- : unit = ()
- : unit = ()
h : heuristic = <heuristic>
- : unit = ()
APPLY CRITERIA (Marked dependency pairs)

TRS termination of:
[1] active(U11(tt,N)) -> mark(N)
[2] active(U21(tt,M,N)) -> mark(s(plus(N,M)))
[3] active(U31(tt)) -> mark(0)
[4] active(U41(tt,M,N)) -> mark(plus(x(N,M),N))
[5] active(and(tt,X)) -> mark(X)
[6] active(isNat(0)) -> mark(tt)
[7] active(isNat(plus(V1,V2))) -> mark(and(isNat(V1),isNat(V2)))
[8] active(isNat(s(V1))) -> mark(isNat(V1))
[9] active(isNat(x(V1,V2))) -> mark(and(isNat(V1),isNat(V2)))
[10] active(plus(N,0)) -> mark(U11(isNat(N),N))
[11] active(plus(N,s(M))) -> mark(U21(and(isNat(M),isNat(N)),M,N))
[12] active(x(N,0)) -> mark(U31(isNat(N)))
[13] active(x(N,s(M))) -> mark(U41(and(isNat(M),isNat(N)),M,N))
[14] active(U11(X1,X2)) -> U11(active(X1),X2)
[15] active(U21(X1,X2,X3)) -> U21(active(X1),X2,X3)
[16] active(s(X)) -> s(active(X))
[17] active(plus(X1,X2)) -> plus(active(X1),X2)
[18] active(plus(X1,X2)) -> plus(X1,active(X2))
[19] active(U31(X)) -> U31(active(X))
[20] active(U41(X1,X2,X3)) -> U41(active(X1),X2,X3)
[21] active(x(X1,X2)) -> x(active(X1),X2)
[22] active(x(X1,X2)) -> x(X1,active(X2))
[23] active(and(X1,X2)) -> and(active(X1),X2)
[24] U11(mark(X1),X2) -> mark(U11(X1,X2))
[25] U21(mark(X1),X2,X3) -> mark(U21(X1,X2,X3))
[26] s(mark(X)) -> mark(s(X))
[27] plus(mark(X1),X2) -> mark(plus(X1,X2))
[28] plus(X1,mark(X2)) -> mark(plus(X1,X2))
[29] U31(mark(X)) -> mark(U31(X))
[30] U41(mark(X1),X2,X3) -> mark(U41(X1,X2,X3))
[31] x(mark(X1),X2) -> mark(x(X1,X2))
[32] x(X1,mark(X2)) -> mark(x(X1,X2))
[33] and(mark(X1),X2) -> mark(and(X1,X2))
[34] proper(U11(X1,X2)) -> U11(proper(X1),proper(X2))
[35] proper(tt) -> ok(tt)
[36] proper(U21(X1,X2,X3)) -> U21(proper(X1),proper(X2),proper(X3))
[37] proper(s(X)) -> s(proper(X))
[38] proper(plus(X1,X2)) -> plus(proper(X1),proper(X2))
[39] proper(U31(X)) -> U31(proper(X))
[40] proper(0) -> ok(0)
[41] proper(U41(X1,X2,X3)) -> U41(proper(X1),proper(X2),proper(X3))
[42] proper(x(X1,X2)) -> x(proper(X1),proper(X2))
[43] proper(and(X1,X2)) -> and(proper(X1),proper(X2))
[44] proper(isNat(X)) -> isNat(proper(X))
[45] U11(ok(X1),ok(X2)) -> ok(U11(X1,X2))
[46] U21(ok(X1),ok(X2),ok(X3)) -> ok(U21(X1,X2,X3))
[47] s(ok(X)) -> ok(s(X))
[48] plus(ok(X1),ok(X2)) -> ok(plus(X1,X2))
[49] U31(ok(X)) -> ok(U31(X))
[50] U41(ok(X1),ok(X2),ok(X3)) -> ok(U41(X1,X2,X3))
[51] x(ok(X1),ok(X2)) -> ok(x(X1,X2))
[52] and(ok(X1),ok(X2)) -> ok(and(X1,X2))
[53] isNat(ok(X)) -> ok(isNat(X))
[54] top(mark(X)) -> top(proper(X))
[55] top(ok(X)) -> top(active(X))


Sub problem: 
guided: 

DP termination of:
<Marked_active(U21(tt,M,N)) , Marked_s(plus(N,M))>
<Marked_active(U21(tt,M,N)) , Marked_plus(N,M)>
<Marked_active(U41(tt,M,N)) , Marked_plus(x(N,M),N)>
<Marked_active(U41(tt,M,N)) , Marked_x(N,M)>
<Marked_active(isNat(plus(V1,V2))) , Marked_and(isNat(V1),isNat(V2))>
<Marked_active(isNat(plus(V1,V2))) , Marked_isNat(V1)>
<Marked_active(isNat(plus(V1,V2))) , Marked_isNat(V2)>
<Marked_active(isNat(s(V1))) , Marked_isNat(V1)>
<Marked_active(isNat(x(V1,V2))) , Marked_and(isNat(V1),isNat(V2))>
<Marked_active(isNat(x(V1,V2))) , Marked_isNat(V1)>
<Marked_active(isNat(x(V1,V2))) , Marked_isNat(V2)>
<Marked_active(plus(N,0)) , Marked_U11(isNat(N),N)>
<Marked_active(plus(N,0)) , Marked_isNat(N)>
<Marked_active(plus(N,s(M))) , Marked_U21(and(isNat(M),isNat(N)),M,N)>
<Marked_active(plus(N,s(M))) , Marked_and(isNat(M),isNat(N))>
<Marked_active(plus(N,s(M))) , Marked_isNat(M)>
<Marked_active(plus(N,s(M))) , Marked_isNat(N)>
<Marked_active(x(N,0)) , Marked_U31(isNat(N))>
<Marked_active(x(N,0)) , Marked_isNat(N)>
<Marked_active(x(N,s(M))) , Marked_U41(and(isNat(M),isNat(N)),M,N)>
<Marked_active(x(N,s(M))) , Marked_and(isNat(M),isNat(N))>
<Marked_active(x(N,s(M))) , Marked_isNat(M)>
<Marked_active(x(N,s(M))) , Marked_isNat(N)>
<Marked_active(U11(X1,X2)) , Marked_U11(active(X1),X2)>
<Marked_active(U11(X1,X2)) , Marked_active(X1)>
<Marked_active(U21(X1,X2,X3)) , Marked_U21(active(X1),X2,X3)>
<Marked_active(U21(X1,X2,X3)) , Marked_active(X1)>
<Marked_active(s(X)) , Marked_s(active(X))>
<Marked_active(s(X)) , Marked_active(X)>
<Marked_active(plus(X1,X2)) , Marked_plus(active(X1),X2)>
<Marked_active(plus(X1,X2)) , Marked_active(X1)>
<Marked_active(plus(X1,X2)) , Marked_plus(X1,active(X2))>
<Marked_active(plus(X1,X2)) , Marked_active(X2)>
<Marked_active(U31(X)) , Marked_U31(active(X))>
<Marked_active(U31(X)) , Marked_active(X)>
<Marked_active(U41(X1,X2,X3)) , Marked_U41(active(X1),X2,X3)>
<Marked_active(U41(X1,X2,X3)) , Marked_active(X1)>
<Marked_active(x(X1,X2)) , Marked_x(active(X1),X2)>
<Marked_active(x(X1,X2)) , Marked_active(X1)>
<Marked_active(x(X1,X2)) , Marked_x(X1,active(X2))>
<Marked_active(x(X1,X2)) , Marked_active(X2)>
<Marked_active(and(X1,X2)) , Marked_and(active(X1),X2)>
<Marked_active(and(X1,X2)) , Marked_active(X1)>
<Marked_U11(mark(X1),X2) , Marked_U11(X1,X2)>
<Marked_U21(mark(X1),X2,X3) , Marked_U21(X1,X2,X3)>
<Marked_s(mark(X)) , Marked_s(X)>
<Marked_plus(mark(X1),X2) , Marked_plus(X1,X2)>
<Marked_plus(X1,mark(X2)) , Marked_plus(X1,X2)>
<Marked_U31(mark(X)) , Marked_U31(X)>
<Marked_U41(mark(X1),X2,X3) , Marked_U41(X1,X2,X3)>
<Marked_x(mark(X1),X2) , Marked_x(X1,X2)>
<Marked_x(X1,mark(X2)) , Marked_x(X1,X2)>
<Marked_and(mark(X1),X2) , Marked_and(X1,X2)>
<Marked_proper(U11(X1,X2)) , Marked_U11(proper(X1),proper(X2))>
<Marked_proper(U11(X1,X2)) , Marked_proper(X1)>
<Marked_proper(U11(X1,X2)) , Marked_proper(X2)>
<Marked_proper(U21(X1,X2,X3)) , Marked_U21(proper(X1),proper(X2),proper(X3))>
<Marked_proper(U21(X1,X2,X3)) , Marked_proper(X1)>
<Marked_proper(U21(X1,X2,X3)) , Marked_proper(X2)>
<Marked_proper(U21(X1,X2,X3)) , Marked_proper(X3)>
<Marked_proper(s(X)) , Marked_s(proper(X))>
<Marked_proper(s(X)) , Marked_proper(X)>
<Marked_proper(plus(X1,X2)) , Marked_plus(proper(X1),proper(X2))>
<Marked_proper(plus(X1,X2)) , Marked_proper(X1)>
<Marked_proper(plus(X1,X2)) , Marked_proper(X2)>
<Marked_proper(U31(X)) , Marked_U31(proper(X))>
<Marked_proper(U31(X)) , Marked_proper(X)>
<Marked_proper(U41(X1,X2,X3)) , Marked_U41(proper(X1),proper(X2),proper(X3))>
<Marked_proper(U41(X1,X2,X3)) , Marked_proper(X1)>
<Marked_proper(U41(X1,X2,X3)) , Marked_proper(X2)>
<Marked_proper(U41(X1,X2,X3)) , Marked_proper(X3)>
<Marked_proper(x(X1,X2)) , Marked_x(proper(X1),proper(X2))>
<Marked_proper(x(X1,X2)) , Marked_proper(X1)>
<Marked_proper(x(X1,X2)) , Marked_proper(X2)>
<Marked_proper(and(X1,X2)) , Marked_and(proper(X1),proper(X2))>
<Marked_proper(and(X1,X2)) , Marked_proper(X1)>
<Marked_proper(and(X1,X2)) , Marked_proper(X2)>
<Marked_proper(isNat(X)) , Marked_isNat(proper(X))>
<Marked_proper(isNat(X)) , Marked_proper(X)>
<Marked_U11(ok(X1),ok(X2)) , Marked_U11(X1,X2)>
<Marked_U21(ok(X1),ok(X2),ok(X3)) , Marked_U21(X1,X2,X3)>
<Marked_s(ok(X)) , Marked_s(X)>
<Marked_plus(ok(X1),ok(X2)) , Marked_plus(X1,X2)>
<Marked_U31(ok(X)) , Marked_U31(X)>
<Marked_U41(ok(X1),ok(X2),ok(X3)) , Marked_U41(X1,X2,X3)>
<Marked_x(ok(X1),ok(X2)) , Marked_x(X1,X2)>
<Marked_and(ok(X1),ok(X2)) , Marked_and(X1,X2)>
<Marked_isNat(ok(X)) , Marked_isNat(X)>
<Marked_top(mark(X)) , Marked_top(proper(X))>
<Marked_top(mark(X)) , Marked_proper(X)>
<Marked_top(ok(X)) , Marked_top(active(X))>
<Marked_top(ok(X)) , Marked_active(X)>
 
END GUIDED
APPLY CRITERIA (Graph splitting)
Found 12 components:

{  <Marked_top(ok(X)) , Marked_top(active(X))> 
      --> <Marked_top(ok(X)) , Marked_top(active(X))>
  <Marked_top(ok(X)) , Marked_top(active(X))> 
     --> <Marked_top(mark(X)) , Marked_top(proper(X))>
  <Marked_top(mark(X)) , Marked_top(proper(X))> 
     --> <Marked_top(ok(X)) , Marked_top(active(X))>
  <Marked_top(mark(X)) , Marked_top(proper(X))> 
     --> <Marked_top(mark(X)) , Marked_top(proper(X))>
}

