Input TRS:
    1: a__fact(X) -> a__if(a__zero(mark(X)),s(0()),prod(X,fact(p(X))))
    2: a__add(0(),X) -> mark(X)
    3: a__add(s(X),Y) -> s(a__add(mark(X),mark(Y)))
    4: a__prod(0(),X) -> 0()
    5: a__prod(s(X),Y) -> a__add(mark(Y),a__prod(mark(X),mark(Y)))
    6: a__if(true(),X,Y) -> mark(X)
    7: a__if(false(),X,Y) -> mark(Y)
    8: a__zero(0()) -> true()
    9: a__zero(s(X)) -> false()
    10: a__p(s(X)) -> mark(X)
    11: mark(fact(X)) -> a__fact(mark(X))
    12: mark(if(X1,X2,X3)) -> a__if(mark(X1),X2,X3)
    13: mark(zero(X)) -> a__zero(mark(X))
    14: mark(prod(X1,X2)) -> a__prod(mark(X1),mark(X2))
    15: mark(p(X)) -> a__p(mark(X))
    16: mark(add(X1,X2)) -> a__add(mark(X1),mark(X2))
    17: mark(s(X)) -> s(mark(X))
    18: mark(0()) -> 0()
    19: mark(true()) -> true()
    20: mark(false()) -> false()
    21: a__fact(X) -> fact(X)
    22: a__if(X1,X2,X3) -> if(X1,X2,X3)
    23: a__zero(X) -> zero(X)
    24: a__prod(X1,X2) -> prod(X1,X2)
    25: a__p(X) -> p(X)
    26: a__add(X1,X2) -> add(X1,X2)
Number of strict rules: 26
Direct Order(PosReal,>,Poly) ... failed.
Freezing ... failed.
Dependency Pairs:
   #1: #a__add(0(),X) -> #mark(X)
   #2: #a__if(true(),X,Y) -> #mark(X)
   #3: #mark(zero(X)) -> #a__zero(mark(X))
   #4: #mark(zero(X)) -> #mark(X)
   #5: #mark(fact(X)) -> #a__fact(mark(X))
   #6: #mark(fact(X)) -> #mark(X)
   #7: #mark(if(X1,X2,X3)) -> #a__if(mark(X1),X2,X3)
   #8: #mark(if(X1,X2,X3)) -> #mark(X1)
   #9: #mark(prod(X1,X2)) -> #a__prod(mark(X1),mark(X2))
   #10: #mark(prod(X1,X2)) -> #mark(X1)
   #11: #mark(prod(X1,X2)) -> #mark(X2)
   #12: #a__if(false(),X,Y) -> #mark(Y)
   #13: #a__p(s(X)) -> #mark(X)
   #14: #a__prod(s(X),Y) -> #a__add(mark(Y),a__prod(mark(X),mark(Y)))
   #15: #a__prod(s(X),Y) -> #mark(Y)
   #16: #a__prod(s(X),Y) -> #a__prod(mark(X),mark(Y))
   #17: #a__prod(s(X),Y) -> #mark(X)
   #18: #a__prod(s(X),Y) -> #mark(Y)
   #19: #mark(s(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__add(s(X),Y) -> #a__add(mark(X),mark(Y))
   #24: #a__add(s(X),Y) -> #mark(X)
   #25: #a__add(s(X),Y) -> #mark(Y)
   #26: #a__fact(X) -> #a__if(a__zero(mark(X)),s(0()),prod(X,fact(p(X))))
   #27: #a__fact(X) -> #a__zero(mark(X))
   #28: #a__fact(X) -> #mark(X)
   #29: #mark(p(X)) -> #a__p(mark(X))
   #30: #mark(p(X)) -> #mark(X)
Number of SCCs: 1, DPs: 28, edges: 295
	SCC { #1 #2 #4..26 #28..30 }
Removing DPs: Order(PosReal,>,Sum)... Order(PosReal,>,Max)... succeeded.
