WORST_CASE(Omega(n^1),?) * Step 1: Sum. WORST_CASE(Omega(n^1),?) + Considered Problem: - Strict TRS: CONV(z0) -> c8(CONVITER(z0,cons(0(),nil()))) CONVITER(z0,z1) -> c9(IF(zero(z0),z0,z1),ZERO(z0)) HALF(0()) -> c() HALF(s(0())) -> c1() HALF(s(s(z0))) -> c2(HALF(z0)) IF(false(),z0,z1) -> c11(CONVITER(half(z0),cons(lastbit(z0),z1)),HALF(z0)) IF(false(),z0,z1) -> c12(CONVITER(half(z0),cons(lastbit(z0),z1)),LASTBIT(z0)) IF(true(),z0,z1) -> c10() LASTBIT(0()) -> c3() LASTBIT(s(0())) -> c4() LASTBIT(s(s(z0))) -> c5(LASTBIT(z0)) ZERO(0()) -> c6() ZERO(s(z0)) -> c7() - Weak TRS: conv(z0) -> conviter(z0,cons(0(),nil())) conviter(z0,z1) -> if(zero(z0),z0,z1) half(0()) -> 0() half(s(0())) -> 0() half(s(s(z0))) -> s(half(z0)) if(false(),z0,z1) -> conviter(half(z0),cons(lastbit(z0),z1)) if(true(),z0,z1) -> z1 lastbit(0()) -> 0() lastbit(s(0())) -> s(0()) lastbit(s(s(z0))) -> lastbit(z0) zero(0()) -> true() zero(s(z0)) -> false() - Signature: {CONV/1,CONVITER/2,HALF/1,IF/3,LASTBIT/1,ZERO/1,conv/1,conviter/2,half/1,if/3,lastbit/1,zero/1} / {0/0,c/0 ,c1/0,c10/0,c11/2,c12/2,c2/1,c3/0,c4/0,c5/1,c6/0,c7/0,c8/1,c9/2,cons/2,false/0,nil/0,s/1,true/0} - Obligation: innermost runtime complexity wrt. defined symbols {CONV,CONVITER,HALF,IF,LASTBIT,ZERO,conv,conviter,half,if ,lastbit,zero} and constructors {0,c,c1,c10,c11,c12,c2,c3,c4,c5,c6,c7,c8,c9,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: CONV(z0) -> c8(CONVITER(z0,cons(0(),nil()))) CONVITER(z0,z1) -> c9(IF(zero(z0),z0,z1),ZERO(z0)) HALF(0()) -> c() HALF(s(0())) -> c1() HALF(s(s(z0))) -> c2(HALF(z0)) IF(false(),z0,z1) -> c11(CONVITER(half(z0),cons(lastbit(z0),z1)),HALF(z0)) IF(false(),z0,z1) -> c12(CONVITER(half(z0),cons(lastbit(z0),z1)),LASTBIT(z0)) IF(true(),z0,z1) -> c10() LASTBIT(0()) -> c3() LASTBIT(s(0())) -> c4() LASTBIT(s(s(z0))) -> c5(LASTBIT(z0)) ZERO(0()) -> c6() ZERO(s(z0)) -> c7() - Weak TRS: conv(z0) -> conviter(z0,cons(0(),nil())) conviter(z0,z1) -> if(zero(z0),z0,z1) half(0()) -> 0() half(s(0())) -> 0() half(s(s(z0))) -> s(half(z0)) if(false(),z0,z1) -> conviter(half(z0),cons(lastbit(z0),z1)) if(true(),z0,z1) -> z1 lastbit(0()) -> 0() lastbit(s(0())) -> s(0()) lastbit(s(s(z0))) -> lastbit(z0) zero(0()) -> true() zero(s(z0)) -> false() - Signature: {CONV/1,CONVITER/2,HALF/1,IF/3,LASTBIT/1,ZERO/1,conv/1,conviter/2,half/1,if/3,lastbit/1,zero/1} / {0/0,c/0 ,c1/0,c10/0,c11/2,c12/2,c2/1,c3/0,c4/0,c5/1,c6/0,c7/0,c8/1,c9/2,cons/2,false/0,nil/0,s/1,true/0} - Obligation: innermost runtime complexity wrt. defined symbols {CONV,CONVITER,HALF,IF,LASTBIT,ZERO,conv,conviter,half,if ,lastbit,zero} and constructors {0,c,c1,c10,c11,c12,c2,c3,c4,c5,c6,c7,c8,c9,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: CONV(z0) -> c8(CONVITER(z0,cons(0(),nil()))) CONVITER(z0,z1) -> c9(IF(zero(z0),z0,z1),ZERO(z0)) HALF(0()) -> c() HALF(s(0())) -> c1() HALF(s(s(z0))) -> c2(HALF(z0)) IF(false(),z0,z1) -> c11(CONVITER(half(z0),cons(lastbit(z0),z1)),HALF(z0)) IF(false(),z0,z1) -> c12(CONVITER(half(z0),cons(lastbit(z0),z1)),LASTBIT(z0)) IF(true(),z0,z1) -> c10() LASTBIT(0()) -> c3() LASTBIT(s(0())) -> c4() LASTBIT(s(s(z0))) -> c5(LASTBIT(z0)) ZERO(0()) -> c6() ZERO(s(z0)) -> c7() - Weak TRS: conv(z0) -> conviter(z0,cons(0(),nil())) conviter(z0,z1) -> if(zero(z0),z0,z1) half(0()) -> 0() half(s(0())) -> 0() half(s(s(z0))) -> s(half(z0)) if(false(),z0,z1) -> conviter(half(z0),cons(lastbit(z0),z1)) if(true(),z0,z1) -> z1 lastbit(0()) -> 0() lastbit(s(0())) -> s(0()) lastbit(s(s(z0))) -> lastbit(z0) zero(0()) -> true() zero(s(z0)) -> false() - Signature: {CONV/1,CONVITER/2,HALF/1,IF/3,LASTBIT/1,ZERO/1,conv/1,conviter/2,half/1,if/3,lastbit/1,zero/1} / {0/0,c/0 ,c1/0,c10/0,c11/2,c12/2,c2/1,c3/0,c4/0,c5/1,c6/0,c7/0,c8/1,c9/2,cons/2,false/0,nil/0,s/1,true/0} - Obligation: innermost runtime complexity wrt. defined symbols {CONV,CONVITER,HALF,IF,LASTBIT,ZERO,conv,conviter,half,if ,lastbit,zero} and constructors {0,c,c1,c10,c11,c12,c2,c3,c4,c5,c6,c7,c8,c9,cons,false,nil,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),?)