## TPTP Problem File: RAL068^1.p

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```%------------------------------------------------------------------------------
% File     : RAL068^1 : TPTP v7.5.0. Released v7.0.0.
% Domain   : Real Algebra (Algebraic curves)
% Problem  : The University of Tokyo, 1991, Science Course, Problem 2
% Version  : [Mat16] axioms : Especial.
% English  : Let a, b, and c be positive real numbers. Consider the plate R
%            consisting of the points (x, y, z) that satisfy |x|le a, |y|le
%            b, and z=c in the xyz space. When the point light source P makes
%            a circuit on the ellipse x^2/a^2+y^2/b^2=1, z=c+1 on the plane
%            z=c+1, draw the figure formed by the locus of the shade of the
%            plate R, and find the area of the figure.

% Refs     : [Mat16] Matsuzaki (2016), Email to Geoff Sutcliffe
%          : [MI+16] Matsuzaki et al. (2016), Race against the Teens - Benc
% Source   : [Mat16]
% Names    : Univ-Tokyo-1991-Ri-2.p [Mat16]

% Status   : Theorem
% Rating   : ? v7.0.0
% Syntax   : Number of formulae    : 3488 (   0 unit;1202 type;   0 defn)
%            Number of atoms       : 45362 (2213 equality;22713 variable)
%            Maximal formula depth :   35 (   9 average)
%            Number of connectives : 39649 ( 104   ~; 233   |;1182   &;36004   @)
%                                         (1095 <=>;1031  =>;   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   : 8061 (  66 sgn;7085   !; 432   ?; 408   ^)
%                                         (8061   :; 136  !>;   0  ?*)
%                                         (   0  @-;   0  @+)
%            Arithmetic symbols    : 1975 (   6 prd;   9 fun;  23 num;1937 var)
% SPC      : TH1_THM_EQU_ARI

% Comments : Theory: RCF; Author: Tomoya Ishii; Generated: 2014-04-15
%            ^ [V_S_dot_0: \$real] :
%              ( V_S_dot_0
%              = ( \$product @ ( \$sum @ ( \$product @ ( \$sum @ 'Pi/0' @ 12.0 ) @ ( '^/2' @ 'c/0' @ 2.0 ) ) @ ( \$sum @ ( \$product @ 16.0 @ 'c/0' ) @ 4.0 ) ) @ ( \$product @ 'a/0' @ 'b/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_R: '3d.Shape'] :
( ( \$lesseq @ 0.0 @ 'a/0' )
& ( \$lesseq @ 0.0 @ 'b/0' )
& ( \$lesseq @ 0.0 @ 'c/0' )
& ( V_R
= ( '3d.shape-of-cpfun/1'
@ ^ [V_p_dot_0: '3d.Point'] :
( ( \$lesseq @ ( 'abs/1' @ ( '3d.x-coord/1' @ V_p_dot_0 ) ) @ 'a/0' )
& ( \$lesseq @ ( 'abs/1' @ ( '3d.y-coord/1' @ V_p_dot_0 ) ) @ 'b/0' )
& ( ( '3d.z-coord/1' @ V_p_dot_0 )
= 'c/0' ) ) ) )
& ( V_S
= ( '3d.area-of/1'
@ ( '3d.shape-of-cpfun/1'
@ ^ [V_p: '3d.Point'] :
( ( '3d.on/2' @ V_p @ '3d.xy-plane/0' )
& ? [V_P: '3d.Point',V_PP: '3d.Point'] :
( ( ( \$sum @ ( \$quotient @ ( '^/2' @ ( '3d.x-coord/1' @ V_P ) @ 2.0 ) @ ( '^/2' @ 'a/0' @ 2.0 ) ) @ ( \$quotient @ ( '^/2' @ ( '3d.y-coord/1' @ V_P ) @ 2.0 ) @ ( '^/2' @ 'a/0' @ 2.0 ) ) )
= 1.0 )
& ( ( '3d.z-coord/1' @ V_P )
= ( \$sum @ 'c/0' @ 1.0 ) )
& ( '3d.on/2' @ V_PP @ V_R )
& ( '3d.on/2' @ V_PP @ ( '3d.line/2' @ V_P @ V_p ) ) ) ) ) ) ) ) )).

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