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Lots of speedups (#116)
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Patrick Stevens
GitHub
68
Fields/FieldOfFractions/Archimedean.agda
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68
Fields/FieldOfFractions/Archimedean.agda
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{-# OPTIONS --safe --warning=error --without-K #-}
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open import Numbers.Naturals.Semiring
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open import Functions
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open import LogicalFormulae
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open import Groups.Definition
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open import Rings.Definition
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open import Rings.IntegralDomains.Definition
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open import Setoids.Setoids
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open import Sets.EquivalenceRelations
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open import Setoids.Orders.Partial.Definition
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open import Setoids.Orders.Total.Definition
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open import Rings.Orders.Partial.Definition
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open import Rings.Orders.Total.Definition
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open import Groups.Orders.Archimedean
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module Fields.FieldOfFractions.Archimedean {a b : _} {A : Set a} {S : Setoid {a} {b} A} {_+_ : A → A → A} {_*_ : A → A → A} {R : Ring S _+_ _*_} (I : IntegralDomain R) {c : _} {_<_ : Rel {_} {c} A} {pOrder : SetoidPartialOrder S _<_} {pRing : PartiallyOrderedRing R pOrder} (order : TotallyOrderedRing pRing) (nonempty : (Setoid._∼_ S (Ring.0R R) (Ring.1R R)) → False) where
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open import Groups.Cyclic.Definition
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open import Fields.FieldOfFractions.Setoid I
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open import Fields.FieldOfFractions.Group I
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open import Fields.FieldOfFractions.Addition I
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open import Fields.FieldOfFractions.Ring I
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open import Fields.FieldOfFractions.Order I order
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open import Fields.FieldOfFractions.Field I
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open import Fields.Orders.Total.Archimedean
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open import Rings.Orders.Partial.Lemmas pRing
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open import Rings.Orders.Total.Lemmas order
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open Setoid S
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open Equivalence eq
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open Ring R
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open Group additiveGroup
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open TotallyOrderedRing order
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open SetoidTotalOrder total
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private
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denomPower : (n : ℕ) → fieldOfFractionsSet.denom (positiveEltPower fieldOfFractionsGroup record { num = 1R ; denom = 1R ; denomNonzero = IntegralDomain.nontrivial I } n) ∼ 1R
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denomPower zero = reflexive
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denomPower (succ n) = transitive identIsIdent (denomPower n)
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denomPlus : {a : A} .(a!=0 : a ∼ 0R → False) (n1 n2 : A) → Setoid._∼_ fieldOfFractionsSetoid (fieldOfFractionsPlus record { num = n1 ; denom = a ; denomNonzero = a!=0 } record { num = n2 ; denom = a ; denomNonzero = a!=0 }) (record { num = n1 + n2 ; denom = a ; denomNonzero = a!=0 })
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denomPlus {a} a!=0 n1 n2 = transitive *Commutative (transitive (*WellDefined reflexive (transitive (+WellDefined *Commutative reflexive) (symmetric *DistributesOver+))) *Associative)
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d : (a : fieldOfFractionsSet) → fieldOfFractionsSet.denom a ∼ 0R → False
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d record { num = num ; denom = denom ; denomNonzero = denomNonzero } bad = exFalso (denomNonzero bad)
