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Comparison on the reals (#55)

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Patrick Stevens
2019-10-26 10:36:24 +01:00
committed by GitHub
parent 4e56b68024
commit 0d68919127
18 changed files with 1343 additions and 691 deletions
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6
Everything/Safe.agda
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@@ -19,6 +19,8 @@ open import Groups.Groups2
open import Groups.SymmetryGroups
open import Fields.Fields
open import Fields.Orders.Definition
open import Fields.Orders.Lemmas
open import Fields.FieldOfFractions
open import Fields.FieldOfFractionsOrder
@@ -53,7 +55,7 @@ open import Monoids.Definition
open import Semirings.Definition
open import Semirings.Solver
open import Fields.CauchyCompletion.Definition
open import Fields.CauchyCompletion.Addition
open import Fields.CauchyCompletion.Group
open import Fields.CauchyCompletion.Comparison
module Everything.Safe where

28
Fields/CauchyCompletion/Addition.agda
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@@ -26,30 +26,17 @@ open Ring R
open Group additiveGroup
open Field F
open import Fields.Lemmas F
open import Fields.CauchyCompletion.Definition order F
open import Rings.Orders.Lemmas(order)
halve : (a : A) → Sg A (λ i → i + i ∼ a)
-- TODO: we need to know the characteristic already
halve a with allInvertible (1R + 1R) charNot2
... | 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))))
where
r : a + a ∼ (1R + 1R) * a
r = Equivalence.symmetric eq (Equivalence.transitive eq *Commutative (Equivalence.transitive eq *DistributesOver+ (+WellDefined (Equivalence.transitive eq *Commutative identIsIdent) (Equivalence.transitive eq *Commutative identIsIdent))))
halvePositive : (a : A) → 0R < (a + a) → 0R < a
halvePositive a 0<2a with totality 0R a
halvePositive a 0<2a | inl (inl x) = x
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))
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))
lemm : (m : ℕ) (a b : Sequence A) → index (apply _+_ a b) m ≡ (index a m) + (index b m)
lemm zero a b = refl
lemm (succ m) a b = lemm m (Sequence.tail a) (Sequence.tail b)
_+C_ : CauchyCompletion → CauchyCompletion → CauchyCompletion
CauchyCompletion.elts (record { elts = a ; converges = convA } +C record { elts = b ; converges = convB }) = apply _+_ a b
CauchyCompletion.converges (record { elts = a ; converges = convA } +C record { elts = b ; converges = convB }) ε 0<e with halve ε
CauchyCompletion.converges (record { elts = a ; converges = convA } +C record { elts = b ; converges = convB }) ε 0<e with halve charNot2 ε
... | e/2 , e/2Pr with convA e/2 (halvePositive e/2 (<WellDefined (Equivalence.reflexive eq) (Equivalence.symmetric eq e/2Pr) 0<e))
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))
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
@@ -60,3 +47,14 @@ CauchyCompletion.converges (record { elts = a ; converges = convA } +C record {
... | bm-bn<e/2 with triangleInequality (index a m + inverse (index a n)) (index b m + inverse (index b n))
... | 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))
... | 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))
inverseDistributes : {r s : A} → inverse (r + s) ∼ inverse r + inverse s
inverseDistributes = Equivalence.transitive eq (invContravariant additiveGroup) groupIsAbelian
-C_ : CauchyCompletion → CauchyCompletion
CauchyCompletion.elts (-C a) = map inverse (CauchyCompletion.elts a)
CauchyCompletion.converges (-C record { elts = elts ; converges = converges }) ε 0<e with converges ε 0<e
CauchyCompletion.converges (-C record { elts = elts ; converges = converges }) ε 0<e | N , prN = N , ans
where
ans : {m n : ℕ} → (N <N m) → (N <N n) → abs ((index (map inverse elts) m) + inverse (index (map inverse elts) n)) < ε
ans {m} {n} N<m N<n = <WellDefined (Equivalence.transitive eq (absWellDefined _ _ (Equivalence.reflexive eq)) (Equivalence.transitive eq (absNegation (index elts m + inverse (index elts n))) (absWellDefined _ _ (Equivalence.transitive eq inverseDistributes (+WellDefined (identityOfIndiscernablesLeft _∼_ (Equivalence.reflexive eq) (equalityCommutative (mapAndIndex elts inverse m))) (inverseWellDefined additiveGroup (identityOfIndiscernablesLeft _∼_ (Equivalence.reflexive eq) (equalityCommutative (mapAndIndex elts inverse n))))))))) (Equivalence.reflexive eq) (prN N<m N<n)

35
Fields/CauchyCompletion/Approximation.agda Normal file
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@@ -0,0 +1,35 @@
{-# OPTIONS --safe --warning=error --without-K --guardedness #-}
open import Agda.Primitive using (Level; lzero; lsuc; _⊔_)
open import Setoids.Setoids
open import Rings.Definition
open import Rings.Lemmas
open import Rings.Order
open import Groups.Definition
open import Groups.Groups
open import Fields.Fields
open import Sets.EquivalenceRelations
open import Sequences
open import Setoids.Orders
open import Functions
open import LogicalFormulae
open import Numbers.Naturals.Naturals
module Fields.CauchyCompletion.Approximation {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
open Setoid S
open SetoidTotalOrder tOrder
open SetoidPartialOrder pOrder
open Equivalence eq
open OrderedRing order
open Ring R
open Group additiveGroup
open Field F
open import Rings.Orders.Lemmas(order)
open import Fields.CauchyCompletion.Definition order F
open import Fields.CauchyCompletion.Addition order F charNot2
open import Fields.CauchyCompletion.Setoid order F charNot2
approximate : (a : CauchyCompletion) → (ε : A) → Sg A (λ b → a +C (-C (injection b)) < ε)
approximate a ε = ?

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Fields/CauchyCompletion/Comparison.agda Normal file
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@@ -0,0 +1,131 @@
{-# OPTIONS --safe --warning=error --without-K --guardedness #-}
open import Agda.Primitive using (Level; lzero; lsuc; _⊔_)
open import Setoids.Setoids
open import Rings.Definition
open import Rings.Lemmas
open import Rings.Order
open import Groups.Definition
open import Groups.Groups
open import Fields.Fields
open import Sets.EquivalenceRelations
open import Sequences
open import Setoids.Orders
open import Functions
open import LogicalFormulae
open import Numbers.Naturals.Naturals
open import Semirings.Definition
module Fields.CauchyCompletion.Comparison {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
open Setoid S
open SetoidTotalOrder tOrder
open SetoidPartialOrder pOrder
open Equivalence eq
open OrderedRing order
open Ring R
open Group additiveGroup
open Field F
open import Fields.Orders.Lemmas {F = F} record { oRing = order }
open import Rings.Orders.Lemmas order
open import Fields.Lemmas F
open import Fields.CauchyCompletion.Definition order F
open import Fields.CauchyCompletion.Setoid order F charNot2
shrinkInequality : {a b c : ℕ} → a +N b <N c → a <N c
shrinkInequality {a} {b} {c} (le x pr) = le (x +N b) (transitivity (applyEquality succ (transitivity (equalityCommutative (Semiring.+Associative ℕSemiring x _ _)) (applyEquality (x +N_) (Semiring.commutative ℕSemiring b a)))) pr)
shrinkInequality' : {a b c : ℕ} → a +N b <N c → b <N c
shrinkInequality' {a} {b} {c} (le x pr) = le (x +N a) (transitivity (applyEquality succ (equalityCommutative (Semiring.+Associative ℕSemiring x _ _))) pr)
_<C'_ : CauchyCompletion → A → Set (m ⊔ o)
a <C' b = Sg A (λ ε → (0G < ε) && Sg ℕ (λ N → ((m : ℕ) → (N<m : N <N m) → (ε + index (CauchyCompletion.elts a) m) < b)))
<C'WellDefinedRight : (a : CauchyCompletion) (b c : A) → b ∼ c → a <C' b → a <C' c
<C'WellDefinedRight a b c b=c (ε , (0<e ,, (N , pr))) = ε , (0<e ,, (N , λ m N<m → <WellDefined (Equivalence.reflexive eq) b=c (pr m N<m)))
halfLess : (e/2 e : A) → (0<e : 0G < e) → (pr : e/2 + e/2 ∼ e) → e/2 < e
halfLess e/2 e 0<e pr with halvePositive e/2 (<WellDefined (Equivalence.reflexive eq) (Equivalence.symmetric eq pr) 0<e)
... | 0<e/2 = <WellDefined identLeft pr (orderRespectsAddition 0<e/2 e/2)
<C'WellDefinedLemma : (a b e/2 e : A) (0<e : 0G < e) (pr : e/2 + e/2 ∼ e) → abs (a + inverse b) < e/2 → (e/2 + b) < (e + a)
<C'WellDefinedLemma a b e/2 e 0<e pr a-b<e with SetoidTotalOrder.totality tOrder 0G (a + inverse b)
<C'WellDefinedLemma a b e/2 e 0<e pr a-b<e | inl (inl 0<a-b) = ringAddInequalities (halfLess e/2 e 0<e pr) (<WellDefined identLeft (Equivalence.transitive eq (Equivalence.symmetric eq +Associative) (Equivalence.transitive eq (+WellDefined (Equivalence.reflexive eq) invLeft) identRight)) (orderRespectsAddition 0<a-b b))
<C'WellDefinedLemma a b e/2 e 0<e pr a-b<e | inl (inr a-b<0) = <WellDefined (Equivalence.transitive eq (+WellDefined (invContravariant (Ring.additiveGroup R)) groupIsAbelian) (Equivalence.transitive eq (Equivalence.transitive eq +Associative (+WellDefined (Equivalence.transitive eq (Equivalence.symmetric eq +Associative) (Equivalence.transitive eq (+WellDefined (invTwice additiveGroup b) invLeft) identRight)) (Equivalence.reflexive eq))) groupIsAbelian)) (Equivalence.transitive eq +Associative (+WellDefined pr (Equivalence.reflexive eq))) (orderRespectsAddition a-b<e (e/2 + a))
<C'WellDefinedLemma a b e/2 e 0<e pr a-b<e | inr 0=a-b = <WellDefined (Equivalence.reflexive eq) (+WellDefined (Equivalence.reflexive eq) (Equivalence.symmetric eq (transferToRight (Ring.additiveGroup R) (Equivalence.symmetric eq 0=a-b)))) (orderRespectsAddition (halfLess e/2 e 0<e pr) b)
<C'WellDefinedLeft : (a b : CauchyCompletion) (c : A) → Setoid._∼_ cauchyCompletionSetoid a b → a <C' c → b <C' c
<C'WellDefinedLeft a b c a=b (ε , (0<e ,, (N , pr))) with halve charNot2 ε
... | e/2 , prE/2 with halvePositive e/2 (<WellDefined (Equivalence.reflexive eq) (Equivalence.symmetric eq prE/2) 0<e)
... | 0<e/2 with a=b e/2 0<e/2
... | N2 , prN2 = e/2 , (0<e/2 ,, ((N +N N2) , ans))
where
ans : (m : ℕ) → (N<m : (N +N N2) <N m) → (e/2 + index (CauchyCompletion.elts b) m) < c
ans m N<m with prN2 {m} (shrinkInequality' N<m)
... | bl rewrite indexAndApply (CauchyCompletion.elts a) (map inverse (CauchyCompletion.elts b)) _+_ {m} | equalityCommutative (mapAndIndex (CauchyCompletion.elts b) inverse m) = SetoidPartialOrder.transitive pOrder (<C'WellDefinedLemma (index (CauchyCompletion.elts a) m) (index (CauchyCompletion.elts b) m) e/2 ε 0<e prE/2 bl) (pr m (shrinkInequality N<m))
<C'WellDefined : (a b : CauchyCompletion) (c d : A) → Setoid._∼_ cauchyCompletionSetoid a b → c ∼ d → a <C' c → b <C' d
<C'WellDefined a b c d a=b c=d a<c = <C'WellDefinedLeft a b d a=b (<C'WellDefinedRight a c d c=d a<c)
_<C''_ : A → CauchyCompletion → Set (m ⊔ o)
a <C'' b = Sg A (λ ε → (0G < ε) && Sg ℕ (λ N → ((m : ℕ) → (N<m : N <N m) → (a + ε) < index (CauchyCompletion.elts b) m)))
<C''WellDefinedLeft : (a b : A) (c : CauchyCompletion) → (a ∼ b) → (a <C'' c) → b <C'' c
<C''WellDefinedLeft a b c a=b (e , (0<e ,, (N , pr))) = e , (0<e ,, (N , λ m N<m → <WellDefined (+WellDefined a=b (Equivalence.reflexive eq)) (Equivalence.reflexive eq) (pr m N<m)))
<C''WellDefinedLemma : (a b c e e/2 : A) (_ : e/2 + e/2 ∼ e) (0<e : 0G < e) (_ : abs (a + inverse b) < e/2) (_ : (c + e) < a) → (c + e/2) < b
<C''WellDefinedLemma a b c e e/2 prE/2 0<e pr c+e<a with SetoidTotalOrder.totality tOrder 0G (a + inverse b)
<C''WellDefinedLemma a b c e e/2 prE/2 0<e pr c+e<a | inl (inl 0<a-b) = SetoidPartialOrder.transitive pOrder (<WellDefined (Equivalence.transitive eq (Equivalence.symmetric eq +Associative) (+WellDefined (Equivalence.reflexive eq) (Equivalence.transitive eq (+WellDefined (Equivalence.symmetric eq prE/2) (Equivalence.reflexive eq)) (Equivalence.transitive eq (Equivalence.transitive eq (Equivalence.symmetric eq +Associative) (+WellDefined (Equivalence.reflexive eq) invRight)) identRight)))) (Equivalence.reflexive eq) (orderRespectsAddition c+e<a (inverse e/2))) (<WellDefined (Equivalence.transitive eq +Associative (+WellDefined (Equivalence.transitive eq (Equivalence.symmetric eq +Associative) (Equivalence.transitive eq (+WellDefined (Equivalence.reflexive eq) invLeft) identRight)) (Equivalence.reflexive eq))) (Equivalence.transitive eq groupIsAbelian (Equivalence.transitive eq (Equivalence.symmetric eq +Associative) (Equivalence.transitive eq (+WellDefined (Equivalence.reflexive eq) (invLeft)) identRight))) (orderRespectsAddition pr (b + inverse e/2)))
<C''WellDefinedLemma a b c e e/2 prE/2 0<e pr c+e<a | inl (inr a-b<0) = SetoidPartialOrder.transitive pOrder (SetoidPartialOrder.transitive pOrder (<WellDefined groupIsAbelian groupIsAbelian (orderRespectsAddition (halfLess e/2 e 0<e prE/2) c)) c+e<a) (<WellDefined (Equivalence.transitive eq (Equivalence.transitive eq (Equivalence.symmetric eq +Associative) (+WellDefined (Equivalence.reflexive eq) invLeft)) identRight) identLeft (orderRespectsAddition a-b<0 b))
<C''WellDefinedLemma a b c e e/2 prE/2 0<e pr c+e<a | inr 0=a-b = SetoidPartialOrder.transitive pOrder (<WellDefined groupIsAbelian (Equivalence.reflexive eq) (orderRespectsAddition (halfLess e/2 e 0<e prE/2) c)) (<WellDefined (groupIsAbelian {c} {e}) (transferToRight additiveGroup (Equivalence.symmetric eq 0=a-b)) c+e<a)
<C''WellDefinedRight : (a b : CauchyCompletion) (c : A) → Setoid._∼_ cauchyCompletionSetoid a b → c <C'' a → c <C'' b
<C''WellDefinedRight a b c a=b (e , (0<e ,, (N , pr))) with halve charNot2 e
... | e/2 , prE/2 with halvePositive e/2 (<WellDefined (Equivalence.reflexive eq) (Equivalence.symmetric eq prE/2) 0<e)
... | 0<e/2 with a=b e/2 0<e/2
... | N2 , prN2 = e/2 , (0<e/2 ,, ((N +N N2) , ans))
where
ans : (m : ℕ) → (N<m : (N +N N2) <N m) → (c + e/2) < index (CauchyCompletion.elts b) m
ans m N<m with prN2 {m} (shrinkInequality' N<m)
... | bl rewrite indexAndApply (CauchyCompletion.elts a) (map inverse (CauchyCompletion.elts b)) _+_ {m} | equalityCommutative (mapAndIndex (CauchyCompletion.elts b) inverse m) = <C''WellDefinedLemma (index (CauchyCompletion.elts a) m) (index (CauchyCompletion.elts b) m) c e e/2 prE/2 0<e bl (pr m (shrinkInequality N<m))
_<C_ : CauchyCompletion → CauchyCompletion → Set (m ⊔ o)
a <C b = Sg A (λ i → (a <C' i) && (i <C'' b))
<CWellDefined : {a b c d : CauchyCompletion} → Setoid._∼_ cauchyCompletionSetoid a b → Setoid._∼_ cauchyCompletionSetoid c d → a <C c → b <C d
<CWellDefined {a} {b} {c} {d} a=b c=d (bound , (a<bound ,, bound<b)) = bound , (<C'WellDefinedLeft a b bound a=b a<bound ,, <C''WellDefinedRight c d bound c=d bound<b)
<C'Preserves : {a b : A} → a < b → injection a <C' b
<C'Preserves {a} {b} a<b with dense charNot2 a<b
... | between , (a<between ,, between<b) = (between + inverse a) , (<WellDefined invRight (Equivalence.reflexive eq) (orderRespectsAddition a<between (inverse a)) ,, (0 , λ m _ → <WellDefined (Equivalence.transitive eq (Equivalence.symmetric eq identRight) (Equivalence.transitive eq (+WellDefined (Equivalence.reflexive eq) (ans a m)) +Associative)) (Equivalence.reflexive eq) between<b))
where
ans : (x : A) (m : ℕ) → 0G ∼ inverse x + index (constSequence x) m
ans x m rewrite indexAndConst x m = Equivalence.symmetric eq invLeft
<C''Preserves : {a b : A} → a < b → a <C'' injection b
<C''Preserves {a} {b} a<b with dense charNot2 a<b
... | between , (a<between ,, between<b) = (between + inverse a) , (<WellDefined invRight (Equivalence.reflexive eq) (orderRespectsAddition a<between (inverse a)) ,, (0 , λ m _ → <WellDefined (Equivalence.transitive eq (Equivalence.transitive eq (Equivalence.transitive eq (Equivalence.symmetric eq identRight) (+WellDefined (Equivalence.reflexive eq) (Equivalence.symmetric eq invLeft))) +Associative) groupIsAbelian) (identityOfIndiscernablesLeft _∼_ (Equivalence.reflexive eq) (indexAndConst b m)) between<b))
<CInherited : {a b : A} → a < b → (injection a) <C (injection b)
<CInherited {a} {b} a<b with dense charNot2 a<b
... | between , (a<between ,, between<b) = between , (<C'Preserves a<between ,, <C''Preserves between<b)
<CInherited' : {a b : A} → (injection a) <C (injection b) → a < b
<CInherited' {a} {b} (bound , ((Na , (0<Na ,, (Na2 , prA))) ,, ((Nb , (0<Nb ,, (Nb2 , prB)))))) with prA (succ Na2) (le 0 refl)
... | bl with prB (succ Nb2) (le 0 refl)
... | bl2 rewrite indexAndConst a Na2 | indexAndConst b Nb2 = SetoidPartialOrder.transitive pOrder (SetoidPartialOrder.transitive pOrder (<WellDefined identLeft (Equivalence.reflexive eq) (orderRespectsAddition 0<Na a)) bl) (SetoidPartialOrder.transitive pOrder (<WellDefined identLeft groupIsAbelian (orderRespectsAddition 0<Nb bound)) bl2)
<CIrreflexive : {a : CauchyCompletion} → a <C a → False
<CIrreflexive {a} (bound , ((bA , (0<bA ,, (Na , prA))) ,, ((bB , (0<bB ,, (Nb , prB)))))) with prA (succ Na +N Nb) (le Nb (applyEquality succ (Semiring.commutative ℕSemiring Nb Na)))
... | bl with prB (succ Na +N Nb) (le Na refl)
... | bl2 with <WellDefined (Equivalence.transitive eq +Associative (+WellDefined (Equivalence.transitive eq (Equivalence.symmetric eq +Associative) groupIsAbelian) (Equivalence.reflexive eq))) groupIsAbelian (ringAddInequalities bl bl2)
... | bad = irreflexive (SetoidPartialOrder.transitive pOrder (<WellDefined identLeft +Associative (<WellDefined (Equivalence.reflexive eq) groupIsAbelian (orderRespectsAddition (<WellDefined identLeft (Equivalence.reflexive eq) (ringAddInequalities 0<bA 0<bB)) (index (CauchyCompletion.elts a) (succ Na +N Nb) + bound)))) bad)
<CTransitive : {a b c : CauchyCompletion} → a <C b → b <C c → a <C c
<CTransitive {a} {b} {c} (boundA , ((eL , (0<eL ,, prL)) ,, (eR , (0<eR ,, (Nr , prR))))) (boundB , ((fL , (0<fL ,, (Nfl , prFl))) ,, (fR , (0<fR ,, (N , prFR))))) = boundA , ((eL , (0<eL ,, prL)) ,, (eR , (0<eR ,, (((Nr +N Nfl) +N N) , λ m N<m → SetoidPartialOrder.transitive pOrder (SetoidPartialOrder.transitive pOrder (SetoidPartialOrder.transitive pOrder (SetoidPartialOrder.transitive pOrder (prR m (shrinkInequality {Nr} (shrinkInequality N<m))) (<WellDefined identLeft (Equivalence.reflexive eq) (orderRespectsAddition 0<fL (index (CauchyCompletion.elts b) m)))) (prFl m (shrinkInequality' {Nr} (shrinkInequality N<m)))) (<WellDefined identLeft groupIsAbelian (orderRespectsAddition 0<fR boundB))) (prFR m (shrinkInequality' N<m))))))
<COrder : SetoidPartialOrder cauchyCompletionSetoid _<C_
SetoidPartialOrder.<WellDefined <COrder {a} {b} {c} {d} a=b c=d a<c = <CWellDefined {a} {b} {c} {d} a=b c=d a<c
SetoidPartialOrder.irreflexive <COrder {a} a<a = <CIrreflexive {a} a<a
SetoidPartialOrder.transitive <COrder {a} {b} {c} a<b b<c = <CTransitive {a} {b} {c} a<b b<c

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@@ -0,0 +1,66 @@
{-# OPTIONS --safe --warning=error --without-K --guardedness #-}
open import Agda.Primitive using (Level; lzero; lsuc; _⊔_)
open import Setoids.Setoids
open import Rings.Definition
open import Rings.Lemmas
open import Rings.Order
open import Groups.Definition
open import Groups.Groups
open import Fields.Fields
open import Sets.EquivalenceRelations
open import Sequences
open import Setoids.Orders
open import Functions
open import LogicalFormulae
open import Numbers.Naturals.Naturals
module Fields.CauchyCompletion.Group {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
open Setoid S
open SetoidTotalOrder tOrder
open SetoidPartialOrder pOrder
open Equivalence eq
open OrderedRing order
open Field F
open Group (Ring.additiveGroup R)
open import Rings.Orders.Lemmas(order)
open import Fields.CauchyCompletion.Definition order F
open import Fields.CauchyCompletion.Addition order F charNot2
open import Fields.CauchyCompletion.Setoid order F charNot2
Cassoc : {a b c : CauchyCompletion} → Setoid._∼_ cauchyCompletionSetoid (a +C (b +C c)) ((a +C b) +C c)
Cassoc {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))) (map inverse (CauchyCompletion.elts ((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) +Associative)) (identityOfIndiscernablesRight _∼_ (Equivalence.reflexive eq) (absZero order)))) (Equivalence.reflexive eq) 0<e
CidentRight : {a : CauchyCompletion} → Setoid._∼_ cauchyCompletionSetoid (a +C injection 0G) a
CidentRight {a} ε 0<e = 0 , ans
where
ans : {m : ℕ} → 0 <N m → abs (index (apply _+_ (CauchyCompletion.elts (a +C injection 0G)) (map inverse (CauchyCompletion.elts a))) m) < ε
ans {m} 0<m rewrite indexAndApply (CauchyCompletion.elts (a +C injection 0G)) (map inverse (CauchyCompletion.elts a)) _+_ {m} | indexAndApply (CauchyCompletion.elts a) (constSequence 0G) _+_ {m} | equalityCommutative (mapAndIndex (CauchyCompletion.elts a) inverse m) | indexAndConst 0G m = <WellDefined (Equivalence.symmetric eq (Equivalence.transitive eq (absWellDefined _ _ (Equivalence.transitive eq (+WellDefined (identRight) (Equivalence.reflexive eq)) (invRight))) (identityOfIndiscernablesRight _∼_ (Equivalence.reflexive eq) (absZero order)))) (Equivalence.reflexive eq) 0<e
CidentLeft : {a : CauchyCompletion} → Setoid._∼_ cauchyCompletionSetoid (injection 0G +C a) a
CidentLeft {a} = Equivalence.transitive (Setoid.eq cauchyCompletionSetoid) {injection 0G +C a} {a +C injection 0G} {a} (+CCommutative {injection 0G} {a}) (CidentRight {a})
CinvRight : {a : CauchyCompletion} → Setoid._∼_ cauchyCompletionSetoid (a +C (-C a)) (injection 0G)
CinvRight {a} ε 0<e = 0 , ans
where
ans : {m : ℕ} → (0 <N m) → abs (index (apply _+_ (CauchyCompletion.elts (a +C (-C a))) (map inverse (CauchyCompletion.elts (injection 0G)))) m) < ε
ans {m} 0<m rewrite indexAndApply (CauchyCompletion.elts (a +C (-C a))) (map inverse (CauchyCompletion.elts (injection 0G))) _+_ {m} | indexAndApply (CauchyCompletion.elts a) (map inverse (CauchyCompletion.elts a)) _+_ {m} | equalityCommutative (mapAndIndex (CauchyCompletion.elts a) inverse m) | equalityCommutative (mapAndIndex (constSequence 0G) inverse m) | indexAndConst 0G m = <WellDefined (Equivalence.symmetric eq (Equivalence.transitive eq (absWellDefined _ _ (Equivalence.transitive eq (+WellDefined invRight (invIdentity (Ring.additiveGroup R))) identRight)) (identityOfIndiscernablesRight _∼_ (Equivalence.reflexive eq) (absZero order)))) (Equivalence.reflexive eq) 0<e
CGroup : Group cauchyCompletionSetoid _+C_
Group.+WellDefined CGroup {a} {b} {c} {d} x y = additionWellDefined {a} {c} {b} {d} x y
Group.0G CGroup = injection 0G
Group.inverse CGroup = -C_
Group.+Associative CGroup {a} {b} {c} = Cassoc {a} {b} {c}
Group.identRight CGroup {a} = CidentRight {a}
Group.identLeft CGroup {a} = CidentLeft {a}
Group.invLeft CGroup {a} = Equivalence.transitive (Setoid.eq cauchyCompletionSetoid) {(-C a) +C a} {a +C (-C a)} {injection 0G} (+CCommutative { -C a} {a}) (CinvRight {a})
Group.invRight CGroup {a} = CinvRight {a}
CInjectionGroupHom : GroupHom (Ring.additiveGroup R) CGroup injection
GroupHom.groupHom CInjectionGroupHom {x} {y} = additionHom x y
GroupHom.wellDefined CInjectionGroupHom {x} {y} x=y = SetoidInjection.wellDefined CInjection {x} {y} x=y

