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Cauchy-completion, first parts (#52)

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Patrick Stevens
2019-10-22 07:51:09 +01:00
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parent 959071214e
commit 6eaa181104
15 changed files with 567 additions and 518 deletions
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76
CauchyCompletion.agda Normal file
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@@ -0,0 +1,76 @@
{-# 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 CauchyCompletion {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)
cauchy : Sequence A → Set (m ⊔ o)
cauchy s = ∀ (ε : A) → (0R < ε) → Sg ℕ (λ N → ∀ {m n : ℕ} → (N <N m) → (N <N n) → abs ((index s m) -R (index s n)) < ε)
record CauchyCompletion : Set (m ⊔ o) where
field
elts : Sequence A
converges : cauchy elts
injection : A → CauchyCompletion
CauchyCompletion.elts (injection a) = constSequence a
CauchyCompletion.converges (injection a) = λ ε 0<e → 0 , λ {m} {n} _ _ → <WellDefined (symmetric (identityOfIndiscernablesRight _∼_ (absWellDefined (index (constSequence a) m + inverse (index (constSequence a) n)) 0R (t m n)) (absZero order))) reflexive 0<e
where
t : (m n : ℕ) → index (constSequence a) m + inverse (index (constSequence a) n) ∼ 0R
t m n = identityOfIndiscernablesLeft _∼_ (identityOfIndiscernablesLeft _∼_ invRight (equalityCommutative (applyEquality (λ i → a + inverse i) (indexAndConst a n)))) (applyEquality (_+ inverse (index (constSequence a) n)) (equalityCommutative (indexAndConst a m)))
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
... | 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
where
t : {m n : ℕ} → Na +N Nb <N m → Na +N Nb <N n → abs ((index (apply _+_ a b) m) + inverse (index (apply _+_ a b) n)) < ε
t {m} {n} <m <n with prA {m} {n} (inequalityShrinkLeft <m) (inequalityShrinkLeft <n)
... | am-an<e/2 with prB {m} {n} (inequalityShrinkRight <m) (inequalityShrinkRight <n)
... | 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))

6
Everything/Safe.agda
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@@ -1,4 +1,4 @@
{-# OPTIONS --warning=error --safe --without-K #-}
{-# OPTIONS --warning=error --safe --without-K --guardedness #-}
-- This file contains everything that can be compiled in --safe mode.
@@ -24,6 +24,8 @@ open import Fields.FieldOfFractionsOrder
open import Rings.Definition
open import Rings.Lemmas
open import Rings.Order
open import Rings.Orders.Lemmas
open import Rings.IntegralDomains
open import Setoids.Setoids
@@ -51,4 +53,6 @@ open import Monoids.Definition
open import Semirings.Definition
open import Semirings.Solver
open import CauchyCompletion
module Everything.Safe where

1
Everything/WithK.agda
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@@ -5,7 +5,6 @@
open import Numbers.Primes.PrimeNumbers
open import Numbers.Primes.IntegerFactorisation
open import Numbers.Rationals
open import Numbers.RationalsLemmas
open import Logic.PropositionalLogic
open import Logic.PropositionalLogicExamples

