Input TRS:
    1: -(x,0()) -> x
    2: -(s(x),s(y)) -> -(x,y)
    3: *(x,0()) -> 0()
    4: *(x,s(y)) -> +(*(x,y),x)
    5: if(true(),x,y) -> x
    6: if(false(),x,y) -> y
    7: odd(0()) -> false()
    8: odd(s(0())) -> true()
    9: odd(s(s(x))) -> odd(x)
    10: half(0()) -> 0()
    11: half(s(0())) -> 0()
    12: half(s(s(x))) -> s(half(x))
    13: if(true(),x,y) -> true()
    14: if(false(),x,y) -> false()
    15: pow(x,y) -> f(x,y,s(0()))
    16: f(x,0(),z) -> z
    17: f(x,s(y),z) -> if(odd(s(y)),f(x,y,*(x,z)),f(*(x,x),half(s(y)),z))
Number of strict rules: 17
Direct Order(PosReal,>,Poly) ... failed.
Freezing half odd
1: -(x,0()) -> x
2: -(s(x),s(y)) -> -(x,y)
3: *(x,0()) -> 0()
4: *(x,s(y)) -> +(*(x,y),x)
5: if(true(),x,y) -> x
6: if(false(),x,y) -> y
7: odd❆1_0() -> false()
8: odd❆1_s(0()) -> true()
9: odd❆1_s(s(x)) -> odd(x)
10: half❆1_0() -> 0()
11: half❆1_s(0()) -> 0()
12: half❆1_s(s(x)) -> s(half(x))
13: if(true(),x,y) -> true()
14: if(false(),x,y) -> false()
15: pow(x,y) -> f(x,y,s(0()))
16: f(x,0(),z) -> z
17: f(x,s(y),z) -> if(odd❆1_s(y),f(x,y,*(x,z)),f(*(x,x),half❆1_s(y),z))
18: odd(0()) ->= odd❆1_0()
19: odd(s(_1)) ->= odd❆1_s(_1)
20: half(0()) ->= half❆1_0()
21: half(s(_1)) ->= half❆1_s(_1)
Number of strict rules: 17
Direct Order(PosReal,>,Poly) ... failed.
Dependency Pairs:
   #1: #-(s(x),s(y)) -> #-(x,y)
   #2: #odd❆1_s(s(x)) -> #odd(x)
   #3: #half❆1_s(s(x)) -> #half(x)
   #4: #half(0()) ->? #half❆1_0()
   #5: #f(x,s(y),z) -> #if(odd❆1_s(y),f(x,y,*(x,z)),f(*(x,x),half❆1_s(y),z))
   #6: #f(x,s(y),z) -> #odd❆1_s(y)
   #7: #f(x,s(y),z) -> #f(x,y,*(x,z))
   #8: #f(x,s(y),z) -> #*(x,z)
   #9: #f(x,s(y),z) -> #f(*(x,x),half❆1_s(y),z)
   #10: #f(x,s(y),z) -> #*(x,x)
   #11: #f(x,s(y),z) -> #half❆1_s(y)
   #12: #odd(s(_1)) ->? #odd❆1_s(_1)
   #13: #half(s(_1)) ->? #half❆1_s(_1)
   #14: #pow(x,y) -> #f(x,y,s(0()))
   #15: #*(x,s(y)) -> #*(x,y)
   #16: #odd(0()) ->? #odd❆1_0()
Number of SCCs: 5, DPs: 8, edges: 10
	SCC { #15 }
Removing DPs: Order(PosReal,>,Sum)... succeeded.
#odd❆1_0()	weight: 0
     s(x1)	weight: (/ 1 2) + x1
#half❆1_s(x1)	weight: 0
   pow(x1,x2)	weight: 0
 false()	weight: 0
   #*(x1,x2)	weight: x2
#half(x1)	weight: 0
  true()	weight: 0
 #pow(x1,x2)	weight: 0
     f(x1,x2,x3)	weight: 0
#half❆1_0()	weight: 0
  half(x1)	weight: 0
half❆1_s(x1)	weight: 0
     0()	weight: 0
    if(x1,x2,x3)	weight: 0
odd❆1_s(x1)	weight: 0
     -(x1,x2)	weight: 0
   #f(x1,x2,x3)	weight: 0
 #odd(x1)	weight: 0
#odd❆1_s(x1)	weight: 0
   odd(x1)	weight: 0
   #-(x1,x2)	weight: 0
odd❆1_0()	weight: 0
  #if(x1,x2,x3)	weight: 0
     +(x1,x2)	weight: 0
half❆1_0()	weight: 0
     *(x1,x2)	weight: 0
    Usable rules: { }
    Removed DPs: #15
Number of SCCs: 4, DPs: 7, edges: 9
	SCC { #1 }
Removing DPs: Order(PosReal,>,Sum)... succeeded.
