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(X1,X2)
    21: activate(n__isNat(X)) -> isNat(X)
    22: activate(n__s(X)) -> s(X)
    23: activate(n__x(X1,X2)) -> x(X1,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(X1,X2)
   #16: #x(N,0()) -> #U31(isNat(N))
   #17: #x(N,0()) -> #isNat(N)
   #18: #activate(n__plus(X1,X2)) -> #plus(X1,X2)
   #19: #isNat(n__plus(V1,V2)) -> #and(isNat(activate(V1)),n__isNat(activate(V2)))
   #20: #isNat(n__plus(V1,V2)) -> #isNat(activate(V1))
   #21: #isNat(n__plus(V1,V2)) -> #activate(V1)
   #22: #isNat(n__plus(V1,V2)) -> #activate(V2)
   #23: #plus(N,0()) -> #U11(isNat(N),N)
   #24: #plus(N,0()) -> #isNat(N)
   #25: #and(tt(),X) -> #activate(X)
   #26: #activate(n__s(X)) -> #s(X)
   #27: #activate(n__0()) -> #0()
   #28: #activate(n__isNat(X)) -> #isNat(X)
   #29: #U31(tt()) -> #0()
   #30: #U11(tt(),N) -> #activate(N)
   #31: #isNat(n__s(V1)) -> #isNat(activate(V1))
   #32: #isNat(n__s(V1)) -> #activate(V1)
   #33: #U41(tt(),M,N) -> #plus(x(activate(N),activate(M)),activate(N))
   #34: #U41(tt(),M,N) -> #x(activate(N),activate(M))
   #35: #U41(tt(),M,N) -> #activate(N)
   #36: #U41(tt(),M,N) -> #activate(M)
   #37: #U41(tt(),M,N) -> #activate(N)
Number of SCCs: 1, DPs: 32, edges: 152
	SCC { #2..15 #17..25 #28 #30..37 }
Removing DPs: Order(PosReal,>,Sum)... Order(PosReal,>,Max)... succeeded.
   #0()	weight: 0
   U21(x1,x2,x3)	weight: max{0, x3, (/ 1 8) + x2}
   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{(/ 1 4) + x2, (/ 1 8) + x1}
#activate(x1)	weight: (/ 1 8) + x1
     x(x1,x2)	weight: max{(/ 1 16) + x2, (/ 1 8) + x1}
  n__s(x1)	weight: x1
     0()	weight: (/ 1 4)
   #x(x1,x2)	weight: max{(/ 3 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: (/ 1 4)
 isNat(x1)	weight: x1
  n__x(x1,x2)	weight: max{(/ 1 16) + x2, (/ 1 8) + x1}
  plus(x1,x2)	weight: max{(/ 1 8) + x2, x1}
 #U11(x1,x2)	weight: max{0, (/ 1 8) + x2}
   U31(x1)	weight: (/ 1 8) + x1
 #U41(x1,x2,x3)	weight: max{0, (/ 1 4) + x3, (/ 3 16) + x2}
 #U21(x1,x2,x3)	weight: max{(/ 1 8) + x3, (/ 1 4) + x2, (/ 1 16) + x1}
    tt()	weight: (/ 1 4)
   U41(x1,x2,x3)	weight: max{0, (/ 1 8) + x3, (/ 1 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 #17 #19 #22 #35..37
Number of SCCs: 1, DPs: 18, edges: 51
	SCC { #2 #3 #5 #12 #13 #15 #18 #20 #21 #23..25 #28 #30..34 }
Removing DPs: Order(PosReal,>,Sum)... Order(PosReal,>,Max)... QLPOpS... succeeded.
