## TPTP Problem File: RAL055^1.p

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```%------------------------------------------------------------------------------
% File     : RAL055^1 : TPTP v7.5.0. Released v7.0.0.
% Domain   : Real Algebra (Complex numbers and complex plane)
% Problem  : Kyushu University, 2001, Science Course, Problem 4
% Version  : [Mat16] axioms : Especial.
% English  : Consider the point z on the complex plane. (1) If the real numbers
%            a and c, and the complex number b satisfy |b|^2 - a c > 0, find
%            the figure drawn by the point z that satisfies a zoverline{z} +
%            overline{b} z + boverline{z} + c = 0 when a != 0, where
%            overline{z} is the complex number conjugate to z. (2) For the
%            complex number d other than 0 and the 2 different points p and q
%            on the complex plane, find the figure drawn by the point z that
%            satisfies d(z - p)(overline{z} -overline{q})=overline{d}(z
%            - q)(overline{z} -overline{p}).

% Refs     : [Mat16] Matsuzaki (2016), Email to Geoff Sutcliffe
%          : [MI+16] Matsuzaki et al. (2016), Race against the Teens - Benc
% Source   : [Mat16]
% Names    : Univ-Kyushu-2001-Ri-4.p [Mat16]

% Status   : Theorem
% Rating   : ? v7.0.0
% Syntax   : Number of formulae    : 3495 (   0 unit;1209 type;   0 defn)
%            Number of atoms       : 45347 (2213 equality;22709 variable)
%            Maximal formula depth :   35 (   9 average)
%            Number of connectives : 39635 ( 105   ~; 233   |;1176   &;35995   @)
%                                         (1095 <=>;1031  =>;   0  <=;   0 <~>)
%                                         (   0  ~|;   0  ~&)
%            Number of type conns  : 2408 (2408   >;   0   *;   0   +;   0  <<)
%            Number of symbols     : 1256 (1209   :;   0   =;   0  @=)
%                                         (   0  !!;   0  ??;   0 @@+;   0 @@-)
%            Number of variables   : 8059 (  66 sgn;7085   !; 431   ?; 407   ^)
%                                         (8059   :; 136  !>;   0  ?*)
%                                         (   0  @-;   0  @+)
%            Arithmetic symbols    : 1974 (   6 prd;   9 fun;  23 num;1936 var)
% SPC      : TH1_THM_EQU_ARI

% Comments : Theory: RCF; Author: Tomoya Ishii; Generated: 2014-05-28
%            ^ [V_Dz_dot_0: '2d.Shape'] :
%              ( ( 'a/0' != 0.0 )
%              & ( \$greater @ ( \$difference @ ( '^/2' @ ( 'complex.abs/1' @ ( 'complex.complex/2' @ 'b1/0' @ 'b2/0' ) ) @ 2.0 ) @ ( \$product @ 'a/0' @ 'c/0' ) ) @ 0.0 )
%              & ( V_Dz_dot_0
%                = ( '2d.circle/2' @ ( 'complex.complex->point/1' @ ( 'complex.//2' @ ( 'complex.complex/2' @ 'b1/0' @ 'b2/0' ) @ ( 'complex.real->complex/1' @ ( \$uminus @ 'a/0' ) ) ) ) @ ( \$quotient @ ( 'sqrt/1' @ ( \$difference @ ( '^/2' @ ( 'complex.abs/1' @ ( 'complex.complex/2' @ 'b1/0' @ 'b2/0' ) ) @ 2.0 ) @ ( \$product @ 'a/0' @ 'c/0' ) ) ) @ ( 'abs/1' @ 'a/0' ) ) ) ) ) )
%------------------------------------------------------------------------------
include('Axioms/MAT001^0.ax').
%------------------------------------------------------------------------------
thf('a/0_type',type,(
'a/0': \$real )).

thf('b1/0_type',type,(
'b1/0': \$real )).

thf('b2/0_type',type,(
'b2/0': \$real )).

thf('c/0_type',type,(
'c/0': \$real )).

thf('d1/0_type',type,(
'd1/0': \$real )).

thf('d2/0_type',type,(
'd2/0': \$real )).

thf('p1/0_type',type,(
'p1/0': \$real )).

thf('p2/0_type',type,(
'p2/0': \$real )).

thf('q1/0_type',type,(
'q1/0': \$real )).

thf('q2/0_type',type,(
'q2/0': \$real )).

thf(p1_D_qustion,conjecture,
( 'find/1' @ '2d.Shape'
@ ( ^ [V_Dz: '2d.Shape'] :
( ( 'a/0' != 0.0 )
& ( \$greater @ ( \$difference @ ( '^/2' @ ( 'complex.abs/1' @ ( 'complex.complex/2' @ 'b1/0' @ 'b2/0' ) ) @ 2.0 ) @ ( \$product @ 'a/0' @ 'c/0' ) ) @ 0.0 )
& ( V_Dz
= ( '2d.shape-of-cpfun/1'
@ ( ^ [V_Pz: '2d.Point'] :
? [V_z: 'complex.Complex',V_b: 'complex.Complex'] :
( ( V_Pz
= ( 'complex.complex->point/1' @ V_z ) )
& ( V_b
= ( 'complex.complex/2' @ 'b1/0' @ 'b2/0' ) )
& ( ( 'complex.+/2' @ ( 'complex.*/2' @ ( 'complex.real->complex/1' @ 'a/0' ) @ ( 'complex.*/2' @ V_z @ ( 'complex.conjugate/1' @ V_z ) ) ) @ ( 'complex.+/2' @ ( 'complex.*/2' @ ( 'complex.conjugate/1' @ V_b ) @ V_z ) @ ( 'complex.+/2' @ ( 'complex.*/2' @ V_b @ ( 'complex.conjugate/1' @ V_z ) ) @ ( 'complex.real->complex/1' @ 'c/0' ) ) ) )
= ( 'complex.complex/2' @ 0.0 @ 0.0 ) ) ) ) ) ) ) ) )).
%------------------------------------------------------------------------------
```