%------------------------------------------------------------------------------
% File     : SYN000_5 : TPTP v8.2.0. Bugfixed v5.5.1.
% Domain   : Syntactic
% Problem  : TF0 syntax with arithmetic
% Version  : Biased.
% English  : 

% Refs     :
% Source   : [TPTP]
% Names    :

% Status   : Theorem
% Rating   : 0.62 v7.5.0, 0.60 v7.4.0, 0.62 v7.3.0, 0.67 v7.0.0, 0.57 v6.4.0, 0.33 v6.3.0, 0.71 v6.2.0, 1.00 v6.0.0
% Syntax   : Number of formulae    :   83 (  70 unt;   6 typ;   0 def)
%            Number of atoms       :   91 (   4 equ)
%            Maximal formula atoms :    7 (   1 avg)
%            Number of connectives :   14 (   0   ~;  10   |;   1   &)
%                                         (   0 <=>;   3  =>;   0  <=;   0 <~>)
%            Maximal formula depth :    7 (   1 avg)
%            Maximal term depth    :    3 (   1 avg)
%            Number arithmetic     :  196 (  19 atm;  54 fun; 109 num;  14 var)
%            Number of types       :    4 (   0 usr;   3 ari)
%            Number of type conns  :    3 (   3   >;   0   *;   0   +;   0  <<)
%            Number of predicates  :   10 (   3 usr;   0 prp; 1-2 aty)
%            Number of functors    :   41 (   3 usr;  24 con; 0-2 aty)
%            Number of variables   :   14 (   3   !;  11   ?;  14   :)
% SPC      : TF0_THM_EQU_ARI

% Comments : 
% Bugfixes : v5.5.1 - Removed $evaleq.
%------------------------------------------------------------------------------
%----Types for what follows
tff(p_int_type,type,
    p_int: $int > $o ).

tff(p_rat_type,type,
    p_rat: $rat > $o ).

tff(p_real_type,type,
    p_real: $real > $o ).

tff(a_int,type,
    a_int: $int ).

tff(a_rat,type,
    a_rat: $rat ).

tff(a_real,type,
    a_real: $real ).

%----Numbers
tff(integers,axiom,
    ( p_int(123)
    | p_int(-123) ) ).

tff(rationals,axiom,
    ( p_rat(123/456)
    | p_rat(-123/456)
    | p_rat(123/456) ) ).

tff(reals,axiom,
    ( p_real(123.456)
    | p_real(-123.456)
    | p_real(123.456E78)
    | p_real(123.456e78)
    | p_real(-123.456E78)
    | p_real(123.456E-78)
    | p_real(-123.456E-78) ) ).

%----Variables
tff(variables_int,axiom,
    ! [X: $int] :
    ? [Y: $int] :
      ( p_int(X)
     => p_int(Y) ) ).

tff(variables_rat,axiom,
    ! [X: $rat] :
    ? [Y: $rat] :
      ( p_rat(X)
     => p_rat(Y) ) ).

tff(variables_real,axiom,
    ! [X: $real] :
    ? [Y: $real] :
      ( p_real(X)
     => p_real(Y) ) ).

%----Arithmetic relations
tff(less_int,axiom,
    $less(a_int,3) ).

tff(less_rat,axiom,
    $less(a_rat,3/9) ).

tff(less_real,axiom,
    $less(a_real,3.3) ).

tff(lesseq_int,axiom,
    $lesseq(a_int,3) ).

tff(lesseq_rat,axiom,
    $lesseq(a_rat,3/9) ).

tff(lesseq_real,axiom,
    $lesseq(a_real,3.3) ).

tff(greater_int,axiom,
    $greater(a_int,-3) ).

tff(greater_rat,axiom,
    $greater(a_rat,-3/9) ).

tff(greater_real,axiom,
    $greater(a_real,-3.3) ).

tff(greatereq_int,axiom,
    $greatereq(a_int,-3) ).

tff(greatereq_rat,axiom,
    $greatereq(a_rat,-3/9) ).

tff(greatereq_real,axiom,
    $greatereq(a_real,-3.3) ).

tff(equal_int,axiom,
    a_int = 0 ).

tff(equal_rat,axiom,
    a_rat = 0/1 ).

tff(equal_real,axiom,
    a_real = 0.0 ).

