tct-trs: Prelude.head: empty list
tct-trs: Prelude.head: empty list
tct-trs: Prelude.head: empty list
tct-trs: Prelude.head: empty list
WORST_CASE(Omega(n^1),?)
* Step 1: Sum.  WORST_CASE(Omega(n^1),?)
    + Considered Problem:
        - Strict TRS:
            F() -> c16()
            F() -> c17()
            HALF(0()) -> c()
            HALF(s(0())) -> c1()
            HALF(s(s(z0))) -> c2(HALF(z0))
            IF(false(),z0,z1,z2) -> c13()
            IF(true(),false(),z0,s(z1)) -> c14()
            IF(true(),true(),z0,z1) -> c15(LOGITER(z0,z1))
            INC(0()) -> c7()
            INC(s(z0)) -> c6(INC(z0))
            LE(0(),z0) -> c3()
            LE(s(z0),0()) -> c4()
            LE(s(z0),s(z1)) -> c5(LE(z0,z1))
            LOGARITHM(z0) -> c8(LOGITER(z0,0()))
            LOGITER(z0,z1) -> c10(IF(le(s(0()),z0),le(s(s(0())),z0),half(z0),inc(z1)),LE(s(s(0())),z0))
            LOGITER(z0,z1) -> c11(IF(le(s(0()),z0),le(s(s(0())),z0),half(z0),inc(z1)),HALF(z0))
            LOGITER(z0,z1) -> c12(IF(le(s(0()),z0),le(s(s(0())),z0),half(z0),inc(z1)),INC(z1))
            LOGITER(z0,z1) -> c9(IF(le(s(0()),z0),le(s(s(0())),z0),half(z0),inc(z1)),LE(s(0()),z0))
        - Weak TRS:
            f() -> g()
            f() -> h()
            half(0()) -> 0()
            half(s(0())) -> 0()
            half(s(s(z0))) -> s(half(z0))
            if(false(),z0,z1,z2) -> logZeroError()
            if(true(),false(),z0,s(z1)) -> z1
            if(true(),true(),z0,z1) -> logIter(z0,z1)
            inc(0()) -> s(0())
            inc(s(z0)) -> s(inc(z0))
            le(0(),z0) -> true()
            le(s(z0),0()) -> false()
            le(s(z0),s(z1)) -> le(z0,z1)
            logIter(z0,z1) -> if(le(s(0()),z0),le(s(s(0())),z0),half(z0),inc(z1))
            logarithm(z0) -> logIter(z0,0())
        - Signature:
            {F/0,HALF/1,IF/4,INC/1,LE/2,LOGARITHM/1,LOGITER/2,f/0,half/1,if/4,inc/1,le/2,logIter/2,logarithm/1} / {0/0
            ,c/0,c1/0,c10/2,c11/2,c12/2,c13/0,c14/0,c15/1,c16/0,c17/0,c2/1,c3/0,c4/0,c5/1,c6/1,c7/0,c8/1,c9/2,false/0
            ,g/0,h/0,logZeroError/0,s/1,true/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {F,HALF,IF,INC,LE,LOGARITHM,LOGITER,f,half,if,inc,le
            ,logIter,logarithm} and constructors {0,c,c1,c10,c11,c12,c13,c14,c15,c16,c17,c2,c3,c4,c5,c6,c7,c8,c9,false,g
            ,h,logZeroError,s,true}
    + Applied Processor:
        Sum {left = someStrategy, right = someStrategy}
    + Details:
        ()
* Step 2: Sum.  WORST_CASE(Omega(n^1),?)
