TPTP Problem File: GEO385^1.p
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%------------------------------------------------------------------------------
% File : GEO385^1 : TPTP v8.2.0. Released v7.0.0.
% Domain : Geometry (Hexahedrons)
% Problem : International Mathematical Olympiad, 1962, Problem 3
% Version : [Mat16] axioms : Especial.
% English : Consider the cube ABCDA'B'C'D' (ABCD and A'B'C'D' are the upper
% and lower bases, respectively, and edges AA', BB', CC', DD' are
% parallel). The point X moves at constant speed along the
% perimeter of the square ABCD in the direction ABCDA, and the
% point Y moves at the same rate along the perimeter of the square
% B'C'CB in the direction B'C'CBB'. Points X and Y begin their
% motion at the same instant from the starting positions A and B',
% respectively. Determine and draw the locus of the midpoints of
% the segments XY.
% Refs : [Mat16] Matsuzaki (2016), Email to Geoff Sutcliffe
% : [MI+16] Matsuzaki et al. (2016), Race against the Teens - Benc
% Source : [Mat16]
% Names : IMO-1962-3.p [Mat16]
% Status : Theorem
% Rating : ? v7.0.0
% Syntax : Number of formulae : 3509 ( 727 unt;1223 typ; 0 def)
% Number of atoms : 6693 (2221 equ; 0 cnn)
% Maximal formula atoms : 31 ( 2 avg)
% Number of connectives : 39705 ( 104 ~; 237 |;1189 &;36049 @)
% (1095 <=>;1031 =>; 0 <=; 0 <~>)
% Maximal formula depth : 33 ( 8 avg)
% Number arithmetic : 4461 ( 371 atm;1203 fun; 951 num;1936 var)
% Number of types : 40 ( 36 usr; 3 ari)
% Number of type conns : 2408 (2408 >; 0 *; 0 +; 0 <<)
% Number of symbols : 1237 (1194 usr; 91 con; 0-9 aty)
% Number of variables : 8066 ( 406 ^;7085 !; 439 ?;8066 :)
% ( 136 !>; 0 ?*; 0 @-; 0 @+)
% SPC : TH1_THM_EQU_ARI
% Comments : Theory: RCF; Score: 8; Author: Jumma Kudo;
% Generated: 2014-12-04
% : Answer
% ^ [V_M_dot_0: '3d.Point'] :
% ? [V_A1_dot_0: '3d.Point',V_B1_dot_0: '3d.Point',V_C1_dot_0: '3d.Point',V_D1_dot_0: '3d.Point',V_A2_dot_0: '3d.Point',V_B2_dot_0: '3d.Point',V_C2_dot_0: '3d.Point',V_D2_dot_0: '3d.Point'] :
% ( ( '3d.is-cube/8' @ V_A1_dot_0 @ V_B1_dot_0 @ V_C1_dot_0 @ V_D1_dot_0 @ V_A2_dot_0 @ V_B2_dot_0 @ V_C2_dot_0 @ V_D2_dot_0 )
% & ( V_A1_dot_0
% = ( '3d.point/3' @ 'A1x/0' @ 'A1y/0' @ 'A1z/0' ) )
% & ( V_B1_dot_0
% = ( '3d.point/3' @ 'B1x/0' @ 'B1y/0' @ 'B1z/0' ) )
% & ( V_C1_dot_0
% = ( '3d.point/3' @ 'C1x/0' @ 'C1y/0' @ 'C1z/0' ) )