#odd❆1_0()	weight: 0
     s(x1)	weight: (/ 1 2) + x1
#half❆1_s(x1)	weight: 0
   pow(x1,x2)	weight: 0
 false()	weight: 0
   #*(x1,x2)	weight: 0
#half(x1)	weight: 0
  true()	weight: 0
 #pow(x1,x2)	weight: 0
     f(x1,x2,x3)	weight: 0
#half❆1_0()	weight: 0
  half(x1)	weight: 0
half❆1_s(x1)	weight: 0
     0()	weight: 0
    if(x1,x2,x3)	weight: 0
odd❆1_s(x1)	weight: 0
     -(x1,x2)	weight: 0
   #f(x1,x2,x3)	weight: 0
 #odd(x1)	weight: 0
#odd❆1_s(x1)	weight: 0
   odd(x1)	weight: 0
   #-(x1,x2)	weight: x2
odd❆1_0()	weight: 0
  #if(x1,x2,x3)	weight: 0
     +(x1,x2)	weight: 0
half❆1_0()	weight: 0
     *(x1,x2)	weight: 0
    Usable rules: { }
    Removed DPs: #1
Number of SCCs: 3, DPs: 6, edges: 8
	SCC { #2 #12 }
Removing DPs: Order(PosReal,>,Sum)... succeeded.
#odd❆1_0()	weight: 0
     s(x1)	weight: (/ 1 2) + x1
#half❆1_s(x1)	weight: 0
   pow(x1,x2)	weight: 0
 false()	weight: 0
   #*(x1,x2)	weight: 0
#half(x1)	weight: 0
  true()	weight: 0
 #pow(x1,x2)	weight: 0
     f(x1,x2,x3)	weight: 0
#half❆1_0()	weight: 0
  half(x1)	weight: 0
half❆1_s(x1)	weight: 0
     0()	weight: 0
    if(x1,x2,x3)	weight: 0
odd❆1_s(x1)	weight: 0
     -(x1,x2)	weight: 0
   #f(x1,x2,x3)	weight: 0
 #odd(x1)	weight: x1
#odd❆1_s(x1)	weight: x1
   odd(x1)	weight: 0
   #-(x1,x2)	weight: 0
odd❆1_0()	weight: 0
  #if(x1,x2,x3)	weight: 0
     +(x1,x2)	weight: 0
half❆1_0()	weight: 0
     *(x1,x2)	weight: 0
    Usable rules: { }
    Removed DPs: #2 #12
Number of SCCs: 2, DPs: 4, edges: 6
	SCC { #3 #13 }
Removing DPs: Order(PosReal,>,Sum)... succeeded.
#odd❆1_0()	weight: 0
     s(x1)	weight: (/ 1 2) + x1
#half❆1_s(x1)	weight: x1
   pow(x1,x2)	weight: 0
 false()	weight: 0
   #*(x1,x2)	weight: 0
#half(x1)	weight: x1
  true()	weight: 0
 #pow(x1,x2)	weight: 0
     f(x1,x2,x3)	weight: 0
#half❆1_0()	weight: 0
  half(x1)	weight: 0
half❆1_s(x1)	weight: 0
     0()	weight: 0
    if(x1,x2,x3)	weight: 0
odd❆1_s(x1)	weight: 0
     -(x1,x2)	weight: 0
   #f(x1,x2,x3)	weight: 0
 #odd(x1)	weight: 0
#odd❆1_s(x1)	weight: 0
   odd(x1)	weight: 0
   #-(x1,x2)	weight: 0
odd❆1_0()	weight: 0
  #if(x1,x2,x3)	weight: 0
     +(x1,x2)	weight: 0
half❆1_0()	weight: 0
     *(x1,x2)	weight: 0
    Usable rules: { }
    Removed DPs: #3 #13
Number of SCCs: 1, DPs: 2, edges: 4
	SCC { #7 #9 }
Removing DPs: Order(PosReal,>,Sum)... succeeded.
#odd❆1_0()	weight: 0
     s(x1)	weight: (/ 1 2) + x1
#half❆1_s(x1)	weight: 0
   pow(x1,x2)	weight: 0
 false()	weight: 0
   #*(x1,x2)	weight: 0
#half(x1)	weight: 0
  true()	weight: 0
 #pow(x1,x2)	weight: 0
     f(x1,x2,x3)	weight: 0
#half❆1_0()	weight: 0
  half(x1)	weight: (/ 1 4) + x1
half❆1_s(x1)	weight: (/ 1 4) + x1
     0()	weight: 0
    if(x1,x2,x3)	weight: 0
odd❆1_s(x1)	weight: 0
     -(x1,x2)	weight: 0
   #f(x1,x2,x3)	weight: x2
 #odd(x1)	weight: 0
#odd❆1_s(x1)	weight: 0
   odd(x1)	weight: 0
   #-(x1,x2)	weight: 0
odd❆1_0()	weight: 0
  #if(x1,x2,x3)	weight: 0
     +(x1,x2)	weight: (/ 1 2)
half❆1_0()	weight: 0
     *(x1,x2)	weight: (/ 1 4)
    Usable rules: { 10..12 20 21 }
    Removed DPs: #7 #9
Number of SCCs: 0, DPs: 0, edges: 0
YES