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Rename some confusing fields (#51)
This commit is contained in:
Patrick Stevens
GitHub
@@ -7,6 +7,7 @@ open import Numbers.Naturals.Naturals
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open import Setoids.Orders
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open import Setoids.Setoids
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open import Functions
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open import Sets.EquivalenceRelations
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open import Agda.Primitive using (Level; lzero; lsuc; _⊔_)
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@@ -17,16 +18,21 @@ module Rings.Definition where
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additiveGroup : Group S _+_
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open Group additiveGroup
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open Setoid S
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open Equivalence eq
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0R : A
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0R = identity
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0R = 0G
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_-R_ : A → A → A
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a -R b = a + (inverse b)
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field
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multWellDefined : {r s t u : A} → (r ∼ t) → (s ∼ u) → r * s ∼ t * u
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*WellDefined : {r s t u : A} → (r ∼ t) → (s ∼ u) → r * s ∼ t * u
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1R : A
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groupIsAbelian : {a b : A} → a + b ∼ b + a
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multAssoc : {a b c : A} → (a * (b * c)) ∼ (a * b) * c
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multCommutative : {a b : A} → a * b ∼ b * a
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multDistributes : {a b c : A} → a * (b + c) ∼ (a * b) + (a * c)
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multIdentIsIdent : {a : A} → 1R * a ∼ a
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*Associative : {a b c : A} → (a * (b * c)) ∼ (a * b) * c
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*Commutative : {a b : A} → a * b ∼ b * a
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*DistributesOver+ : {a b c : A} → a * (b + c) ∼ (a * b) + (a * c)
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identIsIdent : {a : A} → 1R * a ∼ a
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timesZero : {a : A} → a * 0R ∼ 0R
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timesZero {a} = symmetric (transitive (transitive (symmetric invLeft) (+WellDefined reflexive (transitive (*WellDefined {a} {a} reflexive (symmetric identRight)) *DistributesOver+))) (transitive +Associative (transitive (+WellDefined invLeft reflexive) identLeft)))
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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
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open Ring R
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@@ -42,10 +48,10 @@ module Rings.Definition where
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--Ring.multWellDefined (directSumRing r s) (a ,, b) (c ,, d) = Ring.multWellDefined r a c ,, Ring.multWellDefined s b d
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--Ring.1R (directSumRing r s) = Ring.1R r ,, Ring.1R s
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--Ring.groupIsAbelian (directSumRing r s) = Ring.groupIsAbelian r ,, Ring.groupIsAbelian s
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--Ring.multAssoc (directSumRing r s) = Ring.multAssoc r ,, Ring.multAssoc s
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--Ring.assoc (directSumRing r s) = Ring.assoc r ,, Ring.assoc s
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--Ring.multCommutative (directSumRing r s) = Ring.multCommutative r ,, Ring.multCommutative s
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--Ring.multDistributes (directSumRing r s) = Ring.multDistributes r ,, Ring.multDistributes s
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--Ring.multIdentIsIdent (directSumRing r s) = Ring.multIdentIsIdent r ,, Ring.multIdentIsIdent s
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--Ring.identIsIdent (directSumRing r s) = Ring.identIsIdent r ,, Ring.identIsIdent s
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record RingHom {m n o p : _} {A : Set m} {B : Set n} {SA : Setoid {m} {o} A} {SB : Setoid {n} {p} B} {_+A_ : A → A → A} {_*A_ : A → A → A} (R : Ring SA _+A_ _*A_) {_+B_ : B → B → B} {_*B_ : B → B → B} (S : Ring SB _+B_ _*B_) (f : A → B) : Set (m ⊔ n ⊔ o ⊔ p) where
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open Ring S
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@@ -27,7 +27,7 @@ module Rings.IntegralDomains where
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t1 : (a * b) + Group.inverse (Ring.additiveGroup R) (a * c) ∼ Ring.0R R
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t1 = transferToRight'' (Ring.additiveGroup R) ab=ac
