First Git commit

IntegralDomains.agda

@@ -0,0 +1,17 @@
+{-# OPTIONS --safe --warning=error #-}
+
+open import LogicalFormulae
+open import Groups
+open import Naturals
+open import Orders
+open import Setoids
+open import Functions
+open import Rings
+
+open import Agda.Primitive using (Level; lzero; lsuc; _⊔_)
+
+module IntegralDomains where
+  record IntegralDomain {n m : _} {A : Set n} {S : Setoid {n} {m} A} {_+_ : A → A → A} {_*_ : A → A → A} (R : Ring S _+_ _*_) : Set (lsuc m ⊔ n) where
+    field
+      intDom : {a b : A} → Setoid._∼_ S (a * b) (Ring.0R R) → (Setoid._∼_ S a (Ring.0R R)) || (Setoid._∼_ S b (Ring.0R R))
+      nontrivial : Setoid._∼_ S (Ring.1R R) (Ring.0R R) → False