Input TRS:
    1: U11(tt(),V1,V2) -> U12(isNat(activate(V1)),activate(V2))
    2: U12(tt(),V2) -> U13(isNat(activate(V2)))
    3: U13(tt()) -> tt()
    4: U21(tt(),V1) -> U22(isNat(activate(V1)))
    5: U22(tt()) -> tt()
    6: U31(tt(),V1,V2) -> U32(isNat(activate(V1)),activate(V2))
    7: U32(tt(),V2) -> U33(isNat(activate(V2)))
    8: U33(tt()) -> tt()
    9: U41(tt(),N) -> activate(N)
    10: U51(tt(),M,N) -> s(plus(activate(N),activate(M)))
    11: U61(tt()) -> 0()
    12: U71(tt(),M,N) -> plus(x(activate(N),activate(M)),activate(N))
    13: and(tt(),X) -> activate(X)
    14: isNat(n__0()) -> tt()
    15: isNat(n__plus(V1,V2)) -> U11(and(isNatKind(activate(V1)),n__isNatKind(activate(V2))),activate(V1),activate(V2))
    16: isNat(n__s(V1)) -> U21(isNatKind(activate(V1)),activate(V1))
    17: isNat(n__x(V1,V2)) -> U31(and(isNatKind(activate(V1)),n__isNatKind(activate(V2))),activate(V1),activate(V2))
    18: isNatKind(n__0()) -> tt()
    19: isNatKind(n__plus(V1,V2)) -> and(isNatKind(activate(V1)),n__isNatKind(activate(V2)))
    20: isNatKind(n__s(V1)) -> isNatKind(activate(V1))
    21: isNatKind(n__x(V1,V2)) -> and(isNatKind(activate(V1)),n__isNatKind(activate(V2)))
    22: plus(N,0()) -> U41(and(isNat(N),n__isNatKind(N)),N)
    23: plus(N,s(M)) -> U51(and(and(isNat(M),n__isNatKind(M)),n__and(isNat(N),n__isNatKind(N))),M,N)
    24: x(N,0()) -> U61(and(isNat(N),n__isNatKind(N)))
    25: x(N,s(M)) -> U71(and(and(isNat(M),n__isNatKind(M)),n__and(isNat(N),n__isNatKind(N))),M,N)
    26: 0() -> n__0()
    27: plus(X1,X2) -> n__plus(X1,X2)
    28: isNatKind(X) -> n__isNatKind(X)
    29: s(X) -> n__s(X)
    30: x(X1,X2) -> n__x(X1,X2)
    31: and(X1,X2) -> n__and(X1,X2)
    32: activate(n__0()) -> 0()
    33: activate(n__plus(X1,X2)) -> plus(X1,X2)
    34: activate(n__isNatKind(X)) -> isNatKind(X)
    35: activate(n__s(X)) -> s(X)
    36: activate(n__x(X1,X2)) -> x(X1,X2)
    37: activate(n__and(X1,X2)) -> and(X1,X2)
    38: activate(X) -> X
Number of strict rules: 38
Direct Order(PosReal,>,Poly) ... failed.
Freezing ... failed.
