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Minor perf improvements? (#60)
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Patrick Stevens
GitHub
@@ -28,17 +28,19 @@ record OrderedRing {p : _} {_<_ : Rel {_} {p} A} {pOrder : SetoidPartialOrder S
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abs a | inl (inl 0<a) = a
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abs a | inl (inr a<0) = inverse a
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abs a | inr 0=a = a
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absWellDefined : (a b : A) → a ∼ b → abs a ∼ abs b
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absWellDefined a b a=b with SetoidTotalOrder.totality order 0R a
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absWellDefined a b a=b | inl (inl 0<a) with SetoidTotalOrder.totality order 0R b
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absWellDefined a b a=b | inl (inl 0<a) | inl (inl 0<b) = a=b
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absWellDefined a b a=b | inl (inl 0<a) | inl (inr b<0) = exFalso (irreflexive {0G} (transitive 0<a (<WellDefined (Equivalence.symmetric eq a=b) (Equivalence.reflexive eq) b<0)))
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absWellDefined a b a=b | inl (inl 0<a) | inr 0=b = exFalso (irreflexive {0G} (<WellDefined (Equivalence.reflexive eq) (Equivalence.transitive eq a=b (Equivalence.symmetric eq 0=b)) 0<a))
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absWellDefined a b a=b | inl (inr a<0) with SetoidTotalOrder.totality order 0R b
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absWellDefined a b a=b | inl (inr a<0) | inl (inl 0<b) = exFalso (irreflexive {0G} (transitive 0<b (<WellDefined a=b (Equivalence.reflexive eq) a<0)))
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absWellDefined a b a=b | inl (inr a<0) | inl (inr b<0) = inverseWellDefined additiveGroup a=b
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absWellDefined a b a=b | inl (inr a<0) | inr 0=b = exFalso (irreflexive {0G} (<WellDefined (Equivalence.transitive eq a=b (Equivalence.symmetric eq 0=b)) (Equivalence.reflexive eq) a<0))
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absWellDefined a b a=b | inr 0=a with SetoidTotalOrder.totality order 0R b
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absWellDefined a b a=b | inr 0=a | inl (inl 0<b) = exFalso (irreflexive {0G} (<WellDefined (Equivalence.reflexive eq) (Equivalence.symmetric eq (Equivalence.transitive eq 0=a a=b)) 0<b))
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absWellDefined a b a=b | inr 0=a | inl (inr b<0) = exFalso (irreflexive {0G} (<WellDefined (Equivalence.symmetric eq (Equivalence.transitive eq 0=a a=b)) (Equivalence.reflexive eq) b<0))
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absWellDefined a b a=b | inr 0=a | inr 0=b = a=b
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abstract
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absWellDefined : (a b : A) → a ∼ b → abs a ∼ abs b
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absWellDefined a b a=b with SetoidTotalOrder.totality order 0R a
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absWellDefined a b a=b | inl (inl 0<a) with SetoidTotalOrder.totality order 0R b
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absWellDefined a b a=b | inl (inl 0<a) | inl (inl 0<b) = a=b
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absWellDefined a b a=b | inl (inl 0<a) | inl (inr b<0) = exFalso (irreflexive {0G} (transitive 0<a (<WellDefined (Equivalence.symmetric eq a=b) (Equivalence.reflexive eq) b<0)))
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absWellDefined a b a=b | inl (inl 0<a) | inr 0=b = exFalso (irreflexive {0G} (<WellDefined (Equivalence.reflexive eq) (Equivalence.transitive eq a=b (Equivalence.symmetric eq 0=b)) 0<a))
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absWellDefined a b a=b | inl (inr a<0) with SetoidTotalOrder.totality order 0R b
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absWellDefined a b a=b | inl (inr a<0) | inl (inl 0<b) = exFalso (irreflexive {0G} (transitive 0<b (<WellDefined a=b (Equivalence.reflexive eq) a<0)))
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absWellDefined a b a=b | inl (inr a<0) | inl (inr b<0) = inverseWellDefined additiveGroup a=b
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absWellDefined a b a=b | inl (inr a<0) | inr 0=b = exFalso (irreflexive {0G} (<WellDefined (Equivalence.transitive eq a=b (Equivalence.symmetric eq 0=b)) (Equivalence.reflexive eq) a<0))
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absWellDefined a b a=b | inr 0=a with SetoidTotalOrder.totality order 0R b
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absWellDefined a b a=b | inr 0=a | inl (inl 0<b) = exFalso (irreflexive {0G} (<WellDefined (Equivalence.reflexive eq) (Equivalence.symmetric eq (Equivalence.transitive eq 0=a a=b)) 0<b))
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absWellDefined a b a=b | inr 0=a | inl (inr b<0) = exFalso (irreflexive {0G} (<WellDefined (Equivalence.symmetric eq (Equivalence.transitive eq 0=a a=b)) (Equivalence.reflexive eq) b<0))
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absWellDefined a b a=b | inr 0=a | inr 0=b = a=b
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@@ -16,364 +16,372 @@ open import Agda.Primitive using (Level; lzero; lsuc; _⊔_)
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module Rings.Orders.Lemmas {n m p : _} {A : Set n} {S : Setoid {n} {m} A} {_+_ : A → A → A} {_*_ : A → A → A} {_<_ : Rel {_} {p} A} {R : Ring S _+_ _*_} {pOrder : SetoidPartialOrder S _<_} {tOrder : SetoidTotalOrder pOrder} (order : OrderedRing R tOrder) where
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open OrderedRing order
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open Setoid S
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open SetoidPartialOrder pOrder
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open SetoidTotalOrder tOrder
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open Ring R
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open Group additiveGroup
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abstract
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open import Rings.Lemmas R
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open OrderedRing order
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open Setoid S
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open SetoidPartialOrder pOrder
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open SetoidTotalOrder tOrder
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open Ring R
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open Group additiveGroup
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ringAddInequalities : {w x y z : A} → w < x → y < z → (w + y) < (x + z)
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ringAddInequalities {w = w} {x} {y} {z} w<x y<z = transitive (orderRespectsAddition w<x y) (<WellDefined groupIsAbelian groupIsAbelian (orderRespectsAddition y<z x))
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open import Rings.Lemmas R
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lemm2 : (a : A) → a < 0G → 0G < inverse a
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lemm2 a a<0 with SetoidTotalOrder.totality tOrder 0R (inverse a)
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lemm2 a a<0 | inl (inl 0<-a) = 0<-a
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lemm2 a a<0 | inl (inr -a<0) = exFalso (irreflexive {0G} (SetoidPartialOrder.transitive pOrder (<WellDefined (invLeft {a}) (identLeft {a}) (orderRespectsAddition -a<0 a)) a<0))
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lemm2 a a<0 | inr 0=-a = exFalso (irreflexive {0G} (<WellDefined (Equivalence.transitive eq (Equivalence.symmetric eq identRight) t) (Equivalence.reflexive eq) a<0))
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where
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t : a + 0G ∼ 0G
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t = Equivalence.transitive eq (+WellDefined (Equivalence.reflexive eq) 0=-a) (invRight {a})
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ringAddInequalities : {w x y z : A} → w < x → y < z → (w + y) < (x + z)
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ringAddInequalities {w = w} {x} {y} {z} w<x y<z = transitive (orderRespectsAddition w<x y) (<WellDefined groupIsAbelian groupIsAbelian (orderRespectsAddition y<z x))
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lemm2' : (a : A) → 0G < a → inverse a < 0G
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lemm2' a 0<a with SetoidTotalOrder.totality tOrder 0R (inverse a)
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lemm2' a 0<a | inl (inl 0<-a) = exFalso (irreflexive {0G} (SetoidPartialOrder.transitive pOrder 0<a (<WellDefined (identLeft {a}) (invLeft {a}) (orderRespectsAddition 0<-a a))))
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lemm2' a 0<a | inl (inr -a<0) = -a<0
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lemm2' a 0<a | inr 0=-a = exFalso (irreflexive {0G} (<WellDefined (Equivalence.reflexive eq) (Equivalence.transitive eq (Equivalence.symmetric eq identRight) t) 0<a))
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where
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t : a + 0G ∼ 0G
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t = Equivalence.transitive eq (+WellDefined (Equivalence.reflexive eq) 0=-a) (invRight {a})
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lemm2 : (a : A) → a < 0G → 0G < inverse a
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lemm2 a a<0 with SetoidTotalOrder.totality tOrder 0R (inverse a)
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lemm2 a a<0 | inl (inl 0<-a) = 0<-a
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lemm2 a a<0 | inl (inr -a<0) = exFalso (irreflexive {0G} (SetoidPartialOrder.transitive pOrder (<WellDefined (invLeft {a}) (identLeft {a}) (orderRespectsAddition -a<0 a)) a<0))
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lemm2 a a<0 | inr 0=-a = exFalso (irreflexive {0G} (<WellDefined (Equivalence.transitive eq (Equivalence.symmetric eq identRight) t) (Equivalence.reflexive eq) a<0))
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where
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t : a + 0G ∼ 0G
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t = Equivalence.transitive eq (+WellDefined (Equivalence.reflexive eq) 0=-a) (invRight {a})
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lemm3 : (a b : A) → 0G ∼ (a + b) → 0G ∼ a → 0G ∼ b
