Input TRS:
    1: plus(x,0()) -> x
    2: plus(0(),y) -> y
    3: plus(s(x),y) -> s(plus(x,y))
    4: times(0(),y) -> 0()
    5: times(s(0()),y) -> y
    6: times(s(x),y) -> plus(y,times(x,y))
    7: div(0(),y) -> 0()
    8: div(x,y) -> quot(x,y,y)
    9: quot(0(),s(y),z) -> 0()
    10: quot(s(x),s(y),z) -> quot(x,y,z)
    11: quot(x,0(),s(z)) -> s(div(x,s(z)))
    12: div(div(x,y),z) -> div(x,times(y,z))
    13: eq(0(),0()) -> true()
    14: eq(s(x),0()) -> false()
    15: eq(0(),s(y)) -> false()
    16: eq(s(x),s(y)) -> eq(x,y)
    17: divides(y,x) -> eq(x,times(div(x,y),y))
    18: prime(s(s(x))) -> pr(s(s(x)),s(x))
    19: pr(x,s(0())) -> true()
    20: pr(x,s(s(y))) -> if(divides(s(s(y)),x),x,s(y))
    21: if(true(),x,y) -> false()
    22: if(false(),x,y) -> pr(x,y)
Number of strict rules: 22
Direct Order(PosReal,>,Poly) ... failed.
Freezing ... failed.
Dependency Pairs:
   #1: #times(s(x),y) -> #plus(y,times(x,y))
   #2: #times(s(x),y) -> #times(x,y)
   #3: #quot(x,0(),s(z)) -> #div(x,s(z))
   #4: #div(div(x,y),z) -> #div(x,times(y,z))
   #5: #div(div(x,y),z) -> #times(y,z)
   #6: #pr(x,s(s(y))) -> #if(divides(s(s(y)),x),x,s(y))
   #7: #pr(x,s(s(y))) -> #divides(s(s(y)),x)
   #8: #quot(s(x),s(y),z) -> #quot(x,y,z)
   #9: #if(false(),x,y) -> #pr(x,y)
   #10: #divides(y,x) -> #eq(x,times(div(x,y),y))
   #11: #divides(y,x) -> #times(div(x,y),y)
   #12: #divides(y,x) -> #div(x,y)
   #13: #eq(s(x),s(y)) -> #eq(x,y)
   #14: #plus(s(x),y) -> #plus(x,y)
   #15: #div(x,y) -> #quot(x,y,y)
   #16: #prime(s(s(x))) -> #pr(s(s(x)),s(x))
Number of SCCs: 5, DPs: 9, edges: 13
	SCC { #14 }
Removing DPs: Order(PosReal,>,Sum)... succeeded.
 #div(x1,x2)	weight: 0
     s(x1)	weight: (/ 1 2) + x1
 prime(x1)	weight: 0
#plus(x1,x2)	weight: x1
  #pr(x1,x2)	weight: 0
    eq(x1,x2)	weight: 0
 false()	weight: 0
   div(x1,x2)	weight: 0
  true()	weight: 0
  #eq(x1,x2)	weight: 0
#prime(x1)	weight: 0
#times(x1,x2)	weight: 0
     0()	weight: 0
    if(x1,x2,x3)	weight: 0
  quot(x1,x2,x3)	weight: 0
 times(x1,x2)	weight: 0
    pr(x1,x2)	weight: 0
#divides(x1,x2)	weight: 0
  plus(x1,x2)	weight: 0
  #if(x1,x2,x3)	weight: 0
#quot(x1,x2,x3)	weight: 0
divides(x1,x2)	weight: 0
    Usable rules: { }
    Removed DPs: #14
Number of SCCs: 4, DPs: 8, edges: 12
	SCC { #2 }
Removing DPs: Order(PosReal,>,Sum)... succeeded.
 #div(x1,x2)	weight: 0
     s(x1)	weight: (/ 1 2) + x1
 prime(x1)	weight: 0
#plus(x1,x2)	weight: 0
  #pr(x1,x2)	weight: 0
    eq(x1,x2)	weight: 0
 false()	weight: 0
   div(x1,x2)	weight: 0
  true()	weight: 0
  #eq(x1,x2)	weight: 0
#prime(x1)	weight: 0
#times(x1,x2)	weight: x1
     0()	weight: 0
    if(x1,x2,x3)	weight: 0
  quot(x1,x2,x3)	weight: 0
 times(x1,x2)	weight: 0
    pr(x1,x2)	weight: 0
#divides(x1,x2)	weight: 0
  plus(x1,x2)	weight: 0
  #if(x1,x2,x3)	weight: 0
#quot(x1,x2,x3)	weight: 0
divides(x1,x2)	weight: 0
    Usable rules: { }
    Removed DPs: #2
Number of SCCs: 3, DPs: 7, edges: 11
	SCC { #13 }
Removing DPs: Order(PosReal,>,Sum)... succeeded.
