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


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WORST_CASE(Omega(n^1),?)
* Step 1: Sum.  WORST_CASE(Omega(n^1),?)
    + Considered Problem:
        - Strict TRS:
            a__U11(X1,X2) -> U11(X1,X2)
            a__U11(tt(),V2) -> a__U12(a__isNat(V2))
            a__U12(X) -> U12(X)
            a__U12(tt()) -> tt()
            a__U21(X) -> U21(X)
            a__U21(tt()) -> tt()
            a__U31(X1,X2) -> U31(X1,X2)
            a__U31(tt(),N) -> mark(N)
            a__U41(X1,X2,X3) -> U41(X1,X2,X3)
            a__U41(tt(),M,N) -> a__U42(a__isNat(N),M,N)
            a__U42(X1,X2,X3) -> U42(X1,X2,X3)
            a__U42(tt(),M,N) -> s(a__plus(mark(N),mark(M)))
            a__isNat(X) -> isNat(X)
            a__isNat(0()) -> tt()
            a__isNat(plus(V1,V2)) -> a__U11(a__isNat(V1),V2)
            a__isNat(s(V1)) -> a__U21(a__isNat(V1))
            a__plus(N,0()) -> a__U31(a__isNat(N),N)
            a__plus(N,s(M)) -> a__U41(a__isNat(M),M,N)
            a__plus(X1,X2) -> plus(X1,X2)
            mark(0()) -> 0()
            mark(U11(X1,X2)) -> a__U11(mark(X1),X2)
            mark(U12(X)) -> a__U12(mark(X))
            mark(U21(X)) -> a__U21(mark(X))
            mark(U31(X1,X2)) -> a__U31(mark(X1),X2)
            mark(U41(X1,X2,X3)) -> a__U41(mark(X1),X2,X3)
            mark(U42(X1,X2,X3)) -> a__U42(mark(X1),X2,X3)
            mark(isNat(X)) -> a__isNat(X)
            mark(plus(X1,X2)) -> a__plus(mark(X1),mark(X2))
            mark(s(X)) -> s(mark(X))
            mark(tt()) -> tt()
        - Signature:
            {a__U11/2,a__U12/1,a__U21/1,a__U31/2,a__U41/3,a__U42/3,a__isNat/1,a__plus/2,mark/1} / {0/0,U11/2,U12/1,U21/1
            ,U31/2,U41/3,U42/3,isNat/1,plus/2,s/1,tt/0}
        - Obligation:
             runtime complexity wrt. defined symbols {a__U11,a__U12,a__U21,a__U31,a__U41,a__U42,a__isNat,a__plus
            ,mark} and constructors {0,U11,U12,U21,U31,U41,U42,isNat,plus,s,tt}
    + Applied Processor:
        Sum {left = someStrategy, right = someStrategy}
    + Details:
        ()
* Step 2: Sum.  WORST_CASE(Omega(n^1),?)
    + Considered Problem:
        - Strict TRS:
            a__U11(X1,X2) -> U11(X1,X2)
            a__U11(tt(),V2) -> a__U12(a__isNat(V2))
            a__U12(X) -> U12(X)
            a__U12(tt()) -> tt()
            a__U21(X) -> U21(X)
            a__U21(tt()) -> tt()
            a__U31(X1,X2) -> U31(X1,X2)
            a__U31(tt(),N) -> mark(N)
            a__U41(X1,X2,X3) -> U41(X1,X2,X3)
            a__U41(tt(),M,N) -> a__U42(a__isNat(N),M,N)
            a__U42(X1,X2,X3) -> U42(X1,X2,X3)
            a__U42(tt(),M,N) -> s(a__plus(mark(N),mark(M)))
            a__isNat(X) -> isNat(X)
            a__isNat(0()) -> tt()
            a__isNat(plus(V1,V2)) -> a__U11(a__isNat(V1),V2)
            a__isNat(s(V1)) -> a__U21(a__isNat(V1))
            a__plus(N,0()) -> a__U31(a__isNat(N),N)
            a__plus(N,s(M)) -> a__U41(a__isNat(M),M,N)
            a__plus(X1,X2) -> plus(X1,X2)
            mark(0()) -> 0()
            mark(U11(X1,X2)) -> a__U11(mark(X1),X2)