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lemm3 a b pr1 pr2 with transferToRight' additiveGroup (Equivalence.symmetric eq pr1)
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... | a=-b with Equivalence.transitive eq pr2 a=-b
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... | 0=-b with inverseWellDefined additiveGroup 0=-b
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... | -0=--b = Equivalence.transitive eq (Equivalence.symmetric eq (invIdentity additiveGroup)) (Equivalence.transitive eq -0=--b (invTwice additiveGroup b))
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lemm2' : (a : A) → 0G < a → inverse a < 0G
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lemm2' a 0<a with SetoidTotalOrder.totality tOrder 0R (inverse a)
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lemm2' a 0<a | inl (inl 0<-a) = exFalso (irreflexive {0G} (SetoidPartialOrder.transitive pOrder 0<a (<WellDefined (identLeft {a}) (invLeft {a}) (orderRespectsAddition 0<-a a))))
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lemm2' a 0<a | inl (inr -a<0) = -a<0
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lemm2' a 0<a | inr 0=-a = exFalso (irreflexive {0G} (<WellDefined (Equivalence.reflexive eq) (Equivalence.transitive eq (Equivalence.symmetric eq identRight) t) 0<a))
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where
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t : a + 0G ∼ 0G
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t = Equivalence.transitive eq (+WellDefined (Equivalence.reflexive eq) 0=-a) (invRight {a})
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lemm3 : (a b : A) → 0G ∼ (a + b) → 0G ∼ a → 0G ∼ b
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lemm3 a b pr1 pr2 with transferToRight' additiveGroup (Equivalence.symmetric eq pr1)
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... | a=-b with Equivalence.transitive eq pr2 a=-b
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... | 0=-b with inverseWellDefined additiveGroup 0=-b
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... | -0=--b = Equivalence.transitive eq (Equivalence.symmetric eq (invIdentity additiveGroup)) (Equivalence.transitive eq -0=--b (invTwice additiveGroup b))
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triangleInequality : (a b : A) → ((abs (a + b)) < ((abs a) + (abs b))) || (abs (a + b) ∼ (abs a) + (abs b))
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triangleInequality a b with totality 0R (a + b)
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triangleInequality a b | inl (inl 0<a+b) with totality 0R a
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triangleInequality a b | inl (inl 0<a+b) | inl (inl 0<a) with totality 0R b
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triangleInequality a b | inl (inl 0<a+b) | inl (inl 0<a) | inl (inl 0<b) = inr (Equivalence.reflexive eq)
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triangleInequality a b | inl (inl 0<a+b) | inl (inl 0<a) | inl (inr b<0) = inl (<WellDefined groupIsAbelian groupIsAbelian (orderRespectsAddition (SetoidPartialOrder.transitive pOrder b<0 (lemm2 b b<0)) a))
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triangleInequality a b | inl (inl 0<a+b) | inl (inl 0<a) | inr 0=b = inr (Equivalence.reflexive eq)
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triangleInequality a b | inl (inl 0<a+b) | inl (inr a<0) with totality 0R b
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triangleInequality a b | inl (inl 0<a+b) | inl (inr a<0) | inl (inl 0<b) = inl (orderRespectsAddition (SetoidPartialOrder.transitive pOrder a<0 (lemm2 a a<0)) b)
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triangleInequality a b | inl (inl 0<a+b) | inl (inr a<0) | inl (inr b<0) = exFalso (irreflexive {0G} (SetoidPartialOrder.transitive pOrder 0<a+b (<WellDefined (Equivalence.reflexive eq) identLeft (ringAddInequalities a<0 b<0))))
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triangleInequality a b | inl (inl 0<a+b) | inl (inr a<0) | inr 0=b = inl (orderRespectsAddition (SetoidPartialOrder.transitive pOrder a<0 (lemm2 a a<0)) b)
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triangleInequality a b | inl (inl 0<a+b) | inr 0=a with totality 0R b
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triangleInequality a b | inl (inl 0<a+b) | inr 0=a | inl (inl 0<b) = inr (Equivalence.reflexive eq)
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triangleInequality a b | inl (inl 0<a+b) | inr 0=a | inl (inr b<0) = inl (<WellDefined groupIsAbelian groupIsAbelian (orderRespectsAddition (SetoidPartialOrder.transitive pOrder b<0 (lemm2 b b<0)) a))
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triangleInequality a b | inl (inl 0<a+b) | inr 0=a | inr 0=b = inr (Equivalence.reflexive eq)
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triangleInequality a b | inl (inr a+b<0) with totality 0G a
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triangleInequality a b | inl (inr a+b<0) | inl (inl 0<a) with totality 0G b
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triangleInequality a b | inl (inr a+b<0) | inl (inl 0<a) | inl (inl 0<b) = exFalso (irreflexive {0G} (SetoidPartialOrder.transitive pOrder (<WellDefined identLeft (Equivalence.reflexive eq) (ringAddInequalities 0<a 0<b)) a+b<0))
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triangleInequality a b | inl (inr a+b<0) | inl (inl 0<a) | inl (inr b<0) = inl (<WellDefined (Equivalence.transitive eq (Equivalence.symmetric eq (invContravariant additiveGroup)) (inverseWellDefined additiveGroup groupIsAbelian)) (Equivalence.reflexive eq) (orderRespectsAddition (SetoidPartialOrder.transitive pOrder (lemm2' _ 0<a) 0<a) (inverse b)))
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triangleInequality a b | inl (inr a+b<0) | inl (inl 0<a) | inr 0=b = exFalso (irreflexive {0G} (SetoidPartialOrder.transitive pOrder 0<a (<WellDefined (Equivalence.transitive eq (+WellDefined (Equivalence.reflexive eq) (Equivalence.symmetric eq 0=b)) identRight) (Equivalence.reflexive eq) a+b<0)))
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triangleInequality a b | inl (inr a+b<0) | inl (inr a<0) with totality 0G b
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triangleInequality a b | inl (inr a+b<0) | inl (inr a<0) | inl (inl 0<b) = inl (<WellDefined (Equivalence.symmetric eq (invContravariant additiveGroup)) groupIsAbelian (orderRespectsAddition (SetoidPartialOrder.transitive pOrder (lemm2' _ 0<b) 0<b) (inverse a)))
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triangleInequality a b | inl (inr a+b<0) | inl (inr a<0) | inl (inr b<0) = inr (Equivalence.transitive eq (invContravariant additiveGroup) groupIsAbelian)
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triangleInequality a b | inl (inr a+b<0) | inl (inr a<0) | inr 0=b = inr (Equivalence.transitive eq (invContravariant additiveGroup) (Equivalence.transitive eq (Equivalence.transitive eq (Equivalence.transitive eq (+WellDefined (Equivalence.transitive eq (inverseWellDefined additiveGroup (Equivalence.symmetric eq 0=b)) (invIdentity additiveGroup)) (Equivalence.reflexive eq)) identLeft) (Equivalence.symmetric eq identRight)) (+WellDefined (Equivalence.reflexive eq) 0=b)))
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triangleInequality a b | inl (inr a+b<0) | inr 0=a with totality 0G b
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triangleInequality a b | inl (inr a+b<0) | inr 0=a | inl (inl 0<b) = exFalso (irreflexive {0G} (SetoidPartialOrder.transitive pOrder 0<b (<WellDefined (Equivalence.transitive eq (+WellDefined (Equivalence.symmetric eq 0=a) (Equivalence.reflexive eq)) identLeft) (Equivalence.reflexive eq) a+b<0)))
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triangleInequality a b | inl (inr a+b<0) | inr 0=a | inl (inr b<0) = inr (Equivalence.transitive eq (invContravariant additiveGroup) (Equivalence.transitive eq groupIsAbelian (+WellDefined (Equivalence.transitive eq (Equivalence.transitive eq (Equivalence.symmetric eq (inverseWellDefined additiveGroup 0=a)) (invIdentity additiveGroup)) 0=a) (Equivalence.reflexive eq))))
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triangleInequality a b | inl (inr a+b<0) | inr 0=a | inr 0=b = exFalso (irreflexive {0G} (<WellDefined (Equivalence.transitive eq (+WellDefined (Equivalence.symmetric eq 0=a) (Equivalence.symmetric eq 0=b)) identLeft) (Equivalence.reflexive eq) a+b<0))
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triangleInequality a b | inr 0=a+b with totality 0G a
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triangleInequality a b | inr 0=a+b | inl (inl 0<a) with totality 0G b
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triangleInequality a b | inr 0=a+b | inl (inl 0<a) | inl (inl 0<b) = exFalso (irreflexive {0G} (<WellDefined identLeft (Equivalence.symmetric eq 0=a+b) (ringAddInequalities 0<a 0<b)))
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triangleInequality a b | inr 0=a+b | inl (inl 0<a) | inl (inr b<0) = inl (<WellDefined groupIsAbelian groupIsAbelian (orderRespectsAddition (SetoidPartialOrder.transitive pOrder b<0 (lemm2 _ b<0)) a))
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triangleInequality a b | inr 0=a+b | inl (inl 0<a) | inr 0=b = exFalso (irreflexive {0G} (<WellDefined (Equivalence.reflexive eq) (Equivalence.symmetric eq (lemm3 _ _ (Equivalence.transitive eq 0=a+b groupIsAbelian) 0=b)) 0<a))
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triangleInequality a b | inr 0=a+b | inl (inr a<0) with totality 0G b
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triangleInequality a b | inr 0=a+b | inl (inr a<0) | inl (inl 0<b) = inl (orderRespectsAddition (SetoidPartialOrder.transitive pOrder a<0 (lemm2 _ a<0)) b)
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triangleInequality a b | inr 0=a+b | inl (inr a<0) | inl (inr b<0) = exFalso (irreflexive {0G} (<WellDefined (Equivalence.symmetric eq 0=a+b) identLeft (ringAddInequalities a<0 b<0)))
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triangleInequality a b | inr 0=a+b | inl (inr a<0) | inr 0=b = exFalso (irreflexive {0G} (<WellDefined (Equivalence.symmetric eq (lemm3 _ _ (Equivalence.transitive eq 0=a+b groupIsAbelian) 0=b)) (Equivalence.reflexive eq) a<0))
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triangleInequality a b | inr 0=a+b | inr 0=a with totality 0G b
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triangleInequality a b | inr 0=a+b | inr 0=a | inl (inl 0<b) = exFalso (irreflexive {0G} (<WellDefined (Equivalence.reflexive eq) (Equivalence.symmetric eq (lemm3 a b 0=a+b 0=a)) 0<b))
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triangleInequality a b | inr 0=a+b | inr 0=a | inl (inr b<0) = exFalso (irreflexive {0G} (<WellDefined (Equivalence.symmetric eq (lemm3 a b 0=a+b 0=a)) (Equivalence.reflexive eq) b<0))
