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Reals form a ring (#59)
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Patrick Stevens
GitHub
102
Fields/CauchyCompletion/EquivalentMonotone.agda
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102
Fields/CauchyCompletion/EquivalentMonotone.agda
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{-# OPTIONS --safe --warning=error --without-K --guardedness #-}
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open import Agda.Primitive using (Level; lzero; lsuc; _⊔_)
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open import Setoids.Setoids
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open import Rings.Definition
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open import Rings.Lemmas
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open import Rings.Orders.Definition
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open import Groups.Definition
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open import Groups.Groups
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open import Fields.Fields
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open import Sets.EquivalenceRelations
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open import Sequences
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open import Setoids.Orders
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open import Functions
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open import LogicalFormulae
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open import Numbers.Naturals.Naturals
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module Fields.CauchyCompletion.EquivalentMonotone {m n o : _} {A : Set m} {S : Setoid {m} {n} A} {_+_ : A → A → A} {_*_ : A → A → A} {_<_ : Rel {m} {o} A} {pOrder : SetoidPartialOrder S _<_} {tOrder : SetoidTotalOrder {_<_ = _<_} pOrder} {R : Ring S _+_ _*_} (order : OrderedRing R tOrder) (F : Field R) (charNot2 : Setoid._∼_ S ((Ring.1R R) + (Ring.1R R)) (Ring.0R R) → False) where
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open Setoid S
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open SetoidTotalOrder tOrder
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open SetoidPartialOrder pOrder
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open Equivalence eq
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open OrderedRing order
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open Field F
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open Group (Ring.additiveGroup R)
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open Ring R
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open import Fields.Lemmas F
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open import Fields.Orders.Lemmas {F = F} record { oRing = order }
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open import Rings.Orders.Lemmas(order)
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open import Fields.CauchyCompletion.Definition order F
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open import Fields.CauchyCompletion.Multiplication order F charNot2
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open import Fields.CauchyCompletion.Addition order F charNot2
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open import Fields.CauchyCompletion.Setoid order F charNot2
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open import Fields.CauchyCompletion.Group order F charNot2
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open import Fields.CauchyCompletion.Ring order F charNot2
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open import Fields.CauchyCompletion.Comparison order F charNot2
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open import Fields.CauchyCompletion.Approximation order F charNot2
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halvingSequence : (start : A) → Sequence A
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Sequence.head (halvingSequence start) = start
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Sequence.tail (halvingSequence start) with halve charNot2 start
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Sequence.tail (halvingSequence start) | start/2 , _ = halvingSequence start/2
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halvingSequenceMultiple : (start : A) → {n : ℕ} → index (halvingSequence start) n ∼ index (map (start *_) (halvingSequence (Ring.1R R))) n
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halvingSequenceMultiple start {zero} = Equivalence.symmetric eq (Equivalence.transitive eq *Commutative identIsIdent)
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halvingSequenceMultiple start {succ n} with halve charNot2 start
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... | start/2 , _ with allInvertible (1R + 1R) charNot2
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halvingSequenceMultiple start {succ n} | start/2 , b | 1/2 , pr1/2 rewrite equalityCommutative (mapAndIndex (halvingSequence (1R * 1/2)) (_*_ start) n) = Equivalence.transitive eq (halvingSequenceMultiple start/2 {n}) f
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where
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g : (start * (1/2 * index (halvingSequence 1R) n)) ∼ (start * index (map (_*_ (1R * 1/2)) (halvingSequence 1R)) n)
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g rewrite equalityCommutative (mapAndIndex (halvingSequence 1R) (_*_ (1R * 1/2)) n) = *WellDefined (Equivalence.reflexive eq) (*WellDefined (Equivalence.symmetric eq identIsIdent) (Equivalence.reflexive eq))
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f : index (map (_*_ start/2) (halvingSequence 1R)) n ∼ (start * index (halvingSequence (1R * 1/2)) n)
