TPTP Problem File: RNG050+1.p
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%------------------------------------------------------------------------------
% File : RNG050+1 : TPTP v8.2.0. Released v4.0.0.
% Domain : Ring Theory
% Problem : Cauchy-Bouniakowsky-Schwarz inequality 04_04, 00 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_04_04.00 [Pas08]
% Status : Theorem
% Rating : 0.14 v8.1.0, 0.17 v7.5.0, 0.19 v7.4.0, 0.10 v7.2.0, 0.07 v7.1.0, 0.09 v7.0.0, 0.07 v6.4.0, 0.12 v6.3.0, 0.08 v6.2.0, 0.16 v6.1.0, 0.20 v6.0.0, 0.04 v5.5.0, 0.11 v5.4.0, 0.18 v5.3.0, 0.19 v5.2.0, 0.05 v5.1.0, 0.19 v5.0.0, 0.29 v4.1.0, 0.35 v4.0.1, 0.78 v4.0.0
% Syntax : Number of formulae : 39 ( 3 unt; 1 def)
% Number of atoms : 154 ( 41 equ)
% Maximal formula atoms : 9 ( 3 avg)
% Number of connectives : 121 ( 6 ~; 1 |; 60 &)
% ( 1 <=>; 53 =>; 0 <=; 0 <~>)
% Maximal formula depth : 10 ( 5 avg)
% Maximal term depth : 5 ( 1 avg)
% Number of predicates : 7 ( 5 usr; 1 prp; 0-2 aty)
% Number of functors : 11 ( 11 usr; 3 con; 0-2 aty)
% Number of variables : 69 ( 68 !; 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(m__1542,hypothesis,
aVector0(xs) ).
fof(m__1516,hypothesis,
! [W0] :
( aVector0(W0)
=> ( iLess0(aDimensionOf0(W0),aDimensionOf0(xs))
=> sdtlseqdt0(sz0z00,sdtasasdt0(W0,W0)) ) ) ).
fof(m__,conjecture,
( ( aDimensionOf0(xs) != sz00
=> sdtlseqdt0(sz0z00,sdtasasdt0(xs,xs)) )
=> sdtlseqdt0(sz0z00,sdtasasdt0(xs,xs)) ) ).
%------------------------------------------------------------------------------