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simpPower : (n : ℕ) → Setoid._∼_ fieldOfFractionsSetoid (positiveEltPower fieldOfFractionsGroup record { num = Ring.1R R ; denom = Ring.1R R ; denomNonzero = IntegralDomain.nontrivial I} n) record { num = positiveEltPower (Ring.additiveGroup R) (Ring.1R R) n ; denom = Ring.1R R ; denomNonzero = IntegralDomain.nontrivial I }
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simpPower zero = Equivalence.reflexive (Setoid.eq fieldOfFractionsSetoid) {Group.0G fieldOfFractionsGroup}
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simpPower (succ n) = Equivalence.transitive (Setoid.eq fieldOfFractionsSetoid) {record { denomNonzero = d (fieldOfFractionsPlus (record { num = 1R ; denom = 1R ; denomNonzero = λ t → nonempty (symmetric t) }) (positiveEltPower fieldOfFractionsGroup _ n)) }} {record { denomNonzero = λ t → nonempty (symmetric (transitive (symmetric identIsIdent) t)) }} {record { denomNonzero = λ t → nonempty (symmetric t) }} (Group.+WellDefined fieldOfFractionsGroup {record { num = Ring.1R R ; denom = Ring.1R R ; denomNonzero = IntegralDomain.nontrivial I }} {positiveEltPower fieldOfFractionsGroup record { num = Ring.1R R ; denom = Ring.1R R ; denomNonzero = IntegralDomain.nontrivial I } n} {record { num = Ring.1R R ; denom = Ring.1R R ; denomNonzero = IntegralDomain.nontrivial I }} {record { num = positiveEltPower (Ring.additiveGroup R) (Ring.1R R) n ; denom = Ring.1R R ; denomNonzero = IntegralDomain.nontrivial I }} (Equivalence.reflexive (Setoid.eq fieldOfFractionsSetoid) {record { num = Ring.1R R ; denom = Ring.1R R ; denomNonzero = IntegralDomain.nontrivial I}}) (simpPower n)) (transitive (transitive (transitive *Commutative (transitive identIsIdent (+WellDefined identIsIdent identIsIdent))) (symmetric identIsIdent)) (*WellDefined (symmetric identIsIdent) reflexive))
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lemma : (n : ℕ) {num denom : A} .(d!=0 : denom ∼ 0G → False) → (num * denom) < positiveEltPower additiveGroup 1R n → fieldOfFractionsComparison (record { num = num ; denom = denom ; denomNonzero = d!=0}) record { num = positiveEltPower additiveGroup (Ring.1R R) n ; denom = 1R ; denomNonzero = IntegralDomain.nontrivial I }
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lemma n {num} {denom} d!=0 numdenom<n with totality 0G denom
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... | inl (inl 0<denom) with totality 0G 1R
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... | inl (inl 0<1) = {!!}
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... | inl (inr x) = exFalso (1<0False x)
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... | inr x = exFalso (nonempty x)
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lemma n {num} {denom} d!=0 numdenom<n | inl (inr denom<0) with totality 0G 1R
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... | inl (inl 0<1) = {!!}
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... | inl (inr 1<0) = exFalso (1<0False 1<0)
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... | inr 0=1 = exFalso (nonempty 0=1)
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lemma n {num} {denom} d!=0 numdenom<n | inr 0=denom = exFalso (d!=0 (symmetric 0=denom))
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fieldOfFractionsArchimedean : Archimedean (toGroup R pRing) → ArchimedeanField {F = fieldOfFractions} record { oRing = fieldOfFractionsOrderedRing }
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fieldOfFractionsArchimedean arch (record { num = num ; denom = denom ; denomNonzero = denom!=0 }) with arch (num * denom) {!!} {!!} {!!}
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... | bl = {!!}
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--... | N , pr = N , SetoidPartialOrder.<WellDefined fieldOfFractionsOrder {record { denomNonzero = denom!=0 }} {record { denomNonzero = denom!=0 }} {record { denomNonzero = λ t → nonempty (symmetric t) }} {record { denomNonzero = d (positiveEltPower fieldOfFractionsGroup record { num = 1R ; denom = 1R ; denomNonzero = λ t → nonempty (symmetric t) } N) }} (Equivalence.reflexive (Setoid.eq fieldOfFractionsSetoid) { record { denomNonzero = denom!=0 } }) (Equivalence.symmetric (Setoid.eq fieldOfFractionsSetoid) {record { denomNonzero = λ t → d (positiveEltPower fieldOfFractionsGroup record { num = 1R ; denom = 1R ; denomNonzero = λ t → nonempty (symmetric t) } N) t }} {record { denomNonzero = λ t → nonempty (symmetric t) }} (simpPower N)) (lemma N denom!=0 pr)
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@@ -7,7 +7,6 @@ open import Rings.IntegralDomains.Definition
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open import Setoids.Setoids
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open import Sets.EquivalenceRelations
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module Fields.FieldOfFractions.Group {a b : _} {A : Set a} {S : Setoid {a} {b} A} {_+_ : A → A → A} {_*_ : A → A → A} {R : Ring S _+_ _*_} (I : IntegralDomain R) where
