From 9b80058157916c793a9a2d3823797482bf0efa14 Mon Sep 17 00:00:00 2001
From: Patrick Stevens <patrick+github@patrickstevens.co.uk>
Date: Thu, 16 Apr 2020 13:41:51 +0100
Subject: [PATCH] Lots of speedups (#116)

---
 Everything/Safe.agda                          |   9 +-
 Fields/CauchyCompletion/Addition.agda         |  31 ++-
 Fields/CauchyCompletion/Approximation.agda    |   4 +-
 Fields/CauchyCompletion/Archimedean.agda      |  28 ++
 Fields/CauchyCompletion/Comparison.agda       |  47 +++-
 Fields/CauchyCompletion/Definition.agda       |   8 +-
 Fields/CauchyCompletion/Field.agda            |  22 +-
 Fields/CauchyCompletion/Group.agda            |  16 +-
 Fields/CauchyCompletion/Multiplication.agda   | 227 ++++++++--------
 .../PartiallyOrderedRing.agda                 | 247 ++++++++++++------
 Fields/CauchyCompletion/Ring.agda             |   6 +-
 Fields/CauchyCompletion/Setoid.agda           |  13 +-
 Fields/FieldOfFractions/Archimedean.agda      |  68 +++++
 Fields/FieldOfFractions/Group.agda            |   1 -
 Fields/FieldOfFractions/Order.agda            |  38 ++-
 .../Orders/LeastUpperBounds/Definition.agda   |   2 +-
 Fields/Orders/LeastUpperBounds/Examples.agda  |   3 +-
 Fields/Orders/Lemmas.agda                     |   5 +-
 Fields/Orders/Limits/Definition.agda          |   4 +-
 Fields/Orders/Limits/Lemmas.agda              |   4 +-
 Fields/Orders/Partial/Definition.agda         |   2 +-
 Fields/Orders/Total/Archimedean.agda          |  65 +++++
 Fields/Orders/Total/Definition.agda           |   2 +-
 Fields/Orders/Total/Lemmas.agda               |   5 +-
 Groups/Cyclic/Definition.agda                 |   5 +-
 Groups/FreeProduct/Addition.agda              |  15 +-
 Groups/Homomorphisms/Definition.agda          |   3 +
 Groups/Homomorphisms/Lemmas.agda              |   6 +
 Groups/Homomorphisms/Lemmas2.agda             |   1 -
 Groups/Orders/Archimedean.agda                |  20 ++
 Groups/Orders/Partial/Definition.agda         |  17 ++
 Lists/Definition.agda                         |   9 +-
 LogicalFormulae.agda                          |   4 +
 Numbers/ClassicalReals/Examples.agda          |  25 +-
 Numbers/ClassicalReals/RealField.agda         |   2 +-
 Numbers/ClassicalReals/RealField/Lemmas.agda  |   2 +-
 Numbers/Integers/Definition.agda              |   3 +
 Numbers/Integers/Order.agda                   |   2 +-
 .../Integers/RingStructure/Archimedean.agda   |  30 +++
 Numbers/Intervals/Arithmetic.agda             |   2 +-
 Numbers/Intervals/Definition.agda             |   2 +-
 Numbers/Naturals/Addition.agda                |   1 +
 Numbers/Naturals/Multiplication.agda          |   1 +
 Numbers/Rationals/Definition.agda             |   3 +-
 Numbers/Rationals/Lemmas.agda                 |   3 +-
 Numbers/Reals/Definition.agda                 |   2 +-
 Orders/Total/Lemmas.agda                      |   1 -
 Rings/Homomorphisms/Lemmas.agda               |   1 -
 Rings/InitialRing.agda                        |   4 +-
 Rings/Orders/Partial/Bounded.agda             |   2 +-
 Rings/Orders/Partial/Definition.agda          |   6 +-
 Rings/Orders/Partial/Lemmas.agda              |  16 +-
 Rings/Orders/Total/AbsoluteValue.agda         | 217 +++++++++++++++
 Rings/Orders/Total/BaseExpansion.agda         |  17 +-
 Rings/Orders/Total/Bounded.agda               |   3 +-
 Rings/Orders/Total/Cauchy.agda                |  16 +-
 Rings/Orders/Total/Definition.agda            |   3 +-
 Rings/Orders/Total/Lemmas.agda                | 199 +-------------
 Semirings/Solver.agda                         |   5 +-
 Setoids/Orders/Orders.agda                    |  19 ++
 .../Partial/Definition.agda}                  |  22 +-
 Setoids/Orders/Total/Definition.agda          |  33 +++
 Setoids/Orders/Total/Lemmas.agda              |  67 +++++
 63 files changed, 1082 insertions(+), 564 deletions(-)
 create mode 100644 Fields/CauchyCompletion/Archimedean.agda
 create mode 100644 Fields/FieldOfFractions/Archimedean.agda
 create mode 100644 Fields/Orders/Total/Archimedean.agda
 create mode 100644 Groups/Orders/Archimedean.agda
 create mode 100644 Groups/Orders/Partial/Definition.agda
 create mode 100644 Numbers/Integers/RingStructure/Archimedean.agda
 create mode 100644 Rings/Orders/Total/AbsoluteValue.agda
 create mode 100644 Setoids/Orders/Orders.agda
 rename Setoids/{Orders.agda => Orders/Partial/Definition.agda} (57%)
 create mode 100644 Setoids/Orders/Total/Definition.agda
 create mode 100644 Setoids/Orders/Total/Lemmas.agda

diff --git a/Everything/Safe.agda b/Everything/Safe.agda
index 86c8b4d..7887c60 100644
--- a/Everything/Safe.agda
+++ b/Everything/Safe.agda
@@ -16,6 +16,7 @@ open import Numbers.Modulo.Group
 
 open import Numbers.Integers.Integers
 open import Numbers.Integers.RingStructure.EuclideanDomain
+open import Numbers.Integers.RingStructure.Archimedean
 
 open import Numbers.ClassicalReals.Examples
 open import Numbers.ClassicalReals.RealField.Lemmas
@@ -52,12 +53,15 @@ open import Groups.LectureNotes.Lecture1
 open import Fields.Fields
 open import Fields.Orders.Partial.Definition
 open import Fields.Orders.Total.Definition
+open import Fields.Orders.Total.Archimedean
 open import Fields.Orders.LeastUpperBounds.Examples
 open import Fields.Orders.Lemmas
 open import Fields.Orders.Limits.Definition
 open import Fields.FieldOfFractions.Field
 open import Fields.FieldOfFractions.Lemmas
 open import Fields.FieldOfFractions.Order
+open import Fields.FieldOfFractions.Order
+--open import Fields.FieldOfFractions.Archimedean
 
 open import Rings.Definition
 open import Rings.Lemmas
@@ -100,7 +104,7 @@ open import Rings.Orders.Total.BaseExpansion
 open import Setoids.Setoids
 open import Setoids.DirectSum
 open import Setoids.Lists
-open import Setoids.Orders
+open import Setoids.Orders.Total.Lemmas
 open import Setoids.Functions.Definition
 open import Setoids.Functions.Extension
 open import Setoids.Algebra.Lemmas
@@ -159,8 +163,7 @@ open import Modules.PolynomialModule
 open import Modules.Lemmas
 open import Modules.DirectSum
 
-open import Fields.CauchyCompletion.Ring
-open import Fields.CauchyCompletion.Comparison
+open import Fields.CauchyCompletion.Archimedean
 open import Fields.Orders.Limits.Lemmas
 
 open import ProjectEuler.Problem1
diff --git a/Fields/CauchyCompletion/Addition.agda b/Fields/CauchyCompletion/Addition.agda
index 1e8ca63..b955606 100644
--- a/Fields/CauchyCompletion/Addition.agda
+++ b/Fields/CauchyCompletion/Addition.agda
@@ -9,7 +9,8 @@ open import Groups.Lemmas
 open import Fields.Fields
 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
@@ -31,7 +32,9 @@ open import Fields.Lemmas F
 open import Fields.CauchyCompletion.Definition order F
 open import Rings.Orders.Partial.Lemmas pRing
 open import Fields.Orders.Total.Lemmas {F = F} (record { oRing = order })
+open import Rings.Orders.Total.AbsoluteValue order
 open import Rings.Orders.Total.Lemmas order
+open import Rings.Orders.Total.Cauchy order
 open import Rings.InitialRing R
 
 private
@@ -39,19 +42,23 @@ private
   lemm zero a b = refl
   lemm (succ m) a b = lemm m (Sequence.tail a) (Sequence.tail b)
 
+private
+  additionConverges : (a b : CauchyCompletion) → cauchy (apply _+_ (CauchyCompletion.elts a) (CauchyCompletion.elts b))
+  additionConverges record { elts = a ; converges = convA } record { elts = b ; converges = convB } ε 0<e with halve charNot2 ε
+  ... | e/2 , e/2Pr with convA e/2 (halvePositive e/2 (<WellDefined (Equivalence.reflexive eq) (Equivalence.symmetric eq e/2Pr) 0<e))
+  ... | Na , prA with convB e/2 (halvePositive e/2 (<WellDefined (Equivalence.reflexive eq) (Equivalence.symmetric eq e/2Pr) 0<e))
+  ... | Nb , prB = (Na +N Nb) , t
+    where
+      t : {m n : ℕ} → Na +N Nb <N m → Na +N Nb <N n → abs ((index (apply _+_ a b) m) + inverse (index (apply _+_ a b) n)) < ε
+      t {m} {n} <m <n with prA {m} {n} (inequalityShrinkLeft <m) (inequalityShrinkLeft <n)
+      ... | am-an<e/2 with prB {m} {n} (inequalityShrinkRight <m) (inequalityShrinkRight <n)
+      ... | bm-bn<e/2 with triangleInequality (index a m + inverse (index a n)) (index b m + inverse (index b n))
+      ... | inl tri rewrite lemm m a b | lemm n a b = SetoidPartialOrder.<WellDefined pOrder (Equivalence.reflexive eq) e/2Pr (SetoidPartialOrder.<Transitive pOrder {_} {(abs ((index a m) + (inverse (index a n)))) + (abs ((index b m) + (inverse (index b n))))} (<WellDefined (absWellDefined _ _ (Equivalence.transitive eq (Equivalence.symmetric eq (+Associative {index a m})) (Equivalence.transitive eq (+WellDefined (Equivalence.reflexive eq {index a m}) (Equivalence.transitive eq groupIsAbelian (Equivalence.transitive eq (Equivalence.symmetric eq (+Associative {index b m})) (+WellDefined (Equivalence.reflexive eq {index b m}) (Equivalence.symmetric eq (invContravariant additiveGroup)))))) (+Associative {index a m})))) (Equivalence.reflexive eq) tri) (ringAddInequalities am-an<e/2 bm-bn<e/2))
+      ... | inr tri rewrite lemm m a b | lemm n a b = SetoidPartialOrder.<WellDefined pOrder (Equivalence.reflexive eq) e/2Pr (<WellDefined (Equivalence.transitive eq (Equivalence.symmetric eq tri) (absWellDefined _ _ (Equivalence.transitive eq (Equivalence.symmetric eq (+Associative {index a m})) (Equivalence.transitive eq (+WellDefined (Equivalence.reflexive eq {index a m}) (Equivalence.transitive eq groupIsAbelian (Equivalence.transitive eq (Equivalence.symmetric eq (+Associative {index b m})) (+WellDefined (Equivalence.reflexive eq {index b m}) (Equivalence.symmetric eq (invContravariant additiveGroup)))))) (+Associative {index a m}))))) (Equivalence.reflexive eq) (ringAddInequalities am-an<e/2 bm-bn<e/2))
+
 _+C_ : CauchyCompletion → CauchyCompletion → CauchyCompletion
 CauchyCompletion.elts (record { elts = a ; converges = convA } +C record { elts = b ; converges = convB }) = apply _+_ a b
-CauchyCompletion.converges (record { elts = a ; converges = convA } +C record { elts = b ; converges = convB }) ε 0<e with halve charNot2 ε
-... | e/2 , e/2Pr with convA e/2 (halvePositive e/2 (<WellDefined (Equivalence.reflexive eq) (Equivalence.symmetric eq e/2Pr) 0<e))
-CauchyCompletion.converges (record { elts = a ; converges = convA } +C record { elts = b ; converges = convB }) ε 0<e | e/2 , e/2Pr | Na , prA with convB e/2 (halvePositive e/2 (<WellDefined (Equivalence.reflexive eq) (Equivalence.symmetric eq e/2Pr) 0<e))
-CauchyCompletion.converges (record { elts = a ; converges = convA } +C record { elts = b ; converges = convB }) ε 0<e | e/2 , e/2Pr | Na , prA | Nb , prB = (Na +N Nb) , t
-  where
-    t : {m n : ℕ} → Na +N Nb <N m → Na +N Nb <N n → abs ((index (apply _+_ a b) m) + inverse (index (apply _+_ a b) n)) < ε
-    t {m} {n} <m <n with prA {m} {n} (inequalityShrinkLeft <m) (inequalityShrinkLeft <n)
-    ... | am-an<e/2 with prB {m} {n} (inequalityShrinkRight <m) (inequalityShrinkRight <n)
-    ... | bm-bn<e/2 with triangleInequality (index a m + inverse (index a n)) (index b m + inverse (index b n))
-    ... | inl tri rewrite lemm m a b | lemm n a b = SetoidPartialOrder.<WellDefined pOrder (Equivalence.reflexive eq) e/2Pr (SetoidPartialOrder.<Transitive pOrder {_} {(abs ((index a m) + (inverse (index a n)))) + (abs ((index b m) + (inverse (index b n))))} (<WellDefined (absWellDefined _ _ (Equivalence.transitive eq (Equivalence.symmetric eq (+Associative {index a m})) (Equivalence.transitive eq (+WellDefined (Equivalence.reflexive eq {index a m}) (Equivalence.transitive eq groupIsAbelian (Equivalence.transitive eq (Equivalence.symmetric eq (+Associative {index b m})) (+WellDefined (Equivalence.reflexive eq {index b m}) (Equivalence.symmetric eq (invContravariant additiveGroup)))))) (+Associative {index a m})))) (Equivalence.reflexive eq) tri) (ringAddInequalities am-an<e/2 bm-bn<e/2))
-    ... | inr tri rewrite lemm m a b | lemm n a b = SetoidPartialOrder.<WellDefined pOrder (Equivalence.reflexive eq) e/2Pr (<WellDefined (Equivalence.transitive eq (Equivalence.symmetric eq tri) (absWellDefined _ _ (Equivalence.transitive eq (Equivalence.symmetric eq (+Associative {index a m})) (Equivalence.transitive eq (+WellDefined (Equivalence.reflexive eq {index a m}) (Equivalence.transitive eq groupIsAbelian (Equivalence.transitive eq (Equivalence.symmetric eq (+Associative {index b m})) (+WellDefined (Equivalence.reflexive eq {index b m}) (Equivalence.symmetric eq (invContravariant additiveGroup)))))) (+Associative {index a m}))))) (Equivalence.reflexive eq) (ringAddInequalities am-an<e/2 bm-bn<e/2))
+CauchyCompletion.converges (a +C b) = additionConverges a b
 
 inverseDistributes : {r s : A} → inverse (r + s) ∼ inverse r + inverse s
 inverseDistributes = Equivalence.transitive eq (invContravariant additiveGroup) groupIsAbelian
diff --git a/Fields/CauchyCompletion/Approximation.agda b/Fields/CauchyCompletion/Approximation.agda
index 45b2836..33b6bf9 100644
--- a/Fields/CauchyCompletion/Approximation.agda
+++ b/Fields/CauchyCompletion/Approximation.agda
@@ -10,7 +10,8 @@ open import Groups.Lemmas
 open import Fields.Fields
 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
@@ -32,6 +33,7 @@ open Field F
 open import Fields.Lemmas F
 open import Fields.Orders.Lemmas {F = F} record { oRing = order }
 open import Rings.Orders.Total.Lemmas order
+open import Rings.Orders.Total.AbsoluteValue order
 open import Rings.Orders.Partial.Lemmas pRing
 open import Fields.CauchyCompletion.Definition order F
 open import Fields.CauchyCompletion.Addition order F
diff --git a/Fields/CauchyCompletion/Archimedean.agda b/Fields/CauchyCompletion/Archimedean.agda
new file mode 100644
index 0000000..e813719
--- /dev/null
+++ b/Fields/CauchyCompletion/Archimedean.agda
@@ -0,0 +1,28 @@
+{-# OPTIONS --safe --warning=error --without-K --guardedness #-}
+
+open import Agda.Primitive using (Level; lzero; lsuc; _⊔_)
+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 Groups.Orders.Archimedean
+open import Fields.Fields
+open import Sets.EquivalenceRelations
+open import Sequences
+open import Setoids.Orders.Partial.Definition
+open import Functions
+open import LogicalFormulae
+open import Numbers.Naturals.Semiring
+open import Numbers.Naturals.Order
+open import Fields.Orders.Total.Archimedean
+
+module Fields.CauchyCompletion.Archimedean {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) (F : Field R) (arch : ArchimedeanField {F = F} record { oRing = order }) where
+
+open import Fields.CauchyCompletion.Group order F
+open import Fields.CauchyCompletion.Ring order F
+open import Fields.CauchyCompletion.Comparison order F
+--open import Fields.CauchyCompletion.PartiallyOrderedRing order F
+
+--CArchimedean : Archimedean (toGroup CRing CpOrderedRing)
+--CArchimedean = ?
diff --git a/Fields/CauchyCompletion/Comparison.agda b/Fields/CauchyCompletion/Comparison.agda
index 662b3dc..35b9c86 100644
--- a/Fields/CauchyCompletion/Comparison.agda
+++ b/Fields/CauchyCompletion/Comparison.agda
@@ -10,7 +10,8 @@ open import Groups.Lemmas
 open import Fields.Fields
 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
@@ -32,9 +33,11 @@ open import Fields.Orders.Lemmas {F = F} {pRing} (record { oRing = order })
 
 open import Rings.Orders.Partial.Lemmas pRing
 open import Rings.Orders.Total.Lemmas order
+open import Rings.Orders.Total.AbsoluteValue order
 open import Fields.Lemmas F
 open import Fields.CauchyCompletion.Definition order F
 open import Fields.CauchyCompletion.Setoid order F
+open import Fields.CauchyCompletion.Group order F
 open import Fields.Orders.Total.Lemmas {F = F} (record { oRing = order })
 
 -- "<C rational", where the r indicates where the rational number goes
@@ -48,11 +51,12 @@ record _<Cr_ (a : CauchyCompletion) (b : A) : Set (m ⊔ o) where
 <CrWellDefinedRight : (a : CauchyCompletion) (b c : A) → b ∼ c → a <Cr b → a <Cr c
 <CrWellDefinedRight a b c b=c record { e = ε ; 0<e = 0<e ; N = N ; property = pr } = record { e = ε ; 0<e = 0<e ; N = N ; property = λ m N<m → <WellDefined (Equivalence.reflexive eq) b=c (pr m N<m) }
 
-<CrWellDefinedLemma : (a b e/2 e : A) (0<e : 0G < e) (pr : e/2 + e/2 ∼ e) → abs (a + inverse b) < e/2 → (e/2 + b) < (e + a)
-<CrWellDefinedLemma a b e/2 e 0<e pr a-b<e with totality 0G (a + inverse b)
-<CrWellDefinedLemma a b e/2 e 0<e pr a-b<e | inl (inl 0<a-b) = ringAddInequalities (halfLess e/2 e 0<e pr) (<WellDefined identLeft (Equivalence.transitive eq (Equivalence.symmetric eq +Associative) (Equivalence.transitive eq (+WellDefined (Equivalence.reflexive eq) invLeft) identRight)) (orderRespectsAddition 0<a-b b))
-<CrWellDefinedLemma a b e/2 e 0<e pr a-b<e | inl (inr a-b<0) = <WellDefined (Equivalence.transitive eq (+WellDefined (invContravariant (Ring.additiveGroup R)) groupIsAbelian) (Equivalence.transitive eq (Equivalence.transitive eq +Associative (+WellDefined (Equivalence.transitive eq (Equivalence.symmetric eq +Associative) (Equivalence.transitive eq (+WellDefined (invTwice additiveGroup b) invLeft) identRight)) (Equivalence.reflexive eq))) groupIsAbelian)) (Equivalence.transitive eq +Associative (+WellDefined pr (Equivalence.reflexive eq))) (orderRespectsAddition a-b<e (e/2 + a))
-<CrWellDefinedLemma a b e/2 e 0<e pr a-b<e | inr 0=a-b = <WellDefined (Equivalence.reflexive eq) (+WellDefined (Equivalence.reflexive eq) (Equivalence.symmetric eq (transferToRight (Ring.additiveGroup R) (Equivalence.symmetric eq 0=a-b)))) (orderRespectsAddition (halfLess e/2 e 0<e pr) b)
+private
+  <CrWellDefinedLemma : (a b e/2 e : A) (0<e : 0G < e) (pr : e/2 + e/2 ∼ e) → abs (a + inverse b) < e/2 → (e/2 + b) < (e + a)
+  <CrWellDefinedLemma a b e/2 e 0<e pr a-b<e with totality 0G (a + inverse b)
+  <CrWellDefinedLemma a b e/2 e 0<e pr a-b<e | inl (inl 0<a-b) = ringAddInequalities (halfLess e/2 e 0<e pr) (<WellDefined identLeft (Equivalence.transitive eq (Equivalence.symmetric eq +Associative) (Equivalence.transitive eq (+WellDefined (Equivalence.reflexive eq) invLeft) identRight)) (orderRespectsAddition 0<a-b b))
+  <CrWellDefinedLemma a b e/2 e 0<e pr a-b<e | inl (inr a-b<0) = <WellDefined (Equivalence.transitive eq (+WellDefined (invContravariant (Ring.additiveGroup R)) groupIsAbelian) (Equivalence.transitive eq (Equivalence.transitive eq +Associative (+WellDefined (Equivalence.transitive eq (Equivalence.symmetric eq +Associative) (Equivalence.transitive eq (+WellDefined (invTwice additiveGroup b) invLeft) identRight)) (Equivalence.reflexive eq))) groupIsAbelian)) (Equivalence.transitive eq +Associative (+WellDefined pr (Equivalence.reflexive eq))) (orderRespectsAddition a-b<e (e/2 + a))
+  <CrWellDefinedLemma a b e/2 e 0<e pr a-b<e | inr 0=a-b = <WellDefined (Equivalence.reflexive eq) (+WellDefined (Equivalence.reflexive eq) (Equivalence.symmetric eq (transferToRight (Ring.additiveGroup R) (Equivalence.symmetric eq 0=a-b)))) (orderRespectsAddition (halfLess e/2 e 0<e pr) b)
 
 <CrWellDefinedLeft : (a b : CauchyCompletion) (c : A) → Setoid._∼_ cauchyCompletionSetoid a b → a <Cr c → b <Cr c
 <CrWellDefinedLeft a b c a=b record { e = ε ; 0<e = 0<e ; N = N ; property = pr } with halve charNot2 ε
@@ -77,11 +81,12 @@ record _r<C_ (a : A) (b : CauchyCompletion) : Set (m ⊔ o) where
 r<CWellDefinedLeft : (a b : A) (c : CauchyCompletion) → (a ∼ b) → (a r<C c) → b r<C c
 r<CWellDefinedLeft a b c a=b record { e = e ; 0<e = 0<e ; N = N ; pr = pr } = record { e = e ; 0<e = 0<e ; N = N ; pr =  λ m N<m → <WellDefined (+WellDefined a=b (Equivalence.reflexive eq)) (Equivalence.reflexive eq) (pr m N<m) }
 
-r<CWellDefinedLemma : (a b c e e/2 : A) (_ : e/2 + e/2 ∼ e) (0<e : 0G < e) (_ : abs (a + inverse b) < e/2) (_ : (c + e) < a) → (c + e/2) < b
-r<CWellDefinedLemma a b c e e/2 prE/2 0<e pr c+e<a with totality 0G (a + inverse b)
-r<CWellDefinedLemma a b c e e/2 prE/2 0<e pr c+e<a | inl (inl 0<a-b) = SetoidPartialOrder.<Transitive pOrder (<WellDefined (Equivalence.transitive eq (Equivalence.symmetric eq +Associative) (+WellDefined (Equivalence.reflexive eq) (Equivalence.transitive eq (+WellDefined (Equivalence.symmetric eq prE/2) (Equivalence.reflexive eq)) (Equivalence.transitive eq (Equivalence.transitive eq (Equivalence.symmetric eq +Associative) (+WellDefined (Equivalence.reflexive eq) invRight)) identRight)))) (Equivalence.reflexive eq) (orderRespectsAddition c+e<a (inverse e/2))) (<WellDefined (Equivalence.transitive eq +Associative (+WellDefined (Equivalence.transitive eq (Equivalence.symmetric eq +Associative) (Equivalence.transitive eq (+WellDefined (Equivalence.reflexive eq) invLeft) identRight)) (Equivalence.reflexive eq))) (Equivalence.transitive eq groupIsAbelian (Equivalence.transitive eq (Equivalence.symmetric eq +Associative) (Equivalence.transitive eq (+WellDefined (Equivalence.reflexive eq) (invLeft)) identRight))) (orderRespectsAddition pr (b + inverse e/2)))
-r<CWellDefinedLemma a b c e e/2 prE/2 0<e pr c+e<a | inl (inr a-b<0) = SetoidPartialOrder.<Transitive pOrder (SetoidPartialOrder.<Transitive pOrder (<WellDefined groupIsAbelian groupIsAbelian (orderRespectsAddition (halfLess e/2 e 0<e prE/2) c)) c+e<a) (<WellDefined (Equivalence.transitive eq (Equivalence.transitive eq (Equivalence.symmetric eq +Associative) (+WellDefined (Equivalence.reflexive eq) invLeft)) identRight) identLeft (orderRespectsAddition a-b<0 b))
-r<CWellDefinedLemma a b c e e/2 prE/2 0<e pr c+e<a | inr 0=a-b = SetoidPartialOrder.<Transitive pOrder (<WellDefined groupIsAbelian (Equivalence.reflexive eq) (orderRespectsAddition (halfLess e/2 e 0<e prE/2) c)) (<WellDefined (groupIsAbelian {c} {e}) (transferToRight additiveGroup (Equivalence.symmetric eq 0=a-b)) c+e<a)
+private
+  r<CWellDefinedLemma : (a b c e e/2 : A) (_ : e/2 + e/2 ∼ e) (0<e : 0G < e) (_ : abs (a + inverse b) < e/2) (_ : (c + e) < a) → (c + e/2) < b
+  r<CWellDefinedLemma a b c e e/2 prE/2 0<e pr c+e<a with totality 0G (a + inverse b)
+  r<CWellDefinedLemma a b c e e/2 prE/2 0<e pr c+e<a | inl (inl 0<a-b) = SetoidPartialOrder.<Transitive pOrder (<WellDefined (Equivalence.transitive eq (Equivalence.symmetric eq +Associative) (+WellDefined (Equivalence.reflexive eq) (Equivalence.transitive eq (+WellDefined (Equivalence.symmetric eq prE/2) (Equivalence.reflexive eq)) (Equivalence.transitive eq (Equivalence.transitive eq (Equivalence.symmetric eq +Associative) (+WellDefined (Equivalence.reflexive eq) invRight)) identRight)))) (Equivalence.reflexive eq) (orderRespectsAddition c+e<a (inverse e/2))) (<WellDefined (Equivalence.transitive eq +Associative (+WellDefined (Equivalence.transitive eq (Equivalence.symmetric eq +Associative) (Equivalence.transitive eq (+WellDefined (Equivalence.reflexive eq) invLeft) identRight)) (Equivalence.reflexive eq))) (Equivalence.transitive eq groupIsAbelian (Equivalence.transitive eq (Equivalence.symmetric eq +Associative) (Equivalence.transitive eq (+WellDefined (Equivalence.reflexive eq) (invLeft)) identRight))) (orderRespectsAddition pr (b + inverse e/2)))
+  r<CWellDefinedLemma a b c e e/2 prE/2 0<e pr c+e<a | inl (inr a-b<0) = SetoidPartialOrder.<Transitive pOrder (SetoidPartialOrder.<Transitive pOrder (<WellDefined groupIsAbelian groupIsAbelian (orderRespectsAddition (halfLess e/2 e 0<e prE/2) c)) c+e<a) (<WellDefined (Equivalence.transitive eq (Equivalence.transitive eq (Equivalence.symmetric eq +Associative) (+WellDefined (Equivalence.reflexive eq) invLeft)) identRight) identLeft (orderRespectsAddition a-b<0 b))
+  r<CWellDefinedLemma a b c e e/2 prE/2 0<e pr c+e<a | inr 0=a-b = SetoidPartialOrder.<Transitive pOrder (<WellDefined groupIsAbelian (Equivalence.reflexive eq) (orderRespectsAddition (halfLess e/2 e 0<e prE/2) c)) (<WellDefined (groupIsAbelian {c} {e}) (transferToRight additiveGroup (Equivalence.symmetric eq 0=a-b)) c+e<a)
 
 r<CWellDefinedRight : (a b : CauchyCompletion) (c : A) → Setoid._∼_ cauchyCompletionSetoid a b → c r<C a → c r<C b
 r<CWellDefinedRight a b c a=b record { e = e ; 0<e = 0<e ; N = N ; pr = pr } with halve charNot2 e
@@ -142,6 +147,26 @@ record _<C_ (a : CauchyCompletion) (b : CauchyCompletion) : Set (m ⊔ o) where
 <CRelaxR' : {a : CauchyCompletion} {b : A} → a <C injection b → a <Cr b
 <CRelaxR' {a} {b} record { i = i ; a<i = record { e = ε ; 0<e = 0<ε ; N = N ; property = property } ; i<b = i<b } = record { e = ε ; 0<e = 0<ε ; N = N ; property = λ m N<m → <Transitive (property m N<m) (<CCollapsesR _ _ i<b) }
 
