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Tidy a bit

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Smaug123
2020-05-19 07:41:29 +01:00
parent b57cbb8ea6
commit df591cc5f9
3 changed files with 14 additions and 8 deletions
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4
Fields/CauchyCompletion/Archimedean.agda
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@@ -24,5 +24,5 @@ open import Fields.CauchyCompletion.Ring order F
open import Fields.CauchyCompletion.Comparison order F
open import Fields.CauchyCompletion.PartiallyOrderedRing order F
--CArchimedean : Archimedean (toGroup CRing CpOrderedRing)
--CArchimedean = ?
CArchimedean : Archimedean (toGroup CRing CpOrderedRing)
CArchimedean x y x₁ x₂ = {!!}

14
Fields/CauchyCompletion/BaseExpansion.agda
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@@ -10,8 +10,9 @@ open import Groups.Lemmas
open import Fields.Fields
open import Sets.EquivalenceRelations
open import Sequences
open import Setoids.Orders
open import Functions
open import Setoids.Orders.Partial.Definition
open import Setoids.Orders.Total.Definition
open import Functions.Definition
open import LogicalFormulae
open import Numbers.Naturals.Semiring
open import Numbers.Naturals.Order
@@ -34,6 +35,7 @@ open import Fields.Orders.Limits.Definition {F = F} (record { oRing = order })
open import Fields.Orders.Total.Lemmas {F = F} (record { oRing = order })
open import Fields.Orders.Limits.Lemmas {F = F} (record { oRing = order })
open import Rings.Orders.Total.AbsoluteValue order
open import Fields.Lemmas F
open import Fields.Orders.Lemmas {F = F} record { oRing = order }
open import Rings.Orders.Total.Lemmas order
@@ -97,11 +99,11 @@ private
digitExpansionBoundedLemma : {n : ℕ} → .(0<n : 0 <N n) → (seq : Sequence (ℤn n 0<n)) → (m : ℕ) → index (digitExpansionSeq _ seq) m < fromN n
digitExpansionBoundedLemma {n} 0<n seq zero with Sequence.head seq
... | record { x = x ; xLess = xLess } = fromNPreservesOrder nontrivial {x} {n} ((squash<N xLess))
... | record { x = x ; xLess = xLess } = fromNPreservesOrder (0<1 nontrivial) {x} {n} ((squash<N xLess))
digitExpansionBoundedLemma 0<n seq (succ m) = digitExpansionBoundedLemma 0<n (Sequence.tail seq) m
digitExpansionBoundedLemma2 : {n : ℕ} → .(0<n : 0 <N n) → (seq : Sequence (ℤn n 0<n)) → (m : ℕ) → inverse (fromN n) < index (digitExpansionSeq 0<n seq) m
digitExpansionBoundedLemma2 {n} 0<n seq zero = <WellDefined identLeft (transitive (symmetric +Associative) (transitive (+WellDefined reflexive invRight) identRight)) (orderRespectsAddition {_} {fromN (ℤn.x (Sequence.head seq)) + fromN n} (<WellDefined reflexive (fromNPreserves+ (ℤn.x (Sequence.head seq)) n) (fromNPreservesOrder nontrivial {0} {ℤn.x (Sequence.head seq) +N n} (canAddToOneSideOfInequality' _ (squash<N 0<n)))) (inverse (fromN n)))
digitExpansionBoundedLemma2 {n} 0<n seq zero = <WellDefined identLeft (transitive (symmetric +Associative) (transitive (+WellDefined reflexive invRight) identRight)) (orderRespectsAddition {_} {fromN (ℤn.x (Sequence.head seq)) + fromN n} (<WellDefined reflexive (fromNPreserves+ (ℤn.x (Sequence.head seq)) n) (fromNPreservesOrder (0<1 nontrivial) {0} {ℤn.x (Sequence.head seq) +N n} (canAddToOneSideOfInequality' _ (squash<N 0<n)))) (inverse (fromN n)))
digitExpansionBoundedLemma2 0<n seq (succ m) = digitExpansionBoundedLemma2 0<n (Sequence.tail seq) m
digitExpansionBounded : {n : ℕ} → .(0<n : 0 <N n) → (seq : Sequence (ℤn n 0<n)) → Bounded (digitExpansionSeq 0<n seq)
@@ -110,11 +112,11 @@ private
private
1/nPositive : (n : ℕ) → 0R < underlying (allInvertible (fromN (succ n)) (charNotN n))
1/nPositive n with allInvertible (fromN (succ n)) (charNotN n)
... | a , b = reciprocalPositive (fromN (succ n)) a (fromNPreservesOrder nontrivial (succIsPositive n)) (transitive *Commutative b)
... | a , b = reciprocalPositive (fromN (succ n)) a (fromNPreservesOrder (0<1 nontrivial) (succIsPositive n)) (transitive *Commutative b)
1/n<1 : (n : ℕ) → (0 <N n) → underlying (allInvertible (fromN (succ n)) (charNotN n)) < 1R
1/n<1 n 1<n with allInvertible (fromN (succ n)) (charNotN n)
... | a , b = reciprocal<1 (fromN (succ n)) a (<WellDefined identRight reflexive (fromNPreservesOrder nontrivial {1} {succ n} (succPreservesInequality 1<n))) (transitive *Commutative b)
... | a , b = reciprocal<1 (fromN (succ n)) a (<WellDefined identRight reflexive (fromNPreservesOrder (0<1 nontrivial) {1} {succ n} (succPreservesInequality 1<n))) (transitive *Commutative b)
-- Construct the real that is the given base-n expansion between 0 and 1.
ofBaseExpansion : {n : ℕ} → .(1<n : 1 <N n) → (fromN n ∼ 0R → False) → Sequence (ℤn n (0<n 1<n)) → CauchyCompletion

4
Numbers/Naturals/Order.agda
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@@ -213,3 +213,7 @@ squash<N {a} {b} a<b | inr refl = exFalso (lessIrreflexive a<b)
<N'Refl : {a b : ℕ} → (p1 p2 : a <N' b) → p1 ≡ p2
<N'Refl p1 p2 with <NWellDefined (<N'To<N p1) (<N'To<N p2)
... | refl = refl
contractLessSucc : {a b : ℕ} → a <N succ b → a ≤N b
contractLessSucc (le zero proof) = inr (succInjective proof)
contractLessSucc (le (succ x) proof) = inl (le x (succInjective proof))