170
Fields/FieldOfFractionsOrder.agda
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@@ -5,6 +5,8 @@ open import Groups.Groups
open import Groups.Definition
open import Groups.Lemmas
open import Rings.Definition
open import Rings.Order
open import Rings.Orders.Lemmas
open import Rings.Lemmas
open import Rings.IntegralDomains
open import Fields.Fields
@@ -48,11 +50,11 @@ module Fields.FieldOfFractionsOrder where
have : ((numX * denomY) * denomZ) < ((numY * denomX) * denomZ)
have = ringCanMultiplyByPositive order 0<denomZ x<y
p : ((numX * denomZ) * denomY) < ((numY * denomX) * denomZ)
p = SetoidPartialOrder.wellDefined pOrder (transitive (symmetric *Associative) (transitive (*WellDefined reflexive *Commutative) *Associative)) reflexive have
p = SetoidPartialOrder.<WellDefined pOrder (transitive (symmetric *Associative) (transitive (*WellDefined reflexive *Commutative) *Associative)) reflexive have
q : ((denomX * numZ) * denomY) < ((numY * denomX) * denomZ)
q = SetoidPartialOrder.wellDefined pOrder (*WellDefined x=z reflexive) reflexive p
q = SetoidPartialOrder.<WellDefined pOrder (*WellDefined x=z reflexive) reflexive p
r : ((numZ * denomY) * denomX) < ((numY * denomZ) * denomX)
r = SetoidPartialOrder.wellDefined pOrder (transitive (symmetric *Associative) *Commutative) (transitive (symmetric *Associative) (transitive (*WellDefined reflexive *Commutative) *Associative)) q
r = SetoidPartialOrder.<WellDefined pOrder (transitive (symmetric *Associative) *Commutative) (transitive (symmetric *Associative) (transitive (*WellDefined reflexive *Commutative) *Associative)) q
s : (numZ * denomY) < (numY * denomZ)
s = ringCanCancelPositive order 0<denomX r
fieldOfFractionsOrderWellDefinedLeft {S = S} {_*_ = _*_} {R} {pOrder = pOrder} {tOrder = tOrder} I order {numX ,, (denomX , denomX!=0)} {numY ,, (denomY , denomY!=0)} {numZ ,, (denomZ , denomZ!=0)} x<y x=z | inl (inl 0<denomZ) | inl (inl 0<denomY) | inl (inl 0<denomX) | inl (inr x) = exFalso (SetoidPartialOrder.irreflexive pOrder (SetoidPartialOrder.transitive pOrder 0<denomY x))
@@ -65,11 +67,11 @@ module Fields.FieldOfFractionsOrder where
open Ring R
open Equivalence (Setoid.eq S)
p : ((numY * denomX) * denomZ) < ((numX * denomZ) * denomY)
p = SetoidPartialOrder.wellDefined pOrder reflexive (transitive (symmetric *Associative) (transitive (*WellDefined reflexive *Commutative) *Associative)) (ringCanMultiplyByPositive order 0<denomZ x<y)
p = SetoidPartialOrder.<WellDefined pOrder reflexive (transitive (symmetric *Associative) (transitive (*WellDefined reflexive *Commutative) *Associative)) (ringCanMultiplyByPositive order 0<denomZ x<y)
q : ((numY * denomX) * denomZ) < ((denomX * numZ) * denomY)
q = SetoidPartialOrder.wellDefined pOrder reflexive (*WellDefined x=z reflexive) p
q = SetoidPartialOrder.<WellDefined pOrder reflexive (*WellDefined x=z reflexive) p
r : ((numY * denomZ) * denomX) < ((numZ * denomY) * denomX)
r = SetoidPartialOrder.wellDefined pOrder (transitive (symmetric *Associative) (transitive (*WellDefined reflexive *Commutative) *Associative)) (transitive (symmetric *Associative) *Commutative) q
r = SetoidPartialOrder.<WellDefined pOrder (transitive (symmetric *Associative) (transitive (*WellDefined reflexive *Commutative) *Associative)) (transitive (symmetric *Associative) *Commutative) q
fieldOfFractionsOrderWellDefinedLeft {S = S} {_*_ = _*_} {R} {pOrder = pOrder} {tOrder = tOrder} I order {numX ,, (denomX , denomX!=0)} {numY ,, (denomY , denomY!=0)} {numZ ,, (denomZ , denomZ!=0)} x<y x=z | inl (inl 0<denomZ) | inl (inl 0<denomY) | inl (inr denomX<0) | inl (inr x) = exFalso (SetoidPartialOrder.irreflexive pOrder (SetoidPartialOrder.transitive pOrder 0<denomY x))
fieldOfFractionsOrderWellDefinedLeft {S = S} {_*_ = _*_} {R} {tOrder = tOrder} I order {numX ,, (denomX , denomX!=0)} {numY ,, (denomY , denomY!=0)} {numZ ,, (denomZ , denomZ!=0)} x<y x=z | inl (inl 0<denomZ) | inl (inl 0<denomY) | inl (inr denomX<0) | inr x = exFalso (denomY!=0 (Equivalence.symmetric (Setoid.eq S) x))
fieldOfFractionsOrderWellDefinedLeft {S = S} {_*_ = _*_} {R} {tOrder = tOrder} I order {numX ,, (denomX , denomX!=0)} {numY ,, (denomY , denomY!=0)} {numZ ,, (denomZ , denomZ!=0)} x<y x=z | inl (inl 0<denomZ) | inl (inl 0<denomY) | inr 0=denomX = exFalso (denomX!=0 (Equivalence.symmetric (Setoid.eq S) 0=denomX))
@@ -83,9 +85,9 @@ module Fields.FieldOfFractionsOrder where
p : ((numY * denomX) * denomZ) < ((numX * denomY) * denomZ)
p = ringCanMultiplyByPositive order 0<denomZ x<y
q : ((numY * denomX) * denomZ) < ((denomX * numZ) * denomY)
q = SetoidPartialOrder.wellDefined pOrder reflexive (transitive (symmetric *Associative) (transitive (*WellDefined reflexive *Commutative) (transitive *Associative (*WellDefined x=z reflexive)))) p
q = SetoidPartialOrder.<WellDefined pOrder reflexive (transitive (symmetric *Associative) (transitive (*WellDefined reflexive *Commutative) (transitive *Associative (*WellDefined x=z reflexive)))) p
r : ((numY * denomZ) * denomX) < ((numZ * denomY) * denomX)
r = SetoidPartialOrder.wellDefined pOrder (transitive (symmetric *Associative) (transitive (*WellDefined reflexive *Commutative) *Associative)) (transitive (symmetric *Associative) *Commutative) q
r = SetoidPartialOrder.<WellDefined pOrder (transitive (symmetric *Associative) (transitive (*WellDefined reflexive *Commutative) *Associative)) (transitive (symmetric *Associative) *Commutative) q
fieldOfFractionsOrderWellDefinedLeft {S = S} {_*_ = _*_} {R} {tOrder = tOrder} I order {numX ,, (denomX , denomX!=0)} {numY ,, (denomY , denomY!=0)} {numZ ,, (denomZ , denomZ!=0)} x<y x=z | inl (inl 0<denomZ) | inl (inr denomY<0) | inl (inl 0<denomX) | inr x = exFalso (denomY!=0 (Equivalence.symmetric (Setoid.eq S) x))
fieldOfFractionsOrderWellDefinedLeft {S = S} {_*_ = _*_} {R} {tOrder = tOrder} I order {numX ,, (denomX , denomX!=0)} {numY ,, (denomY , denomY!=0)} {numZ ,, (denomZ , denomZ!=0)} x<y x=z | inl (inl 0<denomZ) | inl (inr denomY<0) | inl (inr denomX<0) with SetoidTotalOrder.totality tOrder (Ring.0R R) denomY
fieldOfFractionsOrderWellDefinedLeft {S = S} {_*_ = _*_} {R} {pOrder = pOrder} {tOrder = tOrder} I order {numX ,, (denomX , denomX!=0)} {numY ,, (denomY , denomY!=0)} {numZ ,, (denomZ , denomZ!=0)} x<y x=z | inl (inl 0<denomZ) | inl (inr denomY<0) | inl (inr denomX<0) | inl (inl 0<denomY) = exFalso (SetoidPartialOrder.irreflexive pOrder (SetoidPartialOrder.transitive pOrder 0<denomY denomY<0))
@@ -96,23 +98,23 @@ module Fields.FieldOfFractionsOrder where
p : ((numX * denomY) * denomZ) < ((numY * denomX) * denomZ)
p = ringCanMultiplyByPositive order 0<denomZ x<y
q : ((numZ * denomY) * denomX) < ((numY * denomZ) * denomX)
q = SetoidPartialOrder.wellDefined pOrder (transitive (*WellDefined *Commutative reflexive) (transitive (symmetric *Associative) (transitive (*WellDefined reflexive x=z) (transitive *Commutative (transitive (symmetric *Associative) *Commutative))))) (transitive (symmetric *Associative) (transitive (*WellDefined reflexive *Commutative) *Associative)) p
q = SetoidPartialOrder.<WellDefined pOrder (transitive (*WellDefined *Commutative reflexive) (transitive (symmetric *Associative) (transitive (*WellDefined reflexive x=z) (transitive *Commutative (transitive (symmetric *Associative) *Commutative))))) (transitive (symmetric *Associative) (transitive (*WellDefined reflexive *Commutative) *Associative)) p
fieldOfFractionsOrderWellDefinedLeft {S = S} {_*_ = _*_} {R} {tOrder = tOrder} I order {numX ,, (denomX , denomX!=0)} {numY ,, (denomY , denomY!=0)} {numZ ,, (denomZ , denomZ!=0)} x<y x=z | inl (inl 0<denomZ) | inl (inr denomY<0) | inl (inr denomX<0) | inr x = exFalso (denomY!=0 (Equivalence.symmetric (Setoid.eq S) x))
fieldOfFractionsOrderWellDefinedLeft {S = S} {_*_ = _*_} {R} {tOrder = tOrder} I order {numX ,, (denomX , denomX!=0)} {numY ,, (denomY , denomY!=0)} {numZ ,, (denomZ , denomZ!=0)} x<y x=z | inl (inl 0<denomZ) | inl (inr denomY<0) | inr 0=denomX = exFalso (denomX!=0 (Equivalence.symmetric (Setoid.eq S) 0=denomX))
fieldOfFractionsOrderWellDefinedLeft {S = S} {_*_ = _*_} {R} {tOrder = tOrder} I order {numX ,, (denomX , denomX!=0)} {numY ,, (denomY , denomY!=0)} {numZ ,, (denomZ , denomZ!=0)} x<y x=z | inl (inl 0<denomZ) | inr 0=denomY = exFalso (denomY!=0 (Equivalence.symmetric (Setoid.eq S) 0=denomY))
fieldOfFractionsOrderWellDefinedLeft {S = S} {_*_ = _*_} {R} {tOrder = tOrder} I order {numX ,, (denomX , denomX!=0)} {numY ,, (denomY , denomY!=0)} {numZ ,, (denomZ , denomZ!=0)} x<y x=z | inl (inr denomZ<0) with SetoidTotalOrder.totality tOrder (Ring.0R R) denomY
fieldOfFractionsOrderWellDefinedLeft {S = S} {_*_ = _*_} {R} {tOrder = tOrder} I order {numX ,, (denomX , denomX!=0)} {numY ,, (denomY , denomY!=0)} {numZ ,, (denomZ , denomZ!=0)} x<y x=z | inl (inr denomZ<0) | inl (inl 0<denomY) with SetoidTotalOrder.totality tOrder (Ring.0R R) denomX
fieldOfFractionsOrderWellDefinedLeft {S = S} {_*_ = _*_} {R} {tOrder = tOrder} I order {numX ,, (denomX , denomX!=0)} {numY ,, (denomY , denomY!=0)} {numZ ,, (denomZ , denomZ!=0)} x<y x=z | inl (inr denomZ<0) | inl (inl 0<denomY) | inl (inl 0<denomX) with SetoidTotalOrder.totality tOrder (Ring.0R R) denomY
fieldOfFractionsOrderWellDefinedLeft {S = S} {_*_ = _*_} {R} {_<_ = _<_} {pOrder = pOrder} {tOrder = tOrder} I order {numX ,, (denomX , denomX!=0)} {numY ,, (denomY , denomY!=0)} {numZ ,, (denomZ , denomZ!=0)} x<y x=z | inl (inr denomZ<0) | inl (inl 0<denomY) | inl (inl 0<denomX) | inl (inl _) = ringCanCancelPositive order 0<denomX (SetoidPartialOrder.wellDefined pOrder (transitive (symmetric *Associative) (transitive (*WellDefined reflexive *Commutative) *Associative)) (transitive (symmetric *Associative) (transitive (*WellDefined reflexive x=z) (transitive (*WellDefined reflexive (*Commutative)) (transitive *Associative (*WellDefined *Commutative reflexive))))) p)
fieldOfFractionsOrderWellDefinedLeft {S = S} {_*_ = _*_} {R} {_<_ = _<_} {pOrder = pOrder} {tOrder = tOrder} I order {numX ,, (denomX , denomX!=0)} {numY ,, (denomY , denomY!=0)} {numZ ,, (denomZ , denomZ!=0)} x<y x=z | inl (inr denomZ<0) | inl (inl 0<denomY) | inl (inl 0<denomX) | inl (inl _) = ringCanCancelPositive order 0<denomX (SetoidPartialOrder.<WellDefined pOrder (transitive (symmetric *Associative) (transitive (*WellDefined reflexive *Commutative) *Associative)) (transitive (symmetric *Associative) (transitive (*WellDefined reflexive x=z) (transitive (*WellDefined reflexive (*Commutative)) (transitive *Associative (*WellDefined *Commutative reflexive))))) p)
where
open Ring R
open Equivalence (Setoid.eq S)
p : ((numY * denomX) * denomZ) < ((denomY * numX) * denomZ)
p = ringCanMultiplyByNegative order denomZ<0 (SetoidPartialOrder.wellDefined pOrder *Commutative reflexive x<y)
p = ringCanMultiplyByNegative order denomZ<0 (SetoidPartialOrder.<WellDefined pOrder *Commutative reflexive x<y)
fieldOfFractionsOrderWellDefinedLeft {S = S} {_*_ = _*_} {R} {pOrder = pOrder} {tOrder = tOrder} I order {numX ,, (denomX , denomX!=0)} {numY ,, (denomY , denomY!=0)} {numZ ,, (denomZ , denomZ!=0)} x<y x=z | inl (inr denomZ<0) | inl (inl 0<denomY) | inl (inl 0<denomX) | inl (inr denomY<0) = exFalso (SetoidPartialOrder.irreflexive pOrder (SetoidPartialOrder.transitive pOrder 0<denomY denomY<0))
fieldOfFractionsOrderWellDefinedLeft {S = S} {_*_ = _*_} {R} {pOrder = pOrder} {tOrder = tOrder} I order {numX ,, (denomX , denomX!=0)} {numY ,, (denomY , denomY!=0)} {numZ ,, (denomZ , denomZ!=0)} x<y x=z | inl (inr denomZ<0) | inl (inl 0<denomY) | inl (inl 0<denomX) | inr x = exFalso (denomY!=0 (Equivalence.symmetric (Setoid.eq S) x))
fieldOfFractionsOrderWellDefinedLeft {S = S} {_*_ = _*_} {R} {_<_ = _<_} {pOrder = pOrder} {tOrder = tOrder} I order {numX ,, (denomX , denomX!=0)} {numY ,, (denomY , denomY!=0)} {numZ ,, (denomZ , denomZ!=0)} x<y x=z | inl (inr denomZ<0) | inl (inl 0<denomY) | inl (inr denomX<0) with SetoidTotalOrder.totality tOrder (Ring.0R R) denomY
fieldOfFractionsOrderWellDefinedLeft {S = S} {_*_ = _*_} {R} {_<_} {pOrder} {tOrder} I order {numX ,, (denomX , denomX!=0)} {numY ,, (denomY , denomY!=0)} {numZ ,, (denomZ , denomZ!=0)} x<y x=z | inl (inr denomZ<0) | inl (inl 0<denomY) | inl (inr denomX<0) | inl (inl _) = ringCanCancelNegative order denomX<0 (SetoidPartialOrder.wellDefined pOrder (transitive (symmetric *Associative) (transitive (*WellDefined reflexive *Commutative) (transitive *Associative (transitive (*WellDefined x=z reflexive) (transitive (symmetric *Associative) *Commutative))))) (transitive (symmetric *Associative) (transitive (*WellDefined reflexive *Commutative) *Associative)) p)
fieldOfFractionsOrderWellDefinedLeft {S = S} {_*_ = _*_} {R} {_<_} {pOrder} {tOrder} I order {numX ,, (denomX , denomX!=0)} {numY ,, (denomY , denomY!=0)} {numZ ,, (denomZ , denomZ!=0)} x<y x=z | inl (inr denomZ<0) | inl (inl 0<denomY) | inl (inr denomX<0) | inl (inl _) = ringCanCancelNegative order denomX<0 (SetoidPartialOrder.<WellDefined pOrder (transitive (symmetric *Associative) (transitive (*WellDefined reflexive *Commutative) (transitive *Associative (transitive (*WellDefined x=z reflexive) (transitive (symmetric *Associative) *Commutative))))) (transitive (symmetric *Associative) (transitive (*WellDefined reflexive *Commutative) *Associative)) p)
where
open Ring R
open Equivalence (Setoid.eq S)
@@ -126,14 +128,14 @@ module Fields.FieldOfFractionsOrder where
fieldOfFractionsOrderWellDefinedLeft {S = S} {_*_ = _*_} {R} {tOrder = tOrder} I order {numX ,, (denomX , denomX!=0)} {numY ,, (denomY , denomY!=0)} {numZ ,, (denomZ , denomZ!=0)} x<y x=z | inl (inr denomZ<0) | inl (inr denomY<0) with SetoidTotalOrder.totality tOrder (Ring.0R R) denomX
fieldOfFractionsOrderWellDefinedLeft {S = S} {_*_ = _*_} {R} {tOrder = tOrder} I order {numX ,, (denomX , denomX!=0)} {numY ,, (denomY , denomY!=0)} {numZ ,, (denomZ , denomZ!=0)} x<y x=z | inl (inr denomZ<0) | inl (inr denomY<0) | inl (inl 0<denomX) with SetoidTotalOrder.totality tOrder (Ring.0R R) denomY
fieldOfFractionsOrderWellDefinedLeft {S = S} {_*_ = _*_} {R} {pOrder = pOrder} {tOrder = tOrder} I order {numX ,, (denomX , denomX!=0)} {numY ,, (denomY , denomY!=0)} {numZ ,, (denomZ , denomZ!=0)} x<y x=z | inl (inr denomZ<0) | inl (inr denomY<0) | inl (inl 0<denomX) | inl (inl 0<denomY) = exFalso (SetoidPartialOrder.irreflexive pOrder (SetoidPartialOrder.transitive pOrder 0<denomY denomY<0))
fieldOfFractionsOrderWellDefinedLeft {S = S} {_*_ = _*_} {R} {_<_ = _<_} {pOrder = pOrder} {tOrder = tOrder} I order {numX ,, (denomX , denomX!=0)} {numY ,, (denomY , denomY!=0)} {numZ ,, (denomZ , denomZ!=0)} x<y x=z | inl (inr denomZ<0) | inl (inr denomY<0) | inl (inl 0<denomX) | inl (inr _) = ringCanCancelPositive order 0<denomX (SetoidPartialOrder.wellDefined pOrder (transitive (*WellDefined *Commutative reflexive) (transitive (symmetric *Associative) (transitive (*WellDefined reflexive x=z) (transitive *Associative (transitive *Commutative *Associative))))) (transitive (symmetric *Associative) (transitive (*WellDefined reflexive *Commutative) *Associative)) (ringCanMultiplyByNegative order denomZ<0 x<y))
fieldOfFractionsOrderWellDefinedLeft {S = S} {_*_ = _*_} {R} {_<_ = _<_} {pOrder = pOrder} {tOrder = tOrder} I order {numX ,, (denomX , denomX!=0)} {numY ,, (denomY , denomY!=0)} {numZ ,, (denomZ , denomZ!=0)} x<y x=z | inl (inr denomZ<0) | inl (inr denomY<0) | inl (inl 0<denomX) | inl (inr _) = ringCanCancelPositive order 0<denomX (SetoidPartialOrder.<WellDefined pOrder (transitive (*WellDefined *Commutative reflexive) (transitive (symmetric *Associative) (transitive (*WellDefined reflexive x=z) (transitive *Associative (transitive *Commutative *Associative))))) (transitive (symmetric *Associative) (transitive (*WellDefined reflexive *Commutative) *Associative)) (ringCanMultiplyByNegative order denomZ<0 x<y))
where
open Ring R
open Equivalence (Setoid.eq S)
fieldOfFractionsOrderWellDefinedLeft {S = S} {_*_ = _*_} {R} {tOrder = tOrder} I order {numX ,, (denomX , denomX!=0)} {numY ,, (denomY , denomY!=0)} {numZ ,, (denomZ , denomZ!=0)} x<y x=z | inl (inr denomZ<0) | inl (inr denomY<0) | inl (inl 0<denomX) | inr x = exFalso (denomY!=0 (Equivalence.symmetric (Setoid.eq S) x))
fieldOfFractionsOrderWellDefinedLeft {S = S} {_*_ = _*_} {R} {tOrder = tOrder} I order {numX ,, (denomX , denomX!=0)} {numY ,, (denomY , denomY!=0)} {numZ ,, (denomZ , denomZ!=0)} x<y x=z | inl (inr denomZ<0) | inl (inr denomY<0) | inl (inr denomX<0) with SetoidTotalOrder.totality tOrder (Ring.0R R) denomY
fieldOfFractionsOrderWellDefinedLeft {S = S} {_*_ = _*_} {R} {pOrder = pOrder} {tOrder = tOrder} I order {numX ,, (denomX , denomX!=0)} {numY ,, (denomY , denomY!=0)} {numZ ,, (denomZ , denomZ!=0)} x<y x=z | inl (inr denomZ<0) | inl (inr denomY<0) | inl (inr denomX<0) | inl (inl 0<denomY) = exFalso (SetoidPartialOrder.irreflexive pOrder (SetoidPartialOrder.transitive pOrder 0<denomY denomY<0))
fieldOfFractionsOrderWellDefinedLeft {S = S} {_*_ = _*_} {R} {_<_ = _<_} {pOrder = pOrder} {tOrder = tOrder} I order {numX ,, (denomX , denomX!=0)} {numY ,, (denomY , denomY!=0)} {numZ ,, (denomZ , denomZ!=0)} x<y x=z | inl (inr denomZ<0) | inl (inr denomY<0) | inl (inr denomX<0) | inl (inr _) = ringCanCancelNegative order denomX<0 (SetoidPartialOrder.wellDefined pOrder (transitive (symmetric *Associative) (transitive (*WellDefined reflexive *Commutative) *Associative)) (transitive (*WellDefined *Commutative reflexive) (transitive (symmetric *Associative) (transitive (*WellDefined reflexive x=z) (transitive (*WellDefined reflexive *Commutative) (transitive *Associative (*WellDefined *Commutative reflexive)))))) (ringCanMultiplyByNegative order denomZ<0 x<y))
fieldOfFractionsOrderWellDefinedLeft {S = S} {_*_ = _*_} {R} {_<_ = _<_} {pOrder = pOrder} {tOrder = tOrder} I order {numX ,, (denomX , denomX!=0)} {numY ,, (denomY , denomY!=0)} {numZ ,, (denomZ , denomZ!=0)} x<y x=z | inl (inr denomZ<0) | inl (inr denomY<0) | inl (inr denomX<0) | inl (inr _) = ringCanCancelNegative order denomX<0 (SetoidPartialOrder.<WellDefined pOrder (transitive (symmetric *Associative) (transitive (*WellDefined reflexive *Commutative) *Associative)) (transitive (*WellDefined *Commutative reflexive) (transitive (symmetric *Associative) (transitive (*WellDefined reflexive x=z) (transitive (*WellDefined reflexive *Commutative) (transitive *Associative (*WellDefined *Commutative reflexive)))))) (ringCanMultiplyByNegative order denomZ<0 x<y))
where
open Ring R
open Equivalence (Setoid.eq S)
@@ -150,21 +152,21 @@ module Fields.FieldOfFractionsOrder where
fieldOfFractionsOrderWellDefinedRight {S = S} {_*_ = _*_} {R = R} {pOrder = pOrder} {tOrder = tOrder} I order {numX ,, (denomX , denomX!=0)} {numY ,, (denomY , denomY!=0)} {numZ ,, (denomZ , denomZ!=0)} x<y y=z with SetoidTotalOrder.totality tOrder (Ring.0R R) denomX
fieldOfFractionsOrderWellDefinedRight {S = S} {_*_ = _*_} {R} {pOrder = pOrder} {tOrder} I order {numX ,, (denomX , denomX!=0)} {numY ,, (denomY , denomY!=0)} {numZ ,, (denomZ , denomZ!=0)} x<y y=z | inl (inl 0<denomX) with SetoidTotalOrder.totality tOrder (Ring.0R R) denomZ
fieldOfFractionsOrderWellDefinedRight {S = S} {_*_ = _*_} {R} {pOrder = pOrder} {tOrder} I order {numX ,, (denomX , denomX!=0)} {numY ,, (denomY , denomY!=0)} {numZ ,, (denomZ , denomZ!=0)} x<y y=z | inl (inl 0<denomX) | inl (inl 0<denomZ) with SetoidTotalOrder.totality tOrder (Ring.0R R) denomY
fieldOfFractionsOrderWellDefinedRight {S = S} {_*_ = _*_} {R} {_<_ = _<_} {pOrder = pOrder} {tOrder} I order {numX ,, (denomX , denomX!=0)} {numY ,, (denomY , denomY!=0)} {numZ ,, (denomZ , denomZ!=0)} x<y y=z | inl (inl 0<denomX) | inl (inl 0<denomZ) | inl (inl 0<denomY) = ringCanCancelPositive order 0<denomY (SetoidPartialOrder.wellDefined pOrder (transitive (symmetric *Associative) (transitive (*WellDefined reflexive *Commutative) *Associative)) (transitive (symmetric *Associative) (transitive (*WellDefined reflexive *Commutative) (transitive *Associative (transitive (*WellDefined y=z reflexive) (transitive (symmetric *Associative) *Commutative))))) (ringCanMultiplyByPositive order 0<denomZ x<y))
fieldOfFractionsOrderWellDefinedRight {S = S} {_*_ = _*_} {R} {_<_ = _<_} {pOrder = pOrder} {tOrder} I order {numX ,, (denomX , denomX!=0)} {numY ,, (denomY , denomY!=0)} {numZ ,, (denomZ , denomZ!=0)} x<y y=z | inl (inl 0<denomX) | inl (inl 0<denomZ) | inl (inl 0<denomY) = ringCanCancelPositive order 0<denomY (SetoidPartialOrder.<WellDefined pOrder (transitive (symmetric *Associative) (transitive (*WellDefined reflexive *Commutative) *Associative)) (transitive (symmetric *Associative) (transitive (*WellDefined reflexive *Commutative) (transitive *Associative (transitive (*WellDefined y=z reflexive) (transitive (symmetric *Associative) *Commutative))))) (ringCanMultiplyByPositive order 0<denomZ x<y))
where
open Ring R
open Equivalence (Setoid.eq S)
fieldOfFractionsOrderWellDefinedRight {S = S} {_*_ = _*_} {R} {_<_ = _<_} {pOrder = pOrder} {tOrder} I order {numX ,, (denomX , denomX!=0)} {numY ,, (denomY , denomY!=0)} {numZ ,, (denomZ , denomZ!=0)} x<y y=z | inl (inl 0<denomX) | inl (inl 0<denomZ) | inl (inr denomY<0) = ringCanCancelNegative order denomY<0 (SetoidPartialOrder.wellDefined pOrder (transitive (symmetric *Associative) (transitive (*WellDefined reflexive *Commutative) (transitive *Associative (transitive (*WellDefined y=z reflexive) (transitive (symmetric *Associative) *Commutative))))) (transitive (symmetric *Associative) (transitive (*WellDefined reflexive *Commutative) *Associative)) (ringCanMultiplyByPositive order 0<denomZ x<y))
fieldOfFractionsOrderWellDefinedRight {S = S} {_*_ = _*_} {R} {_<_ = _<_} {pOrder = pOrder} {tOrder} I order {numX ,, (denomX , denomX!=0)} {numY ,, (denomY , denomY!=0)} {numZ ,, (denomZ , denomZ!=0)} x<y y=z | inl (inl 0<denomX) | inl (inl 0<denomZ) | inl (inr denomY<0) = ringCanCancelNegative order denomY<0 (SetoidPartialOrder.<WellDefined pOrder (transitive (symmetric *Associative) (transitive (*WellDefined reflexive *Commutative) (transitive *Associative (transitive (*WellDefined y=z reflexive) (transitive (symmetric *Associative) *Commutative))))) (transitive (symmetric *Associative) (transitive (*WellDefined reflexive *Commutative) *Associative)) (ringCanMultiplyByPositive order 0<denomZ x<y))
where
open Ring R
open Equivalence (Setoid.eq S)
fieldOfFractionsOrderWellDefinedRight {S = S} {_*_ = _*_} {R} {pOrder = pOrder} {tOrder} I order {numX ,, (denomX , denomX!=0)} {numY ,, (denomY , denomY!=0)} {numZ ,, (denomZ , denomZ!=0)} x<y y=z | inl (inl 0<denomX) | inl (inl 0<denomZ) | inr x = exFalso (denomY!=0 (Equivalence.symmetric (Setoid.eq S) x))
fieldOfFractionsOrderWellDefinedRight {S = S} {_*_ = _*_} {R} {pOrder = pOrder} {tOrder} I order {numX ,, (denomX , denomX!=0)} {numY ,, (denomY , denomY!=0)} {numZ ,, (denomZ , denomZ!=0)} x<y y=z | inl (inl 0<denomX) | inl (inr denomZ<0) with SetoidTotalOrder.totality tOrder (Ring.0R R) denomY
fieldOfFractionsOrderWellDefinedRight {S = S} {_*_ = _*_} {R} {_<_ = _<_} {pOrder = pOrder} {tOrder} I order {numX ,, (denomX , denomX!=0)} {numY ,, (denomY , denomY!=0)} {numZ ,, (denomZ , denomZ!=0)} x<y y=z | inl (inl 0<denomX) | inl (inr denomZ<0) | inl (inl 0<denomY) = ringCanCancelPositive order 0<denomY (SetoidPartialOrder.wellDefined pOrder (transitive (symmetric *Associative) (transitive (*WellDefined reflexive *Commutative) (transitive *Associative (transitive (*WellDefined y=z reflexive) (transitive (symmetric *Associative) *Commutative))))) (transitive (symmetric *Associative) (transitive (*WellDefined reflexive *Commutative) *Associative)) (ringCanMultiplyByNegative order denomZ<0 x<y))
fieldOfFractionsOrderWellDefinedRight {S = S} {_*_ = _*_} {R} {_<_ = _<_} {pOrder = pOrder} {tOrder} I order {numX ,, (denomX , denomX!=0)} {numY ,, (denomY , denomY!=0)} {numZ ,, (denomZ , denomZ!=0)} x<y y=z | inl (inl 0<denomX) | inl (inr denomZ<0) | inl (inl 0<denomY) = ringCanCancelPositive order 0<denomY (SetoidPartialOrder.<WellDefined pOrder (transitive (symmetric *Associative) (transitive (*WellDefined reflexive *Commutative) (transitive *Associative (transitive (*WellDefined y=z reflexive) (transitive (symmetric *Associative) *Commutative))))) (transitive (symmetric *Associative) (transitive (*WellDefined reflexive *Commutative) *Associative)) (ringCanMultiplyByNegative order denomZ<0 x<y))
where
open Ring R
open Equivalence (Setoid.eq S)
fieldOfFractionsOrderWellDefinedRight {S = S} {_*_ = _*_} {R} {_<_ = _<_} {pOrder = pOrder} {tOrder} I order {numX ,, (denomX , denomX!=0)} {numY ,, (denomY , denomY!=0)} {numZ ,, (denomZ , denomZ!=0)} x<y y=z | inl (inl 0<denomX) | inl (inr denomZ<0) | inl (inr denomY<0) = ringCanCancelNegative order denomY<0 (SetoidPartialOrder.wellDefined pOrder (transitive (symmetric *Associative) (transitive (*WellDefined reflexive *Commutative) *Associative)) (transitive (symmetric *Associative) (transitive (*WellDefined reflexive *Commutative) (transitive (*Associative) (transitive (*WellDefined y=z reflexive) (transitive (symmetric *Associative) *Commutative))))) (ringCanMultiplyByNegative order denomZ<0 x<y))
fieldOfFractionsOrderWellDefinedRight {S = S} {_*_ = _*_} {R} {_<_ = _<_} {pOrder = pOrder} {tOrder} I order {numX ,, (denomX , denomX!=0)} {numY ,, (denomY , denomY!=0)} {numZ ,, (denomZ , denomZ!=0)} x<y y=z | inl (inl 0<denomX) | inl (inr denomZ<0) | inl (inr denomY<0) = ringCanCancelNegative order denomY<0 (SetoidPartialOrder.<WellDefined pOrder (transitive (symmetric *Associative) (transitive (*WellDefined reflexive *Commutative) *Associative)) (transitive (symmetric *Associative) (transitive (*WellDefined reflexive *Commutative) (transitive (*Associative) (transitive (*WellDefined y=z reflexive) (transitive (symmetric *Associative) *Commutative))))) (ringCanMultiplyByNegative order denomZ<0 x<y))
where
open Ring R
open Equivalence (Setoid.eq S)
@@ -172,21 +174,21 @@ module Fields.FieldOfFractionsOrder where
fieldOfFractionsOrderWellDefinedRight {S = S} {_*_ = _*_} {R} {pOrder = pOrder} {tOrder} I order {numX ,, (denomX , denomX!=0)} {numY ,, (denomY , denomY!=0)} {numZ ,, (denomZ , denomZ!=0)} x<y y=z | inl (inl 0<denomX) | inr x = exFalso (denomZ!=0 (Equivalence.symmetric (Setoid.eq S) x))
fieldOfFractionsOrderWellDefinedRight {S = S} {_*_ = _*_} {R} {pOrder = pOrder} {tOrder} I order {numX ,, (denomX , denomX!=0)} {numY ,, (denomY , denomY!=0)} {numZ ,, (denomZ , denomZ!=0)} x<y y=z | inl (inr denomX<0) with SetoidTotalOrder.totality tOrder (Ring.0R R) denomZ
fieldOfFractionsOrderWellDefinedRight {S = S} {_*_ = _*_} {R} {pOrder = pOrder} {tOrder} I order {numX ,, (denomX , denomX!=0)} {numY ,, (denomY , denomY!=0)} {numZ ,, (denomZ , denomZ!=0)} x<y y=z | inl (inr denomX<0) | inl (inl 0<denomZ) with SetoidTotalOrder.totality tOrder (Ring.0R R) denomY
fieldOfFractionsOrderWellDefinedRight {S = S} {_*_ = _*_} {R} {_<_ = _<_} {pOrder = pOrder} {tOrder} I order {numX ,, (denomX , denomX!=0)} {numY ,, (denomY , denomY!=0)} {numZ ,, (denomZ , denomZ!=0)} x<y y=z | inl (inr denomX<0) | inl (inl 0<denomZ) | inl (inl 0<denomY) = ringCanCancelPositive order 0<denomY (SetoidPartialOrder.wellDefined pOrder (transitive (symmetric *Associative) (transitive (*WellDefined reflexive *Commutative) (transitive *Associative (transitive (*WellDefined y=z reflexive) (transitive (symmetric *Associative) *Commutative))))) (transitive (symmetric *Associative) (transitive (*WellDefined reflexive *Commutative) *Associative)) (ringCanMultiplyByPositive order 0<denomZ x<y))
fieldOfFractionsOrderWellDefinedRight {S = S} {_*_ = _*_} {R} {_<_ = _<_} {pOrder = pOrder} {tOrder} I order {numX ,, (denomX , denomX!=0)} {numY ,, (denomY , denomY!=0)} {numZ ,, (denomZ , denomZ!=0)} x<y y=z | inl (inr denomX<0) | inl (inl 0<denomZ) | inl (inl 0<denomY) = ringCanCancelPositive order 0<denomY (SetoidPartialOrder.<WellDefined pOrder (transitive (symmetric *Associative) (transitive (*WellDefined reflexive *Commutative) (transitive *Associative (transitive (*WellDefined y=z reflexive) (transitive (symmetric *Associative) *Commutative))))) (transitive (symmetric *Associative) (transitive (*WellDefined reflexive *Commutative) *Associative)) (ringCanMultiplyByPositive order 0<denomZ x<y))
where
open Ring R
open Equivalence (Setoid.eq S)
fieldOfFractionsOrderWellDefinedRight {S = S} {_*_ = _*_} {R} {_<_ = _<_} {pOrder = pOrder} {tOrder} I order {numX ,, (denomX , denomX!=0)} {numY ,, (denomY , denomY!=0)} {numZ ,, (denomZ , denomZ!=0)} x<y y=z | inl (inr denomX<0) | inl (inl 0<denomZ) | inl (inr denomY<0) = ringCanCancelNegative order denomY<0 (SetoidPartialOrder.wellDefined pOrder (transitive (symmetric *Associative) (transitive (*WellDefined reflexive *Commutative) *Associative)) (transitive (symmetric *Associative) (transitive (*WellDefined reflexive *Commutative) (transitive *Associative (transitive (*WellDefined y=z reflexive) (transitive (symmetric *Associative) *Commutative))))) (ringCanMultiplyByPositive order 0<denomZ x<y))
fieldOfFractionsOrderWellDefinedRight {S = S} {_*_ = _*_} {R} {_<_ = _<_} {pOrder = pOrder} {tOrder} I order {numX ,, (denomX , denomX!=0)} {numY ,, (denomY , denomY!=0)} {numZ ,, (denomZ , denomZ!=0)} x<y y=z | inl (inr denomX<0) | inl (inl 0<denomZ) | inl (inr denomY<0) = ringCanCancelNegative order denomY<0 (SetoidPartialOrder.<WellDefined pOrder (transitive (symmetric *Associative) (transitive (*WellDefined reflexive *Commutative) *Associative)) (transitive (symmetric *Associative) (transitive (*WellDefined reflexive *Commutative) (transitive *Associative (transitive (*WellDefined y=z reflexive) (transitive (symmetric *Associative) *Commutative))))) (ringCanMultiplyByPositive order 0<denomZ x<y))
where
open Ring R
open Equivalence (Setoid.eq S)
fieldOfFractionsOrderWellDefinedRight {S = S} {_*_ = _*_} {R} {pOrder = pOrder} {tOrder} I order {numX ,, (denomX , denomX!=0)} {numY ,, (denomY , denomY!=0)} {numZ ,, (denomZ , denomZ!=0)} x<y y=z | inl (inr denomX<0) | inl (inl 0<denomZ) | inr x = exFalso (denomY!=0 (Equivalence.symmetric (Setoid.eq S) x))
fieldOfFractionsOrderWellDefinedRight {S = S} {_*_ = _*_} {R} {pOrder = pOrder} {tOrder} I order {numX ,, (denomX , denomX!=0)} {numY ,, (denomY , denomY!=0)} {numZ ,, (denomZ , denomZ!=0)} x<y y=z | inl (inr denomX<0) | inl (inr denomZ<0) with SetoidTotalOrder.totality tOrder (Ring.0R R) denomY
fieldOfFractionsOrderWellDefinedRight {S = S} {_*_ = _*_} {R} {_<_ = _<_} {pOrder = pOrder} {tOrder} I order {numX ,, (denomX , denomX!=0)} {numY ,, (denomY , denomY!=0)} {numZ ,, (denomZ , denomZ!=0)} x<y y=z | inl (inr denomX<0) | inl (inr denomZ<0) | inl (inl 0<denomY) = ringCanCancelPositive order 0<denomY (SetoidPartialOrder.wellDefined pOrder (transitive (symmetric *Associative) (transitive (*WellDefined reflexive *Commutative) *Associative)) (transitive (symmetric *Associative) (transitive (*WellDefined reflexive *Commutative) (transitive *Associative (transitive (*WellDefined y=z reflexive) (transitive (symmetric *Associative) *Commutative))))) (ringCanMultiplyByNegative order denomZ<0 x<y))
fieldOfFractionsOrderWellDefinedRight {S = S} {_*_ = _*_} {R} {_<_ = _<_} {pOrder = pOrder} {tOrder} I order {numX ,, (denomX , denomX!=0)} {numY ,, (denomY , denomY!=0)} {numZ ,, (denomZ , denomZ!=0)} x<y y=z | inl (inr denomX<0) | inl (inr denomZ<0) | inl (inl 0<denomY) = ringCanCancelPositive order 0<denomY (SetoidPartialOrder.<WellDefined pOrder (transitive (symmetric *Associative) (transitive (*WellDefined reflexive *Commutative) *Associative)) (transitive (symmetric *Associative) (transitive (*WellDefined reflexive *Commutative) (transitive *Associative (transitive (*WellDefined y=z reflexive) (transitive (symmetric *Associative) *Commutative))))) (ringCanMultiplyByNegative order denomZ<0 x<y))
where
open Ring R
open Equivalence (Setoid.eq S)
fieldOfFractionsOrderWellDefinedRight {S = S} {_*_ = _*_} {R} {_<_ = _<_} {pOrder = pOrder} {tOrder} I order {numX ,, (denomX , denomX!=0)} {numY ,, (denomY , denomY!=0)} {numZ ,, (denomZ , denomZ!=0)} x<y y=z | inl (inr denomX<0) | inl (inr denomZ<0) | inl (inr denomY<0) = ringCanCancelNegative order denomY<0 (SetoidPartialOrder.wellDefined pOrder (transitive (symmetric *Associative) (transitive (*WellDefined reflexive *Commutative) (transitive *Associative (transitive (*WellDefined y=z reflexive) (transitive (symmetric *Associative) *Commutative))))) (transitive (symmetric *Associative) (transitive (*WellDefined reflexive *Commutative) *Associative)) (ringCanMultiplyByNegative order denomZ<0 x<y))
fieldOfFractionsOrderWellDefinedRight {S = S} {_*_ = _*_} {R} {_<_ = _<_} {pOrder = pOrder} {tOrder} I order {numX ,, (denomX , denomX!=0)} {numY ,, (denomY , denomY!=0)} {numZ ,, (denomZ , denomZ!=0)} x<y y=z | inl (inr denomX<0) | inl (inr denomZ<0) | inl (inr denomY<0) = ringCanCancelNegative order denomY<0 (SetoidPartialOrder.<WellDefined pOrder (transitive (symmetric *Associative) (transitive (*WellDefined reflexive *Commutative) (transitive *Associative (transitive (*WellDefined y=z reflexive) (transitive (symmetric *Associative) *Commutative))))) (transitive (symmetric *Associative) (transitive (*WellDefined reflexive *Commutative) *Associative)) (ringCanMultiplyByNegative order denomZ<0 x<y))
where
open Ring R
open Equivalence (Setoid.eq S)
@@ -202,7 +204,7 @@ module Fields.FieldOfFractionsOrder where
open Equivalence (Setoid.eq S)
fieldOfFractionsOrder : {a b c : _} {A : Set a} {S : Setoid {a} {b} A} {_+_ : A → A → A} {_*_ : A → A → A} {R : Ring S _+_ _*_} {_<_ : Rel {_} {c} A} {pOrder : SetoidPartialOrder S _<_} {tOrder : SetoidTotalOrder pOrder} (I : IntegralDomain R) → (order : OrderedRing R tOrder) → SetoidPartialOrder (fieldOfFractionsSetoid I) (fieldOfFractionsComparison I order)
SetoidPartialOrder.wellDefined (fieldOfFractionsOrder I oRing) {a} {b} {c} {d} a=b c=d a<c = fieldOfFractionsOrderWellDefinedRight I oRing {b} {c} {d} (fieldOfFractionsOrderWellDefinedLeft I oRing {a} {c} {b} a<c a=b) c=d
SetoidPartialOrder.<WellDefined (fieldOfFractionsOrder I oRing) {a} {b} {c} {d} a=b c=d a<c = fieldOfFractionsOrderWellDefinedRight I oRing {b} {c} {d} (fieldOfFractionsOrderWellDefinedLeft I oRing {a} {c} {b} a<c a=b) c=d
SetoidPartialOrder.irreflexive (fieldOfFractionsOrder {S = S} {R = R} {pOrder = pOrder} {tOrder = tOrder} I oRing) {aNum ,, (aDenom , aDenom!=0)} pr with SetoidTotalOrder.totality tOrder (Ring.0R R) aDenom
SetoidPartialOrder.irreflexive (fieldOfFractionsOrder {S = S} {R = R} {pOrder = pOrder} {tOrder} I oRing) {aNum ,, (aDenom , aDenom!=0)} pr | inl (inl 0<aDenom) with SetoidTotalOrder.totality tOrder (Ring.0R R) aDenom
SetoidPartialOrder.irreflexive (fieldOfFractionsOrder {S = S} {R = R} {pOrder = pOrder} {tOrder} I oRing) {aNum ,, (aDenom , aDenom!=0)} pr | inl (inl 0<aDenom) | inl (inl _) = SetoidPartialOrder.irreflexive pOrder pr
@@ -225,11 +227,11 @@ module Fields.FieldOfFractionsOrder where
inter : ((numA * denomB) * denomC) < ((numB * denomA) * denomC)
inter = ringCanMultiplyByPositive oRing 0<denomC a<b
