## TPTP Problem File: RAL035^1.p

View Solutions - Solve Problem

```%------------------------------------------------------------------------------
% File     : RAL035^1 : TPTP v7.5.0. Released v7.0.0.
% Domain   : Real Algebra (Number sequences)
% Problem  : International Mathematical Olympiad, 1991, Problem 6
% Version  : [Mat16] axioms : Especial.
% English  : An infinite sequence x_0, x_1, x_2, ... of real numbers is said to
%            be bounded if there is a constant C such that |x_i| =< C for
%            every i >= 0.  Given any real number a > 1, construct a bounded
%            infinite sequence x_0, x_1, x_2, ... such that
%            |x_i - x_j| |i - j|^a >= 1 for every pair of distinct nonnegative
%            integers i, j.

% Refs     : [Mat16] Matsuzaki (2016), Email to Geoff Sutcliffe
%          : [MI+16] Matsuzaki et al. (2016), Race against the Teens - Benc
% Source   : [Mat16]
% Names    : IMO-1991-6.p [Mat16]

% Status   : Theorem
% Rating   : ? v7.0.0
% Syntax   : Number of formulae    : 3486 (   0 unit;1200 type;   0 defn)
%            Number of atoms       : 45328 (2209 equality;22710 variable)
%            Maximal formula depth :   35 (   9 average)
%            Number of connectives : 39624 ( 105   ~; 233   |;1176   &;35983   @)
%                                         (1095 <=>;1032  =>;   0  <=;   0 <~>)
%                                         (   0  ~|;   0  ~&)
%            Number of type conns  : 2408 (2408   >;   0   *;   0   +;   0  <<)
%            Number of symbols     : 1247 (1200   :;   0   =;   0  @=)
%                                         (   0  !!;   0  ??;   0 @@+;   0 @@-)
%            Number of variables   : 8058 (  66 sgn;7087   !; 430   ?; 405   ^)
%                                         (8058   :; 136  !>;   0  ?*)
%                                         (   0  @-;   0  @+)
%            Arithmetic symbols    : 1976 (   6 prd;   9 fun;  23 num;1938 var)
% SPC      : TH1_THM_EQU_ARI

% Comments : Theory: ZF; Score: 7; Author: Yiyang Zhan;
%            Generated: 2014-12-17
%------------------------------------------------------------------------------
include('Axioms/MAT001^0.ax').
%------------------------------------------------------------------------------
thf('a/0_type',type,(
'a/0': \$real )).

thf(p,conjecture,(
? [V_x: 'seq.Seq'] :
( ( 'seq.is-bounded/1' @ V_x )
& ( \$greater @ 'a/0' @ 1.0 )
& ! [V_i: \$int,V_j: \$int] :
( ( ( V_i != V_j )
& ( \$lesseq @ 0 @ V_i )
& ( \$lesseq @ 0 @ V_j ) )
=> ( \$lesseq @ 1.0 @ ( \$product @ ( 'abs/1' @ ( \$difference @ ( 'seq.nth-term-of/2' @ V_x @ ( 'seq.index/1' @ V_i ) ) @ ( 'seq.nth-term-of/2' @ V_x @ ( 'seq.index/1' @ V_j ) ) ) ) @ ( '^/2' @ ( 'abs/1' @ ( \$to_real @ ( \$difference @ V_i @ V_j ) ) ) @ 'a/0' ) ) ) ) ) )).

%------------------------------------------------------------------------------
```