ceta_eq: termination proof not accepted
1: error below switch to dependency pairs
1.1: error below the dependency graph processor
 1.1.3: error below the reduction pair processor
  1.1.3.1: error when applying the reduction pair processor with usable rules to remove from the DP problem
   pairs:
   
   proper#(s(X)) -> proper#(X)
   proper#(U31(X)) -> proper#(X)
   proper#(isNat(X)) -> proper#(X)
   rules:
   
   U11(mark(X1), X2) -> mark(U11(X1, X2))
   U11(ok(X1), ok(X2)) -> ok(U11(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))
   U41(mark(X1), X2, X3) -> mark(U41(X1, X2, X3))
   U41(ok(X1), ok(X2), ok(X3)) -> ok(U41(X1, X2, X3))
   active(U11(tt, N)) -> mark(N)
   active(U21(tt, M, N)) -> mark(s(plus(N, M)))
   active(U31(tt)) -> mark(0)
   active(U41(tt, M, N)) -> mark(plus(x(N, M), N))
   active(and(tt, X)) -> mark(X)
   active(isNat(0)) -> mark(tt)
   active(isNat(plus(V1, V2))) -> mark(and(isNat(V1), isNat(V2)))
   active(isNat(s(V1))) -> mark(isNat(V1))
   active(isNat(x(V1, V2))) -> mark(and(isNat(V1), isNat(V2)))
   active(plus(N, 0)) -> mark(U11(isNat(N), N))
   active(plus(N, s(M))) -> mark(U21(and(isNat(M), isNat(N)), M, N))
   active(x(N, 0)) -> mark(U31(isNat(N)))
   active(x(N, s(M))) -> mark(U41(and(isNat(M), isNat(N)), M, N))
   active(U11(X1, X2)) -> U11(active(X1), X2)
   active(U21(X1, X2, X3)) -> U21(active(X1), X2, X3)
   active(s(X)) -> s(active(X))
   active(plus(X1, X2)) -> plus(active(X1), X2)
   active(plus(X1, X2)) -> plus(X1, active(X2))
   active(U31(X)) -> U31(active(X))
   active(U41(X1, X2, X3)) -> U41(active(X1), X2, X3)
   active(x(X1, X2)) -> x(active(X1), X2)
   active(x(X1, X2)) -> x(X1, active(X2))
   active(and(X1, X2)) -> and(active(X1), X2)
   and(mark(X1), X2) -> mark(and(X1, X2))
   and(ok(X1), ok(X2)) -> ok(and(X1, X2))
   isNat(ok(X)) -> ok(isNat(X))
   plus(mark(X1), X2) -> mark(plus(X1, X2))
   plus(X1, mark(X2)) -> mark(plus(X1, X2))
   plus(ok(X1), ok(X2)) -> ok(plus(X1, X2))
   proper(U11(X1, X2)) -> U11(proper(X1), proper(X2))
   proper(tt) -> ok(tt)
   proper(U21(X1, X2, X3)) -> U21(proper(X1), proper(X2), proper(X3))
   proper(s(X)) -> s(proper(X))
   proper(plus(X1, X2)) -> plus(proper(X1), proper(X2))
   proper(U31(X)) -> U31(proper(X))
   proper(0) -> ok(0)
   proper(U41(X1, X2, X3)) -> U41(proper(X1), proper(X2), proper(X3))
   proper(x(X1, X2)) -> x(proper(X1), proper(X2))
   proper(and(X1, X2)) -> and(proper(X1), proper(X2))
   proper(isNat(X)) -> isNat(proper(X))
   s(mark(X)) -> mark(s(X))
   s(ok(X)) -> ok(s(X))
   top(mark(X)) -> top(proper(X))
   top(ok(X)) -> top(active(X))
   x(mark(X1), X2) -> mark(x(X1, X2))
   x(X1, mark(X2)) -> mark(x(X1, X2))
   x(ok(X1), ok(X2)) -> ok(x(X1, X2))
   
    the pairs 
   proper#(U31(X)) -> proper#(X)
   
   could not apply the generic root reduction pair processor with the following
   SCNP-version with mu = MS and the level mapping defined by 
   pi(proper#) = [(epsilon,0),(1,0)]
   Argument Filter: 
   pi(proper#/1) = []
   pi(s/1) = 1
   pi(U31/1) = [1]
   pi(isNat/1) = 1
   
   RPO with the following precedence
   precedence(proper#[1]) = 0
   precedence(U31[1]) = 1
   
   precedence(_) = 0
   and the following status
   status(proper#[1]) = lex
   status(U31[1]) = lex
   
   status(_) = lex
   
   problem when orienting DPs
   cannot orient pair proper#(s(X)) -> proper#(X) weakly:
   [(proper#(s(X)),0),(s(X),0)] >=mu [(proper#(X),0),(X,0)] could not be ensured