Input TRS:
    1: p(0()) -> 0()
    2: p(s(x)) -> x
    3: plus(x,0()) -> x
    4: plus(0(),y) -> y
    5: plus(s(x),y) -> s(plus(x,y))
    6: plus(s(x),y) -> s(plus(p(s(x)),y))
    7: plus(x,s(y)) -> s(plus(x,p(s(y))))
    8: times(0(),y) -> 0()
    9: times(s(0()),y) -> y
    10: times(s(x),y) -> plus(y,times(x,y))
    11: div(0(),y) -> 0()
    12: div(x,y) -> quot(x,y,y)
    13: quot(zero(y),s(y),z) -> 0()
    14: quot(s(x),s(y),z) -> quot(x,y,z)
    15: quot(x,0(),s(z)) -> s(div(x,s(z)))
    16: div(div(x,y),z) -> div(x,times(zero(y),z))
    17: eq(0(),0()) -> true()
    18: eq(s(x),0()) -> false()
    19: eq(0(),s(y)) -> false()
    20: eq(s(x),s(y)) -> eq(x,y)
    21: divides(y,x) -> eq(x,times(div(x,y),y))
    22: prime(s(s(x))) -> pr(s(s(x)),s(x))
    23: pr(x,s(0())) -> true()
    24: pr(x,s(s(y))) -> if(divides(s(s(y)),x),x,s(y))
    25: if(true(),x,y) -> false()
    26: if(false(),x,y) -> pr(x,y)
    27: zero(div(x,x)) -> x
    28: zero(divides(x,x)) -> x
    29: zero(times(x,x)) -> x
    30: zero(quot(x,x,x)) -> x
    31: zero(s(x)) -> if(eq(x,s(0())),plus(zero(0()),0()),s(plus(0(),zero(0()))))
Number of strict rules: 31
Direct Order(PosReal,>,Poly) ... failed.
Freezing p
1: p❆1_0() -> 0()
2: p❆1_s(x) -> x
3: plus(x,0()) -> x
4: plus(0(),y) -> y
5: plus(s(x),y) -> s(plus(x,y))
6: plus(s(x),y) -> s(plus(p❆1_s(x),y))
7: plus(x,s(y)) -> s(plus(x,p❆1_s(y)))
8: times(0(),y) -> 0()
9: times(s(0()),y) -> y
10: times(s(x),y) -> plus(y,times(x,y))
11: div(0(),y) -> 0()
12: div(x,y) -> quot(x,y,y)
13: quot(zero(y),s(y),z) -> 0()
14: quot(s(x),s(y),z) -> quot(x,y,z)
15: quot(x,0(),s(z)) -> s(div(x,s(z)))
16: div(div(x,y),z) -> div(x,times(zero(y),z))
17: eq(0(),0()) -> true()
18: eq(s(x),0()) -> false()
19: eq(0(),s(y)) -> false()
20: eq(s(x),s(y)) -> eq(x,y)
21: divides(y,x) -> eq(x,times(div(x,y),y))
22: prime(s(s(x))) -> pr(s(s(x)),s(x))
23: pr(x,s(0())) -> true()
24: pr(x,s(s(y))) -> if(divides(s(s(y)),x),x,s(y))
25: if(true(),x,y) -> false()
26: if(false(),x,y) -> pr(x,y)
27: zero(div(x,x)) -> x
28: zero(divides(x,x)) -> x
29: zero(times(x,x)) -> x
30: zero(quot(x,x,x)) -> x
31: zero(s(x)) -> if(eq(x,s(0())),plus(zero(0()),0()),s(plus(0(),zero(0()))))
32: p(0()) ->= p❆1_0()
33: p(s(_1)) ->= p❆1_s(_1)
Number of strict rules: 31
Direct Order(PosReal,>,Poly) ... failed.
