Input TRS:
    1: inc(s(x)) -> s(inc(x))
    2: inc(|0|()) -> s(|0|())
    3: plus(x,y) -> ifPlus(eq(x,|0|()),minus(x,s(|0|())),x,inc(x))
    4: ifPlus(false(),x,y,z) -> plus(x,z)
    5: ifPlus(true(),x,y,z) -> y
    6: minus(s(x),s(y)) -> minus(x,y)
    7: minus(|0|(),x) -> |0|()
    8: minus(x,|0|()) -> x
    9: minus(x,x) -> |0|()
    10: eq(s(x),s(y)) -> eq(x,y)
    11: eq(|0|(),s(x)) -> false()
    12: eq(s(x),|0|()) -> false()
    13: eq(|0|(),|0|()) -> true()
    14: eq(x,x) -> true()
    15: times(x,y) -> timesIter(x,y,|0|())
    16: timesIter(x,y,z) -> ifTimes(eq(x,|0|()),minus(x,s(|0|())),y,z,plus(y,z))
    17: ifTimes(true(),x,y,z,u) -> z
    18: ifTimes(false(),x,y,z,u) -> timesIter(x,y,u)
    19: f() -> g()
    20: f() -> h()
Number of strict rules: 20
Direct Order(PosReal,>,Poly) ... failed.
Freezing ... failed.
Dependency Pairs:
   #1: #minus(s(x),s(y)) -> #minus(x,y)
   #2: #eq(s(x),s(y)) -> #eq(x,y)
   #3: #timesIter(x,y,z) -> #ifTimes(eq(x,|0|()),minus(x,s(|0|())),y,z,plus(y,z))
   #4: #timesIter(x,y,z) -> #eq(x,|0|())
   #5: #timesIter(x,y,z) -> #minus(x,s(|0|()))
   #6: #timesIter(x,y,z) -> #plus(y,z)
   #7: #plus(x,y) -> #ifPlus(eq(x,|0|()),minus(x,s(|0|())),x,inc(x))
   #8: #plus(x,y) -> #eq(x,|0|())
   #9: #plus(x,y) -> #minus(x,s(|0|()))
   #10: #plus(x,y) -> #inc(x)
   #11: #inc(s(x)) -> #inc(x)
   #12: #times(x,y) -> #timesIter(x,y,|0|())
   #13: #ifPlus(false(),x,y,z) -> #plus(x,z)
   #14: #ifTimes(false(),x,y,z,u) -> #timesIter(x,y,u)
Number of SCCs: 5, DPs: 7, edges: 7
	SCC { #11 }
Removing DPs: Order(PosReal,>,Sum)... succeeded.
   |0|()	weight: 0
     h()	weight: 0
ifTimes(x1,x2,x3,x4,x5)	weight: 0
timesIter(x1,x2,x3)	weight: 0
     s(x1)	weight: (/ 1 2) + x1
 minus(x1,x2)	weight: 0
#timesIter(x1,x2,x3)	weight: 0
#plus(x1,x2)	weight: 0
    eq(x1,x2)	weight: 0
 false()	weight: 0
 #inc(x1)	weight: x1
   inc(x1)	weight: 0
  true()	weight: 0
     f()	weight: 0
  #eq(x1,x2)	weight: 0
#times(x1,x2)	weight: 0
 times(x1,x2)	weight: 0
   #f()	weight: 0
#minus(x1,x2)	weight: 0
ifPlus(x1,x2,x3,x4)	weight: 0
  plus(x1,x2)	weight: 0
#ifPlus(x1,x2,x3,x4)	weight: 0
     g()	weight: 0
#ifTimes(x1,x2,x3,x4,x5)	weight: 0
    Usable rules: { }
    Removed DPs: #11
Number of SCCs: 4, DPs: 6, edges: 6
	SCC { #2 }
Removing DPs: Order(PosReal,>,Sum)... succeeded.
   |0|()	weight: 0
     h()	weight: 0
ifTimes(x1,x2,x3,x4,x5)	weight: 0
timesIter(x1,x2,x3)	weight: 0
     s(x1)	weight: (/ 1 2) + x1
 minus(x1,x2)	weight: 0
#timesIter(x1,x2,x3)	weight: 0
#plus(x1,x2)	weight: 0
    eq(x1,x2)	weight: 0
 false()	weight: 0
 #inc(x1)	weight: 0
   inc(x1)	weight: 0
  true()	weight: 0
     f()	weight: 0
  #eq(x1,x2)	weight: x2
#times(x1,x2)	weight: 0
 times(x1,x2)	weight: 0
   #f()	weight: 0
#minus(x1,x2)	weight: 0
ifPlus(x1,x2,x3,x4)	weight: 0
  plus(x1,x2)	weight: 0
#ifPlus(x1,x2,x3,x4)	weight: 0
     g()	weight: 0
#ifTimes(x1,x2,x3,x4,x5)	weight: 0
    Usable rules: { }
    Removed DPs: #2
Number of SCCs: 3, DPs: 5, edges: 5
	SCC { #1 }
Removing DPs: Order(PosReal,>,Sum)... succeeded.