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t2 : a * (b + Group.inverse (Ring.additiveGroup R) c) ∼ Ring.0R R
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t2 = transitive (transitive (Ring.multDistributes R) (Group.wellDefined (Ring.additiveGroup R) reflexive (transferToRight' (Ring.additiveGroup R) (transitive (symmetric (Ring.multDistributes R)) (transitive (Ring.multWellDefined R reflexive (Group.invLeft (Ring.additiveGroup R))) (ringTimesZero R)))))) t1
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t2 = transitive (transitive (Ring.*DistributesOver+ R) (Group.+WellDefined (Ring.additiveGroup R) reflexive (transferToRight' (Ring.additiveGroup R) (transitive (symmetric (Ring.*DistributesOver+ R)) (transitive (Ring.*WellDefined R reflexive (Group.invLeft (Ring.additiveGroup R))) (Ring.timesZero R)))))) t1
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t3 : (a ∼ Ring.0R R) || ((b + Group.inverse (Ring.additiveGroup R) c) ∼ Ring.0R R)
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t3 = IntegralDomain.intDom I t2
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t4 : (a ∼ Ring.0R R) || (b ∼ c)
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@@ -11,22 +11,9 @@ open import Setoids.Orders
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open import Sets.EquivalenceRelations
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module Rings.Lemmas where
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ringTimesZero : {a b : _} {A : Set a} {S : Setoid {a} {b} A} {_+_ : A → A → A} {_*_ : A → A → A} (R : Ring S _+_ _*_) → {x : A} → Setoid._∼_ S (x * (Ring.0R R)) (Ring.0R R)
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ringTimesZero {S = S} {_+_ = _+_} {_*_ = _*_} R {x} = symmetric (transitive blah'' (transitive (Group.multAssoc additiveGroup) (transitive (wellDefined invLeft reflexive) multIdentLeft)))
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where
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open Ring R
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open Group additiveGroup
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open Setoid S
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open Equivalence eq
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blah : (x * 0R) ∼ (x * 0R) + (x * 0R)
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blah = transitive (multWellDefined reflexive (symmetric multIdentRight)) multDistributes
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blah' : (inverse (x * 0R)) + (x * 0R) ∼ (inverse (x * 0R)) + ((x * 0R) + (x * 0R))
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blah' = wellDefined reflexive blah
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blah'' : 0R ∼ (inverse (x * 0R)) + ((x * 0R) + (x * 0R))
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blah'' = transitive (symmetric invLeft) blah'
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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))
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ringMinusExtracts {S = S} {_+_ = _+_} {_*_ = _*_} R {x = x} {y} = transferToRight' additiveGroup (transitive (symmetric multDistributes) (transitive (multWellDefined reflexive invLeft) (ringTimesZero R)))
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ringMinusExtracts {S = S} {_+_ = _+_} {_*_ = _*_} R {x = x} {y} = transferToRight' additiveGroup (transitive (symmetric *DistributesOver+) (transitive (*WellDefined reflexive invLeft) (Ring.timesZero R)))
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where
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open Setoid S
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open Equivalence eq
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@@ -40,17 +27,17 @@ module Rings.Lemmas where
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open OrderedRing oRing
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open Ring R
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groupLemmaMoveIdentity : {a b : _} → {A : Set a} → {_·_ : A → A → A} → {S : Setoid {a} {b} A} → (G : Group S _·_) → {x : A} → (Setoid._∼_ S (Group.identity G) (Group.inverse G x)) → Setoid._∼_ S x (Group.identity G)
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groupLemmaMoveIdentity {S = S} G {x} pr = transitive (symmetric (invInv G)) (transitive (symmetric p) (invIdent G))
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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)
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groupLemmaMove0G {S = S} G {x} pr = transitive (symmetric (invInv G)) (transitive (symmetric p) (invIdent G))
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where
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open Group G
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open Setoid S
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open Equivalence eq
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p : inverse identity ∼ inverse (inverse x)
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p : inverse 0G ∼ inverse (inverse x)
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p = inverseWellDefined G pr
