Input TRS:
    1: a__U11(tt(),N) -> mark(N)
    2: a__U21(tt(),M,N) -> s(a__plus(mark(N),mark(M)))
    3: a__U31(tt()) -> 0()
    4: a__U41(tt(),M,N) -> a__plus(a__x(mark(N),mark(M)),mark(N))
    5: a__and(tt(),X) -> mark(X)
    6: a__isNat(0()) -> tt()
    7: a__isNat(plus(V1,V2)) -> a__and(a__isNat(V1),isNat(V2))
    8: a__isNat(s(V1)) -> a__isNat(V1)
    9: a__isNat(x(V1,V2)) -> a__and(a__isNat(V1),isNat(V2))
    10: a__plus(N,0()) -> a__U11(a__isNat(N),N)
    11: a__plus(N,s(M)) -> a__U21(a__and(a__isNat(M),isNat(N)),M,N)
    12: a__x(N,0()) -> a__U31(a__isNat(N))
    13: a__x(N,s(M)) -> a__U41(a__and(a__isNat(M),isNat(N)),M,N)
    14: mark(U11(X1,X2)) -> a__U11(mark(X1),X2)
    15: mark(U21(X1,X2,X3)) -> a__U21(mark(X1),X2,X3)
    16: mark(plus(X1,X2)) -> a__plus(mark(X1),mark(X2))
    17: mark(U31(X)) -> a__U31(mark(X))
    18: mark(U41(X1,X2,X3)) -> a__U41(mark(X1),X2,X3)
    19: mark(x(X1,X2)) -> a__x(mark(X1),mark(X2))
    20: mark(and(X1,X2)) -> a__and(mark(X1),X2)
    21: mark(isNat(X)) -> a__isNat(X)
    22: mark(tt()) -> tt()
    23: mark(s(X)) -> s(mark(X))
    24: mark(0()) -> 0()
    25: a__U11(X1,X2) -> U11(X1,X2)
    26: a__U21(X1,X2,X3) -> U21(X1,X2,X3)
    27: a__plus(X1,X2) -> plus(X1,X2)
    28: a__U31(X) -> U31(X)
    29: a__U41(X1,X2,X3) -> U41(X1,X2,X3)
    30: a__x(X1,X2) -> x(X1,X2)
    31: a__and(X1,X2) -> and(X1,X2)
    32: a__isNat(X) -> isNat(X)
Number of strict rules: 32
Direct Order(PosReal,>,Poly) ... failed.
Freezing ... failed.
Dependency Pairs:
   #1: #a__U21(tt(),M,N) -> #a__plus(mark(N),mark(M))
   #2: #a__U21(tt(),M,N) -> #mark(N)
   #3: #a__U21(tt(),M,N) -> #mark(M)
   #4: #a__x(N,s(M)) -> #a__U41(a__and(a__isNat(M),isNat(N)),M,N)
   #5: #a__x(N,s(M)) -> #a__and(a__isNat(M),isNat(N))
   #6: #a__x(N,s(M)) -> #a__isNat(M)
   #7: #a__isNat(x(V1,V2)) -> #a__and(a__isNat(V1),isNat(V2))
   #8: #a__isNat(x(V1,V2)) -> #a__isNat(V1)
   #9: #a__plus(N,s(M)) -> #a__U21(a__and(a__isNat(M),isNat(N)),M,N)
   #10: #a__plus(N,s(M)) -> #a__and(a__isNat(M),isNat(N))
   #11: #a__plus(N,s(M)) -> #a__isNat(M)
   #12: #mark(s(X)) -> #mark(X)
   #13: #a__x(N,0()) -> #a__U31(a__isNat(N))
   #14: #a__x(N,0()) -> #a__isNat(N)
   #15: #mark(U11(X1,X2)) -> #a__U11(mark(X1),X2)
   #16: #mark(U11(X1,X2)) -> #mark(X1)
   #17: #mark(and(X1,X2)) -> #a__and(mark(X1),X2)
   #18: #mark(and(X1,X2)) -> #mark(X1)
   #19: #a__isNat(plus(V1,V2)) -> #a__and(a__isNat(V1),isNat(V2))
   #20: #a__isNat(plus(V1,V2)) -> #a__isNat(V1)
   #21: #a__plus(N,0()) -> #a__U11(a__isNat(N),N)
   #22: #a__plus(N,0()) -> #a__isNat(N)
