## TPTP Problem File: RAL067^1.p

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```%------------------------------------------------------------------------------
% File     : RAL067^1 : TPTP v7.5.0. Released v7.0.0.
% Domain   : Real Algebra (Functions and their graphs)
% Problem  : The University of Tokyo, 1989, Humanities Course, Problem 2
% Version  : [Mat16] axioms : Especial.
% English  : Consider the following two parabolas for a > 0: C_1 : y = x^2
%            +1/a^2 C_2 : y = -(x-a)^2 (1) Prove that there always exist 2
%            straight lines that are in contact with both C_1 and C_2. (2) Find
%            the minimum value of the area S(a) of the quadrilateral defined by
%            the four points determined in (1).

% Refs     : [Mat16] Matsuzaki (2016), Email to Geoff Sutcliffe
%          : [MI+16] Matsuzaki et al. (2016), Race against the Teens - Benc
% Source   : [Mat16]
% Names    : Univ-Tokyo-1989-Bun-2.p [Mat16]

% Status   : Theorem
% Rating   : ? v7.0.0
% Syntax   : Number of formulae    : 3485 (   0 unit;1199 type;   0 defn)
%            Number of atoms       : 45345 (2213 equality;22722 variable)
%            Maximal formula depth :   35 (   9 average)
%            Number of connectives : 39633 ( 105   ~; 233   |;1180   &;35988   @)
%                                         (1095 <=>;1032  =>;   0  <=;   0 <~>)
%                                         (   0  ~|;   0  ~&)
%            Number of type conns  : 2408 (2408   >;   0   *;   0   +;   0  <<)
%            Number of symbols     : 1246 (1199   :;   0   =;   0  @=)
%                                         (   0  !!;   0  ??;   0 @@+;   0 @@-)
%            Number of variables   : 8066 (  66 sgn;7086   !; 437   ?; 407   ^)
%                                         (8066   :; 136  !>;   0  ?*)
%                                         (   0  @-;   0  @+)
%            Arithmetic symbols    : 1977 (   6 prd;   9 fun;  23 num;1939 var)
% SPC      : TH1_THM_EQU_ARI

% Comments : Theory: RCF; Author: Yiyang Zhan; Generated: 2014-03-13
%------------------------------------------------------------------------------
include('Axioms/MAT001^0.ax').
%------------------------------------------------------------------------------
thf(p1,conjecture,(
! [V_a: \$real] :
( ( \$greater @ V_a @ 0.0 )
=> ? [V_C1: '2d.Shape',V_C2: '2d.Shape'] :
( ( V_C1
= ( '2d.graph-of/1'
@ ( 'fun/1'
@ ( ^ [V_x_dot_0: \$real] :
( \$sum @ ( '^/2' @ V_x_dot_0 @ 2.0 ) @ ( \$quotient @ 1.0 @ ( '^/2' @ V_a @ 2.0 ) ) ) ) ) ) )
& ( V_C2
= ( '2d.graph-of/1'
@ ( 'fun/1'
@ ( ^ [V_x: \$real] :
( \$uminus @ ( '^/2' @ ( \$difference @ V_x @ V_a ) @ 2.0 ) ) ) ) ) )
& ? [V_p1: '2d.Point',V_p2: '2d.Point',V_q1: '2d.Point',V_q2: '2d.Point',V_l: '2d.Shape',V_m: '2d.Shape'] :
( ( V_l
= ( '2d.line/2' @ V_p1 @ V_p2 ) )
& ( V_m
= ( '2d.line/2' @ V_q1 @ V_q2 ) )
& ( V_l != V_m )
& ( '2d.tangent/2' @ V_l @ V_C1 )
& ( '2d.tangent/2' @ V_l @ V_C2 )
& ( '2d.tangent/2' @ V_m @ V_C1 )
& ( '2d.tangent/2' @ V_m @ V_C2 ) ) ) ) )).
%------------------------------------------------------------------------------
```