TPTP Problem File: LAT387+4.p
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%------------------------------------------------------------------------------
% File : LAT387+4 : TPTP v8.2.0. Released v4.0.0.
% Domain : Lattice Theory
% Problem : Tarski-Knaster fixed point theorem 03_01_04, 03 expansion
% Version : Especial.
% English :
% Refs : [VLP07] Verchinine et al. (2007), System for Automated Deduction
% : [Pas08] Paskevich (2008), Email to G. Sutcliffe
% : [VL+08] Verchinine et al. (2008), On Correctness of Mathematic
% Source : [Pas08]
% Names : tarski_03_01_04.03 [Pas08]
% Status : Theorem
% Rating : 0.42 v8.2.0, 0.36 v8.1.0, 0.39 v7.5.0, 0.41 v7.4.0, 0.20 v7.3.0, 0.31 v7.2.0, 0.28 v7.1.0, 0.26 v7.0.0, 0.23 v6.3.0, 0.29 v6.2.0, 0.36 v6.1.0, 0.40 v6.0.0, 0.26 v5.5.0, 0.41 v5.4.0, 0.43 v5.3.0, 0.48 v5.2.0, 0.35 v5.1.0, 0.43 v5.0.0, 0.58 v4.1.0, 0.61 v4.0.1, 0.87 v4.0.0
% Syntax : Number of formulae : 30 ( 0 unt; 10 def)
% Number of atoms : 191 ( 13 equ)
% Maximal formula atoms : 37 ( 6 avg)
% Number of connectives : 162 ( 1 ~; 9 |; 74 &)
% ( 10 <=>; 68 =>; 0 <=; 0 <~>)
% Maximal formula depth : 24 ( 8 avg)
% Maximal term depth : 2 ( 1 avg)
% Number of predicates : 17 ( 15 usr; 1 prp; 0-3 aty)
% Number of functors : 11 ( 11 usr; 6 con; 0-3 aty)
% Number of variables : 82 ( 77 !; 5 ?)
% SPC : FOF_THM_RFO_SEQ
% Comments : Problem generated by the SAD system [VLP07]
%------------------------------------------------------------------------------
fof(mSetSort,axiom,
! [W0] :
( aSet0(W0)
=> $true ) ).
fof(mElmSort,axiom,
! [W0] :
( aElement0(W0)
=> $true ) ).
fof(mEOfElem,axiom,
! [W0] :
( aSet0(W0)
=> ! [W1] :
( aElementOf0(W1,W0)
=> aElement0(W1) ) ) ).
fof(mDefEmpty,definition,
! [W0] :
( aSet0(W0)
=> ( isEmpty0(W0)
<=> ~ ? [W1] : aElementOf0(W1,W0) ) ) ).
fof(mDefSub,definition,
! [W0] :
( aSet0(W0)
=> ! [W1] :
( aSubsetOf0(W1,W0)
<=> ( aSet0(W1)
& ! [W2] :
( aElementOf0(W2,W1)
=> aElementOf0(W2,W0) ) ) ) ) ).
fof(mLessRel,axiom,
! [W0,W1] :
( ( aElement0(W0)
& aElement0(W1) )
=> ( sdtlseqdt0(W0,W1)
=> $true ) ) ).
fof(mARefl,axiom,
! [W0] :
( aElement0(W0)
=> sdtlseqdt0(W0,W0) ) ).
fof(mASymm,axiom,
! [W0,W1] :
( ( aElement0(W0)
& aElement0(W1) )
=> ( ( sdtlseqdt0(W0,W1)
& sdtlseqdt0(W1,W0) )
=> W0 = W1 ) ) ).
fof(mTrans,axiom,
! [W0,W1,W2] :
( ( aElement0(W0)
& aElement0(W1)
& aElement0(W2) )
=> ( ( sdtlseqdt0(W0,W1)
& sdtlseqdt0(W1,W2) )
=> sdtlseqdt0(W0,W2) ) ) ).
fof(mDefLB,definition,
! [W0] :
( aSet0(W0)
=> ! [W1] :
( aSubsetOf0(W1,W0)
=> ! [W2] :
( aLowerBoundOfIn0(W2,W1,W0)
<=> ( aElementOf0(W2,W0)
& ! [W3] :
( aElementOf0(W3,W1)
=> sdtlseqdt0(W2,W3) ) ) ) ) ) ).
fof(mDefUB,definition,
! [W0] :
( aSet0(W0)
=> ! [W1] :
( aSubsetOf0(W1,W0)
=> ! [W2] :
( aUpperBoundOfIn0(W2,W1,W0)
<=> ( aElementOf0(W2,W0)
& ! [W3] :
( aElementOf0(W3,W1)
=> sdtlseqdt0(W3,W2) ) ) ) ) ) ).