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{-# OPTIONS --safe --warning=error --without-K --guardedness #-}
open import Agda.Primitive using (Level; lzero; lsuc; _⊔_)
open import Setoids.Setoids
open import Rings.Definition
open import Rings.Lemmas
open import Rings.Order
open import Groups.Definition
open import Groups.Groups
open import Fields.Fields
open import Sets.EquivalenceRelations
open import Sequences
open import Setoids.Orders
open import Functions
open import LogicalFormulae
open import Numbers.Naturals.Naturals
module Fields.CauchyCompletion.Multiplication {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
open Setoid S
open SetoidTotalOrder tOrder
open SetoidPartialOrder pOrder
open Equivalence eq
open OrderedRing order
open Ring R
open Group additiveGroup
open Field F
open import Rings.Orders.Lemmas(order)
open import Fields.CauchyCompletion.Definition order F
open import Fields.CauchyCompletion.Setoid order F charNot2
_*C_ : CauchyCompletion → CauchyCompletion → CauchyCompletion
CauchyCompletion.elts (record { elts = a ; converges = aConv } *C record { elts = b ; converges = bConv }) = apply _*_ a b
CauchyCompletion.converges (record { elts = a ; converges = aConv } *C record { elts = b ; converges = bConv }) ε 0<e = {!!}
*CCommutative : {a b : CauchyCompletion} → Setoid._∼_ cauchyCompletionSetoid (a *C b) (b *C a)
*CCommutative {a} {b} ε 0<e = 0 , ans
where
foo : {x y : A} → (x * y) + inverse (y * x) ∼ 0G
foo = Equivalence.transitive eq (+WellDefined (Equivalence.reflexive eq) (inverseWellDefined additiveGroup *Commutative)) invRight
ans : {m : ℕ} → 0 <N m → abs (index (apply _+_ (CauchyCompletion.elts (a *C b)) (map inverse (CauchyCompletion.elts (b *C a)))) m) < ε
ans {m} 0<m rewrite indexAndApply (apply _*_ (CauchyCompletion.elts a) (CauchyCompletion.elts b)) (map inverse (apply _*_ (CauchyCompletion.elts b) (CauchyCompletion.elts a))) _+_ {m} | indexAndApply (CauchyCompletion.elts a) (CauchyCompletion.elts b) _*_ {m} | equalityCommutative (mapAndIndex (apply _*_ (CauchyCompletion.elts b) (CauchyCompletion.elts a)) inverse m) | indexAndApply (CauchyCompletion.elts b) (CauchyCompletion.elts a) _*_ {m} = <WellDefined (Equivalence.symmetric eq (Equivalence.transitive eq (absWellDefined _ _ foo) (identityOfIndiscernablesRight _∼_ (Equivalence.reflexive eq) (absZero order)))) (Equivalence.reflexive eq) 0<e
multiplicationWellDefinedLeft : (a b c : CauchyCompletion) → Setoid._∼_ cauchyCompletionSetoid a b → Setoid._∼_ cauchyCompletionSetoid (a *C c) (b *C c)
multiplicationWellDefinedLeft a b c a=b ε 0<e = 0 , ans
where
ans : {m : ℕ} → 0 <N m → abs (index (apply _+_ (apply _*_ (CauchyCompletion.elts a) (CauchyCompletion.elts c)) (map inverse (apply _*_ (CauchyCompletion.elts b) (CauchyCompletion.elts c)))) m) < ε
ans {m} N<m rewrite indexAndApply (apply _*_ (CauchyCompletion.elts a) (CauchyCompletion.elts c)) (map inverse (apply _*_ (CauchyCompletion.elts b) (CauchyCompletion.elts c))) _+_ {m} | equalityCommutative (mapAndIndex (apply _*_ (CauchyCompletion.elts b) (CauchyCompletion.elts c)) inverse m) | indexAndApply (CauchyCompletion.elts b) (CauchyCompletion.elts c) _*_ {m} | indexAndApply (CauchyCompletion.elts a) (CauchyCompletion.elts c) _*_ {m} = {!!}
multiplicationPreservedLeft : {a b : A} {c : CauchyCompletion} → (a ∼ b) → Setoid._∼_ cauchyCompletionSetoid (injection a *C c) (injection b *C c)
multiplicationPreservedLeft {a} {b} {c} a=b = multiplicationWellDefinedLeft (injection a) (injection b) c (injectionPreservesSetoid a b a=b)
multiplicationPreservedRight : {a b : A} {c : CauchyCompletion} → (a ∼ b) → Setoid._∼_ cauchyCompletionSetoid (c *C injection a) (c *C injection b)
multiplicationPreservedRight {a} {b} {c} a=b = Equivalence.transitive (Setoid.eq cauchyCompletionSetoid) {c *C injection a} {injection a *C c} {c *C injection b} (*CCommutative {c} {injection a}) (Equivalence.transitive (Setoid.eq cauchyCompletionSetoid) {injection a *C c} {injection b *C c} {c *C injection b} (multiplicationPreservedLeft {a} {b} {c} a=b) (*CCommutative {injection b} {c}))
multiplicationPreserved : {a b c d : A} → (a ∼ b) → (c ∼ d) → Setoid._∼_ cauchyCompletionSetoid (injection a *C injection c) (injection b *C injection d)
multiplicationPreserved {a} {b} {c} {d} a=b c=d = Equivalence.transitive (Setoid.eq cauchyCompletionSetoid) {injection a *C injection c} {injection a *C injection d} {injection b *C injection d} (multiplicationPreservedRight {c} {d} {injection a} c=d) (multiplicationPreservedLeft {a} {b} {injection d} a=b)
multiplicationWellDefinedRight : (a b c : CauchyCompletion) → Setoid._∼_ cauchyCompletionSetoid b c → Setoid._∼_ cauchyCompletionSetoid (a *C b) (a *C c)
multiplicationWellDefinedRight a b c b=c = Equivalence.transitive (Setoid.eq cauchyCompletionSetoid) {a *C b} {b *C a} {a *C c} (*CCommutative {a} {b}) (Equivalence.transitive (Setoid.eq cauchyCompletionSetoid) {b *C a} {c *C a} {a *C c} (multiplicationWellDefinedLeft b c a b=c) (*CCommutative {c} {a}))
multiplicationWellDefined : {a b c d : CauchyCompletion} → Setoid._∼_ cauchyCompletionSetoid a b → Setoid._∼_ cauchyCompletionSetoid c d → Setoid._∼_ cauchyCompletionSetoid (a *C c) (b *C d)
multiplicationWellDefined {a} {b} {c} {d} a=b c=d = Equivalence.transitive (Setoid.eq cauchyCompletionSetoid) {a *C c} {a *C d} {b *C d} (multiplicationWellDefinedRight a c d c=d) (multiplicationWellDefinedLeft a b d a=b)

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{-# OPTIONS --allow-unsolved-metas --warning=error --without-K --guardedness #-}
open import Agda.Primitive using (Level; lzero; lsuc; _⊔_)
open import Setoids.Setoids
open import Rings.Definition
open import Rings.Lemmas
open import Rings.Order
open import Groups.Definition
open import Groups.Groups
open import Fields.Fields
open import Sets.EquivalenceRelations
open import Sequences
open import Setoids.Orders
open import Functions
open import LogicalFormulae
open import Numbers.Naturals.Naturals
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 _<_} {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
open Setoid S
open SetoidTotalOrder tOrder
open SetoidPartialOrder pOrder
open Equivalence eq
open OrderedRing order
open Field F
open Group (Ring.additiveGroup R)
open import Rings.Orders.Lemmas(order)
open import Fields.CauchyCompletion.Definition order F
open import Fields.CauchyCompletion.Multiplication order F charNot2
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 order)))) (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 order)))) (Equivalence.reflexive eq) 0<e
CRing : Ring cauchyCompletionSetoid _+C_ _*C_
Ring.additiveGroup CRing = CGroup
Ring.*WellDefined CRing {a} {b} {c} {d} r=t s=u = multiplicationWellDefined {a} {c} {b} {d} r=t s=u
Ring.1R CRing = injection (Ring.1R R)
Ring.groupIsAbelian CRing {a} {b} = +CCommutative {a} {b}
Ring.*Associative CRing {a} {b} {c} = c*Assoc {a} {b} {c}
Ring.*Commutative CRing {a} {b} = *CCommutative {a} {b}
Ring.*DistributesOver+ CRing = {!!}
Ring.identIsIdent CRing {a} = c*Ident {a}

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{-# OPTIONS --safe --warning=error --without-K --guardedness #-}
open import Agda.Primitive using (Level; lzero; lsuc; _⊔_)
open import Setoids.Setoids
open import Rings.Definition
open import Rings.Lemmas
open import Rings.Order
open import Groups.Definition
open import Groups.Groups
open import Fields.Fields
open import Sets.EquivalenceRelations
open import Sequences
open import Setoids.Orders
open import Functions
open import LogicalFormulae
open import Numbers.Naturals.Naturals
open import Semirings.Definition
module Fields.CauchyCompletion.Setoid {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
open Setoid S
open SetoidTotalOrder tOrder
open SetoidPartialOrder pOrder
open Equivalence eq
open OrderedRing order
open Ring R
open Group additiveGroup
open Field F
open import Fields.Lemmas F
open import Fields.CauchyCompletion.Definition order F
open import Fields.CauchyCompletion.Addition order F charNot2
open import Rings.Orders.Lemmas(order)
isZero : CauchyCompletion → Set (m ⊔ o)
isZero record { elts = elts ; converges = converges } = ∀ ε → 0R < ε → Sg ℕ (λ N → ∀ {m : ℕ} → (N <N m) → (abs (index elts m)) < ε)
transitiveLemma : {a b c e/2 : A} → abs (a + inverse b) < e/2 → abs (b + inverse c) < e/2 → (abs (a + inverse c)) < (e/2 + e/2)
transitiveLemma {a} {b} {c} {e/2} a-b<e/2 b-c<e/2 with triangleInequality (a + inverse b) (b + inverse c)
transitiveLemma {a} {b} {c} {e/2} a-b<e/2 b-c<e/2 | inl x = SetoidPartialOrder.transitive pOrder (<WellDefined (absWellDefined _ _ (Equivalence.transitive eq (Equivalence.symmetric eq +Associative) (+WellDefined (Equivalence.reflexive eq) (Equivalence.transitive eq +Associative (Equivalence.transitive eq (+WellDefined invLeft (Equivalence.reflexive eq)) identLeft))))) (Equivalence.reflexive eq) x) (ringAddInequalities a-b<e/2 b-c<e/2)
transitiveLemma {a} {b} {c} {e/2} a-b<e/2 b-c<e/2 | inr x = <WellDefined (Equivalence.symmetric eq (Equivalence.transitive eq (absWellDefined _ _ (Equivalence.symmetric eq ((Equivalence.transitive eq (Equivalence.symmetric eq +Associative) (+WellDefined (Equivalence.reflexive eq) (Equivalence.transitive eq +Associative (Equivalence.transitive eq (+WellDefined invLeft (Equivalence.reflexive eq)) identLeft))))))) x)) (Equivalence.reflexive eq) (ringAddInequalities a-b<e/2 b-c<e/2)
cauchyCompletionSetoid : Setoid CauchyCompletion
(cauchyCompletionSetoid Setoid.∼ a) b = isZero (a +C (-C b))
Equivalence.reflexive (Setoid.eq cauchyCompletionSetoid) {x} ε 0<e = 0 , t
where
t : {m : ℕ} → (0 <N m) → abs (index (apply _+_ (CauchyCompletion.elts x) (map inverse (CauchyCompletion.elts x))) m) < ε
t {m} 0<m rewrite indexAndApply (CauchyCompletion.elts x) (map inverse (CauchyCompletion.elts x)) _+_ {m} | equalityCommutative (mapAndIndex (CauchyCompletion.elts x) inverse m) = <WellDefined (Equivalence.symmetric eq (Equivalence.transitive eq (absWellDefined _ _ invRight) (identityOfIndiscernablesRight _∼_ (Equivalence.reflexive eq) (absZero order)))) (Equivalence.reflexive eq) 0<e
Equivalence.symmetric (Setoid.eq cauchyCompletionSetoid) {x} {y} x=y ε 0<e with x=y ε 0<e
Equivalence.symmetric (Setoid.eq cauchyCompletionSetoid) {x} {y} x=y ε 0<e | N , pr = N , t
where
t : {m : ℕ} → N <N m → abs (index (apply _+_ (CauchyCompletion.elts y) (map inverse (CauchyCompletion.elts x))) m) < ε
t {m} N<m = <WellDefined (Equivalence.transitive eq (absNegation _) (absWellDefined _ _ (identityOfIndiscernablesRight _∼_ (identityOfIndiscernablesLeft _∼_ (identityOfIndiscernablesLeft _∼_ (Equivalence.transitive eq (inverseDistributes) (Equivalence.transitive eq groupIsAbelian (+WellDefined (Equivalence.transitive eq (inverseWellDefined additiveGroup (identityOfIndiscernablesLeft _∼_ (Equivalence.reflexive eq) (mapAndIndex _ inverse m))) (invTwice additiveGroup (index (CauchyCompletion.elts y) m))) (identityOfIndiscernablesLeft _∼_ (Equivalence.reflexive eq) (equalityCommutative (mapAndIndex (CauchyCompletion.elts x) inverse m)))))) (equalityCommutative (mapAndApply (CauchyCompletion.elts x) (map inverse (CauchyCompletion.elts y)) _+_ inverse m))) (equalityCommutative (mapAndIndex _ inverse m))) (equalityCommutative (indexAndApply (CauchyCompletion.elts y) (map inverse (CauchyCompletion.elts x)) _+_ {m}))))) (Equivalence.reflexive eq) (pr N<m)
Equivalence.transitive (Setoid.eq cauchyCompletionSetoid) {x} {y} {z} x=y y=z ε 0<e with halve charNot2 ε
... | e/2 , prHalf with x=y e/2 (halvePositive e/2 (<WellDefined (Equivalence.reflexive eq) (Equivalence.symmetric eq prHalf) 0<e))
... | Nx , prX with y=z e/2 (halvePositive e/2 (<WellDefined (Equivalence.reflexive eq) (Equivalence.symmetric eq prHalf) 0<e))
... | Ny , prY = (Nx +N Ny) , t
where
t : {m : ℕ} → (Nx +N Ny <N m) → abs (index (apply _+_ (CauchyCompletion.elts x) (map inverse (CauchyCompletion.elts z))) m) < ε
t {m} nsPr with prX {m} (le (_<N_.x nsPr +N Ny) (transitivity (applyEquality succ (transitivity (equalityCommutative (Semiring.+Associative ℕSemiring _ Ny Nx)) (applyEquality ( λ i → (_<N_.x nsPr) +N i) (Semiring.commutative ℕSemiring Ny Nx)))) (_<N_.proof nsPr)))
... | x-y<e/2 with prY {m} (le (_<N_.x nsPr +N Nx) (transitivity (applyEquality succ (equalityCommutative (Semiring.+Associative ℕSemiring _ Nx Ny))) (_<N_.proof nsPr)))
... | y-z<e/2 rewrite indexAndApply (CauchyCompletion.elts x) (map inverse (CauchyCompletion.elts z)) _+_ {m} | indexAndApply (CauchyCompletion.elts x) (map inverse (CauchyCompletion.elts y)) _+_ {m} | indexAndApply (CauchyCompletion.elts y) (map inverse (CauchyCompletion.elts z)) _+_ {m} | equalityCommutative (mapAndIndex (CauchyCompletion.elts z) inverse m) | equalityCommutative (mapAndIndex (CauchyCompletion.elts y) inverse m) = <WellDefined (Equivalence.reflexive eq) prHalf (transitiveLemma x-y<e/2 y-z<e/2)
injectionPreservesSetoid : (a b : A) → (a ∼ b) → Setoid._∼_ cauchyCompletionSetoid (injection a) (injection b)
injectionPreservesSetoid a b a=b ε 0<e = 0 , t
where
t : {m : ℕ} → 0 <N m → abs (index (apply _+_ (constSequence a) (map inverse (constSequence b))) m) < ε
t {m} 0<m = <WellDefined (identityOfIndiscernablesLeft _∼_ (absWellDefined 0G _ (identityOfIndiscernablesRight _∼_ (Equivalence.transitive eq (Equivalence.symmetric eq (transferToRight'' additiveGroup a=b)) (+WellDefined (identityOfIndiscernablesLeft _∼_ (Equivalence.reflexive eq) (indexAndConst a m)) (identityOfIndiscernablesRight _∼_ (Equivalence.reflexive eq) (transitivity (applyEquality inverse (equalityCommutative (indexAndConst _ m))) (mapAndIndex _ inverse m))))) (equalityCommutative (indexAndApply (constSequence a) _ _+_ {m})))) (absZero order)) (Equivalence.reflexive eq) 0<e
infinitesimalImplies0 : (a : A) → ({ε : A} → (0R < ε) → a < ε) → (a ∼ 0R) || (a < 0R)
infinitesimalImplies0 a pr with SetoidTotalOrder.totality tOrder 0R a
infinitesimalImplies0 a pr | inl (inl 0<a) with halve charNot2 a
infinitesimalImplies0 a pr | inl (inl 0<a) | a/2 , prA/2 with pr {a/2} (halvePositive a/2 (<WellDefined (Equivalence.reflexive eq) (Equivalence.symmetric eq prA/2) 0<a))
... | bl with halvePositive a/2 (<WellDefined (Equivalence.reflexive eq) (Equivalence.symmetric eq prA/2) 0<a)
... | 0<a/2 = exFalso (irreflexive {a} (<WellDefined identLeft prA/2 (ringAddInequalities 0<a/2 bl)))
infinitesimalImplies0 a pr | inl (inr a<0) = inr a<0
infinitesimalImplies0 a pr | inr 0=a = inl (Equivalence.symmetric eq 0=a)
injectionPreservesSetoid' : (a b : A) → Setoid._∼_ cauchyCompletionSetoid (injection a) (injection b) → a ∼ b
injectionPreservesSetoid' a b a=b = transferToRight additiveGroup (absZeroImpliesZero ans)
where
infinitesimal : {ε : A} → 0G < ε → abs (a + inverse b) < ε
infinitesimal {ε} 0<e with a=b ε 0<e
... | N , pr with pr {succ N} (le 0 refl)
... | bl rewrite indexAndApply (constSequence a) (map inverse (constSequence b)) _+_ {N} | indexAndConst a N | equalityCommutative (mapAndIndex (constSequence b) inverse N) | indexAndConst b N = bl
ans : (abs (a + inverse b)) ∼ 0G
ans with infinitesimalImplies0 _ infinitesimal
... | inl x = x
... | inr x = exFalso (absNonnegative x)
+CCommutative : {a b : CauchyCompletion} → Setoid._∼_ cauchyCompletionSetoid (a +C b) (b +C a)
+CCommutative {a} {b} ε 0<e = 0 , ans
where
foo : {x y : A} → (x + y) + inverse (y + x) ∼ 0G
foo = Equivalence.transitive eq (+WellDefined (Equivalence.reflexive eq) (inverseWellDefined additiveGroup groupIsAbelian)) invRight
ans : {m : ℕ} → 0 <N m → abs (index (apply _+_ (CauchyCompletion.elts (a +C b)) (map inverse (CauchyCompletion.elts (b +C a)))) m) < ε
ans {m} 0<m rewrite indexAndApply (CauchyCompletion.elts (a +C b)) (map inverse (CauchyCompletion.elts (b +C a))) _+_ {m} | indexAndApply (CauchyCompletion.elts a) (CauchyCompletion.elts b) _+_ {m} | equalityCommutative (mapAndIndex (apply _+_ (CauchyCompletion.elts b) (CauchyCompletion.elts a)) inverse m) | indexAndApply (CauchyCompletion.elts b) (CauchyCompletion.elts a) _+_ {m} = <WellDefined (Equivalence.symmetric eq (Equivalence.transitive eq (absWellDefined _ _ foo) (identityOfIndiscernablesRight _∼_ (Equivalence.reflexive eq) (absZero order)) )) (Equivalence.reflexive eq) 0<e
additionWellDefinedLeft : (a b c : CauchyCompletion) → Setoid._∼_ cauchyCompletionSetoid a b → Setoid._∼_ cauchyCompletionSetoid (a +C c) (b +C c)
additionWellDefinedLeft record { elts = a ; converges = aConv } record { elts = b ; converges = bConv } record { elts = c ; converges = cConv } a=b ε 0<e with a=b ε 0<e
... | Na-b , prA-b = Na-b , ans
where
ans : {m : ℕ} → Na-b <N m → abs (index (apply _+_ (apply _+_ a c) (map inverse (apply _+_ b c))) m) < ε
ans {m} mBig with prA-b {m} mBig
... | bl rewrite indexAndApply (apply _+_ a c) (map inverse (apply _+_ b c)) _+_ {m} | indexAndApply a c _+_ {m} | equalityCommutative (mapAndIndex (apply _+_ b c) inverse m) | indexAndApply b c _+_ {m} = <WellDefined (absWellDefined _ _ t) (Equivalence.reflexive eq) bl
where
t : index (apply _+_ a (map inverse b)) m ∼ ((index a m + index c m) + inverse (index b m + index c m))
t rewrite indexAndApply a (map inverse b) _+_ {m} | equalityCommutative (mapAndIndex b inverse m) = Equivalence.transitive eq (+WellDefined (Equivalence.reflexive eq) (Equivalence.transitive eq (Equivalence.transitive eq (Equivalence.transitive eq (Equivalence.symmetric eq identLeft) (+WellDefined (Equivalence.symmetric eq invRight) (Equivalence.reflexive eq))) (Equivalence.symmetric eq +Associative)) (+WellDefined (Equivalence.reflexive eq) (Equivalence.symmetric eq (invContravariant additiveGroup))))) (+Associative {index a m})
additionPreservedLeft : {a b : A} {c : CauchyCompletion} → (a ∼ b) → Setoid._∼_ cauchyCompletionSetoid (injection a +C c) (injection b +C c)
additionPreservedLeft {a} {b} {c} a=b = additionWellDefinedLeft (injection a) (injection b) c (injectionPreservesSetoid a b a=b)
additionPreservedRight : {a b : A} {c : CauchyCompletion} → (a ∼ b) → Setoid._∼_ cauchyCompletionSetoid (c +C injection a) (c +C injection b)
additionPreservedRight {a} {b} {c} a=b = Equivalence.transitive (Setoid.eq cauchyCompletionSetoid) {c +C injection a} {injection a +C c} {c +C injection b} (+CCommutative {c} {injection a}) (Equivalence.transitive (Setoid.eq cauchyCompletionSetoid) {injection a +C c} {injection b +C c} {c +C injection b} (additionPreservedLeft {a} {b} {c} a=b) (+CCommutative {injection b} {c}))
additionPreserved : {a b c d : A} → (a ∼ b) → (c ∼ d) → Setoid._∼_ cauchyCompletionSetoid (injection a +C injection c) (injection b +C injection d)
additionPreserved {a} {b} {c} {d} a=b c=d = Equivalence.transitive (Setoid.eq cauchyCompletionSetoid) {injection a +C injection c} {injection a +C injection d} {injection b +C injection d} (additionPreservedRight {c} {d} {injection a} c=d) (additionPreservedLeft {a} {b} {injection d} a=b)
additionWellDefinedRight : (a b c : CauchyCompletion) → Setoid._∼_ cauchyCompletionSetoid b c → Setoid._∼_ cauchyCompletionSetoid (a +C b) (a +C c)
additionWellDefinedRight a b c b=c = Equivalence.transitive (Setoid.eq cauchyCompletionSetoid) {a +C b} {b +C a} {a +C c} (+CCommutative {a} {b}) (Equivalence.transitive (Setoid.eq cauchyCompletionSetoid) {b +C a} {c +C a} {a +C c} (additionWellDefinedLeft b c a b=c) (+CCommutative {c} {a}))
additionWellDefined : {a b c d : CauchyCompletion} → Setoid._∼_ cauchyCompletionSetoid a b → Setoid._∼_ cauchyCompletionSetoid c d → Setoid._∼_ cauchyCompletionSetoid (a +C c) (b +C d)
additionWellDefined {a} {b} {c} {d} a=b c=d = Equivalence.transitive (Setoid.eq cauchyCompletionSetoid) {a +C c} {a +C d} {b +C d} (additionWellDefinedRight a c d c=d) (additionWellDefinedLeft a b d a=b)
additionHom : (x y : A) → Setoid._∼_ cauchyCompletionSetoid (injection (x + y)) (injection x +C injection y)
additionHom x y ε 0<e = 0 , ans
where
ans : {m : ℕ} → 0 <N m → abs (index (apply _+_ (CauchyCompletion.elts (injection (x + y))) (map inverse (CauchyCompletion.elts (injection x +C injection y)))) m) < ε
ans {m} 0<m rewrite indexAndApply (CauchyCompletion.elts (injection (x + y))) (map inverse (CauchyCompletion.elts (injection x +C injection y))) _+_ {m} | equalityCommutative (mapAndIndex (apply _+_ (constSequence x) (constSequence y)) inverse m) | indexAndConst (x + y) m | indexAndApply (constSequence x) (constSequence y) _+_ {m} | indexAndConst x m | indexAndConst y m = <WellDefined (Equivalence.symmetric eq (Equivalence.transitive eq (absWellDefined _ _ invRight) (identityOfIndiscernablesRight _∼_ (Equivalence.reflexive eq) (absZero order)))) (Equivalence.reflexive eq) 0<e
CInjection : SetoidInjection S cauchyCompletionSetoid injection
SetoidInjection.wellDefined CInjection {x} {y} x=y = injectionPreservesSetoid x y x=y
SetoidInjection.injective CInjection {x} {y} x=y = injectionPreservesSetoid' x y x=y

