Input TRS:
    1: a__U11(tt(),V2) -> a__U12(a__isNat(V2))
    2: a__U12(tt()) -> tt()
    3: a__U21(tt()) -> tt()
    4: a__U31(tt(),V2) -> a__U32(a__isNat(V2))
    5: a__U32(tt()) -> tt()
    6: a__U41(tt(),N) -> mark(N)
    7: a__U51(tt(),M,N) -> a__U52(a__isNat(N),M,N)
    8: a__U52(tt(),M,N) -> s(a__plus(mark(N),mark(M)))
    9: a__U61(tt()) -> 0()
    10: a__U71(tt(),M,N) -> a__U72(a__isNat(N),M,N)
    11: a__U72(tt(),M,N) -> a__plus(a__x(mark(N),mark(M)),mark(N))
    12: a__isNat(0()) -> tt()
    13: a__isNat(plus(V1,V2)) -> a__U11(a__isNat(V1),V2)
    14: a__isNat(s(V1)) -> a__U21(a__isNat(V1))
    15: a__isNat(x(V1,V2)) -> a__U31(a__isNat(V1),V2)
    16: a__plus(N,0()) -> a__U41(a__isNat(N),N)
    17: a__plus(N,s(M)) -> a__U51(a__isNat(M),M,N)
    18: a__x(N,0()) -> a__U61(a__isNat(N))
    19: a__x(N,s(M)) -> a__U71(a__isNat(M),M,N)
    20: mark(U11(X1,X2)) -> a__U11(mark(X1),X2)
    21: mark(U12(X)) -> a__U12(mark(X))
    22: mark(isNat(X)) -> a__isNat(X)
    23: mark(U21(X)) -> a__U21(mark(X))
    24: mark(U31(X1,X2)) -> a__U31(mark(X1),X2)
    25: mark(U32(X)) -> a__U32(mark(X))
    26: mark(U41(X1,X2)) -> a__U41(mark(X1),X2)
    27: mark(U51(X1,X2,X3)) -> a__U51(mark(X1),X2,X3)
    28: mark(U52(X1,X2,X3)) -> a__U52(mark(X1),X2,X3)
    29: mark(plus(X1,X2)) -> a__plus(mark(X1),mark(X2))
    30: mark(U61(X)) -> a__U61(mark(X))
    31: mark(U71(X1,X2,X3)) -> a__U71(mark(X1),X2,X3)
    32: mark(U72(X1,X2,X3)) -> a__U72(mark(X1),X2,X3)
    33: mark(x(X1,X2)) -> a__x(mark(X1),mark(X2))
    34: mark(tt()) -> tt()
    35: mark(s(X)) -> s(mark(X))
    36: mark(0()) -> 0()
    37: a__U11(X1,X2) -> U11(X1,X2)
    38: a__U12(X) -> U12(X)
    39: a__isNat(X) -> isNat(X)
    40: a__U21(X) -> U21(X)
    41: a__U31(X1,X2) -> U31(X1,X2)
    42: a__U32(X) -> U32(X)
    43: a__U41(X1,X2) -> U41(X1,X2)
    44: a__U51(X1,X2,X3) -> U51(X1,X2,X3)
    45: a__U52(X1,X2,X3) -> U52(X1,X2,X3)
    46: a__plus(X1,X2) -> plus(X1,X2)
    47: a__U61(X) -> U61(X)
    48: a__U71(X1,X2,X3) -> U71(X1,X2,X3)
    49: a__U72(X1,X2,X3) -> U72(X1,X2,X3)
    50: a__x(X1,X2) -> x(X1,X2)
Number of strict rules: 50
Direct Order(PosReal,>,Poly) ... failed.
Freezing ... failed.