p : ((numA * denomC) * denomB) < ((numC * denomA) * denomB)
p = SetoidPartialOrder.transitive pOrder (SetoidPartialOrder.wellDefined pOrder (transitive (symmetric *Associative) (transitive (*WellDefined reflexive *Commutative) *Associative)) reflexive inter) (SetoidPartialOrder.wellDefined pOrder (transitive (symmetric *Associative) (transitive (*WellDefined reflexive *Commutative) *Associative)) (transitive (symmetric *Associative) (transitive (*WellDefined reflexive *Commutative) *Associative)) (ringCanMultiplyByPositive oRing 0<denomA b<c))
p = SetoidPartialOrder.transitive pOrder (SetoidPartialOrder.<WellDefined pOrder (transitive (symmetric *Associative) (transitive (*WellDefined reflexive *Commutative) *Associative)) reflexive inter) (SetoidPartialOrder.<WellDefined pOrder (transitive (symmetric *Associative) (transitive (*WellDefined reflexive *Commutative) *Associative)) (transitive (symmetric *Associative) (transitive (*WellDefined reflexive *Commutative) *Associative)) (ringCanMultiplyByPositive oRing 0<denomA b<c))
SetoidPartialOrder.transitive (fieldOfFractionsOrder {S = S} {R = R} {pOrder = pOrder} {tOrder} I oRing) {numA ,, (denomA , denomA!=0)} {numB ,, (denomB , denomB!=0)} {numC ,, (denomC , denomC!=0)} a<b b<c | inl (inl 0<denomA) | inl (inl 0<denomC) | inl (inl 0<denomB) | inl (inr denomC<0) = exFalso (SetoidPartialOrder.irreflexive pOrder (SetoidPartialOrder.transitive pOrder 0<denomC denomC<0))
SetoidPartialOrder.transitive (fieldOfFractionsOrder {S = S} {R = R} {pOrder = pOrder} {tOrder} I oRing) {numA ,, (denomA , denomA!=0)} {numB ,, (denomB , denomB!=0)} {numC ,, (denomC , denomC!=0)} a<b b<c | inl (inl 0<denomA) | inl (inl 0<denomC) | inl (inl 0<denomB) | inr x = exFalso (denomC!=0 (Equivalence.symmetric (Setoid.eq S) x))
SetoidPartialOrder.transitive (fieldOfFractionsOrder {S = S} {R = R} {pOrder = pOrder} {tOrder} I oRing) {numA ,, (denomA , denomA!=0)} {numB ,, (denomB , denomB!=0)} {numC ,, (denomC , denomC!=0)} a<b b<c | inl (inl 0<denomA) | inl (inl 0<denomC) | inl (inr denomB<0) with SetoidTotalOrder.totality tOrder (Ring.0R R) denomC
SetoidPartialOrder.transitive (fieldOfFractionsOrder {S = S} {_*_ = _*_} {R = R} {_<_ = _<_} {pOrder = pOrder} {tOrder} I oRing) {numA ,, (denomA , denomA!=0)} {numB ,, (denomB , denomB!=0)} {numC ,, (denomC , denomC!=0)} a<b b<c | inl (inl 0<denomA) | inl (inl 0<denomC) | inl (inr denomB<0) | inl (inl _) = ringCanCancelNegative oRing denomB<0 (SetoidPartialOrder.transitive pOrder (SetoidPartialOrder.wellDefined pOrder (transitive (symmetric *Associative) (transitive (*WellDefined reflexive *Commutative) *Associative)) reflexive (ringCanMultiplyByPositive oRing 0<denomA b<c)) (SetoidPartialOrder.wellDefined pOrder (transitive (symmetric *Associative) (transitive (*WellDefined reflexive *Commutative) *Associative)) (transitive (symmetric *Associative) (transitive (*WellDefined reflexive *Commutative) *Associative)) (ringCanMultiplyByPositive oRing 0<denomC a<b)))
SetoidPartialOrder.transitive (fieldOfFractionsOrder {S = S} {_*_ = _*_} {R = R} {_<_ = _<_} {pOrder = pOrder} {tOrder} I oRing) {numA ,, (denomA , denomA!=0)} {numB ,, (denomB , denomB!=0)} {numC ,, (denomC , denomC!=0)} a<b b<c | inl (inl 0<denomA) | inl (inl 0<denomC) | inl (inr denomB<0) | inl (inl _) = ringCanCancelNegative oRing denomB<0 (SetoidPartialOrder.transitive pOrder (SetoidPartialOrder.<WellDefined pOrder (transitive (symmetric *Associative) (transitive (*WellDefined reflexive *Commutative) *Associative)) reflexive (ringCanMultiplyByPositive oRing 0<denomA b<c)) (SetoidPartialOrder.<WellDefined pOrder (transitive (symmetric *Associative) (transitive (*WellDefined reflexive *Commutative) *Associative)) (transitive (symmetric *Associative) (transitive (*WellDefined reflexive *Commutative) *Associative)) (ringCanMultiplyByPositive oRing 0<denomC a<b)))
where
open Setoid S
open Ring R
@@ -240,69 +242,69 @@ module Fields.FieldOfFractionsOrder where
SetoidPartialOrder.transitive (fieldOfFractionsOrder {S = S} {R = R} {pOrder = pOrder} {tOrder} I oRing) {numA ,, (denomA , denomA!=0)} {numB ,, (denomB , denomB!=0)} {numC ,, (denomC , denomC!=0)} a<b b<c | inl (inl 0<denomA) | inl (inr denomC<0) with SetoidTotalOrder.totality tOrder (Ring.0R R) denomB
SetoidPartialOrder.transitive (fieldOfFractionsOrder {S = S} {_*_ = _*_} {R = R} {_<_ = _<_} {pOrder = pOrder} {tOrder} I oRing) {numA ,, (denomA , denomA!=0)} {numB ,, (denomB , denomB!=0)} {numC ,, (denomC , denomC!=0)} a<b b<c | inl (inl 0<denomA) | inl (inr denomC<0) | inl (inl 0<denomB) with SetoidTotalOrder.totality tOrder (Ring.0R R) denomC
SetoidPartialOrder.transitive (fieldOfFractionsOrder {S = S} {_*_ = _*_} {R} {_<_} {pOrder} {tOrder} I oRing) {numA ,, (denomA , denomA!=0)} {numB ,, (denomB , denomB!=0)} {numC ,, (denomC , denomC!=0)} a<b b<c | inl (inl 0<denomA) | inl (inr denomC<0) | inl (inl 0<denomB) | inl (inl 0<denomC) = exFalso (SetoidPartialOrder.irreflexive pOrder (SetoidPartialOrder.transitive pOrder 0<denomC denomC<0))
SetoidPartialOrder.transitive (fieldOfFractionsOrder {S = S} {_*_ = _*_} {R} {_<_} {pOrder} {tOrder} I oRing) {numA ,, (denomA , denomA!=0)} {numB ,, (denomB , denomB!=0)} {numC ,, (denomC , denomC!=0)} a<b b<c | inl (inl 0<denomA) | inl (inr denomC<0) | inl (inl 0<denomB) | inl (inr _) = ringCanCancelPositive oRing 0<denomB (SetoidPartialOrder.transitive pOrder have (SetoidPartialOrder.wellDefined pOrder (swapLemma R) (swapLemma R) (ringCanMultiplyByNegative oRing denomC<0 a<b)))
SetoidPartialOrder.transitive (fieldOfFractionsOrder {S = S} {_*_ = _*_} {R} {_<_} {pOrder} {tOrder} I oRing) {numA ,, (denomA , denomA!=0)} {numB ,, (denomB , denomB!=0)} {numC ,, (denomC , denomC!=0)} a<b b<c | inl (inl 0<denomA) | inl (inr denomC<0) | inl (inl 0<denomB) | inl (inr _) = ringCanCancelPositive oRing 0<denomB (SetoidPartialOrder.transitive pOrder have (SetoidPartialOrder.<WellDefined pOrder (swapLemma R) (swapLemma R) (ringCanMultiplyByNegative oRing denomC<0 a<b)))
where
open Setoid S
open Ring R
open Equivalence (Setoid.eq S)
have : ((numC * denomA) * denomB) < ((numB * denomC) * denomA)
have = SetoidPartialOrder.wellDefined pOrder (swapLemma R) reflexive (ringCanMultiplyByPositive oRing 0<denomA b<c)
have = SetoidPartialOrder.<WellDefined pOrder (swapLemma R) reflexive (ringCanMultiplyByPositive oRing 0<denomA b<c)
SetoidPartialOrder.transitive (fieldOfFractionsOrder {S = S} {_*_ = _*_} {R} {_<_} {pOrder} {tOrder} I oRing) {numA ,, (denomA , denomA!=0)} {numB ,, (denomB , denomB!=0)} {numC ,, (denomC , denomC!=0)} a<b b<c | inl (inl 0<denomA) | inl (inr denomC<0) | inl (inl 0<denomB) | inr x = exFalso (denomC!=0 (Equivalence.symmetric (Setoid.eq S) x))
SetoidPartialOrder.transitive (fieldOfFractionsOrder {S = S} {_*_ = _*_} {R = R} {_<_ = _<_} {pOrder = pOrder} {tOrder} I oRing) {numA ,, (denomA , denomA!=0)} {numB ,, (denomB , denomB!=0)} {numC ,, (denomC , denomC!=0)} a<b b<c | inl (inl 0<denomA) | inl (inr denomC<0) | inl (inr denomB<0) with SetoidTotalOrder.totality tOrder (Ring.0R R) denomC
... | (inl (inl 0<denomC)) = exFalso (SetoidPartialOrder.irreflexive pOrder (SetoidPartialOrder.transitive pOrder 0<denomC denomC<0))
... | (inl (inr _)) = ringCanCancelNegative oRing denomB<0 (SetoidPartialOrder.transitive pOrder have (SetoidPartialOrder.wellDefined pOrder (swapLemma R) (swapLemma R) (ringCanMultiplyByPositive oRing 0<denomA b<c)))
... | (inl (inr _)) = ringCanCancelNegative oRing denomB<0 (SetoidPartialOrder.transitive pOrder have (SetoidPartialOrder.<WellDefined pOrder (swapLemma R) (swapLemma R) (ringCanMultiplyByPositive oRing 0<denomA b<c)))
where
open Setoid S
open Ring R
open Equivalence (Setoid.eq S)
have : ((numA * denomC) * denomB) < ((numB * denomA) * denomC)
have = SetoidPartialOrder.wellDefined pOrder (swapLemma R) reflexive (ringCanMultiplyByNegative oRing denomC<0 a<b)
have = SetoidPartialOrder.<WellDefined pOrder (swapLemma R) reflexive (ringCanMultiplyByNegative oRing denomC<0 a<b)
... | inr x = exFalso (denomC!=0 (Equivalence.symmetric (Setoid.eq S) x))
SetoidPartialOrder.transitive (fieldOfFractionsOrder {S = S} {R = R} {pOrder = pOrder} {tOrder} I oRing) {numA ,, (denomA , denomA!=0)} {numB ,, (denomB , denomB!=0)} {numC ,, (denomC , denomC!=0)} a<b b<c | inl (inl 0<denomA) | inl (inr denomC<0) | inr x = exFalso (denomB!=0 (Equivalence.symmetric (Setoid.eq S) x))
SetoidPartialOrder.transitive (fieldOfFractionsOrder {S = S} {R = R} {pOrder = pOrder} {tOrder} I oRing) {numA ,, (denomA , denomA!=0)} {numB ,, (denomB , denomB!=0)} {numC ,, (denomC , denomC!=0)} a<b b<c | inl (inl 0<denomA) | inr x = exFalso (denomC!=0 (Equivalence.symmetric (Setoid.eq S) x))
SetoidPartialOrder.transitive (fieldOfFractionsOrder {S = S} {R = R} {_<_ = _<_} {pOrder = pOrder} {tOrder} I oRing) {numA ,, (denomA , denomA!=0)} {numB ,, (denomB , denomB!=0)} {numC ,, (denomC , denomC!=0)} a<b b<c | inl (inr denomA<0) with SetoidTotalOrder.totality tOrder (Ring.0R R) denomC
SetoidPartialOrder.transitive (fieldOfFractionsOrder {S = S} {R = R} {_<_} {pOrder} {tOrder} I oRing) {numA ,, (denomA , denomA!=0)} {numB ,, (denomB , denomB!=0)} {numC ,, (denomC , denomC!=0)} a<b b<c | inl (inr denomA<0) | inl (inl 0<denomC) with SetoidTotalOrder.totality tOrder (Ring.0R R) denomB
SetoidPartialOrder.transitive (fieldOfFractionsOrder {a} {b} {c} {A} {S} {_+_} {_*_} {R} {_<_} {pOrder} {tOrder} I oRing) {numA ,, (denomA , denomA!=0)} {numB ,, (denomB , denomB!=0)} {numC ,, (denomC , denomC!=0)} a<b b<c | inl (inr denomA<0) | inl (inl 0<denomC) | inl (inl 0<denomB) with SetoidTotalOrder.totality tOrder (Ring.0R R) denomC
SetoidPartialOrder.transitive (fieldOfFractionsOrder {a} {b} {c} {A} {S} {_+_} {_*_} {R} {_<_} {pOrder} {tOrder} I oRing) {numA ,, (denomA , denomA!=0)} {numB ,, (denomB , denomB!=0)} {numC ,, (denomC , denomC!=0)} a<b b<c | inl (inr denomA<0) | inl (inl 0<denomC) | inl (inl 0<denomB) | inl (inl _) = ringCanCancelPositive oRing 0<denomB (SetoidPartialOrder.transitive pOrder (SetoidPartialOrder.wellDefined pOrder (swapLemma R) (swapLemma R) (ringCanMultiplyByNegative oRing denomA<0 b<c)) have)
SetoidPartialOrder.transitive (fieldOfFractionsOrder {a} {b} {c} {A} {S} {_+_} {_*_} {R} {_<_} {pOrder} {tOrder} I oRing) {numA ,, (denomA , denomA!=0)} {numB ,, (denomB , denomB!=0)} {numC ,, (denomC , denomC!=0)} a<b b<c | inl (inr denomA<0) | inl (inl 0<denomC) | inl (inl 0<denomB) | inl (inl _) = ringCanCancelPositive oRing 0<denomB (SetoidPartialOrder.transitive pOrder (SetoidPartialOrder.<WellDefined pOrder (swapLemma R) (swapLemma R) (ringCanMultiplyByNegative oRing denomA<0 b<c)) have)
where
open Setoid S
open Ring R
open Equivalence (Setoid.eq S)
have : ((numB * denomA) * denomC) < ((numA * denomC) * denomB)
have = SetoidPartialOrder.wellDefined pOrder reflexive (swapLemma R) (ringCanMultiplyByPositive oRing 0<denomC a<b)
have = SetoidPartialOrder.<WellDefined pOrder reflexive (swapLemma R) (ringCanMultiplyByPositive oRing 0<denomC a<b)
SetoidPartialOrder.transitive (fieldOfFractionsOrder {a} {b} {c} {A} {S} {_+_} {_*_} {R} {_<_} {pOrder} {tOrder} I oRing) {numA ,, (denomA , denomA!=0)} {numB ,, (denomB , denomB!=0)} {numC ,, (denomC , denomC!=0)} a<b b<c | inl (inr denomA<0) | inl (inl 0<denomC) | inl (inl 0<denomB) | inl (inr denomC<0) = exFalso (SetoidPartialOrder.irreflexive pOrder (SetoidPartialOrder.transitive pOrder 0<denomC denomC<0))
SetoidPartialOrder.transitive (fieldOfFractionsOrder {a} {b} {c} {A} {S} {_+_} {_*_} {R} {_<_} {pOrder} {tOrder} I oRing) {numA ,, (denomA , denomA!=0)} {numB ,, (denomB , denomB!=0)} {numC ,, (denomC , denomC!=0)} a<b b<c | inl (inr denomA<0) | inl (inl 0<denomC) | inl (inl 0<denomB) | inr x = exFalso (denomC!=0 (Equivalence.symmetric (Setoid.eq S) x))
SetoidPartialOrder.transitive (fieldOfFractionsOrder {a} {b} {c} {A} {S} {_+_} {_*_} {R} {_<_} {pOrder} {tOrder} I oRing) {numA ,, (denomA , denomA!=0)} {numB ,, (denomB , denomB!=0)} {numC ,, (denomC , denomC!=0)} a<b b<c | inl (inr denomA<0) | inl (inl 0<denomC) | inl (inr denomB<0) with SetoidTotalOrder.totality tOrder (Ring.0R R) denomC
SetoidPartialOrder.transitive (fieldOfFractionsOrder {a} {b} {c} {A} {S} {_+_} {_*_} {R} {_<_} {pOrder} {tOrder} I oRing) {numA ,, (denomA , denomA!=0)} {numB ,, (denomB , denomB!=0)} {numC ,, (denomC , denomC!=0)} a<b b<c | inl (inr denomA<0) | inl (inl 0<denomC) | inl (inr denomB<0) | inl (inl _) = ringCanCancelNegative oRing denomB<0 (SetoidPartialOrder.transitive pOrder have (SetoidPartialOrder.wellDefined pOrder (swapLemma R) (swapLemma R) (ringCanMultiplyByNegative oRing denomA<0 b<c)))
SetoidPartialOrder.transitive (fieldOfFractionsOrder {a} {b} {c} {A} {S} {_+_} {_*_} {R} {_<_} {pOrder} {tOrder} I oRing) {numA ,, (denomA , denomA!=0)} {numB ,, (denomB , denomB!=0)} {numC ,, (denomC , denomC!=0)} a<b b<c | inl (inr denomA<0) | inl (inl 0<denomC) | inl (inr denomB<0) | inl (inl _) = ringCanCancelNegative oRing denomB<0 (SetoidPartialOrder.transitive pOrder have (SetoidPartialOrder.<WellDefined pOrder (swapLemma R) (swapLemma R) (ringCanMultiplyByNegative oRing denomA<0 b<c)))
where
open Setoid S
open Ring R
open Equivalence (Setoid.eq S)
have : ((numA * denomC) * denomB) < ((numB * denomA) * denomC)
have = SetoidPartialOrder.wellDefined pOrder (swapLemma R) reflexive (ringCanMultiplyByPositive oRing 0<denomC a<b)
have = SetoidPartialOrder.<WellDefined pOrder (swapLemma R) reflexive (ringCanMultiplyByPositive oRing 0<denomC a<b)
SetoidPartialOrder.transitive (fieldOfFractionsOrder {a} {b} {c} {A} {S} {_+_} {_*_} {R} {_<_} {pOrder} {tOrder} I oRing) {numA ,, (denomA , denomA!=0)} {numB ,, (denomB , denomB!=0)} {numC ,, (denomC , denomC!=0)} a<b b<c | inl (inr denomA<0) | inl (inl 0<denomC) | inl (inr denomB<0) | inl (inr denomC<0) = exFalso (SetoidPartialOrder.irreflexive pOrder (SetoidPartialOrder.transitive pOrder 0<denomC denomC<0))
SetoidPartialOrder.transitive (fieldOfFractionsOrder {a} {b} {c} {A} {S} {_+_} {_*_} {R} {_<_} {pOrder} {tOrder} I oRing) {numA ,, (denomA , denomA!=0)} {numB ,, (denomB , denomB!=0)} {numC ,, (denomC , denomC!=0)} a<b b<c | inl (inr denomA<0) | inl (inl 0<denomC) | inl (inr denomB<0) | inr x = exFalso (denomC!=0 (Equivalence.symmetric (Setoid.eq S) x))
SetoidPartialOrder.transitive (fieldOfFractionsOrder {a} {b} {c} {A} {S} {_+_} {_*_} {R} {_<_} {pOrder} {tOrder} I oRing) {numA ,, (denomA , denomA!=0)} {numB ,, (denomB , denomB!=0)} {numC ,, (denomC , denomC!=0)} a<b b<c | inl (inr denomA<0) | inl (inl 0<denomC) | inr x = exFalso (denomB!=0 (Equivalence.symmetric (Setoid.eq S) x))
SetoidPartialOrder.transitive (fieldOfFractionsOrder {S = S} {R = R} {_<_} {pOrder} {tOrder} I oRing) {numA ,, (denomA , denomA!=0)} {numB ,, (denomB , denomB!=0)} {numC ,, (denomC , denomC!=0)} a<b b<c | inl (inr denomA<0) | inl (inr denomC<0) with SetoidTotalOrder.totality tOrder (Ring.0R R) denomB
SetoidPartialOrder.transitive (fieldOfFractionsOrder {S = S} {R = R} {_<_} {pOrder} {tOrder} I oRing) {numA ,, (denomA , denomA!=0)} {numB ,, (denomB , denomB!=0)} {numC ,, (denomC , denomC!=0)} a<b b<c | inl (inr denomA<0) | inl (inr denomC<0) | inl (inl 0<denomB) with SetoidTotalOrder.totality tOrder (Ring.0R R) denomC
SetoidPartialOrder.transitive (fieldOfFractionsOrder {S = S} {R = R} {_<_} {pOrder} {tOrder} I oRing) {numA ,, (denomA , denomA!=0)} {numB ,, (denomB , denomB!=0)} {numC ,, (denomC , denomC!=0)} a<b b<c | inl (inr denomA<0) | inl (inr denomC<0) | inl (inl 0<denomB) | inl (inl 0<denomC) = exFalso (SetoidPartialOrder.irreflexive pOrder (SetoidPartialOrder.transitive pOrder 0<denomC denomC<0))
SetoidPartialOrder.transitive (fieldOfFractionsOrder {S = S} {_*_ = _*_} {R = R} {_<_} {pOrder} {tOrder} I oRing) {numA ,, (denomA , denomA!=0)} {numB ,, (denomB , denomB!=0)} {numC ,, (denomC , denomC!=0)} a<b b<c | inl (inr denomA<0) | inl (inr denomC<0) | inl (inl 0<denomB) | inl (inr _) = ringCanCancelPositive oRing 0<denomB (SetoidPartialOrder.transitive pOrder have (SetoidPartialOrder.wellDefined pOrder (swapLemma R) (swapLemma R) (ringCanMultiplyByNegative oRing denomA<0 b<c)))
SetoidPartialOrder.transitive (fieldOfFractionsOrder {S = S} {_*_ = _*_} {R = R} {_<_} {pOrder} {tOrder} I oRing) {numA ,, (denomA , denomA!=0)} {numB ,, (denomB , denomB!=0)} {numC ,, (denomC , denomC!=0)} a<b b<c | inl (inr denomA<0) | inl (inr denomC<0) | inl (inl 0<denomB) | inl (inr _) = ringCanCancelPositive oRing 0<denomB (SetoidPartialOrder.transitive pOrder have (SetoidPartialOrder.<WellDefined pOrder (swapLemma R) (swapLemma R) (ringCanMultiplyByNegative oRing denomA<0 b<c)))
where
open Setoid S
open Ring R
open Equivalence (Setoid.eq S)
have : ((numA * denomC) * denomB) < ((numB * denomA) * denomC)
have = SetoidPartialOrder.wellDefined pOrder (swapLemma R) reflexive (ringCanMultiplyByNegative oRing denomC<0 a<b)
have = SetoidPartialOrder.<WellDefined pOrder (swapLemma R) reflexive (ringCanMultiplyByNegative oRing denomC<0 a<b)
SetoidPartialOrder.transitive (fieldOfFractionsOrder {S = S} {R = R} {_<_} {pOrder} {tOrder} I oRing) {numA ,, (denomA , denomA!=0)} {numB ,, (denomB , denomB!=0)} {numC ,, (denomC , denomC!=0)} a<b b<c | inl (inr denomA<0) | inl (inr denomC<0) | inl (inl 0<denomB) | inr x = exFalso (denomC!=0 (Equivalence.symmetric (Setoid.eq S) x))
SetoidPartialOrder.transitive (fieldOfFractionsOrder {S = S} {R = R} {_<_} {pOrder} {tOrder} I oRing) {numA ,, (denomA , denomA!=0)} {numB ,, (denomB , denomB!=0)} {numC ,, (denomC , denomC!=0)} a<b b<c | inl (inr denomA<0) | inl (inr denomC<0) | inl (inr denomB<0) with SetoidTotalOrder.totality tOrder (Ring.0R R) denomC
SetoidPartialOrder.transitive (fieldOfFractionsOrder {S = S} {R = R} {_<_} {pOrder} {tOrder} I oRing) {numA ,, (denomA , denomA!=0)} {numB ,, (denomB , denomB!=0)} {numC ,, (denomC , denomC!=0)} a<b b<c | inl (inr denomA<0) | inl (inr denomC<0) | inl (inr denomB<0) | inl (inl 0<denomC) = exFalso (SetoidPartialOrder.irreflexive pOrder (SetoidPartialOrder.transitive pOrder 0<denomC denomC<0))
SetoidPartialOrder.transitive (fieldOfFractionsOrder {S = S} {_*_ = _*_} {R = R} {_<_} {pOrder} {tOrder} I oRing) {numA ,, (denomA , denomA!=0)} {numB ,, (denomB , denomB!=0)} {numC ,, (denomC , denomC!=0)} a<b b<c | inl (inr denomA<0) | inl (inr denomC<0) | inl (inr denomB<0) | inl (inr _) = ringCanCancelNegative oRing denomB<0 (SetoidPartialOrder.transitive pOrder (SetoidPartialOrder.wellDefined pOrder (swapLemma R) (swapLemma R) (ringCanMultiplyByNegative oRing denomA<0 b<c)) have)
SetoidPartialOrder.transitive (fieldOfFractionsOrder {S = S} {_*_ = _*_} {R = R} {_<_} {pOrder} {tOrder} I oRing) {numA ,, (denomA , denomA!=0)} {numB ,, (denomB , denomB!=0)} {numC ,, (denomC , denomC!=0)} a<b b<c | inl (inr denomA<0) | inl (inr denomC<0) | inl (inr denomB<0) | inl (inr _) = ringCanCancelNegative oRing denomB<0 (SetoidPartialOrder.transitive pOrder (SetoidPartialOrder.<WellDefined pOrder (swapLemma R) (swapLemma R) (ringCanMultiplyByNegative oRing denomA<0 b<c)) have)
where
open Setoid S
open Ring R
open Equivalence (Setoid.eq S)
have : ((numB * denomA) * denomC) < ((numA * denomC) * denomB)
have = SetoidPartialOrder.wellDefined pOrder reflexive (swapLemma R) (ringCanMultiplyByNegative oRing denomC<0 a<b)
have = SetoidPartialOrder.<WellDefined pOrder reflexive (swapLemma R) (ringCanMultiplyByNegative oRing denomC<0 a<b)
SetoidPartialOrder.transitive (fieldOfFractionsOrder {S = S} {R = R} {_<_} {pOrder} {tOrder} I oRing) {numA ,, (denomA , denomA!=0)} {numB ,, (denomB , denomB!=0)} {numC ,, (denomC , denomC!=0)} a<b b<c | inl (inr denomA<0) | inl (inr denomC<0) | inl (inr denomB<0) | inr x = exFalso (denomC!=0 (Equivalence.symmetric (Setoid.eq S) x))
SetoidPartialOrder.transitive (fieldOfFractionsOrder {S = S} {R = R} {_<_} {pOrder} {tOrder} I oRing) {numA ,, (denomA , denomA!=0)} {numB ,, (denomB , denomB!=0)} {numC ,, (denomC , denomC!=0)} a<b b<c | inl (inr denomA<0) | inl (inr denomC<0) | inr x = exFalso (denomB!=0 (Equivalence.symmetric (Setoid.eq S) x))
SetoidPartialOrder.transitive (fieldOfFractionsOrder {S = S} {R = R} {_<_} {pOrder} {tOrder} I oRing) {numA ,, (denomA , denomA!=0)} {numB ,, (denomB , denomB!=0)} {numC ,, (denomC , denomC!=0)} a<b b<c | inl (inr denomA<0) | inr x = exFalso (denomC!=0 (Equivalence.symmetric (Setoid.eq S) x))
@@ -347,12 +349,12 @@ module Fields.FieldOfFractionsOrder where
ineqLemma : {a b c : _} {A : Set a} {S : Setoid {a} {b} A} {_+_ : A → A → A} {_*_ : A → A → A} {R : Ring S _+_ _*_} {_<_ : Rel {_} {c} A} {pOrder : SetoidPartialOrder S _<_} {tOrder : SetoidTotalOrder pOrder} (I : IntegralDomain R) → (order : OrderedRing R tOrder) → {x y : A} → (Ring.0R R) < (x * y) → (Ring.0R R) < x → (Ring.0R R) < y
ineqLemma {S = S} {R = R} {_<_ = _<_} {tOrder = tOrder} I order {x} {y} 0<xy 0<x with SetoidTotalOrder.totality tOrder (Ring.0R R) y
ineqLemma {S = S} {R = R} {_<_} {tOrder = tOrder} I order {x} {y} 0<xy 0<x | inl (inl 0<y) = 0<y
ineqLemma {S = S} {R = R} {_<_} {pOrder = pOrder} {tOrder = tOrder} I order {x} {y} 0<xy 0<x | inl (inr y<0) = exFalso (SetoidPartialOrder.irreflexive pOrder (SetoidPartialOrder.transitive pOrder 0<xy (SetoidPartialOrder.wellDefined pOrder reflexive (transitive *Commutative (Ring.timesZero R)) (ringCanMultiplyByNegative order y<0 0<x))))
ineqLemma {S = S} {R = R} {_<_} {pOrder = pOrder} {tOrder = tOrder} I order {x} {y} 0<xy 0<x | inl (inr y<0) = exFalso (SetoidPartialOrder.irreflexive pOrder (SetoidPartialOrder.transitive pOrder 0<xy (SetoidPartialOrder.<WellDefined pOrder reflexive (transitive *Commutative (Ring.timesZero R)) (ringCanMultiplyByNegative order y<0 0<x))))
where
open Setoid S
open Ring R
open Equivalence (Setoid.eq S)
ineqLemma {S = S} {R = R} {_<_} {pOrder = pOrder} {tOrder = tOrder} I order {x} {y} 0<xy 0<x | inr 0=y = exFalso (SetoidPartialOrder.irreflexive pOrder (SetoidPartialOrder.wellDefined pOrder reflexive (transitive (*WellDefined reflexive (symmetric 0=y)) (Ring.timesZero R)) 0<xy))
ineqLemma {S = S} {R = R} {_<_} {pOrder = pOrder} {tOrder = tOrder} I order {x} {y} 0<xy 0<x | inr 0=y = exFalso (SetoidPartialOrder.irreflexive pOrder (SetoidPartialOrder.<WellDefined pOrder reflexive (transitive (*WellDefined reflexive (symmetric 0=y)) (Ring.timesZero R)) 0<xy))
where
open Setoid S
open Ring R
@@ -360,13 +362,13 @@ module Fields.FieldOfFractionsOrder where
ineqLemma' : {a b c : _} {A : Set a} {S : Setoid {a} {b} A} {_+_ : A → A → A} {_*_ : A → A → A} {R : Ring S _+_ _*_} {_<_ : Rel {_} {c} A} {pOrder : SetoidPartialOrder S _<_} {tOrder : SetoidTotalOrder pOrder} (I : IntegralDomain R) → (order : OrderedRing R tOrder) → {x y : A} → (Ring.0R R) < (x * y) → x < (Ring.0R R) → y < (Ring.0R R)
ineqLemma' {S = S} {R = R} {_<_} {pOrder = pOrder} {tOrder = tOrder} I order {x} {y} 0<xy x<0 with SetoidTotalOrder.totality tOrder (Ring.0R R) y
... | inl (inl 0<y) = exFalso (SetoidPartialOrder.irreflexive pOrder (SetoidPartialOrder.transitive pOrder 0<xy (SetoidPartialOrder.wellDefined pOrder *Commutative (transitive *Commutative (Ring.timesZero R)) (ringCanMultiplyByNegative order x<0 0<y))))
... | inl (inl 0<y) = exFalso (SetoidPartialOrder.irreflexive pOrder (SetoidPartialOrder.transitive pOrder 0<xy (SetoidPartialOrder.<WellDefined pOrder *Commutative (transitive *Commutative (Ring.timesZero R)) (ringCanMultiplyByNegative order x<0 0<y))))
where
open Setoid S
open Ring R
open Equivalence (Setoid.eq S)
... | inl (inr y<0) = y<0
... | (inr 0=y) = exFalso (SetoidPartialOrder.irreflexive pOrder (SetoidPartialOrder.wellDefined pOrder reflexive (transitive (*WellDefined reflexive (symmetric 0=y)) (Ring.timesZero R)) 0<xy))
... | (inr 0=y) = exFalso (SetoidPartialOrder.irreflexive pOrder (SetoidPartialOrder.<WellDefined pOrder reflexive (transitive (*WellDefined reflexive (symmetric 0=y)) (Ring.timesZero R)) 0<xy))
where
open Setoid S
open Ring R
@@ -380,7 +382,7 @@ module Fields.FieldOfFractionsOrder where
open Ring R
open Equivalence (Setoid.eq S)
... | inl (inr y<0) = y<0
... | (inr 0=y) = exFalso (SetoidPartialOrder.irreflexive pOrder (SetoidPartialOrder.wellDefined pOrder (transitive (*WellDefined reflexive (symmetric 0=y)) (Ring.timesZero R)) reflexive xy<0))
... | (inr 0=y) = exFalso (SetoidPartialOrder.irreflexive pOrder (SetoidPartialOrder.<WellDefined pOrder (transitive (*WellDefined reflexive (symmetric 0=y)) (Ring.timesZero R)) reflexive xy<0))
where
open Setoid S
open Ring R
@@ -389,12 +391,12 @@ module Fields.FieldOfFractionsOrder where
ineqLemma''' : {a b c : _} {A : Set a} {S : Setoid {a} {b} A} {_+_ : A → A → A} {_*_ : A → A → A} {R : Ring S _+_ _*_} {_<_ : Rel {_} {c} A} {pOrder : SetoidPartialOrder S _<_} {tOrder : SetoidTotalOrder pOrder} (I : IntegralDomain R) → (order : OrderedRing R tOrder) → {x y : A} → (x * y) < (Ring.0R R) → x < (Ring.0R R) → (Ring.0R R) < y
ineqLemma''' {S = S} {_*_ = _*_} {R = R} {_<_ = _<_} {pOrder} {tOrder} I order {x} {y} xy<0 x<0 with SetoidTotalOrder.totality tOrder (Ring.0R R) y
... | inl (inl 0<y) = 0<y
... | inl (inr y<0) = exFalso (SetoidPartialOrder.irreflexive pOrder (SetoidPartialOrder.transitive pOrder xy<0 (SetoidPartialOrder.wellDefined pOrder (transitive *Commutative (Ring.timesZero R)) reflexive (ringCanMultiplyByNegative order y<0 x<0))))
... | inl (inr y<0) = exFalso (SetoidPartialOrder.irreflexive pOrder (SetoidPartialOrder.transitive pOrder xy<0 (SetoidPartialOrder.<WellDefined pOrder (transitive *Commutative (Ring.timesZero R)) reflexive (ringCanMultiplyByNegative order y<0 x<0))))
where
open Setoid S
open Ring R
open Equivalence (Setoid.eq S)
... | inr 0=y = exFalso (SetoidPartialOrder.irreflexive pOrder (SetoidPartialOrder.wellDefined pOrder (transitive (*WellDefined reflexive (symmetric 0=y)) (Ring.timesZero R)) reflexive xy<0))
... | inr 0=y = exFalso (SetoidPartialOrder.irreflexive pOrder (SetoidPartialOrder.<WellDefined pOrder (transitive (*WellDefined reflexive (symmetric 0=y)) (Ring.timesZero R)) reflexive xy<0))
where
open Setoid S
open Ring R
@@ -405,7 +407,7 @@ module Fields.FieldOfFractionsOrder where
OrderedRing.orderRespectsAddition (fieldOfFractionsOrderedRing {S = S} {_+_} {_*_} {R} {_<_} {pOrder} {tOrder} I oRing) {numA ,, (denomA , denomA!=0)} {numB ,, (denomB , denomB!=0)} a<b (numC ,, (denomC , denomC!=0)) | inl (inl 0<dAdC) with SetoidTotalOrder.totality tOrder (Ring.0R R) (denomB * denomC)
OrderedRing.orderRespectsAddition (fieldOfFractionsOrderedRing {S = S} {_+_} {_*_} {R} {_<_} {pOrder} {tOrder} I oRing) {numA ,, (denomA , denomA!=0)} {numB ,, (denomB , denomB!=0)} a<b (numC ,, (denomC , denomC!=0)) | inl (inl 0<dAdC) | inl (inl 0<dBdC) with SetoidTotalOrder.totality tOrder (Ring.0R R) denomA
OrderedRing.orderRespectsAddition (fieldOfFractionsOrderedRing {S = S} {_+_} {_*_} {R} {_<_} {pOrder} {tOrder} I oRing) {numA ,, (denomA , denomA!=0)} {numB ,, (denomB , denomB!=0)} a<b (numC ,, (denomC , denomC!=0)) | inl (inl 0<dAdC) | inl (inl 0<dBdC) | inl (inl 0<dA) with SetoidTotalOrder.totality tOrder (Ring.0R R) denomB
OrderedRing.orderRespectsAddition (fieldOfFractionsOrderedRing {S = S} {_+_} {_*_} {R} {_<_} {pOrder} {tOrder} I oRing) {numA ,, (denomA , denomA!=0)} {numB ,, (denomB , denomB!=0)} a<b (numC ,, (denomC , denomC!=0)) | inl (inl 0<dAdC) | inl (inl 0<dBdC) | inl (inl 0<dA) | inl (inl 0<dB) = SetoidPartialOrder.wellDefined pOrder (symmetric *Associative) (symmetric *Associative) (ringCanMultiplyByPositive oRing 0<dC (SetoidPartialOrder.wellDefined pOrder (transitive (Group.+WellDefined additiveGroup *Commutative *Commutative) (transitive (symmetric *DistributesOver+) *Commutative)) (transitive (Group.+WellDefined additiveGroup *Commutative *Commutative) (transitive (symmetric *DistributesOver+) *Commutative)) ans))
OrderedRing.orderRespectsAddition (fieldOfFractionsOrderedRing {S = S} {_+_} {_*_} {R} {_<_} {pOrder} {tOrder} I oRing) {numA ,, (denomA , denomA!=0)} {numB ,, (denomB , denomB!=0)} a<b (numC ,, (denomC , denomC!=0)) | inl (inl 0<dAdC) | inl (inl 0<dBdC) | inl (inl 0<dA) | inl (inl 0<dB) = SetoidPartialOrder.<WellDefined pOrder (symmetric *Associative) (symmetric *Associative) (ringCanMultiplyByPositive oRing 0<dC (SetoidPartialOrder.<WellDefined pOrder (transitive (Group.+WellDefined additiveGroup *Commutative *Commutative) (transitive (symmetric *DistributesOver+) *Commutative)) (transitive (Group.+WellDefined additiveGroup *Commutative *Commutative) (transitive (symmetric *DistributesOver+) *Commutative)) ans))
where
open Setoid S
open Equivalence (Setoid.eq S)
@@ -413,12 +415,12 @@ module Fields.FieldOfFractionsOrder where
0<dC : 0R < denomC
0<dC with SetoidTotalOrder.totality tOrder 0R denomC
0<dC | inl (inl x) = x
0<dC | inl (inr dC<0) = exFalso (SetoidPartialOrder.irreflexive pOrder (SetoidPartialOrder.transitive pOrder 0<dBdC (SetoidPartialOrder.wellDefined pOrder reflexive (transitive *Commutative (Ring.timesZero R)) (ringCanMultiplyByNegative oRing dC<0 0<dB))))
0<dC | inl (inr dC<0) = exFalso (SetoidPartialOrder.irreflexive pOrder (SetoidPartialOrder.transitive pOrder 0<dBdC (SetoidPartialOrder.<WellDefined pOrder reflexive (transitive *Commutative (Ring.timesZero R)) (ringCanMultiplyByNegative oRing dC<0 0<dB))))
0<dC | inr x = exFalso (denomC!=0 (Equivalence.symmetric (Setoid.eq S) x))
p : ((numA * denomC) * denomB) < ((numB * denomC) * denomA)
p = SetoidPartialOrder.wellDefined pOrder (swapLemma R) (swapLemma R) (ringCanMultiplyByPositive oRing 0<dC a<b)
p = SetoidPartialOrder.<WellDefined pOrder (swapLemma R) (swapLemma R) (ringCanMultiplyByPositive oRing 0<dC a<b)
ans : ((((numA * denomC) * denomB) + ((denomA * numC) * denomB))) < ((((numB * denomC) * denomA) + ((denomB * numC) * denomA)))
ans = SetoidPartialOrder.wellDefined pOrder reflexive (Group.+WellDefined additiveGroup reflexive (transitive (*WellDefined *Commutative reflexive) (transitive (transitive (symmetric *Associative) (transitive (*WellDefined reflexive *Commutative) *Associative)) (*WellDefined *Commutative reflexive)))) (OrderedRing.orderRespectsAddition oRing p ((denomA * numC) * denomB))
ans = SetoidPartialOrder.<WellDefined pOrder reflexive (Group.+WellDefined additiveGroup reflexive (transitive (*WellDefined *Commutative reflexive) (transitive (transitive (symmetric *Associative) (transitive (*WellDefined reflexive *Commutative) *Associative)) (*WellDefined *Commutative reflexive)))) (OrderedRing.orderRespectsAddition oRing p ((denomA * numC) * denomB))
OrderedRing.orderRespectsAddition (fieldOfFractionsOrderedRing {S = S} {_+_} {_*_} {R} {_<_} {pOrder} {tOrder} I oRing) {numA ,, (denomA , denomA!=0)} {numB ,, (denomB , denomB!=0)} a<b (numC ,, (denomC , denomC!=0)) | inl (inl 0<dAdC) | inl (inl 0<dBdC) | inl (inl 0<dA) | inl (inr dB<0) = exFalso bad
where
open Setoid S
@@ -426,11 +428,11 @@ module Fields.FieldOfFractionsOrder where
open Ring R
dC<0 : denomC < 0R
dC<0 with SetoidTotalOrder.totality tOrder 0R denomC
... | inl (inl x) = exFalso (SetoidPartialOrder.irreflexive pOrder (SetoidPartialOrder.transitive pOrder 0<dBdC (SetoidPartialOrder.wellDefined pOrder reflexive (transitive *Commutative (Ring.timesZero R)) (ringCanMultiplyByPositive oRing x dB<0))))
... | inl (inl x) = exFalso (SetoidPartialOrder.irreflexive pOrder (SetoidPartialOrder.transitive pOrder 0<dBdC (SetoidPartialOrder.<WellDefined pOrder reflexive (transitive *Commutative (Ring.timesZero R)) (ringCanMultiplyByPositive oRing x dB<0))))
... | inl (inr x) = x
... | inr x = exFalso (denomC!=0 (Equivalence.symmetric (Setoid.eq S) x))
bad : False
bad = SetoidPartialOrder.irreflexive pOrder (SetoidPartialOrder.transitive pOrder 0<dAdC (SetoidPartialOrder.wellDefined pOrder reflexive (transitive *Commutative (Ring.timesZero R)) (ringCanMultiplyByNegative oRing dC<0 0<dA)))
bad = SetoidPartialOrder.irreflexive pOrder (SetoidPartialOrder.transitive pOrder 0<dAdC (SetoidPartialOrder.<WellDefined pOrder reflexive (transitive *Commutative (Ring.timesZero R)) (ringCanMultiplyByNegative oRing dC<0 0<dA)))
OrderedRing.orderRespectsAddition (fieldOfFractionsOrderedRing {S = S} {_+_} {_*_} {R} {_<_} {pOrder} {tOrder} I oRing) {numA ,, (denomA , denomA!=0)} {numB ,, (denomB , denomB!=0)} a<b (numC ,, (denomC , denomC!=0)) | inl (inl 0<dAdC) | inl (inl 0<dBdC) | inl (inl 0<dA) | inr x = exFalso (denomB!=0 (Equivalence.symmetric (Setoid.eq S) x))
OrderedRing.orderRespectsAddition (fieldOfFractionsOrderedRing {S = S} {_+_} {_*_} {R} {_<_} {pOrder} {tOrder} I oRing) {numA ,, (denomA , denomA!=0)} {numB ,, (denomB , denomB!=0)} a<b (numC ,, (denomC , denomC!=0)) | inl (inl 0<dAdC) | inl (inl 0<dBdC) | inl (inr dA<0) with SetoidTotalOrder.totality tOrder (Ring.0R R) denomB
OrderedRing.orderRespectsAddition (fieldOfFractionsOrderedRing {S = S} {_+_} {_*_} {R} {_<_} {pOrder} {tOrder} I oRing) {numA ,, (denomA , denomA!=0)} {numB ,, (denomB , denomB!=0)} a<b (numC ,, (denomC , denomC!=0)) | inl (inl 0<dAdC) | inl (inl 0<dBdC) | inl (inr dA<0) | inl (inl 0<dB) = exFalso bad
@@ -441,16 +443,16 @@ module Fields.FieldOfFractionsOrder where
0<dC : 0R < denomC
0<dC with SetoidTotalOrder.totality tOrder 0R denomC
0<dC | inl (inl x) = x
0<dC | inl (inr dC<0) = exFalso (SetoidPartialOrder.irreflexive pOrder (SetoidPartialOrder.transitive pOrder 0<dBdC (SetoidPartialOrder.wellDefined pOrder reflexive (transitive *Commutative (Ring.timesZero R)) (ringCanMultiplyByNegative oRing dC<0 0<dB))))
0<dC | inl (inr dC<0) = exFalso (SetoidPartialOrder.irreflexive pOrder (SetoidPartialOrder.transitive pOrder 0<dBdC (SetoidPartialOrder.<WellDefined pOrder reflexive (transitive *Commutative (Ring.timesZero R)) (ringCanMultiplyByNegative oRing dC<0 0<dB))))
0<dC | inr x = exFalso (denomC!=0 (Equivalence.symmetric (Setoid.eq S) x))
dC<0 : denomC < 0R
dC<0 with SetoidTotalOrder.totality tOrder 0R denomC
dC<0 | inl (inl 0<dC) = exFalso (SetoidPartialOrder.irreflexive pOrder (SetoidPartialOrder.transitive pOrder 0<dAdC (SetoidPartialOrder.wellDefined pOrder *Commutative (transitive *Commutative (Ring.timesZero R)) (ringCanMultiplyByNegative oRing dA<0 0<dC))))
dC<0 | inl (inl 0<dC) = exFalso (SetoidPartialOrder.irreflexive pOrder (SetoidPartialOrder.transitive pOrder 0<dAdC (SetoidPartialOrder.<WellDefined pOrder *Commutative (transitive *Commutative (Ring.timesZero R)) (ringCanMultiplyByNegative oRing dA<0 0<dC))))
dC<0 | inl (inr x) = x
dC<0 | inr x = exFalso (denomC!=0 (Equivalence.symmetric (Setoid.eq S) x))
bad : False
bad = SetoidPartialOrder.irreflexive pOrder (SetoidPartialOrder.transitive pOrder 0<dC dC<0)
OrderedRing.orderRespectsAddition (fieldOfFractionsOrderedRing {S = S} {_+_} {_*_} {R} {_<_} {pOrder} {tOrder} I oRing) {numA ,, (denomA , denomA!=0)} {numB ,, (denomB , denomB!=0)} a<b (numC ,, (denomC , denomC!=0)) | inl (inl 0<dAdC) | inl (inl 0<dBdC) | inl (inr dA<0) | inl (inr dB<0) = SetoidPartialOrder.wellDefined pOrder (symmetric *Associative) (symmetric *Associative) (ringCanMultiplyByNegative oRing dC<0 (SetoidPartialOrder.wellDefined pOrder (transitive (symmetric *DistributesOver+) *Commutative) (transitive (symmetric *DistributesOver+) *Commutative) have''))
