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Move Bool to live on its own (#118)
This commit is contained in:
Patrick Stevens
GitHub
@@ -17,7 +17,7 @@ open import Numbers.Naturals.Semiring
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open import Numbers.Naturals.Order
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open import Fields.Orders.Total.Archimedean
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module Fields.CauchyCompletion.Archimedean {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) (arch : ArchimedeanField {F = F} record { oRing = order }) where
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module Fields.CauchyCompletion.Archimedean {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) (arch : ArchimedeanField {F = F} (record { oRing = pRing })) where
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open import Fields.CauchyCompletion.Group order F
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open import Fields.CauchyCompletion.Ring order F
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@@ -30,6 +30,7 @@ open Group (Ring.additiveGroup R)
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open import Rings.Orders.Total.Cauchy order
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open import Rings.Orders.Total.Lemmas order
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open import Rings.Orders.Total.AbsoluteValue order
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open import Fields.CauchyCompletion.Definition order F
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open import Fields.CauchyCompletion.Multiplication order F
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open import Fields.CauchyCompletion.Addition order F
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@@ -107,42 +107,14 @@ private
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ans : (m : ℕ) → (index (CauchyCompletion.elts (injection b +C a)) m + index (map inverse (CauchyCompletion.elts a)) m) ∼ b
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ans m rewrite indexAndApply (CauchyCompletion.elts (injection b)) (CauchyCompletion.elts a) _+_ {m} | indexAndConst b m | equalityCommutative (mapAndIndex (CauchyCompletion.elts a) inverse m) = transitive (symmetric +Associative) (transitive (+WellDefined reflexive invRight) identRight)
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{-
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minAddLemma : (a b : A) → (0R < a) → (min a b) < (a + b)
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minAddLemma a b 0<a with totality a b
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... | inl (inl x) = <WellDefined identLeft groupIsAbelian (orderRespectsAddition (<Transitive 0<a x) a)
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... | inl (inr x) = <WellDefined identLeft reflexive (orderRespectsAddition 0<a b)
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... | inr x = <WellDefined identLeft groupIsAbelian (orderRespectsAddition (<WellDefined reflexive x 0<a) a)
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addInequalities : {a b : CauchyCompletion} → 0R r<C a → 0R r<C b → 0R r<C (a +C b)
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addInequalities {a} {b} record { e = eA ; 0<e = 0<eA ; N = Na ; pr = prA } record { e = eB ; 0<e = 0<eB ; N = Nb ; pr = prB } = record { e = min eA eB ; 0<e = minInequalitiesR 0<eA 0<eB ; N = N ; pr = λ m N<m → <Transitive (<WellDefined (symmetric identLeft) reflexive (minAddLemma eA eB 0<eA)) (<WellDefined (+WellDefined identLeft identLeft) (identityOfIndiscernablesLeft _∼_ reflexive (indexAndApply (CauchyCompletion.elts a) (CauchyCompletion.elts b) _+_ {m})) (ringAddInequalities (prA m (p2 m N<m)) (prB m (p1 m N<m)))) }
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where
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N = succ (TotalOrder.max ℕTotalOrder Na Nb)
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Nb<N : Nb <N succ (TotalOrder.max ℕTotalOrder Na Nb)
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Nb<N with TotalOrder.totality ℕTotalOrder Na Nb
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... | inl (inl x) = le 0 refl
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... | inl (inr x) = lessTransitive x (le 0 refl)
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... | inr x = le 0 refl
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Na<N : Na <N succ (TotalOrder.max ℕTotalOrder Na Nb)
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Na<N with TotalOrder.totality ℕTotalOrder Na Nb
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... | inl (inl x) = lessTransitive x (le 0 refl)
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... | inl (inr x) = le 0 refl
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... | inr x = le 0 (applyEquality succ x)
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p1 : (m : ℕ) → (N <N m) → Nb <N m
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p1 m N<m = lessTransitive Nb<N N<m
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p2 : (m : ℕ) → (N <N m) → Na <N m
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p2 m N<m = lessTransitive Na<N N<m
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-}
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invInj : (i : A) → Setoid._∼_ cauchyCompletionSetoid (injection (inverse i) +C injection i) (injection 0R)
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invInj i = Equivalence.transitive (Setoid.eq cauchyCompletionSetoid) {record { converges = CauchyCompletion.converges (injection (inverse i) +C injection i) }} {record { converges = CauchyCompletion.converges ((-C (injection i)) +C injection i) }} {record { converges = CauchyCompletion.converges (injection 0R) }} (Group.+WellDefined CGroup {record { converges = CauchyCompletion.converges (injection (inverse i)) }} {record { converges = CauchyCompletion.converges (injection i)}} {record { converges = CauchyCompletion.converges (-C (injection i)) }} {record { converges = CauchyCompletion.converges (injection i) }} homRespectsInverse (Equivalence.reflexive (Setoid.eq cauchyCompletionSetoid) {record { converges = CauchyCompletion.converges (injection i) }})) (Group.invLeft CGroup {injection i})
