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Maybe FieldOfFractions.Order slightly faster (#121)
This commit is contained in:
Patrick Stevens
GitHub
@@ -18,6 +18,7 @@ module Fields.FieldOfFractions.Order {a b c : _} {A : Set a} {S : Setoid {a} {b}
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open import Fields.FieldOfFractions.Setoid I
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open import Fields.FieldOfFractions.Ring I
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open import Fields.FieldOfFractions.Addition I
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open import Fields.FieldOfFractions.Multiplication I
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open import Fields.FieldOfFractions.Lemmas I
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open Ring R
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@@ -37,31 +38,6 @@ fieldOfFractionsComparison (record { num = numA ; denom = denomA ; denomNonzero
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fieldOfFractionsComparison (record { num = numA ; denom = denomA ; denomNonzero = denomA!=0 }) (record { num = numB ; denom = denomB ; denomNonzero = denomB!=0 }) | inl (inr denomA<0) ,, inr 0=denomB = exFalso (denomB!=0 (symmetric 0=denomB))
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fieldOfFractionsComparison (record { num = numA ; denom = denomA ; denomNonzero = denomA!=0 }) (record { num = numB ; denom = denomB ; denomNonzero = denomB!=0 }) | inr 0=denomA ,, _ = exFalso (denomA!=0 (symmetric 0=denomA))
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{-
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fieldOfFractionsComparison : Rel fieldOfFractionsSet
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fieldOfFractionsComparison (record { num = numA ; denom = denomA ; denomNonzero = denomA!=0 }) (record { num = numB ; denom = denomB ; denomNonzero = denomB!=0 }) with totality (Ring.0R R) denomA
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fieldOfFractionsComparison (record { num = numA ; denom = denomA ; denomNonzero = denomA!=0 }) (record { num = numB ; denom = denomB ; denomNonzero = denomB!=0 }) | inl (inl 0<denomA) with totality (Ring.0R R) denomB
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fieldOfFractionsComparison (record { num = numA ; denom = denomA ; denomNonzero = denomA!=0 }) (record { num = numB ; denom = denomB ; denomNonzero = denomB!=0 }) | inl (inl 0<denomA) | inl (inl 0<denomB) = (numA * denomB) < (numB * denomA)
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fieldOfFractionsComparison (record { num = numA ; denom = denomA ; denomNonzero = denomA!=0 }) (record { num = numB ; denom = denomB ; denomNonzero = denomB!=0 }) | inl (inl 0<denomA) | inl (inr denomB<0) = (numB * denomA) < (numA * denomB)
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fieldOfFractionsComparison (record { num = numA ; denom = denomA ; denomNonzero = denomA!=0 }) (record { num = numB ; denom = denomB ; denomNonzero = denomB!=0 }) | inl (inl 0<denomA) | inr 0=denomB = exFalso (denomB!=0 (symmetric 0=denomB))
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fieldOfFractionsComparison (record { num = numA ; denom = denomA ; denomNonzero = denomA!=0 }) (record { num = numB ; denom = denomB ; denomNonzero = denomB!=0 }) | inl (inr denomA<0) with totality (Ring.0R R) denomB
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fieldOfFractionsComparison (record { num = numA ; denom = denomA ; denomNonzero = denomA!=0 }) (record { num = numB ; denom = denomB ; denomNonzero = denomB!=0 }) | inl (inr denomA<0) | inl (inl 0<denomB) = (numB * denomA) < (numA * denomB)
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fieldOfFractionsComparison (record { num = numA ; denom = denomA ; denomNonzero = denomA!=0 }) (record { num = numB ; denom = denomB ; denomNonzero = denomB!=0 }) | inl (inr denomA<0) | inl (inr denomB<0) = (numA * denomB) < (numB * denomA)
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fieldOfFractionsComparison (record { num = numA ; denom = denomA ; denomNonzero = denomA!=0 }) (record { num = numB ; denom = denomB ; denomNonzero = denomB!=0 }) | inl (inr denomA<0) | inr 0=denomB = exFalso (denomB!=0 (symmetric 0=denomB))
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fieldOfFractionsComparison (record { num = numA ; denom = denomA ; denomNonzero = denomA!=0 }) (record { num = numB ; denom = denomB ; denomNonzero = denomB!=0 }) | inr 0=denomA = exFalso (denomA!=0 (symmetric 0=denomA))