   #23: #a__and(tt(),X) -> #mark(X)
   #24: #mark(U31(X)) -> #a__U31(mark(X))
   #25: #mark(U31(X)) -> #mark(X)
   #26: #mark(x(X1,X2)) -> #a__x(mark(X1),mark(X2))
   #27: #mark(x(X1,X2)) -> #mark(X1)
   #28: #mark(x(X1,X2)) -> #mark(X2)
   #29: #mark(isNat(X)) -> #a__isNat(X)
   #30: #mark(plus(X1,X2)) -> #a__plus(mark(X1),mark(X2))
   #31: #mark(plus(X1,X2)) -> #mark(X1)
   #32: #mark(plus(X1,X2)) -> #mark(X2)
   #33: #a__U11(tt(),N) -> #mark(N)
   #34: #a__isNat(s(V1)) -> #a__isNat(V1)
   #35: #mark(U21(X1,X2,X3)) -> #a__U21(mark(X1),X2,X3)
   #36: #mark(U21(X1,X2,X3)) -> #mark(X1)
   #37: #a__U41(tt(),M,N) -> #a__plus(a__x(mark(N),mark(M)),mark(N))
   #38: #a__U41(tt(),M,N) -> #a__x(mark(N),mark(M))
   #39: #a__U41(tt(),M,N) -> #mark(N)
   #40: #a__U41(tt(),M,N) -> #mark(M)
   #41: #a__U41(tt(),M,N) -> #mark(N)
   #42: #mark(U41(X1,X2,X3)) -> #a__U41(mark(X1),X2,X3)
   #43: #mark(U41(X1,X2,X3)) -> #mark(X1)
Number of SCCs: 1, DPs: 41, edges: 375
	SCC { #1..12 #14..23 #25..43 }
Removing DPs: Order(PosReal,>,Sum)... Order(PosReal,>,Max)... succeeded.
a__plus(x1,x2)	weight: max{(/ 29361 2) + x2, x1}
   U21(x1,x2,x3)	weight: max{x3, (/ 29361 2) + x2, x1}
   U11(x1,x2)	weight: max{x2, x1}
     s(x1)	weight: x1
#a__U31(x1)	weight: 0
#a__isNat(x1)	weight: (/ 1 4) + x1
   and(x1,x2)	weight: max{x2, x1}
#a__x(x1,x2)	weight: max{(/ 58723 4) + x2, (/ 58723 4) + x1}
#a__U11(x1,x2)	weight: max{0, (/ 1 4) + x2}
a__U31(x1)	weight: (/ 1 4) + x1
  a__x(x1,x2)	weight: max{(/ 29361 2) + x2, (/ 29361 2) + x1}
a__U41(x1,x2,x3)	weight: max{(/ 29361 2) + x3, (/ 29361 2) + x2, (/ 1 4) + x1}
     x(x1,x2)	weight: max{(/ 29361 2) + x2, (/ 29361 2) + x1}
#a__U21(x1,x2,x3)	weight: max{0, (/ 1 4) + x3, (/ 58723 4) + x2}
#a__plus(x1,x2)	weight: max{(/ 58723 4) + x2, (/ 1 4) + x1}
#mark(x1)	weight: (/ 1 4) + x1
     0()	weight: (/ 1 4)
#a__and(x1,x2)	weight: max{0, (/ 1 4) + x2}
a__U21(x1,x2,x3)	weight: max{x3, (/ 29361 2) + x2, x1}
  mark(x1)	weight: x1
a__U11(x1,x2)	weight: max{x2, x1}
 isNat(x1)	weight: x1
  plus(x1,x2)	weight: max{(/ 29361 2) + x2, x1}
   U31(x1)	weight: (/ 1 4) + x1
#a__U41(x1,x2,x3)	weight: max{0, (/ 58723 4) + x3, (/ 58723 4) + x2}
    tt()	weight: 0
a__isNat(x1)	weight: x1
a__and(x1,x2)	weight: max{x2, x1}
   U41(x1,x2,x3)	weight: max{(/ 29361 2) + x3, (/ 29361 2) + x2, (/ 1 4) + x1}
    Usable rules: { 1..32 }
    Removed DPs: #3 #5..8 #11 #14 #19 #25 #27 #28 #32 #39..41 #43
Number of SCCs: 2, DPs: 23, edges: 115
	SCC { #20 #34 }
Removing DPs: Order(PosReal,>,Sum)... succeeded.
