Input TRS:
    1: active(U11(tt(),V1,V2)) -> mark(U12(isNat(V1),V2))
    2: active(U12(tt(),V2)) -> mark(U13(isNat(V2)))
    3: active(U13(tt())) -> mark(tt())
    4: active(U21(tt(),V1)) -> mark(U22(isNat(V1)))
    5: active(U22(tt())) -> mark(tt())
    6: active(U31(tt(),N)) -> mark(N)
    7: active(U41(tt(),M,N)) -> mark(s(plus(N,M)))
    8: active(and(tt(),X)) -> mark(X)
    9: active(isNat(0())) -> mark(tt())
    10: active(isNat(plus(V1,V2))) -> mark(U11(and(isNatKind(V1),isNatKind(V2)),V1,V2))
    11: active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1))
    12: active(isNatKind(0())) -> mark(tt())
    13: active(isNatKind(plus(V1,V2))) -> mark(and(isNatKind(V1),isNatKind(V2)))
    14: active(isNatKind(s(V1))) -> mark(isNatKind(V1))
    15: active(plus(N,0())) -> mark(U31(and(isNat(N),isNatKind(N)),N))
    16: active(plus(N,s(M))) -> mark(U41(and(and(isNat(M),isNatKind(M)),and(isNat(N),isNatKind(N))),M,N))
    17: active(U11(X1,X2,X3)) -> U11(active(X1),X2,X3)
    18: active(U12(X1,X2)) -> U12(active(X1),X2)
    19: active(U13(X)) -> U13(active(X))
    20: active(U21(X1,X2)) -> U21(active(X1),X2)
    21: active(U22(X)) -> U22(active(X))
    22: active(U31(X1,X2)) -> U31(active(X1),X2)
    23: active(U41(X1,X2,X3)) -> U41(active(X1),X2,X3)
    24: active(s(X)) -> s(active(X))
    25: active(plus(X1,X2)) -> plus(active(X1),X2)
    26: active(plus(X1,X2)) -> plus(X1,active(X2))
    27: active(and(X1,X2)) -> and(active(X1),X2)
    28: U11(mark(X1),X2,X3) -> mark(U11(X1,X2,X3))
    29: U12(mark(X1),X2) -> mark(U12(X1,X2))
    30: U13(mark(X)) -> mark(U13(X))
    31: U21(mark(X1),X2) -> mark(U21(X1,X2))
    32: U22(mark(X)) -> mark(U22(X))
    33: U31(mark(X1),X2) -> mark(U31(X1,X2))
    34: U41(mark(X1),X2,X3) -> mark(U41(X1,X2,X3))
    35: s(mark(X)) -> mark(s(X))
    36: plus(mark(X1),X2) -> mark(plus(X1,X2))
    37: plus(X1,mark(X2)) -> mark(plus(X1,X2))
    38: and(mark(X1),X2) -> mark(and(X1,X2))
    39: proper(U11(X1,X2,X3)) -> U11(proper(X1),proper(X2),proper(X3))
    40: proper(tt()) -> ok(tt())
    41: proper(U12(X1,X2)) -> U12(proper(X1),proper(X2))
    42: proper(isNat(X)) -> isNat(proper(X))
    43: proper(U13(X)) -> U13(proper(X))
    44: proper(U21(X1,X2)) -> U21(proper(X1),proper(X2))
    45: proper(U22(X)) -> U22(proper(X))
    46: proper(U31(X1,X2)) -> U31(proper(X1),proper(X2))
    47: proper(U41(X1,X2,X3)) -> U41(proper(X1),proper(X2),proper(X3))
    48: proper(s(X)) -> s(proper(X))
    49: proper(plus(X1,X2)) -> plus(proper(X1),proper(X2))
    50: proper(and(X1,X2)) -> and(proper(X1),proper(X2))
    51: proper(0()) -> ok(0())
    52: proper(isNatKind(X)) -> isNatKind(proper(X))
    53: U11(ok(X1),ok(X2),ok(X3)) -> ok(U11(X1,X2,X3))
    54: U12(ok(X1),ok(X2)) -> ok(U12(X1,X2))
    55: isNat(ok(X)) -> ok(isNat(X))
    56: U13(ok(X)) -> ok(U13(X))
    57: U21(ok(X1),ok(X2)) -> ok(U21(X1,X2))
    58: U22(ok(X)) -> ok(U22(X))
    59: U31(ok(X1),ok(X2)) -> ok(U31(X1,X2))
    60: U41(ok(X1),ok(X2),ok(X3)) -> ok(U41(X1,X2,X3))
    61: s(ok(X)) -> ok(s(X))
    62: plus(ok(X1),ok(X2)) -> ok(plus(X1,X2))
    63: and(ok(X1),ok(X2)) -> ok(and(X1,X2))
    64: isNatKind(ok(X)) -> ok(isNatKind(X))
    65: top(mark(X)) -> top(proper(X))
    66: top(ok(X)) -> top(active(X))
Number of strict rules: 66
Direct Order(PosReal,>,Poly) ... failed.
Freezing ... failed.