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fieldOfFractionsComparison : Rel fieldOfFractionsSet
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fieldOfFractionsComparison (record { num = numA ; denom = denomA ; denomNonzero = denomA!=0 }) (record { num = numB ; denom = denomB ; denomNonzero = denomB!=0 }) with totality (Ring.0R R) denomA
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fieldOfFractionsComparison (record { num = numA ; denom = denomA ; denomNonzero = denomA!=0 }) (record { num = numB ; denom = denomB ; denomNonzero = denomB!=0 }) | inl (inl 0<denomA) with totality (Ring.0R R) denomB
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fieldOfFractionsComparison (record { num = numA ; denom = denomA ; denomNonzero = denomA!=0 }) (record { num = numB ; denom = denomB ; denomNonzero = denomB!=0 }) | inl (inl 0<denomA) | inl (inl 0<denomB) = (numA * denomB) < (numB * denomA)
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fieldOfFractionsComparison (record { num = numA ; denom = denomA ; denomNonzero = denomA!=0 }) (record { num = numB ; denom = denomB ; denomNonzero = denomB!=0 }) | inl (inl 0<denomA) | inl (inr denomB<0) = (numB * denomA) < (numA * denomB)
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fieldOfFractionsComparison (record { num = numA ; denom = denomA ; denomNonzero = denomA!=0 }) (record { num = numB ; denom = denomB ; denomNonzero = denomB!=0 }) | inl (inl 0<denomA) | inr 0=denomB = exFalso (denomB!=0 (symmetric 0=denomB))
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fieldOfFractionsComparison (record { num = numA ; denom = denomA ; denomNonzero = denomA!=0 }) (record { num = numB ; denom = denomB ; denomNonzero = denomB!=0 }) | inl (inr denomA<0) with totality (Ring.0R R) denomB
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fieldOfFractionsComparison (record { num = numA ; denom = denomA ; denomNonzero = denomA!=0 }) (record { num = numB ; denom = denomB ; denomNonzero = denomB!=0 }) | inl (inr denomA<0) | inl (inl 0<denomB) = (numB * denomA) < (numA * denomB)
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fieldOfFractionsComparison (record { num = numA ; denom = denomA ; denomNonzero = denomA!=0 }) (record { num = numB ; denom = denomB ; denomNonzero = denomB!=0 }) | inl (inr denomA<0) | inl (inr denomB<0) = (numA * denomB) < (numB * denomA)
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fieldOfFractionsComparison (record { num = numA ; denom = denomA ; denomNonzero = denomA!=0 }) (record { num = numB ; denom = denomB ; denomNonzero = denomB!=0 }) | inl (inr denomA<0) | inr 0=denomB = exFalso (denomB!=0 (symmetric 0=denomB))
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fieldOfFractionsComparison (record { num = numA ; denom = denomA ; denomNonzero = denomA!=0 }) (record { num = numB ; denom = denomB ; denomNonzero = denomB!=0 }) | inr 0=denomA = exFalso (denomA!=0 (symmetric 0=denomA))
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-}
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private
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abstract
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@@ -328,29 +304,23 @@ private
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<orderRespectsAddition (record { num = numA ; denom = denomA ; denomNonzero = denomA!=0 }) (record { num = numB ; denom = denomB ; denomNonzero = denomB!=0 }) a<b (record { num = numC ; denom = denomC ; denomNonzero = denomC!=0 }) | inl (inl 0<dAdC) with totality (Ring.0R R) (denomB * denomC)
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<orderRespectsAddition (record { num = numA ; denom = denomA ; denomNonzero = denomA!=0 }) (record { num = numB ; denom = denomB ; denomNonzero = denomB!=0 }) a<b (record { num = numC ; denom = denomC ; denomNonzero = denomC!=0 }) | inl (inl 0<dAdC) | inl (inl 0<dBdC) with totality (Ring.0R R) denomA
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<orderRespectsAddition (record { num = numA ; denom = denomA ; denomNonzero = denomA!=0 }) (record { num = numB ; denom = denomB ; denomNonzero = denomB!=0 }) a<b (record { num = numC ; denom = denomC ; denomNonzero = denomC!=0 }) | inl (inl 0<dAdC) | inl (inl 0<dBdC) | inl (inl 0<dA) with totality (Ring.0R R) denomB
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<orderRespectsAddition (record { num = numA ; denom = denomA ; denomNonzero = denomA!=0 }) (record { num = numB ; denom = denomB ; denomNonzero = denomB!=0 }) a<b (record { num = numC ; denom = denomC ; denomNonzero = 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 pRing 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))
