Input TRS:
    1: a__f(X) -> a__if(mark(X),c(),f(true()))
    2: a__if(true(),X,Y) -> mark(X)
    3: a__if(false(),X,Y) -> mark(Y)
    4: mark(f(X)) -> a__f(mark(X))
    5: mark(if(X1,X2,X3)) -> a__if(mark(X1),mark(X2),X3)
    6: mark(c()) -> c()
    7: mark(true()) -> true()
    8: mark(false()) -> false()
    9: a__f(X) -> f(X)
    10: a__if(X1,X2,X3) -> if(X1,X2,X3)
Number of strict rules: 10
Direct Order(PosReal,>,Poly) ... removes: 9
  a__f(x1)	weight: (/ 1 2) + 2 * x1
 false()	weight: 0
     c()	weight: 0
  true()	weight: 0
     f(x1)	weight: (/ 1 4) + 2 * x1
    if(x1,x2,x3)	weight: x1 + 2 * x2 + x3
  mark(x1)	weight: 2 * x1
 a__if(x1,x2,x3)	weight: x1 + 2 * x2 + 2 * x3
Number of strict rules: 9
Direct Order(PosReal,>,Poly) ... failed.
Freezing ... failed.
Dependency Pairs:
   #1: #a__if(true(),X,Y) -> #mark(X)
   #2: #mark(if(X1,X2,X3)) -> #a__if(mark(X1),mark(X2),X3)
   #3: #mark(if(X1,X2,X3)) -> #mark(X1)
   #4: #mark(if(X1,X2,X3)) -> #mark(X2)
   #5: #a__if(false(),X,Y) -> #mark(Y)
   #6: #a__f(X) -> #a__if(mark(X),c(),f(true()))
   #7: #a__f(X) -> #mark(X)
   #8: #mark(f(X)) -> #a__f(mark(X))
   #9: #mark(f(X)) -> #mark(X)
Number of SCCs: 1, DPs: 9, edges: 36
	SCC { #1..9 }
Removing DPs: Order(PosReal,>,Sum)... succeeded.
  a__f(x1)	weight: (/ 1 2) + x1
 false()	weight: 0
     c()	weight: 0
  true()	weight: 0
     f(x1)	weight: (/ 1 2) + x1
#mark(x1)	weight: x1
    if(x1,x2,x3)	weight: x1 + x2 + x3
  mark(x1)	weight: x1
#a__if(x1,x2,x3)	weight: x2 + x3
#a__f(x1)	weight: (/ 1 2) + x1
 a__if(x1,x2,x3)	weight: x1 + x2 + x3
    Usable rules: { 1..8 10 }
    Removed DPs: #7 #9
Number of SCCs: 1, DPs: 7, edges: 21
	SCC { #1..6 #8 }
Removing DPs: Order(PosReal,>,Sum)... Order(PosReal,>,Max)... succeeded.
  a__f(x1)	weight: (/ 1 2) + x1
 false()	weight: (/ 1 8)
     c()	weight: (/ 1 8)
  true()	weight: 0
     f(x1)	weight: (/ 1 2) + x1
#mark(x1)	weight: (/ 1 8) + x1
    if(x1,x2,x3)	weight: max{x3, (/ 3 8) + x2, (/ 1 4) + x1}
  mark(x1)	weight: x1
#a__if(x1,x2,x3)	weight: max{(/ 1 8) + x3, (/ 1 4) + x2, (/ 3 8) + x1}
#a__f(x1)	weight: (/ 5 8) + x1
 a__if(x1,x2,x3)	weight: max{x3, (/ 3 8) + x2, (/ 1 4) + x1}
    Usable rules: { 1..8 10 }
    Removed DPs: #1 #3 #4
Number of SCCs: 1, DPs: 4, edges: 5
	SCC { #2 #5 #6 #8 }
Removing DPs: Order(PosReal,>,Sum)... succeeded.
  a__f(x1)	weight: (/ 3 8)
 false()	weight: 0
     c()	weight: 0
  true()	weight: 0
     f(x1)	weight: (/ 3 8)
#mark(x1)	weight: (/ 1 8) + x1
    if(x1,x2,x3)	weight: (/ 1 4) + x1 + x2 + x3
  mark(x1)	weight: (/ 1 4)
#a__if(x1,x2,x3)	weight: (/ 1 8) + x3
#a__f(x1)	weight: (/ 1 2)
 a__if(x1,x2,x3)	weight: (/ 1 8) + x2 + x3
    Usable rules: { }
    Removed DPs: #2
Number of SCCs: 1, DPs: 3, edges: 3
	SCC { #5 #6 #8 }
Removing DPs: Order(PosReal,>,Sum)... Order(PosReal,>,Max)... QLPOpS... Order(PosReal,>,MaxSum)... succeeded.
  a__f(x1)	weight: max{0, (/ 1 4) + x1}
 false()	weight: (/ 1 2)
     c()	weight: 0
  true()	weight: (/ 1 8)
     f(x1)	weight: max{0, (/ 1 4) + x1}
#mark(x1)	weight: max{0, (/ 1 2) + x1}
    if(x1,x2,x3)	weight: max{0, (- (/ 1 8)) + x3 + x2 + x1}
  mark(x1)	weight: max{0, x1}
#a__if(x1,x2,x3)	weight: max{0, (/ 1 8) + x3 + x1}
#a__f(x1)	weight: max{0, (/ 5 8) + x1}
 a__if(x1,x2,x3)	weight: max{0, (- (/ 1 8)) + x3 + x2 + x1}
    Usable rules: { 1..8 10 }
    Removed DPs: #5 #6 #8
Number of SCCs: 0, DPs: 0, edges: 0
YES