* Step 1: Sum WORST_CASE(Omega(n^1),O(n^1))
    + Considered Problem:
        - Strict TRS:
            -(x,0()) -> x
            -(s(x),s(y)) -> -(x,y)
            double(0()) -> 0()
            double(s(x)) -> s(s(double(x)))
            half(0()) -> 0()
            half(double(x)) -> x
            half(s(0())) -> 0()
            half(s(s(x))) -> s(half(x))
            if(0(),y,z) -> y
            if(s(x),y,z) -> z
        - Signature:
            {-/2,double/1,half/1,if/3} / {0/0,s/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {-,double,half,if} and constructors {0,s}
    + Applied Processor:
        Sum {left = someStrategy, right = someStrategy}
    + Details:
        ()
** Step 1.a:1: DecreasingLoops WORST_CASE(Omega(n^1),?)
    + Considered Problem:
        - Strict TRS:
            -(x,0()) -> x
            -(s(x),s(y)) -> -(x,y)
            double(0()) -> 0()
            double(s(x)) -> s(s(double(x)))
            half(0()) -> 0()
            half(double(x)) -> x
            half(s(0())) -> 0()
            half(s(s(x))) -> s(half(x))
            if(0(),y,z) -> y
            if(s(x),y,z) -> z
        - Signature:
            {-/2,double/1,half/1,if/3} / {0/0,s/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {-,double,half,if} and constructors {0,s}
    + Applied Processor:
        DecreasingLoops {bound = AnyLoop, narrow = 10}
    + Details:
        The system has following decreasing Loops:
          -(x,y){x -> s(x),y -> s(y)} =
            -(s(x),s(y)) ->^+ -(x,y)
              = C[-(x,y) = -(x,y){}]

** Step 1.b:1: Bounds WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict TRS:
            -(x,0()) -> x
            -(s(x),s(y)) -> -(x,y)
            double(0()) -> 0()
            double(s(x)) -> s(s(double(x)))
            half(0()) -> 0()
            half(double(x)) -> x
            half(s(0())) -> 0()
            half(s(s(x))) -> s(half(x))
            if(0(),y,z) -> y
            if(s(x),y,z) -> z
        - Signature:
            {-/2,double/1,half/1,if/3} / {0/0,s/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {-,double,half,if} and constructors {0,s}
    + Applied Processor:
        Bounds {initialAutomaton = minimal, enrichment = match}
    + Details:
        The problem is match-bounded by 1.
        The enriched problem is compatible with follwoing automaton.
          -_0(2,2) -> 1
          -_1(2,2) -> 1
          0_0() -> 1
          0_0() -> 2
          0_1() -> 1
          0_1() -> 3
          0_1() -> 4
          double_0(2) -> 1
          double_1(2) -> 4
          half_0(2) -> 1
          half_1(2) -> 3
          if_0(2,2,2) -> 1
          s_0(2) -> 1
          s_0(2) -> 2
          s_1(3) -> 1
          s_1(3) -> 3
          s_1(3) -> 4
          s_1(4) -> 3
          2 -> 1
** Step 1.b:2: EmptyProcessor WORST_CASE(?,O(1))
    + Considered Problem:
        - Weak TRS:
            -(x,0()) -> x
            -(s(x),s(y)) -> -(x,y)
            double(0()) -> 0()
            double(s(x)) -> s(s(double(x)))
            half(0()) -> 0()
            half(double(x)) -> x
            half(s(0())) -> 0()
            half(s(s(x))) -> s(half(x))
            if(0(),y,z) -> y
            if(s(x),y,z) -> z
        - Signature:
            {-/2,double/1,half/1,if/3} / {0/0,s/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {-,double,half,if} and constructors {0,s}
    + Applied Processor:
        EmptyProcessor
    + Details:
        The problem is already closed. The intended complexity is O(1).

WORST_CASE(Omega(n^1),O(n^1))