MAYBE
Input TRS:
      1: p(0(x1)) -> s(s(0(s(s(p(x1))))))
      2: p(s(0(x1))) -> 0(x1)
      3: p(s(s(x1))) -> s(p(s(x1)))
      4: f(s(x1)) -> g(q(i(x1)))
      5: g(x1) -> f(p(p(x1)))
      6: q(i(x1)) -> q(s(x1))
      7: q(s(x1)) -> s(s(x1))
      8: i(x1) -> s(x1)
Number of strict rules: 8
Direct POLO(bPol) ... failed.
Uncurrying p
      1: p^1_0(x1) -> s(s(0(s(s(p(x1))))))
      2: p^1_s(0(x1)) -> 0(x1)
      3: p^1_s(s(x1)) -> s(p^1_s(x1))
      4: f(s(x1)) -> g(q(i(x1)))
      5: g(x1) -> f(p(p(x1)))
      6: q(i(x1)) -> q(s(x1))
      7: q(s(x1)) -> s(s(x1))
      8: i(x1) -> s(x1)
      9: p(0(_1)) ->= p^1_0(_1)
      10: p(s(_1)) ->= p^1_s(_1)
Number of strict rules: 8
Direct POLO(bPol) ... failed.
Dependency Pairs:
     #1: #q(i(x1)) -> #q(s(x1))
     #2: #p(0(_1)) ->? #p^1_0(_1)
     #3: #p(s(_1)) ->? #p^1_s(_1)
     #4: #g(x1) -> #f(p(p(x1)))
     #5: #g(x1) -> #p(p(x1))
     #6: #g(x1) -> #p(x1)
     #7: #p^1_s(s(x1)) -> #p^1_s(x1)
     #8: #p^1_0(x1) -> #p(x1)
     #9: #f(s(x1)) -> #g(q(i(x1)))
     #10: #f(s(x1)) -> #q(i(x1))
     #11: #f(s(x1)) -> #i(x1)
Number of SCCs: 3, DPs: 5
  SCC { #7 }
POLO(Sum)... succeeded.
      q 	w: 0
      s 	w: x1 + 1
      #p^1_0	w: 0
      #p^1_s	w: x1
      p^1_0	w: 0
      p^1_s	w: 0
      #p 	w: 0
      f 	w: 0
      p 	w: 0
      0 	w: 0
      #f 	w: 0
      #g 	w: 0
      i 	w: 0
      #q 	w: 0
      g 	w: 0
      #i 	w: 0
    USABLE RULES: { }
    Removed DPs: #7
Number of SCCs: 2, DPs: 4
  SCC { #2 #8 }
POLO(Sum)... succeeded.
      q 	w: 0
      s 	w: 1
      #p^1_0	w: x1 + 1
      #p^1_s	w: 0
      p^1_0	w: 0
      p^1_s	w: 0
      #p 	w: x1
      f 	w: 0
      p 	w: 0
      0 	w: x1 + 2
      #f 	w: 0
      #g 	w: 0
      i 	w: 0
      #q 	w: 0
      g 	w: 0
      #i 	w: 0
    USABLE RULES: { }
    Removed DPs: #2 #8
Number of SCCs: 1, DPs: 2
  SCC { #4 #9 }
POLO(Sum)... POLO(max)... QLPOS... POLO(mSum)... QWPOpS(mSum)... Mat2b... failed.
Finding a loop...  failed.