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Diffstat (limited to 'src/Helium/Algebra/Ordered/StrictTotal/Properties/CommutativeRing.agda')
-rw-r--r-- | src/Helium/Algebra/Ordered/StrictTotal/Properties/CommutativeRing.agda | 27 |
1 files changed, 27 insertions, 0 deletions
diff --git a/src/Helium/Algebra/Ordered/StrictTotal/Properties/CommutativeRing.agda b/src/Helium/Algebra/Ordered/StrictTotal/Properties/CommutativeRing.agda new file mode 100644 index 0000000..24b2663 --- /dev/null +++ b/src/Helium/Algebra/Ordered/StrictTotal/Properties/CommutativeRing.agda @@ -0,0 +1,27 @@ +------------------------------------------------------------------------ +-- Agda Helium +-- +-- Algebraic properties of ordered commutative rings +------------------------------------------------------------------------ + +{-# OPTIONS --safe --without-K #-} + +open import Helium.Algebra.Ordered.StrictTotal.Bundles + +module Helium.Algebra.Ordered.StrictTotal.Properties.CommutativeRing + {ℓ₁ ℓ₂ ℓ₃} + (commutativeRing : CommutativeRing ℓ₁ ℓ₂ ℓ₃) + where + +open CommutativeRing commutativeRing + +open import Algebra.Properties.Ring Unordered.ring public + renaming (-0#≈0# to -0≈0) +open import Algebra.Properties.Semiring.Mult.TCOptimised Unordered.semiring public +open import Algebra.Properties.CommutativeSemiring.Exp.TCOptimised Unordered.commutativeSemiring public +open import Helium.Algebra.Ordered.StrictTotal.Properties.Ring ring public + using + ( <⇒≱; ≤⇒≯; >⇒≉; ≈⇒≯; <⇒≉; ≈⇒≮; ≤∧≉⇒<; ≥∧≉⇒> + ; x<y⇒0<y-x; -‿anti-mono + ; ⊤′; number; negative + ; 0≤1; 1≉0+n≉0⇒0<+n; 1≉0+n≉0⇒-n<0) |