fof(mDefInf,definition,
! [W0] :
( aSet0(W0)
=> ! [W1] :
( aSubsetOf0(W1,W0)
=> ! [W2] :
( aInfimumOfIn0(W2,W1,W0)
<=> ( aElementOf0(W2,W0)
& aLowerBoundOfIn0(W2,W1,W0)
& ! [W3] :
( aLowerBoundOfIn0(W3,W1,W0)
=> sdtlseqdt0(W3,W2) ) ) ) ) ) ).
fof(mDefSup,definition,
! [W0] :
( aSet0(W0)
=> ! [W1] :
( aSubsetOf0(W1,W0)
=> ! [W2] :
( aSupremumOfIn0(W2,W1,W0)
<=> ( aElementOf0(W2,W0)
& aUpperBoundOfIn0(W2,W1,W0)
& ! [W3] :
( aUpperBoundOfIn0(W3,W1,W0)
=> sdtlseqdt0(W2,W3) ) ) ) ) ) ).
fof(mSupUn,axiom,
! [W0] :
( aSet0(W0)
=> ! [W1] :
( aSubsetOf0(W1,W0)
=> ! [W2,W3] :
( ( aSupremumOfIn0(W2,W1,W0)
& aSupremumOfIn0(W3,W1,W0) )
=> W2 = W3 ) ) ) ).
fof(mInfUn,axiom,
! [W0] :
( aSet0(W0)
=> ! [W1] :
( aSubsetOf0(W1,W0)
=> ! [W2,W3] :
( ( aInfimumOfIn0(W2,W1,W0)
& aInfimumOfIn0(W3,W1,W0) )
=> W2 = W3 ) ) ) ).
fof(mDefCLat,definition,
! [W0] :
( aCompleteLattice0(W0)
<=> ( aSet0(W0)
& ! [W1] :
( aSubsetOf0(W1,W0)
=> ? [W2] :
( aInfimumOfIn0(W2,W1,W0)
& ? [W3] : aSupremumOfIn0(W3,W1,W0) ) ) ) ) ).
fof(mConMap,axiom,
! [W0] :
( aFunction0(W0)
=> $true ) ).
fof(mDomSort,axiom,
! [W0] :
( aFunction0(W0)
=> aSet0(szDzozmdt0(W0)) ) ).
fof(mRanSort,axiom,
! [W0] :
( aFunction0(W0)
=> aSet0(szRzazndt0(W0)) ) ).
fof(mDefDom,definition,
! [W0,W1] :
( ( aFunction0(W0)
& aSet0(W1) )
=> ( isOn0(W0,W1)
<=> ( szDzozmdt0(W0) = szRzazndt0(W0)
& szRzazndt0(W0) = W1 ) ) ) ).
fof(mImgSort,axiom,
! [W0] :
( aFunction0(W0)
=> ! [W1] :
( aElementOf0(W1,szDzozmdt0(W0))
=> aElementOf0(sdtlpdtrp0(W0,W1),szRzazndt0(W0)) ) ) ).
fof(mDefFix,definition,
! [W0] :
( aFunction0(W0)
=> ! [W1] :
( aFixedPointOf0(W1,W0)
<=> ( aElementOf0(W1,szDzozmdt0(W0))
& sdtlpdtrp0(W0,W1) = W1 ) ) ) ).
fof(mDefMonot,definition,
! [W0] :
( aFunction0(W0)
=> ( isMonotone0(W0)
<=> ! [W1,W2] :
( ( aElementOf0(W1,szDzozmdt0(W0))
& aElementOf0(W2,szDzozmdt0(W0)) )
=> ( sdtlseqdt0(W1,W2)
=> sdtlseqdt0(sdtlpdtrp0(W0,W1),sdtlpdtrp0(W0,W2)) ) ) ) ) ).