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invInj i = Equivalence.transitive (Setoid.eq cauchyCompletionSetoid) {record { converges = CauchyCompletion.converges (injection (inverse i) +C injection i) }} {record { converges = CauchyCompletion.converges ((-C (injection i)) +C injection i) }} {record { converges = CauchyCompletion.converges (injection 0R) }} (Group.+WellDefinedLeft CGroup {record { converges = CauchyCompletion.converges (injection (inverse i)) }} {record { converges = CauchyCompletion.converges (injection i)}} {record { converges = CauchyCompletion.converges (-C (injection i)) }} homRespectsInverse) (Group.invLeft CGroup {injection i})
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cOrderMove : (a b : A) (c : CauchyCompletion) → (injection a +C c) <Cr b → a r<C (injection b +C (-C c))
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cOrderMove a b c record { e = e ; 0<e = 0<e ; N = N ; property = property } = record { e = e ; 0<e = 0<e ; N = N ; pr = λ m N<m → <WellDefined (transitive (symmetric +Associative) (transitive (+WellDefined reflexive (transitive (+WellDefined (identityOfIndiscernablesRight _∼_ reflexive (indexAndApply (CauchyCompletion.elts (injection a)) (CauchyCompletion.elts c) _+_ {m})) reflexive) (transitive (symmetric +Associative) (transitive (+WellDefined (identityOfIndiscernablesRight _∼_ reflexive (indexAndConst a m)) invRight) identRight)))) groupIsAbelian)) (transitive (+WellDefined (identityOfIndiscernablesLeft _∼_ reflexive (indexAndConst b m)) (identityOfIndiscernablesRight _∼_ reflexive (mapAndIndex (CauchyCompletion.elts c) inverse m))) (identityOfIndiscernablesLeft _∼_ reflexive (indexAndApply (CauchyCompletion.elts (injection b)) (CauchyCompletion.elts (-C c)) _+_ {m}))) (orderRespectsAddition (property m N<m) (inverse (index (CauchyCompletion.elts c) m))) }
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cOrderMove' : (a b : A) (c : CauchyCompletion) → (injection a +C (-C c)) <Cr b → a r<C (injection b +C c)
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cOrderMove' a b c pr = r<CWellDefinedRight _ _ _ (Group.+WellDefined CGroup {record { converges = CauchyCompletion.converges (injection b) }} {record { converges = CauchyCompletion.converges (-C (-C c)) }} {record { converges = CauchyCompletion.converges (injection b) }} {record {converges = CauchyCompletion.converges c }} (Equivalence.reflexive (Setoid.eq cauchyCompletionSetoid) {record { converges = CauchyCompletion.converges (injection b) }}) (invTwice CGroup record { converges = CauchyCompletion.converges c })) (cOrderMove a b (-C c) pr)
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cOrderMove' a b c pr = r<CWellDefinedRight _ _ _ (Group.+WellDefinedRight CGroup {record { converges = CauchyCompletion.converges (injection b) }} {record { converges = CauchyCompletion.converges (-C (-C c)) }} {record {converges = CauchyCompletion.converges c }} (invTwice CGroup record { converges = CauchyCompletion.converges c })) (cOrderMove a b (-C c) pr)
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cOrderMove'' : (a : CauchyCompletion) (b c : A) → (a +C (-C injection b)) <Cr c → a <Cr (b + c)
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cOrderMove'' a b c record { e = e ; 0<e = 0<e ; N = N ; property = property } = record { e = e ; 0<e = 0<e ; N = N ; property = λ m N<m → <WellDefined (transitive (symmetric +Associative) (+WellDefined reflexive (transitive (+WellDefined (identityOfIndiscernablesRight _∼_ reflexive (indexAndApply (CauchyCompletion.elts a) _ _+_ {m})) reflexive) (transitive (transitive (symmetric +Associative) (+WellDefined reflexive (transitive (+WellDefined (transitive (identityOfIndiscernablesLeft _∼_ reflexive (mapAndIndex _ inverse m)) (inverseWellDefined additiveGroup (identityOfIndiscernablesRight _∼_ reflexive (indexAndConst b m)))) reflexive) invLeft))) identRight)))) groupIsAbelian (orderRespectsAddition (property m N<m) b) }
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@@ -157,7 +129,7 @@ private
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g' : ((injection aboveC +C (-C c)) +C injection a) <C (injection (b-a/2 + a))
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g' = <CWellDefined (Equivalence.reflexive (Setoid.eq cauchyCompletionSetoid) {record { converges = CauchyCompletion.converges ((injection aboveC +C (-C c)) +C injection a) }}) (Equivalence.symmetric (Setoid.eq cauchyCompletionSetoid) {record { converges = CauchyCompletion.converges (injection (b-a/2 + a)) }} {record { converges = CauchyCompletion.converges (injection b-a/2 +C injection a) }} (GroupHom.groupHom CInjectionGroupHom)) g
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t : (injection (a + aboveC) +C (-C c)) <Cr (b-a/2 + a)
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t = <CRelaxR' (<CWellDefined (Equivalence.transitive (Setoid.eq cauchyCompletionSetoid) {record { converges = CauchyCompletion.converges (((injection aboveC) +C (-C c)) +C injection a) }} {record { converges = CauchyCompletion.converges (injection a +C ((injection aboveC) +C (-C c))) }} {record { converges = CauchyCompletion.converges (injection (a + aboveC) +C (-C c)) }} (Ring.groupIsAbelian CRing {record { converges = CauchyCompletion.converges (injection aboveC +C (-C c)) }} {record { converges = CauchyCompletion.converges (injection a) }}) (Equivalence.transitive (Setoid.eq cauchyCompletionSetoid) {record { converges = CauchyCompletion.converges (injection a +C ((injection aboveC) +C (-C c))) }} {record { converges = CauchyCompletion.converges (((injection a) +C (injection aboveC)) +C (-C c)) }} {record { converges = CauchyCompletion.converges ((injection (a + aboveC)) +C (-C c)) }} (Group.+Associative CGroup {record { converges = CauchyCompletion.converges (injection a) }} {record { converges = CauchyCompletion.converges (injection aboveC) }} {record { converges = CauchyCompletion.converges (-C c) }}) (Group.+WellDefined CGroup {record { converges = CauchyCompletion.converges (injection a +C injection aboveC) }} {record { converges = CauchyCompletion.converges (-C c) }} {record { converges = CauchyCompletion.converges (injection (a + aboveC)) }} {record { converges = CauchyCompletion.converges (-C c) }} (Equivalence.symmetric (Setoid.eq cauchyCompletionSetoid) {record { converges = CauchyCompletion.converges (injection (a + aboveC)) }} {record { converges = CauchyCompletion.converges (injection a +C injection aboveC) }} (GroupHom.groupHom CInjectionGroupHom)) (Equivalence.reflexive (Setoid.eq cauchyCompletionSetoid) {record { converges = CauchyCompletion.converges (-C c)}})))) (Equivalence.reflexive (Setoid.eq cauchyCompletionSetoid) {record { converges = CauchyCompletion.converges (injection (b-a/2 + a)) }}) g')
