/export/starexec/sandbox/solver/bin/starexec_run_default /export/starexec/sandbox/benchmark/theBenchmark.xml /export/starexec/sandbox/output/output_files


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YES
Input TRS:
    1: a__U11(tt(),V1,V2) -> a__U12(a__isNat(V1),V2)
    2: a__U12(tt(),V2) -> a__U13(a__isNat(V2))
    3: a__U13(tt()) -> tt()
    4: a__U21(tt(),V1) -> a__U22(a__isNat(V1))
    5: a__U22(tt()) -> tt()
    6: a__U31(tt(),N) -> mark(N)
    7: a__U41(tt(),M,N) -> s(a__plus(mark(N),mark(M)))
    8: a__and(tt(),X) -> mark(X)
    9: a__isNat(0()) -> tt()
    10: a__isNat(plus(V1,V2)) -> a__U11(a__and(a__isNatKind(V1),isNatKind(V2)),V1,V2)
    11: a__isNat(s(V1)) -> a__U21(a__isNatKind(V1),V1)
    12: a__isNatKind(0()) -> tt()
    13: a__isNatKind(plus(V1,V2)) -> a__and(a__isNatKind(V1),isNatKind(V2))
    14: a__isNatKind(s(V1)) -> a__isNatKind(V1)
    15: a__plus(N,0()) -> a__U31(a__and(a__isNat(N),isNatKind(N)),N)
    16: a__plus(N,s(M)) -> a__U41(a__and(a__and(a__isNat(M),isNatKind(M)),and(isNat(N),isNatKind(N))),M,N)
    17: mark(U11(X1,X2,X3)) -> a__U11(mark(X1),X2,X3)
    18: mark(U12(X1,X2)) -> a__U12(mark(X1),X2)
    19: mark(isNat(X)) -> a__isNat(X)
    20: mark(U13(X)) -> a__U13(mark(X))
    21: mark(U21(X1,X2)) -> a__U21(mark(X1),X2)
    22: mark(U22(X)) -> a__U22(mark(X))
    23: mark(U31(X1,X2)) -> a__U31(mark(X1),X2)
    24: mark(U41(X1,X2,X3)) -> a__U41(mark(X1),X2,X3)
    25: mark(plus(X1,X2)) -> a__plus(mark(X1),mark(X2))
    26: mark(and(X1,X2)) -> a__and(mark(X1),X2)
    27: mark(isNatKind(X)) -> a__isNatKind(X)
    28: mark(tt()) -> tt()
    29: mark(s(X)) -> s(mark(X))
    30: mark(0()) -> 0()
    31: a__U11(X1,X2,X3) -> U11(X1,X2,X3)
    32: a__U12(X1,X2) -> U12(X1,X2)
    33: a__isNat(X) -> isNat(X)
    34: a__U13(X) -> U13(X)
    35: a__U21(X1,X2) -> U21(X1,X2)
    36: a__U22(X) -> U22(X)
    37: a__U31(X1,X2) -> U31(X1,X2)
    38: a__U41(X1,X2,X3) -> U41(X1,X2,X3)
    39: a__plus(X1,X2) -> plus(X1,X2)
    40: a__and(X1,X2) -> and(X1,X2)
    41: a__isNatKind(X) -> isNatKind(X)
Number of strict rules: 41
Direct poly ... failed.
Freezing ... failed.
