formulas(assumptions).

% This is a problem about an ordering relation.

% Mace4 should produce a model immediately.

% Note that the symbol "<" has no built-in meaning to Mace4 (or to Prover9).

all x exists y (x < y).
all x all y (x < y -> -(y < x)).
all x all y (x < y -> (exists z (x < z & z < y))).

% Here is the source of this problem.

% From: Stefan Kauer <kauer@minet.uni-jena.de>
% Date: Fri, 08 Jan 1999 15:27:29 +0100
% To: mccune@mcs.anl.gov
% 
% Dear Prof. McCune,
% 
% I found the following problem: the following 3 axioms for a relation <
% are given:
%       1. all x exists y: x < y
%       2. all x y: x < y => not y < x 
%       3. all x y: x < y => exists z: x < z and z < y
% 
% One has to find a finite model. It is surprising and not obvious, that a
% finite model exists, and it is not at all to find easily (at least for
% me, may be for you). Since I sometimes work with OTTER, I remembered
% MACE and tried to formulate and solve this problem with MACE, but
% unfortunately was able to do it. 
% Is it possible to formulate and solve this problem with MACE and how?
% 
% This problem was posed as an exercise in a very small german
% mathematical magazin by Dr. Waldmann, a former colleague of mine, who
% works now at another university. We both agree that the smallest model
% has seven elements.
% 
% Sincerely yours
% 
% Stefan Kauer

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