TPTP Problem File: RNG066+2.p

View Solutions - Solve Problem

%------------------------------------------------------------------------------
% File     : RNG066+2 : TPTP v9.0.0. Released v4.0.0.
% Domain   : Ring Theory
% Problem  : Cauchy-Bouniakowsky-Schwarz inequality 05_16_03, 01 expansion
% Version  : Especial.
% English  :

% Refs     : [VLP07] Verchinine et al. (2007), System for Automated Deduction
%          : [Pas08] Paskevich (2008), Email to G. Sutcliffe
% Source   : [Pas08]
% Names    : cauchy_05_16_03.01 [Pas08]

% Status   : Theorem
% Rating   : 0.76 v9.0.0, 0.78 v8.1.0, 0.81 v7.4.0, 0.80 v7.3.0, 0.83 v7.0.0, 0.87 v6.4.0, 0.85 v6.3.0, 0.83 v6.2.0, 0.92 v6.1.0, 1.00 v5.3.0, 0.96 v5.2.0, 1.00 v4.0.0
% Syntax   : Number of formulae    :   59 (   8 unt;   1 def)
%            Number of atoms       :  196 (  61 equ)
%            Maximal formula atoms :    9 (   3 avg)
%            Number of connectives :  143 (   6   ~;   1   |;  80   &)
%                                         (   1 <=>;  55  =>;   0  <=;   0 <~>)
%            Maximal formula depth :   10 (   4 avg)
%            Maximal term depth    :    5 (   1 avg)
%            Number of predicates  :    7 (   5 usr;   1 prp; 0-2 aty)
%            Number of functors    :   26 (  26 usr;  18 con; 0-2 aty)
%            Number of variables   :   73 (  72   !;   1   ?)
% SPC      : FOF_THM_RFO_SEQ

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

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

fof(mSuccNat,axiom,
    ! [W0] :
      ( aNaturalNumber0(W0)
     => ( aNaturalNumber0(szszuzczcdt0(W0))
        & szszuzczcdt0(W0) != sz00 ) ) ).

fof(mNatExtr,axiom,
    ! [W0] :
      ( ( aNaturalNumber0(W0)
        & W0 != sz00 )
     => ? [W1] :
          ( aNaturalNumber0(W1)
          & W0 = szszuzczcdt0(W1) ) ) ).

fof(mSuccEqu,axiom,
    ! [W0,W1] :
      ( ( aNaturalNumber0(W0)
        & aNaturalNumber0(W1) )
     => ( szszuzczcdt0(W0) = szszuzczcdt0(W1)
       => W0 = W1 ) ) ).

fof(mIHOrd,axiom,
    ! [W0,W1] :
      ( ( aNaturalNumber0(W0)
        & aNaturalNumber0(W1) )
     => ( iLess0(W0,W1)
       => $true ) ) ).

fof(mIH,axiom,
    ! [W0] :
      ( aNaturalNumber0(W0)
     => iLess0(W0,szszuzczcdt0(W0)) ) ).

fof(mScSort,axiom,
    ! [W0] :
      ( aScalar0(W0)
     => $true ) ).

fof(mSZeroSc,axiom,
    aScalar0(sz0z00) ).

fof(mSumSc,axiom,
    ! [W0,W1] :
      ( ( aScalar0(W0)
        & aScalar0(W1) )
     => aScalar0(sdtpldt0(W0,W1)) ) ).

fof(mMulSc,axiom,
    ! [W0,W1] :
      ( ( aScalar0(W0)
        & aScalar0(W1) )
     => aScalar0(sdtasdt0(W0,W1)) ) ).

fof(mNegSc,axiom,
    ! [W0] :
      ( aScalar0(W0)
     => aScalar0(smndt0(W0)) ) ).

fof(mScZero,axiom,
    ! [W0] :
      ( aScalar0(W0)
     => ( sdtpldt0(W0,sz0z00) = W0
        & sdtpldt0(sz0z00,W0) = W0
        & sdtasdt0(W0,sz0z00) = sz0z00
        & sdtasdt0(sz0z00,W0) = sz0z00
        & sdtpldt0(W0,smndt0(W0)) = sz0z00
        & sdtpldt0(smndt0(W0),W0) = sz0z00
        & smndt0(smndt0(W0)) = W0
        & smndt0(sz0z00) = sz0z00 ) ) ).