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t = <CRelaxR' (<CWellDefined (Equivalence.transitive (Setoid.eq cauchyCompletionSetoid) {record { converges = CauchyCompletion.converges (((injection aboveC) +C (-C c)) +C injection a) }} {record { converges = CauchyCompletion.converges (injection a +C ((injection aboveC) +C (-C c))) }} {record { converges = CauchyCompletion.converges (injection (a + aboveC) +C (-C c)) }} (Ring.groupIsAbelian CRing {record { converges = CauchyCompletion.converges (injection aboveC +C (-C c)) }} {record { converges = CauchyCompletion.converges (injection a) }}) (Equivalence.transitive (Setoid.eq cauchyCompletionSetoid) {record { converges = CauchyCompletion.converges (injection a +C ((injection aboveC) +C (-C c))) }} {record { converges = CauchyCompletion.converges (((injection a) +C (injection aboveC)) +C (-C c)) }} {record { converges = CauchyCompletion.converges ((injection (a + aboveC)) +C (-C c)) }} (Group.+Associative CGroup {record { converges = CauchyCompletion.converges (injection a) }} {record { converges = CauchyCompletion.converges (injection aboveC) }} {record { converges = CauchyCompletion.converges (-C c) }}) (Group.+WellDefinedLeft CGroup {record { converges = CauchyCompletion.converges (injection a +C injection aboveC) }} {record { converges = CauchyCompletion.converges (-C c) }} {record { converges = CauchyCompletion.converges (injection (a + aboveC)) }} (Equivalence.symmetric (Setoid.eq cauchyCompletionSetoid) {record { converges = CauchyCompletion.converges (injection (a + aboveC)) }} {record { converges = CauchyCompletion.converges (injection a +C injection aboveC) }} (GroupHom.groupHom CInjectionGroupHom))))) (Equivalence.reflexive (Setoid.eq cauchyCompletionSetoid) {record { converges = CauchyCompletion.converges (injection (b-a/2 + a)) }}) g')
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lemm : 0R < b-a/2
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lemm = halvePositive' prDiff (moveInequality a<b)
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u : (b-a/2 + a) < b
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@@ -167,16 +139,7 @@ private
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cOrderRespectsAdditionLeft''Flip a b c a<b = <CWellDefined ((Ring.groupIsAbelian CRing {record { converges = CauchyCompletion.converges (injection a) }} {record { converges = CauchyCompletion.converges c }})) (Ring.groupIsAbelian CRing {record { converges = CauchyCompletion.converges (injection b) }} {record { converges = CauchyCompletion.converges c }}) (cOrderRespectsAdditionLeft'' a b c a<b)
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cOrderRespectsAdditionLeft''' : (a b : CauchyCompletion) (c : A) → (a <C b) → (a +C injection c) <C (b +C injection c)
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cOrderRespectsAdditionLeft''' a b c record { i = i ; a<i = a<i ; i<b = i<b } = <CTransitive (cOrderRespectsAdditionLeft' a i c a<i) (<CWellDefined (Equivalence.transitive (Setoid.eq cauchyCompletionSetoid) {record { converges = CauchyCompletion.converges (injection (c + i)) }} {record { converges = CauchyCompletion.converges (injection c +C injection i) }} {record { converges = CauchyCompletion.converges (injection i +C injection c) }} (GroupHom.groupHom CInjectionGroupHom) (Ring.groupIsAbelian CRing {record { converges = CauchyCompletion.converges (injection c) }} {record { converges = CauchyCompletion.converges (injection i) }})) (Ring.groupIsAbelian CRing {record { converges = CauchyCompletion.converges (injection c) }} {record { converges = CauchyCompletion.converges b }}) (flip<C' {record { converges = CauchyCompletion.converges (injection (c + i)) }} {record { converges = CauchyCompletion.converges (injection c +C b) }} (<CWellDefined (Equivalence.transitive (Setoid.eq cauchyCompletionSetoid) {record { converges = CauchyCompletion.converges ((-C b) +C injection (inverse c)) }} {record { converges = CauchyCompletion.converges ((-C b) +C (-C (injection c))) }} {record { converges = CauchyCompletion.converges (-C (injection c +C b)) }} (Group.+WellDefined CGroup {record { converges = CauchyCompletion.converges (-C b) }} {record { converges = CauchyCompletion.converges (injection (inverse c)) }} {record { converges = CauchyCompletion.converges (-C b) }} {record { converges = CauchyCompletion.converges (-C (injection c)) }} (Equivalence.reflexive (Setoid.eq cauchyCompletionSetoid) {record { converges = CauchyCompletion.converges (-C b) }}) homRespectsInverse) (Equivalence.symmetric (Setoid.eq cauchyCompletionSetoid) {record { converges = CauchyCompletion.converges (-C (injection c +C b)) }} {record { converges = CauchyCompletion.converges ((-C b) +C (-C (injection c))) }} (invContravariant CGroup {record { converges = CauchyCompletion.converges (injection c) }} {record { converges = CauchyCompletion.converges b }}))) homRespectsInverse' (cOrderRespectsAdditionLeft' (-C b) (inverse i) (inverse c) (flipR<C i<b)))))
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{-
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Have 0<a, so 0 < a- < a < a+
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Have c < a- + c, by cOrderRespectsAdditionLeft''
