TPTP Problem File: RNG112+4.p
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- Solve Problem
%------------------------------------------------------------------------------
% File : RNG112+4 : TPTP v8.2.0. Released v4.0.0.
% Domain : Ring Theory
% Problem : Chinese remainder theorem in a ring 07_04_02, 03 expansion
% Version : Especial.
% English :
% Refs : [VLP07] Verchinine et al. (2007), System for Automated Deduction
% : [Pas08] Paskevich (2008), Email to G. Sutcliffe
% Source : [Pas08]
% Names : chines_07_04_02.03 [Pas08]
% Status : Theorem
% Rating : 0.39 v8.2.0, 0.36 v8.1.0, 0.31 v7.5.0, 0.34 v7.4.0, 0.37 v7.3.0, 0.38 v7.2.0, 0.34 v7.1.0, 0.39 v7.0.0, 0.33 v6.4.0, 0.38 v6.3.0, 0.42 v6.2.0, 0.40 v6.1.0, 0.47 v6.0.0, 0.43 v5.5.0, 0.52 v5.4.0, 0.54 v5.3.0, 0.56 v5.2.0, 0.40 v5.1.0, 0.48 v5.0.0, 0.62 v4.1.0, 0.65 v4.0.1, 0.87 v4.0.0
% Syntax : Number of formulae : 46 ( 3 unt; 9 def)
% Number of atoms : 247 ( 60 equ)
% Maximal formula atoms : 23 ( 5 avg)
% Number of connectives : 215 ( 14 ~; 9 |; 120 &)
% ( 17 <=>; 55 =>; 0 <=; 0 <~>)
% Maximal formula depth : 18 ( 7 avg)
% Maximal term depth : 3 ( 1 avg)
% Number of predicates : 13 ( 11 usr; 1 prp; 0-3 aty)
% Number of functors : 13 ( 13 usr; 6 con; 0-2 aty)
% Number of variables : 126 ( 89 !; 37 ?)
% SPC : FOF_THM_RFO_SEQ
% Comments : Problem generated by the SAD system [VLP07]
%------------------------------------------------------------------------------
fof(mElmSort,axiom,
! [W0] :
( aElement0(W0)
=> $true ) ).
fof(mSortsC,axiom,
aElement0(sz00) ).
fof(mSortsC_01,axiom,
aElement0(sz10) ).
fof(mSortsU,axiom,
! [W0] :
( aElement0(W0)
=> aElement0(smndt0(W0)) ) ).
fof(mSortsB,axiom,
! [W0,W1] :
( ( aElement0(W0)
& aElement0(W1) )
=> aElement0(sdtpldt0(W0,W1)) ) ).
fof(mSortsB_02,axiom,
! [W0,W1] :
( ( aElement0(W0)
& aElement0(W1) )
=> aElement0(sdtasdt0(W0,W1)) ) ).
fof(mAddComm,axiom,
! [W0,W1] :
( ( aElement0(W0)
& aElement0(W1) )
=> sdtpldt0(W0,W1) = sdtpldt0(W1,W0) ) ).
fof(mAddAsso,axiom,
! [W0,W1,W2] :
( ( aElement0(W0)
& aElement0(W1)
& aElement0(W2) )
=> sdtpldt0(sdtpldt0(W0,W1),W2) = sdtpldt0(W0,sdtpldt0(W1,W2)) ) ).
fof(mAddZero,axiom,
! [W0] :
( aElement0(W0)
=> ( sdtpldt0(W0,sz00) = W0
& W0 = sdtpldt0(sz00,W0) ) ) ).
fof(mAddInvr,axiom,
! [W0] :
( aElement0(W0)
=> ( sdtpldt0(W0,smndt0(W0)) = sz00
& sz00 = sdtpldt0(smndt0(W0),W0) ) ) ).
fof(mMulComm,axiom,
! [W0,W1] :
( ( aElement0(W0)
& aElement0(W1) )
=> sdtasdt0(W0,W1) = sdtasdt0(W1,W0) ) ).
fof(mMulAsso,axiom,
! [W0,W1,W2] :
( ( aElement0(W0)
& aElement0(W1)
& aElement0(W2) )
=> sdtasdt0(sdtasdt0(W0,W1),W2) = sdtasdt0(W0,sdtasdt0(W1,W2)) ) ).