{  <Marked_proper(isNat(X)) , Marked_proper(X)> 
      --> <Marked_proper(isNat(X)) , Marked_proper(X)>
  <Marked_proper(isNat(X)) , Marked_proper(X)> 
     --> <Marked_proper(and(X1,X2)) , Marked_proper(X2)>
  <Marked_proper(isNat(X)) , Marked_proper(X)> 
     --> <Marked_proper(and(X1,X2)) , Marked_proper(X1)>
  <Marked_proper(isNat(X)) , Marked_proper(X)> 
     --> <Marked_proper(x(X1,X2)) , Marked_proper(X2)>
  <Marked_proper(isNat(X)) , Marked_proper(X)> 
     --> <Marked_proper(x(X1,X2)) , Marked_proper(X1)>
  <Marked_proper(isNat(X)) , Marked_proper(X)> 
     --> <Marked_proper(U41(X1,X2,X3)) , Marked_proper(X3)>
  <Marked_proper(isNat(X)) , Marked_proper(X)> 
     --> <Marked_proper(U41(X1,X2,X3)) , Marked_proper(X2)>
  <Marked_proper(isNat(X)) , Marked_proper(X)> 
     --> <Marked_proper(U41(X1,X2,X3)) , Marked_proper(X1)>
  <Marked_proper(isNat(X)) , Marked_proper(X)> 
     --> <Marked_proper(U31(X)) , Marked_proper(X)>
  <Marked_proper(isNat(X)) , Marked_proper(X)> 
     --> <Marked_proper(plus(X1,X2)) , Marked_proper(X2)>
  <Marked_proper(isNat(X)) , Marked_proper(X)> 
     --> <Marked_proper(plus(X1,X2)) , Marked_proper(X1)>
  <Marked_proper(isNat(X)) , Marked_proper(X)> 
     --> <Marked_proper(s(X)) , Marked_proper(X)>
  <Marked_proper(isNat(X)) , Marked_proper(X)> 
     --> <Marked_proper(U21(X1,X2,X3)) , Marked_proper(X3)>
  <Marked_proper(isNat(X)) , Marked_proper(X)> 
     --> <Marked_proper(U21(X1,X2,X3)) , Marked_proper(X2)>
  <Marked_proper(isNat(X)) , Marked_proper(X)> 
     --> <Marked_proper(U21(X1,X2,X3)) , Marked_proper(X1)>
  <Marked_proper(isNat(X)) , Marked_proper(X)> 
     --> <Marked_proper(U11(X1,X2)) , Marked_proper(X2)>
  <Marked_proper(isNat(X)) , Marked_proper(X)> 
     --> <Marked_proper(U11(X1,X2)) , Marked_proper(X1)>
  <Marked_proper(and(X1,X2)) , Marked_proper(X2)> 
     --> <Marked_proper(isNat(X)) , Marked_proper(X)>
  <Marked_proper(and(X1,X2)) , Marked_proper(X2)> 
     --> <Marked_proper(and(X1,X2)) , Marked_proper(X2)>
  <Marked_proper(and(X1,X2)) , Marked_proper(X2)> 
     --> <Marked_proper(and(X1,X2)) , Marked_proper(X1)>
  <Marked_proper(and(X1,X2)) , Marked_proper(X2)> 
     --> <Marked_proper(x(X1,X2)) , Marked_proper(X2)>
  <Marked_proper(and(X1,X2)) , Marked_proper(X2)> 
     --> <Marked_proper(x(X1,X2)) , Marked_proper(X1)>
  <Marked_proper(and(X1,X2)) , Marked_proper(X2)> 
     --> <Marked_proper(U41(X1,X2,X3)) , Marked_proper(X3)>
  <Marked_proper(and(X1,X2)) , Marked_proper(X2)> 
     --> <Marked_proper(U41(X1,X2,X3)) , Marked_proper(X2)>
  <Marked_proper(and(X1,X2)) , Marked_proper(X2)> 
     --> <Marked_proper(U41(X1,X2,X3)) , Marked_proper(X1)>
  <Marked_proper(and(X1,X2)) , Marked_proper(X2)> 
     --> <Marked_proper(U31(X)) , Marked_proper(X)>
  <Marked_proper(and(X1,X2)) , Marked_proper(X2)> 
     --> <Marked_proper(plus(X1,X2)) , Marked_proper(X2)>
  <Marked_proper(and(X1,X2)) , Marked_proper(X2)> 
     --> <Marked_proper(plus(X1,X2)) , Marked_proper(X1)>
  <Marked_proper(and(X1,X2)) , Marked_proper(X2)> 
     --> <Marked_proper(s(X)) , Marked_proper(X)>
  <Marked_proper(and(X1,X2)) , Marked_proper(X2)> 
     --> <Marked_proper(U21(X1,X2,X3)) , Marked_proper(X3)>
  <Marked_proper(and(X1,X2)) , Marked_proper(X2)> 
     --> <Marked_proper(U21(X1,X2,X3)) , Marked_proper(X2)>
  <Marked_proper(and(X1,X2)) , Marked_proper(X2)> 
     --> <Marked_proper(U21(X1,X2,X3)) , Marked_proper(X1)>
  <Marked_proper(and(X1,X2)) , Marked_proper(X2)> 
     --> <Marked_proper(U11(X1,X2)) , Marked_proper(X2)>
  <Marked_proper(and(X1,X2)) , Marked_proper(X2)> 
     --> <Marked_proper(U11(X1,X2)) , Marked_proper(X1)>
  <Marked_proper(and(X1,X2)) , Marked_proper(X1)> 
     --> <Marked_proper(isNat(X)) , Marked_proper(X)>
  <Marked_proper(and(X1,X2)) , Marked_proper(X1)> 
     --> <Marked_proper(and(X1,X2)) , Marked_proper(X2)>
  <Marked_proper(and(X1,X2)) , Marked_proper(X1)> 
     --> <Marked_proper(and(X1,X2)) , Marked_proper(X1)>
  <Marked_proper(and(X1,X2)) , Marked_proper(X1)> 
     --> <Marked_proper(x(X1,X2)) , Marked_proper(X2)>
  <Marked_proper(and(X1,X2)) , Marked_proper(X1)> 
     --> <Marked_proper(x(X1,X2)) , Marked_proper(X1)>
  <Marked_proper(and(X1,X2)) , Marked_proper(X1)> 
     --> <Marked_proper(U41(X1,X2,X3)) , Marked_proper(X3)>
  <Marked_proper(and(X1,X2)) , Marked_proper(X1)> 
     --> <Marked_proper(U41(X1,X2,X3)) , Marked_proper(X2)>
  <Marked_proper(and(X1,X2)) , Marked_proper(X1)> 
     --> <Marked_proper(U41(X1,X2,X3)) , Marked_proper(X1)>
  <Marked_proper(and(X1,X2)) , Marked_proper(X1)> 
     --> <Marked_proper(U31(X)) , Marked_proper(X)>
  <Marked_proper(and(X1,X2)) , Marked_proper(X1)> 
     --> <Marked_proper(plus(X1,X2)) , Marked_proper(X2)>
  <Marked_proper(and(X1,X2)) , Marked_proper(X1)> 
     --> <Marked_proper(plus(X1,X2)) , Marked_proper(X1)>
  <Marked_proper(and(X1,X2)) , Marked_proper(X1)> 
     --> <Marked_proper(s(X)) , Marked_proper(X)>
  <Marked_proper(and(X1,X2)) , Marked_proper(X1)> 
     --> <Marked_proper(U21(X1,X2,X3)) , Marked_proper(X3)>
  <Marked_proper(and(X1,X2)) , Marked_proper(X1)> 
     --> <Marked_proper(U21(X1,X2,X3)) , Marked_proper(X2)>
  <Marked_proper(and(X1,X2)) , Marked_proper(X1)> 
     --> <Marked_proper(U21(X1,X2,X3)) , Marked_proper(X1)>
  <Marked_proper(and(X1,X2)) , Marked_proper(X1)> 
     --> <Marked_proper(U11(X1,X2)) , Marked_proper(X2)>
  <Marked_proper(and(X1,X2)) , Marked_proper(X1)> 
     --> <Marked_proper(U11(X1,X2)) , Marked_proper(X1)>
  <Marked_proper(x(X1,X2)) , Marked_proper(X2)> 
     --> <Marked_proper(isNat(X)) , Marked_proper(X)>
  <Marked_proper(x(X1,X2)) , Marked_proper(X2)> 
     --> <Marked_proper(and(X1,X2)) , Marked_proper(X2)>
  <Marked_proper(x(X1,X2)) , Marked_proper(X2)> 
     --> <Marked_proper(and(X1,X2)) , Marked_proper(X1)>
  <Marked_proper(x(X1,X2)) , Marked_proper(X2)> 
     --> <Marked_proper(x(X1,X2)) , Marked_proper(X2)>
  <Marked_proper(x(X1,X2)) , Marked_proper(X2)> 
     --> <Marked_proper(x(X1,X2)) , Marked_proper(X1)>
  <Marked_proper(x(X1,X2)) , Marked_proper(X2)> 
     --> <Marked_proper(U41(X1,X2,X3)) , Marked_proper(X3)>
  <Marked_proper(x(X1,X2)) , Marked_proper(X2)> 
     --> <Marked_proper(U41(X1,X2,X3)) , Marked_proper(X2)>
  <Marked_proper(x(X1,X2)) , Marked_proper(X2)> 
     --> <Marked_proper(U41(X1,X2,X3)) , Marked_proper(X1)>
  <Marked_proper(x(X1,X2)) , Marked_proper(X2)> 
     --> <Marked_proper(U31(X)) , Marked_proper(X)>
  <Marked_proper(x(X1,X2)) , Marked_proper(X2)> 
     --> <Marked_proper(plus(X1,X2)) , Marked_proper(X2)>
  <Marked_proper(x(X1,X2)) , Marked_proper(X2)> 
     --> <Marked_proper(plus(X1,X2)) , Marked_proper(X1)>
  <Marked_proper(x(X1,X2)) , Marked_proper(X2)> 
     --> <Marked_proper(s(X)) , Marked_proper(X)>
  <Marked_proper(x(X1,X2)) , Marked_proper(X2)> 
     --> <Marked_proper(U21(X1,X2,X3)) , Marked_proper(X3)>
  <Marked_proper(x(X1,X2)) , Marked_proper(X2)> 
     --> <Marked_proper(U21(X1,X2,X3)) , Marked_proper(X2)>
  <Marked_proper(x(X1,X2)) , Marked_proper(X2)> 
     --> <Marked_proper(U21(X1,X2,X3)) , Marked_proper(X1)>
  <Marked_proper(x(X1,X2)) , Marked_proper(X2)> 
     --> <Marked_proper(U11(X1,X2)) , Marked_proper(X2)>
  <Marked_proper(x(X1,X2)) , Marked_proper(X2)> 
     --> <Marked_proper(U11(X1,X2)) , Marked_proper(X1)>
  <Marked_proper(x(X1,X2)) , Marked_proper(X1)> 
     --> <Marked_proper(isNat(X)) , Marked_proper(X)>
  <Marked_proper(x(X1,X2)) , Marked_proper(X1)> 
     --> <Marked_proper(and(X1,X2)) , Marked_proper(X2)>
  <Marked_proper(x(X1,X2)) , Marked_proper(X1)> 
     --> <Marked_proper(and(X1,X2)) , Marked_proper(X1)>
  <Marked_proper(x(X1,X2)) , Marked_proper(X1)> 
     --> <Marked_proper(x(X1,X2)) , Marked_proper(X2)>
  <Marked_proper(x(X1,X2)) , Marked_proper(X1)> 
     --> <Marked_proper(x(X1,X2)) , Marked_proper(X1)>
  <Marked_proper(x(X1,X2)) , Marked_proper(X1)> 
     --> <Marked_proper(U41(X1,X2,X3)) , Marked_proper(X3)>
  <Marked_proper(x(X1,X2)) , Marked_proper(X1)> 
     --> <Marked_proper(U41(X1,X2,X3)) , Marked_proper(X2)>
  <Marked_proper(x(X1,X2)) , Marked_proper(X1)> 
     --> <Marked_proper(U41(X1,X2,X3)) , Marked_proper(X1)>
  <Marked_proper(x(X1,X2)) , Marked_proper(X1)> 
     --> <Marked_proper(U31(X)) , Marked_proper(X)>
  <Marked_proper(x(X1,X2)) , Marked_proper(X1)> 
     --> <Marked_proper(plus(X1,X2)) , Marked_proper(X2)>
  <Marked_proper(x(X1,X2)) , Marked_proper(X1)> 
     --> <Marked_proper(plus(X1,X2)) , Marked_proper(X1)>
  <Marked_proper(x(X1,X2)) , Marked_proper(X1)> 
     --> <Marked_proper(s(X)) , Marked_proper(X)>
  <Marked_proper(x(X1,X2)) , Marked_proper(X1)> 
     --> <Marked_proper(U21(X1,X2,X3)) , Marked_proper(X3)>
  <Marked_proper(x(X1,X2)) , Marked_proper(X1)> 
     --> <Marked_proper(U21(X1,X2,X3)) , Marked_proper(X2)>
  <Marked_proper(x(X1,X2)) , Marked_proper(X1)> 
     --> <Marked_proper(U21(X1,X2,X3)) , Marked_proper(X1)>
  <Marked_proper(x(X1,X2)) , Marked_proper(X1)> 
     --> <Marked_proper(U11(X1,X2)) , Marked_proper(X2)>
  <Marked_proper(x(X1,X2)) , Marked_proper(X1)> 
     --> <Marked_proper(U11(X1,X2)) , Marked_proper(X1)>
  <Marked_proper(U41(X1,X2,X3)) , Marked_proper(X3)> 
     --> <Marked_proper(isNat(X)) , Marked_proper(X)>
  <Marked_proper(U41(X1,X2,X3)) , Marked_proper(X3)> 
     --> <Marked_proper(and(X1,X2)) , Marked_proper(X2)>
  <Marked_proper(U41(X1,X2,X3)) , Marked_proper(X3)> 
     --> <Marked_proper(and(X1,X2)) , Marked_proper(X1)>
  <Marked_proper(U41(X1,X2,X3)) , Marked_proper(X3)> 
     --> <Marked_proper(x(X1,X2)) , Marked_proper(X2)>
  <Marked_proper(U41(X1,X2,X3)) , Marked_proper(X3)> 
     --> <Marked_proper(x(X1,X2)) , Marked_proper(X1)>
  <Marked_proper(U41(X1,X2,X3)) , Marked_proper(X3)> 
     --> <Marked_proper(U41(X1,X2,X3)) , Marked_proper(X3)>
  <Marked_proper(U41(X1,X2,X3)) , Marked_proper(X3)> 
     --> <Marked_proper(U41(X1,X2,X3)) , Marked_proper(X2)>
  <Marked_proper(U41(X1,X2,X3)) , Marked_proper(X3)> 
     --> <Marked_proper(U41(X1,X2,X3)) , Marked_proper(X1)>
  <Marked_proper(U41(X1,X2,X3)) , Marked_proper(X3)> 
     --> <Marked_proper(U31(X)) , Marked_proper(X)>
  <Marked_proper(U41(X1,X2,X3)) , Marked_proper(X3)> 
     --> <Marked_proper(plus(X1,X2)) , Marked_proper(X2)>
  <Marked_proper(U41(X1,X2,X3)) , Marked_proper(X3)> 
     --> <Marked_proper(plus(X1,X2)) , Marked_proper(X1)>
  <Marked_proper(U41(X1,X2,X3)) , Marked_proper(X3)> 
     --> <Marked_proper(s(X)) , Marked_proper(X)>