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Have a- + c < a+ + c by cOrderRespectsAdditionLeft''
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so a- + c < K < a+ + c
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-}
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cOrderRespectsAdditionLeft''' a b c record { i = i ; a<i = a<i ; i<b = i<b } = <CTransitive (cOrderRespectsAdditionLeft' a i c a<i) (<CWellDefined (Equivalence.transitive (Setoid.eq cauchyCompletionSetoid) {record { converges = CauchyCompletion.converges (injection (c + i)) }} {record { converges = CauchyCompletion.converges (injection c +C injection i) }} {record { converges = CauchyCompletion.converges (injection i +C injection c) }} (GroupHom.groupHom CInjectionGroupHom) (Ring.groupIsAbelian CRing {record { converges = CauchyCompletion.converges (injection c) }} {record { converges = CauchyCompletion.converges (injection i) }})) (Ring.groupIsAbelian CRing {record { converges = CauchyCompletion.converges (injection c) }} {record { converges = CauchyCompletion.converges b }}) (flip<C' {record { converges = CauchyCompletion.converges (injection (c + i)) }} {record { converges = CauchyCompletion.converges (injection c +C b) }} (<CWellDefined (Equivalence.transitive (Setoid.eq cauchyCompletionSetoid) {record { converges = CauchyCompletion.converges ((-C b) +C injection (inverse c)) }} {record { converges = CauchyCompletion.converges ((-C b) +C (-C (injection c))) }} {record { converges = CauchyCompletion.converges (-C (injection c +C b)) }} (Group.+WellDefinedRight CGroup {record { converges = CauchyCompletion.converges (-C b) }} {record { converges = CauchyCompletion.converges (injection (inverse c)) }} {record { converges = CauchyCompletion.converges (-C (injection c)) }} homRespectsInverse) (Equivalence.symmetric (Setoid.eq cauchyCompletionSetoid) {record { converges = CauchyCompletion.converges (-C (injection c +C b)) }} {record { converges = CauchyCompletion.converges ((-C b) +C (-C (injection c))) }} (invContravariant CGroup {record { converges = CauchyCompletion.converges (injection c) }} {record { converges = CauchyCompletion.converges b }}))) homRespectsInverse' (cOrderRespectsAdditionLeft' (-C b) (inverse i) (inverse c) (flipR<C i<b)))))
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cOrderRespectsAdditionRightZero : (a : CauchyCompletion) → (0R r<C a) → (c : CauchyCompletion) → c <C (a +C c)
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cOrderRespectsAdditionRightZero a record { e = e ; 0<e = 0<e ; N = N1 ; pr = pr } c with halve charNot2 e
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@@ -211,7 +174,7 @@ Have a- + c < a+ + c by cOrderRespectsAdditionLeft''
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open Equivalence (Setoid.eq cauchyCompletionSetoid) renaming (transitive to tr)
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cOrderRespectsAdditionRight : (a : A) (b : CauchyCompletion) (c : CauchyCompletion) → (a r<C b) → (injection a +C c) <C (b +C c)
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cOrderRespectsAdditionRight a b c a<b = <CWellDefined (Equivalence.transitive (Setoid.eq cauchyCompletionSetoid) {record { converges = CauchyCompletion.converges (Group.inverse CGroup (injection (inverse a) +C (-C c))) }} {record { converges = CauchyCompletion.converges (Group.inverse CGroup ((-C injection a) +C (-C c))) }} {record { converges = CauchyCompletion.converges (injection a +C c) }} (inverseWellDefined CGroup {record { converges = CauchyCompletion.converges (injection (inverse a) +C (-C c)) }} {record { converges = CauchyCompletion.converges ((-C (injection a)) +C (-C c)) }} (Group.+WellDefined CGroup {record { converges = CauchyCompletion.converges (injection (inverse a)) }} {record { converges = CauchyCompletion.converges (-C c) }} {record { converges = CauchyCompletion.converges (-C (injection a)) }} {record { converges = CauchyCompletion.converges (-C c) }} homRespectsInverse (Equivalence.reflexive (Setoid.eq cauchyCompletionSetoid) {record { converges = CauchyCompletion.converges (-C c) }}))) (lemma2 (injection a) c)) (lemma2 b c) (flip<C (cOrderRespectsAdditionLeft _ _ (Group.inverse CGroup c) (flipR<C a<b)))
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cOrderRespectsAdditionRight a b c a<b = <CWellDefined (Equivalence.transitive (Setoid.eq cauchyCompletionSetoid) {record { converges = CauchyCompletion.converges (Group.inverse CGroup (injection (inverse a) +C (-C c))) }} {record { converges = CauchyCompletion.converges (Group.inverse CGroup ((-C injection a) +C (-C c))) }} {record { converges = CauchyCompletion.converges (injection a +C c) }} (inverseWellDefined CGroup {record { converges = CauchyCompletion.converges (injection (inverse a) +C (-C c)) }} {record { converges = CauchyCompletion.converges ((-C (injection a)) +C (-C c)) }} (Group.+WellDefinedLeft CGroup {record { converges = CauchyCompletion.converges (injection (inverse a)) }} {record { converges = CauchyCompletion.converges (-C c) }} {record { converges = CauchyCompletion.converges (-C (injection a)) }} homRespectsInverse)) (lemma2 (injection a) c)) (lemma2 b c) (flip<C (cOrderRespectsAdditionLeft _ _ (Group.inverse CGroup c) (flipR<C a<b)))
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cOrderRespectsAddition : (a b : CauchyCompletion) → (a <C b) → (c : CauchyCompletion) → (a +C c) <C (b +C c)
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cOrderRespectsAddition a b record { i = i ; a<i = a<i ; i<b = i<b } c = SetoidPartialOrder.<Transitive <COrder (cOrderRespectsAdditionLeft a i c a<i) (cOrderRespectsAdditionRight i b c i<b)
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@@ -13,13 +13,11 @@ module Fields.FieldOfFractions.Addition {a b : _} {A : Set a} {S : Setoid {a} {b
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open import Fields.FieldOfFractions.Setoid I
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fieldOfFractionsPlus : fieldOfFractionsSet → fieldOfFractionsSet → fieldOfFractionsSet
