WORST_CASE(Omega(n^1),?) * Step 1: Sum. WORST_CASE(Omega(n^1),?) + Considered Problem: - Strict TRS: A__PLUS(z0,z1) -> c7() A__PLUS(z0,0()) -> c5(MARK(z0)) A__PLUS(z0,s(z1)) -> c6(A__U11(tt(),z1,z0)) A__U11(z0,z1,z2) -> c1() A__U11(tt(),z0,z1) -> c(A__U12(tt(),z0,z1)) A__U12(z0,z1,z2) -> c4() A__U12(tt(),z0,z1) -> c2(A__PLUS(mark(z1),mark(z0)),MARK(z1)) A__U12(tt(),z0,z1) -> c3(A__PLUS(mark(z1),mark(z0)),MARK(z0)) MARK(0()) -> c14() MARK(U11(z0,z1,z2)) -> c8(A__U11(mark(z0),z1,z2),MARK(z0)) MARK(U12(z0,z1,z2)) -> c9(A__U12(mark(z0),z1,z2),MARK(z0)) MARK(plus(z0,z1)) -> c10(A__PLUS(mark(z0),mark(z1)),MARK(z0)) MARK(plus(z0,z1)) -> c11(A__PLUS(mark(z0),mark(z1)),MARK(z1)) MARK(s(z0)) -> c13(MARK(z0)) MARK(tt()) -> c12() - Weak TRS: a__U11(z0,z1,z2) -> U11(z0,z1,z2) a__U11(tt(),z0,z1) -> a__U12(tt(),z0,z1) a__U12(z0,z1,z2) -> U12(z0,z1,z2) a__U12(tt(),z0,z1) -> s(a__plus(mark(z1),mark(z0))) a__plus(z0,z1) -> plus(z0,z1) a__plus(z0,0()) -> mark(z0) a__plus(z0,s(z1)) -> a__U11(tt(),z1,z0) mark(0()) -> 0() mark(U11(z0,z1,z2)) -> a__U11(mark(z0),z1,z2) mark(U12(z0,z1,z2)) -> a__U12(mark(z0),z1,z2) mark(plus(z0,z1)) -> a__plus(mark(z0),mark(z1)) mark(s(z0)) -> s(mark(z0)) mark(tt()) -> tt() - Signature: {A__PLUS/2,A__U11/3,A__U12/3,MARK/1,a__U11/3,a__U12/3,a__plus/2,mark/1} / {0/0,U11/3,U12/3,c/1,c1/0,c10/2 ,c11/2,c12/0,c13/1,c14/0,c2/2,c3/2,c4/0,c5/1,c6/1,c7/0,c8/2,c9/2,plus/2,s/1,tt/0} - Obligation: innermost runtime complexity wrt. defined symbols {A__PLUS,A__U11,A__U12,MARK,a__U11,a__U12,a__plus ,mark} and constructors {0,U11,U12,c,c1,c10,c11,c12,c13,c14,c2,c3,c4,c5,c6,c7,c8,c9,plus,s,tt} + Applied Processor: Sum {left = someStrategy, right = someStrategy} + Details: () * Step 2: Sum. WORST_CASE(Omega(n^1),?) + Considered Problem: - Strict TRS: A__PLUS(z0,z1) -> c7() A__PLUS(z0,0()) -> c5(MARK(z0)) A__PLUS(z0,s(z1)) -> c6(A__U11(tt(),z1,z0)) A__U11(z0,z1,z2) -> c1() A__U11(tt(),z0,z1) -> c(A__U12(tt(),z0,z1)) A__U12(z0,z1,z2) -> c4() A__U12(tt(),z0,z1) -> c2(A__PLUS(mark(z1),mark(z0)),MARK(z1)) A__U12(tt(),z0,z1) -> c3(A__PLUS(mark(z1),mark(z0)),MARK(z0)) MARK(0()) -> c14() MARK(U11(z0,z1,z2)) -> c8(A__U11(mark(z0),z1,z2),MARK(z0)) MARK(U12(z0,z1,z2)) -> c9(A__U12(mark(z0),z1,z2),MARK(z0)) MARK(plus(z0,z1)) -> c10(A__PLUS(mark(z0),mark(z1)),MARK(z0)) MARK(plus(z0,z1)) -> c11(A__PLUS(mark(z0),mark(z1)),MARK(z1)) MARK(s(z0)) -> c13(MARK(z0)) MARK(tt()) -> c12() - Weak TRS: a__U11(z0,z1,z2) -> U11(z0,z1,z2) a__U11(tt(),z0,z1) -> a__U12(tt(),z0,z1) a__U12(z0,z1,z2) -> U12(z0,z1,z2) a__U12(tt(),z0,z1) -> s(a__plus(mark(z1),mark(z0))) a__plus(z0,z1) -> plus(z0,z1) a__plus(z0,0()) -> mark(z0) a__plus(z0,s(z1)) -> a__U11(tt(),z1,z0) mark(0()) -> 