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Lots of speedups (#116)

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Patrick Stevens
2020-04-16 13:41:51 +01:00
committed by GitHub
parent 1bcb3f8537
commit 9b80058157
63 changed files with 1082 additions and 564 deletions
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Fields/FieldOfFractions/Archimedean.agda Normal file
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{-# OPTIONS --safe --warning=error --without-K #-}
open import Numbers.Naturals.Semiring
open import Functions
open import LogicalFormulae
open import Groups.Definition
open import Rings.Definition
open import Rings.IntegralDomains.Definition
open import Setoids.Setoids
open import Sets.EquivalenceRelations
open import Setoids.Orders.Partial.Definition
open import Setoids.Orders.Total.Definition
open import Rings.Orders.Partial.Definition
open import Rings.Orders.Total.Definition
open import Groups.Orders.Archimedean
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
open import Groups.Cyclic.Definition
open import Fields.FieldOfFractions.Setoid I
open import Fields.FieldOfFractions.Group I
open import Fields.FieldOfFractions.Addition I
open import Fields.FieldOfFractions.Ring I
open import Fields.FieldOfFractions.Order I order
open import Fields.FieldOfFractions.Field I
open import Fields.Orders.Total.Archimedean
open import Rings.Orders.Partial.Lemmas pRing
open import Rings.Orders.Total.Lemmas order
open Setoid S
open Equivalence eq
open Ring R
open Group additiveGroup
open TotallyOrderedRing order
open SetoidTotalOrder total
private
denomPower : (n : ℕ) → fieldOfFractionsSet.denom (positiveEltPower fieldOfFractionsGroup record { num = 1R ; denom = 1R ; denomNonzero = IntegralDomain.nontrivial I } n) ∼ 1R
denomPower zero = reflexive
denomPower (succ n) = transitive identIsIdent (denomPower n)
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 })
denomPlus {a} a!=0 n1 n2 = transitive *Commutative (transitive (*WellDefined reflexive (transitive (+WellDefined *Commutative reflexive) (symmetric *DistributesOver+))) *Associative)
d : (a : fieldOfFractionsSet) → fieldOfFractionsSet.denom a ∼ 0R → False
d record { num = num ; denom = denom ; denomNonzero = denomNonzero } bad = exFalso (denomNonzero bad)
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 }
simpPower zero = Equivalence.reflexive (Setoid.eq fieldOfFractionsSetoid) {Group.0G fieldOfFractionsGroup}
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))
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 }
lemma n {num} {denom} d!=0 numdenom<n with totality 0G denom
... | inl (inl 0<denom) with totality 0G 1R
... | inl (inl 0<1) = {!!}
... | inl (inr x) = exFalso (1<0False x)
... | inr x = exFalso (nonempty x)
lemma n {num} {denom} d!=0 numdenom<n | inl (inr denom<0) with totality 0G 1R
... | inl (inl 0<1) = {!!}
... | inl (inr 1<0) = exFalso (1<0False 1<0)
... | inr 0=1 = exFalso (nonempty 0=1)
lemma n {num} {denom} d!=0 numdenom<n | inr 0=denom = exFalso (d!=0 (symmetric 0=denom))
fieldOfFractionsArchimedean : Archimedean (toGroup R pRing) → ArchimedeanField {F = fieldOfFractions} record { oRing = fieldOfFractionsOrderedRing }
fieldOfFractionsArchimedean arch (record { num = num ; denom = denom ; denomNonzero = denom!=0 }) with arch (num * denom) {!!} {!!} {!!}
... | bl = {!!}
--... | 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)