Input TRS:
    1: a__U11(tt(),V1,V2) -> a__U12(a__isNat(V1),V2)
    2: a__U12(tt(),V2) -> a__U13(a__isNat(V2))
    3: a__U13(tt()) -> tt()
    4: a__U21(tt(),V1) -> a__U22(a__isNat(V1))
    5: a__U22(tt()) -> tt()
    6: a__U31(tt(),N) -> mark(N)
    7: a__U41(tt(),M,N) -> s(a__plus(mark(N),mark(M)))
    8: a__and(tt(),X) -> mark(X)
    9: a__isNat(0()) -> tt()
    10: a__isNat(plus(V1,V2)) -> a__U11(a__and(a__isNatKind(V1),isNatKind(V2)),V1,V2)
    11: a__isNat(s(V1)) -> a__U21(a__isNatKind(V1),V1)
    12: a__isNatKind(0()) -> tt()
    13: a__isNatKind(plus(V1,V2)) -> a__and(a__isNatKind(V1),isNatKind(V2))
    14: a__isNatKind(s(V1)) -> a__isNatKind(V1)
    15: a__plus(N,0()) -> a__U31(a__and(a__isNat(N),isNatKind(N)),N)
    16: a__plus(N,s(M)) -> a__U41(a__and(a__and(a__isNat(M),isNatKind(M)),and(isNat(N),isNatKind(N))),M,N)
    17: mark(U11(X1,X2,X3)) -> a__U11(mark(X1),X2,X3)
    18: mark(U12(X1,X2)) -> a__U12(mark(X1),X2)
    19: mark(isNat(X)) -> a__isNat(X)
    20: mark(U13(X)) -> a__U13(mark(X))
    21: mark(U21(X1,X2)) -> a__U21(mark(X1),X2)
    22: mark(U22(X)) -> a__U22(mark(X))
    23: mark(U31(X1,X2)) -> a__U31(mark(X1),X2)
    24: mark(U41(X1,X2,X3)) -> a__U41(mark(X1),X2,X3)
    25: mark(plus(X1,X2)) -> a__plus(mark(X1),mark(X2))
    26: mark(and(X1,X2)) -> a__and(mark(X1),X2)
    27: mark(isNatKind(X)) -> a__isNatKind(X)
    28: mark(tt()) -> tt()
    29: mark(s(X)) -> s(mark(X))
    30: mark(0()) -> 0()
    31: a__U11(X1,X2,X3) -> U11(X1,X2,X3)
    32: a__U12(X1,X2) -> U12(X1,X2)
    33: a__isNat(X) -> isNat(X)
    34: a__U13(X) -> U13(X)
    35: a__U21(X1,X2) -> U21(X1,X2)
    36: a__U22(X) -> U22(X)
    37: a__U31(X1,X2) -> U31(X1,X2)
    38: a__U41(X1,X2,X3) -> U41(X1,X2,X3)
    39: a__plus(X1,X2) -> plus(X1,X2)
    40: a__and(X1,X2) -> and(X1,X2)
    41: a__isNatKind(X) -> isNatKind(X)
Number of strict rules: 41
Direct Order(PosReal,>,Poly) ... failed.
Freezing ... failed.
Dependency Pairs:
   #1: #a__U12(tt(),V2) -> #a__U13(a__isNat(V2))
   #2: #a__U12(tt(),V2) -> #a__isNat(V2)
   #3: #mark(s(X)) -> #mark(X)
   #4: #a__U31(tt(),N) -> #mark(N)
   #5: #a__isNatKind(plus(V1,V2)) -> #a__and(a__isNatKind(V1),isNatKind(V2))
   #6: #a__isNatKind(plus(V1,V2)) -> #a__isNatKind(V1)
   #7: #a__isNat(s(V1)) -> #a__U21(a__isNatKind(V1),V1)
   #8: #a__isNat(s(V1)) -> #a__isNatKind(V1)
   #9: #mark(U41(X1,X2,X3)) -> #a__U41(mark(X1),X2,X3)
   #10: #mark(U41(X1,X2,X3)) -> #mark(X1)
   #11: #mark(U31(X1,X2)) -> #a__U31(mark(X1),X2)
   #12: #mark(U31(X1,X2)) -> #mark(X1)
   #13: #a__isNatKind(s(V1)) -> #a__isNatKind(V1)
   #14: #mark(plus(X1,X2)) -> #a__plus(mark(X1),mark(X2))
   #15: #mark(plus(X1,X2)) -> #mark(X1)
   #16: #mark(plus(X1,X2)) -> #mark(X2)
   #17: #mark(U13(X)) -> #a__U13(mark(X))
   #18: #mark(U13(X)) -> #mark(X)
   #19: #a__U41(tt(),M,N) -> #a__plus(mark(N),mark(M))
   #20: #a__U41(tt(),M,N) -> #mark(N)
