmetric *Associative) (symmetric *Associative) (ringCanMultiplyByNegative pRing dC<0 (SetoidPartialOrder.<WellDefined pOrder (transitive (symmetric *DistributesOver+) *Commutative) (transitive (symmetric *DistributesOver+) *Commutative) have''))
where
open Setoid S
open Equivalence (Setoid.eq S)
open Ring R
dC<0 : denomC < 0R
dC<0 = ineqLemma' 0<dAdC dA<0
have : ((numB * denomA) * denomC) < ((numA * denomB) * denomC)
@@ -488,9 +367,6 @@ private
<orderRespectsAddition (numA ,, (denomA , denomA!=0)) (numB ,, (denomB , denomB!=0)) a<b (numC ,, (denomC , denomC!=0)) | inl (inl 0<dAdC) | inl (inr dBdC<0) | inl (inl 0<dA) with totality (Ring.0R R) denomB
<orderRespectsAddition (numA ,, (denomA , denomA!=0)) (numB ,, (denomB , denomB!=0)) a<b (numC ,, (denomC , denomC!=0)) | inl (inl 0<dAdC) | inl (inr dBdC<0) | inl (inl 0<dA) | inl (inl 0<dB) = exFalso bad
where
open Setoid S
open Equivalence (Setoid.eq S)
open Ring R
0<dC : 0R < denomC
0<dC = ineqLemma 0<dAdC 0<dA
dC<0 : denomC < 0R
@@ -499,33 +375,24 @@ private
bad = SetoidPartialOrder.irreflexive pOrder (SetoidPartialOrder.<Transitive pOrder 0<dC dC<0)
<orderRespectsAddition (numA ,, (denomA , denomA!=0)) (numB ,, (denomB , denomB!=0)) a<b (numC ,, (denomC , denomC!=0)) | inl (inl 0<dAdC) | inl (inr dBdC<0) | inl (inl 0<dA) | inl (inr dB<0) = SetoidPartialOrder.<WellDefined pOrder (symmetric *Associative) (symmetric *Associative) (ringCanMultiplyByPositive pRing 0<dC ans)
where
open Setoid S
open Equivalence (Setoid.eq S)
open Ring R
0<dC : 0R < denomC
0<dC = ineqLemma 0<dAdC 0<dA
have : ((numB * denomA) * denomC) < ((numA * denomB) * denomC)
have = ringCanMultiplyByPositive pRing 0<dC a<b
have' : (((numB * denomC) * denomA) + ((denomB * numC) * denomA)) < (((numA * denomC) * denomB) + ((denomB * numC) * denomA))
have' = PartiallyOrderedRing.orderRespectsAddition pRing (SetoidPartialOrder.<WellDefined pOrder (swapLemma R) (swapLemma R) have) _
have' = PartiallyOrderedRing.orderRespectsAddition pRing (SetoidPartialOrder.<WellDefined pOrder (swapLemma) (swapLemma) have) _
ans : (((numB * denomC) + (denomB * numC)) * denomA) < (((numA * denomC) + (denomA * numC)) * denomB)
ans = SetoidPartialOrder.<WellDefined pOrder (transitive (Group.+WellDefined additiveGroup *Commutative *Commutative) (transitive (symmetric *DistributesOver+) *Commutative)) (transitive (Group.+WellDefined additiveGroup *Commutative (transitive (symmetric *Associative) (*WellDefined reflexive *Commutative))) (transitive (symmetric *DistributesOver+) *Commutative)) have'
<orderRespectsAddition (numA ,, (denomA , denomA!=0)) (numB ,, (denomB , denomB!=0)) a<b (numC ,, (denomC , denomC!=0)) | inl (inl 0<dAdC) | inl (inr dBdC<0) | inl (inl 0<dA) | inr x = exFalso (denomB!=0 (Equivalence.symmetric (Setoid.eq S) x))
<orderRespectsAddition (numA ,, (denomA , denomA!=0)) (numB ,, (denomB , denomB!=0)) a<b (numC ,, (denomC , denomC!=0)) | inl (inl 0<dAdC) | inl (inr dBdC<0) | inl (inr dA<0) with totality (Ring.0R R) denomB
<orderRespectsAddition (numA ,, (denomA , denomA!=0)) (numB ,, (denomB , denomB!=0)) a<b (numC ,, (denomC , denomC!=0)) | inl (inl 0<dAdC) | inl (inr dBdC<0) | inl (inr dA<0) | inl (inl 0<dB) = SetoidPartialOrder.<WellDefined pOrder (symmetric *Associative) (symmetric *Associative) (ringCanMultiplyByNegative pRing dC<0 (SetoidPartialOrder.<WellDefined pOrder (transitive (Group.+WellDefined additiveGroup *Commutative *Commutative) (transitive (transitive (Group.+WellDefined additiveGroup reflexive (transitive *Associative (transitive (*WellDefined *Commutative reflexive) (symmetric *Associative)))) (symmetric *DistributesOver+)) *Commutative)) (transitive (Group.+WellDefined additiveGroup *Commutative *Commutative) (transitive (symmetric *DistributesOver+) *Commutative)) have))
where
open Setoid S
open Equivalence (Setoid.eq S)
open Ring R
dC<0 : denomC < 0R
dC<0 = ineqLemma'' dBdC<0 0<dB
have : (((numA * denomC) * denomB) + ((denomB * numC) * denomA)) < (((numB * denomC) * denomA) + ((denomB * numC) * denomA))
have = PartiallyOrderedRing.orderRespectsAddition pRing (SetoidPartialOrder.<WellDefined pOrder (swapLemma R) (swapLemma R) (ringCanMultiplyByNegative pRing dC<0 a<b)) _
have = PartiallyOrderedRing.orderRespectsAddition pRing (SetoidPartialOrder.<WellDefined pOrder (swapLemma) (swapLemma) (ringCanMultiplyByNegative pRing dC<0 a<b)) _