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triangleInequality a b | inr 0=a+b | inr 0=a | inr 0=b = inr (Equivalence.reflexive eq)
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triangleInequality : (a b : A) → ((abs (a + b)) < ((abs a) + (abs b))) || (abs (a + b) ∼ (abs a) + (abs b))
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triangleInequality a b with totality 0R (a + b)
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triangleInequality a b | inl (inl 0<a+b) with totality 0R a
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triangleInequality a b | inl (inl 0<a+b) | inl (inl 0<a) with totality 0R b
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triangleInequality a b | inl (inl 0<a+b) | inl (inl 0<a) | inl (inl 0<b) = inr (Equivalence.reflexive eq)
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triangleInequality a b | inl (inl 0<a+b) | inl (inl 0<a) | inl (inr b<0) = inl (<WellDefined groupIsAbelian groupIsAbelian (orderRespectsAddition (SetoidPartialOrder.transitive pOrder b<0 (lemm2 b b<0)) a))
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triangleInequality a b | inl (inl 0<a+b) | inl (inl 0<a) | inr 0=b = inr (Equivalence.reflexive eq)
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triangleInequality a b | inl (inl 0<a+b) | inl (inr a<0) with totality 0R b
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triangleInequality a b | inl (inl 0<a+b) | inl (inr a<0) | inl (inl 0<b) = inl (orderRespectsAddition (SetoidPartialOrder.transitive pOrder a<0 (lemm2 a a<0)) b)
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triangleInequality a b | inl (inl 0<a+b) | inl (inr a<0) | inl (inr b<0) = exFalso (irreflexive {0G} (SetoidPartialOrder.transitive pOrder 0<a+b (<WellDefined (Equivalence.reflexive eq) identLeft (ringAddInequalities a<0 b<0))))
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triangleInequality a b | inl (inl 0<a+b) | inl (inr a<0) | inr 0=b = inl (orderRespectsAddition (SetoidPartialOrder.transitive pOrder a<0 (lemm2 a a<0)) b)
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triangleInequality a b | inl (inl 0<a+b) | inr 0=a with totality 0R b
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triangleInequality a b | inl (inl 0<a+b) | inr 0=a | inl (inl 0<b) = inr (Equivalence.reflexive eq)
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triangleInequality a b | inl (inl 0<a+b) | inr 0=a | inl (inr b<0) = inl (<WellDefined groupIsAbelian groupIsAbelian (orderRespectsAddition (SetoidPartialOrder.transitive pOrder b<0 (lemm2 b b<0)) a))
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triangleInequality a b | inl (inl 0<a+b) | inr 0=a | inr 0=b = inr (Equivalence.reflexive eq)
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triangleInequality a b | inl (inr a+b<0) with totality 0G a
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triangleInequality a b | inl (inr a+b<0) | inl (inl 0<a) with totality 0G b
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triangleInequality a b | inl (inr a+b<0) | inl (inl 0<a) | inl (inl 0<b) = exFalso (irreflexive {0G} (SetoidPartialOrder.transitive pOrder (<WellDefined identLeft (Equivalence.reflexive eq) (ringAddInequalities 0<a 0<b)) a+b<0))
|
||||
triangleInequality a b | inl (inr a+b<0) | inl (inl 0<a) | inl (inr b<0) = inl (<WellDefined (Equivalence.transitive eq (Equivalence.symmetric eq (invContravariant additiveGroup)) (inverseWellDefined additiveGroup groupIsAbelian)) (Equivalence.reflexive eq) (orderRespectsAddition (SetoidPartialOrder.transitive pOrder (lemm2' _ 0<a) 0<a) (inverse b)))
|
||||
triangleInequality a b | inl (inr a+b<0) | inl (inl 0<a) | inr 0=b = exFalso (irreflexive {0G} (SetoidPartialOrder.transitive pOrder 0<a (<WellDefined (Equivalence.transitive eq (+WellDefined (Equivalence.reflexive eq) (Equivalence.symmetric eq 0=b)) identRight) (Equivalence.reflexive eq) a+b<0)))
|
||||
triangleInequality a b | inl (inr a+b<0) | inl (inr a<0) with totality 0G b
|
||||
triangleInequality a b | inl (inr a+b<0) | inl (inr a<0) | inl (inl 0<b) = inl (<WellDefined (Equivalence.symmetric eq (invContravariant additiveGroup)) groupIsAbelian (orderRespectsAddition (SetoidPartialOrder.transitive pOrder (lemm2' _ 0<b) 0<b) (inverse a)))
|
||||
triangleInequality a b | inl (inr a+b<0) | inl (inr a<0) | inl (inr b<0) = inr (Equivalence.transitive eq (invContravariant additiveGroup) groupIsAbelian)
|
||||
triangleInequality a b | inl (inr a+b<0) | inl (inr a<0) | inr 0=b = inr (Equivalence.transitive eq (invContravariant additiveGroup) (Equivalence.transitive eq (Equivalence.transitive eq (Equivalence.transitive eq (+WellDefined (Equivalence.transitive eq (inverseWellDefined additiveGroup (Equivalence.symmetric eq 0=b)) (invIdentity additiveGroup)) (Equivalence.reflexive eq)) identLeft) (Equivalence.symmetric eq identRight)) (+WellDefined (Equivalence.reflexive eq) 0=b)))
|
||||
triangleInequality a b | inl (inr a+b<0) | inr 0=a with totality 0G b
|
||||
triangleInequality a b | inl (inr a+b<0) | inr 0=a | inl (inl 0<b) = exFalso (irreflexive {0G} (SetoidPartialOrder.transitive pOrder 0<b (<WellDefined (Equivalence.transitive eq (+WellDefined (Equivalence.symmetric eq 0=a) (Equivalence.reflexive eq)) identLeft) (Equivalence.reflexive eq) a+b<0)))
|
||||
triangleInequality a b | inl (inr a+b<0) | inr 0=a | inl (inr b<0) = inr (Equivalence.transitive eq (invContravariant additiveGroup) (Equivalence.transitive eq groupIsAbelian (+WellDefined (Equivalence.transitive eq (Equivalence.transitive eq (Equivalence.symmetric eq (inverseWellDefined additiveGroup 0=a)) (invIdentity additiveGroup)) 0=a) (Equivalence.reflexive eq))))
|
||||
triangleInequality a b | inl (inr a+b<0) | inr 0=a | inr 0=b = exFalso (irreflexive {0G} (<WellDefined (Equivalence.transitive eq (+WellDefined (Equivalence.symmetric eq 0=a) (Equivalence.symmetric eq 0=b)) identLeft) (Equivalence.reflexive eq) a+b<0))
|
||||
triangleInequality a b | inr 0=a+b with totality 0G a
|
||||
triangleInequality a b | inr 0=a+b | inl (inl 0<a) with totality 0G b
|
||||
triangleInequality a b | inr 0=a+b | inl (inl 0<a) | inl (inl 0<b) = exFalso (irreflexive {0G} (<WellDefined identLeft (Equivalence.symmetric eq 0=a+b) (ringAddInequalities 0<a 0<b)))
|
||||
triangleInequality a b | inr 0=a+b | inl (inl 0<a) | inl (inr b<0) = inl (<WellDefined groupIsAbelian groupIsAbelian (orderRespectsAddition (SetoidPartialOrder.transitive pOrder b<0 (lemm2 _ b<0)) a))
|
||||
triangleInequality a b | inr 0=a+b | inl (inl 0<a) | inr 0=b = exFalso (irreflexive {0G} (<WellDefined (Equivalence.reflexive eq) (Equivalence.symmetric eq (lemm3 _ _ (Equivalence.transitive eq 0=a+b groupIsAbelian) 0=b)) 0<a))
|
||||
triangleInequality a b | inr 0=a+b | inl (inr a<0) with totality 0G b
|
||||
triangleInequality a b | inr 0=a+b | inl (inr a<0) | inl (inl 0<b) = inl (orderRespectsAddition (SetoidPartialOrder.transitive pOrder a<0 (lemm2 _ a<0)) b)
|
||||
triangleInequality a b | inr 0=a+b | inl (inr a<0) | inl (inr b<0) = exFalso (irreflexive {0G} (<WellDefined (Equivalence.symmetric eq 0=a+b) identLeft (ringAddInequalities a<0 b<0)))
|
||||
triangleInequality a b | inr 0=a+b | inl (inr a<0) | inr 0=b = exFalso (irreflexive {0G} (<WellDefined (Equivalence.symmetric eq (lemm3 _ _ (Equivalence.transitive eq 0=a+b groupIsAbelian) 0=b)) (Equivalence.reflexive eq) a<0))
|
||||
triangleInequality a b | inr 0=a+b | inr 0=a with totality 0G b
|
||||
triangleInequality a b | inr 0=a+b | inr 0=a | inl (inl 0<b) = exFalso (irreflexive {0G} (<WellDefined (Equivalence.reflexive eq) (Equivalence.symmetric eq (lemm3 a b 0=a+b 0=a)) 0<b))
|
||||
triangleInequality a b | inr 0=a+b | inr 0=a | inl (inr b<0) = exFalso (irreflexive {0G} (<WellDefined (Equivalence.symmetric eq (lemm3 a b 0=a+b 0=a)) (Equivalence.reflexive eq) b<0))
|
||||
triangleInequality a b | inr 0=a+b | inr 0=a | inr 0=b = inr (Equivalence.reflexive eq)
|
||||
|
||||
ringMinusFlipsOrder : {x : A} → (Ring.0R R) < x → (Group.inverse (Ring.additiveGroup R) x) < (Ring.0R R)
|
||||
ringMinusFlipsOrder {x = x} 0<x with SetoidTotalOrder.totality tOrder (Ring.0R R) (Group.inverse (Ring.additiveGroup R) x)
|
||||
ringMinusFlipsOrder {x} 0<x | inl (inl 0<inv) = exFalso (SetoidPartialOrder.irreflexive pOrder bad')
|
||||
where
|
||||
bad : (Group.0G (Ring.additiveGroup R) + Group.0G (Ring.additiveGroup R)) < (x + Group.inverse (Ring.additiveGroup R) x)
|
||||
bad = ringAddInequalities 0<x 0<inv
|
||||
bad' : (Group.0G (Ring.additiveGroup R)) < (Group.0G (Ring.additiveGroup R))
|
||||
bad' = SetoidPartialOrder.<WellDefined pOrder (Group.identRight (Ring.additiveGroup R)) (Group.invRight (Ring.additiveGroup R)) bad
|
||||
ringMinusFlipsOrder {x} 0<x | inl (inr inv<0) = inv<0
|
||||
ringMinusFlipsOrder {x} 0<x | inr 0=inv = exFalso (SetoidPartialOrder.irreflexive pOrder (SetoidPartialOrder.<WellDefined pOrder (Equivalence.reflexive (Setoid.eq S)) (groupLemmaMove0G (Ring.additiveGroup R) 0=inv) 0<x))
|
||||
ringMinusFlipsOrder : {x : A} → (Ring.0R R) < x → (Group.inverse (Ring.additiveGroup R) x) < (Ring.0R R)
|
||||
ringMinusFlipsOrder {x = x} 0<x with SetoidTotalOrder.totality tOrder (Ring.0R R) (Group.inverse (Ring.additiveGroup R) x)
|
||||
ringMinusFlipsOrder {x} 0<x | inl (inl 0<inv) = exFalso (SetoidPartialOrder.irreflexive pOrder bad')
|
||||
where
|
||||
bad : (Group.0G (Ring.additiveGroup R) + Group.0G (Ring.additiveGroup R)) < (x + Group.inverse (Ring.additiveGroup R) x)
|
||||
bad = ringAddInequalities 0<x 0<inv
|
||||
bad' : (Group.0G (Ring.additiveGroup R)) < (Group.0G (Ring.additiveGroup R))
|
||||
bad' = SetoidPartialOrder.<WellDefined pOrder (Group.identRight (Ring.additiveGroup R)) (Group.invRight (Ring.additiveGroup R)) bad
|
||||
ringMinusFlipsOrder {x} 0<x | inl (inr inv<0) = inv<0
|
||||
ringMinusFlipsOrder {x} 0<x | inr 0=inv = exFalso (SetoidPartialOrder.irreflexive pOrder (SetoidPartialOrder.<WellDefined pOrder (Equivalence.reflexive (Setoid.eq S)) (groupLemmaMove0G (Ring.additiveGroup R) 0=inv) 0<x))
|
||||
|
||||
ringMinusFlipsOrder' : {x : A} → (Group.inverse (Ring.additiveGroup R) x) < (Ring.0R R) → (Ring.0R R) < x
|
||||
ringMinusFlipsOrder' {x} -x<0 with SetoidTotalOrder.totality tOrder (Ring.0R R) x
|
||||
ringMinusFlipsOrder' {x} -x<0 | inl (inl 0<x) = 0<x
|
||||
ringMinusFlipsOrder' {x} -x<0 | inl (inr x<0) = exFalso (SetoidPartialOrder.irreflexive pOrder (SetoidPartialOrder.<WellDefined pOrder (Group.invLeft (Ring.additiveGroup R)) (Group.identRight (Ring.additiveGroup R)) bad))
|
||||
where
|
||||
bad : ((Group.inverse (Ring.additiveGroup R) x) + x) < (Group.0G (Ring.additiveGroup R) + Group.0G (Ring.additiveGroup R))
|
||||
bad = ringAddInequalities -x<0 x<0
|
||||