% & ( V_D1_dot_0
% = ( '3d.point/3' @ 'D1x/0' @ 'D1y/0' @ 'D1z/0' ) )
% & ( V_A2_dot_0
% = ( '3d.point/3' @ 'A2x/0' @ 'A2y/0' @ 'A2z/0' ) )
% & ( V_B2_dot_0
% = ( '3d.point/3' @ 'B2x/0' @ 'B2y/0' @ 'B2z/0' ) )
% & ( V_C2_dot_0
% = ( '3d.point/3' @ 'C2x/0' @ 'C2y/0' @ 'C2z/0' ) )
% & ( V_D2_dot_0
% = ( '3d.point/3' @ 'D2x/0' @ 'D2y/0' @ 'D2z/0' ) )
% & ( '3d.on/2' @ V_M_dot_0 @ ( '3d.square/4' @ V_C1_dot_0 @ ( '3d.point/3' @ ( $quotient @ ( $sum @ 'A1x/0' @ ( $sum @ 'B1x/0' @ ( $sum @ 'C1x/0' @ 'D1x/0' ) ) ) @ 4.0 ) @ ( $quotient @ ( $sum @ 'A1y/0' @ ( $sum @ 'B1y/0' @ ( $sum @ 'C1y/0' @ 'D1y/0' ) ) ) @ 4.0 ) @ ( $quotient @ ( $sum @ 'A1z/0' @ ( $sum @ 'B1z/0' @ ( $sum @ 'C1z/0' @ 'D1z/0' ) ) ) @ 4.0 ) ) @ ( '3d.point/3' @ ( $quotient @ ( $sum @ 'A1x/0' @ ( $sum @ 'B1x/0' @ ( $sum @ 'A2x/0' @ 'B2x/0' ) ) ) @ 4.0 ) @ ( $quotient @ ( $sum @ 'A1y/0' @ ( $sum @ 'B1y/0' @ ( $sum @ 'A2y/0' @ 'B2y/0' ) ) ) @ 4.0 ) @ ( $quotient @ ( $sum @ 'A1z/0' @ ( $sum @ 'B1z/0' @ ( $sum @ 'A2z/0' @ 'B2z/0' ) ) ) @ 4.0 ) ) @ ( '3d.point/3' @ ( $quotient @ ( $sum @ 'C1x/0' @ ( $sum @ 'B1x/0' @ ( $sum @ 'C2x/0' @ 'B2x/0' ) ) ) @ 4.0 ) @ ( $quotient @ ( $sum @ 'C1y/0' @ ( $sum @ 'B1y/0' @ ( $sum @ 'C2y/0' @ 'B2y/0' ) ) ) @ 4.0 ) @ ( $quotient @ ( $sum @ 'C1z/0' @ ( $sum @ 'B1z/0' @ ( $sum @ 'C2z/0' @ 'B2z/0' ) ) ) @ 4.0 ) ) ) ) ) )
%------------------------------------------------------------------------------
include('Axioms/MAT001^0.ax').
%------------------------------------------------------------------------------
thf('A1x/0_type',type,
'A1x/0': $real ).
thf('A1y/0_type',type,
'A1y/0': $real ).
thf('A1z/0_type',type,
'A1z/0': $real ).
thf('A2x/0_type',type,
'A2x/0': $real ).
thf('A2y/0_type',type,
'A2y/0': $real ).
thf('A2z/0_type',type,
'A2z/0': $real ).
thf('B1x/0_type',type,
'B1x/0': $real ).
thf('B1y/0_type',type,
'B1y/0': $real ).
thf('B1z/0_type',type,
'B1z/0': $real ).
thf('B2x/0_type',type,
'B2x/0': $real ).
thf('B2y/0_type',type,
'B2y/0': $real ).
thf('B2z/0_type',type,
'B2z/0': $real ).
thf('C1x/0_type',type,
'C1x/0': $real ).
thf('C1y/0_type',type,
'C1y/0': $real ).
thf('C1z/0_type',type,
'C1z/0': $real ).
thf('C2x/0_type',type,
'C2x/0': $real ).
thf('C2y/0_type',type,
'C2y/0': $real ).
thf('C2z/0_type',type,
'C2z/0': $real ).
thf('D1x/0_type',type,
'D1x/0': $real ).
thf('D1y/0_type',type,
'D1y/0': $real ).
thf('D1z/0_type',type,
'D1z/0': $real ).