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fieldOfFractionsPlus (record { num = a ; denom = b ; denomNonzero = b!=0 }) (record { num = c ; denom = d ; denomNonzero = d!=0 }) = record { num = ((a * d) + (b * c)) ; denom = b * d ; denomNonzero = ans }
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where
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open Setoid S
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open Ring R
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ans : ((b * d) ∼ Ring.0R R) → False
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ans pr with IntegralDomain.intDom I pr
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ans pr | f = exFalso (d!=0 (f b!=0))
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fieldOfFractionsSet.num (fieldOfFractionsPlus (record { num = a ; denom = b ; denomNonzero = b!=0 }) (record { num = c ; denom = d ; denomNonzero = d!=0 })) = (a * d) + (b * c)
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fieldOfFractionsSet.denom (fieldOfFractionsPlus (record { num = a ; denom = b ; denomNonzero = b!=0 }) (record { num = c ; denom = d ; denomNonzero = d!=0 })) = b * d
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fieldOfFractionsSet.denomNonzero (fieldOfFractionsPlus (record { num = a ; denom = b ; denomNonzero = b!=0 }) (record { num = c ; denom = d ; denomNonzero = d!=0 })) = λ pr → exFalso (d!=0 (IntegralDomain.intDom I pr b!=0))
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--record { num = ((a * d) + (b * c)) ; denom = b * d ; denomNonzero = λ pr → exFalso (d!=0 (IntegralDomain.intDom I pr b!=0)) }
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plusWellDefined : {a b c d : fieldOfFractionsSet} → (Setoid._∼_ fieldOfFractionsSetoid a c) → (Setoid._∼_ fieldOfFractionsSetoid b d) → Setoid._∼_ fieldOfFractionsSetoid (fieldOfFractionsPlus a b) (fieldOfFractionsPlus c d)
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plusWellDefined {record { num = a ; denom = b ; denomNonzero = b!=0 }} {record { num = c ; denom = d ; denomNonzero = d!=0 }} {record { num = e ; denom = f ; denomNonzero = f!=0 }} {record { num = g ; denom = h ; denomNonzero = h!=0 }} af=be ch=dg = need
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@@ -14,7 +14,7 @@ open import Rings.Orders.Partial.Definition
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open import Rings.Orders.Total.Definition
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open import Groups.Orders.Archimedean
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|
||||
module Fields.FieldOfFractions.Archimedean {a b : _} {A : Set a} {S : Setoid {a} {b} A} {_+_ : A → A → A} {_*_ : A → A → A} {R : Ring S _+_ _*_} (I : IntegralDomain R) {c : _} {_<_ : Rel {_} {c} A} {pOrder : SetoidPartialOrder S _<_} {pRing : PartiallyOrderedRing R pOrder} (order : TotallyOrderedRing pRing) (nonempty : (Setoid._∼_ S (Ring.0R R) (Ring.1R R)) → False) where
|
||||
module Fields.FieldOfFractions.Archimedean {a b : _} {A : Set a} {S : Setoid {a} {b} A} {_+_ : A → A → A} {_*_ : A → A → A} {R : Ring S _+_ _*_} (I : IntegralDomain R) {c : _} {_<_ : Rel {_} {c} A} {pOrder : SetoidPartialOrder S _<_} {pRing : PartiallyOrderedRing R pOrder} (order : TotallyOrderedRing pRing) where
|
||||
|
||||
open import Groups.Cyclic.Definition
|
||||
open import Fields.FieldOfFractions.Setoid I
|
||||
@@ -23,9 +23,9 @@ open import Fields.FieldOfFractions.Addition I
|
||||
open import Fields.FieldOfFractions.Ring I
|
||||
open import Fields.FieldOfFractions.Order I order
|
||||
open import Fields.FieldOfFractions.Field I
|
||||
open import Fields.Orders.Total.Archimedean
|
||||
open import Fields.Orders.Partial.Archimedean {F = fieldOfFractions} record { oRing = fieldOfFractionsPOrderedRing }
|
||||
open import Rings.Orders.Partial.Lemmas pRing
|
||||
open import Rings.Orders.Total.Lemmas order
|
||||
open import Rings.Orders.Total.Lemmas
|
||||
|
||||
open Setoid S
|
||||
open Equivalence eq
|
||||
@@ -48,21 +48,33 @@ private
|
||||
|
||||
simpPower : (n : ℕ) → Setoid._∼_ fieldOfFractionsSetoid (positiveEltPower fieldOfFractionsGroup record { num = Ring.1R R ; denom = Ring.1R R ; denomNonzero = IntegralDomain.nontrivial I} n) record { num = positiveEltPower (Ring.additiveGroup R) (Ring.1R R) n ; denom = Ring.1R R ; denomNonzero = IntegralDomain.nontrivial I }
|
||||
simpPower zero = Equivalence.reflexive (Setoid.eq fieldOfFractionsSetoid) {Group.0G fieldOfFractionsGroup}
|
||||
simpPower (succ n) = Equivalence.transitive (Setoid.eq fieldOfFractionsSetoid) {record { denomNonzero = d (fieldOfFractionsPlus (record { num = 1R ; denom = 1R ; denomNonzero = λ t → nonempty (symmetric t) }) (positiveEltPower fieldOfFractionsGroup _ n)) }} {record { denomNonzero = λ t → nonempty (symmetric (transitive (symmetric identIsIdent) t)) }} {record { denomNonzero = λ t → nonempty (symmetric t) }} (Group.+WellDefined fieldOfFractionsGroup {record { num = Ring.1R R ; denom = Ring.1R R ; denomNonzero = IntegralDomain.nontrivial I }} {positiveEltPower fieldOfFractionsGroup record { num = Ring.1R R ; denom = Ring.1R R ; denomNonzero = IntegralDomain.nontrivial I } n} {record { num = Ring.1R R ; denom = Ring.1R R ; denomNonzero = IntegralDomain.nontrivial I }} {record { num = positiveEltPower (Ring.additiveGroup R) (Ring.1R R) n ; denom = Ring.1R R ; denomNonzero = IntegralDomain.nontrivial I }} (Equivalence.reflexive (Setoid.eq fieldOfFractionsSetoid) {record { num = Ring.1R R ; denom = Ring.1R R ; denomNonzero = IntegralDomain.nontrivial I}}) (simpPower n)) (transitive (transitive (transitive *Commutative (transitive identIsIdent (+WellDefined identIsIdent identIsIdent))) (symmetric identIsIdent)) (*WellDefined (symmetric identIsIdent) reflexive))
|
||||