Dependency Pairs:
   #1: #a__U12(tt(),V2) -> #a__U13(a__isNat(V2))
   #2: #a__U12(tt(),V2) -> #a__isNat(V2)
   #3: #mark(s(X)) -> #mark(X)
   #4: #a__U31(tt(),N) -> #mark(N)
   #5: #a__isNatKind(plus(V1,V2)) -> #a__and(a__isNatKind(V1),isNatKind(V2))
   #6: #a__isNatKind(plus(V1,V2)) -> #a__isNatKind(V1)
   #7: #a__isNat(s(V1)) -> #a__U21(a__isNatKind(V1),V1)
   #8: #a__isNat(s(V1)) -> #a__isNatKind(V1)
   #9: #mark(U41(X1,X2,X3)) -> #a__U41(mark(X1),X2,X3)
   #10: #mark(U41(X1,X2,X3)) -> #mark(X1)
   #11: #mark(U31(X1,X2)) -> #a__U31(mark(X1),X2)
   #12: #mark(U31(X1,X2)) -> #mark(X1)
   #13: #a__isNatKind(s(V1)) -> #a__isNatKind(V1)
   #14: #mark(plus(X1,X2)) -> #a__plus(mark(X1),mark(X2))
   #15: #mark(plus(X1,X2)) -> #mark(X1)
   #16: #mark(plus(X1,X2)) -> #mark(X2)
   #17: #mark(U13(X)) -> #a__U13(mark(X))
   #18: #mark(U13(X)) -> #mark(X)
   #19: #a__U41(tt(),M,N) -> #a__plus(mark(N),mark(M))
   #20: #a__U41(tt(),M,N) -> #mark(N)
   #21: #a__U41(tt(),M,N) -> #mark(M)
   #22: #a__isNat(plus(V1,V2)) -> #a__U11(a__and(a__isNatKind(V1),isNatKind(V2)),V1,V2)
   #23: #a__isNat(plus(V1,V2)) -> #a__and(a__isNatKind(V1),isNatKind(V2))
   #24: #a__isNat(plus(V1,V2)) -> #a__isNatKind(V1)
   #25: #mark(U22(X)) -> #a__U22(mark(X))
   #26: #mark(U22(X)) -> #mark(X)
   #27: #mark(isNatKind(X)) -> #a__isNatKind(X)
   #28: #mark(U11(X1,X2,X3)) -> #a__U11(mark(X1),X2,X3)
   #29: #mark(U11(X1,X2,X3)) -> #mark(X1)
   #30: #mark(isNat(X)) -> #a__isNat(X)
   #31: #mark(and(X1,X2)) -> #a__and(mark(X1),X2)
   #32: #mark(and(X1,X2)) -> #mark(X1)
   #33: #mark(U21(X1,X2)) -> #a__U21(mark(X1),X2)
   #34: #mark(U21(X1,X2)) -> #mark(X1)
   #35: #a__plus(N,s(M)) -> #a__U41(a__and(a__and(a__isNat(M),isNatKind(M)),and(isNat(N),isNatKind(N))),M,N)
   #36: #a__plus(N,s(M)) -> #a__and(a__and(a__isNat(M),isNatKind(M)),and(isNat(N),isNatKind(N)))
   #37: #a__plus(N,s(M)) -> #a__and(a__isNat(M),isNatKind(M))
   #38: #a__plus(N,s(M)) -> #a__isNat(M)
   #39: #a__U11(tt(),V1,V2) -> #a__U12(a__isNat(V1),V2)
   #40: #a__U11(tt(),V1,V2) -> #a__isNat(V1)
   #41: #a__and(tt(),X) -> #mark(X)
   #42: #a__plus(N,0()) -> #a__U31(a__and(a__isNat(N),isNatKind(N)),N)
   #43: #a__plus(N,0()) -> #a__and(a__isNat(N),isNatKind(N))
   #44: #a__plus(N,0()) -> #a__isNat(N)
   #45: #a__U21(tt(),V1) -> #a__U22(a__isNat(V1))
   #46: #a__U21(tt(),V1) -> #a__isNat(V1)
   #47: #mark(U12(X1,X2)) -> #a__U12(mark(X1),X2)
   #48: #mark(U12(X1,X2)) -> #mark(X1)
Number of SCCs: 1, DPs: 44
	SCC { #2..16 #18..24 #26..44 #46..48 }
Sum... succeeded.
  #a__isNatKind(x1)		w: (7629)
  isNatKind(x1)		w: (0)
  a__plus(x1,x2)		w: (13507 + x2 + x1)
  U21(x1,x2)		w: (x1)
  U11(x1,x2,x3)		w: (x1)
  s(x1)		w: (29406 + x1)
  #a__U31(x1,x2)		w: (7630 + x2)
  #a__isNat(x1)		w: (7629)
  and(x1,x2)		w: (x2 + x1)
  #a__U13(x1)		w: (0)
  a__U22(x1)		w: (x1)
  #a__U11(x1,x2,x3)		w: (7629)
  a__U31(x1,x2)		w: (2 + x2 + x1)
  U12(x1,x2)		w: (x1)
  a__U41(x1,x2,x3)		w: (42913 + x3 + x2 + x1)
  #a__U12(x1,x2)		w: (7629)
  #a__U21(x1,x2)		w: (7629)
  #a__plus(x1,x2)		w: (7630 + x2 + x1)
  #mark(x1)		w: (7629 + x1)
  0()		w: (1)
  #a__and(x1,x2)		w: (7629 + x2)
  a__U21(x1,x2)		w: (x1)
  mark(x1)		w: (x1)
  a__U11(x1,x2,x3)		w: (x1)
  a__U12(x1,x2)		w: (x1)
  isNat(x1)		w: (0)
  plus(x1,x2)		w: (13507 + x2 + x1)
  #a__U22(x1)		w: (0)
  a__U13(x1)		w: (x1)
  U31(x1,x2)		w: (2 + x2 + x1)
  #a__U41(x1,x2,x3)		w: (7631 + x3 + x2)
  tt()		w: (0)
  a__isNat(x1)		w: (0)
  U13(x1)		w: (x1)
  a__isNatKind(x1)		w: (0)
  U22(x1)		w: (x1)
  a__and(x1,x2)		w: (x2 + x1)
  U41(x1,x2,x3)		w: (42913 + x3 + x2 + x1)
    USABLE RULES: { 1..41 }
    Removed DPs: #3 #4 #9..12 #14..16 #19..21 #35..38 #42..44
Number of SCCs: 1, DPs: 25
	SCC { #2 #5..8 #13 #18 #22..24 #26..34 #39..41 #46..48 }
Sum... succeeded.
