WORST_CASE(?,O(n^2)) * Step 1: Sum. WORST_CASE(?,O(n^2)) + Considered Problem: - Strict TRS: ACTIVATE(z0) -> c13() PLUS(z0,0()) -> c9() PLUS(z0,s(z1)) -> c10(U11'(tt(),z1,z0)) U11'(tt(),z0,z1) -> c(U12'(tt(),activate(z0),activate(z1)),ACTIVATE(z0)) U11'(tt(),z0,z1) -> c1(U12'(tt(),activate(z0),activate(z1)),ACTIVATE(z1)) U12'(tt(),z0,z1) -> c2(PLUS(activate(z1),activate(z0)),ACTIVATE(z1)) U12'(tt(),z0,z1) -> c3(PLUS(activate(z1),activate(z0)),ACTIVATE(z0)) U21'(tt(),z0,z1) -> c4(U22'(tt(),activate(z0),activate(z1)),ACTIVATE(z0)) U21'(tt(),z0,z1) -> c5(U22'(tt(),activate(z0),activate(z1)),ACTIVATE(z1)) U22'(tt(),z0,z1) -> c6(PLUS(x(activate(z1),activate(z0)),activate(z1)) ,X(activate(z1),activate(z0)) ,ACTIVATE(z1)) U22'(tt(),z0,z1) -> c7(PLUS(x(activate(z1),activate(z0)),activate(z1)) ,X(activate(z1),activate(z0)) ,ACTIVATE(z0)) U22'(tt(),z0,z1) -> c8(PLUS(x(activate(z1),activate(z0)),activate(z1)),ACTIVATE(z1)) X(z0,0()) -> c11() X(z0,s(z1)) -> c12(U21'(tt(),z1,z0)) - Weak TRS: U11(tt(),z0,z1) -> U12(tt(),activate(z0),activate(z1)) U12(tt(),z0,z1) -> s(plus(activate(z1),activate(z0))) U21(tt(),z0,z1) -> U22(tt(),activate(z0),activate(z1)) U22(tt(),z0,z1) -> plus(x(activate(z1),activate(z0)),activate(z1)) activate(z0) -> z0 plus(z0,0()) -> z0 plus(z0,s(z1)) -> U11(tt(),z1,z0) x(z0,0()) -> 0() x(z0,s(z1)) -> U21(tt(),z1,z0) - Signature: {ACTIVATE/1,PLUS/2,U11/3,U11'/3,U12/3,U12'/3,U21/3,U21'/3,U22/3,U22'/3,X/2,activate/1,plus/2,x/2} / {0/0,c/2 ,c1/2,c10/1,c11/0,c12/1,c13/0,c2/2,c3/2,c4/2,c5/2,c6/3,c7/3,c8/2,c9/0,s/1,tt/0} - Obligation: innermost runtime complexity wrt. defined symbols {ACTIVATE,PLUS,U11,U11',U12,U12',U21,U21',U22,U22',X ,activate,plus,x} and constructors {0,c,c1,c10,c11,c12,c13,c2,c3,c4,c5,c6,c7,c8,c9,s,tt} + Applied Processor: Sum {left = someStrategy, right = someStrategy} + Details: () * Step 2: DependencyPairs. WORST_CASE(?,O(n^2)) + Considered Problem: - Strict TRS: ACTIVATE(z0) -> c13() PLUS(z0,0()) -> c9() PLUS(z0,s(z1)) -> c10(U11'(tt(),z1,z0)) U11'(tt(),z0,z1) -> c(U12'(tt(),activate(z0),activate(z1)),ACTIVATE(z0)) U11'(tt(),z0,z1) -> c1(U12'(tt(),activate(z0),activate(z1)),ACTIVATE(z1)) U12'(tt(),z0,z1) -> c2(PLUS(activate(z1),activate(z0)),ACTIVATE(z1)) U12'(tt(),z0,z1) -> c3(PLUS(activate(z1),activate(z0)),ACTIVATE(z0)) U21'(tt(),z0,z1) -> c4(U22'(tt