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groupLemmaMoveIdentity' : {a b : _} → {A : Set a} → {_·_ : A → A → A} → {S : Setoid {a} {b} A} → (G : Group S _·_) → {x : A} → Setoid._∼_ S x (Group.identity G) → (Setoid._∼_ S (Group.identity G) (Group.inverse G x))
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groupLemmaMoveIdentity' {S = S} G {x} pr = transferToRight' G (transitive multIdentLeft pr)
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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))
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groupLemmaMove0G' {S = S} G {x} pr = transferToRight' G (transitive identLeft pr)
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where
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open Group G
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open Setoid S
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@@ -60,21 +47,21 @@ module Rings.Lemmas where
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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)
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ringMinusFlipsOrder {S = S} {_+_} {R = R} {_<_} {pOrder} {tOrder} oRing {x} 0<x | inl (inl 0<inv) = exFalso (SetoidPartialOrder.irreflexive pOrder bad')
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where
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bad : (Group.identity (Ring.additiveGroup R) + Group.identity (Ring.additiveGroup R)) < (x + Group.inverse (Ring.additiveGroup R) x)
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bad : (Group.0G (Ring.additiveGroup R) + Group.0G (Ring.additiveGroup R)) < (x + Group.inverse (Ring.additiveGroup R) x)
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bad = ringAddInequalities oRing 0<x 0<inv
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bad' : (Group.identity (Ring.additiveGroup R)) < (Group.identity (Ring.additiveGroup R))
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bad' = SetoidPartialOrder.wellDefined pOrder (Group.multIdentRight (Ring.additiveGroup R)) (Group.invRight (Ring.additiveGroup R)) bad
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bad' : (Group.0G (Ring.additiveGroup R)) < (Group.0G (Ring.additiveGroup R))
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bad' = SetoidPartialOrder.wellDefined pOrder (Group.identRight (Ring.additiveGroup R)) (Group.invRight (Ring.additiveGroup R)) bad
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ringMinusFlipsOrder {S = S} {_+_} {R = R} {_<_} {pOrder} {tOrder} oRing {x} 0<x | inl (inr inv<0) = inv<0
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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)) (groupLemmaMoveIdentity (Ring.additiveGroup R) 0=inv) 0<x))
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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))
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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
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ringMinusFlipsOrder' {R = R} {_<_ = _<_} {tOrder = tOrder} oRing {x} -x<0 with SetoidTotalOrder.totality tOrder (Ring.0R R) x
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ringMinusFlipsOrder' {R = R} {_<_} {tOrder = tOrder} oRing {x} -x<0 | inl (inl 0<x) = 0<x
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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.multIdentRight (Ring.additiveGroup R)) bad))
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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))
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where
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bad : ((Group.inverse (Ring.additiveGroup R) x) + x) < (Group.identity (Ring.additiveGroup R) + Group.identity (Ring.additiveGroup R))
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bad : ((Group.inverse (Ring.additiveGroup R) x) + x) < (Group.0G (Ring.additiveGroup R) + Group.0G (Ring.additiveGroup R))
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bad = ringAddInequalities oRing -x<0 x<0
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ringMinusFlipsOrder' {S = S} {R = R} {_<_} {pOrder = pOrder} {tOrder = tOrder} oRing {x} -x<0 | inr 0=x = exFalso (SetoidPartialOrder.irreflexive pOrder (SetoidPartialOrder.wellDefined pOrder (symmetric (groupLemmaMoveIdentity' (Ring.additiveGroup R) (symmetric 0=x))) (Equivalence.reflexive (Setoid.eq S)) -x<0))
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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))
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where
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open Setoid S
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open Equivalence eq
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@@ -86,23 +73,23 @@ module Rings.Lemmas where
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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)
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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)
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ringCanMultiplyByPositive {S = S} {_+_ = _+_} {_*_ = _*_} {R = R} {_<_ = _<_} {pOrder = pOrder} {tOrder = tOrder} oRing {x} {y} {c} 0<c x<y = SetoidPartialOrder.wellDefined pOrder reflexive (Group.multIdentRight additiveGroup) q'