337
Fields/FieldOfFractions.agda
View File

@@ -15,179 +15,180 @@ open import Sets.EquivalenceRelations
open import Agda.Primitive using (Level; lzero; lsuc; _⊔_)
module Fields.FieldOfFractions where
fieldOfFractionsSet : {a b : _} {A : Set a} {S : Setoid {a} {b} A} {_+_ : A → A → A} {_*_ : A → A → A} → {R : Ring S _+_ _*_} → IntegralDomain R → Set (a ⊔ b)
fieldOfFractionsSet {A = A} {S = S} {R = R} I = (A && (Sg A (λ a → (Setoid._∼_ S a (Ring.0R R) → False))))
fieldOfFractionsSetoid : {a b : _} {A : Set a} {S : Setoid {a} {b} A} {_+_ : A → A → A} {_*_ : A → A → A} → {R : Ring S _+_ _*_} → (I : IntegralDomain R) → Setoid (fieldOfFractionsSet I)
Setoid._∼_ (fieldOfFractionsSetoid {S = S} {_*_ = _*_} I) (a ,, (b , b!=0)) (c ,, (d , d!=0)) = Setoid._∼_ S (a * d) (b * c)
Equivalence.reflexive (Setoid.eq (fieldOfFractionsSetoid {R = R} I)) {a ,, (b , b!=0)} = Ring.*Commutative R
Equivalence.symmetric (Setoid.eq (fieldOfFractionsSetoid {S = S} {R = R} I)) {a ,, (b , b!=0)} {c ,, (d , d!=0)} ad=bc = transitive (Ring.*Commutative R) (transitive (symmetric ad=bc) (Ring.*Commutative R))
where
open Equivalence (Setoid.eq S)
Equivalence.transitive (Setoid.eq (fieldOfFractionsSetoid {S = S} {_+_ = _+_} {_*_ = _*_} {R = R} I)) {a ,, (b , b!=0)} {c ,, (d , d!=0)} {e ,, (f , f!=0)} ad=bc cf=de = p5
where
open Setoid S
open Ring R
open Equivalence eq
p : (a * d) * f ∼ (b * c) * f
p = Ring.*WellDefined R ad=bc reflexive
p2 : (a * f) * d ∼ b * (d * e)
p2 = transitive (transitive (symmetric *Associative) (transitive (*WellDefined reflexive *Commutative) *Associative)) (transitive p (transitive (symmetric *Associative) (*WellDefined reflexive cf=de)))
p3 : (a * f) * d ∼ (b * e) * d
p3 = transitive p2 (transitive (*WellDefined reflexive *Commutative) *Associative)
p4 : (d ∼ 0R) || ((a * f) ∼ (b * e))
p4 = cancelIntDom I (transitive *Commutative (transitive p3 *Commutative))
p5 : (a * f) ∼ (b * e)
p5 with p4
p5 | inl d=0 = exFalso (d!=0 d=0)
p5 | inr x = x
fieldOfFractionsSet : {a b : _} {A : Set a} {S : Setoid {a} {b} A} {_+_ : A → A → A} {_*_ : A → A → A} → {R : Ring S _+_ _*_} → IntegralDomain R → Set (a ⊔ b)
fieldOfFractionsSet {A = A} {S = S} {R = R} I = (A && (Sg A (λ a → (Setoid._∼_ S a (Ring.0R R) → False))))
fieldOfFractionsPlus : {a b : _} {A : Set a} {S : Setoid {a} {b} A} {_+_ : A → A → A} {_*_ : A → A → A} {R : Ring S _+_ _*_} → (I : IntegralDomain R) → fieldOfFractionsSet I → fieldOfFractionsSet I → fieldOfFractionsSet I
fieldOfFractionsPlus {S = S} {_+_ = _+_} {_*_ = _*_} {R = R} I (a ,, (b , b!=0)) (c ,, (d , d!=0)) = (((a * d) + (b * c)) ,, ((b * d) , ans))
where
open Setoid S
open Ring R
ans : ((b * d) ∼ Ring.0R R) → False
ans pr with IntegralDomain.intDom I pr
ans pr | inl x = b!=0 x
ans pr | inr x = d!=0 x
fieldOfFractionsSetoid : {a b : _} {A : Set a} {S : Setoid {a} {b} A} {_+_ : A → A → A} {_*_ : A → A → A} → {R : Ring S _+_ _*_} → (I : IntegralDomain R) → Setoid (fieldOfFractionsSet I)
Setoid._∼_ (fieldOfFractionsSetoid {S = S} {_*_ = _*_} I) (a ,, (b , b!=0)) (c ,, (d , d!=0)) = Setoid._∼_ S (a * d) (b * c)
Equivalence.reflexive (Setoid.eq (fieldOfFractionsSetoid {R = R} I)) {a ,, (b , b!=0)} = Ring.*Commutative R
Equivalence.symmetric (Setoid.eq (fieldOfFractionsSetoid {S = S} {R = R} I)) {a ,, (b , b!=0)} {c ,, (d , d!=0)} ad=bc = transitive (Ring.*Commutative R) (transitive (symmetric ad=bc) (Ring.*Commutative R))
where
open Equivalence (Setoid.eq S)
Equivalence.transitive (Setoid.eq (fieldOfFractionsSetoid {S = S} {_+_ = _+_} {_*_ = _*_} {R = R} I)) {a ,, (b , b!=0)} {c ,, (d , d!=0)} {e ,, (f , f!=0)} ad=bc cf=de = p5
where
open Setoid S
open Ring R
open Equivalence eq
p : (a * d) * f ∼ (b * c) * f
p = Ring.*WellDefined R ad=bc reflexive
p2 : (a * f) * d ∼ b * (d * e)
p2 = transitive (transitive (symmetric *Associative) (transitive (*WellDefined reflexive *Commutative) *Associative)) (transitive p (transitive (symmetric *Associative) (*WellDefined reflexive cf=de)))
p3 : (a * f) * d ∼ (b * e) * d
p3 = transitive p2 (transitive (*WellDefined reflexive *Commutative) *Associative)
p4 : (d ∼ 0R) || ((a * f) ∼ (b * e))
p4 = cancelIntDom I (transitive *Commutative (transitive p3 *Commutative))
p5 : (a * f) ∼ (b * e)
p5 with p4
p5 | inl d=0 = exFalso (d!=0 d=0)
p5 | inr x = x
fieldOfFractionsTimes : {a b : _} {A : Set a} {S : Setoid {a} {b} A} {_+_ : A → A → A} {_*_ : A → A → A} {R : Ring S _+_ _*_} → (I : IntegralDomain R) → fieldOfFractionsSet I → fieldOfFractionsSet I → fieldOfFractionsSet I
fieldOfFractionsTimes {S = S} {_+_ = _+_} {_*_ = _*_} {R = R} I (a ,, (b , b!=0)) (c ,, (d , d!=0)) = (a * c) ,, ((b * d) , ans)
where
open Setoid S
open Ring R
ans : ((b * d) ∼ Ring.0R R) → False
ans pr with IntegralDomain.intDom I pr
ans pr | inl x = b!=0 x
ans pr | inr x = d!=0 x
fieldOfFractionsPlus : {a b : _} {A : Set a} {S : Setoid {a} {b} A} {_+_ : A → A → A} {_*_ : A → A → A} {R : Ring S _+_ _*_} → (I : IntegralDomain R) → fieldOfFractionsSet I → fieldOfFractionsSet I → fieldOfFractionsSet I
fieldOfFractionsPlus {S = S} {_+_ = _+_} {_*_ = _*_} {R = R} I (a ,, (b , b!=0)) (c ,, (d , d!=0)) = (((a * d) + (b * c)) ,, ((b * d) , ans))
where
open Setoid S
open Ring R
ans : ((b * d) ∼ Ring.0R R) → False
ans pr with IntegralDomain.intDom I pr
ans pr | inl x = b!=0 x
ans pr | inr x = d!=0 x
fieldOfFractionsRing : {a b : _} {A : Set a} {S : Setoid {a} {b} A} {_+_ : A → A → A} {_*_ : A → A → A} {R : Ring S _+_ _*_} → (I : IntegralDomain R) → Ring (fieldOfFractionsSetoid I) (fieldOfFractionsPlus I) (fieldOfFractionsTimes I)
Group.+WellDefined (Ring.additiveGroup (fieldOfFractionsRing {S = S} {_+_ = _+_} {_*_ = _*_} {R = R} I)) {a ,, (b , b!=0)} {c ,, (d , d!=0)} {e ,, (f , f!=0)} {g ,, (h , h!=0)} af=be ch=dg = need
where
open Setoid S
open Ring R
open Equivalence eq
have1 : (c * h) ∼ (d * g)
have1 = ch=dg
have2 : (a * f) ∼ (b * e)
have2 = af=be
need : (((a * d) + (b * c)) * (f * h)) ∼ ((b * d) * (((e * h) + (f * g))))
need = transitive (transitive (Ring.*Commutative R) (transitive (Ring.*DistributesOver+ R) (Group.+WellDefined (Ring.additiveGroup R) (transitive *Associative (transitive (*WellDefined (*Commutative) reflexive) (transitive (*WellDefined *Associative reflexive) (transitive (*WellDefined (*WellDefined have2 reflexive) reflexive) (transitive (symmetric *Associative) (transitive (*WellDefined reflexive *Commutative) (transitive *Associative (transitive (*WellDefined (transitive (transitive (symmetric *Associative) (*WellDefined reflexive *Commutative)) *Associative) reflexive) (symmetric *Associative))))))))) (transitive *Commutative (transitive (transitive (symmetric *Associative) (*WellDefined reflexive (transitive (*WellDefined reflexive *Commutative) (transitive *Associative (transitive (*WellDefined have1 reflexive) (transitive (symmetric *Associative) (*WellDefined reflexive *Commutative))))))) *Associative))))) (symmetric (Ring.*DistributesOver+ R))
Group.0G (Ring.additiveGroup (fieldOfFractionsRing {R = R} I)) = Ring.0R R ,, (Ring.1R R , IntegralDomain.nontrivial I)
Group.inverse (Ring.additiveGroup (fieldOfFractionsRing {R = R} I)) (a ,, b) = Group.inverse (Ring.additiveGroup R) a ,, b
Group.+Associative (Ring.additiveGroup (fieldOfFractionsRing {S = S} {_+_ = _+_} {_*_ = _*_} {R = R} I)) {a ,, (b , b!=0)} {c ,, (d , d!=0)} {e ,, (f , f!=0)} = need
where
open Setoid S
open Equivalence eq
need : (((a * (d * f)) + (b * ((c * f) + (d * e)))) * ((b * d) * f)) ∼ ((b * (d * f)) * ((((a * d) + (b * c)) * f) + ((b * d) * e)))
need = transitive (Ring.*Commutative R) (Ring.*WellDefined R (symmetric (Ring.*Associative R)) (transitive (Group.+WellDefined (Ring.additiveGroup R) reflexive (Ring.*DistributesOver+ R)) (transitive (Group.+WellDefined (Ring.additiveGroup R) reflexive (Group.+WellDefined (Ring.additiveGroup R) (Ring.*Associative R) (Ring.*Associative R))) (transitive (Group.+Associative (Ring.additiveGroup R)) (Group.+WellDefined (Ring.additiveGroup R) (transitive (transitive (Group.+WellDefined (Ring.additiveGroup R) (transitive (Ring.*Associative R) (Ring.*Commutative R)) (Ring.*Commutative R)) (symmetric (Ring.*DistributesOver+ R))) (Ring.*Commutative R)) reflexive)))))
Group.identRight (Ring.additiveGroup (fieldOfFractionsRing {S = S} {_+_ = _+_} {_*_ = _*_} {R = R} I)) {a ,, (b , b!=0)} = need
where
open Setoid S
open Equivalence eq
need : (((a * Ring.1R R) + (b * Group.0G (Ring.additiveGroup R))) * b) ∼ ((b * Ring.1R R) * a)
need = transitive (transitive (Ring.*WellDefined R (transitive (Group.+WellDefined (Ring.additiveGroup R) (transitive (Ring.*Commutative R) (Ring.identIsIdent R)) reflexive) (transitive (Group.+WellDefined (Ring.additiveGroup R) reflexive (Ring.timesZero R)) (Group.identRight (Ring.additiveGroup R)))) reflexive) (Ring.*Commutative R)) (symmetric (Ring.*WellDefined R (transitive (Ring.*Commutative R) (Ring.identIsIdent R)) reflexive))
Group.identLeft (Ring.additiveGroup (fieldOfFractionsRing {S = S} {_+_ = _+_} {_*_ = _*_} {R = R} I)) {a ,, (b , _)} = need
where
open Setoid S
open Equivalence eq
need : (((Group.0G (Ring.additiveGroup R) * b) + (Ring.1R R * a)) * b) ∼ ((Ring.1R R * b) * a)
need = transitive (transitive (Ring.*WellDefined R (transitive (Group.+WellDefined (Ring.additiveGroup R) reflexive (Ring.identIsIdent R)) (transitive (Group.+WellDefined (Ring.additiveGroup R) (transitive (Ring.*Commutative R) (Ring.timesZero R)) reflexive) (Group.identLeft (Ring.additiveGroup R)))) reflexive) (Ring.*Commutative R)) (Ring.*WellDefined R (symmetric (Ring.identIsIdent R)) reflexive)
Group.invLeft (Ring.additiveGroup (fieldOfFractionsRing {S = S} {_+_ = _+_} {_*_ = _*_} {R = R} I)) {a ,, (b , _)} = need
where
open Setoid S
open Equivalence eq
need : (((Group.inverse (Ring.additiveGroup R) a * b) + (b * a)) * Ring.1R R) ∼ ((b * b) * Group.0G (Ring.additiveGroup R))
need = transitive (transitive (transitive (Ring.*Commutative R) (Ring.identIsIdent R)) (transitive (Group.+WellDefined (Ring.additiveGroup R) (Ring.*Commutative R) reflexive) (transitive (symmetric (Ring.*DistributesOver+ R)) (transitive (Ring.*WellDefined R reflexive (Group.invLeft (Ring.additiveGroup R))) (Ring.timesZero R))))) (symmetric (Ring.timesZero R))
Group.invRight (Ring.additiveGroup (fieldOfFractionsRing {S = S} {_+_ = _+_} {_*_ = _*_} {R = R} I)) {a ,, (b , _)} = need
where
open Setoid S
open Equivalence eq
need : (((a * b) + (b * Group.inverse (Ring.additiveGroup R) a)) * Ring.1R R) ∼ ((b * b) * Group.0G (Ring.additiveGroup R))
need = transitive (transitive (transitive (Ring.*Commutative R) (Ring.identIsIdent R)) (transitive (Group.+WellDefined (Ring.additiveGroup R) (Ring.*Commutative R) reflexive) (transitive (symmetric (Ring.*DistributesOver+ R)) (transitive (Ring.*WellDefined R reflexive (Group.invRight (Ring.additiveGroup R))) (Ring.timesZero R))))) (symmetric (Ring.timesZero R))
Ring.*WellDefined (fieldOfFractionsRing {S = S} {_+_ = _+_} {_*_ = _*_} {R = R} I) {a ,, (b , _)} {c ,, (d , _)} {e ,, (f , _)} {g ,, (h , _)} af=be ch=dg = need
where
open Setoid S
open Equivalence eq
need : ((a * c) * (f * h)) ∼ ((b * d) * (e * g))
need = transitive (Ring.*WellDefined R reflexive (Ring.*Commutative R)) (transitive (Ring.*Associative R) (transitive (Ring.*WellDefined R (symmetric (Ring.*Associative R)) reflexive) (transitive (Ring.*WellDefined R (Ring.*WellDefined R reflexive ch=dg) reflexive) (transitive (Ring.*Commutative R) (transitive (Ring.*Associative R) (transitive (Ring.*WellDefined R (Ring.*Commutative R) reflexive) (transitive (Ring.*WellDefined R af=be reflexive) (transitive (Ring.*Associative R) (transitive (Ring.*WellDefined R (transitive (symmetric (Ring.*Associative R)) (transitive (Ring.*WellDefined R reflexive (Ring.*Commutative R)) (Ring.*Associative R))) reflexive) (symmetric (Ring.*Associative R)))))))))))
Ring.1R (fieldOfFractionsRing {R = R} I) = Ring.1R R ,, (Ring.1R R , IntegralDomain.nontrivial I)
Ring.groupIsAbelian (fieldOfFractionsRing {S = S} {_+_ = _+_} {_*_ = _*_} {R = R} I) {a ,, (b , _)} {c ,, (d , _)} = need
where
open Setoid S
open Equivalence eq
need : (((a * d) + (b * c)) * (d * b)) ∼ ((b * d) * ((c * b) + (d * a)))
need = transitive (Ring.*Commutative R) (Ring.*WellDefined R (Ring.*Commutative R) (transitive (Group.+WellDefined (Ring.additiveGroup R) (Ring.*Commutative R) (Ring.*Commutative R)) (Ring.groupIsAbelian R)))
Ring.*Associative (fieldOfFractionsRing {S = S} {_+_ = _+_} {_*_ = _*_} {R = R} I) {a ,, (b , _)} {c ,, (d , _)} {e ,, (f , _)} = need
where
open Setoid S
open Equivalence eq
need : ((a * (c * e)) * ((b * d) * f)) ∼ ((b * (d * f)) * ((a * c) * e))
need = transitive (Ring.*WellDefined R (Ring.*Associative R) (symmetric (Ring.*Associative R))) (Ring.*Commutative R)
Ring.*Commutative (fieldOfFractionsRing {S = S} {_+_ = _+_} {_*_ = _*_} {R = R} I) {a ,, (b , _)} {c ,, (d , _)} = need
where
open Setoid S
open Equivalence eq
need : ((a * c) * (d * b)) ∼ ((b * d) * (c * a))
need = transitive (Ring.*Commutative R) (Ring.*WellDefined R (Ring.*Commutative R) (Ring.*Commutative R))
Ring.*DistributesOver+ (fieldOfFractionsRing {S = S} {_+_ = _+_} {_*_ = _*_} {R = R} I) {a ,, (b , _)} {c ,, (d , _)} {e ,, (f , _)} = need
where
open Setoid S
open Ring R
open Equivalence eq
inter : b * (a * ((c * f) + (d * e))) ∼ (((a * c) * (b * f)) + ((b * d) * (a * e)))
inter = transitive *Associative (transitive *DistributesOver+ (Group.+WellDefined additiveGroup (transitive *Associative (transitive (*WellDefined (transitive (*WellDefined (*Commutative) reflexive) (transitive (symmetric *Associative) (transitive (*WellDefined reflexive *Commutative) *Associative))) reflexive) (symmetric *Associative))) (transitive *Associative (transitive (*WellDefined (transitive (symmetric *Associative) (transitive (*WellDefined reflexive *Commutative) *Associative)) reflexive) (symmetric *Associative)))))
need : ((a * ((c * f) + (d * e))) * ((b * d) * (b * f))) ∼ ((b * (d * f)) * (((a * c) * (b * f)) + ((b * d) * (a * e))))
need = transitive (Ring.*WellDefined R reflexive (Ring.*WellDefined R reflexive (Ring.*Commutative R))) (transitive (Ring.*WellDefined R reflexive (Ring.*Associative R)) (transitive (Ring.*Commutative R) (transitive (Ring.*WellDefined R (Ring.*WellDefined R (symmetric (Ring.*Associative R)) reflexive) reflexive) (transitive (symmetric (Ring.*Associative R)) (Ring.*WellDefined R reflexive inter)))))
Ring.identIsIdent (fieldOfFractionsRing {S = S} {_+_ = _+_} {_*_ = _*_} {R = R} I) {a ,, (b , _)} = need
where
open Setoid S
open Equivalence eq
need : (((Ring.1R R) * a) * b) ∼ (((Ring.1R R * b)) * a)
need = transitive (Ring.*WellDefined R (Ring.identIsIdent R) reflexive) (transitive (Ring.*Commutative R) (Ring.*WellDefined R (symmetric (Ring.identIsIdent R)) reflexive))
fieldOfFractionsTimes : {a b : _} {A : Set a} {S : Setoid {a} {b} A} {_+_ : A → A → A} {_*_ : A → A → A} {R : Ring S _+_ _*_} → (I : IntegralDomain R) → fieldOfFractionsSet I → fieldOfFractionsSet I → fieldOfFractionsSet I
fieldOfFractionsTimes {S = S} {_+_ = _+_} {_*_ = _*_} {R = R} I (a ,, (b , b!=0)) (c ,, (d , d!=0)) = (a * c) ,, ((b * d) , ans)
where
open Setoid S
open Ring R
ans : ((b * d) ∼ Ring.0R R) → False
ans pr with IntegralDomain.intDom I pr
ans pr | inl x = b!=0 x
ans pr | inr x = d!=0 x
fieldOfFractions : {a b : _} {A : Set a} {S : Setoid {a} {b} A} {_+_ : A → A → A} {_*_ : A → A → A} {R : Ring S _+_ _*_} → (I : IntegralDomain R) → Field (fieldOfFractionsRing I)
Field.allInvertible (fieldOfFractions {S = S} {_+_ = _+_} {_*_ = _*_} {R = R} I) (fst ,, (b , _)) prA = (b ,, (fst , ans)) , need
where
open Setoid S
open Equivalence eq
need : ((b * fst) * Ring.1R R) ∼ ((fst * b) * Ring.1R R)
need = Ring.*WellDefined R (Ring.*Commutative R) reflexive
ans : fst ∼ Ring.0R R → False
ans pr = prA need'
where
need' : (fst * Ring.1R R) ∼ (b * Ring.0R R)
need' = transitive (Ring.*WellDefined R pr reflexive) (transitive (transitive (Ring.*Commutative R) (Ring.timesZero R)) (symmetric (Ring.timesZero R)))
Field.nontrivial (fieldOfFractions {S = S} {_+_ = _+_} {_*_ = _*_} {R = R} I) pr = IntegralDomain.nontrivial I (symmetric (transitive (symmetric (Ring.timesZero R)) (transitive (Ring.*Commutative R) (transitive pr (Ring.identIsIdent R)))))
where
open Setoid S
open Equivalence eq
pr' : (Ring.0R R) * (Ring.1R R) ∼ (Ring.1R R) * (Ring.1R R)
pr' = pr
fieldOfFractionsRing : {a b : _} {A : Set a} {S : Setoid {a} {b} A} {_+_ : A → A → A} {_*_ : A → A → A} {R : Ring S _+_ _*_} → (I : IntegralDomain R) → Ring (fieldOfFractionsSetoid I) (fieldOfFractionsPlus I) (fieldOfFractionsTimes I)
Group.+WellDefined (Ring.additiveGroup (fieldOfFractionsRing {S = S} {_+_ = _+_} {_*_ = _*_} {R = R} I)) {a ,, (b , b!=0)} {c ,, (d , d!=0)} {e ,, (f , f!=0)} {g ,, (h , h!=0)} af=be ch=dg = need
where
open Setoid S
open Ring R
open Equivalence eq
have1 : (c * h) ∼ (d * g)
have1 = ch=dg
have2 : (a * f) ∼ (b * e)
have2 = af=be
need : (((a * d) + (b * c)) * (f * h)) ∼ ((b * d) * (((e * h) + (f * g))))
need = transitive (transitive (Ring.*Commutative R) (transitive (Ring.*DistributesOver+ R) (Group.+WellDefined (Ring.additiveGroup R) (transitive *Associative (transitive (*WellDefined (*Commutative) reflexive) (transitive (*WellDefined *Associative reflexive) (transitive (*WellDefined (*WellDefined have2 reflexive) reflexive) (transitive (symmetric *Associative) (transitive (*WellDefined reflexive *Commutative) (transitive *Associative (transitive (*WellDefined (transitive (transitive (symmetric *Associative) (*WellDefined reflexive *Commutative)) *Associative) reflexive) (symmetric *Associative))))))))) (transitive *Commutative (transitive (transitive (symmetric *Associative) (*WellDefined reflexive (transitive (*WellDefined reflexive *Commutative) (transitive *Associative (transitive (*WellDefined have1 reflexive) (transitive (symmetric *Associative) (*WellDefined reflexive *Commutative))))))) *Associative))))) (symmetric (Ring.*DistributesOver+ R))
Group.0G (Ring.additiveGroup (fieldOfFractionsRing {R = R} I)) = Ring.0R R ,, (Ring.1R R , IntegralDomain.nontrivial I)
Group.inverse (Ring.additiveGroup (fieldOfFractionsRing {R = R} I)) (a ,, b) = Group.inverse (Ring.additiveGroup R) a ,, b
Group.+Associative (Ring.additiveGroup (fieldOfFractionsRing {S = S} {_+_ = _+_} {_*_ = _*_} {R = R} I)) {a ,, (b , b!=0)} {c ,, (d , d!=0)} {e ,, (f , f!=0)} = need
where
open Setoid S
open Equivalence eq
need : (((a * (d * f)) + (b * ((c * f) + (d * e)))) * ((b * d) * f)) ∼ ((b * (d * f)) * ((((a * d) + (b * c)) * f) + ((b * d) * e)))
need = transitive (Ring.*Commutative R) (Ring.*WellDefined R (symmetric (Ring.*Associative R)) (transitive (Group.+WellDefined (Ring.additiveGroup R) reflexive (Ring.*DistributesOver+ R)) (transitive (Group.+WellDefined (Ring.additiveGroup R) reflexive (Group.+WellDefined (Ring.additiveGroup R) (Ring.*Associative R) (Ring.*Associative R))) (transitive (Group.+Associative (Ring.additiveGroup R)) (Group.+WellDefined (Ring.additiveGroup R) (transitive (transitive (Group.+WellDefined (Ring.additiveGroup R) (transitive (Ring.*Associative R) (Ring.*Commutative R)) (Ring.*Commutative R)) (symmetric (Ring.*DistributesOver+ R))) (Ring.*Commutative R)) reflexive)))))
Group.identRight (Ring.additiveGroup (fieldOfFractionsRing {S = S} {_+_ = _+_} {_*_ = _*_} {R = R} I)) {a ,, (b , b!=0)} = need
where
open Setoid S
open Equivalence eq
need : (((a * Ring.1R R) + (b * Group.0G (Ring.additiveGroup R))) * b) ∼ ((b * Ring.1R R) * a)
need = transitive (transitive (Ring.*WellDefined R (transitive (Group.+WellDefined (Ring.additiveGroup R) (transitive (Ring.*Commutative R) (Ring.identIsIdent R)) reflexive) (transitive (Group.+WellDefined (Ring.additiveGroup R) reflexive (Ring.timesZero R)) (Group.identRight (Ring.additiveGroup R)))) reflexive) (Ring.*Commutative R)) (symmetric (Ring.*WellDefined R (transitive (Ring.*Commutative R) (Ring.identIsIdent R)) reflexive))
Group.identLeft (Ring.additiveGroup (fieldOfFractionsRing {S = S} {_+_ = _+_} {_*_ = _*_} {R = R} I)) {a ,, (b , _)} = need
where
open Setoid S
open Equivalence eq
need : (((Group.0G (Ring.additiveGroup R) * b) + (Ring.1R R * a)) * b) ∼ ((Ring.1R R * b) * a)
need = transitive (transitive (Ring.*WellDefined R (transitive (Group.+WellDefined (Ring.additiveGroup R) reflexive (Ring.identIsIdent R)) (transitive (Group.+WellDefined (Ring.additiveGroup R) (transitive (Ring.*Commutative R) (Ring.timesZero R)) reflexive) (Group.identLeft (Ring.additiveGroup R)))) reflexive) (Ring.*Commutative R)) (Ring.*WellDefined R (symmetric (Ring.identIsIdent R)) reflexive)
Group.invLeft (Ring.additiveGroup (fieldOfFractionsRing {S = S} {_+_ = _+_} {_*_ = _*_} {R = R} I)) {a ,, (b , _)} = need
where
open Setoid S
open Equivalence eq
need : (((Group.inverse (Ring.additiveGroup R) a * b) + (b * a)) * Ring.1R R) ∼ ((b * b) * Group.0G (Ring.additiveGroup R))
need = transitive (transitive (transitive (Ring.*Commutative R) (Ring.identIsIdent R)) (transitive (Group.+WellDefined (Ring.additiveGroup R) (Ring.*Commutative R) reflexive) (transitive (symmetric (Ring.*DistributesOver+ R)) (transitive (Ring.*WellDefined R reflexive (Group.invLeft (Ring.additiveGroup R))) (Ring.timesZero R))))) (symmetric (Ring.timesZero R))
Group.invRight (Ring.additiveGroup (fieldOfFractionsRing {S = S} {_+_ = _+_} {_*_ = _*_} {R = R} I)) {a ,, (b , _)} = need
where
open Setoid S
open Equivalence eq
need : (((a * b) + (b * Group.inverse (Ring.additiveGroup R) a)) * Ring.1R R) ∼ ((b * b) * Group.0G (Ring.additiveGroup R))
need = transitive (transitive (transitive (Ring.*Commutative R) (Ring.identIsIdent R)) (transitive (Group.+WellDefined (Ring.additiveGroup R) (Ring.*Commutative R) reflexive) (transitive (symmetric (Ring.*DistributesOver+ R)) (transitive (Ring.*WellDefined R reflexive (Group.invRight (Ring.additiveGroup R))) (Ring.timesZero R))))) (symmetric (Ring.timesZero R))
Ring.*WellDefined (fieldOfFractionsRing {S = S} {_+_ = _+_} {_*_ = _*_} {R = R} I) {a ,, (b , _)} {c ,, (d , _)} {e ,, (f , _)} {g ,, (h , _)} af=be ch=dg = need
where
open Setoid S
open Equivalence eq
need : ((a * c) * (f * h)) ∼ ((b * d) * (e * g))
need = transitive (Ring.*WellDefined R reflexive (Ring.*Commutative R)) (transitive (Ring.*Associative R) (transitive (Ring.*WellDefined R (symmetric (Ring.*Associative R)) reflexive) (transitive (Ring.*WellDefined R (Ring.*WellDefined R reflexive ch=dg) reflexive) (transitive (Ring.*Commutative R) (transitive (Ring.*Associative R) (transitive (Ring.*WellDefined R (Ring.*Commutative R) reflexive) (transitive (Ring.*WellDefined R af=be reflexive) (transitive (Ring.*Associative R) (transitive (Ring.*WellDefined R (transitive (symmetric (Ring.*Associative R)) (transitive (Ring.*WellDefined R reflexive (Ring.*Commutative R)) (Ring.*Associative R))) reflexive) (symmetric (Ring.*Associative R)))))))))))
Ring.1R (fieldOfFractionsRing {R = R} I) = Ring.1R R ,, (Ring.1R R , IntegralDomain.nontrivial I)
Ring.groupIsAbelian (fieldOfFractionsRing {S = S} {_+_ = _+_} {_*_ = _*_} {R = R} I) {a ,, (b , _)} {c ,, (d , _)} = need
where
open Setoid S
open Equivalence eq
need : (((a * d) + (b * c)) * (d * b)) ∼ ((b * d) * ((c * b) + (d * a)))
need = transitive (Ring.*Commutative R) (Ring.*WellDefined R (Ring.*Commutative R) (transitive (Group.+WellDefined (Ring.additiveGroup R) (Ring.*Commutative R) (Ring.*Commutative R)) (Ring.groupIsAbelian R)))
Ring.*Associative (fieldOfFractionsRing {S = S} {_+_ = _+_} {_*_ = _*_} {R = R} I) {a ,, (b , _)} {c ,, (d , _)} {e ,, (f , _)} = need
where
open Setoid S
open Equivalence eq
need : ((a * (c * e)) * ((b * d) * f)) ∼ ((b * (d * f)) * ((a * c) * e))
need = transitive (Ring.*WellDefined R (Ring.*Associative R) (symmetric (Ring.*Associative R))) (Ring.*Commutative R)
Ring.*Commutative (fieldOfFractionsRing {S = S} {_+_ = _+_} {_*_ = _*_} {R = R} I) {a ,, (b , _)} {c ,, (d , _)} = need
where
open Setoid S
open Equivalence eq
need : ((a * c) * (d * b)) ∼ ((b * d) * (c * a))
need = transitive (Ring.*Commutative R) (Ring.*WellDefined R (Ring.*Commutative R) (Ring.*Commutative R))
Ring.*DistributesOver+ (fieldOfFractionsRing {S = S} {_+_ = _+_} {_*_ = _*_} {R = R} I) {a ,, (b , _)} {c ,, (d , _)} {e ,, (f , _)} = need
where
open Setoid S
open Ring R
open Equivalence eq
inter : b * (a * ((c * f) + (d * e))) ∼ (((a * c) * (b * f)) + ((b * d) * (a * e)))
inter = transitive *Associative (transitive *DistributesOver+ (Group.+WellDefined additiveGroup (transitive *Associative (transitive (*WellDefined (transitive (*WellDefined (*Commutative) reflexive) (transitive (symmetric *Associative) (transitive (*WellDefined reflexive *Commutative) *Associative))) reflexive) (symmetric *Associative))) (transitive *Associative (transitive (*WellDefined (transitive (symmetric *Associative) (transitive (*WellDefined reflexive *Commutative) *Associative)) reflexive) (symmetric *Associative)))))
need : ((a * ((c * f) + (d * e))) * ((b * d) * (b * f))) ∼ ((b * (d * f)) * (((a * c) * (b * f)) + ((b * d) * (a * e))))
need = transitive (Ring.*WellDefined R reflexive (Ring.*WellDefined R reflexive (Ring.*Commutative R))) (transitive (Ring.*WellDefined R reflexive (Ring.*Associative R)) (transitive (Ring.*Commutative R) (transitive (Ring.*WellDefined R (Ring.*WellDefined R (symmetric (Ring.*Associative R)) reflexive) reflexive) (transitive (symmetric (Ring.*Associative R)) (Ring.*WellDefined R reflexive inter)))))
Ring.identIsIdent (fieldOfFractionsRing {S = S} {_+_ = _+_} {_*_ = _*_} {R = R} I) {a ,, (b , _)} = need
where
open Setoid S
open Equivalence eq
need : (((Ring.1R R) * a) * b) ∼ (((Ring.1R R * b)) * a)
need = transitive (Ring.*WellDefined R (Ring.identIsIdent R) reflexive) (transitive (Ring.*Commutative R) (Ring.*WellDefined R (symmetric (Ring.identIsIdent R)) reflexive))
embedIntoFieldOfFractions : {a b : _} {A : Set a} {S : Setoid {a} {b} A} {_+_ : A → A → A} {_*_ : A → A → A} {R : Ring S _+_ _*_} (I : IntegralDomain R) → A → fieldOfFractionsSet I
embedIntoFieldOfFractions {R = R} I a = a ,, (Ring.1R R , IntegralDomain.nontrivial I)
fieldOfFractions : {a b : _} {A : Set a} {S : Setoid {a} {b} A} {_+_ : A → A → A} {_*_ : A → A → A} {R : Ring S _+_ _*_} → (I : IntegralDomain R) → Field (fieldOfFractionsRing I)
Field.allInvertible (fieldOfFractions {S = S} {_+_ = _+_} {_*_ = _*_} {R = R} I) (fst ,, (b , _)) prA = (b ,, (fst , ans)) , need
where
open Setoid S
open Equivalence eq
need : ((b * fst) * Ring.1R R) ∼ ((fst * b) * Ring.1R R)
need = Ring.*WellDefined R (Ring.*Commutative R) reflexive
ans : fst ∼ Ring.0R R → False
ans pr = prA need'
where
need' : (fst * Ring.1R R) ∼ (b * Ring.0R R)
need' = transitive (Ring.*WellDefined R pr reflexive) (transitive (transitive (Ring.*Commutative R) (Ring.timesZero R)) (symmetric (Ring.timesZero R)))
Field.nontrivial (fieldOfFractions {S = S} {_+_ = _+_} {_*_ = _*_} {R = R} I) pr = IntegralDomain.nontrivial I (symmetric (transitive (symmetric (Ring.timesZero R)) (transitive (Ring.*Commutative R) (transitive pr (Ring.identIsIdent R)))))
where
open Setoid S
open Equivalence eq
pr' : (Ring.0R R) * (Ring.1R R) ∼ (Ring.1R R) * (Ring.1R R)
pr' = pr
homIntoFieldOfFractions : {a b : _} {A : Set a} {S : Setoid {a} {b} A} {_+_ : A → A → A} {_*_ : A → A → A} {R : Ring S _+_ _*_} (I : IntegralDomain R) → RingHom R (fieldOfFractionsRing I) (embedIntoFieldOfFractions I)
RingHom.preserves1 (homIntoFieldOfFractions {S = S} I) = Equivalence.reflexive (Setoid.eq S)
RingHom.ringHom (homIntoFieldOfFractions {S = S} {R = R} I) {a} {b} = Equivalence.transitive (Setoid.eq S) (Ring.*WellDefined R (Equivalence.reflexive (Setoid.eq S)) (Ring.identIsIdent R)) (Ring.*Commutative R)
GroupHom.groupHom (RingHom.groupHom (homIntoFieldOfFractions {S = S} {_+_ = _+_} {_*_ = _*_} {R = R} I)) {x} {y} = need
where
open Setoid S
open Equivalence eq
need : ((x + y) * (Ring.1R R * Ring.1R R)) ∼ (Ring.1R R * ((x * Ring.1R R) + (Ring.1R R * y)))
need = transitive (transitive (Ring.*WellDefined R reflexive (Ring.identIsIdent R)) (transitive (Ring.*Commutative R) (transitive (Ring.identIsIdent R) (Group.+WellDefined (Ring.additiveGroup R) (symmetric (transitive (Ring.*Commutative R) (Ring.identIsIdent R))) (symmetric (Ring.identIsIdent R)))))) (symmetric (Ring.identIsIdent R))
GroupHom.wellDefined (RingHom.groupHom (homIntoFieldOfFractions {S = S} {R = R} I)) x=y = transitive (Ring.*Commutative R) (Ring.*WellDefined R reflexive x=y)
where
open Equivalence (Setoid.eq S)
embedIntoFieldOfFractions : {a b : _} {A : Set a} {S : Setoid {a} {b} A} {_+_ : A → A → A} {_*_ : A → A → A} {R : Ring S _+_ _*_} (I : IntegralDomain R) → A → fieldOfFractionsSet I
embedIntoFieldOfFractions {R = R} I a = a ,, (Ring.1R R , IntegralDomain.nontrivial I)
homIntoFieldOfFractionsIsInj : {a b : _} {A : Set a} {S : Setoid {a} {b} A} {_+_ : A → A → A} {_*_ : A → A → A} {R : Ring S _+_ _*_} (I : IntegralDomain R) → SetoidInjection S (fieldOfFractionsSetoid I) (embedIntoFieldOfFractions I)
SetoidInjection.wellDefined (homIntoFieldOfFractionsIsInj {S = S} {R = R} I) x=y = transitive (Ring.*Commutative R) (Ring.*WellDefined R reflexive x=y)
where
open Equivalence (Setoid.eq S)
SetoidInjection.injective (homIntoFieldOfFractionsIsInj {S = S} {R = R} I) x~y = transitive (symmetric identIsIdent) (transitive *Commutative (transitive x~y identIsIdent))
where
open Ring R
open Setoid S
open Equivalence eq
homIntoFieldOfFractions : {a b : _} {A : Set a} {S : Setoid {a} {b} A} {_+_ : A → A → A} {_*_ : A → A → A} {R : Ring S _+_ _*_} (I : IntegralDomain R) → RingHom R (fieldOfFractionsRing I) (embedIntoFieldOfFractions I)
RingHom.preserves1 (homIntoFieldOfFractions {S = S} I) = Equivalence.reflexive (Setoid.eq S)
RingHom.ringHom (homIntoFieldOfFractions {S = S} {R = R} I) {a} {b} = Equivalence.transitive (Setoid.eq S) (Ring.*WellDefined R (Equivalence.reflexive (Setoid.eq S)) (Ring.identIsIdent R)) (Ring.*Commutative R)
GroupHom.groupHom (RingHom.groupHom (homIntoFieldOfFractions {S = S} {_+_ = _+_} {_*_ = _*_} {R = R} I)) {x} {y} = need
where
open Setoid S
open Equivalence eq
need : ((x + y) * (Ring.1R R * Ring.1R R)) ∼ (Ring.1R R * ((x * Ring.1R R) + (Ring.1R R * y)))
need = transitive (transitive (Ring.*WellDefined R reflexive (Ring.identIsIdent R)) (transitive (Ring.*Commutative R) (transitive (Ring.identIsIdent R) (Group.+WellDefined (Ring.additiveGroup R) (symmetric (transitive (Ring.*Commutative R) (Ring.identIsIdent R))) (symmetric (Ring.identIsIdent R)))))) (symmetric (Ring.identIsIdent R))
GroupHom.wellDefined (RingHom.groupHom (homIntoFieldOfFractions {S = S} {R = R} I)) x=y = transitive (Ring.*Commutative R) (Ring.*WellDefined R reflexive x=y)
where
open Equivalence (Setoid.eq S)
homIntoFieldOfFractionsIsInj : {a b : _} {A : Set a} {S : Setoid {a} {b} A} {_+_ : A → A → A} {_*_ : A → A → A} {R : Ring S _+_ _*_} (I : IntegralDomain R) → SetoidInjection S (fieldOfFractionsSetoid I) (embedIntoFieldOfFractions I)
SetoidInjection.wellDefined (homIntoFieldOfFractionsIsInj {S = S} {R = R} I) x=y = transitive (Ring.*Commutative R) (Ring.*WellDefined R reflexive x=y)
where
open Equivalence (Setoid.eq S)
SetoidInjection.injective (homIntoFieldOfFractionsIsInj {S = S} {R = R} I) x~y = transitive (symmetric identIsIdent) (transitive *Commutative (transitive x~y identIsIdent))
where
open Ring R
open Setoid S
open Equivalence eq

