mirror of
https://github.com/Smaug123/agdaproofs
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Lots of speedups (#116)
This commit is contained in:
Patrick Stevens
GitHub
@@ -5,7 +5,6 @@ open import Groups.Groups
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open import Groups.Homomorphisms.Definition
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open import Groups.Definition
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open import Numbers.Naturals.Naturals
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open import Setoids.Orders
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open import Setoids.Setoids
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open import Functions
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open import Sets.EquivalenceRelations
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@@ -25,10 +25,10 @@ open Setoid S
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open Equivalence eq
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open import Rings.Lemmas R
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open import Groups.Lemmas additiveGroup
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open import Groups.Cyclic.Definition additiveGroup
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fromN : ℕ → A
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fromN zero = 0R
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fromN (succ n) = 1R + fromN n
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fromN = positiveEltPower 1R
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fromNPreserves+ : (a b : ℕ) → fromN (a +N b) ∼ (fromN a) + (fromN b)
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fromNPreserves+ zero b = symmetric identLeft
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@@ -7,7 +7,7 @@ open import Rings.Definition
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open import Rings.Orders.Partial.Definition
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open import Sets.EquivalenceRelations
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open import Sequences
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open import Setoids.Orders
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open import Setoids.Orders.Partial.Definition
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open import Functions
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open import LogicalFormulae
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open import Numbers.Naturals.Semiring
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@@ -1,8 +1,8 @@
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{-# OPTIONS --safe --warning=error --without-K #-}
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open import Groups.Definition
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open import Setoids.Orders
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open import Setoids.Setoids
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open import Setoids.Orders.Partial.Definition
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open import Functions
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open import Rings.Definition
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@@ -13,9 +13,13 @@ module Rings.Orders.Partial.Definition {n m : _} {A : Set n} {S : Setoid {n} {m}
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open Ring R
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open Group additiveGroup
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open Setoid S
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open import Groups.Orders.Partial.Definition
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record PartiallyOrderedRing {p : _} {_<_ : Rel {_} {p} A} (pOrder : SetoidPartialOrder S _<_) : Set (lsuc n ⊔ m ⊔ p) where
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field
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orderRespectsAddition : {a b : A} → (a < b) → (c : A) → (a + c) < (b + c)
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orderRespectsMultiplication : {a b : A} → (0R < a) → (0R < b) → (0R < (a * b))
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open SetoidPartialOrder pOrder
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toGroup : {p : _} {_<_ : Rel {_} {p} A} {pOrder : SetoidPartialOrder S _<_} → PartiallyOrderedRing pOrder → PartiallyOrderedGroup additiveGroup pOrder
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PartiallyOrderedGroup.orderRespectsAddition (toGroup p) = PartiallyOrderedRing.orderRespectsAddition p
