/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:
            head(cons(x,xs)) -> x
            if(false(),false(),y,xs,ys,x) -> sumList(ys,x)
            if(false(),true(),y,xs,ys,x) -> sumList(xs,y)
            if(true(),b,y,xs,ys,x) -> y
            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)))
            sum(xs) -> sumList(xs,0())
            sumList(xs,y) -> if(isEmpty(xs),isZero(head(xs)),y,tail(xs),cons(p(head(xs)),tail(xs)),inc(y))
            tail(cons(x,xs)) -> xs
            tail(nil()) -> nil()
        - Signature:
            {head/1,if/6,inc/1,isEmpty/1,isZero/1,p/1,sum/1,sumList/2,tail/1} / {0/0,cons/2,false/0,nil/0,s/1,true/0}
        - Obligation:
             runtime complexity wrt. defined symbols {head,if,inc,isEmpty,isZero,p,sum,sumList,tail} and constructors {0
            ,cons,false,nil,s,true}
    + Applied Processor:
        Sum {left = someStrategy, right = someStrategy}
    + Details:
        ()
* Step 2: Sum.  WORST_CASE(Omega(n^1),?)
    + Considered Problem:
        - Strict TRS:
            head(cons(x,xs)) -> x
            if(false(),false(),y,xs,ys,x) -> sumList(ys,x)
            if(false(),true(),y,xs,ys,x) -> sumList(xs,y)
            if(true(),b,y,xs,ys,x) -> y
            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)))
            sum(xs) -> sumList(xs,0())
            sumList(xs,y) -> if(isEmpty(xs),isZero(head(xs)),y,tail(xs),cons(p(head(xs)),tail(xs)),inc(y))
            tail(cons(x,xs)) -> xs
            tail(nil()) -> nil()
        - Signature:
            {head/1,if/6,inc/1,isEmpty/1,isZero/1,p/1,sum/1,sumList/2,tail/1} / {0/0,cons/2,false/0,nil/0,s/1,true/0}
        - Obligation:
             runtime complexity wrt. defined symbols {head,if,inc,isEmpty,isZero,p,sum,sumList,tail} and constructors {0
            ,cons,false,nil,s,true}
    + Applied Processor:
        Sum {left = someStrategy, right = someStrategy}
    + Details:
        ()
* Step 3: DecreasingLoops.  WORST_CASE(Omega(n^1),?)
    + Considered Problem:
        - Strict TRS:
            head(cons(x,xs)) -> x
            if(false(),false(),y,xs,ys,x) -> sumList(ys,x)
            if(false(),true(),y,xs,ys,x) -> sumList(xs,y)
            if(true(),b,y,xs,ys,x) -> y
            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)))
            sum(xs) -> sumList(xs,0())
            sumList(xs,y) -> if(isEmpty(xs),isZero(head(xs)),y,tail(xs),cons(p(head(xs)),tail(xs)),inc(y))
            tail(cons(x,xs)) -> xs
            tail(nil()) -> nil()
        - Signature:
            {head/1,if/6,inc/1,isEmpty/1,isZero/1,p/1,sum/1,sumList/2,tail/1} / {0/0,cons/2,false/0,nil/0,s/1,true/0}
        - Obligation:
             runtime complexity wrt. defined symbols {head,if,inc,isEmpty,isZero,p,sum,sumList,tail} and constructors {0
            ,cons,false,nil,s,true}
    + Applied Processor:
        DecreasingLoops {bound = AnyLoop, narrow = 10}
    + Details:
        The system has following decreasing Loops:
          inc(x){x -> s(x)} =
            inc(s(x)) ->^+ s(inc(x))
              = C[inc(x) = inc(x){}]

WORST_CASE(Omega(n^1),?)