125
Fields/Fields.agda
View File

@@ -16,73 +16,74 @@ open import Sets.EquivalenceRelations
open import Agda.Primitive using (Level; lzero; lsuc; _⊔_)
module Fields.Fields where
record Field {m n : _} {A : Set m} {S : Setoid {m} {n} A} {_+_ : A → A → A} {_*_ : A → A → A} (R : Ring S _+_ _*_) : Set (lsuc m ⊔ n) where
open Ring R
open Group additiveGroup
record Field {m n : _} {A : Set m} {S : Setoid {m} {n} A} {_+_ : A → A → A} {_*_ : A → A → A} (R : Ring S _+_ _*_) : Set (lsuc m ⊔ n) where
open Ring R
open Group additiveGroup
open Setoid S
field
allInvertible : (a : A) → ((a ∼ Group.0G (Ring.additiveGroup R)) → False) → Sg A (λ t → t * a ∼ 1R)
nontrivial : (0R ∼ 1R) → False
orderedFieldIsIntDom : {a b c : _} {A : Set a} {S : Setoid {a} {b} A} {_+_ : A → A → A} {_*_ : A → A → A} {_<_ : Rel {_} {c} A} {R : Ring S _+_ _*_} {pOrder : SetoidPartialOrder S _<_} {tOrder : SetoidTotalOrder pOrder} (O : OrderedRing R tOrder) (F : Field R) → IntegralDomain R
IntegralDomain.intDom (orderedFieldIsIntDom {S = S} {_*_ = _*_} {R = R} {tOrder = tOrder} O F) {a} {b} ab=0 with SetoidTotalOrder.totality tOrder (Ring.0R R) a
IntegralDomain.intDom (orderedFieldIsIntDom {A = A} {S = S} {_*_ = _*_} {R = R} {pOrder = pOrder} {tOrder = tOrder} O F) {a} {b} ab=0 | inl (inl x) = inr (transitive (transitive (symmetric identIsIdent) (*WellDefined q reflexive)) p')
where
open Setoid S
field
allInvertible : (a : A) → ((a ∼ Group.0G (Ring.additiveGroup R)) → False) → Sg A (λ t → t * a ∼ 1R)
nontrivial : (0R ∼ 1R) → False
open Equivalence eq
open Ring R
a!=0 : (a ∼ Group.0G additiveGroup) → False
a!=0 pr = SetoidPartialOrder.irreflexive pOrder (SetoidPartialOrder.<WellDefined pOrder (symmetric pr) reflexive x)
invA : A
invA = underlying (Field.allInvertible F a a!=0)
q : 1R ∼ (invA * a)
q with Field.allInvertible F a a!=0
... | invA , pr = symmetric pr
p : invA * (a * b) ∼ invA * Group.0G additiveGroup
p = *WellDefined reflexive ab=0
p' : (invA * a) * b ∼ Group.0G additiveGroup
p' = transitive (symmetric *Associative) (transitive p (Ring.timesZero R))
IntegralDomain.intDom (orderedFieldIsIntDom {A = A} {S = S} {_*_ = _*_} {R = R} {pOrder = pOrder} {tOrder = tOrder} O F) {a} {b} ab=0 | inl (inr x) = inr (transitive (transitive (symmetric identIsIdent) (*WellDefined q reflexive)) p')
where
open Setoid S
open Equivalence eq
open Ring R
a!=0 : (a ∼ Group.0G additiveGroup) → False
a!=0 pr = SetoidPartialOrder.irreflexive pOrder (SetoidPartialOrder.<WellDefined pOrder reflexive (symmetric pr) x)
invA : A
invA = underlying (Field.allInvertible F a a!=0)
q : 1R ∼ (invA * a)
q with Field.allInvertible F a a!=0
... | invA , pr = symmetric pr
p : invA * (a * b) ∼ invA * Group.0G additiveGroup
p = *WellDefined reflexive ab=0
p' : (invA * a) * b ∼ Group.0G additiveGroup
p' = transitive (symmetric *Associative) (transitive p (Ring.timesZero R))
IntegralDomain.intDom (orderedFieldIsIntDom {S = S} {_*_ = _*_} {R = R} {tOrder = tOrder} O F) {a} {b} ab=0 | inr x = inl (Equivalence.symmetric (Setoid.eq S) x)
IntegralDomain.nontrivial (orderedFieldIsIntDom {S = S} O F) pr = Field.nontrivial F (Equivalence.symmetric (Setoid.eq S) pr)
orderedFieldIsIntDom : {a b c : _} {A : Set a} {S : Setoid {a} {b} A} {_+_ : A → A → A} {_*_ : A → A → A} {_<_ : Rel {_} {c} A} {R : Ring S _+_ _*_} {pOrder : SetoidPartialOrder S _<_} {tOrder : SetoidTotalOrder pOrder} (O : OrderedRing R tOrder) (F : Field R) → IntegralDomain R
IntegralDomain.intDom (orderedFieldIsIntDom {S = S} {_*_ = _*_} {R = R} {tOrder = tOrder} O F) {a} {b} ab=0 with SetoidTotalOrder.totality tOrder (Ring.0R R) a
IntegralDomain.intDom (orderedFieldIsIntDom {A = A} {S = S} {_*_ = _*_} {R = R} {pOrder = pOrder} {tOrder = tOrder} O F) {a} {b} ab=0 | inl (inl x) = inr (transitive (transitive (symmetric identIsIdent) (*WellDefined q reflexive)) p')
where
open Setoid S
open Equivalence eq
open Ring R
a!=0 : (a ∼ Group.0G additiveGroup) → False
a!=0 pr = SetoidPartialOrder.irreflexive pOrder (SetoidPartialOrder.<WellDefined pOrder (symmetric pr) reflexive x)
invA : A
invA = underlying (Field.allInvertible F a a!=0)
q : 1R ∼ (invA * a)
q with Field.allInvertible F a a!=0
... | invA , pr = symmetric pr
p : invA * (a * b) ∼ invA * Group.0G additiveGroup
p = *WellDefined reflexive ab=0
p' : (invA * a) * b ∼ Group.0G additiveGroup
p' = transitive (symmetric *Associative) (transitive p (Ring.timesZero R))
IntegralDomain.intDom (orderedFieldIsIntDom {A = A} {S = S} {_*_ = _*_} {R = R} {pOrder = pOrder} {tOrder = tOrder} O F) {a} {b} ab=0 | inl (inr x) = inr (transitive (transitive (symmetric identIsIdent) (*WellDefined q reflexive)) p')
where
open Setoid S
open Equivalence eq
open Ring R
a!=0 : (a ∼ Group.0G additiveGroup) → False
a!=0 pr = SetoidPartialOrder.irreflexive pOrder (SetoidPartialOrder.<WellDefined pOrder reflexive (symmetric pr) x)
invA : A
invA = underlying (Field.allInvertible F a a!=0)
q : 1R ∼ (invA * a)
q with Field.allInvertible F a a!=0
... | invA , pr = symmetric pr
p : invA * (a * b) ∼ invA * Group.0G additiveGroup
p = *WellDefined reflexive ab=0
p' : (invA * a) * b ∼ Group.0G additiveGroup
p' = transitive (symmetric *Associative) (transitive p (Ring.timesZero R))
IntegralDomain.intDom (orderedFieldIsIntDom {S = S} {_*_ = _*_} {R = R} {tOrder = tOrder} O F) {a} {b} ab=0 | inr x = inl (Equivalence.symmetric (Setoid.eq S) x)
IntegralDomain.nontrivial (orderedFieldIsIntDom {S = S} O F) pr = Field.nontrivial F (Equivalence.symmetric (Setoid.eq S) pr)
record Field' {m n : _} : Set (lsuc m ⊔ lsuc n) where
field
A : Set m
S : Setoid {m} {n} A
_+_ : A → A → A
_*_ : A → A → A
R : Ring S _+_ _*_
isField : Field R
record Field' {m n : _} : Set (lsuc m ⊔ lsuc n) where
field
A : Set m
S : Setoid {m} {n} A
_+_ : A → A → A
_*_ : A → A → A
R : Ring S _+_ _*_
isField : Field R
encapsulateField : {m n : _} {A : Set m} {S : Setoid {m} {n} A} {_+_ : A → A → A} {_*_ : A → A → A} {R : Ring S _+_ _*_} (F : Field R) → Field'
encapsulateField {A = A} {S = S} {_+_} {_*_} {R} F = record { A = A ; S = S ; _+_ = _+_ ; _*_ = _*_ ; R = R ; isField = F }
encapsulateField : {m n : _} {A : Set m} {S : Setoid {m} {n} A} {_+_ : A → A → A} {_*_ : A → A → A} {R : Ring S _+_ _*_} (F : Field R) → Field'
encapsulateField {A = A} {S = S} {_+_} {_*_} {R} F = record { A = A ; S = S ; _+_ = _+_ ; _*_ = _*_ ; R = R ; isField = F }
{-
record OrderedField {n} {A : Set n} {R : Ring A} (F : Field R) : Set (lsuc n) where
open Field F
field
ord : TotalOrder A
open TotalOrder ord
open Ring R
field
productPos : {a b : A} → (0R < a) → (0R < b) → (0R < (a * b))
orderRespectsAddition : {a b c : A} → (a < b) → (a + c) < (b + c)
record OrderedField {n} {A : Set n} {R : Ring A} (F : Field R) : Set (lsuc n) where
open Field F
field
ord : TotalOrder A
open TotalOrder ord
open Ring R
field
productPos : {a b : A} → (0R < a) → (0R < b) → (0R < (a * b))
orderRespectsAddition : {a b c : A} → (a < b) → (a + c) < (b + c)
-}

