WORST_CASE(Omega(n^1),?) * Step 1: Sum. WORST_CASE(Omega(n^1),?) + Considered Problem: - Strict TRS: A__AND(z0,z1) -> c12() A__AND(tt(),z0) -> c11(MARK(z0)) A__ISNAT(z0) -> c17() A__ISNAT(0()) -> c13() A__ISNAT(plus(z0,z1)) -> c14(A__AND(a__isNat(z0),isNat(z1)),A__ISNAT(z0)) A__ISNAT(s(z0)) -> c15(A__ISNAT(z0)) A__ISNAT(x(z0,z1)) -> c16(A__AND(a__isNat(z0),isNat(z1)),A__ISNAT(z0)) A__PLUS(z0,z1) -> c20() A__PLUS(z0,0()) -> c18(A__U11(a__isNat(z0),z0),A__ISNAT(z0)) A__PLUS(z0,s(z1)) -> c19(A__U21(a__and(a__isNat(z1),isNat(z0)),z1,z0) ,A__AND(a__isNat(z1),isNat(z0)) ,A__ISNAT(z1)) A__U11(z0,z1) -> c1() A__U11(tt(),z0) -> c(MARK(z0)) A__U21(z0,z1,z2) -> c4() A__U21(tt(),z0,z1) -> c2(A__PLUS(mark(z1),mark(z0)),MARK(z1)) A__U21(tt(),z0,z1) -> c3(A__PLUS(mark(z1),mark(z0)),MARK(z0)) A__U31(z0) -> c6() A__U31(tt()) -> c5() A__U41(z0,z1,z2) -> c10() A__U41(tt(),z0,z1) -> c7(A__PLUS(a__x(mark(z1),mark(z0)),mark(z1)),A__X(mark(z1),mark(z0)),MARK(z1)) A__U41(tt(),z0,z1) -> c8(A__PLUS(a__x(mark(z1),mark(z0)),mark(z1)),A__X(mark(z1),mark(z0)),MARK(z0)) A__U41(tt(),z0,z1) -> c9(A__PLUS(a__x(mark(z1),mark(z0)),mark(z1)),MARK(z1)) A__X(z0,z1) -> c23() A__X(z0,0()) -> c21(A__U31(a__isNat(z0)),A__ISNAT(z0)) A__X(z0,s(z1)) -> c22(A__U41(a__and(a__isNat(z1),isNat(z0)),z1,z0) ,A__AND(a__isNat(z1),isNat(z0)) ,A__ISNAT(z1)) MARK(0()) -> c36() MARK(U11(z0,z1)) -> c24(A__U11(mark(z0),z1),MARK(z0)) MARK(U21(z0,z1,z2)) -> c25(A__U21(mark(z0),z1,z2),MARK(z0)) MARK(U31(z0)) -> c28(A__U31(mark(z0)),MARK(z0)) MARK(U41(z0,z1,z2)) -> c29(A__U41(mark(z0),z1,z2),MARK(z0)) MARK(and(z0,z1)) -> c32(A__AND(mark(z0),z1),MARK(z0)) MARK(isNat(z0)) -> c33(A__ISNAT(z0)) MARK(plus(z0,z1)) -> c26(A__PLUS(mark(z0),mark(z1)),MARK(z0)) MARK(plus(z0,z1)) -> c27(A__PLUS(mark(z0),mark(z1)),MARK(z1)) MARK(s(z0)) -> c35(MARK(z0)) MARK(tt()) -> c34() MARK(x(z0,z1)) -> c30(A__X(mark(z0),mark(z1)),MARK(z0)) MARK(x(z0,z1)) -> c31(A__X(mark(z0),mark(z1)),MARK(z1)) - Weak TRS: a__U11(z0,z1) -> U11(z0,z1) a__U11(tt(),z0) -> mark(z0) a__U21(z0,z1,z2) -> U21(z0,z1,z2) a__U21(tt(),z0,z1) -> s(a__plus(mark(z1),mark(z0))) a__U31(z0) -> U31(z0) a__U31(tt()) -> 