   |0|()	weight: 0
     h()	weight: 0
ifTimes(x1,x2,x3,x4,x5)	weight: 0
timesIter(x1,x2,x3)	weight: 0
     s(x1)	weight: (/ 1 2) + x1
 minus(x1,x2)	weight: 0
#timesIter(x1,x2,x3)	weight: 0
#plus(x1,x2)	weight: 0
    eq(x1,x2)	weight: 0
 false()	weight: 0
 #inc(x1)	weight: 0
   inc(x1)	weight: 0
  true()	weight: 0
     f()	weight: 0
  #eq(x1,x2)	weight: 0
#times(x1,x2)	weight: 0
 times(x1,x2)	weight: 0
   #f()	weight: 0
#minus(x1,x2)	weight: x2
ifPlus(x1,x2,x3,x4)	weight: 0
  plus(x1,x2)	weight: 0
#ifPlus(x1,x2,x3,x4)	weight: 0
     g()	weight: 0
#ifTimes(x1,x2,x3,x4,x5)	weight: 0
    Usable rules: { }
    Removed DPs: #1
Number of SCCs: 2, DPs: 4, edges: 4
	SCC { #7 #13 }
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)... succeeded.
   |0|()	weight: 0; (- (/ 1 8))
     h()	weight: 0; 0
ifTimes(x1,x2,x3,x4,x5)	weight: 0; 0
timesIter(x1,x2,x3)	weight: 0; 0
     s(x1)	weight: max{0, (/ 3 8) + x1_1}; (- (/ 3 4))
 minus(x1,x2)	weight: max{0, (/ 1 8) + x2_1 + x2_2 + x1_1}; 0
#timesIter(x1,x2,x3)	weight: 0; 0
#plus(x1,x2)	weight: max{0, (- (/ 1 4)) + x1_1}; 0
    eq(x1,x2)	weight: max{0, (- (/ 1 8)) + x1_1}; (- (/ 1 8))
 false()	weight: (/ 1 4); (- (/ 3 8))
 #inc(x1)	weight: 0; 0
   inc(x1)	weight: max{0, (/ 3 8) + x1_1}; (- (/ 7 8))
  true()	weight: 0; (- (/ 1 8))
     f()	weight: 0; 0
  #eq(x1,x2)	weight: 0; 0
#times(x1,x2)	weight: 0; 0
 times(x1,x2)	weight: 0; 0
   #f()	weight: 0; 0
#minus(x1,x2)	weight: 0; 0
ifPlus(x1,x2,x3,x4)	weight: 0; 0
  plus(x1,x2)	weight: 0; 0
#ifPlus(x1,x2,x3,x4)	weight: max{0, (- (/ 1 8)) + x1_1, x2_1}; 0
     g()	weight: 0; 0
#ifTimes(x1,x2,x3,x4,x5)	weight: 0; 0
    Usable rules: { 6..9 12..14 }
    Removed DPs: #13
Number of SCCs: 1, DPs: 2, edges: 2
	SCC { #3 #14 }
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)... succeeded.
   |0|()	weight: 0; (- (/ 1 8))
     h()	weight: 0; 0
ifTimes(x1,x2,x3,x4,x5)	weight: 0; 0
timesIter(x1,x2,x3)	weight: 0; 0
     s(x1)	weight: max{0, (/ 3 4) + x1_1}; (- (/ 1 4)) + x1_2
 minus(x1,x2)	weight: max{0, (/ 1 8) + x2_2 + x1_1}; 0
#timesIter(x1,x2,x3)	weight: max{0, (- (/ 3 8)) + x1_2, (- (/ 5 8)) + x1_1, (- (/ 1 8)) + x2_2, (- (/ 1 2)) + x3_2}; 0
#plus(x1,x2)	weight: 0; 0
    eq(x1,x2)	weight: max{0, (- (/ 1 4)) + x2_2 + x1_1}; (- (/ 1 8))
 false()	weight: (/ 3 8); (- (/ 5 8))
 #inc(x1)	weight: 0; 0
   inc(x1)	weight: (max (/ 3 4) 0); (- (/ 1 8))
  true()	weight: 0; (- (/ 1 8))
     f()	weight: 0; 0
  #eq(x1,x2)	weight: 0; 0
#times(x1,x2)	weight: 0; 0
 times(x1,x2)	weight: 0; 0
   #f()	weight: 0; 0
#minus(x1,x2)	weight: 0; 0
ifPlus(x1,x2,x3,x4)	weight: max{0, (/ 1 2) + x1_2, (/ 3 4) + x1_1, (/ 3 8) + x2_2, (/ 5 8) + x2_1, (/ 1 4) + x3_2, (- (/ 1 8)) + x4_2, (/ 1 8) + x4_1}; (- (/ 1 8)) + x4_2
  plus(x1,x2)	weight: max{0, (/ 1 4) + x2_1}; (- (/ 3 8))
#ifPlus(x1,x2,x3,x4)	weight: max{0, x1_1, x2_1}; 0
     g()	weight: 0; 0
#ifTimes(x1,x2,x3,x4,x5)	weight: max{0, (- (/ 1 4)) + x1_1, (- (/ 3 8)) + x2_1}; 0
    Usable rules: { 2 6..9 12..14 }
    Removed DPs: #14
Number of SCCs: 0, DPs: 0, edges: 0
YES