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(n__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(n__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: isNat(X) -> n__isNat(X)
    33: activate(n__0()) -> |0|()
    34: activate(n__plus(X1,X2)) -> plus(activate(X1),activate(X2))
    35: activate(n__isNatKind(X)) -> isNatKind(X)
    36: activate(n__s(X)) -> s(activate(X))
    37: activate(n__x(X1,X2)) -> x(activate(X1),activate(X2))
    38: activate(n__and(X1,X2)) -> and(activate(X1),X2)
    39: activate(n__isNat(X)) -> isNat(X)
    40: activate(X) -> X
Number of strict rules: 40
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__isNatKind(X)) -> #isNatKind(X)
   #5: #activate(n__x(X1,X2)) -> #x(activate(X1),activate(X2))
   #6: #activate(n__x(X1,X2)) -> #activate(X1)
   #7: #activate(n__x(X1,X2)) -> #activate(X2)
   #8: #activate(n__and(X1,X2)) -> #and(activate(X1),X2)
   #9: #activate(n__and(X1,X2)) -> #activate(X1)
   #10: #U31(tt(),V1,V2) -> #U32(isNat(activate(V1)),activate(V2))
   #11: #U31(tt(),V1,V2) -> #isNat(activate(V1))
   #12: #U31(tt(),V1,V2) -> #activate(V1)
   #13: #U31(tt(),V1,V2) -> #activate(V2)
   #14: #and(tt(),X) -> #activate(X)
   #15: #U41(tt(),N) -> #activate(N)
   #16: #U61(tt()) -> #|0|()
   #17: #x(N,|0|()) -> #U61(and(isNat(N),n__isNatKind(N)))
   #18: #x(N,|0|()) -> #and(isNat(N),n__isNatKind(N))
   #19: #x(N,|0|()) -> #isNat(N)
   #20: #plus(N,s(M)) -> #U51(and(and(isNat(M),n__isNatKind(M)),n__and(n__isNat(N),n__isNatKind(N))),M,N)
   #21: #plus(N,s(M)) -> #and(and(isNat(M),n__isNatKind(M)),n__and(n__isNat(N),n__isNatKind(N)))
   #22: #plus(N,s(M)) -> #and(isNat(M),n__isNatKind(M))
   #23: #plus(N,s(M)) -> #isNat(M)
   #24: #U71(tt(),M,N) -> #plus(x(activate(N),activate(M)),activate(N))
   #25: #U71(tt(),M,N) -> #x(activate(N),activate(M))
   #26: #U71(tt(),M,N) -> #activate(N)
   #27: #U71(tt(),M,N) -> #activate(M)
   #28: #U71(tt(),M,N) -> #activate(N)
   #29: #x(N,s(M)) -> #U71(and(and(isNat(M),n__isNatKind(M)),n__and(n__isNat(N),n__isNatKind(N))),M,N)
   #30: #x(N,s(M)) -> #and(and(isNat(M),n__isNatKind(M)),n__and(n__isNat(N),n__isNatKind(N)))
   #31: #x(N,s(M)) -> #and(isNat(M),n__isNatKind(M))
   #32: #x(N,s(M)) -> #isNat(M)
   #33: #isNatKind(n__s(V1)) -> #isNatKind(activate(V1))
   #34: #isNatKind(n__s(V1)) -> #activate(V1)
   #35: #U32(tt(),V2) -> #U33(isNat(activate(V2)))
   #36: #U32(tt(),V2) -> #isNat(activate(V2))
   #37: #U32(tt(),V2) -> #activate(V2)
   #38: #activate(n__isNat(X)) -> #isNat(X)
   #39: #U51(tt(),M,N) -> #s(plus(activate(N),activate(M)))
   #40: #U51(tt(),M,N) -> #plus(activate(N),activate(M))
   #41: #U51(tt(),M,N) -> #activate(N)