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open import Fields.FieldOfFractions.Setoid I
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@@ -9,7 +9,8 @@ open import Rings.Orders.Total.Lemmas
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open import Rings.IntegralDomains.Definition
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open import Functions
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open import Setoids.Setoids
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open import Setoids.Orders
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open import Setoids.Orders.Partial.Definition
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open import Setoids.Orders.Total.Definition
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open import Sets.EquivalenceRelations
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module Fields.FieldOfFractions.Order {a b c : _} {A : Set a} {S : Setoid {a} {b} A} {_+_ : A → A → A} {_*_ : A → A → A} {R : Ring S _+_ _*_} {_<_ : Rel {_} {c} A} {pOrder : SetoidPartialOrder S _<_} {pRing : PartiallyOrderedRing R pOrder} (I : IntegralDomain R) (order : TotallyOrderedRing pRing) where
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@@ -171,12 +172,7 @@ private
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<transitive (record { num = numA ; denom = denomA ; denomNonzero = denomA!=0 }) (record { num = numB ; denom = denomB ; denomNonzero = denomB!=0 }) (record { num = numC ; denom = denomC ; denomNonzero = denomC!=0 }) a<b b<c | inl (inl 0<denomA) with totality (Ring.0R R) denomC
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<transitive (record { num = numA ; denom = denomA ; denomNonzero = denomA!=0 }) (record { num = numB ; denom = denomB ; denomNonzero = denomB!=0 }) (record { num = numC ; denom = denomC ; denomNonzero = denomC!=0 }) a<b b<c | inl (inl 0<denomA) | inl (inl 0<denomC) with totality (Ring.0R R) denomB
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<transitive (record { num = numA ; denom = denomA ; denomNonzero = denomA!=0 }) (record { num = numB ; denom = denomB ; denomNonzero = denomB!=0 }) (record { num = numC ; denom = denomC ; denomNonzero = denomC!=0 }) a<b b<c | inl (inl 0<denomA) | inl (inl 0<denomC) | inl (inl x) with totality (Ring.0R R) denomC
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<transitive (record { num = numA ; denom = denomA ; denomNonzero = denomA!=0 }) (record { num = numB ; denom = denomB ; denomNonzero = denomB!=0 }) (record { num = numC ; denom = denomC ; denomNonzero = 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
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where
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inter : ((numA * denomB) * denomC) < ((numB * denomA) * denomC)
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inter = ringCanMultiplyByPositive pRing 0<denomC a<b
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p : ((numA * denomC) * denomB) < ((numC * denomA) * denomB)
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p = SetoidPartialOrder.<Transitive pOrder (SetoidPartialOrder.<WellDefined pOrder (transitive (symmetric *Associative) (transitive (*WellDefined reflexive *Commutative) *Associative)) reflexive inter) (SetoidPartialOrder.<WellDefined pOrder (transitive (symmetric *Associative) (transitive (*WellDefined reflexive *Commutative) *Associative)) (transitive (symmetric *Associative) (transitive (*WellDefined reflexive *Commutative) *Associative)) (ringCanMultiplyByPositive pRing 0<denomA b<c))
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<transitive (record { num = numA ; denom = denomA ; denomNonzero = denomA!=0 }) (record { num = numB ; denom = denomB ; denomNonzero = denomB!=0 }) (record { num = numC ; denom = denomC ; denomNonzero = denomC!=0 }) a<b b<c | inl (inl 0<denomA) | inl (inl 0<denomC) | inl (inl 0<denomB) | inl (inl _) = ringCanCancelPositive order 0<denomB (SetoidPartialOrder.<Transitive pOrder (SetoidPartialOrder.<WellDefined pOrder (transitive (symmetric *Associative) (transitive (*WellDefined reflexive *Commutative) *Associative)) reflexive (ringCanMultiplyByPositive pRing 0<denomC a<b)) (SetoidPartialOrder.<WellDefined pOrder (transitive (symmetric *Associative) (transitive (*WellDefined reflexive *Commutative) *Associative)) (transitive (symmetric *Associative) (transitive (*WellDefined reflexive *Commutative) *Associative)) (ringCanMultiplyByPositive pRing 0<denomA b<c)))
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<transitive (record { num = numA ; denom = denomA ; denomNonzero = denomA!=0 }) (record { num = numB ; denom = denomB ; denomNonzero = denomB!=0 }) (record { num = numC ; denom = denomC ; denomNonzero = denomC!=0 }) a<b b<c | inl (inl 0<denomA) | inl (inl 0<denomC) | inl (inl 0<denomB) | inl (inr denomC<0) = exFalso (SetoidPartialOrder.irreflexive pOrder (SetoidPartialOrder.<Transitive pOrder 0<denomC denomC<0))