Dependency Pairs:
   #1: #mark(plus(X1,X2)) -> #a__plus(mark(X1),mark(X2))
   #2: #mark(plus(X1,X2)) -> #mark(X1)
   #3: #mark(plus(X1,X2)) -> #mark(X2)
   #4: #mark(s(X)) -> #mark(X)
   #5: #a__U41(tt(),N) -> #mark(N)
   #6: #a__isNat(plus(V1,V2)) -> #a__U11(a__isNat(V1),V2)
   #7: #a__isNat(plus(V1,V2)) -> #a__isNat(V1)
   #8: #a__U72(tt(),M,N) -> #a__plus(a__x(mark(N),mark(M)),mark(N))
   #9: #a__U72(tt(),M,N) -> #a__x(mark(N),mark(M))
   #10: #a__U72(tt(),M,N) -> #mark(N)
   #11: #a__U72(tt(),M,N) -> #mark(M)
   #12: #a__U72(tt(),M,N) -> #mark(N)
   #13: #mark(U31(X1,X2)) -> #a__U31(mark(X1),X2)
   #14: #mark(U31(X1,X2)) -> #mark(X1)
   #15: #mark(U21(X)) -> #a__U21(mark(X))
   #16: #mark(U21(X)) -> #mark(X)
   #17: #mark(U71(X1,X2,X3)) -> #a__U71(mark(X1),X2,X3)
   #18: #mark(U71(X1,X2,X3)) -> #mark(X1)
   #19: #a__isNat(s(V1)) -> #a__U21(a__isNat(V1))
   #20: #a__isNat(s(V1)) -> #a__isNat(V1)
   #21: #mark(U61(X)) -> #a__U61(mark(X))
   #22: #mark(U61(X)) -> #mark(X)
   #23: #mark(U32(X)) -> #a__U32(mark(X))
   #24: #mark(U32(X)) -> #mark(X)
   #25: #mark(U11(X1,X2)) -> #a__U11(mark(X1),X2)
   #26: #mark(U11(X1,X2)) -> #mark(X1)
   #27: #a__U51(tt(),M,N) -> #a__U52(a__isNat(N),M,N)
   #28: #a__U51(tt(),M,N) -> #a__isNat(N)
   #29: #a__U71(tt(),M,N) -> #a__U72(a__isNat(N),M,N)
   #30: #a__U71(tt(),M,N) -> #a__isNat(N)
   #31: #mark(x(X1,X2)) -> #a__x(mark(X1),mark(X2))
   #32: #mark(x(X1,X2)) -> #mark(X1)
   #33: #mark(x(X1,X2)) -> #mark(X2)
   #34: #mark(U52(X1,X2,X3)) -> #a__U52(mark(X1),X2,X3)
   #35: #mark(U52(X1,X2,X3)) -> #mark(X1)
   #36: #mark(isNat(X)) -> #a__isNat(X)
   #37: #mark(U51(X1,X2,X3)) -> #a__U51(mark(X1),X2,X3)
   #38: #mark(U51(X1,X2,X3)) -> #mark(X1)
   #39: #a__plus(N,s(M)) -> #a__U51(a__isNat(M),M,N)
   #40: #a__plus(N,s(M)) -> #a__isNat(M)
   #41: #mark(U72(X1,X2,X3)) -> #a__U72(mark(X1),X2,X3)
   #42: #mark(U72(X1,X2,X3)) -> #mark(X1)
   #43: #a__x(N,s(M)) -> #a__U71(a__isNat(M),M,N)
   #44: #a__x(N,s(M)) -> #a__isNat(M)
   #45: #mark(U41(X1,X2)) -> #a__U41(mark(X1),X2)
   #46: #mark(U41(X1,X2)) -> #mark(X1)
   #47: #mark(U12(X)) -> #a__U12(mark(X))
   #48: #mark(U12(X)) -> #mark(X)
   #49: #a__plus(N,0()) -> #a__U41(a__isNat(N),N)
   #50: #a__plus(N,0()) -> #a__isNat(N)
   #51: #a__U11(tt(),V2) -> #a__U12(a__isNat(V2))
   #52: #a__U11(tt(),V2) -> #a__isNat(V2)
   #53: #a__U52(tt(),M,N) -> #a__plus(mark(N),mark(M))
   #54: #a__U52(tt(),M,N) -> #mark(N)
   #55: #a__U52(tt(),M,N) -> #mark(M)
   #56: #a__isNat(x(V1,V2)) -> #a__U31(a__isNat(V1),V2)
   #57: #a__isNat(x(V1,V2)) -> #a__isNat(V1)
   #58: #a__U31(tt(),V2) -> #a__U32(a__isNat(V2))
   #59: #a__U31(tt(),V2) -> #a__isNat(V2)
   #60: #a__x(N,0()) -> #a__U61(a__isNat(N))
   #61: #a__x(N,0()) -> #a__isNat(N)
Number of SCCs: 2, DPs: 44, edges: 563
	SCC { #6 #7 #20 #52 #56 #57 #59 }
Removing DPs: Order(PosReal,>,Sum)... succeeded.