<orderRespectsAddition (numA ,, (denomA , denomA!=0)) (numB ,, (denomB , denomB!=0)) a<b (numC ,, (denomC , denomC!=0)) | inl (inl 0<dAdC) | inl (inr dBdC<0) | inl (inr dA<0) | inl (inr dB<0) = exFalso bad
where
open Setoid S
open Equivalence (Setoid.eq S)
open Ring R
dC<0 : denomC < 0R
dC<0 = ineqLemma' 0<dAdC dA<0
0<dC : 0R < denomC
@@ -541,18 +408,12 @@ private
<orderRespectsAddition (numA ,, (denomA , denomA!=0)) (numB ,, (denomB , denomB!=0)) a<b (numC ,, (denomC , denomC!=0)) | inl (inr dAdC<0) | inl (inl 0<dBdC) | inl (inl 0<dA) with totality (Ring.0R R) denomB
<orderRespectsAddition (numA ,, (denomA , denomA!=0)) (numB ,, (denomB , denomB!=0)) a<b (numC ,, (denomC , denomC!=0)) | inl (inr dAdC<0) | inl (inl 0<dBdC) | inl (inl 0<dA) | inl (inl 0<dB) = exFalso (SetoidPartialOrder.irreflexive pOrder (SetoidPartialOrder.<Transitive pOrder 0<dC dC<0))
where
open Setoid S
open Equivalence (Setoid.eq S)
open Ring R
0<dC : 0R < denomC
0<dC = ineqLemma 0<dBdC 0<dB
dC<0 : denomC < 0R
dC<0 = ineqLemma'' dAdC<0 0<dA
<orderRespectsAddition (numA ,, (denomA , denomA!=0)) (numB ,, (denomB , denomB!=0)) a<b (numC ,, (denomC , denomC!=0)) | inl (inr dAdC<0) | inl (inl 0<dBdC) | inl (inl 0<dA) | inl (inr dB<0) = SetoidPartialOrder.<WellDefined pOrder (symmetric *Associative) (symmetric *Associative) (ringCanMultiplyByNegative pRing dC<0 (SetoidPartialOrder.<WellDefined pOrder (transitive (Group.+WellDefined additiveGroup (transitive (*WellDefined *Commutative reflexive) (symmetric *Associative)) *Commutative) (transitive (symmetric *DistributesOver+) *Commutative)) (transitive (Group.+WellDefined additiveGroup (transitive (*WellDefined *Commutative reflexive) (symmetric *Associative)) (transitive (symmetric *Associative) (*WellDefined reflexive *Commutative))) (transitive (symmetric *DistributesOver+) *Commutative)) have))
where
open Setoid S
open Equivalence (Setoid.eq S)
open Ring R
dC<0 : denomC < 0R
dC<0 = ineqLemma'' dAdC<0 0<dA
have : (((numA * denomB) * denomC) + ((denomA * numC) * denomB)) < (((numB * denomA) * denomC) + ((denomA * numC) * denomB))
@@ -561,18 +422,12 @@ private
<orderRespectsAddition (numA ,, (denomA , denomA!=0)) (numB ,, (denomB , denomB!=0)) a<b (numC ,, (denomC , denomC!=0)) | inl (inr dAdC<0) | inl (inl 0<dBdC) | inl (inr dA<0) with totality (Ring.0R R) denomB
<orderRespectsAddition (numA ,, (denomA , denomA!=0)) (numB ,, (denomB , denomB!=0)) a<b (numC ,, (denomC , denomC!=0)) | inl (inr dAdC<0) | inl (inl 0<dBdC) | inl (inr dA<0) | inl (inl 0<dB) = SetoidPartialOrder.<WellDefined pOrder (symmetric *Associative) (symmetric *Associative) (ringCanMultiplyByPositive pRing 0<dC (SetoidPartialOrder.<WellDefined pOrder (transitive (Group.+WellDefined additiveGroup *Commutative *Commutative) (transitive (transitive (Group.+WellDefined additiveGroup (transitive *Commutative (transitive (transitive (symmetric *Associative) (transitive (*WellDefined reflexive *Commutative) *Associative)) *Commutative)) (transitive *Associative (transitive (*WellDefined *Commutative reflexive) (symmetric *Associative)))) (symmetric *DistributesOver+)) *Commutative)) (transitive (Group.+WellDefined additiveGroup *Commutative *Commutative) (transitive (transitive (Group.+WellDefined additiveGroup (transitive *Commutative (transitive (*WellDefined *Commutative reflexive) (symmetric *Associative))) reflexive) (symmetric *DistributesOver+)) *Commutative)) have))
where
open Setoid S
open Equivalence (Setoid.eq S)
open Ring R
0<dC : 0R < denomC
0<dC = ineqLemma 0<dBdC 0<dB
have : (((numB * denomA) * denomC) + ((denomA * numC) * denomB)) < (((numA * denomB) * denomC) + ((denomA * numC) * denomB))
have = PartiallyOrderedRing.orderRespectsAddition pRing (ringCanMultiplyByPositive pRing 0<dC a<b) _
<orderRespectsAddition (numA ,, (denomA , denomA!=0)) (numB ,, (denomB , denomB!=0)) a<b (numC ,, (denomC , denomC!=0)) | inl (inr dAdC<0) | inl (inl 0<dBdC) | inl (inr dA<0) | inl (inr dB<0) = exFalso (SetoidPartialOrder.irreflexive pOrder (SetoidPartialOrder.<Transitive pOrder 0<dC dC<0))
where
open Setoid S
open Equivalence (Setoid.eq S)
open Ring R
dC<0 : denomC < 0R
dC<0 = ineqLemma' 0<dBdC dB<0
0<dC : 0R < denomC
@@ -583,18 +438,12 @@ private
<orderRespectsAddition (numA ,, (denomA , denomA!=0)) (numB ,, (denomB , denomB!=0)) a<b (numC ,, (denomC , denomC!=0)) | inl (inr dAdC<0) | inl (inr dBdC<0) | inl (inl 0<dA) with totality (Ring.0R R) denomB
