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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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13
Fields/CauchyCompletion/Setoid.agda
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@@ -11,7 +11,8 @@ open import Groups.Lemmas
open import Fields.Fields
open import Sets.EquivalenceRelations
open import Sequences
open import Setoids.Orders
open import Setoids.Orders.Partial.Definition
open import Setoids.Orders.Total.Definition
open import Functions
open import LogicalFormulae
open import Numbers.Naturals.Semiring
@@ -34,16 +35,18 @@ open import Fields.CauchyCompletion.Definition order F
open import Fields.CauchyCompletion.Addition order F
open import Rings.Orders.Partial.Lemmas pRing
open import Rings.Orders.Total.Lemmas order
open import Rings.Orders.Total.AbsoluteValue order
open import Rings.InitialRing R
open import Fields.Orders.Total.Lemmas {F = F} (record { oRing = order })
isZero : CauchyCompletion → Set (m ⊔ o)
isZero record { elts = elts ; converges = converges } = ∀ ε → 0R < ε → Sg ℕ (λ N → ∀ {m : ℕ} → (N <N m) → (abs (index elts m)) < ε)
transitiveLemma : {a b c e/2 : A} → abs (a + inverse b) < e/2 → abs (b + inverse c) < e/2 → (abs (a + inverse c)) < (e/2 + e/2)
transitiveLemma {a} {b} {c} {e/2} a-b<e/2 b-c<e/2 with triangleInequality (a + inverse b) (b + inverse c)
transitiveLemma {a} {b} {c} {e/2} a-b<e/2 b-c<e/2 | inl x = SetoidPartialOrder.<Transitive pOrder (<WellDefined (absWellDefined _ _ (Equivalence.transitive eq (Equivalence.symmetric eq +Associative) (+WellDefined (Equivalence.reflexive eq) (Equivalence.transitive eq +Associative (Equivalence.transitive eq (+WellDefined invLeft (Equivalence.reflexive eq)) identLeft))))) (Equivalence.reflexive eq) x) (ringAddInequalities a-b<e/2 b-c<e/2)
transitiveLemma {a} {b} {c} {e/2} a-b<e/2 b-c<e/2 | inr x = <WellDefined (Equivalence.symmetric eq (Equivalence.transitive eq (absWellDefined _ _ (Equivalence.symmetric eq ((Equivalence.transitive eq (Equivalence.symmetric eq +Associative) (+WellDefined (Equivalence.reflexive eq) (Equivalence.transitive eq +Associative (Equivalence.transitive eq (+WellDefined invLeft (Equivalence.reflexive eq)) identLeft))))))) x)) (Equivalence.reflexive eq) (ringAddInequalities a-b<e/2 b-c<e/2)
private
transitiveLemma : {a b c e/2 : A} → abs (a + inverse b) < e/2 → abs (b + inverse c) < e/2 → (abs (a + inverse c)) < (e/2 + e/2)
transitiveLemma {a} {b} {c} {e/2} a-b<e/2 b-c<e/2 with triangleInequality (a + inverse b) (b + inverse c)
transitiveLemma {a} {b} {c} {e/2} a-b<e/2 b-c<e/2 | inl x = SetoidPartialOrder.<Transitive pOrder (<WellDefined (absWellDefined _ _ (Equivalence.transitive eq (Equivalence.symmetric eq +Associative) (+WellDefined (Equivalence.reflexive eq) (Equivalence.transitive eq +Associative (Equivalence.transitive eq (+WellDefined invLeft (Equivalence.reflexive eq)) identLeft))))) (Equivalence.reflexive eq) x) (ringAddInequalities a-b<e/2 b-c<e/2)
transitiveLemma {a} {b} {c} {e/2} a-b<e/2 b-c<e/2 | inr x = <WellDefined (Equivalence.symmetric eq (Equivalence.transitive eq (absWellDefined _ _ (Equivalence.symmetric eq ((Equivalence.transitive eq (Equivalence.symmetric eq +Associative) (+WellDefined (Equivalence.reflexive eq) (Equivalence.transitive eq +Associative (Equivalence.transitive eq (+WellDefined invLeft (Equivalence.reflexive eq)) identLeft))))))) x)) (Equivalence.reflexive eq) (ringAddInequalities a-b<e/2 b-c<e/2)
cauchyCompletionSetoid : Setoid CauchyCompletion
(cauchyCompletionSetoid Setoid.∼ a) b = isZero (a +C (-C b))