ringMinusFlipsOrder' {x} -x<0 | inr 0=x = exFalso (SetoidPartialOrder.irreflexive pOrder (SetoidPartialOrder.<WellDefined pOrder (symmetric (groupLemmaMove0G' (Ring.additiveGroup R) (symmetric 0=x))) (Equivalence.reflexive (Setoid.eq S)) -x<0))
|
||||
where
|
||||
open Equivalence eq
|
||||
ringMinusFlipsOrder' : {x : A} → (Group.inverse (Ring.additiveGroup R) x) < (Ring.0R R) → (Ring.0R R) < x
|
||||
ringMinusFlipsOrder' {x} -x<0 with SetoidTotalOrder.totality tOrder (Ring.0R R) x
|
||||
ringMinusFlipsOrder' {x} -x<0 | inl (inl 0<x) = 0<x
|
||||
ringMinusFlipsOrder' {x} -x<0 | inl (inr x<0) = exFalso (SetoidPartialOrder.irreflexive pOrder (SetoidPartialOrder.<WellDefined pOrder (Group.invLeft (Ring.additiveGroup R)) (Group.identRight (Ring.additiveGroup R)) bad))
|
||||
where
|
||||
bad : ((Group.inverse (Ring.additiveGroup R) x) + x) < (Group.0G (Ring.additiveGroup R) + Group.0G (Ring.additiveGroup R))
|
||||
bad = ringAddInequalities -x<0 x<0
|
||||
ringMinusFlipsOrder' {x} -x<0 | inr 0=x = exFalso (SetoidPartialOrder.irreflexive pOrder (SetoidPartialOrder.<WellDefined pOrder (symmetric (groupLemmaMove0G' (Ring.additiveGroup R) (symmetric 0=x))) (Equivalence.reflexive (Setoid.eq S)) -x<0))
|
||||
where
|
||||
open Equivalence eq
|
||||
|
||||
ringMinusFlipsOrder'' : {x : A} → x < (Ring.0R R) → (Ring.0R R) < Group.inverse (Ring.additiveGroup R) x
|
||||
ringMinusFlipsOrder'' {x} x<0 = ringMinusFlipsOrder' (SetoidPartialOrder.<WellDefined pOrder {x} {Group.inverse (Ring.additiveGroup R) (Group.inverse (Ring.additiveGroup R) x)} {Ring.0R R} {Ring.0R R} (Equivalence.symmetric (Setoid.eq S) (invInv (Ring.additiveGroup R))) (Equivalence.reflexive (Setoid.eq S)) x<0)
|
||||
ringMinusFlipsOrder'' : {x : A} → x < (Ring.0R R) → (Ring.0R R) < Group.inverse (Ring.additiveGroup R) x
|
||||
ringMinusFlipsOrder'' {x} x<0 = ringMinusFlipsOrder' (SetoidPartialOrder.<WellDefined pOrder {x} {Group.inverse (Ring.additiveGroup R) (Group.inverse (Ring.additiveGroup R) x)} {Ring.0R R} {Ring.0R R} (Equivalence.symmetric (Setoid.eq S) (invInv (Ring.additiveGroup R))) (Equivalence.reflexive (Setoid.eq S)) x<0)
|
||||
|
||||
ringMinusFlipsOrder''' : {x : A} → (Ring.0R R) < (Group.inverse (Ring.additiveGroup R) x) → x < (Ring.0R R)
|
||||
ringMinusFlipsOrder''' {x} 0<-x = SetoidPartialOrder.<WellDefined pOrder (invInv (Ring.additiveGroup R)) (Equivalence.reflexive (Setoid.eq S)) (ringMinusFlipsOrder 0<-x)
|
||||
ringMinusFlipsOrder''' : {x : A} → (Ring.0R R) < (Group.inverse (Ring.additiveGroup R) x) → x < (Ring.0R R)
|
||||
ringMinusFlipsOrder''' {x} 0<-x = SetoidPartialOrder.<WellDefined pOrder (invInv (Ring.additiveGroup R)) (Equivalence.reflexive (Setoid.eq S)) (ringMinusFlipsOrder 0<-x)
|
||||
|
||||
ringCanMultiplyByPositive : {x y c : A} → (Ring.0R R) < c → x < y → (x * c) < (y * c)
|
||||
ringCanMultiplyByPositive {x} {y} {c} 0<c x<y = SetoidPartialOrder.<WellDefined pOrder reflexive (Group.identRight additiveGroup) q'
|
||||
where
|
||||
open Equivalence eq
|
||||
have : 0R < (y + Group.inverse additiveGroup x)
|
||||
have = SetoidPartialOrder.<WellDefined pOrder (Group.invRight additiveGroup) reflexive (OrderedRing.orderRespectsAddition order x<y (Group.inverse additiveGroup x))
|
||||
p1 : 0R < ((y * c) + ((Group.inverse additiveGroup x) * c))
|
||||
p1 = SetoidPartialOrder.<WellDefined pOrder reflexive (Equivalence.transitive eq *Commutative (Equivalence.transitive eq *DistributesOver+ ((Group.+WellDefined additiveGroup) *Commutative *Commutative))) (OrderedRing.orderRespectsMultiplication order have 0<c)
|
||||
p' : 0R < ((y * c) + (Group.inverse additiveGroup (x * c)))
|
||||
p' = SetoidPartialOrder.<WellDefined pOrder reflexive (Group.+WellDefined additiveGroup reflexive (Equivalence.transitive eq (Equivalence.transitive eq *Commutative ringMinusExtracts) (inverseWellDefined additiveGroup *Commutative))) p1
|
||||
q : (0R + (x * c)) < (((y * c) + (Group.inverse additiveGroup (x * c))) + (x * c))
|
||||
q = OrderedRing.orderRespectsAddition order p' (x * c)
|
||||
q' : (x * c) < ((y * c) + 0R)
|
||||
q' = SetoidPartialOrder.<WellDefined pOrder (Group.identLeft additiveGroup) (Equivalence.transitive eq (symmetric (Group.+Associative additiveGroup)) (Group.+WellDefined additiveGroup reflexive (Group.invLeft additiveGroup))) q
|
||||
ringCanMultiplyByPositive : {x y c : A} → (Ring.0R R) < c → x < y → (x * c) < (y * c)
|
||||
ringCanMultiplyByPositive {x} {y} {c} 0<c x<y = SetoidPartialOrder.<WellDefined pOrder reflexive (Group.identRight additiveGroup) q'
|
||||
where
|
||||
open Equivalence eq
|
||||
have : 0R < (y + Group.inverse additiveGroup x)
|
||||
have = SetoidPartialOrder.<WellDefined pOrder (Group.invRight additiveGroup) reflexive (OrderedRing.orderRespectsAddition order x<y (Group.inverse additiveGroup x))
|
||||
p1 : 0R < ((y * c) + ((Group.inverse additiveGroup x) * c))
|
||||
p1 = SetoidPartialOrder.<WellDefined pOrder reflexive (Equivalence.transitive eq *Commutative (Equivalence.transitive eq *DistributesOver+ ((Group.+WellDefined additiveGroup) *Commutative *Commutative))) (OrderedRing.orderRespectsMultiplication order have 0<c)
|
||||
p' : 0R < ((y * c) + (Group.inverse additiveGroup (x * c)))
|
||||
p' = SetoidPartialOrder.<WellDefined pOrder reflexive (Group.+WellDefined additiveGroup reflexive (Equivalence.transitive eq (Equivalence.transitive eq *Commutative ringMinusExtracts) (inverseWellDefined additiveGroup *Commutative))) p1
|
||||
q : (0R + (x * c)) < (((y * c) + (Group.inverse additiveGroup (x * c))) + (x * c))
|
||||
q = OrderedRing.orderRespectsAddition order p' (x * c)
|
||||
q' : (x * c) < ((y * c) + 0R)
|
||||
q' = SetoidPartialOrder.<WellDefined pOrder (Group.identLeft additiveGroup) (Equivalence.transitive eq (symmetric (Group.+Associative additiveGroup)) (Group.+WellDefined additiveGroup reflexive (Group.invLeft additiveGroup))) q
|
||||
|
||||
ringMultiplyPositives : {x y a b : A} → 0R < x → 0R < a → (x < y) → (a < b) → (x * a) < (y * b)
|
||||
ringMultiplyPositives {x} {y} {a} {b} 0<x 0<a x<y a<b = SetoidPartialOrder.transitive pOrder (ringCanMultiplyByPositive 0<a x<y) (<WellDefined *Commutative *Commutative (ringCanMultiplyByPositive (SetoidPartialOrder.transitive pOrder 0<x x<y) a<b))
|
||||
ringMultiplyPositives : {x y a b : A} → 0R < x → 0R < a → (x < y) → (a < b) → (x * a) < (y * b)
|
||||
ringMultiplyPositives {x} {y} {a} {b} 0<x 0<a x<y a<b = SetoidPartialOrder.transitive pOrder (ringCanMultiplyByPositive 0<a x<y) (<WellDefined *Commutative *Commutative (ringCanMultiplyByPositive (SetoidPartialOrder.transitive pOrder 0<x x<y) a<b))
|
||||
|
||||
ringCanCancelPositive : {x y c : A} → (Ring.0R R) < c → (x * c) < (y * c) → x < y
|
||||
ringCanCancelPositive {x} {y} {c} 0<c xc<yc = SetoidPartialOrder.<WellDefined pOrder (Group.identLeft additiveGroup) (Equivalence.transitive eq (symmetric (Group.+Associative additiveGroup)) (Equivalence.transitive eq (Group.+WellDefined additiveGroup reflexive (Group.invLeft additiveGroup)) (Group.identRight additiveGroup))) q''
|
||||
where
|
||||
open Equivalence (Setoid.eq S)
|
||||
have : 0R < ((y * c) + (Group.inverse additiveGroup (x * c)))
|
||||
have = SetoidPartialOrder.<WellDefined pOrder (Group.invRight additiveGroup) reflexive (OrderedRing.orderRespectsAddition order xc<yc (Group.inverse additiveGroup _))
|
||||
p1 : 0R < ((y * c) + ((Group.inverse additiveGroup x) * c))
|
||||
p1 = SetoidPartialOrder.<WellDefined pOrder reflexive (Group.+WellDefined additiveGroup reflexive (symmetric (Equivalence.transitive eq (*Commutative) (Equivalence.transitive eq ringMinusExtracts (inverseWellDefined additiveGroup *Commutative))))) have
|
||||
q : 0R < ((y + Group.inverse additiveGroup x) * c)
|
||||
q = SetoidPartialOrder.<WellDefined pOrder reflexive (Equivalence.transitive eq (Equivalence.transitive eq (Group.+WellDefined additiveGroup *Commutative *Commutative) (symmetric *DistributesOver+)) *Commutative) p1
|
||||
q' : 0R < (y + Group.inverse additiveGroup x)
|
||||
q' with SetoidTotalOrder.totality tOrder 0R (y + Group.inverse additiveGroup x)
|
||||
q' | inl (inl pr) = pr
|
||||
q' | inl (inr y-x<0) = exFalso (SetoidPartialOrder.irreflexive pOrder (SetoidPartialOrder.<WellDefined pOrder reflexive (Equivalence.transitive eq *Commutative (Ring.timesZero R)) k))
|
||||
where
|
||||
f : ((y + inverse x) + (inverse (y + inverse x))) < (0G + inverse (y + inverse x))
|
||||
f = OrderedRing.orderRespectsAddition order y-x<0 _
|
||||
g : 0G < inverse (y + inverse x)
|
||||
g = SetoidPartialOrder.<WellDefined pOrder invRight identLeft f
|
||||
h : (0G * c) < ((inverse (y + inverse x)) * c)
|
||||
h = ringCanMultiplyByPositive 0<c g
|
||||
i : (0R + (0G * c)) < (((y + inverse x) * c) + ((inverse (y + inverse x)) * c))
|
||||
i = ringAddInequalities q h
|
||||
j : 0R < (((y + inverse x) + (inverse (y + inverse x))) * c)
|
||||
j = SetoidPartialOrder.<WellDefined pOrder (Equivalence.transitive eq identLeft (Equivalence.transitive eq *Commutative (Ring.timesZero R))) (symmetric (Equivalence.transitive eq *Commutative (Equivalence.transitive eq *DistributesOver+ (Group.+WellDefined additiveGroup *Commutative *Commutative)))) i
|
||||
k : 0R < (0R * c)
|
||||
k = SetoidPartialOrder.<WellDefined pOrder reflexive (*WellDefined invRight reflexive) j
|
||||
q' | inr 0=y-x = exFalso (SetoidPartialOrder.irreflexive pOrder (SetoidPartialOrder.<WellDefined pOrder (*WellDefined x=y reflexive) reflexive xc<yc))
|
||||
where
|
||||
f : inverse 0G ∼ inverse (y + inverse x)
|
||||
f = inverseWellDefined additiveGroup 0=y-x
|
||||
g : 0G ∼ (inverse y) + x
|
||||
g = Equivalence.transitive eq (symmetric (invIdentity additiveGroup)) (Equivalence.transitive eq f (Equivalence.transitive eq (Equivalence.transitive eq (invContravariant additiveGroup) groupIsAbelian) (+WellDefined reflexive (invInv additiveGroup))))
|
||||
x=y : x ∼ y
|
||||
x=y = transferToRight additiveGroup (symmetric (Equivalence.transitive eq g groupIsAbelian))
|
||||
q'' : (0R + x) < ((y + Group.inverse additiveGroup x) + x)
|
||||
q'' = OrderedRing.orderRespectsAddition order q' x
|
||||
ringCanCancelPositive : {x y c : A} → (Ring.0R R) < c → (x * c) < (y * c) → x < y
|
||||
ringCanCancelPositive {x} {y} {c} 0<c xc<yc = SetoidPartialOrder.<WellDefined pOrder (Group.identLeft additiveGroup) (Equivalence.transitive eq (symmetric (Group.+Associative additiveGroup)) (Equivalence.transitive eq (Group.+WellDefined additiveGroup reflexive (Group.invLeft additiveGroup)) (Group.identRight additiveGroup))) q''
|
||||
where
|
||||
open Equivalence (Setoid.eq S)
|
||||
have : 0R < ((y * c) + (Group.inverse additiveGroup (x * c)))
|
||||
have = SetoidPartialOrder.<WellDefined pOrder (Group.invRight additiveGroup) reflexive (OrderedRing.orderRespectsAddition order xc<yc (Group.inverse additiveGroup _))
|
||||