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@@ -3,7 +3,7 @@
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open import LogicalFormulae
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open import Groups.Lemmas
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open import Groups.Definition
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open import Setoids.Orders
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open import Setoids.Orders.Partial.Definition
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open import Setoids.Setoids
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open import Functions
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open import Sets.EquivalenceRelations
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@@ -91,3 +91,17 @@ abstract
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negativeInequality : {a : A} → a < 0G → 0G < inverse a
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negativeInequality {a} a<0 = <WellDefined invRight identLeft (orderRespectsAddition a<0 (inverse a))
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negativeInequality' : {a : A} → 0G < a → inverse a < 0G
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negativeInequality' {a} 0<a = <WellDefined identLeft invRight (orderRespectsAddition 0<a (inverse a))
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open import Rings.InitialRing R
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fromNIncreasing : (0R < 1R) → (n : ℕ) → (fromN n) < (fromN (succ n))
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fromNIncreasing 0<1 zero = <WellDefined reflexive (symmetric identRight) 0<1
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fromNIncreasing 0<1 (succ n) = <WellDefined groupIsAbelian groupIsAbelian (orderRespectsAddition (fromNIncreasing 0<1 n) 1R)
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fromNPreservesOrder : (0R < 1R) → {a b : ℕ} → (a <N b) → (fromN a) < (fromN b)
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fromNPreservesOrder 0<1 {zero} {succ zero} a<b = fromNIncreasing 0<1 0
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fromNPreservesOrder 0<1 {zero} {succ (succ b)} a<b = <Transitive (fromNPreservesOrder 0<1 (succIsPositive b)) (fromNIncreasing 0<1 (succ b))
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fromNPreservesOrder 0<1 {succ a} {succ b} a<b = <WellDefined groupIsAbelian groupIsAbelian (orderRespectsAddition (fromNPreservesOrder 0<1 (canRemoveSuccFrom<N a<b)) 1R)
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217
Rings/Orders/Total/AbsoluteValue.agda
Normal file
217
Rings/Orders/Total/AbsoluteValue.agda
Normal file
@@ -0,0 +1,217 @@
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{-# OPTIONS --safe --warning=error --without-K #-}
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open import LogicalFormulae
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open import Groups.Lemmas
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open import Groups.Definition
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open import Setoids.Orders.Partial.Definition
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open import Setoids.Orders.Total.Definition
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open import Setoids.Setoids
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open import Functions
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open import Sets.EquivalenceRelations
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open import Rings.Definition
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open import Rings.Orders.Total.Definition
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open import Rings.Orders.Partial.Definition
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open import Numbers.Naturals.Semiring
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open import Numbers.Naturals.Order
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open import Orders.Total.Definition
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open import Rings.IntegralDomains.Definition
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module Rings.Orders.Total.AbsoluteValue {n m p : _} {A : Set n} {S : Setoid {n} {m} A} {_+_ : A → A → A} {_*_ : A → A → A} {R : Ring S _+_ _*_} {_<_ : Rel {_} {p} A} {pOrder : SetoidPartialOrder S _<_} {pOrderRing : PartiallyOrderedRing R pOrder} (order : TotallyOrderedRing pOrderRing) where
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open Ring R
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open Group additiveGroup
