
theorem
  for n being non empty Ordinal, T being admissible connected TermOrder
  of n, L being add-associative right_complementable right_zeroed commutative
  associative well-unital distributive almost_left_invertible non degenerated
  non empty doubleLoopStr ex P be Subset of Polynom-Ring(n,L) st not(P
  is_Groebner_basis_wrt T)
proof
  let n be non empty Ordinal, T be admissible connected TermOrder of n, L be
  add-associative right_complementable right_zeroed commutative associative
  well-unital distributive almost_left_invertible non degenerated non empty
  doubleLoopStr;
  set 1bag = (EmptyBag n)+*({},1);
  reconsider 1bag as Element of Bags n by PRE_POLY:def 12;
  set p = ((1.L) |(n,L))+*(1bag,1.L);
  reconsider p as Function of Bags n,L;
  reconsider p as Series of n,L;
A1: 1.L <> 0.L;
  set q = (0.L |(n,L))+*(1bag,1.L);
  reconsider q as Function of Bags n,L;
  reconsider q as Series of n,L;
A2: now
    let u be bag of n;
    assume that
A3: u <> EmptyBag n and
A4: u <> 1bag;
    p.u = ((1.L) |(n,L)).u by A4,FUNCT_7:32;
    then p.u = (1_(n,L)).u by POLYNOM7:20;
    hence p.u = 0.L by A3,POLYNOM1:25;
  end;
A5: now
    let u9 be object;
    assume
A6: u9 in Support p;
    then reconsider u = u9 as Element of Bags n;
A7: p.u <> 0.L by A6,POLYNOM1:def 4;
    now
      assume not u in {EmptyBag n,1bag};
      then u <> EmptyBag n & u <> 1bag by TARSKI:def 2;
      hence contradiction by A2,A7;
    end;
    hence u9 in {EmptyBag n,1bag};
  end;
  {} in n & dom EmptyBag n = n by ORDINAL1:14,PARTFUN1:def 2;
  then 1bag.{} = 1 by FUNCT_7:31;
  then
A8: EmptyBag n <> 1bag by PBOOLE:5;
  then
A9: q.(EmptyBag n) = (0.L |(n,L)).(EmptyBag n) by FUNCT_7:32
    .= (0_(n,L)).(EmptyBag n) by POLYNOM7:19
    .= 0.L by POLYNOM1:22;
A10: now
    let u be bag of n;
    assume u <> 1bag;
    then q.u = (0.L |(n,L)).u by FUNCT_7:32;
    then q.u = (0_(n,L)).u by POLYNOM7:19;
    hence q.u = 0.L by POLYNOM1:22;
  end;
A11: now
    let u9 be object;
    assume
A12: u9 in Support q;
    then reconsider u = u9 as Element of Bags n;
A13: q.u <> 0.L by A12,POLYNOM1:def 4;
    now
      assume not u in {1bag};
      then u <> 1bag by TARSKI:def 1;
      hence contradiction by A10,A13;
    end;
    hence u9 in {1bag};
  end;
  dom(0.L |(n,L)) = Bags n by FUNCT_2:def 1;
  then
A14: q.(1bag) = 1.L by FUNCT_7:31;
  then
A15: q <> 0_(n,L) by POLYNOM1:22;
  now
    let u be object;
    assume u in {1bag};
    then u = 1bag by TARSKI:def 1;
    hence u in Support q by A14,POLYNOM1:def 4;
