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Lots of refactoring towards partially-ordered ring R (#109)

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Patrick Stevens
2020-04-10 19:00:57 +01:00
committed by GitHub
parent 1cff95c652
commit 412edaf4c7
19 changed files with 1015 additions and 778 deletions
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45
Fields/CauchyCompletion/Ring.agda
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@@ -14,6 +14,7 @@ open import Functions
open import LogicalFormulae
open import Numbers.Naturals.Semiring
open import Numbers.Naturals.Order
open import Rings.Homomorphisms.Definition
module Fields.CauchyCompletion.Ring {m n o : _} {A : Set m} {S : Setoid {m} {n} A} {_+_ : A → A → A} {_*_ : A → A → A} {_<_ : Rel {m} {o} A} {pOrder : SetoidPartialOrder S _<_} {R : Ring S _+_ _*_} {pRing : PartiallyOrderedRing R pOrder} (order : TotallyOrderedRing pRing) (F : Field R) (charNot2 : Setoid._∼_ S ((Ring.1R R) + (Ring.1R R)) (Ring.0R R) → False) where
@@ -32,23 +33,25 @@ open import Fields.CauchyCompletion.Addition order F charNot2
open import Fields.CauchyCompletion.Setoid order F charNot2
open import Fields.CauchyCompletion.Group order F charNot2
c*Assoc : {a b c : CauchyCompletion} → Setoid._∼_ cauchyCompletionSetoid (a *C (b *C c)) ((a *C b) *C c)
c*Assoc {a} {b} {c} ε 0<e = 0 , ans
where
ans : {m : ℕ} → 0 <N m → abs (index (CauchyCompletion.elts ((a *C (b *C c)) +C (-C ((a *C b) *C c)))) m) < ε
ans {m} 0<m rewrite indexAndApply (CauchyCompletion.elts (a *C (b *C c))) (CauchyCompletion.elts (-C ((a *C b) *C c))) _+_ {m} | indexAndApply (CauchyCompletion.elts a) (apply _*_ (CauchyCompletion.elts b) (CauchyCompletion.elts c)) _*_ {m} | equalityCommutative (mapAndIndex (apply _*_ (apply _*_ (CauchyCompletion.elts a) (CauchyCompletion.elts b)) (CauchyCompletion.elts c)) inverse m) | indexAndApply (CauchyCompletion.elts b) (CauchyCompletion.elts c) _*_ {m} | indexAndApply (apply _*_ (CauchyCompletion.elts a) (CauchyCompletion.elts b)) (CauchyCompletion.elts c) _*_ {m} | indexAndApply (CauchyCompletion.elts a) (CauchyCompletion.elts b) _*_ {m} = <WellDefined (Equivalence.symmetric eq (Equivalence.transitive eq (absWellDefined _ _ (transferToRight'' (Ring.additiveGroup R) (Ring.*Associative R))) (identityOfIndiscernablesRight _∼_ (Equivalence.reflexive eq) absZero))) (Equivalence.reflexive eq) 0<e
private
abstract
c*Assoc : {a b c : CauchyCompletion} → Setoid._∼_ cauchyCompletionSetoid (a *C (b *C c)) ((a *C b) *C c)
c*Assoc {a} {b} {c} ε 0<e = 0 , ans
where
ans : {m : ℕ} → 0 <N m → abs (index (CauchyCompletion.elts ((a *C (b *C c)) +C (-C ((a *C b) *C c)))) m) < ε
ans {m} 0<m rewrite indexAndApply (CauchyCompletion.elts (a *C (b *C c))) (CauchyCompletion.elts (-C ((a *C b) *C c))) _+_ {m} | indexAndApply (CauchyCompletion.elts a) (apply _*_ (CauchyCompletion.elts b) (CauchyCompletion.elts c)) _*_ {m} | equalityCommutative (mapAndIndex (apply _*_ (apply _*_ (CauchyCompletion.elts a) (CauchyCompletion.elts b)) (CauchyCompletion.elts c)) inverse m) | indexAndApply (CauchyCompletion.elts b) (CauchyCompletion.elts c) _*_ {m} | indexAndApply (apply _*_ (CauchyCompletion.elts a) (CauchyCompletion.elts b)) (CauchyCompletion.elts c) _*_ {m} | indexAndApply (CauchyCompletion.elts a) (CauchyCompletion.elts b) _*_ {m} = <WellDefined (Equivalence.symmetric eq (Equivalence.transitive eq (absWellDefined _ _ (transferToRight'' (Ring.additiveGroup R) (Ring.*Associative R))) (identityOfIndiscernablesRight _∼_ (Equivalence.reflexive eq) absZero))) (Equivalence.reflexive eq) 0<e