Dependency Pairs:
   #1: #U12(tt(),V2) -> #U13(isNat(activate(V2)))
   #2: #U12(tt(),V2) -> #isNat(activate(V2))
   #3: #U12(tt(),V2) -> #activate(V2)
   #4: #activate(n__s(X)) -> #s(X)
   #5: #activate(n__and(X1,X2)) -> #and(X1,X2)
   #6: #U31(tt(),V1,V2) -> #U32(isNat(activate(V1)),activate(V2))
   #7: #U31(tt(),V1,V2) -> #isNat(activate(V1))
   #8: #U31(tt(),V1,V2) -> #activate(V1)
   #9: #U31(tt(),V1,V2) -> #activate(V2)
   #10: #and(tt(),X) -> #activate(X)
   #11: #U41(tt(),N) -> #activate(N)
   #12: #U61(tt()) -> #0()
   #13: #x(N,0()) -> #U61(and(isNat(N),n__isNatKind(N)))
   #14: #x(N,0()) -> #and(isNat(N),n__isNatKind(N))
   #15: #x(N,0()) -> #isNat(N)
   #16: #plus(N,s(M)) -> #U51(and(and(isNat(M),n__isNatKind(M)),n__and(isNat(N),n__isNatKind(N))),M,N)
   #17: #plus(N,s(M)) -> #and(and(isNat(M),n__isNatKind(M)),n__and(isNat(N),n__isNatKind(N)))
   #18: #plus(N,s(M)) -> #and(isNat(M),n__isNatKind(M))
   #19: #plus(N,s(M)) -> #isNat(M)
   #20: #plus(N,s(M)) -> #isNat(N)
   #21: #U71(tt(),M,N) -> #plus(x(activate(N),activate(M)),activate(N))
   #22: #U71(tt(),M,N) -> #x(activate(N),activate(M))
   #23: #U71(tt(),M,N) -> #activate(N)
   #24: #U71(tt(),M,N) -> #activate(M)
   #25: #U71(tt(),M,N) -> #activate(N)
   #26: #x(N,s(M)) -> #U71(and(and(isNat(M),n__isNatKind(M)),n__and(isNat(N),n__isNatKind(N))),M,N)
   #27: #x(N,s(M)) -> #and(and(isNat(M),n__isNatKind(M)),n__and(isNat(N),n__isNatKind(N)))
   #28: #x(N,s(M)) -> #and(isNat(M),n__isNatKind(M))
   #29: #x(N,s(M)) -> #isNat(M)
   #30: #x(N,s(M)) -> #isNat(N)
   #31: #isNatKind(n__s(V1)) -> #isNatKind(activate(V1))
   #32: #isNatKind(n__s(V1)) -> #activate(V1)
   #33: #U32(tt(),V2) -> #U33(isNat(activate(V2)))
   #34: #U32(tt(),V2) -> #isNat(activate(V2))
   #35: #U32(tt(),V2) -> #activate(V2)
   #36: #U51(tt(),M,N) -> #s(plus(activate(N),activate(M)))
   #37: #U51(tt(),M,N) -> #plus(activate(N),activate(M))
   #38: #U51(tt(),M,N) -> #activate(N)
   #39: #U51(tt(),M,N) -> #activate(M)
   #40: #activate(n__plus(X1,X2)) -> #plus(X1,X2)
   #41: #plus(N,0()) -> #U41(and(isNat(N),n__isNatKind(N)),N)
   #42: #plus(N,0()) -> #and(isNat(N),n__isNatKind(N))
   #43: #plus(N,0()) -> #isNat(N)
   #44: #activate(n__isNatKind(X)) -> #isNatKind(X)
   #45: #isNat(n__x(V1,V2)) -> #U31(and(isNatKind(activate(V1)),n__isNatKind(activate(V2))),activate(V1),activate(V2))
   #46: #isNat(n__x(V1,V2)) -> #and(isNatKind(activate(V1)),n__isNatKind(activate(V2)))
   #47: #isNat(n__x(V1,V2)) -> #isNatKind(activate(V1))
   #48: #isNat(n__x(V1,V2)) -> #activate(V1)
   #49: #isNat(n__x(V1,V2)) -> #activate(V2)
   #50: #isNat(n__x(V1,V2)) -> #activate(V1)
   #51: #isNat(n__x(V1,V2)) -> #activate(V2)
   #52: #activate(n__0()) -> #0()
   #53: #isNatKind(n__plus(V1,V2)) -> #and(isNatKind(activate(V1)),n__isNatKind(activate(V2)))
   #54: #isNatKind(n__plus(V1,V2)) -> #isNatKind(activate(V1))
   #55: #isNatKind(n__plus(V1,V2)) -> #activate(V1)