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<orderRespectsAddition (record { num = numA ; denom = denomA ; denomNonzero = denomA!=0 }) (record { num = numB ; denom = denomB ; denomNonzero = denomB!=0 }) a<b (record { num = numC ; denom = denomC ; denomNonzero = 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 pRing 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)) (SetoidPartialOrder.<WellDefined pOrder reflexive (Group.+WellDefined additiveGroup reflexive (transitive (*WellDefined *Commutative reflexive) (transitive (transitive (symmetric *Associative) (transitive (*WellDefined reflexive *Commutative) *Associative)) (*WellDefined *Commutative reflexive)))) (PartiallyOrderedRing.orderRespectsAddition pRing (SetoidPartialOrder.<WellDefined pOrder (swapLemma) (swapLemma) (ringCanMultiplyByPositive pRing 0<dC a<b)) ((denomA * numC) * denomB)))))
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where
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0<dC : 0R < denomC
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0<dC with totality 0R denomC
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0<dC | inl (inl x) = x
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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 pRing dC<0 0<dB))))
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0<dC | inr x = exFalso (denomC!=0 (Equivalence.symmetric (Setoid.eq S) x))
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p : ((numA * denomC) * denomB) < ((numB * denomC) * denomA)
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p = SetoidPartialOrder.<WellDefined pOrder (swapLemma) (swapLemma) (ringCanMultiplyByPositive pRing 0<dC a<b)
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ans : ((((numA * denomC) * denomB) + ((denomA * numC) * denomB))) < ((((numB * denomC) * denomA) + ((denomB * numC) * denomA)))
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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)))) (PartiallyOrderedRing.orderRespectsAddition pRing p ((denomA * numC) * denomB))
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<orderRespectsAddition (record { num = numA ; denom = denomA ; denomNonzero = denomA!=0 }) (record { num = numB ; denom = denomB ; denomNonzero = denomB!=0 }) a<b (record { num = numC ; denom = denomC ; denomNonzero = denomC!=0 }) | inl (inl 0<dAdC) | inl (inl 0<dBdC) | inl (inl 0<dA) | inl (inr dB<0) = exFalso bad
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<orderRespectsAddition (record { num = numA ; denom = denomA ; denomNonzero = denomA!=0 }) (record { num = numB ; denom = denomB ; denomNonzero = denomB!=0 }) a<b (record { num = numC ; denom = denomC ; denomNonzero = denomC!=0 }) | inl (inl 0<dAdC) | inl (inl 0<dBdC) | inl (inl 0<dA) | inl (inr dB<0) = exFalso (SetoidPartialOrder.irreflexive pOrder (SetoidPartialOrder.<Transitive pOrder 0<dAdC (SetoidPartialOrder.<WellDefined pOrder reflexive (transitive *Commutative (Ring.timesZero R)) (ringCanMultiplyByNegative pRing dC<0 0<dA))))
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where
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dC<0 : denomC < 0R
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dC<0 with totality 0R denomC
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... | inl (inl x) = exFalso (SetoidPartialOrder.irreflexive pOrder (SetoidPartialOrder.<Transitive pOrder 0<dBdC (SetoidPartialOrder.<WellDefined pOrder reflexive (transitive *Commutative (Ring.timesZero R)) (ringCanMultiplyByPositive pRing x dB<0))))
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... | inl (inr x) = x
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... | inr x = exFalso (denomC!=0 (Equivalence.symmetric (Setoid.eq S) x))
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bad : False
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bad = SetoidPartialOrder.irreflexive pOrder (SetoidPartialOrder.<Transitive pOrder 0<dAdC (SetoidPartialOrder.<WellDefined pOrder reflexive (transitive *Commutative (Ring.timesZero R)) (ringCanMultiplyByNegative pRing dC<0 0<dA)))
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<orderRespectsAddition (record { num = numA ; denom = denomA ; denomNonzero = denomA!=0 }) (record { num = numB ; denom = denomB ; denomNonzero = denomB!=0 }) a<b (record { num = numC ; denom = denomC ; denomNonzero = 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))