simpPower (succ n) = Equivalence.transitive (Setoid.eq fieldOfFractionsSetoid) {record { denomNonzero = d (fieldOfFractionsPlus (record { num = 1R ; denom = 1R ; denomNonzero = λ t → IntegralDomain.nontrivial I t }) (positiveEltPower fieldOfFractionsGroup _ n)) }} {record { denomNonzero = λ t → IntegralDomain.nontrivial I (transitive (symmetric identIsIdent) t) }} {record { denomNonzero = IntegralDomain.nontrivial I }} (Group.+WellDefined fieldOfFractionsGroup {record { num = Ring.1R R ; denom = Ring.1R R ; denomNonzero = IntegralDomain.nontrivial I }} {positiveEltPower fieldOfFractionsGroup record { num = Ring.1R R ; denom = Ring.1R R ; denomNonzero = IntegralDomain.nontrivial I } n} {record { num = Ring.1R R ; denom = Ring.1R R ; denomNonzero = IntegralDomain.nontrivial I }} {record { num = positiveEltPower (Ring.additiveGroup R) (Ring.1R R) n ; denom = Ring.1R R ; denomNonzero = IntegralDomain.nontrivial I }} (Equivalence.reflexive (Setoid.eq fieldOfFractionsSetoid) {record { num = Ring.1R R ; denom = Ring.1R R ; denomNonzero = IntegralDomain.nontrivial I}}) (simpPower n)) (transitive (transitive (transitive *Commutative (transitive identIsIdent (+WellDefined identIsIdent identIsIdent))) (symmetric identIsIdent)) (*WellDefined (symmetric identIsIdent) reflexive))
|
||||
|
||||
lemma : (n : ℕ) {num denom : A} .(d!=0 : denom ∼ 0G → False) → (num * denom) < positiveEltPower additiveGroup 1R n → fieldOfFractionsComparison (record { num = num ; denom = denom ; denomNonzero = d!=0}) record { num = positiveEltPower additiveGroup (Ring.1R R) n ; denom = 1R ; denomNonzero = IntegralDomain.nontrivial I }
|
||||
lemma n {num} {denom} d!=0 numdenom<n with totality 0G denom
|
||||
... | inl (inl 0<denom) with totality 0G 1R
|
||||
... | inl (inl 0<1) = {!!}
|
||||
... | inl (inr x) = exFalso (1<0False x)
|
||||
... | inr x = exFalso (nonempty x)
|
||||
... | inl (inr x) = exFalso (1<0False order x)
|
||||
... | inr x = exFalso (IntegralDomain.nontrivial I (symmetric x))
|
||||
lemma n {num} {denom} d!=0 numdenom<n | inl (inr denom<0) with totality 0G 1R
|
||||
... | inl (inl 0<1) = {!!}
|
||||
... | inl (inr 1<0) = exFalso (1<0False 1<0)
|
||||
... | inr 0=1 = exFalso (nonempty 0=1)
|
||||
... | inl (inr 1<0) = exFalso (1<0False order 1<0)
|
||||
... | inr 0=1 = exFalso (IntegralDomain.nontrivial I (symmetric 0=1))
|
||||
lemma n {num} {denom} d!=0 numdenom<n | inr 0=denom = exFalso (d!=0 (symmetric 0=denom))
|
||||
|
||||
fieldOfFractionsArchimedean : Archimedean (toGroup R pRing) → ArchimedeanField {F = fieldOfFractions} record { oRing = fieldOfFractionsOrderedRing }
|
||||
fieldOfFractionsArchimedean arch (record { num = num ; denom = denom ; denomNonzero = denom!=0 }) with arch (num * denom) {!!} {!!} {!!}
|
||||
... | bl = {!!}
|
||||
fieldOfFractionsArchimedean : Archimedean (toGroup R pRing) → ArchimedeanField
|
||||
fieldOfFractionsArchimedean arch (record { num = num ; denom = denom ; denomNonzero = denom!=0 }) 0<q with totality 0G denom ,, totality 0G 1R
|
||||
... | inl (inl 0<denom) ,, inl (inl 0<1) rewrite refl { x = 0} = {!!}
|
||||
... | inl (inl 0<denom) ,, inl (inr x) = exFalso {!!}
|
||||
... | inl (inl 0<denom) ,, inr x = exFalso {!!}
|
||||
... | inl (inr denom<0) ,, inl (inl 0<1) = 0 , {!!}
|
||||
where
|
||||
t : {!!}
|
||||
t = {!!}
|
||||
... | inl (inr denom<0) ,, inl (inr x) = exFalso {!!}
|
||||
... | inl (inr denom<0) ,, inr x = exFalso {!!}
|
||||
... | inr x ,, snd = exFalso (denom!=0 (symmetric x))
|
||||
--... | N , pr = N , SetoidPartialOrder.<WellDefined fieldOfFractionsOrder {record { denomNonzero = denom!=0 }} {record { denomNonzero = denom!=0 }} {record { denomNonzero = λ t → nonempty (symmetric t) }} {record { denomNonzero = d (positiveEltPower fieldOfFractionsGroup record { num = 1R ; denom = 1R ; denomNonzero = λ t → nonempty (symmetric t) } N) }} (Equivalence.reflexive (Setoid.eq fieldOfFractionsSetoid) { record { denomNonzero = denom!=0 } }) (Equivalence.symmetric (Setoid.eq fieldOfFractionsSetoid) {record { denomNonzero = λ t → d (positiveEltPower fieldOfFractionsGroup record { num = 1R ; denom = 1R ; denomNonzero = λ t → nonempty (symmetric t) } N) t }} {record { denomNonzero = λ t → nonempty (symmetric t) }} (simpPower N)) (lemma N denom!=0 pr)
|
||||
|
||||
fieldOfFractionsArchimedean' : Archimedean (toGroup R pRing) → Archimedean (toGroup fieldOfFractionsRing fieldOfFractionsPOrderedRing)
|
||||
fieldOfFractionsArchimedean' arch = archFieldToGrp (reciprocalPositive fieldOfFractionsOrderedRing) (fieldOfFractionsArchimedean arch)
|
||||
|
||||
@@ -27,6 +27,17 @@ open SetoidTotalOrder (TotallyOrderedRing.total order)
|
||||
open import Rings.Orders.Partial.Lemmas
|
||||
open PartiallyOrderedRing pRing
|
||||
|
||||
fieldOfFractionsComparison : Rel fieldOfFractionsSet
|
||||
fieldOfFractionsComparison (record { num = numA ; denom = denomA ; denomNonzero = denomA!=0 }) (record { num = numB ; denom = denomB ; denomNonzero = denomB!=0 }) with (totality (Ring.0R R) denomA ,, totality (Ring.0R R) denomB)
|
||||
fieldOfFractionsComparison (record { num = numA ; denom = denomA ; denomNonzero = denomA!=0 }) (record { num = numB ; denom = denomB ; denomNonzero = denomB!=0 }) | inl (inl 0<denomA) ,, inl (inl 0<denomB) = (numA * denomB) < (numB * denomA)
|
||||
fieldOfFractionsComparison (record { num = numA ; denom = denomA ; denomNonzero = denomA!=0 }) (record { num = numB ; denom = denomB ; denomNonzero = denomB!=0 }) | inl (inl 0<denomA) ,, inl (inr denomB<0) = (numB * denomA) < (numA * denomB)
|
||||
fieldOfFractionsComparison (record { num = numA ; denom = denomA ; denomNonzero = denomA!=0 }) (record { num = numB ; denom = denomB ; denomNonzero = denomB!=0 }) | inl (inl 0<denomA) ,, inr 0=denomB = exFalso (denomB!=0 (symmetric 0=denomB))
|
||||
fieldOfFractionsComparison (record { num = numA ; denom = denomA ; denomNonzero = denomA!=0 }) (record { num = numB ; denom = denomB ; denomNonzero = denomB!=0 }) | inl (inr denomA<0) ,, inl (inl 0<denomB) = (numB * denomA) < (numA * denomB)
|
||||
fieldOfFractionsComparison (record { num = numA ; denom = denomA ; denomNonzero = denomA!=0 }) (record { num = numB ; denom = denomB ; denomNonzero = denomB!=0 }) | inl (inr denomA<0) ,, inl (inr denomB<0) = (numA * denomB) < (numB * denomA)
|
||||
fieldOfFractionsComparison (record { num = numA ; denom = denomA ; denomNonzero = denomA!=0 }) (record { num = numB ; denom = denomB ; denomNonzero = denomB!=0 }) | inl (inr denomA<0) ,, inr 0=denomB = exFalso (denomB!=0 (symmetric 0=denomB))