fof(mArith,axiom,
    ! [W0,W1,W2] :
      ( ( aScalar0(W0)
        & aScalar0(W1)
        & aScalar0(W2) )
     => ( sdtpldt0(sdtpldt0(W0,W1),W2) = sdtpldt0(W0,sdtpldt0(W1,W2))
        & sdtpldt0(W0,W1) = sdtpldt0(W1,W0)
        & sdtasdt0(sdtasdt0(W0,W1),W2) = sdtasdt0(W0,sdtasdt0(W1,W2))
        & sdtasdt0(W0,W1) = sdtasdt0(W1,W0) ) ) ).

fof(mDistr,axiom,
    ! [W0,W1,W2] :
      ( ( aScalar0(W0)
        & aScalar0(W1)
        & aScalar0(W2) )
     => ( sdtasdt0(W0,sdtpldt0(W1,W2)) = sdtpldt0(sdtasdt0(W0,W1),sdtasdt0(W0,W2))
        & sdtasdt0(sdtpldt0(W0,W1),W2) = sdtpldt0(sdtasdt0(W0,W2),sdtasdt0(W1,W2)) ) ) ).

fof(mDistr2,axiom,
    ! [W0,W1,W2,W3] :
      ( ( aScalar0(W0)
        & aScalar0(W1)
        & aScalar0(W2)
        & aScalar0(W3) )
     => sdtasdt0(sdtpldt0(W0,W1),sdtpldt0(W2,W3)) = sdtpldt0(sdtpldt0(sdtasdt0(W0,W2),sdtasdt0(W0,W3)),sdtpldt0(sdtasdt0(W1,W2),sdtasdt0(W1,W3))) ) ).

fof(mMNeg,axiom,
    ! [W0,W1] :
      ( ( aScalar0(W0)
        & aScalar0(W1) )
     => ( sdtasdt0(W0,smndt0(W1)) = smndt0(sdtasdt0(W0,W1))
        & sdtasdt0(smndt0(W0),W1) = smndt0(sdtasdt0(W0,W1)) ) ) ).

fof(mMDNeg,axiom,
    ! [W0,W1] :
      ( ( aScalar0(W0)
        & aScalar0(W1) )
     => sdtasdt0(smndt0(W0),smndt0(W1)) = sdtasdt0(W0,W1) ) ).

fof(mLess,axiom,
    ! [W0,W1] :
      ( ( aScalar0(W0)
        & aScalar0(W1) )
     => ( sdtlseqdt0(W0,W1)
       => $true ) ) ).

fof(mLERef,axiom,
    ! [W0] :
      ( aScalar0(W0)
     => sdtlseqdt0(W0,W0) ) ).

fof(mLEASm,axiom,
    ! [W0,W1] :
      ( ( aScalar0(W0)
        & aScalar0(W1) )
     => ( ( sdtlseqdt0(W0,W1)
          & sdtlseqdt0(W1,W0) )
       => W0 = W1 ) ) ).

fof(mLETrn,axiom,
    ! [W0,W1,W2] :
      ( ( aScalar0(W0)
        & aScalar0(W1)
        & aScalar0(W2) )
     => ( ( sdtlseqdt0(W0,W1)
          & sdtlseqdt0(W1,W2) )
       => sdtlseqdt0(W0,W2) ) ) ).

fof(mLEMon,axiom,
    ! [W0,W1,W2,W3] :
      ( ( aScalar0(W0)
        & aScalar0(W1)
        & aScalar0(W2)
        & aScalar0(W3) )
     => ( ( sdtlseqdt0(W0,W1)
          & sdtlseqdt0(W2,W3) )
       => sdtlseqdt0(sdtpldt0(W0,W2),sdtpldt0(W1,W3)) ) ) ).

fof(mLEMonM,axiom,
    ! [W0,W1,W2,W3] :
      ( ( aScalar0(W0)
        & aScalar0(W1)
        & aScalar0(W2)
        & aScalar0(W3) )
     => ( ( sdtlseqdt0(W0,W1)
          & sdtlseqdt0(sz0z00,W2)
          & sdtlseqdt0(W2,W3) )
       => sdtlseqdt0(sdtasdt0(W0,W2),sdtasdt0(W1,W3)) ) ) ).