fof(mMulUnit,axiom,
! [W0] :
( aElement0(W0)
=> ( sdtasdt0(W0,sz10) = W0
& W0 = sdtasdt0(sz10,W0) ) ) ).
fof(mAMDistr,axiom,
! [W0,W1,W2] :
( ( aElement0(W0)
& aElement0(W1)
& aElement0(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(mMulMnOne,axiom,
! [W0] :
( aElement0(W0)
=> ( sdtasdt0(smndt0(sz10),W0) = smndt0(W0)
& smndt0(W0) = sdtasdt0(W0,smndt0(sz10)) ) ) ).
fof(mMulZero,axiom,
! [W0] :
( aElement0(W0)
=> ( sdtasdt0(W0,sz00) = sz00
& sz00 = sdtasdt0(sz00,W0) ) ) ).
fof(mCancel,axiom,
! [W0,W1] :
( ( aElement0(W0)
& aElement0(W1) )
=> ( sdtasdt0(W0,W1) = sz00
=> ( W0 = sz00
| W1 = sz00 ) ) ) ).
fof(mUnNeZr,axiom,
sz10 != sz00 ).
fof(mSetSort,axiom,
! [W0] :
( aSet0(W0)
=> $true ) ).
fof(mEOfElem,axiom,
! [W0] :
( aSet0(W0)
=> ! [W1] :
( aElementOf0(W1,W0)
=> aElement0(W1) ) ) ).
fof(mSetEq,axiom,
! [W0,W1] :
( ( aSet0(W0)
& aSet0(W1) )
=> ( ( ! [W2] :
( aElementOf0(W2,W0)
=> aElementOf0(W2,W1) )
& ! [W2] :
( aElementOf0(W2,W1)
=> aElementOf0(W2,W0) ) )
=> W0 = W1 ) ) ).
fof(mDefSSum,definition,
! [W0,W1] :
( ( aSet0(W0)
& aSet0(W1) )
=> ! [W2] :
( W2 = sdtpldt1(W0,W1)
<=> ( aSet0(W2)
& ! [W3] :
( aElementOf0(W3,W2)
<=> ? [W4,W5] :
( aElementOf0(W4,W0)
& aElementOf0(W5,W1)
& sdtpldt0(W4,W5) = W3 ) ) ) ) ) ).
fof(mDefSInt,definition,
! [W0,W1] :
( ( aSet0(W0)
& aSet0(W1) )
=> ! [W2] :
( W2 = sdtasasdt0(W0,W1)
<=> ( aSet0(W2)
& ! [W3] :
( aElementOf0(W3,W2)
<=> ( aElementOf0(W3,W0)
& aElementOf0(W3,W1) ) ) ) ) ) ).
fof(mDefIdeal,definition,
! [W0] :
( aIdeal0(W0)
<=> ( aSet0(W0)
& ! [W1] :
( aElementOf0(W1,W0)
=> ( ! [W2] :
( aElementOf0(W2,W0)
=> aElementOf0(sdtpldt0(W1,W2),W0) )
& ! [W2] :
( aElement0(W2)
=> aElementOf0(sdtasdt0(W2,W1),W0) ) ) ) ) ) ).
fof(mIdeSum,axiom,
! [W0,W1] :
( ( aIdeal0(W0)
& aIdeal0(W1) )
=> aIdeal0(sdtpldt1(W0,W1)) ) ).
fof(mIdeInt,axiom,
! [W0,W1] :
( ( aIdeal0(W0)
& aIdeal0(W1) )
=> aIdeal0(sdtasasdt0(W0,W1)) ) ).
fof(mDefMod,definition,
! [W0,W1,W2] :
( ( aElement0(W0)
& aElement0(W1)
& aIdeal0(W2) )
=> ( sdteqdtlpzmzozddtrp0(W0,W1,W2)
<=> aElementOf0(sdtpldt0(W0,smndt0(W1)),W2) ) ) ).
fof(mChineseRemainder,axiom,
! [W0,W1] :
( ( aIdeal0(W0)
& aIdeal0(W1) )
=> ( ! [W2] :
( aElement0(W2)
=> aElementOf0(W2,sdtpldt1(W0,W1)) )
=> ! [W2,W3] :
( ( aElement0(W2)
& aElement0(W3) )
=> ? [W4] :
( aElement0(W4)
& sdteqdtlpzmzozddtrp0(W4,W2,W0)
& sdteqdtlpzmzozddtrp0(W4,W3,W1) ) ) ) ) ).
