Input TRS:
    1: a__U11(tt(),M,N) -> a__U12(tt(),M,N)
    2: a__U12(tt(),M,N) -> s(a__plus(mark(N),mark(M)))
    3: a__U21(tt(),M,N) -> a__U22(tt(),M,N)
    4: a__U22(tt(),M,N) -> a__plus(a__x(mark(N),mark(M)),mark(N))
    5: a__plus(N,0()) -> mark(N)
    6: a__plus(N,s(M)) -> a__U11(tt(),M,N)
    7: a__x(N,0()) -> 0()
    8: a__x(N,s(M)) -> a__U21(tt(),M,N)
    9: mark(U11(X1,X2,X3)) -> a__U11(mark(X1),X2,X3)
    10: mark(U12(X1,X2,X3)) -> a__U12(mark(X1),X2,X3)
    11: mark(plus(X1,X2)) -> a__plus(mark(X1),mark(X2))
    12: mark(U21(X1,X2,X3)) -> a__U21(mark(X1),X2,X3)
    13: mark(U22(X1,X2,X3)) -> a__U22(mark(X1),X2,X3)
    14: mark(x(X1,X2)) -> a__x(mark(X1),mark(X2))
    15: mark(tt()) -> tt()
    16: mark(s(X)) -> s(mark(X))
    17: mark(0()) -> 0()
    18: a__U11(X1,X2,X3) -> U11(X1,X2,X3)
    19: a__U12(X1,X2,X3) -> U12(X1,X2,X3)
    20: a__plus(X1,X2) -> plus(X1,X2)
    21: a__U21(X1,X2,X3) -> U21(X1,X2,X3)
    22: a__U22(X1,X2,X3) -> U22(X1,X2,X3)
    23: a__x(X1,X2) -> x(X1,X2)
Number of strict rules: 23
Direct Order(PosReal,>,Poly) ... failed.
Freezing ... failed.
Dependency Pairs:
   #1: #a__U12(tt(),M,N) -> #a__plus(mark(N),mark(M))
   #2: #a__U12(tt(),M,N) -> #mark(N)
   #3: #a__U12(tt(),M,N) -> #mark(M)
   #4: #a__plus(N,s(M)) -> #a__U11(tt(),M,N)
   #5: #mark(U22(X1,X2,X3)) -> #a__U22(mark(X1),X2,X3)
   #6: #mark(U22(X1,X2,X3)) -> #mark(X1)
   #7: #mark(U11(X1,X2,X3)) -> #a__U11(mark(X1),X2,X3)
   #8: #mark(U11(X1,X2,X3)) -> #mark(X1)
   #9: #mark(plus(X1,X2)) -> #a__plus(mark(X1),mark(X2))
   #10: #mark(plus(X1,X2)) -> #mark(X1)
   #11: #mark(plus(X1,X2)) -> #mark(X2)
   #12: #mark(U21(X1,X2,X3)) -> #a__U21(mark(X1),X2,X3)
   #13: #mark(U21(X1,X2,X3)) -> #mark(X1)
   #14: #mark(x(X1,X2)) -> #a__x(mark(X1),mark(X2))
   #15: #mark(x(X1,X2)) -> #mark(X1)
   #16: #mark(x(X1,X2)) -> #mark(X2)
   #17: #mark(U12(X1,X2,X3)) -> #a__U12(mark(X1),X2,X3)
   #18: #mark(U12(X1,X2,X3)) -> #mark(X1)
   #19: #a__plus(N,0()) -> #mark(N)
   #20: #mark(s(X)) -> #mark(X)
   #21: #a__U21(tt(),M,N) -> #a__U22(tt(),M,N)
   #22: #a__U11(tt(),M,N) -> #a__U12(tt(),M,N)
   #23: #a__x(N,s(M)) -> #a__U21(tt(),M,N)
   #24: #a__U22(tt(),M,N) -> #a__plus(a__x(mark(N),mark(M)),mark(N))
   #25: #a__U22(tt(),M,N) -> #a__x(mark(N),mark(M))
   #26: #a__U22(tt(),M,N) -> #mark(N)
   #27: #a__U22(tt(),M,N) -> #mark(M)
   #28: #a__U22(tt(),M,N) -> #mark(N)
Number of SCCs: 1, DPs: 28, edges: 253
	SCC { #1..28 }
Removing DPs: Order(PosReal,>,Sum)... Order(PosReal,>,Max)... succeeded.