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open Setoid S
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open SetoidPartialOrder pOrder
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open TotallyOrderedRing order
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open SetoidTotalOrder total
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open PartiallyOrderedRing pOrderRing
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open import Rings.Lemmas R
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open import Rings.Orders.Partial.Lemmas pOrderRing
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open import Rings.Orders.Total.Lemmas order
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abs : A → A
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abs a with totality 0R a
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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 totality 0R a
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absWellDefined a b a=b | inl (inl 0<a) with totality 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 totality 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 totality 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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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)) (invIdent 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)) (invIdent 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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absZero : abs (Ring.0R R) ≡ Ring.0R R
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absZero with totality (Ring.0R R) (Ring.0R R)
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absZero | inl (inl x) = exFalso (irreflexive x)
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absZero | inl (inr x) = exFalso (irreflexive x)
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absZero | inr x = refl
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absNegation : (a : A) → (abs a) ∼ (abs (inverse a))
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absNegation a with totality 0R a
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absNegation a | inl (inl 0<a) with totality 0G (inverse a)
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absNegation a | inl (inl 0<a) | inl (inl 0<-a) = exFalso (irreflexive {0G} (SetoidPartialOrder.<Transitive pOrder 0<-a (lemm2' a 0<a)))
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absNegation a | inl (inl 0<a) | inl (inr -a<0) = Equivalence.symmetric eq (invTwice additiveGroup a)
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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))
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absNegation a | inl (inr a<0) with totality 0G (inverse a)
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absNegation a | inl (inr a<0) | inl (inl 0<-a) = Equivalence.reflexive eq
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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))
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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))
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absNegation a | inr 0=a with totality 0G (inverse a)
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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))
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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))
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absNegation a | inr 0=a | inr 0=-a = Equivalence.transitive eq (Equivalence.symmetric eq 0=a) 0=-a
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absRespectsTimes : (a b : A) → abs (a * b) ∼ (abs a) * (abs b)
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absRespectsTimes a b with totality 0R a
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absRespectsTimes a b | inl (inl 0<a) with totality 0R b
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absRespectsTimes a b | inl (inl 0<a) | inl (inl 0<b) with totality 0R (a * b)
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absRespectsTimes a b | inl (inl 0<a) | inl (inl 0<b) | inl (inl 0<ab) = Equivalence.reflexive eq
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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))
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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)))
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absRespectsTimes a b | inl (inl 0<a) | inl (inr b<0) with totality 0R (a * b)
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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))
