%------------------------------------------------------------------------------
% File     : COM023+1 : TPTP v8.2.0. Released v4.0.0.
% Domain   : Computing Theory
% Problem  : Newman's lemma on rewriting systems 03_02, 00 expansion
% Version  : Especial.
% English  :

% Refs     : [VLP07] Verchinine et al. (2007), System for Automated Deduction
%          : [PV+07] Paskevich et al. (2007), Reasoning Inside a Formula an
%          : [Pas08] Paskevich (2008), Email to G. Sutcliffe
% Source   : [Pas08]
% Names    : newman_03_02.00 [Pas08]

% Status   : Theorem
% Rating   : 0.17 v8.2.0, 0.19 v8.1.0, 0.08 v7.5.0, 0.12 v7.4.0, 0.17 v7.3.0, 0.14 v7.2.0, 0.10 v7.1.0, 0.04 v7.0.0, 0.07 v6.4.0, 0.12 v6.3.0, 0.08 v6.1.0, 0.17 v6.0.0, 0.13 v5.5.0, 0.15 v5.4.0, 0.21 v5.3.0, 0.26 v5.2.0, 0.10 v5.1.0, 0.24 v5.0.0, 0.33 v4.1.0, 0.43 v4.0.1, 0.78 v4.0.0
% Syntax   : Number of formulae    :   18 (   2 unt;   6 def)
%            Number of atoms       :   93 (   1 equ)
%            Maximal formula atoms :   10 (   5 avg)
%            Number of connectives :   76 (   1   ~;   2   |;  42   &)
%                                         (   6 <=>;  25  =>;   0  <=;   0 <~>)
%            Maximal formula depth :   12 (   7 avg)
%            Maximal term depth    :    1 (   1 avg)
%            Number of predicates  :   12 (  10 usr;   1 prp; 0-3 aty)
%            Number of functors    :    1 (   1 usr;   1 con; 0-0 aty)
%            Number of variables   :   49 (  43   !;   6   ?)
% SPC      : FOF_THM_RFO_SEQ

% Comments : Problem generated by the SAD system [VLP07]
%------------------------------------------------------------------------------
fof(mElmSort,axiom,
    ! [W0] :
      ( aElement0(W0)
     => $true ) ).

fof(mRelSort,axiom,
    ! [W0] :
      ( aRewritingSystem0(W0)
     => $true ) ).

fof(mReduct,axiom,
    ! [W0,W1] :
      ( ( aElement0(W0)
        & aRewritingSystem0(W1) )
     => ! [W2] :
          ( aReductOfIn0(W2,W0,W1)
         => aElement0(W2) ) ) ).

fof(mWFOrd,axiom,
    ! [W0,W1] :
      ( ( aElement0(W0)
        & aElement0(W1) )
     => ( iLess0(W0,W1)
       => $true ) ) ).

fof(mTCbr,axiom,
    ! [W0,W1,W2] :
      ( ( aElement0(W0)
        & aRewritingSystem0(W1)
        & aElement0(W2) )
     => ( sdtmndtplgtdt0(W0,W1,W2)
       => $true ) ) ).

fof(mTCDef,definition,
    ! [W0,W1,W2] :
      ( ( aElement0(W0)
        & aRewritingSystem0(W1)
        & aElement0(W2) )
     => ( sdtmndtplgtdt0(W0,W1,W2)
      <=> ( aReductOfIn0(W2,W0,W1)
          | ? [W3] :
              ( aElement0(W3)
              & aReductOfIn0(W3,W0,W1)
              & sdtmndtplgtdt0(W3,W1,W2) ) ) ) ) ).

fof(mTCTrans,axiom,
    ! [W0,W1,W2,W3] :
      ( ( aElement0(W0)
        & aRewritingSystem0(W1)
        & aElement0(W2)
        & aElement0(W3) )
     => ( ( sdtmndtplgtdt0(W0,W1,W2)
          & sdtmndtplgtdt0(W2,W1,W3) )
       => sdtmndtplgtdt0(W0,W1,W3) ) ) ).

