TPTP Problem File: RAL064^1.p
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% File : RAL064^1 : TPTP v9.0.0. 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 ( 709 unt;1202 typ; 0 def)
% Number of atoms : 8265 (2214 equ; 0 cnn)
% Maximal formula atoms : 40 ( 3 avg)
% Number of connectives : 39665 ( 105 ~; 233 |;1179 &;36020 @)
% (1095 <=>;1033 =>; 0 <=; 0 <~>)
% Maximal formula depth : 33 ( 8 avg)
% Number arithmetic : 4489 ( 374 atm;1215 fun; 962 num;1938 var)
% Number of types : 40 ( 36 usr; 3 ari)
% Number of type conns : 2408 (2408 >; 0 *; 0 +; 0 <<)
% Number of symbols : 1221 (1178 usr; 75 con; 0-9 aty)
% Number of variables : 8062 ( 408 ^;7085 !; 433 ?;8062 :)
% ( 136 !>; 0 ?*; 0 @-; 0 @+)
% SPC : TH1_THM_EQU_ARI
% Comments : Theory: RCF; Author: Yiyang Zhan; Generated: 2014-04-16
% : Answer
% ^ [V_S_dot_0: $real] :
% ( V_S_dot_0
% = ( $quotient @ 9.0 @ ( $product @ 32.0 @ ( '^/2' @ ( $difference @ 1.0 @ 'a/0' ) @ 2.0 ) ) ) ) )
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include('Axioms/MAT001^0.ax').
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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' ) ) ) ) ) ) ) ) ).
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