  <Marked_proper(U41(X1,X2,X3)) , Marked_proper(X3)> 
     --> <Marked_proper(U21(X1,X2,X3)) , Marked_proper(X3)>
  <Marked_proper(U41(X1,X2,X3)) , Marked_proper(X3)> 
     --> <Marked_proper(U21(X1,X2,X3)) , Marked_proper(X2)>
  <Marked_proper(U41(X1,X2,X3)) , Marked_proper(X3)> 
     --> <Marked_proper(U21(X1,X2,X3)) , Marked_proper(X1)>
  <Marked_proper(U41(X1,X2,X3)) , Marked_proper(X3)> 
     --> <Marked_proper(U11(X1,X2)) , Marked_proper(X2)>
  <Marked_proper(U41(X1,X2,X3)) , Marked_proper(X3)> 
     --> <Marked_proper(U11(X1,X2)) , Marked_proper(X1)>
  <Marked_proper(U41(X1,X2,X3)) , Marked_proper(X2)> 
     --> <Marked_proper(isNat(X)) , Marked_proper(X)>
  <Marked_proper(U41(X1,X2,X3)) , Marked_proper(X2)> 
     --> <Marked_proper(and(X1,X2)) , Marked_proper(X2)>
  <Marked_proper(U41(X1,X2,X3)) , Marked_proper(X2)> 
     --> <Marked_proper(and(X1,X2)) , Marked_proper(X1)>
  <Marked_proper(U41(X1,X2,X3)) , Marked_proper(X2)> 
     --> <Marked_proper(x(X1,X2)) , Marked_proper(X2)>
  <Marked_proper(U41(X1,X2,X3)) , Marked_proper(X2)> 
     --> <Marked_proper(x(X1,X2)) , Marked_proper(X1)>
  <Marked_proper(U41(X1,X2,X3)) , Marked_proper(X2)> 
     --> <Marked_proper(U41(X1,X2,X3)) , Marked_proper(X3)>
  <Marked_proper(U41(X1,X2,X3)) , Marked_proper(X2)> 
     --> <Marked_proper(U41(X1,X2,X3)) , Marked_proper(X2)>
  <Marked_proper(U41(X1,X2,X3)) , Marked_proper(X2)> 
     --> <Marked_proper(U41(X1,X2,X3)) , Marked_proper(X1)>
  <Marked_proper(U41(X1,X2,X3)) , Marked_proper(X2)> 
     --> <Marked_proper(U31(X)) , Marked_proper(X)>
  <Marked_proper(U41(X1,X2,X3)) , Marked_proper(X2)> 
     --> <Marked_proper(plus(X1,X2)) , Marked_proper(X2)>
  <Marked_proper(U41(X1,X2,X3)) , Marked_proper(X2)> 
     --> <Marked_proper(plus(X1,X2)) , Marked_proper(X1)>
  <Marked_proper(U41(X1,X2,X3)) , Marked_proper(X2)> 
     --> <Marked_proper(s(X)) , Marked_proper(X)>
  <Marked_proper(U41(X1,X2,X3)) , Marked_proper(X2)> 
     --> <Marked_proper(U21(X1,X2,X3)) , Marked_proper(X3)>
  <Marked_proper(U41(X1,X2,X3)) , Marked_proper(X2)> 
     --> <Marked_proper(U21(X1,X2,X3)) , Marked_proper(X2)>
  <Marked_proper(U41(X1,X2,X3)) , Marked_proper(X2)> 
     --> <Marked_proper(U21(X1,X2,X3)) , Marked_proper(X1)>
  <Marked_proper(U41(X1,X2,X3)) , Marked_proper(X2)> 
     --> <Marked_proper(U11(X1,X2)) , Marked_proper(X2)>
  <Marked_proper(U41(X1,X2,X3)) , Marked_proper(X2)> 
     --> <Marked_proper(U11(X1,X2)) , Marked_proper(X1)>
  <Marked_proper(U41(X1,X2,X3)) , Marked_proper(X1)> 
     --> <Marked_proper(isNat(X)) , Marked_proper(X)>
  <Marked_proper(U41(X1,X2,X3)) , Marked_proper(X1)> 
     --> <Marked_proper(and(X1,X2)) , Marked_proper(X2)>
  <Marked_proper(U41(X1,X2,X3)) , Marked_proper(X1)> 
     --> <Marked_proper(and(X1,X2)) , Marked_proper(X1)>
  <Marked_proper(U41(X1,X2,X3)) , Marked_proper(X1)> 
     --> <Marked_proper(x(X1,X2)) , Marked_proper(X2)>
  <Marked_proper(U41(X1,X2,X3)) , Marked_proper(X1)> 
     --> <Marked_proper(x(X1,X2)) , Marked_proper(X1)>
  <Marked_proper(U41(X1,X2,X3)) , Marked_proper(X1)> 
     --> <Marked_proper(U41(X1,X2,X3)) , Marked_proper(X3)>
  <Marked_proper(U41(X1,X2,X3)) , Marked_proper(X1)> 
     --> <Marked_proper(U41(X1,X2,X3)) , Marked_proper(X2)>
  <Marked_proper(U41(X1,X2,X3)) , Marked_proper(X1)> 
     --> <Marked_proper(U41(X1,X2,X3)) , Marked_proper(X1)>
  <Marked_proper(U41(X1,X2,X3)) , Marked_proper(X1)> 
     --> <Marked_proper(U31(X)) , Marked_proper(X)>
  <Marked_proper(U41(X1,X2,X3)) , Marked_proper(X1)> 
     --> <Marked_proper(plus(X1,X2)) , Marked_proper(X2)>
  <Marked_proper(U41(X1,X2,X3)) , Marked_proper(X1)> 
     --> <Marked_proper(plus(X1,X2)) , Marked_proper(X1)>
  <Marked_proper(U41(X1,X2,X3)) , Marked_proper(X1)> 
     --> <Marked_proper(s(X)) , Marked_proper(X)>
  <Marked_proper(U41(X1,X2,X3)) , Marked_proper(X1)> 
     --> <Marked_proper(U21(X1,X2,X3)) , Marked_proper(X3)>
  <Marked_proper(U41(X1,X2,X3)) , Marked_proper(X1)> 
     --> <Marked_proper(U21(X1,X2,X3)) , Marked_proper(X2)>
  <Marked_proper(U41(X1,X2,X3)) , Marked_proper(X1)> 
     --> <Marked_proper(U21(X1,X2,X3)) , Marked_proper(X1)>
  <Marked_proper(U41(X1,X2,X3)) , Marked_proper(X1)> 
     --> <Marked_proper(U11(X1,X2)) , Marked_proper(X2)>
  <Marked_proper(U41(X1,X2,X3)) , Marked_proper(X1)> 
     --> <Marked_proper(U11(X1,X2)) , Marked_proper(X1)>
  <Marked_proper(U31(X)) , Marked_proper(X)> 
     --> <Marked_proper(isNat(X)) , Marked_proper(X)>
  <Marked_proper(U31(X)) , Marked_proper(X)> 
     --> <Marked_proper(and(X1,X2)) , Marked_proper(X2)>
  <Marked_proper(U31(X)) , Marked_proper(X)> 
     --> <Marked_proper(and(X1,X2)) , Marked_proper(X1)>
  <Marked_proper(U31(X)) , Marked_proper(X)> 
     --> <Marked_proper(x(X1,X2)) , Marked_proper(X2)>
  <Marked_proper(U31(X)) , Marked_proper(X)> 
     --> <Marked_proper(x(X1,X2)) , Marked_proper(X1)>
  <Marked_proper(U31(X)) , Marked_proper(X)> 
     --> <Marked_proper(U41(X1,X2,X3)) , Marked_proper(X3)>
  <Marked_proper(U31(X)) , Marked_proper(X)> 
     --> <Marked_proper(U41(X1,X2,X3)) , Marked_proper(X2)>
  <Marked_proper(U31(X)) , Marked_proper(X)> 
     --> <Marked_proper(U41(X1,X2,X3)) , Marked_proper(X1)>
  <Marked_proper(U31(X)) , Marked_proper(X)> 
     --> <Marked_proper(U31(X)) , Marked_proper(X)>
  <Marked_proper(U31(X)) , Marked_proper(X)> 
     --> <Marked_proper(plus(X1,X2)) , Marked_proper(X2)>
  <Marked_proper(U31(X)) , Marked_proper(X)> 
     --> <Marked_proper(plus(X1,X2)) , Marked_proper(X1)>
  <Marked_proper(U31(X)) , Marked_proper(X)> 
     --> <Marked_proper(s(X)) , Marked_proper(X)>
  <Marked_proper(U31(X)) , Marked_proper(X)> 
     --> <Marked_proper(U21(X1,X2,X3)) , Marked_proper(X3)>
  <Marked_proper(U31(X)) , Marked_proper(X)> 
     --> <Marked_proper(U21(X1,X2,X3)) , Marked_proper(X2)>
  <Marked_proper(U31(X)) , Marked_proper(X)> 
     --> <Marked_proper(U21(X1,X2,X3)) , Marked_proper(X1)>
  <Marked_proper(U31(X)) , Marked_proper(X)> 
     --> <Marked_proper(U11(X1,X2)) , Marked_proper(X2)>
  <Marked_proper(U31(X)) , Marked_proper(X)> 
     --> <Marked_proper(U11(X1,X2)) , Marked_proper(X1)>
  <Marked_proper(plus(X1,X2)) , Marked_proper(X2)> 
     --> <Marked_proper(isNat(X)) , Marked_proper(X)>
  <Marked_proper(plus(X1,X2)) , Marked_proper(X2)> 
     --> <Marked_proper(and(X1,X2)) , Marked_proper(X2)>
  <Marked_proper(plus(X1,X2)) , Marked_proper(X2)> 
     --> <Marked_proper(and(X1,X2)) , Marked_proper(X1)>
  <Marked_proper(plus(X1,X2)) , Marked_proper(X2)> 
     --> <Marked_proper(x(X1,X2)) , Marked_proper(X2)>
  <Marked_proper(plus(X1,X2)) , Marked_proper(X2)> 
     --> <Marked_proper(x(X1,X2)) , Marked_proper(X1)>
  <Marked_proper(plus(X1,X2)) , Marked_proper(X2)> 
     --> <Marked_proper(U41(X1,X2,X3)) , Marked_proper(X3)>
  <Marked_proper(plus(X1,X2)) , Marked_proper(X2)> 
     --> <Marked_proper(U41(X1,X2,X3)) , Marked_proper(X2)>
  <Marked_proper(plus(X1,X2)) , Marked_proper(X2)> 
     --> <Marked_proper(U41(X1,X2,X3)) , Marked_proper(X1)>
  <Marked_proper(plus(X1,X2)) , Marked_proper(X2)> 
     --> <Marked_proper(U31(X)) , Marked_proper(X)>
  <Marked_proper(plus(X1,X2)) , Marked_proper(X2)> 
     --> <Marked_proper(plus(X1,X2)) , Marked_proper(X2)>
  <Marked_proper(plus(X1,X2)) , Marked_proper(X2)> 
     --> <Marked_proper(plus(X1,X2)) , Marked_proper(X1)>
  <Marked_proper(plus(X1,X2)) , Marked_proper(X2)> 
     --> <Marked_proper(s(X)) , Marked_proper(X)>
  <Marked_proper(plus(X1,X2)) , Marked_proper(X2)> 
     --> <Marked_proper(U21(X1,X2,X3)) , Marked_proper(X3)>
  <Marked_proper(plus(X1,X2)) , Marked_proper(X2)> 
     --> <Marked_proper(U21(X1,X2,X3)) , Marked_proper(X2)>
  <Marked_proper(plus(X1,X2)) , Marked_proper(X2)> 
     --> <Marked_proper(U21(X1,X2,X3)) , Marked_proper(X1)>
  <Marked_proper(plus(X1,X2)) , Marked_proper(X2)> 
     --> <Marked_proper(U11(X1,X2)) , Marked_proper(X2)>
  <Marked_proper(plus(X1,X2)) , Marked_proper(X2)> 
     --> <Marked_proper(U11(X1,X2)) , Marked_proper(X1)>
  <Marked_proper(plus(X1,X2)) , Marked_proper(X1)> 
     --> <Marked_proper(isNat(X)) , Marked_proper(X)>
  <Marked_proper(plus(X1,X2)) , Marked_proper(X1)> 
     --> <Marked_proper(and(X1,X2)) , Marked_proper(X2)>
  <Marked_proper(plus(X1,X2)) , Marked_proper(X1)> 
     --> <Marked_proper(and(X1,X2)) , Marked_proper(X1)>
  <Marked_proper(plus(X1,X2)) , Marked_proper(X1)> 
     --> <Marked_proper(x(X1,X2)) , Marked_proper(X2)>
  <Marked_proper(plus(X1,X2)) , Marked_proper(X1)> 
     --> <Marked_proper(x(X1,X2)) , Marked_proper(X1)>
  <Marked_proper(plus(X1,X2)) , Marked_proper(X1)> 
     --> <Marked_proper(U41(X1,X2,X3)) , Marked_proper(X3)>
  <Marked_proper(plus(X1,X2)) , Marked_proper(X1)> 
     --> <Marked_proper(U41(X1,X2,X3)) , Marked_proper(X2)>
  <Marked_proper(plus(X1,X2)) , Marked_proper(X1)> 
     --> <Marked_proper(U41(X1,X2,X3)) , Marked_proper(X1)>
  <Marked_proper(plus(X1,X2)) , Marked_proper(X1)> 
     --> <Marked_proper(U31(X)) , Marked_proper(X)>
  <Marked_proper(plus(X1,X2)) , Marked_proper(X1)> 
     --> <Marked_proper(plus(X1,X2)) , Marked_proper(X2)>
  <Marked_proper(plus(X1,X2)) , Marked_proper(X1)> 
     --> <Marked_proper(plus(X1,X2)) , Marked_proper(X1)>
  <Marked_proper(plus(X1,X2)) , Marked_proper(X1)> 
     --> <Marked_proper(s(X)) , Marked_proper(X)>
  <Marked_proper(plus(X1,X2)) , Marked_proper(X1)> 
     --> <Marked_proper(U21(X1,X2,X3)) , Marked_proper(X3)>
  <Marked_proper(plus(X1,X2)) , Marked_proper(X1)> 
     --> <Marked_proper(U21(X1,X2,X3)) , Marked_proper(X2)>
  <Marked_proper(plus(X1,X2)) , Marked_proper(X1)> 
     --> <Marked_proper(U21(X1,X2,X3)) , Marked_proper(X1)>
  <Marked_proper(plus(X1,X2)) , Marked_proper(X1)> 
     --> <Marked_proper(U11(X1,X2)) , Marked_proper(X2)>
  <Marked_proper(plus(X1,X2)) , Marked_proper(X1)> 
     --> <Marked_proper(U11(X1,X2)) , Marked_proper(X1)>
  <Marked_proper(s(X)) , Marked_proper(X)> 
     --> <Marked_proper(isNat(X)) , Marked_proper(X)>
  <Marked_proper(s(X)) , Marked_proper(X)> 
     --> <Marked_proper(and(X1,X2)) , Marked_proper(X2)>
  <Marked_proper(s(X)) , Marked_proper(X)> 
     --> <Marked_proper(and(X1,X2)) , Marked_proper(X1)>
  <Marked_proper(s(X)) , Marked_proper(X)> 
     --> <Marked_proper(x(X1,X2)) , Marked_proper(X2)>
  <Marked_proper(s(X)) , Marked_proper(X)> 
     --> <Marked_proper(x(X1,X2)) , Marked_proper(X1)>
  <Marked_proper(s(X)) , Marked_proper(X)> 
     --> <Marked_proper(U41(X1,X2,X3)) , Marked_proper(X3)>
  <Marked_proper(s(X)) , Marked_proper(X)> 
