## TPTP Problem File: RAL045^1.p

View Solutions - Solve Problem

```%------------------------------------------------------------------------------
% File     : RAL045^1 : TPTP v7.5.0. Released v7.0.0.
% Domain   : Real Algebra
% Problem  : International Mathematical Olympiad, 2011, Problem 3
% Version  : [Mat16] axioms : Especial.
% English  : Let f : R -> R be a real-valued function defined on the set of
%            real numbers that satisfies f(x + y) <= yf(x) + f(f(x)) for all
%            real numbers x and y. Prove that f(x) = 0 for all x <= 0.

% Refs     : [Mat16] Matsuzaki (2016), Email to Geoff Sutcliffe
%          : [MI+16] Matsuzaki et al. (2016), Race against the Teens - Benc
% Source   : [Mat16]
% Names    : IMO-2011-3.p [Mat16]

% Status   : Theorem
% Rating   : ? v7.0.0
% Syntax   : Number of formulae    : 3485 (   0 unit;1199 type;   0 defn)
%            Number of atoms       : 45319 (2209 equality;22711 variable)
%            Maximal formula depth :   35 (   9 average)
%            Number of connectives : 39614 ( 104   ~; 233   |;1172   &;35977   @)
%                                         (1095 <=>;1033  =>;   0  <=;   0 <~>)
%                                         (   0  ~|;   0  ~&)
%            Number of type conns  : 2408 (2408   >;   0   *;   0   +;   0  <<)
%            Number of symbols     : 1246 (1199   :;   0   =;   0  @=)
%                                         (   0  !!;   0  ??;   0 @@+;   0 @@-)
%            Number of variables   : 8059 (  66 sgn;7089   !; 429   ?; 405   ^)
%                                         (8059   :; 136  !>;   0  ?*)
%                                         (   0  @-;   0  @+)
%            Arithmetic symbols    : 1977 (   6 prd;   9 fun;  23 num;1939 var)
% SPC      : TH1_THM_EQU_ARI

% Comments : Theory: ZF; Score: 7; Author: Yiyang Zhan;
%            Generated: 2014-10-24
%------------------------------------------------------------------------------
include('Axioms/MAT001^0.ax').
%------------------------------------------------------------------------------
thf(p,conjecture,(
! [V_f: 'R2R'] :
( ! [V_x: \$real,V_y: \$real] :
( \$lesseq @ ( 'funapp/2' @ V_f @ ( \$sum @ V_x @ V_y ) ) @ ( \$sum @ ( \$product @ V_y @ ( 'funapp/2' @ V_f @ V_x ) ) @ ( 'funapp/2' @ V_f @ ( 'funapp/2' @ V_f @ V_x ) ) ) )
=> ! [V_x_dot_0: \$real] :
( ( \$lesseq @ V_x_dot_0 @ 0.0 )
=> ( ( 'funapp/2' @ V_f @ V_x_dot_0 )
= 0.0 ) ) ) )).

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