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... | bl = exFalso (irreflexive {0G} (SetoidPartialOrder.<Transitive pOrder 0<ab (<WellDefined (invTwice additiveGroup _) (Equivalence.reflexive eq) (lemm2' _ bl))))
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absRespectsTimes a b | inl (inl 0<a) | inl (inr b<0) | inl (inr ab<0) = Equivalence.symmetric eq ringMinusExtracts
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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)))
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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 totality 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)
|
||||
|
||||
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
|
||||
|
||||
addingAbsCannotShrink : {a b : A} → 0G < b → 0G < ((abs a) + b)
|
||||
addingAbsCannotShrink {a} {b} 0<b with totality 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
|
||||
|
||||
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))
|
||||
|
||||
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
|
||||
|
||||
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
|
||||
|
||||
absBoundedImpliesBounded : {a b : A} → abs a < b → a < b
|
||||
absBoundedImpliesBounded {a} {b} a<b with totality 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
|
||||
|
||||
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)))
|
||||
@@ -4,7 +4,8 @@ open import Agda.Primitive using (Level; lzero; lsuc; _⊔_)
|
||||
open import LogicalFormulae
|
||||
open import Groups.Lemmas
|
||||
open import Groups.Definition
|
||||
open import Setoids.Orders
|
||||
open import Setoids.Orders.Partial.Definition
|
||||
open import Setoids.Orders.Total.Definition
|
||||
open import Setoids.Setoids
|
||||
open import Functions
|
||||
open import Sets.EquivalenceRelations
|
||||
@@ -48,21 +49,21 @@ FloorIs.prAbove (addOneToFloor record { prBelow = x ; prAbove = prAbove }) = <We
|
||||
|
||||
private
|
||||
0<n : {x y : A} → (x < y) → 0R < fromN n
|
||||
0<n x<y = fromNPreservesOrder (anyComparisonImpliesNontrivial x<y) (lessTransitive (succIsPositive 0) 1<n)
|
||||
0<n x<y = fromNPreservesOrder (0<1 (anyComparisonImpliesNontrivial x<y)) (lessTransitive (succIsPositive 0) 1<n)
|
||||
|
||||
0<aFromN : {a : A} → (0<a : 0R < a) → 0R < (a * fromN n)
|
||||
0<aFromN 0<a = orderRespectsMultiplication 0<a (0<n 0<a)
|
||||
|
||||
floorWellDefinedLemma : {a : A} {n1 n2 : ℕ} → FloorIs a n1 → FloorIs a n2 → n1 <N n2 → False
|
||||
floorWellDefinedLemma {a} {n1} {n2} record { prAbove = prAbove1 } record { prBelow = inl prBelow } n1<n2 with TotalOrder.totality ℕTotalOrder (succ n1) n2
|
||||
... | inl (inl n1+1<n2) = irreflexive (<Transitive prAbove1 (<Transitive (fromNPreservesOrder (anyComparisonImpliesNontrivial prBelow) n1+1<n2) prBelow))
|
||||
... | inl (inl n1+1<n2) = irreflexive (<Transitive prAbove1 (<Transitive (fromNPreservesOrder (0<1 (anyComparisonImpliesNontrivial prBelow)) n1+1<n2) prBelow))
|
||||
... | inl (inr n2<n1+1) = noIntegersBetweenXAndSuccX n1 n1<n2 n2<n1+1
|
||||
... | inr refl = irreflexive (<Transitive prAbove1 prBelow)
|
||||
floorWellDefinedLemma {a} {n1} {n2} record { prBelow = (inl x) ; prAbove = prAbove1 } record { prBelow = (inr eq) ; prAbove = _ } n1<n2 with TotalOrder.totality ℕTotalOrder (succ n1) n2
|
||||
... | inl (inl n1+1<n2) = irreflexive (<Transitive prAbove1 (<WellDefined reflexive eq (fromNPreservesOrder (anyComparisonImpliesNontrivial prAbove1) n1+1<n2)))
|
||||
... | inl (inl n1+1<n2) = irreflexive (<Transitive prAbove1 (<WellDefined reflexive eq (fromNPreservesOrder (0<1 (anyComparisonImpliesNontrivial prAbove1)) n1+1<n2)))
|
||||
... | inl (inr n2<n1+1) = noIntegersBetweenXAndSuccX n1 n1<n2 n2<n1+1
|
||||
... | inr refl = irreflexive (<WellDefined reflexive eq prAbove1)
|
||||
floorWellDefinedLemma {a} {n1} {n2} record { prBelow = (inr x) ; prAbove = prAbove1 } record { prBelow = (inr eq) ; prAbove = _ } n1<n2 = lessIrreflexive {n1} (fromNPreservesOrder' (anyComparisonImpliesNontrivial prAbove1) (<WellDefined reflexive (transitive eq (symmetric x)) (fromNPreservesOrder (anyComparisonImpliesNontrivial prAbove1) n1<n2)))
|
||||
floorWellDefinedLemma {a} {n1} {n2} record { prBelow = (inr x) ; prAbove = prAbove1 } record { prBelow = (inr eq) ; prAbove = _ } n1<n2 = lessIrreflexive {n1} (fromNPreservesOrder' (anyComparisonImpliesNontrivial prAbove1) (<WellDefined reflexive (transitive eq (symmetric x)) (fromNPreservesOrder (0<1 (anyComparisonImpliesNontrivial prAbove1)) n1<n2)))
|
||||
|
||||
floorWellDefined : {a : A} {n1 n2 : ℕ} → FloorIs a n1 → FloorIs a n2 → n1 ≡ n2