+flip<CR : {a : CauchyCompletion} {b : A} → a <Cr b → inverse b r<C Group.inverse CGroup a
+_r<C_.e (flip<CR {a} {b} record { e = e ; 0<e = 0<e ; N = N ; property = property }) = e
+_r<C_.0<e (flip<CR {a} {b} record { e = e ; 0<e = 0<e ; N = N ; property = property }) = 0<e
+_r<C_.N (flip<CR {a} {b} record { e = e ; 0<e = 0<e ; N = N ; property = property }) = N
+_r<C_.pr (flip<CR {a} {b} record { e = e ; 0<e = 0<e ; N = N ; property = property }) m N<m with property m N<m
+... | bl = <WellDefined (transitive (transitive (symmetric +Associative) (+WellDefined reflexive (transitive groupIsAbelian (transitive (symmetric +Associative) (transitive (+WellDefined reflexive invLeft) identRight))))) groupIsAbelian) (transitive +Associative (transitive (+WellDefined invRight reflexive) (transitive identLeft (identityOfIndiscernablesRight _∼_ reflexive (mapAndIndex (CauchyCompletion.elts a) inverse m))))) (orderRespectsAddition bl (inverse b + inverse (index (CauchyCompletion.elts a) m)))
+
+flipR<C : {a : A} {b : CauchyCompletion} → a r<C b → Group.inverse CGroup b <Cr inverse a
+_<Cr_.e (flipR<C record { e = e ; 0<e = 0<e ; N = N ; pr = pr }) = e
+_<Cr_.0<e (flipR<C record { e = e ; 0<e = 0<e ; N = N ; pr = pr }) = 0<e
+_<Cr_.N (flipR<C record { e = e ; 0<e = 0<e ; N = N ; pr = pr }) = N
+_<Cr_.property (flipR<C {a} {b} record { e = e ; 0<e = 0<e ; N = N ; pr = pr }) m N<m with pr m N<m
+... | t = <WellDefined (transitive (+WellDefined groupIsAbelian (+WellDefined reflexive (identityOfIndiscernablesRight _∼_ reflexive (mapAndIndex (CauchyCompletion.elts b) inverse m)))) (transitive (symmetric +Associative) (+WellDefined reflexive (transitive +Associative (transitive (+WellDefined invRight reflexive) identLeft))))) (transitive groupIsAbelian (transitive (symmetric +Associative) (transitive (+WellDefined reflexive invLeft) identRight))) (orderRespectsAddition t (inverse a + inverse (index (CauchyCompletion.elts b) m)))
+
+flip<C : {a b : CauchyCompletion} → a <C b → Group.inverse CGroup b <C Group.inverse CGroup a
+flip<C {a} {b} record { i = i ; a<i = a<i ; i<b = i<b } = record { i = inverse i ; a<i = flipR<C i<b ; i<b = flip<CR a<i }
+
+flip<C' : {a b : CauchyCompletion} → Group.inverse CGroup b <C Group.inverse CGroup a → a <C b
+flip<C' {a} {b} a<b = <CWellDefined (invTwice CGroup a) (invTwice CGroup b) (flip<C a<b)
+
 <CIrreflexive : {a : CauchyCompletion} → a <C a → False
 <CIrreflexive {a} record { i = bound ; a<i = record { e = bA ; 0<e = 0<bA ; N = Na ; property = prA } ; i<b = record { e = bB ; 0<e = 0<bB ; N = Nb ; pr = prB } } with prA (succ Na +N Nb) (le Nb (applyEquality succ (Semiring.commutative ℕSemiring Nb Na)))
 ... | bl with prB (succ Na +N Nb) (le Na refl)
diff --git a/Fields/CauchyCompletion/Definition.agda b/Fields/CauchyCompletion/Definition.agda
index 958741e..dd6bd3c 100644
--- a/Fields/CauchyCompletion/Definition.agda
+++ b/Fields/CauchyCompletion/Definition.agda
@@ -9,11 +9,12 @@ open import Groups.Definition
 open import Fields.Fields
 open import Sets.EquivalenceRelations
 open import Sequences
-open import Setoids.Orders
 open import Functions
 open import LogicalFormulae
 open import Numbers.Naturals.Semiring
 open import Numbers.Naturals.Order
+open import Setoids.Orders.Partial.Definition
+open import Setoids.Orders.Total.Definition
 
 module Fields.CauchyCompletion.Definition {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) (F : Field R) where
 
@@ -27,6 +28,7 @@ open Group additiveGroup
 open Field F
 
 open import Rings.Orders.Total.Lemmas order
+open import Rings.Orders.Total.AbsoluteValue order
 open import Rings.Orders.Total.Cauchy order
 open import Groups.Lemmas additiveGroup
 
@@ -37,11 +39,11 @@ cauchyWellDefined {s1} {s2} prop c e 0<e with c e 0<e
 record CauchyCompletion : Set (m ⊔ o) where
   field
     elts : Sequence A
-    converges : cauchy elts
+    converges : (cauchy elts)
 
 injection : A → CauchyCompletion
 CauchyCompletion.elts (injection a) = constSequence a
-CauchyCompletion.converges (injection a) = λ ε 0<e → 0 , λ {m} {n} _ _ → <WellDefined (symmetric (identityOfIndiscernablesRight _∼_ (absWellDefined (index (constSequence a) m + inverse (index (constSequence a) n)) 0R (t m n)) absZero)) reflexive 0<e
+CauchyCompletion.converges (injection a) = (λ ε 0<e → 0 , λ {m} {n} _ _ → <WellDefined (symmetric (identityOfIndiscernablesRight _∼_ (absWellDefined (index (constSequence a) m + inverse (index (constSequence a) n)) 0R (t m n)) absZero)) reflexive 0<e)
   where
     t : (m n : ℕ) → index (constSequence a) m + inverse (index (constSequence a) n) ∼ 0R
     t m n = identityOfIndiscernablesLeft _∼_ (identityOfIndiscernablesLeft _∼_ invRight (equalityCommutative (applyEquality (λ i → a + inverse i) (indexAndConst a n)))) (applyEquality (_+ inverse (index (constSequence a) n)) (equalityCommutative (indexAndConst a m)))
diff --git a/Fields/CauchyCompletion/Field.agda b/Fields/CauchyCompletion/Field.agda
index f4739f5..1cf70a9 100644
--- a/Fields/CauchyCompletion/Field.agda
+++ b/Fields/CauchyCompletion/Field.agda
@@ -1,4 +1,4 @@
-{-# OPTIONS --safe --warning=error --without-K --guardedness #-}
+{-# OPTIONS --allow-unsolved-metas --warning=error --without-K --guardedness #-}
 
 open import Agda.Primitive using (Level; lzero; lsuc; _⊔_)
 open import Setoids.Setoids
@@ -11,13 +11,14 @@ open import Groups.Groups
 open import Fields.Fields
 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
 open import Numbers.Naturals.Order
 
-module Fields.CauchyCompletion.Field {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) (F : Field R) (charNot2 : Setoid._∼_ S ((Ring.1R R) + (Ring.1R R)) (Ring.0R R) → False) where
+module Fields.CauchyCompletion.Field {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) (F : Field R) where
 
 open Setoid S
 open PartiallyOrderedRing pRing
@@ -27,17 +28,18 @@ open SetoidTotalOrder (TotallyOrderedRing.total order)
 open Field F
 open Group (Ring.additiveGroup R)
 
+open import Rings.Orders.Total.Cauchy order
 open import Rings.Orders.Total.Lemmas order
 open import Fields.CauchyCompletion.Definition order F
-open import Fields.CauchyCompletion.Multiplication order F charNot2
-open import Fields.CauchyCompletion.Addition order F charNot2
-open import Fields.CauchyCompletion.Setoid order F charNot2
-open import Fields.CauchyCompletion.Group order F charNot2
-open import Fields.CauchyCompletion.Ring order F charNot2
-open import Fields.CauchyCompletion.Comparison order F charNot2
+open import Fields.CauchyCompletion.Multiplication order F
+open import Fields.CauchyCompletion.Addition order F
+open import Fields.CauchyCompletion.Setoid order F
+open import Fields.CauchyCompletion.Group order F
+open import Fields.CauchyCompletion.Ring order F
+open import Fields.CauchyCompletion.Comparison order F
 
 Cnontrivial : (pr : Setoid._∼_ cauchyCompletionSetoid (injection (Ring.0R R)) (injection (Ring.1R R))) → False
-Cnontrivial pr with pr (Ring.1R R) (0<1 (charNot2ImpliesNontrivial R charNot2))
+Cnontrivial pr with pr (Ring.1R R) (0<1 nontrivial)
 Cnontrivial pr | N , b with b {succ N} (le 0 refl)
 ... | bl rewrite indexAndApply (constSequence 0G) (map inverse (constSequence (Ring.1R R))) _+_ {N} | indexAndConst 0G N | equalityCommutative (mapAndIndex (constSequence (Ring.1R R)) inverse N) | indexAndConst (Ring.1R R) N = irreflexive {Ring.1R R} (<WellDefined (Equivalence.transitive eq (absWellDefined _ _ identLeft) (Equivalence.transitive eq (Equivalence.symmetric eq (absNegation (Ring.1R R))) abs1Is1)) (Equivalence.reflexive eq) bl)
 
diff --git a/Fields/CauchyCompletion/Group.agda b/Fields/CauchyCompletion/Group.agda
index c321ee5..c4a5f67 100644
--- a/Fields/CauchyCompletion/Group.agda
+++ b/Fields/CauchyCompletion/Group.agda
@@ -10,7 +10,8 @@ open import Groups.Homomorphisms.Definition
 open import Fields.Fields
 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
@@ -28,13 +29,14 @@ open Group (Ring.additiveGroup R)
 open Ring R
 
 open import Rings.Orders.Total.Lemmas order
+open import Rings.Orders.Total.AbsoluteValue order
 open import Fields.CauchyCompletion.Definition order F
 open import Fields.CauchyCompletion.Addition order F
 open import Fields.CauchyCompletion.Setoid order F
 
 abstract
-  +CCommutative : {a b : CauchyCompletion} → Setoid._∼_ cauchyCompletionSetoid (a +C b) (b +C a)
-  +CCommutative {a} {b} ε 0<e = 0 , ans
+  +CCommutative : (a b : CauchyCompletion) → Setoid._∼_ cauchyCompletionSetoid (a +C b) (b +C a)
+  +CCommutative a b ε 0<e = 0 , ans
     where
       foo : {x y : A} → (x + y) + inverse (y + x) ∼ 0G
       foo = Equivalence.transitive eq (+WellDefined (Equivalence.reflexive eq) (inverseWellDefined additiveGroup groupIsAbelian)) invRight
@@ -58,13 +60,13 @@ private
     additionPreservedLeft {a} {b} {c} a=b = additionWellDefinedLeft (injection a) (injection b) c (injectionPreservesSetoid a b a=b)
 
     additionPreservedRight : {a b : A} {c : CauchyCompletion} → (a ∼ b) → Setoid._∼_ cauchyCompletionSetoid (c +C injection a) (c +C injection b)
-    additionPreservedRight {a} {b} {c} a=b = Equivalence.transitive (Setoid.eq cauchyCompletionSetoid) {c +C injection a} {injection a +C c} {c +C injection b} (+CCommutative {c} {injection a}) (Equivalence.transitive (Setoid.eq cauchyCompletionSetoid) {injection a +C c} {injection b +C c} {c +C injection b} (additionPreservedLeft {a} {b} {c} a=b) (+CCommutative {injection b} {c}))
+    additionPreservedRight {a} {b} {c} a=b = Equivalence.transitive (Setoid.eq cauchyCompletionSetoid) {c +C injection a} {injection a +C c} {c +C injection b} (+CCommutative c (injection a)) (Equivalence.transitive (Setoid.eq cauchyCompletionSetoid) {injection a +C c} {injection b +C c} {c +C injection b} (additionPreservedLeft {a} {b} {c} a=b) (+CCommutative (injection b) c))
 
     additionPreserved : {a b c d : A} → (a ∼ b) → (c ∼ d) → Setoid._∼_ cauchyCompletionSetoid (injection a +C injection c) (injection b +C injection d)
     additionPreserved {a} {b} {c} {d} a=b c=d = Equivalence.transitive (Setoid.eq cauchyCompletionSetoid) {injection a +C injection c} {injection a +C injection d} {injection b +C injection d} (additionPreservedRight {c} {d} {injection a} c=d) (additionPreservedLeft {a} {b} {injection d} a=b)
 
     additionWellDefinedRight : (a b c : CauchyCompletion) → Setoid._∼_ cauchyCompletionSetoid b c → Setoid._∼_ cauchyCompletionSetoid (a +C b) (a +C c)
-    additionWellDefinedRight a b c b=c = Equivalence.transitive (Setoid.eq cauchyCompletionSetoid) {a +C b} {b +C a} {a +C c} (+CCommutative {a} {b}) (Equivalence.transitive (Setoid.eq cauchyCompletionSetoid) {b +C a} {c +C a} {a +C c} (additionWellDefinedLeft b c a b=c) (+CCommutative {c} {a}))
+    additionWellDefinedRight a b c b=c = Equivalence.transitive (Setoid.eq cauchyCompletionSetoid) {a +C b} {b +C a} {a +C c} (+CCommutative a b) (Equivalence.transitive (Setoid.eq cauchyCompletionSetoid) {b +C a} {c +C a} {a +C c} (additionWellDefinedLeft b c a b=c) (+CCommutative c a))
 
     additionWellDefined : {a b c d : CauchyCompletion} → Setoid._∼_ cauchyCompletionSetoid a b → Setoid._∼_ cauchyCompletionSetoid c d → Setoid._∼_ cauchyCompletionSetoid (a +C c) (b +C d)
     additionWellDefined {a} {b} {c} {d} a=b c=d = Equivalence.transitive (Setoid.eq cauchyCompletionSetoid) {a +C c} {a +C d} {b +C d} (additionWellDefinedRight a c d c=d) (additionWellDefinedLeft a b d a=b)
@@ -88,7 +90,7 @@ private
         ans {m} 0<m rewrite indexAndApply (CauchyCompletion.elts (a +C injection 0G)) (map inverse (CauchyCompletion.elts a)) _+_ {m} | indexAndApply (CauchyCompletion.elts a) (constSequence 0G) _+_ {m} | equalityCommutative (mapAndIndex (CauchyCompletion.elts a) inverse m) | indexAndConst 0G m = <WellDefined (Equivalence.symmetric eq (Equivalence.transitive eq (absWellDefined _ _ (Equivalence.transitive eq (+WellDefined (identRight) (Equivalence.reflexive eq)) (invRight))) (identityOfIndiscernablesRight _∼_ (Equivalence.reflexive eq) absZero))) (Equivalence.reflexive eq) 0<e
 
     CidentLeft : {a : CauchyCompletion} → Setoid._∼_ cauchyCompletionSetoid (injection 0G +C a) a
-    CidentLeft {a} = Equivalence.transitive (Setoid.eq cauchyCompletionSetoid) {injection 0G +C a} {a +C injection 0G} {a} (+CCommutative {injection 0G} {a}) (CidentRight {a})
+    CidentLeft {a} = Equivalence.transitive (Setoid.eq cauchyCompletionSetoid) {injection 0G +C a} {a +C injection 0G} {a} (+CCommutative (injection 0G) a) (CidentRight {a})
 
     CinvRight : {a : CauchyCompletion} → Setoid._∼_ cauchyCompletionSetoid (a +C (-C a)) (injection 0G)
     CinvRight {a} ε 0<e = 0 , ans
@@ -103,7 +105,7 @@ Group.inverse CGroup = -C_
 Group.+Associative CGroup {a} {b} {c} = Cassoc {a} {b} {c}
 Group.identRight CGroup {a} = CidentRight {a}
 Group.identLeft CGroup {a} = CidentLeft {a}
-Group.invLeft CGroup {a} = Equivalence.transitive (Setoid.eq cauchyCompletionSetoid) {(-C a) +C a} {a +C (-C a)} {injection 0G} (+CCommutative { -C a} {a}) (CinvRight {a})
+Group.invLeft CGroup {a} = Equivalence.transitive (Setoid.eq cauchyCompletionSetoid) {(-C a) +C a} {a +C (-C a)} {injection 0G} (+CCommutative (-C a) a) (CinvRight {a})
 Group.invRight CGroup {a} = CinvRight {a}
 
 CInjectionGroupHom : GroupHom (Ring.additiveGroup R) CGroup injection
diff --git a/Fields/CauchyCompletion/Multiplication.agda b/Fields/CauchyCompletion/Multiplication.agda
index 8e36fe9..c0d0577 100644
--- a/Fields/CauchyCompletion/Multiplication.agda
+++ b/Fields/CauchyCompletion/Multiplication.agda
@@ -10,7 +10,8 @@ open import Groups.Lemmas
 open import Fields.Fields
 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
@@ -33,6 +34,8 @@ open import Fields.Orders.Total.Lemmas {F = F} record { oRing = order }
 open import Fields.Lemmas F
 open import Rings.Orders.Partial.Lemmas pRing
 open import Rings.Orders.Total.Lemmas order
+open import Rings.Orders.Total.Cauchy order
+open import Rings.Orders.Total.AbsoluteValue order
 open import Fields.CauchyCompletion.Definition order F
 open import Fields.CauchyCompletion.Setoid order F
 open import Fields.CauchyCompletion.Approximation order F
@@ -44,112 +47,115 @@ private
   littleLemma : {a b c d : A} → ((a * b) + inverse (c * d)) ∼ ((a * (b + inverse d)) + (d * (a + inverse c)))
   littleLemma {a} {b} {c} {d} = Equivalence.transitive eq (Equivalence.transitive eq (+WellDefined (Equivalence.reflexive eq) (Equivalence.transitive eq (Equivalence.transitive eq (Equivalence.symmetric eq identLeft) (+WellDefined (Equivalence.symmetric eq (Equivalence.transitive eq (+WellDefined (Equivalence.transitive eq (ringMinusExtracts R) (inverseWellDefined additiveGroup *Commutative)) (Equivalence.reflexive eq)) (invLeft {d * a}))) (Equivalence.transitive eq (Equivalence.symmetric eq (ringMinusExtracts' R)) *Commutative))) (Equivalence.symmetric eq +Associative))) (+Associative)) (Equivalence.symmetric eq (+WellDefined (*DistributesOver+) (*DistributesOver+)))
 
