TPTP Problem File: RNG057+2.p
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%------------------------------------------------------------------------------
% File : RNG057+2 : TPTP v9.0.0. Released v4.0.0.
% Domain : Ring Theory
% Problem : Cauchy-Bouniakowsky-Schwarz inequality 05_16_01_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_01_03.01 [Pas08]
% Status : Theorem
% Rating : 0.45 v9.0.0, 0.50 v8.2.0, 0.53 v8.1.0, 0.50 v7.5.0, 0.56 v7.4.0, 0.43 v7.3.0, 0.55 v7.2.0, 0.52 v7.0.0, 0.53 v6.4.0, 0.54 v6.2.0, 0.60 v6.1.0, 0.67 v6.0.0, 0.61 v5.5.0, 0.78 v5.4.0, 0.86 v5.3.0, 0.85 v5.2.0, 0.80 v5.1.0, 0.81 v5.0.0, 0.92 v4.1.0, 0.96 v4.0.1, 1.00 v4.0.0
% Syntax : Number of formulae : 59 ( 8 unt; 1 def)
% Number of atoms : 196 ( 63 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__2027,hypothesis,
sdtasdt0(xP,xP) = sdtasdt0(sdtasdt0(xH,xH),sdtasdt0(xE,xE)) ).
fof(m__2052,hypothesis,
xN = sdtasdt0(sdtasdt0(xH,xH),sdtasdt0(xC,xD)) ).
fof(m__,conjecture,
sdtlseqdt0(sdtasdt0(xP,xP),xN) ).
%------------------------------------------------------------------------------