     --> <Marked_proper(U41(X1,X2,X3)) , Marked_proper(X2)>
  <Marked_proper(s(X)) , Marked_proper(X)> 
     --> <Marked_proper(U41(X1,X2,X3)) , Marked_proper(X1)>
  <Marked_proper(s(X)) , Marked_proper(X)> 
     --> <Marked_proper(U31(X)) , Marked_proper(X)>
  <Marked_proper(s(X)) , Marked_proper(X)> 
     --> <Marked_proper(plus(X1,X2)) , Marked_proper(X2)>
  <Marked_proper(s(X)) , Marked_proper(X)> 
     --> <Marked_proper(plus(X1,X2)) , Marked_proper(X1)>
  <Marked_proper(s(X)) , Marked_proper(X)> 
     --> <Marked_proper(s(X)) , Marked_proper(X)>
  <Marked_proper(s(X)) , Marked_proper(X)> 
     --> <Marked_proper(U21(X1,X2,X3)) , Marked_proper(X3)>
  <Marked_proper(s(X)) , Marked_proper(X)> 
     --> <Marked_proper(U21(X1,X2,X3)) , Marked_proper(X2)>
  <Marked_proper(s(X)) , Marked_proper(X)> 
     --> <Marked_proper(U21(X1,X2,X3)) , Marked_proper(X1)>
  <Marked_proper(s(X)) , Marked_proper(X)> 
     --> <Marked_proper(U11(X1,X2)) , Marked_proper(X2)>
  <Marked_proper(s(X)) , Marked_proper(X)> 
     --> <Marked_proper(U11(X1,X2)) , Marked_proper(X1)>
  <Marked_proper(U21(X1,X2,X3)) , Marked_proper(X3)> 
     --> <Marked_proper(isNat(X)) , Marked_proper(X)>
  <Marked_proper(U21(X1,X2,X3)) , Marked_proper(X3)> 
     --> <Marked_proper(and(X1,X2)) , Marked_proper(X2)>
  <Marked_proper(U21(X1,X2,X3)) , Marked_proper(X3)> 
     --> <Marked_proper(and(X1,X2)) , Marked_proper(X1)>
  <Marked_proper(U21(X1,X2,X3)) , Marked_proper(X3)> 
     --> <Marked_proper(x(X1,X2)) , Marked_proper(X2)>
  <Marked_proper(U21(X1,X2,X3)) , Marked_proper(X3)> 
     --> <Marked_proper(x(X1,X2)) , Marked_proper(X1)>
  <Marked_proper(U21(X1,X2,X3)) , Marked_proper(X3)> 
     --> <Marked_proper(U41(X1,X2,X3)) , Marked_proper(X3)>
  <Marked_proper(U21(X1,X2,X3)) , Marked_proper(X3)> 
     --> <Marked_proper(U41(X1,X2,X3)) , Marked_proper(X2)>
  <Marked_proper(U21(X1,X2,X3)) , Marked_proper(X3)> 
     --> <Marked_proper(U41(X1,X2,X3)) , Marked_proper(X1)>
  <Marked_proper(U21(X1,X2,X3)) , Marked_proper(X3)> 
     --> <Marked_proper(U31(X)) , Marked_proper(X)>
  <Marked_proper(U21(X1,X2,X3)) , Marked_proper(X3)> 
     --> <Marked_proper(plus(X1,X2)) , Marked_proper(X2)>
  <Marked_proper(U21(X1,X2,X3)) , Marked_proper(X3)> 
     --> <Marked_proper(plus(X1,X2)) , Marked_proper(X1)>
  <Marked_proper(U21(X1,X2,X3)) , Marked_proper(X3)> 
     --> <Marked_proper(s(X)) , Marked_proper(X)>
  <Marked_proper(U21(X1,X2,X3)) , Marked_proper(X3)> 
     --> <Marked_proper(U21(X1,X2,X3)) , Marked_proper(X3)>
  <Marked_proper(U21(X1,X2,X3)) , Marked_proper(X3)> 
     --> <Marked_proper(U21(X1,X2,X3)) , Marked_proper(X2)>
  <Marked_proper(U21(X1,X2,X3)) , Marked_proper(X3)> 
     --> <Marked_proper(U21(X1,X2,X3)) , Marked_proper(X1)>
  <Marked_proper(U21(X1,X2,X3)) , Marked_proper(X3)> 
     --> <Marked_proper(U11(X1,X2)) , Marked_proper(X2)>
  <Marked_proper(U21(X1,X2,X3)) , Marked_proper(X3)> 
     --> <Marked_proper(U11(X1,X2)) , Marked_proper(X1)>
  <Marked_proper(U21(X1,X2,X3)) , Marked_proper(X2)> 
     --> <Marked_proper(isNat(X)) , Marked_proper(X)>
  <Marked_proper(U21(X1,X2,X3)) , Marked_proper(X2)> 
     --> <Marked_proper(and(X1,X2)) , Marked_proper(X2)>
  <Marked_proper(U21(X1,X2,X3)) , Marked_proper(X2)> 
     --> <Marked_proper(and(X1,X2)) , Marked_proper(X1)>
  <Marked_proper(U21(X1,X2,X3)) , Marked_proper(X2)> 
     --> <Marked_proper(x(X1,X2)) , Marked_proper(X2)>
  <Marked_proper(U21(X1,X2,X3)) , Marked_proper(X2)> 
     --> <Marked_proper(x(X1,X2)) , Marked_proper(X1)>
  <Marked_proper(U21(X1,X2,X3)) , Marked_proper(X2)> 
     --> <Marked_proper(U41(X1,X2,X3)) , Marked_proper(X3)>
  <Marked_proper(U21(X1,X2,X3)) , Marked_proper(X2)> 
     --> <Marked_proper(U41(X1,X2,X3)) , Marked_proper(X2)>
  <Marked_proper(U21(X1,X2,X3)) , Marked_proper(X2)> 
     --> <Marked_proper(U41(X1,X2,X3)) , Marked_proper(X1)>
  <Marked_proper(U21(X1,X2,X3)) , Marked_proper(X2)> 
     --> <Marked_proper(U31(X)) , Marked_proper(X)>
  <Marked_proper(U21(X1,X2,X3)) , Marked_proper(X2)> 
     --> <Marked_proper(plus(X1,X2)) , Marked_proper(X2)>
  <Marked_proper(U21(X1,X2,X3)) , Marked_proper(X2)> 
     --> <Marked_proper(plus(X1,X2)) , Marked_proper(X1)>
  <Marked_proper(U21(X1,X2,X3)) , Marked_proper(X2)> 
     --> <Marked_proper(s(X)) , Marked_proper(X)>
  <Marked_proper(U21(X1,X2,X3)) , Marked_proper(X2)> 
     --> <Marked_proper(U21(X1,X2,X3)) , Marked_proper(X3)>
  <Marked_proper(U21(X1,X2,X3)) , Marked_proper(X2)> 
     --> <Marked_proper(U21(X1,X2,X3)) , Marked_proper(X2)>
  <Marked_proper(U21(X1,X2,X3)) , Marked_proper(X2)> 
     --> <Marked_proper(U21(X1,X2,X3)) , Marked_proper(X1)>
  <Marked_proper(U21(X1,X2,X3)) , Marked_proper(X2)> 
     --> <Marked_proper(U11(X1,X2)) , Marked_proper(X2)>
  <Marked_proper(U21(X1,X2,X3)) , Marked_proper(X2)> 
     --> <Marked_proper(U11(X1,X2)) , Marked_proper(X1)>
  <Marked_proper(U21(X1,X2,X3)) , Marked_proper(X1)> 
     --> <Marked_proper(isNat(X)) , Marked_proper(X)>
  <Marked_proper(U21(X1,X2,X3)) , Marked_proper(X1)> 
     --> <Marked_proper(and(X1,X2)) , Marked_proper(X2)>
  <Marked_proper(U21(X1,X2,X3)) , Marked_proper(X1)> 
     --> <Marked_proper(and(X1,X2)) , Marked_proper(X1)>
  <Marked_proper(U21(X1,X2,X3)) , Marked_proper(X1)> 
     --> <Marked_proper(x(X1,X2)) , Marked_proper(X2)>
  <Marked_proper(U21(X1,X2,X3)) , Marked_proper(X1)> 
     --> <Marked_proper(x(X1,X2)) , Marked_proper(X1)>
  <Marked_proper(U21(X1,X2,X3)) , Marked_proper(X1)> 
     --> <Marked_proper(U41(X1,X2,X3)) , Marked_proper(X3)>
  <Marked_proper(U21(X1,X2,X3)) , Marked_proper(X1)> 
     --> <Marked_proper(U41(X1,X2,X3)) , Marked_proper(X2)>
  <Marked_proper(U21(X1,X2,X3)) , Marked_proper(X1)> 
     --> <Marked_proper(U41(X1,X2,X3)) , Marked_proper(X1)>
  <Marked_proper(U21(X1,X2,X3)) , Marked_proper(X1)> 
     --> <Marked_proper(U31(X)) , Marked_proper(X)>
  <Marked_proper(U21(X1,X2,X3)) , Marked_proper(X1)> 
     --> <Marked_proper(plus(X1,X2)) , Marked_proper(X2)>
  <Marked_proper(U21(X1,X2,X3)) , Marked_proper(X1)> 
     --> <Marked_proper(plus(X1,X2)) , Marked_proper(X1)>
  <Marked_proper(U21(X1,X2,X3)) , Marked_proper(X1)> 
     --> <Marked_proper(s(X)) , Marked_proper(X)>
  <Marked_proper(U21(X1,X2,X3)) , Marked_proper(X1)> 
     --> <Marked_proper(U21(X1,X2,X3)) , Marked_proper(X3)>
  <Marked_proper(U21(X1,X2,X3)) , Marked_proper(X1)> 
     --> <Marked_proper(U21(X1,X2,X3)) , Marked_proper(X2)>
  <Marked_proper(U21(X1,X2,X3)) , Marked_proper(X1)> 
     --> <Marked_proper(U21(X1,X2,X3)) , Marked_proper(X1)>
  <Marked_proper(U21(X1,X2,X3)) , Marked_proper(X1)> 
     --> <Marked_proper(U11(X1,X2)) , Marked_proper(X2)>
  <Marked_proper(U21(X1,X2,X3)) , Marked_proper(X1)> 
     --> <Marked_proper(U11(X1,X2)) , Marked_proper(X1)>
  <Marked_proper(U11(X1,X2)) , Marked_proper(X2)> 
     --> <Marked_proper(isNat(X)) , Marked_proper(X)>
  <Marked_proper(U11(X1,X2)) , Marked_proper(X2)> 
     --> <Marked_proper(and(X1,X2)) , Marked_proper(X2)>
  <Marked_proper(U11(X1,X2)) , Marked_proper(X2)> 
     --> <Marked_proper(and(X1,X2)) , Marked_proper(X1)>
  <Marked_proper(U11(X1,X2)) , Marked_proper(X2)> 
     --> <Marked_proper(x(X1,X2)) , Marked_proper(X2)>
  <Marked_proper(U11(X1,X2)) , Marked_proper(X2)> 
     --> <Marked_proper(x(X1,X2)) , Marked_proper(X1)>
  <Marked_proper(U11(X1,X2)) , Marked_proper(X2)> 
     --> <Marked_proper(U41(X1,X2,X3)) , Marked_proper(X3)>
  <Marked_proper(U11(X1,X2)) , Marked_proper(X2)> 
     --> <Marked_proper(U41(X1,X2,X3)) , Marked_proper(X2)>
  <Marked_proper(U11(X1,X2)) , Marked_proper(X2)> 
     --> <Marked_proper(U41(X1,X2,X3)) , Marked_proper(X1)>
  <Marked_proper(U11(X1,X2)) , Marked_proper(X2)> 
     --> <Marked_proper(U31(X)) , Marked_proper(X)>
  <Marked_proper(U11(X1,X2)) , Marked_proper(X2)> 
     --> <Marked_proper(plus(X1,X2)) , Marked_proper(X2)>
  <Marked_proper(U11(X1,X2)) , Marked_proper(X2)> 
     --> <Marked_proper(plus(X1,X2)) , Marked_proper(X1)>
  <Marked_proper(U11(X1,X2)) , Marked_proper(X2)> 
     --> <Marked_proper(s(X)) , Marked_proper(X)>
  <Marked_proper(U11(X1,X2)) , Marked_proper(X2)> 
     --> <Marked_proper(U21(X1,X2,X3)) , Marked_proper(X3)>
  <Marked_proper(U11(X1,X2)) , Marked_proper(X2)> 
     --> <Marked_proper(U21(X1,X2,X3)) , Marked_proper(X2)>
  <Marked_proper(U11(X1,X2)) , Marked_proper(X2)> 
     --> <Marked_proper(U21(X1,X2,X3)) , Marked_proper(X1)>
  <Marked_proper(U11(X1,X2)) , Marked_proper(X2)> 
     --> <Marked_proper(U11(X1,X2)) , Marked_proper(X2)>
  <Marked_proper(U11(X1,X2)) , Marked_proper(X2)> 
     --> <Marked_proper(U11(X1,X2)) , Marked_proper(X1)>
  <Marked_proper(U11(X1,X2)) , Marked_proper(X1)> 
     --> <Marked_proper(isNat(X)) , Marked_proper(X)>
  <Marked_proper(U11(X1,X2)) , Marked_proper(X1)> 
     --> <Marked_proper(and(X1,X2)) , Marked_proper(X2)>
  <Marked_proper(U11(X1,X2)) , Marked_proper(X1)> 
     --> <Marked_proper(and(X1,X2)) , Marked_proper(X1)>
  <Marked_proper(U11(X1,X2)) , Marked_proper(X1)> 
     --> <Marked_proper(x(X1,X2)) , Marked_proper(X2)>
  <Marked_proper(U11(X1,X2)) , Marked_proper(X1)> 
     --> <Marked_proper(x(X1,X2)) , Marked_proper(X1)>
  <Marked_proper(U11(X1,X2)) , Marked_proper(X1)> 
     --> <Marked_proper(U41(X1,X2,X3)) , Marked_proper(X3)>
  <Marked_proper(U11(X1,X2)) , Marked_proper(X1)> 
     --> <Marked_proper(U41(X1,X2,X3)) , Marked_proper(X2)>
  <Marked_proper(U11(X1,X2)) , Marked_proper(X1)> 
     --> <Marked_proper(U41(X1,X2,X3)) , Marked_proper(X1)>
  <Marked_proper(U11(X1,X2)) , Marked_proper(X1)> 
     --> <Marked_proper(U31(X)) , Marked_proper(X)>
  <Marked_proper(U11(X1,X2)) , Marked_proper(X1)> 
     --> <Marked_proper(plus(X1,X2)) , Marked_proper(X2)>
  <Marked_proper(U11(X1,X2)) , Marked_proper(X1)> 
     --> <Marked_proper(plus(X1,X2)) , Marked_proper(X1)>
  <Marked_proper(U11(X1,X2)) , Marked_proper(X1)> 
     --> <Marked_proper(s(X)) , Marked_proper(X)>
  <Marked_proper(U11(X1,X2)) , Marked_proper(X1)> 
     --> <Marked_proper(U21(X1,X2,X3)) , Marked_proper(X3)>
  <Marked_proper(U11(X1,X2)) , Marked_proper(X1)> 
     --> <Marked_proper(U21(X1,X2,X3)) , Marked_proper(X2)>
  <Marked_proper(U11(X1,X2)) , Marked_proper(X1)> 
     --> <Marked_proper(U21(X1,X2,X3)) , Marked_proper(X1)>
  <Marked_proper(U11(X1,X2)) , Marked_proper(X1)> 
     --> <Marked_proper(U11(X1,X2)) , Marked_proper(X2)>
  <Marked_proper(U11(X1,X2)) , Marked_proper(X1)> 
     --> <Marked_proper(U11(X1,X2)) , Marked_proper(X1)>
}

