## TPTP Problem File: RAL065^1.p

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```%------------------------------------------------------------------------------
% File     : RAL065^1 : TPTP v7.5.0. Released v7.0.0.
% Domain   : Real Algebra (Quadratic functions)
% Problem  : Tohoku University, 2013, Humanities Course, Problem 1
% Version  : [Mat16] axioms : Especial.
% English  : Let a be a real number. Answer the following questions: (1) Find
%            the range of the value of a such that the quadratic equation x^2
%            - 2(a + 1)x + 3 a = 0 has two different real solutions in the
%            range of - 1le xle 3. (2) When a moves in the range found in
%            (1), find the range of the values of the y coordinate of the
%            vertex of the parabola y = x^2 - 2(a + 1)x + 3 a.

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

% Status   : Theorem
% Rating   : ? v7.0.0
% Syntax   : Number of formulae    : 3485 (   0 unit;1199 type;   0 defn)
%            Number of atoms       : 45359 (2211 equality;22711 variable)
%            Maximal formula depth :   35 (   9 average)
%            Number of connectives : 39651 ( 105   ~; 233   |;1178   &;36009   @)
%                                         (1095 <=>;1031  =>;   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   : 8058 (  66 sgn;7085   !; 431   ?; 406   ^)
%                                         (8058   :; 136  !>;   0  ?*)
%                                         (   0  @-;   0  @+)
%            Arithmetic symbols    : 1977 (   6 prd;   9 fun;  23 num;1939 var)
% SPC      : TH1_THM_EQU_ARI

% Comments : Theory: RCF; Author: Takehiro Naito; Generated: 2015-01-08
%            ^ [V_a_dot_0: \$real] :
%              ( ( \$lesseq @ ( \$quotient @ -3.0 @ 5.0 ) @ V_a_dot_0 )
%              & ( \$lesseq @ V_a_dot_0 @ 1.0 ) ) )
%------------------------------------------------------------------------------
include('Axioms/MAT001^0.ax').
%------------------------------------------------------------------------------
thf(p1_qustion,conjecture,
( 'find/1' @ \$real
@ ( ^ [V_a: \$real] :
? [V_x1: \$real,V_x2: \$real] :
( ( \$lesseq @ ( \$uminus @ 1.0 ) @ V_x1 )
& ( \$lesseq @ V_x1 @ 3.0 )
& ( \$lesseq @ ( \$uminus @ 1.0 ) @ V_x2 )
& ( \$lesseq @ V_x2 @ 3.0 )
& ( V_x1 != V_x2 )
& ( 0.0
= ( 'funapp/2' @ ( 'poly-fun/1' @ ( 'cons/2' @ \$real @ ( \$product @ 3.0 @ V_a ) @ ( 'cons/2' @ \$real @ ( \$uminus @ ( \$product @ 2.0 @ ( \$sum @ V_a @ 1.0 ) ) ) @ ( 'cons/2' @ \$real @ 1.0 @ ( 'nil/0' @ \$real ) ) ) ) ) @ V_x1 ) )
& ( 0.0
= ( 'funapp/2' @ ( 'poly-fun/1' @ ( 'cons/2' @ \$real @ ( \$product @ 3.0 @ V_a ) @ ( 'cons/2' @ \$real @ ( \$uminus @ ( \$product @ 2.0 @ ( \$sum @ V_a @ 1.0 ) ) ) @ ( 'cons/2' @ \$real @ 1.0 @ ( 'nil/0' @ \$real ) ) ) ) ) @ V_x2 ) ) ) ) )).
%------------------------------------------------------------------------------
```