/export/starexec/sandbox/solver/bin/starexec_run_Default /export/starexec/sandbox/benchmark/theBenchmark.xml /export/starexec/sandbox/output/output_files


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YES
Input TRS:
    1: a__terms(N) -> cons(recip(a__sqr(mark(N))),terms(s(N)))
    2: a__sqr(0()) -> 0()
    3: a__sqr(s(X)) -> s(a__add(a__sqr(mark(X)),a__dbl(mark(X))))
    4: a__dbl(0()) -> 0()
    5: a__dbl(s(X)) -> s(s(a__dbl(mark(X))))
    6: a__add(0(),X) -> mark(X)
    7: a__add(s(X),Y) -> s(a__add(mark(X),mark(Y)))
    8: a__first(0(),X) -> nil()
    9: a__first(s(X),cons(Y,Z)) -> cons(mark(Y),first(X,Z))
    10: a__half(0()) -> 0()
    11: a__half(s(0())) -> 0()
    12: a__half(s(s(X))) -> s(a__half(mark(X)))
    13: a__half(dbl(X)) -> mark(X)
    14: mark(terms(X)) -> a__terms(mark(X))
    15: mark(sqr(X)) -> a__sqr(mark(X))
    16: mark(add(X1,X2)) -> a__add(mark(X1),mark(X2))
    17: mark(dbl(X)) -> a__dbl(mark(X))
    18: mark(first(X1,X2)) -> a__first(mark(X1),mark(X2))
    19: mark(half(X)) -> a__half(mark(X))
    20: mark(cons(X1,X2)) -> cons(mark(X1),X2)
    21: mark(recip(X)) -> recip(mark(X))
    22: mark(s(X)) -> s(mark(X))
    23: mark(0()) -> 0()
    24: mark(nil()) -> nil()
    25: a__terms(X) -> terms(X)
    26: a__sqr(X) -> sqr(X)
    27: a__add(X1,X2) -> add(X1,X2)
    28: a__dbl(X) -> dbl(X)
    29: a__first(X1,X2) -> first(X1,X2)
    30: a__half(X) -> half(X)
Number of strict rules: 30
Direct POLO(bPol) ... failed.
Uncurrying ... failed.
Dependency Pairs:
   #1: #a__add(0(),X) -> #mark(X)
   #2: #a__half(dbl(X)) -> #mark(X)
   #3: #a__first(s(X),cons(Y,Z)) -> #mark(Y)
   #4: #a__half(s(s(X))) -> #a__half(mark(X))
   #5: #a__half(s(s(X))) -> #mark(X)
   #6: #mark(terms(X)) -> #a__terms(mark(X))
   #7: #mark(terms(X)) -> #mark(X)
   #8: #mark(cons(X1,X2)) -> #mark(X1)
   #9: #a__add(s(X),Y) -> #a__add(mark(X),mark(Y))
   #10: #a__add(s(X),Y) -> #mark(X)
   #11: #a__add(s(X),Y) -> #mark(Y)
   #12: #a__dbl(s(X)) -> #a__dbl(mark(X))
   #13: #a__dbl(s(X)) -> #mark(X)
   #14: #mark(s(X)) -> #mark(X)
   #15: #mark(dbl(X)) -> #a__dbl(mark(X))
   #16: #mark(dbl(X)) -> #mark(X)
   #17: #mark(half(X)) -> #a__half(mark(X))
   #18: #mark(half(X)) -> #mark(X)
   #19: #mark(recip(X)) -> #mark(X)
   #20: #mark(add(X1,X2)) -> #a__add(mark(X1),mark(X2))
   #21: #mark(add(X1,X2)) -> #mark(X1)
   #22: #mark(add(X1,X2)) -> #mark(X2)
   #23: #a__sqr(s(X)) -> #a__add(a__sqr(mark(X)),a__dbl(mark(X)))
   #24: #a__sqr(s(X)) -> #a__sqr(mark(X))
   #25: #a__sqr(s(X)) -> #mark(X)
   #26: #a__sqr(s(X)) -> #a__dbl(mark(X))
   #27: #a__sqr(s(X)) -> #mark(X)
   #28: #a__terms(N) -> #a__sqr(mark(N))
   #29: #a__terms(N) -> #mark(N)
   #30: #mark(sqr(X)) -> #a__sqr(mark(X))
   #31: #mark(sqr(X)) -> #mark(X)
   #32: #mark(first(X1,X2)) -> #a__first(mark(X1),mark(X2))
   #33: #mark(first(X1,X2)) -> #mark(X1)
   #34: #mark(first(X1,X2)) -> #mark(X2)
Number of SCCs: 1, DPs: 34
  SCC { #1..34 }
POLO(Sum)... POLO(max)... succeeded.