a__plus(x1,x2)	weight: (/ 3 16)
   U21(x1,x2,x3)	weight: (/ 5 16) + x1 + x2 + x3
   U11(x1,x2)	weight: (/ 1 8) + x2
     s(x1)	weight: (/ 1 8) + x1
#a__U31(x1)	weight: 0
#a__isNat(x1)	weight: x1
   and(x1,x2)	weight: (/ 9 16)
#a__x(x1,x2)	weight: 0
#a__U11(x1,x2)	weight: 0
a__U31(x1)	weight: (/ 3 16)
  a__x(x1,x2)	weight: (/ 1 16) + x1
a__U41(x1,x2,x3)	weight: (/ 1 16) + x1 + x2
     x(x1,x2)	weight: (/ 1 8)
#a__U21(x1,x2,x3)	weight: 0
#a__plus(x1,x2)	weight: 0
#mark(x1)	weight: 0
     0()	weight: 0
#a__and(x1,x2)	weight: 0
a__U21(x1,x2,x3)	weight: (/ 1 4)
  mark(x1)	weight: (/ 1 8)
a__U11(x1,x2)	weight: (/ 1 16) + x1
 isNat(x1)	weight: (/ 1 4)
  plus(x1,x2)	weight: (/ 1 4) + x1
   U31(x1)	weight: (/ 1 4) + x1
#a__U41(x1,x2,x3)	weight: 0
    tt()	weight: 0
a__isNat(x1)	weight: (/ 3 16) + x1
a__and(x1,x2)	weight: (/ 1 2)
   U41(x1,x2,x3)	weight: (/ 1 8)
    Usable rules: { }
    Removed DPs: #20 #34
Number of SCCs: 1, DPs: 21, edges: 111
	SCC { #1 #2 #4 #9 #10 #12 #15..18 #21 #23 #26 #30 #31 #33 #35..38 #42 }
Removing DPs: Order(PosReal,>,Sum)... Order(PosReal,>,Max)... succeeded.
