%------------------------------------------------------------------------------
% File     : NUM461+1 : TPTP v9.0.0. Released v4.0.0.
% Domain   : Number Theory
% Problem  : Square root of a prime is irrational 05, 00 expansion
% Version  : Especial.
% English  :

% Refs     : [LPV06] Lyaletski et al. (2006), SAD as a Mathematical Assista
%          : [VLP07] Verchinine et al. (2007), System for Automated Deduction
%          : [Pas08] Paskevich (2008), Email to G. Sutcliffe
% Source   : [Pas08]
% Names    : primes_05.00 [Pas08]

% Status   : Theorem
% Rating   : 0.64 v9.0.0, 0.61 v8.2.0, 0.67 v8.1.0, 0.61 v7.5.0, 0.66 v7.4.0, 0.57 v7.3.0, 0.59 v7.1.0, 0.52 v7.0.0, 0.57 v6.4.0, 0.62 v6.3.0, 0.54 v6.2.0, 0.60 v6.1.0, 0.77 v6.0.0, 0.65 v5.5.0, 0.78 v5.4.0, 0.82 v5.3.0, 0.85 v5.2.0, 0.80 v5.1.0, 0.86 v5.0.0, 0.88 v4.1.0, 0.91 v4.0.1, 1.00 v4.0.0
% Syntax   : Number of formulae    :   27 (   2 unt;   2 def)
%            Number of atoms       :  100 (  34 equ)
%            Maximal formula atoms :    7 (   3 avg)
%            Number of connectives :   79 (   6   ~;   4   |;  37   &)
%                                         (   2 <=>;  30  =>;   0  <=;   0 <~>)
%            Maximal formula depth :    9 (   5 avg)
%            Maximal term depth    :    3 (   1 avg)
%            Number of predicates  :    4 (   2 usr;   1 prp; 0-2 aty)
%            Number of functors    :    8 (   8 usr;   5 con; 0-2 aty)
%            Number of variables   :   45 (  44   !;   1   ?)
% SPC      : FOF_THM_RFO_SEQ

% Comments : Problem generated by the SAD system [VLP07]
%------------------------------------------------------------------------------
fof(mNatSort,axiom,
    ! [W0] :
      ( aNaturalNumber0(W0)
     => $true ) ).

fof(mSortsC,axiom,
    aNaturalNumber0(sz00) ).

fof(mSortsC_01,axiom,
    ( aNaturalNumber0(sz10)
    & sz10 != sz00 ) ).

fof(mSortsB,axiom,
    ! [W0,W1] :
      ( ( aNaturalNumber0(W0)
        & aNaturalNumber0(W1) )
     => aNaturalNumber0(sdtpldt0(W0,W1)) ) ).

fof(mSortsB_02,axiom,
    ! [W0,W1] :
      ( ( aNaturalNumber0(W0)
        & aNaturalNumber0(W1) )
     => aNaturalNumber0(sdtasdt0(W0,W1)) ) ).

fof(mAddComm,axiom,
    ! [W0,W1] :
      ( ( aNaturalNumber0(W0)
        & aNaturalNumber0(W1) )
     => sdtpldt0(W0,W1) = sdtpldt0(W1,W0) ) ).

fof(mAddAsso,axiom,
    ! [W0,W1,W2] :
      ( ( aNaturalNumber0(W0)
        & aNaturalNumber0(W1)
        & aNaturalNumber0(W2) )
     => sdtpldt0(sdtpldt0(W0,W1),W2) = sdtpldt0(W0,sdtpldt0(W1,W2)) ) ).

fof(m_AddZero,axiom,
    ! [W0] :
      ( aNaturalNumber0(W0)
     => ( sdtpldt0(W0,sz00) = W0
        & W0 = sdtpldt0(sz00,W0) ) ) ).

fof(mMulComm,axiom,
    ! [W0,W1] :
      ( ( aNaturalNumber0(W0)
        & aNaturalNumber0(W1) )
     => sdtasdt0(W0,W1) = sdtasdt0(W1,W0) ) ).

fof(mMulAsso,axiom,
    ! [W0,W1,W2] :
      ( ( aNaturalNumber0(W0)
        & aNaturalNumber0(W1)
        & aNaturalNumber0(W2) )
     => sdtasdt0(sdtasdt0(W0,W1),W2) = sdtasdt0(W0,sdtasdt0(W1,W2)) ) ).

fof(m_MulUnit,axiom,
    ! [W0] :
      ( aNaturalNumber0(W0)
     => ( sdtasdt0(W0,sz10) = W0
        & W0 = sdtasdt0(sz10,W0) ) ) ).

fof(m_MulZero,axiom,
    ! [W0] :
      ( aNaturalNumber0(W0)
     => ( sdtasdt0(W0,sz00) = sz00
        & sz00 = sdtasdt0(sz00,W0) ) ) ).

