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Reals form a ring (#59)

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Patrick Stevens
2019-11-02 08:34:09 +00:00
committed by GitHub
parent df1ac6a303
commit a66080b8ae
22 changed files with 155 additions and 51 deletions
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7
Fields/CauchyCompletion/Field.agda
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@@ -45,8 +45,11 @@ boundedMap a | inl (inl x) = underlying (allInvertible a λ pr → irreflexive (
boundedMap a | inl (inr x) = underlying (allInvertible a λ pr → irreflexive (<WellDefined pr (Equivalence.reflexive eq) x))
boundedMap a | inr x = Ring.1R R
aNonzeroImpliesBounded : (a : CauchyCompletion) → (Setoid._∼_ cauchyCompletionSetoid a (injection 0G) → False) → (a <C (injection 0G)) || (injection 0G) <C a
aNonzeroImpliesBounded a a!=0 = ?
-- TODO: make a real which is equivalent by approximating from above;
-- make a real which is equivalent by approximating from below.
-- Use not-zero to show that one of those sequences must pass 0 at some point.
aNonzeroImpliesBounded : (a : CauchyCompletion) → (Setoid._∼_ cauchyCompletionSetoid a (injection 0G) → False) → (a <Cr 0G) || 0G r<C a
aNonzeroImpliesBounded a a!=0 = {!!}
1/aConverges : (a : CauchyCompletion) → (Setoid._∼_ cauchyCompletionSetoid a (injection 0G) → False) → cauchy (map boundedMap (CauchyCompletion.elts a))
1/aConverges a a!=0 e 0<e = {!!}