thf('D2x/0_type',type,
'D2x/0': $real ).
thf('D2y/0_type',type,
'D2y/0': $real ).
thf('D2z/0_type',type,
'D2z/0': $real ).
thf(p_qustion,conjecture,
( 'find/1' @ '3d.Point'
@ ^ [V_M: '3d.Point'] :
? [V_A1: '3d.Point',V_B1: '3d.Point',V_C1: '3d.Point',V_D1: '3d.Point',V_A2: '3d.Point',V_B2: '3d.Point',V_C2: '3d.Point',V_D2: '3d.Point',V_X: '3d.Point',V_Y: '3d.Point'] :
( ( '3d.is-cube/8' @ V_A1 @ V_B1 @ V_C1 @ V_D1 @ V_A2 @ V_B2 @ V_C2 @ V_D2 )
& ( V_A1
= ( '3d.point/3' @ 'A1x/0' @ 'A1y/0' @ 'A1z/0' ) )
& ( V_B1
= ( '3d.point/3' @ 'B1x/0' @ 'B1y/0' @ 'B1z/0' ) )
& ( V_C1
= ( '3d.point/3' @ 'C1x/0' @ 'C1y/0' @ 'C1z/0' ) )
& ( V_D1
= ( '3d.point/3' @ 'D1x/0' @ 'D1y/0' @ 'D1z/0' ) )
& ( V_A2
= ( '3d.point/3' @ 'A2x/0' @ 'A2y/0' @ 'A2z/0' ) )
& ( V_B2
= ( '3d.point/3' @ 'B2x/0' @ 'B2y/0' @ 'B2z/0' ) )
& ( V_C2
= ( '3d.point/3' @ 'C2x/0' @ 'C2y/0' @ 'C2z/0' ) )
& ( V_D2
= ( '3d.point/3' @ 'D2x/0' @ 'D2y/0' @ 'D2z/0' ) )
& ( ( ( '3d.on/2' @ V_X @ ( '3d.seg/2' @ V_A1 @ V_B1 ) )
& ( '3d.on/2' @ V_Y @ ( '3d.seg/2' @ V_B2 @ V_C2 ) )
& ( ( '3d.length-of/1' @ ( '3d.seg/2' @ V_A1 @ V_X ) )
= ( '3d.length-of/1' @ ( '3d.seg/2' @ V_B2 @ V_Y ) ) ) )
| ( ( '3d.on/2' @ V_X @ ( '3d.seg/2' @ V_B1 @ V_C1 ) )
& ( '3d.on/2' @ V_Y @ ( '3d.seg/2' @ V_C2 @ V_C1 ) )
& ( ( '3d.length-of/1' @ ( '3d.seg/2' @ V_B1 @ V_X ) )
= ( '3d.length-of/1' @ ( '3d.seg/2' @ V_C2 @ V_Y ) ) ) )
| ( ( '3d.on/2' @ V_X @ ( '3d.seg/2' @ V_C1 @ V_D1 ) )
& ( '3d.on/2' @ V_Y @ ( '3d.seg/2' @ V_C1 @ V_B1 ) )
& ( ( '3d.length-of/1' @ ( '3d.seg/2' @ V_C1 @ V_X ) )
= ( '3d.length-of/1' @ ( '3d.seg/2' @ V_C1 @ V_Y ) ) ) )
| ( ( '3d.on/2' @ V_X @ ( '3d.seg/2' @ V_D1 @ V_A1 ) )
& ( '3d.on/2' @ V_Y @ ( '3d.seg/2' @ V_B1 @ V_B2 ) )
& ( ( '3d.length-of/1' @ ( '3d.seg/2' @ V_D1 @ V_X ) )
= ( '3d.length-of/1' @ ( '3d.seg/2' @ V_B1 @ V_Y ) ) ) )
| ( V_M
= ( '3d.midpoint-of/2' @ V_X @ V_Y ) ) ) ) ) ).
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