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<transitive (record { num = numA ; denom = denomA ; denomNonzero = denomA!=0 }) (record { num = numB ; denom = denomB ; denomNonzero = denomB!=0 }) (record { num = numC ; denom = denomC ; denomNonzero = 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))
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<transitive (record { num = numA ; denom = denomA ; denomNonzero = denomA!=0 }) (record { num = numB ; denom = denomB ; denomNonzero = denomB!=0 }) (record { num = numC ; denom = denomC ; denomNonzero = denomC!=0 }) a<b b<c | inl (inl 0<denomA) | inl (inl 0<denomC) | inl (inr denomB<0) with totality (Ring.0R R) denomC
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@@ -221,17 +217,11 @@ private
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<transitive (record { num = numA ; denom = denomA ; denomNonzero = denomA!=0 }) (record { num = numB ; denom = denomB ; denomNonzero = denomB!=0 }) (record { num = numC ; denom = denomC ; denomNonzero = denomC!=0 }) a<b b<c | inl (inr denomA<0) | inl (inr denomC<0) with totality (Ring.0R R) denomB
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<transitive (record { num = numA ; denom = denomA ; denomNonzero = denomA!=0 }) (record { num = numB ; denom = denomB ; denomNonzero = denomB!=0 }) (record { num = numC ; denom = denomC ; denomNonzero = denomC!=0 }) a<b b<c | inl (inr denomA<0) | inl (inr denomC<0) | inl (inl 0<denomB) with totality (Ring.0R R) denomC
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<transitive (record { num = numA ; denom = denomA ; denomNonzero = denomA!=0 }) (record { num = numB ; denom = denomB ; denomNonzero = denomB!=0 }) (record { num = numC ; denom = denomC ; denomNonzero = 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))
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<transitive (record { num = numA ; denom = denomA ; denomNonzero = denomA!=0 }) (record { num = numB ; denom = denomB ; denomNonzero = denomB!=0 }) (record { num = numC ; denom = denomC ; denomNonzero = 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)))
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where
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have : ((numA * denomC) * denomB) < ((numB * denomA) * denomC)
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have = SetoidPartialOrder.<WellDefined pOrder (swapLemma) reflexive (ringCanMultiplyByNegative pRing denomC<0 a<b)
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<transitive (record { num = numA ; denom = denomA ; denomNonzero = denomA!=0 }) (record { num = numB ; denom = denomB ; denomNonzero = denomB!=0 }) (record { num = numC ; denom = denomC ; denomNonzero = 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 (SetoidPartialOrder.<WellDefined pOrder swapLemma reflexive (ringCanMultiplyByNegative pRing denomC<0 a<b)) (SetoidPartialOrder.<WellDefined pOrder (swapLemma) (swapLemma) (ringCanMultiplyByNegative pRing denomA<0 b<c)))
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<transitive (record { num = numA ; denom = denomA ; denomNonzero = denomA!=0 }) (record { num = numB ; denom = denomB ; denomNonzero = denomB!=0 }) (record { num = numC ; denom = denomC ; denomNonzero = 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))
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<transitive (record { num = numA ; denom = denomA ; denomNonzero = denomA!=0 }) (record { num = numB ; denom = denomB ; denomNonzero = denomB!=0 }) (record { num = numC ; denom = denomC ; denomNonzero = denomC!=0 }) a<b b<c | inl (inr denomA<0) | inl (inr denomC<0) | inl (inr denomB<0) with totality (Ring.0R R) denomC
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<transitive (record { num = numA ; denom = denomA ; denomNonzero = denomA!=0 }) (record { num = numB ; denom = denomB ; denomNonzero = denomB!=0 }) (record { num = numC ; denom = denomC ; denomNonzero = 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))
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<transitive (record { num = numA ; denom = denomA ; denomNonzero = denomA!=0 }) (record { num = numB ; denom = denomB ; denomNonzero = denomB!=0 }) (record { num = numC ; denom = denomC ; denomNonzero = 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)
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where
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have : ((numB * denomA) * denomC) < ((numA * denomC) * denomB)
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have = SetoidPartialOrder.<WellDefined pOrder reflexive (swapLemma) (ringCanMultiplyByNegative pRing denomC<0 a<b)