|
||||
floorWellDefined {a} {n1} {n2} record { prBelow = prBelow1 ; prAbove = prAbove1 } record { prBelow = prBelow ; prAbove = prAbove } with TotalOrder.totality ℕTotalOrder n1 n2
|
||||
@@ -80,7 +81,7 @@ private
|
||||
computeFloor' {zero} (succ k) pr a 0<a a<f = exFalso (irreflexive (<Transitive 0<a a<f))
|
||||
computeFloor' {succ k} n pr a 0<a a<f with totality 1R a
|
||||
... | inl (inr a<1) = 0 , (record { prAbove = <WellDefined reflexive (symmetric identRight) a<1 ; prBelow = inl 0<a })
|
||||
... | inr 1=a = 1 , (record { prAbove = <WellDefined (transitive identRight 1=a) reflexive (fromNPreservesOrder (anyComparisonImpliesNontrivial 0<a) {1} (le 0 refl)) ; prBelow = inr (transitive identRight 1=a) })
|
||||
... | inr 1=a = 1 , (record { prAbove = <WellDefined (transitive identRight 1=a) reflexive (fromNPreservesOrder (0<1 (anyComparisonImpliesNontrivial 0<a)) {1} (le 0 refl)) ; prBelow = inr (transitive identRight 1=a) })
|
||||
... | inl (inl 1<a) with computeFloor' {k} (succ n) (transitivity (transitivity (Semiring.commutative ℕSemiring k (succ n)) (applyEquality succ (Semiring.commutative ℕSemiring n k))) pr) (a + inverse 1R) (moveInequality 1<a) (<WellDefined reflexive (transitive groupIsAbelian (transitive +Associative (transitive (+WellDefined invLeft reflexive) identLeft))) (orderRespectsAddition a<f (inverse 1R)))
|
||||
... | N , snd = succ N , floorWellDefined' (transitive (symmetric +Associative) (transitive (+WellDefined reflexive invLeft) identRight)) (addOneToFloor snd)
|
||||
|
||||
@@ -179,7 +180,7 @@ approximationComesFromBelow 1/n 1/nPr a 0<a a<1 (succ m) with computeFloor' 0 (S
|
||||
... | x , record { prBelow = inr x=an ; prAbove = an<x+1 } = inr (transitive (partialSums'Translate 0G _ _ m) (transitive (+WellDefined (transitive (*WellDefined x=an reflexive) (transitive (symmetric *Associative) (*WellDefined reflexive (transitive *Commutative 1/nPr)))) reflexive) (transitive (+WellDefined (transitive *Commutative identIsIdent) (transitive (partialSums'WellDefined' 0G _ (constSequence 0G) (λ m → transitive (multiplyZeroSequence (powerSeq 1/n (1/n * 1/n)) m) (identityOfIndiscernablesRight _∼_ reflexive (equalityCommutative (indexAndConst 0G m)))) m) (transitive (partialSumsConst _ _ m) (transitive identLeft timesZero')))) identRight)))
|
||||
... | x , record { prBelow = inl x<an ; prAbove = an<x+1 } with approximationComesFromBelow 1/n 1/nPr ((a * fromN n) + inverse (fromN x)) (moveInequality x<an) (<WellDefined reflexive (transitive (symmetric +Associative) (transitive (+WellDefined reflexive invRight) identRight)) (orderRespectsAddition an<x+1 (inverse (fromN x)))) m
|
||||
... | inl inter<an = inl (<WellDefined (symmetric (partialSums'Translate 0G (fromN x * 1/n) _ m)) reflexive (<WellDefined (+WellDefined reflexive (symmetric (partialSums'WellDefined 0G (fromN n * 0G) (symmetric timesZero) _ m))) reflexive (ringCanCancelPositive (0<n an<x+1) (<WellDefined (symmetric (transitive *DistributesOver+' (+WellDefined (transitive (symmetric *Associative) (transitive (*WellDefined reflexive 1/nPr) (transitive *Commutative identIsIdent))) (transitive *Commutative (transitive (lemma2 (fromN n) (fromN n * 0G) 1/n (1/n * 1/n) _ m) (transitive (partialSums'WellDefined _ 0G (transitive (*WellDefined reflexive timesZero) timesZero) _ m) (partialSums'WellDefined' 0G _ _ (λ m → applyWellDefined _ _ _ dot (λ {x} {y} {s} → dotWellDefined {x} {y} {s}) (λ m → powerSeqWellDefined {_} {1/n} {_} {1/n} reflexive (transitive (symmetric *Associative) (transitive (transitive *Commutative (*WellDefined 1/nPr reflexive)) identIsIdent)) m) m) m))))))) reflexive (<WellDefined groupIsAbelian (transitive (symmetric +Associative) (transitive (+WellDefined reflexive invLeft) identRight)) (orderRespectsAddition inter<an (fromN x)))))))
|
||||
... | inr inter=an = inr (transitive (partialSums'Translate 0G (fromN x * 1/n) _ m) (cancelIntDom' (isIntDom λ t → anyComparisonImpliesNontrivial a<1 (symmetric t)) ans λ t → irreflexive (<WellDefined reflexive t (fromNPreservesOrder (anyComparisonImpliesNontrivial a<1) (lessTransitive (succIsPositive 0) 1<n)))))
|
||||
... | inr inter=an = inr (transitive (partialSums'Translate 0G (fromN x * 1/n) _ m) (cancelIntDom' (isIntDom λ t → anyComparisonImpliesNontrivial a<1 (symmetric t)) ans λ t → irreflexive (<WellDefined reflexive t (fromNPreservesOrder (0<1 (anyComparisonImpliesNontrivial a<1)) (lessTransitive (succIsPositive 0) 1<n)))))
|
||||
where
|
||||