a__plus(x1,x2)	weight: 0
   U21(x1)	weight: (/ 3 8) + x1
#a__U72(x1,x2,x3)	weight: 0
#a__U71(x1,x2,x3)	weight: 0
   U11(x1,x2)	weight: (/ 3 8) + x1 + x2
     s(x1)	weight: (/ 1 8) + x1
#a__U31(x1,x2)	weight: (/ 1 8) + x2
#a__isNat(x1)	weight: x1
   U71(x1,x2,x3)	weight: 0
#a__U51(x1,x2,x3)	weight: 0
#a__x(x1,x2)	weight: 0
   U72(x1,x2,x3)	weight: 0
#a__U11(x1,x2)	weight: (/ 1 8) + x2
a__U31(x1,x2)	weight: (/ 1 8) + x1
a__U51(x1,x2,x3)	weight: 0
  a__x(x1,x2)	weight: 0
   U12(x1)	weight: (/ 1 2) + x1
a__U41(x1,x2)	weight: 0
     x(x1,x2)	weight: (/ 1 4) + x1 + x2
#a__U12(x1)	weight: 0
#a__U21(x1)	weight: 0
#a__U61(x1)	weight: 0
#a__plus(x1,x2)	weight: 0
#mark(x1)	weight: 0
     0()	weight: 0
a__U21(x1)	weight: (/ 1 4)
a__U32(x1)	weight: (/ 1 8) + x1
#a__U52(x1,x2,x3)	weight: 0
  mark(x1)	weight: 0
a__U72(x1,x2,x3)	weight: 0
a__U11(x1,x2)	weight: (/ 1 4)
   U32(x1)	weight: (/ 1 4)
a__U52(x1,x2,x3)	weight: 0
a__U12(x1)	weight: (/ 3 8)
 isNat(x1)	weight: (/ 1 4) + x1
   U52(x1,x2,x3)	weight: 0
  plus(x1,x2)	weight: (/ 1 4) + x1 + x2
   U61(x1)	weight: 0
   U31(x1,x2)	weight: (/ 1 4) + x2
a__U71(x1,x2,x3)	weight: 0
a__U61(x1)	weight: 0
#a__U41(x1,x2)	weight: 0
    tt()	weight: 0
a__isNat(x1)	weight: (/ 1 8)
   U51(x1,x2,x3)	weight: 0
   U41(x1,x2)	weight: 0
#a__U32(x1)	weight: 0
    Usable rules: { }
    Removed DPs: #6 #7 #20 #52 #56 #57 #59
Number of SCCs: 1, DPs: 37, edges: 536
	SCC { #1..5 #8..12 #14 #16..18 #22 #24 #26 #27 #29 #31..35 #37..39 #41..43 #45 #46 #48 #49 #53..55 }
Removing DPs: Order(PosReal,>,Sum)... Order(PosReal,>,Max)... succeeded.