{  <Marked_active(and(X1,X2)) , Marked_active(X1)> 
      --> <Marked_active(and(X1,X2)) , Marked_active(X1)>
  <Marked_active(and(X1,X2)) , Marked_active(X1)> 
     --> <Marked_active(x(X1,X2)) , Marked_active(X2)>
  <Marked_active(and(X1,X2)) , Marked_active(X1)> 
     --> <Marked_active(x(X1,X2)) , Marked_active(X1)>
  <Marked_active(and(X1,X2)) , Marked_active(X1)> 
     --> <Marked_active(U41(X1,X2,X3)) , Marked_active(X1)>
  <Marked_active(and(X1,X2)) , Marked_active(X1)> 
     --> <Marked_active(U31(X)) , Marked_active(X)>
  <Marked_active(and(X1,X2)) , Marked_active(X1)> 
     --> <Marked_active(plus(X1,X2)) , Marked_active(X2)>
  <Marked_active(and(X1,X2)) , Marked_active(X1)> 
     --> <Marked_active(plus(X1,X2)) , Marked_active(X1)>
  <Marked_active(and(X1,X2)) , Marked_active(X1)> 
     --> <Marked_active(s(X)) , Marked_active(X)>
  <Marked_active(and(X1,X2)) , Marked_active(X1)> 
     --> <Marked_active(U21(X1,X2,X3)) , Marked_active(X1)>
  <Marked_active(and(X1,X2)) , Marked_active(X1)> 
     --> <Marked_active(U11(X1,X2)) , Marked_active(X1)>
  <Marked_active(x(X1,X2)) , Marked_active(X2)> 
     --> <Marked_active(and(X1,X2)) , Marked_active(X1)>
  <Marked_active(x(X1,X2)) , Marked_active(X2)> 
     --> <Marked_active(x(X1,X2)) , Marked_active(X2)>
  <Marked_active(x(X1,X2)) , Marked_active(X2)> 
     --> <Marked_active(x(X1,X2)) , Marked_active(X1)>
  <Marked_active(x(X1,X2)) , Marked_active(X2)> 
     --> <Marked_active(U41(X1,X2,X3)) , Marked_active(X1)>
  <Marked_active(x(X1,X2)) , Marked_active(X2)> 
     --> <Marked_active(U31(X)) , Marked_active(X)>
  <Marked_active(x(X1,X2)) , Marked_active(X2)> 
     --> <Marked_active(plus(X1,X2)) , Marked_active(X2)>
  <Marked_active(x(X1,X2)) , Marked_active(X2)> 
     --> <Marked_active(plus(X1,X2)) , Marked_active(X1)>
  <Marked_active(x(X1,X2)) , Marked_active(X2)> 
     --> <Marked_active(s(X)) , Marked_active(X)>
  <Marked_active(x(X1,X2)) , Marked_active(X2)> 
     --> <Marked_active(U21(X1,X2,X3)) , Marked_active(X1)>
  <Marked_active(x(X1,X2)) , Marked_active(X2)> 
     --> <Marked_active(U11(X1,X2)) , Marked_active(X1)>
  <Marked_active(x(X1,X2)) , Marked_active(X1)> 
     --> <Marked_active(and(X1,X2)) , Marked_active(X1)>
  <Marked_active(x(X1,X2)) , Marked_active(X1)> 
     --> <Marked_active(x(X1,X2)) , Marked_active(X2)>
  <Marked_active(x(X1,X2)) , Marked_active(X1)> 
     --> <Marked_active(x(X1,X2)) , Marked_active(X1)>
  <Marked_active(x(X1,X2)) , Marked_active(X1)> 
     --> <Marked_active(U41(X1,X2,X3)) , Marked_active(X1)>
  <Marked_active(x(X1,X2)) , Marked_active(X1)> 
     --> <Marked_active(U31(X)) , Marked_active(X)>
  <Marked_active(x(X1,X2)) , Marked_active(X1)> 
     --> <Marked_active(plus(X1,X2)) , Marked_active(X2)>
  <Marked_active(x(X1,X2)) , Marked_active(X1)> 
     --> <Marked_active(plus(X1,X2)) , Marked_active(X1)>
  <Marked_active(x(X1,X2)) , Marked_active(X1)> 
     --> <Marked_active(s(X)) , Marked_active(X)>
  <Marked_active(x(X1,X2)) , Marked_active(X1)> 
     --> <Marked_active(U21(X1,X2,X3)) , Marked_active(X1)>
  <Marked_active(x(X1,X2)) , Marked_active(X1)> 
     --> <Marked_active(U11(X1,X2)) , Marked_active(X1)>
  <Marked_active(U41(X1,X2,X3)) , Marked_active(X1)> 
     --> <Marked_active(and(X1,X2)) , Marked_active(X1)>
  <Marked_active(U41(X1,X2,X3)) , Marked_active(X1)> 
     --> <Marked_active(x(X1,X2)) , Marked_active(X2)>
  <Marked_active(U41(X1,X2,X3)) , Marked_active(X1)> 
     --> <Marked_active(x(X1,X2)) , Marked_active(X1)>
  <Marked_active(U41(X1,X2,X3)) , Marked_active(X1)> 
     --> <Marked_active(U41(X1,X2,X3)) , Marked_active(X1)>
  <Marked_active(U41(X1,X2,X3)) , Marked_active(X1)> 
     --> <Marked_active(U31(X)) , Marked_active(X)>
  <Marked_active(U41(X1,X2,X3)) , Marked_active(X1)> 
     --> <Marked_active(plus(X1,X2)) , Marked_active(X2)>
  <Marked_active(U41(X1,X2,X3)) , Marked_active(X1)> 
     --> <Marked_active(plus(X1,X2)) , Marked_active(X1)>
  <Marked_active(U41(X1,X2,X3)) , Marked_active(X1)> 
     --> <Marked_active(s(X)) , Marked_active(X)>
  <Marked_active(U41(X1,X2,X3)) , Marked_active(X1)> 
     --> <Marked_active(U21(X1,X2,X3)) , Marked_active(X1)>
  <Marked_active(U41(X1,X2,X3)) , Marked_active(X1)> 
     --> <Marked_active(U11(X1,X2)) , Marked_active(X1)>
  <Marked_active(U31(X)) , Marked_active(X)> 
     --> <Marked_active(and(X1,X2)) , Marked_active(X1)>
  <Marked_active(U31(X)) , Marked_active(X)> 
     --> <Marked_active(x(X1,X2)) , Marked_active(X2)>
  <Marked_active(U31(X)) , Marked_active(X)> 
     --> <Marked_active(x(X1,X2)) , Marked_active(X1)>
  <Marked_active(U31(X)) , Marked_active(X)> 
     --> <Marked_active(U41(X1,X2,X3)) , Marked_active(X1)>
  <Marked_active(U31(X)) , Marked_active(X)> 
     --> <Marked_active(U31(X)) , Marked_active(X)>
  <Marked_active(U31(X)) , Marked_active(X)> 
     --> <Marked_active(plus(X1,X2)) , Marked_active(X2)>
  <Marked_active(U31(X)) , Marked_active(X)> 
     --> <Marked_active(plus(X1,X2)) , Marked_active(X1)>
  <Marked_active(U31(X)) , Marked_active(X)> 
     --> <Marked_active(s(X)) , Marked_active(X)>
  <Marked_active(U31(X)) , Marked_active(X)> 
     --> <Marked_active(U21(X1,X2,X3)) , Marked_active(X1)>
  <Marked_active(U31(X)) , Marked_active(X)> 
     --> <Marked_active(U11(X1,X2)) , Marked_active(X1)>
  <Marked_active(plus(X1,X2)) , Marked_active(X2)> 
     --> <Marked_active(and(X1,X2)) , Marked_active(X1)>
  <Marked_active(plus(X1,X2)) , Marked_active(X2)> 
     --> <Marked_active(x(X1,X2)) , Marked_active(X2)>
  <Marked_active(plus(X1,X2)) , Marked_active(X2)> 
     --> <Marked_active(x(X1,X2)) , Marked_active(X1)>
  <Marked_active(plus(X1,X2)) , Marked_active(X2)> 
     --> <Marked_active(U41(X1,X2,X3)) , Marked_active(X1)>
  <Marked_active(plus(X1,X2)) , Marked_active(X2)> 
     --> <Marked_active(U31(X)) , Marked_active(X)>
  <Marked_active(plus(X1,X2)) , Marked_active(X2)> 
     --> <Marked_active(plus(X1,X2)) , Marked_active(X2)>
  <Marked_active(plus(X1,X2)) , Marked_active(X2)> 
     --> <Marked_active(plus(X1,X2)) , Marked_active(X1)>
  <Marked_active(plus(X1,X2)) , Marked_active(X2)> 
     --> <Marked_active(s(X)) , Marked_active(X)>
  <Marked_active(plus(X1,X2)) , Marked_active(X2)> 
     --> <Marked_active(U21(X1,X2,X3)) , Marked_active(X1)>
  <Marked_active(plus(X1,X2)) , Marked_active(X2)> 
     --> <Marked_active(U11(X1,X2)) , Marked_active(X1)>
  <Marked_active(plus(X1,X2)) , Marked_active(X1)> 
     --> <Marked_active(and(X1,X2)) , Marked_active(X1)>
  <Marked_active(plus(X1,X2)) , Marked_active(X1)> 
     --> <Marked_active(x(X1,X2)) , Marked_active(X2)>
  <Marked_active(plus(X1,X2)) , Marked_active(X1)> 
     --> <Marked_active(x(X1,X2)) , Marked_active(X1)>
  <Marked_active(plus(X1,X2)) , Marked_active(X1)> 
     --> <Marked_active(U41(X1,X2,X3)) , Marked_active(X1)>
  <Marked_active(plus(X1,X2)) , Marked_active(X1)> 
     --> <Marked_active(U31(X)) , Marked_active(X)>
  <Marked_active(plus(X1,X2)) , Marked_active(X1)> 
     --> <Marked_active(plus(X1,X2)) , Marked_active(X2)>
  <Marked_active(plus(X1,X2)) , Marked_active(X1)> 
     --> <Marked_active(plus(X1,X2)) , Marked_active(X1)>
  <Marked_active(plus(X1,X2)) , Marked_active(X1)> 
     --> <Marked_active(s(X)) , Marked_active(X)>
  <Marked_active(plus(X1,X2)) , Marked_active(X1)> 
     --> <Marked_active(U21(X1,X2,X3)) , Marked_active(X1)>
  <Marked_active(plus(X1,X2)) , Marked_active(X1)> 
     --> <Marked_active(U11(X1,X2)) , Marked_active(X1)>
  <Marked_active(s(X)) , Marked_active(X)> 
     --> <Marked_active(and(X1,X2)) , Marked_active(X1)>
  <Marked_active(s(X)) , Marked_active(X)> 
     --> <Marked_active(x(X1,X2)) , Marked_active(X2)>
  <Marked_active(s(X)) , Marked_active(X)> 
     --> <Marked_active(x(X1,X2)) , Marked_active(X1)>
  <Marked_active(s(X)) , Marked_active(X)> 
     --> <Marked_active(U41(X1,X2,X3)) , Marked_active(X1)>
  <Marked_active(s(X)) , Marked_active(X)> 
     --> <Marked_active(U31(X)) , Marked_active(X)>
  <Marked_active(s(X)) , Marked_active(X)> 
     --> <Marked_active(plus(X1,X2)) , Marked_active(X2)>
  <Marked_active(s(X)) , Marked_active(X)> 
     --> <Marked_active(plus(X1,X2)) , Marked_active(X1)>
  <Marked_active(s(X)) , Marked_active(X)> 
     --> <Marked_active(s(X)) , Marked_active(X)>
  <Marked_active(s(X)) , Marked_active(X)> 
     --> <Marked_active(U21(X1,X2,X3)) , Marked_active(X1)>
  <Marked_active(s(X)) , Marked_active(X)> 
     --> <Marked_active(U11(X1,X2)) , Marked_active(X1)>
  <Marked_active(U21(X1,X2,X3)) , Marked_active(X1)> 
     --> <Marked_active(and(X1,X2)) , Marked_active(X1)>
  <Marked_active(U21(X1,X2,X3)) , Marked_active(X1)> 
     --> <Marked_active(x(X1,X2)) , Marked_active(X2)>
  <Marked_active(U21(X1,X2,X3)) , Marked_active(X1)> 
     --> <Marked_active(x(X1,X2)) , Marked_active(X1)>
  <Marked_active(U21(X1,X2,X3)) , Marked_active(X1)> 
     --> <Marked_active(U41(X1,X2,X3)) , Marked_active(X1)>
  <Marked_active(U21(X1,X2,X3)) , Marked_active(X1)> 
     --> <Marked_active(U31(X)) , Marked_active(X)>
  <Marked_active(U21(X1,X2,X3)) , Marked_active(X1)> 
     --> <Marked_active(plus(X1,X2)) , Marked_active(X2)>
  <Marked_active(U21(X1,X2,X3)) , Marked_active(X1)> 
     --> <Marked_active(plus(X1,X2)) , Marked_active(X1)>
  <Marked_active(U21(X1,X2,X3)) , Marked_active(X1)> 
     --> <Marked_active(s(X)) , Marked_active(X)>
  <Marked_active(U21(X1,X2,X3)) , Marked_active(X1)> 
     --> <Marked_active(U21(X1,X2,X3)) , Marked_active(X1)>
  <Marked_active(U21(X1,X2,X3)) , Marked_active(X1)> 
     --> <Marked_active(U11(X1,X2)) , Marked_active(X1)>
  <Marked_active(U11(X1,X2)) , Marked_active(X1)> 
     --> <Marked_active(and(X1,X2)) , Marked_active(X1)>
  <Marked_active(U11(X1,X2)) , Marked_active(X1)> 
     --> <Marked_active(x(X1,X2)) , Marked_active(X2)>
  <Marked_active(U11(X1,X2)) , Marked_active(X1)> 
     --> <Marked_active(x(X1,X2)) , Marked_active(X1)>
  <Marked_active(U11(X1,X2)) , Marked_active(X1)> 
     --> <Marked_active(U41(X1,X2,X3)) , Marked_active(X1)>
  <Marked_active(U11(X1,X2)) , Marked_active(X1)> 
     --> <Marked_active(U31(X)) , Marked_active(X)>
  <Marked_active(U11(X1,X2)) , Marked_active(X1)> 
     --> <Marked_active(plus(X1,X2)) , Marked_active(X2)>
  <Marked_active(U11(X1,X2)) , Marked_active(X1)> 
     --> <Marked_active(plus(X1,X2)) , Marked_active(X1)>
  <Marked_active(U11(X1,X2)) , Marked_active(X1)> 
     --> <Marked_active(s(X)) , Marked_active(X)>
  <Marked_active(U11(X1,X2)) , Marked_active(X1)> 
     --> <Marked_active(U21(X1,X2,X3)) , Marked_active(X1)>
  <Marked_active(U11(X1,X2)) , Marked_active(X1)> 
     --> <Marked_active(U11(X1,X2)) , Marked_active(X1)>
}

