From 038d2109e78d9c688acfc3338140cc908151db76 Mon Sep 17 00:00:00 2001
From: Greg Brown <greg.brown@cl.cam.ac.uk>
Date: Tue, 5 Apr 2022 12:08:00 +0100
Subject: Generalise 0 comparison to ≤.
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---
 .../Ordered/StrictTotal/Properties/AbelianGroup.agda      |  1 +
 .../Ordered/StrictTotal/Properties/CommutativeRing.agda   |  1 +
 .../Ordered/StrictTotal/Properties/DivisionRing.agda      |  1 +
 .../Algebra/Ordered/StrictTotal/Properties/Field.agda     |  1 +
 .../Algebra/Ordered/StrictTotal/Properties/Group.agda     | 15 +++++++++++++++
 .../Algebra/Ordered/StrictTotal/Properties/Ring.agda      |  2 ++
 6 files changed, 21 insertions(+)

diff --git a/src/Helium/Algebra/Ordered/StrictTotal/Properties/AbelianGroup.agda b/src/Helium/Algebra/Ordered/StrictTotal/Properties/AbelianGroup.agda
index aa37d59..7972c4f 100644
--- a/src/Helium/Algebra/Ordered/StrictTotal/Properties/AbelianGroup.agda
+++ b/src/Helium/Algebra/Ordered/StrictTotal/Properties/AbelianGroup.agda
@@ -72,4 +72,5 @@ open import Helium.Algebra.Ordered.StrictTotal.Properties.Group group public
   ; x⁻¹≤ε⇒x≥ε; x⁻¹≥ε⇒x≤ε
 
   ; x<y⇒ε<y∙x⁻¹ ; ε<y∙x⁻¹⇒x<y
+  ; x≤y⇒ε≤y∙x⁻¹ ; ε≤y∙x⁻¹⇒x≤y
   )
diff --git a/src/Helium/Algebra/Ordered/StrictTotal/Properties/CommutativeRing.agda b/src/Helium/Algebra/Ordered/StrictTotal/Properties/CommutativeRing.agda
index 37996b0..6eba0ab 100644
--- a/src/Helium/Algebra/Ordered/StrictTotal/Properties/CommutativeRing.agda
+++ b/src/Helium/Algebra/Ordered/StrictTotal/Properties/CommutativeRing.agda
@@ -65,6 +65,7 @@ open import Helium.Algebra.Ordered.StrictTotal.Properties.Ring ring public
   ; -x≈0⇒x≈0 ; -x<0⇒x>0; -x>0⇒x<0; -x≤0⇒x≥0; -x≥0⇒x≤0
 
   ; x<y⇒0<y-x; 0<y-x⇒x<y
+  ; x≤y⇒0≤y-x; 0≤y-x⇒x≤y
 
   ; 0≤1; 1≈0⇒x≈y; x≉y⇒0<1; x<y⇒0<1
 
diff --git a/src/Helium/Algebra/Ordered/StrictTotal/Properties/DivisionRing.agda b/src/Helium/Algebra/Ordered/StrictTotal/Properties/DivisionRing.agda
index ab8a37b..c0207a3 100644
--- a/src/Helium/Algebra/Ordered/StrictTotal/Properties/DivisionRing.agda
+++ b/src/Helium/Algebra/Ordered/StrictTotal/Properties/DivisionRing.agda
@@ -78,6 +78,7 @@ open import Helium.Algebra.Ordered.StrictTotal.Properties.Ring ring public
   ; -x≈0⇒x≈0 ; -x<0⇒x>0; -x>0⇒x<0; -x≤0⇒x≥0; -x≥0⇒x≤0
 