<orderRespectsAddition (numA ,, (denomA , denomA!=0)) (numB ,, (denomB , denomB!=0)) a<b (numC ,, (denomC , denomC!=0)) | inl (inr dAdC<0) | inl (inr dBdC<0) | inl (inl 0<dA) | inl (inl 0<dB) = SetoidPartialOrder.<WellDefined pOrder (symmetric *Associative) (symmetric *Associative) (ringCanMultiplyByNegative pRing dC<0 (SetoidPartialOrder.<WellDefined pOrder (transitive (transitive (Group.+WellDefined additiveGroup *Commutative *Commutative) (Group.+WellDefined additiveGroup (transitive *Commutative (transitive (*WellDefined *Commutative reflexive) (symmetric *Associative))) reflexive)) (transitive (symmetric *DistributesOver+) *Commutative)) (transitive (transitive (Group.+WellDefined additiveGroup *Commutative *Commutative) (Group.+WellDefined additiveGroup (transitive (transitive *Associative (*WellDefined *Commutative reflexive)) *Commutative) (transitive *Associative (transitive (*WellDefined *Commutative reflexive) (symmetric *Associative))))) (transitive (symmetric *DistributesOver+) *Commutative)) have))
where
open Setoid S
open Equivalence (Setoid.eq S)
open Ring R
dC<0 : denomC < 0R
dC<0 = ineqLemma'' dAdC<0 0<dA
have : (((numB * denomA) * denomC) + ((denomB * numC) * denomA)) < (((numA * denomB) * denomC) + ((denomB * numC) * denomA))
have = PartiallyOrderedRing.orderRespectsAddition pRing (ringCanMultiplyByNegative pRing dC<0 a<b) _
<orderRespectsAddition (numA ,, (denomA , denomA!=0)) (numB ,, (denomB , denomB!=0)) a<b (numC ,, (denomC , denomC!=0)) | inl (inr dAdC<0) | inl (inr dBdC<0) | inl (inl 0<dA) | inl (inr dB<0) = exFalso (SetoidPartialOrder.irreflexive pOrder (SetoidPartialOrder.<Transitive pOrder dC<0 0<dC))
where
open Setoid S
open Equivalence (Setoid.eq S)
open Ring R
dC<0 : denomC < 0R
dC<0 = ineqLemma'' dAdC<0 0<dA
0<dC : 0R < denomC
@@ -603,18 +452,12 @@ private
<orderRespectsAddition (numA ,, (denomA , denomA!=0)) (numB ,, (denomB , denomB!=0)) a<b (numC ,, (denomC , denomC!=0)) | inl (inr dAdC<0) | inl (inr dBdC<0) | inl (inr dA<0) with totality (Ring.0R R) denomB
<orderRespectsAddition (numA ,, (denomA , denomA!=0)) (numB ,, (denomB , denomB!=0)) a<b (numC ,, (denomC , denomC!=0)) | inl (inr dAdC<0) | inl (inr dBdC<0) | inl (inr dA<0) | inl (inl 0<dB) = exFalso (SetoidPartialOrder.irreflexive pOrder (SetoidPartialOrder.<Transitive pOrder 0<dC dC<0))
where
open Setoid S
open Equivalence (Setoid.eq S)
open Ring R
0<dC : 0R < denomC
0<dC = ineqLemma''' dAdC<0 dA<0
dC<0 : denomC < 0R
dC<0 = ineqLemma'' dBdC<0 0<dB
<orderRespectsAddition (numA ,, (denomA , denomA!=0)) (numB ,, (denomB , denomB!=0)) a<b (numC ,, (denomC , denomC!=0)) | inl (inr dAdC<0) | inl (inr dBdC<0) | inl (inr dA<0) | inl (inr dB<0) = SetoidPartialOrder.<WellDefined pOrder (symmetric *Associative) (symmetric *Associative) (ringCanMultiplyByPositive pRing 0<dC (SetoidPartialOrder.<WellDefined pOrder (transitive (Group.+WellDefined additiveGroup *Commutative *Commutative) (transitive (Group.+WellDefined additiveGroup (transitive *Commutative (transitive (*WellDefined *Commutative reflexive) (symmetric *Associative))) reflexive) (transitive (symmetric *DistributesOver+) *Commutative))) (transitive (Group.+WellDefined additiveGroup *Commutative *Commutative) (transitive (transitive (Group.+WellDefined additiveGroup (transitive *Commutative (transitive (*WellDefined *Commutative reflexive) (symmetric *Associative))) (transitive *Commutative (transitive (symmetric *Associative) (*WellDefined reflexive *Commutative)))) (symmetric *DistributesOver+)) *Commutative)) have))
where
open Setoid S
open Equivalence (Setoid.eq S)
open Ring R
0<dC : 0R < denomC
0<dC = ineqLemma''' dAdC<0 dA<0
have : (((numA * denomB) * denomC) + ((denomA * numC) * denomB)) < (((numB * denomA) * denomC) + ((denomA * numC) * denomB))
@@ -634,9 +477,6 @@ PartiallyOrderedRing.orderRespectsMultiplication (fieldOfFractionsPOrderedRing)
PartiallyOrderedRing.orderRespectsMultiplication (fieldOfFractionsPOrderedRing) {numA ,, (denomA , denomA!=0)} {numB ,, (denomB , denomB!=0)} 0<a 0<b | inl (inl 0<1) | inl (inl 0<dAdB) | inl (inl 0<dB) with totality (Ring.0R R) denomA
