## TPTP Problem File: RAL043^1.p

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```%------------------------------------------------------------------------------
% File     : RAL043^1 : TPTP v7.5.0. Released v7.0.0.
% Domain   : Real Algebra (Functional equations)
% Problem  : International Mathematical Olympiad, 2009, Problem 5
% Version  : [Mat16] axioms : Especial.
% English  : Determine all functions f from the set of positive integers to
%            the set of positive integers such that, for all positive
%            integers a and b, there exists a non-degenerate triangle with
%            sides of lengths a, f(b) and f(b + f(a) - 1).  (A triangle is
%            non-degenerate if its vertices are not collinear.)

% Refs     : [Mat16] Matsuzaki (2016), Email to Geoff Sutcliffe
%          : [MI+16] Matsuzaki et al. (2016), Race against the Teens - Benc
% Source   : [Mat16]
% Names    : IMO-2009-5.p [Mat16]

% Status   : Theorem
% Rating   : ? v7.0.0
% Syntax   : Number of formulae    : 3485 (   0 unit;1199 type;   0 defn)
%            Number of atoms       : 45353 (2209 equality;22728 variable)
%            Maximal formula depth :   35 (   9 average)
%            Number of connectives : 39647 ( 104   ~; 233   |;1177   &;36004   @)
%                                         (1095 <=>;1034  =>;   0  <=;   0 <~>)
%                                         (   0  ~|;   0  ~&)
%            Number of type conns  : 2410 (2410   >;   0   *;   0   +;   0  <<)
%            Number of symbols     : 1246 (1199   :;   0   =;   0  @=)
%                                         (   0  !!;   0  ??;   0 @@+;   0 @@-)
%            Number of variables   : 8060 (  66 sgn;7089   !; 429   ?; 406   ^)
%                                         (8060   :; 136  !>;   0  ?*)
%                                         (   0  @-;   0  @+)
%            Arithmetic symbols    : 1978 (   6 prd;   9 fun;  23 num;1940 var)
% SPC      : TH1_THM_EQU_ARI

% Comments : Theory: ZF; Score: 7; Author: Yiyang Zhan;
%            Generated: 2014-10-31
%            ^ [V_f_dot_0: ( \$int > \$int )] :
%            ! [V_x: \$int] :
%              ( ( ( \$greater @ V_x @ 0 )
%               => ( ( V_f_dot_0 @ V_x )
%                  = V_x ) )
%              & ( ( \$lesseq @ V_x @ 0 )
%               => ( ( V_f_dot_0 @ V_x )
%                  = 0 ) ) ) )
%------------------------------------------------------------------------------
include('Axioms/MAT001^0.ax').
%------------------------------------------------------------------------------
thf(p_qustion,conjecture,
( 'find/1' @ ( \$int > \$int )
@ ^ [V_f: ( \$int > \$int )] :
( ! [V_n_dot_0: \$int] :
( ( \$lesseq @ V_n_dot_0 @ 0 )
=> ( ( V_f @ V_n_dot_0 )
= 0 ) )
& ! [V_n: \$int] :
( ( \$less @ 0 @ V_n )
=> ( \$less @ 0 @ ( V_f @ V_n ) ) )
& ! [V_a: \$int,V_b: \$int] :
( ( ( \$less @ 0 @ V_a )
& ( \$less @ 0 @ V_b ) )
=> ( ( \$less @ V_a @ ( \$sum @ ( V_f @ V_b ) @ ( V_f @ ( \$sum @ V_b @ ( \$sum @ ( V_f @ V_a ) @ -1 ) ) ) ) )
& ( \$less @ ( V_f @ V_b ) @ ( \$sum @ ( V_f @ ( \$sum @ V_b @ ( \$sum @ ( V_f @ V_a ) @ -1 ) ) ) @ V_a ) )
& ( \$less @ ( V_f @ ( \$sum @ V_b @ ( \$sum @ ( V_f @ V_a ) @ -1 ) ) ) @ ( \$sum @ V_a @ ( V_f @ V_b ) ) ) ) ) ) )).

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