Input TRS:
    1: +(0(),y) -> y
    2: +(s(x),y) -> s(+(x,y))
    3: sum1(nil()) -> 0()
    4: sum1(cons(x,y)) -> +(x,sum1(y))
    5: sum2(nil(),z) -> z
    6: sum2(cons(x,y),z) -> sum2(y,+(x,z))
    7: tests(0()) -> true()
    8: tests(s(x)) -> and(test(rands(rand(0()),nil())),x)
    9: test(done(y)) -> eq(f(y),g(y))
    10: eq(x,x) -> true()
    11: rands(0(),y) -> done(y)
    12: rands(s(x),y) -> rands(x,::(rand(0()),y))
    13: rand(x) ->= x
    14: rand(x) ->= rand(s(x))
Number of strict rules: 12
Direct Order(PosReal,>,Poly) ... removes: 4 8 1 3 5 10 7 11 9 6
 rands(x1,x2)	weight: (/ 1 4) + x2 + x1
 tests(x1)	weight: (/ 428177 8) + x1
  test(x1)	weight: (/ 86717 4) + 2 * x1
     s(x1)	weight: x1
  done(x1)	weight: (/ 1 8) + x1
   and(x1,x2)	weight: x2 + x1
    eq(x1,x2)	weight: (/ 1 8) + x2 + x1
  sum2(x1,x2)	weight: 2 * x2 + 2 * x1
    ::(x1,x2)	weight: x2 + x1
  true()	weight: 0
  rand(x1)	weight: x1
     f(x1)	weight: (/ 173433 8) + x1
     0()	weight: 0
   nil()	weight: (/ 127369 8)
  cons(x1,x2)	weight: (/ 1 4) + x2 + x1
     +(x1,x2)	weight: (/ 1 8) + x2 + x1
     g(x1)	weight: (/ 1 8) + x1
  sum1(x1)	weight: (/ 2249 8) + x1
Number of strict rules: 2
Direct Order(PosReal,>,Poly) ... failed.
Freezing ... failed.
Dependency Pairs:
   #1: #+(s(x),y) -> #+(x,y)
   #2: #rands(s(x),y) -> #rands(x,::(rand(0()),y))
Number of SCCs: 2, DPs: 2, edges: 2
	SCC { #1 }
Removing DPs: Order(PosReal,>,Sum)... Order(PosReal,>,Max)... QLPOpS... Order(PosReal,>,MaxSum)... QWPOpS(PosReal,>,MaxSum)... succeeded.
 rands(x1,x2)	weight: x1	status: [x1]	precedence above:
 tests(x1)	weight: x1	status: []	precedence above:
  test(x1)	weight: x1	status: []	precedence above:
     s(x1)	weight: x1	status: [x1]	precedence above: rands :: + #+
  done(x1)	weight: (/ 1 4)	status: []	precedence above:
   and(x1,x2)	weight: (/ 1 4) + x2 + x1	status: [x2,x1]	precedence above:
    eq(x1,x2)	weight: x1	status: [x1]	precedence above:
  sum2(x1,x2)	weight: (/ 1 4) + x1	status: [x1]	precedence above:
    ::(x1,x2)	weight: (/ 1 4)	status: []	precedence above: rands s + #+
  true()	weight: 0	status: []	precedence above:
  rand(x1)	weight: (/ 1 4) + x1	status: []	precedence above: rands
     f(x1)	weight: x1	status: []	precedence above:
#rands(x1,x2)	weight: (/ 1 4) + x2 + x1	status: [x1,x2]	precedence above:
     0()	weight: (/ 1 4)	status: []	precedence above:
   nil()	weight: 0	status: []	precedence above:
  cons(x1,x2)	weight: (/ 1 4) + x2	status: [x2]	precedence above:
     +(x1,x2)	weight: x1	status: x1
     g(x1)	weight: x1	status: []	precedence above:
   #+(x1,x2)	weight: x1	status: [x1]	precedence above:
  sum1(x1)	weight: x1	status: []	precedence above:
    Removed DPs: #1
Number of SCCs: 1, DPs: 1, edges: 1
	SCC { #2 }
Removing DPs: Order(PosReal,>,Sum)... Order(PosReal,>,Max)... QLPOpS... Order(PosReal,>,MaxSum)... QWPOpS(PosReal,>,MaxSum)... succeeded.
 rands(x1,x2)	weight: x1	status: [x1]	precedence above:
 tests(x1)	weight: x1	status: []	precedence above:
  test(x1)	weight: x1	status: []	precedence above:
     s(x1)	weight: x1	status: [x1]	precedence above: rands :: #rands 0 + #+
  done(x1)	weight: (/ 1 2)	status: []	precedence above:
   and(x1,x2)	weight: (/ 1 2) + x2 + x1	status: [x2,x1]	precedence above:
    eq(x1,x2)	weight: x1	status: [x1]	precedence above:
  sum2(x1,x2)	weight: (/ 1 2) + x1	status: [x1]	precedence above:
    ::(x1,x2)	weight: (/ 1 2)	status: []	precedence above: rands #+
  true()	weight: 0	status: []	precedence above:
  rand(x1)	weight: (/ 1 2) + x1	status: []	precedence above: rands
     f(x1)	weight: x1	status: []	precedence above:
#rands(x1,x2)	weight: x1	status: [x1]	precedence above: rands :: #+
     0()	weight: 0	status: []	precedence above:
   nil()	weight: 0	status: []	precedence above:
  cons(x1,x2)	weight: (/ 1 2) + x2	status: [x2]	precedence above:
     +(x1,x2)	weight: x1	status: x1
     g(x1)	weight: x1	status: []	precedence above:
   #+(x1,x2)	weight: x1	status: [x1]	precedence above:
  sum1(x1)	weight: x1	status: []	precedence above:
    Removed DPs: #2
Number of SCCs: 0, DPs: 0, edges: 0
YES