+  timesConverges : (a b : CauchyCompletion) → cauchy (apply _*_ (CauchyCompletion.elts a) (CauchyCompletion.elts b))
+  timesConverges record { elts = a ; converges = aConv } record { elts = b ; converges = bConv } e 0<e with boundModulus record { elts = a ; converges = aConv }
+  ... | aBound , (Na , prABound) with boundModulus record { elts = b ; converges = bConv }
+  ... | bBound , (Nb , prBBound) = N , ans
+    where
+      boundBoth : A
+      boundBoth = aBound + bBound
+      abstract
+        ab<bb : aBound < boundBoth
+        ab<bb = <WellDefined identLeft groupIsAbelian (orderRespectsAddition {0R} {bBound} (greaterThanAbsImpliesGreaterThan0 (prBBound (succ Nb) (le 0 refl))) aBound)
+        bb<bb : bBound < boundBoth
+        bb<bb = <WellDefined identLeft (Equivalence.reflexive eq) (orderRespectsAddition {0R} {aBound} (greaterThanAbsImpliesGreaterThan0 (prABound (succ Na) (le 0 refl))) bBound)
+        0<boundBoth : 0R < boundBoth
+        0<boundBoth = <WellDefined identLeft (Equivalence.reflexive eq) (ringAddInequalities (greaterThanAbsImpliesGreaterThan0 (prABound (succ Na) (le 0 refl))) (greaterThanAbsImpliesGreaterThan0 (prBBound (succ Nb) (le 0 refl))))
+      abstract
+        1/boundBoothAndPr : Sg A λ i → i * (aBound + bBound) ∼ 1R
+        1/boundBoothAndPr = allInvertible boundBoth λ pr → irreflexive (<WellDefined (Equivalence.reflexive eq) pr 0<boundBoth)
+        1/boundBooth : A
+        1/boundBooth with 1/boundBoothAndPr
+        ... | a , _ = a
+        1/boundBoothPr : 1/boundBooth * (aBound + bBound) ∼ 1R
+        1/boundBoothPr with 1/boundBoothAndPr
+        ... | _ , pr = pr
+        0<1/boundBooth : 0G < 1/boundBooth
+        0<1/boundBooth = inversePositiveIsPositive 1/boundBoothPr 0<boundBoth
+      abstract
+        miniEAndPr : Sg A (λ i → (i + i) ∼ (e * 1/boundBooth))
+        miniEAndPr = halve charNot2 (e * 1/boundBooth)
+        miniE : A
+        miniE with miniEAndPr
+        ... | a , _ = a
+        miniEPr : (miniE + miniE) ∼ (e * 1/boundBooth)
+        miniEPr with miniEAndPr
+        ... | _ , pr = pr
+        0<miniE : 0R < miniE
+        0<miniE = halvePositive miniE (<WellDefined (Equivalence.reflexive eq) (Equivalence.symmetric eq miniEPr) (orderRespectsMultiplication 0<e 0<1/boundBooth))
+      abstract
+        reallyNAAndPr : Sg ℕ (λ N → {m n : ℕ} → N <N m → N <N n → abs (index a m + inverse (index a n)) < miniE)
+        reallyNAAndPr = aConv miniE 0<miniE
+        reallyNa : ℕ
+        reallyNa with reallyNAAndPr
+        ... | a , _ = a
+        reallyNaPr : {m n : ℕ} → reallyNa <N m → reallyNa <N n → abs (index a m + inverse (index a n)) < miniE
+        reallyNaPr with reallyNAAndPr
+        ... | _ , pr = pr
+        reallyNBAndPr : Sg ℕ (λ N → {m n : ℕ} → N <N m → N <N n → abs (index b m + inverse (index b n)) < miniE)
+        reallyNBAndPr = bConv miniE 0<miniE
+        reallyNb : ℕ
+        reallyNb with reallyNBAndPr
+        ... | a , _ = a
+        reallyNbPr : {m n : ℕ} → reallyNb <N m → reallyNb <N n → abs (index b m + inverse (index b n)) < miniE
+        reallyNbPr with reallyNBAndPr
+        ... | _ , pr = pr
+      N : ℕ
+      N = (Na +N (Nb +N (reallyNa +N reallyNb)))
+      ans : {m : ℕ} {n : ℕ} → N <N m → N <N n → abs (index (apply _*_ a b) m + inverse (index (apply _*_ a b) n)) < e
+      ans {m} {n} N<m N<n rewrite indexAndApply a b _*_ {m} | indexAndApply a b _*_ {n} = ans'''
+        where
+          Na<m : Na <N m
+          Na<m = inequalityShrinkLeft N<m
+          Nb<n : Nb <N n
+          Nb<n = inequalityShrinkLeft (inequalityShrinkRight {Na} N<n)
+          abstract
+            sum : {m n : ℕ} → (reallyNa +N reallyNb) <N m → (reallyNa +N reallyNb) <N n → (boundBoth * ((abs (index b m + inverse (index b n))) + (abs (index a m + inverse (index a n))))) < e
+            sum {m} {n} <m <n = <WellDefined *Commutative (Equivalence.transitive eq (*WellDefined miniEPr (Equivalence.reflexive eq)) (Equivalence.transitive eq (Equivalence.symmetric eq *Associative) (Equivalence.transitive eq (Equivalence.transitive eq (*WellDefined (Equivalence.reflexive eq) 1/boundBoothPr) *Commutative) (identIsIdent)))) (ringCanMultiplyByPositive {c = boundBoth} 0<boundBoth (ringAddInequalities (reallyNbPr {m} {n} (inequalityShrinkRight <m) (inequalityShrinkRight <n)) (reallyNaPr {m} {n} (inequalityShrinkLeft <m) (inequalityShrinkLeft <n))))
+          q : ((boundBoth * (abs (index b m + inverse (index b n)))) + (boundBoth * (abs (index a m + inverse (index a n))))) < e
+          q = <WellDefined *DistributesOver+ (Equivalence.reflexive eq) (sum {m} {n} (inequalityShrinkRight {Nb} (inequalityShrinkRight {Na} N<m)) (inequalityShrinkRight {Nb} (inequalityShrinkRight {Na} N<n)))
+          p : ((abs (index a m) * abs (index b m + inverse (index b n))) + (abs (index b n) * abs (index a m + inverse (index a n)))) < e
+          p with totality 0R (index b m + inverse (index b n))
+          p | inl (inl 0<bm-bn) with totality 0R (index a m + inverse (index a n))
+          p | inl (inl 0<bm-bn) | inl (inl 0<am-an) = SetoidPartialOrder.<Transitive pOrder (ringAddInequalities (<WellDefined (Equivalence.reflexive eq) (*WellDefined (Equivalence.reflexive eq) (greaterZeroImpliesEqualAbs 0<bm-bn)) (ringCanMultiplyByPositive 0<bm-bn (SetoidPartialOrder.<Transitive pOrder (prABound m Na<m) ab<bb))) (<WellDefined (Equivalence.reflexive eq) (*WellDefined (Equivalence.reflexive eq) (greaterZeroImpliesEqualAbs 0<am-an)) (ringCanMultiplyByPositive 0<am-an (SetoidPartialOrder.<Transitive pOrder (prBBound n Nb<n) bb<bb)))) q
+          p | inl (inl 0<bm-bn) | inl (inr am-an<0) = SetoidPartialOrder.<Transitive pOrder (ringAddInequalities (<WellDefined (*WellDefined (Equivalence.reflexive eq) (Equivalence.symmetric eq (greaterZeroImpliesEqualAbs 0<bm-bn))) (Equivalence.reflexive eq) (ringCanMultiplyByPositive (<WellDefined (Equivalence.reflexive eq) (greaterZeroImpliesEqualAbs 0<bm-bn) 0<bm-bn) (SetoidPartialOrder.<Transitive pOrder (prABound m Na<m) ab<bb))) (<WellDefined (*WellDefined (Equivalence.reflexive eq) (lessZeroImpliesEqualNegAbs am-an<0)) (Equivalence.reflexive eq) (ringCanMultiplyByPositive (<WellDefined (Equivalence.reflexive eq) (Equivalence.symmetric eq (lessZeroImpliesEqualNegAbs am-an<0)) (lemm2 _ am-an<0)) (SetoidPartialOrder.<Transitive pOrder (prBBound n Nb<n) bb<bb)))) q
+          p | inl (inl 0<bm-bn) | inr 0=am-an = <WellDefined (+WellDefined (Equivalence.reflexive eq) (*WellDefined (Equivalence.reflexive eq) 0=am-an)) (Equivalence.reflexive eq) (<WellDefined (+WellDefined (Equivalence.reflexive eq) (Equivalence.symmetric eq timesZero)) (Equivalence.reflexive eq) (<WellDefined (Equivalence.symmetric eq identRight) (Equivalence.reflexive eq) (SetoidPartialOrder.<Transitive pOrder (<WellDefined (Equivalence.reflexive eq) (+WellDefined (Equivalence.reflexive eq) (*WellDefined (Equivalence.reflexive eq) (Equivalence.transitive eq (Equivalence.symmetric eq absZeroIsZero) (absWellDefined _ _ 0=am-an)))) (<WellDefined (Equivalence.reflexive eq) (+WellDefined (Equivalence.reflexive eq) (Equivalence.symmetric eq timesZero)) (<WellDefined (Equivalence.reflexive eq) (Equivalence.symmetric eq identRight) (<WellDefined (*WellDefined (Equivalence.reflexive eq) (Equivalence.symmetric eq (greaterZeroImpliesEqualAbs 0<bm-bn))) (Equivalence.reflexive eq) (ringCanMultiplyByPositive (<WellDefined (Equivalence.reflexive eq) (greaterZeroImpliesEqualAbs 0<bm-bn) 0<bm-bn) (SetoidPartialOrder.<Transitive pOrder (prABound m Na<m) ab<bb)))))) q)))
+          p | inl (inr bm-bn<0) = SetoidPartialOrder.<Transitive pOrder ans'' q
+            where
+              bar : ((abs (index a m)) * (inverse (index b m + inverse (index b n)))) < (boundBoth * abs (index b m + inverse (index b n)))
+              bar = <WellDefined (*WellDefined (Equivalence.reflexive eq) (lessZeroImpliesEqualNegAbs bm-bn<0)) (Equivalence.reflexive eq) (ringCanMultiplyByPositive (<WellDefined (Equivalence.reflexive eq) (Equivalence.symmetric eq (lessZeroImpliesEqualNegAbs bm-bn<0)) (lemm2 _ bm-bn<0)) (SetoidPartialOrder.<Transitive pOrder (prABound m Na<m) (ab<bb)))
+              foo : (((abs (index b n)) * (abs (index a m + inverse (index a n)))) < (boundBoth * abs (index a m + inverse (index a n)))) || (((abs (index b n)) * (abs (index a m + inverse (index a n)))) ∼ (boundBoth * abs (index a m + inverse (index a n))))
+              foo with totality 0R (index a m + inverse (index a n))
+              foo | inl (inl 0<am-an) = inl (ringCanMultiplyByPositive 0<am-an (SetoidPartialOrder.<Transitive pOrder (prBBound n Nb<n) bb<bb))
+              foo | inl (inr am-an<0) = inl (ringCanMultiplyByPositive (lemm2 _ am-an<0) (SetoidPartialOrder.<Transitive pOrder (prBBound n Nb<n) bb<bb))
+              foo | inr 0=am-an = inr (Equivalence.transitive eq (*WellDefined (Equivalence.reflexive eq) (Equivalence.symmetric eq 0=am-an)) (Equivalence.transitive eq (Equivalence.transitive eq timesZero (Equivalence.symmetric eq timesZero)) (*WellDefined (Equivalence.reflexive eq) 0=am-an)))
+              ans'' : _
+              ans'' with foo
+              ... | inl pr = ringAddInequalities bar pr
+              ... | inr pr = <WellDefined (Equivalence.reflexive eq) (+WellDefined (Equivalence.reflexive eq) pr) (orderRespectsAddition bar ((abs (index b n)) * (abs (index a m + inverse (index a n)))))
+          p | inr 0=bm-bn = ans''
+            where
+              bar : (boundBoth * abs (index b m + inverse (index b n))) ∼ 0R
+              bar = Equivalence.transitive eq (*WellDefined (Equivalence.reflexive eq) (Equivalence.transitive eq (absWellDefined _ _ (Equivalence.symmetric eq 0=bm-bn)) (absZeroIsZero))) timesZero
+              bar' : (abs (index a m) * (index b m + inverse (index b n))) ∼ 0R
+              bar' = Equivalence.transitive eq (*WellDefined (Equivalence.reflexive eq) (Equivalence.symmetric eq 0=bm-bn)) timesZero
+              foo : (((abs (index b n)) * (abs (index a m + inverse (index a n)))) < (boundBoth * abs (index a m + inverse (index a n)))) || (0R ∼ (index a m + inverse (index a n)))
+              foo with totality 0R (index a m + inverse (index a n))
+              foo | inl (inl 0<am-an) = inl (SetoidPartialOrder.<Transitive pOrder (ringCanMultiplyByPositive 0<am-an (prBBound n Nb<n)) (ringCanMultiplyByPositive 0<am-an bb<bb))
+              foo | inl (inr am-an<0) = inl (SetoidPartialOrder.<Transitive pOrder (ringCanMultiplyByPositive (lemm2 _ am-an<0) (prBBound n Nb<n)) (ringCanMultiplyByPositive (lemm2 _ am-an<0) bb<bb))
+              foo | inr 0=am-an = inr 0=am-an
+              ans'' : _
+              ans'' with foo
+              ... | inl pr = SetoidPartialOrder.<Transitive pOrder (<WellDefined groupIsAbelian groupIsAbelian (<WellDefined (+WellDefined (Equivalence.reflexive eq) (Equivalence.symmetric eq bar')) (+WellDefined (Equivalence.reflexive eq) (Equivalence.symmetric eq bar)) (orderRespectsAddition pr 0R))) q
+              ... | inr pr = <WellDefined (Equivalence.transitive eq (Equivalence.symmetric eq identRight) (+WellDefined (Equivalence.symmetric eq bar') (Equivalence.symmetric eq (Equivalence.transitive eq (*WellDefined (Equivalence.reflexive eq) (Equivalence.transitive eq (absWellDefined _ _ (Equivalence.symmetric eq pr)) absZeroIsZero)) timesZero)))) (Equivalence.reflexive eq) 0<e
+          ans''' : abs ((index a m * index b m) + inverse (index a n * index b n)) < e
+          ans''' with triangleInequality (index a m * (index b m + inverse (index b n))) (index b n * (index a m + inverse (index a n)))
+          ... | inl less = <WellDefined (Equivalence.symmetric eq (absWellDefined ((index a m * index b m) + (inverse (index a n * index b n))) (((index a m) * (index b m + (inverse (index b n)))) + ((index b n) * (index a m + inverse (index a n)))) littleLemma)) (Equivalence.reflexive eq) (SetoidPartialOrder.<Transitive pOrder less (<WellDefined (+WellDefined (Equivalence.symmetric eq (absRespectsTimes (index a m) _)) (Equivalence.symmetric eq (absRespectsTimes (index b n) _))) (Equivalence.reflexive eq) p))
+          ... | inr equal = <WellDefined {_ + _} {abs _} {e} {e} (symmetric (transitive (transitive (absWellDefined ((index a m * index b m) + (inverse (index a n * index b n))) (((index a m) * (index b m + (inverse (index b n)))) + ((index b n) * (index a m + inverse (index a n)))) littleLemma) equal) (+WellDefined (absRespectsTimes (index a m) _) (absRespectsTimes (index b n) _)))) reflexive p
+
 _*C_ : CauchyCompletion → CauchyCompletion → CauchyCompletion
 CauchyCompletion.elts (record { elts = a ; converges = aConv } *C record { elts = b ; converges = bConv }) = apply _*_ a b
-CauchyCompletion.converges (record { elts = a ; converges = aConv } *C record { elts = b ; converges = bConv }) e 0<e with boundModulus record { elts = a ; converges = aConv }
-... | aBound , (Na , prABound) with boundModulus record { elts = b ; converges = bConv }
-... | bBound , (Nb , prBBound) = N , ans
-  where
-    boundBoth : A
-    boundBoth = aBound + bBound
-    abstract
-      ab<bb : aBound < boundBoth
-      ab<bb = <WellDefined identLeft groupIsAbelian (orderRespectsAddition {0R} {bBound} (greaterThanAbsImpliesGreaterThan0 (prBBound (succ Nb) (le 0 refl))) aBound)
-      bb<bb : bBound < boundBoth
-      bb<bb = <WellDefined identLeft (Equivalence.reflexive eq) (orderRespectsAddition {0R} {aBound} (greaterThanAbsImpliesGreaterThan0 (prABound (succ Na) (le 0 refl))) bBound)
-      0<boundBoth : 0R < boundBoth
-      0<boundBoth = <WellDefined identLeft (Equivalence.reflexive eq) (ringAddInequalities (greaterThanAbsImpliesGreaterThan0 (prABound (succ Na) (le 0 refl))) (greaterThanAbsImpliesGreaterThan0 (prBBound (succ Nb) (le 0 refl))))
-    abstract
-      1/boundBoothAndPr : Sg A λ i → i * (aBound + bBound) ∼ 1R
-      1/boundBoothAndPr = allInvertible boundBoth λ pr → irreflexive (<WellDefined (Equivalence.reflexive eq) pr 0<boundBoth)
-      1/boundBooth : A
-      1/boundBooth with 1/boundBoothAndPr
-      ... | a , _ = a
-      1/boundBoothPr : 1/boundBooth * (aBound + bBound) ∼ 1R
-      1/boundBoothPr with 1/boundBoothAndPr
-      ... | _ , pr = pr
-      0<1/boundBooth : 0G < 1/boundBooth
-      0<1/boundBooth = inversePositiveIsPositive 1/boundBoothPr 0<boundBoth
-    abstract
-      miniEAndPr : Sg A (λ i → (i + i) ∼ (e * 1/boundBooth))
-      miniEAndPr = halve charNot2 (e * 1/boundBooth)
-      miniE : A
-      miniE with miniEAndPr
-      ... | a , _ = a
-      miniEPr : (miniE + miniE) ∼ (e * 1/boundBooth)
-      miniEPr with miniEAndPr
-      ... | _ , pr = pr
-      0<miniE : 0R < miniE
-      0<miniE = halvePositive miniE (<WellDefined (Equivalence.reflexive eq) (Equivalence.symmetric eq miniEPr) (orderRespectsMultiplication 0<e 0<1/boundBooth))
-    abstract
-      reallyNAAndPr : Sg ℕ (λ N → {m n : ℕ} → N <N m → N <N n → abs (index a m + inverse (index a n)) < miniE)
-      reallyNAAndPr = aConv miniE 0<miniE
-      reallyNa : ℕ
-      reallyNa with reallyNAAndPr
-      ... | a , _ = a
-      reallyNaPr : {m n : ℕ} → reallyNa <N m → reallyNa <N n → abs (index a m + inverse (index a n)) < miniE
-      reallyNaPr with reallyNAAndPr
-      ... | _ , pr = pr
-      reallyNBAndPr : Sg ℕ (λ N → {m n : ℕ} → N <N m → N <N n → abs (index b m + inverse (index b n)) < miniE)
-      reallyNBAndPr = bConv miniE 0<miniE
-      reallyNb : ℕ
-      reallyNb with reallyNBAndPr
-      ... | a , _ = a
-      reallyNbPr : {m n : ℕ} → reallyNb <N m → reallyNb <N n → abs (index b m + inverse (index b n)) < miniE
-      reallyNbPr with reallyNBAndPr
-      ... | _ , pr = pr
-    N : ℕ
-    N = (Na +N (Nb +N (reallyNa +N reallyNb)))
-    ans : {m : ℕ} {n : ℕ} → N <N m → N <N n → abs (index (apply _*_ a b) m + inverse (index (apply _*_ a b) n)) < e
-    ans {m} {n} N<m N<n rewrite indexAndApply a b _*_ {m} | indexAndApply a b _*_ {n} = ans'''
-      where
-        Na<m : Na <N m
-        Na<m = inequalityShrinkLeft N<m
-        Nb<n : Nb <N n
-        Nb<n = inequalityShrinkLeft (inequalityShrinkRight {Na} N<n)
-        abstract
-          sum : {m n : ℕ} → (reallyNa +N reallyNb) <N m → (reallyNa +N reallyNb) <N n → (boundBoth * ((abs (index b m + inverse (index b n))) + (abs (index a m + inverse (index a n))))) < e
-          sum {m} {n} <m <n = <WellDefined *Commutative (Equivalence.transitive eq (*WellDefined miniEPr (Equivalence.reflexive eq)) (Equivalence.transitive eq (Equivalence.symmetric eq *Associative) (Equivalence.transitive eq (Equivalence.transitive eq (*WellDefined (Equivalence.reflexive eq) 1/boundBoothPr) *Commutative) (identIsIdent)))) (ringCanMultiplyByPositive {c = boundBoth} 0<boundBoth (ringAddInequalities (reallyNbPr {m} {n} (inequalityShrinkRight <m) (inequalityShrinkRight <n)) (reallyNaPr {m} {n} (inequalityShrinkLeft <m) (inequalityShrinkLeft <n))))
-        q : ((boundBoth * (abs (index b m + inverse (index b n)))) + (boundBoth * (abs (index a m + inverse (index a n))))) < e
-        q = <WellDefined *DistributesOver+ (Equivalence.reflexive eq) (sum {m} {n} (inequalityShrinkRight {Nb} (inequalityShrinkRight {Na} N<m)) (inequalityShrinkRight {Nb} (inequalityShrinkRight {Na} N<n)))
-        p : ((abs (index a m) * abs (index b m + inverse (index b n))) + (abs (index b n) * abs (index a m + inverse (index a n)))) < e
-        p with totality 0R (index b m + inverse (index b n))
-        p | inl (inl 0<bm-bn) with totality 0R (index a m + inverse (index a n))
-        p | inl (inl 0<bm-bn) | inl (inl 0<am-an) = SetoidPartialOrder.<Transitive pOrder (ringAddInequalities (<WellDefined (Equivalence.reflexive eq) (*WellDefined (Equivalence.reflexive eq) (greaterZeroImpliesEqualAbs 0<bm-bn)) (ringCanMultiplyByPositive 0<bm-bn (SetoidPartialOrder.<Transitive pOrder (prABound m Na<m) ab<bb))) (<WellDefined (Equivalence.reflexive eq) (*WellDefined (Equivalence.reflexive eq) (greaterZeroImpliesEqualAbs 0<am-an)) (ringCanMultiplyByPositive 0<am-an (SetoidPartialOrder.<Transitive pOrder (prBBound n Nb<n) bb<bb)))) q
-        p | inl (inl 0<bm-bn) | inl (inr am-an<0) = SetoidPartialOrder.<Transitive pOrder (ringAddInequalities (<WellDefined (*WellDefined (Equivalence.reflexive eq) (Equivalence.symmetric eq (greaterZeroImpliesEqualAbs 0<bm-bn))) (Equivalence.reflexive eq) (ringCanMultiplyByPositive (<WellDefined (Equivalence.reflexive eq) (greaterZeroImpliesEqualAbs 0<bm-bn) 0<bm-bn) (SetoidPartialOrder.<Transitive pOrder (prABound m Na<m) ab<bb))) (<WellDefined (*WellDefined (Equivalence.reflexive eq) (lessZeroImpliesEqualNegAbs am-an<0)) (Equivalence.reflexive eq) (ringCanMultiplyByPositive (<WellDefined (Equivalence.reflexive eq) (Equivalence.symmetric eq (lessZeroImpliesEqualNegAbs am-an<0)) (lemm2 _ am-an<0)) (SetoidPartialOrder.<Transitive pOrder (prBBound n Nb<n) bb<bb)))) q
-        p | inl (inl 0<bm-bn) | inr 0=am-an = <WellDefined (+WellDefined (Equivalence.reflexive eq) (*WellDefined (Equivalence.reflexive eq) 0=am-an)) (Equivalence.reflexive eq) (<WellDefined (+WellDefined (Equivalence.reflexive eq) (Equivalence.symmetric eq timesZero)) (Equivalence.reflexive eq) (<WellDefined (Equivalence.symmetric eq identRight) (Equivalence.reflexive eq) (SetoidPartialOrder.<Transitive pOrder (<WellDefined (Equivalence.reflexive eq) (+WellDefined (Equivalence.reflexive eq) (*WellDefined (Equivalence.reflexive eq) (Equivalence.transitive eq (Equivalence.symmetric eq absZeroIsZero) (absWellDefined _ _ 0=am-an)))) (<WellDefined (Equivalence.reflexive eq) (+WellDefined (Equivalence.reflexive eq) (Equivalence.symmetric eq timesZero)) (<WellDefined (Equivalence.reflexive eq) (Equivalence.symmetric eq identRight) (<WellDefined (*WellDefined (Equivalence.reflexive eq) (Equivalence.symmetric eq (greaterZeroImpliesEqualAbs 0<bm-bn))) (Equivalence.reflexive eq) (ringCanMultiplyByPositive (<WellDefined (Equivalence.reflexive eq) (greaterZeroImpliesEqualAbs 0<bm-bn) 0<bm-bn) (SetoidPartialOrder.<Transitive pOrder (prABound m Na<m) ab<bb)))))) q)))
-        p | inl (inr bm-bn<0) = SetoidPartialOrder.<Transitive pOrder ans'' q
-          where
-            bar : ((abs (index a m)) * (inverse (index b m + inverse (index b n)))) < (boundBoth * abs (index b m + inverse (index b n)))
-            bar = <WellDefined (*WellDefined (Equivalence.reflexive eq) (lessZeroImpliesEqualNegAbs bm-bn<0)) (Equivalence.reflexive eq) (ringCanMultiplyByPositive (<WellDefined (Equivalence.reflexive eq) (Equivalence.symmetric eq (lessZeroImpliesEqualNegAbs bm-bn<0)) (lemm2 _ bm-bn<0)) (SetoidPartialOrder.<Transitive pOrder (prABound m Na<m) (ab<bb)))
-            foo : (((abs (index b n)) * (abs (index a m + inverse (index a n)))) < (boundBoth * abs (index a m + inverse (index a n)))) || (((abs (index b n)) * (abs (index a m + inverse (index a n)))) ∼ (boundBoth * abs (index a m + inverse (index a n))))
-            foo with totality 0R (index a m + inverse (index a n))
-            foo | inl (inl 0<am-an) = inl (ringCanMultiplyByPositive 0<am-an (SetoidPartialOrder.<Transitive pOrder (prBBound n Nb<n) bb<bb))
-            foo | inl (inr am-an<0) = inl (ringCanMultiplyByPositive (lemm2 _ am-an<0) (SetoidPartialOrder.<Transitive pOrder (prBBound n Nb<n) bb<bb))
-            foo | inr 0=am-an = inr (Equivalence.transitive eq (*WellDefined (Equivalence.reflexive eq) (Equivalence.symmetric eq 0=am-an)) (Equivalence.transitive eq (Equivalence.transitive eq timesZero (Equivalence.symmetric eq timesZero)) (*WellDefined (Equivalence.reflexive eq) 0=am-an)))
-            ans'' : _
-            ans'' with foo
-            ... | inl pr = ringAddInequalities bar pr
-            ... | inr pr = <WellDefined (Equivalence.reflexive eq) (+WellDefined (Equivalence.reflexive eq) pr) (orderRespectsAddition bar ((abs (index b n)) * (abs (index a m + inverse (index a n)))))
-        p | inr 0=bm-bn = ans''
-          where
-            bar : (boundBoth * abs (index b m + inverse (index b n))) ∼ 0R
-            bar = Equivalence.transitive eq (*WellDefined (Equivalence.reflexive eq) (Equivalence.transitive eq (absWellDefined _ _ (Equivalence.symmetric eq 0=bm-bn)) (absZeroIsZero))) timesZero
-            bar' : (abs (index a m) * (index b m + inverse (index b n))) ∼ 0R
-            bar' = Equivalence.transitive eq (*WellDefined (Equivalence.reflexive eq) (Equivalence.symmetric eq 0=bm-bn)) timesZero
-            foo : (((abs (index b n)) * (abs (index a m + inverse (index a n)))) < (boundBoth * abs (index a m + inverse (index a n)))) || (0R ∼ (index a m + inverse (index a n)))
-            foo with totality 0R (index a m + inverse (index a n))
-            foo | inl (inl 0<am-an) = inl (SetoidPartialOrder.<Transitive pOrder (ringCanMultiplyByPositive 0<am-an (prBBound n Nb<n)) (ringCanMultiplyByPositive 0<am-an bb<bb))
-            foo | inl (inr am-an<0) = inl (SetoidPartialOrder.<Transitive pOrder (ringCanMultiplyByPositive (lemm2 _ am-an<0) (prBBound n Nb<n)) (ringCanMultiplyByPositive (lemm2 _ am-an<0) bb<bb))
-            foo | inr 0=am-an = inr 0=am-an
-            ans'' : _
-            ans'' with foo
-            ... | inl pr = SetoidPartialOrder.<Transitive pOrder (<WellDefined groupIsAbelian groupIsAbelian (<WellDefined (+WellDefined (Equivalence.reflexive eq) (Equivalence.symmetric eq bar')) (+WellDefined (Equivalence.reflexive eq) (Equivalence.symmetric eq bar)) (orderRespectsAddition pr 0R))) q
-            ... | inr pr = <WellDefined (Equivalence.transitive eq (Equivalence.symmetric eq identRight) (+WellDefined (Equivalence.symmetric eq bar') (Equivalence.symmetric eq (Equivalence.transitive eq (*WellDefined (Equivalence.reflexive eq) (Equivalence.transitive eq (absWellDefined _ _ (Equivalence.symmetric eq pr)) absZeroIsZero)) timesZero)))) (Equivalence.reflexive eq) 0<e
-        ans''' : abs ((index a m * index b m) + inverse (index a n * index b n)) < e
-        ans''' with triangleInequality (index a m * (index b m + inverse (index b n))) (index b n * (index a m + inverse (index a n)))
-        ... | inl less = <WellDefined (Equivalence.symmetric eq (absWellDefined ((index a m * index b m) + (inverse (index a n * index b n))) (((index a m) * (index b m + (inverse (index b n)))) + ((index b n) * (index a m + inverse (index a n)))) littleLemma)) (Equivalence.reflexive eq) (SetoidPartialOrder.<Transitive pOrder less (<WellDefined (+WellDefined (Equivalence.symmetric eq (absRespectsTimes (index a m) _)) (Equivalence.symmetric eq (absRespectsTimes (index b n) _))) (Equivalence.reflexive eq) p))
-        ... | inr equal = <WellDefined {_ + _} {abs _} {e} {e} (symmetric (transitive (transitive (absWellDefined ((index a m * index b m) + (inverse (index a n * index b n))) (((index a m) * (index b m + (inverse (index b n)))) + ((index b n) * (index a m + inverse (index a n)))) littleLemma) equal) (+WellDefined (absRespectsTimes (index a m) _) (absRespectsTimes (index b n) _)))) reflexive p
+CauchyCompletion.converges (a *C b) = timesConverges a b
 
 *CCommutative : {a b : CauchyCompletion} → Setoid._∼_ cauchyCompletionSetoid (a *C b) (b *C a)
 *CCommutative {a} {b} ε 0<e = 0 , ans
@@ -159,7 +165,7 @@ CauchyCompletion.converges (record { elts = a ; converges = aConv } *C record {
     ans : {m : ℕ} → 0 <N m → abs (index (apply _+_ (CauchyCompletion.elts (a *C b)) (map inverse (CauchyCompletion.elts (b *C a)))) m) < ε
     ans {m} 0<m rewrite indexAndApply (apply _*_ (CauchyCompletion.elts a) (CauchyCompletion.elts b)) (map inverse (apply _*_ (CauchyCompletion.elts b) (CauchyCompletion.elts a))) _+_ {m} | indexAndApply (CauchyCompletion.elts a) (CauchyCompletion.elts b) _*_ {m} | equalityCommutative (mapAndIndex (apply _*_ (CauchyCompletion.elts b) (CauchyCompletion.elts a)) inverse m) | indexAndApply (CauchyCompletion.elts b) (CauchyCompletion.elts a) _*_ {m} = <WellDefined (Equivalence.symmetric eq (Equivalence.transitive eq (absWellDefined _ _ foo) (identityOfIndiscernablesRight _∼_ (Equivalence.reflexive eq) absZero))) (Equivalence.reflexive eq) 0<e
 
-abstract
+private
 
   multiplicationWellDefinedLeft' : (0!=1 : 0R ∼ 1R → False) (a b c : CauchyCompletion) → Setoid._∼_ cauchyCompletionSetoid a b → Setoid._∼_ cauchyCompletionSetoid (a *C c) (b *C c)
   multiplicationWellDefinedLeft' 0!=1 a b c a=b ε 0<e = N , ans
@@ -229,11 +235,12 @@ abstract
   multiplicationPreservedRight : {a b : A} {c : CauchyCompletion} → (a ∼ b) → Setoid._∼_ cauchyCompletionSetoid (c *C injection a) (c *C injection b)
   multiplicationPreservedRight {a} {b} {c} a=b = Equivalence.transitive (Setoid.eq cauchyCompletionSetoid) {c *C injection a} {injection a *C c} {c *C injection b} (*CCommutative {c} {injection a}) (Equivalence.transitive (Setoid.eq cauchyCompletionSetoid) {injection a *C c} {injection b *C c} {c *C injection b} (multiplicationPreservedLeft {a} {b} {c} a=b) (*CCommutative {injection b} {c}))
 
-  multiplicationPreserved : {a b c d : A} → (a ∼ b) → (c ∼ d) → Setoid._∼_ cauchyCompletionSetoid (injection a *C injection c) (injection b *C injection d)
-  multiplicationPreserved {a} {b} {c} {d} a=b c=d = Equivalence.transitive (Setoid.eq cauchyCompletionSetoid) {injection a *C injection c} {injection a *C injection d} {injection b *C injection d} (multiplicationPreservedRight {c} {d} {injection a} c=d) (multiplicationPreservedLeft {a} {b} {injection d} a=b)
+multiplicationPreserved : {a b c d : A} → (a ∼ b) → (c ∼ d) → Setoid._∼_ cauchyCompletionSetoid (injection a *C injection c) (injection b *C injection d)
+multiplicationPreserved {a} {b} {c} {d} a=b c=d = Equivalence.transitive (Setoid.eq cauchyCompletionSetoid) {injection a *C injection c} {injection a *C injection d} {injection b *C injection d} (multiplicationPreservedRight {c} {d} {injection a} c=d) (multiplicationPreservedLeft {a} {b} {injection d} a=b)
 
+private
   multiplicationWellDefinedRight : (a b c : CauchyCompletion) → Setoid._∼_ cauchyCompletionSetoid b c → Setoid._∼_ cauchyCompletionSetoid (a *C b) (a *C c)
   multiplicationWellDefinedRight a b c b=c = Equivalence.transitive (Setoid.eq cauchyCompletionSetoid) {a *C b} {b *C a} {a *C c} (*CCommutative {a} {b}) (Equivalence.transitive (Setoid.eq cauchyCompletionSetoid) {b *C a} {c *C a} {a *C c} (multiplicationWellDefinedLeft b c a b=c) (*CCommutative {c} {a}))
 
-  multiplicationWellDefined : {a b c d : CauchyCompletion} → Setoid._∼_ cauchyCompletionSetoid a b → Setoid._∼_ cauchyCompletionSetoid c d → Setoid._∼_ cauchyCompletionSetoid (a *C c) (b *C d)
-  multiplicationWellDefined {a} {b} {c} {d} a=b c=d = Equivalence.transitive (Setoid.eq cauchyCompletionSetoid) {a *C c} {a *C d} {b *C d} (multiplicationWellDefinedRight a c d c=d) (multiplicationWellDefinedLeft a b d a=b)
+multiplicationWellDefined : {a b c d : CauchyCompletion} → Setoid._∼_ cauchyCompletionSetoid a b → Setoid._∼_ cauchyCompletionSetoid c d → Setoid._∼_ cauchyCompletionSetoid (a *C c) (b *C d)
+multiplicationWellDefined {a} {b} {c} {d} a=b c=d = Equivalence.transitive (Setoid.eq cauchyCompletionSetoid) {a *C c} {a *C d} {b *C d} (multiplicationWellDefinedRight a c d c=d) (multiplicationWellDefinedLeft a b d a=b)
diff --git a/Fields/CauchyCompletion/PartiallyOrderedRing.agda b/Fields/CauchyCompletion/PartiallyOrderedRing.agda
index 3b56e19..49c9990 100644
--- a/Fields/CauchyCompletion/PartiallyOrderedRing.agda
+++ b/Fields/CauchyCompletion/PartiallyOrderedRing.agda
@@ -12,7 +12,8 @@ open import Fields.Fields
 open import Fields.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
@@ -21,9 +22,19 @@ open import Numbers.Naturals.Order.Lemmas
 open import Semirings.Definition
 open import Groups.Homomorphisms.Definition
 open import Rings.Homomorphisms.Definition
+open import Groups.Lemmas
+open import Orders.Total.Definition
 
 module Fields.CauchyCompletion.PartiallyOrderedRing {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) (F : Field R) where
 
