Input TRS:
    1: p(s(x)) -> x
    2: plus(x,0()) -> x
    3: plus(0(),y) -> y
    4: plus(s(x),y) -> s(plus(x,y))
    5: plus(s(x),y) -> s(plus(p(s(x)),y))
    6: plus(x,s(y)) -> s(plus(x,p(s(y))))
    7: times(0(),y) -> 0()
    8: times(s(0()),y) -> y
    9: times(s(x),y) -> plus(y,times(x,y))
    10: div(0(),y) -> 0()
    11: div(x,y) -> quot(x,y,y)
    12: quot(0(),s(y),z) -> 0()
    13: quot(s(x),s(y),z) -> quot(x,y,z)
    14: quot(x,0(),s(z)) -> s(div(x,s(z)))
    15: div(div(x,y),z) -> div(x,times(y,z))
    16: eq(0(),0()) -> true()
    17: eq(s(x),0()) -> false()
    18: eq(0(),s(y)) -> false()
    19: eq(s(x),s(y)) -> eq(x,y)
    20: divides(y,x) -> eq(x,times(div(x,y),y))
    21: prime(s(s(x))) -> pr(s(s(x)),s(x))
    22: pr(x,s(0())) -> true()
    23: pr(x,s(s(y))) -> if(divides(s(s(y)),x),x,s(y))
    24: if(true(),x,y) -> false()
    25: if(false(),x,y) -> pr(x,y)
Number of strict rules: 25
Direct Order(PosReal,>,Poly) ... failed.
Freezing p
1: p❆1_s(x) -> x
2: plus(x,0()) -> x
3: plus(0(),y) -> y
4: plus(s(x),y) -> s(plus(x,y))
5: plus(s(x),y) -> s(plus(p❆1_s(x),y))
6: plus(x,s(y)) -> s(plus(x,p❆1_s(y)))
7: times(0(),y) -> 0()
8: times(s(0()),y) -> y
9: times(s(x),y) -> plus(y,times(x,y))
10: div(0(),y) -> 0()
11: div(x,y) -> quot(x,y,y)
12: quot(0(),s(y),z) -> 0()
13: quot(s(x),s(y),z) -> quot(x,y,z)
14: quot(x,0(),s(z)) -> s(div(x,s(z)))
15: div(div(x,y),z) -> div(x,times(y,z))
16: eq(0(),0()) -> true()
17: eq(s(x),0()) -> false()
18: eq(0(),s(y)) -> false()
19: eq(s(x),s(y)) -> eq(x,y)
20: divides(y,x) -> eq(x,times(div(x,y),y))
21: prime(s(s(x))) -> pr(s(s(x)),s(x))
22: pr(x,s(0())) -> true()
23: pr(x,s(s(y))) -> if(divides(s(s(y)),x),x,s(y))
24: if(true(),x,y) -> false()
25: if(false(),x,y) -> pr(x,y)
26: p(s(_1)) ->= p❆1_s(_1)
Number of strict rules: 25
Direct Order(PosReal,>,Poly) ... failed.
Dependency Pairs:
   #1: #plus(x,s(y)) -> #plus(x,p❆1_s(y))
   #2: #plus(x,s(y)) -> #p❆1_s(y)
   #3: #quot(s(x),s(y),z) -> #quot(x,y,z)
   #4: #times(s(x),y) -> #plus(y,times(x,y))
   #5: #times(s(x),y) -> #times(x,y)
   #6: #div(x,y) -> #quot(x,y,y)
   #7: #pr(x,s(s(y))) -> #if(divides(s(s(y)),x),x,s(y))
   #8: #pr(x,s(s(y))) -> #divides(s(s(y)),x)
   #9: #quot(x,0(),s(z)) -> #div(x,s(z))
   #10: #if(false(),x,y) -> #pr(x,y)
   #11: #divides(y,x) -> #eq(x,times(div(x,y),y))
   #12: #divides(y,x) -> #times(div(x,y),y)
   #13: #divides(y,x) -> #div(x,y)
   #14: #plus(s(x),y) -> #plus(p❆1_s(x),y)
   #15: #plus(s(x),y) -> #p❆1_s(x)
   #16: #eq(s(x),s(y)) -> #eq(x,y)
   #17: #p(s(_1)) ->? #p❆1_s(_1)
   #18: #prime(s(s(x))) -> #pr(s(s(x)),s(x))
   #19: #div(div(x,y),z) -> #div(x,times(y,z))
   #20: #div(div(x,y),z) -> #times(y,z)
   #21: #plus(s(x),y) -> #plus(x,y)
Number of SCCs: 5, DPs: 11, edges: 21
	SCC { #5 }
Removing DPs: Order(PosReal,>,Sum)... succeeded.
 #div(x1,x2)	weight: 0
     s(x1)	weight: (/ 1 2) + x1
p❆1_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
   #p(x1)	weight: 0
#p❆1_s(x1)	weight: 0
  true()	weight: 0
  #eq(x1,x2)	weight: 0
#prime(x1)	weight: 0
     p(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: #5
Number of SCCs: 4, DPs: 10, edges: 20
	SCC { #16 }
Removing DPs: Order(PosReal,>,Sum)... succeeded.
 #div(x1,x2)	weight: 0
     s(x1)	weight: (/ 1 2) + x1
p❆1_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
   #p(x1)	weight: 0
#p❆1_s(x1)	weight: 0
  true()	weight: 0
  #eq(x1,x2)	weight: x2
#prime(x1)	weight: 0
     p(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: #16
Number of SCCs: 3, DPs: 9, edges: 19
	SCC { #7 #10 }
Removing DPs: Order(PosReal,>,Sum)... succeeded.