a__plus(x1,x2)	weight: max{(/ 1 4) + x2, x1}
   U21(x1)	weight: x1
#a__U72(x1,x2,x3)	weight: max{0, (/ 11 16) + x3, (/ 1 4) + x2}
#a__U71(x1,x2,x3)	weight: max{0, (/ 11 16) + x3, (/ 1 4) + x2}
   U11(x1,x2)	weight: max{(/ 1 4) + x2, x1}
     s(x1)	weight: x1
#a__U31(x1,x2)	weight: 0
#a__isNat(x1)	weight: 0
   U71(x1,x2,x3)	weight: max{(/ 5 8) + x3, (/ 3 16) + x2, (/ 1 8) + x1}
#a__U51(x1,x2,x3)	weight: max{0, (/ 1 16) + x3, (/ 3 16) + x2}
#a__x(x1,x2)	weight: max{(/ 1 4) + x2, (/ 11 16) + x1}
   U72(x1,x2,x3)	weight: max{(/ 5 8) + x3, (/ 3 16) + x2, (/ 3 8) + x1}
#a__U11(x1,x2)	weight: 0
a__U31(x1,x2)	weight: max{(/ 3 16) + x2, (/ 1 4) + x1}
a__U51(x1,x2,x3)	weight: max{x3, (/ 1 4) + x2, (/ 1 16) + x1}
  a__x(x1,x2)	weight: max{(/ 3 16) + x2, (/ 5 8) + x1}
   U12(x1)	weight: (/ 1 4) + x1
a__U41(x1,x2)	weight: max{x2, x1}
     x(x1,x2)	weight: max{(/ 3 16) + x2, (/ 5 8) + x1}
#a__U12(x1)	weight: 0
#a__U21(x1)	weight: 0
#a__U61(x1)	weight: 0
#a__plus(x1,x2)	weight: max{(/ 3 16) + x2, (/ 1 16) + x1}
#mark(x1)	weight: (/ 1 16) + x1
     0()	weight: (/ 3 16)
a__U21(x1)	weight: x1
a__U32(x1)	weight: (/ 3 16) + x1
#a__U52(x1,x2,x3)	weight: max{0, (/ 1 16) + x3, (/ 3 16) + x2}
  mark(x1)	weight: x1
a__U72(x1,x2,x3)	weight: max{(/ 5 8) + x3, (/ 3 16) + x2, (/ 3 8) + x1}
a__U11(x1,x2)	weight: max{(/ 1 4) + x2, x1}
   U32(x1)	weight: (/ 3 16) + x1
a__U52(x1,x2,x3)	weight: max{x3, (/ 1 4) + x2, x1}
a__U12(x1)	weight: (/ 1 4) + x1
 isNat(x1)	weight: x1
   U52(x1,x2,x3)	weight: max{x3, (/ 1 4) + x2, x1}
  plus(x1,x2)	weight: max{(/ 1 4) + x2, x1}
   U61(x1)	weight: (/ 7 16) + x1
   U31(x1,x2)	weight: max{(/ 3 16) + x2, (/ 1 4) + x1}
a__U71(x1,x2,x3)	weight: max{(/ 5 8) + x3, (/ 3 16) + x2, (/ 1 8) + x1}
a__U61(x1)	weight: (/ 7 16) + x1
#a__U41(x1,x2)	weight: max{0, (/ 1 16) + x2}
    tt()	weight: (/ 3 16)
a__isNat(x1)	weight: x1
   U51(x1,x2,x3)	weight: max{x3, (/ 1 4) + x2, (/ 1 16) + x1}
   U41(x1,x2)	weight: max{x2, x1}
#a__U32(x1)	weight: 0
    Usable rules: { 1..50 }
    Removed DPs: #3 #10..12 #14 #18 #22 #24 #32 #33 #38 #42 #48 #55
Number of SCCs: 1, DPs: 23, edges: 126
	SCC { #1 #2 #4 #5 #8 #9 #16 #17 #26 #27 #29 #31 #34 #35 #37 #39 #41 #43 #45 #46 #49 #53 #54 }
Removing DPs: Order(PosReal,>,Sum)... Order(PosReal,>,Max)... succeeded.
