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Base expansions do approximate the number (#114)

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Patrick Stevens
2020-04-13 11:43:35 +01:00
committed by GitHub
parent 269f2aa14f
commit 46fcb023d4
3 changed files with 136 additions and 7 deletions
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14
Rings/Orders/Total/Lemmas.agda
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@@ -13,6 +13,7 @@ open import Rings.Orders.Partial.Definition
open import Numbers.Naturals.Semiring
open import Numbers.Naturals.Order
open import Orders.Total.Definition
open import Rings.IntegralDomains.Definition
module Rings.Orders.Total.Lemmas {n m p : _} {A : Set n} {S : Setoid {n} {m} A} {_+_ : A → A → A} {_*_ : A → A → A} {R : Ring S _+_ _*_} {_<_ : Rel {_} {p} A} {pOrder : SetoidPartialOrder S _<_} {pOrderRing : PartiallyOrderedRing R pOrder} (order : TotallyOrderedRing pOrderRing) where
@@ -382,3 +383,16 @@ 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)))
isIntDom : (nonempty : 1R ∼ 0R → False) → IntegralDomain R
IntegralDomain.intDom (isIntDom n) {a} {b} ab=0 a!=0 with totality 0R b
... | inr 0=b = symmetric 0=b
IntegralDomain.intDom (isIntDom n) {a} {b} ab=0 a!=0 | inl (inl 0<b) with totality 0R a
... | inl (inl x) = exFalso (irreflexive (<WellDefined reflexive ab=0 (orderRespectsMultiplication x 0<b)))
... | inl (inr x) = exFalso (irreflexive (<WellDefined ab=0 timesZero' (ringCanMultiplyByPositive 0<b x)))
... | inr x = exFalso (a!=0 (symmetric x))
IntegralDomain.intDom (isIntDom n) {a} {b} ab=0 a!=0 | inl (inr b<0) with totality 0R a
... | inl (inl x) = exFalso (irreflexive (<WellDefined (transitive *Commutative ab=0) timesZero' (ringCanMultiplyByPositive x b<0)))
... | inl (inr x) = exFalso (irreflexive (<WellDefined reflexive (transitive twoNegativesTimes ab=0) (orderRespectsMultiplication (lemm2 a x) (lemm2 b b<0))))
... | inr x = exFalso (a!=0 (symmetric x))
IntegralDomain.nontrivial (isIntDom n) = n