 #div(x1,x2)	weight: 0
     s(x1)	weight: (/ 1 4) + x1
p❆1_s(x1)	weight: (/ 1 8)
 prime(x1)	weight: 0
#plus(x1,x2)	weight: 0
  #pr(x1,x2)	weight: x2
    eq(x1,x2)	weight: x2
 false()	weight: 0
   div(x1,x2)	weight: (/ 1 8) + x1 + x2
   #p(x1)	weight: 0
#p❆1_s(x1)	weight: 0
  true()	weight: 0
  #eq(x1,x2)	weight: 0
#prime(x1)	weight: 0
     p(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) + x2
    pr(x1,x2)	weight: 0
#divides(x1,x2)	weight: 0
  plus(x1,x2)	weight: (/ 3 8) + x1
  #if(x1,x2,x3)	weight: (/ 1 8) + x3
#quot(x1,x2,x3)	weight: 0
divides(x1,x2)	weight: (/ 1 8)
    Usable rules: { }
    Removed DPs: #7 #10
Number of SCCs: 2, DPs: 7, edges: 17
	SCC { #1 #14 #21 }
Removing DPs: Order(PosReal,>,Sum)... succeeded.
 #div(x1,x2)	weight: 0
     s(x1)	weight: (/ 1 4) + x1
p❆1_s(x1)	weight: (/ 1 8) + x1
 prime(x1)	weight: 0
#plus(x1,x2)	weight: x1
  #pr(x1,x2)	weight: 0
    eq(x1,x2)	weight: x2
 false()	weight: 0
   div(x1,x2)	weight: (/ 1 8) + x1 + x2
   #p(x1)	weight: 0
#p❆1_s(x1)	weight: 0
  true()	weight: 0
  #eq(x1,x2)	weight: 0
#prime(x1)	weight: 0
     p(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) + x2
    pr(x1,x2)	weight: 0
#divides(x1,x2)	weight: 0
  plus(x1,x2)	weight: (/ 3 8) + x1
  #if(x1,x2,x3)	weight: (/ 1 8)
#quot(x1,x2,x3)	weight: 0
divides(x1,x2)	weight: (/ 1 8)
    Usable rules: { 1 }
    Removed DPs: #14 #21
Number of SCCs: 2, DPs: 5, edges: 9
	SCC { #1 }
Removing DPs: Order(PosReal,>,Sum)... succeeded.
 #div(x1,x2)	weight: 0
     s(x1)	weight: (/ 1 4) + x1
p❆1_s(x1)	weight: (/ 1 8) + x1
 prime(x1)	weight: 0
#plus(x1,x2)	weight: x1 + x2
  #pr(x1,x2)	weight: 0
    eq(x1,x2)	weight: x2
 false()	weight: 0
   div(x1,x2)	weight: (/ 1 8) + x1 + x2
   #p(x1)	weight: 0
#p❆1_s(x1)	weight: 0
  true()	weight: 0
  #eq(x1,x2)	weight: 0
#prime(x1)	weight: 0
     p(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) + x2
    pr(x1,x2)	weight: 0
#divides(x1,x2)	weight: 0
  plus(x1,x2)	weight: (/ 3 8) + x1
  #if(x1,x2,x3)	weight: (/ 1 8)
#quot(x1,x2,x3)	weight: 0
divides(x1,x2)	weight: (/ 1 8)
    Usable rules: { 1 }
    Removed DPs: #1
Number of SCCs: 1, DPs: 4, edges: 8
	SCC { #3 #6 #9 #19 }
Removing DPs: Order(PosReal,>,Sum)... succeeded.
 #div(x1,x2)	weight: x1
     s(x1)	weight: (/ 1 8) + x1
p❆1_s(x1)	weight: (/ 1 8) + x1
 prime(x1)	weight: 0
#plus(x1,x2)	weight: x1 + x2
  #pr(x1,x2)	weight: 0
    eq(x1,x2)	weight: x2
 false()	weight: 0
   div(x1,x2)	weight: (/ 1 8) + x1 + x2
   #p(x1)	weight: 0
#p❆1_s(x1)	weight: 0
  true()	weight: 0
  #eq(x1,x2)	weight: 0
#prime(x1)	weight: 0
     p(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) + x2
    pr(x1,x2)	weight: 0
#divides(x1,x2)	weight: 0
  plus(x1,x2)	weight: (/ 3 8) + x1
  #if(x1,x2,x3)	weight: (/ 1 8)
#quot(x1,x2,x3)	weight: x1
divides(x1,x2)	weight: (/ 1 8)
    Usable rules: { 1 }
    Removed DPs: #3 #19
Number of SCCs: 2, DPs: 2, edges: 2
	SCC { #6 #9 }
Removing DPs: Order(PosReal,>,Sum)... Order(PosReal,>,Max)... succeeded.
 #div(x1,x2)	weight: max{0, x2}
     s(x1)	weight: (/ 1 4)
p❆1_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
   #p(x1)	weight: 0
#p❆1_s(x1)	weight: 0
  true()	weight: 0
  #eq(x1,x2)	weight: 0
#prime(x1)	weight: 0
     p(x1)	weight: 0
#times(x1,x2)	weight: 0
     0()	weight: (/ 1 2)
    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, x3, x2}
divides(x1,x2)	weight: 0
    Usable rules: { }
    Removed DPs: #9
Number of SCCs: 1, DPs: 0, edges: 0
YES