p1 : 0R < ((y * c) + ((Group.inverse additiveGroup x) * c))
|
||||
p1 = SetoidPartialOrder.<WellDefined pOrder reflexive (Group.+WellDefined additiveGroup reflexive (symmetric (Equivalence.transitive eq (*Commutative) (Equivalence.transitive eq ringMinusExtracts (inverseWellDefined additiveGroup *Commutative))))) have
|
||||
q : 0R < ((y + Group.inverse additiveGroup x) * c)
|
||||
q = SetoidPartialOrder.<WellDefined pOrder reflexive (Equivalence.transitive eq (Equivalence.transitive eq (Group.+WellDefined additiveGroup *Commutative *Commutative) (symmetric *DistributesOver+)) *Commutative) p1
|
||||
q' : 0R < (y + Group.inverse additiveGroup x)
|
||||
q' with SetoidTotalOrder.totality tOrder 0R (y + Group.inverse additiveGroup x)
|
||||
q' | inl (inl pr) = pr
|
||||
q' | inl (inr y-x<0) = exFalso (SetoidPartialOrder.irreflexive pOrder (SetoidPartialOrder.<WellDefined pOrder reflexive (Equivalence.transitive eq *Commutative (Ring.timesZero R)) k))
|
||||
where
|
||||
f : ((y + inverse x) + (inverse (y + inverse x))) < (0G + inverse (y + inverse x))
|
||||
f = OrderedRing.orderRespectsAddition order y-x<0 _
|
||||
g : 0G < inverse (y + inverse x)
|
||||
g = SetoidPartialOrder.<WellDefined pOrder invRight identLeft f
|
||||
h : (0G * c) < ((inverse (y + inverse x)) * c)
|
||||
h = ringCanMultiplyByPositive 0<c g
|
||||
i : (0R + (0G * c)) < (((y + inverse x) * c) + ((inverse (y + inverse x)) * c))
|
||||
i = ringAddInequalities q h
|
||||
j : 0R < (((y + inverse x) + (inverse (y + inverse x))) * c)
|
||||
j = SetoidPartialOrder.<WellDefined pOrder (Equivalence.transitive eq identLeft (Equivalence.transitive eq *Commutative (Ring.timesZero R))) (symmetric (Equivalence.transitive eq *Commutative (Equivalence.transitive eq *DistributesOver+ (Group.+WellDefined additiveGroup *Commutative *Commutative)))) i
|
||||
k : 0R < (0R * c)
|
||||
k = SetoidPartialOrder.<WellDefined pOrder reflexive (*WellDefined invRight reflexive) j
|
||||
q' | inr 0=y-x = exFalso (SetoidPartialOrder.irreflexive pOrder (SetoidPartialOrder.<WellDefined pOrder (*WellDefined x=y reflexive) reflexive xc<yc))
|
||||
where
|
||||
f : inverse 0G ∼ inverse (y + inverse x)
|
||||
f = inverseWellDefined additiveGroup 0=y-x
|
||||
g : 0G ∼ (inverse y) + x
|
||||
g = Equivalence.transitive eq (symmetric (invIdentity additiveGroup)) (Equivalence.transitive eq f (Equivalence.transitive eq (Equivalence.transitive eq (invContravariant additiveGroup) groupIsAbelian) (+WellDefined reflexive (invInv additiveGroup))))
|
||||
x=y : x ∼ y
|
||||
x=y = transferToRight additiveGroup (symmetric (Equivalence.transitive eq g groupIsAbelian))
|
||||
q'' : (0R + x) < ((y + Group.inverse additiveGroup x) + x)
|
||||
q'' = OrderedRing.orderRespectsAddition order q' x
|
||||
|
||||
ringSwapNegatives : {x y : A} → (Group.inverse (Ring.additiveGroup R) x) < (Group.inverse (Ring.additiveGroup R) y) → y < x
|
||||
ringSwapNegatives {x} {y} -x<-y = SetoidPartialOrder.<WellDefined pOrder (Equivalence.transitive eq (symmetric (Group.+Associative additiveGroup)) (Equivalence.transitive eq (Group.+WellDefined additiveGroup reflexive (Group.invLeft additiveGroup)) (Group.identRight additiveGroup))) (Group.identLeft additiveGroup) v
|
||||
where
|
||||
open Equivalence eq
|
||||
t : ((Group.inverse additiveGroup x) + y) < ((Group.inverse additiveGroup y) + y)
|
||||
t = OrderedRing.orderRespectsAddition order -x<-y y
|
||||
u : (y + (Group.inverse additiveGroup x)) < 0R
|
||||
u = SetoidPartialOrder.<WellDefined pOrder (groupIsAbelian) (Group.invLeft additiveGroup) t
|
||||
v : ((y + (Group.inverse additiveGroup x)) + x) < (0R + x)
|
||||
v = OrderedRing.orderRespectsAddition order u x
|
||||
ringSwapNegatives : {x y : A} → (Group.inverse (Ring.additiveGroup R) x) < (Group.inverse (Ring.additiveGroup R) y) → y < x
|
||||
ringSwapNegatives {x} {y} -x<-y = SetoidPartialOrder.<WellDefined pOrder (Equivalence.transitive eq (symmetric (Group.+Associative additiveGroup)) (Equivalence.transitive eq (Group.+WellDefined additiveGroup reflexive (Group.invLeft additiveGroup)) (Group.identRight additiveGroup))) (Group.identLeft additiveGroup) v
|
||||
where
|
||||
open Equivalence eq
|
||||
t : ((Group.inverse additiveGroup x) + y) < ((Group.inverse additiveGroup y) + y)
|
||||
t = OrderedRing.orderRespectsAddition order -x<-y y
|
||||
u : (y + (Group.inverse additiveGroup x)) < 0R
|
||||
u = SetoidPartialOrder.<WellDefined pOrder (groupIsAbelian) (Group.invLeft additiveGroup) t
|
||||
v : ((y + (Group.inverse additiveGroup x)) + x) < (0R + x)
|
||||
v = OrderedRing.orderRespectsAddition order u x
|
||||
|
||||
ringSwapNegatives' : {x y : A} → x < y → (Group.inverse (Ring.additiveGroup R) y) < (Group.inverse (Ring.additiveGroup R) x)
|
||||
ringSwapNegatives' {x} {y} x<y = ringSwapNegatives (<WellDefined (Equivalence.symmetric eq (invTwice additiveGroup _)) (Equivalence.symmetric eq (invTwice additiveGroup _)) x<y)
|
||||
ringSwapNegatives' : {x y : A} → x < y → (Group.inverse (Ring.additiveGroup R) y) < (Group.inverse (Ring.additiveGroup R) x)
|
||||
ringSwapNegatives' {x} {y} x<y = ringSwapNegatives (<WellDefined (Equivalence.symmetric eq (invTwice additiveGroup _)) (Equivalence.symmetric eq (invTwice additiveGroup _)) x<y)
|
||||
|
||||
ringCanMultiplyByNegative : {x y c : A} → c < (Ring.0R R) → x < y → (y * c) < (x * c)
|
||||
ringCanMultiplyByNegative {x} {y} {c} c<0 x<y = ringSwapNegatives u
|
||||
where
|
||||
open Equivalence eq
|
||||
p1 : (c + Group.inverse additiveGroup c) < (0R + Group.inverse additiveGroup c)
|
||||
p1 = OrderedRing.orderRespectsAddition order c<0 _
|
||||
0<-c : 0R < (Group.inverse additiveGroup c)
|
||||
0<-c = SetoidPartialOrder.<WellDefined pOrder (Group.invRight additiveGroup) (Group.identLeft additiveGroup) p1
|
||||
t : (x * Group.inverse additiveGroup c) < (y * Group.inverse additiveGroup c)
|
||||
t = ringCanMultiplyByPositive 0<-c x<y
|
||||
u : (Group.inverse additiveGroup (x * c)) < Group.inverse additiveGroup (y * c)
|
||||
u = SetoidPartialOrder.<WellDefined pOrder ringMinusExtracts ringMinusExtracts t
|
||||
ringCanMultiplyByNegative : {x y c : A} → c < (Ring.0R R) → x < y → (y * c) < (x * c)
|
||||
ringCanMultiplyByNegative {x} {y} {c} c<0 x<y = ringSwapNegatives u
|
||||
where
|
||||
open Equivalence eq
|
||||
p1 : (c + Group.inverse additiveGroup c) < (0R + Group.inverse additiveGroup c)
|
||||
p1 = OrderedRing.orderRespectsAddition order c<0 _
|
||||
0<-c : 0R < (Group.inverse additiveGroup c)
|
||||
0<-c = SetoidPartialOrder.<WellDefined pOrder (Group.invRight additiveGroup) (Group.identLeft additiveGroup) p1
|
||||
t : (x * Group.inverse additiveGroup c) < (y * Group.inverse additiveGroup c)
|
||||
t = ringCanMultiplyByPositive 0<-c x<y
|
||||
u : (Group.inverse additiveGroup (x * c)) < Group.inverse additiveGroup (y * c)
|
||||
u = SetoidPartialOrder.<WellDefined pOrder ringMinusExtracts ringMinusExtracts t
|
||||
|
||||
ringCanCancelNegative : {x y c : A} → c < (Ring.0R R) → (x * c) < (y * c) → y < x
|
||||
ringCanCancelNegative {x} {y} {c} c<0 xc<yc = r
|
||||
where
|
||||
open Equivalence eq
|
||||
p0 : 0R < ((y * c) + inverse (x * c))
|
||||
p0 = SetoidPartialOrder.<WellDefined pOrder invRight reflexive (OrderedRing.orderRespectsAddition order xc<yc (inverse (x * c)))
|
||||
p1 : 0R < ((y * c) + ((inverse x) * c))
|
||||
p1 = SetoidPartialOrder.<WellDefined pOrder reflexive (Group.+WellDefined additiveGroup reflexive (Equivalence.transitive eq (inverseWellDefined additiveGroup *Commutative) (Equivalence.transitive eq (symmetric ringMinusExtracts) *Commutative))) p0
|
||||
p2 : 0R < ((y + inverse x) * c)
|
||||
p2 = SetoidPartialOrder.<WellDefined pOrder reflexive (Equivalence.transitive eq (Group.+WellDefined additiveGroup *Commutative *Commutative) (Equivalence.transitive eq (symmetric *DistributesOver+) *Commutative)) p1
|
||||
q : (y + inverse x) < 0R
|
||||
q with SetoidTotalOrder.totality tOrder 0R (y + inverse x)
|
||||
q | inl (inl pr) = exFalso (SetoidPartialOrder.irreflexive pOrder (SetoidPartialOrder.transitive pOrder bad c<0))
|
||||
where
|
||||
bad : 0R < c
|
||||
bad = ringCanCancelPositive pr (SetoidPartialOrder.<WellDefined pOrder (symmetric (Equivalence.transitive eq *Commutative (Ring.timesZero R))) *Commutative p2)
|
||||
q | inl (inr pr) = pr
|
||||
q | inr 0=y-x = exFalso (SetoidPartialOrder.irreflexive pOrder (SetoidPartialOrder.<WellDefined pOrder (*WellDefined x=y reflexive) reflexive xc<yc))
|
||||
where
|
||||
x=y : x ∼ y
|
||||
x=y = Equivalence.transitive eq (symmetric identLeft) (Equivalence.transitive eq (Group.+WellDefined additiveGroup 0=y-x reflexive) (Equivalence.transitive eq (symmetric (Group.+Associative additiveGroup)) (Equivalence.transitive eq (Group.+WellDefined additiveGroup reflexive invLeft) identRight)))
|
||||
r : y < x
|
||||
r = SetoidPartialOrder.<WellDefined pOrder (Equivalence.transitive eq (symmetric (Group.+Associative additiveGroup)) (Equivalence.transitive eq (Group.+WellDefined additiveGroup reflexive (invLeft)) identRight)) (Group.identLeft additiveGroup) (OrderedRing.orderRespectsAddition order q x)
|
||||
ringCanCancelNegative : {x y c : A} → c < (Ring.0R R) → (x * c) < (y * c) → y < x
|
||||
ringCanCancelNegative {x} {y} {c} c<0 xc<yc = r
|
||||
where
|
||||
open Equivalence eq
|
||||
p0 : 0R < ((y * c) + inverse (x * c))
|
||||
p0 = SetoidPartialOrder.<WellDefined pOrder invRight reflexive (OrderedRing.orderRespectsAddition order xc<yc (inverse (x * c)))
|
||||
p1 : 0R < ((y * c) + ((inverse x) * c))
|
||||
p1 = SetoidPartialOrder.<WellDefined pOrder reflexive (Group.+WellDefined additiveGroup reflexive (Equivalence.transitive eq (inverseWellDefined additiveGroup *Commutative) (Equivalence.transitive eq (symmetric ringMinusExtracts) *Commutative))) p0
|
||||
p2 : 0R < ((y + inverse x) * c)
|
||||
p2 = SetoidPartialOrder.<WellDefined pOrder reflexive (Equivalence.transitive eq (Group.+WellDefined additiveGroup *Commutative *Commutative) (Equivalence.transitive eq (symmetric *DistributesOver+) *Commutative)) p1
|
||||
q : (y + inverse x) < 0R
|
||||
q with SetoidTotalOrder.totality tOrder 0R (y + inverse x)
|
||||
q | inl (inl pr) = exFalso (SetoidPartialOrder.irreflexive pOrder (SetoidPartialOrder.transitive pOrder bad c<0))
|
||||
where
|
||||
bad : 0R < c
|
||||
bad = ringCanCancelPositive pr (SetoidPartialOrder.<WellDefined pOrder (symmetric (Equivalence.transitive eq *Commutative (Ring.timesZero R))) *Commutative p2)
|
||||
q | inl (inr pr) = pr
|
||||
q | inr 0=y-x = exFalso (SetoidPartialOrder.irreflexive pOrder (SetoidPartialOrder.<WellDefined pOrder (*WellDefined x=y reflexive) reflexive xc<yc))
|
||||
where
|
||||
x=y : x ∼ y
|
||||