0() a__U41(z0,z1,z2) -> U41(z0,z1,z2) a__U41(tt(),z0,z1) -> a__plus(a__x(mark(z1),mark(z0)),mark(z1)) a__and(z0,z1) -> and(z0,z1) a__and(tt(),z0) -> mark(z0) a__isNat(z0) -> isNat(z0) a__isNat(0()) -> tt() a__isNat(plus(z0,z1)) -> a__and(a__isNat(z0),isNat(z1)) a__isNat(s(z0)) -> a__isNat(z0) a__isNat(x(z0,z1)) -> a__and(a__isNat(z0),isNat(z1)) a__plus(z0,z1) -> plus(z0,z1) a__plus(z0,0()) -> a__U11(a__isNat(z0),z0) a__plus(z0,s(z1)) -> a__U21(a__and(a__isNat(z1),isNat(z0)),z1,z0) a__x(z0,z1) -> x(z0,z1) a__x(z0,0()) -> a__U31(a__isNat(z0)) a__x(z0,s(z1)) -> a__U41(a__and(a__isNat(z1),isNat(z0)),z1,z0) mark(0()) -> 0() mark(U11(z0,z1)) -> a__U11(mark(z0),z1) mark(U21(z0,z1,z2)) -> a__U21(mark(z0),z1,z2) mark(U31(z0)) -> a__U31(mark(z0)) mark(U41(z0,z1,z2)) -> a__U41(mark(z0),z1,z2) mark(and(z0,z1)) -> a__and(mark(z0),z1) 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() mark(x(z0,z1)) -> a__x(mark(z0),mark(z1)) - Signature: {A__AND/2,A__ISNAT/1,A__PLUS/2,A__U11/2,A__U21/3,A__U31/1,A__U41/3,A__X/2,MARK/1,a__U11/2,a__U21/3,a__U31/1 ,a__U41/3,a__and/2,a__isNat/1,a__plus/2,a__x/2,mark/1} / {0/0,U11/2,U21/3,U31/1,U41/3,and/2,c/1,c1/0,c10/0 ,c11/1,c12/0,c13/0,c14/2,c15/1,c16/2,c17/0,c18/2,c19/3,c2/2,c20/0,c21/2,c22/3,c23/0,c24/2,c25/2,c26/2,c27/2 ,c28/2,c29/2,c3/2,c30/2,c31/2,c32/2,c33/1,c34/0,c35/1,c36/0,c4/0,c5/0,c6/0,c7/3,c8/3,c9/2,isNat/1,plus/2,s/1 ,tt/0,x/2} - Obligation: innermost runtime complexity wrt. defined symbols {A__AND,A__ISNAT,A__PLUS,A__U11,A__U21,A__U31,A__U41,A__X ,MARK,a__U11,a__U21,a__U31,a__U41,a__and,a__isNat,a__plus,a__x,mark} and constructors {0,U11,U21,U31,U41,and ,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,c32,c33 ,c34,c35,c36,c4,c5,c6,c7,c8,c9,isNat,plus,s,tt,x} + Applied Processor: Sum {left = someStrategy, right = someStrategy} + Details: () * Step 2: Sum. WORST_CASE(Omega(n^1),?) + Considered Problem: - Strict TRS: A__AND(z0,z1) -> c12() A__AND(tt(),z0) -> c11(MARK(z0)) A__ISNAT(z0) -> c17() A__ISNAT(0()) -> c13() A__ISNAT(plus(z0,z1)) -> c14(A__AND(a__isNat(z0),isNat(z1)),A__ISNAT(z0)) A__ISNAT(s(z0)) -> c15(A__ISNAT(z0)) A__ISNAT(x(z0,z1)) -> c16(A__AND(a__isNat(z0),isNat(z1)),A__ISNAT(z0)) A__PLUS(z0,z1) -> c20() A__PLUS(z0,0()) -> c18(A__U11(a__isNat(z0),z0),A__ISNAT(z0)) A__PLUS(z0,s(z1)) -> c19(A__U21(a__and(a__isNat(z1),isNat(z0)),z1,z0) ,A__AND(a__isNat(z1),isNat(z0)) ,A__ISNAT(z1)) A__U11(z0,z1) -> c1() A__U11(tt(),z0) -> c(MARK(z0)) A__U21(z0,z1,z2) -> c4() A__U21(tt(),z0,z1) -> c2(A__PLUS(mark(z1),mark(z0)),MARK(z1)) A__U21(tt(),z0,z1) -> c3(A__PLUS(mark(z1),mark(z0)),MARK(z0)) A__U31(z0) -> c6() A__U31(tt()) -> c5() A__U41(z0,z1,z2) -> c10() A__U41(tt(),z0,z1) -> c7(A__PLUS(a__x(mark(z1),mark(z0)),mark(z1)),A__X(mark(z1),mark(z0)),MARK(z1)) A__U41(tt(),z0,z1) -> c8(A__PLUS(a__x(mark(z1),mark(z0)),mark(z1)),A__X(mark(z1),mark(z0)),MARK(z0)) A__U41(tt(),z0,z1) -> c9(A__PLUS(a__x(mark(z1),mark(z0)),mark(z1)),MARK(z1)) A__X(z0,z1) -> c23() A__X(z0,0()) -> c21(A__U31(a__isNat(z0)),A__ISNAT(z0)) A__X(z0,s(z1)) -> c22(A__U41(a__and(a__isNat(z1),isNat(z0)),z1,z0) ,A__AND(a__isNat(z1),isNat(z0)) ,A__ISNAT(z1)) MARK(0()) -> c36() MARK(U11(z0,z1)) -> c24(A__U11(mark(z0),z1),MARK(z0)) MARK(U21(z0,z1,z2)) -> c25(A__U21(mark(z0),z1,z2),MARK(z0)) MARK(U31(z0)) -> c28(A__U31(mark(z0)),MARK(z0)) MARK(U41(z0,z1,z2)) -> c29(A__U41(mark(z0),z1,z2),MARK(z0)) MARK(and(z0,z1)) -> c32(A__AND(mark(z0),z1),MARK(z0)) MARK(isNat(z0)) -> c33(A__ISNAT(z0)) MARK(plus(z0,z1)) -> c26(A__PLUS(mark(z0),mark(z1)),MARK(z0)) MARK(plus(z0,z1)) -> c27(A__PLUS(mark(z0),mark(z1)),MARK(z1)) MARK(s(z0)) -> c35(MARK(z0)) MARK(tt()) -> c34() MARK(x(z0,z1)) -> c30(A__X(mark(z0),mark(z1)),MARK(z0)) MARK(x(z0,z1)) -> c31(A__X(mark(z0),mark(z1)),MARK(z1)) - Weak TRS: a__U11(z0,z1) -> U11(z0,z1) a__U11(tt(),z0) -> mark(z0) a__U21(z0,z1,z2) -> U21(z0,z1,z2) a__U21(tt(),z0,z1) -> s(a__plus(mark(z1),mark(z0))) a__U31(z0) -> U31(z0) a__U31(tt()) -> 0() a__U41(z0,z1,z2) -> U41(z0,z1,z2) a__U41(tt(),z0,z1) -> a__plus(a__x(mark(z1),mark(z0)),mark(z1)) a__and(z0,z1) -> and(z0,z1) a__and(tt(),z0) -> mark(z0) a__isNat(z0) -> isNat(z0) a__isNat(0()) -> tt() a__isNat(plus(z0,z1)) -> a__and(a__isNat(z0),isNat(z1)) a__isNat(s(z0)) -> a__isNat(z0) a__isNat(x(z0,z1)) -> a__and(a__isNat(z0),isNat(z1)) a__plus(z0,z1) -> plus(z0,z1) a__plus(z0,0()) -> a__U11(a__isNat(z0),z0) a__plus(z0,s(z1)) -> a__U21(a__and(a__isNat(z1),isNat(z0)),z1,z0) a__x(z0,z1) -> x(z0,z1) a__x(z0,0()) -> a__U31(a__isNat(z0)) a__x(z0,s(z1)) -> a__U41(a__and(a__isNat(z1),isNat(z0)),z1,z0) mark(0()) -> 0() mark(U11(z0,z1)) -> a__U11(mark(z0),z1) mark(U21(z0,z1,z2)) -> a__U21(mark(z0),z1,z2) mark(U31(z0)) -> a__U31(mark(z0)) mark(U41(z0,z1,z2)) -> a__U41(mark(z0),z1,z2) mark(and(z0,z1)) -> a__and(mark(z0),z1) 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() mark(x(z0,z1)) -> a__x(mark(z0),mark(z1)) - Signature: {A__AND/2,A__ISNAT/1,A__PLUS/2,A__U11/2,A__U21/3,A__U31/1,A__U41/3,A__X/2,MARK/1,a__U11/2,a__U21/3,a__U31/1 ,a__U41/3,a__and/2,a__isNat/1,a__plus/2,a__x/2,mark/1} / {0/0,U11/2,U21/3,U31/1,U41/3,and/2,c/1,c1/0,c10/0 ,c11/1,c12/0,c13/0,c14/2,c15/1,c16/2,c17/0,c18/2,c19/3,c2/2,c20/0,c21/2,c22/3,c23/0,c24/2,c25/2,c26/2,c27/2 ,c28/2,c29/2,c3/2,c30/2,c31/2,c32/2,c33/1,c34/0,c35/1,c36/0,c4/0,c5/0,c6/0,c7/3,c8/3,c9/2,isNat/1,plus/2,s/1 ,tt/0,x/2} - Obligation: innermost runtime complexity wrt. defined symbols {A__AND,A__ISNAT,A__PLUS,A__U11,A__U21,A__U31,A__U41,A__X ,MARK,a__U11,a__U21,a__U31,a__U41,a__and,a__isNat,a__plus,a__x,mark} and constructors {0,U11,U21,U31,U41,and ,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,c32,c33 ,c34,c35,c36,c4,c5,c6,c7,c8,c9,isNat,plus,s,tt,x} + Applied Processor: Sum {left = someStrategy, right = someStrategy} + Details: () * Step 3: DecreasingLoops. WORST_CASE(Omega(n^1),?) + Considered Problem: - Strict TRS: A__AND(z0,z1) -> c12() A__AND(tt(),z0) -> c11(MARK(z0)) A__ISNAT(z0) -> c17() A__ISNAT(0()) -> c13() A__ISNAT(plus(z0,z1)) -> c14(A__AND(a__isNat(z0),isNat(z1)),A__ISNAT(z0)) A__ISNAT(s(z0)) -> c15(A__ISNAT(z0)) A__ISNAT(x(z0,z1)) -> c16(A__AND(a__isNat(z0),isNat(z1)),A__ISNAT(z0)) A__PLUS(z0,z1) -> c20() A__PLUS(z0,0()) -> c18(A__U11(a__isNat(z0),z0),A__ISNAT(z0)) A__PLUS(z0,s(z1)) -> c19(A__U21(a__and(a__isNat(z1),isNat(z0)),z1,z0) ,A__AND(a__isNat(z1),isNat(z0)) ,A__ISNAT(z1)) A__U11(z0,z1) -> c1() A__U11(tt(),z0) -> c(MARK(z0)) A__U21(z0,z1,z2) -> c4() A__U21(tt(),z0,z1) -> c2(A__PLUS(mark(z1),mark(z0)),MARK(z1)) A__U21(tt(),z0,z1) -> c3(A__PLUS(mark(z1),mark(z0)),MARK(z0)) A__U31(z0) -> c6() A__U31(tt()) -> c5() A__U41(z0,z1,z2) -> c10() A__U41(tt(),z0,z1) -> c7(A__PLUS(a__x(mark(z1),mark(z0)),mark(z1)),A__X(mark(z1),mark(z0)),MARK(z1)) A__U41(tt(),z0,z1) -> c8(A__PLUS(a__x(mark(z1),mark(z0)),mark(z1)),A__X(mark(z1),mark(z0)),MARK(z0)) A__U41(tt(),z0,z1) -> c9(A__PLUS(a__x(mark(z1),mark(z0)),mark(z1)),MARK(z1)) A__X(z0,z1) -> c23() A__X(z0,0()) -> c21(A__U31(a__isNat(z0)),A__ISNAT(z0)) A__X(z0,s(z1)) -> c22(A__U41(a__and(a__isNat(z1),isNat(z0)),z1,z0) ,A__AND(a__isNat(z1),isNat(z0)) ,A__ISNAT(z1)) MARK(0()) -> c36() MARK(U11(z0,z1)) -> c24(A__U11(mark(z0),z1),MARK(z0)) MARK(U21(z0,z1,z2)) -> c25(A__U21(mark(z0),z1,z2),MARK(z0)) MARK(U31(z0)) -> c28(A__U31(mark(z0)),MARK(z0)) MARK(U41(z0,z1,z2)) -> c29(A__U41(mark(z0),z1,z2),MARK(z0)) MARK(and(z0,z1)) -> c32(A__AND(mark(z0),z1),MARK(z0)) MARK(isNat(z0)) -> c33(A__ISNAT(z0)) MARK(plus(z0,z1)) -> c26(A__PLUS(mark(z0),mark(z1)),MARK(z0)) MARK(plus(z0,z1)) -> c27(A__PLUS(mark(z0),mark(z1)),MARK(z1)) MARK(s(z0)) -> c35(MARK(z0)) MARK(tt()) -> c34() MARK(x(z0,z1)) -> c30(A__X(mark(z0),mark(z1)),MARK(z0)) MARK(x(z0,z1)) -> c31(A__X(mark(z0),mark(z1)),MARK(z1)) - Weak