Input TRS:
    1: a__U11(tt(),V2) -> a__U12(a__isNat(V2))
    2: a__U12(tt()) -> tt()
    3: a__U21(tt()) -> tt()
    4: a__U31(tt(),N) -> mark(N)
    5: a__U41(tt(),M,N) -> a__U42(a__isNat(N),M,N)
    6: a__U42(tt(),M,N) -> s(a__plus(mark(N),mark(M)))
    7: a__isNat(0()) -> tt()
    8: a__isNat(plus(V1,V2)) -> a__U11(a__isNat(V1),V2)
    9: a__isNat(s(V1)) -> a__U21(a__isNat(V1))
    10: a__plus(N,0()) -> a__U31(a__isNat(N),N)
    11: a__plus(N,s(M)) -> a__U41(a__isNat(M),M,N)
    12: mark(U11(X1,X2)) -> a__U11(mark(X1),X2)
    13: mark(U12(X)) -> a__U12(mark(X))
    14: mark(isNat(X)) -> a__isNat(X)
    15: mark(U21(X)) -> a__U21(mark(X))
    16: mark(U31(X1,X2)) -> a__U31(mark(X1),X2)
    17: mark(U41(X1,X2,X3)) -> a__U41(mark(X1),X2,X3)
    18: mark(U42(X1,X2,X3)) -> a__U42(mark(X1),X2,X3)
    19: mark(plus(X1,X2)) -> a__plus(mark(X1),mark(X2))
    20: mark(tt()) -> tt()
    21: mark(s(X)) -> s(mark(X))
    22: mark(0()) -> 0()
    23: a__U11(X1,X2) -> U11(X1,X2)
    24: a__U12(X) -> U12(X)
    25: a__isNat(X) -> isNat(X)
    26: a__U21(X) -> U21(X)
    27: a__U31(X1,X2) -> U31(X1,X2)
    28: a__U41(X1,X2,X3) -> U41(X1,X2,X3)
    29: a__U42(X1,X2,X3) -> U42(X1,X2,X3)
    30: a__plus(X1,X2) -> plus(X1,X2)
Number of strict rules: 30
Direct Order(PosReal,>,Poly) ... failed.
Freezing ... failed.
Dependency Pairs:
   #1: #a__U42(tt(),M,N) -> #a__plus(mark(N),mark(M))
   #2: #a__U42(tt(),M,N) -> #mark(N)
   #3: #a__U42(tt(),M,N) -> #mark(M)
   #4: #mark(U12(X)) -> #a__U12(mark(X))
   #5: #mark(U12(X)) -> #mark(X)
   #6: #a__isNat(s(V1)) -> #a__U21(a__isNat(V1))
   #7: #a__isNat(s(V1)) -> #a__isNat(V1)
   #8: #a__plus(N,s(M)) -> #a__U41(a__isNat(M),M,N)
   #9: #a__plus(N,s(M)) -> #a__isNat(M)
   #10: #mark(U11(X1,X2)) -> #a__U11(mark(X1),X2)
   #11: #mark(U11(X1,X2)) -> #mark(X1)
   #12: #mark(isNat(X)) -> #a__isNat(X)
   #13: #a__plus(N,0()) -> #a__U31(a__isNat(N),N)
   #14: #a__plus(N,0()) -> #a__isNat(N)
   #15: #a__U41(tt(),M,N) -> #a__U42(a__isNat(N),M,N)
   #16: #a__U41(tt(),M,N) -> #a__isNat(N)
   #17: #mark(U41(X1,X2,X3)) -> #a__U41(mark(X1),X2,X3)
   #18: #mark(U41(X1,X2,X3)) -> #mark(X1)
   #19: #mark(plus(X1,X2)) -> #a__plus(mark(X1),mark(X2))
   #20: #mark(plus(X1,X2)) -> #mark(X1)
   #21: #mark(plus(X1,X2)) -> #mark(X2)
   #22: #mark(s(X)) -> #mark(X)
   #23: #mark(U31(X1,X2)) -> #a__U31(mark(X1),X2)
   #24: #mark(U31(X1,X2)) -> #mark(X1)
   #25: #a__U11(tt(),V2) -> #a__U12(a__isNat(V2))
   #26: #a__U11(tt(),V2) -> #a__isNat(V2)
   #27: #a__isNat(plus(V1,V2)) -> #a__U11(a__isNat(V1),V2)
   #28: #a__isNat(plus(V1,V2)) -> #a__isNat(V1)
   #29: #mark(U21(X)) -> #a__U21(mark(X))
   #30: #mark(U21(X)) -> #mark(X)
   #31: #a__U31(tt(),N) -> #mark(N)
   #32: #mark(U42(X1,X2,X3)) -> #a__U42(mark(X1),X2,X3)
   #33: #mark(U42(X1,X2,X3)) -> #mark(X1)
Number of SCCs: 2, DPs: 24, edges: 180
	SCC { #7 #26..28 }
Removing DPs: Order(PosReal,>,Sum)... succeeded.