OrderedRing.orderRespectsAddition (fieldOfFractionsOrderedRing {S = S} {_+_} {_*_} {R} {_<_} {pOrder} {tOrder} I oRing) {numA ,, (denomA , denomA!=0)} {numB ,, (denomB , denomB!=0)} a<b (numC ,, (denomC , denomC!=0)) | inl (inl 0<dAdC) | inl (inl 0<dBdC) | inl (inr dA<0) | inl (inr dB<0) = SetoidPartialOrder.<WellDefined pOrder (symmetric *Associative) (symmetric *Associative) (ringCanMultiplyByNegative oRing dC<0 (SetoidPartialOrder.<WellDefined pOrder (transitive (symmetric *DistributesOver+) *Commutative) (transitive (symmetric *DistributesOver+) *Commutative) have''))
where
open Setoid S
open Equivalence (Setoid.eq S)
@@ -460,9 +462,9 @@ module Fields.FieldOfFractionsOrder where
have : ((numB * denomA) * denomC) < ((numA * denomB) * denomC)
have = ringCanMultiplyByNegative oRing dC<0 a<b
have' : (denomA * (numB * denomC)) < (denomB * (numA * denomC))
have' = SetoidPartialOrder.wellDefined pOrder (transitive (*WellDefined *Commutative reflexive) (symmetric *Associative)) (transitive (*WellDefined *Commutative reflexive) (symmetric *Associative)) have
have' = SetoidPartialOrder.<WellDefined pOrder (transitive (*WellDefined *Commutative reflexive) (symmetric *Associative)) (transitive (*WellDefined *Commutative reflexive) (symmetric *Associative)) have
have'' : ((denomA * (numB * denomC)) + (denomA * (denomB * numC))) < ((denomB * (numA * denomC)) + (denomB * (denomA * numC)))
have'' = SetoidPartialOrder.wellDefined pOrder reflexive (Group.+WellDefined additiveGroup reflexive (transitive *Associative (transitive (*WellDefined *Commutative reflexive) (symmetric *Associative)))) (OrderedRing.orderRespectsAddition oRing have' (denomA * (denomB * numC)))
have'' = SetoidPartialOrder.<WellDefined pOrder reflexive (Group.+WellDefined additiveGroup reflexive (transitive *Associative (transitive (*WellDefined *Commutative reflexive) (symmetric *Associative)))) (OrderedRing.orderRespectsAddition oRing have' (denomA * (denomB * numC)))
OrderedRing.orderRespectsAddition (fieldOfFractionsOrderedRing {S = S} {_+_} {_*_} {R} {_<_} {pOrder} {tOrder} I oRing) {numA ,, (denomA , denomA!=0)} {numB ,, (denomB , denomB!=0)} a<b (numC ,, (denomC , denomC!=0)) | inl (inl 0<dAdC) | inl (inl 0<dBdC) | inl (inr dA<0) | inr x = exFalso (denomB!=0 (Equivalence.symmetric (Setoid.eq S) x))
OrderedRing.orderRespectsAddition (fieldOfFractionsOrderedRing {S = S} {_+_} {_*_} {R} {_<_} {pOrder} {tOrder} I oRing) {numA ,, (denomA , denomA!=0)} {numB ,, (denomB , denomB!=0)} a<b (numC ,, (denomC , denomC!=0)) | inl (inl 0<dAdC) | inl (inl 0<dBdC) | inr x = exFalso (denomA!=0 (Equivalence.symmetric (Setoid.eq S) x))
OrderedRing.orderRespectsAddition (fieldOfFractionsOrderedRing {S = S} {_+_} {_*_} {R} {_<_} {pOrder} {tOrder} I oRing) {numA ,, (denomA , denomA!=0)} {numB ,, (denomB , denomB!=0)} a<b (numC ,, (denomC , denomC!=0)) | inl (inl 0<dAdC) | inl (inr dBdC<0) with SetoidTotalOrder.totality tOrder (Ring.0R R) denomA
@@ -478,7 +480,7 @@ module Fields.FieldOfFractionsOrder where
dC<0 = ineqLemma'' I oRing dBdC<0 0<dB
bad : False
bad = SetoidPartialOrder.irreflexive pOrder (SetoidPartialOrder.transitive pOrder 0<dC dC<0)
OrderedRing.orderRespectsAddition (fieldOfFractionsOrderedRing {S = S} {_+_} {_*_} {R} {_<_} {pOrder} {tOrder} I oRing) {numA ,, (denomA , denomA!=0)} {numB ,, (denomB , denomB!=0)} a<b (numC ,, (denomC , denomC!=0)) | inl (inl 0<dAdC) | inl (inr dBdC<0) | inl (inl 0<dA) | inl (inr dB<0) = SetoidPartialOrder.wellDefined pOrder (symmetric *Associative) (symmetric *Associative) (ringCanMultiplyByPositive oRing 0<dC ans)
OrderedRing.orderRespectsAddition (fieldOfFractionsOrderedRing {S = S} {_+_} {_*_} {R} {_<_} {pOrder} {tOrder} I oRing) {numA ,, (denomA , denomA!=0)} {numB ,, (denomB , denomB!=0)} a<b (numC ,, (denomC , denomC!=0)) | inl (inl 0<dAdC) | inl (inr dBdC<0) | inl (inl 0<dA) | inl (inr dB<0) = SetoidPartialOrder.<WellDefined pOrder (symmetric *Associative) (symmetric *Associative) (ringCanMultiplyByPositive oRing 0<dC ans)
where
open Setoid S
open Equivalence (Setoid.eq S)
@@ -488,12 +490,12 @@ module Fields.FieldOfFractionsOrder where
have : ((numB * denomA) * denomC) < ((numA * denomB) * denomC)
have = ringCanMultiplyByPositive oRing 0<dC a<b
have' : (((numB * denomC) * denomA) + ((denomB * numC) * denomA)) < (((numA * denomC) * denomB) + ((denomB * numC) * denomA))
have' = OrderedRing.orderRespectsAddition oRing (SetoidPartialOrder.wellDefined pOrder (swapLemma R) (swapLemma R) have) _
have' = OrderedRing.orderRespectsAddition oRing (SetoidPartialOrder.<WellDefined pOrder (swapLemma R) (swapLemma R) have) _
ans : (((numB * denomC) + (denomB * numC)) * denomA) < (((numA * denomC) + (denomA * numC)) * denomB)
ans = SetoidPartialOrder.wellDefined pOrder (transitive (Group.+WellDefined additiveGroup *Commutative *Commutative) (transitive (symmetric *DistributesOver+) *Commutative)) (transitive (Group.+WellDefined additiveGroup *Commutative (transitive (symmetric *Associative) (*WellDefined reflexive *Commutative))) (transitive (symmetric *DistributesOver+) *Commutative)) have'
ans = SetoidPartialOrder.<WellDefined pOrder (transitive (Group.+WellDefined additiveGroup *Commutative *Commutative) (transitive (symmetric *DistributesOver+) *Commutative)) (transitive (Group.+WellDefined additiveGroup *Commutative (transitive (symmetric *Associative) (*WellDefined reflexive *Commutative))) (transitive (symmetric *DistributesOver+) *Commutative)) have'
OrderedRing.orderRespectsAddition (fieldOfFractionsOrderedRing {S = S} {_+_} {_*_} {R} {_<_} {pOrder} {tOrder} I oRing) {numA ,, (denomA , denomA!=0)} {numB ,, (denomB , denomB!=0)} a<b (numC ,, (denomC , denomC!=0)) | inl (inl 0<dAdC) | inl (inr dBdC<0) | inl (inl 0<dA) | inr x = exFalso (denomB!=0 (Equivalence.symmetric (Setoid.eq S) x))
OrderedRing.orderRespectsAddition (fieldOfFractionsOrderedRing {S = S} {_+_} {_*_} {R} {_<_} {pOrder} {tOrder} I oRing) {numA ,, (denomA , denomA!=0)} {numB ,, (denomB , denomB!=0)} a<b (numC ,, (denomC , denomC!=0)) | inl (inl 0<dAdC) | inl (inr dBdC<0) | inl (inr dA<0) with SetoidTotalOrder.totality tOrder (Ring.0R R) denomB
OrderedRing.orderRespectsAddition (fieldOfFractionsOrderedRing {S = S} {_+_} {_*_} {R} {_<_} {pOrder} {tOrder} I oRing) {numA ,, (denomA , denomA!=0)} {numB ,, (denomB , denomB!=0)} a<b (numC ,, (denomC , denomC!=0)) | inl (inl 0<dAdC) | inl (inr dBdC<0) | inl (inr dA<0) | inl (inl 0<dB) = SetoidPartialOrder.wellDefined pOrder (symmetric *Associative) (symmetric *Associative) (ringCanMultiplyByNegative oRing dC<0 (SetoidPartialOrder.wellDefined pOrder (transitive (Group.+WellDefined additiveGroup *Commutative *Commutative) (transitive (transitive (Group.+WellDefined additiveGroup reflexive (transitive *Associative (transitive (*WellDefined *Commutative reflexive) (symmetric *Associative)))) (symmetric *DistributesOver+)) *Commutative)) (transitive (Group.+WellDefined additiveGroup *Commutative *Commutative) (transitive (symmetric *DistributesOver+) *Commutative)) have))
OrderedRing.orderRespectsAddition (fieldOfFractionsOrderedRing {S = S} {_+_} {_*_} {R} {_<_} {pOrder} {tOrder} I oRing) {numA ,, (denomA , denomA!=0)} {numB ,, (denomB , denomB!=0)} a<b (numC ,, (denomC , denomC!=0)) | inl (inl 0<dAdC) | inl (inr dBdC<0) | inl (inr dA<0) | inl (inl 0<dB) = SetoidPartialOrder.<WellDefined pOrder (symmetric *Associative) (symmetric *Associative) (ringCanMultiplyByNegative oRing dC<0 (SetoidPartialOrder.<WellDefined pOrder (transitive (Group.+WellDefined additiveGroup *Commutative *Commutative) (transitive (transitive (Group.+WellDefined additiveGroup reflexive (transitive *Associative (transitive (*WellDefined *Commutative reflexive) (symmetric *Associative)))) (symmetric *DistributesOver+)) *Commutative)) (transitive (Group.+WellDefined additiveGroup *Commutative *Commutative) (transitive (symmetric *DistributesOver+) *Commutative)) have))
where
open Setoid S
open Equivalence (Setoid.eq S)
@@ -501,7 +503,7 @@ module Fields.FieldOfFractionsOrder where
dC<0 : denomC < 0R
dC<0 = ineqLemma'' I oRing dBdC<0 0<dB
have : (((numA * denomC) * denomB) + ((denomB * numC) * denomA)) < (((numB * denomC) * denomA) + ((denomB * numC) * denomA))
have = OrderedRing.orderRespectsAddition oRing (SetoidPartialOrder.wellDefined pOrder (swapLemma R) (swapLemma R) (ringCanMultiplyByNegative oRing dC<0 a<b)) _
have = OrderedRing.orderRespectsAddition oRing (SetoidPartialOrder.<WellDefined pOrder (swapLemma R) (swapLemma R) (ringCanMultiplyByNegative oRing dC<0 a<b)) _
OrderedRing.orderRespectsAddition (fieldOfFractionsOrderedRing {S = S} {_+_} {_*_} {R} {_<_} {pOrder} {tOrder} I oRing) {numA ,, (denomA , denomA!=0)} {numB ,, (denomB , denomB!=0)} a<b (numC ,, (denomC , denomC!=0)) | inl (inl 0<dAdC) | inl (inr dBdC<0) | inl (inr dA<0) | inl (inr dB<0) = exFalso bad
where
open Setoid S
@@ -530,7 +532,7 @@ module Fields.FieldOfFractionsOrder where
0<dC = ineqLemma I oRing 0<dBdC 0<dB
dC<0 : denomC < 0R
dC<0 = ineqLemma'' I oRing dAdC<0 0<dA
OrderedRing.orderRespectsAddition (fieldOfFractionsOrderedRing {S = S} {_+_} {_*_} {R} {_<_} {pOrder} {tOrder} I oRing) {numA ,, (denomA , denomA!=0)} {numB ,, (denomB , denomB!=0)} a<b (numC ,, (denomC , denomC!=0)) | inl (inr dAdC<0) | inl (inl 0<dBdC) | inl (inl 0<dA) | inl (inr dB<0) = SetoidPartialOrder.wellDefined pOrder (symmetric *Associative) (symmetric *Associative) (ringCanMultiplyByNegative oRing dC<0 (SetoidPartialOrder.wellDefined pOrder (transitive (Group.+WellDefined additiveGroup (transitive (*WellDefined *Commutative reflexive) (symmetric *Associative)) *Commutative) (transitive (symmetric *DistributesOver+) *Commutative)) (transitive (Group.+WellDefined additiveGroup (transitive (*WellDefined *Commutative reflexive) (symmetric *Associative)) (transitive (symmetric *Associative) (*WellDefined reflexive *Commutative))) (transitive (symmetric *DistributesOver+) *Commutative)) have))
OrderedRing.orderRespectsAddition (fieldOfFractionsOrderedRing {S = S} {_+_} {_*_} {R} {_<_} {pOrder} {tOrder} I oRing) {numA ,, (denomA , denomA!=0)} {numB ,, (denomB , denomB!=0)} a<b (numC ,, (denomC , denomC!=0)) | inl (inr dAdC<0) | inl (inl 0<dBdC) | inl (inl 0<dA) | inl (inr dB<0) = SetoidPartialOrder.<WellDefined pOrder (symmetric *Associative) (symmetric *Associative) (ringCanMultiplyByNegative oRing dC<0 (SetoidPartialOrder.<WellDefined pOrder (transitive (Group.+WellDefined additiveGroup (transitive (*WellDefined *Commutative reflexive) (symmetric *Associative)) *Commutative) (transitive (symmetric *DistributesOver+) *Commutative)) (transitive (Group.+WellDefined additiveGroup (transitive (*WellDefined *Commutative reflexive) (symmetric *Associative)) (transitive (symmetric *Associative) (*WellDefined reflexive *Commutative))) (transitive (symmetric *DistributesOver+) *Commutative)) have))
where
open Setoid S
open Equivalence (Setoid.eq S)
@@ -541,7 +543,7 @@ module Fields.FieldOfFractionsOrder where
have = OrderedRing.orderRespectsAddition oRing (ringCanMultiplyByNegative oRing dC<0 a<b) _
OrderedRing.orderRespectsAddition (fieldOfFractionsOrderedRing {S = S} {_+_} {_*_} {R} {_<_} {pOrder} {tOrder} I oRing) {numA ,, (denomA , denomA!=0)} {numB ,, (denomB , denomB!=0)} a<b (numC ,, (denomC , denomC!=0)) | inl (inr dAdC<0) | inl (inl 0<dBdC) | inl (inl 0<dA) | inr 0=dB = exFalso (denomB!=0 (Equivalence.symmetric (Setoid.eq S) 0=dB))
OrderedRing.orderRespectsAddition (fieldOfFractionsOrderedRing {S = S} {_+_} {_*_} {R} {_<_} {pOrder} {tOrder} I oRing) {numA ,, (denomA , denomA!=0)} {numB ,, (denomB , denomB!=0)} a<b (numC ,, (denomC , denomC!=0)) | inl (inr dAdC<0) | inl (inl 0<dBdC) | inl (inr dA<0) with SetoidTotalOrder.totality tOrder (Ring.0R R) denomB
OrderedRing.orderRespectsAddition (fieldOfFractionsOrderedRing {S = S} {_+_} {_*_} {R} {_<_} {pOrder} {tOrder} I oRing) {numA ,, (denomA , denomA!=0)} {numB ,, (denomB , denomB!=0)} a<b (numC ,, (denomC , denomC!=0)) | inl (inr dAdC<0) | inl (inl 0<dBdC) | inl (inr dA<0) | inl (inl 0<dB) = SetoidPartialOrder.wellDefined pOrder (symmetric *Associative) (symmetric *Associative) (ringCanMultiplyByPositive oRing 0<dC (SetoidPartialOrder.wellDefined pOrder (transitive (Group.+WellDefined additiveGroup *Commutative *Commutative) (transitive (transitive (Group.+WellDefined additiveGroup (transitive *Commutative (transitive (transitive (symmetric *Associative) (transitive (*WellDefined reflexive *Commutative) *Associative)) *Commutative)) (transitive *Associative (transitive (*WellDefined *Commutative reflexive) (symmetric *Associative)))) (symmetric *DistributesOver+)) *Commutative)) (transitive (Group.+WellDefined additiveGroup *Commutative *Commutative) (transitive (transitive (Group.+WellDefined additiveGroup (transitive *Commutative (transitive (*WellDefined *Commutative reflexive) (symmetric *Associative))) reflexive) (symmetric *DistributesOver+)) *Commutative)) have))
OrderedRing.orderRespectsAddition (fieldOfFractionsOrderedRing {S = S} {_+_} {_*_} {R} {_<_} {pOrder} {tOrder} I oRing) {numA ,, (denomA , denomA!=0)} {numB ,, (denomB , denomB!=0)} a<b (numC ,, (denomC , denomC!=0)) | inl (inr dAdC<0) | inl (inl 0<dBdC) | inl (inr dA<0) | inl (inl 0<dB) = SetoidPartialOrder.<WellDefined pOrder (symmetric *Associative) (symmetric *Associative) (ringCanMultiplyByPositive oRing 0<dC (SetoidPartialOrder.<WellDefined pOrder (transitive (Group.+WellDefined additiveGroup *Commutative *Commutative) (transitive (transitive (Group.+WellDefined additiveGroup (transitive *Commutative (transitive (transitive (symmetric *Associative) (transitive (*WellDefined reflexive *Commutative) *Associative)) *Commutative)) (transitive *Associative (transitive (*WellDefined *Commutative reflexive) (symmetric *Associative)))) (symmetric *DistributesOver+)) *Commutative)) (transitive (Group.+WellDefined additiveGroup *Commutative *Commutative) (transitive (transitive (Group.+WellDefined additiveGroup (transitive *Commutative (transitive (*WellDefined *Commutative reflexive) (symmetric *Associative))) reflexive) (symmetric *DistributesOver+)) *Commutative)) have))
where
open Setoid S
open Equivalence (Setoid.eq S)
@@ -563,7 +565,7 @@ module Fields.FieldOfFractionsOrder where
OrderedRing.orderRespectsAddition (fieldOfFractionsOrderedRing {S = S} {_+_} {_*_} {R} {_<_} {pOrder} {tOrder} I oRing) {numA ,, (denomA , denomA!=0)} {numB ,, (denomB , denomB!=0)} a<b (numC ,, (denomC , denomC!=0)) | inl (inr dAdC<0) | inl (inl 0<dBdC) | inr 0=dA = exFalso (denomA!=0 (Equivalence.symmetric (Setoid.eq S) 0=dA))
OrderedRing.orderRespectsAddition (fieldOfFractionsOrderedRing {S = S} {_+_} {_*_} {R} {_<_} {pOrder} {tOrder} I oRing) {numA ,, (denomA , denomA!=0)} {numB ,, (denomB , denomB!=0)} a<b (numC ,, (denomC , denomC!=0)) | inl (inr dAdC<0) | inl (inr dBdC<0) with SetoidTotalOrder.totality tOrder (Ring.0R R) denomA
OrderedRing.orderRespectsAddition (fieldOfFractionsOrderedRing {S = S} {_+_} {_*_} {R} {_<_} {pOrder} {tOrder} I oRing) {numA ,, (denomA , denomA!=0)} {numB ,, (denomB , denomB!=0)} a<b (numC ,, (denomC , denomC!=0)) | inl (inr dAdC<0) | inl (inr dBdC<0) | inl (inl 0<dA) with SetoidTotalOrder.totality tOrder (Ring.0R R) denomB
OrderedRing.orderRespectsAddition (fieldOfFractionsOrderedRing {S = S} {_+_} {_*_} {R} {_<_} {pOrder} {tOrder} I oRing) {numA ,, (denomA , denomA!=0)} {numB ,, (denomB , denomB!=0)} a<b (numC ,, (denomC , denomC!=0)) | inl (inr dAdC<0) | inl (inr dBdC<0) | inl (inl 0<dA) | inl (inl 0<dB) = SetoidPartialOrder.wellDefined pOrder (symmetric *Associative) (symmetric *Associative) (ringCanMultiplyByNegative oRing dC<0 (SetoidPartialOrder.wellDefined pOrder (transitive (transitive (Group.+WellDefined additiveGroup *Commutative *Commutative) (Group.+WellDefined additiveGroup (transitive *Commutative (transitive (*WellDefined *Commutative reflexive) (symmetric *Associative))) reflexive)) (transitive (symmetric *DistributesOver+) *Commutative)) (transitive (transitive (Group.+WellDefined additiveGroup *Commutative *Commutative) (Group.+WellDefined additiveGroup (transitive (transitive *Associative (*WellDefined *Commutative reflexive)) *Commutative) (transitive *Associative (transitive (*WellDefined *Commutative reflexive) (symmetric *Associative))))) (transitive (symmetric *DistributesOver+) *Commutative)) have))
OrderedRing.orderRespectsAddition (fieldOfFractionsOrderedRing {S = S} {_+_} {_*_} {R} {_<_} {pOrder} {tOrder} I oRing) {numA ,, (denomA , denomA!=0)} {numB ,, (denomB , denomB!=0)} a<b (numC ,, (denomC , denomC!=0)) | inl (inr dAdC<0) | inl (inr dBdC<0) | inl (inl 0<dA) | inl (inl 0<dB) = SetoidPartialOrder.<WellDefined pOrder (symmetric *Associative) (symmetric *Associative) (ringCanMultiplyByNegative oRing dC<0 (SetoidPartialOrder.<WellDefined pOrder (transitive (transitive (Group.+WellDefined additiveGroup *Commutative *Commutative) (Group.+WellDefined additiveGroup (transitive *Commutative (transitive (*WellDefined *Commutative reflexive) (symmetric *Associative))) reflexive)) (transitive (symmetric *DistributesOver+) *Commutative)) (transitive (transitive (Group.+WellDefined additiveGroup *Commutative *Commutative) (Group.+WellDefined additiveGroup (transitive (transitive *Associative (*WellDefined *Commutative reflexive)) *Commutative) (transitive *Associative (transitive (*WellDefined *Commutative reflexive) (symmetric *Associative))))) (transitive (symmetric *DistributesOver+) *Commutative)) have))
where
open Setoid S
open Equivalence (Setoid.eq S)
@@ -592,7 +594,7 @@ module Fields.FieldOfFractionsOrder where
0<dC = ineqLemma''' I oRing dAdC<0 dA<0
dC<0 : denomC < 0R
dC<0 = ineqLemma'' I oRing dBdC<0 0<dB
OrderedRing.orderRespectsAddition (fieldOfFractionsOrderedRing {S = S} {_+_} {_*_} {R} {_<_} {pOrder} {tOrder} I oRing) {numA ,, (denomA , denomA!=0)} {numB ,, (denomB , denomB!=0)} a<b (numC ,, (denomC , denomC!=0)) | inl (inr dAdC<0) | inl (inr dBdC<0) | inl (inr dA<0) | inl (inr dB<0) = SetoidPartialOrder.wellDefined pOrder (symmetric *Associative) (symmetric *Associative) (ringCanMultiplyByPositive oRing 0<dC (SetoidPartialOrder.wellDefined pOrder (transitive (Group.+WellDefined additiveGroup *Commutative *Commutative) (transitive (Group.+WellDefined additiveGroup (transitive *Commutative (transitive (*WellDefined *Commutative reflexive) (symmetric *Associative))) reflexive) (transitive (symmetric *DistributesOver+) *Commutative))) (transitive (Group.+WellDefined additiveGroup *Commutative *Commutative) (transitive (transitive (Group.+WellDefined additiveGroup (transitive *Commutative (transitive (*WellDefined *Commutative reflexive) (symmetric *Associative))) (transitive *Commutative (transitive (symmetric *Associative) (*WellDefined reflexive *Commutative)))) (symmetric *DistributesOver+)) *Commutative)) have))
OrderedRing.orderRespectsAddition (fieldOfFractionsOrderedRing {S = S} {_+_} {_*_} {R} {_<_} {pOrder} {tOrder} I oRing) {numA ,, (denomA , denomA!=0)} {numB ,, (denomB , denomB!=0)} a<b (numC ,, (denomC , denomC!=0)) | inl (inr dAdC<0) | inl (inr dBdC<0) | inl (inr dA<0) | inl (inr dB<0) = SetoidPartialOrder.<WellDefined pOrder (symmetric *Associative) (symmetric *Associative) (ringCanMultiplyByPositive oRing 0<dC (SetoidPartialOrder.<WellDefined pOrder (transitive (Group.+WellDefined additiveGroup *Commutative *Commutative) (transitive (Group.+WellDefined additiveGroup (transitive *Commutative (transitive (*WellDefined *Commutative reflexive) (symmetric *Associative))) reflexive) (transitive (symmetric *DistributesOver+) *Commutative))) (transitive (Group.+WellDefined additiveGroup *Commutative *Commutative) (transitive (transitive (Group.+WellDefined additiveGroup (transitive *Commutative (transitive (*WellDefined *Commutative reflexive) (symmetric *Associative))) (transitive *Commutative (transitive (symmetric *Associative) (*WellDefined reflexive *Commutative)))) (symmetric *DistributesOver+)) *Commutative)) have))
where
open Setoid S
open Equivalence (Setoid.eq S)
@@ -613,40 +615,40 @@ module Fields.FieldOfFractionsOrder where
OrderedRing.orderRespectsMultiplication (fieldOfFractionsOrderedRing {S = S} {_+_} {_*_} {R} {_<_} {pOrder} {tOrder} I oRing) {numA ,, (denomA , denomA!=0)} {numB ,, (denomB , denomB!=0)} 0<a 0<b | inl (inl 0<1) with SetoidTotalOrder.totality tOrder (Ring.0R R) (denomA * denomB)
OrderedRing.orderRespectsMultiplication (fieldOfFractionsOrderedRing {S = S} {_+_} {_*_} {R} {_<_} {pOrder} {tOrder} I oRing) {numA ,, (denomA , denomA!=0)} {numB ,, (denomB , denomB!=0)} 0<a 0<b | inl (inl 0<1) | inl (inl 0<dAdB) with SetoidTotalOrder.totality tOrder (Ring.0R R) denomB
OrderedRing.orderRespectsMultiplication (fieldOfFractionsOrderedRing {S = S} {_+_} {_*_} {R} {_<_} {pOrder} {tOrder} I oRing) {numA ,, (denomA , denomA!=0)} {numB ,, (denomB , denomB!=0)} 0<a 0<b | inl (inl 0<1) | inl (inl 0<dAdB) | inl (inl 0<dB) with SetoidTotalOrder.totality tOrder (Ring.0R R) denomA
OrderedRing.orderRespectsMultiplication (fieldOfFractionsOrderedRing {S = S} {_+_} {_*_} {R} {_<_} {pOrder} {tOrder} I oRing) {numA ,, (denomA , denomA!=0)} {numB ,, (denomB , denomB!=0)} 0<a 0<b | inl (inl 0<1) | inl (inl 0<dAdB) | inl (inl 0<dB) | inl (inl 0<dA) = SetoidPartialOrder.wellDefined pOrder (symmetric (transitive *Commutative (Ring.timesZero R))) (symmetric (transitive *Commutative identIsIdent)) 0<nAnB
OrderedRing.orderRespectsMultiplication (fieldOfFractionsOrderedRing {S = S} {_+_} {_*_} {R} {_<_} {pOrder} {tOrder} I oRing) {numA ,, (denomA , denomA!=0)} {numB ,, (denomB , denomB!=0)} 0<a 0<b | inl (inl 0<1) | inl (inl 0<dAdB) | inl (inl 0<dB) | inl (inl 0<dA) = SetoidPartialOrder.<WellDefined pOrder (symmetric (transitive *Commutative (Ring.timesZero R))) (symmetric (transitive *Commutative identIsIdent)) 0<nAnB
where
open Setoid S
open Equivalence (Setoid.eq S)
open Ring R
0<nA : 0R < numA
0<nA = SetoidPartialOrder.wellDefined pOrder (transitive *Commutative (Ring.timesZero R)) (transitive *Commutative identIsIdent) 0<a
0<nA = SetoidPartialOrder.<WellDefined pOrder (transitive *Commutative (Ring.timesZero R)) (transitive *Commutative identIsIdent) 0<a
0<nB : 0R < numB
0<nB = SetoidPartialOrder.wellDefined pOrder (transitive *Commutative (Ring.timesZero R)) (transitive *Commutative identIsIdent) 0<b
0<nB = SetoidPartialOrder.<WellDefined pOrder (transitive *Commutative (Ring.timesZero R)) (transitive *Commutative identIsIdent) 0<b
0<nAnB : 0R < (numA * numB)
0<nAnB = SetoidPartialOrder.wellDefined pOrder (transitive *Commutative (Ring.timesZero R)) reflexive (ringCanMultiplyByPositive oRing 0<nB 0<nA)
OrderedRing.orderRespectsMultiplication (fieldOfFractionsOrderedRing {S = S} {_+_} {_*_} {R} {_<_} {pOrder} {tOrder} I oRing) {numA ,, (denomA , denomA!=0)} {numB ,, (denomB , denomB!=0)} 0<a 0<b | inl (inl 0<1) | inl (inl 0<dAdB) | inl (inl 0<dB) | inl (inr dA<0) = exFalso (SetoidPartialOrder.irreflexive pOrder (SetoidPartialOrder.transitive pOrder 0<dAdB (SetoidPartialOrder.wellDefined pOrder *Commutative (transitive *Commutative (Ring.timesZero R)) (ringCanMultiplyByNegative oRing dA<0 0<dB))))
0<nAnB = SetoidPartialOrder.<WellDefined pOrder (transitive *Commutative (Ring.timesZero R)) reflexive (ringCanMultiplyByPositive oRing 0<nB 0<nA)
OrderedRing.orderRespectsMultiplication (fieldOfFractionsOrderedRing {S = S} {_+_} {_*_} {R} {_<_} {pOrder} {tOrder} I oRing) {numA ,, (denomA , denomA!=0)} {numB ,, (denomB , denomB!=0)} 0<a 0<b | inl (inl 0<1) | inl (inl 0<dAdB) | inl (inl 0<dB) | inl (inr dA<0) = exFalso (SetoidPartialOrder.irreflexive pOrder (SetoidPartialOrder.transitive pOrder 0<dAdB (SetoidPartialOrder.<WellDefined pOrder *Commutative (transitive *Commutative (Ring.timesZero R)) (ringCanMultiplyByNegative oRing dA<0 0<dB))))
where
open Setoid S
open Equivalence (Setoid.eq S)
open Ring R
OrderedRing.orderRespectsMultiplication (fieldOfFractionsOrderedRing {S = S} {_+_} {_*_} {R} {_<_} {pOrder} {tOrder} I oRing) {numA ,, (denomA , denomA!=0)} {numB ,, (denomB , denomB!=0)} 0<a 0<b | inl (inl 0<1) | inl (inl 0<dAdB) | inl (inl 0<dB) | inr x = exFalso (denomA!=0 (Equivalence.symmetric (Setoid.eq S) x))
OrderedRing.orderRespectsMultiplication (fieldOfFractionsOrderedRing {S = S} {_+_} {_*_} {R} {_<_} {pOrder} {tOrder} I oRing) {numA ,, (denomA , denomA!=0)} {numB ,, (denomB , denomB!=0)} 0<a 0<b | inl (inl 0<1) | inl (inl 0<dAdB) | inl (inr dB<0) with SetoidTotalOrder.totality tOrder (Ring.0R R) denomA
OrderedRing.orderRespectsMultiplication (fieldOfFractionsOrderedRing {S = S} {_+_} {_*_} {R} {_<_} {pOrder} {tOrder} I oRing) {numA ,, (denomA , denomA!=0)} {numB ,, (denomB , denomB!=0)} 0<a 0<b | inl (inl 0<1) | inl (inl 0<dAdB) | inl (inr dB<0) | inl (inl 0<dA) = exFalso (SetoidPartialOrder.irreflexive pOrder (SetoidPartialOrder.transitive pOrder 0<dAdB (SetoidPartialOrder.wellDefined pOrder reflexive (transitive *Commutative (Ring.timesZero R)) (ringCanMultiplyByNegative oRing dB<0 0<dA))))
OrderedRing.orderRespectsMultiplication (fieldOfFractionsOrderedRing {S = S} {_+_} {_*_} {R} {_<_} {pOrder} {tOrder} I oRing) {numA ,, (denomA , denomA!=0)} {numB ,, (denomB , denomB!=0)} 0<a 0<b | inl (inl 0<1) | inl (inl 0<dAdB) | inl (inr dB<0) | inl (inl 0<dA) = exFalso (SetoidPartialOrder.irreflexive pOrder (SetoidPartialOrder.transitive pOrder 0<dAdB (SetoidPartialOrder.<WellDefined pOrder reflexive (transitive *Commutative (Ring.timesZero R)) (ringCanMultiplyByNegative oRing dB<0 0<dA))))
where
open Setoid S
open Equivalence (Setoid.eq S)
open Ring R
OrderedRing.orderRespectsMultiplication (fieldOfFractionsOrderedRing {S = S} {_+_} {_*_} {R} {_<_} {pOrder} {tOrder} I oRing) {numA ,, (denomA , denomA!=0)} {numB ,, (denomB , denomB!=0)} 0<a 0<b | inl (inl 0<1) | inl (inl 0<dAdB) | inl (inr dB<0) | inl (inr dA<0) = SetoidPartialOrder.wellDefined pOrder (symmetric (transitive *Commutative (Ring.timesZero R))) (symmetric (transitive *Commutative identIsIdent)) 0<nAnB
OrderedRing.orderRespectsMultiplication (fieldOfFractionsOrderedRing {S = S} {_+_} {_*_} {R} {_<_} {pOrder} {tOrder} I oRing) {numA ,, (denomA , denomA!=0)} {numB ,, (denomB , denomB!=0)} 0<a 0<b | inl (inl 0<1) | inl (inl 0<dAdB) | inl (inr dB<0) | inl (inr dA<0) = SetoidPartialOrder.<WellDefined pOrder (symmetric (transitive *Commutative (Ring.timesZero R))) (symmetric (transitive *Commutative identIsIdent)) 0<nAnB
where
open Setoid S
open Equivalence (Setoid.eq S)
open Ring R
nB<0 : numB < 0R
nB<0 = SetoidPartialOrder.wellDefined pOrder (transitive *Commutative identIsIdent) (transitive *Commutative (Ring.timesZero R)) 0<b
nB<0 = SetoidPartialOrder.<WellDefined pOrder (transitive *Commutative identIsIdent) (transitive *Commutative (Ring.timesZero R)) 0<b
nA<0 : numA < 0R
nA<0 = SetoidPartialOrder.wellDefined pOrder (transitive *Commutative identIsIdent) (transitive *Commutative (Ring.timesZero R)) 0<a
nA<0 = SetoidPartialOrder.<WellDefined pOrder (transitive *Commutative identIsIdent) (transitive *Commutative (Ring.timesZero R)) 0<a
0<nAnB : 0R < (numA * numB)
0<nAnB = SetoidPartialOrder.wellDefined pOrder (transitive *Commutative (Ring.timesZero R)) *Commutative (ringCanMultiplyByNegative oRing nA<0 nB<0)
0<nAnB = SetoidPartialOrder.<WellDefined pOrder (transitive *Commutative (Ring.timesZero R)) *Commutative (ringCanMultiplyByNegative oRing nA<0 nB<0)
OrderedRing.orderRespectsMultiplication (fieldOfFractionsOrderedRing {S = S} {_+_} {_*_} {R} {_<_} {pOrder} {tOrder} I oRing) {numA ,, (denomA , denomA!=0)} {numB ,, (denomB , denomB!=0)} 0<a 0<b | inl (inl 0<1) | inl (inl 0<dAdB) | inl (inr dB<0) | inr x = exFalso (denomA!=0 (Equivalence.symmetric (Setoid.eq S) x))
OrderedRing.orderRespectsMultiplication (fieldOfFractionsOrderedRing {S = S} {_+_} {_*_} {R} {_<_} {pOrder} {tOrder} I oRing) {numA ,, (denomA , denomA!=0)} {numB ,, (denomB , denomB!=0)} 0<a 0<b | inl (inl 0<1) | inl (inl 0<dAdB) | inr x = exFalso (denomB!=0 (Equivalence.symmetric (Setoid.eq S) x))
OrderedRing.orderRespectsMultiplication (fieldOfFractionsOrderedRing {S = S} {_+_} {_*_} {R} {_<_} {pOrder} {tOrder} I oRing) {numA ,, (denomA , denomA!=0)} {numB ,, (denomB , denomB!=0)} 0<a 0<b | inl (inl 0<1) | inl (inr dAdB<0) with SetoidTotalOrder.totality tOrder (Ring.0R R) denomB
@@ -659,37 +661,37 @@ module Fields.FieldOfFractionsOrder where
f : False
f with OrderedRing.orderRespectsMultiplication oRing 0<denomA 0<denomB
... | bl = SetoidPartialOrder.irreflexive pOrder (SetoidPartialOrder.transitive pOrder bl dAdB<0)
OrderedRing.orderRespectsMultiplication (fieldOfFractionsOrderedRing {S = S} {_+_} {_*_} {R} {_<_} {pOrder} {tOrder} I oRing) {numA ,, (denomA , denomA!=0)} {numB ,, (denomB , denomB!=0)} 0<a 0<b | inl (inl 0<1) | inl (inr dAdB<0) | inl (inl 0<denomB) | inl (inr denomA<0) = SetoidPartialOrder.wellDefined pOrder (symmetric (transitive *Commutative identIsIdent)) (symmetric (transitive *Commutative (Ring.timesZero R))) ans
OrderedRing.orderRespectsMultiplication (fieldOfFractionsOrderedRing {S = S} {_+_} {_*_} {R} {_<_} {pOrder} {tOrder} I oRing) {numA ,, (denomA , denomA!=0)} {numB ,, (denomB , denomB!=0)} 0<a 0<b | inl (inl 0<1) | inl (inr dAdB<0) | inl (inl 0<denomB) | inl (inr denomA<0) = SetoidPartialOrder.<WellDefined pOrder (symmetric (transitive *Commutative identIsIdent)) (symmetric (transitive *Commutative (Ring.timesZero R))) ans
where
open Setoid S
open Equivalence (Setoid.eq S)
open Ring R
0<nB : 0R < numB
0<nB = SetoidPartialOrder.wellDefined pOrder (transitive *Commutative (Ring.timesZero R)) (transitive *Commutative identIsIdent) 0<b
0<nB = SetoidPartialOrder.<WellDefined pOrder (transitive *Commutative (Ring.timesZero R)) (transitive *Commutative identIsIdent) 0<b
nA<0 : numA < 0R
nA<0 = SetoidPartialOrder.wellDefined pOrder (transitive *Commutative identIsIdent) (transitive *Commutative (Ring.timesZero R)) 0<a
nA<0 = SetoidPartialOrder.<WellDefined pOrder (transitive *Commutative identIsIdent) (transitive *Commutative (Ring.timesZero R)) 0<a
ans : (numA * numB) < 0R
ans = SetoidPartialOrder.wellDefined pOrder *Commutative (transitive *Commutative (Ring.timesZero R)) (ringCanMultiplyByNegative oRing nA<0 0<nB)
ans = SetoidPartialOrder.<WellDefined pOrder *Commutative (transitive *Commutative (Ring.timesZero R)) (ringCanMultiplyByNegative oRing nA<0 0<nB)
OrderedRing.orderRespectsMultiplication (fieldOfFractionsOrderedRing {S = S} {_+_} {_*_} {R} {_<_} {pOrder} {tOrder} I oRing) {numA ,, (denomA , denomA!=0)} {numB ,, (denomB , denomB!=0)} 0<a 0<b | inl (inl 0<1) | inl (inr dAdB<0) | inl (inl 0<denomB) | inr x = exFalso (denomA!=0 (Equivalence.symmetric (Setoid.eq S) x))
OrderedRing.orderRespectsMultiplication (fieldOfFractionsOrderedRing {S = S} {_+_} {_*_} {R} {_<_} {pOrder} {tOrder} I oRing) {numA ,, (denomA , denomA!=0)} {numB ,, (denomB , denomB!=0)} 0<a 0<b | inl (inl 0<1) | inl (inr dAdB<0) | inl (inr denomB<0) with SetoidTotalOrder.totality tOrder (Ring.0R R) denomA
OrderedRing.orderRespectsMultiplication (fieldOfFractionsOrderedRing {S = S} {_+_} {_*_} {R} {_<_} {pOrder} {tOrder} I oRing) {numA ,, (denomA , denomA!=0)} {numB ,, (denomB , denomB!=0)} 0<a 0<b | inl (inl 0<1) | inl (inr dAdB<0) | inl (inr denomB<0) | inl (inl 0<denomA) = SetoidPartialOrder.wellDefined pOrder (symmetric (transitive *Commutative identIsIdent)) (symmetric (transitive *Commutative (Ring.timesZero R))) nAnB<0
OrderedRing.orderRespectsMultiplication (fieldOfFractionsOrderedRing {S = S} {_+_} {_*_} {R} {_<_} {pOrder} {tOrder} I oRing) {numA ,, (denomA , denomA!=0)} {numB ,, (denomB , denomB!=0)} 0<a 0<b | inl (inl 0<1) | inl (inr dAdB<0) | inl (inr denomB<0) | inl (inl 0<denomA) = SetoidPartialOrder.<WellDefined pOrder (symmetric (transitive *Commutative identIsIdent)) (symmetric (transitive *Commutative (Ring.timesZero R))) nAnB<0
where
open Setoid S
open Equivalence (Setoid.eq S)
open Ring R
nB<0 : numB < 0R
nB<0 = SetoidPartialOrder.wellDefined pOrder (transitive *Commutative identIsIdent) (transitive *Commutative (Ring.timesZero R)) 0<b
nB<0 = SetoidPartialOrder.<WellDefined pOrder (transitive *Commutative identIsIdent) (transitive *Commutative (Ring.timesZero R)) 0<b
0<nA : 0R < numA
0<nA = SetoidPartialOrder.wellDefined pOrder (transitive *Commutative (Ring.timesZero R)) (transitive *Commutative identIsIdent) 0<a
0<nA = SetoidPartialOrder.<WellDefined pOrder (transitive *Commutative (Ring.timesZero R)) (transitive *Commutative identIsIdent) 0<a
nAnB<0 : (numA * numB) < 0R
nAnB<0 = SetoidPartialOrder.wellDefined pOrder reflexive (transitive *Commutative (Ring.timesZero R)) (ringCanMultiplyByNegative oRing nB<0 0<nA)
nAnB<0 = SetoidPartialOrder.<WellDefined pOrder reflexive (transitive *Commutative (Ring.timesZero R)) (ringCanMultiplyByNegative oRing nB<0 0<nA)
OrderedRing.orderRespectsMultiplication (fieldOfFractionsOrderedRing {S = S} {_+_} {_*_} {R} {_<_} {pOrder} {tOrder} I oRing) {numA ,, (denomA , denomA!=0)} {numB ,, (denomB , denomB!=0)} 0<a 0<b | inl (inl 0<1) | inl (inr dAdB<0) | inl (inr denomB<0) | inl (inr denomA<0) = exFalso f
where
open Setoid S
open Equivalence (Setoid.eq S)
open Ring R
h : 0R < (denomA * denomB)
h = SetoidPartialOrder.wellDefined pOrder (transitive *Commutative (Ring.timesZero R)) reflexive (ringCanMultiplyByNegative oRing denomB<0 denomA<0)
h = SetoidPartialOrder.<WellDefined pOrder (transitive *Commutative (Ring.timesZero R)) reflexive (ringCanMultiplyByNegative oRing denomB<0 denomA<0)
f : False
f = SetoidPartialOrder.irreflexive pOrder (SetoidPartialOrder.transitive pOrder dAdB<0 h)
OrderedRing.orderRespectsMultiplication (fieldOfFractionsOrderedRing {S = S} {_+_} {_*_} {R} {_<_} {pOrder} {tOrder} I oRing) {numA ,, (denomA , denomA!=0)} {numB ,, (denomB , denomB!=0)} 0<a 0<b | inl (inl 0<1) | inl (inr dAdB<0) | inl (inr denomB<0) | inr x = exFalso (denomA!=0 (Equivalence.symmetric (Setoid.eq S) x))
@@ -697,7 +699,7 @@ module Fields.FieldOfFractionsOrder where
OrderedRing.orderRespectsMultiplication (fieldOfFractionsOrderedRing {S = S} {_+_} {_*_} {R} {_<_} {pOrder} {tOrder} I oRing) {numA ,, (denomA , denomA!=0)} {numB ,, (denomB , denomB!=0)} 0<a 0<b | inl (inl 0<1) | inr 0=dAdB with IntegralDomain.intDom I (Equivalence.symmetric (Setoid.eq S) 0=dAdB)
... | inl x = exFalso (denomA!=0 x)
... | inr x = exFalso (denomB!=0 x)
OrderedRing.orderRespectsMultiplication (fieldOfFractionsOrderedRing {S = S} {_+_} {_*_} {R} {_<_} {pOrder} {tOrder} I oRing) {numA ,, (denomA , denomA!=0)} {numB ,, (denomB , denomB!=0)} 0<a 0<b | inl (inr 1<0) = exFalso (SetoidPartialOrder.irreflexive pOrder (SetoidPartialOrder.transitive pOrder 1<0 (SetoidPartialOrder.wellDefined pOrder (transitive *Commutative (Ring.timesZero R)) identIsIdent (ringCanMultiplyByNegative oRing 1<0 1<0))))
OrderedRing.orderRespectsMultiplication (fieldOfFractionsOrderedRing {S = S} {_+_} {_*_} {R} {_<_} {pOrder} {tOrder} I oRing) {numA ,, (denomA , denomA!=0)} {numB ,, (denomB , denomB!=0)} 0<a 0<b | inl (inr 1<0) = exFalso (SetoidPartialOrder.irreflexive pOrder (SetoidPartialOrder.transitive pOrder 1<0 (SetoidPartialOrder.<WellDefined pOrder (transitive *Commutative (Ring.timesZero R)) identIsIdent (ringCanMultiplyByNegative oRing 1<0 1<0))))
where
open Setoid S
open Equivalence (Setoid.eq S)