  end;
  then
A16: Support q = {1bag} by A11,TARSKI:2;
  then reconsider q as Polynomial of n,L by POLYNOM1:def 5;
  reconsider q as non-zero Polynomial of n,L by A15,POLYNOM7:def 1;
  set q1 = q - (q.HT(q,T)/HC(q,T)) * ((EmptyBag n) *' q);
  Support q <> {} by A15,POLYNOM7:1;
  then
A17: HT(q,T) in Support q by TERMORD:def 6;
  EmptyBag n + HT(q,T) = HT(q,T) by PRE_POLY:53;
  then q reduces_to q1,q,HT(q,T),T by A15,A17,POLYRED:def 5;
  then
A18: q reduces_to q1,q,T by POLYRED:def 6;
A19: q1 = q - (HC(q,T)/HC(q,T)) * ((EmptyBag n) *' q) by TERMORD:def 7
    .= q - (HC(q,T)*(HC(q,T))") * ((EmptyBag n) *' q)
    .= q - 1.L * ((EmptyBag n) *' q) by VECTSP_1:def 10
    .= q - 1.L * q by POLYRED:17
    .= q - (1.L) |(n,L) *' q by POLYNOM7:27
    .= q - 1_(n,L) *' q by POLYNOM7:20
    .= q - q by POLYNOM1:30
    .= 0_(n,L) by POLYNOM1:24;
A20: dom((1.L) |(n,L)) = Bags n by FUNCT_2:def 1;
  then
A21: p.(1bag) = 1.L by FUNCT_7:31;
  then
A22: p <> 0_(n,L) by A1,POLYNOM1:22;
A23: p.(EmptyBag n) = ((1.L) |(n,L)).(EmptyBag n) by A8,FUNCT_7:32
    .= (1_(n,L)).(EmptyBag n) by POLYNOM7:20
    .= 1.L by POLYNOM1:25;
  now
    let u be object;
    assume
A24: u in {EmptyBag n,1bag};
    now
      per cases by A24,TARSKI:def 2;
      case
        u = EmptyBag n;
        hence u in Support p by A1,A23,POLYNOM1:def 4;
      end;
      case
        u = 1bag;
        hence u in Support p by A1,A21,POLYNOM1:def 4;
      end;
    end;
    hence u in Support p;
  end;
  then
A25: Support p = {EmptyBag n,1bag} by A5,TARSKI:2;
  then reconsider p as Polynomial of n,L by POLYNOM1:def 5;
  reconsider p as non-zero Polynomial of n,L by A22,POLYNOM7:def 1;
A26: EmptyBag n + HT(p,T) = HT(p,T) by PRE_POLY:53;
A27: now
A28: EmptyBag n <= 1bag,T by TERMORD:9;
    assume
A29: HT(p,T) = EmptyBag n;
    1bag in Support p by A25,TARSKI:def 2;
    then 1bag <= EmptyBag n,T by A29,TERMORD:def 6;
    hence contradiction by A8,A28,TERMORD:7;
  end;
  set p1 = q - (q.HT(p,T)/HC(p,T)) * ((EmptyBag n) *' p);
  Support p <> {} by A22,POLYNOM7:1;
  then
A30: HT(p,T) in Support p by TERMORD:def 6;
  then
A31: HT(p,T) = 1bag by A25,A27,TARSKI:def 2;
  then HT(p,T) in Support q by A16,TARSKI:def 1;
  then q reduces_to p1,p,HT(p,T),T by A22,A15,A26,POLYRED:def 5;
  then
A32: q reduces_to p1,p,T by POLYRED:def 6;
A33: now
    assume Support q = Support p;
    then EmptyBag n in {1bag} by A25,A16,TARSKI:def 2;
    hence contradiction by A8,TARSKI:def 1;
  end;
A34: now
    assume q - p = 0_(n,L);
    then p = (q-p)+p by POLYRED:2
