## TPTP Problem File: RAL070^1.p

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%------------------------------------------------------------------------------
% File     : RAL070^1 : TPTP v7.5.0. Released v7.0.0.
% Domain   : Real Algebra (Functions and their graphs)
% Problem  : The University of Tokyo, 2013, Humanities Course, Problem 1
% Version  : [Mat16] axioms : Especial.
% English  : Let C be the graph of the function y=x(x - 1)(x - 3), let l be
%            the straight line with the inclination t passing through the
%            origin O, and assume that C and l have common points other than
%            O.  Let O, P, and Q be the common points of C and l, and g(t),
%            the product of |vec{OP}| and |vec{OQ}|. However, if one of such
%            common points is a point of contact, the point is regarded as
%            two among O, P, and Q. Investigate the increase/decrease of the
%            function g(t), and find the extremums of the function.

% Refs     : [Mat16] Matsuzaki (2016), Email to Geoff Sutcliffe
%          : [MI+16] Matsuzaki et al. (2016), Race against the Teens - Benc
% Source   : [Mat16]
% Names    : Univ-Tokyo-2013-Bun-1.p [Mat16]

% Status   : Theorem
% Rating   : ? v7.0.0
% Syntax   : Number of formulae    : 3486 (   0 unit;1200 type;   0 defn)
%            Number of atoms       : 45368 (2216 equality;22739 variable)
%            Maximal formula depth :   35 (   9 average)
%            Number of connectives : 39651 ( 106   ~; 236   |;1185   &;35997   @)
%                                         (1095 <=>;1032  =>;   0  <=;   0 <~>)
%                                         (   0  ~|;   0  ~&)
%            Number of type conns  : 2408 (2408   >;   0   *;   0   +;   0  <<)
%            Number of symbols     : 1247 (1200   :;   0   =;   0  @=)
%                                         (   0  !!;   0  ??;   0 @@+;   0 @@-)
%            Number of variables   : 8066 (  66 sgn;7087   !; 436   ?; 407   ^)
%                                         (8066   :; 136  !>;   0  ?*)
%                                         (   0  @-;   0  @+)
%            Arithmetic symbols    : 1978 (   6 prd;   9 fun;  23 num;1940 var)
% SPC      : TH1_THM_EQU_ARI

% Comments : Theory: RCF; Author: Takuya Matsuzaki; Generated: 2013-11-11
%            ^ [V_y_opt_dot_0: \$real] :
%              ( ( V_y_opt_dot_0
%                = ( \$quotient @ ( \$sum @ 36.0 @ ( \$product @ 4.0 @ ( 'sqrt/1' @ 6.0 ) ) ) @ 9.0 ) )
%              | ( V_y_opt_dot_0
%                = ( \$quotient @ ( \$difference @ 36.0 @ ( \$product @ 4.0 @ ( 'sqrt/1' @ 6.0 ) ) ) @ 9.0 ) ) ) )
%------------------------------------------------------------------------------
include('Axioms/MAT001^0.ax').
%------------------------------------------------------------------------------
thf('t/0_type',type,(
't/0': \$real )).

thf(p_qustion,conjecture,
( 'find/1' @ \$real
@ ^ [V_y_opt: \$real] :
? [V_g: 'R2R',V_C: '2d.Shape',V_O: '2d.Point',V_l: '2d.Shape',V_P: '2d.Point',V_Q: '2d.Point',V_x_opt: \$real] :
( ( V_C
= ( '2d.graph-of/1'
@ ( 'fun/1'
@ ^ [V_x: \$real] :
( \$product @ V_x @ ( \$product @ ( \$difference @ V_x @ 1.0 ) @ ( \$difference @ V_x @ 3.0 ) ) ) ) ) )
& ( V_O = '2d.origin/0' )
& ( '2d.line-type/1' @ V_l )
& ( '2d.is-slope-of/2' @ 't/0' @ V_l )
& ( '2d.on/2' @ V_O @ V_l )
& ( '2d.on/2' @ V_P @ V_l )
& ( '2d.on/2' @ V_Q @ V_l )
& ( '2d.on/2' @ V_P @ V_C )
& ( '2d.on/2' @ V_Q @ V_C )
& ( ( V_O != V_P )
| ( V_O != V_Q ) )
& ! [V_R: '2d.Point'] :
( ( ( '2d.on/2' @ V_R @ V_C )
& ( '2d.on/2' @ V_R @ V_l ) )
=> ( ( V_R = V_O )
| ( V_R = V_P )
| ( V_R = V_Q ) ) )
& ! [V_t_dot_0: \$real] :
( ( 'funapp/2' @ V_g @ V_t_dot_0 )
= ( \$product @ ( '2d.distance/2' @ V_O @ V_P ) @ ( '2d.distance/2' @ V_O @ V_Q ) ) )
& ( 'func-extreme/3' @ V_g @ V_x_opt @ V_y_opt ) ) )).

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