a__plus(x1,x2)	weight: 0
   U21(x1)	weight: (/ 3 4) + x1
   U11(x1,x2)	weight: 0
     s(x1)	weight: (/ 1 4) + x1
#a__U31(x1,x2)	weight: 0
   U42(x1,x2,x3)	weight: 0
#a__isNat(x1)	weight: x1
#a__U11(x1,x2)	weight: (/ 1 4) + x2
a__U31(x1,x2)	weight: 0
   U12(x1)	weight: 0
#a__U42(x1,x2,x3)	weight: 0
a__U41(x1,x2,x3)	weight: 0
#a__U12(x1)	weight: 0
#a__U21(x1)	weight: 0
#a__plus(x1,x2)	weight: 0
#mark(x1)	weight: 0
     0()	weight: 0
a__U21(x1)	weight: (/ 1 2)
  mark(x1)	weight: 0
a__U11(x1,x2)	weight: 0
a__U42(x1,x2,x3)	weight: 0
a__U12(x1)	weight: 0
 isNat(x1)	weight: 0
  plus(x1,x2)	weight: (/ 1 2) + x1 + x2
   U31(x1,x2)	weight: 0
#a__U41(x1,x2,x3)	weight: 0
    tt()	weight: 0
a__isNat(x1)	weight: (/ 1 4)
   U41(x1,x2,x3)	weight: 0
    Usable rules: { 1 2 7 8 23..25 }
    Removed DPs: #7 #26..28
Number of SCCs: 1, DPs: 20, edges: 170
	SCC { #1..3 #5 #8 #11 #13 #15 #17..24 #30..33 }
Removing DPs: Order(PosReal,>,Sum)... succeeded.
a__plus(x1,x2)	weight: (/ 3 8) + x1 + x2
   U21(x1)	weight: x1
   U11(x1,x2)	weight: x1
     s(x1)	weight: (/ 3 8) + x1
#a__U31(x1,x2)	weight: (/ 1 8) + x2
   U42(x1,x2,x3)	weight: (/ 3 4) + x1 + x2 + x3
#a__isNat(x1)	weight: 0
#a__U11(x1,x2)	weight: (/ 1 8)
a__U31(x1,x2)	weight: (/ 1 4) + x1 + x2
   U12(x1)	weight: x1
#a__U42(x1,x2,x3)	weight: (/ 3 8) + x2 + x3
a__U41(x1,x2,x3)	weight: (/ 3 4) + x1 + x2 + x3
#a__U12(x1)	weight: 0
#a__U21(x1)	weight: 0
#a__plus(x1,x2)	weight: (/ 1 4) + x1 + x2
#mark(x1)	weight: x1
     0()	weight: 0
a__U21(x1)	weight: x1
  mark(x1)	weight: x1
a__U11(x1,x2)	weight: x1
a__U42(x1,x2,x3)	weight: (/ 3 4) + x1 + x2 + x3
a__U12(x1)	weight: x1
 isNat(x1)	weight: 0
  plus(x1,x2)	weight: (/ 3 8) + x1 + x2
   U31(x1,x2)	weight: (/ 1 4) + x1 + x2
#a__U41(x1,x2,x3)	weight: (/ 1 2) + x2 + x3
    tt()	weight: 0
a__isNat(x1)	weight: 0
   U41(x1,x2,x3)	weight: (/ 3 4) + x1 + x2 + x3
    Usable rules: { 1..30 }
    Removed DPs: #1..3 #8 #13 #15 #17..24 #31..33
Number of SCCs: 1, DPs: 3, edges: 9
	SCC { #5 #11 #30 }
Removing DPs: Order(PosReal,>,Sum)... succeeded.