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{-# OPTIONS --safe --warning=error --without-K #-}
open import Agda.Primitive using (Level; lzero; lsuc; _⊔_)
open import Setoids.Setoids
open import Rings.Definition
open import Rings.Lemmas
open import Rings.Order
open import Groups.Definition
open import Groups.Groups
open import Fields.Fields
open import Sets.EquivalenceRelations
open import Setoids.Orders
open import Functions
open import LogicalFormulae
open import Numbers.Naturals.Naturals
module Fields.Lemmas {m n : _} {A : Set m} {S : Setoid {m} {n} A} {_+_ : A → A → A} {_*_ : A → A → A} {R : Ring S _+_ _*_} (F : Field R) where
open Setoid S
open Field F
open Ring R
open Group additiveGroup
halve : (charNot2 : ((1R + 1R) ∼ 0R) → False) → (a : A) → Sg A (λ i → i + i ∼ a)
halve charNot2 a with allInvertible (1R + 1R) charNot2
... | 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))))
where
r : a + a ∼ (1R + 1R) * a
r = Equivalence.symmetric eq (Equivalence.transitive eq *Commutative (Equivalence.transitive eq *DistributesOver+ (+WellDefined (Equivalence.transitive eq *Commutative identIsIdent) (Equivalence.transitive eq *Commutative identIsIdent))))
halfHalves : {x : A} (1/2 : A) (pr : 1/2 + 1/2 ∼ 1R) → (x + x) * 1/2 ∼ x
halfHalves {x} 1/2 pr = Equivalence.transitive eq (Equivalence.transitive eq (Equivalence.transitive eq *Commutative (Equivalence.transitive eq (Equivalence.transitive eq *DistributesOver+ (Equivalence.transitive eq (+WellDefined *Commutative *Commutative) (Equivalence.symmetric eq *DistributesOver+))) *Commutative)) (*WellDefined pr (Equivalence.reflexive eq))) identIsIdent

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{-# OPTIONS --safe --warning=error --without-K #-}
open import LogicalFormulae
open import Groups.Groups
open import Groups.Definition
open import Rings.Definition
open import Rings.Order
open import Rings.Lemmas
open import Setoids.Setoids
open import Setoids.Orders
open import Orders
open import Rings.IntegralDomains
open import Functions
open import Sets.EquivalenceRelations
open import Fields.Fields
open import Agda.Primitive using (Level; lzero; lsuc; _⊔_)
module Fields.Orders.Definition {m n : _} {A : Set m} {S : Setoid {m} {n} A} {_+_ : A → A → A} {_*_ : A → A → A} {R : Ring S _+_ _*_} (F : Field R) where
open Ring R
open import Fields.Lemmas F
record OrderedField {p} {_<_ : Rel {_} {p} A} {pOrder : SetoidPartialOrder S _<_} (order : SetoidTotalOrder pOrder) : Set (lsuc (m ⊔ n ⊔ p)) where
field
oRing : OrderedRing R order

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{-# OPTIONS --safe --warning=error --without-K #-}
open import LogicalFormulae
open import Groups.Groups
open import Groups.Definition
open import Rings.Definition
open import Rings.Order
open import Rings.Lemmas
open import Setoids.Setoids
open import Setoids.Orders
open import Orders
open import Rings.IntegralDomains
open import Functions
open import Sets.EquivalenceRelations
open import Fields.Fields
open import Fields.Orders.Definition
open import Agda.Primitive using (Level; lzero; lsuc; _⊔_)
module Fields.Orders.Lemmas {m n o : _} {A : Set m} {S : Setoid {m} {n} A} {_+_ : A → A → A} {_*_ : A → A → A} {_<_ : Rel {_} {o} A} {R : Ring S _+_ _*_} {pOrder : SetoidPartialOrder S _<_} {tOrder : SetoidTotalOrder pOrder} {F : Field R} (oF : OrderedField F tOrder) where
open Ring R
open Group additiveGroup
open OrderedRing (OrderedField.oRing oF)
open import Rings.Orders.Lemmas (OrderedField.oRing oF)
open import Fields.Lemmas F
open Setoid S
open SetoidPartialOrder pOrder
clearDenominatorHalf : (x y 1/2 : A) → (1/2 + 1/2 ∼ 1R) → x < (y * 1/2) → (x + x) < y
clearDenominatorHalf x y 1/2 pr1/2 x<1/2y = <WellDefined (Equivalence.reflexive eq) (Equivalence.transitive eq (Equivalence.transitive eq (Equivalence.symmetric eq *DistributesOver+) (Equivalence.transitive eq *Commutative (*WellDefined pr1/2 (Equivalence.reflexive eq)))) identIsIdent) (ringAddInequalities x<1/2y x<1/2y)
clearDenominatorHalf' : (x y 1/2 : A) → (1/2 + 1/2 ∼ 1R) → (x * 1/2) < y → x < (y + y)
clearDenominatorHalf' x y 1/2 pr1/2 1/2x<y = <WellDefined (Equivalence.transitive eq (Equivalence.symmetric eq *DistributesOver+) (Equivalence.transitive eq (Equivalence.transitive eq *Commutative (*WellDefined pr1/2 (Equivalence.reflexive eq))) identIsIdent)) (Equivalence.reflexive eq) (ringAddInequalities 1/2x<y 1/2x<y)
halveInequality : (x y 1/2 : A) → (1/2 + 1/2 ∼ 1R) → (x + x) < y → x < (y * 1/2)
halveInequality x y 1/2 pr1/2 2x<y with SetoidTotalOrder.totality tOrder 0R 1R
... | inl (inl 0<1') = <WellDefined (halfHalves 1/2 pr1/2) (Equivalence.reflexive eq) (ringCanMultiplyByPositive {_} {_} {1/2} (halvePositive 1/2 (<WellDefined (Equivalence.reflexive eq) (Equivalence.symmetric eq pr1/2) (0<1 λ bad → irreflexive {0R} (<WellDefined (Equivalence.reflexive eq) (Equivalence.symmetric eq bad) 0<1')))) 2x<y)
... | inl (inr 1<0) = <WellDefined (halfHalves 1/2 pr1/2) (Equivalence.reflexive eq) (ringCanMultiplyByPositive {_} {_} {1/2} (halvePositive 1/2 (<WellDefined (Equivalence.reflexive eq) (Equivalence.symmetric eq pr1/2) (0<1 λ bad → irreflexive {0R} (<WellDefined (Equivalence.symmetric eq bad) (Equivalence.reflexive eq) 1<0)))) 2x<y)
... | inr 0=1 = exFalso (irreflexive {0R} (<WellDefined (oneZeroImpliesAllZero R 0=1) (oneZeroImpliesAllZero R 0=1) 2x<y))
halveInequality' : (x y 1/2 : A) → (1/2 + 1/2 ∼ 1R) → x < (y + y) → (x * 1/2) < y
halveInequality' x y 1/2 pr1/2 x<2y with halveInequality (inverse y) (inverse x) 1/2 pr1/2 (<WellDefined (invContravariant additiveGroup) (Equivalence.reflexive eq) (ringSwapNegatives' x<2y))
... | bl = ringSwapNegatives (<WellDefined (Equivalence.reflexive eq) (ringMinusExtracts' R) bl)
dense : (charNot2 : ((1R + 1R) ∼ 0R) → False) {x y : A} → (x < y) → Sg A (λ i → (x < i) && (i < y))
dense charNot2 {x} {y} x<y with halve charNot2 1R
dense charNot2 {x} {y} x<y | 1/2 , pr1/2 = ((x + y) * 1/2) , (halveInequality x (x + y) 1/2 pr1/2 (<WellDefined (Equivalence.reflexive eq) groupIsAbelian (orderRespectsAddition x<y x)) ,, halveInequality' (x + y) y 1/2 pr1/2 (orderRespectsAddition x<y y))

