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Comparison on the reals (#55)

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Patrick Stevens
2019-10-26 10:36:24 +01:00
committed by GitHub
parent 4e56b68024
commit 0d68919127
18 changed files with 1343 additions and 691 deletions
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5
Rings/Lemmas.agda
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@@ -21,7 +21,7 @@ ringMinusExtracts {x = x} {y} = transferToRight' additiveGroup (transitive (symm
where
open Equivalence eq
ringMinusExtracts' : {x y : A} → Setoid._∼_ S ((Group.inverse (Ring.additiveGroup R) x) * y) (Group.inverse (Ring.additiveGroup R) (x * y))
ringMinusExtracts' : {x y : A} → ((inverse x) * y) ∼ inverse (x * y)
ringMinusExtracts' {x = x} {y} = transitive *Commutative (transitive ringMinusExtracts (inverseWellDefined additiveGroup *Commutative))
where
open Equivalence eq
@@ -40,3 +40,6 @@ groupLemmaMove0G {S = S} G {x} pr = transitive (symmetric (invInv G)) (transitiv
groupLemmaMove0G' : {a b : _} → {A : Set a} → {_·_ : A → A → A} → {S : Setoid {a} {b} A} → (G : Group S _·_) → {x : A} → Setoid._∼_ S x (Group.0G G) → (Setoid._∼_ S (Group.0G G) (Group.inverse G x))
groupLemmaMove0G' {S = S} G {x} pr = transferToRight' G (Equivalence.transitive (Setoid.eq S) (Group.identLeft G) pr)
oneZeroImpliesAllZero : 0R ∼ 1R → {x : A} → x ∼ 0R
oneZeroImpliesAllZero 0=1 = Equivalence.transitive eq (Equivalence.symmetric eq identIsIdent) (Equivalence.transitive eq (*WellDefined (Equivalence.symmetric eq 0=1) (Equivalence.reflexive eq)) (Equivalence.transitive eq *Commutative timesZero))

52
Rings/Orders/Lemmas.agda
View File

@@ -25,7 +25,6 @@ open Group additiveGroup
open import Rings.Lemmas R
ringAddInequalities : {w x y z : A} → w < x → y < z → (w + y) < (x + z)
ringAddInequalities {w = w} {x} {y} {z} w<x y<z = transitive (orderRespectsAddition w<x y) (<WellDefined groupIsAbelian groupIsAbelian (orderRespectsAddition y<z x))
@@ -188,6 +187,9 @@ ringSwapNegatives {x} {y} -x<-y = SetoidPartialOrder.<WellDefined pOrder (Equiva
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)
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
@@ -246,12 +248,12 @@ absNegation a | inr 0=a | inl (inl 0<-a) = exFalso (irreflexive {0G} (<WellDefin
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
lemm4 : (a b : A) → (0G < a) → (b < 0G) → (a * b) < 0G
lemm4 a b 0<a b<0 with orderRespectsMultiplication 0<a (lemm2 _ b<0)
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))
lemm5 : (a b : A) → (a < 0G) → (b < 0G) → 0G < (a * b)
lemm5 a b a<0 b<0 with orderRespectsMultiplication (lemm2 _ a<0) (lemm2 _ b<0)
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)
@@ -265,20 +267,20 @@ 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) (lemm4 a b 0<a b<0)))
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) (lemm4 b a 0<b a<0))))
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) (lemm4 b a 0<b a<0)))
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 (lemm5 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) (lemm5 a b a<0 b<0))))
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))
@@ -296,3 +298,33 @@ 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)
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
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)