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Split partial and total order of rings (#61)

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Patrick Stevens
2019-11-02 18:42:37 +00:00
committed by GitHub
parent 55995ea801
commit 763ddb8dbb
26 changed files with 768 additions and 618 deletions
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18
Fields/CauchyCompletion/EquivalentMonotone.agda
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@@ -67,6 +67,24 @@ decreasingHalving N with halve charNot2 1R
decreasingHalving N | 1/2 , pr1/2 with (halfLess 1/2 1R (0<1 (charNot2ImpliesNontrivial charNot2)) pr1/2)
... | 1/2<1 = <WellDefined (Equivalence.transitive eq (identityOfIndiscernablesLeft _∼_ (Equivalence.reflexive eq) (equalityCommutative (mapAndIndex (halvingSequence 1R) (_*_ 1/2) N))) (Equivalence.symmetric eq (halvingSequenceMultiple 1/2 {N}))) identIsIdent (ringCanMultiplyByPositive {c = index (halvingSequence 1R) N} (halvingSequencePositive N) 1/2<1)
imageOfN : ℕ → A
imageOfN zero = 0R
imageOfN (succ x) = 1R + imageOfN x
nextImageOfN : (a : A) → 0R < a → ℕ
nextImageOfN a 0<a = ?
halvingToZero : (a : A) → (0G < a) → Sg ℕ (λ N → (index (halvingSequence 1R) N) < a)
halvingToZero a 0<a with SetoidTotalOrder.totality tOrder a 1R
halvingToZero a 0<a | inl (inl a<1) = {!!}
halvingToZero a 0<a | inl (inr 1<a) = 0 , 1<a
halvingToZero a 0<a | inr a=1 with halve charNot2 1R
... | 1/2 , pr1/2 = 1 , <WellDefined ans (Equivalence.symmetric eq a=1) (halfLess 1/2 1R (0<1 (charNot2ImpliesNontrivial charNot2)) pr1/2)
where
ans : 1/2 ∼ Sequence.head (Sequence.tail (halvingSequence 1R))
ans with halve charNot2 1R
ans | 1/2' , pr1/2' = halvesEqual charNot2 1/2 1/2' pr1/2 pr1/2'
halvesCauchy : cauchy (halvingSequence 1R)
halvesCauchy e 0<e = {!!}