## TPTP Problem File: RAL064^1.p

View Solutions - Solve Problem

```%------------------------------------------------------------------------------
% File     : RAL064^1 : TPTP v7.5.0. Released v7.0.0.
% Domain   : Real Algebra (Functions and their graphs)
% Problem  : Tohoku University, 2005, Science Course, Problem 4
% Version  : [Mat16] axioms : Especial.
% English  : Let a be a negative real number, and consider the parabola C_1:
%            y = a x^2 + b x + c. When C_1 is in contact with the curve C_2:
%            y = x^2-x+3/4 & (if x > 0) x^2+2 x+3/4 & (if xle 0) at two points,
%            represent the area of the region enclosed by C_1 and C_2 using 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-2005-Ri-4.p [Mat16]

% Status   : Theorem
% Rating   : ? v7.0.0
% Syntax   : Number of formulae    : 3488 (   0 unit;1202 type;   0 defn)
%            Number of atoms       : 45379 (2214 equality;22722 variable)
%            Maximal formula depth :   35 (   9 average)
%            Number of connectives : 39665 ( 105   ~; 233   |;1179   &;36020   @)
%                                         (1095 <=>;1033  =>;   0  <=;   0 <~>)
%                                         (   0  ~|;   0  ~&)
%            Number of type conns  : 2408 (2408   >;   0   *;   0   +;   0  <<)
%            Number of symbols     : 1249 (1202   :;   0   =;   0  @=)
%                                         (   0  !!;   0  ??;   0 @@+;   0 @@-)
%            Number of variables   : 8062 (  66 sgn;7085   !; 433   ?; 408   ^)
%                                         (8062   :; 136  !>;   0  ?*)
%                                         (   0  @-;   0  @+)
%            Arithmetic symbols    : 1976 (   6 prd;   9 fun;  23 num;1938 var)
% SPC      : TH1_THM_EQU_ARI

% Comments : Theory: RCF; Author: Yiyang Zhan; Generated: 2014-04-16
%            ^ [V_S_dot_0: \$real] :
%              ( V_S_dot_0
%              = ( \$quotient @ 9.0 @ ( \$product @ 32.0 @ ( '^/2' @ ( \$difference @ 1.0 @ 'a/0' ) @ 2.0 ) ) ) ) )
%------------------------------------------------------------------------------
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(p_qustion,conjecture,
( 'find/1' @ \$real
@ ^ [V_S: \$real] :
? [V_C1: '2d.Shape',V_C2: '2d.Shape'] :
( ( \$less @ 'a/0' @ 0.0 )
& ( V_C1
= ( '2d.graph-of/1'
@ ( 'fun/1'
@ ^ [V_x: \$real] :
( \$sum @ ( \$product @ 'a/0' @ ( '^/2' @ V_x @ 2.0 ) ) @ ( \$sum @ ( \$product @ 'b/0' @ V_x ) @ 'c/0' ) ) ) ) )
& ( V_C2
= ( '2d.shape-of-cpfun/1'
@ ^ [V_p: '2d.Point'] :
( ( ( \$greater @ ( '2d.x-coord/1' @ V_p ) @ 0.0 )
=> ( ( '2d.y-coord/1' @ V_p )
= ( \$sum @ ( '^/2' @ ( '2d.x-coord/1' @ V_p ) @ 2.0 ) @ ( \$sum @ ( \$uminus @ ( '2d.x-coord/1' @ V_p ) ) @ ( \$quotient @ 3.0 @ 4.0 ) ) ) ) )
& ( ( \$lesseq @ ( '2d.x-coord/1' @ V_p ) @ 0.0 )
=> ( ( '2d.y-coord/1' @ V_p )
= ( \$sum @ ( '^/2' @ ( '2d.x-coord/1' @ V_p ) @ 2.0 ) @ ( \$sum @ ( \$product @ 2.0 @ ( '2d.x-coord/1' @ V_p ) ) @ ( \$quotient @ 3.0 @ 4.0 ) ) ) ) ) ) ) )
& ? [V_P: '2d.Point',V_Q: '2d.Point'] :
( ( V_P != V_Q )
& ( '2d.tangent/3' @ V_C1 @ V_C2 @ V_P )
& ( '2d.tangent/3' @ V_C1 @ V_C2 @ V_Q ) )
& ( V_S
= ( '2d.area-of/1' @ ( '2d.shape-enclosed-by/1' @ ( 'cons/2' @ '2d.Shape' @ V_C1 @ ( 'cons/2' @ '2d.Shape' @ V_C2 @ ( 'nil/0' @ '2d.Shape' ) ) ) ) ) ) ) )).

%------------------------------------------------------------------------------
```