Dependency Pairs:
   #1: #active(U12(tt(),V2)) -> #U13(isNat(V2))
   #2: #active(U12(tt(),V2)) -> #isNat(V2)
   #3: #proper(U13(X)) -> #U13(proper(X))
   #4: #proper(U13(X)) -> #proper(X)
   #5: #U12(mark(X1),X2) -> #U12(X1,X2)
   #6: #s(mark(X)) -> #s(X)
   #7: #top(ok(X)) -> #top(active(X))
   #8: #top(ok(X)) -> #active(X)
   #9: #proper(U31(X1,X2)) -> #U31(proper(X1),proper(X2))
   #10: #proper(U31(X1,X2)) -> #proper(X1)
   #11: #proper(U31(X1,X2)) -> #proper(X2)
   #12: #proper(isNat(X)) -> #isNat(proper(X))
   #13: #proper(isNat(X)) -> #proper(X)
   #14: #proper(U12(X1,X2)) -> #U12(proper(X1),proper(X2))
   #15: #proper(U12(X1,X2)) -> #proper(X1)
   #16: #proper(U12(X1,X2)) -> #proper(X2)
   #17: #plus(X1,mark(X2)) -> #plus(X1,X2)
   #18: #proper(U41(X1,X2,X3)) -> #U41(proper(X1),proper(X2),proper(X3))
   #19: #proper(U41(X1,X2,X3)) -> #proper(X1)
   #20: #proper(U41(X1,X2,X3)) -> #proper(X2)
   #21: #proper(U41(X1,X2,X3)) -> #proper(X3)
   #22: #U11(ok(X1),ok(X2),ok(X3)) -> #U11(X1,X2,X3)
   #23: #proper(s(X)) -> #s(proper(X))
   #24: #proper(s(X)) -> #proper(X)
   #25: #U22(ok(X)) -> #U22(X)
   #26: #s(ok(X)) -> #s(X)
   #27: #and(mark(X1),X2) -> #and(X1,X2)
   #28: #U31(ok(X1),ok(X2)) -> #U31(X1,X2)
   #29: #isNat(ok(X)) -> #isNat(X)
   #30: #active(isNatKind(plus(V1,V2))) -> #and(isNatKind(V1),isNatKind(V2))
   #31: #active(isNatKind(plus(V1,V2))) -> #isNatKind(V1)
   #32: #active(isNatKind(plus(V1,V2))) -> #isNatKind(V2)
   #33: #active(isNat(s(V1))) -> #U21(isNatKind(V1),V1)
   #34: #active(isNat(s(V1))) -> #isNatKind(V1)
   #35: #U21(ok(X1),ok(X2)) -> #U21(X1,X2)
   #36: #active(s(X)) -> #s(active(X))
   #37: #active(s(X)) -> #active(X)
   #38: #active(U41(X1,X2,X3)) -> #U41(active(X1),X2,X3)
   #39: #active(U41(X1,X2,X3)) -> #active(X1)
   #40: #proper(U22(X)) -> #U22(proper(X))
   #41: #proper(U22(X)) -> #proper(X)
   #42: #U21(mark(X1),X2) -> #U21(X1,X2)
   #43: #U13(ok(X)) -> #U13(X)
   #44: #active(isNatKind(s(V1))) -> #isNatKind(V1)
   #45: #plus(ok(X1),ok(X2)) -> #plus(X1,X2)
   #46: #U13(mark(X)) -> #U13(X)
   #47: #proper(isNatKind(X)) -> #isNatKind(proper(X))
   #48: #proper(isNatKind(X)) -> #proper(X)
   #49: #proper(plus(X1,X2)) -> #plus(proper(X1),proper(X2))
   #50: #proper(plus(X1,X2)) -> #proper(X1)
   #51: #proper(plus(X1,X2)) -> #proper(X2)
   #52: #active(plus(X1,X2)) -> #plus(active(X1),X2)
   #53: #active(plus(X1,X2)) -> #active(X1)
   #54: #active(U21(X1,X2)) -> #U21(active(X1),X2)
   #55: #active(U21(X1,X2)) -> #active(X1)
   #56: #active(U41(tt(),M,N)) -> #s(plus(N,M))
   #57: #active(U41(tt(),M,N)) -> #plus(N,M)
   #58: #proper(U11(X1,X2,X3)) -> #U11(proper(X1),proper(X2),proper(X3))
   #59: #proper(U11(X1,X2,X3)) -> #proper(X1)
   #60: #proper(U11(X1,X2,X3)) -> #proper(X2)
   #61: #proper(U11(X1,X2,X3)) -> #proper(X3)
   #62: #active(isNat(plus(V1,V2))) -> #U11(and(isNatKind(V1),isNatKind(V2)),V1,V2)
   #63: #active(isNat(plus(V1,V2))) -> #and(isNatKind(V1),isNatKind(V2))
   #64: #active(isNat(plus(V1,V2))) -> #isNatKind(V1)
   #65: #active(isNat(plus(V1,V2))) -> #isNatKind(V2)
   #66: #isNatKind(ok(X)) -> #isNatKind(X)
   #67: #U31(mark(X1),X2) -> #U31(X1,X2)
   #68: #proper(U21(X1,X2)) -> #U21(proper(X1),proper(X2))
   #69: #proper(U21(X1,X2)) -> #proper(X1)
   #70: #proper(U21(X1,X2)) -> #proper(X2)
   #71: #top(mark(X)) -> #top(proper(X))
   #72: #top(mark(X)) -> #proper(X)
   #73: #U11(mark(X1),X2,X3) -> #U11(X1,X2,X3)
   #74: #active(U31(X1,X2)) -> #U31(active(X1),X2)
   #75: #active(U31(X1,X2)) -> #active(X1)
   #76: #U41(mark(X1),X2,X3) -> #U41(X1,X2,X3)
   #77: #active(and(X1,X2)) -> #and(active(X1),X2)
   #78: #active(and(X1,X2)) -> #active(X1)
   #79: #U41(ok(X1),ok(X2),ok(X3)) -> #U41(X1,X2,X3)
   #80: #active(U11(X1,X2,X3)) -> #U11(active(X1),X2,X3)
   #81: #active(U11(X1,X2,X3)) -> #active(X1)
   #82: #U22(mark(X)) -> #U22(X)
   #83: #active(U13(X)) -> #U13(active(X))
   #84: #active(U13(X)) -> #active(X)
   #85: #and(ok(X1),ok(X2)) -> #and(X1,X2)
   #86: #active(plus(X1,X2)) -> #plus(X1,active(X2))
   #87: #active(plus(X1,X2)) -> #active(X2)
   #88: #plus(mark(X1),X2) -> #plus(X1,X2)
   #89: #active(U22(X)) -> #U22(active(X))