ans : (((fromN x * 1/n) + index (partialSums' 0G (apply dot (powerSeq 1/n (1/n * 1/n)) (baseNExpansion ((a * fromN n) + inverse (fromN x)) (moveInequality x<an) (<WellDefined reflexive (transitive (symmetric +Associative) (transitive (+WellDefined reflexive invRight) identRight)) (orderRespectsAddition an<x+1 (inverse (fromN x))))))) m) * fromN n) ∼ a * fromN n
|
||||
ans = transitive (transitive *DistributesOver+' (+WellDefined (transitive (symmetric *Associative) (transitive (*WellDefined reflexive 1/nPr) (transitive *Commutative identIsIdent))) *Commutative)) (transitive (+WellDefined reflexive (transitive (lemma2 (fromN n) 0G 1/n (1/n * 1/n) _ m) (transitive (partialSums'WellDefined _ _ timesZero _ m) (partialSums'WellDefined' _ _ _ (λ m → applyWellDefined _ _ _ dot (λ {x} {y} {s} → dotWellDefined {x} {y} {s}) (λ m → powerSeqWellDefined reflexive (transitive (symmetric *Associative) (transitive (*WellDefined reflexive 1/nPr) (transitive *Commutative identIsIdent))) m) m) m)))) (transitive (+WellDefined reflexive inter=an) (transitive groupIsAbelian (transitive (symmetric +Associative) (transitive (+WellDefined reflexive invLeft) identRight)))))
|
||||
@@ -194,7 +195,7 @@ powPos (succ m) 0<a = orderRespectsMultiplication 0<a (powPos m 0<a)
|
||||
|
||||
approximationIsNear : (1/n : A) → (1/nPr : 1/n * (fromN n) ∼ 1R) → (a : A) → (0<a : 0R < a) → (a<1 : a < 1R) → (m : ℕ) → a < ((index (approximations 1/n 1/nPr (baseNExpansion a 0<a a<1)) m) + pow m 1/n)
|
||||
approximationIsNear 1/n 1/npr a 0<a a<1 zero with computeFloor' 0 (Semiring.sumZeroRight ℕSemiring n) (a * fromN n) (0<aFromN 0<a) (<WellDefined reflexive identIsIdent (ringCanMultiplyByPositive (0<n 0<a) a<1))
|
||||
... | x , record { prBelow = inl prBelow ; prAbove = prAbove } = <WellDefined reflexive (transitive (symmetric identLeft) +Associative) (ringCanCancelPositive (0<n prAbove) (<Transitive prAbove (<WellDefined reflexive (transitive (+WellDefined (transitive (*WellDefined reflexive (symmetric 1/npr)) *Associative) (symmetric identIsIdent)) (symmetric *DistributesOver+')) (<WellDefined reflexive (transitive groupIsAbelian (+WellDefined (transitive (symmetric identIsIdent) *Commutative) reflexive)) (orderRespectsAddition (<WellDefined identRight reflexive (fromNPreservesOrder (anyComparisonImpliesNontrivial prAbove) 1<n)) (fromN x))))))
|
||||
... | x , record { prBelow = inl prBelow ; prAbove = prAbove } = <WellDefined reflexive (transitive (symmetric identLeft) +Associative) (ringCanCancelPositive (0<n prAbove) (<Transitive prAbove (<WellDefined reflexive (transitive (+WellDefined (transitive (*WellDefined reflexive (symmetric 1/npr)) *Associative) (symmetric identIsIdent)) (symmetric *DistributesOver+')) (<WellDefined reflexive (transitive groupIsAbelian (+WellDefined (transitive (symmetric identIsIdent) *Commutative) reflexive)) (orderRespectsAddition (<WellDefined identRight reflexive (fromNPreservesOrder (0<1 (anyComparisonImpliesNontrivial prAbove)) 1<n)) (fromN x))))))
|
||||
... | x , record { prBelow = inr prBelow ; prAbove = prAbove } = <WellDefined reflexive (symmetric (transitive (symmetric +Associative) identLeft)) (ringCanCancelPositive (0<n prAbove) (<WellDefined prBelow (symmetric (transitive *DistributesOver+' (+WellDefined (transitive (symmetric *Associative) (transitive *Commutative (transitive (*WellDefined 1/npr reflexive) identIsIdent))) identIsIdent))) (<WellDefined identLeft groupIsAbelian (orderRespectsAddition (0<n prAbove) (fromN x)))))
|
||||
approximationIsNear 1/n 1/npr a 0<a a<1 (succ m) with computeFloor' 0 (Semiring.sumZeroRight ℕSemiring n) (a * fromN n) (0<aFromN 0<a) (<WellDefined reflexive identIsIdent (ringCanMultiplyByPositive (0<n 0<a) a<1))
|
||||
... | x , record { prBelow = inl prBelow ; prAbove = prAbove } = <WellDefined reflexive (+WellDefined (symmetric (partialSums'Translate 0G (fromN x * 1/n) _ m)) reflexive) (ringCanCancelPositive (0<n prAbove) (<WellDefined reflexive (transitive (+WellDefined (transitive (+WellDefined (transitive (transitive (symmetric identIsIdent) (transitive *Commutative (*WellDefined reflexive (symmetric 1/npr))) ) *Associative) *Commutative) (symmetric *DistributesOver+')) (transitive (transitive (transitive (symmetric identIsIdent) (*WellDefined (transitive (symmetric 1/npr) *Commutative) reflexive)) (symmetric *Associative)) *Commutative)) (symmetric *DistributesOver+')) (<WellDefined reflexive (+WellDefined (+WellDefined reflexive (transitive (partialSums'WellDefined' _ _ _ (λ m → applyWellDefined _ _ _ dot (λ {_} {_} {s} → dotWellDefined {_} {_} {s}) (λ m → powerSeqWellDefined reflexive (transitive (transitive (transitive (symmetric identIsIdent) *Commutative) (*WellDefined reflexive (symmetric 1/npr))) *Associative) m) m) m) (symmetric (lemma2 (fromN n) 0G 1/n (1/n * 1/n) _ m)))) reflexive) (<WellDefined reflexive (+WellDefined (+WellDefined reflexive (partialSums'WellDefined _ _ (symmetric timesZero) _ m)) reflexive) (<WellDefined reflexive +Associative u)))))