   #21: #a__U41(tt(),M,N) -> #mark(M)
   #22: #a__isNat(plus(V1,V2)) -> #a__U11(a__and(a__isNatKind(V1),isNatKind(V2)),V1,V2)
   #23: #a__isNat(plus(V1,V2)) -> #a__and(a__isNatKind(V1),isNatKind(V2))
   #24: #a__isNat(plus(V1,V2)) -> #a__isNatKind(V1)
   #25: #mark(U22(X)) -> #a__U22(mark(X))
   #26: #mark(U22(X)) -> #mark(X)
   #27: #mark(isNatKind(X)) -> #a__isNatKind(X)
   #28: #mark(U11(X1,X2,X3)) -> #a__U11(mark(X1),X2,X3)
   #29: #mark(U11(X1,X2,X3)) -> #mark(X1)
   #30: #mark(isNat(X)) -> #a__isNat(X)
   #31: #mark(and(X1,X2)) -> #a__and(mark(X1),X2)
   #32: #mark(and(X1,X2)) -> #mark(X1)
   #33: #mark(U21(X1,X2)) -> #a__U21(mark(X1),X2)
   #34: #mark(U21(X1,X2)) -> #mark(X1)
   #35: #a__plus(N,s(M)) -> #a__U41(a__and(a__and(a__isNat(M),isNatKind(M)),and(isNat(N),isNatKind(N))),M,N)
   #36: #a__plus(N,s(M)) -> #a__and(a__and(a__isNat(M),isNatKind(M)),and(isNat(N),isNatKind(N)))
   #37: #a__plus(N,s(M)) -> #a__and(a__isNat(M),isNatKind(M))
   #38: #a__plus(N,s(M)) -> #a__isNat(M)
   #39: #a__U11(tt(),V1,V2) -> #a__U12(a__isNat(V1),V2)
   #40: #a__U11(tt(),V1,V2) -> #a__isNat(V1)
   #41: #a__and(tt(),X) -> #mark(X)
   #42: #a__plus(N,0()) -> #a__U31(a__and(a__isNat(N),isNatKind(N)),N)
   #43: #a__plus(N,0()) -> #a__and(a__isNat(N),isNatKind(N))
   #44: #a__plus(N,0()) -> #a__isNat(N)
   #45: #a__U21(tt(),V1) -> #a__U22(a__isNat(V1))
   #46: #a__U21(tt(),V1) -> #a__isNat(V1)
   #47: #mark(U12(X1,X2)) -> #a__U12(mark(X1),X2)
   #48: #mark(U12(X1,X2)) -> #mark(X1)
Number of SCCs: 1, DPs: 44, edges: 381
	SCC { #2..16 #18..24 #26..44 #46..48 }
Removing DPs: Order(PosReal,>,Sum)... succeeded.
#a__isNatKind(x1)	weight: 0
isNatKind(x1)	weight: 0
a__plus(x1,x2)	weight: (/ 1 4) + x1 + x2
   U21(x1,x2)	weight: x1
   U11(x1,x2,x3)	weight: x1
     s(x1)	weight: (/ 1 4) + x1
#a__U31(x1,x2)	weight: x2
#a__isNat(x1)	weight: 0
   and(x1,x2)	weight: x1 + x2
#a__U13(x1)	weight: 0
a__U22(x1)	weight: x1
#a__U11(x1,x2,x3)	weight: 0
a__U31(x1,x2)	weight: x1 + x2
   U12(x1,x2)	weight: x1
a__U41(x1,x2,x3)	weight: (/ 1 2) + x1 + x2 + x3
#a__U12(x1,x2)	weight: 0
#a__U21(x1,x2)	weight: 0
#a__plus(x1,x2)	weight: x1 + x2
#mark(x1)	weight: x1
     0()	weight: 0
#a__and(x1,x2)	weight: x2
a__U21(x1,x2)	weight: x1
  mark(x1)	weight: x1
a__U11(x1,x2,x3)	weight: x1
a__U12(x1,x2)	weight: x1
 isNat(x1)	weight: 0
  plus(x1,x2)	weight: (/ 1 4) + x1 + x2
#a__U22(x1)	weight: 0
a__U13(x1)	weight: x1
   U31(x1,x2)	weight: x1 + x2
#a__U41(x1,x2,x3)	weight: x2 + x3
    tt()	weight: 0
a__isNat(x1)	weight: 0
   U13(x1)	weight: x1
a__isNatKind(x1)	weight: 0
   U22(x1)	weight: x1
a__and(x1,x2)	weight: x1 + x2
   U41(x1,x2,x3)	weight: (/ 1 2) + x1 + x2 + x3
    Usable rules: { 1..41 }
    Removed DPs: #3 #9 #10 #14..16 #35..38
Number of SCCs: 1, DPs: 28, edges: 173
	SCC { #2 #4..8 #11..13 #18 #22..24 #26..34 #39..41 #46..48 }
Removing DPs: Order(PosReal,>,Sum)... succeeded.
