Input TRS:
    1: minus(x,0()) -> x
    2: minus(0(),y) -> 0()
    3: minus(s(x),s(y)) -> minus(p(s(x)),p(s(y)))
    4: minus(x,plus(y,z)) -> minus(minus(x,y),z)
    5: p(s(s(x))) -> s(p(s(x)))
    6: p(0()) -> s(s(0()))
    7: div(s(x),s(y)) -> s(div(minus(x,y),s(y)))
    8: div(plus(x,y),z) -> plus(div(x,z),div(y,z))
    9: plus(0(),y) -> y
    10: plus(s(x),y) -> s(plus(y,minus(s(x),s(0()))))
Number of strict rules: 10
Direct Order(PosReal,>,Poly) ... failed.
Freezing p
1: minus(x,0()) -> x
2: minus(0(),y) -> 0()
3: minus(s(x),s(y)) -> minus(p❆1_s(x),p❆1_s(y))
4: minus(x,plus(y,z)) -> minus(minus(x,y),z)
5: p❆1_s(s(x)) -> s(p❆1_s(x))
6: p❆1_0() -> s(s(0()))
7: div(s(x),s(y)) -> s(div(minus(x,y),s(y)))
8: div(plus(x,y),z) -> plus(div(x,z),div(y,z))
9: plus(0(),y) -> y
10: plus(s(x),y) -> s(plus(y,minus(s(x),s(0()))))
11: p(0()) ->= p❆1_0()
12: p(s(_1)) ->= p❆1_s(_1)
Number of strict rules: 10
Direct Order(PosReal,>,Poly) ... failed.
Dependency Pairs:
   #1: #p(0()) ->? #p❆1_0()
   #2: #p(s(_1)) ->? #p❆1_s(_1)
   #3: #div(s(x),s(y)) -> #div(minus(x,y),s(y))
   #4: #div(s(x),s(y)) -> #minus(x,y)
   #5: #plus(s(x),y) -> #plus(y,minus(s(x),s(0())))
   #6: #plus(s(x),y) -> #minus(s(x),s(0()))
   #7: #p❆1_s(s(x)) -> #p❆1_s(x)
   #8: #minus(s(x),s(y)) -> #minus(p❆1_s(x),p❆1_s(y))
   #9: #minus(s(x),s(y)) -> #p❆1_s(x)
   #10: #minus(s(x),s(y)) -> #p❆1_s(y)
   #11: #div(plus(x,y),z) -> #plus(div(x,z),div(y,z))
   #12: #div(plus(x,y),z) -> #div(x,z)
   #13: #div(plus(x,y),z) -> #div(y,z)
   #14: #minus(x,plus(y,z)) -> #minus(minus(x,y),z)
   #15: #minus(x,plus(y,z)) -> #minus(x,y)
Number of SCCs: 5, DPs: 8, edges: 16
	SCC { #7 }
Removing DPs: Order(PosReal,>,Sum)... succeeded.
 #div(x1,x2)	weight: 0
     s(x1)	weight: (/ 1 2) + x1
 minus(x1,x2)	weight: 0
p❆1_s(x1)	weight: 0
#p❆1_0()	weight: 0
#plus(x1,x2)	weight: 0
   div(x1,x2)	weight: 0
   #p(x1)	weight: 0
#p❆1_s(x1)	weight: x1
     p(x1)	weight: 0
     0()	weight: 0
p❆1_0()	weight: 0
#minus(x1,x2)	weight: 0
  plus(x1,x2)	weight: 0
    Usable rules: { }
    Removed DPs: #7
Number of SCCs: 4, DPs: 7, edges: 15
	SCC { #8 }
Removing DPs: Order(PosReal,>,Sum)... succeeded.
 #div(x1,x2)	weight: 0
     s(x1)	weight: (/ 1 2) + x1
 minus(x1,x2)	weight: 0
p❆1_s(x1)	weight: (/ 1 4) + x1
#p❆1_0()	weight: 0
#plus(x1,x2)	weight: 0
   div(x1,x2)	weight: 0
   #p(x1)	weight: 0
#p❆1_s(x1)	weight: 0
     p(x1)	weight: 0
     0()	weight: 0
p❆1_0()	weight: 0
#minus(x1,x2)	weight: x2
  plus(x1,x2)	weight: 0
    Usable rules: { 5 }
    Removed DPs: #8
Number of SCCs: 3, DPs: 6, edges: 14
	SCC { #5 }
Removing DPs: Order(PosReal,>,Sum)... succeeded.
 #div(x1,x2)	weight: 0
     s(x1)	weight: (/ 1 2)
 minus(x1,x2)	weight: (/ 1 4)
p❆1_s(x1)	weight: (/ 1 4) + x1
#p❆1_0()	weight: 0
#plus(x1,x2)	weight: x1 + x2
   div(x1,x2)	weight: 0
   #p(x1)	weight: 0
#p❆1_s(x1)	weight: 0
     p(x1)	weight: 0
     0()	weight: 0
p❆1_0()	weight: 0
#minus(x1,x2)	weight: 0
  plus(x1,x2)	weight: 0
    Usable rules: { 3 }
    Removed DPs: #5
Number of SCCs: 2, DPs: 5, edges: 13
	SCC { #14 #15 }
Removing DPs: Order(PosReal,>,Sum)... succeeded.
 #div(x1,x2)	weight: 0
     s(x1)	weight: (/ 1 2)
 minus(x1,x2)	weight: (/ 1 4)
p❆1_s(x1)	weight: (/ 1 4) + x1
#p❆1_0()	weight: 0
#plus(x1,x2)	weight: x2
   div(x1,x2)	weight: 0
   #p(x1)	weight: 0
#p❆1_s(x1)	weight: 0
     p(x1)	weight: 0
     0()	weight: 0
p❆1_0()	weight: 0
#minus(x1,x2)	weight: x2
  plus(x1,x2)	weight: (/ 1 4) + x1 + x2
    Usable rules: { 3 5 }
    Removed DPs: #14 #15
Number of SCCs: 1, DPs: 3, edges: 9
	SCC { #3 #12 #13 }
Removing DPs: Order(PosReal,>,Sum)... succeeded.
 #div(x1,x2)	weight: x1
     s(x1)	weight: (/ 1 2) + x1
 minus(x1,x2)	weight: x1
p❆1_s(x1)	weight: (/ 1 4) + x1
#p❆1_0()	weight: 0
#plus(x1,x2)	weight: x2
   div(x1,x2)	weight: 0
   #p(x1)	weight: 0
#p❆1_s(x1)	weight: 0
     p(x1)	weight: 0
     0()	weight: 0
p❆1_0()	weight: 0
#minus(x1,x2)	weight: x2
  plus(x1,x2)	weight: (/ 1 4) + x1 + x2
    Usable rules: { 1..5 }
    Removed DPs: #3 #12 #13
Number of SCCs: 0, DPs: 0, edges: 0
YES