a__plus(x1,x2)	weight: max{(/ 5 16) + x2, x1}
   U21(x1,x2,x3)	weight: max{x3, (/ 5 16) + x2, (/ 3 16) + x1}
   U11(x1,x2)	weight: max{x2, (/ 1 16) + x1}
     s(x1)	weight: x1
#a__U31(x1)	weight: 0
#a__isNat(x1)	weight: (/ 1 16)
   and(x1,x2)	weight: max{x2, x1}
#a__x(x1,x2)	weight: max{(/ 9 16) + x2, (/ 1 2) + x1}
#a__U11(x1,x2)	weight: max{0, (/ 3 16) + x2}
a__U31(x1)	weight: (/ 1 16)
  a__x(x1,x2)	weight: max{(/ 3 8) + x2, (/ 5 16) + x1}
a__U41(x1,x2,x3)	weight: max{(/ 5 16) + x3, (/ 3 8) + x2, (/ 3 16) + x1}
     x(x1,x2)	weight: max{(/ 3 8) + x2, (/ 5 16) + x1}
#a__U21(x1,x2,x3)	weight: max{(/ 3 16) + x3, (/ 5 16) + x2, (/ 3 16) + x1}
#a__plus(x1,x2)	weight: max{(/ 5 16) + x2, (/ 3 16) + x1}
#mark(x1)	weight: (/ 3 16) + x1
     0()	weight: (/ 1 16)
#a__and(x1,x2)	weight: max{(/ 3 16) + x2, (/ 1 16) + x1}
a__U21(x1,x2,x3)	weight: max{x3, (/ 5 16) + x2, (/ 3 16) + x1}
  mark(x1)	weight: x1
a__U11(x1,x2)	weight: max{x2, (/ 1 16) + x1}
 isNat(x1)	weight: (/ 1 16)
  plus(x1,x2)	weight: max{(/ 5 16) + x2, x1}
   U31(x1)	weight: (/ 1 16)
#a__U41(x1,x2,x3)	weight: max{0, (/ 1 2) + x3, (/ 9 16) + x2}
    tt()	weight: (/ 1 16)
a__isNat(x1)	weight: (/ 1 16)
a__and(x1,x2)	weight: max{x2, x1}
   U41(x1,x2,x3)	weight: max{(/ 5 16) + x3, (/ 3 8) + x2, (/ 3 16) + x1}
    Usable rules: { 1..32 }
    Removed DPs: #10 #16 #36
Number of SCCs: 1, DPs: 18, edges: 73
	SCC { #1 #2 #4 #9 #12 #15 #17 #18 #21 #23 #26 #30 #31 #33 #35 #37 #38 #42 }
Removing DPs: Order(PosReal,>,Sum)... Order(PosReal,>,Max)... succeeded.
a__plus(x1,x2)	weight: max{(/ 5 16) + x2, x1}
   U21(x1,x2,x3)	weight: max{0, x3, (/ 5 16) + x2}
   U11(x1,x2)	weight: max{x2, x1}
     s(x1)	weight: x1
#a__U31(x1)	weight: 0
#a__isNat(x1)	weight: (/ 1 16)
   and(x1,x2)	weight: max{(/ 1 8) + x2, x1}
#a__x(x1,x2)	weight: max{(/ 5 16) + x2, (/ 7 16) + x1}
#a__U11(x1,x2)	weight: max{(/ 1 8) + x2, (/ 1 8) + x1}
a__U31(x1)	weight: x1
  a__x(x1,x2)	weight: max{(/ 3 16) + x2, (/ 5 16) + x1}
a__U41(x1,x2,x3)	weight: max{(/ 5 16) + x3, (/ 3 16) + x2, (/ 1 16) + x1}
     x(x1,x2)	weight: max{(/ 3 16) + x2, (/ 5 16) + x1}
#a__U21(x1,x2,x3)	weight: max{0, (/ 1 8) + x3, (/ 3 8) + x2}
#a__plus(x1,x2)	weight: max{(/ 3 8) + x2, (/ 1 8) + x1}
#mark(x1)	weight: (/ 1 8) + x1
     0()	weight: (/ 1 4)
#a__and(x1,x2)	weight: max{(/ 3 16) + x2, (/ 1 16) + x1}
a__U21(x1,x2,x3)	weight: max{0, x3, (/ 5 16) + x2}
  mark(x1)	weight: x1
a__U11(x1,x2)	weight: max{x2, x1}
 isNat(x1)	weight: x1
  plus(x1,x2)	weight: max{(/ 5 16) + x2, x1}
   U31(x1)	weight: x1
#a__U41(x1,x2,x3)	weight: max{0, (/ 7 16) + x3, (/ 5 16) + x2}
    tt()	weight: (/ 1 4)
a__isNat(x1)	weight: x1
a__and(x1,x2)	weight: max{(/ 1 8) + x2, x1}
   U41(x1,x2,x3)	weight: max{(/ 5 16) + x3, (/ 3 16) + x2, (/ 1 16) + x1}
    Usable rules: { 1..32 }
    Removed DPs: #17 #23
Number of SCCs: 1, DPs: 16, edges: 58
	SCC { #1 #2 #4 #9 #12 #15 #18 #21 #26 #30 #31 #33 #35 #37 #38 #42 }
Removing DPs: Order(PosReal,>,Sum)... Order(PosReal,>,Max)... QLPOpS... succeeded.