Dependency Pairs:
   #1: #plus(s(x),y) -> #plus(p❆1_s(x),y)
   #2: #plus(s(x),y) -> #p❆1_s(x)
   #3: #pr(x,s(s(y))) -> #if(divides(s(s(y)),x),x,s(y))
   #4: #pr(x,s(s(y))) -> #divides(s(s(y)),x)
   #5: #div(x,y) -> #quot(x,y,y)
   #6: #zero(s(x)) -> #if(eq(x,s(0())),plus(zero(0()),0()),s(plus(0(),zero(0()))))
   #7: #zero(s(x)) -> #eq(x,s(0()))
   #8: #zero(s(x)) -> #plus(zero(0()),0())
   #9: #zero(s(x)) -> #zero(0())
   #10: #zero(s(x)) -> #plus(0(),zero(0()))
   #11: #zero(s(x)) -> #zero(0())
   #12: #quot(s(x),s(y),z) -> #quot(x,y,z)
   #13: #eq(s(x),s(y)) -> #eq(x,y)
   #14: #plus(x,s(y)) -> #plus(x,p❆1_s(y))
   #15: #plus(x,s(y)) -> #p❆1_s(y)
   #16: #times(s(x),y) -> #plus(y,times(x,y))
   #17: #times(s(x),y) -> #times(x,y)
   #18: #p(s(_1)) ->? #p❆1_s(_1)
   #19: #plus(s(x),y) -> #plus(x,y)
   #20: #prime(s(s(x))) -> #pr(s(s(x)),s(x))
   #21: #p(0()) ->? #p❆1_0()
   #22: #if(false(),x,y) -> #pr(x,y)
   #23: #divides(y,x) -> #eq(x,times(div(x,y),y))
   #24: #divides(y,x) -> #times(div(x,y),y)
   #25: #divides(y,x) -> #div(x,y)
   #26: #div(div(x,y),z) -> #div(x,times(zero(y),z))
   #27: #div(div(x,y),z) -> #times(zero(y),z)
   #28: #div(div(x,y),z) -> #zero(y)
   #29: #quot(x,0(),s(z)) -> #div(x,s(z))
Number of SCCs: 4, DPs: 15, edges: 30
	SCC { #17 }
Removing DPs: Order(PosReal,>,Sum)... succeeded.
  zero(x1)	weight: 0
 #div(x1,x2)	weight: 0
     s(x1)	weight: (/ 1 2) + x1
p❆1_s(x1)	weight: 0
 prime(x1)	weight: 0
#p❆1_0()	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
p❆1_0()	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
#zero(x1)	weight: 0
divides(x1,x2)	weight: 0
    Usable rules: { }
    Removed DPs: #17
Number of SCCs: 3, DPs: 14, edges: 29
	SCC { #13 }
Removing DPs: Order(PosReal,>,Sum)... succeeded.
  zero(x1)	weight: 0
 #div(x1,x2)	weight: 0
     s(x1)	weight: (/ 1 2) + x1
p❆1_s(x1)	weight: 0
 prime(x1)	weight: 0
#p❆1_0()	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
p❆1_0()	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
#zero(x1)	weight: 0
divides(x1,x2)	weight: 0
    Usable rules: { }
    Removed DPs: #13
Number of SCCs: 2, DPs: 13, edges: 28
	SCC { #1 #14 #19 }
Removing DPs: Order(PosReal,>,Sum)... succeeded.
  zero(x1)	weight: 0
 #div(x1,x2)	weight: 0
     s(x1)	weight: (/ 1 2) + x1
p❆1_s(x1)	weight: (/ 1 4) + x1
 prime(x1)	weight: 0
#p❆1_0()	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
   #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: 0
 times(x1,x2)	weight: 0
p❆1_0()	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
#zero(x1)	weight: 0
divides(x1,x2)	weight: 0
    Usable rules: { 2 }
    Removed DPs: #1 #19
Number of SCCs: 2, DPs: 11, edges: 20
	SCC { #14 }
Removing DPs: Order(PosReal,>,Sum)... succeeded.