TRS: a__U11(z0,z1) -> U11(z0,z1) a__U11(tt(),z0) -> mark(z0) a__U21(z0,z1,z2) -> U21(z0,z1,z2) a__U21(tt(),z0,z1) -> s(a__plus(mark(z1),mark(z0))) a__U31(z0) -> U31(z0) a__U31(tt()) -> 0() a__U41(z0,z1,z2) -> U41(z0,z1,z2) a__U41(tt(),z0,z1) -> a__plus(a__x(mark(z1),mark(z0)),mark(z1)) a__and(z0,z1) -> and(z0,z1) a__and(tt(),z0) -> mark(z0) a__isNat(z0) -> isNat(z0) a__isNat(0()) -> tt() a__isNat(plus(z0,z1)) -> a__and(a__isNat(z0),isNat(z1)) a__isNat(s(z0)) -> a__isNat(z0) a__isNat(x(z0,z1)) -> a__and(a__isNat(z0),isNat(z1)) a__plus(z0,z1) -> plus(z0,z1) a__plus(z0,0()) -> a__U11(a__isNat(z0),z0) a__plus(z0,s(z1)) -> a__U21(a__and(a__isNat(z1),isNat(z0)),z1,z0) a__x(z0,z1) -> x(z0,z1) a__x(z0,0()) -> a__U31(a__isNat(z0)) a__x(z0,s(z1)) -> a__U41(a__and(a__isNat(z1),isNat(z0)),z1,z0) mark(0()) -> 0() mark(U11(z0,z1)) -> a__U11(mark(z0),z1) mark(U21(z0,z1,z2)) -> a__U21(mark(z0),z1,z2) mark(U31(z0)) -> a__U31(mark(z0)) mark(U41(z0,z1,z2)) -> a__U41(mark(z0),z1,z2) mark(and(z0,z1)) -> a__and(mark(z0),z1) 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() mark(x(z0,z1)) -> a__x(mark(z0),mark(z1)) - Signature: {A__AND/2,A__ISNAT/1,A__PLUS/2,A__U11/2,A__U21/3,A__U31/1,A__U41/3,A__X/2,MARK/1,a__U11/2,a__U21/3,a__U31/1 ,a__U41/3,a__and/2,a__isNat/1,a__plus/2,a__x/2,mark/1} / {0/0,U11/2,U21/3,U31/1,U41/3,and/2,c/1,c1/0,c10/0 ,c11/1,c12/0,c13/0,c14/2,c15/1,c16/2,c17/0,c18/2,c19/3,c2/2,c20/0,c21/2,c22/3,c23/0,c24/2,c25/2,c26/2,c27/2 ,c28/2,c29/2,c3/2,c30/2,c31/2,c32/2,c33/1,c34/0,c35/1,c36/0,c4/0,c5/0,c6/0,c7/3,c8/3,c9/2,isNat/1,plus/2,s/1 ,tt/0,x/2} - Obligation: innermost runtime complexity wrt. defined symbols {A__AND,A__ISNAT,A__PLUS,A__U11,A__U21,A__U31,A__U41,A__X ,MARK,a__U11,a__U21,a__U31,a__U41,a__and,a__isNat,a__plus,a__x,mark} and constructors {0,U11,U21,U31,U41,and ,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,c32,c33 ,c34,c35,c36,c4,c5,c6,c7,c8,c9,isNat,plus,s,tt,x} + 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__AND(a__isNat(x),isNat(y)),A__ISNAT(x)) = C[A__ISNAT(x) = A__ISNAT(x){}] WORST_CASE(Omega(n^1),?)