{  <Marked_isNat(ok(X)) , Marked_isNat(X)> 
      --> <Marked_isNat(ok(X)) , Marked_isNat(X)>
}

{  <Marked_U11(ok(X1),ok(X2)) , Marked_U11(X1,X2)> 
      --> <Marked_U11(ok(X1),ok(X2)) , Marked_U11(X1,X2)>
  <Marked_U11(ok(X1),ok(X2)) , Marked_U11(X1,X2)> 
     --> <Marked_U11(mark(X1),X2) , Marked_U11(X1,X2)>
  <Marked_U11(mark(X1),X2) , Marked_U11(X1,X2)> 
     --> <Marked_U11(ok(X1),ok(X2)) , Marked_U11(X1,X2)>
  <Marked_U11(mark(X1),X2) , Marked_U11(X1,X2)> 
     --> <Marked_U11(mark(X1),X2) , Marked_U11(X1,X2)>
}

{  <Marked_U21(ok(X1),ok(X2),ok(X3)) , Marked_U21(X1,X2,X3)> 
      --> <Marked_U21(ok(X1),ok(X2),ok(X3)) , Marked_U21(X1,X2,X3)>
  <Marked_U21(ok(X1),ok(X2),ok(X3)) , Marked_U21(X1,X2,X3)> 
     --> <Marked_U21(mark(X1),X2,X3) , Marked_U21(X1,X2,X3)>
  <Marked_U21(mark(X1),X2,X3) , Marked_U21(X1,X2,X3)> 
     --> <Marked_U21(ok(X1),ok(X2),ok(X3)) , Marked_U21(X1,X2,X3)>
  <Marked_U21(mark(X1),X2,X3) , Marked_U21(X1,X2,X3)> 
     --> <Marked_U21(mark(X1),X2,X3) , Marked_U21(X1,X2,X3)>
}