PartiallyOrderedRing.orderRespectsMultiplication (fieldOfFractionsPOrderedRing) {numA ,, (denomA , denomA!=0)} {numB ,, (denomB , denomB!=0)} 0<a 0<b | inl (inl 0<1) | inl (inl 0<dAdB) | inl (inl 0<dB) | inl (inl 0<dA) = SetoidPartialOrder.<WellDefined pOrder (symmetric (transitive *Commutative (Ring.timesZero R))) (symmetric (transitive *Commutative identIsIdent)) 0<nAnB
where
open Setoid S
open Equivalence (Setoid.eq S)
open Ring R
0<nA : 0R < numA
0<nA = SetoidPartialOrder.<WellDefined pOrder (transitive *Commutative (Ring.timesZero R)) (transitive *Commutative identIsIdent) 0<a
0<nB : 0R < numB
@@ -644,22 +484,11 @@ PartiallyOrderedRing.orderRespectsMultiplication (fieldOfFractionsPOrderedRing)
0<nAnB : 0R < (numA * numB)
0<nAnB = SetoidPartialOrder.<WellDefined pOrder (transitive *Commutative (Ring.timesZero R)) reflexive (ringCanMultiplyByPositive pRing 0<nB 0<nA)
PartiallyOrderedRing.orderRespectsMultiplication (fieldOfFractionsPOrderedRing) {numA ,, (denomA , denomA!=0)} {numB ,, (denomB , denomB!=0)} 0<a 0<b | inl (inl 0<1) | inl (inl 0<dAdB) | inl (inl 0<dB) | inl (inr dA<0) = exFalso (SetoidPartialOrder.irreflexive pOrder (SetoidPartialOrder.<Transitive pOrder 0<dAdB (SetoidPartialOrder.<WellDefined pOrder *Commutative (transitive *Commutative (Ring.timesZero R)) (ringCanMultiplyByNegative pRing dA<0 0<dB))))
where
open Setoid S
open Equivalence (Setoid.eq S)
open Ring R
PartiallyOrderedRing.orderRespectsMultiplication (fieldOfFractionsPOrderedRing) {numA ,, (denomA , denomA!=0)} {numB ,, (denomB , denomB!=0)} 0<a 0<b | inl (inl 0<1) | inl (inl 0<dAdB) | inl (inl 0<dB) | inr x = exFalso (denomA!=0 (Equivalence.symmetric (Setoid.eq S) x))
PartiallyOrderedRing.orderRespectsMultiplication (fieldOfFractionsPOrderedRing) {numA ,, (denomA , denomA!=0)} {numB ,, (denomB , denomB!=0)} 0<a 0<b | inl (inl 0<1) | inl (inl 0<dAdB) | inl (inr dB<0) with totality (Ring.0R R) denomA
PartiallyOrderedRing.orderRespectsMultiplication (fieldOfFractionsPOrderedRing) {numA ,, (denomA , denomA!=0)} {numB ,, (denomB , denomB!=0)} 0<a 0<b | inl (inl 0<1) | inl (inl 0<dAdB) | inl (inr dB<0) | inl (inl 0<dA) = exFalso (SetoidPartialOrder.irreflexive pOrder (SetoidPartialOrder.<Transitive pOrder 0<dAdB (SetoidPartialOrder.<WellDefined pOrder reflexive (transitive *Commutative (Ring.timesZero R)) (ringCanMultiplyByNegative pRing dB<0 0<dA))))
where
open Setoid S
open Equivalence (Setoid.eq S)
open Ring R
PartiallyOrderedRing.orderRespectsMultiplication (fieldOfFractionsPOrderedRing) {numA ,, (denomA , denomA!=0)} {numB ,, (denomB , denomB!=0)} 0<a 0<b | inl (inl 0<1) | inl (inl 0<dAdB) | inl (inr dB<0) | inl (inr dA<0) = SetoidPartialOrder.<WellDefined pOrder (symmetric (transitive *Commutative (Ring.timesZero R))) (symmetric (transitive *Commutative identIsIdent)) 0<nAnB
where
open Setoid S
open Equivalence (Setoid.eq S)
open Ring R
nB<0 : numB < 0R
nB<0 = SetoidPartialOrder.<WellDefined pOrder (transitive *Commutative identIsIdent) (transitive *Commutative (Ring.timesZero R)) 0<b
nA<0 : numA < 0R
@@ -672,17 +501,11 @@ PartiallyOrderedRing.orderRespectsMultiplication (fieldOfFractionsPOrderedRing)
PartiallyOrderedRing.orderRespectsMultiplication (fieldOfFractionsPOrderedRing) {numA ,, (denomA , denomA!=0)} {numB ,, (denomB , denomB!=0)} 0<a 0<b | inl (inl 0<1) | inl (inr dAdB<0) | inl (inl 0<denomB) with totality (Ring.0R R) denomA
PartiallyOrderedRing.orderRespectsMultiplication (fieldOfFractionsPOrderedRing) {numA ,, (denomA , denomA!=0)} {numB ,, (denomB , denomB!=0)} 0<a 0<b | inl (inl 0<1) | inl (inr dAdB<0) | inl (inl 0<denomB) | inl (inl 0<denomA) = exFalso f
where
open Setoid S
open Equivalence (Setoid.eq S)
open Ring R
f : False
f with PartiallyOrderedRing.orderRespectsMultiplication pRing 0<denomA 0<denomB
... | bl = SetoidPartialOrder.irreflexive pOrder (SetoidPartialOrder.<Transitive pOrder bl dAdB<0)
PartiallyOrderedRing.orderRespectsMultiplication (fieldOfFractionsPOrderedRing) {numA ,, (denomA , denomA!=0)} {numB ,, (denomB , denomB!=0)} 0<a 0<b | inl (inl 0<1) | inl (inr dAdB<0) | inl (inl 0<denomB) | inl (inr denomA<0) = SetoidPartialOrder.<WellDefined pOrder (symmetric (transitive *Commutative identIsIdent)) (symmetric (transitive *Commutative (Ring.timesZero R))) ans
where
open Setoid S
open Equivalence (Setoid.eq S)
open Ring R
0<nB : 0R < numB
0<nB = SetoidPartialOrder.<WellDefined pOrder (transitive *Commutative (Ring.timesZero R)) (transitive *Commutative identIsIdent) 0<b