   #90: #active(U22(X)) -> #active(X)
   #91: #active(plus(N,s(M))) -> #U41(and(and(isNat(M),isNatKind(M)),and(isNat(N),isNatKind(N))),M,N)
   #92: #active(plus(N,s(M))) -> #and(and(isNat(M),isNatKind(M)),and(isNat(N),isNatKind(N)))
   #93: #active(plus(N,s(M))) -> #and(isNat(M),isNatKind(M))
   #94: #active(plus(N,s(M))) -> #isNat(M)
   #95: #active(plus(N,s(M))) -> #isNatKind(M)
   #96: #active(plus(N,s(M))) -> #and(isNat(N),isNatKind(N))
   #97: #active(plus(N,s(M))) -> #isNat(N)
   #98: #active(plus(N,s(M))) -> #isNatKind(N)
   #99: #active(U11(tt(),V1,V2)) -> #U12(isNat(V1),V2)
   #100: #active(U11(tt(),V1,V2)) -> #isNat(V1)
   #101: #U12(ok(X1),ok(X2)) -> #U12(X1,X2)
   #102: #active(plus(N,0())) -> #U31(and(isNat(N),isNatKind(N)),N)
   #103: #active(plus(N,0())) -> #and(isNat(N),isNatKind(N))
   #104: #active(plus(N,0())) -> #isNat(N)
   #105: #active(plus(N,0())) -> #isNatKind(N)
   #106: #active(U21(tt(),V1)) -> #U22(isNat(V1))
   #107: #active(U21(tt(),V1)) -> #isNat(V1)
   #108: #proper(and(X1,X2)) -> #and(proper(X1),proper(X2))
   #109: #proper(and(X1,X2)) -> #proper(X1)
   #110: #proper(and(X1,X2)) -> #proper(X2)
   #111: #active(U12(X1,X2)) -> #U12(active(X1),X2)
   #112: #active(U12(X1,X2)) -> #active(X1)
Number of SCCs: 15, DPs: 57, edges: 613
	SCC { #66 }
Removing DPs: Order(PosReal,>,Sum)... succeeded.
isNatKind(x1)	weight: 0
   U21(x1,x2)	weight: 0
   U11(x1,x2,x3)	weight: 0
     s(x1)	weight: 0
#isNat(x1)	weight: 0
   top(x1)	weight: 0
   and(x1,x2)	weight: 0
#plus(x1,x2)	weight: 0
 #top(x1)	weight: 0
 #U13(x1)	weight: 0
   U12(x1,x2)	weight: 0
 #U12(x1,x2)	weight: 0
proper(x1)	weight: 0
    ok(x1)	weight: (/ 1 2) + x1
     0()	weight: 0
   #s(x1)	weight: 0
  mark(x1)	weight: 0
#proper(x1)	weight: 0
 isNat(x1)	weight: 0
  plus(x1,x2)	weight: 0
 #U11(x1,x2,x3)	weight: 0
active(x1)	weight: 0
   U31(x1,x2)	weight: 0
 #U41(x1,x2,x3)	weight: 0
#active(x1)	weight: 0
 #U21(x1,x2)	weight: 0
 #U22(x1)	weight: 0
    tt()	weight: 0
   U13(x1)	weight: 0
   U22(x1)	weight: 0
#isNatKind(x1)	weight: x1
   U41(x1,x2,x3)	weight: 0
 #U31(x1,x2)	weight: 0
 #and(x1,x2)	weight: 0
    Usable rules: { }
    Removed DPs: #66
Number of SCCs: 14, DPs: 56, edges: 612
	SCC { #29 }
Removing DPs: Order(PosReal,>,Sum)... succeeded.
isNatKind(x1)	weight: 0
   U21(x1,x2)	weight: 0
   U11(x1,x2,x3)	weight: 0
     s(x1)	weight: 0
#isNat(x1)	weight: x1
   top(x1)	weight: 0
   and(x1,x2)	weight: 0
#plus(x1,x2)	weight: 0
 #top(x1)	weight: 0
 #U13(x1)	weight: 0
   U12(x1,x2)	weight: 0
 #U12(x1,x2)	weight: 0
proper(x1)	weight: 0
    ok(x1)	weight: (/ 1 2) + x1
     0()	weight: 0
   #s(x1)	weight: 0
  mark(x1)	weight: 0
#proper(x1)	weight: 0
 isNat(x1)	weight: 0
  plus(x1,x2)	weight: 0
 #U11(x1,x2,x3)	weight: 0
active(x1)	weight: 0
   U31(x1,x2)	weight: 0
 #U41(x1,x2,x3)	weight: 0
#active(x1)	weight: 0
 #U21(x1,x2)	weight: 0
 #U22(x1)	weight: 0
    tt()	weight: 0
   U13(x1)	weight: 0
   U22(x1)	weight: 0
#isNatKind(x1)	weight: 0
   U41(x1,x2,x3)	weight: 0
 #U31(x1,x2)	weight: 0
 #and(x1,x2)	weight: 0
    Usable rules: { }
    Removed DPs: #29
Number of SCCs: 13, DPs: 55, edges: 611
	SCC { #25 #82 }
Removing DPs: Order(PosReal,>,Sum)... succeeded.
isNatKind(x1)	weight: 0
   U21(x1,x2)	weight: 0
   U11(x1,x2,x3)	weight: 0
     s(x1)	weight: 0
#isNat(x1)	weight: 0
   top(x1)	weight: 0
   and(x1,x2)	weight: 0
#plus(x1,x2)	weight: 0
 #top(x1)	weight: 0
 #U13(x1)	weight: 0
   U12(x1,x2)	weight: 0
 #U12(x1,x2)	weight: 0
proper(x1)	weight: 0
    ok(x1)	weight: (/ 1 2) + x1
     0()	weight: 0
   #s(x1)	weight: 0
  mark(x1)	weight: (/ 1 2) + x1
#proper(x1)	weight: 0
 isNat(x1)	weight: 0
  plus(x1,x2)	weight: 0
 #U11(x1,x2,x3)	weight: 0
active(x1)	weight: 0
   U31(x1,x2)	weight: 0
 #U41(x1,x2,x3)	weight: 0
#active(x1)	weight: 0
 #U21(x1,x2)	weight: 0
 #U22(x1)	weight: x1
    tt()	weight: 0
   U13(x1)	weight: 0
   U22(x1)	weight: 0
#isNatKind(x1)	weight: 0
   U41(x1,x2,x3)	weight: 0
 #U31(x1,x2)	weight: 0
 #and(x1,x2)	weight: 0
    Usable rules: { }
    Removed DPs: #25 #82
Number of SCCs: 12, DPs: 53, edges: 607
	SCC { #6 #26 }
Removing DPs: Order(PosReal,>,Sum)... succeeded.