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<orderRespectsAddition (record { num = numA ; denom = denomA ; denomNonzero = denomA!=0 }) (record { num = numB ; denom = denomB ; denomNonzero = denomB!=0 }) a<b (record { num = numC ; denom = denomC ; denomNonzero = denomC!=0 }) | inl (inl 0<dAdC) | inl (inl 0<dBdC) | inl (inr dA<0) with totality (Ring.0R R) denomB
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<orderRespectsAddition (record { num = numA ; denom = denomA ; denomNonzero = denomA!=0 }) (record { num = numB ; denom = denomB ; denomNonzero = denomB!=0 }) a<b (record { num = numC ; denom = denomC ; denomNonzero = denomC!=0 }) | inl (inl 0<dAdC) | inl (inl 0<dBdC) | inl (inr dA<0) | inl (inl 0<dB) = exFalso bad
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<orderRespectsAddition (record { num = numA ; denom = denomA ; denomNonzero = denomA!=0 }) (record { num = numB ; denom = denomB ; denomNonzero = denomB!=0 }) a<b (record { num = numC ; denom = denomC ; denomNonzero = denomC!=0 }) | inl (inl 0<dAdC) | inl (inl 0<dBdC) | inl (inr dA<0) | inl (inl 0<dB) = exFalso (SetoidPartialOrder.irreflexive pOrder (SetoidPartialOrder.<Transitive pOrder 0<dC dC<0))
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where
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0<dC : 0R < denomC
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0<dC with totality 0R denomC
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@@ -362,14 +332,13 @@ private
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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 pRing dA<0 0<dC))))
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dC<0 | inl (inr x) = x
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dC<0 | inr x = exFalso (denomC!=0 (Equivalence.symmetric (Setoid.eq S) x))
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bad : False
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bad = SetoidPartialOrder.irreflexive pOrder (SetoidPartialOrder.<Transitive pOrder 0<dC dC<0)
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<orderRespectsAddition (record { num = numA ; denom = denomA ; denomNonzero = denomA!=0 }) (record { num = numB ; denom = denomB ; denomNonzero = denomB!=0 }) a<b (record { num = numC ; denom = denomC ; denomNonzero = 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 pRing dC<0 (SetoidPartialOrder.<WellDefined pOrder (transitive (symmetric *DistributesOver+) *Commutative) (transitive (symmetric *DistributesOver+) *Commutative) have''))
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where
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dC<0 : denomC < 0R
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dC<0 = ineqLemma' 0<dAdC dA<0
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have : ((numB * denomA) * denomC) < ((numA * denomB) * denomC)
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have = ringCanMultiplyByNegative pRing dC<0 a<b
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have' : (denomA * (numB * denomC)) < (denomB * (numA * denomC))
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have' = SetoidPartialOrder.<WellDefined pOrder (transitive (*WellDefined *Commutative reflexive) (symmetric *Associative)) (transitive (*WellDefined *Commutative reflexive) (symmetric *Associative)) have
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have'' : ((denomA * (numB * denomC)) + (denomA * (denomB * numC))) < ((denomB * (numA * denomC)) + (denomB * (denomA * numC)))
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@@ -482,71 +451,37 @@ private
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<orderRespectsAddition (record { num = numA ; denom = denomA ; denomNonzero = denomA!=0 }) (record { num = numB ; denom = denomB ; denomNonzero = denomB!=0 }) a<b (record { num = numC ; denom = denomC ; denomNonzero = denomC!=0 }) | inr (0=dAdC) with IntegralDomain.intDom I (Equivalence.symmetric (Setoid.eq S) 0=dAdC)
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<orderRespectsAddition (record { num = numA ; denom = denomA ; denomNonzero = denomA!=0 }) (record { num = numB ; denom = denomB ; denomNonzero = denomB!=0 }) a<b (record { num = numC ; denom = denomC ; denomNonzero = denomC!=0 }) | inr 0=dAdC | f = exFalso (denomC!=0 (f denomA!=0))
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private