   #0()	status: []	precedence above:
   U21(x1,x2,x3)	status: [x2,x3,x1]	precedence above: U11 s and n__s n__plus plus
   U11(x1,x2)	status: x2
     s(x1)	status: [x1]	precedence above: n__s
#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: [x1,x2]	precedence above: U21 U11 s activate and n__s 0 #x n__plus n__0 n__x plus U31 #U41 tt U41
  n__s(x1)	status: [x1]	precedence above: s
     0()	status: []	precedence above: U11 s n__s n__0 tt
   #x(x1,x2)	status: [x1,x2]	precedence above: U21 U11 s activate and x n__s 0 n__plus n__0 n__x plus U31 #U41 tt U41
   #s(x1)	status: []	precedence above:
n__isNat(x1)	status: x1
n__plus(x1,x2)	status: [x2,x1]	precedence above: U21 U11 s and n__s plus
  n__0()	status: []	precedence above: U11 s n__s 0 tt
 isNat(x1)	status: x1
  n__x(x1,x2)	status: [x1,x2]	precedence above: U21 U11 s activate and x n__s 0 #x n__plus n__0 plus U31 #U41 tt U41
  plus(x1,x2)	status: [x2,x1]	precedence above: U21 U11 s and n__s n__plus
 #U11(x1,x2)	status: x2
   U31(x1)	status: [x1]	precedence above: U11 s n__s 0 n__0 tt
 #U41(x1,x2,x3)	status: [x3,x2]	precedence above: U21 U11 s activate and x n__s 0 #x n__plus n__0 n__x plus U31 tt U41
 #U21(x1,x2,x3)	status: x3
    tt()	status: []	precedence above: s n__s
   U41(x1,x2,x3)	status: [x3,x2,x1]	precedence above: U21 U11 s activate and x n__s 0 #x n__plus n__0 n__x plus U31 #U41 tt
 #U31(x1)	status: []	precedence above:
 #and(x1,x2)	status: x2
    Usable rules: { 1..24 }
    Removed DPs: #5 #18 #20 #21 #31 #32
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:
   U21(x1,x2,x3)	status: [x2,x3,x1]	precedence above: U11 s and n__s n__plus plus
   U11(x1,x2)	status: x2
     s(x1)	status: [x1]	precedence above: n__s
#isNat(x1)	status: x1
activate(x1)	status: x1
   and(x1,x2)	status: x2
#plus(x1,x2)	status: [x2,x1]	precedence above: and #activate n__isNat isNat #U11 #U21 #and
#activate(x1)	status: x1
     x(x1,x2)	status: [x1,x2]	precedence above: U21 U11 s activate and n__s 0 #x n__plus n__0 n__x plus U31 #U41 tt U41
  n__s(x1)	status: [x1]	precedence above: s
     0()	status: []	precedence above: U11 s n__s n__0 tt
   #x(x1,x2)	status: [x1,x2]	precedence above: U21 U11 s activate and x n__s 0 n__plus n__0 n__x plus U31 #U41 tt U41
   #s(x1)	status: []	precedence above:
n__isNat(x1)	status: x1
n__plus(x1,x2)	status: [x2,x1]	precedence above: U21 U11 s and n__s plus
  n__0()	status: []	precedence above: U11 s n__s 0 tt
 isNat(x1)	status: x1
  n__x(x1,x2)	status: [x1,x2]	precedence above: U21 U11 s activate and x n__s 0 #x n__plus n__0 plus U31 #U41 tt U41
  plus(x1,x2)	status: [x2,x1]	precedence above: U21 U11 s and n__s n__plus
 #U11(x1,x2)	status: x2
   U31(x1)	status: [x1]	precedence above: U11 s n__s 0 n__0 tt
 #U41(x1,x2,x3)	status: [x3,x2]	precedence above: U21 U11 s activate and x n__s 0 #x n__plus n__0 n__x plus U31 tt U41
 #U21(x1,x2,x3)	status: [x2,x3]	precedence above: and #plus #activate n__isNat isNat #U11 #and
    tt()	status: []	precedence above: s n__s
   U41(x1,x2,x3)	status: [x3,x2,x1]	precedence above: U21 U11 s activate and x n__s 0 #x n__plus n__0 n__x plus U31 #U41 tt
 #U31(x1)	status: []	precedence above:
 #and(x1,x2)	status: x2
    Usable rules: { 1..24 }
    Removed DPs: #12
Number of SCCs: 0, DPs: 0, edges: 0
YES