a__plus(x1,x2)	status: [x2,x1]	precedence above: U21 U11 s and #a__U21 a__U21 mark a__U11 plus a__isNat a__and
   U21(x1,x2,x3)	status: [x2,x3,x1]	precedence above: a__plus U11 s and #a__U21 a__U21 mark a__U11 plus a__isNat a__and
   U11(x1,x2)	status: [x1,x2]	precedence above: and #a__U21 mark a__U11 a__isNat a__and
     s(x1)	status: [x1]	precedence above: and #a__U21 mark a__isNat a__and
#a__U31(x1)	status: []	precedence above:
#a__isNat(x1)	status: []	precedence above:
   and(x1,x2)	status: [x1,x2]	precedence above: a__and
#a__x(x1,x2)	status: [x2,x1]	precedence above: a__plus U21 U11 s and a__U31 a__x a__U41 x #a__U21 0 a__U21 mark a__U11 plus U31 #a__U41 tt a__isNat a__and U41
#a__U11(x1,x2)	status: x2
a__U31(x1)	status: [x1]	precedence above: 0 U31 tt
  a__x(x1,x2)	status: [x2,x1]	precedence above: a__plus U21 U11 s and #a__x a__U31 a__U41 x #a__U21 0 a__U21 mark a__U11 plus U31 #a__U41 tt a__isNat a__and U41
a__U41(x1,x2,x3)	status: [x2,x3,x1]	precedence above: a__plus U21 U11 s and #a__x a__U31 a__x x #a__U21 0 a__U21 mark a__U11 plus U31 #a__U41 tt a__isNat a__and U41
     x(x1,x2)	status: [x2,x1]	precedence above: a__plus U21 U11 s and #a__x a__U31 a__x a__U41 #a__U21 0 a__U21 mark a__U11 plus U31 #a__U41 tt a__isNat a__and U41
#a__U21(x1,x2,x3)	status: x3
#a__plus(x1,x2)	status: x1
#mark(x1)	status: x1
     0()	status: []	precedence above: tt
#a__and(x1,x2)	status: [x2,x1]	precedence above:
a__U21(x1,x2,x3)	status: [x2,x3,x1]	precedence above: a__plus U21 U11 s and #a__U21 mark a__U11 plus a__isNat a__and
  mark(x1)	status: x1
a__U11(x1,x2)	status: [x1,x2]	precedence above: U11 and #a__U21 mark a__isNat a__and
 isNat(x1)	status: x1
  plus(x1,x2)	status: [x2,x1]	precedence above: a__plus U21 U11 s and #a__U21 a__U21 mark a__U11 a__isNat a__and
   U31(x1)	status: [x1]	precedence above: a__U31 0 tt
#a__U41(x1,x2,x3)	status: [x2,x3,x1]	precedence above: a__plus U21 U11 s and #a__x a__U31 a__x a__U41 x #a__U21 0 a__U21 mark a__U11 plus U31 tt a__isNat a__and U41
    tt()	status: []	precedence above:
a__isNat(x1)	status: x1
a__and(x1,x2)	status: [x1,x2]	precedence above: and
   U41(x1,x2,x3)	status: [x2,x3,x1]	precedence above: a__plus U21 U11 s and #a__x a__U31 a__x a__U41 x #a__U21 0 a__U21 mark a__U11 plus U31 #a__U41 tt a__isNat a__and
    Usable rules: { 1..32 }
    Removed DPs: #4 #12 #15 #18 #30 #31 #35 #37 #38
Number of SCCs: 1, DPs: 2, edges: 2
	SCC { #1 #9 }
Removing DPs: Order(PosReal,>,Sum)... Order(PosReal,>,Max)... QLPOpS... succeeded.