{  <Marked_s(ok(X)) , Marked_s(X)> --> <Marked_s(ok(X)) , Marked_s(X)>
  <Marked_s(ok(X)) , Marked_s(X)> --> <Marked_s(mark(X)) , Marked_s(X)>
  <Marked_s(mark(X)) , Marked_s(X)> --> <Marked_s(ok(X)) , Marked_s(X)>
  <Marked_s(mark(X)) , Marked_s(X)> --> <Marked_s(mark(X)) , Marked_s(X)>
}

{  <Marked_plus(ok(X1),ok(X2)) , Marked_plus(X1,X2)> 
      --> <Marked_plus(ok(X1),ok(X2)) , Marked_plus(X1,X2)>
  <Marked_plus(ok(X1),ok(X2)) , Marked_plus(X1,X2)> 
     --> <Marked_plus(X1,mark(X2)) , Marked_plus(X1,X2)>
  <Marked_plus(ok(X1),ok(X2)) , Marked_plus(X1,X2)> 
     --> <Marked_plus(mark(X1),X2) , Marked_plus(X1,X2)>
  <Marked_plus(X1,mark(X2)) , Marked_plus(X1,X2)> 
     --> <Marked_plus(ok(X1),ok(X2)) , Marked_plus(X1,X2)>
  <Marked_plus(X1,mark(X2)) , Marked_plus(X1,X2)> 
     --> <Marked_plus(X1,mark(X2)) , Marked_plus(X1,X2)>
  <Marked_plus(X1,mark(X2)) , Marked_plus(X1,X2)> 
     --> <Marked_plus(mark(X1),X2) , Marked_plus(X1,X2)>
  <Marked_plus(mark(X1),X2) , Marked_plus(X1,X2)> 
     --> <Marked_plus(ok(X1),ok(X2)) , Marked_plus(X1,X2)>
  <Marked_plus(mark(X1),X2) , Marked_plus(X1,X2)> 
     --> <Marked_plus(X1,mark(X2)) , Marked_plus(X1,X2)>
  <Marked_plus(mark(X1),X2) , Marked_plus(X1,X2)> 
     --> <Marked_plus(mark(X1),X2) , Marked_plus(X1,X2)>
}

{  <Marked_U31(ok(X)) , Marked_U31(X)> --> <Marked_U31(ok(X)) , Marked_U31(X)>
  <Marked_U31(ok(X)) , Marked_U31(X)> --> <Marked_U31(mark(X)) , Marked_U31(X)>
  <Marked_U31(mark(X)) , Marked_U31(X)> --> <Marked_U31(ok(X)) , Marked_U31(X)>
  <Marked_U31(mark(X)) , Marked_U31(X)> 
     --> <Marked_U31(mark(X)) , Marked_U31(X)>
}

{  <Marked_U41(ok(X1),ok(X2),ok(X3)) , Marked_U41(X1,X2,X3)> 
      --> <Marked_U41(ok(X1),ok(X2),ok(X3)) , Marked_U41(X1,X2,X3)>
  <Marked_U41(ok(X1),ok(X2),ok(X3)) , Marked_U41(X1,X2,X3)> 
     --> <Marked_U41(mark(X1),X2,X3) , Marked_U41(X1,X2,X3)>
  <Marked_U41(mark(X1),X2,X3) , Marked_U41(X1,X2,X3)> 
     --> <Marked_U41(ok(X1),ok(X2),ok(X3)) , Marked_U41(X1,X2,X3)>
  <Marked_U41(mark(X1),X2,X3) , Marked_U41(X1,X2,X3)> 
     --> <Marked_U41(mark(X1),X2,X3) , Marked_U41(X1,X2,X3)>
}

{  <Marked_x(ok(X1),ok(X2)) , Marked_x(X1,X2)> 
      --> <Marked_x(ok(X1),ok(X2)) , Marked_x(X1,X2)>
  <Marked_x(ok(X1),ok(X2)) , Marked_x(X1,X2)> 
     --> <Marked_x(X1,mark(X2)) , Marked_x(X1,X2)>
  <Marked_x(ok(X1),ok(X2)) , Marked_x(X1,X2)> 
     --> <Marked_x(mark(X1),X2) , Marked_x(X1,X2)>
  <Marked_x(X1,mark(X2)) , Marked_x(X1,X2)> 
     --> <Marked_x(ok(X1),ok(X2)) , Marked_x(X1,X2)>
  <Marked_x(X1,mark(X2)) , Marked_x(X1,X2)> 
     --> <Marked_x(X1,mark(X2)) , Marked_x(X1,X2)>
  <Marked_x(X1,mark(X2)) , Marked_x(X1,X2)> 
     --> <Marked_x(mark(X1),X2) , Marked_x(X1,X2)>
  <Marked_x(mark(X1),X2) , Marked_x(X1,X2)> 
     --> <Marked_x(ok(X1),ok(X2)) , Marked_x(X1,X2)>
  <Marked_x(mark(X1),X2) , Marked_x(X1,X2)> 
     --> <Marked_x(X1,mark(X2)) , Marked_x(X1,X2)>
  <Marked_x(mark(X1),X2) , Marked_x(X1,X2)> 
     --> <Marked_x(mark(X1),X2) , Marked_x(X1,X2)>
}

