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Rename some confusing fields (#51)

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This commit is contained in:
Patrick Stevens
2019-10-13 09:31:54 +01:00
committed by GitHub
parent 96d15c6017
commit 959071214e
21 changed files with 441 additions and 441 deletions
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14
Numbers/Integers/Addition.agda
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@@ -76,14 +76,14 @@ additiveInverse (nonneg (succ x)) = negSucc x
additiveInverse (negSucc x) = nonneg (succ x)
ℤGroup : Group (reflSetoid ℤ) (_+Z_)
Group.wellDefined ℤGroup refl refl = refl
Group.identity ℤGroup = nonneg 0
Group.+WellDefined ℤGroup refl refl = refl
Group.0G ℤGroup = nonneg 0
Group.inverse ℤGroup = additiveInverse
Group.multAssoc ℤGroup {a} {b} {c} = +ZAssociative a b c
Group.multIdentRight ℤGroup {nonneg zero} = refl
Group.multIdentRight ℤGroup {nonneg (succ x)} = applyEquality (λ i → nonneg (succ i)) (Semiring.commutative ℕSemiring x 0)
Group.multIdentRight ℤGroup {negSucc x} = refl
Group.multIdentLeft ℤGroup = refl
Group.+Associative ℤGroup {a} {b} {c} = +ZAssociative a b c
Group.identRight ℤGroup {nonneg zero} = refl
Group.identRight ℤGroup {nonneg (succ x)} = applyEquality (λ i → nonneg (succ i)) (Semiring.commutative ℕSemiring x 0)
Group.identRight ℤGroup {negSucc x} = refl
Group.identLeft ℤGroup = refl
Group.invLeft ℤGroup {nonneg zero} = refl
Group.invLeft ℤGroup {nonneg (succ x)} = additiveInverseExists x
Group.invLeft ℤGroup {negSucc x} = transitivity (+ZCommutative (nonneg (succ x)) (negSucc x)) (additiveInverseExists x)

5
Numbers/Integers/Integers.agda
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@@ -1,5 +1,6 @@
{-# OPTIONS --safe --warning=error --without-K #-}
open import Numbers.Naturals.Naturals
open import Numbers.Integers.Definition
open import Numbers.Integers.Addition
open import Numbers.Integers.Multiplication
@@ -15,3 +16,7 @@ open Numbers.Integers.Order using (_<Z_ ; ℤOrderedRing) public
_-Z_ : ℤ → ℤ → ℤ
a -Z b = a +Z (Group.inverse ℤGroup b)
_^Z_ : ℤ → ℕ → ℤ
a ^Z zero = nonneg 1
a ^Z succ b = a *Z (a ^Z b)

10
Numbers/Integers/Multiplication.agda
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@@ -180,13 +180,13 @@ distLemma a b rewrite Semiring.commutative ℕSemiring (a +N b *N a) b | Semirin
ℤRing : Ring (reflSetoid ℤ) _+Z_ _*Z_
Ring.additiveGroup ℤRing = ℤGroup
Ring.multWellDefined ℤRing refl refl = refl
Ring.*WellDefined ℤRing refl refl = refl
Ring.1R ℤRing = nonneg 1
Ring.groupIsAbelian ℤRing {a} {b} = +ZCommutative a b
Ring.multAssoc ℤRing {a} {b} {c} = *ZAssociative a b c
Ring.multCommutative ℤRing {a} {b} = *ZCommutative a b
Ring.multDistributes ℤRing {a} {b} {c} = *ZDistributesOver+Z a b c
Ring.multIdentIsIdent ℤRing {a} = *ZleftIdent a
Ring.*Associative ℤRing {a} {b} {c} = *ZAssociative a b c
Ring.*Commutative ℤRing {a} {b} = *ZCommutative a b
Ring.*DistributesOver+ ℤRing {a} {b} {c} = *ZDistributesOver+Z a b c
Ring.identIsIdent ℤRing {a} = *ZleftIdent a
intDom : (a b : ℤ) → a *Z b ≡ nonneg 0 → (a ≡ nonneg 0) || (b ≡ nonneg 0)
intDom (nonneg zero) (nonneg b) pr = inl refl

