TPTP Problem File: ARI741=1.p

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%------------------------------------------------------------------------------
% File     : ARI741=1 : TPTP v7.4.0. Released v7.0.0.
% Domain   : Arithmetic
% Problem  : Real power of real
% Version  : Especial.
% English  :

% Refs     : [Pas16] Paskevich (2016), Email to Geoff Sutcliffe
% Source   : [Pas16]
% Names    : real-PowerRealTest-Pow_2_2.p [Pas16]

% Status   : Theorem
% Rating   : 0.30 v7.4.0, 0.38 v7.3.0, 0.33 v7.0.0
% Syntax   : Number of formulae    :   31 (   9 unit;   7 type)
%            Number of atoms       :   42 (  21 equality)
%            Maximal formula depth :    5 (   3 average)
%            Number of connectives :   18 (   0   ~;   0   |;   3   &)
%                                         (   0 <=>;  15  =>;   0  <=;   0 <~>)
%                                         (   0  ~|;   0  ~&)
%            Number of type conns  :    8 (   7   >;   1   *;   0   +;   0  <<)
%            Number of predicates  :   11 (   8 propositional; 0-2 arity)
%            Number of functors    :   16 (   6 constant; 0-2 arity)
%            Number of variables   :   31 (   0 sgn;  31   !;   0   ?)
%                                         (  31   :;   0  !>;   0  ?*)
%            Maximal term depth    :    4 (   2 average)
%            Arithmetic symbols    :   42 (   2 prd;   3 fun;   6 num;  31 var)
% SPC      : TF0_THM_EQU_ARI

% Comments :
%------------------------------------------------------------------------------
tff(exp,type,(
    exp: $real > $real )).

tff(exp_zero,axiom,(
    exp(0.0) = 1.0 )).

tff(exp_sum,axiom,(
    ! [X: $real,Y: $real] : exp($sum(X,Y)) = $product(exp(X),exp(Y)) )).

tff(log,type,(
    log: $real > $real )).

tff(log_one,axiom,(
    log(1.0) = 0.0 )).

tff(log_mul,axiom,(
    ! [X: $real,Y: $real] :
      ( ( $less(0.0,X)
        & $less(0.0,Y) )
     => log($product(X,Y)) = $sum(log(X),log(Y)) ) )).

tff(log_exp,axiom,(
    ! [X: $real] : log(exp(X)) = X )).

tff(exp_log,axiom,(
    ! [X: $real] :
      ( $less(0.0,X)
     => exp(log(X)) = X ) )).

tff(log2,type,(
    log2: $real > $real )).

tff(log2_def,axiom,(
    ! [X: $real] : log2(X) = $quotient(log(X),log(2.0)) )).

tff(log10,type,(
    log10: $real > $real )).

tff(log10_def,axiom,(
    ! [X: $real] : log10(X) = $quotient(log(X),log(10.0)) )).

tff(pow,type,(
    pow: ( $real * $real ) > $real )).

tff(pow_def,axiom,(
    ! [X: $real,Y: $real] :
      ( $less(0.0,X)
     => pow(X,Y) = exp($product(Y,log(X))) ) )).

tff(pow_pos,axiom,(
    ! [X: $real,Y: $real] :
      ( $less(0.0,X)
     => $less(0.0,pow(X,Y)) ) )).

tff(pow_plus,axiom,(
    ! [X: $real,Y: $real,Z: $real] :
      ( $less(0.0,Z)
     => pow(Z,$sum(X,Y)) = $product(pow(Z,X),pow(Z,Y)) ) )).

tff(pow_mult,axiom,(
    ! [X: $real,Y: $real,Z: $real] :
      ( $less(0.0,X)
     => pow(pow(X,Y),Z) = pow(X,$product(Y,Z)) ) )).

tff(pow_x_zero,axiom,(
    ! [X: $real] :
      ( $less(0.0,X)
     => pow(X,0.0) = 1.0 ) )).

tff(pow_x_one,axiom,(
    ! [X: $real] :
      ( $less(0.0,X)
     => pow(X,1.0) = X ) )).

tff(pow_one_y,axiom,(
    ! [Y: $real] : pow(1.0,Y) = 1.0 )).

tff(sqr,type,(
    sqr: $real > $real )).

tff(sqr_def,axiom,(
    ! [X: $real] : sqr(X) = $product(X,X) )).

tff(sqrt,type,(
    sqrt: $real > $real )).

tff(sqrt_positive,axiom,(
    ! [X: $real] :
      ( $lesseq(0.0,X)
     => $lesseq(0.0,sqrt(X)) ) )).

tff(sqrt_square,axiom,(
    ! [X: $real] :
      ( $lesseq(0.0,X)
     => sqr(sqrt(X)) = X ) )).

tff(square_sqrt,axiom,(
    ! [X: $real] :
      ( $lesseq(0.0,X)
     => sqrt($product(X,X)) = X ) )).

tff(sqrt_mul,axiom,(
    ! [X: $real,Y: $real] :
      ( ( $lesseq(0.0,X)
        & $lesseq(0.0,Y) )
     => sqrt($product(X,Y)) = $product(sqrt(X),sqrt(Y)) ) )).

tff(sqrt_le,axiom,(
    ! [X: $real,Y: $real] :
      ( ( $lesseq(0.0,X)
        & $lesseq(X,Y) )
     => $lesseq(sqrt(X),sqrt(Y)) ) )).

tff(pow_x_two,axiom,(
    ! [X: $real] :
      ( $less(0.0,X)
     => pow(X,2.0) = sqr(X) ) )).

tff(pow_half,axiom,(
    ! [X: $real] :
      ( $less(0.0,X)
     => pow(X,0.5) = sqrt(X) ) )).

tff(pow_2_21,conjecture,(
    pow(2.0,2.0) = 4.0 )).

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