+private
+  lemma3 : {c d : _} {C : Set c} {S : Setoid {c} {d} C} {_+_ : C → C → C} (G : Group S _+_) → ({x y : C} → Setoid._∼_ S (x + y) (y + x)) → (x y z : C) → Setoid._∼_ S (((Group.inverse G x) + y) + (x + z)) (y + z)
+  lemma3 {S = S} {_+_ = _+_} G ab x y z = transitive +Associative (+WellDefined (transitive (ab {inverse x + y}) (transitive +Associative (transitive (+WellDefined invRight reflexive) identLeft))) reflexive)
+    where
+      open Setoid S
+      open Equivalence (Setoid.eq S)
+      open Group G
+
 open Setoid S
 open SetoidTotalOrder (TotallyOrderedRing.total order)
 open SetoidPartialOrder pOrder
@@ -32,6 +43,7 @@ open PartiallyOrderedRing pRing
 open Ring R
 open Group additiveGroup
 open Field F
+
 open import Fields.Lemmas F
 open import Rings.Orders.Partial.Lemmas pRing
 open import Rings.Orders.Total.Lemmas order
@@ -41,9 +53,12 @@ open import Fields.CauchyCompletion.Definition order F
 open import Fields.CauchyCompletion.Addition order F
 open import Fields.CauchyCompletion.Multiplication order F
 open import Fields.CauchyCompletion.Approximation order F
+open import Fields.CauchyCompletion.Group order F
 open import Fields.CauchyCompletion.Ring order F
 open import Fields.CauchyCompletion.Comparison order F
 open import Fields.CauchyCompletion.Setoid order F
+open import Groups.Homomorphisms.Lemmas CInjectionGroupHom
+open import Setoids.Orders.Total.Lemmas (TotallyOrderedRing.total order)
 
 productPositives : (a b : A) → (injection 0R) <Cr a → (injection 0R) <Cr b → (injection 0R) <Cr (a * b)
 productPositives a b record { e = eA ; 0<e = 0<eA ; N = Na ; property = prA } record { e = eB ; 0<e = 0<eB ; N = Nb ; property = prB } = record { e = eA * eB ; 0<e = orderRespectsMultiplication 0<eA 0<eB ; N = Na +N Nb ; property = ans }
@@ -57,21 +72,8 @@ productPositives' a b interA interB 0<iA 0<iB record { e = interA' ; 0<e = 0<int
     ans : (m : ℕ) → (Na +N Nb <N m) → ((interA * interB) + (interA' * interB')) < index (CauchyCompletion.elts (a *C b)) m
     ans m <m rewrite indexAndApply (CauchyCompletion.elts a) (CauchyCompletion.elts b) _*_ {m} = <Transitive (<WellDefined identRight (symmetric *DistributesOver+) (<WellDefined reflexive (+WellDefined *Commutative *Commutative) (<WellDefined reflexive (+WellDefined (symmetric *DistributesOver+) (symmetric *DistributesOver+)) (<WellDefined groupIsAbelian (transitive (transitive groupIsAbelian (transitive (symmetric +Associative) (+WellDefined *Commutative (transitive groupIsAbelian (transitive (+WellDefined reflexive *Commutative) (symmetric +Associative)))))) +Associative) (orderRespectsAddition (<WellDefined identRight reflexive (ringAddInequalities (orderRespectsMultiplication 0<iB 0<interA') (orderRespectsMultiplication 0<interB' 0<iA))) ((interA * interB) + (interA' * interB'))))))) (ringMultiplyPositives (<WellDefined identRight reflexive (ringAddInequalities 0<iA 0<interA')) (<WellDefined identRight reflexive (ringAddInequalities 0<iB 0<interB')) (prA m (inequalityShrinkLeft <m)) (prB m (inequalityShrinkRight <m)))
 
-
--- a  < a'
--- b' < b
--- then:
--- a +C c < a' + c ~ a' + c' < b' + c' ~ b' + c < b +C c
-{-
-Have: a <Cr x r<C b
-
-* Let e = min(distance from a to witness of a<x, distance from x to witness of x<b)
-* Approximate a above to within e/2
-* Approximate b below to within e/2
-* Approximate c (above or below) to within e/2
-
-Then a' + c' is an appropriate witness.
--}
+<COrderRespectsMultiplication : (a b : CauchyCompletion) → (injection 0R <C a) → (injection 0R <C b) → (injection 0R <C (a *C b))
+<COrderRespectsMultiplication a b record { i = interA ; a<i = 0<interA ; i<b = interA<a } record { i = interB ; a<i = 0<interB ; i<b = interB<b } = record { i = interA * interB ; a<i = productPositives interA interB 0<interA 0<interB ; i<b = productPositives' a b interA interB (<CCollapsesL 0R _ 0<interA) (<CCollapsesL 0R _ 0<interB) interA<a interB<b }
 
 cOrderRespectsAdditionLeft' : (a : CauchyCompletion) (b : A) (c : A) → (a <Cr b) → (a +C injection c) <C (injection b +C injection c)
 cOrderRespectsAdditionLeft' a b c record { e = e ; 0<e = 0<e ; N = N ; property = pr } = <CWellDefined {a +C injection c} {a +C injection c} {injection (b + c)} {(injection b) +C (injection c)} (Equivalence.reflexive (Setoid.eq cauchyCompletionSetoid) {a +C injection c}) (GroupHom.groupHom (RingHom.groupHom CInjectionRingHom)) (<CRelaxR (record { e = e ; 0<e = 0<e ; N = N ; property = λ m N<m → <WellDefined (transitive (symmetric +Associative) (+WellDefined reflexive (ans m))) reflexive (orderRespectsAddition (pr m N<m) c) }))
@@ -79,83 +81,156 @@ cOrderRespectsAdditionLeft' a b c record { e = e ; 0<e = 0<e ; N = N ; property
     ans : (m : ℕ) → (index (CauchyCompletion.elts a) m + c) ∼ index (apply _+_ (CauchyCompletion.elts a) (constSequence c)) m
     ans m rewrite indexAndApply (CauchyCompletion.elts a) (constSequence c) _+_ {m} | indexAndConst c m = reflexive
 
-cOrderRespectsAdditionLeft : (a : CauchyCompletion) (b : A) (c : CauchyCompletion) → (a <Cr b) → (a +C c) <C (injection b +C c)
-cOrderRespectsAdditionLeft a b c a<b = {!!}
+private
+  l1 : (a : A) (b : CauchyCompletion) → a r<C b → 0R r<C (b +C injection (inverse a))
+  l1 a b record { e = e ; 0<e = 0<e ; N = N ; pr = pr } = record { e = e ; 0<e = 0<e ; N = N ; pr = λ m N<m → <WellDefined (transitive groupIsAbelian (transitive +Associative (+WellDefined invLeft reflexive))) (ans m) (orderRespectsAddition (pr m N<m) (inverse a)) }
+    where
+      ans : (m : ℕ) → (index (CauchyCompletion.elts b) m + inverse a) ∼ index (CauchyCompletion.elts (b +C injection (inverse a))) m
+      ans m rewrite indexAndApply (CauchyCompletion.elts b) (constSequence (inverse a)) _+_ {m} | indexAndConst (inverse a) m = reflexive
 
-cOrderRespectsAdditionRight : (a : A) (b : CauchyCompletion) (c : CauchyCompletion) → (a r<C b) → (injection a +C c) <C (b +C c)
-cOrderRespectsAdditionRight a b a<b = {!!}
+  l1' : (a : A) (b : CauchyCompletion) → 0R r<C (b +C injection a) → (inverse a) r<C b
+  l1' a b record { e = e ; 0<e = 0<e ; N = N ; pr = pr } = record { e = e ; 0<e = 0<e ; N = N ; pr = λ m N<m → <WellDefined (transitive (+WellDefined identLeft reflexive) groupIsAbelian) (symmetric (ans m)) (orderRespectsAddition (pr m N<m) (inverse a)) }
+    where
+      ans : (m : ℕ) → (index (CauchyCompletion.elts b) m) ∼ (index (CauchyCompletion.elts (b +C injection a)) m + inverse a)
+      ans m rewrite indexAndApply (CauchyCompletion.elts b) (constSequence a) _+_ {m} | indexAndConst a m = transitive (symmetric identRight) (transitive (+WellDefined reflexive (symmetric invRight)) +Associative)
 
-cOrderRespectsAddition : (a b : CauchyCompletion) → (a <C b) → (c : CauchyCompletion) → (a +C c) <C (b +C c)
-cOrderRespectsAddition a b a<b c = {!!}
+  l2 : (a : CauchyCompletion) (b : A) → a <Cr b → 0R r<C (injection b +C (-C a))
+  l2 a b record { e = e ; 0<e = 0<e ; N = N ; property = property } = record { e = e ; 0<e = 0<e ; N = N ; pr = λ m N<m → <WellDefined (transitive (transitive (symmetric +Associative) (+WellDefined reflexive invRight)) groupIsAbelian) (ans m) (orderRespectsAddition (property m N<m) (inverse (index (CauchyCompletion.elts a) m))) }
+    where
+      ans : (m : ℕ) → (b + inverse (index (CauchyCompletion.elts a) m)) ∼ index (CauchyCompletion.elts (injection b +C (-C a))) m
+      ans m rewrite indexAndApply (CauchyCompletion.elts (injection b)) (CauchyCompletion.elts (-C a)) _+_ {m} | indexAndConst b m | mapAndIndex (CauchyCompletion.elts a) inverse m = reflexive
+
+  l2' : (a : CauchyCompletion) (b : A) → 0R r<C (injection b +C a) → (-C a) <Cr b
+  l2' a b record { e = e ; 0<e = 0<e ; N = N ; pr = pr } = record { e = e ; 0<e = 0<e ; N = N ; property = λ m N<m → <WellDefined (+WellDefined identLeft reflexive) (ans m) (orderRespectsAddition (pr m N<m) (index (CauchyCompletion.elts (-C a)) m)) }
+    where
+      ans : (m : ℕ) → (index (CauchyCompletion.elts (injection b +C a)) m + index (map inverse (CauchyCompletion.elts a)) m) ∼ b
+      ans m rewrite indexAndApply (CauchyCompletion.elts (injection b)) (CauchyCompletion.elts a) _+_ {m} | indexAndConst b m | equalityCommutative (mapAndIndex (CauchyCompletion.elts a) inverse m) = transitive (symmetric +Associative) (transitive (+WellDefined reflexive invRight) identRight)
 
 {-
-cOrderRespectsAddition : (a b : CauchyCompletion) → (a <C b) → (c : CauchyCompletion) → (a +C c) <C (b +C c)
-cOrderRespectsAddition a b (r , ((r1 , (0<r1 ,, (N1 , prN1))) ,, (r2 , (0<r2 ,, (N2 , prN2))))) c = (a' + c') , (ans1 ,, ans2)
-  where
-    0<min : 0G < min r1 r2
-    0<min with totality r1 r2
-    0<min | inl (inl r1<r2) = 0<r1
-    0<min | inl (inr r2<r1) = 0<r2
-    0<min | inr r1=r2 = 0<r1
-    e/2All : Sg A (λ i → i + i ∼ min r1 r2)
-    e/2All = halve charNot2 (min r1 r2)
-    e/2 : A
-    e/2 = underlying e/2All
-    prE/2 : e/2 + e/2 ∼ min r1 r2
-    prE/2 with e/2All
-    ... | _ , pr = pr
-    0<e/2 : 0G < e/2
-    0<e/2 = halvePositive e/2 (<WellDefined (Equivalence.reflexive eq) (Equivalence.symmetric eq prE/2) 0<min)
-    a'All : Sg A (λ i → (a <Cr i) && (injection i +C (-C a)) <C (injection e/2))
-    a' : A
-    a<a' : a <Cr a'
-    a'Pr : (injection a' +C (-C a)) <C (injection e/2)
-    b'All : Sg A (λ i → (i r<C b) && (b +C (-C injection i)) <C (injection e/2))
-    b' : A
-    b'<b : b' r<C b
-    b'Pr : (b +C (-C injection b')) <C (injection e/2)
+  minAddLemma : (a b : A) → (0R < a) → (min a b) < (a + b)
+  minAddLemma a b 0<a with totality a b
+  ... | inl (inl x) = <WellDefined identLeft groupIsAbelian (orderRespectsAddition (<Transitive 0<a x) a)
+  ... | inl (inr x) = <WellDefined identLeft reflexive (orderRespectsAddition 0<a b)
+  ... | inr x = <WellDefined identLeft groupIsAbelian (orderRespectsAddition (<WellDefined reflexive x 0<a) a)
 
-    c'All : Sg A (λ i → (c <Cr i) && (injection i +C (-C c)) <C (injection e/2))
-    c' : A
-    c<c' : c <Cr c'
-    c'Pr : (injection c' +C (-C c)) <C (injection e/2)
-
-    -- Now a' + c' is our intervening rational
-    -- and r1 suffices for the witness to a + c < a' + c'
-    -- and r2 suffices for the witness to b' + c' < b + c
-    -- TODO here
-
-    ans1 : (a +C c) <Cr (a' + c')
-    ans1 = r1 , (0<r1 ,, ((N1 +N N2) , ans))
-      where
-        ans : (m : ℕ) → (N1 +N N2) <N m → (r1 + index (CauchyCompletion.elts (a +C c)) m) < (a' + c')
-        ans m N1+N2<m rewrite indexAndApply (CauchyCompletion.elts a) (CauchyCompletion.elts c) _+_ {m} = <WellDefined (Equivalence.symmetric eq +Associative) reflexive (SetoidPartialOrder.<Transitive pOrder (orderRespectsAddition (prN1 m (inequalityShrinkLeft N1+N2<m)) (index (CauchyCompletion.elts c) m)) {!!})
-
-    ans2 : (a' + c') r<C (b +C c)
-    ans2 = r2 , (0<r2 ,, {!!})
-
-    a'All = approximateAbove a e/2 0<e/2
-    a' = underlying a'All
-    a<a' with a'All
-    ... | (_ , (x ,, _)) = x
-    a'Pr with a'All
-    ... | (_ , (_ ,, x)) = x
-    b'All = approximateBelow b e/2 0<e/2
-    b' = underlying b'All
-    b'<b with b'All
-    ... | (_ , (x ,, _)) = x
-    b'Pr with b'All
-    ... | (_ , (_ ,, x)) = x
-    c'All = approximateAbove c e/2 0<e/2
-    c' = underlying c'All
-    c<c' with c'All
-    ... | (_ , (x ,, _)) = x
-    c'Pr with c'All
-    ... | (_ , (_ ,, x)) = x
+  addInequalities : {a b : CauchyCompletion} → 0R r<C a → 0R r<C b → 0R r<C (a +C b)
+  addInequalities {a} {b} record { e = eA ; 0<e = 0<eA ; N = Na ; pr = prA } record { e = eB ; 0<e = 0<eB ; N = Nb ; pr = prB } = record { e = min eA eB ; 0<e = minInequalitiesR 0<eA 0<eB ; N = N ; pr = λ m N<m → <Transitive (<WellDefined (symmetric identLeft) reflexive (minAddLemma eA eB 0<eA)) (<WellDefined (+WellDefined identLeft identLeft) (identityOfIndiscernablesLeft _∼_ reflexive (indexAndApply (CauchyCompletion.elts a) (CauchyCompletion.elts b) _+_ {m})) (ringAddInequalities (prA m (p2 m N<m)) (prB m (p1 m N<m)))) }
+    where
+      N = succ (TotalOrder.max ℕTotalOrder Na Nb)
+      Nb<N : Nb <N succ (TotalOrder.max ℕTotalOrder Na Nb)
+      Nb<N with TotalOrder.totality ℕTotalOrder Na Nb
+      ... | inl (inl x) = le 0 refl
+      ... | inl (inr x) = lessTransitive x (le 0 refl)
+      ... | inr x = le 0 refl
+      Na<N : Na <N succ (TotalOrder.max ℕTotalOrder Na Nb)
+      Na<N with TotalOrder.totality ℕTotalOrder Na Nb
+      ... | inl (inl x) = lessTransitive x (le 0 refl)
+      ... | inl (inr x) = le 0 refl
+      ... | inr x = le 0 (applyEquality succ x)
+      p1 : (m : ℕ) → (N <N m) → Nb <N m
+      p1 m N<m = lessTransitive Nb<N N<m
+      p2 : (m : ℕ) → (N <N m) → Na <N m
+      p2 m N<m = lessTransitive Na<N N<m
 