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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'
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where
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open Ring R
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open Equivalence (Setoid.eq S)
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have : 0R < (y + Group.inverse additiveGroup x)
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have = SetoidPartialOrder.wellDefined pOrder (Group.invRight additiveGroup) reflexive (OrderedRing.orderRespectsAddition oRing x<y (Group.inverse additiveGroup x))
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p : 0R < ((y * c) + ((Group.inverse additiveGroup x) * c))
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p = SetoidPartialOrder.wellDefined pOrder reflexive (transitive multCommutative (transitive multDistributes ((Group.wellDefined additiveGroup) multCommutative multCommutative))) (OrderedRing.orderRespectsMultiplication oRing have 0<c)
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p = SetoidPartialOrder.wellDefined pOrder reflexive (transitive *Commutative (transitive *DistributesOver+ ((Group.+WellDefined additiveGroup) *Commutative *Commutative))) (OrderedRing.orderRespectsMultiplication oRing have 0<c)
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p' : 0R < ((y * c) + (Group.inverse additiveGroup (x * c)))
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p' = SetoidPartialOrder.wellDefined pOrder reflexive (Group.wellDefined additiveGroup reflexive (transitive (transitive multCommutative (ringMinusExtracts R)) (inverseWellDefined additiveGroup multCommutative))) p
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p' = SetoidPartialOrder.wellDefined pOrder reflexive (Group.+WellDefined additiveGroup reflexive (transitive (transitive *Commutative (ringMinusExtracts R)) (inverseWellDefined additiveGroup *Commutative))) p
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q : (0R + (x * c)) < (((y * c) + (Group.inverse additiveGroup (x * c))) + (x * c))
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q = OrderedRing.orderRespectsAddition oRing p' (x * c)
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q' : (x * c) < ((y * c) + 0R)
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q' = SetoidPartialOrder.wellDefined pOrder (Group.multIdentLeft additiveGroup) (transitive (symmetric (Group.multAssoc additiveGroup)) (Group.wellDefined additiveGroup reflexive (Group.invLeft additiveGroup))) q
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q' = SetoidPartialOrder.wellDefined pOrder (Group.identLeft additiveGroup) (transitive (symmetric (Group.+Associative additiveGroup)) (Group.+WellDefined additiveGroup reflexive (Group.invLeft additiveGroup))) q
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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
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ringCanCancelPositive {S = S} {_+_ = _+_} {_*_ = _*_} {R = R} {_<_ = _<_} {pOrder = pOrder} {tOrder = tOrder} oRing {x} {y} {c} 0<c xc<yc = SetoidPartialOrder.wellDefined pOrder (Group.multIdentLeft additiveGroup) (transitive (symmetric (Group.multAssoc additiveGroup)) (transitive (Group.wellDefined additiveGroup reflexive (Group.invLeft additiveGroup)) (Group.multIdentRight additiveGroup))) q''
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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''
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where
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open Ring R
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open Setoid S
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@@ -110,41 +97,41 @@ module Rings.Lemmas where
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have : 0R < ((y * c) + (Group.inverse additiveGroup (x * c)))
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have = SetoidPartialOrder.wellDefined pOrder (Group.invRight additiveGroup) reflexive (OrderedRing.orderRespectsAddition oRing xc<yc (Group.inverse additiveGroup _))
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p : 0R < ((y * c) + ((Group.inverse additiveGroup x) * c))
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p = SetoidPartialOrder.wellDefined pOrder reflexive (Group.wellDefined additiveGroup reflexive (symmetric (transitive (multCommutative) (transitive (ringMinusExtracts R) (inverseWellDefined additiveGroup multCommutative))))) have
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p = SetoidPartialOrder.wellDefined pOrder reflexive (Group.+WellDefined additiveGroup reflexive (symmetric (transitive (*Commutative) (transitive (ringMinusExtracts R) (inverseWellDefined additiveGroup *Commutative))))) have
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q : 0R < ((y + Group.inverse additiveGroup x) * c)
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q = SetoidPartialOrder.wellDefined pOrder reflexive (transitive (transitive (Group.wellDefined additiveGroup multCommutative multCommutative) (symmetric multDistributes)) multCommutative) p
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q = SetoidPartialOrder.wellDefined pOrder reflexive (transitive (transitive (Group.+WellDefined additiveGroup *Commutative *Commutative) (symmetric *DistributesOver+)) *Commutative) p