fof(m__1123,hypothesis,
( aSet0(xU)
& ! [W0] :
( ( ( aSet0(W0)
& ! [W1] :
( aElementOf0(W1,W0)
=> aElementOf0(W1,xU) ) )
| aSubsetOf0(W0,xU) )
=> ? [W1] :
( aElementOf0(W1,xU)
& aElementOf0(W1,xU)
& ! [W2] :
( aElementOf0(W2,W0)
=> sdtlseqdt0(W1,W2) )
& aLowerBoundOfIn0(W1,W0,xU)
& ! [W2] :
( ( ( aElementOf0(W2,xU)
& ! [W3] :
( aElementOf0(W3,W0)
=> sdtlseqdt0(W2,W3) ) )
| aLowerBoundOfIn0(W2,W0,xU) )
=> sdtlseqdt0(W2,W1) )
& aInfimumOfIn0(W1,W0,xU)
& ? [W2] :
( aElementOf0(W2,xU)
& aElementOf0(W2,xU)
& ! [W3] :
( aElementOf0(W3,W0)
=> sdtlseqdt0(W3,W2) )
& aUpperBoundOfIn0(W2,W0,xU)
& ! [W3] :
( ( ( aElementOf0(W3,xU)
& ! [W4] :
( aElementOf0(W4,W0)
=> sdtlseqdt0(W4,W3) ) )
| aUpperBoundOfIn0(W3,W0,xU) )
=> sdtlseqdt0(W2,W3) )
& aSupremumOfIn0(W2,W0,xU) ) ) )
& aCompleteLattice0(xU)
& aFunction0(xf)
& ! [W0,W1] :
( ( aElementOf0(W0,szDzozmdt0(xf))
& aElementOf0(W1,szDzozmdt0(xf)) )
=> ( sdtlseqdt0(W0,W1)
=> sdtlseqdt0(sdtlpdtrp0(xf,W0),sdtlpdtrp0(xf,W1)) ) )
& isMonotone0(xf)
& szDzozmdt0(xf) = szRzazndt0(xf)
& szRzazndt0(xf) = xU
& isOn0(xf,xU) ) ).
fof(m__1144,hypothesis,
( aSet0(xS)
& ! [W0] :
( ( aElementOf0(W0,xS)
=> ( aElementOf0(W0,szDzozmdt0(xf))
& sdtlpdtrp0(xf,W0) = W0
& aFixedPointOf0(W0,xf) ) )
& ( ( ( aElementOf0(W0,szDzozmdt0(xf))
& sdtlpdtrp0(xf,W0) = W0 )
| aFixedPointOf0(W0,xf) )
=> aElementOf0(W0,xS) ) )
& xS = cS1142(xf) ) ).
fof(m__1173,hypothesis,
( aSet0(xT)
& ! [W0] :
( aElementOf0(W0,xT)
=> aElementOf0(W0,xS) )
& aSubsetOf0(xT,xS) ) ).
fof(m__1244,hypothesis,
( aSet0(xP)
& ! [W0] :
( ( aElementOf0(W0,xP)
=> ( aElementOf0(W0,xU)
& sdtlseqdt0(sdtlpdtrp0(xf,W0),W0)
& ! [W1] :
( aElementOf0(W1,xT)
=> sdtlseqdt0(W1,W0) )
& aUpperBoundOfIn0(W0,xT,xU) ) )
& ( ( aElementOf0(W0,xU)
& sdtlseqdt0(sdtlpdtrp0(xf,W0),W0)
& ( ! [W1] :
( aElementOf0(W1,xT)
=> sdtlseqdt0(W1,W0) )
| aUpperBoundOfIn0(W0,xT,xU) ) )
=> aElementOf0(W0,xP) ) )
& xP = cS1241(xU,xf,xT) ) ).
fof(m__1261,hypothesis,
( aElementOf0(xp,xU)
& aElementOf0(xp,xU)
& ! [W0] :
( aElementOf0(W0,xP)
=> sdtlseqdt0(xp,W0) )
& aLowerBoundOfIn0(xp,xP,xU)
& ! [W0] :
( ( ( aElementOf0(W0,xU)
& ! [W1] :
( aElementOf0(W1,xP)
=> sdtlseqdt0(W0,W1) ) )
| aLowerBoundOfIn0(W0,xP,xU) )
=> sdtlseqdt0(W0,xp) )
& aInfimumOfIn0(xp,xP,xU) ) ).
fof(m__1299,hypothesis,
( ! [W0] :
( aElementOf0(W0,xP)
=> sdtlseqdt0(sdtlpdtrp0(xf,xp),W0) )
& aLowerBoundOfIn0(sdtlpdtrp0(xf,xp),xP,xU)
& ! [W0] :
( aElementOf0(W0,xT)
=> sdtlseqdt0(W0,sdtlpdtrp0(xf,xp)) )
& aUpperBoundOfIn0(sdtlpdtrp0(xf,xp),xT,xU) ) ).
fof(m__,conjecture,
( ( ( aElementOf0(xp,szDzozmdt0(xf))
& sdtlpdtrp0(xf,xp) = xp )
| aFixedPointOf0(xp,xf) )
& ( ( ( ! [W0] :
( aElementOf0(W0,xT)
=> sdtlseqdt0(W0,xp) )
| aUpperBoundOfIn0(xp,xT,xS) )
& ! [W0] :
( ( aElementOf0(W0,xS)
& ! [W1] :
( aElementOf0(W1,xT)
=> sdtlseqdt0(W1,W0) )
& aUpperBoundOfIn0(W0,xT,xS) )
=> sdtlseqdt0(xp,W0) ) )
| aSupremumOfIn0(xp,xT,xS) ) ) ).
%------------------------------------------------------------------------------