TPTP Problem File: RAL058^1.p
View Solutions
- Solve Problem
%------------------------------------------------------------------------------
% File : RAL058^1 : TPTP v9.0.0. Released v7.0.0.
% Domain : Real Algebra (Functions and their graphs)
% Problem : Nagoya University, 2003, Humanities Course, Problem 2
% Version : [Mat16] axioms : Especial.
% English : Consider the parabola C: y = a x^2 ( a > 0). Let l be the straight
% line passing through P and orthogonal to the tangent to the
% parabola C at the point P(p, a p^2) (p != 0). Let S(P) be the
% area of the region enclosed by the straight line l and the
% parabola C. (1) Find the equation of the straight line l.
% (2) Move the point P in the range of p > 0. Then, find the minimum
% value of S(P) and the inclination m of the straight line l at
% that time.
% Refs : [Mat16] Matsuzaki (2016), Email to Geoff Sutcliffe
% : [MI+16] Matsuzaki et al. (2016), Race against the Teens - Benc
% Source : [Mat16]
% Names : Univ-Nagoya-2003-Bun-2.p [Mat16]
% Status : Theorem
% Rating : ? v7.0.0
% Syntax : Number of formulae : 3492 ( 710 unt;1206 typ; 0 def)
% Number of atoms : 8216 (2211 equ; 0 cnn)
% Maximal formula atoms : 41 ( 3 avg)
% Number of connectives : 39632 ( 105 ~; 233 |;1180 &;35988 @)
% (1095 <=>;1031 =>; 0 <=; 0 <~>)
% Maximal formula depth : 33 ( 8 avg)
% Number arithmetic : 4468 ( 372 atm;1204 fun; 956 num;1936 var)
% Number of types : 40 ( 36 usr; 3 ari)
% Number of type conns : 2408 (2408 >; 0 *; 0 +; 0 <<)
% Number of symbols : 1224 (1181 usr; 78 con; 0-9 aty)
% Number of variables : 8059 ( 406 ^;7085 !; 432 ?;8059 :)
% ( 136 !>; 0 ?*; 0 @-; 0 @+)
% SPC : TH1_THM_EQU_ARI
% Comments : Theory: RCF; Author: Ukyo Suzuki; Generated: 2014-07-30
% : Answer
% ^ [V_l_dot_0: '2d.Shape'] :
% ( ( $less @ 0.0 @ 'a/0' )
% & ( 'p/0' != 0.0 )
% & ( V_l_dot_0
% = ( '2d.shape-of-cpfun/1'
% @ ^ [V_P_dot_0: '2d.Point'] :
% ( ( '2d.y-coord/1' @ V_P_dot_0 )
% = ( $sum @ ( $product @ ( $uminus @ ( $quotient @ 1.0 @ ( $product @ 2.0 @ ( $product @ 'a/0' @ 'p/0' ) ) ) ) @ ( '2d.x-coord/1' @ V_P_dot_0 ) ) @ ( $sum @ ( $quotient @ 1.0 @ ( $product @ 2.0 @ 'a/0' ) ) @ ( $product @ 'a/0' @ ( '^/2' @ 'p/0' @ 2.0 ) ) ) ) ) ) ) ) )
%------------------------------------------------------------------------------
include('Axioms/MAT001^0.ax').
%------------------------------------------------------------------------------
thf('P/0_type',type,
'P/0': '2d.Point' ).
thf('a/0_type',type,
'a/0': $real ).
thf('l/0_type',type,
'l/0': '2d.Shape' ).
thf('l2/0_type',type,
'l2/0': '2d.Shape' ).
thf('min-sp/0_type',type,
'min-sp/0': $real ).
thf('min_sp/0_type',type,
'min_sp/0': $real ).
thf('p/0_type',type,
'p/0': $real ).
thf(p1_qustion,conjecture,
( 'find/1' @ '2d.Shape'
@ ^ [V_l: '2d.Shape'] :
? [V_P: '2d.Point',V_C: '2d.Shape',V_l2: '2d.Shape'] :
( ( '2d.line-type/1' @ V_l2 )
& ( '2d.line-type/1' @ V_l )
& ( V_C
= ( '2d.graph-of/1' @ ( 'poly-fun/1' @ ( 'cons/2' @ $real @ 0.0 @ ( 'cons/2' @ $real @ 0.0 @ ( 'cons/2' @ $real @ 'a/0' @ ( 'nil/0' @ $real ) ) ) ) ) ) )
& ( $less @ 0.0 @ 'a/0' )
& ( V_P
= ( '2d.point/2' @ 'p/0' @ ( $product @ 'a/0' @ ( '^/2' @ 'p/0' @ 2.0 ) ) ) )
& ( 'p/0' != 0.0 )
& ( '2d.tangent/3' @ V_C @ V_l2 @ V_P )
& ( '2d.on/2' @ V_P @ V_l )
& ( '2d.perpendicular/2' @ V_l @ V_l2 ) ) ) ).
%------------------------------------------------------------------------------