Input TRS:
    1: U11(tt(),N) -> activate(N)
    2: U21(tt(),M,N) -> s(plus(activate(N),activate(M)))
    3: U31(tt()) -> |0|()
    4: U41(tt(),M,N) -> plus(x(activate(N),activate(M)),activate(N))
    5: and(tt(),X) -> activate(X)
    6: isNat(n__0()) -> tt()
    7: isNat(n__plus(V1,V2)) -> and(isNat(activate(V1)),n__isNat(activate(V2)))
    8: isNat(n__s(V1)) -> isNat(activate(V1))
    9: isNat(n__x(V1,V2)) -> and(isNat(activate(V1)),n__isNat(activate(V2)))
    10: plus(N,|0|()) -> U11(isNat(N),N)
    11: plus(N,s(M)) -> U21(and(isNat(M),n__isNat(N)),M,N)
    12: x(N,|0|()) -> U31(isNat(N))
    13: x(N,s(M)) -> U41(and(isNat(M),n__isNat(N)),M,N)
    14: |0|() -> n__0()
    15: plus(X1,X2) -> n__plus(X1,X2)
    16: isNat(X) -> n__isNat(X)
    17: s(X) -> n__s(X)
    18: x(X1,X2) -> n__x(X1,X2)
    19: activate(n__0()) -> |0|()
    20: activate(n__plus(X1,X2)) -> plus(activate(X1),activate(X2))
    21: activate(n__isNat(X)) -> isNat(X)
    22: activate(n__s(X)) -> s(activate(X))
    23: activate(n__x(X1,X2)) -> x(activate(X1),activate(X2))
    24: activate(X) -> X
Number of strict rules: 24
Direct Order(PosReal,>,Poly) ... failed.
Freezing ... failed.
Dependency Pairs:
   #1: #U21(tt(),M,N) -> #s(plus(activate(N),activate(M)))
   #2: #U21(tt(),M,N) -> #plus(activate(N),activate(M))
   #3: #U21(tt(),M,N) -> #activate(N)
   #4: #U21(tt(),M,N) -> #activate(M)
   #5: #x(N,s(M)) -> #U41(and(isNat(M),n__isNat(N)),M,N)
   #6: #x(N,s(M)) -> #and(isNat(M),n__isNat(N))
   #7: #x(N,s(M)) -> #isNat(M)
   #8: #isNat(n__x(V1,V2)) -> #and(isNat(activate(V1)),n__isNat(activate(V2)))
   #9: #isNat(n__x(V1,V2)) -> #isNat(activate(V1))
   #10: #isNat(n__x(V1,V2)) -> #activate(V1)
   #11: #isNat(n__x(V1,V2)) -> #activate(V2)
   #12: #plus(N,s(M)) -> #U21(and(isNat(M),n__isNat(N)),M,N)
   #13: #plus(N,s(M)) -> #and(isNat(M),n__isNat(N))
   #14: #plus(N,s(M)) -> #isNat(M)
   #15: #activate(n__x(X1,X2)) -> #x(activate(X1),activate(X2))
   #16: #activate(n__x(X1,X2)) -> #activate(X1)
   #17: #activate(n__x(X1,X2)) -> #activate(X2)
   #18: #x(N,|0|()) -> #U31(isNat(N))
   #19: #x(N,|0|()) -> #isNat(N)
   #20: #activate(n__plus(X1,X2)) -> #plus(activate(X1),activate(X2))
   #21: #activate(n__plus(X1,X2)) -> #activate(X1)
   #22: #activate(n__plus(X1,X2)) -> #activate(X2)
   #23: #isNat(n__plus(V1,V2)) -> #and(isNat(activate(V1)),n__isNat(activate(V2)))
   #24: #isNat(n__plus(V1,V2)) -> #isNat(activate(V1))
   #25: #isNat(n__plus(V1,V2)) -> #activate(V1)
   #26: #isNat(n__plus(V1,V2)) -> #activate(V2)
   #27: #plus(N,|0|()) -> #U11(isNat(N),N)
   #28: #plus(N,|0|()) -> #isNat(N)
   #29: #and(tt(),X) -> #activate(X)
   #30: #activate(n__s(X)) -> #s(activate(X))
   #31: #activate(n__s(X)) -> #activate(X)
   #32: #activate(n__0()) -> #|0|()
   #33: #activate(n__isNat(X)) -> #isNat(X)
   #34: #U31(tt()) -> #|0|()
   #35: #U11(tt(),N) -> #activate(N)
   #36: #isNat(n__s(V1)) -> #isNat(activate(V1))
   #37: #isNat(n__s(V1)) -> #activate(V1)
   #38: #U41(tt(),M,N) -> #plus(x(activate(N),activate(M)),activate(N))
   #39: #U41(tt(),M,N) -> #x(activate(N),activate(M))
   #40: #U41(tt(),M,N) -> #activate(N)
   #41: #U41(tt(),M,N) -> #activate(M)
   #42: #U41(tt(),M,N) -> #activate(N)
Number of SCCs: 1, DPs: 37, edges: 252
	SCC { #2..17 #19..29 #31 #33 #35..42 }
Removing DPs: Order(PosReal,>,Sum)... Order(PosReal,>,Max)... succeeded.