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q' : 0R < (y + Group.inverse additiveGroup x)
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q' with SetoidTotalOrder.totality tOrder 0R (y + Group.inverse additiveGroup x)
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q' | inl (inl pr) = pr
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q' | inl (inr y-x<0) = exFalso (SetoidPartialOrder.irreflexive pOrder (SetoidPartialOrder.wellDefined pOrder reflexive (transitive multCommutative (ringTimesZero R)) k))
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q' | inl (inr y-x<0) = exFalso (SetoidPartialOrder.irreflexive pOrder (SetoidPartialOrder.wellDefined pOrder reflexive (transitive *Commutative (Ring.timesZero R)) k))
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where
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open Group additiveGroup
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f : ((y + inverse x) + (inverse (y + inverse x))) < (identity + inverse (y + inverse x))
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f : ((y + inverse x) + (inverse (y + inverse x))) < (0G + inverse (y + inverse x))
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f = OrderedRing.orderRespectsAddition oRing y-x<0 _
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g : identity < inverse (y + inverse x)
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g = SetoidPartialOrder.wellDefined pOrder invRight multIdentLeft f
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h : (identity * c) < ((inverse (y + inverse x)) * c)
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g : 0G < inverse (y + inverse x)
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g = SetoidPartialOrder.wellDefined pOrder invRight identLeft f
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h : (0G * c) < ((inverse (y + inverse x)) * c)
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h = ringCanMultiplyByPositive oRing 0<c g
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i : (0R + (identity * c)) < (((y + inverse x) * c) + ((inverse (y + inverse x)) * c))
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i : (0R + (0G * c)) < (((y + inverse x) * c) + ((inverse (y + inverse x)) * c))
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i = ringAddInequalities oRing q h
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j : 0R < (((y + inverse x) + (inverse (y + inverse x))) * c)
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j = SetoidPartialOrder.wellDefined pOrder (transitive multIdentLeft (transitive multCommutative (ringTimesZero R))) (symmetric (transitive multCommutative (transitive multDistributes (Group.wellDefined additiveGroup multCommutative multCommutative)))) i
|
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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 (multWellDefined invRight reflexive) j
|
||||
q' | inr 0=y-x = exFalso (SetoidPartialOrder.irreflexive pOrder (SetoidPartialOrder.wellDefined pOrder (multWellDefined x=y reflexive) reflexive xc<yc))
|
||||
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 identity ∼ inverse (y + inverse x)
|
||||
f : inverse 0G ∼ inverse (y + inverse x)
|
||||
f = inverseWellDefined additiveGroup 0=y-x
|
||||
g : identity ∼ (inverse y) + x
|
||||
g = transitive (symmetric (invIdentity additiveGroup)) (transitive f (transitive (transitive (invContravariant additiveGroup) groupIsAbelian) (wellDefined reflexive (invInv additiveGroup))))
|
||||
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.multAssoc additiveGroup)) (transitive (Group.wellDefined additiveGroup reflexive (Group.invLeft additiveGroup)) (Group.multIdentRight additiveGroup))) (Group.multIdentLeft additiveGroup) v
|
||||
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
|
||||
@@ -165,7 +152,7 @@ module Rings.Lemmas where
|
||||
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.multIdentLeft additiveGroup) p
|
||||
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)
|
||||
@@ -181,19 +168,19 @@ module Rings.Lemmas where
|
||||
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 multCommutative) (transitive (symmetric (ringMinusExtracts R)) multCommutative))) p
|
||||
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 multCommutative multCommutative) (transitive (symmetric multDistributes) multCommutative)) p1
|
||||
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 multCommutative (ringTimesZero R))) multCommutative p2)
|
||||
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 (multWellDefined x=y reflexive) reflexive xc<yc))
|
||||
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 multIdentLeft) (transitive (Group.wellDefined additiveGroup 0=y-x reflexive) (transitive (symmetric (Group.multAssoc additiveGroup)) (transitive (Group.wellDefined additiveGroup reflexive invLeft) multIdentRight)))
|
||||
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.multAssoc additiveGroup)) (transitive (Group.wellDefined additiveGroup reflexive (invLeft)) multIdentRight)) (Group.multIdentLeft additiveGroup) (OrderedRing.orderRespectsAddition oRing q 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)
|
||||
|
||||
Reference in New Issue
Block a user