isNatKind(x1)	weight: 0
   U21(x1,x2)	weight: 0
   U11(x1,x2,x3)	weight: 0
     s(x1)	weight: 0
#isNat(x1)	weight: 0
   top(x1)	weight: 0
   and(x1,x2)	weight: 0
#plus(x1,x2)	weight: 0
 #top(x1)	weight: 0
 #U13(x1)	weight: 0
   U12(x1,x2)	weight: 0
 #U12(x1,x2)	weight: 0
proper(x1)	weight: 0
    ok(x1)	weight: (/ 1 2) + x1
     0()	weight: 0
   #s(x1)	weight: x1
  mark(x1)	weight: (/ 1 2) + x1
#proper(x1)	weight: 0
 isNat(x1)	weight: 0
  plus(x1,x2)	weight: 0
 #U11(x1,x2,x3)	weight: 0
active(x1)	weight: 0
   U31(x1,x2)	weight: 0
 #U41(x1,x2,x3)	weight: 0
#active(x1)	weight: 0
 #U21(x1,x2)	weight: 0
 #U22(x1)	weight: 0
    tt()	weight: 0
   U13(x1)	weight: 0
   U22(x1)	weight: 0
#isNatKind(x1)	weight: 0
   U41(x1,x2,x3)	weight: 0
 #U31(x1,x2)	weight: 0
 #and(x1,x2)	weight: 0
    Usable rules: { }
    Removed DPs: #6 #26
Number of SCCs: 11, DPs: 51, edges: 603
	SCC { #43 #46 }
Removing DPs: Order(PosReal,>,Sum)... succeeded.
isNatKind(x1)	weight: 0
   U21(x1,x2)	weight: 0
   U11(x1,x2,x3)	weight: 0
     s(x1)	weight: 0
#isNat(x1)	weight: 0
   top(x1)	weight: 0
   and(x1,x2)	weight: 0
#plus(x1,x2)	weight: 0
 #top(x1)	weight: 0
 #U13(x1)	weight: x1
   U12(x1,x2)	weight: 0
 #U12(x1,x2)	weight: 0
proper(x1)	weight: 0
    ok(x1)	weight: (/ 1 2) + x1
     0()	weight: 0
   #s(x1)	weight: 0
  mark(x1)	weight: (/ 1 2) + x1
#proper(x1)	weight: 0
 isNat(x1)	weight: 0
  plus(x1,x2)	weight: 0
 #U11(x1,x2,x3)	weight: 0
active(x1)	weight: 0
   U31(x1,x2)	weight: 0
 #U41(x1,x2,x3)	weight: 0
#active(x1)	weight: 0
 #U21(x1,x2)	weight: 0
 #U22(x1)	weight: 0
    tt()	weight: 0
   U13(x1)	weight: 0
   U22(x1)	weight: 0
#isNatKind(x1)	weight: 0
   U41(x1,x2,x3)	weight: 0
 #U31(x1,x2)	weight: 0
 #and(x1,x2)	weight: 0
    Usable rules: { }
    Removed DPs: #43 #46
Number of SCCs: 10, DPs: 49, edges: 599
	SCC { #7 #71 }
Removing DPs: Order(PosReal,>,Sum)... Order(PosReal,>,Max)... QLPOpS... succeeded.
isNatKind(x1)	status: [x1]	precedence above: ok mark active tt
   U21(x1,x2)	status: [x1,x2]	precedence above: ok mark active tt U22
   U11(x1,x2,x3)	status: [x2,x1,x3]	precedence above: U12 mark active tt U13
     s(x1)	status: [x1]	precedence above: isNatKind U21 U12 proper ok mark active tt U13 U22
#isNat(x1)	status: []	precedence above:
   top(x1)	status: []	precedence above:
   and(x1,x2)	status: [x2,x1]	precedence above: proper mark active
#plus(x1,x2)	status: [x1,x2]	precedence above:
 #top(x1)	status: [x1]	precedence above:
 #U13(x1)	status: []	precedence above:
   U12(x1,x2)	status: [x2,x1]	precedence above: mark active tt U13
 #U12(x1,x2)	status: []	precedence above:
proper(x1)	status: x1
    ok(x1)	status: x1
     0()	status: []	precedence above: mark active tt
   #s(x1)	status: []	precedence above:
  mark(x1)	status: [x1]	precedence above: active
#proper(x1)	status: []	precedence above:
 isNat(x1)	status: x1
  plus(x1,x2)	status: [x1,x2]	precedence above: isNatKind U21 U11 s and U12 proper ok mark active U31 tt U13 U22 U41
 #U11(x1,x2,x3)	status: [x3,x1,x2]	precedence above:
active(x1)	status: x1
   U31(x1,x2)	status: [x2,x1]	precedence above: ok mark active
 #U41(x1,x2,x3)	status: [x2]	precedence above:
#active(x1)	status: []	precedence above:
 #U21(x1,x2)	status: [x2]	precedence above:
 #U22(x1)	status: []	precedence above:
    tt()	status: []	precedence above:
   U13(x1)	status: [x1]	precedence above: U12 mark active tt
   U22(x1)	status: [x1]	precedence above: ok mark active tt
#isNatKind(x1)	status: []	precedence above:
   U41(x1,x2,x3)	status: [x3,x2,x1]	precedence above: isNatKind U21 U11 s and U12 proper ok mark plus active U31 tt U13 U22
 #U31(x1,x2)	status: [x2]	precedence above:
 #and(x1,x2)	status: [x2]	precedence above:
    Usable rules: { 1..64 }
    Removed DPs: #71
Number of SCCs: 10, DPs: 48, edges: 596
	SCC { #7 }
Removing DPs: Order(PosReal,>,Sum)... succeeded.