5
Fields/Fields.agda
View File

@@ -4,6 +4,7 @@ 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
@@ -31,7 +32,7 @@ module Fields.Fields where
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)
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)
@@ -47,7 +48,7 @@ module Fields.Fields where
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)
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)

1
Numbers/Integers/Order.agda
View File

@@ -7,6 +7,7 @@ open import Numbers.Integers.Addition
open import Numbers.Integers.Multiplication
open import Semirings.Definition
open import Rings.Definition
open import Rings.Order
open import Setoids.Setoids
open import Setoids.Orders
open import Orders

6
Numbers/Naturals/Naturals.agda
View File

@@ -554,3 +554,9 @@ module Numbers.Naturals.Naturals where
sumZeroImpliesOperandsZero : (a : ℕ) {b : ℕ} → a +N b ≡ 0 → (a ≡ 0) && (b ≡ 0)
sumZeroImpliesOperandsZero zero {zero} pr = refl ,, refl
inequalityShrinkRight : {a b c : ℕ} → a +N b <N c → b <N c
inequalityShrinkRight {a} {b} {c} (le x proof) = le (x +N a) (transitivity (applyEquality succ (additionNIsAssociative x a b)) proof)
inequalityShrinkLeft : {a b c : ℕ} → a +N b <N c → a <N c
inequalityShrinkLeft {a} {b} {c} (le x proof) = le (x +N b) (transitivity (applyEquality succ (transitivity (additionNIsAssociative x b a) (applyEquality (x +N_) (additionNIsCommutative b a)))) proof)

75
Numbers/Rationals.agda
View File

@@ -6,6 +6,7 @@ open import Numbers.Integers.Integers
open import Groups.Groups
open import Groups.Definition
open import Rings.Definition
open import Rings.Order
open import Fields.Fields
open import Numbers.Primes.PrimeNumbers
open import Setoids.Setoids
@@ -17,53 +18,57 @@ open import Sets.EquivalenceRelations
module Numbers.Rationals where
ℚ : Set
ℚ = fieldOfFractionsSet ℤIntDom
ℚ : Set
ℚ = fieldOfFractionsSet ℤIntDom
_+Q_ : ℚ → ℚ → ℚ
a +Q b = fieldOfFractionsPlus ℤIntDom a b
_+Q_ : ℚ → ℚ → ℚ
a +Q b = fieldOfFractionsPlus ℤIntDom a b
_*Q_ : ℚ → ℚ → ℚ
a *Q b = fieldOfFractionsTimes ℤIntDom a b
_*Q_ : ℚ → ℚ → ℚ
a *Q b = fieldOfFractionsTimes ℤIntDom a b
ℚRing : Ring (fieldOfFractionsSetoid ℤIntDom) _+Q_ _*Q_
ℚRing = fieldOfFractionsRing ℤIntDom
ℚRing : Ring (fieldOfFractionsSetoid ℤIntDom) _+Q_ _*Q_
ℚRing = fieldOfFractionsRing ℤIntDom
0Q : ℚ
0Q = Ring.0R ℚRing
0Q : ℚ
0Q = Ring.0R ℚRing
ℚField : Field ℚRing
ℚField = fieldOfFractions ℤIntDom
ℚField : Field ℚRing
ℚField = fieldOfFractions ℤIntDom
_<Q_ : ℚ → ℚ → Set
_<Q_ = fieldOfFractionsComparison ℤIntDom ℤOrderedRing
_<Q_ : ℚ → ℚ → Set
_<Q_ = fieldOfFractionsComparison ℤIntDom ℤOrderedRing
_=Q_ : ℚ → ℚ → Set
a =Q b = Setoid._∼_ (fieldOfFractionsSetoid ℤIntDom) a b
_=Q_ : ℚ → ℚ → Set
a =Q b = Setoid._∼_ (fieldOfFractionsSetoid ℤIntDom) a b
reflQ : {x : ℚ} → (x =Q x)
reflQ {x} = Equivalence.reflexive (Setoid.eq (fieldOfFractionsSetoid ℤIntDom)) {x}
reflQ : {x : ℚ} → (x =Q x)
reflQ {x} = Equivalence.reflexive (Setoid.eq (fieldOfFractionsSetoid ℤIntDom)) {x}
_≤Q_ : ℚ → ℚ → Set
a ≤Q b = (a <Q b) || (a =Q b)
_≤Q_ : ℚ → ℚ → Set
a ≤Q b = (a <Q b) || (a =Q b)
negateQ : ℚ → ℚ
negateQ a = Group.inverse (Ring.additiveGroup ℚRing) a
negateQ : ℚ → ℚ
negateQ a = Group.inverse (Ring.additiveGroup ℚRing) a
_-Q_ : ℚ → ℚ → ℚ
a -Q b = a +Q negateQ b
_-Q_ : ℚ → ℚ → ℚ
a -Q b = a +Q negateQ b
ℚPartialOrder : SetoidPartialOrder (fieldOfFractionsSetoid ℤIntDom) (fieldOfFractionsComparison ℤIntDom ℤOrderedRing)
ℚPartialOrder = fieldOfFractionsOrder ℤIntDom ℤOrderedRing
a-A : (a : ℚ) → (a -Q a) =Q 0Q
a-A a = Group.invRight (Ring.additiveGroup ℚRing) {a}
ℚTotalOrder : SetoidTotalOrder (fieldOfFractionsOrder ℤIntDom ℤOrderedRing)
ℚTotalOrder = fieldOfFractionsTotalOrder ℤIntDom ℤOrderedRing
ℚPartialOrder : SetoidPartialOrder (fieldOfFractionsSetoid ℤIntDom) (fieldOfFractionsComparison ℤIntDom ℤOrderedRing)
ℚPartialOrder = fieldOfFractionsOrder ℤIntDom ℤOrderedRing
absQ : ℚ → ℚ
absQ q with SetoidTotalOrder.totality (fieldOfFractionsTotalOrder ℤIntDom ℤOrderedRing) 0Q q
absQ q | inl (inl 0<q) = q
absQ q | inl (inr q<0) = Group.inverse (Ring.additiveGroup ℚRing) q
absQ q | inr x = 0Q
ℚTotalOrder : SetoidTotalOrder (fieldOfFractionsOrder ℤIntDom ℤOrderedRing)
ℚTotalOrder = fieldOfFractionsTotalOrder ℤIntDom ℤOrderedRing
ℚOrdered : OrderedRing ℚRing (fieldOfFractionsTotalOrder ℤIntDom ℤOrderedRing)
ℚOrdered = fieldOfFractionsOrderedRing ℤIntDom ℤOrderedRing
open SetoidTotalOrder (fieldOfFractionsTotalOrder ℤIntDom ℤOrderedRing)
open SetoidPartialOrder partial
open Setoid (fieldOfFractionsSetoid ℤIntDom)
negateQWellDefined : (a b : ℚ) → (a =Q b) → (negateQ a) =Q (negateQ b)
negateQWellDefined a b a=b = inverseWellDefined (Ring.additiveGroup ℚRing) {a} {b} a=b
ℚOrdered : OrderedRing ℚRing (fieldOfFractionsTotalOrder ℤIntDom ℤOrderedRing)
ℚOrdered = fieldOfFractionsOrderedRing ℤIntDom ℤOrderedRing