      .= (q+-p)+p by POLYNOM1:def 7
      .= q+(-p+p) by POLYNOM1:21
      .= q + 0_(n,L) by POLYRED:3;
    hence contradiction by A33,POLYNOM1:23;
  end;
  set P = {p,q};
  now
    let u be object;
    assume u in P;
    then u = p or u = q by TARSKI:def 2;
    hence u in the carrier of Polynom-Ring(n,L) by POLYNOM1:def 11;
  end;
  then reconsider P as Subset of Polynom-Ring(n,L) by TARSKI:def 3;
  reconsider P as Subset of Polynom-Ring(n,L);
  set R = PolyRedRel(P,T);
  take P;
A35: p in P by TARSKI:def 2;
  q in P by TARSKI:def 2;
  then q reduces_to 0_(n,L),P,T by A18,A19,POLYRED:def 7;
  then
A36: [q,0_(n,L)] in R by POLYRED:def 13;
  p1 = q - (1.L/p.1bag) * ((EmptyBag n) *' p) by A14,A31,TERMORD:def 7
    .= q - (1.L/1.L) * ((EmptyBag n) *' p) by A20,FUNCT_7:31
    .= q - (1.L*(1.L)") * ((EmptyBag n) *' p)
    .= q - 1.L * ((EmptyBag n) *' p) by VECTSP_1:def 10
    .= q - 1.L * p by POLYRED:17
    .= q - (1.L) |(n,L) *' p by POLYNOM7:27
    .= q - 1_(n,L) *' p by POLYNOM7:20
    .= q - p by POLYNOM1:30;
  then q reduces_to q-p,P,T by A32,A35,POLYRED:def 7;
  then
A37: [q,q-p] in R by POLYRED:def 13;
  now
A38: now
      let u be object;
      now
        let u be object;
        assume u in {1bag};
        then u = 1bag by TARSKI:def 1;
        hence u in {EmptyBag n,1bag} by TARSKI:def 2;
      end;
      then {1bag} c= {EmptyBag n,1bag};
      then
A39:  {1bag} \/ {EmptyBag n,1bag} = {EmptyBag n,1bag} by XBOOLE_1:12;
A40:  (q-p).1bag = (q+-p).1bag by POLYNOM1:def 7
        .= q.1bag + (-p).1bag by POLYNOM1:15
        .= q.1bag + -(p.1bag) by POLYNOM1:17
        .= 1.L + -1.L by A20,A14,FUNCT_7:31
        .= 0.L by RLVECT_1:5;
      Support(q-p) = Support (q+-p) by POLYNOM1:def 7;
      then Support(q-p) c= Support(q) \/ Support(-p) by POLYNOM1:20;
      then
A41:  Support(q-p) c= {1bag} \/ {EmptyBag n,1bag} by A25,A16,Th5;
      assume
A42:  u in Support(q-p);
      then (q-p).u <> 0.L by POLYNOM1:def 4;
      then u = EmptyBag n by A42,A41,A39,A40,TARSKI:def 2;
      hence u in {EmptyBag n} by TARSKI:def 1;
    end;
    assume R is locally-confluent;
    then 0_(n,L),q-p are_convergent_wrt R by A37,A36,REWRITE1:def 24;
    then consider c being object such that
A43: R reduces 0_(n,L),c and
A44: R reduces q-p,c by REWRITE1:def 7;
    reconsider c as Polynomial of n,L by A43,Lm4;
    consider s being FinSequence such that
A45: len s > 0 and
A46: s.1 = 0_(n,L) and
A47: s.len s = c and
A48: for i being Nat st i in dom s & i+1 in dom s holds [s.