   |0|()	weight: (/ 3 16)
   U21(x1,x2,x3)	weight: max{0, x3, (/ 1 8) + x2}
 #|0|()	weight: 0
   U11(x1,x2)	weight: max{0, x2}
     s(x1)	weight: x1
#isNat(x1)	weight: (/ 1 8) + x1
activate(x1)	weight: x1
   and(x1,x2)	weight: max{0, (/ 1 16) + x2}
#plus(x1,x2)	weight: max{(/ 3 16) + x2, (/ 1 8) + x1}
#activate(x1)	weight: (/ 1 8) + x1
     x(x1,x2)	weight: max{(/ 3 16) + x2, (/ 1 8) + x1}
  n__s(x1)	weight: x1
   #x(x1,x2)	weight: max{(/ 5 16) + x2, (/ 1 4) + x1}
   #s(x1)	weight: 0
n__isNat(x1)	weight: x1
n__plus(x1,x2)	weight: max{(/ 1 8) + x2, x1}
  n__0()	weight: (/ 3 16)
 isNat(x1)	weight: x1
  n__x(x1,x2)	weight: max{(/ 3 16) + x2, (/ 1 8) + x1}
  plus(x1,x2)	weight: max{(/ 1 8) + x2, x1}
 #U11(x1,x2)	weight: max{(/ 1 8) + x2, (/ 1 8) + x1}
   U31(x1)	weight: (/ 3 16)
 #U41(x1,x2,x3)	weight: max{(/ 1 4) + x3, (/ 5 16) + x2, (/ 3 16) + x1}
 #U21(x1,x2,x3)	weight: max{(/ 1 8) + x3, (/ 3 16) + x2, (/ 1 16) + x1}
    tt()	weight: (/ 3 16)
   U41(x1,x2,x3)	weight: max{0, (/ 1 8) + x3, (/ 3 16) + x2}
 #U31(x1)	weight: 0
 #and(x1,x2)	weight: max{0, (/ 1 8) + x2}
    Usable rules: { 1..24 }
    Removed DPs: #4 #6..11 #14 #16 #17 #19 #22 #23 #26 #40..42
Number of SCCs: 1, DPs: 20, edges: 71
	SCC { #2 #3 #5 #12 #13 #15 #20 #21 #24 #25 #27..29 #31 #33 #35..39 }
Removing DPs: Order(PosReal,>,Sum)... Order(PosReal,>,Max)... QLPOpS... succeeded.