209
Numbers/RationalsLemmas.agda
View File

@@ -1,209 +0,0 @@
{-# OPTIONS --safe --warning=error #-}
open import LogicalFormulae
open import Numbers.Naturals.Naturals
open import Numbers.Integers.Integers
open import Numbers.Rationals
open import Groups.Groups
open import Groups.Lemmas
open import Groups.Definition
open import Rings.Definition
open import Fields.Fields
open import Numbers.Primes.PrimeNumbers
open import Setoids.Setoids
open import Setoids.Orders
open import Functions
open import Fields.FieldOfFractions
open import Fields.FieldOfFractionsOrder
open import Rings.IntegralDomains
open import Rings.Lemmas
open import Sets.EquivalenceRelations
module Numbers.RationalsLemmas where
triangleInequality : {a b : ℚ} → (absQ (a +Q b)) ≤Q ((absQ a) +Q (absQ b))
triangleInequality {a} {b} with SetoidTotalOrder.totality ℚTotalOrder 0Q a
triangleInequality {a} {b} | inl (inl 0<a) with SetoidTotalOrder.totality ℚTotalOrder 0Q b
triangleInequality {a} {b} | inl (inl 0<a) | inl (inl 0<b) with SetoidTotalOrder.totality ℚTotalOrder 0Q (a +Q b)
triangleInequality {a} {b} | inl (inl 0<a) | inl (inl 0<b) | inl (inl 0<a+b) = inr (reflexive {a +Q b})
where
open Equivalence (Setoid.eq (fieldOfFractionsSetoid ℤIntDom))
triangleInequality {a} {b} | inl (inl 0<a) | inl (inl 0<b) | inl (inr a+b<0) = exFalso (exFalso (SetoidPartialOrder.irreflexive (fieldOfFractionsOrder ℤIntDom ℤOrderedRing) {a +Q b} (SetoidPartialOrder.transitive (fieldOfFractionsOrder ℤIntDom ℤOrderedRing) {a +Q b} {0Q} {a +Q b} a+b<0 (SetoidPartialOrder.wellDefined (fieldOfFractionsOrder ℤIntDom ℤOrderedRing) {0Q +Q 0Q} {0Q} {a +Q b} {a +Q b} (symmetric {0Q} {0Q +Q 0Q} (Group.identRight (Ring.additiveGroup ℚRing) {0Q})) (reflexive {a +Q b}) (ringAddInequalities ℚOrdered {0Q} {a} {0Q} {b} 0<a 0<b)))))
where
open Group (Ring.additiveGroup ℚRing)
open Setoid (fieldOfFractionsSetoid ℤIntDom)
open Equivalence eq
triangleInequality {a} {b} | inl (inl 0<a) | inl (inl 0<b) | inr 0=a+b = exFalso (SetoidPartialOrder.irreflexive (fieldOfFractionsOrder ℤIntDom ℤOrderedRing) {0Q} (SetoidPartialOrder.wellDefined (fieldOfFractionsOrder ℤIntDom ℤOrderedRing) {0Q +Q 0Q} {0Q} {a +Q b} {0Q} (identRight {0Q}) (symmetric {0Q} {a +Q b} 0=a+b) (ringAddInequalities ℚOrdered {0Q} {a} {0Q} {b} 0<a 0<b)))
where
open Group (Ring.additiveGroup ℚRing)
open Setoid (fieldOfFractionsSetoid ℤIntDom)
open Equivalence eq
triangleInequality {a} {b} | inl (inl 0<a) | inl (inr b<0) with SetoidTotalOrder.totality ℚTotalOrder 0Q (a +Q b)
triangleInequality {a} {b} | inl (inl 0<a) | inl (inr b<0) | inl (inl 0<a+b) = inl (SetoidPartialOrder.wellDefined (fieldOfFractionsOrder ℤIntDom ℤOrderedRing) {b +Q a} {a +Q b} {inverse b +Q a} {a +Q inverse b} (Ring.groupIsAbelian ℚRing {b} {a}) (Ring.groupIsAbelian ℚRing {inverse b} {a}) (OrderedRing.orderRespectsAddition ℚOrdered {b} {inverse b} (SetoidPartialOrder.transitive (fieldOfFractionsOrder ℤIntDom ℤOrderedRing) {b} {0Q} {inverse b} b<0 (ringMinusFlipsOrder'' ℚOrdered {b} b<0)) a))
where
open Group (Ring.additiveGroup ℚRing)
open Setoid (fieldOfFractionsSetoid ℤIntDom)
open Equivalence eq
triangleInequality {a} {b} | inl (inl 0<a) | inl (inr b<0) | inl (inr a+b<0) = inl (SetoidPartialOrder.wellDefined (fieldOfFractionsOrder ℤIntDom ℤOrderedRing) {inverse a +Q inverse b} {inverse (a +Q b)} {a +Q inverse b} {a +Q inverse b} (transitive {inverse a +Q inverse b} {inverse b +Q inverse a} {inverse (a +Q b)} (Ring.groupIsAbelian ℚRing {inverse a} {inverse b}) (symmetric {inverse (a +Q b)} {inverse b +Q inverse a} (invContravariant (Ring.additiveGroup ℚRing) {a} {b}))) (reflexive {a +Q inverse b}) (OrderedRing.orderRespectsAddition ℚOrdered {inverse a} {a} (SetoidPartialOrder.transitive (fieldOfFractionsOrder ℤIntDom ℤOrderedRing) {inverse a} {0Q} {a} (ringMinusFlipsOrder ℚOrdered {a} 0<a) 0<a) (inverse b)))
where
open Group (Ring.additiveGroup ℚRing)
open Setoid (fieldOfFractionsSetoid ℤIntDom)
open Equivalence eq
triangleInequality {a} {b} | inl (inl 0<a) | inl (inr b<0) | inr 0=a+b = inl (SetoidPartialOrder.transitive (fieldOfFractionsOrder ℤIntDom ℤOrderedRing) {0Q} {a} {a +Q (inverse b)} 0<a (SetoidPartialOrder.wellDefined (fieldOfFractionsOrder ℤIntDom ℤOrderedRing) {0Q +Q a} {a} {inverse b +Q a} {a +Q inverse b} (identLeft {a}) (Ring.groupIsAbelian ℚRing {inverse b} {a}) (OrderedRing.orderRespectsAddition ℚOrdered {0Q} {inverse b} (ringMinusFlipsOrder'' ℚOrdered {b} b<0) a)))
where
open Group (Ring.additiveGroup ℚRing)
open Setoid (fieldOfFractionsSetoid ℤIntDom)
open Equivalence eq
triangleInequality {a} {b} | inl (inl 0<a) | inr 0=b with SetoidTotalOrder.totality ℚTotalOrder 0Q (a +Q b)
triangleInequality {a} {b} | inl (inl 0<a) | inr 0=b | inl (inl 0<a+b) = inr (+WellDefined {a} {b} {a} {0Q} (reflexive {a}) (symmetric {0Q} {b} 0=b))
where
open Group (Ring.additiveGroup ℚRing)
open Setoid (fieldOfFractionsSetoid ℤIntDom)
open Equivalence eq
triangleInequality {a} {b} | inl (inl 0<a) | inr 0=b | inl (inr a+b<0) = exFalso (SetoidPartialOrder.irreflexive (fieldOfFractionsOrder ℤIntDom ℤOrderedRing) {a +Q b} (SetoidPartialOrder.transitive (fieldOfFractionsOrder ℤIntDom ℤOrderedRing) {a +Q b} {0Q} {a +Q b} a+b<0 (SetoidPartialOrder.wellDefined (fieldOfFractionsOrder ℤIntDom ℤOrderedRing) {0Q} {0Q} {a} {a +Q b} (reflexive {0Q}) (transitive {a} {a +Q 0Q} {a +Q b} (symmetric {a +Q 0Q} {a} (identRight {a})) (+WellDefined {a} {0Q} {a} {b} (reflexive {a}) 0=b)) 0<a)))
where
open Group (Ring.additiveGroup ℚRing)
open Setoid (fieldOfFractionsSetoid ℤIntDom)
open Equivalence eq
triangleInequality {a} {b} | inl (inl 0<a) | inr 0=b | inr 0=a+b = exFalso (SetoidPartialOrder.irreflexive (fieldOfFractionsOrder ℤIntDom ℤOrderedRing) {a +Q b} (SetoidPartialOrder.wellDefined (fieldOfFractionsOrder ℤIntDom ℤOrderedRing) {0Q} {a +Q b} {a} {a +Q b} 0=a+b (transitive {a} {a +Q 0Q} {a +Q b} (symmetric {a +Q 0Q} {a} (identRight {a})) (+WellDefined {a} {0Q} {a} {b} (reflexive {a}) 0=b)) 0<a))
where
open Group (Ring.additiveGroup ℚRing)
open Setoid (fieldOfFractionsSetoid ℤIntDom)
open Equivalence eq
triangleInequality {a} {b} | inl (inr a<0) with SetoidTotalOrder.totality ℚTotalOrder 0Q b
triangleInequality {a} {b} | inl (inr a<0) | inl (inl 0<b) with SetoidTotalOrder.totality ℚTotalOrder 0Q (a +Q b)
triangleInequality {a} {b} | inl (inr a<0) | inl (inl 0<b) | inl (inl 0<a+b) = inl (OrderedRing.orderRespectsAddition ℚOrdered {a} {inverse a} (SetoidPartialOrder.transitive (fieldOfFractionsOrder ℤIntDom ℤOrderedRing) {a} {0Q} {inverse a} a<0 (ringMinusFlipsOrder'' ℚOrdered {a} a<0)) b)
where
open Group (Ring.additiveGroup ℚRing)
open Setoid (fieldOfFractionsSetoid ℤIntDom)
open Equivalence eq
triangleInequality {a} {b} | inl (inr a<0) | inl (inl 0<b) | inl (inr a+b<0) = inl (SetoidPartialOrder.wellDefined (fieldOfFractionsOrder ℤIntDom ℤOrderedRing) {inverse a +Q inverse b} {inverse (a +Q b)} {inverse a +Q b} {inverse a +Q b} (transitive {inverse a +Q inverse b} {inverse b +Q inverse a} {inverse (a +Q b)} (Ring.groupIsAbelian ℚRing {inverse a} {inverse b}) (symmetric {inverse (a +Q b)} {inverse b +Q inverse a} (invContravariant (Ring.additiveGroup ℚRing) {a} {b}))) (reflexive {inverse a +Q b}) blah)
where
open Group (Ring.additiveGroup ℚRing)
open Setoid (fieldOfFractionsSetoid ℤIntDom)
open Equivalence eq
blah : (inverse a +Q inverse b) <Q (inverse a +Q b)
blah = SetoidPartialOrder.wellDefined (fieldOfFractionsOrder ℤIntDom ℤOrderedRing) {inverse b +Q inverse a} {inverse a +Q inverse b} {b +Q inverse a} {inverse a +Q b} (Ring.groupIsAbelian ℚRing {inverse b} {inverse a}) (Ring.groupIsAbelian ℚRing {b} {inverse a}) (OrderedRing.orderRespectsAddition ℚOrdered {inverse b} {b} (SetoidPartialOrder.transitive (fieldOfFractionsOrder ℤIntDom ℤOrderedRing) {inverse b} {0Q} {b} (ringMinusFlipsOrder ℚOrdered {b} 0<b) 0<b) (inverse a))
triangleInequality {a} {b} | inl (inr a<0) | inl (inl 0<b) | inr 0=a+b = inl (SetoidPartialOrder.transitive (fieldOfFractionsOrder ℤIntDom ℤOrderedRing) {0Q} {b} {inverse a +Q b} 0<b (SetoidPartialOrder.wellDefined (fieldOfFractionsOrder ℤIntDom ℤOrderedRing) {b +Q 0Q} {b} {b +Q inverse a} {inverse a +Q b} (identRight {b}) (Ring.groupIsAbelian ℚRing {b} {inverse a}) (SetoidPartialOrder.wellDefined (fieldOfFractionsOrder ℤIntDom ℤOrderedRing) {0Q +Q b} {b +Q 0Q} {inverse a +Q b} {b +Q inverse a} (Ring.groupIsAbelian ℚRing {0Q} {b}) (Ring.groupIsAbelian ℚRing {inverse a} {b}) (OrderedRing.orderRespectsAddition ℚOrdered {0Q} {inverse a} (ringMinusFlipsOrder'' ℚOrdered {a} a<0) b))))
where
open Group (Ring.additiveGroup ℚRing)
open Setoid (fieldOfFractionsSetoid ℤIntDom)
open Equivalence eq
triangleInequality {a} {b} | inl (inr a<0) | inl (inr b<0) with SetoidTotalOrder.totality ℚTotalOrder 0Q (a +Q b)
triangleInequality {a} {b} | inl (inr a<0) | inl (inr b<0) | inl (inl 0<a+b) = exFalso (SetoidPartialOrder.irreflexive (fieldOfFractionsOrder ℤIntDom ℤOrderedRing) {a +Q b} (SetoidPartialOrder.transitive (fieldOfFractionsOrder ℤIntDom ℤOrderedRing) {a +Q b} {0Q} {a +Q b} (SetoidPartialOrder.wellDefined (fieldOfFractionsOrder ℤIntDom ℤOrderedRing) {a +Q b} {a +Q b} {0Q +Q 0Q} {0Q} (reflexive {a +Q b}) (identRight {0Q}) (ringAddInequalities ℚOrdered {a} {0Q} {b} {0Q} a<0 b<0)) 0<a+b))
where
open Group (Ring.additiveGroup ℚRing)
open Setoid (fieldOfFractionsSetoid ℤIntDom)
open Equivalence eq
triangleInequality {a} {b} | inl (inr a<0) | inl (inr b<0) | inl (inr a+b<0) = inr (transitive {inverse (fieldOfFractionsPlus ℤIntDom a b)} {fieldOfFractionsPlus ℤIntDom (inverse b) (inverse a)} {fieldOfFractionsPlus ℤIntDom (inverse a) (inverse b)} (invContravariant (Ring.additiveGroup ℚRing) {a} {b}) (Ring.groupIsAbelian ℚRing {inverse b} {inverse a}))
where
open Group (Ring.additiveGroup ℚRing)
open Setoid (fieldOfFractionsSetoid ℤIntDom)
open Equivalence eq
triangleInequality {a} {b} | inl (inr a<0) | inl (inr b<0) | inr 0=a+b = exFalso (SetoidPartialOrder.irreflexive (fieldOfFractionsOrder ℤIntDom ℤOrderedRing) {0Q} (SetoidPartialOrder.wellDefined (fieldOfFractionsOrder ℤIntDom ℤOrderedRing) {a +Q b} {0Q} {0Q} {0Q} (symmetric {0Q} {a +Q b} 0=a+b) (reflexive {0Q}) (SetoidPartialOrder.wellDefined (fieldOfFractionsOrder ℤIntDom ℤOrderedRing) {a +Q b} {a +Q b} {0Q +Q 0Q} {0Q} (reflexive {a +Q b}) (identLeft {0Q}) (ringAddInequalities ℚOrdered {a} {0Q} {b} {0Q} a<0 b<0))))
where
open Group (Ring.additiveGroup ℚRing)
open Setoid (fieldOfFractionsSetoid ℤIntDom)
open Equivalence eq
triangleInequality {a} {b} | inl (inr a<0) | inr 0=b with SetoidTotalOrder.totality ℚTotalOrder 0Q (a +Q b)
triangleInequality {a} {b} | inl (inr a<0) | inr 0=b | inl (inl 0<a+b) = exFalso (SetoidPartialOrder.irreflexive (fieldOfFractionsOrder ℤIntDom ℤOrderedRing) {a} (SetoidPartialOrder.transitive (fieldOfFractionsOrder ℤIntDom ℤOrderedRing) {a} {0Q} {a} a<0 (SetoidPartialOrder.wellDefined (fieldOfFractionsOrder ℤIntDom ℤOrderedRing) {0Q} {0Q} {a +Q b} {a} (reflexive {0Q}) (transitive {a +Q b} {a +Q 0Q} {a} (+WellDefined {a} {b} {a} {0Q} (reflexive {a}) (symmetric {0Q} {b} 0=b)) (identRight {a})) 0<a+b)))
where
open Group (Ring.additiveGroup ℚRing)
open Setoid (fieldOfFractionsSetoid ℤIntDom)
open Equivalence eq
triangleInequality {a} {b} | inl (inr a<0) | inr 0=b | inl (inr a+b<0) = inr (transitive {inverse (a +Q b)} {(inverse b) +Q (inverse a)} {(inverse a) +Q 0Q} (invContravariant (Ring.additiveGroup ℚRing) {a} {b}) (transitive {inverse b +Q inverse a} {inverse a +Q inverse b} {inverse a +Q 0Q} (Ring.groupIsAbelian ℚRing {inverse b} {inverse a}) (+WellDefined {inverse a} {inverse b} {inverse a} {0Q} (reflexive {inverse a}) (transitive {inverse b} {inverse 0Q} {0Q} (inverseWellDefined (Ring.additiveGroup ℚRing) {b} {0Q} (symmetric {0Q} {b} 0=b)) (symmetric {0Q} {inverse 0Q} (invIdentity (Ring.additiveGroup ℚRing)))))))
where
open Group (Ring.additiveGroup ℚRing)
open Setoid (fieldOfFractionsSetoid ℤIntDom)
open Equivalence eq
triangleInequality {a} {b} | inl (inr a<0) | inr 0=b | inr 0=a+b = exFalso (SetoidPartialOrder.irreflexive (fieldOfFractionsOrder ℤIntDom ℤOrderedRing) {0Q} (SetoidPartialOrder.wellDefined (fieldOfFractionsOrder ℤIntDom ℤOrderedRing) {a} {0Q} {0Q} {0Q} (transitive {a} {a +Q b} {0Q} (transitive {a} {a +Q 0Q} {a +Q b} (symmetric {a +Q 0Q} {a} (identRight {a})) (+WellDefined {a} {0Q} {a} {b} (reflexive {a}) 0=b)) (symmetric {0Q} {a +Q b} 0=a+b)) (reflexive {0Q}) a<0))
where
open Group (Ring.additiveGroup ℚRing)
open Setoid (fieldOfFractionsSetoid ℤIntDom)
open Equivalence eq
triangleInequality {a} {b} | inr 0=a with SetoidTotalOrder.totality ℚTotalOrder 0Q b
triangleInequality {a} {b} | inr 0=a | inl (inl 0<b) with SetoidTotalOrder.totality ℚTotalOrder 0Q (a +Q b)
triangleInequality {a} {b} | inr 0=a | inl (inl 0<b) | inl (inl 0<a+b) = inr (Group.+WellDefined (Ring.additiveGroup ℚRing) {a} {b} {0Q} {b} (symmetric {0Q} {a} 0=a) (reflexive {b}))
where
open Setoid (fieldOfFractionsSetoid ℤIntDom)
open Equivalence eq
triangleInequality {a} {b} | inr 0=a | inl (inl 0<b) | inl (inr a+b<0) = exFalso (SetoidPartialOrder.irreflexive (fieldOfFractionsOrder ℤIntDom ℤOrderedRing) {a +Q b} (SetoidPartialOrder.transitive (fieldOfFractionsOrder ℤIntDom ℤOrderedRing) {a +Q b} {0Q} {a +Q b} a+b<0 (SetoidPartialOrder.wellDefined (fieldOfFractionsOrder ℤIntDom ℤOrderedRing) {0Q} {0Q} {b} {a +Q b} (reflexive {0Q}) (transitive {b} {0Q +Q b} {a +Q b} (symmetric {0Q +Q b} {b} (Group.identLeft (Ring.additiveGroup ℚRing) {b})) (Group.+WellDefined (Ring.additiveGroup ℚRing) {0Q} {b} {a} {b} 0=a (reflexive {b}))) 0<b)))
where
open Setoid (fieldOfFractionsSetoid ℤIntDom)
open Equivalence eq
triangleInequality {a} {b} | inr 0=a | inl (inl 0<b) | inr 0=a+b = exFalso (SetoidPartialOrder.irreflexive (fieldOfFractionsOrder ℤIntDom ℤOrderedRing) {b} (SetoidPartialOrder.wellDefined (fieldOfFractionsOrder ℤIntDom ℤOrderedRing) {0Q} {b} {b} {b} (transitive {0Q} {a +Q b} {b} 0=a+b (transitive {a +Q b} {0Q +Q b} {b} (+WellDefined {a} {b} {0Q} {b} (symmetric {0Q} {a} 0=a) (reflexive {b})) (identLeft {b}))) (reflexive {b}) 0<b))
where
open Group (Ring.additiveGroup ℚRing)
open Setoid (fieldOfFractionsSetoid ℤIntDom)
open Equivalence eq
triangleInequality {a} {b} | inr 0=a | inl (inr b<0) with SetoidTotalOrder.totality ℚTotalOrder 0Q (a +Q b)
triangleInequality {a} {b} | inr 0=a | inl (inr b<0) | inl (inl 0<a+b) = exFalso (SetoidPartialOrder.irreflexive (fieldOfFractionsOrder ℤIntDom ℤOrderedRing) {b} (SetoidPartialOrder.transitive (fieldOfFractionsOrder ℤIntDom ℤOrderedRing) {b} {0Q} {b} b<0 (SetoidPartialOrder.wellDefined (fieldOfFractionsOrder ℤIntDom ℤOrderedRing) {0Q} {0Q} {a +Q b} {b} (reflexive {0Q}) (transitive {a +Q b} {0Q +Q b} {b} (+WellDefined {a} {b} {0Q} {b} (symmetric {0Q} {a} 0=a) (reflexive {b})) (identLeft {b})) 0<a+b)))
where
open Group (Ring.additiveGroup ℚRing)
open Setoid (fieldOfFractionsSetoid ℤIntDom)
open Equivalence eq
triangleInequality {a} {b} | inr 0=a | inl (inr b<0) | inl (inr a+b<0) = inr (transitive {inverse (a +Q b)} {inverse b +Q inverse a} {0Q +Q inverse b} (invContravariant (Ring.additiveGroup ℚRing) {a} {b}) (transitive {inverse b +Q inverse a} {inverse a +Q inverse b} {0Q +Q inverse b} (Ring.groupIsAbelian ℚRing {inverse b} {inverse a}) (+WellDefined {inverse a} {inverse b} {0Q} {inverse b} (transitive {inverse a} {inverse 0Q} {0Q} (symmetric {inverse 0Q} {inverse a} (inverseWellDefined (Ring.additiveGroup ℚRing) {0Q} {a} 0=a)) (invIdentity (Ring.additiveGroup ℚRing))) (reflexive {inverse b}))))
where
open Group (Ring.additiveGroup ℚRing)
open Setoid (fieldOfFractionsSetoid ℤIntDom)
open Equivalence eq
triangleInequality {a} {b} | inr 0=a | inl (inr b<0) | inr 0=a+b = exFalso (SetoidPartialOrder.irreflexive (fieldOfFractionsOrder ℤIntDom ℤOrderedRing) {b} (SetoidPartialOrder.wellDefined (fieldOfFractionsOrder ℤIntDom ℤOrderedRing) {b} {b} {0Q} {b} (reflexive {b}) (transitive {0Q} {a +Q b} {b} 0=a+b (transitive {a +Q b} {0Q +Q b} {b} (+WellDefined {a} {b} {0Q} {b} (symmetric {0Q} {a} 0=a) (reflexive {b})) (identLeft {b}))) b<0))
where
open Group (Ring.additiveGroup ℚRing)
open Setoid (fieldOfFractionsSetoid ℤIntDom)
open Equivalence eq
triangleInequality {a} {b} | inr 0=a | inr 0=b with SetoidTotalOrder.totality ℚTotalOrder 0Q (a +Q b)
triangleInequality {a} {b} | inr 0=a | inr 0=b | inl (inl 0<a+b) = exFalso (SetoidPartialOrder.irreflexive (fieldOfFractionsOrder ℤIntDom ℤOrderedRing) {0Q} (SetoidPartialOrder.wellDefined (fieldOfFractionsOrder ℤIntDom ℤOrderedRing) {0Q} {0Q} {a +Q b} {0Q} (reflexive {0Q}) (transitive {a +Q b} {0Q +Q 0Q} {0Q} (+WellDefined {a} {b} {0Q} {0Q} (symmetric {0Q} {a} 0=a) (symmetric {0Q} {b} 0=b)) (identRight {0Q})) 0<a+b))
where
open Group (Ring.additiveGroup ℚRing)
open Setoid (fieldOfFractionsSetoid ℤIntDom)
open Equivalence eq
triangleInequality {a} {b} | inr 0=a | inr 0=b | inl (inr a+b<0) = exFalso (SetoidPartialOrder.irreflexive (fieldOfFractionsOrder ℤIntDom ℤOrderedRing) {0Q} (SetoidPartialOrder.wellDefined (fieldOfFractionsOrder ℤIntDom ℤOrderedRing) {a +Q b} {0Q} {0Q} {0Q} (transitive {a +Q b} {0Q +Q 0Q} {0Q} (+WellDefined {a} {b} {0Q} {0Q} (symmetric {0Q} {a} 0=a) (symmetric {0Q} {b} 0=b)) (identRight {0Q})) (reflexive {0Q}) a+b<0))
where
open Group (Ring.additiveGroup ℚRing)
open Setoid (fieldOfFractionsSetoid ℤIntDom)
open Equivalence eq
triangleInequality {a} {b} | inr 0=a | inr 0=b | inr 0=a+b = inr refl
absNegation : (a : ℚ) → ((absQ a) =Q (absQ (negateQ a)))
absNegation a with SetoidTotalOrder.totality ℚTotalOrder 0Q a
absNegation a | inl (inl 0<a) with SetoidTotalOrder.totality ℚTotalOrder 0Q (negateQ a)
absNegation a | inl (inl 0<a) | inl (inl 0<-a) = exFalso (SetoidPartialOrder.irreflexive (fieldOfFractionsOrder ℤIntDom ℤOrderedRing) {a} (SetoidPartialOrder.transitive (fieldOfFractionsOrder ℤIntDom ℤOrderedRing) {a} {0Q} {a} (ringMinusFlipsOrder''' ℚOrdered {a} 0<-a) 0<a))
where
open Group (Ring.additiveGroup ℚRing)
open Setoid (fieldOfFractionsSetoid ℤIntDom)
open Equivalence eq
absNegation a | inl (inl 0<a) | inl (inr _) = symmetric {inverse (inverse a)} {a} (invTwice (Ring.additiveGroup ℚRing) a)
where
open Group (Ring.additiveGroup ℚRing)
open Setoid (fieldOfFractionsSetoid ℤIntDom)
open Equivalence eq
absNegation a | inl (inl 0<a) | inr 0=-a = exFalso (SetoidPartialOrder.irreflexive (fieldOfFractionsOrder ℤIntDom ℤOrderedRing) {0Q} (SetoidPartialOrder.wellDefined (fieldOfFractionsOrder ℤIntDom ℤOrderedRing) {0Q} {0Q} {a} {0Q} (reflexive {0Q}) (transitive {a} {inverse 0Q} {0Q} (swapInv (Ring.additiveGroup ℚRing) {a} {0Q} (symmetric {0Q} {inverse a} 0=-a)) (invIdentity (Ring.additiveGroup ℚRing))) 0<a))
where
open Group (Ring.additiveGroup ℚRing)
open Setoid (fieldOfFractionsSetoid ℤIntDom)
open Equivalence eq
absNegation a | inl (inr a<0) with SetoidTotalOrder.totality ℚTotalOrder 0Q (negateQ a)
absNegation a | inl (inr a<0) | inl (inl _) = Equivalence.reflexive (Setoid.eq (fieldOfFractionsSetoid ℤIntDom)) {Group.inverse (Ring.additiveGroup ℚRing) a}
absNegation a | inl (inr a<0) | inl (inr -a<0) = exFalso (SetoidPartialOrder.irreflexive (fieldOfFractionsOrder ℤIntDom ℤOrderedRing) {0Q} (SetoidPartialOrder.transitive (fieldOfFractionsOrder ℤIntDom ℤOrderedRing) {0Q} {a} {0Q} (ringMinusFlipsOrder' ℚOrdered {a} -a<0) a<0))
where
open Group (Ring.additiveGroup ℚRing)
open Setoid (fieldOfFractionsSetoid ℤIntDom)
open Equivalence eq
absNegation a | inl (inr a<0) | inr -a=0 = exFalso (SetoidPartialOrder.irreflexive (fieldOfFractionsOrder ℤIntDom ℤOrderedRing) {0Q} (SetoidPartialOrder.wellDefined (fieldOfFractionsOrder ℤIntDom ℤOrderedRing) {a} {0Q} {0Q} {0Q} (transitive {a} {inverse 0Q} {0Q} (swapInv (Ring.additiveGroup ℚRing) {a} {0Q} (symmetric {0Q} {inverse a} -a=0)) (invIdentity (Ring.additiveGroup ℚRing))) (reflexive {0Q}) a<0))
where
open Group (Ring.additiveGroup ℚRing)
open Setoid (fieldOfFractionsSetoid ℤIntDom)
open Equivalence eq
absNegation a | inr 0=a with SetoidTotalOrder.totality ℚTotalOrder 0Q (negateQ a)
absNegation a | inr 0=a | inl (inl 0<-a) = exFalso (SetoidPartialOrder.irreflexive (fieldOfFractionsOrder ℤIntDom ℤOrderedRing) {0Q} (SetoidPartialOrder.wellDefined (fieldOfFractionsOrder ℤIntDom ℤOrderedRing) {0Q} {0Q} {inverse a} {0Q} (reflexive {0Q}) (transitive {inverse a} {inverse 0Q} {0Q} (symmetric {inverse 0Q} {inverse a} (inverseWellDefined (Ring.additiveGroup ℚRing) {0Q} {a} 0=a)) (invIdentity (Ring.additiveGroup ℚRing))) 0<-a))
where
open Group (Ring.additiveGroup ℚRing)
open Setoid (fieldOfFractionsSetoid ℤIntDom)
open Equivalence eq
absNegation a | inr 0=a | inl (inr -a<0) = exFalso (SetoidPartialOrder.irreflexive (fieldOfFractionsOrder ℤIntDom ℤOrderedRing) {0Q} (SetoidPartialOrder.wellDefined (fieldOfFractionsOrder ℤIntDom ℤOrderedRing) {inverse a} {0Q} {0Q} {0Q} (transitive {inverse a} {inverse 0Q} {0Q} (symmetric {inverse 0Q} {inverse a} (inverseWellDefined (Ring.additiveGroup ℚRing) {0Q} {a} 0=a)) (invIdentity (Ring.additiveGroup ℚRing))) (reflexive {0Q}) -a<0))
where
open Group (Ring.additiveGroup ℚRing)
open Setoid (fieldOfFractionsSetoid ℤIntDom)
open Equivalence eq
absNegation a | inr 0=a | inr _ = refl