  zero(x1)	weight: (/ 1 8) + x1
  prod(x1,x2)	weight: max{x2, (/ 1 8) + x1}
     s(x1)	weight: x1
a__prod(x1,x2)	weight: max{x2, (/ 1 8) + x1}
#a__zero(x1)	weight: 0
a__fact(x1)	weight: (/ 1 2) + x1
#a__add(x1,x2)	weight: max{(/ 1 8) + x2, (/ 1 8) + x1}
a__add(x1,x2)	weight: max{x2, x1}
 false()	weight: (/ 1 8)
  a__p(x1)	weight: x1
  true()	weight: (/ 1 8)
a__zero(x1)	weight: (/ 1 8) + x1
     p(x1)	weight: x1
#mark(x1)	weight: (/ 1 8) + x1
     0()	weight: (/ 1 8)
    if(x1,x2,x3)	weight: max{x3, (/ 3 8) + x2, (/ 1 4) + x1}
#a__fact(x1)	weight: (/ 5 8) + x1
  fact(x1)	weight: (/ 1 2) + x1
  mark(x1)	weight: x1
#a__p(x1)	weight: (/ 1 8) + x1
#a__if(x1,x2,x3)	weight: max{0, (/ 1 8) + x3, (/ 1 4) + x2}
#a__prod(x1,x2)	weight: max{(/ 1 8) + x2, (/ 1 4) + x1}
   add(x1,x2)	weight: max{x2, x1}
 a__if(x1,x2,x3)	weight: max{x3, (/ 3 8) + x2, (/ 1 4) + x1}
    Usable rules: { 1..26 }
    Removed DPs: #2 #4 #6 #8 #10 #17 #28
Number of SCCs: 1, DPs: 21, edges: 144
	SCC { #1 #5 #7 #9 #11..16 #18..26 #29 #30 }
Removing DPs: Order(PosReal,>,Sum)... Order(PosReal,>,Max)... succeeded.
  zero(x1)	weight: (/ 1 8)
  prod(x1,x2)	weight: max{0, x2}
     s(x1)	weight: x1
a__prod(x1,x2)	weight: max{0, x2}
#a__zero(x1)	weight: 0
a__fact(x1)	weight: (/ 1 4)
#a__add(x1,x2)	weight: max{(/ 1 4) + x2, (/ 1 4) + x1}
a__add(x1,x2)	weight: max{x2, x1}
 false()	weight: (/ 1 8)
  a__p(x1)	weight: (/ 1 4) + x1
  true()	weight: (/ 1 8)
a__zero(x1)	weight: (/ 1 8)
     p(x1)	weight: (/ 1 4) + x1
#mark(x1)	weight: (/ 1 4) + x1
     0()	weight: 0
    if(x1,x2,x3)	weight: max{x3, (/ 1 8) + x2, (/ 1 8) + x1}
#a__fact(x1)	weight: (/ 1 2)
  fact(x1)	weight: (/ 1 4)
  mark(x1)	weight: x1
#a__p(x1)	weight: (/ 3 8) + x1
#a__if(x1,x2,x3)	weight: max{0, (/ 1 4) + x3, (/ 1 8) + x2}
#a__prod(x1,x2)	weight: max{0, (/ 1 4) + x2}
   add(x1,x2)	weight: max{x2, x1}
 a__if(x1,x2,x3)	weight: max{x3, (/ 1 8) + x2, (/ 1 8) + x1}
    Usable rules: { 1..26 }
    Removed DPs: #13 #29 #30
Number of SCCs: 1, DPs: 18, edges: 103
	SCC { #1 #5 #7 #9 #11 #12 #14..16 #18..26 }
Removing DPs: Order(PosReal,>,Sum)... Order(PosReal,>,Max)... QLPOpS... Order(PosReal,>,MaxSum)... QWPOpS(PosReal,>,MaxSum)... Order(PosReal,>,Sum-Sum; PosReal,≥,Sum-Sum)... Order(PosReal,>,Sum-Sum; NegReal,≥,Sum)... Order(PosReal,>,MaxSum-Sum; NegReal,≥,Sum)... failed.
Removing edges: failed.
Finding a loop...  failed.
MAYBE