{  <Marked_and(ok(X1),ok(X2)) , Marked_and(X1,X2)> 
      --> <Marked_and(ok(X1),ok(X2)) , Marked_and(X1,X2)>
  <Marked_and(ok(X1),ok(X2)) , Marked_and(X1,X2)> 
     --> <Marked_and(mark(X1),X2) , Marked_and(X1,X2)>
  <Marked_and(mark(X1),X2) , Marked_and(X1,X2)> 
     --> <Marked_and(ok(X1),ok(X2)) , Marked_and(X1,X2)>
  <Marked_and(mark(X1),X2) , Marked_and(X1,X2)> 
     --> <Marked_and(mark(X1),X2) , Marked_and(X1,X2)>
}
APPLY CRITERIA (Subterm criterion)
APPLY CRITERIA (Choosing graph)
Trying to solve the following constraints:
{
  active(U11(tt,N)) >= mark(N) ;
  active(U11(X1,X2)) >= U11(active(X1),X2) ;
  active(s(X)) >= s(active(X)) ;
  active(plus(N,s(M))) >= mark(U21(and(isNat(M),isNat(N)),M,N)) ;
  active(plus(N,0)) >= mark(U11(isNat(N),N)) ;
  active(plus(X1,X2)) >= plus(active(X1),X2) ;
  active(plus(X1,X2)) >= plus(X1,active(X2)) ;
  active(U21(tt,M,N)) >= mark(s(plus(N,M))) ;
  active(U21(X1,X2,X3)) >= U21(active(X1),X2,X3) ;
  active(U31(tt)) >= mark(0) ;
  active(U31(X)) >= U31(active(X)) ;
  active(x(N,s(M))) >= mark(U41(and(isNat(M),isNat(N)),M,N)) ;
  active(x(N,0)) >= mark(U31(isNat(N))) ;
  active(x(X1,X2)) >= x(active(X1),X2) ;
  active(x(X1,X2)) >= x(X1,active(X2)) ;
  active(U41(tt,M,N)) >= mark(plus(x(N,M),N)) ;
  active(U41(X1,X2,X3)) >= U41(active(X1),X2,X3) ;
  active(and(tt,X)) >= mark(X) ;
  active(and(X1,X2)) >= and(active(X1),X2) ;
  active(isNat(s(V1))) >= mark(isNat(V1)) ;
  active(isNat(plus(V1,V2))) >= mark(and(isNat(V1),isNat(V2))) ;
  active(isNat(0)) >= mark(tt) ;
  active(isNat(x(V1,V2))) >= mark(and(isNat(V1),isNat(V2))) ;
  U11(mark(X1),X2) >= mark(U11(X1,X2)) ;
  U11(ok(X1),ok(X2)) >= ok(U11(X1,X2)) ;
  s(mark(X)) >= mark(s(X)) ;
  s(ok(X)) >= ok(s(X)) ;
  plus(mark(X1),X2) >= mark(plus(X1,X2)) ;
  plus(ok(X1),ok(X2)) >= ok(plus(X1,X2)) ;
  plus(X1,mark(X2)) >= mark(plus(X1,X2)) ;
  U21(mark(X1),X2,X3) >= mark(U21(X1,X2,X3)) ;
  U21(ok(X1),ok(X2),ok(X3)) >= ok(U21(X1,X2,X3)) ;
  U31(mark(X)) >= mark(U31(X)) ;
  U31(ok(X)) >= ok(U31(X)) ;
  x(mark(X1),X2) >= mark(x(X1,X2)) ;
  x(ok(X1),ok(X2)) >= ok(x(X1,X2)) ;
  x(X1,mark(X2)) >= mark(x(X1,X2)) ;
  U41(mark(X1),X2,X3) >= mark(U41(X1,X2,X3)) ;
  U41(ok(X1),ok(X2),ok(X3)) >= ok(U41(X1,X2,X3)) ;
  and(mark(X1),X2) >= mark(and(X1,X2)) ;
  and(ok(X1),ok(X2)) >= ok(and(X1,X2)) ;
  isNat(ok(X)) >= ok(isNat(X)) ;
  proper(U11(X1,X2)) >= U11(proper(X1),proper(X2)) ;
  proper(tt) >= ok(tt) ;
  proper(s(X)) >= s(proper(X)) ;
  proper(plus(X1,X2)) >= plus(proper(X1),proper(X2)) ;
  proper(U21(X1,X2,X3)) >= U21(proper(X1),proper(X2),proper(X3)) ;
  proper(0) >= ok(0) ;
  proper(U31(X)) >= U31(proper(X)) ;
  proper(x(X1,X2)) >= x(proper(X1),proper(X2)) ;
  proper(U41(X1,X2,X3)) >= U41(proper(X1),proper(X2),proper(X3)) ;
  proper(and(X1,X2)) >= and(proper(X1),proper(X2)) ;
  proper(isNat(X)) >= isNat(proper(X)) ;
  top(mark(X)) >= top(proper(X)) ;
  top(ok(X)) >= top(active(X)) ;
  Marked_top(mark(X)) >= Marked_top(proper(X)) ;
  Marked_top(ok(X)) >= Marked_top(active(X)) ;
 }
 + Disjunctions:{ 
{
  Marked_top(mark(X)) > Marked_top(proper(X)) ;
 }
{
  Marked_top(ok(X)) > Marked_top(active(X)) ;
 }
 } 
=== TIMER virtual : 10.000000 ===
Entering poly_solver
Starting Sat solver initialization
Calling Sat solver...
=== STOPING TIMER virtual ===
=== TIMER real : 10.000000 ===
=== STOPING TIMER real ===
Sat solver returned
=== STOPING TIMER real ===
=== STOPING TIMER virtual ===
No solution found for these parameters.
Entering rpo_solver
=== TIMER virtual : 25.000000 ===
Search parameters: AFS type: 2 ; time limit: 25.. 
=== STOPING TIMER virtual ===
=== TIMER virtual : 25.000000 ===
Search parameters: AFS type: 2 ; time limit: 25.. 
=== STOPING TIMER virtual ===
=== TIMER virtual : 15.000000 ===
Entering poly_solver
Starting Sat solver initialization
Calling Sat solver...
=== STOPING TIMER virtual ===
=== TIMER real : 15.000000 ===
=== STOPING TIMER real ===
Sat solver returned
=== STOPING TIMER real ===
=== STOPING TIMER virtual ===
No solution found for these parameters.
=== TIMER virtual : 50.000000 ===
trying sub matrices of size: 1
Matrix interpretation constraints generated.
Search parameters: LINEAR MATRIX 3x3 (strict=1x1) ; time limit: 50.. 
Termination constraints generated.
Starting Sat solver initialization
Calling Sat solver...
=== STOPING TIMER virtual ===
=== TIMER real : 50.000000 ===
=== STOPING TIMER real ===
Sat timeout reached
=== STOPING TIMER virtual ===
No solution found for these parameters.
No solution found for these constraints.
APPLY CRITERIA (Simple graph)
Found the following constraints:
{
  active(U11(tt,N)) >= mark(N) ;
  active(U11(X1,X2)) >= U11(active(X1),X2) ;
  active(s(X)) >= s(active(X)) ;
  active(plus(N,s(M))) >= mark(U21(and(isNat(M),isNat(N)),M,N)) ;
  active(plus(N,0)) >= mark(U11(isNat(N),N)) ;
  active(plus(X1,X2)) >= plus(active(X1),X2) ;
  active(plus(X1,X2)) >= plus(X1,active(X2)) ;
  active(U21(tt,M,N)) >= mark(s(plus(N,M))) ;
  active(U21(X1,X2,X3)) >= U21(active(X1),X2,X3) ;
  active(U31(tt)) >= mark(0) ;
  active(U31(X)) >= U31(active(X)) ;
  active(x(N,s(M))) >= mark(U41(and(isNat(M),isNat(N)),M,N)) ;
  active(x(N,0)) >= mark(U31(isNat(N))) ;
  active(x(X1,X2)) >= x(active(X1),X2) ;
  active(x(X1,X2)) >= x(X1,active(X2)) ;
  active(U41(tt,M,N)) >= mark(plus(x(N,M),N)) ;
  active(U41(X1,X2,X3)) >= U41(active(X1),X2,X3) ;
  active(and(tt,X)) >= mark(X) ;
  active(and(X1,X2)) >= and(active(X1),X2) ;
  active(isNat(s(V1))) >= mark(isNat(V1)) ;
  active(isNat(plus(V1,V2))) >= mark(and(isNat(V1),isNat(V2))) ;
  active(isNat(0)) >= mark(tt) ;
  active(isNat(x(V1,V2))) >= mark(and(isNat(V1),isNat(V2))) ;
  U11(mark(X1),X2) >= mark(U11(X1,X2)) ;
  U11(ok(X1),ok(X2)) >= ok(U11(X1,X2)) ;
  s(mark(X)) >= mark(s(X)) ;
  s(ok(X)) >= ok(s(X)) ;
  plus(mark(X1),X2) >= mark(plus(X1,X2)) ;
  plus(ok(X1),ok(X2)) >= ok(plus(X1,X2)) ;
  plus(X1,mark(X2)) >= mark(plus(X1,X2)) ;
  U21(mark(X1),X2,X3) >= mark(U21(X1,X2,X3)) ;
  U21(ok(X1),ok(X2),ok(X3)) >= ok(U21(X1,X2,X3)) ;
  U31(mark(X)) >= mark(U31(X)) ;
  U31(ok(X)) >= ok(U31(X)) ;
  x(mark(X1),X2) >= mark(x(X1,X2)) ;
  x(ok(X1),ok(X2)) >= ok(x(X1,X2)) ;
  x(X1,mark(X2)) >= mark(x(X1,X2)) ;
  U41(mark(X1),X2,X3) >= mark(U41(X1,X2,X3)) ;
  U41(ok(X1),ok(X2),ok(X3)) >= ok(U41(X1,X2,X3)) ;
  and(mark(X1),X2) >= mark(and(X1,X2)) ;
  and(ok(X1),ok(X2)) >= ok(and(X1,X2)) ;
  isNat(ok(X)) >= ok(isNat(X)) ;
  proper(U11(X1,X2)) >= U11(proper(X1),proper(X2)) ;
  proper(tt) >= ok(tt) ;
  proper(s(X)) >= s(proper(X)) ;
  proper(plus(X1,X2)) >= plus(proper(X1),proper(X2)) ;
  proper(U21(X1,X2,X3)) >= U21(proper(X1),proper(X2),proper(X3)) ;
  proper(0) >= ok(0) ;
  proper(U31(X)) >= U31(proper(X)) ;
  proper(x(X1,X2)) >= x(proper(X1),proper(X2)) ;
  proper(U41(X1,X2,X3)) >= U41(proper(X1),proper(X2),proper(X3)) ;
  proper(and(X1,X2)) >= and(proper(X1),proper(X2)) ;
  proper(isNat(X)) >= isNat(proper(X)) ;
  top(mark(X)) >= top(proper(X)) ;
  top(ok(X)) >= top(active(X)) ;
  Marked_top(mark(X)) > Marked_top(proper(X)) ;
  Marked_top(ok(X)) > Marked_top(active(X)) ;
 }

APPLY CRITERIA (SOLVE_ORD)
Trying to solve the following constraints:
{
  active(U11(tt,N)) >= mark(N) ;
  active(U11(X1,X2)) >= U11(active(X1),X2) ;
  active(s(X)) >= s(active(X)) ;
  active(plus(N,s(M))) >= mark(U21(and(isNat(M),isNat(N)),M,N)) ;
  active(plus(N,0)) >= mark(U11(isNat(N),N)) ;
  active(plus(X1,X2)) >= plus(active(X1),X2) ;
  active(plus(X1,X2)) >= plus(X1,active(X2)) ;
  active(U21(tt,M,N)) >= mark(s(plus(N,M))) ;
  active(U21(X1,X2,X3)) >= U21(active(X1),X2,X3) ;
  active(U31(tt)) >= mark(0) ;
  active(U31(X)) >= U31(active(X)) ;
  active(x(N,s(M))) >= mark(U41(and(isNat(M),isNat(N)),M,N)) ;
  active(x(N,0)) >= mark(U31(isNat(N))) ;
  active(x(X1,X2)) >= x(active(X1),X2) ;
  active(x(X1,X2)) >= x(X1,active(X2)) ;
  active(U41(tt,M,N)) >= mark(plus(x(N,M),N)) ;
  active(U41(X1,X2,X3)) >= U41(active(X1),X2,X3) ;
  active(and(tt,X)) >= mark(X) ;
  active(and(X1,X2)) >= and(active(X1),X2) ;
  active(isNat(s(V1))) >= mark(isNat(V1)) ;
  active(isNat(plus(V1,V2))) >= mark(and(isNat(V1),isNat(V2))) ;
  active(isNat(0)) >= mark(tt) ;
  active(isNat(x(V1,V2))) >= mark(and(isNat(V1),isNat(V2))) ;
  U11(mark(X1),X2) >= mark(U11(X1,X2)) ;
  U11(ok(X1),ok(X2)) >= ok(U11(X1,X2)) ;
  s(mark(X)) >= mark(s(X)) ;
  s(ok(X)) >= ok(s(X)) ;
  plus(mark(X1),X2) >= mark(plus(X1,X2)) ;
  plus(ok(X1),ok(X2)) >= ok(plus(X1,X2)) ;
  plus(X1,mark(X2)) >= mark(plus(X1,X2)) ;
  U21(mark(X1),X2,X3) >= mark(U21(X1,X2,X3)) ;
  U21(ok(X1),ok(X2),ok(X3)) >= ok(U21(X1,X2,X3)) ;
  U31(mark(X)) >= mark(U31(X)) ;
  U31(ok(X)) >= ok(U31(X)) ;
  x(mark(X1),X2) >= mark(x(X1,X2)) ;
  x(ok(X1),ok(X2)) >= ok(x(X1,X2)) ;
  x(X1,mark(X2)) >= mark(x(X1,X2)) ;
  U41(mark(X1),X2,X3) >= mark(U41(X1,X2,X3)) ;
  U41(ok(X1),ok(X2),ok(X3)) >= ok(U41(X1,X2,X3)) ;
  and(mark(X1),X2) >= mark(and(X1,X2)) ;
  and(ok(X1),ok(X2)) >= ok(and(X1,X2)) ;
  isNat(ok(X)) >= ok(isNat(X)) ;
  proper(U11(X1,X2)) >= U11(proper(X1),proper(X2)) ;
  proper(tt) >= ok(tt) ;
  proper(s(X)) >= s(proper(X)) ;
  proper(plus(X1,X2)) >= plus(proper(X1),proper(X2)) ;
  proper(U21(X1,X2,X3)) >= U21(proper(X1),proper(X2),proper(X3)) ;
  proper(0) >= ok(0) ;
  proper(U31(X)) >= U31(proper(X)) ;
  proper(x(X1,X2)) >= x(proper(X1),proper(X2)) ;
  proper(U41(X1,X2,X3)) >= U41(proper(X1),proper(X2),proper(X3)) ;
  proper(and(X1,X2)) >= and(proper(X1),proper(X2)) ;
  proper(isNat(X)) >= isNat(proper(X)) ;
  top(mark(X)) >= top(proper(X)) ;
  top(ok(X)) >= top(active(X)) ;
  Marked_top(mark(X)) > Marked_top(proper(X)) ;
  Marked_top(ok(X)) > Marked_top(active(X)) ;
 }
 + Disjunctions:{ 
 } 
=== TIMER virtual : 10.000000 ===
Entering poly_solver
Starting Sat solver initialization
Calling Sat solver...
=== STOPING TIMER virtual ===
=== TIMER real : 10.000000 ===
=== STOPING TIMER real ===
Sat solver returned
=== STOPING TIMER real ===
=== STOPING TIMER virtual ===
No solution found for these parameters.
Entering rpo_solver
=== TIMER virtual : 25.000000 ===
Search parameters: AFS type: 2 ; time limit: 25.. 
=== STOPING TIMER virtual ===
No solution found for these parameters.(9796 bt (13580) [313])
=== TIMER virtual : 15.000000 ===
Entering poly_solver
Starting Sat solver initialization
Calling Sat solver...
=== STOPING TIMER virtual ===
=== TIMER real : 15.000000 ===
=== STOPING TIMER real ===
Sat solver returned
=== STOPING TIMER real ===
=== STOPING TIMER virtual ===
No solution found for these parameters.
=== TIMER virtual : 50.000000 ===
trying sub matrices of size: 1
Matrix interpretation constraints generated.
Search parameters: LINEAR MATRIX 3x3 (strict=1x1) ; time limit: 50.. 
Termination constraints generated.
Starting Sat solver initialization
Calling Sat solver...
=== STOPING TIMER virtual ===
=== TIMER real : 50.000000 ===
=== STOPING TIMER real ===
Sat timeout reached
=== STOPING TIMER virtual ===
No solution found for these parameters.
No solution found for these constraints.
APPLY CRITERIA (ID_CRIT)
NOT SOLVED
No proof found
Cime worked for 116.853436 seconds (real time)
Cime Exit Status: 0