From dc2c15ebc9c494b45e8f6c8435752fcb0a66bd3a Mon Sep 17 00:00:00 2001
From: Greg Brown <greg.brown@cl.cam.ac.uk>
Date: Mon, 4 Apr 2022 12:39:22 +0100
Subject: Generalise the precondition for 0 < 1 in rings.

---
 .../Algebra/Ordered/StrictTotal/Properties/CommutativeRing.agda      | 2 +-
 src/Helium/Algebra/Ordered/StrictTotal/Properties/DivisionRing.agda  | 2 +-
 src/Helium/Algebra/Ordered/StrictTotal/Properties/Field.agda         | 2 +-
 src/Helium/Algebra/Ordered/StrictTotal/Properties/Ring.agda          | 5 ++++-
 4 files changed, 7 insertions(+), 4 deletions(-)

(limited to 'src')

diff --git a/src/Helium/Algebra/Ordered/StrictTotal/Properties/CommutativeRing.agda b/src/Helium/Algebra/Ordered/StrictTotal/Properties/CommutativeRing.agda
index aa265c8..1ccde6f 100644
--- a/src/Helium/Algebra/Ordered/StrictTotal/Properties/CommutativeRing.agda
+++ b/src/Helium/Algebra/Ordered/StrictTotal/Properties/CommutativeRing.agda
@@ -66,7 +66,7 @@ open import Helium.Algebra.Ordered.StrictTotal.Properties.Ring ring public
 
   ; x<y⇒0<y-x; 0<y-x⇒x<y
 
-  ; 0≤1; 1≈0⇒x≈y; x<y⇒0<1
+  ; 0≤1; 1≈0⇒x≈y; x≉y⇒0<1; x<y⇒0<1
 
   ; x>0⇒*-monoˡ-<; x>0⇒*-monoʳ-<; x<0⇒*-anti-monoˡ-<; x<0⇒*-anti-monoʳ-<
   ; x≥0⇒*-monoˡ-≤; x≥0⇒*-monoʳ-≤; x≤0⇒*-anti-monoˡ-≤; x≤0⇒*-anti-monoʳ-≤
diff --git a/src/Helium/Algebra/Ordered/StrictTotal/Properties/DivisionRing.agda b/src/Helium/Algebra/Ordered/StrictTotal/Properties/DivisionRing.agda
index 4531605..3961ca1 100644
--- a/src/Helium/Algebra/Ordered/StrictTotal/Properties/DivisionRing.agda
+++ b/src/Helium/Algebra/Ordered/StrictTotal/Properties/DivisionRing.agda
@@ -66,7 +66,7 @@ open import Helium.Algebra.Ordered.StrictTotal.Properties.Ring ring public
 
   ; x<y⇒0<y-x; 0<y-x⇒x<y
 
-  ; 0≤1; 1≈0⇒x≈y; x<y⇒0<1
+  ; 0≤1; 1≈0⇒x≈y; x≉y⇒0<1; x<y⇒0<1
 
   ; x>0⇒*-monoˡ-<; x>0⇒*-monoʳ-<; x<0⇒*-anti-monoˡ-<; x<0⇒*-anti-monoʳ-<
   ; x≥0⇒*-monoˡ-≤; x≥0⇒*-monoʳ-≤; x≤0⇒*-anti-monoˡ-≤; x≤0⇒*-anti-monoʳ-≤
diff --git a/src/Helium/Algebra/Ordered/StrictTotal/Properties/Field.agda b/src/Helium/Algebra/Ordered/StrictTotal/Properties/Field.agda
index 337947a..f0ae47d 100644
--- a/src/Helium/Algebra/Ordered/StrictTotal/Properties/Field.agda
+++ b/src/Helium/Algebra/Ordered/StrictTotal/Properties/Field.agda
@@ -66,7 +66,7 @@ open import Helium.Algebra.Ordered.StrictTotal.Properties.DivisionRing divisionR
 
   ; x<y⇒0<y-x; 0<y-x⇒x<y
 
-  ; 0≤1; 1≈0⇒x≈y; x<y⇒0<1
+  ; 0≤1; 1≈0⇒x≈y; x≉y⇒0<1; x<y⇒0<1
 
   ; x>0⇒*-monoˡ-<; x>0⇒*-monoʳ-<; x<0⇒*-anti-monoˡ-<; x<0⇒*-anti-monoʳ-<
   ; x≥0⇒*-monoˡ-≤; x≥0⇒*-monoʳ-≤; x≤0⇒*-anti-monoˡ-≤; x≤0⇒*-anti-monoʳ-≤
diff --git a/src/Helium/Algebra/Ordered/StrictTotal/Properties/Ring.agda b/src/Helium/Algebra/Ordered/StrictTotal/Properties/Ring.agda
index eff2d37..db070e0 100644
--- a/src/Helium/Algebra/Ordered/StrictTotal/Properties/Ring.agda
+++ b/src/Helium/Algebra/Ordered/StrictTotal/Properties/Ring.agda
@@ -365,8 +365,11 @@ x≤0∧y≤0⇒x*y≥0 {x} {y} (inj₂ x≈0) y≤0        = ≤-reflexive (Eq.
   y * 1# ≈⟨  *-identityʳ y ⟩
   y      ∎
 
+x≉y⇒0<1 : ∀ {x y} → x ≉ y → 0# < 1#
+x≉y⇒0<1 x≉y = ≤∧≉⇒< 0≤1 (x≉y ∘ 1≈0⇒x≈y ∘ Eq.sym)
+
 x<y⇒0<1 : ∀ {x y} → x < y → 0# < 1#
-x<y⇒0<1 x<y = ≤∧≉⇒< 0≤1 (<⇒≉ x<y ∘ 1≈0⇒x≈y ∘ Eq.sym)
+x<y⇒0<1 = x≉y⇒0<1 ∘ <⇒≉
 
 --------------------------------------------------------------------------------
 ---- Properties of _*_ (again)
-- 
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