14
Rings/Definition.agda
View File

@@ -34,13 +34,6 @@ module Rings.Definition where
timesZero : {a : A} → a * 0R ∼ 0R
timesZero {a} = symmetric (transitive (transitive (symmetric invLeft) (+WellDefined reflexive (transitive (*WellDefined {a} {a} reflexive (symmetric identRight)) *DistributesOver+))) (transitive +Associative (transitive (+WellDefined invLeft reflexive) identLeft)))
record OrderedRing {n m p} {A : Set n} {S : Setoid {n} {m} A} {_+_ : A → A → A} {_*_ : A → A → A} {_<_ : Rel {_} {p} A} {pOrder : SetoidPartialOrder S _<_} (R : Ring S _+_ _*_) (order : SetoidTotalOrder pOrder) : Set (lsuc n ⊔ m ⊔ p) where
open Ring R
open Group additiveGroup
open Setoid S
field
orderRespectsAddition : {a b : A} → (a < b) → (c : A) → (a + c) < (b + c)
orderRespectsMultiplication : {a b : A} → (0R < a) → (0R < b) → (0R < (a * b))
--directSumRing : {m n : _} → {A : Set m} {B : Set n} (r : Ring A) (s : Ring B) → Ring (A && B)
--Ring.additiveGroup (directSumRing r s) = directSumGroup (Ring.additiveGroup r) (Ring.additiveGroup s)
@@ -66,10 +59,3 @@ module Rings.Definition where
-- field
-- ringHom : RingHom R S f
-- bijective : Bijection f
abs : {a b c : _} {A : Set a} {S : Setoid {a} {b} A} {_+_ : A → A → A} {_*_ : A → A → A} {_<_ : Rel {_} {c} A} {pOrder : SetoidPartialOrder S _<_} (R : Ring S _+_ _*_) (order : SetoidTotalOrder pOrder) (x : A) → A
abs R order x with SetoidTotalOrder.totality order (Ring.0R R) x
... | inl (inl 0<x) = x
... | inl (inr x<0) = Group.inverse (Ring.additiveGroup R) x
... | inr 0=x = Ring.0R R

191
Rings/Lemmas.agda
View File

@@ -12,175 +12,26 @@ open import Sets.EquivalenceRelations
module Rings.Lemmas where
ringMinusExtracts : {a b : _} {A : Set a} {S : Setoid {a} {b} A} {_+_ : A → A → A} {_*_ : A → A → A} (R : Ring S _+_ _*_) → {x y : A} → Setoid._∼_ S (x * Group.inverse (Ring.additiveGroup R) y) (Group.inverse (Ring.additiveGroup R) (x * y))
ringMinusExtracts {S = S} {_+_ = _+_} {_*_ = _*_} R {x = x} {y} = transferToRight' additiveGroup (transitive (symmetric *DistributesOver+) (transitive (*WellDefined reflexive invLeft) (Ring.timesZero R)))
where
open Setoid S
open Equivalence eq
open Ring R
open Group additiveGroup
ringMinusExtracts : {a b : _} {A : Set a} {S : Setoid {a} {b} A} {_+_ : A → A → A} {_*_ : A → A → A} (R : Ring S _+_ _*_) → {x y : A} → Setoid._∼_ S (x * Group.inverse (Ring.additiveGroup R) y) (Group.inverse (Ring.additiveGroup R) (x * y))
ringMinusExtracts {S = S} {_+_ = _+_} {_*_ = _*_} R {x = x} {y} = transferToRight' additiveGroup (transitive (symmetric *DistributesOver+) (transitive (*WellDefined reflexive invLeft) (Ring.timesZero R)))
where
open Setoid S
open Equivalence eq
open Ring R
open Group additiveGroup
ringAddInequalities : {a b c : _} {A : Set a} {S : Setoid {a} {b} A} {_+_ : A → A → A} {_*_ : A → A → A} {R : Ring S _+_ _*_} {_<_ : Rel {_} {c} A} {pOrder : SetoidPartialOrder S _<_} {tOrder : SetoidTotalOrder pOrder} (oRing : OrderedRing R tOrder) → {w x y z : A} → w < x → y < z → (w + y) < (x + z)
ringAddInequalities {S = S} {_+_ = _+_} {R = R} {_<_ = _<_} {pOrder = pOrder} oRing {w = w} {x} {y} {z} w<x y<z = transitive (orderRespectsAddition w<x y) (wellDefined groupIsAbelian groupIsAbelian (orderRespectsAddition y<z x))
where
open SetoidPartialOrder pOrder
open OrderedRing oRing
open Ring R
groupLemmaMove0G : {a b : _} → {A : Set a} → {_·_ : A → A → A} → {S : Setoid {a} {b} A} → (G : Group S _·_) → {x : A} → (Setoid._∼_ S (Group.0G G) (Group.inverse G x)) → Setoid._∼_ S x (Group.0G G)
groupLemmaMove0G {S = S} G {x} pr = transitive (symmetric (invInv G)) (transitive (symmetric p) (invIdent G))
where
open Group G
open Setoid S
open Equivalence eq
p : inverse 0G ∼ inverse (inverse x)
p = inverseWellDefined G pr
groupLemmaMove0G : {a b : _} → {A : Set a} → {_·_ : A → A → A} → {S : Setoid {a} {b} A} → (G : Group S _·_) → {x : A} → (Setoid._∼_ S (Group.0G G) (Group.inverse G x)) → Setoid._∼_ S x (Group.0G G)
groupLemmaMove0G {S = S} G {x} pr = transitive (symmetric (invInv G)) (transitive (symmetric p) (invIdent G))
where
open Group G
open Setoid S
open Equivalence eq
p : inverse 0G ∼ inverse (inverse x)
p = inverseWellDefined G pr
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 (transitive identLeft pr)
where
open Group G
open Setoid S
open Equivalence eq
ringMinusFlipsOrder : {a b c : _} {A : Set a} {S : Setoid {a} {b} A} {_+_ : A → A → A} {_*_ : A → A → A} {R : Ring S _+_ _*_} {_<_ : Rel {_} {c} A} {pOrder : SetoidPartialOrder S _<_} {tOrder : SetoidTotalOrder pOrder} (oRing : OrderedRing R tOrder) → {x : A} → (Ring.0R R) < x → (Group.inverse (Ring.additiveGroup R) x) < (Ring.0R R)
ringMinusFlipsOrder {S = S} {_+_ = _+_} {R = R} {_<_ = _<_} {pOrder = pOrder} {tOrder = tOrder} oRing {x = x} 0<x with SetoidTotalOrder.totality tOrder (Ring.0R R) (Group.inverse (Ring.additiveGroup R) x)
ringMinusFlipsOrder {S = S} {_+_} {R = R} {_<_} {pOrder} {tOrder} oRing {x} 0<x | inl (inl 0<inv) = exFalso (SetoidPartialOrder.irreflexive pOrder bad')
where
bad : (Group.0G (Ring.additiveGroup R) + Group.0G (Ring.additiveGroup R)) < (x + Group.inverse (Ring.additiveGroup R) x)
bad = ringAddInequalities oRing 0<x 0<inv
bad' : (Group.0G (Ring.additiveGroup R)) < (Group.0G (Ring.additiveGroup R))
bad' = SetoidPartialOrder.wellDefined pOrder (Group.identRight (Ring.additiveGroup R)) (Group.invRight (Ring.additiveGroup R)) bad
ringMinusFlipsOrder {S = S} {_+_} {R = R} {_<_} {pOrder} {tOrder} oRing {x} 0<x | inl (inr inv<0) = inv<0
ringMinusFlipsOrder {S = S} {_+_} {R = R} {_<_} {pOrder} {tOrder} oRing {x} 0<x | inr 0=inv = exFalso (SetoidPartialOrder.irreflexive pOrder (SetoidPartialOrder.wellDefined pOrder (Equivalence.reflexive (Setoid.eq S)) (groupLemmaMove0G (Ring.additiveGroup R) 0=inv) 0<x))
ringMinusFlipsOrder' : {a b c : _} {A : Set a} {S : Setoid {a} {b} A} {_+_ : A → A → A} {_*_ : A → A → A} {R : Ring S _+_ _*_} {_<_ : Rel {_} {c} A} {pOrder : SetoidPartialOrder S _<_} {tOrder : SetoidTotalOrder pOrder} (oRing : OrderedRing R tOrder) → {x : A} → (Group.inverse (Ring.additiveGroup R) x) < (Ring.0R R) → (Ring.0R R) < x
ringMinusFlipsOrder' {R = R} {_<_ = _<_} {tOrder = tOrder} oRing {x} -x<0 with SetoidTotalOrder.totality tOrder (Ring.0R R) x
ringMinusFlipsOrder' {R = R} {_<_} {tOrder = tOrder} oRing {x} -x<0 | inl (inl 0<x) = 0<x
ringMinusFlipsOrder' {_+_ = _+_} {R = R} {_<_} {pOrder = pOrder} {tOrder = tOrder} oRing {x} -x<0 | inl (inr x<0) = exFalso (SetoidPartialOrder.irreflexive pOrder (SetoidPartialOrder.wellDefined pOrder (Group.invLeft (Ring.additiveGroup R)) (Group.identRight (Ring.additiveGroup R)) bad))
where
bad : ((Group.inverse (Ring.additiveGroup R) x) + x) < (Group.0G (Ring.additiveGroup R) + Group.0G (Ring.additiveGroup R))
bad = ringAddInequalities oRing -x<0 x<0
ringMinusFlipsOrder' {S = S} {R = R} {_<_} {pOrder = pOrder} {tOrder = tOrder} oRing {x} -x<0 | inr 0=x = exFalso (SetoidPartialOrder.irreflexive pOrder (SetoidPartialOrder.wellDefined pOrder (symmetric (groupLemmaMove0G' (Ring.additiveGroup R) (symmetric 0=x))) (Equivalence.reflexive (Setoid.eq S)) -x<0))
where
open Setoid S
open Equivalence eq
ringMinusFlipsOrder'' : {a b c : _} {A : Set a} {S : Setoid {a} {b} A} {_+_ : A → A → A} {_*_ : A → A → A} {R : Ring S _+_ _*_} {_<_ : Rel {_} {c} A} {pOrder : SetoidPartialOrder S _<_} {tOrder : SetoidTotalOrder pOrder} (oRing : OrderedRing R tOrder) → {x : A} → x < (Ring.0R R) → (Ring.0R R) < Group.inverse (Ring.additiveGroup R) x
ringMinusFlipsOrder'' {S = S} {R = R} {pOrder = pOrder} oRing {x} x<0 = ringMinusFlipsOrder' oRing (SetoidPartialOrder.wellDefined pOrder {x} {Group.inverse (Ring.additiveGroup R) (Group.inverse (Ring.additiveGroup R) x)} {Ring.0R R} {Ring.0R R} (Equivalence.symmetric (Setoid.eq S) (invInv (Ring.additiveGroup R))) (Equivalence.reflexive (Setoid.eq S)) x<0)
ringMinusFlipsOrder''' : {a b c : _} {A : Set a} {S : Setoid {a} {b} A} {_+_ : A → A → A} {_*_ : A → A → A} {R : Ring S _+_ _*_} {_<_ : Rel {_} {c} A} {pOrder : SetoidPartialOrder S _<_} {tOrder : SetoidTotalOrder pOrder} (oRing : OrderedRing R tOrder) → {x : A} → (Ring.0R R) < (Group.inverse (Ring.additiveGroup R) x) → x < (Ring.0R R)
ringMinusFlipsOrder''' {S = S} {R = R} {pOrder = pOrder} oRing {x} 0<-x = SetoidPartialOrder.wellDefined pOrder (invInv (Ring.additiveGroup R)) (Equivalence.reflexive (Setoid.eq S)) (ringMinusFlipsOrder oRing 0<-x)
ringCanMultiplyByPositive : {a b c : _} {A : Set a} {S : Setoid {a} {b} A} {_+_ : A → A → A} {_*_ : A → A → A} {R : Ring S _+_ _*_} {_<_ : Rel {_} {c} A} {pOrder : SetoidPartialOrder S _<_} {tOrder : SetoidTotalOrder pOrder} (oRing : OrderedRing R tOrder) → {x y c : A} → (Ring.0R R) < c → x < y → (x * c) < (y * c)
ringCanMultiplyByPositive {S = S} {_+_ = _+_} {_*_ = _*_} {R = R} {_<_ = _<_} {pOrder = pOrder} {tOrder = tOrder} oRing {x} {y} {c} 0<c x<y = SetoidPartialOrder.wellDefined pOrder reflexive (Group.identRight additiveGroup) q'
where
open Ring R
open Equivalence (Setoid.eq S)
have : 0R < (y + Group.inverse additiveGroup x)
have = SetoidPartialOrder.wellDefined pOrder (Group.invRight additiveGroup) reflexive (OrderedRing.orderRespectsAddition oRing x<y (Group.inverse additiveGroup x))
p : 0R < ((y * c) + ((Group.inverse additiveGroup x) * c))
p = SetoidPartialOrder.wellDefined pOrder reflexive (transitive *Commutative (transitive *DistributesOver+ ((Group.+WellDefined additiveGroup) *Commutative *Commutative))) (OrderedRing.orderRespectsMultiplication oRing have 0<c)
p' : 0R < ((y * c) + (Group.inverse additiveGroup (x * c)))
p' = SetoidPartialOrder.wellDefined pOrder reflexive (Group.+WellDefined additiveGroup reflexive (transitive (transitive *Commutative (ringMinusExtracts R)) (inverseWellDefined additiveGroup *Commutative))) p
q : (0R + (x * c)) < (((y * c) + (Group.inverse additiveGroup (x * c))) + (x * c))
q = OrderedRing.orderRespectsAddition oRing p' (x * c)
q' : (x * c) < ((y * c) + 0R)
q' = SetoidPartialOrder.wellDefined pOrder (Group.identLeft additiveGroup) (transitive (symmetric (Group.+Associative additiveGroup)) (Group.+WellDefined additiveGroup reflexive (Group.invLeft additiveGroup))) q
ringCanCancelPositive : {a b c : _} {A : Set a} {S : Setoid {a} {b} A} {_+_ : A → A → A} {_*_ : A → A → A} {R : Ring S _+_ _*_} {_<_ : Rel {_} {c} A} {pOrder : SetoidPartialOrder S _<_} {tOrder : SetoidTotalOrder pOrder} (oRing : OrderedRing R tOrder) → {x y c : A} → (Ring.0R R) < c → (x * c) < (y * c) → x < y
ringCanCancelPositive {S = S} {_+_ = _+_} {_*_ = _*_} {R = R} {_<_ = _<_} {pOrder = pOrder} {tOrder = tOrder} oRing {x} {y} {c} 0<c xc<yc = SetoidPartialOrder.wellDefined pOrder (Group.identLeft additiveGroup) (transitive (symmetric (Group.+Associative additiveGroup)) (transitive (Group.+WellDefined additiveGroup reflexive (Group.invLeft additiveGroup)) (Group.identRight additiveGroup))) q''
where
open Ring R
open Setoid S
open Equivalence (Setoid.eq S)
have : 0R < ((y * c) + (Group.inverse additiveGroup (x * c)))
have = SetoidPartialOrder.wellDefined pOrder (Group.invRight additiveGroup) reflexive (OrderedRing.orderRespectsAddition oRing xc<yc (Group.inverse additiveGroup _))
p : 0R < ((y * c) + ((Group.inverse additiveGroup x) * c))
p = SetoidPartialOrder.wellDefined pOrder reflexive (Group.+WellDefined additiveGroup reflexive (symmetric (transitive (*Commutative) (transitive (ringMinusExtracts R) (inverseWellDefined additiveGroup *Commutative))))) have
q : 0R < ((y + Group.inverse additiveGroup x) * c)
q = SetoidPartialOrder.wellDefined pOrder reflexive (transitive (transitive (Group.+WellDefined additiveGroup *Commutative *Commutative) (symmetric *DistributesOver+)) *Commutative) p
q' : 0R < (y + Group.inverse additiveGroup x)
q' with SetoidTotalOrder.totality tOrder 0R (y + Group.inverse additiveGroup x)
q' | inl (inl pr) = pr
q' | inl (inr y-x<0) = exFalso (SetoidPartialOrder.irreflexive pOrder (SetoidPartialOrder.wellDefined pOrder reflexive (transitive *Commutative (Ring.timesZero R)) k))
where
open Group additiveGroup
f : ((y + inverse x) + (inverse (y + inverse x))) < (0G + inverse (y + inverse x))
f = OrderedRing.orderRespectsAddition oRing y-x<0 _
g : 0G < inverse (y + inverse x)
g = SetoidPartialOrder.wellDefined pOrder invRight identLeft f
h : (0G * c) < ((inverse (y + inverse x)) * c)
h = ringCanMultiplyByPositive oRing 0<c g
i : (0R + (0G * c)) < (((y + inverse x) * c) + ((inverse (y + inverse x)) * c))
i = ringAddInequalities oRing q h
j : 0R < (((y + inverse x) + (inverse (y + inverse x))) * c)
j = SetoidPartialOrder.wellDefined pOrder (transitive identLeft (transitive *Commutative (Ring.timesZero R))) (symmetric (transitive *Commutative (transitive *DistributesOver+ (Group.+WellDefined additiveGroup *Commutative *Commutative)))) i
k : 0R < (0R * c)
k = SetoidPartialOrder.wellDefined pOrder reflexive (*WellDefined invRight reflexive) j
q' | inr 0=y-x = exFalso (SetoidPartialOrder.irreflexive pOrder (SetoidPartialOrder.wellDefined pOrder (*WellDefined x=y reflexive) reflexive xc<yc))
where
open Group additiveGroup
f : inverse 0G ∼ inverse (y + inverse x)
f = inverseWellDefined additiveGroup 0=y-x
g : 0G ∼ (inverse y) + x
g = transitive (symmetric (invIdentity additiveGroup)) (transitive f (transitive (transitive (invContravariant additiveGroup) groupIsAbelian) (+WellDefined reflexive (invInv additiveGroup))))
x=y : x ∼ y
x=y = transferToRight additiveGroup (symmetric (transitive g groupIsAbelian))
q'' : (0R + x) < ((y + Group.inverse additiveGroup x) + x)
q'' = OrderedRing.orderRespectsAddition oRing q' x
ringSwapNegatives : {a b c : _} {A : Set a} {S : Setoid {a} {b} A} {_+_ : A → A → A} {_*_ : A → A → A} {R : Ring S _+_ _*_} {_<_ : Rel {_} {c} A} {pOrder : SetoidPartialOrder S _<_} {tOrder : SetoidTotalOrder pOrder} (oRing : OrderedRing R tOrder) → {x y : A} → (Group.inverse (Ring.additiveGroup R) x) < (Group.inverse (Ring.additiveGroup R) y) → y < x
ringSwapNegatives {S = S} {_+_ = _+_} {_*_ = _*_} {R = R} {_<_ = _<_} {pOrder = pOrder} oRing {x} {y} -x<-y = SetoidPartialOrder.wellDefined pOrder (transitive (symmetric (Group.+Associative additiveGroup)) (transitive (Group.+WellDefined additiveGroup reflexive (Group.invLeft additiveGroup)) (Group.identRight additiveGroup))) (Group.identLeft additiveGroup) v
where
open Ring R
open Setoid S
open Equivalence eq
t : ((Group.inverse additiveGroup x) + y) < ((Group.inverse additiveGroup y) + y)
t = OrderedRing.orderRespectsAddition oRing -x<-y y
u : (y + (Group.inverse additiveGroup x)) < 0R
u = SetoidPartialOrder.wellDefined pOrder (groupIsAbelian) (Group.invLeft additiveGroup) t
v : ((y + (Group.inverse additiveGroup x)) + x) < (0R + x)
v = OrderedRing.orderRespectsAddition oRing u x
ringCanMultiplyByNegative : {a b c : _} {A : Set a} {S : Setoid {a} {b} A} {_+_ : A → A → A} {_*_ : A → A → A} {R : Ring S _+_ _*_} {_<_ : Rel {_} {c} A} {pOrder : SetoidPartialOrder S _<_} {tOrder : SetoidTotalOrder pOrder} (oRing : OrderedRing R tOrder) → {x y c : A} → c < (Ring.0R R) → x < y → (y * c) < (x * c)
ringCanMultiplyByNegative {S = S} {_+_ = _+_} {_*_ = _*_} {R = R} {_<_ = _<_} {pOrder = pOrder} {tOrder = tOrder} oRing {x} {y} {c} c<0 x<y = ringSwapNegatives oRing u
where
open Ring R
open Setoid S
open Equivalence eq
p : (c + Group.inverse additiveGroup c) < (0R + Group.inverse additiveGroup c)
p = OrderedRing.orderRespectsAddition oRing c<0 _
0<-c : 0R < (Group.inverse additiveGroup c)
0<-c = SetoidPartialOrder.wellDefined pOrder (Group.invRight additiveGroup) (Group.identLeft additiveGroup) p
t : (x * Group.inverse additiveGroup c) < (y * Group.inverse additiveGroup c)
t = ringCanMultiplyByPositive oRing 0<-c x<y
u : (Group.inverse additiveGroup (x * c)) < Group.inverse additiveGroup (y * c)
u = SetoidPartialOrder.wellDefined pOrder (ringMinusExtracts R) (ringMinusExtracts R) t
ringCanCancelNegative : {a b c : _} {A : Set a} {S : Setoid {a} {b} A} {_+_ : A → A → A} {_*_ : A → A → A} {R : Ring S _+_ _*_} {_<_ : Rel {_} {c} A} {pOrder : SetoidPartialOrder S _<_} {tOrder : SetoidTotalOrder pOrder} (oRing : OrderedRing R tOrder) → {x y c : A} → c < (Ring.0R R) → (x * c) < (y * c) → y < x
ringCanCancelNegative {S = S} {_+_ = _+_} {_*_ = _*_} {R = R} {_<_ = _<_} {pOrder = pOrder} {tOrder = tOrder} oRing {x} {y} {c} c<0 xc<yc = r
where
open Ring R
open Setoid S
open Group additiveGroup
open Equivalence eq
p : 0R < ((y * c) + inverse (x * c))
p = SetoidPartialOrder.wellDefined pOrder invRight reflexive (OrderedRing.orderRespectsAddition oRing xc<yc (inverse (x * c)))
p1 : 0R < ((y * c) + ((inverse x) * c))
p1 = SetoidPartialOrder.wellDefined pOrder reflexive (Group.+WellDefined additiveGroup reflexive (transitive (inverseWellDefined additiveGroup *Commutative) (transitive (symmetric (ringMinusExtracts R)) *Commutative))) p
p2 : 0R < ((y + inverse x) * c)
p2 = SetoidPartialOrder.wellDefined pOrder reflexive (transitive (Group.+WellDefined additiveGroup *Commutative *Commutative) (transitive (symmetric *DistributesOver+) *Commutative)) p1
q : (y + inverse x) < 0R
q with SetoidTotalOrder.totality tOrder 0R (y + inverse x)
q | inl (inl pr) = exFalso (SetoidPartialOrder.irreflexive pOrder (SetoidPartialOrder.transitive pOrder bad c<0))
where
bad : 0R < c
bad = ringCanCancelPositive oRing pr (SetoidPartialOrder.wellDefined pOrder (symmetric (transitive *Commutative (Ring.timesZero R))) *Commutative p2)
q | inl (inr pr) = pr
q | inr 0=y-x = exFalso (SetoidPartialOrder.irreflexive pOrder (SetoidPartialOrder.wellDefined pOrder (*WellDefined x=y reflexive) reflexive xc<yc))
where
x=y : x ∼ y
x=y = transitive (symmetric identLeft) (transitive (Group.+WellDefined additiveGroup 0=y-x reflexive) (transitive (symmetric (Group.+Associative additiveGroup)) (transitive (Group.+WellDefined additiveGroup reflexive invLeft) identRight)))
r : y < x
r = SetoidPartialOrder.wellDefined pOrder (transitive (symmetric (Group.+Associative additiveGroup)) (transitive (Group.+WellDefined additiveGroup reflexive (invLeft)) identRight)) (Group.identLeft additiveGroup) (OrderedRing.orderRespectsAddition oRing q x)
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 (transitive identLeft pr)
where
open Group G
open Setoid S
open Equivalence eq

44
Rings/Order.agda Normal file
View File

@@ -0,0 +1,44 @@
{-# OPTIONS --safe --warning=error --without-K #-}
open import LogicalFormulae
open import Groups.Groups
open import Groups.Definition
open import Numbers.Naturals.Naturals
open import Setoids.Orders
open import Setoids.Setoids
open import Functions
open import Sets.EquivalenceRelations
open import Rings.Definition
open import Agda.Primitive using (Level; lzero; lsuc; _⊔_)
module Rings.Order {n m : _} {A : Set n} {S : Setoid {n} {m} A} {_+_ : A → A → A} {_*_ : A → A → A} (R : Ring S _+_ _*_) where
open Ring R
open Group additiveGroup
open Setoid S
record OrderedRing {p : _} {_<_ : Rel {_} {p} A} {pOrder : SetoidPartialOrder S _<_} (order : SetoidTotalOrder pOrder) : Set (lsuc n ⊔ m ⊔ p) where
field
orderRespectsAddition : {a b : A} → (a < b) → (c : A) → (a + c) < (b + c)
orderRespectsMultiplication : {a b : A} → (0R < a) → (0R < b) → (0R < (a * b))
open SetoidPartialOrder pOrder
abs : A → A
abs a with SetoidTotalOrder.totality order 0R a
abs a | inl (inl 0<a) = a
abs a | inl (inr a<0) = inverse a
abs a | inr 0=a = a
absWellDefined : (a b : A) → a ∼ b → abs a ∼ abs b
absWellDefined a b a=b with SetoidTotalOrder.totality order 0R a
absWellDefined a b a=b | inl (inl 0<a) with SetoidTotalOrder.totality order 0R b
absWellDefined a b a=b | inl (inl 0<a) | inl (inl 0<b) = a=b
absWellDefined a b a=b | inl (inl 0<a) | inl (inr b<0) = exFalso (irreflexive {0G} (transitive 0<a (<WellDefined (Equivalence.symmetric eq a=b) (Equivalence.reflexive eq) b<0)))
absWellDefined a b a=b | inl (inl 0<a) | inr 0=b = exFalso (irreflexive {0G} (<WellDefined (Equivalence.reflexive eq) (Equivalence.transitive eq a=b (Equivalence.symmetric eq 0=b)) 0<a))
absWellDefined a b a=b | inl (inr a<0) with SetoidTotalOrder.totality order 0R b
absWellDefined a b a=b | inl (inr a<0) | inl (inl 0<b) = exFalso (irreflexive {0G} (transitive 0<b (<WellDefined a=b (Equivalence.reflexive eq) a<0)))
absWellDefined a b a=b | inl (inr a<0) | inl (inr b<0) = inverseWellDefined additiveGroup a=b
absWellDefined a b a=b | inl (inr a<0) | inr 0=b = exFalso (irreflexive {0G} (<WellDefined (Equivalence.transitive eq a=b (Equivalence.symmetric eq 0=b)) (Equivalence.reflexive eq) a<0))
absWellDefined a b a=b | inr 0=a with SetoidTotalOrder.totality order 0R b
absWellDefined a b a=b | inr 0=a | inl (inl 0<b) = exFalso (irreflexive {0G} (<WellDefined (Equivalence.reflexive eq) (Equivalence.symmetric eq (Equivalence.transitive eq 0=a a=b)) 0<b))
absWellDefined a b a=b | inr 0=a | inl (inr b<0) = exFalso (irreflexive {0G} (<WellDefined (Equivalence.symmetric eq (Equivalence.transitive eq 0=a a=b)) (Equivalence.reflexive eq) b<0))
absWellDefined a b a=b | inr 0=a | inr 0=b = a=b

