mirror of
https://github.com/Smaug123/agdaproofs
synced 2025-10-12 23:28:39 +00:00
Total order on the rationals (#17)
This commit is contained in:
Patrick Stevens
GitHub
@@ -2,6 +2,7 @@
|
||||
|
||||
open import LogicalFormulae
|
||||
open import Groups.Groups
|
||||
open import Groups.GroupDefinition
|
||||
open import Numbers.Naturals
|
||||
open import Orders
|
||||
open import Setoids.Setoids
|
||||
|
||||
@@ -2,6 +2,7 @@
|
||||
|
||||
open import LogicalFormulae
|
||||
open import Groups.Groups
|
||||
open import Groups.GroupDefinition
|
||||
open import Numbers.Naturals
|
||||
open import Setoids.Orders
|
||||
open import Setoids.Setoids
|
||||
|
||||
@@ -3,6 +3,7 @@
|
||||
open import LogicalFormulae
|
||||
open import Functions
|
||||
open import Groups.Groups
|
||||
open import Groups.GroupDefinition
|
||||
open import Groups.GroupsLemmas
|
||||
open import Rings.RingDefinition
|
||||
open import Setoids.Setoids
|
||||
@@ -86,3 +87,126 @@ module Rings.RingLemmas where
|
||||
open Transitive (Equivalence.transitiveEq (Setoid.eq S))
|
||||
open Reflexive (Equivalence.reflexiveEq (Setoid.eq S))
|
||||
open Symmetric (Equivalence.symmetricEq (Setoid.eq S))
|
||||
|
||||
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
|
||||
open Ring R
|
||||
open Transitive (Equivalence.transitiveEq (Setoid.eq S))
|
||||
open Reflexive (Equivalence.reflexiveEq (Setoid.eq S))
|
||||
open Symmetric (Equivalence.symmetricEq (Setoid.eq S))
|
||||
have : 0R < (y + Group.inverse additiveGroup x)
|
||||
have = SetoidPartialOrder.wellDefined pOrder (Group.invRight additiveGroup) reflexive (OrderedRing.orderRespectsAddition oRing x<y (Group.inverse additiveGroup x))
|
||||
p : 0R < ((y * c) + ((Group.inverse additiveGroup x) * c))
|
||||
p = SetoidPartialOrder.wellDefined pOrder reflexive (transitive multCommutative (transitive multDistributes ((Group.wellDefined additiveGroup) multCommutative multCommutative))) (OrderedRing.orderRespectsMultiplication oRing have 0<c)
|
||||
p' : 0R < ((y * c) + (Group.inverse additiveGroup (x * c)))
|
||||
p' = SetoidPartialOrder.wellDefined pOrder reflexive (Group.wellDefined additiveGroup reflexive (transitive (transitive multCommutative (ringMinusExtracts R)) (inverseWellDefined additiveGroup multCommutative))) p
|
||||
q : (0R + (x * c)) < (((y * c) + (Group.inverse additiveGroup (x * c))) + (x * c))
|
||||
q = OrderedRing.orderRespectsAddition oRing p' (x * c)
|
||||
q' : (x * c) < ((y * c) + 0R)
|
||||
q' = SetoidPartialOrder.wellDefined pOrder (Group.multIdentLeft additiveGroup) (transitive (symmetric (Group.multAssoc additiveGroup)) (Group.wellDefined additiveGroup reflexive (Group.invLeft additiveGroup))) q
|
||||
|
||||
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
|
||||
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''
|
||||
where
|
||||
open Ring R
|
||||
open Setoid S
|
||||
open Transitive (Equivalence.transitiveEq (Setoid.eq S))
|
||||
open Reflexive (Equivalence.reflexiveEq (Setoid.eq S))
|
||||
open Symmetric (Equivalence.symmetricEq (Setoid.eq S))
|
||||
have : 0R < ((y * c) + (Group.inverse additiveGroup (x * c)))
|
||||
have = SetoidPartialOrder.wellDefined pOrder (Group.invRight additiveGroup) reflexive (OrderedRing.orderRespectsAddition oRing xc<yc (Group.inverse additiveGroup _))
|
||||
p : 0R < ((y * c) + ((Group.inverse additiveGroup x) * c))
|
||||
p = SetoidPartialOrder.wellDefined pOrder reflexive (Group.wellDefined additiveGroup reflexive (symmetric (transitive (multCommutative) (transitive (ringMinusExtracts R) (inverseWellDefined additiveGroup multCommutative))))) have
|
||||
q : 0R < ((y + Group.inverse additiveGroup x) * c)
|
||||
q = SetoidPartialOrder.wellDefined pOrder reflexive (transitive (transitive (Group.wellDefined additiveGroup multCommutative multCommutative) (symmetric multDistributes)) multCommutative) p