|
||||
fieldOfFractionsComparison (record { num = numA ; denom = denomA ; denomNonzero = denomA!=0 }) (record { num = numB ; denom = denomB ; denomNonzero = denomB!=0 }) | inr 0=denomA ,, _ = exFalso (denomA!=0 (symmetric 0=denomA))
|
||||
|
||||
{-
|
||||
fieldOfFractionsComparison : Rel fieldOfFractionsSet
|
||||
fieldOfFractionsComparison (record { num = numA ; denom = denomA ; denomNonzero = denomA!=0 }) (record { num = numB ; denom = denomB ; denomNonzero = denomB!=0 }) with totality (Ring.0R R) denomA
|
||||
fieldOfFractionsComparison (record { num = numA ; denom = denomA ; denomNonzero = denomA!=0 }) (record { num = numB ; denom = denomB ; denomNonzero = denomB!=0 }) | inl (inl 0<denomA) with totality (Ring.0R R) denomB
|
||||
@@ -38,6 +49,18 @@ fieldOfFractionsComparison (record { num = numA ; denom = denomA ; denomNonzero
|
||||
fieldOfFractionsComparison (record { num = numA ; denom = denomA ; denomNonzero = denomA!=0 }) (record { num = numB ; denom = denomB ; denomNonzero = denomB!=0 }) | inl (inr denomA<0) | inl (inr denomB<0) = (numA * denomB) < (numB * denomA)
|
||||
fieldOfFractionsComparison (record { num = numA ; denom = denomA ; denomNonzero = denomA!=0 }) (record { num = numB ; denom = denomB ; denomNonzero = denomB!=0 }) | inl (inr denomA<0) | inr 0=denomB = exFalso (denomB!=0 (symmetric 0=denomB))
|
||||
fieldOfFractionsComparison (record { num = numA ; denom = denomA ; denomNonzero = denomA!=0 }) (record { num = numB ; denom = denomB ; denomNonzero = denomB!=0 }) | inr 0=denomA = exFalso (denomA!=0 (symmetric 0=denomA))
|
||||
fieldOfFractionsComparison : Rel fieldOfFractionsSet
|
||||
fieldOfFractionsComparison (record { num = numA ; denom = denomA ; denomNonzero = denomA!=0 }) (record { num = numB ; denom = denomB ; denomNonzero = denomB!=0 }) with totality (Ring.0R R) denomA
|
||||
fieldOfFractionsComparison (record { num = numA ; denom = denomA ; denomNonzero = denomA!=0 }) (record { num = numB ; denom = denomB ; denomNonzero = denomB!=0 }) | inl (inl 0<denomA) with totality (Ring.0R R) denomB
|
||||
fieldOfFractionsComparison (record { num = numA ; denom = denomA ; denomNonzero = denomA!=0 }) (record { num = numB ; denom = denomB ; denomNonzero = denomB!=0 }) | inl (inl 0<denomA) | inl (inl 0<denomB) = (numA * denomB) < (numB * denomA)
|
||||
fieldOfFractionsComparison (record { num = numA ; denom = denomA ; denomNonzero = denomA!=0 }) (record { num = numB ; denom = denomB ; denomNonzero = denomB!=0 }) | inl (inl 0<denomA) | inl (inr denomB<0) = (numB * denomA) < (numA * denomB)
|
||||
fieldOfFractionsComparison (record { num = numA ; denom = denomA ; denomNonzero = denomA!=0 }) (record { num = numB ; denom = denomB ; denomNonzero = denomB!=0 }) | inl (inl 0<denomA) | inr 0=denomB = exFalso (denomB!=0 (symmetric 0=denomB))
|
||||
fieldOfFractionsComparison (record { num = numA ; denom = denomA ; denomNonzero = denomA!=0 }) (record { num = numB ; denom = denomB ; denomNonzero = denomB!=0 }) | inl (inr denomA<0) with totality (Ring.0R R) denomB
|
||||
fieldOfFractionsComparison (record { num = numA ; denom = denomA ; denomNonzero = denomA!=0 }) (record { num = numB ; denom = denomB ; denomNonzero = denomB!=0 }) | inl (inr denomA<0) | inl (inl 0<denomB) = (numB * denomA) < (numA * denomB)
|
||||
fieldOfFractionsComparison (record { num = numA ; denom = denomA ; denomNonzero = denomA!=0 }) (record { num = numB ; denom = denomB ; denomNonzero = denomB!=0 }) | inl (inr denomA<0) | inl (inr denomB<0) = (numA * denomB) < (numB * denomA)
|
||||
fieldOfFractionsComparison (record { num = numA ; denom = denomA ; denomNonzero = denomA!=0 }) (record { num = numB ; denom = denomB ; denomNonzero = denomB!=0 }) | inl (inr denomA<0) | inr 0=denomB = exFalso (denomB!=0 (symmetric 0=denomB))
|
||||
fieldOfFractionsComparison (record { num = numA ; denom = denomA ; denomNonzero = denomA!=0 }) (record { num = numB ; denom = denomB ; denomNonzero = denomB!=0 }) | inr 0=denomA = exFalso (denomA!=0 (symmetric 0=denomA))
|
||||
-}
|
||||
|
||||
private
|
||||
abstract
|
||||
|
||||
58
Fields/Orders/Partial/Archimedean.agda
Normal file
58
Fields/Orders/Partial/Archimedean.agda
Normal file
@@ -0,0 +1,58 @@
|
||||
{-# OPTIONS --safe --warning=error --without-K #-}
|
||||
|
||||
open import Sets.EquivalenceRelations
|
||||
open import Numbers.Naturals.Semiring
|
||||
open import Numbers.Integers.Definition
|
||||
open import Numbers.Naturals.Order
|
||||
open import LogicalFormulae
|
||||
open import Groups.Definition
|
||||
open import Rings.Orders.Partial.Definition
|
||||
open import Setoids.Orders.Partial.Definition
|
||||
open import Setoids.Setoids
|
||||
open import Functions
|
||||
open import Rings.Definition
|
||||
open import Fields.Fields
|
||||
open import Fields.Orders.Partial.Definition
|
||||
|
||||
open import Agda.Primitive using (Level; lzero; lsuc; _⊔_)
|
||||
|
||||
module Fields.Orders.Partial.Archimedean {a b : _} {A : Set a} {S : Setoid {a} {b} A} {_+_ _*_ : A → A → A} {R : Ring S _+_ _*_} {c : _} {_<_ : A → A → Set c} {pOrder : SetoidPartialOrder S _<_} {F : Field R} (p : PartiallyOrderedField F pOrder) where
|
||||
|
||||
open Setoid S
|
||||
open Equivalence eq
|
||||
open Ring R
|
||||
open Group additiveGroup
|
||||
open import Groups.Lemmas additiveGroup
|
||||
open import Groups.Cyclic.Definition additiveGroup
|
||||
open import Groups.Cyclic.DefinitionLemmas additiveGroup
|
||||
open import Groups.Orders.Archimedean (toGroup R (PartiallyOrderedField.oRing p))
|
||||
open import Rings.Orders.Partial.Lemmas (PartiallyOrderedField.oRing p)
|
||||
open import Rings.InitialRing R
|
||||
open SetoidPartialOrder pOrder
|
||||
open PartiallyOrderedRing (PartiallyOrderedField.oRing p)
|
||||
open Field F
|
||||
|
||||
ArchimedeanField : Set (a ⊔ c)
|
||||
ArchimedeanField = (x : A) → (0R < x) → Sg ℕ (λ n → x < (fromN n))
|
||||
|
||||
private
|
||||
lemma : (r : A) (N : ℕ) → (positiveEltPower 1R N * r) ∼ positiveEltPower r N
|
||||
lemma r zero = timesZero'
|
||||
