WORST_CASE(Omega(n^1),?)
* Step 1: Sum.  WORST_CASE(Omega(n^1),?)
    + Considered Problem:
        - Strict TRS:
            HALF(0()) -> c()
            HALF(s(0())) -> c1()
            HALF(s(s(z0))) -> c2(HALF(z0))
            IF(false(),z0,z1) -> c12(LOG2(half(z0),z1),HALF(z0))
            IF(true(),z0,s(z1)) -> c11()
            INC(0()) -> c6()
            INC(s(z0)) -> c7(INC(z0))
            LE(0(),z0) -> c3()
            LE(s(z0),0()) -> c4()
            LE(s(z0),s(z1)) -> c5(LE(z0,z1))
            LOG(z0) -> c8(LOG2(z0,0()))
            LOG2(z0,z1) -> c10(IF(le(z0,s(0())),z0,inc(z1)),INC(z1))
            LOG2(z0,z1) -> c9(IF(le(z0,s(0())),z0,inc(z1)),LE(z0,s(0())))
        - Weak TRS:
            half(0()) -> 0()
            half(s(0())) -> 0()
            half(s(s(z0))) -> s(half(z0))
            if(false(),z0,z1) -> log2(half(z0),z1)
            if(true(),z0,s(z1)) -> z1
            inc(0()) -> 0()
            inc(s(z0)) -> s(inc(z0))
            le(0(),z0) -> true()
            le(s(z0),0()) -> false()
            le(s(z0),s(z1)) -> le(z0,z1)
            log(z0) -> log2(z0,0())
            log2(z0,z1) -> if(le(z0,s(0())),z0,inc(z1))
        - Signature:
            {HALF/1,IF/3,INC/1,LE/2,LOG/1,LOG2/2,half/1,if/3,inc/1,le/2,log/1,log2/2} / {0/0,c/0,c1/0,c10/2,c11/0,c12/2
            ,c2/1,c3/0,c4/0,c5/1,c6/0,c7/1,c8/1,c9/2,false/0,s/1,true/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {HALF,IF,INC,LE,LOG,LOG2,half,if,inc,le,log
            ,log2} and constructors {0,c,c1,c10,c11,c12,c2,c3,c4,c5,c6,c7,c8,c9,false,s,true}
    + Applied Processor:
        Sum {left = someStrategy, right = someStrategy}
    + Details:
        ()
* Step 2: Sum.  WORST_CASE(Omega(n^1),?)
    + Considered Problem:
        - Strict TRS:
            HALF(0()) -> c()
            HALF(s(0())) -> c1()
            HALF(s(s(z0))) -> c2(HALF(z0))
            IF(false(),z0,z1) -> c12(LOG2(half(z0),z1),HALF(z0))
            IF(true(),z0,s(z1)) -> c11()
            INC(0()) -> c6()
            INC(s(z0)) -> c7(INC(z0))
            LE(0(),z0) -> c3()
            LE(s(z0),0()) -> c4()
            LE(s(z0),s(z1)) -> c5(LE(z0,z1))
            LOG(z0) -> c8(LOG2(z0,0()))
            LOG2(z0,z1) -> c10(IF(le(z0,s(0())),z0,inc(z1)),INC(z1))
            LOG2(z0,z1) -> c9(IF(le(z0,s(0())),z0,inc(z1)),LE(z0,s(0())))
        - Weak TRS:
            half(0()) -> 0()
            half(s(0())) -> 0()
            half(s(s(z0))) -> s(half(z0))
            if(false(),z0,z1) -> log2(half(z0),z1)
            if(true(),z0,s(z1)) -> z1
            inc(0()) -> 0()
            inc(s(z0)) -> s(inc(z0))
            le(0(),z0) -> true()
            le(s(z0),0()) -> false()
            le(s(z0),s(z1)) -> le(z0,z1)
            log(z0) -> log2(z0,0())
            log2(z0,z1) -> if(le(z0,s(0())),z0,inc(z1))
        - Signature:
            {HALF/1,IF/3,INC/1,LE/2,LOG/1,LOG2/2,half/1,if/3,inc/1,le/2,log/1,log2/2} / {0/0,c/0,c1/0,c10/2,c11/0,c12/2
            ,c2/1,c3/0,c4/0,c5/1,c6/0,c7/1,c8/1,c9/2,false/0,s/1,true/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {HALF,IF,INC,LE,LOG,LOG2,half,if,inc,le,log
            ,log2} and constructors {0,c,c1,c10,c11,c12,c2,c3,c4,c5,c6,c7,c8,c9,false,s,true}
    + Applied Processor:
        Sum {left = someStrategy, right = someStrategy}
    + Details:
        ()
* Step 3: DecreasingLoops.  WORST_CASE(Omega(n^1),?)
    + Considered Problem:
        - Strict TRS:
            HALF(0()) -> c()
            HALF(s(0())) -> c1()
            HALF(s(s(z0))) -> c2(HALF(z0))
            IF(false(),z0,z1) -> c12(LOG2(half(z0),z1),HALF(z0))
            IF(true(),z0,s(z1)) -> c11()
            INC(0()) -> c6()
            INC(s(z0)) -> c7(INC(z0))
            LE(0(),z0) -> c3()
            LE(s(z0),0()) -> c4()
            LE(s(z0),s(z1)) -> c5(LE(z0,z1))
            LOG(z0) -> c8(LOG2(z0,0()))
            LOG2(z0,z1) -> c10(IF(le(z0,s(0())),z0,inc(z1)),INC(z1))
            LOG2(z0,z1) -> c9(IF(le(z0,s(0())),z0,inc(z1)),LE(z0,s(0())))
        - Weak TRS:
            half(0()) -> 0()
            half(s(0())) -> 0()
            half(s(s(z0))) -> s(half(z0))
            if(false(),z0,z1) -> log2(half(z0),z1)
            if(true(),z0,s(z1)) -> z1
            inc(0()) -> 0()
            inc(s(z0)) -> s(inc(z0))
            le(0(),z0) -> true()
            le(s(z0),0()) -> false()
            le(s(z0),s(z1)) -> le(z0,z1)
            log(z0) -> log2(z0,0())
            log2(z0,z1) -> if(le(z0,s(0())),z0,inc(z1))
        - Signature:
            {HALF/1,IF/3,INC/1,LE/2,LOG/1,LOG2/2,half/1,if/3,inc/1,le/2,log/1,log2/2} / {0/0,c/0,c1/0,c10/2,c11/0,c12/2
            ,c2/1,c3/0,c4/0,c5/1,c6/0,c7/1,c8/1,c9/2,false/0,s/1,true/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {HALF,IF,INC,LE,LOG,LOG2,half,if,inc,le,log
            ,log2} and constructors {0,c,c1,c10,c11,c12,c2,c3,c4,c5,c6,c7,c8,c9,false,s,true}
    + Applied Processor:
        DecreasingLoops {bound = AnyLoop, narrow = 10}
    + Details:
        The system has following decreasing Loops:
          HALF(x){x -> s(s(x))} =
            HALF(s(s(x))) ->^+ c2(HALF(x))
              = C[HALF(x) = HALF(x){}]

WORST_CASE(Omega(n^1),?)