fof(mNatSort,axiom,
! [W0] :
( aNaturalNumber0(W0)
=> $true ) ).
fof(mEucSort,axiom,
! [W0] :
( ( aElement0(W0)
& W0 != sz00 )
=> aNaturalNumber0(sbrdtbr0(W0)) ) ).
fof(mNatLess,axiom,
! [W0,W1] :
( ( aNaturalNumber0(W0)
& aNaturalNumber0(W1) )
=> ( iLess0(W0,W1)
=> $true ) ) ).
fof(mDivision,axiom,
! [W0,W1] :
( ( aElement0(W0)
& aElement0(W1)
& W1 != sz00 )
=> ? [W2,W3] :
( aElement0(W2)
& aElement0(W3)
& W0 = sdtpldt0(sdtasdt0(W2,W1),W3)
& ( W3 != sz00
=> iLess0(sbrdtbr0(W3),sbrdtbr0(W1)) ) ) ) ).
fof(mDefDiv,definition,
! [W0,W1] :
( ( aElement0(W0)
& aElement0(W1) )
=> ( doDivides0(W0,W1)
<=> ? [W2] :
( aElement0(W2)
& sdtasdt0(W0,W2) = W1 ) ) ) ).
fof(mDefDvs,definition,
! [W0] :
( aElement0(W0)
=> ! [W1] :
( aDivisorOf0(W1,W0)
<=> ( aElement0(W1)
& doDivides0(W1,W0) ) ) ) ).
fof(mDefGCD,definition,
! [W0,W1] :
( ( aElement0(W0)
& aElement0(W1) )
=> ! [W2] :
( aGcdOfAnd0(W2,W0,W1)
<=> ( aDivisorOf0(W2,W0)
& aDivisorOf0(W2,W1)
& ! [W3] :
( ( aDivisorOf0(W3,W0)
& aDivisorOf0(W3,W1) )
=> doDivides0(W3,W2) ) ) ) ) ).
fof(mDefRel,definition,
! [W0,W1] :
( ( aElement0(W0)
& aElement0(W1) )
=> ( misRelativelyPrime0(W0,W1)
<=> aGcdOfAnd0(sz10,W0,W1) ) ) ).
fof(mDefPrIdeal,definition,
! [W0] :
( aElement0(W0)
=> ! [W1] :
( W1 = slsdtgt0(W0)
<=> ( aSet0(W1)
& ! [W2] :
( aElementOf0(W2,W1)
<=> ? [W3] :
( aElement0(W3)
& sdtasdt0(W0,W3) = W2 ) ) ) ) ) ).
fof(mPrIdeal,axiom,
! [W0] :
( aElement0(W0)
=> aIdeal0(slsdtgt0(W0)) ) ).
fof(m__2091,hypothesis,
( aElement0(xa)
& aElement0(xb) ) ).
fof(m__2110,hypothesis,
( xa != sz00
| xb != sz00 ) ).
fof(m__2129,hypothesis,
( aElement0(xc)
& ? [W0] :
( aElement0(W0)
& sdtasdt0(xc,W0) = xa )
& doDivides0(xc,xa)
& aDivisorOf0(xc,xa)
& aElement0(xc)
& ? [W0] :
( aElement0(W0)
& sdtasdt0(xc,W0) = xb )
& doDivides0(xc,xb)
& aDivisorOf0(xc,xb)
& ! [W0] :
( ( ( ( aElement0(W0)
& ( ? [W1] :
( aElement0(W1)
& sdtasdt0(W0,W1) = xa )
| doDivides0(W0,xa) ) )
| aDivisorOf0(W0,xa) )
& ( ? [W1] :
( aElement0(W1)
& sdtasdt0(W0,W1) = xb )
| doDivides0(W0,xb)
| aDivisorOf0(W0,xb) ) )
=> ( ? [W1] :
( aElement0(W1)
& sdtasdt0(W0,W1) = xc )
& doDivides0(W0,xc) ) )
& aGcdOfAnd0(xc,xa,xb) ) ).
