%------------------------------------------------------------------------------
% File     : SWC465_1 : TPTP v9.0.0. Released v9.0.0.
% Domain   : Software Creation
% Problem  : Prove equivalence of small and fast program for sequence A167389
% Version  : Especial.
% English  : Terms: 2 3 5 6 8 9 10 12 13 15 16 18 19 21 22 23 25 26 28 29
%            Small: (((((1+(((x/(2+(2*(2+2))))-x)/(2+2)))/2)+x)/2)+2)+x
%            Fast : (((((1+2)*(1+(2+x)))+(x/(2+(2*(2+2)))))/
%                   (1+(2+(2+2))))+1)+x

% Refs     : [GB+23] Gauthier et al. (2023), A Mathematical Benchmark for I
%          : [Git23] https://github.com/ai4reason/oeis-atp-benchmark
% Source   : [Git23]
% Names    : A167389 [Git23]

% Status   : CounterSatisfiable
% Rating   : 1.00 v9.0.0
% Syntax   : Number of formulae    :    5 (   2 unt;   2 typ;   0 def)
%            Number of atoms       :   10 (   3 equ)
%            Maximal formula atoms :    2 (   3 avg)
%            Number of connectives :    3 (   2   ~;   0   |;   1   &)
%                                         (   0 <=>;   0  =>;   0  <=;   0 <~>)
%            Maximal formula depth :    5 (   3 avg)
%            Maximal term depth    :    9 (   2 avg)
%            Number of FOOLs       :    6 (   6 fml;   0 var)
%            Number arithmetic     :   50 (   1 atm;  22 fun;  24 num;   3 var)
%            Number of types       :    1 (   0 usr;   1 ari)
%            Number of type conns  :    2 (   2   >;   0   *;   0   +;   0  <<)
%            Number of predicates  :    3 (   1 usr;   0 prp; 2-2 aty)
%            Number of functors    :    8 (   2 usr;   3 con; 0-2 aty)
%            Number of variables   :    3 (;   2   !;   1   ?;   3   :)
% SPC      : TX0_CSA_EQU_ARI

% Comments : Not in an "aind_*" subset, i.e., unlikely to require induction.
%------------------------------------------------------------------------------
tff(fast,type,
    fast: $int > $int ).

tff(small,type,
    small: $int > $int ).

%----∀ x:Int (small(x) = ((((((1 + (((x div (2 + (2 * (2 + 2)))) - x) div (2 +
%----2))) div 2) + x) div 2) + 2) + x))
tff(formula_1,axiom,
    ! [X: $int] : ( small(X) = $sum($sum('div:(Int*Int)>Int'($sum('div:(Int*Int)>Int'($sum(1,'div:(Int*Int)>Int'($difference('div:(Int*Int)>Int'(X,$sum(2,$product(2,$sum(2,2)))),X),$sum(2,2))),2),X),2),2),X) ) ).

%----∀ x:Int (fast(x) = ((((((1 + 2) * (1 + (2 + x))) + (x div (2 + (2 * (2 +
%----2))))) div (1 + (2 + (2 + 2)))) + 1) + x))
tff(formula_2,axiom,
    ! [X: $int] : ( fast(X) = $sum($sum('div:(Int*Int)>Int'($sum($product($sum(1,2),$sum(1,$sum(2,X))),'div:(Int*Int)>Int'(X,$sum(2,$product(2,$sum(2,2))))),$sum(1,$sum(2,$sum(2,2)))),1),X) ) ).

%----∃ c:Int ((c ≥ 0) ∧ ¬(small(c) = fast(c)))
tff(conjecture_1,conjecture,
    ~ ? [C: $int] :
        ( $greatereq(C,0)
        & ( small(C) != fast(C) ) ) ).

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