2
Numbers/Modulo/IntegersModN.agda
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@@ -680,7 +680,7 @@ module Numbers.Modulo.IntegersModN where
... | (inr n-a+sa=sn) = equalityZn _ _ refl
ℤnGroup : (n : ℕ) → (pr : 0 <N n) → Group (reflSetoid (ℤn n pr)) _+n_
ℤnGroup n pr = record { identity = record { x = 0 ; xLess = pr } ; inverse = λ a → underlying (inverseZn a) ; multAssoc = λ {a} {b} {c} → plusZnAssociative a b c ; multIdentRight = λ {a} → plusZnIdentityRight a ; multIdentLeft = λ {a} → plusZnIdentityLeft a ; invLeft = λ {a} → helpInvLeft a ; invRight = λ {a} → helpInvRight a ; wellDefined = reflGroupWellDefined }
ℤnGroup n pr = record { 0G = record { x = 0 ; xLess = pr } ; inverse = λ a → underlying (inverseZn a) ; +Associative = λ {a} {b} {c} → plusZnAssociative a b c ; identRight = λ {a} → plusZnIdentityRight a ; identLeft = λ {a} → plusZnIdentityLeft a ; invLeft = λ {a} → helpInvLeft a ; invRight = λ {a} → helpInvRight a ; +WellDefined = reflGroupWellDefined }
where
helpInvLeft : (a : ℤn n pr) → underlying (inverseZn a) +n a ≡ record { x = 0 ; xLess = pr }
helpInvLeft a with inverseZn a

40
Numbers/RationalsLemmas.agda
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@@ -28,12 +28,12 @@ module Numbers.RationalsLemmas where
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.multIdentRight (Ring.additiveGroup ℚRing) {0Q})) (reflexive {a +Q b}) (ringAddInequalities ℚOrdered {0Q} {a} {0Q} {b} 0<a 0<b)))))
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} (multIdentRight {0Q}) (symmetric {0Q} {a +Q b} 0=a+b) (ringAddInequalities ℚOrdered {0Q} {a} {0Q} {b} 0<a 0<b)))
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)
@@ -49,23 +49,23 @@ module Numbers.RationalsLemmas 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} (multIdentLeft {a}) (Ring.groupIsAbelian ℚRing {inverse b} {a}) (OrderedRing.orderRespectsAddition ℚOrdered {0Q} {inverse b} (ringMinusFlipsOrder'' ℚOrdered {b} b<0) a)))
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))
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} (multIdentRight {a})) (wellDefined {a} {0Q} {a} {b} (reflexive {a}) 0=b)) 0<a)))
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} (multIdentRight {a})) (wellDefined {a} {0Q} {a} {b} (reflexive {a}) 0=b)) 0<a))
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)
@@ -84,13 +84,13 @@ module Numbers.RationalsLemmas where
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} (multIdentRight {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))))
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}) (multIdentRight {0Q}) (ringAddInequalities ℚOrdered {a} {0Q} {b} {0Q} a<0 b<0)) 0<a+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)
@@ -100,65 +100,65 @@ module Numbers.RationalsLemmas 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}) (multIdentLeft {0Q}) (ringAddInequalities ℚOrdered {a} {0Q} {b} {0Q} a<0 b<0))))
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)) (multIdentRight {a})) 0<a+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)))))))
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} (multIdentRight {a})) (wellDefined {a} {0Q} {a} {b} (reflexive {a}) 0=b)) (symmetric {0Q} {a +Q b} 0=a+b)) (reflexive {0Q}) a<0))
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}))
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.multIdentLeft (Ring.additiveGroup ℚRing) {b})) (Group.wellDefined (Ring.additiveGroup ℚRing) {0Q} {b} {a} {b} 0=a (reflexive {b}))) 0<b)))
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})) (multIdentLeft {b}))) (reflexive {b}) 0<b))
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})) (multIdentLeft {b})) 0<a+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}))))
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})) (multIdentLeft {b}))) b<0))
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)) (multIdentRight {0Q})) 0<a+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)) (multIdentRight {0Q})) (reflexive {0Q}) a+b<0))
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)