isNatKind(x1)	weight: x1
   U21(x1,x2)	weight: x1
   U11(x1,x2,x3)	weight: x3
     s(x1)	weight: x1
#isNat(x1)	weight: 0
   top(x1)	weight: 0
   and(x1,x2)	weight: x1
#plus(x1,x2)	weight: 0
 #top(x1)	weight: x1
 #U13(x1)	weight: 0
   U12(x1,x2)	weight: x2
 #U12(x1,x2)	weight: 0
proper(x1)	weight: (/ 1 2)
    ok(x1)	weight: (/ 1 2) + x1
     0()	weight: 0
   #s(x1)	weight: 0
  mark(x1)	weight: 0
#proper(x1)	weight: 0
 isNat(x1)	weight: x1
  plus(x1,x2)	weight: x1
 #U11(x1,x2,x3)	weight: 0
active(x1)	weight: x1
   U31(x1,x2)	weight: x2
 #U41(x1,x2,x3)	weight: 0
#active(x1)	weight: 0
 #U21(x1,x2)	weight: 0
 #U22(x1)	weight: 0
    tt()	weight: 0
   U13(x1)	weight: x1
   U22(x1)	weight: x1
#isNatKind(x1)	weight: 0
   U41(x1,x2,x3)	weight: x2
 #U31(x1,x2)	weight: 0
 #and(x1,x2)	weight: 0
    Usable rules: { 1..38 53..64 }
    Removed DPs: #7
Number of SCCs: 9, DPs: 47, edges: 595
	SCC { #27 #85 }
Removing DPs: Order(PosReal,>,Sum)... succeeded.
isNatKind(x1)	weight: x1
   U21(x1,x2)	weight: x1
   U11(x1,x2,x3)	weight: x3
     s(x1)	weight: x1
#isNat(x1)	weight: 0
   top(x1)	weight: 0
   and(x1,x2)	weight: x1
#plus(x1,x2)	weight: 0
 #top(x1)	weight: x1
 #U13(x1)	weight: 0
   U12(x1,x2)	weight: x2
 #U12(x1,x2)	weight: 0
proper(x1)	weight: (/ 1 2)
    ok(x1)	weight: (/ 1 2) + x1
     0()	weight: 0
   #s(x1)	weight: 0
  mark(x1)	weight: 0
#proper(x1)	weight: 0
 isNat(x1)	weight: x1
  plus(x1,x2)	weight: x1
 #U11(x1,x2,x3)	weight: 0
active(x1)	weight: x1
   U31(x1,x2)	weight: x2
 #U41(x1,x2,x3)	weight: 0
#active(x1)	weight: 0
 #U21(x1,x2)	weight: 0
 #U22(x1)	weight: 0
    tt()	weight: 0
   U13(x1)	weight: x1
   U22(x1)	weight: x1
#isNatKind(x1)	weight: 0
   U41(x1,x2,x3)	weight: x2
 #U31(x1,x2)	weight: 0
 #and(x1,x2)	weight: x2
    Usable rules: { }
    Removed DPs: #85
Number of SCCs: 10, DPs: 46, edges: 592
	SCC { #27 }
Removing DPs: Order(PosReal,>,Sum)... succeeded.
isNatKind(x1)	weight: x1
   U21(x1,x2)	weight: x1
   U11(x1,x2,x3)	weight: x3
     s(x1)	weight: x1
#isNat(x1)	weight: 0
   top(x1)	weight: 0
   and(x1,x2)	weight: x1
#plus(x1,x2)	weight: 0
 #top(x1)	weight: 0
 #U13(x1)	weight: 0
   U12(x1,x2)	weight: x2
 #U12(x1,x2)	weight: 0
proper(x1)	weight: (/ 1 2)
    ok(x1)	weight: (/ 1 2) + x1
     0()	weight: 0
   #s(x1)	weight: 0
  mark(x1)	weight: (/ 1 2) + x1
#proper(x1)	weight: 0
 isNat(x1)	weight: x1
  plus(x1,x2)	weight: 0
 #U11(x1,x2,x3)	weight: 0
active(x1)	weight: x1
   U31(x1,x2)	weight: 0
 #U41(x1,x2,x3)	weight: 0
#active(x1)	weight: 0
 #U21(x1,x2)	weight: 0
 #U22(x1)	weight: 0
    tt()	weight: 0
   U13(x1)	weight: x1
   U22(x1)	weight: 0
#isNatKind(x1)	weight: 0
   U41(x1,x2,x3)	weight: x1 + x2
 #U31(x1,x2)	weight: 0
 #and(x1,x2)	weight: x1
    Usable rules: { }
    Removed DPs: #27
Number of SCCs: 9, DPs: 45, edges: 591
	SCC { #35 #42 }
Removing DPs: Order(PosReal,>,Sum)... succeeded.