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<RespectsMultiplication : {a b : fieldOfFractionsSet} → fieldOfFractionsComparison (Ring.0R fieldOfFractionsRing) a → fieldOfFractionsComparison (Ring.0R fieldOfFractionsRing) b → fieldOfFractionsComparison (Ring.0R fieldOfFractionsRing) (fieldOfFractionsTimes a b)
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<RespectsMultiplication {record { num = numA ; denom = denomA ; denomNonzero = denomA!=0 }} {record { num = numB ; denom = denomB ; denomNonzero = denomB!=0 }} t u with totality (Ring.0R R) (Ring.1R R)
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<RespectsMultiplication {record { num = numA ; denom = denomA ; denomNonzero = denomA!=0 }} {record { num = numB ; denom = denomB ; denomNonzero = denomB!=0 }} 0<a 0<b | inl (inl 0<1) with totality (Ring.0R R) (denomA * denomB)
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<RespectsMultiplication {record { num = numA ; denom = denomA ; denomNonzero = denomA!=0 }} {record { num = numB ; denom = denomB ; denomNonzero = denomB!=0 }} 0<a 0<b | inl (inl 0<1) | inl (inl 0<dAdB) with totality (Ring.0R R) denomB
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<RespectsMultiplication {record { num = numA ; denom = denomA ; denomNonzero = denomA!=0 }} {record { num = numB ; denom = denomB ; denomNonzero = denomB!=0 }} 0<a 0<b | inl (inl 0<1) | inl (inl 0<dAdB) | inl (inl 0<dB) with totality (Ring.0R R) denomA
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<RespectsMultiplication {record { num = numA ; denom = denomA ; denomNonzero = denomA!=0 }} {record { num = numB ; denom = denomB ; denomNonzero = 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)) (SetoidPartialOrder.<WellDefined pOrder (transitive *Commutative (Ring.timesZero R)) reflexive (ringCanMultiplyByPositive pRing (SetoidPartialOrder.<WellDefined pOrder (transitive *Commutative (Ring.timesZero R)) (transitive *Commutative identIsIdent) 0<b) (SetoidPartialOrder.<WellDefined pOrder (transitive *Commutative (Ring.timesZero R)) (transitive *Commutative identIsIdent) 0<a)))
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<RespectsMultiplication {record { num = numA ; denom = denomA ; denomNonzero = denomA!=0 }} {record { num = numB ; denom = denomB ; denomNonzero = 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 pRing dA<0 0<dB))))
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<RespectsMultiplication {record { num = numA ; denom = denomA ; denomNonzero = denomA!=0 }} {record { num = numB ; denom = denomB ; denomNonzero = 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))
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<RespectsMultiplication {record { num = numA ; denom = denomA ; denomNonzero = denomA!=0 }} {record { num = numB ; denom = denomB ; denomNonzero = denomB!=0 }} 0<a 0<b | inl (inl 0<1) | inl (inl 0<dAdB) | inl (inr dB<0) with totality (Ring.0R R) denomA
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<RespectsMultiplication {record { num = numA ; denom = denomA ; denomNonzero = denomA!=0 }} {record { num = numB ; denom = denomB ; denomNonzero = 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 pRing dB<0 0<dA))))
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<RespectsMultiplication {record { num = numA ; denom = denomA ; denomNonzero = denomA!=0 }} {record { num = numB ; denom = denomB ; denomNonzero = 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)) (SetoidPartialOrder.<WellDefined pOrder (transitive *Commutative (Ring.timesZero R)) *Commutative (ringCanMultiplyByNegative pRing (SetoidPartialOrder.<WellDefined pOrder (transitive *Commutative identIsIdent) (transitive *Commutative (Ring.timesZero R)) 0<a) (SetoidPartialOrder.<WellDefined pOrder (transitive *Commutative identIsIdent) (transitive *Commutative (Ring.timesZero R)) 0<b)))
|
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<RespectsMultiplication {record { num = numA ; denom = denomA ; denomNonzero = denomA!=0 }} {record { num = numB ; denom = denomB ; denomNonzero = 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))
|
||||
<RespectsMultiplication {record { num = numA ; denom = denomA ; denomNonzero = denomA!=0 }} {record { num = numB ; denom = denomB ; denomNonzero = 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))
|
||||
<RespectsMultiplication {record { num = numA ; denom = denomA ; denomNonzero = denomA!=0 }} {record { num = numB ; denom = denomB ; denomNonzero = denomB!=0 }} 0<a 0<b | inl (inl 0<1) | inl (inr dAdB<0) with totality (Ring.0R R) denomB
|
||||