fof(mLETot,axiom,
    ! [W0,W1] :
      ( ( aScalar0(W0)
        & aScalar0(W1) )
     => ( sdtlseqdt0(W0,W1)
        | sdtlseqdt0(W1,W0) ) ) ).

fof(mPosMon,axiom,
    ! [W0,W1] :
      ( ( aScalar0(W0)
        & aScalar0(W1) )
     => ( ( sdtlseqdt0(sz0z00,W0)
          & sdtlseqdt0(sz0z00,W1) )
       => ( sdtlseqdt0(sz0z00,sdtpldt0(W0,W1))
          & sdtlseqdt0(sz0z00,sdtasdt0(W0,W1)) ) ) ) ).

fof(mSqPos,axiom,
    ! [W0] :
      ( aScalar0(W0)
     => sdtlseqdt0(sz0z00,sdtasdt0(W0,W0)) ) ).

fof(mSqrt,axiom,
    ! [W0,W1] :
      ( ( aScalar0(W0)
        & aScalar0(W1) )
     => ( ( sdtlseqdt0(sz0z00,W0)
          & sdtlseqdt0(sz0z00,W1)
          & sdtasdt0(W0,W0) = sdtasdt0(W1,W1) )
       => W0 = W1 ) ) ).

fof(mVcSort,axiom,
    ! [W0] :
      ( aVector0(W0)
     => $true ) ).

fof(mDimNat,axiom,
    ! [W0] :
      ( aVector0(W0)
     => aNaturalNumber0(aDimensionOf0(W0)) ) ).

fof(mElmSc,axiom,
    ! [W0,W1] :
      ( ( aVector0(W0)
        & aNaturalNumber0(W1) )
     => aScalar0(sdtlbdtrb0(W0,W1)) ) ).

fof(mDefInit,definition,
    ! [W0] :
      ( aVector0(W0)
     => ( aDimensionOf0(W0) != sz00
       => ! [W1] :
            ( W1 = sziznziztdt0(W0)
          <=> ( aVector0(W1)
              & szszuzczcdt0(aDimensionOf0(W1)) = aDimensionOf0(W0)
              & ! [W2] :
                  ( aNaturalNumber0(W2)
                 => sdtlbdtrb0(W1,W2) = sdtlbdtrb0(W0,W2) ) ) ) ) ) ).

fof(mEqInit,axiom,
    ! [W0,W1] :
      ( ( aVector0(W0)
        & aVector0(W1) )
     => ( ( aDimensionOf0(W0) = aDimensionOf0(W1)
          & aDimensionOf0(W1) != sz00 )
       => aDimensionOf0(sziznziztdt0(W0)) = aDimensionOf0(sziznziztdt0(W1)) ) ) ).

fof(mScPr,axiom,
    ! [W0,W1] :
      ( ( aVector0(W0)
        & aVector0(W1) )
     => ( aDimensionOf0(W0) = aDimensionOf0(W1)
       => aScalar0(sdtasasdt0(W0,W1)) ) ) ).

fof(mDefSPZ,axiom,
    ! [W0,W1] :
      ( ( aVector0(W0)
        & aVector0(W1) )
     => ( ( aDimensionOf0(W0) = aDimensionOf0(W1)
          & aDimensionOf0(W1) = sz00 )
       => sdtasasdt0(W0,W1) = sz0z00 ) ) ).

fof(mDefSPN,axiom,
    ! [W0,W1] :
      ( ( aVector0(W0)
        & aVector0(W1) )
     => ( ( aDimensionOf0(W0) = aDimensionOf0(W1)
          & aDimensionOf0(W1) != sz00 )
       => sdtasasdt0(W0,W1) = sdtpldt0(sdtasasdt0(sziznziztdt0(W0),sziznziztdt0(W1)),sdtasdt0(sdtlbdtrb0(W0,aDimensionOf0(W0)),sdtlbdtrb0(W1,aDimensionOf0(W1)))) ) ) ).