|
||||
|
||||
@@ -8,7 +8,8 @@ open import Rings.Orders.Partial.Definition
|
||||
open import Rings.Orders.Total.Definition
|
||||
open import Sets.EquivalenceRelations
|
||||
open import Sequences
|
||||
open import Setoids.Orders
|
||||
open import Setoids.Orders.Partial.Definition
|
||||
open import Setoids.Orders.Total.Definition
|
||||
open import Functions
|
||||
open import LogicalFormulae
|
||||
open import Numbers.Naturals.Semiring
|
||||
|
||||
@@ -5,10 +5,9 @@ open import Setoids.Setoids
|
||||
open import Rings.Definition
|
||||
open import Rings.Orders.Partial.Definition
|
||||
open import Rings.Orders.Total.Definition
|
||||
open import Groups.Definition
|
||||
open import Sets.EquivalenceRelations
|
||||
open import Sequences
|
||||
open import Setoids.Orders
|
||||
open import Setoids.Orders.Partial.Definition
|
||||
open import Setoids.Orders.Total.Definition
|
||||
open import Functions
|
||||
open import LogicalFormulae
|
||||
open import Numbers.Naturals.Semiring
|
||||
@@ -16,15 +15,8 @@ open import Numbers.Naturals.Order
|
||||
|
||||
module Rings.Orders.Total.Cauchy {m n o : _} {A : Set m} {S : Setoid {m} {n} A} {_+_ : A → A → A} {_*_ : A → A → A} {_<_ : Rel {m} {o} A} {pOrder : SetoidPartialOrder S _<_} {R : Ring S _+_ _*_} {pRing : PartiallyOrderedRing R pOrder} (order : TotallyOrderedRing pRing) where
|
||||
|
||||
open Setoid S
|
||||
open SetoidTotalOrder (TotallyOrderedRing.total order)
|
||||
open SetoidPartialOrder pOrder
|
||||
open Equivalence eq
|
||||
open TotallyOrderedRing order
|
||||
open Ring R
|
||||
open Group additiveGroup
|
||||
|
||||
open import Rings.Orders.Total.Lemmas order
|
||||
open import Rings.Orders.Total.AbsoluteValue order
|
||||
|
||||
cauchy : Sequence A → Set (m ⊔ o)
|
||||
cauchy s = ∀ (ε : A) → (0R < ε) → Sg ℕ (λ N → ∀ {m n : ℕ} → (N <N m) → (N <N n) → abs ((index s m) -R (index s n)) < ε)
|
||||
cauchy s = ∀ (ε : A) → (Ring.0R R < ε) → Sg ℕ (λ N → ∀ {m n : ℕ} → (N <N m) → (N <N n) → abs (Ring._-R_ R (index s m) (index s n)) < ε)
|
||||
|
||||
@@ -1,7 +1,8 @@
|
||||
{-# OPTIONS --safe --warning=error --without-K #-}
|
||||
|
||||
open import Groups.Definition
|
||||
open import Setoids.Orders
|
||||
open import Setoids.Orders.Partial.Definition
|
||||
open import Setoids.Orders.Total.Definition
|
||||
open import Setoids.Setoids
|
||||
open import Functions
|
||||
open import Rings.Definition
|
||||
|
||||
@@ -3,7 +3,8 @@
|
||||
open import LogicalFormulae
|
||||
open import Groups.Lemmas
|
||||
open import Groups.Definition
|
||||
open import Setoids.Orders
|
||||
open import Setoids.Orders.Partial.Definition
|
||||
open import Setoids.Orders.Total.Definition
|
||||
open import Setoids.Setoids
|
||||
open import Functions
|
||||
open import Sets.EquivalenceRelations
|
||||
@@ -27,27 +28,7 @@ open PartiallyOrderedRing pOrderRing
|
||||
open import Rings.Lemmas R
|
||||
open import Rings.Orders.Partial.Lemmas pOrderRing
|
||||
|
||||
abs : A → A
|
||||
abs a with totality 0R a
|
||||
abs a | inl (inl 0<a) = a
|
||||
abs a | inl (inr a<0) = inverse a
|
||||
abs a | inr 0=a = a
|
||||
|
||||
abstract
|
||||
absWellDefined : (a b : A) → a ∼ b → abs a ∼ abs b
|
||||
absWellDefined a b a=b with totality 0R a
|
||||
absWellDefined a b a=b | inl (inl 0<a) with totality 0R b
|
||||
absWellDefined a b a=b | inl (inl 0<a) | inl (inl 0<b) = a=b
|
||||
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)))
|
||||
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))
|
||||
absWellDefined a b a=b | inl (inr a<0) with totality 0R b
|
||||
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)))
|
||||
absWellDefined a b a=b | inl (inr a<0) | inl (inr b<0) = inverseWellDefined additiveGroup a=b
|
||||
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))
|
||||
absWellDefined a b a=b | inr 0=a with totality 0R b
|
||||
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))
|
||||
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))
|
||||
absWellDefined a b a=b | inr 0=a | inr 0=b = a=b
|
||||
|
||||
lemm2 : (a : A) → a < 0G → 0G < inverse a
|
||||
lemm2 a a<0 with totality 0R (inverse a)
|
||||
@@ -67,48 +48,6 @@ abstract
|
||||
t : a + 0G ∼ 0G
|
||||
t = Equivalence.transitive eq (+WellDefined (Equivalence.reflexive eq) 0=-a) (invRight {a})
|
||||
|
||||
triangleInequality : (a b : A) → ((abs (a + b)) < ((abs a) + (abs b))) || (abs (a + b) ∼ (abs a) + (abs b))
|
||||
triangleInequality a b with totality 0R (a + b)
