WORST_CASE(Omega(n^1),?) * Step 1: Sum. WORST_CASE(Omega(n^1),?) + Considered Problem: - Strict TRS: A__ISNAT(z0) -> c16() A__ISNAT(0()) -> c13() A__ISNAT(plus(z0,z1)) -> c14(A__U11(a__isNat(z0),z1),A__ISNAT(z0)) A__ISNAT(s(z0)) -> c15(A__U21(a__isNat(z0)),A__ISNAT(z0)) A__PLUS(z0,z1) -> c19() A__PLUS(z0,0()) -> c17(A__U31(a__isNat(z0),z0),A__ISNAT(z0)) A__PLUS(z0,s(z1)) -> c18(A__U41(a__isNat(z1),z1,z0),A__ISNAT(z1)) A__U11(z0,z1) -> c1() A__U11(tt(),z0) -> c(A__U12(a__isNat(z0)),A__ISNAT(z0)) A__U12(z0) -> c3() A__U12(tt()) -> c2() A__U21(z0) -> c5() A__U21(tt()) -> c4() A__U31(z0,z1) -> c7() A__U31(tt(),z0) -> c6(MARK(z0)) A__U41(z0,z1,z2) -> c9() A__U41(tt(),z0,z1) -> c8(A__U42(a__isNat(z1),z0,z1),A__ISNAT(z1)) A__U42(z0,z1,z2) -> c12() A__U42(tt(),z0,z1) -> c10(A__PLUS(mark(z1),mark(z0)),MARK(z1)) A__U42(tt(),z0,z1) -> c11(A__PLUS(mark(z1),mark(z0)),MARK(z0)) MARK(0()) -> c31() MARK(U11(z0,z1)) -> c20(A__U11(mark(z0),z1),MARK(z0)) MARK(U12(z0)) -> c21(A__U12(mark(z0)),MARK(z0)) MARK(U21(z0)) -> c23(A__U21(mark(z0)),MARK(z0)) MARK(U31(z0,z1)) -> c24(A__U31(mark(z0),z1),MARK(z0)) MARK(U41(z0,z1,z2)) -> c25(A__U41(mark(z0),z1,z2),MARK(z0)) MARK(U42(z0,z1,z2)) -> c26(A__U42(mark(z0),z1,z2),MARK(z0)) MARK(isNat(z0)) -> c22(A__ISNAT(z0)) MARK(plus(z0,z1)) -> c27(A__PLUS(mark(z0),mark(z1)),MARK(z0)) MARK(plus(z0,z1)) -> c28(A__PLUS(mark(z0),mark(z1)),MARK(z1)) MARK(s(z0)) -> c30(MARK(z0)) MARK(tt()) -> c29() - Weak TRS: a__U11(z0,z1) -> U11(z0,z1) a__U11(tt(),z0) -> a__U12(a__isNat(z0)) a__U12(z0) -> U12(z0) a__U12(tt()) -> tt() a__U21(z0) -> U21(z0) a__U21(tt()) -> tt() a__U31(z0,z1) -> U31(z0,z1) a__U31(tt(),z0) -> mark(z0) a__U41(z0,z1,z2) -> U41(z0,z1,z2) a__U41(tt(),z0,z1) -> a__U42(a__isNat(z1),z0,z1) a__U42(z0,z1,z2) -> U42(z0,z1,z2) a__U42(tt(),z0,z1) -> s(a__plus(mark(z1),mark(z0))) a__isNat(z0) -> isNat(z0) a__isNat(0()) -> tt() a__isNat(plus(z0,z1)) -> a__U11(a__isNat(z0),z1) a__isNat(s(z0)) -> a__U21(a__isNat(z0)) a__plus(z0,z1) -> plus(z0,z1) a__plus(z0,0()) -> a__U31(a__isNat(z0),z0) a__plus(z0,s(z1)) -> a__U41(a__isNat(z1),z1,z0) mark(0()) -> 0() mark(U11(z0,z1)) -> a__U11(mark(z0),z1) mark(U12(z0)) -> a__U12(mark(z0)) mark(U21(z0)) -> a__U21(mark(z0)) mark(U31(z0,z1)) -> a__U31(mark(z0),z1) mark(U41(z0,z1,z2)) -> a__U41(mark(z0),z1,z2) mark(U42(z0,z1,z2)) -> a__U42(mark(z0),z1,z2) mark(isNat(z0)) -> a__isNat(z0) mark(plus(z0,z1)) -> a__plus(mark(z0),mark(z1)) mark(s(z0)) -> s(mark(z0)) mark(tt()) -> tt() - Signature: {A__ISNAT/1,A__PLUS/2,A__U11/2,A__U12/1,A__U21/1,A__U31/2,A__U41/3,A__U42/3,MARK/1,a__U11/2,a__U12/1 ,a__U21/1,a__U31/2,a__U41/3,a__U42/3,a__isNat/1,a__plus/2,mark/1} / {0/0,U11/2,U12/1,U21/1,U31/2,U41/3,U42/3 ,c/2,c1/0,c10/2,c11/2,c12/0,c13/0,c14/2,c15/2,c16/0,c17/2,c18/2,c19/0,c2/0,c20/2,c21/2,c22/1,c23/2,c24/2 ,c25/2,c26/2,c27/2,c28/2,c29/0,c3/0,c30/1,c31/0,c4/0,c5/0,c6/1,c7/0,c8/2,c9/0,isNat/1,plus/2,s/1,tt/0} - Obligation: innermost runtime complexity wrt. defined symbols {A__ISNAT,A__PLUS,A__U11,A__U12,A__U21,A__U31,A__U41 ,A__U42,MARK,a__U11,a__U12,a__U21,a__U31,a__U41,a__U42,a__isNat,a__plus,mark} and constructors {0,U11,U12 ,U21,U31,U41,U42,c,c1,c10,c11,c12,c13,c14,c15,c16,c17,c18,c19,c2,c20,c21,c22,c23,c24,c25,c26,c27,c28,c29,c3 ,c30,c31,c4,c5,c6,c7,c8,c9,isNat,plus,s,tt} + Applied Processor: Sum {left = someStrategy, right = someStrategy} + Details: () * Step 2: Sum. WORST_CASE(Omega(n^1),?) + Considered Problem: - Strict TRS: A__ISNAT(z0) -> c16() A__ISNAT(0()) -> c13() A__ISNAT(plus(z0,z1)) -> c14(A__U11(a__isNat(z0),z1),A__ISNAT(z0)) A__ISNAT(s(z0)) -> c15(A__U21(a__isNat(z0)),A__ISNAT(z0)) A__PLUS(z0,z1) -> c19() A__PLUS(z0,0()) -> c17(A__U31(a__isNat(z0),z0),A__ISNAT(z0)) A__PLUS(z0,s(z1)) -> c18(A__U41(a__isNat(z1),z1,z0),A__ISNAT(z1)) A__U11(z0,z1) -> c1() A__U11(tt(),z0) -> c(A__U12(a__isNat(z0)),A__ISNAT(z0)) A__U12(z0) -> c3() A__U12(tt()) -> c2() A__U21(z0) -> c5() A__U21(tt()) -> c4() A__U31(z0,z1) -> c7() A__U31(tt(),z0) -> c6(MARK(z0)) A__U41(z0,z1,z2) -> c9() A__U41(tt(),z0,z1) -> c8(A__U42(a__isNat(z1),z0,z1),A__ISNAT(z1)) A__U42(z0,z1,z2) -> c12() A__U42(tt(),z0,z1) -> c10(A__PLUS(mark(z1),mark(z0)),MARK(z1)) A__U42(tt(),z0,z1) -> c11(A__PLUS(mark(z1),mark(z0)),MARK(z0)) MARK(0()) -> c31() MARK(U11(z0,z1)) -> c20(A__U11(mark(z0),z1),MARK(z0)) MARK(U12(z0)) -> c21(A__U12(mark(z0)),MARK(z0)) MARK(U21(z0)) -> c23(A__U21(mark(z0)),MARK(z0)) MARK(U31(z0,z1)) -> c24(A__U31(mark(z0),z1),MARK(z0)) MARK(U41(z0,z1,z2)) -> c25(A__U41(mark(z0),z1,z2),MARK(z0)) MARK(U42(z0,z1,z2)) -> c26(A__U42(mark(z0),z1,z2),MARK(z0)) MARK(isNat(z0)) -> c22(A__ISNAT(z0)) MARK(plus(z0,z1)) -> c27(A__PLUS(mark(z0),mark(z1)),MARK(z0)) MARK(plus(z0,z1)) -> c28(A__PLUS(mark(z0),mark(z1)),MARK(z1)) MARK(s(z0)) -> c30(MARK(z0)) MARK(tt()) -> c29() - Weak TRS: a__U11(z0,z1) -> U11(z0,z1) a__U11(tt(),z0) -> a__U12(a__isNat(z0)) a__U12(z0) -> U12(z0) a__U12(tt()) -> tt() a__U21(z0) -> U21(z0) a__U21(tt()) -> tt() a__U31(z0,z1) -> U31(z0,z1) a__U31(tt(),z0) -> mark(z0) a__U41(z0,z1,z2) -> U41(z0,z1,z2) a__U41(tt(),z0,z1) -> a__U42(a__isNat(z1),z0,z1) a__U42(z0,z1,z2) -> U42(z0,z1,z2) a__U42(tt(),z0,z1) -> s(a__plus(mark(z1),mark(z0))) a__isNat(z0) -> isNat(z0) a__isNat(0()) -> tt() a__isNat(plus(z0,z1)) -> a__U11(a__isNat(z0),z1) a__isNat(s(z0)) -> a__U21(a__isNat(z0)) a__plus(z0,z1) -> plus(z0,z1) a__plus(z0,0()) -> a__U31(a__isNat(z0),z0) a__plus(z0,s(z1)) -> a__U41(a__isNat(z1),z1,z0) mark(0()) -> 