fof(mAMDistr,axiom,
    ! [W0,W1,W2] :
      ( ( aNaturalNumber0(W0)
        & aNaturalNumber0(W1)
        & aNaturalNumber0(W2) )
     => ( sdtasdt0(W0,sdtpldt0(W1,W2)) = sdtpldt0(sdtasdt0(W0,W1),sdtasdt0(W0,W2))
        & sdtasdt0(sdtpldt0(W1,W2),W0) = sdtpldt0(sdtasdt0(W1,W0),sdtasdt0(W2,W0)) ) ) ).

fof(mAddCanc,axiom,
    ! [W0,W1,W2] :
      ( ( aNaturalNumber0(W0)
        & aNaturalNumber0(W1)
        & aNaturalNumber0(W2) )
     => ( ( sdtpldt0(W0,W1) = sdtpldt0(W0,W2)
          | sdtpldt0(W1,W0) = sdtpldt0(W2,W0) )
       => W1 = W2 ) ) ).

fof(mMulCanc,axiom,
    ! [W0] :
      ( aNaturalNumber0(W0)
     => ( W0 != sz00
       => ! [W1,W2] :
            ( ( aNaturalNumber0(W1)
              & aNaturalNumber0(W2) )
           => ( ( sdtasdt0(W0,W1) = sdtasdt0(W0,W2)
                | sdtasdt0(W1,W0) = sdtasdt0(W2,W0) )
             => W1 = W2 ) ) ) ) ).

fof(mZeroAdd,axiom,
    ! [W0,W1] :
      ( ( aNaturalNumber0(W0)
        & aNaturalNumber0(W1) )
     => ( sdtpldt0(W0,W1) = sz00
       => ( W0 = sz00
          & W1 = sz00 ) ) ) ).

fof(mZeroMul,axiom,
    ! [W0,W1] :
      ( ( aNaturalNumber0(W0)
        & aNaturalNumber0(W1) )
     => ( sdtasdt0(W0,W1) = sz00
       => ( W0 = sz00
          | W1 = sz00 ) ) ) ).

fof(mDefLE,definition,
    ! [W0,W1] :
      ( ( aNaturalNumber0(W0)
        & aNaturalNumber0(W1) )
     => ( sdtlseqdt0(W0,W1)
      <=> ? [W2] :
            ( aNaturalNumber0(W2)
            & sdtpldt0(W0,W2) = W1 ) ) ) ).

fof(mDefDiff,definition,
    ! [W0,W1] :
      ( ( aNaturalNumber0(W0)
        & aNaturalNumber0(W1) )
     => ( sdtlseqdt0(W0,W1)
       => ! [W2] :
            ( W2 = sdtmndt0(W1,W0)
          <=> ( aNaturalNumber0(W2)
              & sdtpldt0(W0,W2) = W1 ) ) ) ) ).

fof(mLERefl,axiom,
    ! [W0] :
      ( aNaturalNumber0(W0)
     => sdtlseqdt0(W0,W0) ) ).

fof(mLEAsym,axiom,
    ! [W0,W1] :
      ( ( aNaturalNumber0(W0)
        & aNaturalNumber0(W1) )
     => ( ( sdtlseqdt0(W0,W1)
          & sdtlseqdt0(W1,W0) )
       => W0 = W1 ) ) ).

fof(mLETran,axiom,
    ! [W0,W1,W2] :
      ( ( aNaturalNumber0(W0)
        & aNaturalNumber0(W1)
        & aNaturalNumber0(W2) )
     => ( ( sdtlseqdt0(W0,W1)
          & sdtlseqdt0(W1,W2) )
       => sdtlseqdt0(W0,W2) ) ) ).

fof(mLETotal,axiom,
    ! [W0,W1] :
      ( ( aNaturalNumber0(W0)
        & aNaturalNumber0(W1) )
     => ( sdtlseqdt0(W0,W1)
        | ( W1 != W0
          & sdtlseqdt0(W1,W0) ) ) ) ).

fof(m__840,hypothesis,
    ( aNaturalNumber0(xl)
    & aNaturalNumber0(xn) ) ).

fof(m__840_03,hypothesis,
    ( xl != xn
    & sdtlseqdt0(xl,xn) ) ).

fof(m__873,hypothesis,
    aNaturalNumber0(xm) ).

fof(m__,conjecture,
    ( sdtpldt0(xm,xl) != sdtpldt0(xm,xn)
    & sdtlseqdt0(sdtpldt0(xm,xl),sdtpldt0(xm,xn))
    & sdtpldt0(xl,xm) != sdtpldt0(xn,xm)
    & sdtlseqdt0(sdtpldt0(xl,xm),sdtpldt0(xn,xm)) ) ).

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