810
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@@ -11,422 +11,422 @@ open import Sets.EquivalenceRelations
module Groups.Groups where
reflGroupWellDefined : {lvl : _} {A : Set lvl} {m n x y : A} {op : A → A → A} → m ≡ x → n ≡ y → (op m n) ≡ (op x y)
reflGroupWellDefined {lvl} {A} {m} {n} {.m} {.n} {op} refl refl = refl
reflGroupWellDefined : {lvl : _} {A : Set lvl} {m n x y : A} {op : A → A → A} → m ≡ x → n ≡ y → (op m n) ≡ (op x y)
reflGroupWellDefined {lvl} {A} {m} {n} {.m} {.n} {op} refl refl = refl
record AbelianGroup {a} {b} {A : Set a} {S : Setoid {a} {b} A} {_·_ : A → A → A} (G : Group S _·_) : Set (lsuc a ⊔ b) where
open Setoid S
field
commutative : {a b : A} → (a · b) ∼ (b · a)
record AbelianGroup {a} {b} {A : Set a} {S : Setoid {a} {b} A} {_·_ : A → A → A} (G : Group S _·_) : Set (lsuc a ⊔ b) where
open Setoid S
field
commutative : {a b : A} → (a · b) ∼ (b · a)
groupsHaveLeftCancellation : {a b : _} → {A : Set a} → {_·_ : A → A → A} → {S : Setoid {a} {b} A} → (G : Group S _·_) → (x y z : A) → (Setoid._∼_ S (x · y) (x · z)) → (Setoid._∼_ S y z)
groupsHaveLeftCancellation {_·_ = _·_} {S} g x y z pr = o
groupsHaveLeftCancellation : {a b : _} → {A : Set a} → {_·_ : A → A → A} → {S : Setoid {a} {b} A} → (G : Group S _·_) → (x y z : A) → (Setoid._∼_ S (x · y) (x · z)) → (Setoid._∼_ S y z)
groupsHaveLeftCancellation {_·_ = _·_} {S} g x y z pr = o
where
open Group g renaming (inverse to _^-1)
open Setoid S
transitive = Equivalence.transitive (Setoid.eq S)
symmetric = Equivalence.symmetric (Setoid.eq S)
j : ((x ^-1) · x) · y ∼ (x ^-1) · (x · z)
j = Equivalence.transitive eq (symmetric (+Associative {x ^-1} {x} {y})) (+WellDefined ~refl pr)
k : ((x ^-1) · x) · y ∼ ((x ^-1) · x) · z
k = transitive j +Associative
l : 0G · y ∼ ((x ^-1) · x) · z
l = transitive (+WellDefined (symmetric invLeft) ~refl) k
m : 0G · y ∼ 0G · z
m = transitive l (+WellDefined invLeft ~refl)
n : y ∼ 0G · z
n = transitive (symmetric identLeft) m
o : y ∼ z
o = transitive n identLeft
groupsHaveRightCancellation : {a b : _} → {A : Set a} → {_·_ : A → A → A} → {S : Setoid {a} {b} A} → (G : Group S _·_) → (x y z : A) → (Setoid._∼_ S (y · x) (z · x)) → (Setoid._∼_ S y z)
groupsHaveRightCancellation {_·_ = _·_} {S} g x y z pr = transitive m identRight
where
open Group g renaming (inverse to _^-1)
open Setoid S
transitive = Equivalence.transitive (Setoid.eq S)
symmetric = Equivalence.symmetric (Setoid.eq S)
i : (y · x) · (x ^-1) ∼ (z · x) · (x ^-1)
i = +WellDefined pr ~refl
j : y · (x · (x ^-1)) ∼ (z · x) · (x ^-1)
j = transitive +Associative i
j' : y · 0G ∼ (z · x) · (x ^-1)
j' = transitive (+WellDefined ~refl (symmetric invRight)) j
k : y ∼ (z · x) · (x ^-1)
k = transitive (symmetric identRight) j'
l : y ∼ z · (x · (x ^-1))
l = transitive k (symmetric +Associative)
m : y ∼ z · 0G
m = transitive l (+WellDefined ~refl invRight)
replaceGroupOp : {l m : _} {A : Set l} {S : Setoid {l} {m} A} {_·_ : A → A → A} → (G : Group S _·_) → {a b c d w x y z : A} → (Setoid._∼_ S a c) → (Setoid._∼_ S b d) → (Setoid._∼_ S w y) → (Setoid._∼_ S x z) → Setoid._∼_ S (a · b) (w · x) → Setoid._∼_ S (c · d) (y · z)
replaceGroupOp {S = S} {_·_} G a~c b~d w~y x~z pr = transitive (symmetric (+WellDefined a~c b~d)) (transitive pr (+WellDefined w~y x~z))
where
open Group G
open Setoid S
open Equivalence eq
replaceGroupOpRight : {l m : _} {A : Set l} {S : Setoid {l} {m} A} {_·_ : A → A → A} → (G : Group S _·_) → {a b c x : A} → (Setoid._∼_ S a (b · c)) → (Setoid._∼_ S c x) → (Setoid._∼_ S a (b · x))
replaceGroupOpRight {S = S} {_·_} G a~bc c~x = transitive a~bc (+WellDefined reflexive c~x)
where
open Group G
open Setoid S
open Equivalence eq
inverseWellDefined : {l m : _} {A : Set l} {S : Setoid {l} {m} A} {_·_ : A → A → A} (G : Group S _·_) → {x y : A} → (Setoid._∼_ S x y) → Setoid._∼_ S (Group.inverse G x) (Group.inverse G y)
inverseWellDefined {S = S} {_·_} G {x} {y} x~y = groupsHaveRightCancellation G x (inverse x) (inverse y) q
where
open Group G
open Setoid S
open Equivalence eq
p : inverse x · x ∼ inverse y · y
p = transitive invLeft (symmetric invLeft)
q : inverse x · x ∼ inverse y · x
q = replaceGroupOpRight G {_·_ (inverse x) x} {inverse y} {y} {x} p (symmetric x~y)
rightInversesAreUnique : {a b : _} → {A : Set a} → {S : Setoid {a} {b} A} → {_·_ : A → A → A} → (G : Group S _·_) → (x : A) → (y : A) → Setoid._∼_ S (y · x) (Group.0G G) → Setoid._∼_ S y (Group.inverse G x)
rightInversesAreUnique {S = S} {_·_} G x y f = transitive i (transitive j (transitive k (transitive l m)))
where
open Group G renaming (inverse to _^-1)
open Setoid S
open Equivalence eq
i : y ∼ y · 0G
j : y · 0G ∼ y · (x · (x ^-1))
k : y · (x · (x ^-1)) ∼ (y · x) · (x ^-1)
l : (y · x) · (x ^-1) ∼ 0G · (x ^-1)
m : 0G · (x ^-1) ∼ x ^-1
i = symmetric identRight
j = +WellDefined ~refl (symmetric invRight)
k = +Associative
l = +WellDefined f ~refl
m = identLeft
leftInversesAreUnique : {a b : _} → {A : Set a} → {S : Setoid {a} {b} A} → {_·_ : A → A → A} → (G : Group S _·_) → {x : A} → {y : A} → Setoid._∼_ S (x · y) (Group.0G G) → Setoid._∼_ S y (Group.inverse G x)
leftInversesAreUnique {S = S} {_·_} G {x} {y} f = rightInversesAreUnique G x y l
where
open Group G renaming (inverse to _^-1)
open Setoid S
open Equivalence eq
i : y · (x · y) ∼ y · 0G
i' : y · (x · y) ∼ y
j : (y · x) · y ∼ y
k : (y · x) · y ∼ 0G · y
l : y · x ∼ 0G
i = +WellDefined ~refl f
i' = transitive i identRight
j = transitive (symmetric +Associative) i'
k = transitive j (symmetric identLeft)
l = groupsHaveRightCancellation G y (y · x) 0G k
invTwice : {a b : _} → {A : Set a} → {S : Setoid {a} {b} A} → {_·_ : A → A → A} → (G : Group S _·_) → (x : A) → Setoid._∼_ S (Group.inverse G (Group.inverse G x)) x
invTwice {S = S} {_·_} G x = symmetric (rightInversesAreUnique G (x ^-1) x invRight)
where
open Group G renaming (inverse to _^-1)
open Setoid S
open Equivalence eq
fourWay+Associative : {a b : _} → {A : Set a} → {S : Setoid {a} {b} A} → {_·_ : A → A → A} → (G : Group S _·_) → {r s t u : A} → (Setoid._∼_ S) (r · ((s · t) · u)) ((r · s) · (t · u))
fourWay+Associative {S = S} {_·_} G {r} {s} {t} {u} = transitive p1 (transitive p2 p3)
where
open Group G renaming (inverse to _^-1)
open Setoid S
open Equivalence eq
p1 : r · ((s · t) · u) ∼ (r · (s · t)) · u
p2 : (r · (s · t)) · u ∼ ((r · s) · t) · u
p3 : ((r · s) · t) · u ∼ (r · s) · (t · u)
p1 = Group.+Associative G
p2 = Group.+WellDefined G (Group.+Associative G) reflexive
p3 = symmetric (Group.+Associative G)
fourWay+Associative' : {m n : _} {A : Set m} {S : Setoid {m} {n} A} {_·_ : A → A → A} (G : Group S _·_) {a b c d : A} → (Setoid._∼_ S (((a · b) · c) · d) (a · ((b · c) · d)))
fourWay+Associative' {S = S} G = transitive (symmetric +Associative) (symmetric (fourWay+Associative G))
where
open Group G
open Setoid S
open Equivalence eq
fourWay+Associative'' : {m n : _} {A : Set m} {S : Setoid {m} {n} A} {_·_ : A → A → A} (G : Group S _·_) {a b c d : A} → (Setoid._∼_ S (a · (b · (c · d))) (a · ((b · c) · d)))
fourWay+Associative'' {S = S} {_·_ = _·_} G = transitive +Associative (symmetric (fourWay+Associative G))
where
open Group G
open Setoid S
open Equivalence eq
invContravariant : {a b : _} → {A : Set a} → {S : Setoid {a} {b} A} → {_·_ : A → A → A} → (G : Group S _·_) → {x y : A} → (Setoid._∼_ S (Group.inverse G (x · y)) ((Group.inverse G y) · (Group.inverse G x)))
invContravariant {S = S} {_·_} G {x} {y} = ans
where
open Group G renaming (inverse to _^-1)
open Setoid S
open Equivalence eq
otherInv = (y ^-1) · (x ^-1)
many+Associatives : x · ((y · (y ^-1)) · (x ^-1)) ∼ (x · y) · ((y ^-1) · (x ^-1))
oneMult : (x · y) · otherInv ∼ x · (x ^-1)
many+Associatives = fourWay+Associative G
oneMult = symmetric (transitive (Group.+WellDefined G reflexive (transitive (symmetric (Group.identLeft G)) (Group.+WellDefined G (symmetric (Group.invRight G)) reflexive))) many+Associatives)
otherInvIsInverse : (x · y) · otherInv ∼ 0G
otherInvIsInverse = transitive oneMult (Group.invRight G)
ans : (x · y) ^-1 ∼ (y ^-1) · (x ^-1)
ans = symmetric (leftInversesAreUnique G otherInvIsInverse)
invIdentity : {l m : _} → {A : Set l} → {S : Setoid {l} {m} A} → {_·_ : A → A → A} → (G : Group S _·_) → Setoid._∼_ S ((Group.inverse G) (Group.0G G)) (Group.0G G)
invIdentity {S = S} G = transitive (symmetric identLeft) invRight
where
open Group G
open Setoid S
open Equivalence eq
transferToRight : {m n : _} {A : Set m} {S : Setoid {m} {n} A} {_·_ : A → A → A} → (G : Group S _·_) → {a b : A} → Setoid._∼_ S (a · (Group.inverse G b)) (Group.0G G) → Setoid._∼_ S a b
transferToRight {S = S} {_·_ = _·_} G {a} {b} ab-1 = transitive (symmetric (invTwice G a)) (transitive u (invTwice G b))
where
open Setoid S
open Group G
open Equivalence eq
t : inverse a ∼ inverse b
t = symmetric (leftInversesAreUnique G ab-1)
u : inverse (inverse a) ∼ inverse (inverse b)
u = inverseWellDefined G t
transferToRight' : {m n : _} {A : Set m} {S : Setoid {m} {n} A} {_·_ : A → A → A} → (G : Group S _·_) → {a b : A} → Setoid._∼_ S (a · b) (Group.0G G) → Setoid._∼_ S a (Group.inverse G b)
transferToRight' {S = S} {_·_ = _·_} G {a} {b} ab-1 = transferToRight G lemma
where
open Setoid S
open Group G
open Equivalence eq
lemma : a · (inverse (inverse b)) ∼ 0G
lemma = transitive (+WellDefined reflexive (invTwice G b)) ab-1
transferToRight'' : {m n : _} {A : Set m} {S : Setoid {m} {n} A} {_·_ : A → A → A} → (G : Group S _·_) → {a b : A} → Setoid._∼_ S a b → Setoid._∼_ S (a · (Group.inverse G b)) (Group.0G G)
transferToRight'' {S = S} {_·_ = _·_} G {a} {b} a~b = transitive (Group.+WellDefined G a~b reflexive) invRight
where
open Group G
open Setoid S
open Equivalence eq
directSumGroup : {m n o p : _} → {A : Set m} {S : Setoid {m} {o} A} {_·A_ : A → A → A} {B : Set n} {T : Setoid {n} {p} B} {_·B_ : B → B → B} (G : Group S _·A_) (h : Group T _·B_) → Group (directSumSetoid S T) (λ x1 y1 → (((_&&_.fst x1) ·A (_&&_.fst y1)) ,, ((_&&_.snd x1) ·B (_&&_.snd y1))))
Group.+WellDefined (directSumGroup {A = A} {B} g h) (s ,, t) (u ,, v) = Group.+WellDefined g s u ,, Group.+WellDefined h t v
Group.0G (directSumGroup {A = A} {B} g h) = (Group.0G g ,, Group.0G h)
Group.inverse (directSumGroup {A = A} {B} g h) (g1 ,, h1) = (Group.inverse g g1) ,, (Group.inverse h h1)
Group.+Associative (directSumGroup {A = A} {B} g h) = Group.+Associative g ,, Group.+Associative h
Group.identRight (directSumGroup {A = A} {B} g h) = Group.identRight g ,, Group.identRight h
Group.identLeft (directSumGroup {A = A} {B} g h) = Group.identLeft g ,, Group.identLeft h
Group.invLeft (directSumGroup {A = A} {B} g h) = Group.invLeft g ,, Group.invLeft h
Group.invRight (directSumGroup {A = A} {B} g h) = Group.invRight g ,, Group.invRight h
directSumAbelianGroup : {m n o p : _} → {A : Set m} {S : Setoid {m} {o} A} {_·A_ : A → A → A} {B : Set n} {T : Setoid {n} {p} B} {_·B_ : B → B → B} {underG : Group S _·A_} {underH : Group T _·B_} (G : AbelianGroup underG) (h : AbelianGroup underH) → (AbelianGroup (directSumGroup underG underH))
AbelianGroup.commutative (directSumAbelianGroup {A = A} {B} G H) = AbelianGroup.commutative G ,, AbelianGroup.commutative H
record GroupHom {m n o p : _} {A : Set m} {S : Setoid {m} {o} A} {_·A_ : A → A → A} {B : Set n} {T : Setoid {n} {p} B} {_·B_ : B → B → B} (G : Group S _·A_) (H : Group T _·B_) (f : A → B) : Set (m ⊔ n ⊔ o ⊔ p) where
open Group H
open Setoid T
field
groupHom : {x y : A} → f (x ·A y) ∼ (f x) ·B (f y)
wellDefined : {x y : A} → Setoid._∼_ S x y → f x ∼ f y
record InjectiveGroupHom {m n o p : _} {A : Set m} {S : Setoid {m} {o} A} {_·A_ : A → A → A} {B : Set n} {T : Setoid {n} {p} B} {_·B_ : B → B → B} {G : Group S _·A_} {H : Group T _·B_} {underf : A → B} (f : GroupHom G H underf) : Set (m ⊔ n ⊔ o ⊔ p) where
open Setoid S renaming (_∼_ to _∼A_)
open Setoid T renaming (_∼_ to _∼B_)
field
injective : SetoidInjection S T underf
imageOfIdentityIsIdentity : {m n o p : _} {A : Set m} {S : Setoid {m} {o} A} {B : Set n} {T : Setoid {n} {p} B} {_·A_ : A → A → A} {_·B_ : B → B → B} {G : Group S _·A_} {H : Group T _·B_} {f : A → B} → (hom : GroupHom G H f) → Setoid._∼_ T (f (Group.0G G)) (Group.0G H)
imageOfIdentityIsIdentity {S = S} {T = T} {_·A_ = _·A_} {_·B_ = _·B_} {G = G} {H = H} {f = f} hom = Equivalence.symmetric (Setoid.eq T) t
where
open Group H
open Setoid T
id2 : Setoid._∼_ S (Group.0G G) ((Group.0G G) ·A (Group.0G G))
id2 = Equivalence.symmetric (Setoid.eq S) (Group.identRight G)
r : f (Group.0G G) ∼ f (Group.0G G) ·B f (Group.0G G)
s : 0G ·B f (Group.0G G) ∼ f (Group.0G G) ·B f (Group.0G G)
t : 0G ∼ f (Group.0G G)
t = groupsHaveRightCancellation H (f (Group.0G G)) 0G (f (Group.0G G)) s
s = Equivalence.transitive (Setoid.eq T) identLeft r
r = Equivalence.transitive (Setoid.eq T) (GroupHom.wellDefined hom id2) (GroupHom.groupHom hom)
groupHomsCompose : {m n o r s t : _} {A : Set m} {S : Setoid {m} {r} A} {_+A_ : A → A → A} {B : Set n} {T : Setoid {n} {s} B} {_+B_ : B → B → B} {C : Set o} {U : Setoid {o} {t} C} {_+C_ : C → C → C} {G : Group S _+A_} {H : Group T _+B_} {I : Group U _+C_} {f : A → B} {g : B → C} (fHom : GroupHom G H f) (gHom : GroupHom H I g) → GroupHom G I (g ∘ f)
GroupHom.wellDefined (groupHomsCompose {G = G} {H} {I} {f} {g} fHom gHom) {x} {y} pr = GroupHom.wellDefined gHom (GroupHom.wellDefined fHom pr)
GroupHom.groupHom (groupHomsCompose {S = S} {_+A_ = _·A_} {T = T} {U = U} {_+C_ = _·C_} {G = G} {H} {I} {f} {g} fHom gHom) {x} {y} = answer
where
open Group I
answer : (Setoid._∼_ U) ((g ∘ f) (x ·A y)) ((g ∘ f) x ·C (g ∘ f) y)
answer = (Equivalence.transitive (Setoid.eq U)) (GroupHom.wellDefined gHom (GroupHom.groupHom fHom {x} {y}) ) (GroupHom.groupHom gHom {f x} {f y})
record GroupIso {m n o p : _} {A : Set m} {S : Setoid {m} {o} A} {_·A_ : A → A → A} {B : Set n} {T : Setoid {n} {p} B} {_·B_ : B → B → B} (G : Group S _·A_) (H : Group T _·B_) (f : A → B) : Set (m ⊔ n ⊔ o ⊔ p) where
open Setoid S renaming (_∼_ to _∼G_)
open Setoid T renaming (_∼_ to _∼H_)
field
groupHom : GroupHom G H f
bij : SetoidBijection S T f
record GroupsIsomorphic {m n o p : _} {A : Set m} {S : Setoid {m} {o} A} {_·A_ : A → A → A} {B : Set n} {T : Setoid {n} {p} B} {_·B_ : B → B → B} (G : Group S _·A_) (H : Group T _·B_) : Set (m ⊔ n ⊔ o ⊔ p) where
field
isomorphism : A → B
proof : GroupIso G H isomorphism
groupIsosCompose : {m n o r s t : _} {A : Set m} {S : Setoid {m} {r} A} {_+A_ : A → A → A} {B : Set n} {T : Setoid {n} {s} B} {_+B_ : B → B → B} {C : Set o} {U : Setoid {o} {t} C} {_+C_ : C → C → C} {G : Group S _+A_} {H : Group T _+B_} {I : Group U _+C_} {f : A → B} {g : B → C} (fHom : GroupIso G H f) (gHom : GroupIso H I g) → GroupIso G I (g ∘ f)
GroupIso.groupHom (groupIsosCompose fHom gHom) = groupHomsCompose (GroupIso.groupHom fHom) (GroupIso.groupHom gHom)
GroupIso.bij (groupIsosCompose {A = A} {S = S} {T = T} {C = C} {U = U} {f = f} {g = g} fHom gHom) = record { inj = record { injective = λ pr → (SetoidInjection.injective (SetoidBijection.inj (GroupIso.bij fHom))) (SetoidInjection.injective (SetoidBijection.inj (GroupIso.bij gHom)) pr) ; wellDefined = +WellDefined } ; surj = record { surjective = surj ; wellDefined = +WellDefined } }
where
open Setoid S renaming (_∼_ to _∼A_)
open Setoid U renaming (_∼_ to _∼C_)
+WellDefined : {x y : A} → (x ∼A y) → (g (f x) ∼C g (f y))
+WellDefined = GroupHom.wellDefined (groupHomsCompose (GroupIso.groupHom fHom) (GroupIso.groupHom gHom))
surj : {x : C} → Sg A (λ a → (g (f a) ∼C x))
surj {x} with SetoidSurjection.surjective (SetoidBijection.surj (GroupIso.bij gHom)) {x}
surj {x} | a , prA with SetoidSurjection.surjective (SetoidBijection.surj (GroupIso.bij fHom)) {a}
... | b , prB = b , transitive (GroupHom.wellDefined (GroupIso.groupHom gHom) prB) prA
where
open Group g renaming (inverse to _^-1)
open Setoid S
transitive = Equivalence.transitive (Setoid.eq S)
symmetric = Equivalence.symmetric (Setoid.eq S)
j : ((x ^-1) · x) · y ∼ (x ^-1) · (x · z)
j = Equivalence.transitive eq (symmetric (+Associative {x ^-1} {x} {y})) (+WellDefined ~refl pr)
k : ((x ^-1) · x) · y ∼ ((x ^-1) · x) · z
k = transitive j +Associative
l : 0G · y ∼ ((x ^-1) · x) · z
l = transitive (+WellDefined (symmetric invLeft) ~refl) k
m : 0G · y ∼ 0G · z
m = transitive l (+WellDefined invLeft ~refl)
n : y ∼ 0G · z
n = transitive (symmetric identLeft) m
o : y ∼ z
o = transitive n identLeft
open Equivalence (Setoid.eq U)
groupsHaveRightCancellation : {a b : _} → {A : Set a} → {_·_ : A → A → A} → {S : Setoid {a} {b} A} → (G : Group S _·_) → (x y z : A) → (Setoid._∼_ S (y · x) (z · x)) → (Setoid._∼_ S y z)
groupsHaveRightCancellation {_·_ = _·_} {S} g x y z pr = transitive m identRight
record FiniteGroup {a b : _} {A : Set a} {S : Setoid {a} {b} A} {_·_ : A → A → A} (G : Group S _·_) {c : _} (quotientSet : Set c) : Set (a ⊔ b ⊔ c) where
field
toSet : SetoidToSet S quotientSet
finite : FiniteSet quotientSet
groupOrder : {a b : _} {A : Set a} {S : Setoid {a} {b} A} {_·_ : A → A → A} {underG : Group S _·_} {c : _} {quotientSet : Set c} (G : FiniteGroup underG quotientSet) → ℕ
groupOrder record { toSet = toSet ; finite = record { size = size ; mapping = mapping ; bij = bij } } = size
--groupIsoInvertible : {a b c d : _} {A : Set a} {B : Set b} {S : Setoid {a} {c} A} {T : Setoid {b} {d}} {_+A_ : A → A → A} {_+B_ : B → B → B} {G : Group S _+A_} {H : Group T _+B_} {f : A → B} → (iso : GroupIso G H f) → GroupIso H G (Invertible.inverse (bijectionImpliesInvertible (GroupIso.bijective iso)))
--GroupHom.groupHom (GroupIso.groupHom (groupIsoInvertible {G = G} {H} {f} iso)) {x} {y} = {!!}
-- where
-- open Group G renaming (_·_ to _+G_)
-- open Group H renaming (_·_ to _+H_)
--GroupHom.wellDefined (GroupIso.groupHom (groupIsoInvertible {G = G} {H} {f} iso)) {x} {y} x∼y = {!GroupHom.wellDefined x∼y!}
--GroupIso.bijective (groupIsoInvertible {G = G} {H} {f} iso) = invertibleImpliesBijection (inverseIsInvertible (bijectionImpliesInvertible (GroupIso.bijective iso)))
homRespectsInverse : {a b c d : _} {A : Set a} {B : Set b} {S : Setoid {a} {c} A} {T : Setoid {b} {d} B} {_·A_ : A → A → A} {_·B_ : B → B → B} {G : Group S _·A_} {H : Group T _·B_} {underF : A → B} → (f : GroupHom G H underF) → {x : A} → Setoid._∼_ T (underF (Group.inverse G x)) (Group.inverse H (underF x))
homRespectsInverse {T = T} {_·A_ = _·A_} {_·B_ = _·B_} {G = G} {H = H} {underF = f} fHom {x} = rightInversesAreUnique H (f x) (f (Group.inverse G x)) (transitive (symmetric (GroupHom.groupHom fHom)) (transitive (GroupHom.wellDefined fHom (Group.invLeft G)) (imageOfIdentityIsIdentity fHom)))
where
open Setoid T
open Equivalence eq
quotientGroupSetoid : {a b c d : _} {A : Set a} {B : Set b} {S : Setoid {a} {c} A} {T : Setoid {b} {d} B} {_·A_ : A → A → A} {_·B_ : B → B → B} (G : Group S _·A_) {H : Group T _·B_} → {underf : A → B} → (f : GroupHom G H underf) → (Setoid {a} {d} A)
quotientGroupSetoid {A = A} {S = S} {T = T} {_·A_ = _·A_} {_·B_ = _·B_} G {H} {f} fHom = ansSetoid
where
open Setoid T
open Group H
open Equivalence eq
ansSetoid : Setoid A
Setoid._∼_ ansSetoid r s = (f (r ·A (Group.inverse G s))) ∼ 0G
Equivalence.reflexive (Setoid.eq ansSetoid) {b} = transitive (GroupHom.wellDefined fHom (Group.invRight G)) (imageOfIdentityIsIdentity fHom)
Equivalence.symmetric (Setoid.eq ansSetoid) {m} {n} pr = i
where
open Group g renaming (inverse to _^-1)
open Setoid S
transitive = Equivalence.transitive (Setoid.eq S)
symmetric = Equivalence.symmetric (Setoid.eq S)
i : (y · x) · (x ^-1) ∼ (z · x) · (x ^-1)
i = +WellDefined pr ~refl
j : y · (x · (x ^-1)) ∼ (z · x) · (x ^-1)
j = transitive +Associative i
j' : y · 0G ∼ (z · x) · (x ^-1)
j' = transitive (+WellDefined ~refl (symmetric invRight)) j
k : y ∼ (z · x) · (x ^-1)
k = transitive (symmetric identRight) j'
l : y ∼ z · (x · (x ^-1))
l = transitive k (symmetric +Associative)
m : y ∼ z · 0G
m = transitive l (+WellDefined ~refl invRight)
replaceGroupOp : {l m : _} {A : Set l} {S : Setoid {l} {m} A} {_·_ : A → A → A} → (G : Group S _·_) → {a b c d w x y z : A} → (Setoid._∼_ S a c) → (Setoid._∼_ S b d) → (Setoid._∼_ S w y) → (Setoid._∼_ S x z) → Setoid._∼_ S (a · b) (w · x) → Setoid._∼_ S (c · d) (y · z)
replaceGroupOp {S = S} {_·_} G a~c b~d w~y x~z pr = transitive (symmetric (+WellDefined a~c b~d)) (transitive pr (+WellDefined w~y x~z))
g : f (Group.inverse G (m ·A Group.inverse G n)) ∼ 0G
g = transitive (homRespectsInverse fHom {m ·A Group.inverse G n}) (transitive (inverseWellDefined H pr) (invIdentity H))
h : f (Group.inverse G (Group.inverse G n) ·A Group.inverse G m) ∼ 0G
h = transitive (GroupHom.wellDefined fHom (Equivalence.symmetric (Setoid.eq S) (invContravariant G))) g
i : f (n ·A Group.inverse G m) ∼ 0G
i = transitive (GroupHom.wellDefined fHom (Group.+WellDefined G (Equivalence.symmetric (Setoid.eq S) (invTwice G n)) (Equivalence.reflexive (Setoid.eq S)))) h
Equivalence.transitive (Setoid.eq ansSetoid) {m} {n} {o} prmn prno = transitive (GroupHom.wellDefined fHom (Group.+WellDefined G (Equivalence.reflexive (Setoid.eq S)) (Equivalence.symmetric (Setoid.eq S) (Group.identLeft G)))) k
where
open Group G
open Setoid S
open Equivalence eq
g : f (m ·A Group.inverse G n) ·B f (n ·A Group.inverse G o) ∼ 0G ·B 0G
g = replaceGroupOp H reflexive reflexive prmn prno reflexive
h : f (m ·A Group.inverse G n) ·B f (n ·A Group.inverse G o) ∼ 0G
h = transitive g identLeft
i : f ((m ·A Group.inverse G n) ·A (n ·A Group.inverse G o)) ∼ 0G
i = transitive (GroupHom.groupHom fHom) h
j : f (m ·A (((Group.inverse G n) ·A n) ·A Group.inverse G o)) ∼ 0G
j = transitive (GroupHom.wellDefined fHom (fourWay+Associative G)) i
k : f (m ·A ((Group.0G G) ·A Group.inverse G o)) ∼ 0G
k = transitive (GroupHom.wellDefined fHom (Group.+WellDefined G (Equivalence.reflexive (Setoid.eq S)) (Group.+WellDefined G (Equivalence.symmetric (Setoid.eq S) (Group.invLeft G)) (Equivalence.reflexive (Setoid.eq S))))) j
replaceGroupOpRight : {l m : _} {A : Set l} {S : Setoid {l} {m} A} {_·_ : A → A → A} → (G : Group S _·_) → {a b c x : A} → (Setoid._∼_ S a (b · c)) → (Setoid._∼_ S c x) → (Setoid._∼_ S a (b · x))
replaceGroupOpRight {S = S} {_·_} G a~bc c~x = transitive a~bc (+WellDefined reflexive c~x)
where
open Group G
open Setoid S
open Equivalence eq
quotientGroup : {a b c d : _} {A : Set a} {B : Set b} {S : Setoid {a} {c} A} {T : Setoid {b} {d} B} {_·A_ : A → A → A} {_·B_ : B → B → B} (G : Group S _·A_) {H : Group T _·B_} → {underf : A → B} → (f : GroupHom G H underf) → Group (quotientGroupSetoid G f) _·A_
Group.+WellDefined (quotientGroup {S = S} {T = T} {_·A_ = _·A_} {_·B_ = _·B_} G {H = H} {underf = f} fHom) {x} {y} {m} {n} x~m y~n = ans
where
open Setoid T
open Equivalence (Setoid.eq T)
p1 : f ((x ·A y) ·A (Group.inverse G (m ·A n))) ∼ f ((x ·A y) ·A ((Group.inverse G n) ·A (Group.inverse G m)))
p2 : f ((x ·A y) ·A ((Group.inverse G n) ·A (Group.inverse G m))) ∼ f (x ·A ((y ·A (Group.inverse G n)) ·A (Group.inverse G m)))
p3 : f (x ·A ((y ·A (Group.inverse G n)) ·A (Group.inverse G m))) ∼ (f x) ·B f ((y ·A (Group.inverse G n)) ·A (Group.inverse G m))
p4 : (f x) ·B f ((y ·A (Group.inverse G n)) ·A (Group.inverse G m)) ∼ (f x) ·B (f (y ·A (Group.inverse G n)) ·B f (Group.inverse G m))
p5 : (f x) ·B (f (y ·A (Group.inverse G n)) ·B f (Group.inverse G m)) ∼ (f x) ·B (Group.0G H ·B f (Group.inverse G m))
p6 : (f x) ·B (Group.0G H ·B f (Group.inverse G m)) ∼ (f x) ·B f (Group.inverse G m)
p7 : (f x) ·B f (Group.inverse G m) ∼ f (x ·A (Group.inverse G m))
p8 : f (x ·A (Group.inverse G m)) ∼ Group.0G H
p1 = GroupHom.wellDefined fHom (Group.+WellDefined G (Equivalence.reflexive (Setoid.eq S)) (invContravariant G))
p2 = GroupHom.wellDefined fHom (Equivalence.symmetric (Setoid.eq S) (fourWay+Associative G))
p3 = GroupHom.groupHom fHom
p4 = Group.+WellDefined H reflexive (GroupHom.groupHom fHom)
p5 = Group.+WellDefined H reflexive (replaceGroupOp H (symmetric y~n) reflexive reflexive reflexive reflexive)
p6 = Group.+WellDefined H reflexive (Group.identLeft H)
p7 = symmetric (GroupHom.groupHom fHom)
p8 = x~m
ans : f ((x ·A y) ·A (Group.inverse G (m ·A n))) ∼ Group.0G H
ans = transitive p1 (transitive p2 (transitive p3 (transitive p4 (transitive p5 (transitive p6 (transitive p7 p8))))))
Group.0G (quotientGroup G fHom) = Group.0G G
Group.inverse (quotientGroup G fHom) = Group.inverse G
Group.+Associative (quotientGroup {S = S} {T = T} {_·A_ = _·A_} G {H = H} {underf = f} fHom) {a} {b} {c} = ans
where
open Setoid T
open Equivalence (Setoid.eq T)
ans : f ((a ·A (b ·A c)) ·A (Group.inverse G ((a ·A b) ·A c))) ∼ Group.0G H
ans = transitive (GroupHom.wellDefined fHom (transferToRight'' G (Group.+Associative G))) (imageOfIdentityIsIdentity fHom)
Group.identRight (quotientGroup {S = S} {T = T} {_·A_ = _·A_} G {H = H} {underf = f} fHom) {a} = ans
where
open Group G
open Setoid T
open Equivalence eq
transitiveG = Equivalence.transitive (Setoid.eq S)
ans : f ((a ·A 0G) ·A inverse a) ∼ Group.0G H
ans = transitive (GroupHom.wellDefined fHom (transitiveG (Group.+WellDefined G (Group.identRight G) (Equivalence.reflexive (Setoid.eq S))) (Group.invRight G))) (imageOfIdentityIsIdentity fHom)
Group.identLeft (quotientGroup {S = S} {T = T} {_·A_ = _·A_} G {H = H} {underf = f} fHom) {a} = ans
where
open Group G
open Setoid T
open Equivalence eq
transitiveG = Equivalence.transitive (Setoid.eq S)
ans : f ((0G ·A a) ·A (inverse a)) ∼ Group.0G H
ans = transitive (GroupHom.wellDefined fHom (transitiveG (Group.+WellDefined G (Group.identLeft G) (Equivalence.reflexive (Setoid.eq S))) (Group.invRight G))) (imageOfIdentityIsIdentity fHom)
Group.invLeft (quotientGroup {S = S} {T = T} {_·A_ = _·A_} G {H = H} {underf = f} fHom) {x} = ans
where
open Group G
open Setoid T
open Equivalence eq
ans : f ((inverse x ·A x) ·A (inverse 0G)) ∼ (Group.0G H)
ans = transitive (GroupHom.wellDefined fHom (Equivalence.transitive (Setoid.eq S) (replaceGroupOp G (Equivalence.symmetric (Setoid.eq S) (Group.invLeft G)) (Equivalence.symmetric (Setoid.eq S) (invIdentity G)) (Equivalence.reflexive (Setoid.eq S)) ((Equivalence.reflexive (Setoid.eq S))) ((Equivalence.reflexive (Setoid.eq S)))) (identRight {0G}))) (imageOfIdentityIsIdentity fHom)
Group.invRight (quotientGroup {S = S} {T = T} {_·A_ = _·A_} G {H = H} {underf = f} fHom) {x} = ans
where
open Group G
open Setoid T
open Equivalence eq
ans : f ((x ·A inverse x) ·A (inverse 0G)) ∼ (Group.0G H)
ans = transitive (GroupHom.wellDefined fHom (Equivalence.transitive (Setoid.eq S) (replaceGroupOp G (Equivalence.symmetric (Setoid.eq S) (Group.invRight G)) (Equivalence.symmetric (Setoid.eq S) (invIdentity G)) (Equivalence.reflexive (Setoid.eq S)) (Equivalence.reflexive (Setoid.eq S)) (Equivalence.reflexive (Setoid.eq S))) (identRight {0G}))) (imageOfIdentityIsIdentity fHom)