lemma r (succ n) = transitive *DistributesOver+' (+WellDefined identIsIdent (lemma r n))
|
||||
|
||||
findBound : (0<1 : 0R < 1R) → (r s : A) (N : ℕ) → (1R < r) → (s < positiveEltPower 1R N) → s < positiveEltPower r N
|
||||
findBound _ r s zero 1<r s<N = s<N
|
||||
findBound 0<1 r s (succ N) 1<r s<N = <Transitive s<N (<WellDefined (transitive *Commutative identIsIdent) (lemma r (succ N)) (ringCanMultiplyByPositive' (fromNPreservesOrder 0<1 (succIsPositive N)) 1<r))
|
||||
|
||||
archFieldToGrp : ((x y : A) → .(0R < x) → (x * y) ∼ 1R → 0R < y) → ArchimedeanField → Archimedean
|
||||
archFieldToGrp reciprocalPositive a r s 0<r 0<s with allInvertible r (λ r=0 → irreflexive (<WellDefined reflexive r=0 0<r))
|
||||
... | inv , prInv with a inv (reciprocalPositive r inv 0<r (transitive *Commutative prInv))
|
||||
... | N , invBound with a s 0<s
|
||||
... | Ns , nSBound = (Ns *N N) , <WellDefined reflexive (symmetric (elementPowerMultiplies (nonneg Ns) (nonneg N) r)) (findBound (<WellDefined reflexive (transitive *Commutative prInv) (orderRespectsMultiplication 0<r (reciprocalPositive r inv 0<r (transitive *Commutative prInv)))) (positiveEltPower r N) s Ns m nSBound)
|
||||
where
|
||||
m : 1R < positiveEltPower r N
|
||||
m = <WellDefined prInv (lemma r N) (ringCanMultiplyByPositive 0<r invBound)
|
||||
|
||||
archToArchField : 0R < 1R → Archimedean → ArchimedeanField
|
||||
archToArchField 0<1 arch a 0<a with arch 1R a 0<1 0<a
|
||||
... | N , pr = N , pr
|
||||
@@ -14,11 +14,11 @@ open import Setoids.Setoids
|
||||
open import Functions
|
||||
open import Rings.Definition
|
||||
open import Fields.Fields
|
||||
open import Fields.Orders.Total.Definition
|
||||
open import Fields.Orders.Partial.Definition
|
||||
|
||||
open import Agda.Primitive using (Level; lzero; lsuc; _⊔_)
|
||||
|
||||
module Fields.Orders.Total.Archimedean {a b : _} {A : Set a} {S : Setoid {a} {b} A} {_+_ _*_ : A → A → A} {R : Ring S _+_ _*_} {c : _} {_<_ : A → A → Set c} {pOrder : SetoidPartialOrder S _<_} {p : PartiallyOrderedRing R pOrder} {F : Field R} (t : TotallyOrderedField F p) where
|
||||
module Fields.Orders.Total.Archimedean {a b : _} {A : Set a} {S : Setoid {a} {b} A} {_+_ _*_ : A → A → A} {R : Ring S _+_ _*_} {c : _} {_<_ : A → A → Set c} {pOrder : SetoidPartialOrder S _<_} {F : Field R} (p : PartiallyOrderedField F pOrder) where
|
||||
|
||||
open Setoid S
|
||||
open Equivalence eq
|
||||
@@ -27,13 +27,11 @@ open Group additiveGroup
|
||||
open import Groups.Lemmas additiveGroup
|
||||
open import Groups.Cyclic.Definition additiveGroup
|
||||
open import Groups.Cyclic.DefinitionLemmas additiveGroup
|
||||
open import Groups.Orders.Archimedean (toGroup R p)
|
||||
open import Rings.Orders.Partial.Lemmas p
|
||||
open import Groups.Orders.Archimedean (toGroup R (PartiallyOrderedField.oRing p))
|
||||
open import Rings.Orders.Partial.Lemmas (PartiallyOrderedField.oRing p)
|
||||
open import Rings.InitialRing R
|
||||
open SetoidPartialOrder pOrder
|
||||
open PartiallyOrderedRing p
|
||||
open SetoidTotalOrder (TotallyOrderedRing.total (TotallyOrderedField.oRing t))
|
||||
open import Rings.Orders.Total.Lemmas (TotallyOrderedField.oRing t)
|
||||
open PartiallyOrderedRing (PartiallyOrderedField.oRing p)
|
||||
open Field F
|
||||
|
||||
ArchimedeanField : Set (a ⊔ c)
|
||||
@@ -44,22 +42,19 @@ private
|
||||
lemma r zero = timesZero'
|
||||
lemma r (succ n) = transitive *DistributesOver+' (+WellDefined identIsIdent (lemma r n))
|
||||
|
||||
findBound : (r s : A) (N : ℕ) → (1R < r) → (s < positiveEltPower 1R N) → s < positiveEltPower r N
|
||||
findBound r s zero 1<r s<N = s<N
|
||||
findBound r s (succ N) 1<r s<N = <Transitive s<N (<WellDefined (transitive *Commutative identIsIdent) (lemma r (succ N)) (ringCanMultiplyByPositive' (fromNPreservesOrder (0<1 (anyComparisonImpliesNontrivial 1<r)) (succIsPositive N)) 1<r))
|
||||
findBound : (0<1 : 0R < 1R) → (r s : A) (N : ℕ) → (1R < r) → (s < positiveEltPower 1R N) → s < positiveEltPower r N
|
||||
findBound _ r s zero 1<r s<N = s<N
|
||||
findBound 0<1 r s (succ N) 1<r s<N = <Transitive s<N (<WellDefined (transitive *Commutative identIsIdent) (lemma r (succ N)) (ringCanMultiplyByPositive' (fromNPreservesOrder 0<1 (succIsPositive N)) 1<r))
|
||||
|
||||
archFieldToGrp : ArchimedeanField → Archimedean
|
||||
archFieldToGrp a r s 0<r 0<s with totality r s
|
||||
archFieldToGrp a r s 0<r 0<s | inl (inr s<r) = 1 , <WellDefined reflexive (symmetric identRight) s<r
|
||||
archFieldToGrp a r s 0<r 0<s | inr r=s = 2 , <WellDefined (transitive identLeft r=s) (+WellDefined reflexive (symmetric identRight)) (orderRespectsAddition 0<r r)
|
||||
... | inl (inl r<s) with allInvertible r (λ r=0 → irreflexive (<WellDefined reflexive r=0 0<r))
|
||||
archFieldToGrp : ((x y : A) → 0R < x → (x * y) ∼ 1R → 0R < y) → ArchimedeanField → Archimedean
|
||||
archFieldToGrp reciprocalPositive a r s 0<r 0<s with allInvertible r (λ r=0 → irreflexive (<WellDefined reflexive r=0 0<r))
|
||||
... | inv , prInv with a inv (reciprocalPositive r inv 0<r (transitive *Commutative prInv))
|
||||
... | N , invBound with a s 0<s
|
||||
... | Ns , nSBound = (Ns *N N) , <WellDefined reflexive (symmetric (elementPowerMultiplies (nonneg Ns) (nonneg N) r)) (findBound (positiveEltPower r N) s Ns m nSBound)
|
||||
... | Ns , nSBound = (Ns *N N) , <WellDefined reflexive (symmetric (elementPowerMultiplies (nonneg Ns) (nonneg N) r)) (findBound (<WellDefined reflexive (transitive *Commutative prInv) (orderRespectsMultiplication 0<r (reciprocalPositive r inv 0<r (transitive *Commutative prInv)))) (positiveEltPower r N) s Ns m nSBound)
|
||||
where
|
||||
m : 1R < positiveEltPower r N
|
||||
m = <WellDefined prInv (lemma r N) (ringCanMultiplyByPositive 0<r invBound)
|
||||
|
||||
archToArchField : Archimedean → ArchimedeanField
|
||||
archToArchField arch a 0<a with arch 1R a (0<1 (anyComparisonImpliesNontrivial 0<a)) 0<a
|
||||
archToArchField : 0R < 1R → Archimedean → ArchimedeanField
|
||||
archToArchField 0<1 arch a 0<a with arch 1R a 0<1 0<a
|
||||
... | N , pr = N , pr
|
||||
|
||||
Reference in New Issue
Block a user