x=y = Equivalence.transitive eq (symmetric identLeft) (Equivalence.transitive eq (Group.+WellDefined additiveGroup 0=y-x reflexive) (Equivalence.transitive eq (symmetric (Group.+Associative additiveGroup)) (Equivalence.transitive eq (Group.+WellDefined additiveGroup reflexive invLeft) identRight)))
|
||||
r : y < x
|
||||
r = SetoidPartialOrder.<WellDefined pOrder (Equivalence.transitive eq (symmetric (Group.+Associative additiveGroup)) (Equivalence.transitive eq (Group.+WellDefined additiveGroup reflexive (invLeft)) identRight)) (Group.identLeft additiveGroup) (OrderedRing.orderRespectsAddition order q x)
|
||||
|
||||
absZero : {a b c : _} {A : Set a} {S : Setoid {a} {b} A} {_+_ : A → A → A} {_*_ : A → A → A} {R : Ring S _+_ _*_} {_<_ : Rel {a} {c} A} {p : SetoidPartialOrder S _<_} {t : SetoidTotalOrder p} (o : OrderedRing R t) → OrderedRing.abs o (Ring.0R R) ≡ Ring.0R R
|
||||
absZero {R = R} {t = t} oR with SetoidTotalOrder.totality t (Ring.0R R) (Ring.0R R)
|
||||
absZero {R = R} {t = t} oR | inl (inl x) = exFalso (SetoidPartialOrder.irreflexive (SetoidTotalOrder.partial t) x)
|
||||
absZero {R = R} {t = t} oR | inl (inr x) = exFalso (SetoidPartialOrder.irreflexive (SetoidTotalOrder.partial t) x)
|
||||
absZero {R = R} {t = t} oR | inr x = refl
|
||||
absZero : {a b c : _} {A : Set a} {S : Setoid {a} {b} A} {_+_ : A → A → A} {_*_ : A → A → A} {R : Ring S _+_ _*_} {_<_ : Rel {a} {c} A} {p : SetoidPartialOrder S _<_} {t : SetoidTotalOrder p} (o : OrderedRing R t) → OrderedRing.abs o (Ring.0R R) ≡ Ring.0R R
|
||||
absZero {R = R} {t = t} oR with SetoidTotalOrder.totality t (Ring.0R R) (Ring.0R R)
|
||||
absZero {R = R} {t = t} oR | inl (inl x) = exFalso (SetoidPartialOrder.irreflexive (SetoidTotalOrder.partial t) x)
|
||||
absZero {R = R} {t = t} oR | inl (inr x) = exFalso (SetoidPartialOrder.irreflexive (SetoidTotalOrder.partial t) x)
|
||||
absZero {R = R} {t = t} oR | inr x = refl
|
||||
|
||||
absNegation : (a : A) → (abs a) ∼ (abs (inverse a))
|
||||
absNegation a with totality 0R a
|
||||
absNegation a | inl (inl 0<a) with totality 0G (inverse a)
|
||||
absNegation a | inl (inl 0<a) | inl (inl 0<-a) = exFalso (irreflexive {0G} (SetoidPartialOrder.transitive pOrder 0<-a (lemm2' a 0<a)))
|
||||
absNegation a | inl (inl 0<a) | inl (inr -a<0) = Equivalence.symmetric eq (invTwice additiveGroup a)
|
||||
absNegation a | inl (inl 0<a) | inr 0=-a = exFalso (irreflexive {0G} (<WellDefined (Equivalence.reflexive eq) (Equivalence.transitive eq (Equivalence.transitive eq (Equivalence.symmetric eq (invTwice additiveGroup a)) (inverseWellDefined additiveGroup (Equivalence.symmetric eq 0=-a))) (invIdent additiveGroup)) 0<a))
|
||||
absNegation a | inl (inr a<0) with totality 0G (inverse a)
|
||||
absNegation a | inl (inr a<0) | inl (inl 0<-a) = Equivalence.reflexive eq
|
||||
absNegation a | inl (inr a<0) | inl (inr -a<0) = exFalso (irreflexive {0G} (SetoidPartialOrder.transitive pOrder (<WellDefined (Equivalence.reflexive eq) (invTwice additiveGroup a) (lemm2 (inverse a) -a<0)) a<0))
|
||||
absNegation a | inl (inr a<0) | inr 0=-a = exFalso (irreflexive {0G} (<WellDefined (Equivalence.transitive eq (Equivalence.symmetric eq (Equivalence.transitive eq (inverseWellDefined additiveGroup 0=-a) (invTwice additiveGroup a))) (invIdent additiveGroup)) (Equivalence.reflexive eq) a<0))
|
||||
absNegation a | inr 0=a with totality 0G (inverse a)
|
||||
absNegation a | inr 0=a | inl (inl 0<-a) = exFalso (irreflexive {0G} (<WellDefined (Equivalence.reflexive eq) (Equivalence.transitive eq (inverseWellDefined additiveGroup (Equivalence.symmetric eq 0=a)) (invIdent additiveGroup)) 0<-a))
|
||||
absNegation a | inr 0=a | inl (inr -a<0) = exFalso (irreflexive {0G} (<WellDefined (Equivalence.transitive eq (inverseWellDefined additiveGroup (Equivalence.symmetric eq 0=a)) (invIdent additiveGroup)) (Equivalence.reflexive eq) -a<0))
|
||||
absNegation a | inr 0=a | inr 0=-a = Equivalence.transitive eq (Equivalence.symmetric eq 0=a) 0=-a
|
||||
absNegation : (a : A) → (abs a) ∼ (abs (inverse a))
|
||||
absNegation a with totality 0R a
|
||||
absNegation a | inl (inl 0<a) with totality 0G (inverse a)
|
||||
absNegation a | inl (inl 0<a) | inl (inl 0<-a) = exFalso (irreflexive {0G} (SetoidPartialOrder.transitive pOrder 0<-a (lemm2' a 0<a)))
|
||||
absNegation a | inl (inl 0<a) | inl (inr -a<0) = Equivalence.symmetric eq (invTwice additiveGroup a)
|
||||
absNegation a | inl (inl 0<a) | inr 0=-a = exFalso (irreflexive {0G} (<WellDefined (Equivalence.reflexive eq) (Equivalence.transitive eq (Equivalence.transitive eq (Equivalence.symmetric eq (invTwice additiveGroup a)) (inverseWellDefined additiveGroup (Equivalence.symmetric eq 0=-a))) (invIdent additiveGroup)) 0<a))
|
||||
absNegation a | inl (inr a<0) with totality 0G (inverse a)
|
||||
absNegation a | inl (inr a<0) | inl (inl 0<-a) = Equivalence.reflexive eq
|
||||
absNegation a | inl (inr a<0) | inl (inr -a<0) = exFalso (irreflexive {0G} (SetoidPartialOrder.transitive pOrder (<WellDefined (Equivalence.reflexive eq) (invTwice additiveGroup a) (lemm2 (inverse a) -a<0)) a<0))
|
||||
absNegation a | inl (inr a<0) | inr 0=-a = exFalso (irreflexive {0G} (<WellDefined (Equivalence.transitive eq (Equivalence.symmetric eq (Equivalence.transitive eq (inverseWellDefined additiveGroup 0=-a) (invTwice additiveGroup a))) (invIdent additiveGroup)) (Equivalence.reflexive eq) a<0))
|
||||
absNegation a | inr 0=a with totality 0G (inverse a)
|
||||
absNegation a | inr 0=a | inl (inl 0<-a) = exFalso (irreflexive {0G} (<WellDefined (Equivalence.reflexive eq) (Equivalence.transitive eq (inverseWellDefined additiveGroup (Equivalence.symmetric eq 0=a)) (invIdent additiveGroup)) 0<-a))
|
||||
absNegation a | inr 0=a | inl (inr -a<0) = exFalso (irreflexive {0G} (<WellDefined (Equivalence.transitive eq (inverseWellDefined additiveGroup (Equivalence.symmetric eq 0=a)) (invIdent additiveGroup)) (Equivalence.reflexive eq) -a<0))
|
||||
absNegation a | inr 0=a | inr 0=-a = Equivalence.transitive eq (Equivalence.symmetric eq 0=a) 0=-a
|
||||
|
||||
posTimesNeg : (a b : A) → (0G < a) → (b < 0G) → (a * b) < 0G
|
||||
posTimesNeg a b 0<a b<0 with orderRespectsMultiplication 0<a (lemm2 _ b<0)
|
||||
... | bl = <WellDefined (invTwice additiveGroup _) (Equivalence.reflexive eq) (lemm2' _ (<WellDefined (Equivalence.reflexive eq) ringMinusExtracts bl))
|
||||
posTimesNeg : (a b : A) → (0G < a) → (b < 0G) → (a * b) < 0G
|
||||
posTimesNeg a b 0<a b<0 with orderRespectsMultiplication 0<a (lemm2 _ b<0)
|
||||
... | bl = <WellDefined (invTwice additiveGroup _) (Equivalence.reflexive eq) (lemm2' _ (<WellDefined (Equivalence.reflexive eq) ringMinusExtracts bl))
|
||||
|
||||
negTimesPos : (a b : A) → (a < 0G) → (b < 0G) → 0G < (a * b)
|
||||
negTimesPos a b a<0 b<0 with orderRespectsMultiplication (lemm2 _ a<0) (lemm2 _ b<0)
|
||||
... | bl = <WellDefined (Equivalence.reflexive eq) twoNegativesTimes bl
|
||||
negTimesPos : (a b : A) → (a < 0G) → (b < 0G) → 0G < (a * b)
|
||||
negTimesPos a b a<0 b<0 with orderRespectsMultiplication (lemm2 _ a<0) (lemm2 _ b<0)
|
||||
... | bl = <WellDefined (Equivalence.reflexive eq) twoNegativesTimes bl
|
||||
|
||||
absRespectsTimes : (a b : A) → abs (a * b) ∼ (abs a) * (abs b)
|
||||
absRespectsTimes a b with totality 0R a
|
||||
absRespectsTimes a b | inl (inl 0<a) with totality 0R b
|
||||
absRespectsTimes a b | inl (inl 0<a) | inl (inl 0<b) with totality 0R (a * b)
|
||||
absRespectsTimes a b | inl (inl 0<a) | inl (inl 0<b) | inl (inl 0<ab) = Equivalence.reflexive eq
|
||||
absRespectsTimes a b | inl (inl 0<a) | inl (inl 0<b) | inl (inr ab<0) = exFalso (irreflexive {0G} (SetoidPartialOrder.transitive pOrder (orderRespectsMultiplication 0<a 0<b) ab<0))
|
||||
absRespectsTimes a b | inl (inl 0<a) | inl (inl 0<b) | inr 0=ab = exFalso (irreflexive {0G} (<WellDefined (Equivalence.reflexive eq) (Equivalence.symmetric eq 0=ab) (orderRespectsMultiplication 0<a 0<b)))
|
||||
absRespectsTimes a b | inl (inl 0<a) | inl (inr b<0) with totality 0R (a * b)
|
||||
absRespectsTimes a b | inl (inl 0<a) | inl (inr b<0) | inl (inl 0<ab) with <WellDefined (Equivalence.reflexive eq) ringMinusExtracts (orderRespectsMultiplication 0<a (lemm2 b b<0))
|
||||
... | bl = exFalso (irreflexive {0G} (SetoidPartialOrder.transitive pOrder 0<ab (<WellDefined (invTwice additiveGroup _) (Equivalence.reflexive eq) (lemm2' _ bl))))
|
||||
absRespectsTimes a b | inl (inl 0<a) | inl (inr b<0) | inl (inr ab<0) = Equivalence.symmetric eq ringMinusExtracts
|
||||
absRespectsTimes a b | inl (inl 0<a) | inl (inr b<0) | inr 0=ab = exFalso (irreflexive {0G} (<WellDefined (Equivalence.symmetric eq 0=ab) (Equivalence.reflexive eq) (posTimesNeg a b 0<a b<0)))
|
||||
absRespectsTimes a b | inl (inl 0<a) | inr 0=b with totality 0R (a * b)
|
||||
absRespectsTimes a b | inl (inl 0<a) | inr 0=b | inl (inl 0<ab) = exFalso (irreflexive {0G} (<WellDefined (Equivalence.reflexive eq) (Equivalence.transitive eq (*WellDefined (Equivalence.reflexive eq) (Equivalence.symmetric eq 0=b)) (timesZero {a})) 0<ab))
|
||||
absRespectsTimes a b | inl (inl 0<a) | inr 0=b | inl (inr ab<0) = exFalso ((irreflexive {0G} (<WellDefined (Equivalence.transitive eq (*WellDefined (Equivalence.reflexive eq) (Equivalence.symmetric eq 0=b)) (timesZero {a})) (Equivalence.reflexive eq) ab<0)))
|
||||
absRespectsTimes a b | inl (inl 0<a) | inr 0=b | inr 0=ab = Equivalence.reflexive eq
|
||||
absRespectsTimes a b | inl (inr a<0) with totality 0R b
|
||||
absRespectsTimes a b | inl (inr a<0) | inl (inl 0<b) with totality 0R (a * b)
|
||||
absRespectsTimes a b | inl (inr a<0) | inl (inl 0<b) | inl (inl 0<ab) = exFalso (irreflexive {0G} (SetoidPartialOrder.transitive pOrder 0<ab (<WellDefined *Commutative (Equivalence.reflexive eq) (posTimesNeg b a 0<b a<0))))
|
||||
absRespectsTimes a b | inl (inr a<0) | inl (inl 0<b) | inl (inr ab<0) = Equivalence.symmetric eq ringMinusExtracts'
|
||||
absRespectsTimes a b | inl (inr a<0) | inl (inl 0<b) | inr 0=ab = exFalso (irreflexive {0G} (<WellDefined (Equivalence.symmetric eq (Equivalence.transitive eq 0=ab *Commutative)) (Equivalence.reflexive eq) (posTimesNeg b a 0<b a<0)))
|
||||
absRespectsTimes a b | inl (inr a<0) | inl (inr b<0) with totality 0R (a * b)
|
||||
absRespectsTimes a b | inl (inr a<0) | inl (inr b<0) | inl (inl 0<ab) = Equivalence.symmetric eq twoNegativesTimes
|
||||
absRespectsTimes a b | inl (inr a<0) | inl (inr b<0) | inl (inr ab<0) = exFalso (irreflexive {0G} (SetoidPartialOrder.transitive pOrder (negTimesPos a b a<0 b<0) ab<0))
|
||||
absRespectsTimes a b | inl (inr a<0) | inl (inr b<0) | inr 0=ab = exFalso (exFalso (irreflexive {0G} (<WellDefined (Equivalence.reflexive eq) (Equivalence.symmetric eq 0=ab) (negTimesPos a b a<0 b<0))))
|
||||
absRespectsTimes a b | inl (inr a<0) | inr 0=b with totality 0R (a * b)
|
||||
absRespectsTimes a b | inl (inr a<0) | inr 0=b | inl (inl 0<ab) = exFalso (irreflexive {0R} (<WellDefined (Equivalence.reflexive eq) (Equivalence.transitive eq (*WellDefined (Equivalence.reflexive eq) (Equivalence.symmetric eq 0=b)) (timesZero {a})) 0<ab))
