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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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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)))