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Triangle inequality on the rationals (#20)

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Patrick Stevens
2019-01-18 13:00:15 +00:00
committed by GitHub
parent 20b31da82d
commit 9cccd51cbf
6 changed files with 356 additions and 22 deletions
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6
Rings/RingDefinition.agda
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@@ -61,3 +61,9 @@ module Rings.RingDefinition where
-- ringHom : RingHom R S f
-- bijective : Bijection f
abs : {a b c : _} {A : Set a} {S : Setoid {a} {b} A} {_+_ : A → A → A} {_*_ : A → A → A} {_<_ : Rel {_} {c} A} {pOrder : SetoidPartialOrder S _<_} (R : Ring S _+_ _*_) (order : SetoidTotalOrder pOrder) (x : A) → A
abs R order x with SetoidTotalOrder.totality order (Ring.0R R) x
... | inl (inl 0<x) = x
... | inl (inr x<0) = Group.inverse (Ring.additiveGroup R) x
... | inr 0=x = Ring.0R R

6
Rings/RingLemmas.agda
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@@ -88,6 +88,12 @@ module Rings.RingLemmas where
open Reflexive (Equivalence.reflexiveEq (Setoid.eq S))
open Symmetric (Equivalence.symmetricEq (Setoid.eq S))
ringMinusFlipsOrder'' : {a b c : _} {A : Set a} {S : Setoid {a} {b} A} {_+_ : A → A → A} {_*_ : A → A → A} {R : Ring S _+_ _*_} {_<_ : Rel {_} {c} A} {pOrder : SetoidPartialOrder S _<_} {tOrder : SetoidTotalOrder pOrder} (oRing : OrderedRing R tOrder) → {x : A} → x < (Ring.0R R) → (Ring.0R R) < Group.inverse (Ring.additiveGroup R) x
ringMinusFlipsOrder'' {S = S} {R = R} {pOrder = pOrder} oRing {x} x<0 = ringMinusFlipsOrder' oRing (SetoidPartialOrder.wellDefined pOrder {x} {Group.inverse (Ring.additiveGroup R) (Group.inverse (Ring.additiveGroup R) x)} {Ring.0R R} {Ring.0R R} (Symmetric.symmetric (Equivalence.symmetricEq (Setoid.eq S)) (invInv (Ring.additiveGroup R))) (Reflexive.reflexive (Equivalence.reflexiveEq (Setoid.eq S))) x<0)
ringMinusFlipsOrder''' : {a b c : _} {A : Set a} {S : Setoid {a} {b} A} {_+_ : A → A → A} {_*_ : A → A → A} {R : Ring S _+_ _*_} {_<_ : Rel {_} {c} A} {pOrder : SetoidPartialOrder S _<_} {tOrder : SetoidTotalOrder pOrder} (oRing : OrderedRing R tOrder) → {x : A} → (Ring.0R R) < (Group.inverse (Ring.additiveGroup R) x) → x < (Ring.0R R)
ringMinusFlipsOrder''' {S = S} {R = R} {pOrder = pOrder} oRing {x} 0<-x = SetoidPartialOrder.wellDefined pOrder (invInv (Ring.additiveGroup R)) (Reflexive.reflexive (Equivalence.reflexiveEq (Setoid.eq S))) (ringMinusFlipsOrder oRing 0<-x)
ringCanMultiplyByPositive : {a b c : _} {A : Set a} {S : Setoid {a} {b} A} {_+_ : A → A → A} {_*_ : A → A → A} {R : Ring S _+_ _*_} {_<_ : Rel {_} {c} A} {pOrder : SetoidPartialOrder S _<_} {tOrder : SetoidTotalOrder pOrder} (oRing : OrderedRing R tOrder) → {x y c : A} → (Ring.0R R) < c → x < y → (x * c) < (y * c)
ringCanMultiplyByPositive {S = S} {_+_ = _+_} {_*_ = _*_} {R = R} {_<_ = _<_} {pOrder = pOrder} {tOrder = tOrder} oRing {x} {y} {c} 0<c x<y = SetoidPartialOrder.wellDefined pOrder reflexive (Group.multIdentRight additiveGroup) q'
where