   |0|()	status: []	precedence above: n__0 tt
   U21(x1,x2,x3)	status: [x3,x2,x1]	precedence above: U11 s n__s n__isNat n__plus isNat plus
 #|0|()	status: []	precedence above:
   U11(x1,x2)	status: [x1,x2]	precedence above:
     s(x1)	status: [x1]	precedence above: n__s n__isNat isNat
#isNat(x1)	status: x1
activate(x1)	status: x1
   and(x1,x2)	status: x2
#plus(x1,x2)	status: x1
#activate(x1)	status: x1
     x(x1,x2)	status: [x2,x1]	precedence above: |0| U21 U11 s n__s #x n__isNat n__plus n__0 isNat n__x plus U31 #U41 tt U41
  n__s(x1)	status: [x1]	precedence above: s n__isNat isNat
   #x(x1,x2)	status: [x2,x1]	precedence above: |0| U21 U11 s x n__s n__isNat n__plus n__0 isNat n__x plus U31 #U41 tt U41
   #s(x1)	status: []	precedence above:
n__isNat(x1)	status: x1
n__plus(x1,x2)	status: [x1,x2]	precedence above: U21 U11 s n__s n__isNat isNat plus
  n__0()	status: []	precedence above: |0| tt
 isNat(x1)	status: x1
  n__x(x1,x2)	status: [x2,x1]	precedence above: |0| U21 U11 s x n__s #x n__isNat n__plus n__0 isNat plus U31 #U41 tt U41
  plus(x1,x2)	status: [x1,x2]	precedence above: U21 U11 s n__s n__isNat n__plus isNat
 #U11(x1,x2)	status: x2
   U31(x1)	status: [x1]	precedence above: |0| n__0 tt
 #U41(x1,x2,x3)	status: [x2,x3]	precedence above: |0| U21 U11 s x n__s #x n__isNat n__plus n__0 isNat n__x plus U31 tt U41
 #U21(x1,x2,x3)	status: x3
    tt()	status: []	precedence above:
   U41(x1,x2,x3)	status: [x2,x3,x1]	precedence above: |0| U21 U11 s x n__s #x n__isNat n__plus n__0 isNat n__x plus U31 #U41 tt
 #U31(x1)	status: []	precedence above:
 #and(x1,x2)	status: x2
    Usable rules: { 1..24 }
    Removed DPs: #20 #21 #24 #25 #31 #36 #37
Number of SCCs: 1, DPs: 11, edges: 17
	SCC { #2 #3 #5 #12 #13 #15 #27 #29 #35 #38 #39 }
Removing DPs: Order(PosReal,>,Sum)... Order(PosReal,>,Max)... succeeded.
   |0|()	weight: (/ 1 8)
   U21(x1,x2,x3)	weight: max{x3, (/ 1 8) + x2, x1}
 #|0|()	weight: 0
   U11(x1,x2)	weight: max{x2, x1}
     s(x1)	weight: x1
#isNat(x1)	weight: 0
activate(x1)	weight: x1
   and(x1,x2)	weight: max{x2, x1}
#plus(x1,x2)	weight: max{(/ 3 8) + x2, x1}
#activate(x1)	weight: x1
     x(x1,x2)	weight: max{0, (/ 5 8) + x1}
  n__s(x1)	weight: x1
   #x(x1,x2)	weight: max{0, (/ 5 8) + x1}
   #s(x1)	weight: 0
n__isNat(x1)	weight: (/ 1 8)
n__plus(x1,x2)	weight: max{(/ 1 8) + x2, x1}
  n__0()	weight: (/ 1 8)
 isNat(x1)	weight: (/ 1 8)
  n__x(x1,x2)	weight: max{0, (/ 5 8) + x1}
  plus(x1,x2)	weight: max{(/ 1 8) + x2, x1}
 #U11(x1,x2)	weight: max{x2, (/ 3 8) + x1}
   U31(x1)	weight: x1
 #U41(x1,x2,x3)	weight: max{0, (/ 5 8) + x3, (/ 3 8) + x1}
 #U21(x1,x2,x3)	weight: max{x3, (/ 3 8) + x2, (/ 1 4) + x1}
    tt()	weight: (/ 1 8)
   U41(x1,x2,x3)	weight: max{0, (/ 5 8) + x3, x1}
 #U31(x1)	weight: 0
 #and(x1,x2)	weight: max{0, (/ 1 8) + x2}
    Usable rules: { 1..24 }
    Removed DPs: #13 #29
Number of SCCs: 1, DPs: 9, edges: 13
	SCC { #2 #3 #5 #12 #15 #27 #35 #38 #39 }
Removing DPs: Order(PosReal,>,Sum)... Order(PosReal,>,Max)... QLPOpS... succeeded.