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{-# OPTIONS --safe --warning=error --without-K #-}
open import LogicalFormulae
open import Groups.Groups
open import Groups.Lemmas
open import Groups.Definition
open import Numbers.Naturals.Naturals
open import Setoids.Orders
open import Setoids.Setoids
open import Functions
open import Sets.EquivalenceRelations
open import Rings.Definition
open import Rings.Lemmas
open import Rings.Order
open import Agda.Primitive using (Level; lzero; lsuc; _⊔_)
module Rings.Orders.Lemmas {n m p : _} {A : Set n} {S : Setoid {n} {m} A} {_+_ : A → A → A} {_*_ : A → A → A} {_<_ : Rel {_} {p} A} {R : Ring S _+_ _*_} {pOrder : SetoidPartialOrder S _<_} {tOrder : SetoidTotalOrder pOrder} (order : OrderedRing R tOrder) where
open OrderedRing order
open Setoid S
open SetoidPartialOrder pOrder
open SetoidTotalOrder tOrder
open Ring R
open Group additiveGroup
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))
lemm2 : (a : A) → a < 0G → 0G < inverse a
lemm2 a a<0 with SetoidTotalOrder.totality tOrder 0R (inverse a)
lemm2 a a<0 | inl (inl 0<-a) = 0<-a
lemm2 a a<0 | inl (inr -a<0) = exFalso (irreflexive {0G} (SetoidPartialOrder.transitive pOrder (<WellDefined (invLeft {a}) (identLeft {a}) (orderRespectsAddition -a<0 a)) a<0))
lemm2 a a<0 | inr 0=-a = exFalso (irreflexive {0G} (<WellDefined (Equivalence.transitive eq (Equivalence.symmetric eq identRight) t) (Equivalence.reflexive eq) a<0))
where
t : a + 0G ∼ 0G
t = Equivalence.transitive eq (+WellDefined (Equivalence.reflexive eq) 0=-a) (invRight {a})
lemm2' : (a : A) → 0G < a → inverse a < 0G
lemm2' a 0<a with SetoidTotalOrder.totality tOrder 0R (inverse a)
lemm2' a 0<a | inl (inl 0<-a) = exFalso (irreflexive {0G} (SetoidPartialOrder.transitive pOrder 0<a (<WellDefined (identLeft {a}) (invLeft {a}) (orderRespectsAddition 0<-a a))))
lemm2' a 0<a | inl (inr -a<0) = -a<0
lemm2' a 0<a | inr 0=-a = exFalso (irreflexive {0G} (<WellDefined (Equivalence.reflexive eq) (Equivalence.transitive eq (Equivalence.symmetric eq identRight) t) 0<a))
where
t : a + 0G ∼ 0G
t = Equivalence.transitive eq (+WellDefined (Equivalence.reflexive eq) 0=-a) (invRight {a})
lemm3 : (a b : A) → 0G ∼ (a + b) → 0G ∼ a → 0G ∼ b
lemm3 a b pr1 pr2 with transferToRight' additiveGroup (Equivalence.symmetric eq pr1)
... | a=-b with Equivalence.transitive eq pr2 a=-b
... | 0=-b with inverseWellDefined additiveGroup 0=-b
... | -0=--b = Equivalence.transitive eq (Equivalence.symmetric eq (invIdentity additiveGroup)) (Equivalence.transitive eq -0=--b (invTwice additiveGroup b))
triangleInequality : (a b : A) → ((abs (a + b)) < ((abs a) + (abs b))) || (abs (a + b) ∼ (abs a) + (abs b))
triangleInequality a b with totality 0R (a + b)
triangleInequality a b | inl (inl 0<a+b) with totality 0R a
triangleInequality a b | inl (inl 0<a+b) | inl (inl 0<a) with totality 0R b
triangleInequality a b | inl (inl 0<a+b) | inl (inl 0<a) | inl (inl 0<b) = inr (Equivalence.reflexive eq)
triangleInequality a b | inl (inl 0<a+b) | inl (inl 0<a) | inl (inr b<0) = inl (<WellDefined groupIsAbelian groupIsAbelian (orderRespectsAddition (SetoidPartialOrder.transitive pOrder b<0 (lemm2 b b<0)) a))
triangleInequality a b | inl (inl 0<a+b) | inl (inl 0<a) | inr 0=b = inr (Equivalence.reflexive eq)
triangleInequality a b | inl (inl 0<a+b) | inl (inr a<0) with totality 0R b
triangleInequality a b | inl (inl 0<a+b) | inl (inr a<0) | inl (inl 0<b) = inl (orderRespectsAddition (SetoidPartialOrder.transitive pOrder a<0 (lemm2 a a<0)) b)
triangleInequality a b | inl (inl 0<a+b) | inl (inr a<0) | inl (inr b<0) = exFalso (irreflexive {0G} (SetoidPartialOrder.transitive pOrder 0<a+b (<WellDefined (Equivalence.reflexive eq) identLeft (ringAddInequalities a<0 b<0))))
triangleInequality a b | inl (inl 0<a+b) | inl (inr a<0) | inr 0=b = inl (orderRespectsAddition (SetoidPartialOrder.transitive pOrder a<0 (lemm2 a a<0)) b)
triangleInequality a b | inl (inl 0<a+b) | inr 0=a with totality 0R b
triangleInequality a b | inl (inl 0<a+b) | inr 0=a | inl (inl 0<b) = inr (Equivalence.reflexive eq)
triangleInequality a b | inl (inl 0<a+b) | inr 0=a | inl (inr b<0) = inl (<WellDefined groupIsAbelian groupIsAbelian (orderRespectsAddition (SetoidPartialOrder.transitive pOrder b<0 (lemm2 b b<0)) a))
triangleInequality a b | inl (inl 0<a+b) | inr 0=a | inr 0=b = inr (Equivalence.reflexive eq)
triangleInequality a b | inl (inr a+b<0) with totality 0G a
triangleInequality a b | inl (inr a+b<0) | inl (inl 0<a) with totality 0G b
triangleInequality a b | inl (inr a+b<0) | inl (inl 0<a) | inl (inl 0<b) = exFalso (irreflexive {0G} (SetoidPartialOrder.transitive pOrder (<WellDefined identLeft (Equivalence.reflexive eq) (ringAddInequalities 0<a 0<b)) a+b<0))
triangleInequality a b | inl (inr a+b<0) | inl (inl 0<a) | inl (inr b<0) = inl (<WellDefined (Equivalence.transitive eq (Equivalence.symmetric eq (invContravariant additiveGroup)) (inverseWellDefined additiveGroup groupIsAbelian)) (Equivalence.reflexive eq) (orderRespectsAddition (SetoidPartialOrder.transitive pOrder (lemm2' _ 0<a) 0<a) (inverse b)))
triangleInequality a b | inl (inr a+b<0) | inl (inl 0<a) | inr 0=b = exFalso (irreflexive {0G} (SetoidPartialOrder.transitive pOrder 0<a (<WellDefined (Equivalence.transitive eq (+WellDefined (Equivalence.reflexive eq) (Equivalence.symmetric eq 0=b)) identRight) (Equivalence.reflexive eq) a+b<0)))
triangleInequality a b | inl (inr a+b<0) | inl (inr a<0) with totality 0G b
triangleInequality a b | inl (inr a+b<0) | inl (inr a<0) | inl (inl 0<b) = inl (<WellDefined (Equivalence.symmetric eq (invContravariant additiveGroup)) groupIsAbelian (orderRespectsAddition (SetoidPartialOrder.transitive pOrder (lemm2' _ 0<b) 0<b) (inverse a)))
triangleInequality a b | inl (inr a+b<0) | inl (inr a<0) | inl (inr b<0) = inr (Equivalence.transitive eq (invContravariant additiveGroup) groupIsAbelian)
triangleInequality a b | inl (inr a+b<0) | inl (inr a<0) | inr 0=b = inr (Equivalence.transitive eq (invContravariant additiveGroup) (Equivalence.transitive eq (Equivalence.transitive eq (Equivalence.transitive eq (+WellDefined (Equivalence.transitive eq (inverseWellDefined additiveGroup (Equivalence.symmetric eq 0=b)) (invIdentity additiveGroup)) (Equivalence.reflexive eq)) identLeft) (Equivalence.symmetric eq identRight)) (+WellDefined (Equivalence.reflexive eq) 0=b)))
triangleInequality a b | inl (inr a+b<0) | inr 0=a with totality 0G b
triangleInequality a b | inl (inr a+b<0) | inr 0=a | inl (inl 0<b) = exFalso (irreflexive {0G} (SetoidPartialOrder.transitive pOrder 0<b (<WellDefined (Equivalence.transitive eq (+WellDefined (Equivalence.symmetric eq 0=a) (Equivalence.reflexive eq)) identLeft) (Equivalence.reflexive eq) a+b<0)))
triangleInequality a b | inl (inr a+b<0) | inr 0=a | inl (inr b<0) = inr (Equivalence.transitive eq (invContravariant additiveGroup) (Equivalence.transitive eq groupIsAbelian (+WellDefined (Equivalence.transitive eq (Equivalence.transitive eq (Equivalence.symmetric eq (inverseWellDefined additiveGroup 0=a)) (invIdentity additiveGroup)) 0=a) (Equivalence.reflexive eq))))
triangleInequality a b | inl (inr a+b<0) | inr 0=a | inr 0=b = exFalso (irreflexive {0G} (<WellDefined (Equivalence.transitive eq (+WellDefined (Equivalence.symmetric eq 0=a) (Equivalence.symmetric eq 0=b)) identLeft) (Equivalence.reflexive eq) a+b<0))
triangleInequality a b | inr 0=a+b with totality 0G a
triangleInequality a b | inr 0=a+b | inl (inl 0<a) with totality 0G b
triangleInequality a b | inr 0=a+b | inl (inl 0<a) | inl (inl 0<b) = exFalso (irreflexive {0G} (<WellDefined identLeft (Equivalence.symmetric eq 0=a+b) (ringAddInequalities 0<a 0<b)))
triangleInequality a b | inr 0=a+b | inl (inl 0<a) | inl (inr b<0) = inl (<WellDefined groupIsAbelian groupIsAbelian (orderRespectsAddition (SetoidPartialOrder.transitive pOrder b<0 (lemm2 _ b<0)) a))
triangleInequality a b | inr 0=a+b | inl (inl 0<a) | inr 0=b = exFalso (irreflexive {0G} (<WellDefined (Equivalence.reflexive eq) (Equivalence.symmetric eq (lemm3 _ _ (Equivalence.transitive eq 0=a+b groupIsAbelian) 0=b)) 0<a))
triangleInequality a b | inr 0=a+b | inl (inr a<0) with totality 0G b
triangleInequality a b | inr 0=a+b | inl (inr a<0) | inl (inl 0<b) = inl (orderRespectsAddition (SetoidPartialOrder.transitive pOrder a<0 (lemm2 _ a<0)) b)
triangleInequality a b | inr 0=a+b | inl (inr a<0) | inl (inr b<0) = exFalso (irreflexive {0G} (<WellDefined (Equivalence.symmetric eq 0=a+b) identLeft (ringAddInequalities a<0 b<0)))
triangleInequality a b | inr 0=a+b | inl (inr a<0) | inr 0=b = exFalso (irreflexive {0G} (<WellDefined (Equivalence.symmetric eq (lemm3 _ _ (Equivalence.transitive eq 0=a+b groupIsAbelian) 0=b)) (Equivalence.reflexive eq) a<0))
triangleInequality a b | inr 0=a+b | inr 0=a with totality 0G b
triangleInequality a b | inr 0=a+b | inr 0=a | inl (inl 0<b) = exFalso (irreflexive {0G} (<WellDefined (Equivalence.reflexive eq) (Equivalence.symmetric eq (lemm3 a b 0=a+b 0=a)) 0<b))
triangleInequality a b | inr 0=a+b | inr 0=a | inl (inr b<0) = exFalso (irreflexive {0G} (<WellDefined (Equivalence.symmetric eq (lemm3 a b 0=a+b 0=a)) (Equivalence.reflexive eq) b<0))
triangleInequality a b | inr 0=a+b | inr 0=a | inr 0=b = inr (Equivalence.reflexive eq)
ringMinusFlipsOrder : {x : A} → (Ring.0R R) < x → (Group.inverse (Ring.additiveGroup R) x) < (Ring.0R R)
ringMinusFlipsOrder {x = x} 0<x with SetoidTotalOrder.totality tOrder (Ring.0R R) (Group.inverse (Ring.additiveGroup R) x)
ringMinusFlipsOrder {x} 0<x | inl (inl 0<inv) = exFalso (SetoidPartialOrder.irreflexive pOrder bad')
where
bad : (Group.0G (Ring.additiveGroup R) + Group.0G (Ring.additiveGroup R)) < (x + Group.inverse (Ring.additiveGroup R) x)
bad = ringAddInequalities 0<x 0<inv
bad' : (Group.0G (Ring.additiveGroup R)) < (Group.0G (Ring.additiveGroup R))
bad' = SetoidPartialOrder.<WellDefined pOrder (Group.identRight (Ring.additiveGroup R)) (Group.invRight (Ring.additiveGroup R)) bad
ringMinusFlipsOrder {x} 0<x | inl (inr inv<0) = inv<0
ringMinusFlipsOrder {x} 0<x | inr 0=inv = exFalso (SetoidPartialOrder.irreflexive pOrder (SetoidPartialOrder.<WellDefined pOrder (Equivalence.reflexive (Setoid.eq S)) (groupLemmaMove0G (Ring.additiveGroup R) 0=inv) 0<x))
ringMinusFlipsOrder' : {x : A} → (Group.inverse (Ring.additiveGroup R) x) < (Ring.0R R) → (Ring.0R R) < x
ringMinusFlipsOrder' {x} -x<0 with SetoidTotalOrder.totality tOrder (Ring.0R R) x
ringMinusFlipsOrder' {x} -x<0 | inl (inl 0<x) = 0<x
ringMinusFlipsOrder' {x} -x<0 | inl (inr x<0) = exFalso (SetoidPartialOrder.irreflexive pOrder (SetoidPartialOrder.<WellDefined pOrder (Group.invLeft (Ring.additiveGroup R)) (Group.identRight (Ring.additiveGroup R)) bad))
where
bad : ((Group.inverse (Ring.additiveGroup R) x) + x) < (Group.0G (Ring.additiveGroup R) + Group.0G (Ring.additiveGroup R))
bad = ringAddInequalities -x<0 x<0
ringMinusFlipsOrder' {x} -x<0 | inr 0=x = exFalso (SetoidPartialOrder.irreflexive pOrder (SetoidPartialOrder.<WellDefined pOrder (symmetric (groupLemmaMove0G' (Ring.additiveGroup R) (symmetric 0=x))) (Equivalence.reflexive (Setoid.eq S)) -x<0))
where
open Equivalence eq
ringMinusFlipsOrder'' : {x : A} → x < (Ring.0R R) → (Ring.0R R) < Group.inverse (Ring.additiveGroup R) x
ringMinusFlipsOrder'' {x} x<0 = ringMinusFlipsOrder' (SetoidPartialOrder.<WellDefined pOrder {x} {Group.inverse (Ring.additiveGroup R) (Group.inverse (Ring.additiveGroup R) x)} {Ring.0R R} {Ring.0R R} (Equivalence.symmetric (Setoid.eq S) (invInv (Ring.additiveGroup R))) (Equivalence.reflexive (Setoid.eq S)) x<0)
ringMinusFlipsOrder''' : {x : A} → (Ring.0R R) < (Group.inverse (Ring.additiveGroup R) x) → x < (Ring.0R R)
ringMinusFlipsOrder''' {x} 0<-x = SetoidPartialOrder.<WellDefined pOrder (invInv (Ring.additiveGroup R)) (Equivalence.reflexive (Setoid.eq S)) (ringMinusFlipsOrder 0<-x)
ringCanMultiplyByPositive : {x y c : A} → (Ring.0R R) < c → x < y → (x * c) < (y * c)
ringCanMultiplyByPositive {x} {y} {c} 0<c x<y = SetoidPartialOrder.<WellDefined pOrder reflexive (Group.identRight additiveGroup) q'
where
open Equivalence eq
have : 0R < (y + Group.inverse additiveGroup x)
have = SetoidPartialOrder.<WellDefined pOrder (Group.invRight additiveGroup) reflexive (OrderedRing.orderRespectsAddition order x<y (Group.inverse additiveGroup x))
p1 : 0R < ((y * c) + ((Group.inverse additiveGroup x) * c))
p1 = SetoidPartialOrder.<WellDefined pOrder reflexive (Equivalence.transitive eq *Commutative (Equivalence.transitive eq *DistributesOver+ ((Group.+WellDefined additiveGroup) *Commutative *Commutative))) (OrderedRing.orderRespectsMultiplication order have 0<c)
p' : 0R < ((y * c) + (Group.inverse additiveGroup (x * c)))
p' = SetoidPartialOrder.<WellDefined pOrder reflexive (Group.+WellDefined additiveGroup reflexive (Equivalence.transitive eq (Equivalence.transitive eq *Commutative (ringMinusExtracts R)) (inverseWellDefined additiveGroup *Commutative))) p1
q : (0R + (x * c)) < (((y * c) + (Group.inverse additiveGroup (x * c))) + (x * c))
q = OrderedRing.orderRespectsAddition order p' (x * c)
q' : (x * c) < ((y * c) + 0R)
q' = SetoidPartialOrder.<WellDefined pOrder (Group.identLeft additiveGroup) (Equivalence.transitive eq (symmetric (Group.+Associative additiveGroup)) (Group.+WellDefined additiveGroup reflexive (Group.invLeft additiveGroup))) q
ringCanCancelPositive : {x y c : A} → (Ring.0R R) < c → (x * c) < (y * c) → x < y
ringCanCancelPositive {x} {y} {c} 0<c xc<yc = SetoidPartialOrder.<WellDefined pOrder (Group.identLeft additiveGroup) (Equivalence.transitive eq (symmetric (Group.+Associative additiveGroup)) (Equivalence.transitive eq (Group.+WellDefined additiveGroup reflexive (Group.invLeft additiveGroup)) (Group.identRight additiveGroup))) q''
where
open Equivalence (Setoid.eq S)
have : 0R < ((y * c) + (Group.inverse additiveGroup (x * c)))
have = SetoidPartialOrder.<WellDefined pOrder (Group.invRight additiveGroup) reflexive (OrderedRing.orderRespectsAddition order xc<yc (Group.inverse additiveGroup _))
p1 : 0R < ((y * c) + ((Group.inverse additiveGroup x) * c))
p1 = SetoidPartialOrder.<WellDefined pOrder reflexive (Group.+WellDefined additiveGroup reflexive (symmetric (Equivalence.transitive eq (*Commutative) (Equivalence.transitive eq (ringMinusExtracts R) (inverseWellDefined additiveGroup *Commutative))))) have
q : 0R < ((y + Group.inverse additiveGroup x) * c)
q = SetoidPartialOrder.<WellDefined pOrder reflexive (Equivalence.transitive eq (Equivalence.transitive eq (Group.+WellDefined additiveGroup *Commutative *Commutative) (symmetric *DistributesOver+)) *Commutative) p1
q' : 0R < (y + Group.inverse additiveGroup x)
q' with SetoidTotalOrder.totality tOrder 0R (y + Group.inverse additiveGroup x)
q' | inl (inl pr) = pr
q' | inl (inr y-x<0) = exFalso (SetoidPartialOrder.irreflexive pOrder (SetoidPartialOrder.<WellDefined pOrder reflexive (Equivalence.transitive eq *Commutative (Ring.timesZero R)) k))
where
f : ((y + inverse x) + (inverse (y + inverse x))) < (0G + inverse (y + inverse x))
f = OrderedRing.orderRespectsAddition order y-x<0 _
g : 0G < inverse (y + inverse x)
g = SetoidPartialOrder.<WellDefined pOrder invRight identLeft f
h : (0G * c) < ((inverse (y + inverse x)) * c)
h = ringCanMultiplyByPositive 0<c g
i : (0R + (0G * c)) < (((y + inverse x) * c) + ((inverse (y + inverse x)) * c))
i = ringAddInequalities q h
j : 0R < (((y + inverse x) + (inverse (y + inverse x))) * c)
j = SetoidPartialOrder.<WellDefined pOrder (Equivalence.transitive eq identLeft (Equivalence.transitive eq *Commutative (Ring.timesZero R))) (symmetric (Equivalence.transitive eq *Commutative (Equivalence.transitive eq *DistributesOver+ (Group.+WellDefined additiveGroup *Commutative *Commutative)))) i
k : 0R < (0R * c)
k = SetoidPartialOrder.<WellDefined pOrder reflexive (*WellDefined invRight reflexive) j
q' | inr 0=y-x = exFalso (SetoidPartialOrder.irreflexive pOrder (SetoidPartialOrder.<WellDefined pOrder (*WellDefined x=y reflexive) reflexive xc<yc))
where
f : inverse 0G ∼ inverse (y + inverse x)
f = inverseWellDefined additiveGroup 0=y-x
g : 0G ∼ (inverse y) + x
g = Equivalence.transitive eq (symmetric (invIdentity additiveGroup)) (Equivalence.transitive eq f (Equivalence.transitive eq (Equivalence.transitive eq (invContravariant additiveGroup) groupIsAbelian) (+WellDefined reflexive (invInv additiveGroup))))
x=y : x ∼ y
x=y = transferToRight additiveGroup (symmetric (Equivalence.transitive eq g groupIsAbelian))
q'' : (0R + x) < ((y + Group.inverse additiveGroup x) + x)
q'' = OrderedRing.orderRespectsAddition order q' x
ringSwapNegatives : {x y : A} → (Group.inverse (Ring.additiveGroup R) x) < (Group.inverse (Ring.additiveGroup R) y) → y < x
ringSwapNegatives {x} {y} -x<-y = SetoidPartialOrder.<WellDefined pOrder (Equivalence.transitive eq (symmetric (Group.+Associative additiveGroup)) (Equivalence.transitive eq (Group.+WellDefined additiveGroup reflexive (Group.invLeft additiveGroup)) (Group.identRight additiveGroup))) (Group.identLeft additiveGroup) v
where
open Equivalence eq
t : ((Group.inverse additiveGroup x) + y) < ((Group.inverse additiveGroup y) + y)
t = OrderedRing.orderRespectsAddition order -x<-y y
u : (y + (Group.inverse additiveGroup x)) < 0R
u = SetoidPartialOrder.<WellDefined pOrder (groupIsAbelian) (Group.invLeft additiveGroup) t
v : ((y + (Group.inverse additiveGroup x)) + x) < (0R + x)
v = OrderedRing.orderRespectsAddition order u x
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
open Equivalence eq
p1 : (c + Group.inverse additiveGroup c) < (0R + Group.inverse additiveGroup c)
p1 = OrderedRing.orderRespectsAddition order c<0 _
0<-c : 0R < (Group.inverse additiveGroup c)
0<-c = SetoidPartialOrder.<WellDefined pOrder (Group.invRight additiveGroup) (Group.identLeft additiveGroup) p1
t : (x * Group.inverse additiveGroup c) < (y * Group.inverse additiveGroup c)
t = ringCanMultiplyByPositive 0<-c x<y
u : (Group.inverse additiveGroup (x * c)) < Group.inverse additiveGroup (y * c)
u = SetoidPartialOrder.<WellDefined pOrder (ringMinusExtracts R) (ringMinusExtracts R) t
ringCanCancelNegative : {x y c : A} → c < (Ring.0R R) → (x * c) < (y * c) → y < x
ringCanCancelNegative {x} {y} {c} c<0 xc<yc = r
where
open Equivalence eq
p0 : 0R < ((y * c) + inverse (x * c))
p0 = SetoidPartialOrder.<WellDefined pOrder invRight reflexive (OrderedRing.orderRespectsAddition order xc<yc (inverse (x * c)))
p1 : 0R < ((y * c) + ((inverse x) * c))
p1 = SetoidPartialOrder.<WellDefined pOrder reflexive (Group.+WellDefined additiveGroup reflexive (Equivalence.transitive eq (inverseWellDefined additiveGroup *Commutative) (Equivalence.transitive eq (symmetric (ringMinusExtracts R)) *Commutative))) p0
p2 : 0R < ((y + inverse x) * c)
p2 = SetoidPartialOrder.<WellDefined pOrder reflexive (Equivalence.transitive eq (Group.+WellDefined additiveGroup *Commutative *Commutative) (Equivalence.transitive eq (symmetric *DistributesOver+) *Commutative)) p1
q : (y + inverse x) < 0R
q with SetoidTotalOrder.totality tOrder 0R (y + inverse x)
q | inl (inl pr) = exFalso (SetoidPartialOrder.irreflexive pOrder (SetoidPartialOrder.transitive pOrder bad c<0))
where
bad : 0R < c
bad = ringCanCancelPositive pr (SetoidPartialOrder.<WellDefined pOrder (symmetric (Equivalence.transitive eq *Commutative (Ring.timesZero R))) *Commutative p2)
q | inl (inr pr) = pr
q | inr 0=y-x = exFalso (SetoidPartialOrder.irreflexive pOrder (SetoidPartialOrder.<WellDefined pOrder (*WellDefined x=y reflexive) reflexive xc<yc))
where
x=y : x ∼ y
x=y = Equivalence.transitive eq (symmetric identLeft) (Equivalence.transitive eq (Group.+WellDefined additiveGroup 0=y-x reflexive) (Equivalence.transitive eq (symmetric (Group.+Associative additiveGroup)) (Equivalence.transitive eq (Group.+WellDefined additiveGroup reflexive invLeft) identRight)))
r : y < x
r = SetoidPartialOrder.<WellDefined pOrder (Equivalence.transitive eq (symmetric (Group.+Associative additiveGroup)) (Equivalence.transitive eq (Group.+WellDefined additiveGroup reflexive (invLeft)) identRight)) (Group.identLeft additiveGroup) (OrderedRing.orderRespectsAddition order q x)
absZero : {a b c : _} {A : Set a} {S : Setoid {a} {b} A} {_+_ : A → A → A} {_*_ : A → A → A} {R : Ring S _+_ _*_} {_<_ : Rel {a} {c} A} {p : SetoidPartialOrder S _<_} {t : SetoidTotalOrder p} (o : OrderedRing R t) → OrderedRing.abs o (Ring.0R R) ≡ Ring.0R R
absZero {R = R} {t = t} oR with SetoidTotalOrder.totality t (Ring.0R R) (Ring.0R R)
absZero {R = R} {t = t} oR | inl (inl x) = exFalso (SetoidPartialOrder.irreflexive (SetoidTotalOrder.partial t) x)
absZero {R = R} {t = t} oR | inl (inr x) = exFalso (SetoidPartialOrder.irreflexive (SetoidTotalOrder.partial t) x)
absZero {R = R} {t = t} oR | inr x = refl
absNegation : (a : A) → (abs a) ∼ (abs (inverse a))
absNegation a with totality 0R a
absNegation a | inl (inl 0<a) with totality 0G (inverse a)
absNegation a | inl (inl 0<a) | inl (inl 0<-a) = exFalso (irreflexive {0G} (SetoidPartialOrder.transitive pOrder 0<-a (lemm2' a 0<a)))
absNegation a | inl (inl 0<a) | inl (inr -a<0) = Equivalence.symmetric eq (invTwice additiveGroup a)
absNegation a | inl (inl 0<a) | inr 0=-a = exFalso (irreflexive {0G} (<WellDefined (Equivalence.reflexive eq) (Equivalence.transitive eq (Equivalence.transitive eq (Equivalence.symmetric eq (invTwice additiveGroup a)) (inverseWellDefined additiveGroup (Equivalence.symmetric eq 0=-a))) (invIdent additiveGroup)) 0<a))
absNegation a | inl (inr a<0) with totality 0G (inverse a)
absNegation a | inl (inr a<0) | inl (inl 0<-a) = Equivalence.reflexive eq
absNegation a | inl (inr a<0) | inl (inr -a<0) = exFalso (irreflexive {0G} (SetoidPartialOrder.transitive pOrder (<WellDefined (Equivalence.reflexive eq) (invTwice additiveGroup a) (lemm2 (inverse a) -a<0)) a<0))
absNegation a | inl (inr a<0) | inr 0=-a = exFalso (irreflexive {0G} (<WellDefined (Equivalence.transitive eq (Equivalence.symmetric eq (Equivalence.transitive eq (inverseWellDefined additiveGroup 0=-a) (invTwice additiveGroup a))) (invIdent additiveGroup)) (Equivalence.reflexive eq) a<0))
absNegation a | inr 0=a with totality 0G (inverse a)
absNegation a | inr 0=a | inl (inl 0<-a) = exFalso (irreflexive {0G} (<WellDefined (Equivalence.reflexive eq) (Equivalence.transitive eq (inverseWellDefined additiveGroup (Equivalence.symmetric eq 0=a)) (invIdent additiveGroup)) 0<-a))
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

36
Sequences.agda Normal file
View File

@@ -0,0 +1,36 @@
{-# OPTIONS --warning=error --without-K --guardedness --safe #-}
open import LogicalFormulae
open import Numbers.Naturals.Definition
module Sequences where
record Sequence {a : _} (A : Set a) : Set a where
coinductive
field
head : A
tail : Sequence A
constSequence : {a : _} {A : Set a} (k : A) → Sequence A
Sequence.head (constSequence k) = k
Sequence.tail (constSequence k) = constSequence k
index : {a : _} {A : Set a} (s : Sequence A) (n : ℕ) → A
index s zero = Sequence.head s
index s (succ n) = index (Sequence.tail s) n
funcToSequence : {a : _} {A : Set a} (f : ℕ → A) → Sequence A
Sequence.head (funcToSequence f) = f 0
Sequence.tail (funcToSequence f) = funcToSequence (λ i → f (succ i))
funcToSequenceReversible : {a : _} {A : Set a} (f : ℕ → A) → (n : ℕ) → index (funcToSequence f) n ≡ f n
funcToSequenceReversible f zero = refl
funcToSequenceReversible f (succ n) = funcToSequenceReversible (λ i → f (succ i)) n
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)
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

6
Setoids/Orders.agda
View File

@@ -12,7 +12,7 @@ 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
<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)
@@ -20,6 +20,8 @@ module Setoids.Orders 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
@@ -32,7 +34,7 @@ module Setoids.Orders where
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.<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