|
||||
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 (transitive multCommutative (ringTimesZero R)) k))
|
||||
where
|
||||
open Group additiveGroup
|
||||
f : ((y + inverse x) + (inverse (y + inverse x))) < (identity + inverse (y + inverse x))
|
||||
f = OrderedRing.orderRespectsAddition oRing y-x<0 _
|
||||
g : identity < inverse (y + inverse x)
|
||||
g = SetoidPartialOrder.wellDefined pOrder invRight multIdentLeft f
|
||||
h : (identity * c) < ((inverse (y + inverse x)) * c)
|
||||
h = ringCanMultiplyByPositive oRing 0<c g
|
||||
i : (0R + (identity * c)) < (((y + inverse x) * c) + ((inverse (y + inverse x)) * c))
|
||||
i = ringAddInequalities oRing q h
|
||||
j : 0R < (((y + inverse x) + (inverse (y + inverse x))) * c)
|
||||
j = SetoidPartialOrder.wellDefined pOrder (transitive multIdentLeft (transitive multCommutative (ringTimesZero R))) (symmetric (transitive multCommutative (transitive multDistributes (Group.wellDefined additiveGroup multCommutative multCommutative)))) 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))
|
||||
where
|
||||
open Group additiveGroup
|
||||
f : inverse identity ∼ 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))))
|
||||
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
|
||||
where
|
||||
open Ring R
|
||||
open Setoid S
|
||||
open Transitive (Equivalence.transitiveEq (Setoid.eq S))
|
||||
open Reflexive (Equivalence.reflexiveEq (Setoid.eq S))
|
||||
open Symmetric (Equivalence.symmetricEq (Setoid.eq S))
|
||||
t : ((Group.inverse additiveGroup x) + y) < ((Group.inverse additiveGroup y) + y)
|
||||
t = OrderedRing.orderRespectsAddition oRing -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 oRing u x
|
||||
|
||||
ringCanMultiplyByNegative : {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} → c < (Ring.0R R) → x < y → (y * c) < (x * c)
|
||||
ringCanMultiplyByNegative {S = S} {_+_ = _+_} {_*_ = _*_} {R = R} {_<_ = _<_} {pOrder = pOrder} {tOrder = tOrder} oRing {x} {y} {c} c<0 x<y = ringSwapNegatives oRing u
|
||||
where
|
||||
open Ring R
|
||||
open Setoid S
|
||||
open Transitive (Equivalence.transitiveEq (Setoid.eq S))
|
||||
open Reflexive (Equivalence.reflexiveEq (Setoid.eq S))
|
||||
open Symmetric (Equivalence.symmetricEq (Setoid.eq S))
|
||||
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
|
||||
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)
|
||||
u = SetoidPartialOrder.wellDefined pOrder (ringMinusExtracts R) (ringMinusExtracts R) t
|
||||
|
||||
ringCanCancelNegative : {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} → c < (Ring.0R R) → (x * c) < (y * c) → y < x
|
||||
ringCanCancelNegative {S = S} {_+_ = _+_} {_*_ = _*_} {R = R} {_<_ = _<_} {pOrder = pOrder} {tOrder = tOrder} oRing {x} {y} {c} c<0 xc<yc = r
|
||||
where
|
||||
open Ring R
|
||||
open Setoid S
|
||||
open Group additiveGroup
|
||||
open Transitive (Equivalence.transitiveEq (Setoid.eq S))
|
||||
open Reflexive (Equivalence.reflexiveEq (Setoid.eq S))
|
||||
open Symmetric (Equivalence.symmetricEq (Setoid.eq S))
|
||||
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
|
||||
p2 : 0R < ((y + inverse x) * c)
|
||||
p2 = SetoidPartialOrder.wellDefined pOrder reflexive (transitive (Group.wellDefined additiveGroup multCommutative multCommutative) (transitive (symmetric multDistributes) multCommutative)) 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)
|
||||
q | inl (inr pr) = pr
|
||||
q | inr 0=y-x = exFalso (SetoidPartialOrder.irreflexive pOrder (SetoidPartialOrder.wellDefined pOrder (multWellDefined 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)))
|
||||
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)
|
||||
|
||||
Reference in New Issue
Block a user