   #56: #isNatKind(n__plus(V1,V2)) -> #activate(V2)
   #57: #activate(n__x(X1,X2)) -> #x(X1,X2)
   #58: #isNatKind(n__x(V1,V2)) -> #and(isNatKind(activate(V1)),n__isNatKind(activate(V2)))
   #59: #isNatKind(n__x(V1,V2)) -> #isNatKind(activate(V1))
   #60: #isNatKind(n__x(V1,V2)) -> #activate(V1)
   #61: #isNatKind(n__x(V1,V2)) -> #activate(V2)
   #62: #isNat(n__s(V1)) -> #U21(isNatKind(activate(V1)),activate(V1))
   #63: #isNat(n__s(V1)) -> #isNatKind(activate(V1))
   #64: #isNat(n__s(V1)) -> #activate(V1)
   #65: #isNat(n__s(V1)) -> #activate(V1)
   #66: #U11(tt(),V1,V2) -> #U12(isNat(activate(V1)),activate(V2))
   #67: #U11(tt(),V1,V2) -> #isNat(activate(V1))
   #68: #U11(tt(),V1,V2) -> #activate(V1)
   #69: #U11(tt(),V1,V2) -> #activate(V2)
   #70: #isNat(n__plus(V1,V2)) -> #U11(and(isNatKind(activate(V1)),n__isNatKind(activate(V2))),activate(V1),activate(V2))
   #71: #isNat(n__plus(V1,V2)) -> #and(isNatKind(activate(V1)),n__isNatKind(activate(V2)))
   #72: #isNat(n__plus(V1,V2)) -> #isNatKind(activate(V1))
   #73: #isNat(n__plus(V1,V2)) -> #activate(V1)
   #74: #isNat(n__plus(V1,V2)) -> #activate(V2)
   #75: #isNat(n__plus(V1,V2)) -> #activate(V1)
   #76: #isNat(n__plus(V1,V2)) -> #activate(V2)
   #77: #U21(tt(),V1) -> #U22(isNat(activate(V1)))
   #78: #U21(tt(),V1) -> #isNat(activate(V1))
   #79: #U21(tt(),V1) -> #activate(V1)
Number of SCCs: 1, DPs: 71, edges: 456
	SCC { #2 #3 #5..11 #14..32 #34 #35 #37..51 #53..76 #78 #79 }
Removing DPs: Order(PosReal,>,Sum)... Order(PosReal,>,Max)... succeeded.
   #0()	weight: 0
 #U32(x1,x2)	weight: max{(/ 3 16) + x2, (/ 1 4) + x1}
isNatKind(x1)	weight: x1
   U21(x1,x2)	weight: max{0, (/ 1 16) + x1}
   U11(x1,x2,x3)	weight: max{0, (/ 1 8) + x3, (/ 1 8) + x1}
     s(x1)	weight: x1
#isNat(x1)	weight: (/ 1 8) + x1
activate(x1)	weight: x1
   U71(x1,x2,x3)	weight: max{0, (/ 3 8) + x3, (/ 5 16) + x2}
n__isNatKind(x1)	weight: x1
   and(x1,x2)	weight: max{0, x2}
#plus(x1,x2)	weight: max{(/ 3 8) + x2, (/ 1 8) + x1}
#activate(x1)	weight: (/ 1 8) + x1
 #U13(x1)	weight: 0
   U12(x1,x2)	weight: 0
 #U33(x1)	weight: 0
     x(x1,x2)	weight: max{(/ 5 16) + x2, (/ 3 8) + x1}
  n__s(x1)	weight: x1
 #U12(x1,x2)	weight: max{(/ 1 8) + x2, x1}
     0()	weight: (/ 1 16)
   #x(x1,x2)	weight: max{(/ 7 16) + x2, (/ 1 2) + x1}
   #s(x1)	weight: 0
n__plus(x1,x2)	weight: max{(/ 1 4) + x2, x1}
   U32(x1,x2)	weight: max{0, (/ 3 8) + x1}
   U33(x1)	weight: 0
  n__0()	weight: (/ 1 16)
 isNat(x1)	weight: (/ 1 16) + x1
  n__x(x1,x2)	weight: max{(/ 5 16) + x2, (/ 3 8) + x1}
  plus(x1,x2)	weight: max{(/ 1 4) + x2, x1}
   U61(x1)	weight: (/ 1 16)
 #U51(x1,x2,x3)	weight: max{(/ 1 8) + x3, (/ 3 8) + x2, (/ 1 16) + x1}
 #U11(x1,x2,x3)	weight: max{(/ 3 8) + x3, (/ 1 8) + x2, (/ 3 16) + x1}
   U31(x1,x2,x3)	weight: max{(/ 3 8) + x3, (/ 7 16) + x2, (/ 3 8) + x1}
 #U41(x1,x2)	weight: max{(/ 1 8) + x2, (/ 1 16) + x1}