a__plus(x1,x2)	weight: max{(/ 3 16) + x2, x1}
   U21(x1,x2,x3)	weight: max{(/ 3 16) + x3, (/ 9 32) + x2, (/ 1 8) + x1}
   U11(x1,x2,x3)	weight: max{x3, (/ 3 16) + x2, (/ 1 32) + x1}
     s(x1)	weight: x1
#a__x(x1,x2)	weight: max{(/ 5 16) + x2, (/ 7 32) + x1}
a__U22(x1,x2,x3)	weight: max{(/ 3 16) + x3, (/ 9 32) + x2, (/ 5 32) + x1}
#a__U11(x1,x2,x3)	weight: max{0, (/ 1 32) + x3, (/ 3 16) + x2}
  a__x(x1,x2)	weight: max{(/ 9 32) + x2, (/ 3 16) + x1}
   U12(x1,x2,x3)	weight: max{x3, (/ 3 16) + x2, (/ 1 16) + x1}
     x(x1,x2)	weight: max{(/ 9 32) + x2, (/ 3 16) + x1}
#a__U12(x1,x2,x3)	weight: max{0, (/ 1 32) + x3, (/ 3 16) + x2}
#a__U21(x1,x2,x3)	weight: max{0, (/ 7 32) + x3, (/ 5 16) + x2}
#a__plus(x1,x2)	weight: max{(/ 3 16) + x2, (/ 1 32) + x1}
#mark(x1)	weight: (/ 1 32) + x1
     0()	weight: (/ 7 32)
a__U21(x1,x2,x3)	weight: max{(/ 3 16) + x3, (/ 9 32) + x2, (/ 1 8) + x1}
  mark(x1)	weight: x1
a__U11(x1,x2,x3)	weight: max{x3, (/ 3 16) + x2, (/ 1 32) + x1}
a__U12(x1,x2,x3)	weight: max{x3, (/ 3 16) + x2, (/ 1 16) + x1}
  plus(x1,x2)	weight: max{(/ 3 16) + x2, x1}
#a__U22(x1,x2,x3)	weight: max{0, (/ 7 32) + x3, (/ 5 16) + x2}
    tt()	weight: (/ 3 32)
   U22(x1,x2,x3)	weight: max{(/ 3 16) + x3, (/ 9 32) + x2, (/ 5 32) + x1}
    Usable rules: { 1..23 }
    Removed DPs: #3 #6 #8 #11 #13 #15 #16 #18 #26..28
Number of SCCs: 1, DPs: 17, edges: 52
	SCC { #1 #2 #4 #5 #7 #9 #10 #12 #14 #17 #19..25 }
Removing DPs: Order(PosReal,>,Sum)... Order(PosReal,>,Max)... QLPOpS... succeeded.
a__plus(x1,x2)	status: [x1,x2]	precedence above: U11 s #a__U11 U12 #a__U12 a__U11 a__U12 plus tt
   U21(x1,x2,x3)	status: [x2,x3,x1]	precedence above: a__plus U11 s #a__x a__U22 #a__U11 a__x U12 x #a__U12 #a__U21 a__U21 mark a__U11 a__U12 plus #a__U22 tt U22
   U11(x1,x2,x3)	status: [x3,x2,x1]	precedence above: a__plus s #a__U11 U12 #a__U12 a__U11 a__U12 plus tt
     s(x1)	status: [x1]	precedence above: #a__U11
#a__x(x1,x2)	status: [x2,x1]	precedence above: a__plus U21 U11 s a__U22 #a__U11 a__x U12 x #a__U12 #a__U21 a__U21 mark a__U11 a__U12 plus #a__U22 tt U22
a__U22(x1,x2,x3)	status: [x2,x3,x1]	precedence above: a__plus U21 U11 s #a__x #a__U11 a__x U12 x #a__U12 #a__U21 a__U21 mark a__U11 a__U12 plus #a__U22 tt U22
#a__U11(x1,x2,x3)	status: x3
  a__x(x1,x2)	status: [x2,x1]	precedence above: a__plus U21 U11 s #a__x a__U22 #a__U11 U12 x #a__U12 #a__U21 a__U21 mark a__U11 a__U12 plus #a__U22 tt U22
   U12(x1,x2,x3)	status: [x3,x2,x1]	precedence above: a__plus U11 s #a__U11 #a__U12 a__U11 a__U12 plus tt
     x(x1,x2)	status: [x2,x1]	precedence above: a__plus U21 U11 s #a__x a__U22 #a__U11 a__x U12 #a__U12 #a__U21 a__U21 mark a__U11 a__U12 plus #a__U22 tt U22
#a__U12(x1,x2,x3)	status: x3
#a__U21(x1,x2,x3)	status: [x2,x3,x1]	precedence above: a__plus U21 U11 s #a__x a__U22 #a__U11 a__x U12 x #a__U12 a__U21 mark a__U11 a__U12 plus #a__U22 tt U22
#a__plus(x1,x2)	status: x1
#mark(x1)	status: x1
     0()	status: []	precedence above: mark
a__U21(x1,x2,x3)	status: [x2,x3,x1]	precedence above: a__plus U21 U11 s #a__x a__U22 #a__U11 a__x U12 x #a__U12 #a__U21 mark a__U11 a__U12 plus #a__U22 tt U22
  mark(x1)	status: x1
a__U11(x1,x2,x3)	status: [x3,x2,x1]	precedence above: a__plus U11 s #a__U11 U12 #a__U12 a__U12 plus tt
a__U12(x1,x2,x3)	status: [x3,x2,x1]	precedence above: a__plus U11 s #a__U11 U12 #a__U12 a__U11 plus tt
  plus(x1,x2)	status: [x1,x2]	precedence above: a__plus U11 s #a__U11 U12 #a__U12 a__U11 a__U12 tt
#a__U22(x1,x2,x3)	status: [x2,x3,x1]	precedence above: a__plus U21 U11 s #a__x a__U22 #a__U11 a__x U12 x #a__U12 #a__U21 a__U21 mark a__U11 a__U12 plus tt U22
    tt()	status: []	precedence above:
   U22(x1,x2,x3)	status: [x2,x3,x1]	precedence above: a__plus U21 U11 s #a__x a__U22 #a__U11 a__x U12 x #a__U12 #a__U21 a__U21 mark a__U11 a__U12 plus #a__U22 tt
    Usable rules: { 1..23 }
    Removed DPs: #7 #9 #10 #17 #20 #23..25
Number of SCCs: 1, DPs: 3, edges: 3
	SCC { #1 #4 #22 }
Removing DPs: Order(PosReal,>,Sum)... Order(PosReal,>,Max)... QLPOpS... succeeded.