   #42: #U51(tt(),M,N) -> #activate(M)
   #43: #activate(n__0()) -> #|0|()
   #44: #plus(N,|0|()) -> #U41(and(isNat(N),n__isNatKind(N)),N)
   #45: #plus(N,|0|()) -> #and(isNat(N),n__isNatKind(N))
   #46: #plus(N,|0|()) -> #isNat(N)
   #47: #activate(n__plus(X1,X2)) -> #plus(activate(X1),activate(X2))
   #48: #activate(n__plus(X1,X2)) -> #activate(X1)
   #49: #activate(n__plus(X1,X2)) -> #activate(X2)
   #50: #isNat(n__x(V1,V2)) -> #U31(and(isNatKind(activate(V1)),n__isNatKind(activate(V2))),activate(V1),activate(V2))
   #51: #isNat(n__x(V1,V2)) -> #and(isNatKind(activate(V1)),n__isNatKind(activate(V2)))
   #52: #isNat(n__x(V1,V2)) -> #isNatKind(activate(V1))
   #53: #isNat(n__x(V1,V2)) -> #activate(V1)
   #54: #isNat(n__x(V1,V2)) -> #activate(V2)
   #55: #isNat(n__x(V1,V2)) -> #activate(V1)
   #56: #isNat(n__x(V1,V2)) -> #activate(V2)
   #57: #isNatKind(n__plus(V1,V2)) -> #and(isNatKind(activate(V1)),n__isNatKind(activate(V2)))
   #58: #isNatKind(n__plus(V1,V2)) -> #isNatKind(activate(V1))
   #59: #isNatKind(n__plus(V1,V2)) -> #activate(V1)
   #60: #isNatKind(n__plus(V1,V2)) -> #activate(V2)
   #61: #activate(n__s(X)) -> #s(activate(X))
   #62: #activate(n__s(X)) -> #activate(X)
   #63: #isNatKind(n__x(V1,V2)) -> #and(isNatKind(activate(V1)),n__isNatKind(activate(V2)))
   #64: #isNatKind(n__x(V1,V2)) -> #isNatKind(activate(V1))
   #65: #isNatKind(n__x(V1,V2)) -> #activate(V1)
   #66: #isNatKind(n__x(V1,V2)) -> #activate(V2)
   #67: #isNat(n__s(V1)) -> #U21(isNatKind(activate(V1)),activate(V1))
   #68: #isNat(n__s(V1)) -> #isNatKind(activate(V1))
   #69: #isNat(n__s(V1)) -> #activate(V1)
   #70: #isNat(n__s(V1)) -> #activate(V1)
   #71: #U11(tt(),V1,V2) -> #U12(isNat(activate(V1)),activate(V2))
   #72: #U11(tt(),V1,V2) -> #isNat(activate(V1))
   #73: #U11(tt(),V1,V2) -> #activate(V1)
   #74: #U11(tt(),V1,V2) -> #activate(V2)
   #75: #isNat(n__plus(V1,V2)) -> #U11(and(isNatKind(activate(V1)),n__isNatKind(activate(V2))),activate(V1),activate(V2))
   #76: #isNat(n__plus(V1,V2)) -> #and(isNatKind(activate(V1)),n__isNatKind(activate(V2)))
   #77: #isNat(n__plus(V1,V2)) -> #isNatKind(activate(V1))
   #78: #isNat(n__plus(V1,V2)) -> #activate(V1)
   #79: #isNat(n__plus(V1,V2)) -> #activate(V2)
   #80: #isNat(n__plus(V1,V2)) -> #activate(V1)
   #81: #isNat(n__plus(V1,V2)) -> #activate(V2)
   #82: #U21(tt(),V1) -> #U22(isNat(activate(V1)))
   #83: #U21(tt(),V1) -> #isNat(activate(V1))
   #84: #U21(tt(),V1) -> #activate(V1)
Number of SCCs: 1, DPs: 76, edges: 702
	SCC { #2..15 #18..34 #36..38 #40..42 #44..60 #62..81 #83 #84 }
Removing DPs: Order(PosReal,>,Sum)... Order(PosReal,>,Max)... succeeded.