   ; x<y⇒0<y-x; 0<y-x⇒x<y
+  ; x≤y⇒0≤y-x; 0≤y-x⇒x≤y
 
   ; 0≤1; 1≈0⇒x≈y; x≉y⇒0<1; x<y⇒0<1
 
diff --git a/src/Helium/Algebra/Ordered/StrictTotal/Properties/Field.agda b/src/Helium/Algebra/Ordered/StrictTotal/Properties/Field.agda
index 33578e0..f3ab1c1 100644
--- a/src/Helium/Algebra/Ordered/StrictTotal/Properties/Field.agda
+++ b/src/Helium/Algebra/Ordered/StrictTotal/Properties/Field.agda
@@ -75,6 +75,7 @@ open import Helium.Algebra.Ordered.StrictTotal.Properties.DivisionRing divisionR
   ; -x≈0⇒x≈0 ; -x<0⇒x>0; -x>0⇒x<0; -x≤0⇒x≥0; -x≥0⇒x≤0
 
   ; x<y⇒0<y-x; 0<y-x⇒x<y
+  ; x≤y⇒0≤y-x; 0≤y-x⇒x≤y
 
   ; 0≤1; 1≈0⇒x≈y; x≉y⇒0<1; x<y⇒0<1
 
diff --git a/src/Helium/Algebra/Ordered/StrictTotal/Properties/Group.agda b/src/Helium/Algebra/Ordered/StrictTotal/Properties/Group.agda
index 5cf2d5b..81c57ad 100644
--- a/src/Helium/Algebra/Ordered/StrictTotal/Properties/Group.agda
+++ b/src/Helium/Algebra/Ordered/StrictTotal/Properties/Group.agda
@@ -213,3 +213,18 @@ x<y⇒ε<y∙x⁻¹ {x} {y} x<y = begin-strict
   y ∙ (x ⁻¹ ∙ x) ≈⟨  ∙-congˡ (inverseˡ x) ⟩
   y ∙ ε          ≈⟨  identityʳ y ⟩
   y              ∎
+
+x≤y⇒ε≤y∙x⁻¹ : ∀ {x y} → x ≤ y → ε ≤ y ∙ x ⁻¹
+x≤y⇒ε≤y∙x⁻¹ {x} {y} x≤y = begin
+  ε        ≈˘⟨ inverseʳ x ⟩
+  x ∙ x ⁻¹ ≤⟨  ∙-monoʳ-≤ x≤y ⟩
+  y ∙ x ⁻¹ ∎
+
+ε≤y∙x⁻¹⇒x≤y : ∀ {x y} → ε ≤ y ∙ x ⁻¹ → x ≤ y
+ε≤y∙x⁻¹⇒x≤y {x} {y} ε≤y∙x⁻¹ = begin
+  x              ≈˘⟨ identityˡ x ⟩
+  ε ∙ x          ≤⟨  ∙-monoʳ-≤ ε≤y∙x⁻¹ ⟩
+  y ∙ x ⁻¹ ∙ x   ≈⟨  assoc y (x ⁻¹) x ⟩
+  y ∙ (x ⁻¹ ∙ x) ≈⟨  ∙-congˡ (inverseˡ x) ⟩
+  y ∙ ε          ≈⟨  identityʳ y ⟩
+  y              ∎
diff --git a/src/Helium/Algebra/Ordered/StrictTotal/Properties/Ring.agda b/src/Helium/Algebra/Ordered/StrictTotal/Properties/Ring.agda
index c83f683..5c35176 100644
--- a/src/Helium/Algebra/Ordered/StrictTotal/Properties/Ring.agda
+++ b/src/Helium/Algebra/Ordered/StrictTotal/Properties/Ring.agda
@@ -119,6 +119,8 @@ open import Helium.Algebra.Ordered.StrictTotal.Properties.AbelianGroup +-abelian
 
   ; x<y⇒ε<y∙x⁻¹ to x<y⇒0<y-x
   ; ε<y∙x⁻¹⇒x<y to 0<y-x⇒x<y
+  ; x≤y⇒ε≤y∙x⁻¹ to x≤y⇒0≤y-x
+  ; ε≤y∙x⁻¹⇒x≤y to 0≤y-x⇒x≤y
   )
 
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