<RespectsMultiplication {record { num = numA ; denom = denomA ; denomNonzero = denomA!=0 }} {record { num = numB ; denom = denomB ; denomNonzero = denomB!=0 }} 0<a 0<b | inl (inl 0<1) | inl (inr dAdB<0) | inl (inl 0<denomB) with totality (Ring.0R R) denomA
|
||||
<RespectsMultiplication {record { num = numA ; denom = denomA ; denomNonzero = denomA!=0 }} {record { num = numB ; denom = denomB ; denomNonzero = denomB!=0 }} 0<a 0<b | inl (inl 0<1) | inl (inr dAdB<0) | inl (inl 0<denomB) | inl (inl 0<denomA) = exFalso (SetoidPartialOrder.irreflexive pOrder (SetoidPartialOrder.<Transitive pOrder (PartiallyOrderedRing.orderRespectsMultiplication pRing 0<denomA 0<denomB) dAdB<0))
|
||||
<RespectsMultiplication {record { num = numA ; denom = denomA ; denomNonzero = denomA!=0 }} {record { num = numB ; denom = denomB ; denomNonzero = 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))) (SetoidPartialOrder.<WellDefined pOrder *Commutative (transitive *Commutative (Ring.timesZero R)) (ringCanMultiplyByNegative pRing (SetoidPartialOrder.<WellDefined pOrder (transitive *Commutative identIsIdent) (transitive *Commutative (Ring.timesZero R)) 0<a) (SetoidPartialOrder.<WellDefined pOrder (transitive *Commutative (Ring.timesZero R)) (transitive *Commutative identIsIdent) 0<b)))
|
||||
<RespectsMultiplication {record { num = numA ; denom = denomA ; denomNonzero = denomA!=0 }} {record { num = numB ; denom = denomB ; denomNonzero = 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))
|
||||
<RespectsMultiplication {record { num = numA ; denom = denomA ; denomNonzero = denomA!=0 }} {record { num = numB ; denom = denomB ; denomNonzero = denomB!=0 }} 0<a 0<b | inl (inl 0<1) | inl (inr dAdB<0) | inl (inr denomB<0) with totality (Ring.0R R) denomA
|
||||
<RespectsMultiplication {record { num = numA ; denom = denomA ; denomNonzero = denomA!=0 }} {record { num = numB ; denom = denomB ; denomNonzero = 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))) (SetoidPartialOrder.<WellDefined pOrder reflexive (transitive *Commutative (Ring.timesZero R)) (ringCanMultiplyByNegative pRing (SetoidPartialOrder.<WellDefined pOrder (transitive *Commutative identIsIdent) (transitive *Commutative (Ring.timesZero R)) 0<b) (SetoidPartialOrder.<WellDefined pOrder (transitive *Commutative (Ring.timesZero R)) (transitive *Commutative identIsIdent) 0<a)))
|
||||
<RespectsMultiplication {record { num = numA ; denom = denomA ; denomNonzero = denomA!=0 }} {record { num = numB ; denom = denomB ; denomNonzero = denomB!=0 }} 0<a 0<b | inl (inl 0<1) | inl (inr dAdB<0) | inl (inr denomB<0) | inl (inr denomA<0) = exFalso (SetoidPartialOrder.irreflexive pOrder (SetoidPartialOrder.<Transitive pOrder dAdB<0 (SetoidPartialOrder.<WellDefined pOrder (transitive *Commutative (Ring.timesZero R)) reflexive (ringCanMultiplyByNegative pRing denomB<0 denomA<0))))
|
||||
<RespectsMultiplication {record { num = numA ; denom = denomA ; denomNonzero = denomA!=0 }} {record { num = numB ; denom = denomB ; denomNonzero = 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))
|
||||
<RespectsMultiplication {record { num = numA ; denom = denomA ; denomNonzero = denomA!=0 }} {record { num = numB ; denom = denomB ; denomNonzero = denomB!=0 }} 0<a 0<b | inl (inl 0<1) | inl (inr dAdB<0) | inr x = exFalso (denomB!=0 (Equivalence.symmetric (Setoid.eq S) x))
|
||||
<RespectsMultiplication {record { num = numA ; denom = denomA ; denomNonzero = denomA!=0 }} {record { num = numB ; denom = denomB ; denomNonzero = denomB!=0 }} 0<a 0<b | inl (inl 0<1) | inr 0=dAdB = exFalso (denomB!=0 (IntegralDomain.intDom I (Equivalence.symmetric (Setoid.eq S) 0=dAdB) denomA!=0))
|
||||
<RespectsMultiplication {record { num = numA ; denom = denomA ; denomNonzero = denomA!=0 }} {record { num = numB ; denom = denomB ; denomNonzero = 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 pRing 1<0 1<0))))
|
||||
<RespectsMultiplication {record { num = numA ; denom = denomA ; denomNonzero = denomA!=0 }} {record { num = numB ; denom = denomB ; denomNonzero = denomB!=0 }} 0<a 0<b | inr x = exFalso (IntegralDomain.nontrivial I (Equivalence.symmetric (Setoid.eq S) x))
|
||||
|
||||
fieldOfFractionsPOrderedRing : PartiallyOrderedRing fieldOfFractionsRing (SetoidTotalOrder.partial fieldOfFractionsTotalOrder)
|
||||
PartiallyOrderedRing.orderRespectsAddition fieldOfFractionsPOrderedRing {a} {b} a<b c = <orderRespectsAddition a b a<b c
|
||||
PartiallyOrderedRing.orderRespectsMultiplication (fieldOfFractionsPOrderedRing) {record { num = numA ; denom = denomA ; denomNonzero = denomA!=0 }} {record { num = numB ; denom = denomB ; denomNonzero = denomB!=0 }} t u with totality (Ring.0R R) (Ring.1R R)
