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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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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 = {!!}