 #div(x1,x2)	weight: 0
     s(x1)	weight: (/ 1 2) + x1
 prime(x1)	weight: 0
#plus(x1,x2)	weight: 0
  #pr(x1,x2)	weight: 0
    eq(x1,x2)	weight: 0
 false()	weight: 0
   div(x1,x2)	weight: 0
  true()	weight: 0
  #eq(x1,x2)	weight: x2
#prime(x1)	weight: 0
#times(x1,x2)	weight: 0
     0()	weight: 0
    if(x1,x2,x3)	weight: 0
  quot(x1,x2,x3)	weight: 0
 times(x1,x2)	weight: 0
    pr(x1,x2)	weight: 0
#divides(x1,x2)	weight: 0
  plus(x1,x2)	weight: 0
  #if(x1,x2,x3)	weight: 0
#quot(x1,x2,x3)	weight: 0
divides(x1,x2)	weight: 0
    Usable rules: { }
    Removed DPs: #13
Number of SCCs: 2, DPs: 6, edges: 10
	SCC { #6 #9 }
Removing DPs: Order(PosReal,>,Sum)... succeeded.
 #div(x1,x2)	weight: 0
     s(x1)	weight: (/ 1 4) + x1
 prime(x1)	weight: 0
#plus(x1,x2)	weight: 0
  #pr(x1,x2)	weight: x2
    eq(x1,x2)	weight: 0
 false()	weight: 0
   div(x1,x2)	weight: (/ 1 8) + x2
  true()	weight: 0
  #eq(x1,x2)	weight: 0
#prime(x1)	weight: 0
#times(x1,x2)	weight: 0
     0()	weight: 0
    if(x1,x2,x3)	weight: 0
  quot(x1,x2,x3)	weight: (/ 1 4)
 times(x1,x2)	weight: (/ 1 8) + x1 + x2
    pr(x1,x2)	weight: 0
#divides(x1,x2)	weight: 0
  plus(x1,x2)	weight: (/ 3 8) + x2
  #if(x1,x2,x3)	weight: (/ 1 8) + x3
#quot(x1,x2,x3)	weight: 0
divides(x1,x2)	weight: (/ 1 8)
    Usable rules: { 13..16 }
    Removed DPs: #6 #9
Number of SCCs: 1, DPs: 4, edges: 8
	SCC { #3 #4 #8 #15 }
Removing DPs: Order(PosReal,>,Sum)... succeeded.
 #div(x1,x2)	weight: x1
     s(x1)	weight: (/ 1 4) + x1
 prime(x1)	weight: 0
#plus(x1,x2)	weight: 0
  #pr(x1,x2)	weight: 0
    eq(x1,x2)	weight: 0
 false()	weight: 0
   div(x1,x2)	weight: (/ 1 8) + x1 + x2
  true()	weight: 0
  #eq(x1,x2)	weight: 0
#prime(x1)	weight: 0
#times(x1,x2)	weight: 0
     0()	weight: 0
    if(x1,x2,x3)	weight: 0
  quot(x1,x2,x3)	weight: (/ 1 4)
 times(x1,x2)	weight: (/ 1 4) + x1
    pr(x1,x2)	weight: 0
#divides(x1,x2)	weight: 0
  plus(x1,x2)	weight: (/ 5 8)
  #if(x1,x2,x3)	weight: (/ 1 8)
#quot(x1,x2,x3)	weight: x1
divides(x1,x2)	weight: (/ 1 8)
    Usable rules: { }
    Removed DPs: #4 #8
Number of SCCs: 1, DPs: 2, edges: 2
	SCC { #3 #15 }
Removing DPs: Order(PosReal,>,Sum)... Order(PosReal,>,Max)... succeeded.
 #div(x1,x2)	weight: max{0, (/ 1 2) + x2}
     s(x1)	weight: 0
 prime(x1)	weight: 0
#plus(x1,x2)	weight: 0
  #pr(x1,x2)	weight: 0
    eq(x1,x2)	weight: 0
 false()	weight: 0
   div(x1,x2)	weight: 0
  true()	weight: 0
  #eq(x1,x2)	weight: 0
#prime(x1)	weight: 0
#times(x1,x2)	weight: 0
     0()	weight: (/ 1 4)
    if(x1,x2,x3)	weight: 0
  quot(x1,x2,x3)	weight: 0
 times(x1,x2)	weight: 0
    pr(x1,x2)	weight: 0
#divides(x1,x2)	weight: 0
  plus(x1,x2)	weight: 0
  #if(x1,x2,x3)	weight: 0
#quot(x1,x2,x3)	weight: max{0, (/ 1 4) + x2}
divides(x1,x2)	weight: 0
    Usable rules: { }
    Removed DPs: #15
Number of SCCs: 0, DPs: 0, edges: 0
YES