inverseWellDefined : {l m : _} {A : Set l} {S : Setoid {l} {m} A} {_·_ : A → A → A} (G : Group S _·_) → {x y : A} → (Setoid._∼_ S x y) → Setoid._∼_ S (Group.inverse G x) (Group.inverse G y)
inverseWellDefined {S = S} {_·_} G {x} {y} x~y = groupsHaveRightCancellation G x (inverse x) (inverse y) q
where
open Group G
open Setoid S
open Equivalence eq
p : inverse x · x ∼ inverse y · y
p = transitive invLeft (symmetric invLeft)
q : inverse x · x ∼ inverse y · x
q = replaceGroupOpRight G {_·_ (inverse x) x} {inverse y} {y} {x} p (symmetric x~y)
rightInversesAreUnique : {a b : _} → {A : Set a} → {S : Setoid {a} {b} A} → {_·_ : A → A → A} → (G : Group S _·_) → (x : A) → (y : A) → Setoid._∼_ S (y · x) (Group.0G G) → Setoid._∼_ S y (Group.inverse G x)
rightInversesAreUnique {S = S} {_·_} G x y f = transitive i (transitive j (transitive k (transitive l m)))
where
open Group G renaming (inverse to _^-1)
open Setoid S
open Equivalence eq
i : y ∼ y · 0G
j : y · 0G ∼ y · (x · (x ^-1))
k : y · (x · (x ^-1)) ∼ (y · x) · (x ^-1)
l : (y · x) · (x ^-1) ∼ 0G · (x ^-1)
m : 0G · (x ^-1) ∼ x ^-1
i = symmetric identRight
j = +WellDefined ~refl (symmetric invRight)
k = +Associative
l = +WellDefined f ~refl
m = identLeft
leftInversesAreUnique : {a b : _} → {A : Set a} → {S : Setoid {a} {b} A} → {_·_ : A → A → A} → (G : Group S _·_) → {x : A} → {y : A} → Setoid._∼_ S (x · y) (Group.0G G) → Setoid._∼_ S y (Group.inverse G x)
leftInversesAreUnique {S = S} {_·_} G {x} {y} f = rightInversesAreUnique G x y l
where
open Group G renaming (inverse to _^-1)
open Setoid S
open Equivalence eq
i : y · (x · y) ∼ y · 0G
i' : y · (x · y) ∼ y
j : (y · x) · y ∼ y
k : (y · x) · y ∼ 0G · y
l : y · x ∼ 0G
i = +WellDefined ~refl f
i' = transitive i identRight
j = transitive (symmetric +Associative) i'
k = transitive j (symmetric identLeft)
l = groupsHaveRightCancellation G y (y · x) 0G k
invTwice : {a b : _} → {A : Set a} → {S : Setoid {a} {b} A} → {_·_ : A → A → A} → (G : Group S _·_) → (x : A) → Setoid._∼_ S (Group.inverse G (Group.inverse G x)) x
invTwice {S = S} {_·_} G x = symmetric (rightInversesAreUnique G (x ^-1) x invRight)
where
open Group G renaming (inverse to _^-1)
open Setoid S
open Equivalence eq
fourWay+Associative : {a b : _} → {A : Set a} → {S : Setoid {a} {b} A} → {_·_ : A → A → A} → (G : Group S _·_) → {r s t u : A} → (Setoid._∼_ S) (r · ((s · t) · u)) ((r · s) · (t · u))
fourWay+Associative {S = S} {_·_} G {r} {s} {t} {u} = transitive p1 (transitive p2 p3)
where
open Group G renaming (inverse to _^-1)
open Setoid S
open Equivalence eq
p1 : r · ((s · t) · u) ∼ (r · (s · t)) · u
p2 : (r · (s · t)) · u ∼ ((r · s) · t) · u
p3 : ((r · s) · t) · u ∼ (r · s) · (t · u)
p1 = Group.+Associative G
p2 = Group.+WellDefined G (Group.+Associative G) reflexive
p3 = symmetric (Group.+Associative G)
fourWay+Associative' : {m n : _} {A : Set m} {S : Setoid {m} {n} A} {_·_ : A → A → A} (G : Group S _·_) {a b c d : A} → (Setoid._∼_ S (((a · b) · c) · d) (a · ((b · c) · d)))
fourWay+Associative' {S = S} G = transitive (symmetric +Associative) (symmetric (fourWay+Associative G))
where
open Group G
open Setoid S
open Equivalence eq
fourWay+Associative'' : {m n : _} {A : Set m} {S : Setoid {m} {n} A} {_·_ : A → A → A} (G : Group S _·_) {a b c d : A} → (Setoid._∼_ S (a · (b · (c · d))) (a · ((b · c) · d)))
fourWay+Associative'' {S = S} {_·_ = _·_} G = transitive +Associative (symmetric (fourWay+Associative G))
where
open Group G
open Setoid S
open Equivalence eq
invContravariant : {a b : _} → {A : Set a} → {S : Setoid {a} {b} A} → {_·_ : A → A → A} → (G : Group S _·_) → {x y : A} → (Setoid._∼_ S (Group.inverse G (x · y)) ((Group.inverse G y) · (Group.inverse G x)))
invContravariant {S = S} {_·_} G {x} {y} = ans
where
open Group G renaming (inverse to _^-1)
open Setoid S
open Equivalence eq
otherInv = (y ^-1) · (x ^-1)
many+Associatives : x · ((y · (y ^-1)) · (x ^-1)) ∼ (x · y) · ((y ^-1) · (x ^-1))
oneMult : (x · y) · otherInv ∼ x · (x ^-1)
many+Associatives = fourWay+Associative G
oneMult = symmetric (transitive (Group.+WellDefined G reflexive (transitive (symmetric (Group.identLeft G)) (Group.+WellDefined G (symmetric (Group.invRight G)) reflexive))) many+Associatives)
otherInvIsInverse : (x · y) · otherInv ∼ 0G
otherInvIsInverse = transitive oneMult (Group.invRight G)
ans : (x · y) ^-1 ∼ (y ^-1) · (x ^-1)
ans = symmetric (leftInversesAreUnique G otherInvIsInverse)
invIdentity : {l m : _} → {A : Set l} → {S : Setoid {l} {m} A} → {_·_ : A → A → A} → (G : Group S _·_) → Setoid._∼_ S ((Group.inverse G) (Group.0G G)) (Group.0G G)
invIdentity {S = S} G = transitive (symmetric identLeft) invRight
where
open Group G
open Setoid S
open Equivalence eq
transferToRight : {m n : _} {A : Set m} {S : Setoid {m} {n} A} {_·_ : A → A → A} → (G : Group S _·_) → {a b : A} → Setoid._∼_ S (a · (Group.inverse G b)) (Group.0G G) → Setoid._∼_ S a b
transferToRight {S = S} {_·_ = _·_} G {a} {b} ab-1 = transitive (symmetric (invTwice G a)) (transitive u (invTwice G b))
where
open Setoid S
open Group G
open Equivalence eq
t : inverse a ∼ inverse b
t = symmetric (leftInversesAreUnique G ab-1)
u : inverse (inverse a) ∼ inverse (inverse b)
u = inverseWellDefined G t
transferToRight' : {m n : _} {A : Set m} {S : Setoid {m} {n} A} {_·_ : A → A → A} → (G : Group S _·_) → {a b : A} → Setoid._∼_ S (a · b) (Group.0G G) → Setoid._∼_ S a (Group.inverse G b)
transferToRight' {S = S} {_·_ = _·_} G {a} {b} ab-1 = transferToRight G lemma
where
open Setoid S
open Group G
open Equivalence eq
lemma : a · (inverse (inverse b)) ∼ 0G
lemma = transitive (+WellDefined reflexive (invTwice G b)) ab-1
transferToRight'' : {m n : _} {A : Set m} {S : Setoid {m} {n} A} {_·_ : A → A → A} → (G : Group S _·_) → {a b : A} → Setoid._∼_ S a b → Setoid._∼_ S (a · (Group.inverse G b)) (Group.0G G)
transferToRight'' {S = S} {_·_ = _·_} G {a} {b} a~b = transitive (Group.+WellDefined G a~b reflexive) invRight
where
open Group G
open Setoid S
open Equivalence eq
directSumGroup : {m n o p : _} → {A : Set m} {S : Setoid {m} {o} A} {_·A_ : A → A → A} {B : Set n} {T : Setoid {n} {p} B} {_·B_ : B → B → B} (G : Group S _·A_) (h : Group T _·B_) → Group (directSumSetoid S T) (λ x1 y1 → (((_&&_.fst x1) ·A (_&&_.fst y1)) ,, ((_&&_.snd x1) ·B (_&&_.snd y1))))
Group.+WellDefined (directSumGroup {A = A} {B} g h) (s ,, t) (u ,, v) = Group.+WellDefined g s u ,, Group.+WellDefined h t v
Group.0G (directSumGroup {A = A} {B} g h) = (Group.0G g ,, Group.0G h)
Group.inverse (directSumGroup {A = A} {B} g h) (g1 ,, h1) = (Group.inverse g g1) ,, (Group.inverse h h1)
Group.+Associative (directSumGroup {A = A} {B} g h) = Group.+Associative g ,, Group.+Associative h
Group.identRight (directSumGroup {A = A} {B} g h) = Group.identRight g ,, Group.identRight h
Group.identLeft (directSumGroup {A = A} {B} g h) = Group.identLeft g ,, Group.identLeft h
Group.invLeft (directSumGroup {A = A} {B} g h) = Group.invLeft g ,, Group.invLeft h
Group.invRight (directSumGroup {A = A} {B} g h) = Group.invRight g ,, Group.invRight h
directSumAbelianGroup : {m n o p : _} → {A : Set m} {S : Setoid {m} {o} A} {_·A_ : A → A → A} {B : Set n} {T : Setoid {n} {p} B} {_·B_ : B → B → B} {underG : Group S _·A_} {underH : Group T _·B_} (G : AbelianGroup underG) (h : AbelianGroup underH) → (AbelianGroup (directSumGroup underG underH))
AbelianGroup.commutative (directSumAbelianGroup {A = A} {B} G H) = AbelianGroup.commutative G ,, AbelianGroup.commutative H
record GroupHom {m n o p : _} {A : Set m} {S : Setoid {m} {o} A} {_·A_ : A → A → A} {B : Set n} {T : Setoid {n} {p} B} {_·B_ : B → B → B} (G : Group S _·A_) (H : Group T _·B_) (f : A → B) : Set (m ⊔ n ⊔ o ⊔ p) where
open Group H
open Setoid T
field
groupHom : {x y : A} → f (x ·A y) ∼ (f x) ·B (f y)
wellDefined : {x y : A} → Setoid._∼_ S x y → f x ∼ f y
record InjectiveGroupHom {m n o p : _} {A : Set m} {S : Setoid {m} {o} A} {_·A_ : A → A → A} {B : Set n} {T : Setoid {n} {p} B} {_·B_ : B → B → B} {G : Group S _·A_} {H : Group T _·B_} {underf : A → B} (f : GroupHom G H underf) : Set (m ⊔ n ⊔ o ⊔ p) where
open Setoid S renaming (_∼_ to _∼A_)
open Setoid T renaming (_∼_ to _∼B_)
field
injective : SetoidInjection S T underf
imageOfIdentityIsIdentity : {m n o p : _} {A : Set m} {S : Setoid {m} {o} A} {B : Set n} {T : Setoid {n} {p} B} {_·A_ : A → A → A} {_·B_ : B → B → B} {G : Group S _·A_} {H : Group T _·B_} {f : A → B} → (hom : GroupHom G H f) → Setoid._∼_ T (f (Group.0G G)) (Group.0G H)
imageOfIdentityIsIdentity {S = S} {T = T} {_·A_ = _·A_} {_·B_ = _·B_} {G = G} {H = H} {f = f} hom = Equivalence.symmetric (Setoid.eq T) t
where
open Group H
open Setoid T
id2 : Setoid._∼_ S (Group.0G G) ((Group.0G G) ·A (Group.0G G))
id2 = Equivalence.symmetric (Setoid.eq S) (Group.identRight G)
r : f (Group.0G G) ∼ f (Group.0G G) ·B f (Group.0G G)
s : 0G ·B f (Group.0G G) ∼ f (Group.0G G) ·B f (Group.0G G)
t : 0G ∼ f (Group.0G G)
t = groupsHaveRightCancellation H (f (Group.0G G)) 0G (f (Group.0G G)) s
s = Equivalence.transitive (Setoid.eq T) identLeft r
r = Equivalence.transitive (Setoid.eq T) (GroupHom.wellDefined hom id2) (GroupHom.groupHom hom)
groupHomsCompose : {m n o r s t : _} {A : Set m} {S : Setoid {m} {r} A} {_+A_ : A → A → A} {B : Set n} {T : Setoid {n} {s} B} {_+B_ : B → B → B} {C : Set o} {U : Setoid {o} {t} C} {_+C_ : C → C → C} {G : Group S _+A_} {H : Group T _+B_} {I : Group U _+C_} {f : A → B} {g : B → C} (fHom : GroupHom G H f) (gHom : GroupHom H I g) → GroupHom G I (g ∘ f)
GroupHom.wellDefined (groupHomsCompose {G = G} {H} {I} {f} {g} fHom gHom) {x} {y} pr = GroupHom.wellDefined gHom (GroupHom.wellDefined fHom pr)
GroupHom.groupHom (groupHomsCompose {S = S} {_+A_ = _·A_} {T = T} {U = U} {_+C_ = _·C_} {G = G} {H} {I} {f} {g} fHom gHom) {x} {y} = answer
where
open Group I
answer : (Setoid._∼_ U) ((g ∘ f) (x ·A y)) ((g ∘ f) x ·C (g ∘ f) y)
answer = (Equivalence.transitive (Setoid.eq U)) (GroupHom.wellDefined gHom (GroupHom.groupHom fHom {x} {y}) ) (GroupHom.groupHom gHom {f x} {f y})
record GroupIso {m n o p : _} {A : Set m} {S : Setoid {m} {o} A} {_·A_ : A → A → A} {B : Set n} {T : Setoid {n} {p} B} {_·B_ : B → B → B} (G : Group S _·A_) (H : Group T _·B_) (f : A → B) : Set (m ⊔ n ⊔ o ⊔ p) where
open Setoid S renaming (_∼_ to _∼G_)
open Setoid T renaming (_∼_ to _∼H_)
field
groupHom : GroupHom G H f
bij : SetoidBijection S T f
record GroupsIsomorphic {m n o p : _} {A : Set m} {S : Setoid {m} {o} A} {_·A_ : A → A → A} {B : Set n} {T : Setoid {n} {p} B} {_·B_ : B → B → B} (G : Group S _·A_) (H : Group T _·B_) : Set (m ⊔ n ⊔ o ⊔ p) where
field
isomorphism : A → B
proof : GroupIso G H isomorphism
groupIsosCompose : {m n o r s t : _} {A : Set m} {S : Setoid {m} {r} A} {_+A_ : A → A → A} {B : Set n} {T : Setoid {n} {s} B} {_+B_ : B → B → B} {C : Set o} {U : Setoid {o} {t} C} {_+C_ : C → C → C} {G : Group S _+A_} {H : Group T _+B_} {I : Group U _+C_} {f : A → B} {g : B → C} (fHom : GroupIso G H f) (gHom : GroupIso H I g) → GroupIso G I (g ∘ f)
GroupIso.groupHom (groupIsosCompose fHom gHom) = groupHomsCompose (GroupIso.groupHom fHom) (GroupIso.groupHom gHom)
GroupIso.bij (groupIsosCompose {A = A} {S = S} {T = T} {C = C} {U = U} {f = f} {g = g} fHom gHom) = record { inj = record { injective = λ pr → (SetoidInjection.injective (SetoidBijection.inj (GroupIso.bij fHom))) (SetoidInjection.injective (SetoidBijection.inj (GroupIso.bij gHom)) pr) ; wellDefined = +WellDefined } ; surj = record { surjective = surj ; wellDefined = +WellDefined } }
where
open Setoid S renaming (_∼_ to _∼A_)
open Setoid U renaming (_∼_ to _∼C_)
+WellDefined : {x y : A} → (x ∼A y) → (g (f x) ∼C g (f y))
+WellDefined = GroupHom.wellDefined (groupHomsCompose (GroupIso.groupHom fHom) (GroupIso.groupHom gHom))
surj : {x : C} → Sg A (λ a → (g (f a) ∼C x))
surj {x} with SetoidSurjection.surjective (SetoidBijection.surj (GroupIso.bij gHom)) {x}
surj {x} | a , prA with SetoidSurjection.surjective (SetoidBijection.surj (GroupIso.bij fHom)) {a}
... | b , prB = b , transitive (GroupHom.wellDefined (GroupIso.groupHom gHom) prB) prA
where
open Equivalence (Setoid.eq U)
record FiniteGroup {a b : _} {A : Set a} {S : Setoid {a} {b} A} {_·_ : A → A → A} (G : Group S _·_) {c : _} (quotientSet : Set c) : Set (a ⊔ b ⊔ c) where
field
toSet : SetoidToSet S quotientSet
finite : FiniteSet quotientSet
groupOrder : {a b : _} {A : Set a} {S : Setoid {a} {b} A} {_·_ : A → A → A} {underG : Group S _·_} {c : _} {quotientSet : Set c} (G : FiniteGroup underG quotientSet) → ℕ
groupOrder record { toSet = toSet ; finite = record { size = size ; mapping = mapping ; bij = bij } } = size
--groupIsoInvertible : {a b c d : _} {A : Set a} {B : Set b} {S : Setoid {a} {c} A} {T : Setoid {b} {d}} {_+A_ : A → A → A} {_+B_ : B → B → B} {G : Group S _+A_} {H : Group T _+B_} {f : A → B} → (iso : GroupIso G H f) → GroupIso H G (Invertible.inverse (bijectionImpliesInvertible (GroupIso.bijective iso)))
--GroupHom.groupHom (GroupIso.groupHom (groupIsoInvertible {G = G} {H} {f} iso)) {x} {y} = {!!}
-- where
-- open Group G renaming (_·_ to _+G_)
-- open Group H renaming (_·_ to _+H_)
--GroupHom.wellDefined (GroupIso.groupHom (groupIsoInvertible {G = G} {H} {f} iso)) {x} {y} x∼y = {!GroupHom.wellDefined x∼y!}
--GroupIso.bijective (groupIsoInvertible {G = G} {H} {f} iso) = invertibleImpliesBijection (inverseIsInvertible (bijectionImpliesInvertible (GroupIso.bijective iso)))
homRespectsInverse : {a b c d : _} {A : Set a} {B : Set b} {S : Setoid {a} {c} A} {T : Setoid {b} {d} B} {_·A_ : A → A → A} {_·B_ : B → B → B} {G : Group S _·A_} {H : Group T _·B_} {underF : A → B} → (f : GroupHom G H underF) → {x : A} → Setoid._∼_ T (underF (Group.inverse G x)) (Group.inverse H (underF x))
homRespectsInverse {T = T} {_·A_ = _·A_} {_·B_ = _·B_} {G = G} {H = H} {underF = f} fHom {x} = rightInversesAreUnique H (f x) (f (Group.inverse G x)) (transitive (symmetric (GroupHom.groupHom fHom)) (transitive (GroupHom.wellDefined fHom (Group.invLeft G)) (imageOfIdentityIsIdentity fHom)))
where
open Setoid T
open Equivalence eq
quotientGroupSetoid : {a b c d : _} {A : Set a} {B : Set b} {S : Setoid {a} {c} A} {T : Setoid {b} {d} B} {_·A_ : A → A → A} {_·B_ : B → B → B} (G : Group S _·A_) {H : Group T _·B_} → {underf : A → B} → (f : GroupHom G H underf) → (Setoid {a} {d} A)
quotientGroupSetoid {A = A} {S = S} {T = T} {_·A_ = _·A_} {_·B_ = _·B_} G {H} {f} fHom = ansSetoid
where
open Setoid T
open Group H
open Equivalence eq
ansSetoid : Setoid A
Setoid._∼_ ansSetoid r s = (f (r ·A (Group.inverse G s))) ∼ 0G
Equivalence.reflexive (Setoid.eq ansSetoid) {b} = transitive (GroupHom.wellDefined fHom (Group.invRight G)) (imageOfIdentityIsIdentity fHom)
Equivalence.symmetric (Setoid.eq ansSetoid) {m} {n} pr = i
where
g : f (Group.inverse G (m ·A Group.inverse G n)) ∼ 0G
g = transitive (homRespectsInverse fHom {m ·A Group.inverse G n}) (transitive (inverseWellDefined H pr) (invIdentity H))
h : f (Group.inverse G (Group.inverse G n) ·A Group.inverse G m) ∼ 0G
h = transitive (GroupHom.wellDefined fHom (Equivalence.symmetric (Setoid.eq S) (invContravariant G))) g
i : f (n ·A Group.inverse G m) ∼ 0G
i = transitive (GroupHom.wellDefined fHom (Group.+WellDefined G (Equivalence.symmetric (Setoid.eq S) (invTwice G n)) (Equivalence.reflexive (Setoid.eq S)))) h
Equivalence.transitive (Setoid.eq ansSetoid) {m} {n} {o} prmn prno = transitive (GroupHom.wellDefined fHom (Group.+WellDefined G (Equivalence.reflexive (Setoid.eq S)) (Equivalence.symmetric (Setoid.eq S) (Group.identLeft G)))) k
where
g : f (m ·A Group.inverse G n) ·B f (n ·A Group.inverse G o) ∼ 0G ·B 0G
g = replaceGroupOp H reflexive reflexive prmn prno reflexive
h : f (m ·A Group.inverse G n) ·B f (n ·A Group.inverse G o) ∼ 0G
h = transitive g identLeft
i : f ((m ·A Group.inverse G n) ·A (n ·A Group.inverse G o)) ∼ 0G
i = transitive (GroupHom.groupHom fHom) h
j : f (m ·A (((Group.inverse G n) ·A n) ·A Group.inverse G o)) ∼ 0G
j = transitive (GroupHom.wellDefined fHom (fourWay+Associative G)) i
k : f (m ·A ((Group.0G G) ·A Group.inverse G o)) ∼ 0G
k = transitive (GroupHom.wellDefined fHom (Group.+WellDefined G (Equivalence.reflexive (Setoid.eq S)) (Group.+WellDefined G (Equivalence.symmetric (Setoid.eq S) (Group.invLeft G)) (Equivalence.reflexive (Setoid.eq S))))) j
quotientGroup : {a b c d : _} {A : Set a} {B : Set b} {S : Setoid {a} {c} A} {T : Setoid {b} {d} B} {_·A_ : A → A → A} {_·B_ : B → B → B} (G : Group S _·A_) {H : Group T _·B_} → {underf : A → B} → (f : GroupHom G H underf) → Group (quotientGroupSetoid G f) _·A_
Group.+WellDefined (quotientGroup {S = S} {T = T} {_·A_ = _·A_} {_·B_ = _·B_} G {H = H} {underf = f} fHom) {x} {y} {m} {n} x~m y~n = ans
where
open Setoid T
open Equivalence (Setoid.eq T)
p1 : f ((x ·A y) ·A (Group.inverse G (m ·A n))) ∼ f ((x ·A y) ·A ((Group.inverse G n) ·A (Group.inverse G m)))
p2 : f ((x ·A y) ·A ((Group.inverse G n) ·A (Group.inverse G m))) ∼ f (x ·A ((y ·A (Group.inverse G n)) ·A (Group.inverse G m)))
p3 : f (x ·A ((y ·A (Group.inverse G n)) ·A (Group.inverse G m))) ∼ (f x) ·B f ((y ·A (Group.inverse G n)) ·A (Group.inverse G m))
p4 : (f x) ·B f ((y ·A (Group.inverse G n)) ·A (Group.inverse G m)) ∼ (f x) ·B (f (y ·A (Group.inverse G n)) ·B f (Group.inverse G m))
p5 : (f x) ·B (f (y ·A (Group.inverse G n)) ·B f (Group.inverse G m)) ∼ (f x) ·B (Group.0G H ·B f (Group.inverse G m))
p6 : (f x) ·B (Group.0G H ·B f (Group.inverse G m)) ∼ (f x) ·B f (Group.inverse G m)
p7 : (f x) ·B f (Group.inverse G m) ∼ f (x ·A (Group.inverse G m))
p8 : f (x ·A (Group.inverse G m)) ∼ Group.0G H
p1 = GroupHom.wellDefined fHom (Group.+WellDefined G (Equivalence.reflexive (Setoid.eq S)) (invContravariant G))
p2 = GroupHom.wellDefined fHom (Equivalence.symmetric (Setoid.eq S) (fourWay+Associative G))
p3 = GroupHom.groupHom fHom
p4 = Group.+WellDefined H reflexive (GroupHom.groupHom fHom)
p5 = Group.+WellDefined H reflexive (replaceGroupOp H (symmetric y~n) reflexive reflexive reflexive reflexive)
p6 = Group.+WellDefined H reflexive (Group.identLeft H)
p7 = symmetric (GroupHom.groupHom fHom)
p8 = x~m
ans : f ((x ·A y) ·A (Group.inverse G (m ·A n))) ∼ Group.0G H
ans = transitive p1 (transitive p2 (transitive p3 (transitive p4 (transitive p5 (transitive p6 (transitive p7 p8))))))
Group.0G (quotientGroup G fHom) = Group.0G G
Group.inverse (quotientGroup G fHom) = Group.inverse G
Group.+Associative (quotientGroup {S = S} {T = T} {_·A_ = _·A_} G {H = H} {underf = f} fHom) {a} {b} {c} = ans
where
open Setoid T
open Equivalence (Setoid.eq T)
ans : f ((a ·A (b ·A c)) ·A (Group.inverse G ((a ·A b) ·A c))) ∼ Group.0G H
ans = transitive (GroupHom.wellDefined fHom (transferToRight'' G (Group.+Associative G))) (imageOfIdentityIsIdentity fHom)
Group.identRight (quotientGroup {S = S} {T = T} {_·A_ = _·A_} G {H = H} {underf = f} fHom) {a} = ans
where
open Group G
open Setoid T
open Equivalence eq
transitiveG = Equivalence.transitive (Setoid.eq S)
ans : f ((a ·A 0G) ·A inverse a) ∼ Group.0G H
ans = transitive (GroupHom.wellDefined fHom (transitiveG (Group.+WellDefined G (Group.identRight G) (Equivalence.reflexive (Setoid.eq S))) (Group.invRight G))) (imageOfIdentityIsIdentity fHom)
Group.identLeft (quotientGroup {S = S} {T = T} {_·A_ = _·A_} G {H = H} {underf = f} fHom) {a} = ans
where
open Group G
open Setoid T
open Equivalence eq
transitiveG = Equivalence.transitive (Setoid.eq S)
ans : f ((0G ·A a) ·A (inverse a)) ∼ Group.0G H
ans = transitive (GroupHom.wellDefined fHom (transitiveG (Group.+WellDefined G (Group.identLeft G) (Equivalence.reflexive (Setoid.eq S))) (Group.invRight G))) (imageOfIdentityIsIdentity fHom)
Group.invLeft (quotientGroup {S = S} {T = T} {_·A_ = _·A_} G {H = H} {underf = f} fHom) {x} = ans
where
open Group G
open Setoid T
open Equivalence eq
ans : f ((inverse x ·A x) ·A (inverse 0G)) ∼ (Group.0G H)
ans = transitive (GroupHom.wellDefined fHom (Equivalence.transitive (Setoid.eq S) (replaceGroupOp G (Equivalence.symmetric (Setoid.eq S) (Group.invLeft G)) (Equivalence.symmetric (Setoid.eq S) (invIdentity G)) (Equivalence.reflexive (Setoid.eq S)) ((Equivalence.reflexive (Setoid.eq S))) ((Equivalence.reflexive (Setoid.eq S)))) (identRight {0G}))) (imageOfIdentityIsIdentity fHom)
Group.invRight (quotientGroup {S = S} {T = T} {_·A_ = _·A_} G {H = H} {underf = f} fHom) {x} = ans
where
open Group G
open Setoid T
open Equivalence eq
ans : f ((x ·A inverse x) ·A (inverse 0G)) ∼ (Group.0G H)
ans = transitive (GroupHom.wellDefined fHom (Equivalence.transitive (Setoid.eq S) (replaceGroupOp G (Equivalence.symmetric (Setoid.eq S) (Group.invRight G)) (Equivalence.symmetric (Setoid.eq S) (invIdentity G)) (Equivalence.reflexive (Setoid.eq S)) (Equivalence.reflexive (Setoid.eq S)) (Equivalence.reflexive (Setoid.eq S))) (identRight {0G}))) (imageOfIdentityIsIdentity fHom)
record Subgroup {a} {b} {c} {d} {A : Set a} {B : Set c} {S : Setoid {a} {b} A} {T : Setoid {c} {d} B} {_·A_ : A → A → A} {_·B_ : B → B → B} (G : Group S _·A_) (H : Group T _·B_) {f : B → A} (hom : GroupHom H G f) : Set (a ⊔ b ⊔ c ⊔ d) where
open Setoid T renaming (_∼_ to _∼G_)
open Setoid S renaming (_∼_ to _∼H_)
field
fInj : SetoidInjection T S f
record Subgroup {a} {b} {c} {d} {A : Set a} {B : Set c} {S : Setoid {a} {b} A} {T : Setoid {c} {d} B} {_·A_ : A → A → A} {_·B_ : B → B → B} (G : Group S _·A_) (H : Group T _·B_) {f : B → A} (hom : GroupHom H G f) : Set (a ⊔ b ⊔ c ⊔ d) where
open Setoid T renaming (_∼_ to _∼G_)
open Setoid S renaming (_∼_ to _∼H_)
field
fInj : SetoidInjection T S f
{-
quotientHom : {a b c d : _} {A : Set a} {B : Set b} {S : Setoid {a} {c} A} {T : Setoid {b} {d} B} {_·A_ : A → A → A} {_·B_ : B → B → B} (G : Group S _·A_) {H : Group T _·B_} → {f : A → B} → (fHom : GroupHom G H f) → A → A
quotientHom {S = S} {T = T} {_·A_ = _·A_} {_·B_ = _·B_} G {f = f} fHom a = {!!}
quotientHom : {a b c d : _} {A : Set a} {B : Set b} {S : Setoid {a} {c} A} {T : Setoid {b} {d} B} {_·A_ : A → A → A} {_·B_ : B → B → B} (G : Group S _·A_) {H : Group T _·B_} → {f : A → B} → (fHom : GroupHom G H f) → A → A
quotientHom {S = S} {T = T} {_·A_ = _·A_} {_·B_ = _·B_} G {f = f} fHom a = {!!}
quotientInjection : {a b c d : _} {A : Set a} {B : Set b} {S : Setoid {a} {c} A} {T : Setoid {b} {d} B} {_·A_ : A → A → A} {_·B_ : B → B → B} (G : Group S _·A_) {H : Group T _·B_} → {f : A → B} → (fHom : GroupHom G H f) → GroupHom (quotientGroup G fHom) G (quotientHom G fHom)
GroupHom.groupHom (quotientInjection {S = S} {T = T} {_·A_ = _·A_} {_·B_ = _·B_} G {f = f} fHom) {x} {y} = {!!}
where
open Setoid S
open Equivalence eq
open Reflexive reflexiveEq
GroupHom.wellDefined (quotientInjection {S = S} {T = T} {_·A_ = _·A_} G {H = H} {f = f} fHom) {x} {y} x~y = {!!}
where
open Group G
open Setoid S
open Setoid T renaming (_∼_ to _∼T_)
open Equivalence (Setoid.eq S)
open Reflexive reflexiveEq
have : f (x ·A inverse y) ∼T Group.0G H
have = x~y
need : x ∼ y
need = {!!}
quotientInjection : {a b c d : _} {A : Set a} {B : Set b} {S : Setoid {a} {c} A} {T : Setoid {b} {d} B} {_·A_ : A → A → A} {_·B_ : B → B → B} (G : Group S _·A_) {H : Group T _·B_} → {f : A → B} → (fHom : GroupHom G H f) → GroupHom (quotientGroup G fHom) G (quotientHom G fHom)
GroupHom.groupHom (quotientInjection {S = S} {T = T} {_·A_ = _·A_} {_·B_ = _·B_} G {f = f} fHom) {x} {y} = {!!}
where
open Setoid S
open Equivalence eq
open Reflexive reflexiveEq
GroupHom.wellDefined (quotientInjection {S = S} {T = T} {_·A_ = _·A_} G {H = H} {f = f} fHom) {x} {y} x~y = {!!}
where
open Group G
open Setoid S
open Setoid T renaming (_∼_ to _∼T_)
open Equivalence (Setoid.eq S)
open Reflexive reflexiveEq
have : f (x ·A inverse y) ∼T Group.0G H
have = x~y
need : x ∼ y
need = {!!}
quotientIsSubgroup : {a b c d : _} {A : Set a} {B : Set b} {S : Setoid {a} {c} A} {T : Setoid {b} {d} B} {_·A_ : A → A → A} {_·B_ : B → B → B} {G : Group S _·A_} {H : Group T _·B_} → {f : A → B} → {fHom : GroupHom G H f} → Subgroup G (quotientGroup G fHom) (quotientInjection G fHom)
quotientIsSubgroup = {!!}
quotientIsSubgroup : {a b c d : _} {A : Set a} {B : Set b} {S : Setoid {a} {c} A} {T : Setoid {b} {d} B} {_·A_ : A → A → A} {_·B_ : B → B → B} {G : Group S _·A_} {H : Group T _·B_} → {f : A → B} → {fHom : GroupHom G H f} → Subgroup G (quotientGroup G fHom) (quotientInjection G fHom)
quotientIsSubgroup = {!!}
-}
subgroupOfAbelianIsAbelian : {a b c d : _} {A : Set a} {B : Set b} {S : Setoid {a} {c} A} {T : Setoid {b} {d} B} {_+A_ : A → A → A} {_+B_ : B → B → B} {G : Group S _+A_} {H : Group T _+B_} {f : B → A} {fHom : GroupHom H G f} → Subgroup G H fHom → AbelianGroup G → AbelianGroup H
AbelianGroup.commutative (subgroupOfAbelianIsAbelian {S = S} {_+B_ = _+B_} {fHom = fHom} record { fInj = fInj } record { commutative = commutative }) {x} {y} = SetoidInjection.injective fInj (transitive (GroupHom.groupHom fHom) (transitive commutative (symmetric (GroupHom.groupHom fHom))))
where
open Setoid S
open Equivalence eq
subgroupOfAbelianIsAbelian : {a b c d : _} {A : Set a} {B : Set b} {S : Setoid {a} {c} A} {T : Setoid {b} {d} B} {_+A_ : A → A → A} {_+B_ : B → B → B} {G : Group S _+A_} {H : Group T _+B_} {f : B → A} {fHom : GroupHom H G f} → Subgroup G H fHom → AbelianGroup G → AbelianGroup H
AbelianGroup.commutative (subgroupOfAbelianIsAbelian {S = S} {_+B_ = _+B_} {fHom = fHom} record { fInj = fInj } record { commutative = commutative }) {x} {y} = SetoidInjection.injective fInj (transitive (GroupHom.groupHom fHom) (transitive commutative (symmetric (GroupHom.groupHom fHom))))
where
open Setoid S
open Equivalence eq
abelianIsGroupProperty : {a b c d : _} {A : Set a} {B : Set b} {S : Setoid {a} {c} A} {T : Setoid {b} {d} B} {_+A_ : A → A → A} {_+B_ : B → B → B} {G : Group S _+A_} {H : Group T _+B_} → GroupsIsomorphic G H → AbelianGroup H → AbelianGroup G
abelianIsGroupProperty iso abH = subgroupOfAbelianIsAbelian {fHom = GroupIso.groupHom (GroupsIsomorphic.proof iso)} (record { fInj = SetoidBijection.inj (GroupIso.bij (GroupsIsomorphic.proof iso)) }) abH
abelianIsGroupProperty : {a b c d : _} {A : Set a} {B : Set b} {S : Setoid {a} {c} A} {T : Setoid {b} {d} B} {_+A_ : A → A → A} {_+B_ : B → B → B} {G : Group S _+A_} {H : Group T _+B_} → GroupsIsomorphic G H → AbelianGroup H → AbelianGroup G
abelianIsGroupProperty iso abH = subgroupOfAbelianIsAbelian {fHom = GroupIso.groupHom (GroupsIsomorphic.proof iso)} (record { fInj = SetoidBijection.inj (GroupIso.bij (GroupsIsomorphic.proof iso)) }) abH