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<transitive (record { num = numA ; denom = denomA ; denomNonzero = denomA!=0 }) (record { num = numB ; denom = denomB ; denomNonzero = denomB!=0 }) (record { num = numC ; denom = denomC ; denomNonzero = 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)) (SetoidPartialOrder.<WellDefined pOrder reflexive (swapLemma) (ringCanMultiplyByNegative pRing denomC<0 a<b)))
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<transitive (record { num = numA ; denom = denomA ; denomNonzero = denomA!=0 }) (record { num = numB ; denom = denomB ; denomNonzero = denomB!=0 }) (record { num = numC ; denom = denomC ; denomNonzero = 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))
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<transitive (record { num = numA ; denom = denomA ; denomNonzero = denomA!=0 }) (record { num = numB ; denom = denomB ; denomNonzero = denomB!=0 }) (record { num = numC ; denom = denomC ; denomNonzero = 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))
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<transitive (record { num = numA ; denom = denomA ; denomNonzero = denomA!=0 }) (record { num = numB ; denom = denomB ; denomNonzero = denomB!=0 }) (record { num = numC ; denom = denomC ; denomNonzero = denomC!=0 }) a<b b<c | inl (inr denomA<0) | inr x = exFalso (denomC!=0 (Equivalence.symmetric (Setoid.eq S) x))
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@@ -538,11 +528,15 @@ PartiallyOrderedRing.orderRespectsMultiplication (fieldOfFractionsPOrderedRing)
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fieldOfFractionsOrderedRing : TotallyOrderedRing fieldOfFractionsPOrderedRing
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TotallyOrderedRing.total fieldOfFractionsOrderedRing = fieldOfFractionsTotalOrder
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private
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fieldOfFractionsOrderInherited' : {x y : A} → x < y → fieldOfFractionsComparison (embedIntoFieldOfFractions x) (embedIntoFieldOfFractions y)
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fieldOfFractionsOrderInherited' {x} {y} x<y with totality 0R 1R
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fieldOfFractionsOrderInherited' {x} {y} x<y | inl (inl 0<1) with totality 0R 1R
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fieldOfFractionsOrderInherited' {x} {y} x<y | inl (inl 0<1) | inl (inl _) = SetoidPartialOrder.<WellDefined pOrder (symmetric (transitive *Commutative identIsIdent)) (symmetric (transitive *Commutative identIsIdent)) x<y
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fieldOfFractionsOrderInherited' {x} {y} x<y | inl (inl 0<1) | inl (inr 1<0) = exFalso (SetoidPartialOrder.irreflexive pOrder (SetoidPartialOrder.<Transitive pOrder 0<1 1<0))
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fieldOfFractionsOrderInherited' {x} {y} x<y | inl (inl 0<1) | inr 0=1 = exFalso (SetoidPartialOrder.irreflexive pOrder (SetoidPartialOrder.<WellDefined pOrder 0=1 reflexive 0<1))
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fieldOfFractionsOrderInherited' {x} {y} x<y | inl (inr 1<0) = exFalso (1<0False order 1<0)
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fieldOfFractionsOrderInherited' {x} {y} x<y | inr 0=1 = exFalso (anyComparisonImpliesNontrivial pRing x<y 0=1)
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fieldOfFractionsOrderInherited : {x y : A} → x < y → fieldOfFractionsComparison (embedIntoFieldOfFractions x) (embedIntoFieldOfFractions y)
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fieldOfFractionsOrderInherited {x} {y} x<y with totality 0R 1R
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fieldOfFractionsOrderInherited {x} {y} x<y | inl (inl 0<1) with totality 0R 1R
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fieldOfFractionsOrderInherited {x} {y} x<y | inl (inl 0<1) | inl (inl _) = SetoidPartialOrder.<WellDefined pOrder (symmetric (transitive *Commutative identIsIdent)) (symmetric (transitive *Commutative identIsIdent)) x<y
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fieldOfFractionsOrderInherited {x} {y} x<y | inl (inl 0<1) | inl (inr 1<0) = exFalso (SetoidPartialOrder.irreflexive pOrder (SetoidPartialOrder.<Transitive pOrder 0<1 1<0))
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fieldOfFractionsOrderInherited {x} {y} x<y | inl (inl 0<1) | inr 0=1 = exFalso (SetoidPartialOrder.irreflexive pOrder (SetoidPartialOrder.<WellDefined pOrder 0=1 reflexive 0<1))
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fieldOfFractionsOrderInherited {x} {y} x<y | inl (inr 1<0) = exFalso (1<0False order 1<0)
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fieldOfFractionsOrderInherited {x} {y} x<y | inr 0=1 = exFalso (anyComparisonImpliesNontrivial pRing x<y 0=1)
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fieldOfFractionsOrderInherited {x} {y} x<y = fieldOfFractionsOrderInherited' {x} {y} x<y
|
||||
|
||||
Reference in New Issue
Block a user