  zero(x1)	weight: 0
 #div(x1,x2)	weight: 0
     s(x1)	weight: (/ 1 2) + x1
p❆1_s(x1)	weight: (/ 1 4) + x1
 prime(x1)	weight: 0
#p❆1_0()	weight: 0
#plus(x1,x2)	weight: x1 + x2
  #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: 0
    if(x1,x2,x3)	weight: 0
  quot(x1,x2,x3)	weight: 0
 times(x1,x2)	weight: 0
p❆1_0()	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
#zero(x1)	weight: 0
divides(x1,x2)	weight: 0
    Usable rules: { 2 }
    Removed DPs: #14
Number of SCCs: 1, DPs: 10, edges: 19
	SCC { #3..6 #12 #22 #25 #26 #28 #29 }
Removing DPs: Order(PosReal,>,Sum)... succeeded.
  zero(x1)	weight: 0
 #div(x1,x2)	weight: (/ 1 2)
     s(x1)	weight: (/ 1 4) + x1
p❆1_s(x1)	weight: x1
 prime(x1)	weight: 0
#p❆1_0()	weight: 0
#plus(x1,x2)	weight: x1 + x2
  #pr(x1,x2)	weight: x2
    eq(x1,x2)	weight: (/ 3 8) + x1 + x2
 false()	weight: 0
   div(x1,x2)	weight: (/ 1 8)
   #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: x1 + x3
  quot(x1,x2,x3)	weight: (/ 1 4)
 times(x1,x2)	weight: (/ 1 8) + x1
p❆1_0()	weight: 0
    pr(x1,x2)	weight: (/ 1 8) + x1 + x2
#divides(x1,x2)	weight: (/ 1 2)
  plus(x1,x2)	weight: x1 + x2
  #if(x1,x2,x3)	weight: (/ 1 4) + x3
#quot(x1,x2,x3)	weight: (/ 1 2)
#zero(x1)	weight: (/ 1 2)
divides(x1,x2)	weight: (/ 1 2)
    Usable rules: { 2..7 }
    Removed DPs: #22
Number of SCCs: 2, DPs: 4, edges: 8
	SCC { #5 #12 #26 #29 }
Removing DPs: Order(PosReal,>,Sum)... succeeded.
  zero(x1)	weight: 0
 #div(x1,x2)	weight: (/ 1 4) + x1
     s(x1)	weight: (/ 1 8) + x1
p❆1_s(x1)	weight: x1
 prime(x1)	weight: 0
#p❆1_0()	weight: 0
#plus(x1,x2)	weight: x1 + x2
  #pr(x1,x2)	weight: 0
    eq(x1,x2)	weight: (/ 1 4) + x1 + x2
 false()	weight: 0
   div(x1,x2)	weight: (/ 1 16) + x1
   #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: x1 + x3
  quot(x1,x2,x3)	weight: (/ 1 8)
 times(x1,x2)	weight: (/ 1 16)
p❆1_0()	weight: 0
    pr(x1,x2)	weight: (/ 1 16) + x1 + x2
#divides(x1,x2)	weight: (/ 1 4)
  plus(x1,x2)	weight: x1 + x2
  #if(x1,x2,x3)	weight: (/ 1 8) + x3
#quot(x1,x2,x3)	weight: (/ 1 4) + x1
#zero(x1)	weight: (/ 1 4)
divides(x1,x2)	weight: (/ 1 4)
    Usable rules: { 2..7 }
    Removed DPs: #12 #26
Number of SCCs: 2, DPs: 2, edges: 2
	SCC { #5 #29 }
Removing DPs: Order(PosReal,>,Sum)... Order(PosReal,>,Max)... succeeded.
  zero(x1)	weight: 0
 #div(x1,x2)	weight: max{0, (/ 1 2) + x2}
     s(x1)	weight: 0
p❆1_s(x1)	weight: 0
 prime(x1)	weight: 0
#p❆1_0()	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
p❆1_0()	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) + x3, (/ 1 4) + x2}
#zero(x1)	weight: 0
divides(x1,x2)	weight: 0
    Usable rules: { }
    Removed DPs: #5 #29
Number of SCCs: 1, DPs: 0, edges: 0
YES