|
||||
absRespectsTimes a b | inl (inr a<0) | inr 0=b | inl (inr ab<0) = exFalso (irreflexive {0R} (<WellDefined (Equivalence.transitive eq (*WellDefined (Equivalence.reflexive eq) (Equivalence.symmetric eq 0=b)) timesZero) (Equivalence.reflexive eq) ab<0))
|
||||
absRespectsTimes a b | inl (inr a<0) | inr 0=b | inr 0=ab = Equivalence.transitive eq (*WellDefined (Equivalence.reflexive eq) (Equivalence.symmetric eq 0=b)) (Equivalence.transitive eq (Equivalence.transitive eq timesZero (Equivalence.symmetric eq timesZero)) (*WellDefined (Equivalence.reflexive eq) 0=b))
|
||||
absRespectsTimes a b | inr 0=a with totality 0R b
|
||||
absRespectsTimes a b | inr 0=a | inl (inl 0<b) with totality 0R (a * b)
|
||||
absRespectsTimes a b | inr 0=a | inl (inl 0<b) | inl (inl 0<ab) = Equivalence.reflexive eq
|
||||
absRespectsTimes a b | inr 0=a | inl (inl 0<b) | inl (inr ab<0) = Equivalence.transitive eq (Equivalence.transitive eq (Equivalence.transitive eq (inverseWellDefined additiveGroup (Equivalence.transitive eq (*WellDefined (Equivalence.symmetric eq 0=a) (Equivalence.reflexive eq)) (Equivalence.transitive eq *Commutative timesZero))) (invIdent additiveGroup)) (Equivalence.transitive eq (Equivalence.symmetric eq timesZero) *Commutative)) (*WellDefined 0=a (Equivalence.reflexive eq))
|
||||
absRespectsTimes a b | inr 0=a | inl (inl 0<b) | inr 0=ab = Equivalence.reflexive eq
|
||||
absRespectsTimes a b | inr 0=a | inl (inr b<0) with totality 0R (a * b)
|
||||
absRespectsTimes a b | inr 0=a | inl (inr b<0) | inl (inl 0<ab) = Equivalence.transitive eq (Equivalence.transitive eq (*WellDefined (Equivalence.symmetric eq 0=a) (Equivalence.reflexive eq)) *Commutative) (Equivalence.transitive eq timesZero (Equivalence.transitive eq (Equivalence.symmetric eq (Equivalence.transitive eq *Commutative timesZero)) (*WellDefined 0=a (Equivalence.reflexive eq))))
|
||||
absRespectsTimes a b | inr 0=a | inl (inr b<0) | inl (inr ab<0) = Equivalence.symmetric eq ringMinusExtracts
|
||||
absRespectsTimes a b | inr 0=a | inl (inr b<0) | inr 0=ab = Equivalence.transitive eq (Equivalence.transitive eq (*WellDefined (Equivalence.symmetric eq 0=a) (Equivalence.reflexive eq)) *Commutative) (Equivalence.transitive eq timesZero (Equivalence.transitive eq (Equivalence.symmetric eq (Equivalence.transitive eq *Commutative timesZero)) (*WellDefined 0=a (Equivalence.reflexive eq))))
|
||||
absRespectsTimes a b | inr 0=a | inr 0=b with totality 0R (a * b)
|
||||
absRespectsTimes a b | inr 0=a | inr 0=b | inl (inl 0<ab) = exFalso (irreflexive {0R} (<WellDefined (Equivalence.reflexive eq) (Equivalence.transitive eq (*WellDefined (Equivalence.reflexive eq) (Equivalence.symmetric eq 0=b)) timesZero) 0<ab))
|
||||
absRespectsTimes a b | inr 0=a | inr 0=b | inl (inr ab<0) = exFalso (irreflexive {0R} (<WellDefined (Equivalence.transitive eq (*WellDefined (Equivalence.reflexive eq) (Equivalence.symmetric eq 0=b)) timesZero) (Equivalence.reflexive eq) ab<0))
|
||||
absRespectsTimes a b | inr 0=a | inr 0=b | inr 0=ab = Equivalence.reflexive eq
|
||||
absRespectsTimes : (a b : A) → abs (a * b) ∼ (abs a) * (abs b)
|
||||
absRespectsTimes a b with totality 0R a
|
||||
absRespectsTimes a b | inl (inl 0<a) with totality 0R b
|
||||
absRespectsTimes a b | inl (inl 0<a) | inl (inl 0<b) with totality 0R (a * b)
|
||||
absRespectsTimes a b | inl (inl 0<a) | inl (inl 0<b) | inl (inl 0<ab) = Equivalence.reflexive eq
|
||||
absRespectsTimes a b | inl (inl 0<a) | inl (inl 0<b) | inl (inr ab<0) = exFalso (irreflexive {0G} (SetoidPartialOrder.transitive pOrder (orderRespectsMultiplication 0<a 0<b) ab<0))
|
||||
absRespectsTimes a b | inl (inl 0<a) | inl (inl 0<b) | inr 0=ab = exFalso (irreflexive {0G} (<WellDefined (Equivalence.reflexive eq) (Equivalence.symmetric eq 0=ab) (orderRespectsMultiplication 0<a 0<b)))
|
||||
absRespectsTimes a b | inl (inl 0<a) | inl (inr b<0) with totality 0R (a * b)
|
||||
absRespectsTimes a b | inl (inl 0<a) | inl (inr b<0) | inl (inl 0<ab) with <WellDefined (Equivalence.reflexive eq) ringMinusExtracts (orderRespectsMultiplication 0<a (lemm2 b b<0))
|
||||
... | bl = exFalso (irreflexive {0G} (SetoidPartialOrder.transitive pOrder 0<ab (<WellDefined (invTwice additiveGroup _) (Equivalence.reflexive eq) (lemm2' _ bl))))
|
||||
absRespectsTimes a b | inl (inl 0<a) | inl (inr b<0) | inl (inr ab<0) = Equivalence.symmetric eq ringMinusExtracts
|
||||
absRespectsTimes a b | inl (inl 0<a) | inl (inr b<0) | inr 0=ab = exFalso (irreflexive {0G} (<WellDefined (Equivalence.symmetric eq 0=ab) (Equivalence.reflexive eq) (posTimesNeg a b 0<a b<0)))
|
||||
absRespectsTimes a b | inl (inl 0<a) | inr 0=b with totality 0R (a * b)
|
||||
absRespectsTimes a b | inl (inl 0<a) | inr 0=b | inl (inl 0<ab) = exFalso (irreflexive {0G} (<WellDefined (Equivalence.reflexive eq) (Equivalence.transitive eq (*WellDefined (Equivalence.reflexive eq) (Equivalence.symmetric eq 0=b)) (timesZero {a})) 0<ab))
|
||||
absRespectsTimes a b | inl (inl 0<a) | inr 0=b | inl (inr ab<0) = exFalso ((irreflexive {0G} (<WellDefined (Equivalence.transitive eq (*WellDefined (Equivalence.reflexive eq) (Equivalence.symmetric eq 0=b)) (timesZero {a})) (Equivalence.reflexive eq) ab<0)))
|
||||
absRespectsTimes a b | inl (inl 0<a) | inr 0=b | inr 0=ab = Equivalence.reflexive eq
|
||||
absRespectsTimes a b | inl (inr a<0) with totality 0R b
|
||||
absRespectsTimes a b | inl (inr a<0) | inl (inl 0<b) with totality 0R (a * b)
|
||||
absRespectsTimes a b | inl (inr a<0) | inl (inl 0<b) | inl (inl 0<ab) = exFalso (irreflexive {0G} (SetoidPartialOrder.transitive pOrder 0<ab (<WellDefined *Commutative (Equivalence.reflexive eq) (posTimesNeg b a 0<b a<0))))
|
||||
absRespectsTimes a b | inl (inr a<0) | inl (inl 0<b) | inl (inr ab<0) = Equivalence.symmetric eq ringMinusExtracts'
|
||||
absRespectsTimes a b | inl (inr a<0) | inl (inl 0<b) | inr 0=ab = exFalso (irreflexive {0G} (<WellDefined (Equivalence.symmetric eq (Equivalence.transitive eq 0=ab *Commutative)) (Equivalence.reflexive eq) (posTimesNeg b a 0<b a<0)))
|
||||
absRespectsTimes a b | inl (inr a<0) | inl (inr b<0) with totality 0R (a * b)
|
||||
absRespectsTimes a b | inl (inr a<0) | inl (inr b<0) | inl (inl 0<ab) = Equivalence.symmetric eq twoNegativesTimes
|
||||
absRespectsTimes a b | inl (inr a<0) | inl (inr b<0) | inl (inr ab<0) = exFalso (irreflexive {0G} (SetoidPartialOrder.transitive pOrder (negTimesPos a b a<0 b<0) ab<0))
|
||||
absRespectsTimes a b | inl (inr a<0) | inl (inr b<0) | inr 0=ab = exFalso (exFalso (irreflexive {0G} (<WellDefined (Equivalence.reflexive eq) (Equivalence.symmetric eq 0=ab) (negTimesPos a b a<0 b<0))))
|
||||
absRespectsTimes a b | inl (inr a<0) | inr 0=b with totality 0R (a * b)
|
||||
absRespectsTimes a b | inl (inr a<0) | inr 0=b | inl (inl 0<ab) = exFalso (irreflexive {0R} (<WellDefined (Equivalence.reflexive eq) (Equivalence.transitive eq (*WellDefined (Equivalence.reflexive eq) (Equivalence.symmetric eq 0=b)) (timesZero {a})) 0<ab))
|
||||
absRespectsTimes a b | inl (inr a<0) | inr 0=b | inl (inr ab<0) = exFalso (irreflexive {0R} (<WellDefined (Equivalence.transitive eq (*WellDefined (Equivalence.reflexive eq) (Equivalence.symmetric eq 0=b)) timesZero) (Equivalence.reflexive eq) ab<0))
|
||||
absRespectsTimes a b | inl (inr a<0) | inr 0=b | inr 0=ab = Equivalence.transitive eq (*WellDefined (Equivalence.reflexive eq) (Equivalence.symmetric eq 0=b)) (Equivalence.transitive eq (Equivalence.transitive eq timesZero (Equivalence.symmetric eq timesZero)) (*WellDefined (Equivalence.reflexive eq) 0=b))
|
||||
absRespectsTimes a b | inr 0=a with totality 0R b
|
||||
absRespectsTimes a b | inr 0=a | inl (inl 0<b) with totality 0R (a * b)
|
||||
absRespectsTimes a b | inr 0=a | inl (inl 0<b) | inl (inl 0<ab) = Equivalence.reflexive eq
|
||||
absRespectsTimes a b | inr 0=a | inl (inl 0<b) | inl (inr ab<0) = Equivalence.transitive eq (Equivalence.transitive eq (Equivalence.transitive eq (inverseWellDefined additiveGroup (Equivalence.transitive eq (*WellDefined (Equivalence.symmetric eq 0=a) (Equivalence.reflexive eq)) (Equivalence.transitive eq *Commutative timesZero))) (invIdent additiveGroup)) (Equivalence.transitive eq (Equivalence.symmetric eq timesZero) *Commutative)) (*WellDefined 0=a (Equivalence.reflexive eq))
|
||||
absRespectsTimes a b | inr 0=a | inl (inl 0<b) | inr 0=ab = Equivalence.reflexive eq
|
||||
absRespectsTimes a b | inr 0=a | inl (inr b<0) with totality 0R (a * b)
|
||||
absRespectsTimes a b | inr 0=a | inl (inr b<0) | inl (inl 0<ab) = Equivalence.transitive eq (Equivalence.transitive eq (*WellDefined (Equivalence.symmetric eq 0=a) (Equivalence.reflexive eq)) *Commutative) (Equivalence.transitive eq timesZero (Equivalence.transitive eq (Equivalence.symmetric eq (Equivalence.transitive eq *Commutative timesZero)) (*WellDefined 0=a (Equivalence.reflexive eq))))
|
||||
absRespectsTimes a b | inr 0=a | inl (inr b<0) | inl (inr ab<0) = Equivalence.symmetric eq ringMinusExtracts
|
||||
absRespectsTimes a b | inr 0=a | inl (inr b<0) | inr 0=ab = Equivalence.transitive eq (Equivalence.transitive eq (*WellDefined (Equivalence.symmetric eq 0=a) (Equivalence.reflexive eq)) *Commutative) (Equivalence.transitive eq timesZero (Equivalence.transitive eq (Equivalence.symmetric eq (Equivalence.transitive eq *Commutative timesZero)) (*WellDefined 0=a (Equivalence.reflexive eq))))
|
||||
absRespectsTimes a b | inr 0=a | inr 0=b with totality 0R (a * b)
|
||||
absRespectsTimes a b | inr 0=a | inr 0=b | inl (inl 0<ab) = exFalso (irreflexive {0R} (<WellDefined (Equivalence.reflexive eq) (Equivalence.transitive eq (*WellDefined (Equivalence.reflexive eq) (Equivalence.symmetric eq 0=b)) timesZero) 0<ab))
|
||||
absRespectsTimes a b | inr 0=a | inr 0=b | inl (inr ab<0) = exFalso (irreflexive {0R} (<WellDefined (Equivalence.transitive eq (*WellDefined (Equivalence.reflexive eq) (Equivalence.symmetric eq 0=b)) timesZero) (Equivalence.reflexive eq) ab<0))
|
||||
absRespectsTimes a b | inr 0=a | inr 0=b | inr 0=ab = Equivalence.reflexive eq
|
||||
|
||||
absNonnegative : {a : A} → (abs a < 0R) → False
|
||||
absNonnegative {a} pr with SetoidTotalOrder.totality tOrder 0R a
|
||||
absNonnegative {a} pr | inl (inl x) = irreflexive {0G} (SetoidPartialOrder.transitive pOrder x pr)
|
||||
absNonnegative {a} pr | inl (inr x) = irreflexive {0G} (SetoidPartialOrder.transitive pOrder (<WellDefined (Equivalence.reflexive eq) (invTwice additiveGroup a) (lemm2 (inverse a) pr)) x)
|
||||
absNonnegative {a} pr | inr x = irreflexive {0G} (<WellDefined (Equivalence.symmetric eq x) (Equivalence.reflexive eq) pr)
|
||||
absNonnegative : {a : A} → (abs a < 0R) → False
|
||||
absNonnegative {a} pr with SetoidTotalOrder.totality tOrder 0R a