   |0|()	status: []	precedence above: n__0 tt
   U21(x1,x2,x3)	status: [x3,x2,x1]	precedence above: U11 s n__s n__isNat n__plus isNat plus
 #|0|()	status: []	precedence above:
   U11(x1,x2)	status: [x1,x2]	precedence above:
     s(x1)	status: [x1]	precedence above: n__s n__isNat isNat
#isNat(x1)	status: x1
activate(x1)	status: x1
   and(x1,x2)	status: x2
#plus(x1,x2)	status: x1
#activate(x1)	status: x1
     x(x1,x2)	status: [x2,x1]	precedence above: |0| U21 U11 s n__s #x n__isNat n__plus n__0 isNat n__x plus U31 #U41 tt U41
  n__s(x1)	status: [x1]	precedence above: s n__isNat isNat
   #x(x1,x2)	status: [x2,x1]	precedence above: |0| U21 U11 s x n__s n__isNat n__plus n__0 isNat n__x plus U31 #U41 tt U41
   #s(x1)	status: []	precedence above:
n__isNat(x1)	status: x1
n__plus(x1,x2)	status: [x1,x2]	precedence above: U21 U11 s n__s n__isNat isNat plus
  n__0()	status: []	precedence above: |0| tt
 isNat(x1)	status: x1
  n__x(x1,x2)	status: [x2,x1]	precedence above: |0| U21 U11 s x n__s #x n__isNat n__plus n__0 isNat plus U31 #U41 tt U41
  plus(x1,x2)	status: [x1,x2]	precedence above: U21 U11 s n__s n__isNat n__plus isNat
 #U11(x1,x2)	status: x2
   U31(x1)	status: [x1]	precedence above: |0| n__0 tt
 #U41(x1,x2,x3)	status: [x2,x3]	precedence above: |0| U21 U11 s x n__s #x n__isNat n__plus n__0 isNat n__x plus U31 tt U41
 #U21(x1,x2,x3)	status: x3
    tt()	status: []	precedence above:
   U41(x1,x2,x3)	status: [x2,x3,x1]	precedence above: |0| U21 U11 s x n__s #x n__isNat n__plus n__0 isNat n__x plus U31 #U41 tt
 #U31(x1)	status: []	precedence above:
 #and(x1,x2)	status: x2
    Usable rules: { 1..24 }
    Removed DPs: #5
Number of SCCs: 1, DPs: 2, edges: 2
	SCC { #2 #12 }
Removing DPs: Order(PosReal,>,Sum)... Order(PosReal,>,Max)... QLPOpS... succeeded.
   |0|()	status: []	precedence above: n__0 tt
   U21(x1,x2,x3)	status: [x3,x2,x1]	precedence above: U11 s and n__s n__isNat n__plus isNat plus
 #|0|()	status: []	precedence above:
   U11(x1,x2)	status: [x1,x2]	precedence above:
     s(x1)	status: [x1]	precedence above: n__s
#isNat(x1)	status: x1
activate(x1)	status: x1
   and(x1,x2)	status: [x2,x1]	precedence above: s n__s
#plus(x1,x2)	status: [x1,x2]	precedence above: s and n__s n__isNat isNat #U21
#activate(x1)	status: x1
     x(x1,x2)	status: [x2,x1]	precedence above: |0| U21 U11 s and n__s #x n__isNat n__plus n__0 isNat n__x plus U31 #U41 tt U41
  n__s(x1)	status: [x1]	precedence above: s
   #x(x1,x2)	status: [x2,x1]	precedence above: |0| U21 U11 s and x n__s n__isNat n__plus n__0 isNat n__x plus U31 #U41 tt U41
   #s(x1)	status: []	precedence above:
n__isNat(x1)	status: [x1]	precedence above: s and n__s isNat
n__plus(x1,x2)	status: [x1,x2]	precedence above: U21 U11 s and n__s n__isNat isNat plus
  n__0()	status: []	precedence above: |0| tt
 isNat(x1)	status: [x1]	precedence above: s and n__s n__isNat
  n__x(x1,x2)	status: [x2,x1]	precedence above: |0| U21 U11 s and x n__s #x n__isNat n__plus n__0 isNat plus U31 #U41 tt U41
  plus(x1,x2)	status: [x1,x2]	precedence above: U21 U11 s and n__s n__isNat n__plus isNat
 #U11(x1,x2)	status: x2
   U31(x1)	status: []	precedence above: |0| n__0 tt
 #U41(x1,x2,x3)	status: [x2,x3]	precedence above: |0| U21 U11 s and x n__s #x n__isNat n__plus n__0 isNat n__x plus U31 tt U41
 #U21(x1,x2,x3)	status: [x3,x2,x1]	precedence above: s and #plus n__s n__isNat isNat
    tt()	status: []	precedence above:
   U41(x1,x2,x3)	status: [x2,x3,x1]	precedence above: |0| U21 U11 s and x n__s #x n__isNat n__plus n__0 isNat n__x plus U31 #U41 tt
 #U31(x1)	status: []	precedence above:
 #and(x1,x2)	status: x2
    Usable rules: { 1..24 }
    Removed DPs: #2 #12
Number of SCCs: 0, DPs: 0, edges: 0
YES