isNatKind(x1)	weight: x1
   U21(x1,x2)	weight: x1
   U11(x1,x2,x3)	weight: x3
     s(x1)	weight: x1
#isNat(x1)	weight: 0
   top(x1)	weight: 0
   and(x1,x2)	weight: x1
#plus(x1,x2)	weight: 0
 #top(x1)	weight: 0
 #U13(x1)	weight: 0
   U12(x1,x2)	weight: x2
 #U12(x1,x2)	weight: 0
proper(x1)	weight: (/ 1 2)
    ok(x1)	weight: (/ 1 2) + x1
     0()	weight: 0
   #s(x1)	weight: 0
  mark(x1)	weight: (/ 1 2) + x1
#proper(x1)	weight: 0
 isNat(x1)	weight: x1
  plus(x1,x2)	weight: 0
 #U11(x1,x2,x3)	weight: 0
active(x1)	weight: x1
   U31(x1,x2)	weight: 0
 #U41(x1,x2,x3)	weight: 0
#active(x1)	weight: 0
 #U21(x1,x2)	weight: x2
 #U22(x1)	weight: 0
    tt()	weight: 0
   U13(x1)	weight: x1
   U22(x1)	weight: 0
#isNatKind(x1)	weight: 0
   U41(x1,x2,x3)	weight: x1 + x2
 #U31(x1,x2)	weight: 0
 #and(x1,x2)	weight: 0
    Usable rules: { }
    Removed DPs: #35
Number of SCCs: 10, DPs: 44, edges: 588
	SCC { #42 }
Removing DPs: Order(PosReal,>,Sum)... succeeded.
isNatKind(x1)	weight: x1
   U21(x1,x2)	weight: x1
   U11(x1,x2,x3)	weight: x3
     s(x1)	weight: x1
#isNat(x1)	weight: 0
   top(x1)	weight: 0
   and(x1,x2)	weight: x1
#plus(x1,x2)	weight: 0
 #top(x1)	weight: 0
 #U13(x1)	weight: 0
   U12(x1,x2)	weight: x2
 #U12(x1,x2)	weight: 0
proper(x1)	weight: (/ 1 2)
    ok(x1)	weight: (/ 1 2) + x1
     0()	weight: 0
   #s(x1)	weight: 0
  mark(x1)	weight: (/ 1 2) + x1
#proper(x1)	weight: 0
 isNat(x1)	weight: x1
  plus(x1,x2)	weight: 0
 #U11(x1,x2,x3)	weight: 0
active(x1)	weight: x1
   U31(x1,x2)	weight: 0
 #U41(x1,x2,x3)	weight: 0
#active(x1)	weight: 0
 #U21(x1,x2)	weight: x1
 #U22(x1)	weight: 0
    tt()	weight: 0
   U13(x1)	weight: x1
   U22(x1)	weight: 0
#isNatKind(x1)	weight: 0
   U41(x1,x2,x3)	weight: x1 + x2
 #U31(x1,x2)	weight: 0
 #and(x1,x2)	weight: 0
    Usable rules: { }
    Removed DPs: #42
Number of SCCs: 9, DPs: 43, edges: 587
	SCC { #5 #101 }
Removing DPs: Order(PosReal,>,Sum)... succeeded.
isNatKind(x1)	weight: x1
   U21(x1,x2)	weight: x1
   U11(x1,x2,x3)	weight: x3
     s(x1)	weight: x1
#isNat(x1)	weight: 0
   top(x1)	weight: 0
   and(x1,x2)	weight: x1
#plus(x1,x2)	weight: 0
 #top(x1)	weight: 0
 #U13(x1)	weight: 0
   U12(x1,x2)	weight: x2
 #U12(x1,x2)	weight: x1 + x2
proper(x1)	weight: (/ 1 4)
    ok(x1)	weight: (/ 1 4) + x1
     0()	weight: 0
   #s(x1)	weight: 0
  mark(x1)	weight: (/ 1 4) + x1
#proper(x1)	weight: 0
 isNat(x1)	weight: x1
  plus(x1,x2)	weight: 0
 #U11(x1,x2,x3)	weight: 0
active(x1)	weight: x1
   U31(x1,x2)	weight: 0
 #U41(x1,x2,x3)	weight: 0
#active(x1)	weight: 0
 #U21(x1,x2)	weight: 0
 #U22(x1)	weight: 0
    tt()	weight: 0
   U13(x1)	weight: x1
   U22(x1)	weight: 0
#isNatKind(x1)	weight: 0
   U41(x1,x2,x3)	weight: x1 + x2
 #U31(x1,x2)	weight: 0
 #and(x1,x2)	weight: 0
    Usable rules: { }
    Removed DPs: #5 #101
Number of SCCs: 9, DPs: 41, edges: 583
	SCC { #28 #67 }
Removing DPs: Order(PosReal,>,Sum)... succeeded.
isNatKind(x1)	weight: x1
   U21(x1,x2)	weight: x1
   U11(x1,x2,x3)	weight: x3
     s(x1)	weight: x1
#isNat(x1)	weight: 0
   top(x1)	weight: 0
   and(x1,x2)	weight: x1
#plus(x1,x2)	weight: 0
 #top(x1)	weight: 0
 #U13(x1)	weight: 0
   U12(x1,x2)	weight: x2
 #U12(x1,x2)	weight: 0
proper(x1)	weight: (/ 1 2)
    ok(x1)	weight: (/ 1 2) + x1
     0()	weight: 0
   #s(x1)	weight: 0
  mark(x1)	weight: (/ 1 2) + x1
#proper(x1)	weight: 0
 isNat(x1)	weight: x1
  plus(x1,x2)	weight: 0
 #U11(x1,x2,x3)	weight: 0
active(x1)	weight: x1
   U31(x1,x2)	weight: 0
 #U41(x1,x2,x3)	weight: 0
#active(x1)	weight: 0
 #U21(x1,x2)	weight: 0
 #U22(x1)	weight: 0
    tt()	weight: 0
   U13(x1)	weight: x1
   U22(x1)	weight: 0
#isNatKind(x1)	weight: 0
   U41(x1,x2,x3)	weight: x1 + x2
 #U31(x1,x2)	weight: x1
 #and(x1,x2)	weight: 0
    Usable rules: { }
    Removed DPs: #28 #67
Number of SCCs: 8, DPs: 39, edges: 579
	SCC { #22 #73 }
Removing DPs: Order(PosReal,>,Sum)... succeeded.