 #U21(x1,x2)	weight: max{0, (/ 1 8) + x2}
 #U22(x1)	weight: 0
    tt()	weight: 0
n__and(x1,x2)	weight: max{0, x2}
 #U71(x1,x2,x3)	weight: max{0, (/ 1 2) + x3, (/ 7 16) + x2}
   U13(x1)	weight: 0
   U22(x1)	weight: 0
   U51(x1,x2,x3)	weight: max{0, x3, (/ 1 4) + x2}
#isNatKind(x1)	weight: (/ 1 8) + x1
   U41(x1,x2)	weight: max{0, x2}
 #U31(x1,x2,x3)	weight: max{(/ 1 4) + x3, (/ 3 8) + x2, (/ 3 8) + x1}
 #and(x1,x2)	weight: max{0, (/ 1 8) + x2}
 #U61(x1)	weight: 0
    Usable rules: { 1..38 }
    Removed DPs: #6..9 #14 #15 #18 #19 #23..25 #27..30 #34 #35 #39 #45..51 #53 #56 #58..61 #66 #69 #71 #74 #76
Number of SCCs: 1, DPs: 33, edges: 128
	SCC { #5 #10 #11 #16 #17 #20..22 #26 #31 #32 #37 #38 #40..44 #54 #55 #57 #62..65 #67 #68 #70 #72 #73 #75 #78 #79 }
Removing DPs: Order(PosReal,>,Sum)... Order(PosReal,>,Max)... QLPOpS... succeeded.
   #0()	status: []	precedence above:
 #U32(x1,x2)	status: []	precedence above:
isNatKind(x1)	status: [x1]	precedence above: s #isNat n__isNatKind and #activate n__s #U21 n__and #isNatKind #and
   U21(x1,x2)	status: x1
   U11(x1,x2,x3)	status: [x2]	precedence above: isNatKind s #isNat n__isNatKind and #activate U12 n__s isNat #U21 tt n__and U13 U22 #isNatKind #and
     s(x1)	status: [x1]	precedence above: #isNat #activate n__s #U21 #isNatKind #and
#isNat(x1)	status: [x1]	precedence above: #activate #isNatKind
activate(x1)	status: x1
   U71(x1,x2,x3)	status: [x3,x2,x1]	precedence above: isNatKind U11 s #isNat n__isNatKind and #plus #activate U12 x n__s #x n__plus U32 U33 isNat n__x plus #U51 #U11 U31 #U41 #U21 tt n__and #U71 U13 U22 U51 #isNatKind U41 #and
n__isNatKind(x1)	status: [x1]	precedence above: isNatKind s #isNat and #activate n__s #U21 n__and #isNatKind #and
   and(x1,x2)	status: [x1,x2]	precedence above: isNatKind s #isNat n__isNatKind #activate n__s #U21 n__and #isNatKind #and
#plus(x1,x2)	status: [x2,x1]	precedence above: isNatKind s #isNat n__isNatKind and #activate n__s isNat #U51 #U11 #U41 #U21 n__and #isNatKind #and
#activate(x1)	status: [x1]	precedence above: #isNat #isNatKind
 #U13(x1)	status: []	precedence above:
   U12(x1,x2)	status: [x1]	precedence above: isNatKind s #isNat n__isNatKind and #activate n__s isNat #U21 tt n__and U13 U22 #isNatKind #and
 #U33(x1)	status: []	precedence above:
     x(x1,x2)	status: [x1,x2]	precedence above: isNatKind U11 s #isNat U71 n__isNatKind and #plus #activate U12 n__s #x n__plus U32 U33 isNat n__x plus #U51 #U11 U31 #U41 #U21 tt n__and #U71 U13 U22 U51 #isNatKind U41 #and
  n__s(x1)	status: [x1]	precedence above: s #isNat #activate #U21 #isNatKind #and
 #U12(x1,x2)	status: [x1]	precedence above:
     0()	status: []	precedence above: n__0 U61 tt U13 U22
   #x(x1,x2)	status: [x1,x2]	precedence above: isNatKind U11 s #isNat U71 n__isNatKind and #plus #activate U12 x n__s n__plus U32 U33 isNat n__x plus #U51 #U11 U31 #U41 #U21 tt n__and #U71 U13 U22 U51 #isNatKind U41 #and