  #a__isNatKind(x1)		w: (7065 + x1)
  isNatKind(x1)		w: (1 + x1)
  a__plus(x1,x2)		w: (2 + x1)
  U21(x1,x2)		w: (7 + x2 + x1)
  U11(x1,x2,x3)		w: (11 + x3 + x2 + x1)
  s(x1)		w: (x1)
  #a__U31(x1,x2)		w: (7630)
  #a__isNat(x1)		w: (7065 + x1)
  and(x1,x2)		w: (2 + x2 + x1)
  #a__U13(x1)		w: (0)
  a__U22(x1)		w: (7)
  #a__U11(x1,x2,x3)		w: (7067 + x3 + x2)
  a__U31(x1,x2)		w: (3 + x2)
  U12(x1,x2)		w: (x2 + x1)
  a__U41(x1,x2,x3)		w: (5 + x3 + x2)
  #a__U12(x1,x2)		w: (7066 + x2)
  #a__U21(x1,x2)		w: (7065 + x2)
  #a__plus(x1,x2)		w: (7630)
  #mark(x1)		w: (7066 + x1)
  0()		w: (12835)
  #a__and(x1,x2)		w: (7067 + x2)
  a__U21(x1,x2)		w: (6 + x2)
  mark(x1)		w: (4)
  a__U11(x1,x2,x3)		w: (10 + x3 + x2)
  a__U12(x1,x2)		w: (11 + x2)
  isNat(x1)		w: (x1)
  plus(x1,x2)		w: (4 + x2 + x1)
  #a__U22(x1)		w: (0)
  a__U13(x1)		w: (12)
  U31(x1,x2)		w: (x1)
  #a__U41(x1,x2,x3)		w: (7631)
  tt()		w: (8)
  a__isNat(x1)		w: (5 + x1)
  U13(x1)		w: (13 + x1)
  a__isNatKind(x1)		w: (2)
  U22(x1)		w: (8 + x1)
  a__and(x1,x2)		w: (27574)
  U41(x1,x2,x3)		w: (x3 + x1)
    USABLE RULES: { }
    Removed DPs: #2 #5 #6 #18 #22..24 #26..34 #39..41
Number of SCCs: 3, DPs: 4
	SCC { #13 }
Sum... succeeded.
  #a__isNatKind(x1)		w: (7065 + x1)
  isNatKind(x1)		w: (1 + x1)
  a__plus(x1,x2)		w: (1365 + x1)
  U21(x1,x2)		w: (12851 + x2 + x1)
  U11(x1,x2,x3)		w: (12851 + x3 + x2 + x1)
  s(x1)		w: (12701 + x1)
  #a__U31(x1,x2)		w: (7630)
  #a__isNat(x1)		w: (7065)
  and(x1,x2)		w: (2 + x2 + x1)
  #a__U13(x1)		w: (0)
  a__U22(x1)		w: (1368)
  #a__U11(x1,x2,x3)		w: (7067)
  a__U31(x1,x2)		w: (1366 + x2)
  U12(x1,x2)		w: (x2 + x1)
  a__U41(x1,x2,x3)		w: (15432 + x3 + x2)
  #a__U12(x1,x2)		w: (7066)
  #a__U21(x1,x2)		w: (7065)
  #a__plus(x1,x2)		w: (7630)
  #mark(x1)		w: (7066)
  0()		w: (1368)
  #a__and(x1,x2)		w: (7067)
  a__U21(x1,x2)		w: (12850 + x2)
  mark(x1)		w: (1367)
  a__U11(x1,x2,x3)		w: (12850 + x3 + x2)
  a__U12(x1,x2)		w: (12851 + x2)
  isNat(x1)		w: (x1)
  plus(x1,x2)		w: (1366 + x2 + x1)
  #a__U22(x1)		w: (0)
  a__U13(x1)		w: (12852)
  U31(x1,x2)		w: (x1)
  #a__U41(x1,x2,x3)		w: (7631)
  tt()		w: (1370)
  a__isNat(x1)		w: (12849)
  U13(x1)		w: (12853 + x1)
  a__isNatKind(x1)		w: (1 + x1)
  U22(x1)		w: (1369 + x1)
  a__and(x1,x2)		w: (27574)
  U41(x1,x2,x3)		w: (x3 + x1)
    USABLE RULES: { }
    Removed DPs: #13
Number of SCCs: 2, DPs: 3
	SCC { #48 }
Sum... succeeded.
