diff options
Diffstat (limited to 'src/Helium')
4 files changed, 7 insertions, 4 deletions
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) |