## TPTP Problem File: RAL063^1.p

View Solutions - Solve Problem

```%------------------------------------------------------------------------------
% File     : RAL063^1 : TPTP v7.5.0. Released v7.0.0.
% Domain   : Real Algebra (Functions and their graphs)
% Problem  : Tohoku University, 1999, Science Course, Problem 3
% Version  : [Mat16] axioms : Especial.
% English  : Let l be the tangent to the curve y = x^2 at the point (a, a^2).
%            Let P and Q be the points on l for which x coordinates are a - 1
%            and a + 1, respectively. When a moves in the range of
%            -1 =< a =< 1, find the area of the moving range of the line
%            segment PQ.

% Refs     : [Mat16] Matsuzaki (2016), Email to Geoff Sutcliffe
%          : [MI+16] Matsuzaki et al. (2016), Race against the Teens - Benc
% Source   : [Mat16]
% Names    : Univ-Tohoku-1999-Ri-3.p [Mat16]

% Status   : Theorem
% Rating   : ? v7.0.0
% Syntax   : Number of formulae    : 3485 (   0 unit;1199 type;   0 defn)
%            Number of atoms       : 45355 (2212 equality;22719 variable)
%            Maximal formula depth :   35 (   9 average)
%            Number of connectives : 39644 ( 104   ~; 233   |;1181   &;36000   @)
%                                         (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   : 8062 (  66 sgn;7085   !; 434   ?; 407   ^)
%                                         (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: Hidenao Iwane; Generated: 2014-01-10
%            ^ [V_S_dot_0: \$real] :
%              ( V_S_dot_0
%              = ( \$quotient @ 10.0 @ 3.0 ) ) )
%------------------------------------------------------------------------------
include('Axioms/MAT001^0.ax').
%------------------------------------------------------------------------------
thf(p1_qustion,conjecture,
( 'find/1' @ \$real
@ ^ [V_S: \$real] :
( V_S
= ( '2d.area-of/1'
@ ( '2d.shape-of-cpfun/1'
@ ^ [V_p: '2d.Point'] :
? [V_P: '2d.Point',V_Q: '2d.Point',V_l: '2d.Shape',V_a: \$real,V_C: '2d.Shape'] :
( ( V_C
= ( '2d.graph-of/1' @ ( 'poly-fun/1' @ ( 'cons/2' @ \$real @ 0.0 @ ( 'cons/2' @ \$real @ 0.0 @ ( 'cons/2' @ \$real @ 1.0 @ ( 'nil/0' @ \$real ) ) ) ) ) ) )
& ( '2d.tangent/3' @ V_l @ V_C @ ( '2d.point/2' @ V_a @ ( '^/2' @ V_a @ 2.0 ) ) )
& ( '2d.line-type/1' @ V_l )
& ( '2d.on/2' @ V_P @ V_l )
& ( '2d.on/2' @ V_Q @ V_l )
& ( ( '2d.x-coord/1' @ V_P )
= ( \$sum @ V_a @ ( \$uminus @ 1.0 ) ) )
& ( ( '2d.x-coord/1' @ V_Q )
= ( \$sum @ V_a @ 1.0 ) )
& ( \$lesseq @ -1.0 @ V_a )
& ( \$lesseq @ V_a @ 1.0 )
& ( '2d.on/2' @ V_p @ ( '2d.seg/2' @ V_P @ V_Q ) ) ) ) ) ) )).

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