      s 	w: x1
      #a__first	w: max(x2 + 1)
      recip	w: x1 + 1
      dbl	w: x1 + 1
      a__half	w: x1 + 2
      #a__add	w: max(x1 + 2, x2 + 3)
      a__add	w: max(x1, x2 + 1)
      a__dbl	w: x1 + 1
      #a__half	w: x1 + 3
      a__sqr	w: x1 + 3
      half	w: x1 + 2
      #a__terms	w: x1 + 6
      #mark	w: x1 + 2
      0 	w: 2
      nil	w: 3
      #a__dbl	w: x1 + 3
      mark	w: x1
      first	w: max(x1 + 1, x2 + 2)
      a__first	w: max(x1 + 1, x2 + 2)
      cons	w: max(x1 + 3)
      add	w: max(x1, x2 + 1)
      sqr	w: x1 + 3
      a__terms	w: x1 + 7
      #a__sqr	w: x1 + 5
      terms	w: x1 + 7
    USABLE RULES: { 1..30 }
    Removed DPs: #1..3 #5..8 #11 #13 #16..19 #22 #25..29 #31..34
Number of SCCs: 3, DPs: 10
  SCC { #12 }
POLO(Sum)... POLO(max)... QLPOS... succeeded.
      s 	s: [1]	p: 1
      #a__first	s: []	p: 0
      recip	s: []	p: 1
      dbl	s: [1]	p: 3
      a__half	s: [1]	p: 1
      #a__add	s: []	p: 0
      a__add	s: [2,1]	p: 2
      a__dbl	s: [1]	p: 3
      #a__half	s: []	p: 0
      a__sqr	s: [1]	p: 3
      half	s: [1]	p: 1
      #a__terms	s: []	p: 0
      #mark	s: []	p: 0
      0 	s: []	p: 1
      nil	s: []	p: 1
      #a__dbl	s: [1]	p: 0
      mark	s: 1
      first	s: [2]	p: 1
      a__first	s: [2]	p: 1
      cons	s: [1]	p: 0
      add	s: [2,1]	p: 2
      sqr	s: [1]	p: 3
      a__terms	s: [1]	p: 1
      #a__sqr	s: []	p: 0
      terms	s: [1]	p: 1
    USABLE RULES: { 1..30 }
    Removed DPs: #12
Number of SCCs: 2, DPs: 9
  SCC { #4 }
POLO(Sum)... POLO(max)... QLPOS... succeeded.
      s 	s: [1]	p: 1
      #a__first	s: []	p: 0
      recip	s: []	p: 1
      dbl	s: [1]	p: 3
      a__half	s: [1]	p: 1
      #a__add	s: []	p: 0
      a__add	s: [2,1]	p: 2
      a__dbl	s: [1]	p: 3
      #a__half	s: [1]	p: 0
      a__sqr	s: [1]	p: 3
      half	s: [1]	p: 1
      #a__terms	s: []	p: 0
      #mark	s: []	p: 0
      0 	s: []	p: 1
      nil	s: []	p: 1
      #a__dbl	s: [1]	p: 0
      mark	s: 1
      first	s: [2]	p: 1
      a__first	s: [2]	p: 1
      cons	s: [1]	p: 0
      add	s: [2,1]	p: 2
      sqr	s: [1]	p: 3
      a__terms	s: [1]	p: 1
      #a__sqr	s: []	p: 0
      terms	s: [1]	p: 1
    USABLE RULES: { 1..30 }
    Removed DPs: #4
Number of SCCs: 1, DPs: 8
  SCC { #9 #10 #14 #20 #21 #23 #24 #30 }
POLO(Sum)... POLO(max)... QLPOS... succeeded.
      s 	s: [1]	p: 3
      #a__first	s: []	p: 0
      recip	s: []	p: 7
      dbl	s: [1]	p: 5
      a__half	s: [1]	p: 3
      #a__add	s: 1
      a__add	s: [2,1]	p: 4
      a__dbl	s: [1]	p: 5
      #a__half	s: []	p: 0
      a__sqr	s: [1]	p: 5
      half	s: [1]	p: 3
      #a__terms	s: []	p: 0
      #mark	s: 1
      0 	s: []	p: 3
      nil	s: []	p: 2
      #a__dbl	s: []	p: 0
      mark	s: 1
      first	s: [2,1]	p: 1
      a__first	s: [2,1]	p: 1
      cons	s: [1]	p: 0
      add	s: [2,1]	p: 4
      sqr	s: [1]	p: 5
      a__terms	s: [1]	p: 7
      #a__sqr	s: [1]	p: 5
      terms	s: [1]	p: 7
    USABLE RULES: { 1..30 }
    Removed DPs: #9 #10 #14 #20 #21 #23 #24
Number of SCCs: 0, DPs: 0