Input TRS:
    1: minus(x,0()) -> x
    2: minus(s(x),s(y)) -> minus(x,y)
    3: quot(0(),s(y)) -> 0()
    4: quot(s(x),s(y)) -> s(quot(minus(x,y),s(y)))
    5: plus(0(),y) -> y
    6: plus(s(x),y) -> s(plus(x,y))
    7: plus(minus(x,s(0())),minus(y,s(s(z)))) -> plus(minus(y,s(s(z))),minus(x,s(0())))
Number of strict rules: 7
Direct Order(PosReal,>,Poly) ... failed.
Freezing ... failed.
Dependency Pairs:
   #1: #minus(s(x),s(y)) -> #minus(x,y)
   #2: #plus(s(x),y) -> #plus(x,y)
   #3: #plus(minus(x,s(0())),minus(y,s(s(z)))) -> #plus(minus(y,s(s(z))),minus(x,s(0())))
   #4: #quot(s(x),s(y)) -> #quot(minus(x,y),s(y))
   #5: #quot(s(x),s(y)) -> #minus(x,y)
Number of SCCs: 3, DPs: 4, edges: 6
	SCC { #1 }
Removing DPs: Order(PosReal,>,Sum)... succeeded.
     s(x1)	weight: (/ 1 2) + x1
 minus(x1,x2)	weight: 0
#plus(x1,x2)	weight: 0
     0()	weight: 0
  quot(x1,x2)	weight: 0
#minus(x1,x2)	weight: x2
  plus(x1,x2)	weight: 0
#quot(x1,x2)	weight: 0
    Usable rules: { }
    Removed DPs: #1
Number of SCCs: 2, DPs: 3, edges: 5
	SCC { #4 }
Removing DPs: Order(PosReal,>,Sum)... succeeded.
     s(x1)	weight: (/ 1 2) + x1
 minus(x1,x2)	weight: (/ 1 4) + x1
#plus(x1,x2)	weight: 0
     0()	weight: 0
  quot(x1,x2)	weight: 0
#minus(x1,x2)	weight: 0
  plus(x1,x2)	weight: 0
#quot(x1,x2)	weight: x1
    Usable rules: { 1 2 }
    Removed DPs: #4
Number of SCCs: 1, DPs: 2, edges: 4
	SCC { #2 #3 }
Removing DPs: Order(PosReal,>,Sum)... succeeded.
     s(x1)	weight: (/ 1 4) + x1
 minus(x1,x2)	weight: (/ 1 4) + x1
#plus(x1,x2)	weight: x1 + x2
     0()	weight: 0
  quot(x1,x2)	weight: 0
#minus(x1,x2)	weight: 0
  plus(x1,x2)	weight: 0
#quot(x1,x2)	weight: x1
    Usable rules: { 1 2 }
    Removed DPs: #2
Number of SCCs: 1, DPs: 1, edges: 1
	SCC { #3 }
Removing DPs: Order(PosReal,>,Sum)... Order(PosReal,>,Max)... QLPOpS... Order(PosReal,>,MaxSum)... QWPOpS(PosReal,>,MaxSum)... Order(PosReal,>,Sum-Sum; PosReal,≥,Sum-Sum)... succeeded.
     s(x1)	weight: (/ 1 4); (/ 1 8) + x1_1 + x1_2
 minus(x1,x2)	weight: (/ 1 8) + x1_1 + x1_2; x2_2 + x1_1 + x1_2
#plus(x1,x2)	weight: x2_2 + x1_1; x2_2 + x1_2
     0()	weight: (/ 1 8); (/ 1 8)
  quot(x1,x2)	weight: 0; 0
#minus(x1,x2)	weight: 0; 0
  plus(x1,x2)	weight: 0; 0
#quot(x1,x2)	weight: 0; 0
    Usable rules: { 1 2 }
    Removed DPs: #3
Number of SCCs: 0, DPs: 0, edges: 0
YES