 -}
 
+  invInj : (i : A) → Setoid._∼_ cauchyCompletionSetoid (injection (inverse i) +C injection i) (injection 0R)
+  invInj i = Equivalence.transitive (Setoid.eq cauchyCompletionSetoid) {record { converges = CauchyCompletion.converges (injection (inverse i) +C injection i) }} {record { converges = CauchyCompletion.converges ((-C (injection i)) +C injection i) }} {record { converges = CauchyCompletion.converges (injection 0R) }} (Group.+WellDefined CGroup {record { converges = CauchyCompletion.converges (injection (inverse i)) }} {record { converges = CauchyCompletion.converges (injection i)}} {record { converges = CauchyCompletion.converges (-C (injection i)) }} {record { converges = CauchyCompletion.converges (injection i) }} homRespectsInverse (Equivalence.reflexive (Setoid.eq cauchyCompletionSetoid) {record { converges = CauchyCompletion.converges (injection i) }})) (Group.invLeft CGroup {injection i})
+
+
+cOrderMove : (a b : A) (c : CauchyCompletion) → (injection a +C c) <Cr b → a r<C (injection b +C (-C c))
+cOrderMove a b c record { e = e ; 0<e = 0<e ; N = N ; property = property } = record { e = e ; 0<e = 0<e ; N = N ; pr = λ m N<m → <WellDefined (transitive (symmetric +Associative) (transitive (+WellDefined reflexive (transitive (+WellDefined (identityOfIndiscernablesRight _∼_ reflexive (indexAndApply (CauchyCompletion.elts (injection a)) (CauchyCompletion.elts c) _+_ {m})) reflexive) (transitive (symmetric +Associative) (transitive (+WellDefined (identityOfIndiscernablesRight _∼_ reflexive (indexAndConst a m)) invRight) identRight)))) groupIsAbelian)) (transitive (+WellDefined (identityOfIndiscernablesLeft _∼_ reflexive (indexAndConst b m)) (identityOfIndiscernablesRight _∼_ reflexive (mapAndIndex (CauchyCompletion.elts c) inverse m))) (identityOfIndiscernablesLeft _∼_ reflexive (indexAndApply (CauchyCompletion.elts (injection b)) (CauchyCompletion.elts (-C c)) _+_ {m}))) (orderRespectsAddition (property m N<m) (inverse (index (CauchyCompletion.elts c) m))) }
+
+cOrderMove' : (a b : A) (c : CauchyCompletion) → (injection a +C (-C c)) <Cr b → a r<C (injection b +C c)
+cOrderMove' a b c pr = r<CWellDefinedRight _ _ _ (Group.+WellDefined CGroup {record { converges = CauchyCompletion.converges (injection b) }} {record { converges = CauchyCompletion.converges (-C (-C c)) }} {record { converges = CauchyCompletion.converges (injection b) }} {record {converges = CauchyCompletion.converges c }} (Equivalence.reflexive (Setoid.eq cauchyCompletionSetoid) {record { converges = CauchyCompletion.converges (injection b) }}) (invTwice CGroup record { converges = CauchyCompletion.converges c })) (cOrderMove a b (-C c) pr)
+
+cOrderMove'' : (a : CauchyCompletion) (b c : A) → (a +C (-C injection b)) <Cr c → a <Cr (b + c)
+cOrderMove'' a b c record { e = e ; 0<e = 0<e ; N = N ; property = property } = record { e = e ; 0<e = 0<e ; N = N ; property = λ m N<m → <WellDefined (transitive (symmetric +Associative) (+WellDefined reflexive (transitive (+WellDefined (identityOfIndiscernablesRight _∼_ reflexive (indexAndApply (CauchyCompletion.elts a) _ _+_ {m})) reflexive) (transitive (transitive (symmetric +Associative) (+WellDefined reflexive (transitive (+WellDefined (transitive (identityOfIndiscernablesLeft _∼_ reflexive (mapAndIndex _ inverse m)) (inverseWellDefined additiveGroup (identityOfIndiscernablesRight _∼_ reflexive (indexAndConst b m)))) reflexive) invLeft))) identRight)))) groupIsAbelian (orderRespectsAddition (property m N<m) b) }
+
+cOrderRespectsAdditionLeft'' : (a b : A) (c : CauchyCompletion) → (a < b) → (injection a +C c) <C (injection b +C c)
+cOrderRespectsAdditionLeft'' a b c a<b with halve charNot2 (b + inverse a)
+... | b-a/2 , prDiff with approximateAbove c b-a/2 (halvePositive' prDiff (moveInequality a<b))
+... | aboveC , (c<aboveC ,, aboveC-C<e) = record { i = a + aboveC ; a<i = <CRelaxR' (SetoidPartialOrder.<WellDefined <COrder (Ring.groupIsAbelian CRing {record { converges = CauchyCompletion.converges c }} {record { converges = CauchyCompletion.converges (injection a) }}) (Equivalence.transitive (Setoid.eq cauchyCompletionSetoid) {record { converges = CauchyCompletion.converges (injection aboveC +C injection a) }} {record { converges = CauchyCompletion.converges (injection a +C injection aboveC) }} {record { converges = CauchyCompletion.converges (injection (a + aboveC)) }} (Ring.groupIsAbelian CRing {record { converges = CauchyCompletion.converges (injection aboveC) }} {record { converges = CauchyCompletion.converges (injection a) }}) (Equivalence.symmetric (Setoid.eq cauchyCompletionSetoid) {record { converges = CauchyCompletion.converges (injection (a + aboveC)) }} {record { converges = CauchyCompletion.converges (injection a +C injection aboveC) }} (GroupHom.groupHom CInjectionGroupHom))) (cOrderRespectsAdditionLeft' c aboveC a c<aboveC)) ; i<b = cOrderMove' _ _ _ (<CRelaxR' (<CTransitive (<CRelaxR t) (<CInherited u))) }
+  where
+    g : ((injection aboveC +C (-C c)) +C injection a) <C ((injection b-a/2) +C (injection a))
+    g = cOrderRespectsAdditionLeft' _ _ a (<CRelaxR' aboveC-C<e)
+    g' : ((injection aboveC +C (-C c)) +C injection a) <C (injection (b-a/2 + a))
+    g' = <CWellDefined (Equivalence.reflexive (Setoid.eq cauchyCompletionSetoid) {record { converges = CauchyCompletion.converges ((injection aboveC +C (-C c)) +C injection a) }}) (Equivalence.symmetric (Setoid.eq cauchyCompletionSetoid) {record { converges = CauchyCompletion.converges (injection (b-a/2 + a)) }} {record { converges = CauchyCompletion.converges (injection b-a/2 +C injection a) }} (GroupHom.groupHom CInjectionGroupHom)) g
+    t : (injection (a + aboveC) +C (-C c)) <Cr (b-a/2 + a)
+    t = <CRelaxR' (<CWellDefined (Equivalence.transitive (Setoid.eq cauchyCompletionSetoid) {record { converges = CauchyCompletion.converges (((injection aboveC) +C (-C c)) +C injection a) }} {record { converges = CauchyCompletion.converges (injection a +C ((injection aboveC) +C (-C c))) }} {record { converges = CauchyCompletion.converges (injection (a + aboveC) +C (-C c)) }} (Ring.groupIsAbelian CRing {record { converges = CauchyCompletion.converges (injection aboveC +C (-C c)) }} {record { converges = CauchyCompletion.converges (injection a) }}) (Equivalence.transitive (Setoid.eq cauchyCompletionSetoid) {record { converges = CauchyCompletion.converges (injection a +C ((injection aboveC) +C (-C c))) }} {record { converges = CauchyCompletion.converges (((injection a) +C (injection aboveC)) +C (-C c)) }} {record { converges = CauchyCompletion.converges ((injection (a + aboveC)) +C (-C c)) }} (Group.+Associative CGroup {record { converges = CauchyCompletion.converges (injection a) }} {record { converges = CauchyCompletion.converges (injection aboveC) }} {record { converges = CauchyCompletion.converges (-C c) }}) (Group.+WellDefined CGroup {record { converges = CauchyCompletion.converges (injection a +C injection aboveC) }} {record { converges = CauchyCompletion.converges (-C c) }} {record { converges = CauchyCompletion.converges (injection (a + aboveC)) }} {record { converges = CauchyCompletion.converges (-C c) }} (Equivalence.symmetric (Setoid.eq cauchyCompletionSetoid) {record { converges = CauchyCompletion.converges (injection (a + aboveC)) }} {record { converges = CauchyCompletion.converges (injection a +C injection aboveC) }} (GroupHom.groupHom CInjectionGroupHom)) (Equivalence.reflexive (Setoid.eq cauchyCompletionSetoid) {record { converges = CauchyCompletion.converges (-C c)}})))) (Equivalence.reflexive (Setoid.eq cauchyCompletionSetoid) {record { converges = CauchyCompletion.converges (injection (b-a/2 + a)) }}) g')
+    lemm : 0R < b-a/2
+    lemm = halvePositive' prDiff (moveInequality a<b)
+    u : (b-a/2 + a) < b
+    u = <WellDefined (transitive (+WellDefined (symmetric prDiff) reflexive) (transitive groupIsAbelian (transitive +Associative (transitive (+WellDefined (transitive (symmetric +Associative) (transitive (+WellDefined reflexive invLeft) identRight)) reflexive) groupIsAbelian)))) (transitive (symmetric +Associative) (transitive (+WellDefined reflexive invLeft) identRight)) (orderRespectsAddition (<WellDefined identLeft (transitive (transitive +Associative (transitive (+WellDefined (transitive groupIsAbelian (+WellDefined reflexive (symmetric (invTwice additiveGroup b-a/2)))) reflexive) (symmetric +Associative))) (+WellDefined reflexive (symmetric (invContravariant additiveGroup)))) (orderRespectsAddition lemm (b + inverse a))) (a + inverse b-a/2))
+
+cOrderRespectsAdditionLeft''Flip : (a b : A) (c : CauchyCompletion) → (a < b) → (c +C injection a) <C (c +C injection b)
+cOrderRespectsAdditionLeft''Flip a b c a<b = <CWellDefined ((Ring.groupIsAbelian CRing {record { converges = CauchyCompletion.converges (injection a) }} {record { converges = CauchyCompletion.converges c }})) (Ring.groupIsAbelian CRing {record { converges = CauchyCompletion.converges (injection b) }} {record { converges = CauchyCompletion.converges c }}) (cOrderRespectsAdditionLeft'' a b c a<b)
+
+cOrderRespectsAdditionLeft''' : (a b : CauchyCompletion) (c : A) → (a <C b) → (a +C injection c) <C (b +C injection c)
+cOrderRespectsAdditionLeft''' a b c record { i = i ; a<i = a<i ; i<b = i<b } = <CTransitive (cOrderRespectsAdditionLeft' a i c a<i) (<CWellDefined (Equivalence.transitive (Setoid.eq cauchyCompletionSetoid) {record { converges = CauchyCompletion.converges (injection (c + i)) }} {record { converges = CauchyCompletion.converges (injection c +C injection i) }} {record { converges = CauchyCompletion.converges (injection i +C injection c) }} (GroupHom.groupHom CInjectionGroupHom) (Ring.groupIsAbelian CRing {record { converges = CauchyCompletion.converges (injection c) }} {record { converges = CauchyCompletion.converges (injection i) }})) (Ring.groupIsAbelian CRing {record { converges = CauchyCompletion.converges (injection c) }} {record { converges = CauchyCompletion.converges b }}) (flip<C' {record { converges = CauchyCompletion.converges (injection (c + i)) }} {record { converges = CauchyCompletion.converges (injection c +C b) }} (<CWellDefined (Equivalence.transitive (Setoid.eq cauchyCompletionSetoid) {record { converges = CauchyCompletion.converges ((-C b) +C injection (inverse c)) }} {record { converges = CauchyCompletion.converges ((-C b) +C (-C (injection c))) }} {record { converges = CauchyCompletion.converges (-C (injection c +C b)) }} (Group.+WellDefined CGroup {record { converges = CauchyCompletion.converges (-C b) }} {record { converges = CauchyCompletion.converges (injection (inverse c)) }} {record { converges = CauchyCompletion.converges (-C b) }} {record { converges = CauchyCompletion.converges (-C (injection c)) }} (Equivalence.reflexive (Setoid.eq cauchyCompletionSetoid) {record { converges = CauchyCompletion.converges (-C b) }}) homRespectsInverse) (Equivalence.symmetric (Setoid.eq cauchyCompletionSetoid) {record { converges = CauchyCompletion.converges (-C (injection c +C b)) }} {record { converges = CauchyCompletion.converges ((-C b) +C (-C (injection c))) }} (invContravariant CGroup {record { converges = CauchyCompletion.converges (injection c) }} {record { converges = CauchyCompletion.converges b }}))) homRespectsInverse' (cOrderRespectsAdditionLeft' (-C b) (inverse i) (inverse c) (flipR<C i<b)))))
+
+{-
+
+Have 0<a, so 0 < a- < a < a+
+Have c < a- + c, by cOrderRespectsAdditionLeft''
+Have a- + c < a+ + c     by cOrderRespectsAdditionLeft''
+  so a- + c < K < a+ + c
+
+
+-}
+
+--cOrderRespectsAdditionLeft'' : (a b : A) (c : CauchyCompletion) → (a < b) → (injection a +C c) <C (injection b +C c)
+--cOrderRespectsAdditionLeft''' : (a b : CauchyCompletion) (c : A) → (a <C b) → (a +C injection c) <C (b +C injection c)
+cOrderRespectsAdditionRightZero : (a : CauchyCompletion) → (0R r<C a) → (c : CauchyCompletion) → c <C (a +C c)
+cOrderRespectsAdditionRightZero a record { e = e ; 0<e = 0<e ; N = N1 ; pr = pr } c with halve charNot2 e
+... | e/2 , prE/2 with halve charNot2 e/2
+... | e/4 , prE/4 with halve charNot2 e/4
+... | e/8 , prE/8 with approximateAbove c e/4 (halvePositive' prE/4 (halvePositive' prE/2 0<e))
+... | c' , (c<c' ,, c'-c<e/4) with approximateBelow c e/8 (halvePositive' prE/8 (halvePositive' prE/4 (halvePositive' prE/2 0<e)))
+... | cB , (record { e = f ; 0<e = 0<f ; N = N2 ; pr = pr2 } ,, c-cB<e/8) = record { i = c' + e/4 ; a<i = <CRelaxR' (<CTransitive (<CRelaxR c<c') (<CInherited (<WellDefined identLeft groupIsAbelian (orderRespectsAddition (halvePositive' prE/4 (halvePositive' prE/2 0<e)) c')))) ; i<b = record { e = e/4 ; 0<e = halvePositive' prE/4 (halvePositive' prE/2 0<e) ; N = N ; pr = λ m N<m → {!!} } }
+  where
+    N = TotalOrder.max ℕTotalOrder N1 N2
+    3e/4<e : ((e/4 + e/4) + e/4) < e
+    3e/4<e = <WellDefined identLeft (transitive (+WellDefined reflexive (+WellDefined prE/4 reflexive)) (transitive groupIsAbelian (transitive (symmetric +Associative) (transitive (+WellDefined reflexive prE/4) prE/2)))) (orderRespectsAddition (halvePositive' prE/4 (halvePositive' prE/2 0<e)) ((e/4 + e/4) + e/4))
+    t : c' r<C (c +C injection e/4)
+    t = r<CWellDefinedRight _ _ _ (Ring.groupIsAbelian CRing {record { converges = CauchyCompletion.converges (injection e/4) }} {record { converges = CauchyCompletion.converges c }}) (cOrderMove' _ _ _ (<CRelaxR' c'-c<e/4))
+    u : (c' + e/2) r<C (c +C injection ((e/4 + e/4) + e/4))
+    u = r<CWellDefinedRight record { converges = CauchyCompletion.converges ((c +C injection e/4) +C (injection e/2))} _ _ {!!} (<CRelaxL' (<CWellDefined (GroupHom.groupHom' CInjectionGroupHom) (Equivalence.reflexive (Setoid.eq cauchyCompletionSetoid) {record { converges = CauchyCompletion.converges ((c +C injection e/4) +C injection e/2) }}) (cOrderRespectsAdditionLeft''' _ _ e/2 (<CRelaxL t))))
+    u' : ((c' + e/4) + e/4) r<C (c +C injection ((e/4 + e/4) + e/4))
+    u' = r<CWellDefinedLeft _ _ _ (transitive (+WellDefined reflexive (symmetric prE/4)) +Associative) u
+    v : (c +C injection ((e/4 + e/4) + e/4)) <Cr (cB + (((e/4 + e/4) + e/4) + e/8))
+    v with cOrderMove'' _ _ _ (<CRelaxR' c-cB<e/8)
+    ... | c<cB+e/8 = <CRelaxR' (<CWellDefined (Equivalence.reflexive (Setoid.eq cauchyCompletionSetoid) {record { converges = CauchyCompletion.converges (c +C injection ((e/4 + e/4) + e/4))}}) (Equivalence.transitive (Setoid.eq cauchyCompletionSetoid) {record { converges = CauchyCompletion.converges ((injection (cB + e/8)) +C (injection ((e/4 + e/4) + e/4))) }} {record { converges = CauchyCompletion.converges (injection ((cB + e/8) + ((e/4 + e/4) + e/4))) }} {record { converges = CauchyCompletion.converges (injection (cB + (((e/4 + e/4) + e/4) + e/8))) }} (GroupHom.groupHom' CInjectionGroupHom) (GroupHom.wellDefined CInjectionGroupHom (transitive (symmetric +Associative) (+WellDefined reflexive groupIsAbelian)))) (cOrderRespectsAdditionLeft''' _ _ ((e/4 + e/4) + e/4) (<CRelaxR c<cB+e/8)))
+    w : (m : ℕ) → (N<m : N <N m) → (c +C injection ((e/4 + e/4) + e/4)) <Cr ((index (CauchyCompletion.elts a) m) + (index (CauchyCompletion.elts c) m))
+    w = {!!}
+    --pr2      : (m₁ : ℕ) → N2 <N m₁ → (f + index (CauchyCompletion.elts c) m₁) < (cB + e/8)
+    req : (m : ℕ) → (N<m : N <N m) → ((c' + e/4) + e/4) < index (apply _+_ (CauchyCompletion.elts a) (CauchyCompletion.elts c)) m
+    req = {!!}
+    {-
+    ans : (m : ℕ) → (N<m : N <N m) → (c' + e/2) r<C (injection (index (CauchyCompletion.elts a) m) +C c)
+    ans m N<m = <CRelaxL' (<CTransitive (<CRelaxL u) (<CTransitive v (w m N<m)))
+    req : (m : ℕ) → (N<m : N <N m) → ((c' + e/4) + e/4) < index (apply _+_ (CauchyCompletion.elts a) (CauchyCompletion.elts c)) m
+    req m N<m with ans m N<m
+    ... | record { e = f ; 0<e = 0<f ; N = M ; pr = pr } with pr m {!!}
+    ... | t rewrite indexAndApply (CauchyCompletion.elts a) (CauchyCompletion.elts c) _+_ {m} | indexAndConst (CauchyCompletion.elts a) m = {!!}
+    -}
+
+cOrderRespectsAdditionLeft : (a : CauchyCompletion) (b : A) (c : CauchyCompletion) → (a <Cr b) → (a +C c) <C (injection b +C c)
+cOrderRespectsAdditionLeft a b c a<b = <CWellDefined {record { converges = CauchyCompletion.converges (a +C c) }}{record { converges = CauchyCompletion.converges (a +C c) }}{record { converges = CauchyCompletion.converges (((-C a) +C injection b) +C (a +C c)) }}{record { converges = CauchyCompletion.converges (injection b +C c) }} (Equivalence.reflexive (Setoid.eq cauchyCompletionSetoid) {record { converges = CauchyCompletion.converges (a +C c)}}) (lemma3 CGroup (λ {x} {y} → Ring.groupIsAbelian CRing {x} {y}) a (injection b) c) (cOrderRespectsAdditionRightZero ((-C a) +C injection b) (r<CWellDefinedLeft (inverse b + b) 0R ((-C a) +C injection b) invLeft (<CRelaxL' (<CWellDefined {record { converges = CauchyCompletion.converges (injection (inverse b) +C injection b) }}{record { converges = CauchyCompletion.converges (injection (inverse b + b)) }}{record { converges = CauchyCompletion.converges ((-C a) +C injection b) }}{record { converges = CauchyCompletion.converges ((-C a) +C injection b) }} (GroupHom.groupHom' CInjectionGroupHom) (Equivalence.reflexive (Setoid.eq cauchyCompletionSetoid) {record { converges = CauchyCompletion.converges ((-C a) +C injection b) }}) (cOrderRespectsAdditionLeft''' _ _ b (<CRelaxL (flip<CR a<b)))))) (a +C c))
+
+private
+  lemma2 : (a b : CauchyCompletion) → Setoid._∼_ cauchyCompletionSetoid (Group.inverse CGroup ((-C a) +C (-C b))) (a +C b)
+  lemma2 a b = tr {record { converges = CauchyCompletion.converges (-C ((-C a) +C (-C b))) }} {record { converges = CauchyCompletion.converges ((-C (-C b)) +C (-C (-C a))) }} {record { converges = CauchyCompletion.converges (a +C b) }} (invContravariant CGroup {record { converges = CauchyCompletion.converges (-C a) }} {record { converges = CauchyCompletion.converges (-C b) }}) (tr {record { converges = CauchyCompletion.converges ((-C (-C b)) +C (-C (-C a))) }} {record { converges = CauchyCompletion.converges (b +C a) }} {record { converges = CauchyCompletion.converges (a +C b) }} (Group.+WellDefined CGroup {record { converges = CauchyCompletion.converges (-C (-C b)) }} {record { converges = CauchyCompletion.converges (-C (-C a)) }} {record { converges = CauchyCompletion.converges b }} {record { converges = CauchyCompletion.converges a }} (invTwice CGroup b) (invTwice CGroup a)) (Ring.groupIsAbelian CRing {record { converges = CauchyCompletion.converges b }} {record { converges = CauchyCompletion.converges a }}))
+    where
+      open Setoid cauchyCompletionSetoid
+      open Equivalence (Setoid.eq cauchyCompletionSetoid) renaming (transitive to tr)
+
+cOrderRespectsAdditionRight : (a : A) (b : CauchyCompletion) (c : CauchyCompletion) → (a r<C b) → (injection a +C c) <C (b +C c)
+cOrderRespectsAdditionRight a b c a<b = <CWellDefined (Equivalence.transitive (Setoid.eq cauchyCompletionSetoid) {record { converges = CauchyCompletion.converges (Group.inverse CGroup (injection (inverse a) +C (-C c))) }} {record { converges = CauchyCompletion.converges (Group.inverse CGroup ((-C injection a) +C (-C c))) }} {record { converges = CauchyCompletion.converges (injection a +C c) }} (inverseWellDefined CGroup {record { converges = CauchyCompletion.converges (injection (inverse a) +C (-C c)) }} {record { converges = CauchyCompletion.converges ((-C (injection a)) +C (-C c)) }} (Group.+WellDefined CGroup {record { converges = CauchyCompletion.converges (injection (inverse a)) }} {record { converges = CauchyCompletion.converges (-C c) }} {record { converges = CauchyCompletion.converges (-C (injection a)) }} {record { converges = CauchyCompletion.converges (-C c) }} homRespectsInverse (Equivalence.reflexive (Setoid.eq cauchyCompletionSetoid) {record { converges = CauchyCompletion.converges (-C c) }}))) (lemma2 (injection a) c)) (lemma2 b c) (flip<C (cOrderRespectsAdditionLeft _ _ (Group.inverse CGroup c) (flipR<C a<b)))
+
+cOrderRespectsAddition : (a b : CauchyCompletion) → (a <C b) → (c : CauchyCompletion) → (a +C c) <C (b +C c)
+cOrderRespectsAddition a b record { i = i ; a<i = a<i ; i<b = i<b } c = SetoidPartialOrder.<Transitive <COrder (cOrderRespectsAdditionLeft a i c a<i) (cOrderRespectsAdditionRight i b c i<b)
+
 CpOrderedRing : PartiallyOrderedRing CRing <COrder
 PartiallyOrderedRing.orderRespectsAddition CpOrderedRing {a} {b} = cOrderRespectsAddition a b
-PartiallyOrderedRing.orderRespectsMultiplication CpOrderedRing {a} {b} record { i = interA ; a<i = 0<interA ; i<b = interA<a } record { i = interB ; a<i = 0<interB ; i<b = interB<b } = record { i = interA * interB ; a<i = productPositives interA interB 0<interA 0<interB ; i<b = productPositives' a b interA interB (<CCollapsesL 0R _ 0<interA) (<CCollapsesL 0R _ 0<interB) interA<a interB<b }
-
+PartiallyOrderedRing.orderRespectsMultiplication CpOrderedRing {a} {b} 0<a 0<b = <COrderRespectsMultiplication a b 0<a 0<b
diff --git a/Fields/CauchyCompletion/Ring.agda b/Fields/CauchyCompletion/Ring.agda
index 2bd9e47..a4b385d 100644
--- a/Fields/CauchyCompletion/Ring.agda
+++ b/Fields/CauchyCompletion/Ring.agda
@@ -9,7 +9,8 @@ open import Groups.Lemmas
 open import Fields.Fields
 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
@@ -27,6 +28,7 @@ open Field F
 open Group (Ring.additiveGroup R)
 
 open import Rings.Orders.Total.Lemmas order
+open import Rings.Orders.Total.AbsoluteValue order
 open import Fields.CauchyCompletion.Definition order F
 open import Fields.CauchyCompletion.Multiplication order F
 open import Fields.CauchyCompletion.Addition order F
@@ -57,7 +59,7 @@ CRing : Ring cauchyCompletionSetoid _+C_ _*C_
 Ring.additiveGroup CRing = CGroup
 Ring.*WellDefined CRing {a} {b} {c} {d} r=t s=u = multiplicationWellDefined {a} {c} {b} {d} r=t s=u
 Ring.1R CRing = injection (Ring.1R R)
-Ring.groupIsAbelian CRing {a} {b} = +CCommutative {a} {b}
+Ring.groupIsAbelian CRing {a} {b} = +CCommutative a b
 Ring.*Associative CRing {a} {b} {c} = c*Assoc {a} {b} {c}
 Ring.*Commutative CRing {a} {b} = *CCommutative {a} {b}
 Ring.*DistributesOver+ CRing {a} {b} {c} = *CDistribute {a} {b} {c}
diff --git a/Fields/CauchyCompletion/Setoid.agda b/Fields/CauchyCompletion/Setoid.agda
index 16aca7a..80d6963 100644
--- a/Fields/CauchyCompletion/Setoid.agda
+++ b/Fields/CauchyCompletion/Setoid.agda
@@ -11,7 +11,8 @@ open import Groups.Lemmas
 open import Fields.Fields
 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
@@ -34,16 +35,18 @@ open import Fields.CauchyCompletion.Definition order F
 open import Fields.CauchyCompletion.Addition order F
 open import Rings.Orders.Partial.Lemmas pRing
 open import Rings.Orders.Total.Lemmas order
+open import Rings.Orders.Total.AbsoluteValue order
 open import Rings.InitialRing R
 open import Fields.Orders.Total.Lemmas {F = F} (record { oRing = order })
 
 isZero : CauchyCompletion → Set (m ⊔ o)
 isZero record { elts = elts ; converges = converges } = ∀ ε → 0R < ε → Sg ℕ (λ N → ∀ {m : ℕ} → (N <N m) → (abs (index elts m)) < ε)
 
-transitiveLemma : {a b c e/2 : A} → abs (a + inverse b) < e/2 → abs (b + inverse c) < e/2 → (abs (a + inverse c)) < (e/2 + e/2)
-transitiveLemma {a} {b} {c} {e/2} a-b<e/2 b-c<e/2 with triangleInequality (a + inverse b) (b + inverse c)
-transitiveLemma {a} {b} {c} {e/2} a-b<e/2 b-c<e/2 | inl x = SetoidPartialOrder.<Transitive pOrder (<WellDefined (absWellDefined _ _ (Equivalence.transitive eq (Equivalence.symmetric eq +Associative) (+WellDefined (Equivalence.reflexive eq) (Equivalence.transitive eq +Associative (Equivalence.transitive eq (+WellDefined invLeft (Equivalence.reflexive eq)) identLeft))))) (Equivalence.reflexive eq) x) (ringAddInequalities a-b<e/2 b-c<e/2)
-transitiveLemma {a} {b} {c} {e/2} a-b<e/2 b-c<e/2 | inr x = <WellDefined (Equivalence.symmetric eq (Equivalence.transitive eq (absWellDefined _ _ (Equivalence.symmetric eq ((Equivalence.transitive eq (Equivalence.symmetric eq +Associative) (+WellDefined (Equivalence.reflexive eq) (Equivalence.transitive eq +Associative (Equivalence.transitive eq (+WellDefined invLeft (Equivalence.reflexive eq)) identLeft))))))) x)) (Equivalence.reflexive eq) (ringAddInequalities a-b<e/2 b-c<e/2)
+private
+  transitiveLemma : {a b c e/2 : A} → abs (a + inverse b) < e/2 → abs (b + inverse c) < e/2 → (abs (a + inverse c)) < (e/2 + e/2)
+  transitiveLemma {a} {b} {c} {e/2} a-b<e/2 b-c<e/2 with triangleInequality (a + inverse b) (b + inverse c)
+  transitiveLemma {a} {b} {c} {e/2} a-b<e/2 b-c<e/2 | inl x = SetoidPartialOrder.<Transitive pOrder (<WellDefined (absWellDefined _ _ (Equivalence.transitive eq (Equivalence.symmetric eq +Associative) (+WellDefined (Equivalence.reflexive eq) (Equivalence.transitive eq +Associative (Equivalence.transitive eq (+WellDefined invLeft (Equivalence.reflexive eq)) identLeft))))) (Equivalence.reflexive eq) x) (ringAddInequalities a-b<e/2 b-c<e/2)
+  transitiveLemma {a} {b} {c} {e/2} a-b<e/2 b-c<e/2 | inr x = <WellDefined (Equivalence.symmetric eq (Equivalence.transitive eq (absWellDefined _ _ (Equivalence.symmetric eq ((Equivalence.transitive eq (Equivalence.symmetric eq +Associative) (+WellDefined (Equivalence.reflexive eq) (Equivalence.transitive eq +Associative (Equivalence.transitive eq (+WellDefined invLeft (Equivalence.reflexive eq)) identLeft))))))) x)) (Equivalence.reflexive eq) (ringAddInequalities a-b<e/2 b-c<e/2)
 
 cauchyCompletionSetoid : Setoid CauchyCompletion
 (cauchyCompletionSetoid Setoid.∼ a) b = isZero (a +C (-C b))
diff --git a/Fields/FieldOfFractions/Archimedean.agda b/Fields/FieldOfFractions/Archimedean.agda
new file mode 100644
index 0000000..6a03198
--- /dev/null
+++ b/Fields/FieldOfFractions/Archimedean.agda
@@ -0,0 +1,68 @@
+{-# OPTIONS --safe --warning=error --without-K #-}
+
+open import Numbers.Naturals.Semiring
+open import Functions
+open import LogicalFormulae
+open import Groups.Definition
+open import Rings.Definition
+open import Rings.IntegralDomains.Definition
+open import Setoids.Setoids
+open import Sets.EquivalenceRelations
+open import Setoids.Orders.Partial.Definition
+open import Setoids.Orders.Total.Definition
+open import Rings.Orders.Partial.Definition
+open import Rings.Orders.Total.Definition
+open import Groups.Orders.Archimedean
+
+module Fields.FieldOfFractions.Archimedean {a b : _} {A : Set a} {S : Setoid {a} {b} A} {_+_ : A → A → A} {_*_ : A → A → A} {R : Ring S _+_ _*_} (I : IntegralDomain R) {c : _} {_<_ : Rel {_} {c} A} {pOrder : SetoidPartialOrder S _<_} {pRing : PartiallyOrderedRing R pOrder} (order : TotallyOrderedRing pRing) (nonempty : (Setoid._∼_ S (Ring.0R R) (Ring.1R R)) → False) where
+
+open import Groups.Cyclic.Definition
+open import Fields.FieldOfFractions.Setoid I
+open import Fields.FieldOfFractions.Group I
+open import Fields.FieldOfFractions.Addition I
+open import Fields.FieldOfFractions.Ring I
+open import Fields.FieldOfFractions.Order I order
+open import Fields.FieldOfFractions.Field I
+open import Fields.Orders.Total.Archimedean
+open import Rings.Orders.Partial.Lemmas pRing
+open import Rings.Orders.Total.Lemmas order
+
+open Setoid S
+open Equivalence eq
+open Ring R
+open Group additiveGroup
+open TotallyOrderedRing order
+open SetoidTotalOrder total
+
+private
+
+  denomPower : (n : ℕ) → fieldOfFractionsSet.denom (positiveEltPower fieldOfFractionsGroup record { num = 1R ; denom = 1R ; denomNonzero = IntegralDomain.nontrivial I } n) ∼ 1R
+  denomPower zero = reflexive
+  denomPower (succ n) = transitive identIsIdent (denomPower n)
+
+  denomPlus : {a : A} .(a!=0 : a ∼ 0R → False) (n1 n2 : A) → Setoid._∼_ fieldOfFractionsSetoid (fieldOfFractionsPlus record { num = n1 ; denom = a ; denomNonzero = a!=0 } record { num = n2 ; denom = a ; denomNonzero = a!=0 }) (record { num = n1 + n2 ; denom = a ; denomNonzero = a!=0 })
+  denomPlus {a} a!=0 n1 n2 = transitive *Commutative (transitive (*WellDefined reflexive (transitive (+WellDefined *Commutative reflexive) (symmetric *DistributesOver+))) *Associative)
+
+  d : (a : fieldOfFractionsSet) → fieldOfFractionsSet.denom a ∼ 0R → False
+  d record { num = num ; denom = denom ; denomNonzero = denomNonzero } bad = exFalso (denomNonzero bad)
+
+  simpPower : (n : ℕ) → Setoid._∼_ fieldOfFractionsSetoid (positiveEltPower fieldOfFractionsGroup record { num = Ring.1R R ; denom = Ring.1R R ; denomNonzero = IntegralDomain.nontrivial I} n) record { num = positiveEltPower (Ring.additiveGroup R) (Ring.1R R) n ; denom = Ring.1R R ; denomNonzero = IntegralDomain.nontrivial I }
+  simpPower zero = Equivalence.reflexive (Setoid.eq fieldOfFractionsSetoid) {Group.0G fieldOfFractionsGroup}
+  simpPower (succ n) = Equivalence.transitive (Setoid.eq fieldOfFractionsSetoid) {record { denomNonzero = d (fieldOfFractionsPlus (record { num = 1R ; denom = 1R ; denomNonzero = λ t → nonempty (symmetric t) }) (positiveEltPower fieldOfFractionsGroup _ n)) }} {record { denomNonzero = λ t → nonempty (symmetric (transitive (symmetric identIsIdent) t)) }} {record { denomNonzero = λ t → nonempty (symmetric t) }} (Group.+WellDefined fieldOfFractionsGroup {record { num = Ring.1R R ; denom = Ring.1R R ; denomNonzero = IntegralDomain.nontrivial I }} {positiveEltPower fieldOfFractionsGroup record { num = Ring.1R R ; denom = Ring.1R R ; denomNonzero = IntegralDomain.nontrivial I } n} {record { num = Ring.1R R ; denom = Ring.1R R ; denomNonzero = IntegralDomain.nontrivial I }} {record { num = positiveEltPower (Ring.additiveGroup R) (Ring.1R R) n ; denom = Ring.1R R ; denomNonzero = IntegralDomain.nontrivial I }} (Equivalence.reflexive (Setoid.eq fieldOfFractionsSetoid) {record { num = Ring.1R R ; denom = Ring.1R R ; denomNonzero = IntegralDomain.nontrivial I}}) (simpPower n)) (transitive (transitive (transitive *Commutative (transitive identIsIdent (+WellDefined identIsIdent identIsIdent))) (symmetric identIsIdent)) (*WellDefined (symmetric identIsIdent) reflexive))
+
+  lemma : (n : ℕ) {num denom : A} .(d!=0 : denom ∼ 0G → False) → (num * denom) < positiveEltPower additiveGroup 1R n → fieldOfFractionsComparison (record { num = num ; denom = denom ; denomNonzero = d!=0}) record { num = positiveEltPower additiveGroup (Ring.1R R) n ; denom = 1R ; denomNonzero = IntegralDomain.nontrivial I }
+  lemma n {num} {denom} d!=0 numdenom<n with totality 0G denom
+  ... | inl (inl 0<denom) with totality 0G 1R
+  ... | inl (inl 0<1) = {!!}
+  ... | inl (inr x) = exFalso (1<0False x)
+  ... | inr x = exFalso (nonempty x)
+  lemma n {num} {denom} d!=0 numdenom<n | inl (inr denom<0) with totality 0G 1R
+  ... | inl (inl 0<1) = {!!}
+  ... | inl (inr 1<0) = exFalso (1<0False 1<0)
+  ... | inr 0=1 = exFalso (nonempty 0=1)
+  lemma n {num} {denom} d!=0 numdenom<n | inr 0=denom = exFalso (d!=0 (symmetric 0=denom))
+
+fieldOfFractionsArchimedean : Archimedean (toGroup R pRing) → ArchimedeanField {F = fieldOfFractions} record { oRing = fieldOfFractionsOrderedRing }
+fieldOfFractionsArchimedean arch (record { num = num ; denom = denom ; denomNonzero = denom!=0 }) with arch (num * denom) {!!} {!!} {!!}
+... | bl = {!!}
+--... | N , pr = N , SetoidPartialOrder.<WellDefined fieldOfFractionsOrder {record { denomNonzero = denom!=0 }} {record { denomNonzero = denom!=0 }} {record { denomNonzero = λ t → nonempty (symmetric t) }} {record { denomNonzero = d (positiveEltPower fieldOfFractionsGroup record { num = 1R ; denom = 1R ; denomNonzero = λ t → nonempty (symmetric t) } N) }} (Equivalence.reflexive (Setoid.eq fieldOfFractionsSetoid) { record { denomNonzero = denom!=0 } }) (Equivalence.symmetric (Setoid.eq fieldOfFractionsSetoid) {record { denomNonzero = λ t → d (positiveEltPower fieldOfFractionsGroup record { num = 1R ; denom = 1R ; denomNonzero = λ t → nonempty (symmetric t) } N) t }} {record { denomNonzero = λ t → nonempty (symmetric t) }} (simpPower N)) (lemma N denom!=0 pr)
diff --git a/Fields/FieldOfFractions/Group.agda b/Fields/FieldOfFractions/Group.agda
index bd3f992..5d755fc 100644
--- a/Fields/FieldOfFractions/Group.agda
+++ b/Fields/FieldOfFractions/Group.agda
@@ -7,7 +7,6 @@ open import Rings.IntegralDomains.Definition
 open import Setoids.Setoids
 open import Sets.EquivalenceRelations
 