c*Ident : {a : CauchyCompletion} → Setoid._∼_ cauchyCompletionSetoid (injection (Ring.1R R) *C a) a
c*Ident {a} ε 0<e = 0 , ans
where
ans : {m : ℕ} → 0 <N m → abs (index (apply _+_ (CauchyCompletion.elts (injection (Ring.1R R) *C a)) (map inverse (CauchyCompletion.elts a))) m) < ε
ans {m} 0<m rewrite indexAndApply (CauchyCompletion.elts (injection (Ring.1R R) *C a)) (map inverse (CauchyCompletion.elts a)) _+_ {m} | indexAndApply (constSequence (Ring.1R R)) (CauchyCompletion.elts a) _*_ {m} | equalityCommutative (mapAndIndex (CauchyCompletion.elts a) inverse m) | indexAndConst (Ring.1R R) m = <WellDefined (Equivalence.symmetric eq (Equivalence.transitive eq (absWellDefined _ _ (transferToRight'' (Ring.additiveGroup R) (Ring.identIsIdent R))) (identityOfIndiscernablesRight _∼_ (Equivalence.reflexive eq) (absZero)))) (Equivalence.reflexive eq) 0<e
c*Ident : {a : CauchyCompletion} → Setoid._∼_ cauchyCompletionSetoid (injection (Ring.1R R) *C a) a
c*Ident {a} ε 0<e = 0 , ans
where
ans : {m : ℕ} → 0 <N m → abs (index (apply _+_ (CauchyCompletion.elts (injection (Ring.1R R) *C a)) (map inverse (CauchyCompletion.elts a))) m) < ε
ans {m} 0<m rewrite indexAndApply (CauchyCompletion.elts (injection (Ring.1R R) *C a)) (map inverse (CauchyCompletion.elts a)) _+_ {m} | indexAndApply (constSequence (Ring.1R R)) (CauchyCompletion.elts a) _*_ {m} | equalityCommutative (mapAndIndex (CauchyCompletion.elts a) inverse m) | indexAndConst (Ring.1R R) m = <WellDefined (Equivalence.symmetric eq (Equivalence.transitive eq (absWellDefined _ _ (transferToRight'' (Ring.additiveGroup R) (Ring.identIsIdent R))) (identityOfIndiscernablesRight _∼_ (Equivalence.reflexive eq) (absZero)))) (Equivalence.reflexive eq) 0<e
*CDistribute : {a b c : CauchyCompletion} → Setoid._∼_ cauchyCompletionSetoid (a *C (b +C c)) ((a *C b) +C (a *C c))
*CDistribute {a} {b} {c} e 0<e = 0 , ans
where
ans : {m : ℕ} → 0 <N m → abs (index (apply _+_ (CauchyCompletion.elts (a *C (b +C c))) (map inverse (CauchyCompletion.elts ((a *C b) +C (a *C c))))) m) < e
ans {m} N<m rewrite indexAndApply (CauchyCompletion.elts (a *C (b +C c))) (map inverse (CauchyCompletion.elts ((a *C b) +C (a *C c)))) _+_ {m} | indexAndApply (CauchyCompletion.elts a) (apply _+_ (CauchyCompletion.elts b) (CauchyCompletion.elts c)) _*_ {m} | equalityCommutative (mapAndIndex (apply _+_ (apply _*_ (CauchyCompletion.elts a) (CauchyCompletion.elts b)) (apply _*_ (CauchyCompletion.elts a) (CauchyCompletion.elts c))) inverse m) | indexAndApply (CauchyCompletion.elts b) (CauchyCompletion.elts c) _+_ {m} | indexAndApply (apply _*_ (CauchyCompletion.elts a) (CauchyCompletion.elts b)) (apply _*_ (CauchyCompletion.elts a) (CauchyCompletion.elts c)) _+_ {m} | indexAndApply (CauchyCompletion.elts a) (CauchyCompletion.elts b) _*_ {m} | indexAndApply (CauchyCompletion.elts a) (CauchyCompletion.elts c) _*_ {m} = <WellDefined (Equivalence.symmetric eq (Equivalence.transitive eq (absWellDefined _ _ (transferToRight'' (Ring.additiveGroup R) (Ring.*DistributesOver+ R))) (absZeroIsZero))) (Equivalence.reflexive eq) 0<e