|
||||
triangleInequality a b | inl (inl 0<a+b) with totality 0R a
|
||||
triangleInequality a b | inl (inl 0<a+b) | inl (inl 0<a) with totality 0R b
|
||||
triangleInequality a b | inl (inl 0<a+b) | inl (inl 0<a) | inl (inl 0<b) = inr (Equivalence.reflexive eq)
|
||||
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))
|
||||
triangleInequality a b | inl (inl 0<a+b) | inl (inl 0<a) | inr 0=b = inr (Equivalence.reflexive eq)
|
||||
triangleInequality a b | inl (inl 0<a+b) | inl (inr a<0) with totality 0R b
|
||||
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)
|
||||
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))))
|
||||
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)
|
||||
triangleInequality a b | inl (inl 0<a+b) | inr 0=a with totality 0R b
|
||||
triangleInequality a b | inl (inl 0<a+b) | inr 0=a | inl (inl 0<b) = inr (Equivalence.reflexive eq)
|
||||
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))
|
||||
triangleInequality a b | inl (inl 0<a+b) | inr 0=a | inr 0=b = inr (Equivalence.reflexive eq)
|
||||
triangleInequality a b | inl (inr a+b<0) with totality 0G a
|
||||
triangleInequality a b | inl (inr a+b<0) | inl (inl 0<a) with totality 0G b
|
||||
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)) (invIdent 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)) (invIdent 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 totality (Ring.0R R) (Group.inverse (Ring.additiveGroup R) x)
|
||||
ringMinusFlipsOrder {x} 0<x | inl (inl 0<inv) = exFalso (SetoidPartialOrder.irreflexive pOrder bad')
|
||||
@@ -200,27 +139,6 @@ abstract
|
||||
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) (orderRespectsAddition q x)
|
||||
|
||||
absZero : abs (Ring.0R R) ≡ Ring.0R R
|
||||
absZero with totality (Ring.0R R) (Ring.0R R)
|
||||
absZero | inl (inl x) = exFalso (irreflexive x)
|
||||
absZero | inl (inr x) = exFalso (irreflexive x)
|
||||
absZero | 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
|
||||
|
||||
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))
|
||||
@@ -229,67 +147,6 @@ abstract
|
||||
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
|
||||
|
||||
absNonnegative : {a : A} → (abs a < 0R) → False
|
||||
absNonnegative {a} pr with totality 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
|
||||
@@ -305,71 +162,21 @@ abstract
|
||||
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 totality 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)))
|
||||
|
||||
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))
|
||||
|
||||
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
|
||||
|
||||
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
|
||||
|
||||
absBoundedImpliesBounded : {a b : A} → abs a < b → a < b
|
||||
absBoundedImpliesBounded {a} {b} a<b with totality 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
|
||||
|
||||
orderedImpliesCharNot2 : (0R ∼ 1R → False) → 1R + 1R ∼ 0R → False
|
||||
orderedImpliesCharNot2 0!=1 x = irreflexive (<WellDefined (identRight {0R}) x (ringAddInequalities (0<1 0!=1) (0<1 0!=1)))
|
||||
|
||||
open import Rings.InitialRing R
|
||||
open Equivalence eq
|
||||
|
||||
fromNIncreasing : (0R ∼ 1R → False) → (n : ℕ) → (fromN n) < (fromN (succ n))
|
||||
fromNIncreasing 0!=1 zero = <WellDefined reflexive (symmetric identRight) (0<1 0!=1)
|
||||
fromNIncreasing 0!=1 (succ n) = <WellDefined groupIsAbelian groupIsAbelian (orderRespectsAddition (fromNIncreasing 0!=1 n) 1R)
|
||||
|
||||
fromNPreservesOrder : (0R ∼ 1R → False) → {a b : ℕ} → (a <N b) → (fromN a) < (fromN b)
|
||||
fromNPreservesOrder 0!=1 {zero} {succ zero} a<b = fromNIncreasing 0!=1 0
|
||||
fromNPreservesOrder 0!=1 {zero} {succ (succ b)} a<b = <Transitive (fromNPreservesOrder 0!=1 (succIsPositive b)) (fromNIncreasing 0!=1 (succ b))
|
||||
fromNPreservesOrder 0!=1 {succ a} {succ b} a<b = <WellDefined groupIsAbelian groupIsAbelian (orderRespectsAddition (fromNPreservesOrder 0!=1 (canRemoveSuccFrom<N a<b)) 1R)
|
||||
|
||||
fromNPreservesOrder' : (0R ∼ 1R → False) → {a b : ℕ} → (fromN a) < (fromN b) → a <N b
|
||||
fromNPreservesOrder' nontrivial {a} {b} a<b with TotalOrder.totality ℕTotalOrder a b
|
||||
... | inl (inl x) = x
|
||||
... | inl (inr x) = exFalso (irreflexive (<Transitive a<b (fromNPreservesOrder nontrivial x)))
|
||||
... | inl (inr x) = exFalso (irreflexive (<Transitive a<b (fromNPreservesOrder (0<1 nontrivial) x)))
|
||||
... | inr x = exFalso (irreflexive (<WellDefined (fromNWellDefined x) reflexive a<b))
|
||||
|
||||
reciprocalPositive : (a b : A) → .(0<a : 0R < a) → (a * b ∼ 1R) → 0R < b
|
||||
|
||||
Reference in New Issue
Block a user