fof(m__2174,hypothesis,
( aSet0(xI)
& ! [W0] :
( aElementOf0(W0,xI)
=> ( ! [W1] :
( aElementOf0(W1,xI)
=> aElementOf0(sdtpldt0(W0,W1),xI) )
& ! [W1] :
( aElement0(W1)
=> aElementOf0(sdtasdt0(W1,W0),xI) ) ) )
& aIdeal0(xI)
& ! [W0] :
( aElementOf0(W0,slsdtgt0(xa))
<=> ? [W1] :
( aElement0(W1)
& sdtasdt0(xa,W1) = W0 ) )
& ! [W0] :
( aElementOf0(W0,slsdtgt0(xb))
<=> ? [W1] :
( aElement0(W1)
& sdtasdt0(xb,W1) = W0 ) )
& ! [W0] :
( aElementOf0(W0,xI)
<=> ? [W1,W2] :
( aElementOf0(W1,slsdtgt0(xa))
& aElementOf0(W2,slsdtgt0(xb))
& sdtpldt0(W1,W2) = W0 ) )
& xI = sdtpldt1(slsdtgt0(xa),slsdtgt0(xb)) ) ).
fof(m__2203,hypothesis,
( ? [W0] :
( aElement0(W0)
& sdtasdt0(xa,W0) = sz00 )
& aElementOf0(sz00,slsdtgt0(xa))
& ? [W0] :
( aElement0(W0)
& sdtasdt0(xa,W0) = xa )
& aElementOf0(xa,slsdtgt0(xa))
& ? [W0] :
( aElement0(W0)
& sdtasdt0(xb,W0) = sz00 )
& aElementOf0(sz00,slsdtgt0(xb))
& ? [W0] :
( aElement0(W0)
& sdtasdt0(xb,W0) = xb )
& aElementOf0(xb,slsdtgt0(xb)) ) ).
fof(m__2228,hypothesis,
? [W0] :
( ! [W1] :
( aElementOf0(W1,slsdtgt0(xa))
<=> ? [W2] :
( aElement0(W2)
& sdtasdt0(xa,W2) = W1 ) )
& ! [W1] :
( aElementOf0(W1,slsdtgt0(xb))
<=> ? [W2] :
( aElement0(W2)
& sdtasdt0(xb,W2) = W1 ) )
& ? [W1,W2] :
( aElementOf0(W1,slsdtgt0(xa))
& aElementOf0(W2,slsdtgt0(xb))
& sdtpldt0(W1,W2) = W0 )
& aElementOf0(W0,sdtpldt1(slsdtgt0(xa),slsdtgt0(xb)))
& W0 != sz00 ) ).
fof(m__2351,hypothesis,
! [W0] :
( ( ( ? [W1,W2] :
( aElementOf0(W1,slsdtgt0(xa))
& aElementOf0(W2,slsdtgt0(xb))
& sdtpldt0(W1,W2) = W0 )
| aElementOf0(W0,xI) )
& W0 != sz00 )
=> ? [W1] :
( ? [W2,W3] :
( aElementOf0(W2,slsdtgt0(xa))
& aElementOf0(W3,slsdtgt0(xb))
& sdtpldt0(W2,W3) = W1 )
& aElementOf0(W1,xI)
& W1 != sz00
& ! [W2] :
( ( ( ? [W3,W4] :
( aElementOf0(W3,slsdtgt0(xa))
& aElementOf0(W4,slsdtgt0(xb))
& sdtpldt0(W3,W4) = W2 )
| aElementOf0(W2,xI) )
& W2 != sz00 )
=> ~ iLess0(sbrdtbr0(W2),sbrdtbr0(W1)) ) ) ) ).
fof(m__,conjecture,
? [W0] :
( ( ? [W1,W2] :
( aElementOf0(W1,slsdtgt0(xa))
& aElementOf0(W2,slsdtgt0(xb))
& sdtpldt0(W1,W2) = W0 )
| aElementOf0(W0,xI) )
& W0 != sz00
& ! [W1] :
( ( ? [W2,W3] :
( aElementOf0(W2,slsdtgt0(xa))
& aElementOf0(W3,slsdtgt0(xb))
& sdtpldt0(W2,W3) = W1 )
& aElementOf0(W1,xI)
& W1 != sz00 )
=> ~ iLess0(sbrdtbr0(W1),sbrdtbr0(W0)) ) ) ).
%------------------------------------------------------------------------------