## TPTP Problem File: KLE010+4.p

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```%------------------------------------------------------------------------------
% File     : KLE010+4 : TPTP v7.5.0. Released v4.0.0.
% Domain   : Kleene Algebra (Idempotent Test Semirings)
% Problem  : Split 1 into all combinations of p,q and their complements
% Version  : [Hoe08] axioms : Augmented.
% English  :

% Refs     : [Hoe08] Hoefner (2008), Email to G. Sutcliffe
% Source   : [Hoe08]
% Names    :

% Status   : Theorem
% Rating   : 0.36 v7.5.0, 0.41 v7.4.0, 0.30 v7.3.0, 0.34 v7.2.0, 0.38 v7.1.0, 0.35 v7.0.0, 0.43 v6.4.0, 0.42 v6.3.0, 0.33 v6.2.0, 0.40 v6.1.0, 0.57 v5.5.0, 0.63 v5.4.0, 0.61 v5.3.0, 0.67 v5.2.0, 0.50 v5.1.0, 0.52 v5.0.0, 0.54 v4.1.0, 0.52 v4.0.0
% Syntax   : Number of formulae    :   19 (  11 unit)
%            Number of atoms       :   34 (  19 equality)
%            Maximal formula depth :    6 (   4 average)
%            Number of connectives :   16 (   1   ~;   0   |;   6   &)
%                                         (   4 <=>;   5  =>;   0  <=;   0 <~>)
%                                         (   0  ~|;   0  ~&)
%            Number of predicates  :    4 (   0 propositional; 1-2 arity)
%            Number of functors    :    5 (   2 constant; 0-2 arity)
%            Number of variables   :   35 (   0 sgn;  34   !;   1   ?)
%            Maximal term depth    :    7 (   2 average)
% SPC      : FOF_THM_RFO_SEQ

% Comments : Inequational encoding : additionally the de Morgan laws : proof
%            goal is split into 2 inequations.
%------------------------------------------------------------------------------
%---Include axioms for idempotent test semiring
include('Axioms/KLE001+0.ax').
%---Include test axioms
include('Axioms/KLE001+1.ax').
%---Include additionally axioms
include('Axioms/KLE001+2.ax').
%------------------------------------------------------------------------------
fof(goals,conjecture,(
! [X0,X1] :
( ( test(X1)
& test(X0) )
=> ( leq(one,addition(addition(addition(addition(multiplication(X1,X0),multiplication(c(X1),X0)),multiplication(X0,X1)),multiplication(c(X0),X1)),multiplication(c(X0),c(X1))))
& leq(addition(addition(addition(addition(multiplication(X1,X0),multiplication(c(X1),X0)),multiplication(X0,X1)),multiplication(c(X0),X1)),multiplication(c(X0),c(X1))),one) ) ) )).

%------------------------------------------------------------------------------
```