Move towards base-n expansions (#112)

This commit is contained in:
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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@@ -34,11 +34,11 @@ open import Rings.Orders.Partial.Lemmas oR
open import Rings.Lemmas R
open import Groups.Lemmas additiveGroup
1/nPositive : (n : ) (charNotN : (fromN (succ n) 0R) False) 0R < underlying (allInvertible _ charNotN)
1/nPositive 0 nNot0 with allInvertible (fromN 1) nNot0
... | 1/1 , pr1 = <WellDefined reflexive (transitive (symmetric pr1) (transitive (transitive (*WellDefined reflexive identRight) *Commutative) identIsIdent)) (0<1 λ i nNot0 (transitive identRight (symmetric i)))
1/nPositive (succ n) nNot0 with allInvertible (fromN (succ (succ n))) nNot0
... | 1/n , pr1/n with totality 0R 1/n
... | inr x = exFalso (nNot0 (oneZeroImpliesAllZero (transitive (symmetric (transitive (*WellDefined (symmetric x) reflexive) timesZero')) pr1/n)))
... | inl (inl x) = x
... | inl (inr x) = exFalso (1<0False (<WellDefined pr1/n timesZero' (ringCanMultiplyByPositive (fromNPreservesOrder (anyComparisonImpliesNontrivial x) (succIsPositive (succ n))) x)))
charNotN : (n : ) fromN (succ n) 0R False
charNotN n pr = irreflexive (<WellDefined reflexive pr t)
where
t : 0R < fromN (succ n)
t = fromNPreservesOrder (Field.nontrivial F) (succIsPositive n)
charNot2 : Setoid.__ S ((Ring.1R R) + (Ring.1R R)) (Ring.0R R) False
charNot2 pr = charNotN 1 (transitive (transitive +Associative identRight) pr)