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f rewrite equalityCommutative (mapAndIndex (halvingSequence 1R) (_*_ start/2) n) = Equivalence.transitive eq (Equivalence.transitive eq (Equivalence.transitive eq (*WellDefined (Equivalence.symmetric eq (Equivalence.transitive eq (*WellDefined (Equivalence.symmetric eq b) (Equivalence.reflexive eq)) (halfHalves 1/2 (Equivalence.transitive eq (Equivalence.transitive eq (+WellDefined (Equivalence.symmetric eq (Equivalence.transitive eq *Commutative identIsIdent)) (Equivalence.symmetric eq (Equivalence.transitive eq *Commutative identIsIdent))) (Equivalence.symmetric eq *DistributesOver+)) pr1/2)))) (Equivalence.reflexive eq)) (Equivalence.symmetric eq *Associative)) g) (Equivalence.symmetric eq (*WellDefined (Equivalence.reflexive eq) (halvingSequenceMultiple (1R * 1/2) {n})))
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Decreasing : Sequence A → Set o
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Decreasing seq = ∀ (N : ℕ) → (index seq (succ N)) < (index seq N)
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halvingSequencePositive : (n : ℕ) → 0G < index (halvingSequence 1R) n
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halvingSequencePositive zero = 0<1 (charNot2ImpliesNontrivial charNot2)
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halvingSequencePositive (succ n) with halve charNot2 1R
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halvingSequencePositive (succ n) | 1/2 , pr1/2 = <WellDefined (Equivalence.reflexive eq) (Equivalence.symmetric eq (Equivalence.transitive eq (halvingSequenceMultiple 1/2 {n}) (Equivalence.transitive eq (identityOfIndiscernablesLeft _∼_ (Equivalence.reflexive eq) (mapAndIndex (halvingSequence 1R) (_*_ 1/2) n)) *Commutative))) (orderRespectsMultiplication (halvingSequencePositive n) (halvePositive 1/2 (<WellDefined (Equivalence.reflexive eq) (Equivalence.symmetric eq pr1/2) (0<1 (charNot2ImpliesNontrivial charNot2)))))
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decreasingHalving : Decreasing (halvingSequence 1R)
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decreasingHalving N with halve charNot2 1R
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decreasingHalving N | 1/2 , pr1/2 with (halfLess 1/2 1R (0<1 (charNot2ImpliesNontrivial charNot2)) pr1/2)
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... | 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)
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halvesCauchy : cauchy (halvingSequence 1R)
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halvesCauchy e 0<e = {!!}
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multipleOfCauchyIsCauchy : (mult : A) → (seq : Sequence A) → cauchy seq → cauchy (map (mult *_) seq)
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multipleOfCauchyIsCauchy mult seq seqCauchy e 0<e with SetoidTotalOrder.totality tOrder 0R mult
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multipleOfCauchyIsCauchy mult seq seqCauchy e 0<e | inl (inl x) with allInvertible mult λ pr → irreflexive (<WellDefined (Equivalence.reflexive eq) pr x)
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... | 1/mult , pr1/mult = {!!}
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multipleOfCauchyIsCauchy mult seq seqCauchy e 0<e | inl (inr x) with allInvertible mult λ pr → irreflexive (<WellDefined pr (Equivalence.reflexive eq) x)
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... | 1/mult , pr1/mult = {!!}
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multipleOfCauchyIsCauchy mult seq seqCauchy e 0<e | inr 0=mult = 0 , ans
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where
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ans : {m n : ℕ} → (0 <N m) → (0 <N n) → abs (index (map (_*_ mult) seq) m + inverse (index (map (_*_ mult) seq) n)) < e
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ans {m} {n} _ _ rewrite equalityCommutative (mapAndIndex seq (_*_ mult) m) | equalityCommutative (mapAndIndex seq (_*_ mult) n) = <WellDefined {!!} (Equivalence.reflexive eq) 0<e
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halvingSequenceCauchy : (start : A) → cauchy (halvingSequence start)
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halvingSequenceCauchy start = {!!}
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sequenceAllAbove : (a : CauchyCompletion) → Sequence A
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sequenceAllAbove a = go (Ring.1R R) (0<1 (charNot2ImpliesNontrivial charNot2))
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where
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go : (e : A) → (0G < e) → Sequence A
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Sequence.head (go e 0<e) = rationalApproximatelyAbove a e 0<e
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Sequence.tail (go e 0<e) with halve charNot2 e
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... | e/2 , prE/2 = go e/2 (halvePositive e/2 (<WellDefined (Equivalence.reflexive eq) (Equivalence.symmetric eq prE/2) 0<e))
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sequenceAllAboveCauchy : (a : CauchyCompletion) → cauchy (sequenceAllAbove a)
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sequenceAllAboveCauchy a e 0<e = {!!}
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-- find N such that 1/2^N < e
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-- show that this N is enough
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-- show that sequenceAllAbove ∼ a
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-- monotonify sequenceAllAbove
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-- show that monotonify of a sequence which is all above its limit still converges to that limit
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