TPTP Problem File: GEO438^1.p
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% File : GEO438^1 : TPTP v8.2.0. Released v7.0.0.
% Domain : Geometry (Geometric figures and equations)
% Problem : The University of Tokyo, 2013, Humanities Course, Problem 3
% Version : [Mat16] axioms : Especial.
% English : Let a and b be constant real numbers. When the real numbers x
% and y satisfy both x^2+y^2 =< 25 and 2 x+y =< 5, find the minimum
% value of z=x^2+y^2-2 a x-2 b y.
% Refs : [Mat16] Matsuzaki (2016), Email to Geoff Sutcliffe
% : [MI+16] Matsuzaki et al. (2016), Race against the Teens - Benc
% Source : [Mat16]
% Names : Univ-Tokyo-2013-Bun-3.p [Mat16]
% Status : Theorem
% Rating : ? v7.0.0
% Syntax : Number of formulae : 3487 ( 727 unt;1201 typ; 0 def)
% Number of atoms : 6578 (2209 equ; 0 cnn)
% Maximal formula atoms : 20 ( 2 avg)
% Number of connectives : 39634 ( 104 ~; 233 |;1174 &;35997 @)
% (1095 <=>;1031 =>; 0 <=; 0 <~>)
% Maximal formula depth : 33 ( 8 avg)
% Number arithmetic : 4488 ( 373 atm;1215 fun; 960 num;1940 var)
% Number of types : 40 ( 36 usr; 3 ari)
% Number of type conns : 2408 (2408 >; 0 *; 0 +; 0 <<)
% Number of symbols : 1213 (1169 usr; 67 con; 0-9 aty)
% Number of variables : 8059 ( 407 ^;7085 !; 431 ?;8059 :)
% ( 136 !>; 0 ?*; 0 @-; 0 @+)
% SPC : TH1_THM_EQU_ARI
% Comments : Theory: RCF; Author: Takuya Matsuzaki; Generated: 2013-11-11
% : Answer
% ^ [V_m_dot_0: $real] :
% ( ( ( $lesseq @ ( $sum @ ( '^/2' @ 'a/0' @ 2.0 ) @ ( '^/2' @ 'b/0' @ 2.0 ) ) @ 25.0 )
% & ( $lesseq @ 'b/0' @ ( $difference @ 5.0 @ ( $product @ 2.0 @ 'a/0' ) ) )
% & ( V_m_dot_0
% = ( $uminus @ ( $sum @ ( '^/2' @ 'a/0' @ 2.0 ) @ ( '^/2' @ 'b/0' @ 2.0 ) ) ) ) )
% | ( ( $greatereq @ ( $sum @ ( '^/2' @ 'a/0' @ 2.0 ) @ ( '^/2' @ 'b/0' @ 2.0 ) ) @ 25.0 )
% & ( ( $lesseq @ 'a/0' @ 0.0 )
% | ( $lesseq @ 'b/0' @ ( $product @ ( $uminus @ ( $quotient @ 3.0 @ 4.0 ) ) @ 'a/0' ) ) )
% & ( V_m_dot_0
% = ( $difference @ 25.0 @ ( $product @ 10.0 @ ( 'sqrt/1' @ ( $sum @ ( '^/2' @ 'a/0' @ 2.0 ) @ ( '^/2' @ 'b/0' @ 2.0 ) ) ) ) ) ) )
% | ( ( $greatereq @ 'b/0' @ ( $difference @ 5.0 @ ( $product @ 2.0 @ 'a/0' ) ) )
% & ( $lesseq @ 'b/0' @ ( $sum @ ( $product @ ( $quotient @ 1.0 @ 2.0 ) @ 'a/0' ) @ 5.0 ) )
% & ( $greatereq @ 'b/0' @ ( $difference @ 5.0 @ ( $product @ ( $quotient @ 1.0 @ 2.0 ) @ 'a/0' ) ) )
% & ( V_m_dot_0
% = ( $product @ ( $quotient @ 1.0 @ 5.0 ) @ ( $sum @ ( $uminus @ ( '^/2' @ 'a/0' @ 2.0 ) ) @ ( $sum @ ( $uminus @ ( $product @ 4.0 @ ( '^/2' @ 'b/0' @ 2.0 ) ) ) @ ( $sum @ ( $product @ 4.0 @ ( $product @ 'a/0' @ 'b/0' ) ) @ ( $sum @ ( $uminus @ ( $product @ 20.0 @ 'a/0' ) ) @ ( $sum @ ( $uminus @ ( $product @ 10.0 @ 'b/0' ) ) @ 25.0 ) ) ) ) ) ) ) )
% | ( ( $greatereq @ 'a/0' @ 0.0 )
% & ( $greatereq @ 'b/0' @ ( $sum @ ( $product @ ( $quotient @ 1.0 @ 2.0 ) @ 'a/0' ) @ 5.0 ) )
% & ( V_m_dot_0
% = ( $difference @ 25.0 @ ( $product @ 10.0 @ 'b/0' ) ) ) )
% | ( ( $greatereq @ 'b/0' @ ( $product @ ( $uminus @ ( $quotient @ 3.0 @ 4.0 ) ) @ 'a/0' ) )
% & ( $lesseq @ 'b/0' @ ( $difference @ ( $product @ ( $quotient @ 1.0 @ 2.0 ) @ 'a/0' ) @ 5.0 ) )
% & ( V_m_dot_0
% = ( $sum @ 25.0 @ ( $sum @ ( $uminus @ ( $product @ 8.0 @ 'a/0' ) ) @ ( $product @ 6.0 @ 'b/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(p_qustion,conjecture,
( 'find/1' @ $real
@ ^ [V_m: $real] :
( 'minimum/2'
@ ( 'set-by-def/1' @ $real
@ ^ [V_z: $real] :
? [V_x: $real,V_y: $real] :
( ( $lesseq @ ( $sum @ ( '^/2' @ V_x @ 2.0 ) @ ( '^/2' @ V_y @ 2.0 ) ) @ 25.0 )
& ( $lesseq @ ( $sum @ ( $product @ 2.0 @ V_x ) @ V_y ) @ 5.0 )
& ( V_z
= ( $sum @ ( '^/2' @ V_x @ 2.0 ) @ ( $sum @ ( '^/2' @ V_y @ 2.0 ) @ ( $sum @ ( $uminus @ ( $product @ 2.0 @ ( $product @ 'a/0' @ V_x ) ) ) @ ( $uminus @ ( $product @ 2.0 @ ( $product @ 'b/0' @ V_y ) ) ) ) ) ) ) ) )
@ V_m ) ) ).
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