a__plus(x1,x2)	status: [x2,x1]	precedence above: U21 U11 s and a__U21 mark a__U11 plus a__isNat a__and
   U21(x1,x2,x3)	status: [x2,x3,x1]	precedence above: a__plus U11 s and a__U21 mark a__U11 plus a__isNat a__and
   U11(x1,x2)	status: [x1,x2]	precedence above: and mark a__U11 a__isNat a__and
     s(x1)	status: [x1]	precedence above: and mark a__isNat a__and
#a__U31(x1)	status: []	precedence above:
#a__isNat(x1)	status: []	precedence above:
   and(x1,x2)	status: [x1,x2]	precedence above: a__and
#a__x(x1,x2)	status: [x2,x1]	precedence above: a__plus U21 U11 s and a__U31 a__x a__U41 x 0 a__U21 mark a__U11 plus U31 #a__U41 tt a__isNat a__and U41
#a__U11(x1,x2)	status: x2
a__U31(x1)	status: [x1]	precedence above: 0 U31 tt
  a__x(x1,x2)	status: [x2,x1]	precedence above: a__plus U21 U11 s and #a__x a__U31 a__U41 x 0 a__U21 mark a__U11 plus U31 #a__U41 tt a__isNat a__and U41
a__U41(x1,x2,x3)	status: [x2,x3,x1]	precedence above: a__plus U21 U11 s and #a__x a__U31 a__x x 0 a__U21 mark a__U11 plus U31 #a__U41 tt a__isNat a__and U41
     x(x1,x2)	status: [x2,x1]	precedence above: a__plus U21 U11 s and #a__x a__U31 a__x a__U41 0 a__U21 mark a__U11 plus U31 #a__U41 tt a__isNat a__and U41
#a__U21(x1,x2,x3)	status: [x2,x3]	precedence above: and #a__plus a__and
#a__plus(x1,x2)	status: [x2,x1]	precedence above: and #a__U21 a__and
#mark(x1)	status: x1
     0()	status: []	precedence above: tt
#a__and(x1,x2)	status: [x2,x1]	precedence above:
a__U21(x1,x2,x3)	status: [x2,x3,x1]	precedence above: a__plus U21 U11 s and mark a__U11 plus a__isNat a__and
  mark(x1)	status: x1
a__U11(x1,x2)	status: [x1,x2]	precedence above: U11 and mark a__isNat a__and
 isNat(x1)	status: x1
  plus(x1,x2)	status: [x2,x1]	precedence above: a__plus U21 U11 s and a__U21 mark a__U11 a__isNat a__and
   U31(x1)	status: [x1]	precedence above: a__U31 0 tt
#a__U41(x1,x2,x3)	status: [x2,x3,x1]	precedence above: a__plus U21 U11 s and #a__x a__U31 a__x a__U41 x 0 a__U21 mark a__U11 plus U31 tt a__isNat a__and U41
    tt()	status: []	precedence above:
a__isNat(x1)	status: x1
a__and(x1,x2)	status: [x1,x2]	precedence above: and
   U41(x1,x2,x3)	status: [x2,x3,x1]	precedence above: a__plus U21 U11 s and #a__x a__U31 a__x a__U41 x 0 a__U21 mark a__U11 plus U31 #a__U41 tt a__isNat a__and
    Usable rules: { 1..32 }
    Removed DPs: #9
Number of SCCs: 0, DPs: 0, edges: 0
YES