0() mark(U11(z0,z1,z2)) -> a__U11(mark(z0),z1,z2) mark(U12(z0,z1,z2)) -> a__U12(mark(z0),z1,z2) mark(plus(z0,z1)) -> a__plus(mark(z0),mark(z1)) mark(s(z0)) -> s(mark(z0)) mark(tt()) -> tt() - Signature: {A__PLUS/2,A__U11/3,A__U12/3,MARK/1,a__U11/3,a__U12/3,a__plus/2,mark/1} / {0/0,U11/3,U12/3,c/1,c1/0,c10/2 ,c11/2,c12/0,c13/1,c14/0,c2/2,c3/2,c4/0,c5/1,c6/1,c7/0,c8/2,c9/2,plus/2,s/1,tt/0} - Obligation: innermost runtime complexity wrt. defined symbols {A__PLUS,A__U11,A__U12,MARK,a__U11,a__U12,a__plus ,mark} and constructors {0,U11,U12,c,c1,c10,c11,c12,c13,c14,c2,c3,c4,c5,c6,c7,c8,c9,plus,s,tt} + Applied Processor: Sum {left = someStrategy, right = someStrategy} + Details: () * Step 3: DecreasingLoops. WORST_CASE(Omega(n^1),?) + Considered Problem: - Strict TRS: A__PLUS(z0,z1) -> c7() A__PLUS(z0,0()) -> c5(MARK(z0)) A__PLUS(z0,s(z1)) -> c6(A__U11(tt(),z1,z0)) A__U11(z0,z1,z2) -> c1() A__U11(tt(),z0,z1) -> c(A__U12(tt(),z0,z1)) A__U12(z0,z1,z2) -> c4() A__U12(tt(),z0,z1) -> c2(A__PLUS(mark(z1),mark(z0)),MARK(z1)) A__U12(tt(),z0,z1) -> c3(A__PLUS(mark(z1),mark(z0)),MARK(z0)) MARK(0()) -> c14() MARK(U11(z0,z1,z2)) -> c8(A__U11(mark(z0),z1,z2),MARK(z0)) MARK(U12(z0,z1,z2)) -> c9(A__U12(mark(z0),z1,z2),MARK(z0)) MARK(plus(z0,z1)) -> c10(A__PLUS(mark(z0),mark(z1)),MARK(z0)) MARK(plus(z0,z1)) -> c11(A__PLUS(mark(z0),mark(z1)),MARK(z1)) MARK(s(z0)) -> c13(MARK(z0)) MARK(tt()) -> c12() - Weak TRS: a__U11(z0,z1,z2) -> U11(z0,z1,z2) a__U11(tt(),z0,z1) -> a__U12(tt(),z0,z1) a__U12(z0,z1,z2) -> U12(z0,z1,z2) a__U12(tt(),z0,z1) -> s(a__plus(mark(z1),mark(z0))) a__plus(z0,z1) -> plus(z0,z1) a__plus(z0,0()) -> mark(z0) a__plus(z0,s(z1)) -> a__U11(tt(),z1,z0) mark(0()) -> 0() mark(U11(z0,z1,z2)) -> a__U11(mark(z0),z1,z2) mark(U12(z0,z1,z2)) -> a__U12(mark(z0),z1,z2) mark(plus(z0,z1)) -> a__plus(mark(z0),mark(z1)) mark(s(z0)) -> s(mark(z0)) mark(tt()) -> tt() - Signature: {A__PLUS/2,A__U11/3,A__U12/3,MARK/1,a__U11/3,a__U12/3,a__plus/2,mark/1} / {0/0,U11/3,U12/3,c/1,c1/0,c10/2 ,c11/2,c12/0,c13/1,c14/0,c2/2,c3/2,c4/0,c5/1,c6/1,c7/0,c8/2,c9/2,plus/2,s/1,tt/0} - Obligation: innermost runtime complexity wrt. defined symbols {A__PLUS,A__U11,A__U12,MARK,a__U11,a__U12,a__plus ,mark} and constructors {0,U11,U12,c,c1,c10,c11,c12,c13,c14,c2,c3,c4,c5,c6,c7,c8,c9,plus,s,tt} + Applied Processor: DecreasingLoops {bound = AnyLoop, narrow = 10} + Details: The system has following decreasing Loops: MARK(x){x -> U11(x,y,z)} = MARK(U11(x,y,z)) ->^+ c8(A__U11(mark(x),y,z),MARK(x)) = C[MARK(x) = MARK(x){}] WORST_CASE(Omega(n^1),?)