fof(mTCRDef,definition,
    ! [W0,W1,W2] :
      ( ( aElement0(W0)
        & aRewritingSystem0(W1)
        & aElement0(W2) )
     => ( sdtmndtasgtdt0(W0,W1,W2)
      <=> ( W0 = W2
          | sdtmndtplgtdt0(W0,W1,W2) ) ) ) ).

fof(mTCRTrans,axiom,
    ! [W0,W1,W2,W3] :
      ( ( aElement0(W0)
        & aRewritingSystem0(W1)
        & aElement0(W2)
        & aElement0(W3) )
     => ( ( sdtmndtasgtdt0(W0,W1,W2)
          & sdtmndtasgtdt0(W2,W1,W3) )
       => sdtmndtasgtdt0(W0,W1,W3) ) ) ).

fof(mCRDef,definition,
    ! [W0] :
      ( aRewritingSystem0(W0)
     => ( isConfluent0(W0)
      <=> ! [W1,W2,W3] :
            ( ( aElement0(W1)
              & aElement0(W2)
              & aElement0(W3)
              & sdtmndtasgtdt0(W1,W0,W2)
              & sdtmndtasgtdt0(W1,W0,W3) )
           => ? [W4] :
                ( aElement0(W4)
                & sdtmndtasgtdt0(W2,W0,W4)
                & sdtmndtasgtdt0(W3,W0,W4) ) ) ) ) ).

fof(mWCRDef,definition,
    ! [W0] :
      ( aRewritingSystem0(W0)
     => ( isLocallyConfluent0(W0)
      <=> ! [W1,W2,W3] :
            ( ( aElement0(W1)
              & aElement0(W2)
              & aElement0(W3)
              & aReductOfIn0(W2,W1,W0)
              & aReductOfIn0(W3,W1,W0) )
           => ? [W4] :
                ( aElement0(W4)
                & sdtmndtasgtdt0(W2,W0,W4)
                & sdtmndtasgtdt0(W3,W0,W4) ) ) ) ) ).

fof(mTermin,definition,
    ! [W0] :
      ( aRewritingSystem0(W0)
     => ( isTerminating0(W0)
      <=> ! [W1,W2] :
            ( ( aElement0(W1)
              & aElement0(W2) )
           => ( sdtmndtplgtdt0(W1,W0,W2)
             => iLess0(W2,W1) ) ) ) ) ).

fof(mNFRDef,definition,
    ! [W0,W1] :
      ( ( aElement0(W0)
        & aRewritingSystem0(W1) )
     => ! [W2] :
          ( aNormalFormOfIn0(W2,W0,W1)
        <=> ( aElement0(W2)
            & sdtmndtasgtdt0(W0,W1,W2)
            & ~ ? [W3] : aReductOfIn0(W3,W2,W1) ) ) ) ).

fof(mTermNF,axiom,
    ! [W0] :
      ( ( aRewritingSystem0(W0)
        & isTerminating0(W0) )
     => ! [W1] :
          ( aElement0(W1)
         => ? [W2] : aNormalFormOfIn0(W2,W1,W0) ) ) ).

fof(m__656,hypothesis,
    aRewritingSystem0(xR) ).

fof(m__656_01,hypothesis,
    ( isLocallyConfluent0(xR)
    & isTerminating0(xR) ) ).

fof(m__715,hypothesis,
    ! [W0,W1,W2] :
      ( ( aElement0(W0)
        & aElement0(W1)
        & aElement0(W2)
        & sdtmndtasgtdt0(W0,xR,W1)
        & sdtmndtasgtdt0(W0,xR,W2) )
     => ? [W3] :
          ( aElement0(W3)
          & sdtmndtasgtdt0(W1,xR,W3)
          & sdtmndtasgtdt0(W2,xR,W3) ) ) ).

fof(m__,conjecture,
    isConfluent0(xR) ).

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