From c6aee22d68aa614619cecc08a15332294a9de0de Mon Sep 17 00:00:00 2001 From: Greg Brown Date: Sat, 9 Apr 2022 16:24:18 +0100 Subject: Add some more algebraic properties. --- .../Algebra/Ordered/StrictTotal/Properties/DivisionRing.agda | 10 ++++++++++ src/Helium/Algebra/Ordered/StrictTotal/Properties/Field.agda | 2 +- 2 files changed, 11 insertions(+), 1 deletion(-) (limited to 'src/Helium/Algebra/Ordered/StrictTotal') diff --git a/src/Helium/Algebra/Ordered/StrictTotal/Properties/DivisionRing.agda b/src/Helium/Algebra/Ordered/StrictTotal/Properties/DivisionRing.agda index 3915f07..4469c03 100644 --- a/src/Helium/Algebra/Ordered/StrictTotal/Properties/DivisionRing.agda +++ b/src/Helium/Algebra/Ordered/StrictTotal/Properties/DivisionRing.agda @@ -199,6 +199,16 @@ y>0∧x0 x0) (beg 1# * y ∎) where y≉0 = <⇒≉ y>0 ∘ Eq.sym +y>0∧x≤y⇒x*y⁻¹≤1 : ∀ {x y} (y>0 : y > 0#) → x ≤ y → x * (<⇒≉ y>0 ∘ Eq.sym) ⁻¹ ≤ 1# +y>0∧x≤y⇒x*y⁻¹≤1 {x} {y} y>0 x≤y = x>0⇒*-cancelʳ-≤ y>0 (begin + x * y≉0 ⁻¹ * y ≈⟨ *-assoc x _ y ⟩ + x * (y≉0 ⁻¹ * y) ≈⟨ *-congˡ (⁻¹-inverseˡ y≉0) ⟩ + x * 1# ≈⟨ *-identityʳ x ⟩ + x ≤⟨ x≤y ⟩ + y ≈˘⟨ *-identityˡ y ⟩ + 1# * y ∎) + where y≉0 = <⇒≉ y>0 ∘ Eq.sym + -------------------------------------------------------------------------------- ---- Properties of -_ and _⁻¹ diff --git a/src/Helium/Algebra/Ordered/StrictTotal/Properties/Field.agda b/src/Helium/Algebra/Ordered/StrictTotal/Properties/Field.agda index 79d3b2d..eb46e27 100644 --- a/src/Helium/Algebra/Ordered/StrictTotal/Properties/Field.agda +++ b/src/Helium/Algebra/Ordered/StrictTotal/Properties/Field.agda @@ -126,7 +126,7 @@ open import Helium.Algebra.Ordered.StrictTotal.Properties.DivisionRing divisionR ; -‿⁻¹-comm ; x≉0⇒x⁻¹≉0 - ; y>0∧x0∧x0∧x≤y⇒x*y⁻¹≤1 ) -------------------------------------------------------------------------------- -- cgit v1.2.3