nA<0 : numA < 0R
@@ -693,9 +516,6 @@ PartiallyOrderedRing.orderRespectsMultiplication (fieldOfFractionsPOrderedRing)
PartiallyOrderedRing.orderRespectsMultiplication (fieldOfFractionsPOrderedRing) {numA ,, (denomA , denomA!=0)} {numB ,, (denomB , denomB!=0)} 0<a 0<b | inl (inl 0<1) | inl (inr dAdB<0) | inl (inr denomB<0) with totality (Ring.0R R) denomA
PartiallyOrderedRing.orderRespectsMultiplication (fieldOfFractionsPOrderedRing) {numA ,, (denomA , denomA!=0)} {numB ,, (denomB , denomB!=0)} 0<a 0<b | inl (inl 0<1) | inl (inr dAdB<0) | inl (inr denomB<0) | inl (inl 0<denomA) = SetoidPartialOrder.<WellDefined pOrder (symmetric (transitive *Commutative identIsIdent)) (symmetric (transitive *Commutative (Ring.timesZero R))) nAnB<0
where
open Setoid S
open Equivalence (Setoid.eq S)
open Ring R
nB<0 : numB < 0R
nB<0 = SetoidPartialOrder.<WellDefined pOrder (transitive *Commutative identIsIdent) (transitive *Commutative (Ring.timesZero R)) 0<b
0<nA : 0R < numA
@@ -704,9 +524,6 @@ PartiallyOrderedRing.orderRespectsMultiplication (fieldOfFractionsPOrderedRing)
nAnB<0 = SetoidPartialOrder.<WellDefined pOrder reflexive (transitive *Commutative (Ring.timesZero R)) (ringCanMultiplyByNegative pRing nB<0 0<nA)
PartiallyOrderedRing.orderRespectsMultiplication (fieldOfFractionsPOrderedRing) {numA ,, (denomA , denomA!=0)} {numB ,, (denomB , denomB!=0)} 0<a 0<b | inl (inl 0<1) | inl (inr dAdB<0) | inl (inr denomB<0) | inl (inr denomA<0) = exFalso f
where
open Setoid S
open Equivalence (Setoid.eq S)
open Ring R
h : 0R < (denomA * denomB)
h = SetoidPartialOrder.<WellDefined pOrder (transitive *Commutative (Ring.timesZero R)) reflexive (ringCanMultiplyByNegative pRing denomB<0 denomA<0)
f : False
@@ -716,11 +533,16 @@ PartiallyOrderedRing.orderRespectsMultiplication (fieldOfFractionsPOrderedRing)
PartiallyOrderedRing.orderRespectsMultiplication (fieldOfFractionsPOrderedRing) {numA ,, (denomA , denomA!=0)} {numB ,, (denomB , denomB!=0)} 0<a 0<b | inl (inl 0<1) | inr 0=dAdB with IntegralDomain.intDom I (Equivalence.symmetric (Setoid.eq S) 0=dAdB)
... | f = exFalso (denomB!=0 (f denomA!=0))
PartiallyOrderedRing.orderRespectsMultiplication (fieldOfFractionsPOrderedRing) {numA ,, (denomA , denomA!=0)} {numB ,, (denomB , denomB!=0)} 0<a 0<b | inl (inr 1<0) = exFalso (SetoidPartialOrder.irreflexive pOrder (SetoidPartialOrder.<Transitive pOrder 1<0 (SetoidPartialOrder.<WellDefined pOrder (transitive *Commutative (Ring.timesZero R)) identIsIdent (ringCanMultiplyByNegative pRing 1<0 1<0))))
where
open Setoid S
open Equivalence (Setoid.eq S)
open Ring R
PartiallyOrderedRing.orderRespectsMultiplication (fieldOfFractionsPOrderedRing) {numA ,, (denomA , denomA!=0)} {numB ,, (denomB , denomB!=0)} 0<a 0<b | inr x = exFalso (IntegralDomain.nontrivial I (Equivalence.symmetric (Setoid.eq S) x))
fieldOfFractionsOrderedRing : TotallyOrderedRing fieldOfFractionsPOrderedRing
TotallyOrderedRing.total fieldOfFractionsOrderedRing = fieldOfFractionsTotalOrder
fieldOfFractionsOrderInherited : {x y : A} → x < y → fieldOfFractionsComparison (embedIntoFieldOfFractions x) (embedIntoFieldOfFractions y)
fieldOfFractionsOrderInherited {x} {y} x<y with totality 0R 1R
fieldOfFractionsOrderInherited {x} {y} x<y | inl (inl 0<1) with totality 0R 1R
fieldOfFractionsOrderInherited {x} {y} x<y | inl (inl 0<1) | inl (inl _) = SetoidPartialOrder.<WellDefined pOrder (symmetric (transitive *Commutative identIsIdent)) (symmetric (transitive *Commutative identIsIdent)) x<y
fieldOfFractionsOrderInherited {x} {y} x<y | inl (inl 0<1) | inl (inr 1<0) = exFalso (SetoidPartialOrder.irreflexive pOrder (SetoidPartialOrder.<Transitive pOrder 0<1 1<0))
fieldOfFractionsOrderInherited {x} {y} x<y | inl (inl 0<1) | inr 0=1 = exFalso (SetoidPartialOrder.irreflexive pOrder (SetoidPartialOrder.<WellDefined pOrder 0=1 reflexive 0<1))
fieldOfFractionsOrderInherited {x} {y} x<y | inl (inr 1<0) = exFalso (1<0False order 1<0)
fieldOfFractionsOrderInherited {x} {y} x<y | inr 0=1 = exFalso (anyComparisonImpliesNontrivial pRing x<y 0=1)

4
Numbers/ClassicalReals/Sequences.agda
View File

@@ -1,4 +1,3 @@
{-# OPTIONS --safe --warning=error --without-K --guardedness #-}
open import Agda.Primitive using (Level; lzero; lsuc; _⊔_)
@@ -53,9 +52,6 @@ StrictlyIncreasing x = (n : ℕ) → (index x n) < (index x (succ n))
Increasing : Sequence A → Set (b ⊔ c)
