Lots of speedups (#116)

Setoids/Orders/Total/Definition.agda

@@ -0,0 +1,33 @@
+{-# OPTIONS --safe --warning=error --without-K #-}
+
+open import LogicalFormulae
+open import Orders.Total.Definition
+open import Orders.Partial.Definition
+open import Setoids.Setoids
+open import Functions
+
+open import Agda.Primitive using (Level; lzero; lsuc; _⊔_)
+
+module Setoids.Orders.Total.Definition where
+
+open import Setoids.Orders.Partial.Definition
+
+record SetoidTotalOrder {a b c : _} {A : Set a} {S : Setoid {a} {b} A} {_<_ : Rel {a} {c} A} (P : SetoidPartialOrder S _<_) : Set (a ⊔ b ⊔ c) where
+  open Setoid S
+  field
+    totality : (a b : A) → ((a < b) || (b < a)) || (a ∼ b)
+  partial : SetoidPartialOrder S _<_
+  partial = P
+  min : A → A → A
+  min a b with totality a b
+  min a b | inl (inl a<b) = a
+  min a b | inl (inr b<a) = b
+  min a b | inr a=b = a
+  max : A → A → A
+  max a b with totality a b
+  max a b | inl (inl a<b) = b
+  max a b | inl (inr b<a) = a
+  max a b | inr a=b = b
+
+totalOrderToSetoidTotalOrder : {a b : _} {A : Set a} (T : TotalOrder {a} A {b}) → SetoidTotalOrder (partialOrderToSetoidPartialOrder (TotalOrder.order T))
+SetoidTotalOrder.totality (totalOrderToSetoidTotalOrder T) = TotalOrder.totality T