isNatKind(x1)	weight: x1
   U21(x1,x2)	weight: x1
   U11(x1,x2,x3)	weight: x3
     s(x1)	weight: x1
#isNat(x1)	weight: 0
   top(x1)	weight: 0
   and(x1,x2)	weight: x1
#plus(x1,x2)	weight: 0
 #top(x1)	weight: 0
 #U13(x1)	weight: 0
   U12(x1,x2)	weight: x2
 #U12(x1,x2)	weight: 0
proper(x1)	weight: (/ 1 2)
    ok(x1)	weight: (/ 1 2) + x1
     0()	weight: 0
   #s(x1)	weight: 0
  mark(x1)	weight: (/ 1 2) + x1
#proper(x1)	weight: 0
 isNat(x1)	weight: x1
  plus(x1,x2)	weight: 0
 #U11(x1,x2,x3)	weight: x3
active(x1)	weight: x1
   U31(x1,x2)	weight: 0
 #U41(x1,x2,x3)	weight: 0
#active(x1)	weight: 0
 #U21(x1,x2)	weight: 0
 #U22(x1)	weight: 0
    tt()	weight: 0
   U13(x1)	weight: x1
   U22(x1)	weight: 0
#isNatKind(x1)	weight: 0
   U41(x1,x2,x3)	weight: x1 + x2
 #U31(x1,x2)	weight: 0
 #and(x1,x2)	weight: 0
    Usable rules: { }
    Removed DPs: #22
Number of SCCs: 8, DPs: 38, edges: 576
	SCC { #73 }
Removing DPs: Order(PosReal,>,Sum)... succeeded.
isNatKind(x1)	weight: x1
   U21(x1,x2)	weight: x1
   U11(x1,x2,x3)	weight: x3
     s(x1)	weight: x1
#isNat(x1)	weight: 0
   top(x1)	weight: 0
   and(x1,x2)	weight: x1
#plus(x1,x2)	weight: 0
 #top(x1)	weight: 0
 #U13(x1)	weight: 0
   U12(x1,x2)	weight: x2
 #U12(x1,x2)	weight: 0
proper(x1)	weight: (/ 1 2)
    ok(x1)	weight: (/ 1 2) + x1
     0()	weight: 0
   #s(x1)	weight: 0
  mark(x1)	weight: (/ 1 2) + x1
#proper(x1)	weight: 0
 isNat(x1)	weight: x1
  plus(x1,x2)	weight: 0
 #U11(x1,x2,x3)	weight: x1
active(x1)	weight: x1
   U31(x1,x2)	weight: 0
 #U41(x1,x2,x3)	weight: 0
#active(x1)	weight: 0
 #U21(x1,x2)	weight: 0
 #U22(x1)	weight: 0
    tt()	weight: 0
   U13(x1)	weight: x1
   U22(x1)	weight: 0
#isNatKind(x1)	weight: 0
   U41(x1,x2,x3)	weight: x1 + x2
 #U31(x1,x2)	weight: 0
 #and(x1,x2)	weight: 0
    Usable rules: { }
    Removed DPs: #73
Number of SCCs: 7, DPs: 37, edges: 575
	SCC { #76 #79 }
Removing DPs: Order(PosReal,>,Sum)... succeeded.
isNatKind(x1)	weight: x1
   U21(x1,x2)	weight: x1
   U11(x1,x2,x3)	weight: x3
     s(x1)	weight: x1
#isNat(x1)	weight: 0
   top(x1)	weight: 0
   and(x1,x2)	weight: x1
#plus(x1,x2)	weight: 0
 #top(x1)	weight: 0
 #U13(x1)	weight: 0
   U12(x1,x2)	weight: x2
 #U12(x1,x2)	weight: 0
proper(x1)	weight: (/ 1 4)
    ok(x1)	weight: (/ 1 4) + x1
     0()	weight: 0
   #s(x1)	weight: 0
  mark(x1)	weight: (/ 1 4) + x1
#proper(x1)	weight: 0
 isNat(x1)	weight: x1
  plus(x1,x2)	weight: 0
 #U11(x1,x2,x3)	weight: 0
active(x1)	weight: x1
   U31(x1,x2)	weight: 0
 #U41(x1,x2,x3)	weight: x1 + x3
#active(x1)	weight: 0
 #U21(x1,x2)	weight: 0
 #U22(x1)	weight: 0
    tt()	weight: 0
   U13(x1)	weight: x1
   U22(x1)	weight: 0
#isNatKind(x1)	weight: 0
   U41(x1,x2,x3)	weight: x1 + x2
 #U31(x1,x2)	weight: 0
 #and(x1,x2)	weight: 0
    Usable rules: { }
    Removed DPs: #76 #79
Number of SCCs: 7, DPs: 35, edges: 571
	SCC { #17 #45 #88 }
Removing DPs: Order(PosReal,>,Sum)... succeeded.
isNatKind(x1)	weight: x1
   U21(x1,x2)	weight: x1
   U11(x1,x2,x3)	weight: x3
     s(x1)	weight: x1
#isNat(x1)	weight: 0
   top(x1)	weight: 0
   and(x1,x2)	weight: x1
#plus(x1,x2)	weight: x2
 #top(x1)	weight: 0
 #U13(x1)	weight: 0
   U12(x1,x2)	weight: x2
 #U12(x1,x2)	weight: 0
proper(x1)	weight: (/ 1 2)
    ok(x1)	weight: (/ 1 2) + x1
     0()	weight: 0
   #s(x1)	weight: 0
  mark(x1)	weight: (/ 1 2) + x1
#proper(x1)	weight: 0
 isNat(x1)	weight: x1
  plus(x1,x2)	weight: 0
 #U11(x1,x2,x3)	weight: 0
active(x1)	weight: x1
   U31(x1,x2)	weight: 0
 #U41(x1,x2,x3)	weight: 0
#active(x1)	weight: 0
 #U21(x1,x2)	weight: 0
 #U22(x1)	weight: 0
    tt()	weight: 0
   U13(x1)	weight: x1
   U22(x1)	weight: 0
#isNatKind(x1)	weight: 0
   U41(x1,x2,x3)	weight: x1 + x2
 #U31(x1,x2)	weight: 0
 #and(x1,x2)	weight: 0
    Usable rules: { }
    Removed DPs: #17 #45
Number of SCCs: 7, DPs: 33, edges: 563
	SCC { #88 }
Removing DPs: Order(PosReal,>,Sum)... succeeded.