|
||||
absNonnegative {a} pr | inl (inl x) = irreflexive {0G} (SetoidPartialOrder.transitive pOrder x pr)
|
||||
absNonnegative {a} pr | inl (inr x) = irreflexive {0G} (SetoidPartialOrder.transitive pOrder (<WellDefined (Equivalence.reflexive eq) (invTwice additiveGroup a) (lemm2 (inverse a) pr)) x)
|
||||
absNonnegative {a} pr | inr x = irreflexive {0G} (<WellDefined (Equivalence.symmetric eq x) (Equivalence.reflexive eq) pr)
|
||||
|
||||
a-bPos : {a b : A} → ((a ∼ b) → False) → 0R < abs (a + inverse b)
|
||||
a-bPos {a} {b} a!=b with totality 0R (a + inverse b)
|
||||
a-bPos {a} {b} a!=b | inl (inl x) = x
|
||||
a-bPos {a} {b} a!=b | inl (inr x) = lemm2 _ x
|
||||
a-bPos {a} {b} a!=b | inr x = exFalso (a!=b (transferToRight additiveGroup (Equivalence.symmetric eq x)))
|
||||
a-bPos : {a b : A} → ((a ∼ b) → False) → 0R < abs (a + inverse b)
|
||||
a-bPos {a} {b} a!=b with totality 0R (a + inverse b)
|
||||
a-bPos {a} {b} a!=b | inl (inl x) = x
|
||||
a-bPos {a} {b} a!=b | inl (inr x) = lemm2 _ x
|
||||
a-bPos {a} {b} a!=b | inr x = exFalso (a!=b (transferToRight additiveGroup (Equivalence.symmetric eq x)))
|
||||
|
||||
absZeroImpliesZero : {a : A} → abs a ∼ 0R → a ∼ 0R
|
||||
absZeroImpliesZero {a} a=0 with totality 0R a
|
||||
absZeroImpliesZero {a} a=0 | inl (inl 0<a) = exFalso (irreflexive {0G} (<WellDefined (Equivalence.reflexive eq) a=0 0<a))
|
||||
absZeroImpliesZero {a} a=0 | inl (inr a<0) = Equivalence.symmetric eq (lemm3 (inverse a) a (Equivalence.symmetric eq invLeft) (Equivalence.symmetric eq a=0))
|
||||
absZeroImpliesZero {a} a=0 | inr 0=a = a=0
|
||||
absZeroImpliesZero : {a : A} → abs a ∼ 0R → a ∼ 0R
|
||||
absZeroImpliesZero {a} a=0 with totality 0R a
|
||||
absZeroImpliesZero {a} a=0 | inl (inl 0<a) = exFalso (irreflexive {0G} (<WellDefined (Equivalence.reflexive eq) a=0 0<a))
|
||||
absZeroImpliesZero {a} a=0 | inl (inr a<0) = Equivalence.symmetric eq (lemm3 (inverse a) a (Equivalence.symmetric eq invLeft) (Equivalence.symmetric eq a=0))
|
||||
absZeroImpliesZero {a} a=0 | inr 0=a = a=0
|
||||
|
||||
halvePositive : (a : A) → 0R < (a + a) → 0R < a
|
||||
halvePositive a 0<2a with totality 0R a
|
||||
halvePositive a 0<2a | inl (inl x) = x
|
||||
halvePositive a 0<2a | inl (inr a<0) = exFalso (irreflexive {a + a} (SetoidPartialOrder.transitive pOrder (<WellDefined (Equivalence.reflexive eq) identRight (ringAddInequalities a<0 a<0)) 0<2a))
|
||||
halvePositive a 0<2a | inr x = exFalso (irreflexive {0G} (<WellDefined (Equivalence.reflexive eq) (Equivalence.transitive eq (+WellDefined (Equivalence.symmetric eq x) (Equivalence.symmetric eq x)) identRight) 0<2a))
|
||||
halvePositive : (a : A) → 0R < (a + a) → 0R < a
|
||||
halvePositive a 0<2a with totality 0R a
|
||||
halvePositive a 0<2a | inl (inl x) = x
|
||||
halvePositive a 0<2a | inl (inr a<0) = exFalso (irreflexive {a + a} (SetoidPartialOrder.transitive pOrder (<WellDefined (Equivalence.reflexive eq) identRight (ringAddInequalities a<0 a<0)) 0<2a))
|
||||
halvePositive a 0<2a | inr x = exFalso (irreflexive {0G} (<WellDefined (Equivalence.reflexive eq) (Equivalence.transitive eq (+WellDefined (Equivalence.symmetric eq x) (Equivalence.symmetric eq x)) identRight) 0<2a))
|
||||
|
||||
0<1 : (0R ∼ 1R → False) → 0R < 1R
|
||||
0<1 0!=1 with SetoidTotalOrder.totality tOrder 0R 1R
|
||||
0<1 0!=1 | inl (inl x) = x
|
||||
0<1 0!=1 | inl (inr x) = <WellDefined (Equivalence.reflexive eq) (Equivalence.transitive eq twoNegativesTimes identIsIdent) (orderRespectsMultiplication (lemm2 1R x) (lemm2 1R x))
|
||||
0<1 0!=1 | inr x = exFalso (0!=1 x)
|
||||
0<1 : (0R ∼ 1R → False) → 0R < 1R
|
||||
0<1 0!=1 with SetoidTotalOrder.totality tOrder 0R 1R
|
||||
0<1 0!=1 | inl (inl x) = x
|
||||
0<1 0!=1 | inl (inr x) = <WellDefined (Equivalence.reflexive eq) (Equivalence.transitive eq twoNegativesTimes identIsIdent) (orderRespectsMultiplication (lemm2 1R x) (lemm2 1R x))
|
||||
0<1 0!=1 | inr x = exFalso (0!=1 x)
|
||||
|
||||
addingAbsCannotShrink : {a b : A} → 0G < b → 0G < ((abs a) + b)
|
||||
addingAbsCannotShrink {a} {b} 0<b with SetoidTotalOrder.totality tOrder 0G a
|
||||
addingAbsCannotShrink {a} {b} 0<b | inl (inl x) = <WellDefined identLeft (Equivalence.reflexive eq) (ringAddInequalities x 0<b)
|
||||
addingAbsCannotShrink {a} {b} 0<b | inl (inr x) = <WellDefined identLeft (Equivalence.reflexive eq) (ringAddInequalities (lemm2 a x) 0<b)
|
||||
addingAbsCannotShrink {a} {b} 0<b | inr x = <WellDefined (Equivalence.reflexive eq) (Equivalence.transitive eq (Equivalence.symmetric eq identLeft) (+WellDefined x (Equivalence.reflexive eq))) 0<b
|
||||
addingAbsCannotShrink : {a b : A} → 0G < b → 0G < ((abs a) + b)
|
||||
addingAbsCannotShrink {a} {b} 0<b with SetoidTotalOrder.totality tOrder 0G a
|
||||
addingAbsCannotShrink {a} {b} 0<b | inl (inl x) = <WellDefined identLeft (Equivalence.reflexive eq) (ringAddInequalities x 0<b)
|
||||
addingAbsCannotShrink {a} {b} 0<b | inl (inr x) = <WellDefined identLeft (Equivalence.reflexive eq) (ringAddInequalities (lemm2 a x) 0<b)
|
||||
addingAbsCannotShrink {a} {b} 0<b | inr x = <WellDefined (Equivalence.reflexive eq) (Equivalence.transitive eq (Equivalence.symmetric eq identLeft) (+WellDefined x (Equivalence.reflexive eq))) 0<b
|
||||
|
||||
1<0False : (1R < 0R) → False
|
||||
1<0False 1<0 with orderRespectsMultiplication (lemm2 _ 1<0) (lemm2 _ 1<0)
|
||||
... | bl = exFalso (irreflexive (SetoidPartialOrder.transitive pOrder 1<0 (<WellDefined (Equivalence.reflexive eq) (Equivalence.transitive eq (twoNegativesTimes) identIsIdent) bl)))
|
||||
1<0False : (1R < 0R) → False
|
||||
1<0False 1<0 with orderRespectsMultiplication (lemm2 _ 1<0) (lemm2 _ 1<0)
|
||||
... | bl = exFalso (irreflexive (SetoidPartialOrder.transitive pOrder 1<0 (<WellDefined (Equivalence.reflexive eq) (Equivalence.transitive eq (twoNegativesTimes) identIsIdent) bl)))
|
||||
|
||||
greaterZeroImpliesEqualAbs : {a : A} → 0R < a → a ∼ abs a
|
||||
greaterZeroImpliesEqualAbs {a} 0<a with totality 0R a
|
||||
... | inl (inl _) = Equivalence.reflexive eq
|
||||
... | inl (inr a<0) = exFalso (irreflexive (SetoidPartialOrder.transitive pOrder a<0 0<a))
|
||||
... | inr 0=a = exFalso (irreflexive (<WellDefined 0=a (Equivalence.reflexive eq) 0<a))
|
||||
greaterZeroImpliesEqualAbs : {a : A} → 0R < a → a ∼ abs a
|
||||
greaterZeroImpliesEqualAbs {a} 0<a with totality 0R a
|
||||
... | inl (inl _) = Equivalence.reflexive eq
|
||||
... | inl (inr a<0) = exFalso (irreflexive (SetoidPartialOrder.transitive pOrder a<0 0<a))
|
||||
... | inr 0=a = exFalso (irreflexive (<WellDefined 0=a (Equivalence.reflexive eq) 0<a))
|
||||
|
||||
lessZeroImpliesEqualNegAbs : {a : A} → a < 0R → abs a ∼ inverse a
|
||||
lessZeroImpliesEqualNegAbs {a} a<0 with totality 0R a
|
||||
... | inl (inr _) = Equivalence.reflexive eq
|
||||
... | inl (inl 0<a) = exFalso (irreflexive (SetoidPartialOrder.transitive pOrder a<0 0<a))
|
||||
... | inr 0=a = exFalso (irreflexive (<WellDefined (Equivalence.reflexive eq) 0=a a<0))
|
||||
lessZeroImpliesEqualNegAbs : {a : A} → a < 0R → abs a ∼ inverse a
|
||||
lessZeroImpliesEqualNegAbs {a} a<0 with totality 0R a
|
||||
... | inl (inr _) = Equivalence.reflexive eq
|
||||
... | inl (inl 0<a) = exFalso (irreflexive (SetoidPartialOrder.transitive pOrder a<0 0<a))
|
||||
... | inr 0=a = exFalso (irreflexive (<WellDefined (Equivalence.reflexive eq) 0=a a<0))
|
||||
|
||||
absZeroIsZero : abs 0R ∼ 0R
|
||||
absZeroIsZero with totality 0R 0R
|
||||
... | inl (inl bad) = exFalso (irreflexive bad)
|
||||
... | inl (inr bad) = exFalso (irreflexive bad)
|
||||
... | inr _ = Equivalence.reflexive eq
|
||||
absZeroIsZero : abs 0R ∼ 0R
|
||||
absZeroIsZero with totality 0R 0R
|
||||
... | inl (inl bad) = exFalso (irreflexive bad)
|
||||
... | inl (inr bad) = exFalso (irreflexive bad)
|
||||
... | inr _ = Equivalence.reflexive eq
|
||||
|
||||
greaterThanAbsImpliesGreaterThan0 : {a b : A} → (abs a) < b → 0R < b
|
||||
greaterThanAbsImpliesGreaterThan0 {a} {b} a<b with totality 0R a
|
||||
greaterThanAbsImpliesGreaterThan0 {a} {b} a<b | inl (inl 0<a) = SetoidPartialOrder.transitive pOrder 0<a a<b
|
||||
greaterThanAbsImpliesGreaterThan0 {a} {b} a<b | inl (inr a<0) = SetoidPartialOrder.transitive pOrder (lemm2 _ a<0) a<b
|
||||
greaterThanAbsImpliesGreaterThan0 {a} {b} a<b | inr 0=a = <WellDefined (Equivalence.symmetric eq 0=a) (Equivalence.reflexive eq) a<b
|
||||
greaterThanAbsImpliesGreaterThan0 : {a b : A} → (abs a) < b → 0R < b
|
||||
greaterThanAbsImpliesGreaterThan0 {a} {b} a<b with totality 0R a
|
||||
greaterThanAbsImpliesGreaterThan0 {a} {b} a<b | inl (inl 0<a) = SetoidPartialOrder.transitive pOrder 0<a a<b
|
||||
greaterThanAbsImpliesGreaterThan0 {a} {b} a<b | inl (inr a<0) = SetoidPartialOrder.transitive pOrder (lemm2 _ a<0) a<b
|
||||
greaterThanAbsImpliesGreaterThan0 {a} {b} a<b | inr 0=a = <WellDefined (Equivalence.symmetric eq 0=a) (Equivalence.reflexive eq) a<b
|
||||
|
||||
anyComparisonImpliesNontrivial : {a b : A} → a < b → (0R ∼ 1R) → False
|
||||
anyComparisonImpliesNontrivial {a} {b} a<b 0=1 = irreflexive (<WellDefined (oneZeroImpliesAllZero 0=1) (oneZeroImpliesAllZero 0=1) a<b)
|
||||
anyComparisonImpliesNontrivial : {a b : A} → a < b → (0R ∼ 1R) → False
|
||||
anyComparisonImpliesNontrivial {a} {b} a<b 0=1 = irreflexive (<WellDefined (oneZeroImpliesAllZero 0=1) (oneZeroImpliesAllZero 0=1) a<b)
|
||||
|
||||
abs1Is1 : abs 1R ∼ 1R
|
||||
abs1Is1 with totality 0R 1R
|
||||
abs1Is1 | inl (inl 0<1) = Equivalence.reflexive eq
|
||||
abs1Is1 | inl (inr 1<0) = exFalso (1<0False 1<0)
|
||||
abs1Is1 | inr 0=1 = Equivalence.reflexive eq
|
||||
abs1Is1 : abs 1R ∼ 1R
|
||||
abs1Is1 with totality 0R 1R
|
||||
abs1Is1 | inl (inl 0<1) = Equivalence.reflexive eq
|
||||
abs1Is1 | inl (inr 1<0) = exFalso (1<0False 1<0)
|
||||
abs1Is1 | inr 0=1 = Equivalence.reflexive eq
|
||||
|
||||
charNot2ImpliesNontrivial : ((1R + 1R) ∼ 0R → False) → (0R ∼ 1R) → False
|
||||
charNot2ImpliesNontrivial charNot2 0=1 = charNot2 (Equivalence.transitive eq (+WellDefined (Equivalence.symmetric eq 0=1) (Equivalence.symmetric eq 0=1)) identRight)
|
||||
charNot2ImpliesNontrivial : ((1R + 1R) ∼ 0R → False) → (0R ∼ 1R) → False
|
||||
charNot2ImpliesNontrivial charNot2 0=1 = charNot2 (Equivalence.transitive eq (+WellDefined (Equivalence.symmetric eq 0=1) (Equivalence.symmetric eq 0=1)) identRight)
|
||||
|
||||
absBoundedImpliesBounded : {a b : A} → abs a < b → a < b
|
||||
absBoundedImpliesBounded {a} {b} a<b with SetoidTotalOrder.totality tOrder 0G a
|
||||
absBoundedImpliesBounded {a} {b} a<b | inl (inl x) = a<b
|
||||
absBoundedImpliesBounded {a} {b} a<b | inl (inr x) = SetoidPartialOrder.transitive pOrder x (SetoidPartialOrder.transitive pOrder (lemm2 a x) a<b)
|
||||
absBoundedImpliesBounded {a} {b} a<b | inr x = a<b
|
||||
|
||||
Reference in New Issue
Block a user