5
Rings/Lemmas.agda
View File

@@ -21,7 +21,7 @@ ringMinusExtracts {x = x} {y} = transferToRight' additiveGroup (transitive (symm
where
open Equivalence eq
ringMinusExtracts' : {x y : A} → Setoid._∼_ S ((Group.inverse (Ring.additiveGroup R) x) * y) (Group.inverse (Ring.additiveGroup R) (x * y))
ringMinusExtracts' : {x y : A} → ((inverse x) * y) ∼ inverse (x * y)
ringMinusExtracts' {x = x} {y} = transitive *Commutative (transitive ringMinusExtracts (inverseWellDefined additiveGroup *Commutative))
where
open Equivalence eq
@@ -40,3 +40,6 @@ groupLemmaMove0G {S = S} G {x} pr = transitive (symmetric (invInv G)) (transitiv
groupLemmaMove0G' : {a b : _} → {A : Set a} → {_·_ : A → A → A} → {S : Setoid {a} {b} A} → (G : Group S _·_) → {x : A} → Setoid._∼_ S x (Group.0G G) → (Setoid._∼_ S (Group.0G G) (Group.inverse G x))
groupLemmaMove0G' {S = S} G {x} pr = transferToRight' G (Equivalence.transitive (Setoid.eq S) (Group.identLeft G) pr)
oneZeroImpliesAllZero : 0R ∼ 1R → {x : A} → x ∼ 0R
oneZeroImpliesAllZero 0=1 = Equivalence.transitive eq (Equivalence.symmetric eq identIsIdent) (Equivalence.transitive eq (*WellDefined (Equivalence.symmetric eq 0=1) (Equivalence.reflexive eq)) (Equivalence.transitive eq *Commutative timesZero))

52
Rings/Orders/Lemmas.agda
View File

@@ -25,7 +25,6 @@ open Group additiveGroup
open import Rings.Lemmas R
ringAddInequalities : {w x y z : A} → w < x → y < z → (w + y) < (x + z)
ringAddInequalities {w = w} {x} {y} {z} w<x y<z = transitive (orderRespectsAddition w<x y) (<WellDefined groupIsAbelian groupIsAbelian (orderRespectsAddition y<z x))
@@ -188,6 +187,9 @@ ringSwapNegatives {x} {y} -x<-y = SetoidPartialOrder.<WellDefined pOrder (Equiva
v : ((y + (Group.inverse additiveGroup x)) + x) < (0R + x)
v = OrderedRing.orderRespectsAddition order u x
ringSwapNegatives' : {x y : A} → x < y → (Group.inverse (Ring.additiveGroup R) y) < (Group.inverse (Ring.additiveGroup R) x)
ringSwapNegatives' {x} {y} x<y = ringSwapNegatives (<WellDefined (Equivalence.symmetric eq (invTwice additiveGroup _)) (Equivalence.symmetric eq (invTwice additiveGroup _)) x<y)
ringCanMultiplyByNegative : {x y c : A} → c < (Ring.0R R) → x < y → (y * c) < (x * c)
ringCanMultiplyByNegative {x} {y} {c} c<0 x<y = ringSwapNegatives u
where
@@ -246,12 +248,12 @@ absNegation a | inr 0=a | inl (inl 0<-a) = exFalso (irreflexive {0G} (<WellDefin
absNegation a | inr 0=a | inl (inr -a<0) = exFalso (irreflexive {0G} (<WellDefined (Equivalence.transitive eq (inverseWellDefined additiveGroup (Equivalence.symmetric eq 0=a)) (invIdent additiveGroup)) (Equivalence.reflexive eq) -a<0))
absNegation a | inr 0=a | inr 0=-a = Equivalence.transitive eq (Equivalence.symmetric eq 0=a) 0=-a
lemm4 : (a b : A) → (0G < a) → (b < 0G) → (a * b) < 0G
lemm4 a b 0<a b<0 with orderRespectsMultiplication 0<a (lemm2 _ b<0)
posTimesNeg : (a b : A) → (0G < a) → (b < 0G) → (a * b) < 0G
posTimesNeg a b 0<a b<0 with orderRespectsMultiplication 0<a (lemm2 _ b<0)
... | bl = <WellDefined (invTwice additiveGroup _) (Equivalence.reflexive eq) (lemm2' _ (<WellDefined (Equivalence.reflexive eq) ringMinusExtracts bl))
lemm5 : (a b : A) → (a < 0G) → (b < 0G) → 0G < (a * b)
lemm5 a b a<0 b<0 with orderRespectsMultiplication (lemm2 _ a<0) (lemm2 _ b<0)
negTimesPos : (a b : A) → (a < 0G) → (b < 0G) → 0G < (a * b)
negTimesPos a b a<0 b<0 with orderRespectsMultiplication (lemm2 _ a<0) (lemm2 _ b<0)
... | bl = <WellDefined (Equivalence.reflexive eq) twoNegativesTimes bl
absRespectsTimes : (a b : A) → abs (a * b) ∼ (abs a) * (abs b)
@@ -265,20 +267,20 @@ absRespectsTimes a b | inl (inl 0<a) | inl (inr b<0) with totality 0R (a * b)
absRespectsTimes a b | inl (inl 0<a) | inl (inr b<0) | inl (inl 0<ab) with <WellDefined (Equivalence.reflexive eq) ringMinusExtracts (orderRespectsMultiplication 0<a (lemm2 b b<0))
... | bl = exFalso (irreflexive {0G} (SetoidPartialOrder.transitive pOrder 0<ab (<WellDefined (invTwice additiveGroup _) (Equivalence.reflexive eq) (lemm2' _ bl))))
absRespectsTimes a b | inl (inl 0<a) | inl (inr b<0) | inl (inr ab<0) = Equivalence.symmetric eq ringMinusExtracts
absRespectsTimes a b | inl (inl 0<a) | inl (inr b<0) | inr 0=ab = exFalso (irreflexive {0G} (<WellDefined (Equivalence.symmetric eq 0=ab) (Equivalence.reflexive eq) (lemm4 a b 0<a b<0)))
absRespectsTimes a b | inl (inl 0<a) | inl (inr b<0) | inr 0=ab = exFalso (irreflexive {0G} (<WellDefined (Equivalence.symmetric eq 0=ab) (Equivalence.reflexive eq) (posTimesNeg a b 0<a b<0)))
absRespectsTimes a b | inl (inl 0<a) | inr 0=b with totality 0R (a * b)
absRespectsTimes a b | inl (inl 0<a) | inr 0=b | inl (inl 0<ab) = exFalso (irreflexive {0G} (<WellDefined (Equivalence.reflexive eq) (Equivalence.transitive eq (*WellDefined (Equivalence.reflexive eq) (Equivalence.symmetric eq 0=b)) (timesZero {a})) 0<ab))
absRespectsTimes a b | inl (inl 0<a) | inr 0=b | inl (inr ab<0) = exFalso ((irreflexive {0G} (<WellDefined (Equivalence.transitive eq (*WellDefined (Equivalence.reflexive eq) (Equivalence.symmetric eq 0=b)) (timesZero {a})) (Equivalence.reflexive eq) ab<0)))
absRespectsTimes a b | inl (inl 0<a) | inr 0=b | inr 0=ab = Equivalence.reflexive eq
absRespectsTimes a b | inl (inr a<0) with totality 0R b
absRespectsTimes a b | inl (inr a<0) | inl (inl 0<b) with totality 0R (a * b)
absRespectsTimes a b | inl (inr a<0) | inl (inl 0<b) | inl (inl 0<ab) = exFalso (irreflexive {0G} (SetoidPartialOrder.transitive pOrder 0<ab (<WellDefined *Commutative (Equivalence.reflexive eq) (lemm4 b a 0<b a<0))))
absRespectsTimes a b | inl (inr a<0) | inl (inl 0<b) | inl (inl 0<ab) = exFalso (irreflexive {0G} (SetoidPartialOrder.transitive pOrder 0<ab (<WellDefined *Commutative (Equivalence.reflexive eq) (posTimesNeg b a 0<b a<0))))
absRespectsTimes a b | inl (inr a<0) | inl (inl 0<b) | inl (inr ab<0) = Equivalence.symmetric eq ringMinusExtracts'
absRespectsTimes a b | inl (inr a<0) | inl (inl 0<b) | inr 0=ab = exFalso (irreflexive {0G} (<WellDefined (Equivalence.symmetric eq (Equivalence.transitive eq 0=ab *Commutative)) (Equivalence.reflexive eq) (lemm4 b a 0<b a<0)))
absRespectsTimes a b | inl (inr a<0) | inl (inl 0<b) | inr 0=ab = exFalso (irreflexive {0G} (<WellDefined (Equivalence.symmetric eq (Equivalence.transitive eq 0=ab *Commutative)) (Equivalence.reflexive eq) (posTimesNeg b a 0<b a<0)))
absRespectsTimes a b | inl (inr a<0) | inl (inr b<0) with totality 0R (a * b)
absRespectsTimes a b | inl (inr a<0) | inl (inr b<0) | inl (inl 0<ab) = Equivalence.symmetric eq twoNegativesTimes
absRespectsTimes a b | inl (inr a<0) | inl (inr b<0) | inl (inr ab<0) = exFalso (irreflexive {0G} (SetoidPartialOrder.transitive pOrder (lemm5 a b a<0 b<0) ab<0))
absRespectsTimes a b | inl (inr a<0) | inl (inr b<0) | inr 0=ab = exFalso (exFalso (irreflexive {0G} (<WellDefined (Equivalence.reflexive eq) (Equivalence.symmetric eq 0=ab) (lemm5 a b a<0 b<0))))
absRespectsTimes a b | inl (inr a<0) | inl (inr b<0) | inl (inr ab<0) = exFalso (irreflexive {0G} (SetoidPartialOrder.transitive pOrder (negTimesPos a b a<0 b<0) ab<0))
absRespectsTimes a b | inl (inr a<0) | inl (inr b<0) | inr 0=ab = exFalso (exFalso (irreflexive {0G} (<WellDefined (Equivalence.reflexive eq) (Equivalence.symmetric eq 0=ab) (negTimesPos a b a<0 b<0))))
absRespectsTimes a b | inl (inr a<0) | inr 0=b with totality 0R (a * b)
absRespectsTimes a b | inl (inr a<0) | inr 0=b | inl (inl 0<ab) = exFalso (irreflexive {0R} (<WellDefined (Equivalence.reflexive eq) (Equivalence.transitive eq (*WellDefined (Equivalence.reflexive eq) (Equivalence.symmetric eq 0=b)) (timesZero {a})) 0<ab))
absRespectsTimes a b | inl (inr a<0) | inr 0=b | inl (inr ab<0) = exFalso (irreflexive {0R} (<WellDefined (Equivalence.transitive eq (*WellDefined (Equivalence.reflexive eq) (Equivalence.symmetric eq 0=b)) timesZero) (Equivalence.reflexive eq) ab<0))
@@ -296,3 +298,33 @@ absRespectsTimes a b | inr 0=a | inr 0=b with totality 0R (a * b)
absRespectsTimes a b | inr 0=a | inr 0=b | inl (inl 0<ab) = exFalso (irreflexive {0R} (<WellDefined (Equivalence.reflexive eq) (Equivalence.transitive eq (*WellDefined (Equivalence.reflexive eq) (Equivalence.symmetric eq 0=b)) timesZero) 0<ab))
absRespectsTimes a b | inr 0=a | inr 0=b | inl (inr ab<0) = exFalso (irreflexive {0R} (<WellDefined (Equivalence.transitive eq (*WellDefined (Equivalence.reflexive eq) (Equivalence.symmetric eq 0=b)) timesZero) (Equivalence.reflexive eq) ab<0))
absRespectsTimes a b | inr 0=a | inr 0=b | inr 0=ab = Equivalence.reflexive eq
absNonnegative : {a : A} → (abs a < 0R) → False
absNonnegative {a} pr with SetoidTotalOrder.totality tOrder 0R a
absNonnegative {a} pr | inl (inl x) = irreflexive {0G} (SetoidPartialOrder.transitive pOrder x pr)
absNonnegative {a} pr | inl (inr x) = irreflexive {0G} (SetoidPartialOrder.transitive pOrder (<WellDefined (Equivalence.reflexive eq) (invTwice additiveGroup a) (lemm2 (inverse a) pr)) x)
absNonnegative {a} pr | inr x = irreflexive {0G} (<WellDefined (Equivalence.symmetric eq x) (Equivalence.reflexive eq) pr)
a-bPos : {a b : A} → ((a ∼ b) → False) → 0R < abs (a + inverse b)
a-bPos {a} {b} a!=b with totality 0R (a + inverse b)
a-bPos {a} {b} a!=b | inl (inl x) = x
a-bPos {a} {b} a!=b | inl (inr x) = lemm2 _ x
a-bPos {a} {b} a!=b | inr x = exFalso (a!=b (transferToRight additiveGroup (Equivalence.symmetric eq x)))
absZeroImpliesZero : {a : A} → abs a ∼ 0R → a ∼ 0R
absZeroImpliesZero {a} a=0 with totality 0R a
absZeroImpliesZero {a} a=0 | inl (inl 0<a) = exFalso (irreflexive {0G} (<WellDefined (Equivalence.reflexive eq) a=0 0<a))
absZeroImpliesZero {a} a=0 | inl (inr a<0) = Equivalence.symmetric eq (lemm3 (inverse a) a (Equivalence.symmetric eq invLeft) (Equivalence.symmetric eq a=0))
absZeroImpliesZero {a} a=0 | inr 0=a = a=0
halvePositive : (a : A) → 0R < (a + a) → 0R < a
halvePositive a 0<2a with totality 0R a
halvePositive a 0<2a | inl (inl x) = x
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))
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))
0<1 : (0R ∼ 1R → False) → 0R < 1R
0<1 0!=1 with SetoidTotalOrder.totality tOrder 0R 1R
0<1 0!=1 | inl (inl x) = x
0<1 0!=1 | inl (inr x) = <WellDefined (Equivalence.reflexive eq) (Equivalence.transitive eq twoNegativesTimes identIsIdent) (orderRespectsMultiplication (lemm2 1R x) (lemm2 1R x))
0<1 0!=1 | inr x = exFalso (0!=1 x)

25
Sequences.agda
View File

@@ -2,6 +2,7 @@
open import LogicalFormulae
open import Numbers.Naturals.Definition
open import Setoids.Setoids
module Sequences where
@@ -27,6 +28,10 @@ funcToSequenceReversible : {a : _} {A : Set a} (f : ℕ → A) → (n : ℕ) →
funcToSequenceReversible f zero = refl
funcToSequenceReversible f (succ n) = funcToSequenceReversible (λ i → f (succ i)) n
map : {a b : _} {A : Set a} {B : Set b} (f : A → B) (s : Sequence A) → Sequence B
Sequence.head (map f s) = f (Sequence.head s)
Sequence.tail (map f s) = map f (Sequence.tail s)
apply : {a b c : _} {A : Set a} {B : Set b} {C : Set c} (f : A → B → C) (s1 : Sequence A) (s2 : Sequence B) → Sequence C
Sequence.head (apply f s1 s2) = f (Sequence.head s1) (Sequence.head s2)
Sequence.tail (apply f s1 s2) = apply f (Sequence.tail s1) (Sequence.tail s2)
@@ -34,3 +39,23 @@ Sequence.tail (apply f s1 s2) = apply f (Sequence.tail s1) (Sequence.tail s2)
indexAndConst : {a : _} {A : Set a} (a : A) (n : ℕ) → index (constSequence a) n ≡ a
indexAndConst a zero = refl
indexAndConst a (succ n) = indexAndConst a n
mapTwice : {a b c : _} {A : Set a} {B : Set b} {C : Set c} (f : A → B) (g : B → C) (s : Sequence A) → {n : ℕ} → index (map g (map f s)) n ≡ index (map (λ i → g (f i)) s) n
mapTwice f g s {zero} = refl
mapTwice f g s {succ n} = mapTwice f g (Sequence.tail s) {n}
mapAndIndex : {a b : _} {A : Set a} {B : Set b} (s : Sequence A) (f : A → B) (n : ℕ) → f (index s n) ≡ index (map f s) n
mapAndIndex s f zero = refl
mapAndIndex s f (succ n) = mapAndIndex (Sequence.tail s) f n
indexExtensional : {a b c : _} {A : Set a} {B : Set b} (T : Setoid {_} {c} B) (s : Sequence A) (f g : A → B) → (extension : ∀ {x} → (Setoid._∼_ T (f x) (g x))) → {n : ℕ} → Setoid._∼_ T (index (map f s) n) (index (map g s) n)
indexExtensional T s f g extension {zero} = extension
indexExtensional T s f g extension {succ n} = indexExtensional T (Sequence.tail s) f g extension {n}
indexAndApply : {a b c : _} {A : Set a} {B : Set b} {C : Set c} (s1 : Sequence A) (s2 : Sequence B) (f : A → B → C) → {m : ℕ} → index (apply f s1 s2) m ≡ f (index s1 m) (index s2 m)
indexAndApply s1 s2 f {zero} = refl
indexAndApply s1 s2 f {succ m} = indexAndApply (Sequence.tail s1) (Sequence.tail s2) f {m}
mapAndApply : {a b c d : _} {A : Set a} {B : Set b} {C : Set c} {D : Set d} (s1 : Sequence A) (s2 : Sequence B) (f : A → B → C) (g : C → D) → (m : ℕ) → index (map g (apply f s1 s2)) m ≡ g (f (index s1 m) (index s2 m))
mapAndApply s1 s2 f g zero = refl
mapAndApply s1 s2 f g (succ m) = mapAndApply (Sequence.tail s1) (Sequence.tail s2) f g m

56
Setoids/Orders.agda
View File

@@ -9,34 +9,34 @@ open import Agda.Primitive using (Level; lzero; lsuc; _⊔_)
module Setoids.Orders where
record SetoidPartialOrder {a b c : _} {A : Set a} (S : Setoid {a} {b} A) (_<_ : Rel {a} {c} A) : Set (a ⊔ b ⊔ c) where
open Setoid S
field
<WellDefined : {a b c d : A} → (a ∼ b) → (c ∼ d) → a < c → b < d
irreflexive : {x : A} → (x < x) → False
transitive : {a b c : A} → (a < b) → (b < c) → (a < c)
record SetoidPartialOrder {a b c : _} {A : Set a} (S : Setoid {a} {b} A) (_<_ : Rel {a} {c} A) : Set (a ⊔ b ⊔ c) where
open Setoid S
field
<WellDefined : {a b c d : A} → (a ∼ b) → (c ∼ d) → a < c → b < d
irreflexive : {x : A} → (x < x) → False
transitive : {a b c : A} → (a < b) → (b < c) → (a < c)
record SetoidTotalOrder {a b c : _} {A : Set a} {S : Setoid {a} {b} A} {_<_ : Rel {a} {c} A} (P : SetoidPartialOrder S _<_) : Set (a ⊔ b ⊔ c) where
open Setoid S
field
totality : (a b : A) → ((a < b) || (b < a)) || (a ∼ b)
partial : SetoidPartialOrder S _<_
partial = P
min : A → A → A
min a b with totality a b
min a b | inl (inl a<b) = a
min a b | inl (inr b<a) = b
min a b | inr a=b = a
max : A → A → A
max a b with totality a b
max a b | inl (inl a<b) = b
max a b | inl (inr b<a) = a
max a b | inr a=b = b
record SetoidTotalOrder {a b c : _} {A : Set a} {S : Setoid {a} {b} A} {_<_ : Rel {a} {c} A} (P : SetoidPartialOrder S _<_) : Set (a ⊔ b ⊔ c) where
open Setoid S
field
totality : (a b : A) → ((a < b) || (b < a)) || (a ∼ b)
partial : SetoidPartialOrder S _<_
partial = P
min : A → A → A
min a b with totality a b
min a b | inl (inl a<b) = a
min a b | inl (inr b<a) = b
min a b | inr a=b = a
max : A → A → A
max a b with totality a b
max a b | inl (inl a<b) = b
max a b | inl (inr b<a) = a
max a b | inr a=b = b
partialOrderToSetoidPartialOrder : {a b : _} {A : Set a} (P : PartialOrder {a} {b} A) → SetoidPartialOrder (reflSetoid A) (PartialOrder._<_ P)
SetoidPartialOrder.<WellDefined (partialOrderToSetoidPartialOrder P) a=b c=d a<c rewrite a=b | c=d = a<c
SetoidPartialOrder.irreflexive (partialOrderToSetoidPartialOrder P) = PartialOrder.irreflexive P
SetoidPartialOrder.transitive (partialOrderToSetoidPartialOrder P) = PartialOrder.transitive P
partialOrderToSetoidPartialOrder : {a b : _} {A : Set a} (P : PartialOrder {a} {b} A) → SetoidPartialOrder (reflSetoid A) (PartialOrder._<_ P)
SetoidPartialOrder.<WellDefined (partialOrderToSetoidPartialOrder P) a=b c=d a<c rewrite a=b | c=d = a<c
SetoidPartialOrder.irreflexive (partialOrderToSetoidPartialOrder P) = PartialOrder.irreflexive P
SetoidPartialOrder.transitive (partialOrderToSetoidPartialOrder P) = PartialOrder.transitive P
totalOrderToSetoidTotalOrder : {a b : _} {A : Set a} (T : TotalOrder {a} {b} A) → SetoidTotalOrder (partialOrderToSetoidPartialOrder (TotalOrder.order T))
SetoidTotalOrder.totality (totalOrderToSetoidTotalOrder T) = TotalOrder.totality T
totalOrderToSetoidTotalOrder : {a b : _} {A : Set a} (T : TotalOrder {a} {b} A) → SetoidTotalOrder (partialOrderToSetoidPartialOrder (TotalOrder.order T))
SetoidTotalOrder.totality (totalOrderToSetoidTotalOrder T) = TotalOrder.totality T