isNatKind(x1)	weight: x1
   U21(x1,x2)	weight: x1
   U11(x1,x2,x3)	weight: x3
     s(x1)	weight: x1
#isNat(x1)	weight: 0
   top(x1)	weight: 0
   and(x1,x2)	weight: x1
#plus(x1,x2)	weight: x1
 #top(x1)	weight: 0
 #U13(x1)	weight: 0
   U12(x1,x2)	weight: x2
 #U12(x1,x2)	weight: 0
proper(x1)	weight: (/ 1 2)
    ok(x1)	weight: (/ 1 2) + x1
     0()	weight: 0
   #s(x1)	weight: 0
  mark(x1)	weight: (/ 1 2) + x1
#proper(x1)	weight: 0
 isNat(x1)	weight: x1
  plus(x1,x2)	weight: 0
 #U11(x1,x2,x3)	weight: 0
active(x1)	weight: x1
   U31(x1,x2)	weight: 0
 #U41(x1,x2,x3)	weight: 0
#active(x1)	weight: 0
 #U21(x1,x2)	weight: 0
 #U22(x1)	weight: 0
    tt()	weight: 0
   U13(x1)	weight: x1
   U22(x1)	weight: 0
#isNatKind(x1)	weight: 0
   U41(x1,x2,x3)	weight: x1 + x2
 #U31(x1,x2)	weight: 0
 #and(x1,x2)	weight: 0
    Usable rules: { }
    Removed DPs: #88
Number of SCCs: 6, DPs: 32, edges: 562
	SCC { #37 #39 #53 #55 #75 #78 #81 #84 #87 #90 #112 }
Removing DPs: Order(PosReal,>,Sum)... succeeded.
isNatKind(x1)	weight: x1
   U21(x1,x2)	weight: (/ 1 8) + x1
   U11(x1,x2,x3)	weight: (/ 1 8) + x1 + x3
     s(x1)	weight: (/ 1 8) + x1
#isNat(x1)	weight: 0
   top(x1)	weight: 0
   and(x1,x2)	weight: (/ 1 8) + x1
#plus(x1,x2)	weight: 0
 #top(x1)	weight: 0
 #U13(x1)	weight: 0
   U12(x1,x2)	weight: (/ 1 8) + x1 + x2
 #U12(x1,x2)	weight: 0
proper(x1)	weight: (/ 1 8)
    ok(x1)	weight: (/ 1 8) + x1
     0()	weight: 0
   #s(x1)	weight: 0
  mark(x1)	weight: (/ 1 8) + x1
#proper(x1)	weight: 0
 isNat(x1)	weight: x1
  plus(x1,x2)	weight: (/ 1 8) + x1 + x2
 #U11(x1,x2,x3)	weight: 0
active(x1)	weight: 0
   U31(x1,x2)	weight: (/ 1 8) + x1
 #U41(x1,x2,x3)	weight: 0
#active(x1)	weight: x1
 #U21(x1,x2)	weight: 0
 #U22(x1)	weight: 0
    tt()	weight: 0
   U13(x1)	weight: (/ 1 8) + x1
   U22(x1)	weight: (/ 1 8) + x1
#isNatKind(x1)	weight: 0
   U41(x1,x2,x3)	weight: (/ 1 8) + x1 + x2
 #U31(x1,x2)	weight: 0
 #and(x1,x2)	weight: 0
    Usable rules: { }
    Removed DPs: #37 #39 #53 #55 #75 #78 #81 #84 #87 #90 #112
Number of SCCs: 6, DPs: 21, edges: 441
	SCC { #4 #10 #11 #13 #15 #16 #19..21 #24 #41 #48 #50 #51 #59..61 #69 #70 #109 #110 }
Removing DPs: Order(PosReal,>,Sum)... succeeded.
isNatKind(x1)	weight: (/ 1 16) + x1
   U21(x1,x2)	weight: (/ 1 16) + x1 + x2
   U11(x1,x2,x3)	weight: (/ 1 16) + x1 + x2 + x3
     s(x1)	weight: (/ 1 16) + x1
#isNat(x1)	weight: 0
   top(x1)	weight: 0
   and(x1,x2)	weight: (/ 1 16) + x1 + x2
#plus(x1,x2)	weight: 0
 #top(x1)	weight: 0
 #U13(x1)	weight: 0
   U12(x1,x2)	weight: (/ 1 16) + x1 + x2
 #U12(x1,x2)	weight: 0
proper(x1)	weight: (/ 1 16)
    ok(x1)	weight: (/ 1 16) + x1
     0()	weight: 0
   #s(x1)	weight: 0
  mark(x1)	weight: (/ 1 16) + x1
#proper(x1)	weight: x1
 isNat(x1)	weight: (/ 1 16) + x1
  plus(x1,x2)	weight: (/ 1 16) + x1 + x2
 #U11(x1,x2,x3)	weight: 0
active(x1)	weight: 0
   U31(x1,x2)	weight: (/ 1 16) + x1 + x2
 #U41(x1,x2,x3)	weight: 0
#active(x1)	weight: 0
 #U21(x1,x2)	weight: 0
 #U22(x1)	weight: 0
    tt()	weight: 0
   U13(x1)	weight: (/ 1 16) + x1
   U22(x1)	weight: (/ 1 16) + x1
#isNatKind(x1)	weight: 0
   U41(x1,x2,x3)	weight: (/ 1 16) + x1 + x2 + x3
 #U31(x1,x2)	weight: 0
 #and(x1,x2)	weight: 0
    Usable rules: { }
    Removed DPs: #4 #10 #11 #13 #15 #16 #19..21 #24 #41 #48 #50 #51 #59..61 #69 #70 #109 #110
Number of SCCs: 5, DPs: 0, edges: 0
YES