   |0|()	weight: (/ 1 16)
 #U32(x1,x2)	weight: max{0, (/ 1 4) + x2}
isNatKind(x1)	weight: x1
   U21(x1,x2)	weight: 0
 #|0|()	weight: 0
   U11(x1,x2,x3)	weight: max{0, (/ 1 16) + x3}
     s(x1)	weight: x1
#isNat(x1)	weight: (/ 1 8) + x1
activate(x1)	weight: x1
   U71(x1,x2,x3)	weight: max{0, (/ 5 16) + x3, (/ 1 4) + x2}
n__isNatKind(x1)	weight: x1
   and(x1,x2)	weight: max{x2, x1}
#plus(x1,x2)	weight: max{(/ 7 16) + x2, (/ 1 8) + x1}
#activate(x1)	weight: (/ 1 8) + x1
 #U13(x1)	weight: 0
   U12(x1,x2)	weight: max{0, (/ 1 16) + x2}
 #U33(x1)	weight: 0
     x(x1,x2)	weight: max{(/ 1 4) + x2, (/ 5 16) + x1}
  n__s(x1)	weight: x1
 #U12(x1,x2)	weight: max{0, (/ 3 16) + x2}
   #x(x1,x2)	weight: max{(/ 3 8) + x2, (/ 7 16) + x1}
   #s(x1)	weight: 0
n__isNat(x1)	weight: x1
n__plus(x1,x2)	weight: max{(/ 5 16) + x2, x1}
   U32(x1,x2)	weight: 0
   U33(x1)	weight: 0
  n__0()	weight: (/ 1 16)
 isNat(x1)	weight: x1
  n__x(x1,x2)	weight: max{(/ 1 4) + x2, (/ 5 16) + x1}
  plus(x1,x2)	weight: max{(/ 5 16) + x2, x1}
   U61(x1)	weight: (/ 1 16)
 #U51(x1,x2,x3)	weight: max{(/ 1 8) + x3, (/ 7 16) + x2, (/ 1 16) + x1}
 #U11(x1,x2,x3)	weight: max{0, (/ 7 16) + x3, (/ 1 8) + x2}
   U31(x1,x2,x3)	weight: 0
 #U41(x1,x2)	weight: max{0, (/ 1 8) + x2}
 #U21(x1,x2)	weight: max{0, (/ 1 8) + x2}
 #U22(x1)	weight: 0
    tt()	weight: 0
n__and(x1,x2)	weight: max{x2, x1}
 #U71(x1,x2,x3)	weight: max{0, (/ 7 16) + x3, (/ 3 8) + x2}
   U13(x1)	weight: (/ 1 16)
   U22(x1)	weight: 0
   U51(x1,x2,x3)	weight: max{0, x3, (/ 5 16) + x2}
#isNatKind(x1)	weight: (/ 1 8) + x1
   U41(x1,x2)	weight: max{0, x2}
 #U31(x1,x2,x3)	weight: max{0, (/ 5 16) + x3, (/ 3 16) + x2}
 #and(x1,x2)	weight: max{0, (/ 1 8) + x2}
 #U61(x1)	weight: 0
    Usable rules: { 1..40 }
    Removed DPs: #2 #3 #6 #7 #10..13 #18 #19 #22 #23 #26..28 #30..32 #36 #37 #42 #49..57 #60 #63..66 #71 #74 #76 #79 #81
Number of SCCs: 1, DPs: 36, edges: 193
	SCC { #4 #5 #8 #9 #14 #15 #20 #21 #24 #25 #29 #33 #34 #38 #40 #41 #44..48 #58 #59 #62 #67..70 #72 #73 #75 #77 #78 #80 #83 #84 }
Removing DPs: Order(PosReal,>,Sum)... Order(PosReal,>,Max)... QLPOpS... succeeded.
   |0|()	status: []	precedence above: U12 U32 U33 n__0 tt U13 U22
 #U32(x1,x2)	status: [x2]	precedence above:
isNatKind(x1)	status: x1
   U21(x1,x2)	status: x1
 #|0|()	status: []	precedence above:
   U11(x1,x2,x3)	status: x1
     s(x1)	status: [x1]	precedence above: #activate n__s #and
#isNat(x1)	status: [x1]	precedence above: isNatKind s n__isNatKind and #activate n__s #U11 #U21 n__and #and
activate(x1)	status: x1
   U71(x1,x2,x3)	status: [x3,x2,x1]	precedence above: |0| isNatKind s #isNat n__isNatKind and #plus #activate U12 x n__s #x n__isNat n__plus U32 U33 n__0 isNat n__x plus U61 #U51 #U11 U31 #U41 #U21 tt n__and #U71 U13 U22 U51 #and
n__isNatKind(x1)	status: x1
   and(x1,x2)	status: [x2,x1]	precedence above: n__and
#plus(x1,x2)	status: [x2,x1]	precedence above: isNatKind s #isNat n__isNatKind and #activate n__s n__isNat isNat #U51 #U11 U31 #U41 #U21 n__and #and
#activate(x1)	status: [x1]	precedence above: s n__s #and
 #U13(x1)	status: []	precedence above:
   U12(x1,x2)	status: []	precedence above: U32 U33 tt U13 U22
 #U33(x1)	status: []	precedence above:
     x(x1,x2)	status: [x1,x2]	precedence above: |0| isNatKind s #isNat U71 n__isNatKind and #plus #activate U12 n__s #x n__isNat n__plus U32 U33 n__0 isNat n__x plus U61 #U51 #U11 U31 #U41 #U21 tt n__and #U71 U13 U22 U51 #and
  n__s(x1)	status: [x1]	precedence above: s #activate #and
 #U12(x1,x2)	status: [x2]	precedence above:
   #x(x1,x2)	status: [x1,x2]	precedence above: |0| isNatKind s #isNat U71 n__isNatKind and #plus #activate U12 x n__s n__isNat n__plus U32 U33 n__0 isNat n__x plus U61 #U51 #U11 U31 #U41 #U21 tt n__and #U71 U13 U22 U51 #and
   #s(x1)	status: []	precedence above:
n__isNat(x1)	status: [x1]	precedence above: isNatKind s #isNat n__isNatKind and #activate n__s isNat #U11 U31 #U21 n__and #and
n__plus(x1,x2)	status: [x2,x1]	precedence above: isNatKind s #isNat n__isNatKind and #plus #activate n__s n__isNat isNat plus #U51 #U11 U31 #U41 #U21 n__and U51 #and
   U32(x1,x2)	status: []	precedence above: U12 U33 tt U13 U22
   U33(x1)	status: []	precedence above: U12 U32 tt U13 U22
  n__0()	status: []	precedence above: |0| U12 U32 U33 tt U13 U22
 isNat(x1)	status: [x1]	precedence above: isNatKind s #isNat n__isNatKind and #activate n__s n__isNat #U11 U31 #U21 n__and #and
  n__x(x1,x2)	status: [x1,x2]	precedence above: |0| isNatKind s #isNat U71 n__isNatKind and #plus #activate U12 x n__s #x n__isNat n__plus U32 U33 n__0 isNat plus U61 #U51 #U11 U31 #U41 #U21 tt n__and #U71 U13 U22 U51 #and
  plus(x1,x2)	status: [x2,x1]	precedence above: isNatKind s #isNat n__isNatKind and #plus #activate n__s n__isNat n__plus isNat #U51 #U11 U31 #U41 #U21 n__and U51 #and
   U61(x1)	status: [x1]	precedence above: |0| U12 U32 U33 n__0 tt U13 U22
 #U51(x1,x2,x3)	status: [x2,x3]	precedence above: isNatKind s #isNat n__isNatKind and #plus #activate n__s n__isNat isNat #U11 U31 #U41 #U21 n__and #and
 #U11(x1,x2,x3)	status: [x2,x3]	precedence above: isNatKind s #isNat n__isNatKind and #activate n__s #U21 n__and #and
   U31(x1,x2,x3)	status: [x2,x1]	precedence above:
 #U41(x1,x2)	status: [x1,x2]	precedence above: s and #activate n__s n__and #and
 #U21(x1,x2)	status: [x2,x1]	precedence above: isNatKind s #isNat n__isNatKind and #activate n__s #U11 n__and #and
 #U22(x1)	status: []	precedence above:
    tt()	status: []	precedence above: U12 U32 U33 U13 U22
n__and(x1,x2)	status: [x2,x1]	precedence above: and
 #U71(x1,x2,x3)	status: [x3,x2,x1]	precedence above: |0| isNatKind s #isNat U71 n__isNatKind and #plus #activate U12 x n__s #x n__isNat n__plus U32 U33 n__0 isNat n__x plus U61 #U51 #U11 U31 #U41 #U21 tt n__and U13 U22 U51 #and
   U13(x1)	status: []	precedence above: U12 U32 U33 tt U22
   U22(x1)	status: []	precedence above: U12 U32 U33 tt U13
   U51(x1,x2,x3)	status: [x2,x3,x1]	precedence above: isNatKind s #isNat n__isNatKind and #plus #activate n__s n__isNat n__plus isNat plus #U51 #U11 U31 #U41 #U21 n__and #and
#isNatKind(x1)	status: x1
   U41(x1,x2)	status: x2
 #U31(x1,x2,x3)	status: [x3,x2,x1]	precedence above:
 #and(x1,x2)	status: [x2,x1]	precedence above: s #activate n__s
 #U61(x1)	status: []	precedence above:
    Usable rules: { 1..40 }
    Removed DPs: #4 #5 #8 #9 #14 #15 #20 #21 #24 #25 #29 #33 #38 #41 #44..48 #58 #59 #62 #67..70 #72 #73 #75 #77 #78 #80 #83 #84
Number of SCCs: 0, DPs: 0, edges: 0
YES