    i, s.(i+1)] in R by A43,REWRITE1:11;
    now
A49:  0 + 1 <= len s by A45,NAT_1:13;
A50:  dom s = Seg(len s) by FINSEQ_1:def 3;
      assume len s <> 1;
      then 1 < len s by A49,XXREAL_0:1;
      then 1+1 <= len s by NAT_1:13;
      then
A51:  1+1 in dom s by A50,FINSEQ_1:1;
A52:  1 in dom s by A49,A50,FINSEQ_1:1;
      then consider f9,h9 being object such that
A53:  [0_(n,L),s.2] = [f9,h9] and
      f9 in NonZero Polynom-Ring(n,L) and
A54:  h9 in (the carrier of Polynom-Ring(n,L)) by A46,A48,A51,RELSET_1:2;
      s.2 = h9 by A53,XTUPLE_0:1;
      then reconsider c9 = s.2 as Polynomial of n,L by A54,POLYNOM1:def 11;
      [s.1,s.2] in R by A48,A52,A51;
      then 0_(n,L) reduces_to c9,P,T by A46,POLYRED:def 13;
      then consider g being Polynomial of n,L such that
      g in P and
A55:  0_(n,L) reduces_to c9,g,T by POLYRED:def 7;
      0_(n,L) is_reducible_wrt g,T by A55,POLYRED:def 8;
      hence contradiction by POLYRED:37;
    end;
    then consider s being FinSequence such that
A56: len s > 0 and
A57: s.1 = q-p and
A58: s.len s = 0_(n,L) and
A59: for i being Nat st i in dom s & i+1 in dom s holds [s
    .i, s.(i+1)] in R by A44,A46,A47,REWRITE1:11;
A60: now
      assume -1.L = 0.L;
      then --1.L = 0.L by RLVECT_1:12;
      hence contradiction by RLVECT_1:17;
    end;
    now
      let u be object;
      assume
A61:  u in {EmptyBag n};
      then reconsider u9 = u as Element of Bags n by TARSKI:def 1;
      (q-p).u9 = (q+-p).u9 by POLYNOM1:def 7
        .= q.u9 + (-p).u9 by POLYNOM1:15
        .= q.u9 + -(p.u9) by POLYNOM1:17
        .= 0.L + -(p.u9) by A9,A61,TARSKI:def 1
        .= 0.L + -1.L by A23,A61,TARSKI:def 1
        .= -1.L by ALGSTR_1:def 2;
      hence u in Support(q-p) by A60,POLYNOM1:def 4;
    end;
    then
A62: Support(q-p) = {EmptyBag n} by A38,TARSKI:2;
A63: now
      assume q-p is_reducible_wrt P,T;
      then consider g being Polynomial of n,L such that
A64:  q-p reduces_to g,P,T by POLYRED:def 9;
      consider h being Polynomial of n,L such that
A65:  h in P and
A66:  q-p reduces_to g,h,T by A64,POLYRED:def 7;
      ex b being bag of n st q-p reduces_to g,h,b,T by A66,POLYRED:def 6;
      then h <> 0_(n,L) by POLYRED:def 5;
      then reconsider h as non-zero Polynomial of n,L by POLYNOM7:def 1;
      q-p is_reducible_wrt h,T by A66,POLYRED:def 8;
      then consider b9 being bag of n such that
A67:  b9 in Support(q-p) and
A68:  HT(h,T) divides b9 by POLYRED:36;
A69:  HT(h,T) = 1bag
      proof
        now
          per cases by A65,TARSKI:def 2;
          case
            h = p;
            hence thesis by A25,A30,A27,TARSKI:def 2;
          end;
          case
            h = q;
            hence thesis by A16,A17,TARSKI:def 1;
          end;
        end;
        hence thesis;
      end;
      b9 = EmptyBag n by A62,A67,TARSKI:def 1;
      hence contradiction by A8,A68,A69,PRE_POLY:58;
    end;
    now
A70:  0 + 1 <= len s by A56,NAT_1:13;
A71:  dom s = Seg(len s) by FINSEQ_1:def 3;
      assume len s <> 1;
      then 1 < len s by A70,XXREAL_0:1;
      then 1+1 <= len s by NAT_1:13;
      then
A72:  1+1 in dom s by A71,FINSEQ_1:1;
A73:  1 in dom s by A70,A71,FINSEQ_1:1;
      then consider f9,h9 being object such that
A74:  [q-p,s.2] = [f9,h9] and
      f9 in NonZero Polynom-Ring(n,L) and
A75:  h9 in (the carrier of Polynom-Ring(n,L)) by A57,A59,A72,RELSET_1:2;
      s.2 = h9 by A74,XTUPLE_0:1;
      then reconsider c9 = s.2 as Polynomial of n,L by A75,POLYNOM1:def 11;
      [q-p,s.2] in R by A57,A59,A73,A72;
      then q-p reduces_to c9,P,T by POLYRED:def 13;
      hence contradiction by A63,POLYRED:def 9;
    end;
    hence contradiction by A34,A57,A58;
  end;
  hence thesis;
end;