   #s(x1)	status: []	precedence above:
n__plus(x1,x2)	status: [x1,x2]	precedence above: isNatKind U11 s #isNat n__isNatKind and #plus #activate U12 n__s isNat plus #U51 #U11 #U41 #U21 tt n__and U13 U22 U51 #isNatKind U41 #and
   U32(x1,x2)	status: []	precedence above: isNatKind U11 s #isNat U71 n__isNatKind and #plus #activate U12 x n__s #x n__plus U33 isNat n__x plus #U51 #U11 U31 #U41 #U21 tt n__and #U71 U13 U22 U51 #isNatKind U41 #and
   U33(x1)	status: []	precedence above: tt U13 U22
  n__0()	status: []	precedence above: 0 U61 tt U13 U22
 isNat(x1)	status: [x1]	precedence above: isNatKind s #isNat n__isNatKind and #activate n__s #U21 n__and #isNatKind #and
  n__x(x1,x2)	status: [x1,x2]	precedence above: isNatKind U11 s #isNat U71 n__isNatKind and #plus #activate U12 x n__s #x n__plus U32 U33 isNat plus #U51 #U11 U31 #U41 #U21 tt n__and #U71 U13 U22 U51 #isNatKind U41 #and
  plus(x1,x2)	status: [x1,x2]	precedence above: isNatKind U11 s #isNat n__isNatKind and #plus #activate U12 n__s n__plus isNat #U51 #U11 #U41 #U21 tt n__and U13 U22 U51 #isNatKind U41 #and
   U61(x1)	status: []	precedence above: 0 n__0 tt U13 U22
 #U51(x1,x2,x3)	status: [x2,x3,x1]	precedence above: isNatKind s #isNat n__isNatKind and #plus #activate n__s isNat #U11 #U41 #U21 n__and #isNatKind #and
 #U11(x1,x2,x3)	status: [x1,x3,x2]	precedence above: #isNat #activate #isNatKind #and
   U31(x1,x2,x3)	status: [x2]	precedence above: isNatKind U11 s #isNat U71 n__isNatKind and #plus #activate U12 x n__s #x n__plus U32 U33 isNat n__x plus #U51 #U11 #U41 #U21 tt n__and #U71 U13 U22 U51 #isNatKind U41 #and
 #U41(x1,x2)	status: [x2]	precedence above: #isNat #activate #U11 #U21 #isNatKind #and
 #U21(x1,x2)	status: [x2]	precedence above: #isNat #activate #isNatKind #and
 #U22(x1)	status: []	precedence above:
    tt()	status: []	precedence above: U13 U22
n__and(x1,x2)	status: [x1,x2]	precedence above: isNatKind s #isNat n__isNatKind and #activate n__s #U21 #isNatKind #and
 #U71(x1,x2,x3)	status: [x3,x2,x1]	precedence above: isNatKind U11 s #isNat U71 n__isNatKind and #plus #activate U12 x n__s #x n__plus U32 U33 isNat n__x plus #U51 #U11 U31 #U41 #U21 tt n__and U13 U22 U51 #isNatKind U41 #and
   U13(x1)	status: []	precedence above: tt U22
   U22(x1)	status: []	precedence above: tt U13
   U51(x1,x2,x3)	status: [x3,x2,x1]	precedence above: isNatKind U11 s #isNat n__isNatKind and #plus #activate U12 n__s n__plus isNat plus #U51 #U11 #U41 #U21 tt n__and U13 U22 #isNatKind U41 #and
#isNatKind(x1)	status: x1
   U41(x1,x2)	status: [x1,x2]	precedence above:
 #U31(x1,x2,x3)	status: [x3,x2,x1]	precedence above:
 #and(x1,x2)	status: [x1,x2]	precedence above: #isNat #activate #isNatKind
 #U61(x1)	status: []	precedence above:
    Usable rules: { 1..38 }
    Removed DPs: #5 #10 #11 #16 #17 #20..22 #26 #31 #32 #37 #38 #40..44 #54 #55 #57 #62..65 #67 #68 #70 #72 #73 #75 #78 #79
Number of SCCs: 0, DPs: 0, edges: 0
YES