|
||||
PartiallyOrderedRing.orderRespectsMultiplication (fieldOfFractionsPOrderedRing) {record { num = numA ; denom = denomA ; denomNonzero = denomA!=0 }} {record { num = numB ; denom = denomB ; denomNonzero = denomB!=0 }} 0<a 0<b | inl (inl 0<1) with totality (Ring.0R R) (denomA * denomB)
|
||||
PartiallyOrderedRing.orderRespectsMultiplication (fieldOfFractionsPOrderedRing) {record { num = numA ; denom = denomA ; denomNonzero = denomA!=0 }} {record { num = numB ; denom = denomB ; denomNonzero = denomB!=0 }} 0<a 0<b | inl (inl 0<1) | inl (inl 0<dAdB) with totality (Ring.0R R) denomB
|
||||
PartiallyOrderedRing.orderRespectsMultiplication (fieldOfFractionsPOrderedRing) {record { num = numA ; denom = denomA ; denomNonzero = denomA!=0 }} {record { num = numB ; denom = denomB ; denomNonzero = denomB!=0 }} 0<a 0<b | inl (inl 0<1) | inl (inl 0<dAdB) | inl (inl 0<dB) with totality (Ring.0R R) denomA
|
||||
PartiallyOrderedRing.orderRespectsMultiplication (fieldOfFractionsPOrderedRing) {record { num = numA ; denom = denomA ; denomNonzero = denomA!=0 }} {record { num = numB ; denom = denomB ; denomNonzero = 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
|
||||
0<nA : 0R < numA
|
||||
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<nAnB : 0R < (numA * numB)
|
||||
0<nAnB = SetoidPartialOrder.<WellDefined pOrder (transitive *Commutative (Ring.timesZero R)) reflexive (ringCanMultiplyByPositive pRing 0<nB 0<nA)
|
||||
PartiallyOrderedRing.orderRespectsMultiplication (fieldOfFractionsPOrderedRing) {record { num = numA ; denom = denomA ; denomNonzero = denomA!=0 }} {record { num = numB ; denom = denomB ; denomNonzero = 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 pRing dA<0 0<dB))))
|
||||
PartiallyOrderedRing.orderRespectsMultiplication (fieldOfFractionsPOrderedRing) {record { num = numA ; denom = denomA ; denomNonzero = denomA!=0 }} {record { num = numB ; denom = denomB ; denomNonzero = 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))
|
||||
PartiallyOrderedRing.orderRespectsMultiplication (fieldOfFractionsPOrderedRing) {record { num = numA ; denom = denomA ; denomNonzero = denomA!=0 }} {record { num = numB ; denom = denomB ; denomNonzero = denomB!=0 }} 0<a 0<b | inl (inl 0<1) | inl (inl 0<dAdB) | inl (inr dB<0) with totality (Ring.0R R) denomA
|
||||
PartiallyOrderedRing.orderRespectsMultiplication (fieldOfFractionsPOrderedRing) {record { num = numA ; denom = denomA ; denomNonzero = denomA!=0 }} {record { num = numB ; denom = denomB ; denomNonzero = 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 pRing dB<0 0<dA))))
|
||||
PartiallyOrderedRing.orderRespectsMultiplication (fieldOfFractionsPOrderedRing) {record { num = numA ; denom = denomA ; denomNonzero = denomA!=0 }} {record { num = numB ; denom = denomB ; denomNonzero = 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
|
||||
nB<0 : numB < 0R
|
||||
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
|
||||
0<nAnB : 0R < (numA * numB)
|
||||
0<nAnB = SetoidPartialOrder.<WellDefined pOrder (transitive *Commutative (Ring.timesZero R)) *Commutative (ringCanMultiplyByNegative pRing nA<0 nB<0)
|
||||
PartiallyOrderedRing.orderRespectsMultiplication (fieldOfFractionsPOrderedRing) {record { num = numA ; denom = denomA ; denomNonzero = denomA!=0 }} {record { num = numB ; denom = denomB ; denomNonzero = 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))
|
||||
PartiallyOrderedRing.orderRespectsMultiplication (fieldOfFractionsPOrderedRing) {record { num = numA ; denom = denomA ; denomNonzero = denomA!=0 }} {record { num = numB ; denom = denomB ; denomNonzero = 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))
|
||||
PartiallyOrderedRing.orderRespectsMultiplication (fieldOfFractionsPOrderedRing) {record { num = numA ; denom = denomA ; denomNonzero = denomA!=0 }} {record { num = numB ; denom = denomB ; denomNonzero = denomB!=0 }} 0<a 0<b | inl (inl 0<1) | inl (inr dAdB<0) with totality (Ring.0R R) denomB
|
||||
PartiallyOrderedRing.orderRespectsMultiplication (fieldOfFractionsPOrderedRing) {record { num = numA ; denom = denomA ; denomNonzero = denomA!=0 }} {record { num = numB ; denom = denomB ; denomNonzero = denomB!=0 }} 0<a 0<b | inl (inl 0<1) | inl (inr dAdB<0) | inl (inl 0<denomB) with totality (Ring.0R R) denomA
|
||||
PartiallyOrderedRing.orderRespectsMultiplication (fieldOfFractionsPOrderedRing) {record { num = numA ; denom = denomA ; denomNonzero = denomA!=0 }} {record { num = numB ; denom = denomB ; denomNonzero = denomB!=0 }} 0<a 0<b | inl (inl 0<1) | inl (inr dAdB<0) | inl (inl 0<denomB) | inl (inl 0<denomA) = exFalso f
