Interface:First-order linear order defined via nonstrict inequality

A linear order or total order has an ordering predicate which lets us compare any two elements for order.

Here we define a linear order in terms of what is called non-strict inequality, or "less than or equal to".

This is a first-order presentation (that is, not built on set theory).

We start by importing propositional and predicate logic.  param (PROPOSITIONAL Interface:Classical_propositional_calculus ) param (FIRSTORDER Interface:First-order_logic_with_quantifiability (PROPOSITIONAL) ) 

Non-strict order
There is an order which obeys the usual equality axioms.  var (object x y z x0 y0 z0 x1 y1 z1)

term (formula (≤ object object)) stmt (LessEqualBuilder  (((x0 = x1) ∧ (y0 = y1)) → ((x0 ≤ y0) ↔ (x1 ≤ y1)))) 

It is antisymmetric, transitive, and total.  stmt (LessEqualAntisymmetry  (((x ≤ y) ∧ (y ≤ x)) → (x = y))) stmt (LessEqualTransitivity  (((x ≤ y) ∧ (y ≤ z)) → (x ≤ z))) stmt (LessEqualTotality  ((x ≤ y) ∨ (y ≤ x))) 

That is sufficient. These properties imply the theorems in Interface:First-order linear order.