mirror of
https://github.com/Smaug123/agdaproofs
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63 lines
5.4 KiB
Agda
63 lines
5.4 KiB
Agda
{-# 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.Order
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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.Addition {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 Ring R
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open Group additiveGroup
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open Field F
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open import Fields.CauchyCompletion.Definition order F
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open import Rings.Orders.Lemmas(order)
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halve : (a : A) → Sg A (λ i → i + i ∼ a)
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-- TODO: we need to know the characteristic already
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halve a with allInvertible (1R + 1R) charNot2
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... | 1/2 , pr1/2 = (a * 1/2) , Equivalence.transitive eq (+WellDefined *Commutative *Commutative) (Equivalence.transitive eq (Equivalence.symmetric eq (*DistributesOver+ {1/2} {a} {a})) (Equivalence.transitive eq (*WellDefined (Equivalence.reflexive eq) r) (Equivalence.transitive eq (*Associative) (Equivalence.transitive eq (*WellDefined pr1/2 (Equivalence.reflexive eq)) identIsIdent))))
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where
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r : a + a ∼ (1R + 1R) * a
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r = Equivalence.symmetric eq (Equivalence.transitive eq *Commutative (Equivalence.transitive eq *DistributesOver+ (+WellDefined (Equivalence.transitive eq *Commutative identIsIdent) (Equivalence.transitive eq *Commutative identIsIdent))))
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halvePositive : (a : A) → 0R < (a + a) → 0R < a
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halvePositive a 0<2a with totality 0R a
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halvePositive a 0<2a | inl (inl x) = x
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halvePositive a 0<2a | inl (inr a<0) = exFalso (irreflexive {a + a} (SetoidPartialOrder.transitive pOrder (<WellDefined (Equivalence.reflexive eq) identRight (ringAddInequalities a<0 a<0)) 0<2a))
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halvePositive a 0<2a | inr x = exFalso (irreflexive {0G} (<WellDefined (Equivalence.reflexive eq) (Equivalence.transitive eq (+WellDefined (Equivalence.symmetric eq x) (Equivalence.symmetric eq x)) identRight) 0<2a))
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lemm : (m : ℕ) (a b : Sequence A) → index (apply _+_ a b) m ≡ (index a m) + (index b m)
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lemm zero a b = refl
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lemm (succ m) a b = lemm m (Sequence.tail a) (Sequence.tail b)
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_+C_ : CauchyCompletion → CauchyCompletion → CauchyCompletion
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CauchyCompletion.elts (record { elts = a ; converges = convA } +C record { elts = b ; converges = convB }) = apply _+_ a b
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CauchyCompletion.converges (record { elts = a ; converges = convA } +C record { elts = b ; converges = convB }) ε 0<e with halve ε
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... | e/2 , e/2Pr with convA e/2 (halvePositive e/2 (<WellDefined (Equivalence.reflexive eq) (Equivalence.symmetric eq e/2Pr) 0<e))
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CauchyCompletion.converges (record { elts = a ; converges = convA } +C record { elts = b ; converges = convB }) ε 0<e | e/2 , e/2Pr | Na , prA with convB e/2 (halvePositive e/2 (<WellDefined (Equivalence.reflexive eq) (Equivalence.symmetric eq e/2Pr) 0<e))
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CauchyCompletion.converges (record { elts = a ; converges = convA } +C record { elts = b ; converges = convB }) ε 0<e | e/2 , e/2Pr | Na , prA | Nb , prB = (Na +N Nb) , t
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where
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t : {m n : ℕ} → Na +N Nb <N m → Na +N Nb <N n → abs ((index (apply _+_ a b) m) + inverse (index (apply _+_ a b) n)) < ε
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t {m} {n} <m <n with prA {m} {n} (inequalityShrinkLeft <m) (inequalityShrinkLeft <n)
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... | am-an<e/2 with prB {m} {n} (inequalityShrinkRight <m) (inequalityShrinkRight <n)
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... | bm-bn<e/2 with triangleInequality (index a m + inverse (index a n)) (index b m + inverse (index b n))
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... | inl tri rewrite lemm m a b | lemm n a b = SetoidPartialOrder.<WellDefined pOrder (Equivalence.reflexive eq) e/2Pr (SetoidPartialOrder.transitive pOrder {_} {(abs ((index a m) + (inverse (index a n)))) + (abs ((index b m) + (inverse (index b n))))} (<WellDefined (absWellDefined _ _ (Equivalence.transitive eq (Equivalence.symmetric eq (+Associative {index a m})) (Equivalence.transitive eq (+WellDefined (Equivalence.reflexive eq {index a m}) (Equivalence.transitive eq groupIsAbelian (Equivalence.transitive eq (Equivalence.symmetric eq (+Associative {index b m})) (+WellDefined (Equivalence.reflexive eq {index b m}) (Equivalence.symmetric eq (invContravariant additiveGroup)))))) (+Associative {index a m})))) (Equivalence.reflexive eq) tri) (ringAddInequalities am-an<e/2 bm-bn<e/2))
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... | inr tri rewrite lemm m a b | lemm n a b = SetoidPartialOrder.<WellDefined pOrder (Equivalence.reflexive eq) e/2Pr (<WellDefined (Equivalence.transitive eq (Equivalence.symmetric eq tri) (absWellDefined _ _ (Equivalence.transitive eq (Equivalence.symmetric eq (+Associative {index a m})) (Equivalence.transitive eq (+WellDefined (Equivalence.reflexive eq {index a m}) (Equivalence.transitive eq groupIsAbelian (Equivalence.transitive eq (Equivalence.symmetric eq (+Associative {index b m})) (+WellDefined (Equivalence.reflexive eq {index b m}) (Equivalence.symmetric eq (invContravariant additiveGroup)))))) (+Associative {index a m}))))) (Equivalence.reflexive eq) (ringAddInequalities am-an<e/2 bm-bn<e/2))
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