TPTP Problem File: MSC033^1.p
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% File : MSC033^1 : TPTP v8.2.0. Released v7.0.0.
% Domain : Miscellaneous
% Problem : International Mathematical Olympiad, 2014, Problem 2
% Version : [Mat16] axioms : Especial.
% English : Let n geq 2 be an integer. Consider an n times n chessboard
% consisting of n^2 unit squares. A configuration of n rooks on
% this board is peaceful if every row and every column contains
% exactly one rook. Find the greatest positive integer k such
% that, for each peaceful configuration of n rooks, there is a k
% times k square which does not contain a rook on any of its k^2
% unit squares.
% Refs : [Mat16] Matsuzaki (2016), Email to Geoff Sutcliffe
% : [MI+16] Matsuzaki et al. (2016), Race against the Teens - Benc
% Source : [Mat16]
% Names : IMO-2014-2.p [Mat16]
% Status : Theorem
% Rating : ? v7.0.0
% Syntax : Number of formulae : 3486 ( 710 unt;1200 typ; 0 def)
% Number of atoms : 7952 (2211 equ; 0 cnn)
% Maximal formula atoms : 44 ( 3 avg)
% Number of connectives : 39728 ( 107 ~; 233 |;1190 &;36068 @)
% (1095 <=>;1035 =>; 0 <=; 0 <~>)
% Maximal formula depth : 33 ( 8 avg)
% Number arithmetic : 4499 ( 389 atm;1207 fun; 958 num;1945 var)
% Number of types : 40 ( 36 usr; 3 ari)
% Number of type conns : 2408 (2408 >; 0 *; 0 +; 0 <<)
% Number of symbols : 1219 (1176 usr; 73 con; 0-9 aty)
% Number of variables : 8065 ( 407 ^;7091 !; 431 ?;8065 :)
% ( 136 !>; 0 ?*; 0 @-; 0 @+)
% SPC : TH1_THM_EQU_ARI
% Comments : Theory: PA(comb); Score: 7; Author: Jumma Kudo;
% Generated: 2014-10-15
% : Answer
% ^ [V_k_dot_1: $int] :
% ( ( $less @ 'n/0' @ ( 'int.^/2' @ ( $sum @ V_k_dot_1 @ 1 ) @ 2 ) )
% & ( $lesseq @ ( 'int.^/2' @ V_k_dot_1 @ 2 ) @ 'n/0' ) ) )
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include('Axioms/MAT001^0.ax').
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thf('n/0_type',type,
'n/0': $int ).
thf(p_qustion,conjecture,
( 'find/1' @ $int
@ ^ [V_max_k: $int] :
( 'int.maximum/2'
@ ( 'set-by-def/1' @ $int
@ ^ [V_k: $int] :
! [V_ps: 'ListOf' @ ( 'ListOf' @ $int )] :
( ( ( ( 'list-len/1' @ ( 'ListOf' @ $int ) @ V_ps )
= 'n/0' )
& ! [V_i: $int] :
( ( ( $lesseq @ 0 @ V_i )
& ( $less @ V_i @ 'n/0' ) )
=> ( ( 'int.is-lattice-point/1' @ ( 'nth/2' @ ( 'ListOf' @ $int ) @ V_i @ V_ps ) )
& ( $lesseq @ 1 @ ( 'int.x-coord/1' @ ( 'nth/2' @ ( 'ListOf' @ $int ) @ V_i @ V_ps ) ) )
& ( $lesseq @ ( 'int.x-coord/1' @ ( 'nth/2' @ ( 'ListOf' @ $int ) @ V_i @ V_ps ) ) @ 'n/0' )
& ( $lesseq @ 1 @ ( 'int.y-coord/1' @ ( 'nth/2' @ ( 'ListOf' @ $int ) @ V_i @ V_ps ) ) )
& ( $lesseq @ ( 'int.y-coord/1' @ ( 'nth/2' @ ( 'ListOf' @ $int ) @ V_i @ V_ps ) ) @ 'n/0' ) ) )
& ! [V_k_dot_0: $int,V_l: $int] :
( ( ( $lesseq @ 0 @ V_k_dot_0 )
& ( $less @ V_k_dot_0 @ 'n/0' )
& ( $lesseq @ 0 @ V_l )
& ( $less @ V_l @ 'n/0' ) )
=> ( ( ( 'int.x-coord/1' @ ( 'nth/2' @ ( 'ListOf' @ $int ) @ V_k_dot_0 @ V_ps ) )
!= ( 'int.x-coord/1' @ ( 'nth/2' @ ( 'ListOf' @ $int ) @ V_l @ V_ps ) ) )
& ( ( 'int.y-coord/1' @ ( 'nth/2' @ ( 'ListOf' @ $int ) @ V_k_dot_0 @ V_ps ) )
!= ( 'int.y-coord/1' @ ( 'nth/2' @ ( 'ListOf' @ $int ) @ V_l @ V_ps ) ) ) ) ) )
=> ? [V_x: $int,V_y: $int] :
( ( $lesseq @ 0 @ V_x )
& ( $less @ ( $sum @ V_x @ V_k ) @ 'n/0' )
& ( $lesseq @ 0 @ V_y )
& ( $less @ ( $sum @ V_y @ V_k ) @ 'n/0' )
& ! [V_i_dot_0: $int,V_j: $int] :
( ( ( $lesseq @ V_x @ V_i_dot_0 )
& ( $less @ V_i_dot_0 @ ( $sum @ V_x @ V_k ) )
& ( $lesseq @ V_y @ V_j )
& ( $less @ V_j @ ( $sum @ V_y @ V_k ) ) )
=> ~ ( 'member/2' @ ( 'ListOf' @ $int ) @ ( 'cons/2' @ $int @ V_i_dot_0 @ ( 'cons/2' @ $int @ V_j @ ( 'nil/0' @ $int ) ) ) @ V_ps ) ) ) ) )
@ V_max_k ) ) ).
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