%--------------------------------------------------------------------------
% File     : MGT002+0 : TPTP v9.3.1. Released v9.3.0.
% Domain   : Management
% Axioms   : Strict total order for interval reasoning
% Version  : [Han98] axioms.
% English  : Foundation axioms for axis-specific constraint reasoning.
%            Defines a strict total order (less/2) and derived non-strict
%            order (leq/2). Works for both discrete (N) and continuous (R)
%            domains. 

% Refs     : [MC+26] Mustafa et al. (2026), Axis Decomposition for ODRL
% Source   : [MC+26]
% Names    : ORD000-0.ax [MC+26]

% Status   : Satisfiable
% Syntax   : Number of formulae    :    9 (   1 unt;   0 def)
%            Number of atoms       :   24 (   3 equ)
%            Maximal formula atoms :    3 (   2 avg)
%            Number of connectives :   16 (   1   ~;   3   |;   5   &)
%                                         (   1 <=>;   6  =>;   0  <=;   0 <~>)
%            Maximal formula depth :    6 (   5 avg)
%            Maximal term depth    :    1 (   1 avg)
%            Number of predicates  :    3 (   2 usr;   0 prp; 2-2 aty)
%            Number of functors    :    0 (   0 usr;   0 con; --- aty)
%            Number of variables   :   21 (  21   !;   0   ?)
% SPC      : FOF_SAT_RFO_SEQ

% Comments : 
%--------------------------------------------------------------------------
%----Core: Strict total order (3 axioms)
%----Irreflexivity: nothing is less than itself
fof(irreflexive,axiom,
    ! [X] : ~ less(X,X) ).

%----Transitivity: less-than chains
fof(transitive,axiom,
    ! [X,Y,Z] :
      ( ( less(X,Y)
        & less(Y,Z) )
     => less(X,Z) ) ).

%----Trichotomy: for any two elements, exactly one of <, =, > holds
fof(trichotomy,axiom,
    ! [X,Y] :
      ( less(X,Y)
      | X = Y
      | less(Y,X) ) ).

%----Derived: Non-strict order (1 axiom)
%----leq defined as the reflexive closure of less
fof(leq_def,axiom,
    ! [X,Y] :
      ( leq(X,Y)
    <=> ( less(X,Y)
        | X = Y ) ) ).

%----Helper lemmas (derivable but aid Vampire/E performance)
%----Antisymmetry of leq
fof(leq_antisym,axiom,
    ! [X,Y] :
      ( ( leq(X,Y)
        & leq(Y,X) )
     => X = Y ) ).

%----less implies leq
fof(less_implies_leq,axiom,
    ! [X,Y] :
      ( less(X,Y)
     => leq(X,Y) ) ).

%----Mixed transitivity
fof(leq_less_trans,axiom,
    ! [X,Y,Z] :
      ( ( leq(X,Y)
        & less(Y,Z) )
     => less(X,Z) ) ).

fof(less_leq_trans,axiom,
    ! [X,Y,Z] :
      ( ( less(X,Y)
        & leq(Y,Z) )
     => less(X,Z) ) ).

fof(leq_trans,axiom,
    ! [X,Y,Z] :
      ( ( leq(X,Y)
        & leq(Y,Z) )
     => leq(X,Z) ) ).

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