%--------------------------------------------------------------------------
% File     : MGT002+1 : TPTP v9.3.1. Released v9.3.0.
% Domain   : Management
% Axioms   : Density axiom for continuous (real-valued) domains
% Version  : [Han98] axioms.
% English  : Layer 1 axioms for the ODRL Axis Decomposition benchmark.

% Refs     : [MC+26] Mustafa et al. (2026), Axis Decomposition for ODRL
% Source   : [MC+26]
% Names    : ORD001-0.ax [MC+26]

% Status   : Satisfiable
% Syntax   : Number of formulae    :    1 (   0 unt;   0 def)
%            Number of atoms       :    3 (   0 equ)
%            Maximal formula atoms :    3 (   3 avg)
%            Number of connectives :    2 (   0   ~;   0   |;   1   &)
%                                         (   0 <=>;   1  =>;   0  <=;   0 <~>)
%            Maximal formula depth :    6 (   6 avg)
%            Maximal term depth    :    1 (   1 avg)
%            Number of predicates  :    1 (   1 usr;   0 prp; 2-2 aty)
%            Number of functors    :    0 (   0 usr;   0 con; --- aty)
%            Number of variables   :    3 (   2   !;   1   ?)
% SPC      : FOF_SAT_RFO_NEQ

% Comments : Requires ORD000-0.ax.
%--------------------------------------------------------------------------
%----Density: between any two distinct values exists a third.
%----Required for open-interval witnesses over continuous domains (R, Q).
%----Example where density matters:
%----  Policy 1: width gt 599  -> denotation (599, inf)
%----  Policy 2: width lt 601  -> denotation (-inf, 601)
%----  Intersection: (599, 601) -- non-empty only if dense.
fof(dense,axiom,
    ! [X,Y] :
      ( less(X,Y)
     => ? [Z] :
          ( less(X,Z)
          & less(Z,Y) ) ) ).

%--------------------------------------------------------------------------