fof(mScSqPos,axiom,
    ! [W0] :
      ( aVector0(W0)
     => sdtlseqdt0(sz0z00,sdtasasdt0(W0,W0)) ) ).

fof(m__1678,hypothesis,
    ( aVector0(xs)
    & aVector0(xt) ) ).

fof(m__1652,hypothesis,
    ! [W0,W1] :
      ( ( aVector0(W0)
        & aVector0(W1) )
     => ( aDimensionOf0(W0) = aDimensionOf0(W1)
       => ( iLess0(aDimensionOf0(W0),aDimensionOf0(xs))
         => sdtlseqdt0(sdtasdt0(sdtasasdt0(W0,W1),sdtasasdt0(W0,W1)),sdtasdt0(sdtasasdt0(W0,W0),sdtasasdt0(W1,W1))) ) ) ) ).

fof(m__1678_01,hypothesis,
    aDimensionOf0(xs) = aDimensionOf0(xt) ).

fof(m__1692,hypothesis,
    aDimensionOf0(xs) != sz00 ).

fof(m__1709,hypothesis,
    ( aVector0(xp)
    & szszuzczcdt0(aDimensionOf0(xp)) = aDimensionOf0(xs)
    & ! [W0] :
        ( aNaturalNumber0(W0)
       => sdtlbdtrb0(xp,W0) = sdtlbdtrb0(xs,W0) )
    & xp = sziznziztdt0(xs) ) ).

fof(m__1726,hypothesis,
    ( aVector0(xq)
    & szszuzczcdt0(aDimensionOf0(xq)) = aDimensionOf0(xt)
    & ! [W0] :
        ( aNaturalNumber0(W0)
       => sdtlbdtrb0(xq,W0) = sdtlbdtrb0(xt,W0) )
    & xq = sziznziztdt0(xt) ) ).

fof(m__1746,hypothesis,
    ( aScalar0(xA)
    & xA = sdtlbdtrb0(xs,aDimensionOf0(xs)) ) ).

fof(m__1766,hypothesis,
    ( aScalar0(xB)
    & xB = sdtlbdtrb0(xt,aDimensionOf0(xt)) ) ).

fof(m__1783,hypothesis,
    ( aScalar0(xC)
    & xC = sdtasasdt0(xp,xp) ) ).

fof(m__1800,hypothesis,
    ( aScalar0(xD)
    & xD = sdtasasdt0(xq,xq) ) ).

fof(m__1820,hypothesis,
    ( aScalar0(xE)
    & xE = sdtasasdt0(xp,xq) ) ).

fof(m__1837,hypothesis,
    ( aScalar0(xF)
    & xF = sdtasdt0(xA,xA) ) ).

fof(m__1854,hypothesis,
    ( aScalar0(xG)
    & xG = sdtasdt0(xB,xB) ) ).

fof(m__1873,hypothesis,
    ( aScalar0(xH)
    & xH = sdtasdt0(xA,xB) ) ).

fof(m__1892,hypothesis,
    ( aScalar0(xR)
    & xR = sdtasdt0(xC,xG) ) ).

fof(m__1911,hypothesis,
    ( aScalar0(xP)
    & xP = sdtasdt0(xE,xH) ) ).

fof(m__1930,hypothesis,
    ( aScalar0(xS)
    & xS = sdtasdt0(xF,xD) ) ).

fof(m__1949,hypothesis,
    ( aScalar0(xN)
    & xN = sdtasdt0(xR,xS) ) ).

fof(m__1967,hypothesis,
    sdtlseqdt0(sdtasdt0(xE,xE),sdtasdt0(xC,xD)) ).

fof(m__2004,hypothesis,
    sdtlseqdt0(sdtasdt0(xP,xP),xN) ).

fof(m__2104,hypothesis,
    sdtlseqdt0(sdtpldt0(sdtasdt0(xP,xP),sdtasdt0(xP,xP)),sdtpldt0(sdtasdt0(xR,xR),sdtasdt0(xS,xS))) ).

fof(m__,conjecture,
    sdtlseqdt0(sdtasdt0(sdtpldt0(xP,xP),sdtpldt0(xP,xP)),sdtasdt0(sdtpldt0(xR,xS),sdtpldt0(xR,xS))) ).

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