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Move towards base-n expansions (#112)

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Patrick Stevens
2020-04-11 19:46:26 +01:00
committed by GitHub
parent e9aa1bcc05
commit 380548134d
22 changed files with 312 additions and 102 deletions
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35
Rings/Orders/Partial/Bounded.agda Normal file
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{-# OPTIONS --safe --warning=error --without-K --guardedness #-}
open import Agda.Primitive using (Level; lzero; lsuc; _⊔_)
open import Setoids.Setoids
open import Rings.Definition
open import Rings.Orders.Partial.Definition
open import Sets.EquivalenceRelations
open import Sequences
open import Setoids.Orders
open import Functions
open import LogicalFormulae
open import Numbers.Naturals.Semiring
open import Groups.Definition
module Rings.Orders.Partial.Bounded {m n o : _} {A : Set m} {S : Setoid {m} {n} A} {_+_ : A → A → A} {_*_ : A → A → A} {_<_ : Rel {m} {o} A} {pOrder : SetoidPartialOrder S _<_} {R : Ring S _+_ _*_} (pRing : PartiallyOrderedRing R pOrder) where
open Group (Ring.additiveGroup R)
open import Groups.Lemmas (Ring.additiveGroup R)
open Setoid S
open Equivalence eq
open SetoidPartialOrder pOrder
BoundedAbove : Sequence A → Set (m ⊔ o)
BoundedAbove x = Sg A (λ K → (n : ℕ) → index x n < K)
BoundedBelow : Sequence A → Set (m ⊔ o)
BoundedBelow x = Sg A (λ K → (n : ℕ) → K < index x n)
Bounded : Sequence A → Set (m ⊔ o)
Bounded x = Sg A (λ K → (n : ℕ) → ((Group.inverse (Ring.additiveGroup R) K) < index x n) && (index x n < K))
boundNonzero : {s : Sequence A} → (b : Bounded s) → underlying b ∼ 0G → False
boundNonzero {s} (a , b) isEq with b 0
... | bad1 ,, bad2 = irreflexive (<Transitive bad1 (<WellDefined reflexive (transitive isEq (symmetric (transitive (inverseWellDefined isEq) invIdent))) bad2))

31
Rings/Orders/Total/Bounded.agda Normal file
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{-# OPTIONS --safe --warning=error --without-K --guardedness #-}
open import Agda.Primitive using (Level; lzero; lsuc; _⊔_)
open import Setoids.Setoids
open import Rings.Definition
open import Rings.Orders.Partial.Definition
open import Rings.Orders.Total.Definition
open import Sets.EquivalenceRelations
open import Sequences
open import Setoids.Orders
open import Functions
open import LogicalFormulae
open import Numbers.Naturals.Semiring
open import Groups.Definition
module Rings.Orders.Total.Bounded {m n o : _} {A : Set m} {S : Setoid {m} {n} A} {_+_ : A → A → A} {_*_ : A → A → A} {_<_ : Rel {m} {o} A} {pOrder : SetoidPartialOrder S _<_} {R : Ring S _+_ _*_} {pRing : PartiallyOrderedRing R pOrder} (tRing : TotallyOrderedRing pRing) where
open import Rings.Orders.Partial.Bounded pRing
open Ring R
open Group additiveGroup
open import Groups.Lemmas (Ring.additiveGroup R)
open Setoid S
open Equivalence eq
open SetoidPartialOrder pOrder
open import Rings.Orders.Total.Lemmas tRing
open PartiallyOrderedRing pRing
boundGreaterThanZero : {s : Sequence A} → (b : Bounded s) → 0G < underlying b
boundGreaterThanZero {s} (a , b) with b 0
... | (l ,, r) = halvePositive a (<WellDefined invLeft reflexive (orderRespectsAddition (<Transitive l r) a))

30
Rings/Orders/Total/Cauchy.agda Normal file
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{-# OPTIONS --safe --warning=error --without-K --guardedness #-}
open import Agda.Primitive using (Level; lzero; lsuc; _⊔_)
open import Setoids.Setoids
open import Rings.Definition
open import Rings.Orders.Partial.Definition
open import Rings.Orders.Total.Definition
open import Groups.Definition
open import Sets.EquivalenceRelations
open import Sequences
open import Setoids.Orders
open import Functions
open import LogicalFormulae
open import Numbers.Naturals.Semiring
open import Numbers.Naturals.Order
module Rings.Orders.Total.Cauchy {m n o : _} {A : Set m} {S : Setoid {m} {n} A} {_+_ : A → A → A} {_*_ : A → A → A} {_<_ : Rel {m} {o} A} {pOrder : SetoidPartialOrder S _<_} {R : Ring S _+_ _*_} {pRing : PartiallyOrderedRing R pOrder} (order : TotallyOrderedRing pRing) where
open Setoid S
open SetoidTotalOrder (TotallyOrderedRing.total order)
open SetoidPartialOrder pOrder
open Equivalence eq
open TotallyOrderedRing order
open Ring R
open Group additiveGroup
open import Rings.Orders.Total.Lemmas order
cauchy : Sequence A → Set (m ⊔ o)
cauchy s = ∀ (ε : A) → (0R < ε) → Sg ℕ (λ N → ∀ {m n : ℕ} → (N <N m) → (N <N n) → abs ((index s m) -R (index s n)) < ε)

12
Rings/Orders/Total/Lemmas.agda
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@@ -363,3 +363,15 @@ fromNPreservesOrder : (0R ∼ 1R → False) → {a b : ℕ} → (a <N b) → (fr
fromNPreservesOrder 0!=1 {zero} {succ zero} a<b = fromNIncreasing 0!=1 0
fromNPreservesOrder 0!=1 {zero} {succ (succ b)} a<b = <Transitive (fromNPreservesOrder 0!=1 (succIsPositive b)) (fromNIncreasing 0!=1 (succ b))
fromNPreservesOrder 0!=1 {succ a} {succ b} a<b = <WellDefined groupIsAbelian groupIsAbelian (orderRespectsAddition (fromNPreservesOrder 0!=1 (canRemoveSuccFrom<N a<b)) 1R)
reciprocalPositive : (a b : A) → .(0<a : 0R < a) → (a * b ∼ 1R) → 0R < b
reciprocalPositive a 1/a 0<a ab=1 with totality 0G 1/a
... | inl (inl x) = x
... | inl (inr x) = exFalso (1<0False (<WellDefined (transitive *Commutative ab=1) timesZero' (ringCanMultiplyByPositive 0<a x)))
... | inr x = exFalso (anyComparisonImpliesNontrivial 0<a (transitive (transitive (symmetric timesZero) (*WellDefined reflexive x)) ab=1))
reciprocal<1 : (a b : A) → .(1<a : 1R < a) → (a * b ∼ 1R) → b < 1R
reciprocal<1 a b 0<a ab=1 with totality b 1R
... | inl (inl x) = x
... | inr b=1 = exFalso (irreflexive (<WellDefined (symmetric ab=1) (transitive (symmetric identIsIdent) (transitive *Commutative ((*WellDefined reflexive (symmetric b=1))))) 0<a))
... | inl (inr x) = exFalso (irreflexive (<WellDefined identIsIdent ab=1 (ringMultiplyPositives (0<1 (anyComparisonImpliesNontrivial 0<a)) (0<1 (anyComparisonImpliesNontrivial 0<a)) 0<a x)))