%----Arithmetic functions
tff(uminus_int,axiom,
    p_int($uminus(3)) ).

tff(uminus_rat,axiom,
    p_rat($uminus(3/9)) ).

tff(uminus_real,axiom,
    p_real($uminus(3.3)) ).

tff(sum_int,axiom,
    p_int($sum(3,3)) ).

tff(sum_rat,axiom,
    p_rat($sum(3/9,3/9)) ).

tff(sum_real,axiom,
    p_real($sum(3.3,3.3)) ).

tff(difference_int,axiom,
    p_int($difference(3,3)) ).

tff(difference_rat,axiom,
    p_rat($difference(3/9,3/9)) ).

tff(difference_real,axiom,
    p_real($difference(3.3,3.3)) ).

tff(product_int,axiom,
    p_int($product(3,3)) ).

tff(product_rat,axiom,
    p_rat($product(3/9,3/9)) ).

tff(product_real,axiom,
    p_real($product(3.3,3.3)) ).

tff(quotient_rat,axiom,
    p_rat($quotient(3/9,3/9)) ).

tff(quotient_real,axiom,
    p_real($quotient(3.3,3.3)) ).

tff(quotient_e_int,axiom,
    p_int($quotient_e(3,3)) ).

tff(quotient_e_rat,axiom,
    p_rat($quotient_e(3/9,3/9)) ).

tff(quotient_e_real,axiom,
    p_real($quotient_e(3.3,3.3)) ).

tff(quotient_t_int,axiom,
    p_int($quotient_t(3,3)) ).

tff(quotient_t_rat,axiom,
    p_rat($quotient_t(3/9,3/9)) ).

tff(quotient_t_real,axiom,
    p_real($quotient_t(3.3,3.3)) ).

tff(quotient_f_int,axiom,
    p_int($quotient_f(3,3)) ).

tff(quotient_f_rat,axiom,
    p_rat($quotient_f(3/9,3/9)) ).

tff(quotient_f_real,axiom,
    p_real($quotient_f(3.3,3.3)) ).

tff(remainder_e_int,axiom,
    p_int($remainder_e(3,3)) ).

tff(remainder_e_rat,axiom,
    p_rat($remainder_e(3/9,3/9)) ).

tff(remainder_e_real,axiom,
    p_real($remainder_e(3.3,3.3)) ).

tff(remainder_t_int,axiom,
    p_int($remainder_t(3,3)) ).

tff(remainder_t_rat,axiom,
    p_rat($remainder_t(3/9,3/9)) ).

tff(remainder_t_real,axiom,
    p_real($remainder_t(3.3,3.3)) ).

tff(remainder_f_int,axiom,
    p_int($remainder_f(3,3)) ).

tff(remainder_f_rat,axiom,
    p_rat($remainder_f(3/9,3/9)) ).

tff(remainder_f_real,axiom,
    p_real($remainder_f(3.3,3.3)) ).

tff(floor_int,axiom,
    p_int($floor(3)) ).

tff(floor_rat,axiom,
    p_rat($floor(3/9)) ).

tff(floor_real,axiom,
    p_real($floor(3.3)) ).

tff(ceiling_int,axiom,
    p_int($ceiling(3)) ).

tff(ceiling_rat,axiom,
    p_rat($ceiling(3/9)) ).

tff(ceiling_real,axiom,
    p_real($ceiling(3.3)) ).

tff(truncate_int,axiom,
    p_int($truncate(3)) ).

tff(truncate_rat,axiom,
    p_rat($truncate(3/9)) ).

tff(truncate_real,axiom,
    p_real($truncate(3.3)) ).

%----Recognizing numbers
tff(is_int_int,axiom,
    ? [X: $int] : $is_int(X) ).

tff(is_int_rat,axiom,
    ? [X: $rat] : $is_int(X) ).

tff(is_int_real,axiom,
    ? [X: $real] : $is_int(X) ).

tff(is_rat_rat,axiom,
    ? [X: $rat] : $is_rat(X) ).

tff(is_rat_real,axiom,
    ? [X: $real] : $is_rat(X) ).

%----Coercion
tff(to_int_int,axiom,
    p_int($to_int(3)) ).

tff(to_int_rat,axiom,
    p_int($to_int(3/9)) ).

tff(to_int_real,axiom,
    p_int($to_int(3.3)) ).

tff(to_rat_int,axiom,
    p_rat($to_rat(3)) ).

tff(to_rat_rat,axiom,
    p_rat($to_rat(3/9)) ).

tff(to_rat_real,axiom,
    p_rat($to_rat(3.3)) ).

tff(to_real_int,axiom,
    p_real($to_real(3)) ).

tff(to_real_rat,axiom,
    p_real($to_real(3/9)) ).

tff(to_real_real,axiom,
    p_real($to_real(3.3)) ).

%----A conjecture to prove
tff(mixed,conjecture,
    ? [X: $int,Y: $rat,Z: $real] :
      ( ( Y = $to_rat($sum(X,2)) )
      & ( $less($to_int(Y),3)
        | $greater($to_real(Y),3.3) ) ) ).

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