ceta_eq: termination proof not accepted
1: error below switch to dependency pairs
1.1: error below the dependency graph processor
 1.1.1: error when applying the reduction pair processor with usable rules to remove from the DP problem
  pairs:
  
  top#(mark(X)) -> top#(proper(X))
  top#(ok(X)) -> top#(active(X))
  rules:
  
  active(terms(N)) -> mark(cons(recip(sqr(N)), terms(s(N))))
  active(sqr(0)) -> mark(0)
  active(sqr(s(X))) -> mark(s(add(sqr(X), dbl(X))))
  active(dbl(0)) -> mark(0)
  active(dbl(s(X))) -> mark(s(s(dbl(X))))
  active(add(0, X)) -> mark(X)
  active(add(s(X), Y)) -> mark(s(add(X, Y)))
  active(first(0, X)) -> mark(nil)
  active(first(s(X), cons(Y, Z))) -> mark(cons(Y, first(X, Z)))
  active(terms(X)) -> terms(active(X))
  active(cons(X1, X2)) -> cons(active(X1), X2)
  active(recip(X)) -> recip(active(X))
  active(sqr(X)) -> sqr(active(X))
  active(add(X1, X2)) -> add(active(X1), X2)
  active(add(X1, X2)) -> add(X1, active(X2))
  active(dbl(X)) -> dbl(active(X))
  active(first(X1, X2)) -> first(active(X1), X2)
  active(first(X1, X2)) -> first(X1, active(X2))
  add(mark(X1), X2) -> mark(add(X1, X2))
  add(X1, mark(X2)) -> mark(add(X1, X2))
  add(ok(X1), ok(X2)) -> ok(add(X1, X2))
  cons(mark(X1), X2) -> mark(cons(X1, X2))
  cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
  dbl(mark(X)) -> mark(dbl(X))
  dbl(ok(X)) -> ok(dbl(X))
  first(mark(X1), X2) -> mark(first(X1, X2))
  first(X1, mark(X2)) -> mark(first(X1, X2))
  first(ok(X1), ok(X2)) -> ok(first(X1, X2))
  proper(terms(X)) -> terms(proper(X))
  proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
  proper(recip(X)) -> recip(proper(X))
  proper(sqr(X)) -> sqr(proper(X))
  proper(s(X)) -> s(proper(X))
  proper(0) -> ok(0)
  proper(add(X1, X2)) -> add(proper(X1), proper(X2))
  proper(dbl(X)) -> dbl(proper(X))
  proper(first(X1, X2)) -> first(proper(X1), proper(X2))
  proper(nil) -> ok(nil)
  recip(mark(X)) -> mark(recip(X))
  recip(ok(X)) -> ok(recip(X))
  s(ok(X)) -> ok(s(X))
  sqr(mark(X)) -> mark(sqr(X))
  sqr(ok(X)) -> ok(sqr(X))
  terms(mark(X)) -> mark(terms(X))
  terms(ok(X)) -> ok(terms(X))
  top(mark(X)) -> top(proper(X))
  top(ok(X)) -> top(active(X))
  
   the pairs 
  top#(mark(X)) -> top#(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(top#) = [(epsilon,0),(1,1)]
  Argument Filter: 
  pi(top#/1) = []
  pi(ok/1) = 1
  pi(active/1) = 1
  pi(mark/1) = [1]
  pi(proper/1) = 1
  pi(terms/1) = [1]
  pi(cons/2) = 1
  pi(recip/1) = [1]
  pi(sqr/1) = [1]
  pi(s/1) = []
  pi(0/0) = []
  pi(add/2) = [2,1]
  pi(dbl/1) = [1]
  pi(first/2) = [1,2]
  pi(nil/0) = []
  
  RPO with the following precedence
  precedence(top#[1]) = 0
  precedence(mark[1]) = 1
  precedence(terms[1]) = 3
  precedence(recip[1]) = 1
  precedence(sqr[1]) = 2
  precedence(s[1]) = 4
  precedence(0[0]) = 6
  precedence(add[2]) = 7
  precedence(dbl[1]) = 1
  precedence(first[2]) = 8
  precedence(nil[0]) = 5
  
  precedence(_) = 0
  and the following status
  status(top#[1]) = lex
  status(mark[1]) = lex
  status(terms[1]) = lex
  status(recip[1]) = lex
  status(sqr[1]) = lex
  status(s[1]) = lex
  status(0[0]) = lex
  status(add[2]) = lex
  status(dbl[1]) = lex
  status(first[2]) = lex
  status(nil[0]) = lex
  
  status(_) = lex
  
  problem when orienting DPs
  cannot orient pair top#(ok(X)) -> top#(active(X)) weakly:
  [(top#(ok(X)),0),(ok(X),1)] >=mu [(top#(active(X)),0),(active(X),1)] could not be ensured