Increasing x = (n : ℕ) → ((index x n) < (index x (succ n))) || ((index x n) ∼ (index x (succ n)))
Bounded : Sequence A → Set (a ⊔ c)
Bounded x = Sg A (λ K → (n : ℕ) → index x n < K)
sequencePredicate : (x : Sequence A) → A → Set b
sequencePredicate x a = Sg ℕ (λ n → index x n ∼ a)

10
Numbers/Rationals/Definition.agda
View File

@@ -1,6 +1,7 @@
{-# OPTIONS --safe --warning=error --without-K #-}
open import LogicalFormulae
open import Numbers.Naturals.Definition
open import Numbers.Naturals.Naturals
open import Numbers.Integers.Integers
open import Groups.Groups
@@ -22,6 +23,7 @@ open import Fields.FieldOfFractions.Addition ℤIntDom
open import Fields.FieldOfFractions.Multiplication ℤIntDom
open import Fields.FieldOfFractions.Ring ℤIntDom
open import Fields.FieldOfFractions.Field ℤIntDom
open import Fields.FieldOfFractions.Lemmas ℤIntDom
open import Fields.FieldOfFractions.Order ℤIntDom ℤOrderedRing
ℚ : Set
@@ -43,7 +45,10 @@ a *Q b = fieldOfFractionsTimes a b
0Q = Ring.0R ℚRing
injectionQ : ℤ → ℚ
injectionQ z = z ,, (nonneg 1 , λ ())
injectionQ = embedIntoFieldOfFractions
injectionNQ : ℕ → ℚ
injectionNQ n = injectionQ (nonneg n)
injectionQInjective : Injection injectionQ
injectionQInjective pr = equalityLeft pr
@@ -78,6 +83,9 @@ a-A a = Group.invRight (Ring.additiveGroup ℚRing) {a}
ℚTotalOrder : SetoidTotalOrder fieldOfFractionsOrder
ℚTotalOrder = fieldOfFractionsTotalOrder
ℚOrderInherited : (a b : ℤ) → a <Z b → injectionQ a <Q injectionQ b
ℚOrderInherited a b a<b = fieldOfFractionsOrderInherited a<b
open SetoidTotalOrder fieldOfFractionsTotalOrder
open SetoidPartialOrder partial
open Setoid fieldOfFractionsSetoid

3
Rings/Orders/Partial/Lemmas.agda
View File

@@ -42,6 +42,9 @@ abstract
q' : (x * c) < ((y * c) + 0R)
q' = SetoidPartialOrder.<WellDefined pOrder (Group.identLeft additiveGroup) (transitive (symmetric (Group.+Associative additiveGroup)) (Group.+WellDefined additiveGroup reflexive (Group.invLeft additiveGroup))) q
ringCanMultiplyByPositive' : {x y c : A} → (Ring.0R R) < c → x < y → (c * x) < (c * y)
ringCanMultiplyByPositive' {x} {y} {c} 0<c x<y = SetoidPartialOrder.<WellDefined pOrder *Commutative *Commutative (ringCanMultiplyByPositive 0<c x<y)
ringMultiplyPositives : {x y a b : A} → 0R < x → 0R < a → (x < y) → (a < b) → (x * a) < (y * b)
ringMultiplyPositives {x} {y} {a} {b} 0<x 0<a x<y a<b = SetoidPartialOrder.<Transitive pOrder (ringCanMultiplyByPositive 0<a x<y) (<WellDefined *Commutative *Commutative (ringCanMultiplyByPositive (SetoidPartialOrder.<Transitive pOrder 0<x x<y) a<b))

93
Rings/Orders/Total/BaseExpansion.agda Normal file
View File

@@ -0,0 +1,93 @@
{-# OPTIONS --safe --warning=error --without-K --guardedness #-}
open import Agda.Primitive using (Level; lzero; lsuc; _⊔_)
open import LogicalFormulae
open import Groups.Lemmas
open import Groups.Definition
open import Setoids.Orders
open import Setoids.Setoids
open import Functions
open import Sets.EquivalenceRelations
open import Rings.Definition
open import Rings.Orders.Total.Definition
open import Rings.Orders.Partial.Definition
open import Numbers.Naturals.Semiring
open import Numbers.Naturals.Order
open import Numbers.Modulo.Definition
open import Semirings.Definition
open import Orders.Total.Definition
open import Sequences
module Rings.Orders.Total.BaseExpansion {a m p : _} {A : Set a} {S : Setoid {a} {m} A} {_+_ : A → A → A} {_*_ : A → A → A} {R : Ring S _+_ _*_} {_<_ : Rel {_} {p} A} {pOrder : SetoidPartialOrder S _<_} {pOrderRing : PartiallyOrderedRing R pOrder} (order : TotallyOrderedRing pOrderRing) {n : ℕ} (1<n : 1 <N n) where
open Ring R
open Group additiveGroup
open Setoid S
open Equivalence eq
open SetoidPartialOrder pOrder
open TotallyOrderedRing order
open SetoidTotalOrder total
open PartiallyOrderedRing pOrderRing
open import Rings.Lemmas R
open import Rings.Orders.Partial.Lemmas pOrderRing
open import Rings.Orders.Total.Lemmas order
open import Rings.InitialRing R
record FloorIs (a : A) (n : ℕ) : Set (m ⊔ p) where
field
prBelow : fromN n <= a
prAbove : a < fromN (succ n)
addOneToFloor : {a : A} {n : ℕ} → FloorIs a n → FloorIs (a + 1R) (succ n)
FloorIs.prBelow (addOneToFloor record { prBelow = (inl x) ; prAbove = prAbove }) = inl (<WellDefined groupIsAbelian reflexive (orderRespectsAddition x 1R))