|
||||
where
|
||||
f : False
|
||||
f with PartiallyOrderedRing.orderRespectsMultiplication pRing 0<denomA 0<denomB
|
||||
... | bl = SetoidPartialOrder.irreflexive pOrder (SetoidPartialOrder.<Transitive pOrder bl dAdB<0)
|
||||
PartiallyOrderedRing.orderRespectsMultiplication (fieldOfFractionsPOrderedRing) {record { num = numA ; denom = denomA ; denomNonzero = denomA!=0 }} {record { num = numB ; denom = denomB ; denomNonzero = 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
|
||||
0<nB : 0R < numB
|
||||
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
|
||||
ans : (numA * numB) < 0R
|
||||
ans = SetoidPartialOrder.<WellDefined pOrder *Commutative (transitive *Commutative (Ring.timesZero R)) (ringCanMultiplyByNegative pRing nA<0 0<nB)
|
||||
PartiallyOrderedRing.orderRespectsMultiplication (fieldOfFractionsPOrderedRing) {record { num = numA ; denom = denomA ; denomNonzero = denomA!=0 }} {record { num = numB ; denom = denomB ; denomNonzero = 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))
|
||||
PartiallyOrderedRing.orderRespectsMultiplication (fieldOfFractionsPOrderedRing) {record { num = numA ; denom = denomA ; denomNonzero = denomA!=0 }} {record { num = numB ; denom = denomB ; denomNonzero = denomB!=0 }} 0<a 0<b | inl (inl 0<1) | inl (inr dAdB<0) | inl (inr denomB<0) with totality (Ring.0R R) denomA
|
||||
PartiallyOrderedRing.orderRespectsMultiplication (fieldOfFractionsPOrderedRing) {record { num = numA ; denom = denomA ; denomNonzero = denomA!=0 }} {record { num = numB ; denom = denomB ; denomNonzero = 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
|
||||
nB<0 : numB < 0R
|
||||
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
|
||||
nAnB<0 : (numA * numB) < 0R
|
||||
nAnB<0 = SetoidPartialOrder.<WellDefined pOrder reflexive (transitive *Commutative (Ring.timesZero R)) (ringCanMultiplyByNegative pRing nB<0 0<nA)
|
||||
PartiallyOrderedRing.orderRespectsMultiplication (fieldOfFractionsPOrderedRing) {record { num = numA ; denom = denomA ; denomNonzero = denomA!=0 }} {record { num = numB ; denom = denomB ; denomNonzero = 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
|
||||
h : 0R < (denomA * denomB)
|
||||
h = SetoidPartialOrder.<WellDefined pOrder (transitive *Commutative (Ring.timesZero R)) reflexive (ringCanMultiplyByNegative pRing denomB<0 denomA<0)
|
||||
f : False
|
||||
f = SetoidPartialOrder.irreflexive pOrder (SetoidPartialOrder.<Transitive pOrder dAdB<0 h)
|
||||
PartiallyOrderedRing.orderRespectsMultiplication (fieldOfFractionsPOrderedRing) {record { num = numA ; denom = denomA ; denomNonzero = denomA!=0 }} {record { num = numB ; denom = denomB ; denomNonzero = 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))
|
||||
PartiallyOrderedRing.orderRespectsMultiplication (fieldOfFractionsPOrderedRing) {record { num = numA ; denom = denomA ; denomNonzero = denomA!=0 }} {record { num = numB ; denom = denomB ; denomNonzero = denomB!=0 }} 0<a 0<b | inl (inl 0<1) | inl (inr dAdB<0) | inr x = exFalso (denomB!=0 (Equivalence.symmetric (Setoid.eq S) x))
|
||||
PartiallyOrderedRing.orderRespectsMultiplication (fieldOfFractionsPOrderedRing) {record { num = numA ; denom = denomA ; denomNonzero = denomA!=0 }} {record { num = numB ; denom = denomB ; denomNonzero = denomB!=0 }} 0<a 0<b | inl (inl 0<1) | inr 0=dAdB with IntegralDomain.intDom I (Equivalence.symmetric (Setoid.eq S) 0=dAdB)
|
||||
... | f = exFalso (denomB!=0 (f denomA!=0))
|
||||
PartiallyOrderedRing.orderRespectsMultiplication (fieldOfFractionsPOrderedRing) {record { num = numA ; denom = denomA ; denomNonzero = denomA!=0 }} {record { num = numB ; denom = denomB ; denomNonzero = 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 pRing 1<0 1<0))))
|
||||
PartiallyOrderedRing.orderRespectsMultiplication (fieldOfFractionsPOrderedRing) {record { num = numA ; denom = denomA ; denomNonzero = denomA!=0 }} {record { num = numB ; denom = denomB ; denomNonzero = denomB!=0 }} 0<a 0<b | inr x = exFalso (IntegralDomain.nontrivial I (Equivalence.symmetric (Setoid.eq S) x))
|
||||
PartiallyOrderedRing.orderRespectsMultiplication fieldOfFractionsPOrderedRing {a} {b} 0<a 0<b = <RespectsMultiplication {a} {b} 0<a 0<b
|
||||
|
||||
fieldOfFractionsOrderedRing : TotallyOrderedRing fieldOfFractionsPOrderedRing
|
||||
TotallyOrderedRing.total fieldOfFractionsOrderedRing = fieldOfFractionsTotalOrder
|
||||
|
||||
Reference in New Issue
Block a user