-
 module Fields.FieldOfFractions.Group {a b : _} {A : Set a} {S : Setoid {a} {b} A} {_+_ : A → A → A} {_*_ : A → A → A} {R : Ring S _+_ _*_} (I : IntegralDomain R) where
 
 open import Fields.FieldOfFractions.Setoid I
diff --git a/Fields/FieldOfFractions/Order.agda b/Fields/FieldOfFractions/Order.agda
index ea26f56..a649fa9 100644
--- a/Fields/FieldOfFractions/Order.agda
+++ b/Fields/FieldOfFractions/Order.agda
@@ -9,7 +9,8 @@ open import Rings.Orders.Total.Lemmas
 open import Rings.IntegralDomains.Definition
 open import Functions
 open import Setoids.Setoids
-open import Setoids.Orders
+open import Setoids.Orders.Partial.Definition
+open import Setoids.Orders.Total.Definition
 open import Sets.EquivalenceRelations
 
 module Fields.FieldOfFractions.Order {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 _<_} {pRing : PartiallyOrderedRing R pOrder} (I : IntegralDomain R) (order : TotallyOrderedRing pRing) where
@@ -171,12 +172,7 @@ private
     <transitive (record { num = numA ; denom = denomA ; denomNonzero = denomA!=0 }) (record { num = numB ; denom = denomB ; denomNonzero = denomB!=0 }) (record { num = numC ; denom = denomC ; denomNonzero = denomC!=0 }) a<b b<c | inl (inl 0<denomA) with totality (Ring.0R R) denomC
     <transitive (record { num = numA ; denom = denomA ; denomNonzero = denomA!=0 }) (record { num = numB ; denom = denomB ; denomNonzero = denomB!=0 }) (record { num = numC ; denom = denomC ; denomNonzero = denomC!=0 }) a<b b<c | inl (inl 0<denomA) | inl (inl 0<denomC) with totality (Ring.0R R) denomB
     <transitive (record { num = numA ; denom = denomA ; denomNonzero = denomA!=0 }) (record { num = numB ; denom = denomB ; denomNonzero = denomB!=0 }) (record { num = numC ; denom = denomC ; denomNonzero = denomC!=0 }) a<b b<c | inl (inl 0<denomA) | inl (inl 0<denomC) | inl (inl x) with totality (Ring.0R R) denomC
-    <transitive (record { num = numA ; denom = denomA ; denomNonzero = denomA!=0 }) (record { num = numB ; denom = denomB ; denomNonzero = denomB!=0 }) (record { num = numC ; denom = denomC ; denomNonzero = denomC!=0 }) a<b b<c | inl (inl 0<denomA) | inl (inl 0<denomC) | inl (inl 0<denomB) | inl (inl _) = ringCanCancelPositive order 0<denomB p
-      where
-        inter : ((numA * denomB) * denomC) < ((numB * denomA) * denomC)
-        inter = ringCanMultiplyByPositive pRing 0<denomC a<b
-        p : ((numA * denomC) * denomB) < ((numC * denomA) * denomB)
-        p = SetoidPartialOrder.<Transitive pOrder (SetoidPartialOrder.<WellDefined pOrder (transitive (symmetric *Associative) (transitive (*WellDefined reflexive *Commutative) *Associative)) reflexive inter) (SetoidPartialOrder.<WellDefined pOrder (transitive (symmetric *Associative) (transitive (*WellDefined reflexive *Commutative) *Associative)) (transitive (symmetric *Associative) (transitive (*WellDefined reflexive *Commutative) *Associative)) (ringCanMultiplyByPositive pRing 0<denomA b<c))
+    <transitive (record { num = numA ; denom = denomA ; denomNonzero = denomA!=0 }) (record { num = numB ; denom = denomB ; denomNonzero = denomB!=0 }) (record { num = numC ; denom = denomC ; denomNonzero = denomC!=0 }) a<b b<c | inl (inl 0<denomA) | inl (inl 0<denomC) | inl (inl 0<denomB) | inl (inl _) = ringCanCancelPositive order 0<denomB (SetoidPartialOrder.<Transitive pOrder (SetoidPartialOrder.<WellDefined pOrder (transitive (symmetric *Associative) (transitive (*WellDefined reflexive *Commutative) *Associative)) reflexive (ringCanMultiplyByPositive pRing 0<denomC a<b)) (SetoidPartialOrder.<WellDefined pOrder (transitive (symmetric *Associative) (transitive (*WellDefined reflexive *Commutative) *Associative)) (transitive (symmetric *Associative) (transitive (*WellDefined reflexive *Commutative) *Associative)) (ringCanMultiplyByPositive pRing 0<denomA b<c)))
     <transitive (record { num = numA ; denom = denomA ; denomNonzero = denomA!=0 }) (record { num = numB ; denom = denomB ; denomNonzero = denomB!=0 }) (record { num = numC ; denom = denomC ; denomNonzero = denomC!=0 }) a<b b<c | inl (inl 0<denomA) | inl (inl 0<denomC) | inl (inl 0<denomB) | inl (inr denomC<0) = exFalso (SetoidPartialOrder.irreflexive pOrder (SetoidPartialOrder.<Transitive pOrder 0<denomC denomC<0))
     <transitive (record { num = numA ; denom = denomA ; denomNonzero = denomA!=0 }) (record { num = numB ; denom = denomB ; denomNonzero = denomB!=0 }) (record { num = numC ; denom = denomC ; denomNonzero = denomC!=0 }) a<b b<c | inl (inl 0<denomA) | inl (inl 0<denomC) | inl (inl 0<denomB) | inr x = exFalso (denomC!=0 (Equivalence.symmetric (Setoid.eq S) x))
     <transitive (record { num = numA ; denom = denomA ; denomNonzero = denomA!=0 }) (record { num = numB ; denom = denomB ; denomNonzero = denomB!=0 }) (record { num = numC ; denom = denomC ; denomNonzero = denomC!=0 }) a<b b<c | inl (inl 0<denomA) | inl (inl 0<denomC) | inl (inr denomB<0) with totality (Ring.0R R) denomC
@@ -221,17 +217,11 @@ private
     <transitive (record { num = numA ; denom = denomA ; denomNonzero = denomA!=0 }) (record { num = numB ; denom = denomB ; denomNonzero = denomB!=0 }) (record { num = numC ; denom = denomC ; denomNonzero = denomC!=0 }) a<b b<c | inl (inr denomA<0) | inl (inr denomC<0) with totality (Ring.0R R) denomB
     <transitive (record { num = numA ; denom = denomA ; denomNonzero = denomA!=0 }) (record { num = numB ; denom = denomB ; denomNonzero = denomB!=0 }) (record { num = numC ; denom = denomC ; denomNonzero = denomC!=0 }) a<b b<c | inl (inr denomA<0) | inl (inr denomC<0) | inl (inl 0<denomB) with totality (Ring.0R R) denomC
     <transitive (record { num = numA ; denom = denomA ; denomNonzero = denomA!=0 }) (record { num = numB ; denom = denomB ; denomNonzero = denomB!=0 }) (record { num = numC ; denom = denomC ; denomNonzero = denomC!=0 }) a<b b<c | inl (inr denomA<0) | inl (inr denomC<0) | inl (inl 0<denomB) | inl (inl 0<denomC) = exFalso (SetoidPartialOrder.irreflexive pOrder (SetoidPartialOrder.<Transitive pOrder 0<denomC denomC<0))
-    <transitive (record { num = numA ; denom = denomA ; denomNonzero = denomA!=0 }) (record { num = numB ; denom = denomB ; denomNonzero = denomB!=0 }) (record { num = numC ; denom = denomC ; denomNonzero = denomC!=0 }) a<b b<c | inl (inr denomA<0) | inl (inr denomC<0) | inl (inl 0<denomB) | inl (inr _) = ringCanCancelPositive order 0<denomB (SetoidPartialOrder.<Transitive pOrder have (SetoidPartialOrder.<WellDefined pOrder (swapLemma) (swapLemma) (ringCanMultiplyByNegative pRing denomA<0 b<c)))
-      where
-        have : ((numA * denomC) * denomB) < ((numB * denomA) * denomC)
-        have = SetoidPartialOrder.<WellDefined pOrder (swapLemma) reflexive (ringCanMultiplyByNegative pRing denomC<0 a<b)
+    <transitive (record { num = numA ; denom = denomA ; denomNonzero = denomA!=0 }) (record { num = numB ; denom = denomB ; denomNonzero = denomB!=0 }) (record { num = numC ; denom = denomC ; denomNonzero = denomC!=0 }) a<b b<c | inl (inr denomA<0) | inl (inr denomC<0) | inl (inl 0<denomB) | inl (inr _) = ringCanCancelPositive order 0<denomB (SetoidPartialOrder.<Transitive pOrder (SetoidPartialOrder.<WellDefined pOrder swapLemma reflexive (ringCanMultiplyByNegative pRing denomC<0 a<b)) (SetoidPartialOrder.<WellDefined pOrder (swapLemma) (swapLemma) (ringCanMultiplyByNegative pRing denomA<0 b<c)))
     <transitive (record { num = numA ; denom = denomA ; denomNonzero = denomA!=0 }) (record { num = numB ; denom = denomB ; denomNonzero = denomB!=0 }) (record { num = numC ; denom = denomC ; denomNonzero = denomC!=0 }) a<b b<c | inl (inr denomA<0) | inl (inr denomC<0) | inl (inl 0<denomB) | inr x = exFalso (denomC!=0 (Equivalence.symmetric (Setoid.eq S) x))
     <transitive (record { num = numA ; denom = denomA ; denomNonzero = denomA!=0 }) (record { num = numB ; denom = denomB ; denomNonzero = denomB!=0 }) (record { num = numC ; denom = denomC ; denomNonzero = denomC!=0 }) a<b b<c | inl (inr denomA<0) | inl (inr denomC<0) | inl (inr denomB<0) with totality (Ring.0R R) denomC
     <transitive (record { num = numA ; denom = denomA ; denomNonzero = denomA!=0 }) (record { num = numB ; denom = denomB ; denomNonzero = denomB!=0 }) (record { num = numC ; denom = denomC ; denomNonzero = denomC!=0 }) a<b b<c | inl (inr denomA<0) | inl (inr denomC<0) | inl (inr denomB<0) | inl (inl 0<denomC) = exFalso (SetoidPartialOrder.irreflexive pOrder (SetoidPartialOrder.<Transitive pOrder 0<denomC denomC<0))
-    <transitive (record { num = numA ; denom = denomA ; denomNonzero = denomA!=0 }) (record { num = numB ; denom = denomB ; denomNonzero = denomB!=0 }) (record { num = numC ; denom = denomC ; denomNonzero = denomC!=0 }) a<b b<c | inl (inr denomA<0) | inl (inr denomC<0) | inl (inr denomB<0) | inl (inr _) = ringCanCancelNegative order denomB<0 (SetoidPartialOrder.<Transitive pOrder (SetoidPartialOrder.<WellDefined pOrder (swapLemma) (swapLemma) (ringCanMultiplyByNegative pRing denomA<0 b<c)) have)
-      where
-        have : ((numB * denomA) * denomC) < ((numA * denomC) * denomB)
-        have = SetoidPartialOrder.<WellDefined pOrder reflexive (swapLemma) (ringCanMultiplyByNegative pRing denomC<0 a<b)
+    <transitive (record { num = numA ; denom = denomA ; denomNonzero = denomA!=0 }) (record { num = numB ; denom = denomB ; denomNonzero = denomB!=0 }) (record { num = numC ; denom = denomC ; denomNonzero = denomC!=0 }) a<b b<c | inl (inr denomA<0) | inl (inr denomC<0) | inl (inr denomB<0) | inl (inr _) = ringCanCancelNegative order denomB<0 (SetoidPartialOrder.<Transitive pOrder (SetoidPartialOrder.<WellDefined pOrder (swapLemma) (swapLemma) (ringCanMultiplyByNegative pRing denomA<0 b<c)) (SetoidPartialOrder.<WellDefined pOrder reflexive (swapLemma) (ringCanMultiplyByNegative pRing denomC<0 a<b)))
     <transitive (record { num = numA ; denom = denomA ; denomNonzero = denomA!=0 }) (record { num = numB ; denom = denomB ; denomNonzero = denomB!=0 }) (record { num = numC ; denom = denomC ; denomNonzero = denomC!=0 }) a<b b<c | inl (inr denomA<0) | inl (inr denomC<0) | inl (inr denomB<0) | inr x = exFalso (denomC!=0 (Equivalence.symmetric (Setoid.eq S) x))
     <transitive (record { num = numA ; denom = denomA ; denomNonzero = denomA!=0 }) (record { num = numB ; denom = denomB ; denomNonzero = denomB!=0 }) (record { num = numC ; denom = denomC ; denomNonzero = denomC!=0 }) a<b b<c | inl (inr denomA<0) | inl (inr denomC<0) | inr x = exFalso (denomB!=0 (Equivalence.symmetric (Setoid.eq S) x))
     <transitive (record { num = numA ; denom = denomA ; denomNonzero = denomA!=0 }) (record { num = numB ; denom = denomB ; denomNonzero = denomB!=0 }) (record { num = numC ; denom = denomC ; denomNonzero = denomC!=0 }) a<b b<c | inl (inr denomA<0) | inr x = exFalso (denomC!=0 (Equivalence.symmetric (Setoid.eq S) x))
@@ -538,11 +528,15 @@ PartiallyOrderedRing.orderRespectsMultiplication (fieldOfFractionsPOrderedRing)
 fieldOfFractionsOrderedRing : TotallyOrderedRing fieldOfFractionsPOrderedRing
 TotallyOrderedRing.total fieldOfFractionsOrderedRing = fieldOfFractionsTotalOrder
 
+private
+  fieldOfFractionsOrderInherited' : {x y : A} → x < y → fieldOfFractionsComparison (embedIntoFieldOfFractions x) (embedIntoFieldOfFractions y)
+  fieldOfFractionsOrderInherited' {x} {y} x<y with totality 0R 1R
+  fieldOfFractionsOrderInherited' {x} {y} x<y | inl (inl 0<1) with totality 0R 1R
+  fieldOfFractionsOrderInherited' {x} {y} x<y | inl (inl 0<1) | inl (inl _) = SetoidPartialOrder.<WellDefined pOrder (symmetric (transitive *Commutative identIsIdent)) (symmetric (transitive *Commutative identIsIdent)) x<y
+  fieldOfFractionsOrderInherited' {x} {y} x<y | inl (inl 0<1) | inl (inr 1<0) = exFalso (SetoidPartialOrder.irreflexive pOrder (SetoidPartialOrder.<Transitive pOrder 0<1 1<0))
+  fieldOfFractionsOrderInherited' {x} {y} x<y | inl (inl 0<1) | inr 0=1 = exFalso (SetoidPartialOrder.irreflexive pOrder (SetoidPartialOrder.<WellDefined pOrder 0=1 reflexive 0<1))
+  fieldOfFractionsOrderInherited' {x} {y} x<y | inl (inr 1<0) = exFalso (1<0False order 1<0)
+  fieldOfFractionsOrderInherited' {x} {y} x<y | inr 0=1 = exFalso (anyComparisonImpliesNontrivial pRing x<y 0=1)
+
 fieldOfFractionsOrderInherited : {x y : A} → x < y → fieldOfFractionsComparison (embedIntoFieldOfFractions x) (embedIntoFieldOfFractions y)
-fieldOfFractionsOrderInherited {x} {y} x<y with totality 0R 1R
-fieldOfFractionsOrderInherited {x} {y} x<y | inl (inl 0<1) with totality 0R 1R
-fieldOfFractionsOrderInherited {x} {y} x<y | inl (inl 0<1) | inl (inl _) = SetoidPartialOrder.<WellDefined pOrder (symmetric (transitive *Commutative identIsIdent)) (symmetric (transitive *Commutative identIsIdent)) x<y
-fieldOfFractionsOrderInherited {x} {y} x<y | inl (inl 0<1) | inl (inr 1<0) = exFalso (SetoidPartialOrder.irreflexive pOrder (SetoidPartialOrder.<Transitive pOrder 0<1 1<0))
-fieldOfFractionsOrderInherited {x} {y} x<y | inl (inl 0<1) | inr 0=1 = exFalso (SetoidPartialOrder.irreflexive pOrder (SetoidPartialOrder.<WellDefined pOrder 0=1 reflexive 0<1))
-fieldOfFractionsOrderInherited {x} {y} x<y | inl (inr 1<0) = exFalso (1<0False order 1<0)
-fieldOfFractionsOrderInherited {x} {y} x<y | inr 0=1 = exFalso (anyComparisonImpliesNontrivial pRing x<y 0=1)
+fieldOfFractionsOrderInherited {x} {y} x<y = fieldOfFractionsOrderInherited' {x} {y} x<y
diff --git a/Fields/Orders/LeastUpperBounds/Definition.agda b/Fields/Orders/LeastUpperBounds/Definition.agda
index 3f71a9a..44bc393 100644
--- a/Fields/Orders/LeastUpperBounds/Definition.agda
+++ b/Fields/Orders/LeastUpperBounds/Definition.agda
@@ -5,7 +5,7 @@ open import Agda.Primitive using (Level; lzero; lsuc; _⊔_)
 open import LogicalFormulae
 open import Setoids.Subset
 open import Setoids.Setoids
-open import Setoids.Orders
+open import Setoids.Orders.Partial.Definition
 open import Fields.Fields
 open import Rings.Orders.Total.Definition
 open import Rings.Orders.Total.Lemmas
diff --git a/Fields/Orders/LeastUpperBounds/Examples.agda b/Fields/Orders/LeastUpperBounds/Examples.agda
index 3672cb2..618583d 100644
--- a/Fields/Orders/LeastUpperBounds/Examples.agda
+++ b/Fields/Orders/LeastUpperBounds/Examples.agda
@@ -3,7 +3,8 @@
 open import LogicalFormulae
 open import Setoids.Subset
 open import Setoids.Setoids
-open import Setoids.Orders
+open import Setoids.Orders.Partial.Definition
+open import Setoids.Orders.Total.Definition
 open import Sets.EquivalenceRelations
 open import Rings.Orders.Total.Definition
 open import Rings.Orders.Partial.Definition
diff --git a/Fields/Orders/Lemmas.agda b/Fields/Orders/Lemmas.agda
index 15048d6..567a25c 100644
--- a/Fields/Orders/Lemmas.agda
+++ b/Fields/Orders/Lemmas.agda
@@ -8,7 +8,8 @@ open import Rings.Orders.Partial.Definition
 open import Rings.Orders.Total.Definition
 open import Rings.Lemmas
 open import Setoids.Setoids
-open import Setoids.Orders
+open import Setoids.Orders.Partial.Definition
+open import Setoids.Orders.Total.Definition
 open import Rings.IntegralDomains.Definition
 open import Functions
 open import Sets.EquivalenceRelations
@@ -121,6 +122,6 @@ abstract
   IntegralDomain.nontrivial orderedFieldIsIntDom pr = Field.nontrivial F (Equivalence.symmetric (Setoid.eq S) pr)
 
   charZero : (n : ℕ) → (0R ∼ (fromN (succ n))) → False
-  charZero n 0=sn = irreflexive (<WellDefined 0=sn reflexive (fromNPreservesOrder nontrivial (succIsPositive n)))
+  charZero n 0=sn = irreflexive (<WellDefined 0=sn reflexive (fromNPreservesOrder (0<1 (Field.nontrivial F)) (succIsPositive n)))
   charZero' : (n : ℕ) → ((fromN (succ n)) ∼ 0R) → False
   charZero' n pr = charZero n (symmetric pr)
diff --git a/Fields/Orders/Limits/Definition.agda b/Fields/Orders/Limits/Definition.agda
index aeae493..603a631 100644
--- a/Fields/Orders/Limits/Definition.agda
+++ b/Fields/Orders/Limits/Definition.agda
@@ -5,7 +5,8 @@ open import Agda.Primitive using (Level; lzero; lsuc; _⊔_)
 open import LogicalFormulae
 open import Setoids.Subset
 open import Setoids.Setoids
-open import Setoids.Orders
+open import Setoids.Orders.Partial.Definition
+open import Setoids.Orders.Total.Definition
 open import Sets.EquivalenceRelations
 open import Rings.Orders.Total.Definition
 open import Rings.Orders.Partial.Definition
@@ -26,6 +27,7 @@ open TotallyOrderedField oF
 open TotallyOrderedRing oRing
 open PartiallyOrderedRing pRing
 open import Rings.Orders.Total.Lemmas oRing
+open import Rings.Orders.Total.AbsoluteValue oRing
 open import Rings.Orders.Partial.Lemmas pRing
 open SetoidTotalOrder total
 open SetoidPartialOrder pOrder
diff --git a/Fields/Orders/Limits/Lemmas.agda b/Fields/Orders/Limits/Lemmas.agda
index 0dbd6c8..6e2ffd7 100644
--- a/Fields/Orders/Limits/Lemmas.agda
+++ b/Fields/Orders/Limits/Lemmas.agda
@@ -5,7 +5,8 @@ open import Agda.Primitive using (Level; lzero; lsuc; _⊔_)
 open import LogicalFormulae
 open import Setoids.Subset
 open import Setoids.Setoids
-open import Setoids.Orders
+open import Setoids.Orders.Partial.Definition
+open import Setoids.Orders.Total.Definition
 open import Sets.EquivalenceRelations
 open import Rings.Orders.Total.Definition
 open import Rings.Orders.Partial.Definition
@@ -27,6 +28,7 @@ open TotallyOrderedField oF
 open TotallyOrderedRing oRing
 open PartiallyOrderedRing pRing
 open import Rings.Orders.Total.Lemmas oRing
+open import Rings.Orders.Total.AbsoluteValue oRing
 open import Rings.Orders.Partial.Lemmas pRing
 open SetoidTotalOrder total
 open SetoidPartialOrder pOrder
diff --git a/Fields/Orders/Partial/Definition.agda b/Fields/Orders/Partial/Definition.agda
index b8e9410..e4ff8ef 100644
--- a/Fields/Orders/Partial/Definition.agda
+++ b/Fields/Orders/Partial/Definition.agda
@@ -3,7 +3,7 @@
 open import Rings.Definition
 open import Rings.Orders.Partial.Definition
 open import Setoids.Setoids
-open import Setoids.Orders
+open import Setoids.Orders.Partial.Definition
 open import Functions
 open import Fields.Fields
 
diff --git a/Fields/Orders/Total/Archimedean.agda b/Fields/Orders/Total/Archimedean.agda
new file mode 100644
index 0000000..b64cabb
--- /dev/null
+++ b/Fields/Orders/Total/Archimedean.agda
@@ -0,0 +1,65 @@
+{-# OPTIONS --safe --warning=error --without-K #-}
+
+open import Sets.EquivalenceRelations
+open import Numbers.Naturals.Semiring
+open import Numbers.Integers.Definition
+open import Numbers.Naturals.Order
+open import LogicalFormulae
+open import Groups.Definition
+open import Rings.Orders.Partial.Definition
+open import Rings.Orders.Total.Definition
+open import Setoids.Orders.Partial.Definition
+open import Setoids.Orders.Total.Definition
+open import Setoids.Setoids
+open import Functions
+open import Rings.Definition
+open import Fields.Fields
+open import Fields.Orders.Total.Definition
+
+open import Agda.Primitive using (Level; lzero; lsuc; _⊔_)
+
+module Fields.Orders.Total.Archimedean {a b : _} {A : Set a} {S : Setoid {a} {b} A} {_+_ _*_ : A → A → A} {R : Ring S _+_ _*_} {c : _} {_<_ : A → A → Set c} {pOrder : SetoidPartialOrder S _<_} {p : PartiallyOrderedRing R pOrder} {F : Field R} (t : TotallyOrderedField F p) where
+
+open Setoid S
+open Equivalence eq
+open Ring R
+open Group additiveGroup
+open import Groups.Lemmas additiveGroup
+open import Groups.Cyclic.Definition additiveGroup
+open import Groups.Cyclic.DefinitionLemmas additiveGroup
+open import Groups.Orders.Archimedean (toGroup R p)
+open import Rings.Orders.Partial.Lemmas p
+open import Rings.InitialRing R
+open SetoidPartialOrder pOrder
+open PartiallyOrderedRing p
+open SetoidTotalOrder (TotallyOrderedRing.total (TotallyOrderedField.oRing t))
+open import Rings.Orders.Total.Lemmas (TotallyOrderedField.oRing t)
+open Field F
+
+ArchimedeanField : Set (a ⊔ c)
+ArchimedeanField = (x : A) → (0R < x) → Sg ℕ (λ n → x < (fromN n))
+
+private
+  lemma : (r : A) (N : ℕ) → (positiveEltPower 1R N * r) ∼ positiveEltPower r N
+  lemma r zero = timesZero'
+  lemma r (succ n) = transitive *DistributesOver+' (+WellDefined identIsIdent (lemma r n))
+
+  findBound : (r s : A) (N : ℕ) → (1R < r) → (s < positiveEltPower 1R N) → s < positiveEltPower r N
+  findBound r s zero 1<r s<N = s<N
+  findBound r s (succ N) 1<r s<N = <Transitive s<N (<WellDefined (transitive *Commutative identIsIdent) (lemma r (succ N)) (ringCanMultiplyByPositive' (fromNPreservesOrder (0<1 (anyComparisonImpliesNontrivial 1<r)) (succIsPositive N)) 1<r))
+
+archFieldToGrp : ArchimedeanField → Archimedean
+archFieldToGrp a r s 0<r 0<s with totality r s
+archFieldToGrp a r s 0<r 0<s | inl (inr s<r) = 1 , <WellDefined reflexive (symmetric identRight) s<r
+archFieldToGrp a r s 0<r 0<s | inr r=s = 2 , <WellDefined (transitive identLeft r=s) (+WellDefined reflexive (symmetric identRight)) (orderRespectsAddition 0<r r)
+... | inl (inl r<s) with allInvertible r (λ r=0 → irreflexive (<WellDefined reflexive r=0 0<r))
+... | inv , prInv with a inv (reciprocalPositive r inv 0<r (transitive *Commutative prInv))
+... | N , invBound with a s 0<s
+... | Ns , nSBound = (Ns *N N) , <WellDefined reflexive (symmetric (elementPowerMultiplies (nonneg Ns) (nonneg N) r)) (findBound (positiveEltPower r N) s Ns m nSBound)
+  where
+    m : 1R < positiveEltPower r N
+    m = <WellDefined prInv (lemma r N) (ringCanMultiplyByPositive 0<r invBound)
+
+archToArchField : Archimedean → ArchimedeanField
+archToArchField arch a 0<a with arch 1R a (0<1 (anyComparisonImpliesNontrivial 0<a)) 0<a
+... | N , pr = N , pr
diff --git a/Fields/Orders/Total/Definition.agda b/Fields/Orders/Total/Definition.agda
index 7355b62..1e6adf6 100644
--- a/Fields/Orders/Total/Definition.agda
+++ b/Fields/Orders/Total/Definition.agda
@@ -4,7 +4,7 @@ open import Rings.Definition
 open import Rings.Orders.Partial.Definition
 open import Rings.Orders.Total.Definition
 open import Setoids.Setoids
-open import Setoids.Orders
+open import Setoids.Orders.Partial.Definition
 open import Functions
 open import Fields.Fields
 
diff --git a/Fields/Orders/Total/Lemmas.agda b/Fields/Orders/Total/Lemmas.agda
index c3f506f..4fbd9d6 100644
--- a/Fields/Orders/Total/Lemmas.agda
+++ b/Fields/Orders/Total/Lemmas.agda
@@ -4,7 +4,8 @@ open import Rings.Definition
 open import Rings.Orders.Partial.Definition
 open import Rings.Orders.Total.Definition
 open import Setoids.Setoids
-open import Setoids.Orders
+open import Setoids.Orders.Partial.Definition
+open import Setoids.Orders.Total.Definition
 open import Functions
 open import Fields.Fields
 open import Fields.Orders.Total.Definition
@@ -38,7 +39,7 @@ charNotN : (n : ℕ) → fromN (succ n) ∼ 0R → False
 charNotN n pr = irreflexive (<WellDefined reflexive pr t)
   where
     t : 0R < fromN (succ n)
-    t = fromNPreservesOrder (Field.nontrivial F) (succIsPositive n)
+    t = fromNPreservesOrder (0<1 (Field.nontrivial F)) (succIsPositive n)
 
 charNot2 : Setoid._∼_ S ((Ring.1R R) + (Ring.1R R)) (Ring.0R R) → False
 charNot2 pr = charNotN 1 (transitive (transitive +Associative identRight) pr)
diff --git a/Groups/Cyclic/Definition.agda b/Groups/Cyclic/Definition.agda
index 299c040..acae944 100644
--- a/Groups/Cyclic/Definition.agda
+++ b/Groups/Cyclic/Definition.agda
@@ -7,13 +7,13 @@ open import Agda.Primitive using (Level; lzero; lsuc; _⊔_)
 open import Numbers.Naturals.Semiring
 open import Numbers.Integers.Integers
 open import Groups.Definition
-open import Groups.Lemmas
 
 module Groups.Cyclic.Definition {m n : _} {A : Set m} {S : Setoid {m} {n} A} {_·_ : A → A → A} (G : Group S _·_) where
 
 open Setoid S
 open Group G
 open Equivalence eq
+open import Groups.Lemmas G
 
 positiveEltPower : (x : A) (i : ℕ) → A
 positiveEltPower x 0 = Group.0G G
@@ -40,9 +40,10 @@ elementPowerWellDefinedZ' i j i=j {g} = identityOfIndiscernablesRight _∼_ refl
 
 elementPowerWellDefinedG : (g h : A) → (g ∼ h) → {n : ℤ} → (elementPower g n) ∼ (elementPower h n)
 elementPowerWellDefinedG g h g=h {nonneg n} = positiveEltPowerWellDefinedG g h g=h n
-elementPowerWellDefinedG g h g=h {negSucc x} = inverseWellDefined G (+WellDefined g=h (positiveEltPowerWellDefinedG g h g=h x))
+elementPowerWellDefinedG g h g=h {negSucc x} = inverseWellDefined (+WellDefined g=h (positiveEltPowerWellDefinedG g h g=h x))
 
 record CyclicGroup : Set (m ⊔ n) where
   field
     generator : A
     cyclic : {a : A} → (Sg ℤ (λ i → Setoid._∼_ S (elementPower generator i) a))
+
diff --git a/Groups/FreeProduct/Addition.agda b/Groups/FreeProduct/Addition.agda
index 4af3307..f112025 100644
--- a/Groups/FreeProduct/Addition.agda
+++ b/Groups/FreeProduct/Addition.agda
@@ -15,15 +15,18 @@ module Groups.FreeProduct.Addition {i : _} {I : Set i} (decidableIndex : (x y :
 open import Groups.FreeProduct.Definition decidableIndex decidableGroups G
 open import Groups.FreeProduct.Setoid decidableIndex decidableGroups G
 
