## TPTP Problem File: RAL069^1.p

View Solutions - Solve Problem

```%------------------------------------------------------------------------------
% File     : RAL069^1 : TPTP v7.5.0. Released v7.0.0.
% Domain   : Real Algebra (Functions and their graphs)
% Problem  : The University of Tokyo, 2011, Humanities Course, Problem 1
% Version  : [Mat16] axioms : Especial.
% English  : Assume that the cubic function f(x)=a x^3+b x^2+c x+d satisfies
%            all the three conditions, f(1)=1, f(-, 1)=-1, and int_{-1}^1(b
%            x^2 + c x + d)dx=1. Find f(x) that give the minimum value of the
%            definite integral I=int_{-1}^{1/2}{f''(x)}^2 dx@ and the value of
%            I that gives the minimum value, where f''(x) is a derivative of
%            f'(x).

% Refs     : [Mat16] Matsuzaki (2016), Email to Geoff Sutcliffe
%          : [MI+16] Matsuzaki et al. (2016), Race against the Teens - Benc
% Source   : [Mat16]
% Names    : Univ-Tokyo-2011-Bun-1.p [Mat16]

% Status   : Theorem
% Rating   : ? v7.0.0
% Syntax   : Number of formulae    : 3489 (   0 unit;1203 type;   0 defn)
%            Number of atoms       : 45417 (2219 equality;22725 variable)
%            Maximal formula depth :   35 (   9 average)
%            Number of connectives : 39693 ( 104   ~; 233   |;1182   &;36048   @)
%                                         (1095 <=>;1031  =>;   0  <=;   0 <~>)
%                                         (   0  ~|;   0  ~&)
%            Number of type conns  : 2408 (2408   >;   0   *;   0   +;   0  <<)
%            Number of symbols     : 1250 (1203   :;   0   =;   0  @=)
%                                         (   0  !!;   0  ??;   0 @@+;   0 @@-)
%            Number of variables   : 8068 (  66 sgn;7085   !; 436   ?; 411   ^)
%                                         (8068   :; 136  !>;   0  ?*)
%                                         (   0  @-;   0  @+)
%            Arithmetic symbols    : 1984 (   6 prd;   9 fun;  23 num;1946 var)
% SPC      : TH1_THM_EQU_ARI

% Comments : Theory: RCF; Author: Yiyang Zhan; Generated: 2013-11-21
%            ^ [V_f_dot_1: 'R2R'] :
%              ( ( V_f_dot_1
%                = ( 'poly-fun/1' @ ( 'cons/2' @ \$real @ ( \$quotient @ 3.0 @ 4.0 ) @ ( 'cons/2' @ \$real @ ( \$quotient @ 5.0 @ 4.0 ) @ ( 'cons/2' @ \$real @ ( \$uminus @ ( \$quotient @ 3.0 @ 4.0 ) ) @ ( 'cons/2' @ \$real @ ( \$uminus @ ( \$quotient @ 1.0 @ 4.0 ) ) @ ( 'nil/0' @ \$real ) ) ) ) ) ) )
%              & ( 'a/0'
%                = ( \$uminus @ ( \$quotient @ 1.0 @ 4.0 ) ) )
%              & ( 'b/0'
%                = ( \$uminus @ ( \$quotient @ 3.0 @ 4.0 ) ) )
%              & ( 'c/0'
%                = ( \$quotient @ 5.0 @ 4.0 ) )
%              & ( 'd/0'
%                = ( \$quotient @ 3.0 @ 4.0 ) ) ) )
%------------------------------------------------------------------------------
include('Axioms/MAT001^0.ax').
%------------------------------------------------------------------------------
thf('a/0_type',type,(
'a/0': \$real )).

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

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

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

thf(p1_qustion,conjecture,
( 'find/1' @ 'R2R'
@ ( ^ [V_f: 'R2R'] :
? [V_I_set: ( 'SetOf' @ \$real ),V_I_min: \$real] :
( ( 'minimum/2' @ V_I_set @ V_I_min )
& ( V_I_set
= ( 'set-by-def/1' @ \$real
@ ( ^ [V_I: \$real] :
? [V_a_dot_0: \$real,V_b_dot_0: \$real,V_c_dot_0: \$real,V_d_dot_0: \$real,V_f_dot_0: 'R2R'] :
( ( V_f_dot_0
= ( 'poly-fun/1' @ ( 'cons/2' @ \$real @ V_d_dot_0 @ ( 'cons/2' @ \$real @ V_c_dot_0 @ ( 'cons/2' @ \$real @ V_b_dot_0 @ ( 'cons/2' @ \$real @ V_a_dot_0 @ ( 'nil/0' @ \$real ) ) ) ) ) ) )
& ( ( 'funapp/2' @ V_f_dot_0 @ 1.0 )
= 1.0 )
& ( ( 'funapp/2' @ V_f_dot_0 @ -1.0 )
= -1.0 )
& ( ( 'integral/3'
@ ( ^ [V_x_dot_2: \$real] :
( \$sum @ ( \$product @ V_b_dot_0 @ ( '^/2' @ V_x_dot_2 @ 2.0 ) ) @ ( \$sum @ ( \$product @ V_c_dot_0 @ V_x_dot_2 ) @ V_d_dot_0 ) ) ) @ -1.0 @ 1.0 )
= 1.0 )
& ( V_I
= ( 'integral/3'
@ ( ^ [V_x_dot_1: \$real] :
( '^/2' @ ( 'funapp/2' @ ( 'derivative/1' @ ( 'derivative/1' @ V_f_dot_0 ) ) @ V_x_dot_1 ) @ 2.0 ) ) @ -1.0 @ ( \$quotient @ 1.0 @ 2.0 ) ) ) ) ) ) )
& ( V_f
= ( 'poly-fun/1' @ ( 'cons/2' @ \$real @ 'd/0' @ ( 'cons/2' @ \$real @ 'c/0' @ ( 'cons/2' @ \$real @ 'b/0' @ ( 'cons/2' @ \$real @ 'a/0' @ ( 'nil/0' @ \$real ) ) ) ) ) ) )
& ( ( 'funapp/2' @ V_f @ 1.0 )
= 1.0 )
& ( ( 'funapp/2' @ V_f @ -1.0 )
= -1.0 )
& ( ( 'integral/3'
@ ( ^ [V_x_dot_0: \$real] :
( \$sum @ ( \$product @ 'b/0' @ ( '^/2' @ V_x_dot_0 @ 2.0 ) ) @ ( \$sum @ ( \$product @ 'c/0' @ V_x_dot_0 ) @ 'd/0' ) ) ) @ -1.0 @ 1.0 )
= 1.0 )
& ( V_I_min
= ( 'integral/3'
@ ( ^ [V_x: \$real] :
( '^/2' @ ( 'funapp/2' @ ( 'derivative/1' @ ( 'derivative/1' @ V_f ) ) @ V_x ) @ 2.0 ) ) @ -1.0 @ ( \$quotient @ 1.0 @ 2.0 ) ) ) ) ) )).
%------------------------------------------------------------------------------
```