  #a__isNatKind(x1)		w: (7065)
  isNatKind(x1)		w: (1 + x1)
  a__plus(x1,x2)		w: (1 + x1)
  U21(x1,x2)		w: (12851 + x2 + x1)
  U11(x1,x2,x3)		w: (12851 + x3 + x2 + x1)
  s(x1)		w: (12701 + x1)
  #a__U31(x1,x2)		w: (7630)
  #a__isNat(x1)		w: (7065)
  and(x1,x2)		w: (2 + x2 + x1)
  #a__U13(x1)		w: (0)
  a__U22(x1)		w: (5)
  #a__U11(x1,x2,x3)		w: (7067)
  a__U31(x1,x2)		w: (2 + x2)
  U12(x1,x2)		w: (1 + x2 + x1)
  a__U41(x1,x2,x3)		w: (12704 + x3 + x2)
  #a__U12(x1,x2)		w: (7066)
  #a__U21(x1,x2)		w: (7065)
  #a__plus(x1,x2)		w: (7630)
  #mark(x1)		w: (7066 + x1)
  0()		w: (4)
  #a__and(x1,x2)		w: (7067)
  a__U21(x1,x2)		w: (12850 + x2)
  mark(x1)		w: (3)
  a__U11(x1,x2,x3)		w: (12850 + x3 + x2)
  a__U12(x1,x2)		w: (12851 + x2)
  isNat(x1)		w: (x1)
  plus(x1,x2)		w: (5970 + x2 + x1)
  #a__U22(x1)		w: (0)
  a__U13(x1)		w: (12852)
  U31(x1,x2)		w: (x1)
  #a__U41(x1,x2,x3)		w: (7631)
  tt()		w: (6)
  a__isNat(x1)		w: (6)
  U13(x1)		w: (12853 + x1)
  a__isNatKind(x1)		w: (1 + x1)
  U22(x1)		w: (6 + x1)
  a__and(x1,x2)		w: (27574)
  U41(x1,x2,x3)		w: (x3 + x1)
    USABLE RULES: { }
    Removed DPs: #48
Number of SCCs: 1, DPs: 2
	SCC { #7 #46 }
Sum... succeeded.
  #a__isNatKind(x1)		w: (7065)
  isNatKind(x1)		w: (1 + x1)
  a__plus(x1,x2)		w: (1 + x1)
  U21(x1,x2)		w: (12851 + x2 + x1)
  U11(x1,x2,x3)		w: (11149 + x3 + x2 + x1)
  s(x1)		w: (12701 + x1)
  #a__U31(x1,x2)		w: (7630)
  #a__isNat(x1)		w: (x1)
  and(x1,x2)		w: (2 + x2 + x1)
  #a__U13(x1)		w: (0)
  a__U22(x1)		w: (4)
  #a__U11(x1,x2,x3)		w: (7067)
  a__U31(x1,x2)		w: (2 + x2)
  U12(x1,x2)		w: (1 + x2 + x1)
  a__U41(x1,x2,x3)		w: (12704 + x3 + x2)
  #a__U12(x1,x2)		w: (7066)
  #a__U21(x1,x2)		w: (1 + x2)
  #a__plus(x1,x2)		w: (7630)
  #mark(x1)		w: (7066)
  0()		w: (4)
  #a__and(x1,x2)		w: (7067)
  a__U21(x1,x2)		w: (12850 + x2)
  mark(x1)		w: (3)
  a__U11(x1,x2,x3)		w: (11148 + x3 + x2)
  a__U12(x1,x2)		w: (11149 + x2)
  isNat(x1)		w: (x1)
  plus(x1,x2)		w: (29274 + x2 + x1)
  #a__U22(x1)		w: (0)
  a__U13(x1)		w: (11150)
  U31(x1,x2)		w: (x1)
  #a__U41(x1,x2,x3)		w: (7631)
  tt()		w: (6)
  a__isNat(x1)		w: (6)
  U13(x1)		w: (11151 + x1)
  a__isNatKind(x1)		w: (1 + x1)
  U22(x1)		w: (5 + x1)
  a__and(x1,x2)		w: (29276)
  U41(x1,x2,x3)		w: (x3 + x1)
    USABLE RULES: { }
    Removed DPs: #7 #46
Number of SCCs: 0, DPs: 0