            mark(U12(X)) -> a__U12(mark(X))
            mark(U21(X)) -> a__U21(mark(X))
            mark(U31(X1,X2)) -> a__U31(mark(X1),X2)
            mark(U41(X1,X2,X3)) -> a__U41(mark(X1),X2,X3)
            mark(U42(X1,X2,X3)) -> a__U42(mark(X1),X2,X3)
            mark(isNat(X)) -> a__isNat(X)
            mark(plus(X1,X2)) -> a__plus(mark(X1),mark(X2))
            mark(s(X)) -> s(mark(X))
            mark(tt()) -> tt()
        - Signature:
            {a__U11/2,a__U12/1,a__U21/1,a__U31/2,a__U41/3,a__U42/3,a__isNat/1,a__plus/2,mark/1} / {0/0,U11/2,U12/1,U21/1
            ,U31/2,U41/3,U42/3,isNat/1,plus/2,s/1,tt/0}
        - Obligation:
             runtime complexity wrt. defined symbols {a__U11,a__U12,a__U21,a__U31,a__U41,a__U42,a__isNat,a__plus
            ,mark} and constructors {0,U11,U12,U21,U31,U41,U42,isNat,plus,s,tt}
    + Applied Processor:
        Sum {left = someStrategy, right = someStrategy}
    + Details:
        ()
* Step 3: DecreasingLoops.  WORST_CASE(Omega(n^1),?)
    + Considered Problem:
        - Strict TRS:
            a__U11(X1,X2) -> U11(X1,X2)
            a__U11(tt(),V2) -> a__U12(a__isNat(V2))
            a__U12(X) -> U12(X)
            a__U12(tt()) -> tt()
            a__U21(X) -> U21(X)
            a__U21(tt()) -> tt()
            a__U31(X1,X2) -> U31(X1,X2)
            a__U31(tt(),N) -> mark(N)
            a__U41(X1,X2,X3) -> U41(X1,X2,X3)
            a__U41(tt(),M,N) -> a__U42(a__isNat(N),M,N)
            a__U42(X1,X2,X3) -> U42(X1,X2,X3)
            a__U42(tt(),M,N) -> s(a__plus(mark(N),mark(M)))
            a__isNat(X) -> isNat(X)
            a__isNat(0()) -> tt()
            a__isNat(plus(V1,V2)) -> a__U11(a__isNat(V1),V2)
            a__isNat(s(V1)) -> a__U21(a__isNat(V1))
            a__plus(N,0()) -> a__U31(a__isNat(N),N)
            a__plus(N,s(M)) -> a__U41(a__isNat(M),M,N)
            a__plus(X1,X2) -> plus(X1,X2)
            mark(0()) -> 0()
            mark(U11(X1,X2)) -> a__U11(mark(X1),X2)
            mark(U12(X)) -> a__U12(mark(X))
            mark(U21(X)) -> a__U21(mark(X))
            mark(U31(X1,X2)) -> a__U31(mark(X1),X2)
            mark(U41(X1,X2,X3)) -> a__U41(mark(X1),X2,X3)
            mark(U42(X1,X2,X3)) -> a__U42(mark(X1),X2,X3)
            mark(isNat(X)) -> a__isNat(X)
            mark(plus(X1,X2)) -> a__plus(mark(X1),mark(X2))
            mark(s(X)) -> s(mark(X))
            mark(tt()) -> tt()
        - Signature:
            {a__U11/2,a__U12/1,a__U21/1,a__U31/2,a__U41/3,a__U42/3,a__isNat/1,a__plus/2,mark/1} / {0/0,U11/2,U12/1,U21/1
            ,U31/2,U41/3,U42/3,isNat/1,plus/2,s/1,tt/0}
        - Obligation:
             runtime complexity wrt. defined symbols {a__U11,a__U12,a__U21,a__U31,a__U41,a__U42,a__isNat,a__plus
            ,mark} and constructors {0,U11,U12,U21,U31,U41,U42,isNat,plus,s,tt}
    + Applied Processor:
        DecreasingLoops {bound = AnyLoop, narrow = 10}
    + Details:
        The system has following decreasing Loops:
          a__isNat(x){x -> plus(x,y)} =
            a__isNat(plus(x,y)) ->^+ a__U11(a__isNat(x),y)
              = C[a__isNat(x) = a__isNat(x){}]

WORST_CASE(Omega(n^1),?)