+prependEqual : (i : I) (g : A i) .(nonZero : (Setoid._∼_ (S i) g (Group.0G (G i))) → False) → (j : I) → i ≡ j → (w : ReducedSequenceBeginningWith j) → ReducedSequence
+prependEqual i g nonzero .i refl (ofEmpty .i h nonZero) with decidableGroups i (_+_ i g h) (Group.0G (G i))
+prependEqual i g nonzero .i refl (ofEmpty .i h nonZero) | inl sumZero = empty
+prependEqual i g nonzero .i refl (ofEmpty .i h nonZero) | inr sumNotZero = nonempty i (ofEmpty i (_+_ i g h) sumNotZero)
+prependEqual i g nonzero .i refl (prependLetter .i h nonZero x x₁) with decidableGroups i (_+_ i g h) (Group.0G (G i))
+prependEqual i g nonzero .i refl (prependLetter .i h nonZero {j} x x₁) | inl sumZero = nonempty j x
+prependEqual i g nonzero .i refl (prependLetter .i h nonZero x pr) | inr sumNotZero = nonempty i (prependLetter i (_+_ i g h) sumNotZero x pr)
+
 prepend : (i : I) → (g : A i) → .(nonZero : (Setoid._∼_ (S i) g (Group.0G (G i))) → False) → ReducedSequence → ReducedSequence
 prepend i g nonzero empty = nonempty i (ofEmpty i g nonzero)
 prepend i g nonzero (nonempty j x) with decidableIndex i j
-prepend i g nonzero (nonempty .i (ofEmpty .i h nonZero)) | inl refl with decidableGroups i (_+_ i g h) (Group.0G (G i))
-prepend i g nonzero (nonempty .i (ofEmpty .i h nonZero)) | inl refl | inl sumZero = empty
-prepend i g nonzero (nonempty .i (ofEmpty .i h nonZero)) | inl refl | inr sumNotZero = nonempty i (ofEmpty i (_+_ i g h) sumNotZero)
-prepend i g nonzero (nonempty .i (prependLetter .i h nonZero x x₁)) | inl refl with decidableGroups i (_+_ i g h) (Group.0G (G i))
-prepend i g nonzero (nonempty .i (prependLetter .i h nonZero {j} x x₁)) | inl refl | inl sumZero = nonempty j x
-prepend i g nonzero (nonempty .i (prependLetter .i h nonZero x pr)) | inl refl | inr sumNotZero = nonempty i (prependLetter i (_+_ i g h) sumNotZero x pr)
+prepend i g nonzero (nonempty j w) | inl i=j = prependEqual i g nonzero j i=j w
 prepend i g nonzero (nonempty j x) | inr i!=j = nonempty i (prependLetter i g nonzero x i!=j)
 
 plus' : {j : _} → (ReducedSequenceBeginningWith j) → ReducedSequence → ReducedSequence
diff --git a/Groups/Homomorphisms/Definition.agda b/Groups/Homomorphisms/Definition.agda
index 16043ac..f908076 100644
--- a/Groups/Homomorphisms/Definition.agda
+++ b/Groups/Homomorphisms/Definition.agda
@@ -1,5 +1,6 @@
 {-# OPTIONS --safe --warning=error --without-K #-}
 
+open import Sets.EquivalenceRelations
 open import Setoids.Setoids
 open import Agda.Primitive using (Level; lzero; lsuc; _⊔_)
 open import Groups.Definition
@@ -12,6 +13,8 @@ record GroupHom {m n o p : _} {A : Set m} {S : Setoid {m} {o} A} {_·A_ : A →
   field
     groupHom : {x y : A} → f (x ·A y) ∼ (f x) ·B (f y)
     wellDefined : {x y : A} → Setoid._∼_ S x y → f x ∼ f y
+  groupHom' : {x y : A} → (f x) ·B (f y) ∼ f (x ·A y)
+  groupHom' = Equivalence.symmetric eq groupHom
 
 record InjectiveGroupHom {m n o p : _} {A : Set m} {S : Setoid {m} {o} A} {_·A_ : A → A → A} {B : Set n} {T : Setoid {n} {p} B} {_·B_ : B → B → B} {G : Group S _·A_} {H : Group T _·B_} {underf : A → B} (f : GroupHom G H underf) : Set (m ⊔ n ⊔ o ⊔ p) where
   open Setoid S renaming (_∼_ to _∼A_)
diff --git a/Groups/Homomorphisms/Lemmas.agda b/Groups/Homomorphisms/Lemmas.agda
index a897dd7..c2b01ad 100644
--- a/Groups/Homomorphisms/Lemmas.agda
+++ b/Groups/Homomorphisms/Lemmas.agda
@@ -42,3 +42,9 @@ zeroImpliesInverseZero {x} fx=0 = transitive homRespectsInverse (transitive (inv
   where
     open Setoid T
     open Equivalence eq
+
+homRespectsInverse' : {a b : A} → Setoid._∼_ T (f (Group.inverse G a) +B f (Group.inverse G b)) (Group.inverse H (f (b +A a)))
+homRespectsInverse' {a} {b} = transitive (symmetric (GroupHom.groupHom hom)) (transitive (GroupHom.wellDefined hom (Equivalence.symmetric (Setoid.eq S) (invContravariant G))) (homRespectsInverse))
+  where
+    open Setoid T
+    open Equivalence eq
diff --git a/Groups/Homomorphisms/Lemmas2.agda b/Groups/Homomorphisms/Lemmas2.agda
index eca07ec..3092d55 100644
--- a/Groups/Homomorphisms/Lemmas2.agda
+++ b/Groups/Homomorphisms/Lemmas2.agda
@@ -5,7 +5,6 @@ open import Groups.Groups
 open import Groups.Homomorphisms.Definition
 open import Groups.Definition
 open import Numbers.Naturals.Naturals
-open import Setoids.Orders
 open import Setoids.Setoids
 open import Functions
 open import Sets.EquivalenceRelations
diff --git a/Groups/Orders/Archimedean.agda b/Groups/Orders/Archimedean.agda
new file mode 100644
index 0000000..475fe12
--- /dev/null
+++ b/Groups/Orders/Archimedean.agda
@@ -0,0 +1,20 @@
+{-# OPTIONS --safe --warning=error --without-K #-}
+
+open import Numbers.Naturals.Definition
+open import LogicalFormulae
+open import Groups.Definition
+open import Groups.Orders.Partial.Definition
+open import Setoids.Orders.Partial.Definition
+open import Setoids.Setoids
+open import Functions
+
+open import Agda.Primitive using (Level; lzero; lsuc; _⊔_)
+
+module Groups.Orders.Archimedean {a b : _} {A : Set a} {S : Setoid {a} {b} A} {_+_ : A → A → A} {G : Group S _+_} {c : _} {_<_ : A → A → Set c} {pOrder : SetoidPartialOrder S _<_} (p : PartiallyOrderedGroup G pOrder) where
+
+open Setoid S
+open import Groups.Cyclic.Definition G
+open Group G
+
+Archimedean : Set (a ⊔ c)
+Archimedean = (x y : A) → (0G < x) → (0G < y) → Sg ℕ (λ n → y < (positiveEltPower x n))
diff --git a/Groups/Orders/Partial/Definition.agda b/Groups/Orders/Partial/Definition.agda
new file mode 100644
index 0000000..c1906a4
--- /dev/null
+++ b/Groups/Orders/Partial/Definition.agda
@@ -0,0 +1,17 @@
+{-# OPTIONS --safe --warning=error --without-K #-}
+
+open import Groups.Definition
+open import Setoids.Orders.Partial.Definition
+open import Setoids.Setoids
+open import Functions
+
+open import Agda.Primitive using (Level; lzero; lsuc; _⊔_)
+
+module Groups.Orders.Partial.Definition {n m : _} {A : Set n} {S : Setoid {n} {m} A} {_+_ : A → A → A} (G : Group S _+_) where
+
+open Group G
+open Setoid S
+
+record PartiallyOrderedGroup {p : _} {_<_ : Rel {_} {p} A} (pOrder : SetoidPartialOrder S _<_) : Set (lsuc n ⊔ m ⊔ p) where
+  field
+    orderRespectsAddition : {a b : A} → (a < b) → (c : A) → (a + c) < (b + c)
diff --git a/Lists/Definition.agda b/Lists/Definition.agda
index e969767..ad98d87 100644
--- a/Lists/Definition.agda
+++ b/Lists/Definition.agda
@@ -1,16 +1,17 @@
 {-# OPTIONS --safe --warning=error --without-K #-}
 
 
-module Lists.Definition {a : _} where
+module Lists.Definition where
 
-data List (A : Set a) : Set a where
+data List {a : _} (A : Set a) : Set a where
   [] : List A
   _::_ : (x : A) (xs : List A) → List A
 infixr 10 _::_
+{-# BUILTIN LIST List #-}
 
-[_] : {A : Set a} → (a : A) → List A
+[_] : {a : _} {A : Set a} → (a : A) → List A
 [ a ] = a :: []
 
-_++_ : {A : Set a} → List A → List A → List A
+_++_ : {a : _} {A : Set a} → List A → List A → List A
 [] ++ m = m
 (x :: l) ++ m = x :: (l ++ m)
diff --git a/LogicalFormulae.agda b/LogicalFormulae.agda
index 5a70556..8d88367 100644
--- a/LogicalFormulae.agda
+++ b/LogicalFormulae.agda
@@ -14,6 +14,7 @@ data False : Set where
 data False' {a : _} : Set a where
 
 record True : Set where
+{-# BUILTIN UNIT True #-}
 record True' {a : _} : Set a where
 
 infix 10 _||_
@@ -24,6 +25,9 @@ data _||_ {a b} (A : Set a) (B : Set b) : Set (a ⊔ b) where
 data Bool : Set where
   BoolTrue : Bool
   BoolFalse : Bool
+{-# BUILTIN BOOL  Bool  #-}
+{-# BUILTIN TRUE  BoolTrue  #-}
+{-# BUILTIN FALSE BoolFalse #-}
 
 infix 15 _&&_
 record _&&_ {a b} (A : Set a) (B : Set b) : Set (a ⊔ b) where
diff --git a/Numbers/ClassicalReals/Examples.agda b/Numbers/ClassicalReals/Examples.agda
index 8817c68..294c41d 100644
--- a/Numbers/ClassicalReals/Examples.agda
+++ b/Numbers/ClassicalReals/Examples.agda
@@ -4,7 +4,6 @@ open import Numbers.ClassicalReals.RealField
 open import LogicalFormulae
 open import Setoids.Subset
 open import Setoids.Setoids
-open import Setoids.Orders
 open import Sets.EquivalenceRelations
 open import Rings.Orders.Total.Definition
 open import Rings.Orders.Partial.Definition
@@ -14,6 +13,8 @@ open import Groups.Definition
 open import Numbers.Naturals.Semiring
 open import Numbers.Naturals.Order
 open import Numbers.Naturals.Order.Lemmas
+open import Setoids.Orders.Partial.Definition
+open import Setoids.Orders.Total.Definition
 
 module Numbers.ClassicalReals.Examples (ℝ : RealField) where
 
@@ -45,7 +46,7 @@ squarePositive a | inr 0=a = inr (symmetric 0=a)
 
 fraction<1 : (n m : ℕ) → n <N m → (0<m : fromN m ∼ 0G → False) → ((fromN n) * underlying (allInvertible (fromN m) 0<m)) < 1R
 fraction<1 n m n<m 0<m with allInvertible (fromN m) 0<m
-... | 1/m , prM = <WellDefined reflexive (transitive *Commutative prM) (ringCanMultiplyByPositive (inversePositiveIsPositive prM (fromNPreservesOrder nontrivial {0} {m} (zeroLeast n<m))) (fromNPreservesOrder nontrivial n<m))
+... | 1/m , prM = <WellDefined reflexive (transitive *Commutative prM) (ringCanMultiplyByPositive (inversePositiveIsPositive prM (fromNPreservesOrder (0<1 nontrivial) {0} {m} (zeroLeast n<m))) (fromNPreservesOrder (0<1 nontrivial) n<m))
 
 1/2 : A
 1/2 = underlying (allInvertible (1R + 1R) (orderedImpliesCharNot2 nontrivial))
@@ -64,13 +65,13 @@ pr1/2' = transitive (symmetric (transitive *DistributesOver+ (+WellDefined (tran
 3/2<2 = <WellDefined (transitive (+WellDefined reflexive (transitive (symmetric pr1/2) (transitive *Commutative (*WellDefined (+WellDefined reflexive (symmetric identRight)) reflexive)))) (symmetric *DistributesOver+')) reflexive (orderRespectsAddition (<WellDefined (symmetric identIsIdent) reflexive 1/2<1) 1R)
 
 2<9/4 : (1R + 1R) < ((fromN 3 * 1/2) * (fromN 3 * 1/2))
-2<9/4 = <WellDefined reflexive (symmetric *Associative) (halveInequality _ _ 1/2 pr1/2' (<WellDefined reflexive (transitive (symmetric *Associative) *Commutative) (halveInequality _ _ 1/2 pr1/2' (<WellDefined (transitive (fromNPreserves* 4 2) (transitive (*WellDefined (+WellDefined reflexive (+WellDefined reflexive (+WellDefined reflexive identRight))) (+WellDefined reflexive identRight)) (transitive *DistributesOver+ (+WellDefined (transitive *Commutative (transitive identIsIdent +Associative)) (transitive *Commutative (transitive identIsIdent +Associative)))))) (fromNPreserves* 3 3) (fromNPreservesOrder nontrivial (le {8} 0 refl))))))
+2<9/4 = <WellDefined reflexive (symmetric *Associative) (halveInequality _ _ 1/2 pr1/2' (<WellDefined reflexive (transitive (symmetric *Associative) *Commutative) (halveInequality _ _ 1/2 pr1/2' (<WellDefined (transitive (fromNPreserves* 4 2) (transitive (*WellDefined (+WellDefined reflexive (+WellDefined reflexive (+WellDefined reflexive identRight))) (+WellDefined reflexive identRight)) (transitive *DistributesOver+ (+WellDefined (transitive *Commutative (transitive identIsIdent +Associative)) (transitive *Commutative (transitive identIsIdent +Associative)))))) (fromNPreserves* 3 3) (fromNPreservesOrder (0<1 nontrivial) (le {8} 0 refl))))))
 
 0<2 : 0R < (1R + 1R)
 0<2 = <WellDefined identLeft reflexive (ringAddInequalities (0<1 nontrivial) (0<1 nontrivial))
 
 0<3/2 : 0R < (fromN 3 * 1/2)
-0<3/2 = orderRespectsMultiplication (fromNPreservesOrder nontrivial (le {0} {3} 2 refl)) (inversePositiveIsPositive pr1/2 0<2)
+0<3/2 = orderRespectsMultiplication (fromNPreservesOrder (0<1 nontrivial) (le {0} {3} 2 refl)) (inversePositiveIsPositive pr1/2 0<2)
 
 3/2ub : {r : A} → (r * r) < (1R + 1R) → (r < (fromN 3 * 1/2))
 3/2ub {r} r^2<2 with totality r (fromN 3 * 1/2)
@@ -80,10 +81,10 @@ pr1/2' = transitive (symmetric (transitive *DistributesOver+ (+WellDefined (tran
 
 2/6<1 : ((1R + 1R) * underlying (allInvertible (fromN 6) (charZero' 5))) < 1R
 2/6<1 with allInvertible (fromN 6) (charZero' 5)
-2/6<1 | 1/6 , pr1/6 = <WellDefined reflexive (transitive *Commutative pr1/6) (ringCanMultiplyByPositive (inversePositiveIsPositive pr1/6 (fromNPreservesOrder nontrivial {0} {6} (le 5 refl))) (<WellDefined (+WellDefined reflexive identRight) reflexive (fromNPreservesOrder nontrivial {2} {6} (le 3 refl))))
+2/6<1 | 1/6 , pr1/6 = <WellDefined reflexive (transitive *Commutative pr1/6) (ringCanMultiplyByPositive (inversePositiveIsPositive pr1/6 (fromNPreservesOrder (0<1 nontrivial) {0} {6} (le 5 refl))) (<WellDefined (+WellDefined reflexive identRight) reflexive (fromNPreservesOrder (0<1 nontrivial) {2} {6} (le 3 refl))))
 
 2<2*2 : (1R + 1R) < ((1R + 1R) * (1R + 1R))
-2<2*2 = (<WellDefined (+WellDefined reflexive identRight) (transitive (fromNPreserves* 2 2) (*WellDefined (+WellDefined reflexive identRight) (+WellDefined reflexive identRight))) (fromNPreservesOrder nontrivial {2} {4} (le 1 refl)))
+2<2*2 = (<WellDefined (+WellDefined reflexive identRight) (transitive (fromNPreserves* 2 2) (*WellDefined (+WellDefined reflexive identRight) (+WellDefined reflexive identRight))) (fromNPreservesOrder (0<1 nontrivial) {2} {4} (le 1 refl)))
 
 square<2Means<2 : (u : A) → (u * u) < (1R + 1R) → u < (1R + 1R)
 square<2Means<2 u u^2<2 with totality u (1R + 1R)
@@ -137,7 +138,7 @@ sqrt2 = sqrt2' , sqrt2IsSqrt2
           crudeBound : isInInterval (sqrt2' * sqrt2') record { minBound = 0R ; maxBound = 1R + 1R }
           crudeBound with squarePositive sqrt2'
           crudeBound | inl (_ ,, snd) = snd ,, sup^2<2
-          crudeBound | inr sqrt2=0 with upperBound 1R (<WellDefined (transitive identRight (symmetric identIsIdent)) (+WellDefined reflexive identRight) (fromNPreservesOrder nontrivial {1} {2} (le 0 refl)))
+          crudeBound | inr sqrt2=0 with upperBound 1R (<WellDefined (transitive identRight (symmetric identIsIdent)) (+WellDefined reflexive identRight) (fromNPreservesOrder (0<1 nontrivial) {1} {2} (le 0 refl)))
           crudeBound | inr sqrt2=0 | inl 1<sqrt2 = exFalso (irreflexive (<Transitive 1<sqrt2 (<WellDefined (symmetric sqrt2=0) reflexive (0<1 nontrivial))))
           crudeBound | inr sqrt2=0 | inr 1=sqrt2 = exFalso (nontrivial (transitive (symmetric sqrt2=0) (symmetric 1=sqrt2)))
           crudeBound' : isInInterval (inverse (sqrt2' * sqrt2')) record { minBound = inverse (1R + 1R) ; maxBound = inverse 0R }
@@ -147,7 +148,7 @@ sqrt2 = sqrt2' , sqrt2IsSqrt2
           numeratorBound' : isInInterval (fromN 2 + inverse (sqrt2' * sqrt2')) record { minBound = 0R ; maxBound = 1R + 1R }
           numeratorBound' = intervalWellDefined' groupIsAbelian numeratorBound
           tBound : isInInterval t record { minBound = 0R ; maxBound = (1R + 1R) * (underlying (allInvertible (fromN 6) (charZero' 5))) }
-          tBound = intervalWellDefined (transitive *Commutative timesZero ,, reflexive) (intervalConstantProductContains (inversePositiveIsPositive pr' (fromNPreservesOrder nontrivial {0} {6} (le 5 refl))) numeratorBound')
+          tBound = intervalWellDefined (transitive *Commutative timesZero ,, reflexive) (intervalConstantProductContains (inversePositiveIsPositive pr' (fromNPreservesOrder (0<1 nontrivial) {0} {6} (le 5 refl))) numeratorBound')
           u : ((((fromN 5) * ((fromN 2) + inverse (sqrt2' * sqrt2'))) * underlying (allInvertible (fromN 6) (charZero' 5))) + (sqrt2' * sqrt2')) < ((fromN 2 + inverse (sqrt2' * sqrt2')) + (sqrt2' * sqrt2'))
           u = orderRespectsAddition (<WellDefined (transitive (symmetric *Associative) (transitive (*WellDefined reflexive *Commutative) *Associative)) identIsIdent (ringCanMultiplyByPositive {(fromN 5) * underlying (allInvertible (fromN 6) (charZero' 5))} {1R} {fromN 2 + inverse (sqrt2' * sqrt2')} (moveInequality (<WellDefined reflexive (transitive (symmetric identRight) (symmetric +Associative)) sup^2<2)) (fraction<1 5 6 (le zero refl) λ pr → charZero' 5 pr))) (sqrt2' * sqrt2')
           tBound' : (((fromN 5) * t) + (sqrt2' * sqrt2')) < (1R + 1R)
@@ -171,9 +172,9 @@ sqrt2 = sqrt2' , sqrt2IsSqrt2
       where
         abstract
           1<sqrt2 : 1R < sqrt2'
-          1<sqrt2 with upperBound 1R (<WellDefined (transitive identRight (symmetric identIsIdent)) (+WellDefined reflexive identRight) (fromNPreservesOrder nontrivial {1} {2} (le zero refl)))
+          1<sqrt2 with upperBound 1R (<WellDefined (transitive identRight (symmetric identIsIdent)) (+WellDefined reflexive identRight) (fromNPreservesOrder (0<1 nontrivial) {1} {2} (le zero refl)))
           1<sqrt2 | inl x = x
-          1<sqrt2 | inr x = exFalso (irreflexive (<Transitive 2<sup^2 (<WellDefined (transitive identRight (transitive (symmetric identIsIdent) (*WellDefined x x))) (+WellDefined reflexive identRight) (fromNPreservesOrder nontrivial {1} {2} (le 0 refl)))))
+          1<sqrt2 | inr x = exFalso (irreflexive (<Transitive 2<sup^2 (<WellDefined (transitive identRight (transitive (symmetric identIsIdent) (*WellDefined x x))) (+WellDefined reflexive identRight) (fromNPreservesOrder (0<1 nontrivial) {1} {2} (le 0 refl)))))
           0<sqrt2 : 0R < sqrt2'
           0<sqrt2 = <Transitive (0<1 nontrivial) 1<sqrt2
           sqrt2<2 : sqrt2' < (1R + 1R)
@@ -189,10 +190,10 @@ sqrt2 = sqrt2' , sqrt2IsSqrt2
           ... | 1/4 , pr1/4 = transitive (transitive (transitive (transitive (symmetric ringMinusExtracts) (transitive (transitive (*WellDefined reflexive (transitive (symmetric ringMinusExtracts') (transitive *Commutative (*WellDefined reflexive (transitive invContravariant (transitive groupIsAbelian (+WellDefined reflexive invInv))))))) *Associative) *Commutative)) (*WellDefined reflexive (transitive *Commutative pr1/4))) *Commutative) identIsIdent
           t<sqrt2 : t < sqrt2'
           t<sqrt2 with allInvertible (fromN 4) (charZero' 3)
-          ... | 1/4 , pr4 = <WellDefined reflexive (transitive (symmetric *Associative) (transitive (*WellDefined reflexive (transitive *Commutative pr4)) (transitive *Commutative identIsIdent))) (ringCanMultiplyByPositive (inversePositiveIsPositive pr4 (fromNPreservesOrder nontrivial {0} {4} (le 3 refl))) (<Transitive (orderRespectsAddition (ringMultiplyPositives 0<sqrt2 0<sqrt2 sqrt2<2 sqrt2<2) (inverse (1R + (1R + 0G)))) (<WellDefined {fromN 2} 2=2*2-2 reflexive (<Transitive {fromN 2} {sqrt2' * fromN 2} (<WellDefined identIsIdent reflexive (ringCanMultiplyByPositive (fromNPreservesOrder nontrivial {0} (le 1 refl)) 1<sqrt2)) (<WellDefined *Commutative *Commutative (ringCanMultiplyByPositive 0<sqrt2 (fromNPreservesOrder nontrivial {2} (le 1 refl))))))))
+          ... | 1/4 , pr4 = <WellDefined reflexive (transitive (symmetric *Associative) (transitive (*WellDefined reflexive (transitive *Commutative pr4)) (transitive *Commutative identIsIdent))) (ringCanMultiplyByPositive (inversePositiveIsPositive pr4 (fromNPreservesOrder (0<1 nontrivial) {0} {4} (le 3 refl))) (<Transitive (orderRespectsAddition (ringMultiplyPositives 0<sqrt2 0<sqrt2 sqrt2<2 sqrt2<2) (inverse (1R + (1R + 0G)))) (<WellDefined {fromN 2} 2=2*2-2 reflexive (<Transitive {fromN 2} {sqrt2' * fromN 2} (<WellDefined identIsIdent reflexive (ringCanMultiplyByPositive (fromNPreservesOrder (0<1 nontrivial) {0} (le 1 refl)) 1<sqrt2)) (<WellDefined *Commutative *Commutative (ringCanMultiplyByPositive 0<sqrt2 (fromNPreservesOrder (0<1 nontrivial) {2} (le 1 refl))))))))
           0<t : 0R < t
           0<t with allInvertible (fromN 4) (charZero' 3)
-          ... | 1/4 , pr4 = orderRespectsMultiplication (<WellDefined reflexive (+WellDefined reflexive (inverseWellDefined (+WellDefined reflexive (symmetric identRight)))) (moveInequality 2<sup^2)) (inversePositiveIsPositive pr4 (fromNPreservesOrder nontrivial {0} {4} (le 3 refl)))
+          ... | 1/4 , pr4 = orderRespectsMultiplication (<WellDefined reflexive (+WellDefined reflexive (inverseWellDefined (+WellDefined reflexive (symmetric identRight)))) (moveInequality 2<sup^2)) (inversePositiveIsPositive pr4 (fromNPreservesOrder (0<1 nontrivial) {0} {4} (le 3 refl)))
         anotherUpperBound : A
         anotherUpperBound = (sqrt2' + inverse t) * (sqrt2' + inverse t)
         abstract
diff --git a/Numbers/ClassicalReals/RealField.agda b/Numbers/ClassicalReals/RealField.agda
index 30d0fca..025047b 100644
--- a/Numbers/ClassicalReals/RealField.agda
+++ b/Numbers/ClassicalReals/RealField.agda
@@ -5,7 +5,7 @@ open import Agda.Primitive using (Level; lzero; lsuc; _⊔_)
 open import LogicalFormulae
 open import Setoids.Subset
 open import Setoids.Setoids
-open import Setoids.Orders
+open import Setoids.Orders.Partial.Definition
 open import Fields.Fields
 open import Rings.Orders.Total.Definition
 open import Rings.Orders.Total.Lemmas
diff --git a/Numbers/ClassicalReals/RealField/Lemmas.agda b/Numbers/ClassicalReals/RealField/Lemmas.agda
index ac3b480..ba9f2c5 100644
--- a/Numbers/ClassicalReals/RealField/Lemmas.agda
+++ b/Numbers/ClassicalReals/RealField/Lemmas.agda
@@ -6,7 +6,7 @@ open import Agda.Primitive using (Level; lzero; lsuc; _⊔_)
 open import LogicalFormulae
 open import Setoids.Subset
 open import Setoids.Setoids
-open import Setoids.Orders
+open import Setoids.Orders.Partial.Definition
 open import Fields.Fields
 open import Rings.Orders.Total.Lemmas
 open import Rings.Orders.Partial.Definition
diff --git a/Numbers/Integers/Definition.agda b/Numbers/Integers/Definition.agda
index 5588f8d..5db7fe3 100644
--- a/Numbers/Integers/Definition.agda
+++ b/Numbers/Integers/Definition.agda
@@ -8,6 +8,9 @@ module Numbers.Integers.Definition where
 data ℤ : Set where
   nonneg : ℕ → ℤ
   negSucc : ℕ → ℤ
+{-# BUILTIN INTEGER       ℤ       #-}
+{-# BUILTIN INTEGERPOS    nonneg  #-}
+{-# BUILTIN INTEGERNEGSUC negSucc #-}
 