a__plus(x1,x2)	weight: max{(/ 1 2) + x2, x1}
   U21(x1)	weight: x1
#a__U72(x1,x2,x3)	weight: max{0, (/ 1 2) + x3}
#a__U71(x1,x2,x3)	weight: max{0, (/ 1 2) + x3}
   U11(x1,x2)	weight: max{0, x1}
     s(x1)	weight: x1
#a__U31(x1,x2)	weight: 0
#a__isNat(x1)	weight: 0
   U71(x1,x2,x3)	weight: max{0, (/ 1 2) + x3}
#a__U51(x1,x2,x3)	weight: max{0, x3, (/ 3 8) + x2}
#a__x(x1,x2)	weight: max{0, (/ 1 2) + x1}
   U72(x1,x2,x3)	weight: max{0, (/ 1 2) + x3}
#a__U11(x1,x2)	weight: 0
a__U31(x1,x2)	weight: 0
a__U51(x1,x2,x3)	weight: max{x3, (/ 1 2) + x2, (/ 1 4) + x1}
  a__x(x1,x2)	weight: max{0, (/ 1 2) + x1}
   U12(x1)	weight: 0
a__U41(x1,x2)	weight: max{x2, (/ 1 8) + x1}
     x(x1,x2)	weight: max{0, (/ 1 2) + x1}
#a__U12(x1)	weight: 0
#a__U21(x1)	weight: 0
#a__U61(x1)	weight: 0
#a__plus(x1,x2)	weight: max{(/ 3 8) + x2, x1}
#mark(x1)	weight: x1
     0()	weight: (/ 1 8)
a__U21(x1)	weight: x1
a__U32(x1)	weight: 0
#a__U52(x1,x2,x3)	weight: max{0, x3, (/ 3 8) + x2}
  mark(x1)	weight: x1
a__U72(x1,x2,x3)	weight: max{0, (/ 1 2) + x3}
a__U11(x1,x2)	weight: max{0, x1}
   U32(x1)	weight: 0
a__U52(x1,x2,x3)	weight: max{x3, (/ 1 2) + x2, (/ 1 8) + x1}
a__U12(x1)	weight: 0
 isNat(x1)	weight: (/ 1 8)
   U52(x1,x2,x3)	weight: max{x3, (/ 1 2) + x2, (/ 1 8) + x1}
  plus(x1,x2)	weight: max{(/ 1 2) + x2, x1}
   U61(x1)	weight: (/ 1 8) + x1
   U31(x1,x2)	weight: 0
a__U71(x1,x2,x3)	weight: max{0, (/ 1 2) + x3}
a__U61(x1)	weight: (/ 1 8) + x1
#a__U41(x1,x2)	weight: max{0, x2}
    tt()	weight: 0
a__isNat(x1)	weight: (/ 1 8)
   U51(x1,x2,x3)	weight: max{x3, (/ 1 2) + x2, (/ 1 4) + x1}
   U41(x1,x2)	weight: max{x2, (/ 1 8) + x1}
#a__U32(x1)	weight: 0
    Usable rules: { 1..50 }
    Removed DPs: #35 #46
Number of SCCs: 1, DPs: 21, edges: 88
	SCC { #1 #2 #4 #5 #8 #9 #16 #17 #26 #27 #29 #31 #34 #37 #39 #41 #43 #45 #49 #53 #54 }
Removing DPs: Order(PosReal,>,Sum)... Order(PosReal,>,Max)... QLPOpS... succeeded.