(),activate(z0),activate(z1)),ACTIVATE(z0)) U21'(tt(),z0,z1) -> c5(U22'(tt(),activate(z0),activate(z1)),ACTIVATE(z1)) U22'(tt(),z0,z1) -> c6(PLUS(x(activate(z1),activate(z0)),activate(z1)) ,X(activate(z1),activate(z0)) ,ACTIVATE(z1)) U22'(tt(),z0,z1) -> c7(PLUS(x(activate(z1),activate(z0)),activate(z1)) ,X(activate(z1),activate(z0)) ,ACTIVATE(z0)) U22'(tt(),z0,z1) -> c8(PLUS(x(activate(z1),activate(z0)),activate(z1)),ACTIVATE(z1)) X(z0,0()) -> c11() X(z0,s(z1)) -> c12(U21'(tt(),z1,z0)) - Weak TRS: U11(tt(),z0,z1) -> U12(tt(),activate(z0),activate(z1)) U12(tt(),z0,z1) -> s(plus(activate(z1),activate(z0))) U21(tt(),z0,z1) -> U22(tt(),activate(z0),activate(z1)) U22(tt(),z0,z1) -> plus(x(activate(z1),activate(z0)),activate(z1)) activate(z0) -> z0 plus(z0,0()) -> z0 plus(z0,s(z1)) -> U11(tt(),z1,z0) x(z0,0()) -> 0() x(z0,s(z1)) -> U21(tt(),z1,z0) - Signature: {ACTIVATE/1,PLUS/2,U11/3,U11'/3,U12/3,U12'/3,U21/3,U21'/3,U22/3,U22'/3,X/2,activate/1,plus/2,x/2} / {0/0,c/2 ,c1/2,c10/1,c11/0,c12/1,c13/0,c2/2,c3/2,c4/2,c5/2,c6/3,c7/3,c8/2,c9/0,s/1,tt/0} - Obligation: innermost runtime complexity wrt. defined symbols {ACTIVATE,PLUS,U11,U11',U12,U12',U21,U21',U22,U22',X ,activate,plus,x} and constructors {0,c,c1,c10,c11,c12,c13,c2,c3,c4,c5,c6,c7,c8,c9,s,tt} + Applied Processor: DependencyPairs {dpKind_ = DT} + Details: We add the following dependency tuples: Strict DPs ACTIVATE#(z0) -> c_1() PLUS#(z0,0()) -> c_2() PLUS#(z0,s(z1)) -> c_3(U11'#(tt(),z1,z0)) U11'#(tt(),z0,z1) -> c_4(U12'#(tt(),activate(z0),activate(z1)),activate#(z0),activate#(z1),ACTIVATE#(z0)) U11'#(tt(),z0,z1) -> c_5(U12'#(tt(),activate(z0),activate(z1)),activate#(z0),activate#(z1),ACTIVATE#(z1)) U12'#(tt(),z0,z1) -> c_6(PLUS#(activate(z1),activate(z0)),activate#(z1),activate#(z0),ACTIVATE#(z1)) U12'#(tt(),z0,z1) -> c_7(PLUS#(activate(z1),activate(z0)),activate#(z1),activate#(z0),ACTIVATE#(z0)) U21'#(tt(),z0,z1) -> c_8(U22'#(tt(),activate(z0),activate(z1)),activate#(z0),activate#(z1),ACTIVATE#(z0)) U21'#(tt(),z0,z1) -> c_9(U22'#(tt(),activate(z0),activate(z1)),activate#(z0),activate#(z1),ACTIVATE#(z1)) U22'#(tt(),z0,z1) -> c_10(PLUS#(x(activate(z1),activate(z0)),activate(z1)) ,x#(activate(z1),activate(z0)) ,activate#(z1) ,activate#(z0) ,activate#(z1) ,X#(activate(z1),activate(z0)) ,activate#(z1) ,activate#(z0) ,ACTIVATE#(z1)) U22'#(tt(),z0,z1) -> c_11(PLUS#(x(activate(z1),activate(z0)),activate(z1)) ,x#(activate(z1),activate(z0)) ,activate#(z1) ,activate#(z0) ,activate#(z1) ,X#(activate(z1),activate(z0)) ,activate#(z1) ,activate#(z0) ,ACTIVATE#(z0)) U22'#(tt(),z0,z1) -> c_12(PLUS#(x(activate(z1),activate(z0)),activate(z1)) ,x#(activate(z1),activate(z0)) ,activate#(z1) ,activate#(z0) ,activate#(z1) ,ACTIVATE#(z1)) X#(z0,0()) -> c_13() X#(z0,s(z1)) -> c_14(U21'#(tt(),z1,z0)) Weak DPs U11#(tt(),z0,z1) -> c_15(U12#(tt(),activate(z0),activate(z1)),activate#(z0),activate#(z1)) U12#(tt(),z0,z1) -> c_16(plus#(activate(z1),activate(z0)),activate#(z1),activate#(z0)) U21#(tt(),z0,z1) -> c_17(U22#(tt(),activate(z0),activate(z1)),activate#(z0),activate#(z1)) U22#(tt(),z0,z1) -> c_18(plus#(x(activate(z1),activate(z0)),activate(z1)) ,x#(activate(z1),activate(z0)) ,activate#(z1) ,activate#(z0) ,activate#(z1)) activate#(z0) -> c_19() plus#(z0,0()) -> c_20() plus#(z0,s(z1)) -> c_21(U11#(tt(),z1,z0)) x#(z0,0()) -> c_22() x#(z0,s(z1)) -> c_23(U21#(tt(),z1,z0)) and mark the set of starting terms. * Step 3: PredecessorEstimation. WORST_CASE(?,O(n^2)) + Considered Problem: - Strict DPs: ACTIVATE#(z0) -> c_1() PLUS#(z0,0()) -> c_2() PLUS#(z0,s(z1)) -> c_3(U11'#(tt(),z1,z0)) U11'#(tt(),z0,z1) -> c_4(U12'#(tt(),activate(z0),activate(z1)),activate#(z0),activate#(z1),ACTIVATE#(z0)) U11'#(tt(),z0,z1) -> c_5(U12'#(tt(),activate(z0),activate(z1)),activate#(z0),activate#(z1),ACTIVATE#(z1)) U12'#(tt(),z0,z1) -> c_6(PLUS#(activate(z1),activate(z0)),activate#(z1),activate#(z0),ACTIVATE#(z1)) U12'#(tt(),z0,z1) -> c_7(PLUS#(activate(z1),activate(z0)),activate#(z1),activate#(z0),ACTIVATE#(z0)) U21'#(tt(),z0,z1) -> c_8(U22'#(tt(),activate(z0),activate(z1)),activate#(z0),activate#(z1),ACTIVATE#(z0)) U21'#(tt(),z0,z1) -> c_9(U22'#(tt(),activate(z0),activate(z1)),activate#(z0),activate#(z1),ACTIVATE#(z1)) U22'#(tt(),z0,z1) -> c_10(PLUS#(x(activate(z1),activate(z0)),activate(z1)) ,x#(activate(z1),activate(z0)) ,activate#(z1) ,activate#(z0) ,activate#(z1) ,X#(activate(z1),activate(z0)) ,activate#(z1) ,activate#(z0) ,ACTIVATE#(z1)) U22'#(tt(),z0,z1) -> c_11(PLUS#(x(activate(z1),activate(z0)),activate(z1)) ,x#(activate(z1),activate(z0)) ,activate#(z1) ,activate#(z0) ,activate#(z1) ,X#(activate(z1),activate(z0)) ,activate#(z1) ,activate#(z0) ,ACTIVATE#(z0)) U22'#(tt(),z0,z1) -> c_12(PLUS#(x(activate(z1),activate(z0)),activate(z1)) ,x#(activate(z1),activate(z0)) ,activate#(z1) ,activate#(z0) ,activate#(z1) ,ACTIVATE#(z1)) X#(z0,0()) -> c_13() X#(z0,s(z1)) -> c_14(U21'#(tt(),z1,z0)) - Weak