%------------------------------------------------------------------------------
% File     : SWC452_1 : TPTP v9.0.0. Released v9.0.0.
% Domain   : Software Creation
% Problem  : Prove equivalence of small and fast program for sequence A80460
% Version  : Especial.
% English  : Terms: 2 2 6 10 14 14 18 22 26 26 30 34 38 38 42 46 50 50 54 58
%            Small: 2+(2*(2*(((x/2)+x)/2)))
%            Fast : 2+((((x%(0-(2+2)))+x)+x)+x)

% Refs     : [GB+23] Gauthier et al. (2023), A Mathematical Benchmark for I
%          : [Git23] https://github.com/ai4reason/oeis-atp-benchmark
% Source   : [Git23]
% Names    : A80460 [Git23]

% Status   : CounterSatisfiable
% Rating   : 1.00 v9.0.0
% Syntax   : Number of formulae    :    6 (   2 unt;   3 typ;   0 def)
%            Number of atoms       :    6 (   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    :    8 (   2 avg)
%            Number of FOOLs       :    2 (   2 fml;   0 var)
%            Number arithmetic     :   24 (   1 atm;  10 fun;  10 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    :    8 (   3 usr;   2 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) = (2 + (2 * (2 * (((x div 2) + x) div 2)))))
tff(formula_1,axiom,
    ! [X: $int] : ( small(X) = $sum(2,$product(2,$product(2,'div:(Int*Int)>Int'($sum('div:(Int*Int)>Int'(X,2),X),2)))) ) ).

%----∀ x:Int (fast(x) = (2 + (((mod(x, (0 - (2 + 2))) + x) + x) + x)))
tff(formula_2,axiom,
    ! [X: $int] : ( fast(X) = $sum(2,$sum($sum($sum('mod:(Int*Int)>Int'(X,$difference(0,$sum(2,2))),X),X),X)) ) ).

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

%------------------------------------------------------------------------------