a__plus(x1,x2)	status: [x1,x2]	precedence above: s #a__U51 a__U31 a__U51 U12 a__U41 #a__plus #mark a__U32 #a__U52 U32 a__U52 a__U12 isNat U52 plus U31 #a__U41 tt a__isNat U51 U41
   U21(x1)	status: x1
#a__U72(x1,x2,x3)	status: [x2,x3,x1]	precedence above: a__plus #a__U71 s U71 #a__U51 #a__x U72 a__U31 a__U51 a__x U12 a__U41 x #a__plus #mark a__U32 #a__U52 mark a__U72 U32 a__U52 a__U12 isNat U52 plus U31 a__U71 #a__U41 tt a__isNat U51 U41
#a__U71(x1,x2,x3)	status: [x2,x3,x1]	precedence above: a__plus #a__U72 s U71 #a__U51 #a__x U72 a__U31 a__U51 a__x U12 a__U41 x #a__plus #mark a__U32 #a__U52 mark a__U72 U32 a__U52 a__U12 isNat U52 plus U31 a__U71 #a__U41 tt a__isNat U51 U41
   U11(x1,x2)	status: x1
     s(x1)	status: [x1]	precedence above: a__U31 U12 a__U32 U32 a__U12 isNat U31 tt a__isNat
#a__U31(x1,x2)	status: [x2,x1]	precedence above:
#a__isNat(x1)	status: []	precedence above:
   U71(x1,x2,x3)	status: [x2,x3,x1]	precedence above: a__plus #a__U72 #a__U71 s #a__U51 #a__x U72 a__U31 a__U51 a__x U12 a__U41 x #a__plus #mark a__U32 #a__U52 mark a__U72 U32 a__U52 a__U12 isNat U52 plus U31 a__U71 #a__U41 tt a__isNat U51 U41
#a__U51(x1,x2,x3)	status: [x3]	precedence above: a__plus s a__U31 a__U51 U12 a__U41 #a__plus #mark a__U32 #a__U52 U32 a__U52 a__U12 isNat U52 plus U31 #a__U41 tt a__isNat U51 U41
#a__x(x1,x2)	status: [x2,x1]	precedence above: a__plus #a__U72 #a__U71 s U71 #a__U51 U72 a__U31 a__U51 a__x U12 a__U41 x #a__plus #mark a__U32 #a__U52 mark a__U72 U32 a__U52 a__U12 isNat U52 plus U31 a__U71 #a__U41 tt a__isNat U51 U41
   U72(x1,x2,x3)	status: [x2,x3,x1]	precedence above: a__plus #a__U72 #a__U71 s U71 #a__U51 #a__x a__U31 a__U51 a__x U12 a__U41 x #a__plus #mark a__U32 #a__U52 mark a__U72 U32 a__U52 a__U12 isNat U52 plus U31 a__U71 #a__U41 tt a__isNat U51 U41
#a__U11(x1,x2)	status: [x2]	precedence above:
a__U31(x1,x2)	status: []	precedence above: U12 a__U32 U32 a__U12 isNat U31 tt a__isNat
a__U51(x1,x2,x3)	status: [x3,x2,x1]	precedence above: a__plus s #a__U51 a__U31 U12 a__U41 #a__plus #mark a__U32 #a__U52 U32 a__U52 a__U12 isNat U52 plus U31 #a__U41 tt a__isNat U51 U41
  a__x(x1,x2)	status: [x2,x1]	precedence above: a__plus #a__U72 #a__U71 s U71 #a__U51 #a__x U72 a__U31 a__U51 U12 a__U41 x #a__plus #mark a__U32 #a__U52 mark a__U72 U32 a__U52 a__U12 isNat U52 plus U31 a__U71 #a__U41 tt a__isNat U51 U41
   U12(x1)	status: []	precedence above: a__U31 a__U32 U32 a__U12 isNat U31 tt a__isNat
a__U41(x1,x2)	status: [x2]	precedence above: U41
     x(x1,x2)	status: [x2,x1]	precedence above: a__plus #a__U72 #a__U71 s U71 #a__U51 #a__x U72 a__U31 a__U51 a__x U12 a__U41 #a__plus #mark a__U32 #a__U52 mark a__U72 U32 a__U52 a__U12 isNat U52 plus U31 a__U71 #a__U41 tt a__isNat U51 U41
#a__U12(x1)	status: []	precedence above:
#a__U21(x1)	status: []	precedence above:
#a__U61(x1)	status: []	precedence above:
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    Removed DPs: #1 #2 #4 #8 #9 #17 #31 #34 #37 #41 #43 #45
Number of SCCs: 2, DPs: 5, edges: 7
	SCC { #16 #26 }
Removing DPs: Order(PosReal,>,Sum)... succeeded.
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    Usable rules: { }
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	SCC { #27 #39 #53 }
Removing DPs: Order(PosReal,>,Sum)... Order(PosReal,>,Max)... QLPOpS... succeeded.
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   U41(x1,x2)	status: [x2]	precedence above: a__U41
#a__U32(x1)	status: []	precedence above:
    Usable rules: { 1..50 }
    Removed DPs: #27 #39
Number of SCCs: 0, DPs: 0, edges: 0
YES