    + Considered Problem:
        - Strict TRS:
            F() -> c16()
            F() -> c17()
            HALF(0()) -> c()
            HALF(s(0())) -> c1()
            HALF(s(s(z0))) -> c2(HALF(z0))
            IF(false(),z0,z1,z2) -> c13()
            IF(true(),false(),z0,s(z1)) -> c14()
            IF(true(),true(),z0,z1) -> c15(LOGITER(z0,z1))
            INC(0()) -> c7()
            INC(s(z0)) -> c6(INC(z0))
            LE(0(),z0) -> c3()
            LE(s(z0),0()) -> c4()
            LE(s(z0),s(z1)) -> c5(LE(z0,z1))
            LOGARITHM(z0) -> c8(LOGITER(z0,0()))
            LOGITER(z0,z1) -> c10(IF(le(s(0()),z0),le(s(s(0())),z0),half(z0),inc(z1)),LE(s(s(0())),z0))
            LOGITER(z0,z1) -> c11(IF(le(s(0()),z0),le(s(s(0())),z0),half(z0),inc(z1)),HALF(z0))
            LOGITER(z0,z1) -> c12(IF(le(s(0()),z0),le(s(s(0())),z0),half(z0),inc(z1)),INC(z1))
            LOGITER(z0,z1) -> c9(IF(le(s(0()),z0),le(s(s(0())),z0),half(z0),inc(z1)),LE(s(0()),z0))
        - Weak TRS:
            f() -> g()
            f() -> h()
            half(0()) -> 0()
            half(s(0())) -> 0()
            half(s(s(z0))) -> s(half(z0))
            if(false(),z0,z1,z2) -> logZeroError()
            if(true(),false(),z0,s(z1)) -> z1
            if(true(),true(),z0,z1) -> logIter(z0,z1)
            inc(0()) -> s(0())
            inc(s(z0)) -> s(inc(z0))
            le(0(),z0) -> true()
            le(s(z0),0()) -> false()
            le(s(z0),s(z1)) -> le(z0,z1)
            logIter(z0,z1) -> if(le(s(0()),z0),le(s(s(0())),z0),half(z0),inc(z1))
            logarithm(z0) -> logIter(z0,0())
        - Signature:
            {F/0,HALF/1,IF/4,INC/1,LE/2,LOGARITHM/1,LOGITER/2,f/0,half/1,if/4,inc/1,le/2,logIter/2,logarithm/1} / {0/0
            ,c/0,c1/0,c10/2,c11/2,c12/2,c13/0,c14/0,c15/1,c16/0,c17/0,c2/1,c3/0,c4/0,c5/1,c6/1,c7/0,c8/1,c9/2,false/0
            ,g/0,h/0,logZeroError/0,s/1,true/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {F,HALF,IF,INC,LE,LOGARITHM,LOGITER,f,half,if,inc,le
            ,logIter,logarithm} and constructors {0,c,c1,c10,c11,c12,c13,c14,c15,c16,c17,c2,c3,c4,c5,c6,c7,c8,c9,false,g
            ,h,logZeroError,s,true}
    + Applied Processor:
        Sum {left = someStrategy, right = someStrategy}
    + Details:
        ()
* Step 3: DecreasingLoops.  WORST_CASE(Omega(n^1),?)
    + Considered Problem:
        - Strict TRS:
            F() -> c16()
            F() -> c17()
            HALF(0()) -> c()
            HALF(s(0())) -> c1()
            HALF(s(s(z0))) -> c2(HALF(z0))
            IF(false(),z0,z1,z2) -> c13()
            IF(true(),false(),z0,s(z1)) -> c14()
            IF(true(),true(),z0,z1) -> c15(LOGITER(z0,z1))
            INC(0()) -> c7()
            INC(s(z0)) -> c6(INC(z0))
            LE(0(),z0) -> c3()
            LE(s(z0),0()) -> c4()
            LE(s(z0),s(z1)) -> c5(LE(z0,z1))
            LOGARITHM(z0) -> c8(LOGITER(z0,0()))
            LOGITER(z0,z1) -> c10(IF(le(s(0()),z0),le(s(s(0())),z0),half(z0),inc(z1)),LE(s(s(0())),z0))
            LOGITER(z0,z1) -> c11(IF(le(s(0()),z0),le(s(s(0())),z0),half(z0),inc(z1)),HALF(z0))
            LOGITER(z0,z1) -> c12(IF(le(s(0()),z0),le(s(s(0())),z0),half(z0),inc(z1)),INC(z1))
            LOGITER(z0,z1) -> c9(IF(le(s(0()),z0),le(s(s(0())),z0),half(z0),inc(z1)),LE(s(0()),z0))
        - Weak TRS:
            f() -> g()
            f() -> h()
            half(0()) -> 0()
            half(s(0())) -> 0()
            half(s(s(z0))) -> s(half(z0))
            if(false(),z0,z1,z2) -> logZeroError()
            if(true(),false(),z0,s(z1)) -> z1
            if(true(),true(),z0,z1) -> logIter(z0,z1)
            inc(0()) -> s(0())
            inc(s(z0)) -> s(inc(z0))
            le(0(),z0) -> true()
            le(s(z0),0()) -> false()
            le(s(z0),s(z1)) -> le(z0,z1)
            logIter(z0,z1) -> if(le(s(0()),z0),le(s(s(0())),z0),half(z0),inc(z1))
            logarithm(z0) -> logIter(z0,0())
        - Signature:
            {F/0,HALF/1,IF/4,INC/1,LE/2,LOGARITHM/1,LOGITER/2,f/0,half/1,if/4,inc/1,le/2,logIter/2,logarithm/1} / {0/0
            ,c/0,c1/0,c10/2,c11/2,c12/2,c13/0,c14/0,c15/1,c16/0,c17/0,c2/1,c3/0,c4/0,c5/1,c6/1,c7/0,c8/1,c9/2,false/0
            ,g/0,h/0,logZeroError/0,s/1,true/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {F,HALF,IF,INC,LE,LOGARITHM,LOGITER,f,half,if,inc,le
            ,logIter,logarithm} and constructors {0,c,c1,c10,c11,c12,c13,c14,c15,c16,c17,c2,c3,c4,c5,c6,c7,c8,c9,false,g
            ,h,logZeroError,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),?)