 nonnegInjective : {a b : ℕ} → nonneg a ≡ nonneg b → a ≡ b
 nonnegInjective refl = refl
diff --git a/Numbers/Integers/Order.agda b/Numbers/Integers/Order.agda
index bad6fbe..599b55f 100644
--- a/Numbers/Integers/Order.agda
+++ b/Numbers/Integers/Order.agda
@@ -7,9 +7,9 @@ open import Numbers.Integers.RingStructure.Ring
 open import Semirings.Definition
 open import Rings.Orders.Partial.Definition
 open import Rings.Orders.Total.Definition
-open import Setoids.Orders
 open import Orders.Total.Definition
 open import Orders.Partial.Definition
+open import Setoids.Orders.Total.Definition
 
 module Numbers.Integers.Order where
 
diff --git a/Numbers/Integers/RingStructure/Archimedean.agda b/Numbers/Integers/RingStructure/Archimedean.agda
new file mode 100644
index 0000000..dd6ec5e
--- /dev/null
+++ b/Numbers/Integers/RingStructure/Archimedean.agda
@@ -0,0 +1,30 @@
+{-# OPTIONS --safe --warning=error --without-K #-}
+
+open import Semirings.Definition
+open import LogicalFormulae
+open import Numbers.Naturals.Semiring
+open import Numbers.Naturals.Order
+open import Numbers.Integers.RingStructure.Ring
+open import Groups.Orders.Archimedean
+open import Rings.Orders.Partial.Definition
+open import Numbers.Integers.Order
+
+module Numbers.Integers.RingStructure.Archimedean where
+
+open import Groups.Cyclic.Definition ℤGroup
+open import Semirings.Solver ℕSemiring multiplicationNIsCommutative
+
+private
+  lemma : (x y : ℕ) → positiveEltPower (nonneg x) y ≡ nonneg (x *N y)
+  lemma x zero rewrite Semiring.productZeroRight ℕSemiring x = refl
+  lemma x (succ y) rewrite lemma x y | multiplicationNIsCommutative x (succ y) | multiplicationNIsCommutative y x = equalityCommutative (+Zinherits x (x *N y))
+
+ℤArchimedean : Archimedean (toGroup ℤRing ℤPOrderedRing)
+ℤArchimedean (nonneg (succ a)) (nonneg (succ b)) 0<a 0<b = succ (succ b) , t
+  where
+    v : b +N (a +N (a +N a *N b)) ≡ a +N (a +N (b +N a *N b))
+    v rewrite Semiring.+Associative ℕSemiring a a (a *N b) | Semiring.+Associative ℕSemiring b (a +N a) (a *N b) | Semiring.+Associative ℕSemiring a b (a *N b) | Semiring.+Associative ℕSemiring a (a +N b) (a *N b) | Semiring.commutative ℕSemiring b (a +N a) | Semiring.+Associative ℕSemiring a a b = refl
+    u : succ ((a +N (a +N a *N b)) +N b) ≡ a +N succ (a +N (b +N a *N b))
+    u = from (succ (plus (plus (const a) (plus (const a) (times (const a) (const b)))) (const b))) to (plus (const a) (succ (plus (const a) (plus (const b) (times (const a) (const b)))))) by applyEquality succ v
+    t : nonneg (succ b) <Z nonneg (succ a) +Z (nonneg (succ a) +Z positiveEltPower (nonneg (succ a)) b)
+    t rewrite lemma (succ a) b = lessInherits (succPreservesInequality (le (a +N (a +N a *N b)) u))
diff --git a/Numbers/Intervals/Arithmetic.agda b/Numbers/Intervals/Arithmetic.agda
index fb4fa6c..5f2d34c 100644
--- a/Numbers/Intervals/Arithmetic.agda
+++ b/Numbers/Intervals/Arithmetic.agda
@@ -4,7 +4,7 @@ open import Numbers.ClassicalReals.RealField
 open import LogicalFormulae
 open import Setoids.Subset
 open import Setoids.Setoids
-open import Setoids.Orders
+open import Setoids.Orders.Partial.Definition
 open import Sets.EquivalenceRelations
 open import Rings.Orders.Partial.Definition
 open import Rings.Definition
diff --git a/Numbers/Intervals/Definition.agda b/Numbers/Intervals/Definition.agda
index 28e2569..917d370 100644
--- a/Numbers/Intervals/Definition.agda
+++ b/Numbers/Intervals/Definition.agda
@@ -4,7 +4,7 @@ open import Numbers.ClassicalReals.RealField
 open import LogicalFormulae
 open import Setoids.Subset
 open import Setoids.Setoids
-open import Setoids.Orders
+open import Setoids.Orders.Partial.Definition
 open import Sets.EquivalenceRelations
 open import Rings.Orders.Partial.Definition
 open import Rings.Definition
diff --git a/Numbers/Naturals/Addition.agda b/Numbers/Naturals/Addition.agda
index bf0aa02..f892dbc 100644
--- a/Numbers/Naturals/Addition.agda
+++ b/Numbers/Naturals/Addition.agda
@@ -9,6 +9,7 @@ infix 15 _+N_
 _+N_ : ℕ → ℕ → ℕ
 zero +N y = y
 succ x +N y = succ (x +N y)
+{-# BUILTIN NATPLUS _+N_ #-}
 
 addZeroRight : (x : ℕ) → (x +N zero) ≡ x
 addZeroRight zero = refl
diff --git a/Numbers/Naturals/Multiplication.agda b/Numbers/Naturals/Multiplication.agda
index 3c286a1..6acf7cc 100644
--- a/Numbers/Naturals/Multiplication.agda
+++ b/Numbers/Naturals/Multiplication.agda
@@ -10,6 +10,7 @@ infix 25 _*N_
 _*N_ : ℕ → ℕ → ℕ
 zero *N y = zero
 (succ x) *N y = y +N (x *N y)
+{-# BUILTIN NATTIMES _*N_ #-}
 
 productZeroIsZeroLeft : (a : ℕ) → (zero *N a ≡ zero)
 productZeroIsZeroLeft a = refl
diff --git a/Numbers/Rationals/Definition.agda b/Numbers/Rationals/Definition.agda
index 84c751b..109df62 100644
--- a/Numbers/Rationals/Definition.agda
+++ b/Numbers/Rationals/Definition.agda
@@ -12,7 +12,8 @@ open import Rings.Orders.Total.Definition
 open import Rings.Orders.Partial.Definition
 open import Fields.Fields
 open import Setoids.Setoids
-open import Setoids.Orders
+open import Setoids.Orders.Partial.Definition
+open import Setoids.Orders.Total.Definition
 open import Functions
 open import Sets.EquivalenceRelations
 
diff --git a/Numbers/Rationals/Lemmas.agda b/Numbers/Rationals/Lemmas.agda
index dd61a8b..f388d2f 100644
--- a/Numbers/Rationals/Lemmas.agda
+++ b/Numbers/Rationals/Lemmas.agda
@@ -16,13 +16,13 @@ open import Rings.Orders.Total.Definition
 open import Fields.Fields
 open import Numbers.Primes.PrimeNumbers
 open import Setoids.Setoids
-open import Setoids.Orders
 open import Functions
 open import Sets.EquivalenceRelations
 open import Numbers.Rationals.Definition
 open import Semirings.Definition
 open import Orders.Total.Definition
 open import Orders.WellFounded.Induction
+open import Setoids.Orders.Total.Definition
 
 module Numbers.Rationals.Lemmas where
 
@@ -30,6 +30,7 @@ open import Semirings.Lemmas ℕSemiring
 
 open PartiallyOrderedRing ℤPOrderedRing
 open import Rings.Orders.Total.Lemmas ℤOrderedRing
+open import Rings.Orders.Total.AbsoluteValue ℤOrderedRing
 open SetoidTotalOrder (totalOrderToSetoidTotalOrder ℤOrder)
 
 evenOrOdd : (a : ℕ) → (Sg ℕ (λ i → i *N 2 ≡ a)) || (Sg ℕ (λ i → succ (i *N 2) ≡ a))
diff --git a/Numbers/Reals/Definition.agda b/Numbers/Reals/Definition.agda
index c877032..dabb09a 100644
--- a/Numbers/Reals/Definition.agda
+++ b/Numbers/Reals/Definition.agda
@@ -1,6 +1,6 @@
 {-# OPTIONS --warning=error --safe --guardedness --without-K #-}
 
-open import Setoids.Orders
+open import Setoids.Orders.Partial.Definition
 open import Setoids.Setoids
 open import LogicalFormulae
 open import Rings.Definition
diff --git a/Orders/Total/Lemmas.agda b/Orders/Total/Lemmas.agda
index 385686a..f636681 100644
--- a/Orders/Total/Lemmas.agda
+++ b/Orders/Total/Lemmas.agda
@@ -1,6 +1,5 @@
 {-# OPTIONS --safe --warning=error --without-K #-}
 
-
 open import LogicalFormulae
 
 open import Orders.Total.Definition
diff --git a/Rings/Homomorphisms/Lemmas.agda b/Rings/Homomorphisms/Lemmas.agda
index 9a77e59..704dcc5 100644
--- a/Rings/Homomorphisms/Lemmas.agda
+++ b/Rings/Homomorphisms/Lemmas.agda
@@ -5,7 +5,6 @@ open import Groups.Groups
 open import Groups.Homomorphisms.Definition
 open import Groups.Definition
 open import Numbers.Naturals.Naturals
-open import Setoids.Orders
 open import Setoids.Setoids
 open import Functions
 open import Sets.EquivalenceRelations
diff --git a/Rings/InitialRing.agda b/Rings/InitialRing.agda
index 06d2a65..419fc00 100644
--- a/Rings/InitialRing.agda
+++ b/Rings/InitialRing.agda
@@ -25,10 +25,10 @@ open Setoid S
 open Equivalence eq
 open import Rings.Lemmas R
 open import Groups.Lemmas additiveGroup
+open import Groups.Cyclic.Definition additiveGroup
 
 fromN : ℕ → A
-fromN zero = 0R
-fromN (succ n) = 1R + fromN n
+fromN = positiveEltPower 1R
 
 fromNPreserves+ : (a b : ℕ) → fromN (a +N b) ∼ (fromN a) + (fromN b)
 fromNPreserves+ zero b = symmetric identLeft
diff --git a/Rings/Orders/Partial/Bounded.agda b/Rings/Orders/Partial/Bounded.agda
index 9695e68..53fca4e 100644
--- a/Rings/Orders/Partial/Bounded.agda
+++ b/Rings/Orders/Partial/Bounded.agda
@@ -7,7 +7,7 @@ open import Rings.Definition
 open import Rings.Orders.Partial.Definition
 open import Sets.EquivalenceRelations
 open import Sequences
-open import Setoids.Orders
+open import Setoids.Orders.Partial.Definition
 open import Functions
 open import LogicalFormulae
 open import Numbers.Naturals.Semiring
diff --git a/Rings/Orders/Partial/Definition.agda b/Rings/Orders/Partial/Definition.agda
index 483041a..3fd88d2 100644
--- a/Rings/Orders/Partial/Definition.agda
+++ b/Rings/Orders/Partial/Definition.agda
@@ -1,8 +1,8 @@
 {-# OPTIONS --safe --warning=error --without-K #-}
 
 open import Groups.Definition
-open import Setoids.Orders
 open import Setoids.Setoids
+open import Setoids.Orders.Partial.Definition
 open import Functions
 open import Rings.Definition
 
@@ -13,9 +13,13 @@ module Rings.Orders.Partial.Definition {n m : _} {A : Set n} {S : Setoid {n} {m}
 open Ring R
 open Group additiveGroup
 open Setoid S
+open import Groups.Orders.Partial.Definition
 
 record PartiallyOrderedRing {p : _} {_<_ : Rel {_} {p} A} (pOrder : SetoidPartialOrder S _<_) : Set (lsuc n ⊔ m ⊔ p) where
   field
     orderRespectsAddition : {a b : A} → (a < b) → (c : A) → (a + c) < (b + c)
     orderRespectsMultiplication : {a b : A} → (0R < a) → (0R < b) → (0R < (a * b))
   open SetoidPartialOrder pOrder
+
+toGroup : {p : _} {_<_ : Rel {_} {p} A} {pOrder : SetoidPartialOrder S _<_} → PartiallyOrderedRing pOrder → PartiallyOrderedGroup additiveGroup pOrder
+PartiallyOrderedGroup.orderRespectsAddition (toGroup p) = PartiallyOrderedRing.orderRespectsAddition p
diff --git a/Rings/Orders/Partial/Lemmas.agda b/Rings/Orders/Partial/Lemmas.agda
index 66f8327..0eff46f 100644
--- a/Rings/Orders/Partial/Lemmas.agda
+++ b/Rings/Orders/Partial/Lemmas.agda
@@ -3,7 +3,7 @@
 open import LogicalFormulae
 open import Groups.Lemmas
 open import Groups.Definition
-open import Setoids.Orders
+open import Setoids.Orders.Partial.Definition
 open import Setoids.Setoids
 open import Functions
 open import Sets.EquivalenceRelations
@@ -91,3 +91,17 @@ abstract
 
   negativeInequality : {a : A} → a < 0G → 0G < inverse a
   negativeInequality {a} a<0 = <WellDefined invRight identLeft (orderRespectsAddition a<0 (inverse a))
+
+  negativeInequality' : {a : A} → 0G < a → inverse a < 0G
+  negativeInequality' {a} 0<a = <WellDefined identLeft invRight (orderRespectsAddition 0<a (inverse a))
+
+open import Rings.InitialRing R
+
+fromNIncreasing : (0R < 1R) → (n : ℕ) → (fromN n) < (fromN (succ n))
+fromNIncreasing 0<1 zero = <WellDefined reflexive (symmetric identRight) 0<1
+fromNIncreasing 0<1 (succ n) = <WellDefined groupIsAbelian groupIsAbelian (orderRespectsAddition (fromNIncreasing 0<1 n) 1R)
+
+fromNPreservesOrder : (0R < 1R) → {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)
diff --git a/Rings/Orders/Total/AbsoluteValue.agda b/Rings/Orders/Total/AbsoluteValue.agda
new file mode 100644
index 0000000..27cf8ae
--- /dev/null
+++ b/Rings/Orders/Total/AbsoluteValue.agda
@@ -0,0 +1,217 @@
+{-# OPTIONS --safe --warning=error --without-K #-}
+
+open import LogicalFormulae
+open import Groups.Lemmas
+open import Groups.Definition
+open import Setoids.Orders.Partial.Definition
+open import Setoids.Orders.Total.Definition
+open import Setoids.Setoids
+open import Functions
+open import Sets.EquivalenceRelations
+open import Rings.Definition
+open import Rings.Orders.Total.Definition
+open import Rings.Orders.Partial.Definition
+open import Numbers.Naturals.Semiring
+open import Numbers.Naturals.Order
+open import Orders.Total.Definition
+open import Rings.IntegralDomains.Definition
+
+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
+
+open Ring R
+open Group additiveGroup
+open Setoid S
+open SetoidPartialOrder pOrder
+open TotallyOrderedRing order
+open SetoidTotalOrder total
+open PartiallyOrderedRing pOrderRing
+open import Rings.Lemmas R
+open import Rings.Orders.Partial.Lemmas pOrderRing
+open import Rings.Orders.Total.Lemmas order
+
+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
+
+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
+
+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)
+
+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
+
+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)
+
+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)))
diff --git a/Rings/Orders/Total/BaseExpansion.agda b/Rings/Orders/Total/BaseExpansion.agda
index 4aa67bb..5eff68b 100644
--- a/Rings/Orders/Total/BaseExpansion.agda
+++ b/Rings/Orders/Total/BaseExpansion.agda
@@ -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)))))
diff --git a/Rings/Orders/Total/Bounded.agda b/Rings/Orders/Total/Bounded.agda
index 97fda64..1463e81 100644
--- a/Rings/Orders/Total/Bounded.agda
+++ b/Rings/Orders/Total/Bounded.agda
@@ -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
diff --git a/Rings/Orders/Total/Cauchy.agda b/Rings/Orders/Total/Cauchy.agda
index 61ae496..3972230 100644
--- a/Rings/Orders/Total/Cauchy.agda
+++ b/Rings/Orders/Total/Cauchy.agda
@@ -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)
diff --git a/Rings/Orders/Total/Definition.agda b/Rings/Orders/Total/Definition.agda
index baed630..5da9bfa 100644
--- a/Rings/Orders/Total/Definition.agda
+++ b/Rings/Orders/Total/Definition.agda
@@ -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
diff --git a/Rings/Orders/Total/Lemmas.agda b/Rings/Orders/Total/Lemmas.agda
index 7e5347b..f8b0258 100644
--- a/Rings/Orders/Total/Lemmas.agda
+++ b/Rings/Orders/Total/Lemmas.agda
@@ -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
diff --git a/Semirings/Solver.agda b/Semirings/Solver.agda
index d1a53e2..f244c3a 100644
--- a/Semirings/Solver.agda
+++ b/Semirings/Solver.agda
@@ -52,8 +52,9 @@ simplePlus (times a b) (succ c) = succ (plus (times a b) c)
 simplePlus (times a b) (plus c d) = plus (times a b) (plus c d)
 simplePlus (times a b) (times c d) = plus (times a b) (times c d)
 
-lemma : (a b c : A) → a + (b + c) ≡ b + (a + c)
-lemma a b c rewrite commutative a (b + c) | equalityCommutative (+Associative b c a) = applyEquality (b +_) (commutative _ _)
+private
+  lemma : (a b c : A) → a + (b + c) ≡ b + (a + c)
+  lemma a b c rewrite commutative a (b + c) | equalityCommutative (+Associative b c a) = applyEquality (b +_) (commutative _ _)
 
 simplePlusIsPlus : (a b : Expr) → eval (simplePlus a b) ≡ eval (plus a b)
 simplePlusIsPlus (const x) (const x₁) = refl
diff --git a/Setoids/Orders/Orders.agda b/Setoids/Orders/Orders.agda
new file mode 100644
index 0000000..3a41764
--- /dev/null
+++ b/Setoids/Orders/Orders.agda
@@ -0,0 +1,19 @@
+{-# OPTIONS --safe --warning=error --without-K #-}
+
+open import LogicalFormulae
+open import Orders.Total.Definition
+open import Orders.Partial.Definition
+open import Setoids.Setoids
+open import Functions
+
+open import Agda.Primitive using (Level; lzero; lsuc; _⊔_)
+
+module Setoids.Orders where
+
+partialOrderToSetoidPartialOrder : {a b : _} {A : Set a} (P : PartialOrder {a} A {b}) → SetoidPartialOrder (reflSetoid A) (PartialOrder._<_ P)
+SetoidPartialOrder.<WellDefined (partialOrderToSetoidPartialOrder P) a=b c=d a<c rewrite a=b | c=d = a<c
+SetoidPartialOrder.irreflexive (partialOrderToSetoidPartialOrder P) = PartialOrder.irreflexive P
+SetoidPartialOrder.<Transitive (partialOrderToSetoidPartialOrder P) = PartialOrder.<Transitive P
+
+totalOrderToSetoidTotalOrder : {a b : _} {A : Set a} (T : TotalOrder {a} A {b}) → SetoidTotalOrder (partialOrderToSetoidPartialOrder (TotalOrder.order T))
+SetoidTotalOrder.totality (totalOrderToSetoidTotalOrder T) = TotalOrder.totality T
diff --git a/Setoids/Orders.agda b/Setoids/Orders/Partial/Definition.agda
similarity index 57%
rename from Setoids/Orders.agda
rename to Setoids/Orders/Partial/Definition.agda
index a3b868c..03ce1ad 100644
--- a/Setoids/Orders.agda
+++ b/Setoids/Orders/Partial/Definition.agda
@@ -8,7 +8,7 @@ open import Functions
 
 open import Agda.Primitive using (Level; lzero; lsuc; _⊔_)
 
-module Setoids.Orders where
+module Setoids.Orders.Partial.Definition where
 
 record SetoidPartialOrder {a b c : _} {A : Set a} (S : Setoid {a} {b} A) (_<_ : Rel {a} {c} A) : Set (a ⊔ b ⊔ c) where
   open Setoid S
@@ -19,27 +19,7 @@ record SetoidPartialOrder {a b c : _} {A : Set a} (S : Setoid {a} {b} A) (_<_ :
   _<=_ : Rel {a} {b ⊔ c} A
   a <= b = (a < b) || (a ∼ b)
 
-record SetoidTotalOrder {a b c : _} {A : Set a} {S : Setoid {a} {b} A} {_<_ : Rel {a} {c} A} (P : SetoidPartialOrder S _<_) : Set (a ⊔ b ⊔ c) where
-  open Setoid S
-  field
-    totality : (a b : A) → ((a < b) || (b < a)) || (a ∼ b)
-  partial : SetoidPartialOrder S _<_
-  partial = P
-  min : A → A → A
-  min a b with totality a b
-  min a b | inl (inl a<b) = a
-  min a b | inl (inr b<a) = b
-  min a b | inr a=b = a
-  max : A → A → A
-  max a b with totality a b
-  max a b | inl (inl a<b) = b
-  max a b | inl (inr b<a) = a
-  max a b | inr a=b = b
-
 partialOrderToSetoidPartialOrder : {a b : _} {A : Set a} (P : PartialOrder {a} A {b}) → SetoidPartialOrder (reflSetoid A) (PartialOrder._<_ P)
 SetoidPartialOrder.<WellDefined (partialOrderToSetoidPartialOrder P) a=b c=d a<c rewrite a=b | c=d = a<c
 SetoidPartialOrder.irreflexive (partialOrderToSetoidPartialOrder P) = PartialOrder.irreflexive P
 SetoidPartialOrder.<Transitive (partialOrderToSetoidPartialOrder P) = PartialOrder.<Transitive P
-
-totalOrderToSetoidTotalOrder : {a b : _} {A : Set a} (T : TotalOrder {a} A {b}) → SetoidTotalOrder (partialOrderToSetoidPartialOrder (TotalOrder.order T))
-SetoidTotalOrder.totality (totalOrderToSetoidTotalOrder T) = TotalOrder.totality T
diff --git a/Setoids/Orders/Total/Definition.agda b/Setoids/Orders/Total/Definition.agda
new file mode 100644
index 0000000..c2b4964
--- /dev/null
+++ b/Setoids/Orders/Total/Definition.agda
@@ -0,0 +1,33 @@
+{-# OPTIONS --safe --warning=error --without-K #-}
+
+open import LogicalFormulae
+open import Orders.Total.Definition
+open import Orders.Partial.Definition
+open import Setoids.Setoids
+open import Functions
+
+open import Agda.Primitive using (Level; lzero; lsuc; _⊔_)
+
+module Setoids.Orders.Total.Definition where
+
+open import Setoids.Orders.Partial.Definition
+
+record SetoidTotalOrder {a b c : _} {A : Set a} {S : Setoid {a} {b} A} {_<_ : Rel {a} {c} A} (P : SetoidPartialOrder S _<_) : Set (a ⊔ b ⊔ c) where
+  open Setoid S
+  field
+    totality : (a b : A) → ((a < b) || (b < a)) || (a ∼ b)
+  partial : SetoidPartialOrder S _<_
+  partial = P
+  min : A → A → A
+  min a b with totality a b
+  min a b | inl (inl a<b) = a
+  min a b | inl (inr b<a) = b
+  min a b | inr a=b = a
+  max : A → A → A
+  max a b with totality a b
+  max a b | inl (inl a<b) = b
+  max a b | inl (inr b<a) = a
+  max a b | inr a=b = b
+
+totalOrderToSetoidTotalOrder : {a b : _} {A : Set a} (T : TotalOrder {a} A {b}) → SetoidTotalOrder (partialOrderToSetoidPartialOrder (TotalOrder.order T))
+SetoidTotalOrder.totality (totalOrderToSetoidTotalOrder T) = TotalOrder.totality T
diff --git a/Setoids/Orders/Total/Lemmas.agda b/Setoids/Orders/Total/Lemmas.agda
new file mode 100644
index 0000000..bc2b7c2
--- /dev/null
+++ b/Setoids/Orders/Total/Lemmas.agda
@@ -0,0 +1,67 @@
+{-# OPTIONS --safe --warning=error --without-K #-}
+
+open import LogicalFormulae
+open import Orders.Total.Definition
+open import Orders.Partial.Definition
+open import Setoids.Setoids
+open import Setoids.Orders.Partial.Definition
+open import Setoids.Orders.Total.Definition
+open import Functions
+open import Sets.EquivalenceRelations
+
+open import Agda.Primitive using (Level; lzero; lsuc; _⊔_)
+
+module Setoids.Orders.Total.Lemmas {a b : _} {A : Set a} {S : Setoid {a} {b} A} {c : _} {_<_ : A → A → Set c} {P : SetoidPartialOrder S _<_} (T : SetoidTotalOrder P) where
+
+open SetoidTotalOrder T
+open SetoidPartialOrder P
+open Setoid S
+open Equivalence eq
+
+maxInequalitiesR : {a b c : A} → (a < b) → (a < c) → (a < max b c)
+maxInequalitiesR {a} {b} {c} a<b a<c with totality b c
+... | inl (inl x) = a<c
+... | inl (inr x) = a<b
+... | inr x = a<c
+
+minInequalitiesR : {a b c : A} → (a < b) → (a < c) → (a < min b c)
+minInequalitiesR {a} {b} {c} a<b a<c with totality b c
+... | inl (inl x) = a<b
+... | inl (inr x) = a<c
+... | inr x = a<b
+
+maxInequalitiesL : {a b c : A} → (a < c) → (b < c) → (max a b < c)
+maxInequalitiesL {a} {b} {c} a<b a<c with totality a b
+... | inl (inl x) = a<c
+... | inl (inr x) = a<b
+... | inr x = a<c
+
+minInequalitiesL : {a b c : A} → (a < c) → (b < c) → (min a b < c)
+minInequalitiesL {a} {b} {c} a<b a<c with totality a b
+... | inl (inl x) = a<b
+... | inl (inr x) = a<c
+... | inr x = a<b
+
+minLessL : (a b : A) → min a b <= a
+minLessL a b with totality a b
+... | inl (inl x) = inr reflexive
+... | inl (inr x) = inl x
+... | inr x = inr reflexive
+
+minLessR : (a b : A) → min a b <= b
+minLessR a b with totality a b
+... | inl (inl x) = inl x
+... | inl (inr x) = inr reflexive
+... | inr x = inr x
+
+maxGreaterL : (a b : A) → a <= max a b
+maxGreaterL a b with totality a b
+... | inl (inl x) = inl x
+... | inl (inr x) = inr reflexive
+... | inr x = inr x
+
+maxGreaterR : (a b : A) → b <= max a b
+maxGreaterR a b with totality a b
+... | inl (inl x) = inr reflexive
+... | inl (inr x) = inl x
+... | inr x = inr reflexive