TPTP Problem File: RAL057^1.p
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% File : RAL057^1 : TPTP v9.0.0. Released v7.0.0.
% Domain : Real Algebra (Functions and their graphs)
% Problem : Kyushu University, 2005, Humanities Course, Problem 1
% Version : [Mat16] axioms : Especial.
% English : Let a be a positive real number, and consider the point A(0,
% a+1/2 a) and the curve C: y = a x^2 ( xge 0). Let P the point on
% the curve C that is the closest to the point A. Answer the
% following questions: (1) Find the coordinates of the point P and
% the length of the line segment AP. (2) Find the area S(a) of the
% region enclosed by the curve C, the y axis, and the line segment
% AP. (3) When a > 0, find the minimum value of the area S(a) and
% the value of a that gives the minimum value.
% Refs : [Mat16] Matsuzaki (2016), Email to Geoff Sutcliffe
% : [MI+16] Matsuzaki et al. (2016), Race against the Teens - Benc
% Source : [Mat16]
% Names : Univ-Kyushu-2005-Bun-1.p [Mat16]
% Status : Theorem
% Rating : ? v7.0.0
% Syntax : Number of formulae : 3486 ( 726 unt;1200 typ; 0 def)
% Number of atoms : 7004 (2211 equ; 0 cnn)
% Maximal formula atoms : 20 ( 3 avg)
% Number of connectives : 39639 ( 104 ~; 233 |;1179 &;35996 @)
% (1095 <=>;1032 =>; 0 <=; 0 <~>)
% Maximal formula depth : 33 ( 8 avg)
% Number arithmetic : 4478 ( 375 atm;1207 fun; 958 num;1938 var)
% Number of types : 40 ( 36 usr; 3 ari)
% Number of type conns : 2408 (2408 >; 0 *; 0 +; 0 <<)
% Number of symbols : 1216 (1173 usr; 70 con; 0-9 aty)
% Number of variables : 8061 ( 407 ^;7086 !; 432 ?;8061 :)
% ( 136 !>; 0 ?*; 0 @-; 0 @+)
% SPC : TH1_THM_EQU_ARI
% Comments : Theory: RCF; Author: Tomoya Ishii; Generated: 2014-05-21
% : Answer
% ^ [V_P_dot_0: '2d.Point'] :
% ( ( $less @ 0.0 @ 'a/0' )
% & ( V_P_dot_0
% = ( '2d.point/2' @ 1.0 @ 'a/0' ) ) ) )
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include('Axioms/MAT001^0.ax').
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thf('a/0_type',type,
'a/0': $real ).
thf(p1P_qustion,conjecture,
( 'find/1' @ '2d.Point'
@ ^ [V_P: '2d.Point'] :
? [V_A: '2d.Point',V_C: '2d.Shape',V_AP: $real] :
( ( $less @ 0.0 @ 'a/0' )
& ( V_A
= ( '2d.point/2' @ 0.0 @ ( $sum @ 'a/0' @ ( $quotient @ 1.0 @ ( $product @ 2.0 @ 'a/0' ) ) ) ) )
& ( V_C
= ( '2d.graph-of/1'
@ ( 'fun/1'
@ ^ [V_x: $real] : ( $product @ 'a/0' @ ( '^/2' @ V_x @ 2.0 ) ) ) ) )
& ( '2d.on/2' @ V_P @ V_C )
& ( $lesseq @ 0.0 @ ( '2d.x-coord/1' @ V_P ) )
& ( V_AP
= ( '2d.length-of/1' @ ( '2d.seg/2' @ V_A @ V_P ) ) )
& ! [V_Q: '2d.Point'] :
( ( ( '2d.on/2' @ V_Q @ V_C )
& ( $lesseq @ 0.0 @ ( '2d.x-coord/1' @ V_Q ) ) )
=> ( $lesseq @ ( '2d.length-of/1' @ ( '2d.seg/2' @ V_A @ V_P ) ) @ ( '2d.length-of/1' @ ( '2d.seg/2' @ V_A @ V_Q ) ) ) ) ) ) ).
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