%------------------------------------------------------------------------------
% File     : SWC464_1 : TPTP v9.0.0. Released v9.0.0.
% Domain   : Software Creation
% Problem  : Prove equivalence of small and fast program for sequence A163584
% Version  : Especial.
% English  : Terms: 0 1 0 0 1 0 0 1 0 0 1 0 0 0 1 0 0 1 0 0
%            Small: (((((1+(x/(1-(2*(2+(2+2))))))/2)+1)+x)%(1+2))/2
%            Fast : (((1+((2*(2+2))+x))%(2+(2*(2+(2*(2+2))))))%(1+2))%2

% Refs     : [GB+23] Gauthier et al. (2023), A Mathematical Benchmark for I
%          : [Git23] https://github.com/ai4reason/oeis-atp-benchmark
% Source   : [Git23]
% Names    : A163584 [Git23]

% Status   : CounterSatisfiable
% Rating   : 1.00 v9.0.0
% Syntax   : Number of formulae    :    6 (   2 unt;   3 typ;   0 def)
%            Number of atoms       :    7 (   3 equ)
%            Maximal formula atoms :    2 (   2 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       :    3 (   3 fml;   0 var)
%            Number arithmetic     :   47 (   1 atm;  18 fun;  25 num;   3 var)
%            Number of types       :    1 (   0 usr;   1 ari)
%            Number of type conns  :    4 (   3   >;   1   *;   0   +;   0  <<)
%            Number of predicates  :    3 (   1 usr;   0 prp; 2-2 aty)
%            Number of functors    :    9 (   3 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(mod,type,
    'mod:(Int*Int)>Int': ( $int * $int ) > $int ).

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

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

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

%----∃ 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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