## TPTP Problem File: LCL729^5.p

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```%------------------------------------------------------------------------------
% File     : LCL729^5 : TPTP v7.2.0. Released v4.0.0.
% Domain   : Logical Calculi
% Problem  : TPS problem THM560
% Version  : Especial.
% English  : AC1(A) equiv AC3(OA,A) from [RR93]. Note that AC3 usually refers
%            to 'relations' where here we are using it for relations on OA x A.

% Refs     : [RR93]  Rubin & Rubin (1993), Weak Forms of the Axiom of Choic
%          : [Bro09] Brown (2009), Email to Geoff Sutcliffe
% Source   : [Bro09]
% Names    : tps_0513 [Bro09]
%          : THM560 [TPS]

% Status   : Theorem
% Rating   : 0.80 v7.2.0, 0.75 v7.1.0, 0.86 v7.0.0, 0.88 v6.4.0, 0.86 v6.3.0, 0.83 v5.5.0, 0.80 v5.4.0, 0.75 v5.3.0, 1.00 v5.2.0, 0.75 v4.1.0, 0.67 v4.0.0
% Syntax   : Number of formulae    :    2 (   0 unit;   1 type;   0 defn)
%            Number of atoms       :   16 (   0 equality;  16 variable)
%            Maximal formula depth :    9 (   6 average)
%            Number of connectives :   15 (   0   ~;   0   |;   0   &;  10   @)
%                                         (   1 <=>;   4  =>;   0  <=;   0 <~>)
%                                         (   0  ~|;   0  ~&)
%            Number of type conns  :   12 (  12   >;   0   *;   0   +;   0  <<)
%            Number of symbols     :    3 (   1   :;   0   =)
%            Number of variables   :    9 (   0 sgn;   5   !;   4   ?;   0   ^)
%                                         (   9   :;   0  !>;   0  ?*)
%                                         (   0  @-;   0  @+)
% SPC      : TH0_THM_NEQ_NAR

% Comments : This problem is from the TPS library. Copyright (c) 2009 The TPS
%            project in the Department of Mathematical Sciences at Carnegie
%          :
%------------------------------------------------------------------------------
thf(a_type,type,(
a: \$tType )).

thf(cTHM560,conjecture,
( ! [Xr: ( a > \$o ) > a > \$o] :
? [Xg: ( a > \$o ) > a] :
! [Xx: a > \$o] :
( ? [Xy: a] :
( Xr @ Xx @ Xy )
=> ( Xr @ Xx @ ( Xg @ Xx ) ) )
<=> ! [Xs: ( a > \$o ) > \$o] :
( ! [X: a > \$o] :
( ( Xs @ X )
=> ? [Xt: a] :
( X @ Xt ) )
=> ? [Xf: ( a > \$o ) > a] :
! [X: a > \$o] :
( ( Xs @ X )
=> ( X @ ( Xf @ X ) ) ) ) )).

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