FloorIs.prBelow (addOneToFloor record { prBelow = (inr x) ; prAbove = prAbove }) = inr (transitive groupIsAbelian (+WellDefined x reflexive))
FloorIs.prAbove (addOneToFloor record { prBelow = x ; prAbove = prAbove }) = <WellDefined reflexive groupIsAbelian (orderRespectsAddition prAbove 1R)
private
0<n : {x y : A} → (x < y) → 0R < fromN n
0<n x<y = fromNPreservesOrder (anyComparisonImpliesNontrivial x<y) (lessTransitive (succIsPositive 0) 1<n)
floorWellDefinedLemma : {a : A} {n1 n2 : ℕ} → FloorIs a n1 → FloorIs a n2 → n1 <N n2 → False
floorWellDefinedLemma {a} {n1} {n2} record { prAbove = prAbove1 } record { prBelow = inl prBelow } n1<n2 with TotalOrder.totality ℕTotalOrder (succ n1) n2
... | inl (inl n1+1<n2) = irreflexive (<Transitive prAbove1 (<Transitive (fromNPreservesOrder (anyComparisonImpliesNontrivial prBelow) n1+1<n2) prBelow))
... | inl (inr n2<n1+1) = noIntegersBetweenXAndSuccX n1 n1<n2 n2<n1+1
... | inr refl = irreflexive (<Transitive prAbove1 prBelow)
floorWellDefinedLemma {a} {n1} {n2} record { prBelow = (inl x) ; prAbove = prAbove1 } record { prBelow = (inr eq) ; prAbove = _ } n1<n2 with TotalOrder.totality ℕTotalOrder (succ n1) n2
... | inl (inl n1+1<n2) = irreflexive (<Transitive prAbove1 (<WellDefined reflexive eq (fromNPreservesOrder (anyComparisonImpliesNontrivial prAbove1) n1+1<n2)))
... | inl (inr n2<n1+1) = noIntegersBetweenXAndSuccX n1 n1<n2 n2<n1+1
... | inr refl = irreflexive (<WellDefined reflexive eq prAbove1)
floorWellDefinedLemma {a} {n1} {n2} record { prBelow = (inr x) ; prAbove = prAbove1 } record { prBelow = (inr eq) ; prAbove = _ } n1<n2 = lessIrreflexive {n1} (fromNPreservesOrder' (anyComparisonImpliesNontrivial prAbove1) (<WellDefined reflexive (transitive eq (symmetric x)) (fromNPreservesOrder (anyComparisonImpliesNontrivial prAbove1) n1<n2)))
floorWellDefined : {a : A} {n1 n2 : ℕ} → FloorIs a n1 → FloorIs a n2 → n1 ≡ n2
floorWellDefined {a} {n1} {n2} record { prBelow = prBelow1 ; prAbove = prAbove1 } record { prBelow = prBelow ; prAbove = prAbove } with TotalOrder.totality ℕTotalOrder n1 n2
... | inr x = x
floorWellDefined {a} {n1} {n2} f1 f2 | inl (inl x) = exFalso (floorWellDefinedLemma f1 f2 x)
floorWellDefined {a} {n1} {n2} f1 f2 | inl (inr x) = exFalso (floorWellDefinedLemma f2 f1 x)
floorWellDefined' : {a b : A} {n : ℕ} → (a ∼ b) → FloorIs a n → FloorIs b n
FloorIs.prBelow (floorWellDefined' {a} {b} {n} a=b record { prBelow = (inl x) ; prAbove = s }) = inl (<WellDefined reflexive a=b x)
FloorIs.prBelow (floorWellDefined' {a} {b} {n} a=b record { prBelow = (inr x) ; prAbove = s }) = inr (transitive x a=b)
FloorIs.prAbove (floorWellDefined' {a} {b} {n} a=b record { prBelow = t ; prAbove = s }) = <WellDefined a=b reflexive s
computeFloor' : {k : ℕ} → (fuel : ℕ) → k +N fuel ≡ n → (a : A) → 0R < a → a < fromN k → Sg ℕ (FloorIs a)
computeFloor' {zero} zero refl a 0<a a<f = exFalso (lessIrreflexive (lessTransitive 1<n (succIsPositive 0)))
computeFloor' {zero} (succ k) pr a 0<a a<f = exFalso (irreflexive (<Transitive 0<a a<f))
computeFloor' {succ k} n pr a 0<a a<f with totality 1R a
... | inl (inr a<1) = 0 , (record { prAbove = <WellDefined reflexive (symmetric identRight) a<1 ; prBelow = inl 0<a })
... | inr 1=a = 1 , (record { prAbove = <WellDefined (transitive identRight 1=a) reflexive (fromNPreservesOrder (anyComparisonImpliesNontrivial 0<a) {1} (le 0 refl)) ; prBelow = inr (transitive identRight 1=a) })
... | inl (inl 1<a) with computeFloor' {k} (succ n) (transitivity (transitivity (Semiring.commutative ℕSemiring k (succ n)) (applyEquality succ (Semiring.commutative ℕSemiring n k))) pr) (a + inverse 1R) (moveInequality 1<a) (<WellDefined reflexive (transitive groupIsAbelian (transitive +Associative (transitive (+WellDefined invLeft reflexive) identLeft))) (orderRespectsAddition a<f (inverse 1R)))
... | N , snd = succ N , floorWellDefined' (transitive (symmetric +Associative) (transitive (+WellDefined reflexive invLeft) identRight)) (addOneToFloor snd)
computeFloor : (a : A) → 0R < a → a < fromN n → Sg (ℤn n (lessTransitive (succIsPositive 0) 1<n)) (λ z → FloorIs a (ℤn.x z))
computeFloor a 0<a a<n with computeFloor' {n} 0 (Semiring.sumZeroRight ℕSemiring n) a 0<a a<n
... | floor , record { prBelow = inl p ; prAbove = prAbove } = record { x = floor ; xLess = fromNPreservesOrder' (anyComparisonImpliesNontrivial 0<a) (<Transitive p a<n) } , record { prBelow = inl p ; prAbove = prAbove }
