Split partial and total order of rings (#61)

Rings/Orders/Partial/Lemmas.agda

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+{-# OPTIONS --safe --warning=error --without-K #-}
+
+open import LogicalFormulae
+open import Groups.Groups
+open import Groups.Lemmas
+open import Groups.Definition
+open import Numbers.Naturals.Naturals
+open import Setoids.Orders
+open import Setoids.Setoids
+open import Functions
+open import Sets.EquivalenceRelations
+open import Rings.Definition
+open import Rings.Orders.Partial.Definition
+
+open import Agda.Primitive using (Level; lzero; lsuc; _⊔_)
+
+module Rings.Orders.Partial.Lemmas {n m p : _} {A : Set n} {S : Setoid {n} {m} A} {_+_ : A → A → A} {_*_ : A → A → A} {_<_ : Rel {_} {p} A} {R : Ring S _+_ _*_} {pOrder : SetoidPartialOrder S _<_} (pRing : PartiallyOrderedRing R pOrder) where
+
+abstract
+
+  open PartiallyOrderedRing pRing
+  open Setoid S
+  open SetoidPartialOrder pOrder
+  open Ring R
+  open Group additiveGroup
+
+  open import Rings.Lemmas R
+
+  ringAddInequalities : {w x y z : A} → w < x → y < z → (w + y) < (x + z)
+  ringAddInequalities {w = w} {x} {y} {z} w<x y<z = transitive (orderRespectsAddition w<x y) (<WellDefined groupIsAbelian groupIsAbelian (orderRespectsAddition y<z x))
+
+  ringCanMultiplyByPositive : {x y c : A} → (Ring.0R R) < c → x < y → (x * c) < (y * c)
+  ringCanMultiplyByPositive {x} {y} {c} 0<c x<y = SetoidPartialOrder.<WellDefined pOrder reflexive (Group.identRight additiveGroup) q'
+    where
+      open Equivalence eq
+      have : 0R < (y + Group.inverse additiveGroup x)
+      have = SetoidPartialOrder.<WellDefined pOrder (Group.invRight additiveGroup) reflexive (orderRespectsAddition x<y (Group.inverse additiveGroup x))
+      p1 : 0R < ((y * c) + ((Group.inverse additiveGroup x) * c))
+      p1 = SetoidPartialOrder.<WellDefined pOrder reflexive (Equivalence.transitive eq *Commutative (Equivalence.transitive eq *DistributesOver+ ((Group.+WellDefined additiveGroup) *Commutative *Commutative))) (orderRespectsMultiplication have 0<c)
+      p' : 0R < ((y * c) + (Group.inverse additiveGroup (x * c)))
+      p' = SetoidPartialOrder.<WellDefined pOrder reflexive (Group.+WellDefined additiveGroup reflexive (Equivalence.transitive eq (Equivalence.transitive eq *Commutative ringMinusExtracts) (inverseWellDefined additiveGroup *Commutative))) p1
+      q : (0R + (x * c)) < (((y * c) + (Group.inverse additiveGroup (x * c))) + (x * c))
+      q = orderRespectsAddition p' (x * c)
+      q' : (x * c) < ((y * c) + 0R)
+      q' = SetoidPartialOrder.<WellDefined pOrder (Group.identLeft additiveGroup) (Equivalence.transitive eq (symmetric (Group.+Associative additiveGroup)) (Group.+WellDefined additiveGroup reflexive (Group.invLeft additiveGroup))) q
+
+  ringMultiplyPositives : {x y a b : A} → 0R < x → 0R < a → (x < y) → (a < b) → (x * a) < (y * b)
+  ringMultiplyPositives {x} {y} {a} {b} 0<x 0<a x<y a<b = SetoidPartialOrder.transitive pOrder (ringCanMultiplyByPositive 0<a x<y) (<WellDefined *Commutative *Commutative (ringCanMultiplyByPositive (SetoidPartialOrder.transitive pOrder 0<x x<y) a<b))
+
+  ringSwapNegatives : {x y : A} → (Group.inverse (Ring.additiveGroup R) x) < (Group.inverse (Ring.additiveGroup R) y) → y < x
+  ringSwapNegatives {x} {y} -x<-y = SetoidPartialOrder.<WellDefined pOrder (Equivalence.transitive eq (symmetric (Group.+Associative additiveGroup)) (Equivalence.transitive eq (Group.+WellDefined additiveGroup reflexive (Group.invLeft additiveGroup)) (Group.identRight additiveGroup))) (Group.identLeft additiveGroup) v
+    where
+      open Equivalence eq
+      t : ((Group.inverse additiveGroup x) + y) < ((Group.inverse additiveGroup y) + y)
+      t = orderRespectsAddition -x<-y y
+      u : (y + (Group.inverse additiveGroup x)) < 0R
+      u = SetoidPartialOrder.<WellDefined pOrder (groupIsAbelian) (Group.invLeft additiveGroup) t
+      v : ((y + (Group.inverse additiveGroup x)) + x) < (0R + x)
+      v = orderRespectsAddition u x
+
+  ringSwapNegatives' : {x y : A} → x < y → (Group.inverse (Ring.additiveGroup R) y) < (Group.inverse (Ring.additiveGroup R) x)
+  ringSwapNegatives' {x} {y} x<y = ringSwapNegatives (<WellDefined (Equivalence.symmetric eq (invTwice additiveGroup _)) (Equivalence.symmetric eq (invTwice additiveGroup _)) x<y)
+
+  ringCanMultiplyByNegative : {x y c : A} → c < (Ring.0R R) → x < y → (y * c) < (x * c)
+  ringCanMultiplyByNegative {x} {y} {c} c<0 x<y = ringSwapNegatives u
+    where
+    open Equivalence eq
+    p1 : (c + Group.inverse additiveGroup c) < (0R + Group.inverse additiveGroup c)
+    p1 = orderRespectsAddition c<0 _
+    0<-c : 0R < (Group.inverse additiveGroup c)
+    0<-c = SetoidPartialOrder.<WellDefined pOrder (Group.invRight additiveGroup) (Group.identLeft additiveGroup) p1
+    t : (x * Group.inverse additiveGroup c) < (y * Group.inverse additiveGroup c)
+    t = ringCanMultiplyByPositive 0<-c x<y
+    u : (Group.inverse additiveGroup (x * c)) < Group.inverse additiveGroup (y * c)
+    u = SetoidPartialOrder.<WellDefined pOrder ringMinusExtracts ringMinusExtracts t
+
+  anyComparisonImpliesNontrivial : {a b : A} → a < b → (0R ∼ 1R) → False
+  anyComparisonImpliesNontrivial {a} {b} a<b 0=1 = irreflexive (<WellDefined (oneZeroImpliesAllZero 0=1) (oneZeroImpliesAllZero 0=1) a<b)