*CDistribute : {a b c : CauchyCompletion} → Setoid._∼_ cauchyCompletionSetoid (a *C (b +C c)) ((a *C b) +C (a *C c))
*CDistribute {a} {b} {c} e 0<e = 0 , ans
where
ans : {m : ℕ} → 0 <N m → abs (index (apply _+_ (CauchyCompletion.elts (a *C (b +C c))) (map inverse (CauchyCompletion.elts ((a *C b) +C (a *C c))))) m) < e
ans {m} N<m rewrite indexAndApply (CauchyCompletion.elts (a *C (b +C c))) (map inverse (CauchyCompletion.elts ((a *C b) +C (a *C c)))) _+_ {m} | indexAndApply (CauchyCompletion.elts a) (apply _+_ (CauchyCompletion.elts b) (CauchyCompletion.elts c)) _*_ {m} | equalityCommutative (mapAndIndex (apply _+_ (apply _*_ (CauchyCompletion.elts a) (CauchyCompletion.elts b)) (apply _*_ (CauchyCompletion.elts a) (CauchyCompletion.elts c))) inverse m) | indexAndApply (CauchyCompletion.elts b) (CauchyCompletion.elts c) _+_ {m} | indexAndApply (apply _*_ (CauchyCompletion.elts a) (CauchyCompletion.elts b)) (apply _*_ (CauchyCompletion.elts a) (CauchyCompletion.elts c)) _+_ {m} | indexAndApply (CauchyCompletion.elts a) (CauchyCompletion.elts b) _*_ {m} | indexAndApply (CauchyCompletion.elts a) (CauchyCompletion.elts c) _*_ {m} = <WellDefined (Equivalence.symmetric eq (Equivalence.transitive eq (absWellDefined _ _ (transferToRight'' (Ring.additiveGroup R) (Ring.*DistributesOver+ R))) (absZeroIsZero))) (Equivalence.reflexive eq) 0<e
CRing : Ring cauchyCompletionSetoid _+C_ _*C_
Ring.additiveGroup CRing = CGroup
@@ -59,3 +62,15 @@ Ring.*Associative CRing {a} {b} {c} = c*Assoc {a} {b} {c}
Ring.*Commutative CRing {a} {b} = *CCommutative {a} {b}
Ring.*DistributesOver+ CRing {a} {b} {c} = *CDistribute {a} {b} {c}
Ring.identIsIdent CRing {a} = c*Ident {a}
private
injectionIsRingHom : (a b : A) → Setoid._∼_ cauchyCompletionSetoid (injection (a * b)) (injection a *C injection b)
injectionIsRingHom a b ε 0<e = 0 , ans
where
ans : {m : ℕ} → 0 <N m → abs (index (apply _+_ (CauchyCompletion.elts (injection (a * b))) (map inverse (CauchyCompletion.elts (injection a *C injection b)))) m) < ε
ans {m} 0<m rewrite indexAndApply (constSequence (a * b)) (map inverse (apply _*_ (constSequence a) (constSequence b))) _+_ {m} | indexAndConst (a * b) m | equalityCommutative (mapAndIndex (apply _*_ (constSequence a) (constSequence b)) inverse m) | indexAndApply (constSequence a) (constSequence b) _*_ {m} | indexAndConst a m | indexAndConst b m = <WellDefined (symmetric (transitive (absWellDefined _ _ invRight) absZeroIsZero)) reflexive 0<e
CInjectionRingHom : RingHom R CRing injection
RingHom.preserves1 CInjectionRingHom = Equivalence.reflexive (Setoid.eq cauchyCompletionSetoid) {injection (Ring.1R R)}
RingHom.ringHom CInjectionRingHom {a} {b} = injectionIsRingHom a b
RingHom.groupHom CInjectionRingHom = CInjectionGroupHom