#a__isNatKind(x1)	weight: x1
isNatKind(x1)	weight: (/ 1 16) + x1
a__plus(x1,x2)	weight: (/ 3 8) + x1 + x2
   U21(x1,x2)	weight: (/ 1 16) + x1 + x2
   U11(x1,x2,x3)	weight: x1 + x2 + x3
     s(x1)	weight: (/ 1 8) + x1
#a__U31(x1,x2)	weight: (/ 1 4) + x2
#a__isNat(x1)	weight: x1
   and(x1,x2)	weight: (/ 1 8) + x1 + x2
#a__U13(x1)	weight: 0
a__U22(x1)	weight: (/ 1 16)
#a__U11(x1,x2,x3)	weight: (/ 1 8) + x2 + x3
a__U31(x1,x2)	weight: (/ 3 16) + x2
   U12(x1,x2)	weight: (/ 1 16) + x1 + x2
a__U41(x1,x2,x3)	weight: (/ 5 8) + x1 + x2 + x3
#a__U12(x1,x2)	weight: (/ 1 16) + x2
#a__U21(x1,x2)	weight: (/ 1 16) + x2
#a__plus(x1,x2)	weight: 0
#mark(x1)	weight: (/ 3 16) + x1
     0()	weight: 0
#a__and(x1,x2)	weight: (/ 1 4) + x2
a__U21(x1,x2)	weight: (/ 1 16) + x1
  mark(x1)	weight: (/ 1 16) + x1
a__U11(x1,x2,x3)	weight: (/ 1 16) + x1 + x2 + x3
a__U12(x1,x2)	weight: (/ 1 16)
 isNat(x1)	weight: (/ 1 4) + x1
  plus(x1,x2)	weight: (/ 3 8) + x1 + x2
#a__U22(x1)	weight: 0
a__U13(x1)	weight: (/ 1 16)
   U31(x1,x2)	weight: (/ 1 8) + x1 + x2
#a__U41(x1,x2,x3)	weight: 0
    tt()	weight: 0
a__isNat(x1)	weight: (/ 5 16)
   U13(x1)	weight: (/ 1 16) + x1
a__isNatKind(x1)	weight: (/ 1 16) + x1
   U22(x1)	weight: (/ 1 16) + x1
a__and(x1,x2)	weight: (/ 1 8) + x1 + x2
   U41(x1,x2,x3)	weight: (/ 5 8) + x1 + x2 + x3
    Usable rules: { 3 5 9 12 28 30 31 38..41 }
    Removed DPs: #2 #4..8 #11..13 #18 #22..24 #26..28 #30..34 #39..41 #46..48
Number of SCCs: 1, DPs: 1, edges: 1
	SCC { #29 }
Removing DPs: Order(PosReal,>,Sum)... succeeded.
#a__isNatKind(x1)	weight: 0
isNatKind(x1)	weight: (/ 1 16) + x1
a__plus(x1,x2)	weight: (/ 3 8) + x1 + x2
   U21(x1,x2)	weight: (/ 1 16) + x1 + x2
   U11(x1,x2,x3)	weight: (/ 1 16) + x1 + x2 + x3
     s(x1)	weight: (/ 1 8) + x1
#a__U31(x1,x2)	weight: (/ 1 4)
#a__isNat(x1)	weight: 0
   and(x1,x2)	weight: (/ 1 8) + x1 + x2
#a__U13(x1)	weight: 0
a__U22(x1)	weight: (/ 1 16)
#a__U11(x1,x2,x3)	weight: (/ 1 8)
a__U31(x1,x2)	weight: (/ 3 16) + x2
   U12(x1,x2)	weight: (/ 1 8) + x1 + x2
a__U41(x1,x2,x3)	weight: (/ 5 8) + x1 + x2 + x3
#a__U12(x1,x2)	weight: (/ 1 16)
#a__U21(x1,x2)	weight: (/ 1 16)
#a__plus(x1,x2)	weight: 0
#mark(x1)	weight: (/ 3 16) + x1
     0()	weight: 0
#a__and(x1,x2)	weight: (/ 1 4)
a__U21(x1,x2)	weight: (/ 1 16) + x1
  mark(x1)	weight: (/ 1 16) + x1
a__U11(x1,x2,x3)	weight: (/ 1 8) + x1 + x2 + x3
a__U12(x1,x2)	weight: (/ 1 8)
 isNat(x1)	weight: (/ 5 16) + x1
  plus(x1,x2)	weight: (/ 3 8) + x1 + x2
#a__U22(x1)	weight: 0
a__U13(x1)	weight: (/ 1 16)
   U31(x1,x2)	weight: (/ 1 8) + x1 + x2
#a__U41(x1,x2,x3)	weight: 0
    tt()	weight: 0
a__isNat(x1)	weight: (/ 3 8)
   U13(x1)	weight: (/ 1 16) + x1
a__isNatKind(x1)	weight: (/ 1 16) + x1
   U22(x1)	weight: (/ 1 16) + x1
a__and(x1,x2)	weight: (/ 1 8) + x1 + x2
   U41(x1,x2,x3)	weight: (/ 5 8) + x1 + x2 + x3
    Usable rules: { }
    Removed DPs: #29
Number of SCCs: 0, DPs: 0, edges: 0
YES