a__plus(x1,x2)	weight: x1 + x2
   U21(x1)	weight: (/ 1 8) + x1
   U11(x1,x2)	weight: x1 + x2
     s(x1)	weight: (/ 3 4) + x1
#a__U31(x1,x2)	weight: (/ 1 8)
   U42(x1,x2,x3)	weight: (/ 3 4) + x2 + x3
#a__isNat(x1)	weight: 0
#a__U11(x1,x2)	weight: (/ 1 8)
a__U31(x1,x2)	weight: x2
   U12(x1)	weight: x1
#a__U42(x1,x2,x3)	weight: (/ 3 8)
a__U41(x1,x2,x3)	weight: (/ 3 4) + x2 + x3
#a__U12(x1)	weight: 0
#a__U21(x1)	weight: 0
#a__plus(x1,x2)	weight: (/ 1 4) + x1 + x2
#mark(x1)	weight: x1
     0()	weight: 0
a__U21(x1)	weight: (/ 1 8) + x1
  mark(x1)	weight: x1
a__U11(x1,x2)	weight: x1 + x2
a__U42(x1,x2,x3)	weight: (/ 3 4) + x2 + x3
a__U12(x1)	weight: x1
 isNat(x1)	weight: x1
  plus(x1,x2)	weight: x1 + x2
   U31(x1,x2)	weight: x2
#a__U41(x1,x2,x3)	weight: (/ 1 2)
    tt()	weight: 0
a__isNat(x1)	weight: x1
   U41(x1,x2,x3)	weight: (/ 3 4) + x2 + x3
    Usable rules: { }
    Removed DPs: #30
Number of SCCs: 1, DPs: 2, edges: 4
	SCC { #5 #11 }
Removing DPs: Order(PosReal,>,Sum)... succeeded.
a__plus(x1,x2)	weight: x1 + x2
   U21(x1)	weight: (/ 1 4) + x1
   U11(x1,x2)	weight: (/ 1 8) + x1 + x2
     s(x1)	weight: (/ 1 8) + x1
#a__U31(x1,x2)	weight: (/ 1 8)
   U42(x1,x2,x3)	weight: (/ 3 4) + x1 + x2 + x3
#a__isNat(x1)	weight: 0
#a__U11(x1,x2)	weight: (/ 1 8)
a__U31(x1,x2)	weight: (/ 1 8) + x1 + x2
   U12(x1)	weight: x1
#a__U42(x1,x2,x3)	weight: (/ 3 8)
a__U41(x1,x2,x3)	weight: (/ 3 8) + x1 + x3
#a__U12(x1)	weight: 0
#a__U21(x1)	weight: 0
#a__plus(x1,x2)	weight: (/ 1 4)
#mark(x1)	weight: x1
     0()	weight: 0
a__U21(x1)	weight: (/ 1 8)
  mark(x1)	weight: (/ 1 4) + x1
a__U11(x1,x2)	weight: 0
a__U42(x1,x2,x3)	weight: (/ 1 2) + x2
a__U12(x1)	weight: 0
 isNat(x1)	weight: x1
  plus(x1,x2)	weight: (/ 1 4) + x1 + x2
   U31(x1,x2)	weight: (/ 1 4) + x2
#a__U41(x1,x2,x3)	weight: (/ 1 2)
    tt()	weight: 0
a__isNat(x1)	weight: 0
   U41(x1,x2,x3)	weight: (/ 1 2) + x2
    Usable rules: { }
    Removed DPs: #11
Number of SCCs: 1, DPs: 1, edges: 1
	SCC { #5 }
Removing DPs: Order(PosReal,>,Sum)... succeeded.
a__plus(x1,x2)	weight: x1 + x2
   U21(x1)	weight: (/ 1 4) + x1
   U11(x1,x2)	weight: (/ 1 8) + x1 + x2
     s(x1)	weight: (/ 1 8) + x1
#a__U31(x1,x2)	weight: (/ 1 8)
   U42(x1,x2,x3)	weight: (/ 3 4) + x1 + x2 + x3
#a__isNat(x1)	weight: 0
#a__U11(x1,x2)	weight: (/ 1 8)
a__U31(x1,x2)	weight: (/ 1 8) + x1 + x2
   U12(x1)	weight: (/ 1 8) + x1
#a__U42(x1,x2,x3)	weight: (/ 3 8)
a__U41(x1,x2,x3)	weight: (/ 3 8) + x1 + x3
#a__U12(x1)	weight: 0
#a__U21(x1)	weight: 0
#a__plus(x1,x2)	weight: (/ 1 4)
#mark(x1)	weight: x1
     0()	weight: 0
a__U21(x1)	weight: (/ 1 8)
  mark(x1)	weight: (/ 1 4) + x1
a__U11(x1,x2)	weight: 0
a__U42(x1,x2,x3)	weight: (/ 1 2) + x2
a__U12(x1)	weight: 0
 isNat(x1)	weight: x1
  plus(x1,x2)	weight: (/ 1 4) + x1 + x2
   U31(x1,x2)	weight: (/ 1 4) + x2
#a__U41(x1,x2,x3)	weight: (/ 1 2)
    tt()	weight: 0
a__isNat(x1)	weight: 0
   U41(x1,x2,x3)	weight: (/ 1 2) + x2
    Usable rules: { }
    Removed DPs: #5
Number of SCCs: 0, DPs: 0, edges: 0
YES