/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:
            append(add(N,X),Y) -> add(N,append(X,Y))
            append(nil(),Y) -> Y
            f_1(pair(X,Z),N,M,Y) -> f_2(lt(N,M),N,M,Y,X,Z)
            f_2(false(),N,M,Y,X,Z) -> pair(add(M,X),Z)
            f_2(true(),N,M,Y,X,Z) -> pair(X,add(M,Z))
            f_3(pair(Y,Z),N,X) -> append(qsort(Y),add(X,qsort(Z)))
            lt(0(),s(X)) -> true()
            lt(s(X),0()) -> false()
            lt(s(X),s(Y)) -> lt(X,Y)
            qsort(add(N,X)) -> f_3(split(N,X),N,X)
            qsort(nil()) -> nil()
            split(N,add(M,Y)) -> f_1(split(N,Y),N,M,Y)
            split(N,nil()) -> pair(nil(),nil())
        - Signature:
            {append/2,f_1/4,f_2/6,f_3/3,lt/2,qsort/1,split/2} / {0/0,add/2,false/0,nil/0,pair/2,s/1,true/0}
        - Obligation:
             runtime complexity wrt. defined symbols {append,f_1,f_2,f_3,lt,qsort,split} and constructors {0,add,false
            ,nil,pair,s,true}
    + Applied Processor:
        Sum {left = someStrategy, right = someStrategy}
    + Details:
        ()
* Step 2: Sum.  WORST_CASE(Omega(n^1),?)
    + Considered Problem:
        - Strict TRS:
            append(add(N,X),Y) -> add(N,append(X,Y))
            append(nil(),Y) -> Y
            f_1(pair(X,Z),N,M,Y) -> f_2(lt(N,M),N,M,Y,X,Z)
            f_2(false(),N,M,Y,X,Z) -> pair(add(M,X),Z)
            f_2(true(),N,M,Y,X,Z) -> pair(X,add(M,Z))
            f_3(pair(Y,Z),N,X) -> append(qsort(Y),add(X,qsort(Z)))
            lt(0(),s(X)) -> true()
            lt(s(X),0()) -> false()
            lt(s(X),s(Y)) -> lt(X,Y)
            qsort(add(N,X)) -> f_3(split(N,X),N,X)
            qsort(nil()) -> nil()
            split(N,add(M,Y)) -> f_1(split(N,Y),N,M,Y)
            split(N,nil()) -> pair(nil(),nil())
        - Signature:
            {append/2,f_1/4,f_2/6,f_3/3,lt/2,qsort/1,split/2} / {0/0,add/2,false/0,nil/0,pair/2,s/1,true/0}
        - Obligation:
             runtime complexity wrt. defined symbols {append,f_1,f_2,f_3,lt,qsort,split} and constructors {0,add,false
            ,nil,pair,s,true}
    + Applied Processor:
        Sum {left = someStrategy, right = someStrategy}
    + Details:
        ()
* Step 3: DecreasingLoops.  WORST_CASE(Omega(n^1),?)
    + Considered Problem:
        - Strict TRS:
            append(add(N,X),Y) -> add(N,append(X,Y))
            append(nil(),Y) -> Y
            f_1(pair(X,Z),N,M,Y) -> f_2(lt(N,M),N,M,Y,X,Z)
            f_2(false(),N,M,Y,X,Z) -> pair(add(M,X),Z)
            f_2(true(),N,M,Y,X,Z) -> pair(X,add(M,Z))
            f_3(pair(Y,Z),N,X) -> append(qsort(Y),add(X,qsort(Z)))
            lt(0(),s(X)) -> true()
            lt(s(X),0()) -> false()
            lt(s(X),s(Y)) -> lt(X,Y)
            qsort(add(N,X)) -> f_3(split(N,X),N,X)
            qsort(nil()) -> nil()
            split(N,add(M,Y)) -> f_1(split(N,Y),N,M,Y)
            split(N,nil()) -> pair(nil(),nil())
        - Signature:
            {append/2,f_1/4,f_2/6,f_3/3,lt/2,qsort/1,split/2} / {0/0,add/2,false/0,nil/0,pair/2,s/1,true/0}
        - Obligation:
             runtime complexity wrt. defined symbols {append,f_1,f_2,f_3,lt,qsort,split} and constructors {0,add,false
            ,nil,pair,s,true}
    + Applied Processor:
        DecreasingLoops {bound = AnyLoop, narrow = 10}
    + Details:
        The system has following decreasing Loops:
          append(y,z){y -> add(x,y)} =
            append(add(x,y),z) ->^+ add(x,append(y,z))
              = C[append(y,z) = append(y,z){}]

WORST_CASE(Omega(n^1),?)