/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:
            addList(xs,ys) -> addLists(xs,ys,nil())
            addLists(xs,ys,zs) -> if(isEmpty(xs)
                                    ,isEmpty(ys)
                                    ,isZero(head(xs))
                                    ,tail(xs)
                                    ,tail(ys)
                                    ,cons(p(head(xs)),tail(xs))
                                    ,cons(inc(head(ys)),tail(ys))
                                    ,zs
                                    ,append(zs,head(ys)))
            append(cons(y,ys),x) -> cons(y,append(ys,x))
            append(nil(),x) -> cons(x,nil())
            head(cons(x,xs)) -> x
            if(false(),false(),false(),xs,ys,xs2,ys2,zs,zs2) -> addLists(xs2,ys2,zs)
            if(false(),false(),true(),xs,ys,xs2,ys2,zs,zs2) -> addLists(xs,ys,zs2)
            if(false(),true(),b,xs,ys,xs2,ys2,zs,zs2) -> differentLengthError()
            if(true(),false(),b,xs,ys,xs2,ys2,zs,zs2) -> differentLengthError()
            if(true(),true(),b,xs,ys,xs2,ys2,zs,zs2) -> zs
            inc(0()) -> s(0())
            inc(s(x)) -> s(inc(x))
            isEmpty(cons(x,xs)) -> false()
            isEmpty(nil()) -> true()
            isZero(0()) -> true()
            isZero(s(x)) -> false()
            p(0()) -> 0()
            p(s(0())) -> 0()
            p(s(s(x))) -> s(p(s(x)))
            tail(cons(x,xs)) -> xs
            tail(nil()) -> nil()
        - Signature:
            {addList/2,addLists/3,append/2,head/1,if/9,inc/1,isEmpty/1,isZero/1,p/1,tail/1} / {0/0,cons/2
            ,differentLengthError/0,false/0,nil/0,s/1,true/0}
        - Obligation:
             runtime complexity wrt. defined symbols {addList,addLists,append,head,if,inc,isEmpty,isZero,p
            ,tail} and constructors {0,cons,differentLengthError,false,nil,s,true}
    + Applied Processor:
        Sum {left = someStrategy, right = someStrategy}
    + Details:
        ()
* Step 2: Sum.  WORST_CASE(Omega(n^1),?)
    + Considered Problem:
        - Strict TRS:
            addList(xs,ys) -> addLists(xs,ys,nil())
            addLists(xs,ys,zs) -> if(isEmpty(xs)
                                    ,isEmpty(ys)
                                    ,isZero(head(xs))
                                    ,tail(xs)
                                    ,tail(ys)
                                    ,cons(p(head(xs)),tail(xs))
                                    ,cons(inc(head(ys)),tail(ys))
                                    ,zs
                                    ,append(zs,head(ys)))
            append(cons(y,ys),x) -> cons(y,append(ys,x))
            append(nil(),x) -> cons(x,nil())
            head(cons(x,xs)) -> x
            if(false(),false(),false(),xs,ys,xs2,ys2,zs,zs2) -> addLists(xs2,ys2,zs)
            if(false(),false(),true(),xs,ys,xs2,ys2,zs,zs2) -> addLists(xs,ys,zs2)
            if(false(),true(),b,xs,ys,xs2,ys2,zs,zs2) -> differentLengthError()
            if(true(),false(),b,xs,ys,xs2,ys2,zs,zs2) -> differentLengthError()
            if(true(),true(),b,xs,ys,xs2,ys2,zs,zs2) -> zs
            inc(0()) -> s(0())
            inc(s(x)) -> s(inc(x))
            isEmpty(cons(x,xs)) -> false()
            isEmpty(nil()) -> true()
            isZero(0()) -> true()
            isZero(s(x)) -> false()
            p(0()) -> 0()
            p(s(0())) -> 0()
            p(s(s(x))) -> s(p(s(x)))
            tail(cons(x,xs)) -> xs
            tail(nil()) -> nil()
        - Signature:
            {addList/2,addLists/3,append/2,head/1,if/9,inc/1,isEmpty/1,isZero/1,p/1,tail/1} / {0/0,cons/2
            ,differentLengthError/0,false/0,nil/0,s/1,true/0}
        - Obligation:
             runtime complexity wrt. defined symbols {addList,addLists,append,head,if,inc,isEmpty,isZero,p
            ,tail} and constructors {0,cons,differentLengthError,false,nil,s,true}
    + Applied Processor:
        Sum {left = someStrategy, right = someStrategy}
    + Details:
        ()
* Step 3: DecreasingLoops.  WORST_CASE(Omega(n^1),?)
    + Considered Problem:
        - Strict TRS:
            addList(xs,ys) -> addLists(xs,ys,nil())
            addLists(xs,ys,zs) -> if(isEmpty(xs)
                                    ,isEmpty(ys)
                                    ,isZero(head(xs))
                                    ,tail(xs)
                                    ,tail(ys)
                                    ,cons(p(head(xs)),tail(xs))
                                    ,cons(inc(head(ys)),tail(ys))
                                    ,zs
                                    ,append(zs,head(ys)))
            append(cons(y,ys),x) -> cons(y,append(ys,x))
            append(nil(),x) -> cons(x,nil())
            head(cons(x,xs)) -> x
            if(false(),false(),false(),xs,ys,xs2,ys2,zs,zs2) -> addLists(xs2,ys2,zs)
            if(false(),false(),true(),xs,ys,xs2,ys2,zs,zs2) -> addLists(xs,ys,zs2)
            if(false(),true(),b,xs,ys,xs2,ys2,zs,zs2) -> differentLengthError()
            if(true(),false(),b,xs,ys,xs2,ys2,zs,zs2) -> differentLengthError()
            if(true(),true(),b,xs,ys,xs2,ys2,zs,zs2) -> zs
            inc(0()) -> s(0())
            inc(s(x)) -> s(inc(x))
            isEmpty(cons(x,xs)) -> false()
            isEmpty(nil()) -> true()
            isZero(0()) -> true()
            isZero(s(x)) -> false()
            p(0()) -> 0()
            p(s(0())) -> 0()
            p(s(s(x))) -> s(p(s(x)))
            tail(cons(x,xs)) -> xs
            tail(nil()) -> nil()
        - Signature:
            {addList/2,addLists/3,append/2,head/1,if/9,inc/1,isEmpty/1,isZero/1,p/1,tail/1} / {0/0,cons/2
            ,differentLengthError/0,false/0,nil/0,s/1,true/0}
        - Obligation:
             runtime complexity wrt. defined symbols {addList,addLists,append,head,if,inc,isEmpty,isZero,p
            ,tail} and constructors {0,cons,differentLengthError,false,nil,s,true}
    + Applied Processor:
        DecreasingLoops {bound = AnyLoop, narrow = 10}
    + Details:
        The system has following decreasing Loops:
          append(y,z){y -> cons(x,y)} =
            append(cons(x,y),z) ->^+ cons(x,append(y,z))
              = C[append(y,z) = append(y,z){}]

WORST_CASE(Omega(n^1),?)