a__plus(x1,x2)	status: [x1,x2]	precedence above: U11 s #a__U11 U12 #a__U12 a__U11 a__U12 plus tt
   U21(x1,x2,x3)	status: [x2,x3,x1]	precedence above: a__plus U11 s #a__x a__U22 #a__U11 a__x U12 x #a__U12 #a__U21 a__U21 mark a__U11 a__U12 plus #a__U22 tt U22
   U11(x1,x2,x3)	status: [x3,x2,x1]	precedence above: a__plus s #a__U11 U12 #a__U12 a__U11 a__U12 plus tt
     s(x1)	status: [x1]	precedence above: #a__U11 #a__U12 tt
#a__x(x1,x2)	status: [x2,x1]	precedence above: a__plus U21 U11 s a__U22 #a__U11 a__x U12 x #a__U12 #a__U21 a__U21 mark a__U11 a__U12 plus #a__U22 tt U22
a__U22(x1,x2,x3)	status: [x2,x3,x1]	precedence above: a__plus U21 U11 s #a__x #a__U11 a__x U12 x #a__U12 #a__U21 a__U21 mark a__U11 a__U12 plus #a__U22 tt U22
#a__U11(x1,x2,x3)	status: [x2]	precedence above: s #a__U12 tt
  a__x(x1,x2)	status: [x2,x1]	precedence above: a__plus U21 U11 s #a__x a__U22 #a__U11 U12 x #a__U12 #a__U21 a__U21 mark a__U11 a__U12 plus #a__U22 tt U22
   U12(x1,x2,x3)	status: [x3,x2,x1]	precedence above: a__plus U11 s #a__U11 #a__U12 a__U11 a__U12 plus tt
     x(x1,x2)	status: [x2,x1]	precedence above: a__plus U21 U11 s #a__x a__U22 #a__U11 a__x U12 #a__U12 #a__U21 a__U21 mark a__U11 a__U12 plus #a__U22 tt U22
#a__U12(x1,x2,x3)	status: [x2]	precedence above:
#a__U21(x1,x2,x3)	status: [x2,x3,x1]	precedence above: a__plus U21 U11 s #a__x a__U22 #a__U11 a__x U12 x #a__U12 a__U21 mark a__U11 a__U12 plus #a__U22 tt U22
#a__plus(x1,x2)	status: x2
#mark(x1)	status: x1
     0()	status: []	precedence above: mark
a__U21(x1,x2,x3)	status: [x2,x3,x1]	precedence above: a__plus U21 U11 s #a__x a__U22 #a__U11 a__x U12 x #a__U12 #a__U21 mark a__U11 a__U12 plus #a__U22 tt U22
  mark(x1)	status: x1
a__U11(x1,x2,x3)	status: [x3,x2,x1]	precedence above: a__plus U11 s #a__U11 U12 #a__U12 a__U12 plus tt
a__U12(x1,x2,x3)	status: [x3,x2,x1]	precedence above: a__plus U11 s #a__U11 U12 #a__U12 a__U11 plus tt
  plus(x1,x2)	status: [x1,x2]	precedence above: a__plus U11 s #a__U11 U12 #a__U12 a__U11 a__U12 tt
#a__U22(x1,x2,x3)	status: [x2,x3,x1]	precedence above: a__plus U21 U11 s #a__x a__U22 #a__U11 a__x U12 x #a__U12 #a__U21 a__U21 mark a__U11 a__U12 plus tt U22
    tt()	status: []	precedence above:
   U22(x1,x2,x3)	status: [x2,x3,x1]	precedence above: a__plus U21 U11 s #a__x a__U22 #a__U11 a__x U12 x #a__U12 #a__U21 a__U21 mark a__U11 a__U12 plus #a__U22 tt
    Usable rules: { 1..23 }
    Removed DPs: #1 #22
Number of SCCs: 0, DPs: 0, edges: 0
YES