0() mark(U11(z0,z1)) -> a__U11(mark(z0),z1) mark(U12(z0)) -> a__U12(mark(z0)) mark(U21(z0)) -> a__U21(mark(z0)) mark(U31(z0,z1)) -> a__U31(mark(z0),z1) mark(U41(z0,z1,z2)) -> a__U41(mark(z0),z1,z2) mark(U42(z0,z1,z2)) -> a__U42(mark(z0),z1,z2) mark(isNat(z0)) -> a__isNat(z0) mark(plus(z0,z1)) -> a__plus(mark(z0),mark(z1)) mark(s(z0)) -> s(mark(z0)) mark(tt()) -> tt() - Signature: {A__ISNAT/1,A__PLUS/2,A__U11/2,A__U12/1,A__U21/1,A__U31/2,A__U41/3,A__U42/3,MARK/1,a__U11/2,a__U12/1 ,a__U21/1,a__U31/2,a__U41/3,a__U42/3,a__isNat/1,a__plus/2,mark/1} / {0/0,U11/2,U12/1,U21/1,U31/2,U41/3,U42/3 ,c/2,c1/0,c10/2,c11/2,c12/0,c13/0,c14/2,c15/2,c16/0,c17/2,c18/2,c19/0,c2/0,c20/2,c21/2,c22/1,c23/2,c24/2 ,c25/2,c26/2,c27/2,c28/2,c29/0,c3/0,c30/1,c31/0,c4/0,c5/0,c6/1,c7/0,c8/2,c9/0,isNat/1,plus/2,s/1,tt/0} - Obligation: innermost runtime complexity wrt. defined symbols {A__ISNAT,A__PLUS,A__U11,A__U12,A__U21,A__U31,A__U41 ,A__U42,MARK,a__U11,a__U12,a__U21,a__U31,a__U41,a__U42,a__isNat,a__plus,mark} and constructors {0,U11,U12 ,U21,U31,U41,U42,c,c1,c10,c11,c12,c13,c14,c15,c16,c17,c18,c19,c2,c20,c21,c22,c23,c24,c25,c26,c27,c28,c29,c3 ,c30,c31,c4,c5,c6,c7,c8,c9,isNat,plus,s,tt} + Applied Processor: Sum {left = someStrategy, right = someStrategy} + Details: () * Step 3: DecreasingLoops. WORST_CASE(Omega(n^1),?) + Considered Problem: - Strict TRS: A__ISNAT(z0) -> c16() A__ISNAT(0()) -> c13() A__ISNAT(plus(z0,z1)) -> c14(A__U11(a__isNat(z0),z1),A__ISNAT(z0)) A__ISNAT(s(z0)) -> c15(A__U21(a__isNat(z0)),A__ISNAT(z0)) A__PLUS(z0,z1) -> c19() A__PLUS(z0,0()) -> c17(A__U31(a__isNat(z0),z0),A__ISNAT(z0)) A__PLUS(z0,s(z1)) -> c18(A__U41(a__isNat(z1),z1,z0),A__ISNAT(z1)) A__U11(z0,z1) -> c1() A__U11(tt(),z0) -> c(A__U12(a__isNat(z0)),A__ISNAT(z0)) A__U12(z0) -> c3() A__U12(tt()) -> c2() A__U21(z0) -> c5() A__U21(tt()) -> c4() A__U31(z0,z1) -> c7() A__U31(tt(),z0) -> c6(MARK(z0)) A__U41(z0,z1,z2) -> c9() A__U41(tt(),z0,z1) -> c8(A__U42(a__isNat(z1),z0,z1),A__ISNAT(z1)) A__U42(z0,z1,z2) -> c12() A__U42(tt(),z0,z1) -> c10(A__PLUS(mark(z1),mark(z0)),MARK(z1)) A__U42(tt(),z0,z1) -> c11(A__PLUS(mark(z1),mark(z0)),MARK(z0)) MARK(0()) -> c31() MARK(U11(z0,z1)) -> c20(A__U11(mark(z0),z1),MARK(z0)) MARK(U12(z0)) -> c21(A__U12(mark(z0)),MARK(z0)) MARK(U21(z0)) -> c23(A__U21(mark(z0)),MARK(z0)) MARK(U31(z0,z1)) -> c24(A__U31(mark(z0),z1),MARK(z0)) MARK(U41(z0,z1,z2)) -> c25(A__U41(mark(z0),z1,z2),MARK(z0)) MARK(U42(z0,z1,z2)) -> c26(A__U42(mark(z0),z1,z2),MARK(z0)) MARK(isNat(z0)) -> c22(A__ISNAT(z0)) MARK(plus(z0,z1)) -> c27(A__PLUS(mark(z0),mark(z1)),MARK(z0)) MARK(plus(z0,z1)) -> c28(A__PLUS(mark(z0),mark(z1)),MARK(z1)) MARK(s(z0)) -> c30(MARK(z0)) MARK(tt()) -> c29() - Weak TRS: a__U11(z0,z1) -> U11(z0,z1) a__U11(tt(),z0) -> a__U12(a__isNat(z0)) a__U12(z0) -> U12(z0) a__U12(tt()) -> tt() a__U21(z0) -> U21(z0) a__U21(tt()) -> tt() a__U31(z0,z1) -> U31(z0,z1) a__U31(tt(),z0) -> mark(z0) a__U41(z0,z1,z2) -> U41(z0,z1,z2) a__U41(tt(),z0,z1) -> a__U42(a__isNat(z1),z0,z1) a__U42(z0,z1,z2) -> U42(z0,z1,z2) a__U42(tt(),z0,z1) -> s(a__plus(mark(z1),mark(z0))) a__isNat(z0) -> isNat(z0) a__isNat(0()) -> tt() a__isNat(plus(z0,z1)) -> a__U11(a__isNat(z0),z1) a__isNat(s(z0)) -> a__U21(a__isNat(z0)) a__plus(z0,z1) -> plus(z0,z1) a__plus(z0,0()) -> a__U31(a__isNat(z0),z0) a__plus(z0,s(z1)) -> a__U41(a__isNat(z1),z1,z0) mark(0()) -> 0() mark(U11(z0,z1)) -> a__U11(mark(z0),z1) mark(U12(z0)) -> a__U12(mark(z0)) mark(U21(z0)) -> a__U21(mark(z0)) mark(U31(z0,z1)) -> a__U31(mark(z0),z1) mark(U41(z0,z1,z2)) -> a__U41(mark(z0),z1,z2) mark(U42(z0,z1,z2)) -> a__U42(mark(z0),z1,z2) mark(isNat(z0)) -> a__isNat(z0) mark(plus(z0,z1)) -> a__plus(mark(z0),mark(z1)) mark(s(z0)) -> s(mark(z0)) mark(tt()) -> tt() - Signature: {A__ISNAT/1,A__PLUS/2,A__U11/2,A__U12/1,A__U21/1,A__U31/2,A__U41/3,A__U42/3,MARK/1,a__U11/2,a__U12/1 ,a__U21/1,a__U31/2,a__U41/3,a__U42/3,a__isNat/1,a__plus/2,mark/1} / {0/0,U11/2,U12/1,U21/1,U31/2,U41/3,U42/3 ,c/2,c1/0,c10/2,c11/2,c12/0,c13/0,c14/2,c15/2,c16/0,c17/2,c18/2,c19/0,c2/0,c20/2,c21/2,c22/1,c23/2,c24/2 ,c25/2,c26/2,c27/2,c28/2,c29/0,c3/0,c30/1,c31/0,c4/0,c5/0,c6/1,c7/0,c8/2,c9/0,isNat/1,plus/2,s/1,tt/0} - Obligation: innermost runtime complexity wrt. defined symbols {A__ISNAT,A__PLUS,A__U11,A__U12,A__U21,A__U31,A__U41 ,A__U42,MARK,a__U11,a__U12,a__U21,a__U31,a__U41,a__U42,a__isNat,a__plus,mark} and constructors {0,U11,U12 ,U21,U31,U41,U42,c,c1,c10,c11,c12,c13,c14,c15,c16,c17,c18,c19,c2,c20,c21,c22,c23,c24,c25,c26,c27,c28,c29,c3 ,c30,c31,c4,c5,c6,c7,c8,c9,isNat,plus,s,tt} + Applied Processor: DecreasingLoops {bound = AnyLoop, narrow = 10} + Details: The system has following decreasing Loops: A__ISNAT(x){x -> plus(x,y)} = A__ISNAT(plus(x,y)) ->^+ c14(A__U11(a__isNat(x),y),A__ISNAT(x)) = C[A__ISNAT(x) = A__ISNAT(x){}] WORST_CASE(Omega(n^1),?)