... | floor , record { prBelow = inr p ; prAbove = prAbove } = record { x = floor ; xLess = fromNPreservesOrder' (anyComparisonImpliesNontrivial 0<a) (<WellDefined (symmetric p) reflexive a<n) } , record { prBelow = inr p ; prAbove = prAbove }
firstDigit : (a : A) → 0R < a → a < 1R → ℤn n (lessTransitive (succIsPositive 0) 1<n)
firstDigit a 0<a a<1 = underlying (computeFloor (a * fromN n) (orderRespectsMultiplication 0<a (0<n 0<a)) (<WellDefined reflexive identIsIdent (ringCanMultiplyByPositive (0<n 0<a) a<1)))
baseNExpansion : (a : A) → 0R < a → a < 1R → Sequence (ℤn n (lessTransitive (succIsPositive 0) 1<n))
Sequence.head (baseNExpansion a 0<a a<1) = firstDigit a 0<a a<1
Sequence.tail (baseNExpansion a 0<a a<1) with computeFloor (a * fromN n) (orderRespectsMultiplication 0<a (0<n 0<a)) (<WellDefined reflexive identIsIdent (ringCanMultiplyByPositive (0<n 0<a) a<1))
... | record { x = x ; xLess = xLess } , record { prBelow = inl prB ; prAbove = prA } = baseNExpansion ((a * fromN n) + inverse (fromN x)) (moveInequality prB) (<WellDefined reflexive (transitive (symmetric +Associative) (transitive (+WellDefined reflexive invRight) identRight)) (orderRespectsAddition prA (inverse (fromN x))))
... | record { x = x ; xLess = xLess } , record { prBelow = inr prB ; prAbove = prA } = constSequence (record { x = 0 ; xLess = lessTransitive (succIsPositive 0) 1<n })

7
Rings/Orders/Total/Lemmas.agda
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@@ -12,6 +12,7 @@ open import Rings.Orders.Total.Definition
open import Rings.Orders.Partial.Definition
open import Numbers.Naturals.Semiring
open import Numbers.Naturals.Order
open import Orders.Total.Definition
module Rings.Orders.Total.Lemmas {n m p : _} {A : Set n} {S : Setoid {n} {m} A} {_+_ : A → A → A} {_*_ : A → A → A} {R : Ring S _+_ _*_} {_<_ : Rel {_} {p} A} {pOrder : SetoidPartialOrder S _<_} {pOrderRing : PartiallyOrderedRing R pOrder} (order : TotallyOrderedRing pOrderRing) where
@@ -364,6 +365,12 @@ fromNPreservesOrder 0!=1 {zero} {succ zero} a<b = fromNIncreasing 0!=1 0
fromNPreservesOrder 0!=1 {zero} {succ (succ b)} a<b = <Transitive (fromNPreservesOrder 0!=1 (succIsPositive b)) (fromNIncreasing 0!=1 (succ b))
fromNPreservesOrder 0!=1 {succ a} {succ b} a<b = <WellDefined groupIsAbelian groupIsAbelian (orderRespectsAddition (fromNPreservesOrder 0!=1 (canRemoveSuccFrom<N a<b)) 1R)
fromNPreservesOrder' : (0R ∼ 1R → False) → {a b : ℕ} → (fromN a) < (fromN b) → a <N b
fromNPreservesOrder' nontrivial {a} {b} a<b with TotalOrder.totality ℕTotalOrder a b
... | inl (inl x) = x
... | inl (inr x) = exFalso (irreflexive (<Transitive a<b (fromNPreservesOrder nontrivial x)))
... | inr x = exFalso (irreflexive (<WellDefined (fromNWellDefined x) reflexive a<b))
reciprocalPositive : (a b : A) → .(0<a : 0R < a) → (a * b ∼ 1R) → 0R < b
reciprocalPositive a 1/a 0<a ab=1 with totality 0G 1/a
... | inl (inl x) = x

5
Sequences.agda
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@@ -4,6 +4,7 @@ open import LogicalFormulae
open import Numbers.Naturals.Definition
open import Setoids.Setoids
open import Numbers.Naturals.Order
open import Vectors
module Sequences where
@@ -78,3 +79,7 @@ tailFrom (succ n) s = tailFrom n (Sequence.tail s)
subsequence : {a : _} {A : Set a} (x : Sequence A) → (indices : Sequence ℕ) → ((n : ℕ) → index indices n <N index indices (succ n)) → Sequence A
Sequence.head (subsequence x selector increasing) = index x (Sequence.head selector)
Sequence.tail (subsequence x selector increasing) = subsequence (tailFrom (succ (Sequence.head selector)) x) (Sequence.tail selector) λ n → increasing (succ n)
take : {a : _} {A : Set a} (n : ℕ) (s : Sequence A) → Vec A n
take zero s = []
take (succ n) s = Sequence.head s ,- take n (Sequence.tail s)

2
Setoids/Orders.agda
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@@ -16,6 +16,8 @@ record SetoidPartialOrder {a b c : _} {A : Set a} (S : Setoid {a} {b} A) (_<_ :
<WellDefined : {a b c d : A} → (a ∼ b) → (c ∼ d) → a < c → b < d
irreflexive : {x : A} → (x < x) → False
<Transitive : {a b c : A} → (a < b) → (b < c) → (a < c)
_<=_ : Rel {a} {b ⊔ c} A
a <= b = (a < b) || (a ∼ b)
record SetoidTotalOrder {a b c : _} {A : Set a} {S : Setoid {a} {b} A} {_<_ : Rel {a} {c} A} (P : SetoidPartialOrder S _<_) : Set (a ⊔ b ⊔ c) where
open Setoid S