%------------------------------------------------------------------------------
% File     : SWC480_1 : TPTP v9.1.0. Bugfixed v9.1.0.
% Domain   : Software Creation
% Problem  : Prove equivalence of small and fast program for sequence A258048
% Version  : Especial.
% English  : Terms: 1 3 4 6 7 9 10 12 13 14 16 17 19 20 22 23 25 26 28 29
%            Small: (((2+(((1+(x+x))/(1+(2+2)))+x))/(1+2))+1)+x
%            Fast : ((((2*(2+(2*(x+x))))/(1-(2*(2*(2+2)))))+2)+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    : A258048 [Git23]

% Status   : CounterSatisfiable
% Rating   : ? v9.1.0
% Syntax   : Number of formulae    :    6 (   2 unt;   3 typ;   0 def)
%            Number of atoms       :    4 (   3 equ)
%            Maximal formula atoms :    2 (   1 avg)
%            Number of connectives :    3 (   2   ~;   0   |;   1   &)
%                                         (   0 <=>;   0  =>;   0  <=;   0 <~>)
%            Maximal formula depth :    5 (   3 avg)
%            Maximal term depth    :    9 (   3 avg)
%            Number arithmetic     :   42 (   1 atm;  20 fun;  18 num;   3 var)
%            Number of types       :    1 (   0 usr;   1 ari)
%            Number of type conns  :    4 (   3   >;   1   *;   0   +;   0  <<)
%            Number of predicates  :    2 (   0 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.
% Bugfixes : v9.1.0 - Added missing type declarations.
%------------------------------------------------------------------------------
tff(fast,type,
    fast: $int > $int ).

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

tff(div,type,
    'div:(Int*Int)>Int': ( $int * $int ) > $int ).

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

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

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

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