DPs: U11#(tt(),z0,z1) -> c_15(U12#(tt(),activate(z0),activate(z1)),activate#(z0),activate#(z1)) U12#(tt(),z0,z1) -> c_16(plus#(activate(z1),activate(z0)),activate#(z1),activate#(z0)) U21#(tt(),z0,z1) -> c_17(U22#(tt(),activate(z0),activate(z1)),activate#(z0),activate#(z1)) U22#(tt(),z0,z1) -> c_18(plus#(x(activate(z1),activate(z0)),activate(z1)) ,x#(activate(z1),activate(z0)) ,activate#(z1) ,activate#(z0) ,activate#(z1)) activate#(z0) -> c_19() plus#(z0,0()) -> c_20() plus#(z0,s(z1)) -> c_21(U11#(tt(),z1,z0)) x#(z0,0()) -> c_22() x#(z0,s(z1)) -> c_23(U21#(tt(),z1,z0)) - Weak TRS: ACTIVATE(z0) -> c13() PLUS(z0,0()) -> c9() PLUS(z0,s(z1)) -> c10(U11'(tt(),z1,z0)) U11(tt(),z0,z1) -> U12(tt(),activate(z0),activate(z1)) U11'(tt(),z0,z1) -> c(U12'(tt(),activate(z0),activate(z1)),ACTIVATE(z0)) U11'(tt(),z0,z1) -> c1(U12'(tt(),activate(z0),activate(z1)),ACTIVATE(z1)) U12(tt(),z0,z1) -> s(plus(activate(z1),activate(z0))) U12'(tt(),z0,z1) -> c2(PLUS(activate(z1),activate(z0)),ACTIVATE(z1)) U12'(tt(),z0,z1) -> c3(PLUS(activate(z1),activate(z0)),ACTIVATE(z0)) U21(tt(),z0,z1) -> U22(tt(),activate(z0),activate(z1)) U21'(tt(),z0,z1) -> c4(U22'(tt(),activate(z0),activate(z1)),ACTIVATE(z0)) U21'(tt(),z0,z1) -> c5(U22'(tt(),activate(z0),activate(z1)),ACTIVATE(z1)) U22(tt(),z0,z1) -> plus(x(activate(z1),activate(z0)),activate(z1)) U22'(tt(),z0,z1) -> c6(PLUS(x(activate(z1),activate(z0)),activate(z1)) ,X(activate(z1),activate(z0)) ,ACTIVATE(z1)) U22'(tt(),z0,z1) -> c7(PLUS(x(activate(z1),activate(z0)),activate(z1)) ,X(activate(z1),activate(z0)) ,ACTIVATE(z0)) U22'(tt(),z0,z1) -> c8(PLUS(x(activate(z1),activate(z0)),activate(z1)),ACTIVATE(z1)) X(z0,0()) -> c11() X(z0,s(z1)) -> c12(U21'(tt(),z1,z0)) activate(z0) -> z0 plus(z0,0()) -> z0 plus(z0,s(z1)) -> U11(tt(),z1,z0) x(z0,0()) -> 0() x(z0,s(z1)) -> U21(tt(),z1,z0) - Signature: {ACTIVATE/1,PLUS/2,U11/3,U11'/3,U12/3,U12'/3,U21/3,U21'/3,U22/3,U22'/3,X/2,activate/1,plus/2,x/2,ACTIVATE#/1 ,PLUS#/2,U11#/3,U11'#/3,U12#/3,U12'#/3,U21#/3,U21'#/3,U22#/3,U22'#/3,X#/2,activate#/1,plus#/2,x#/2} / {0/0 ,c/2,c1/2,c10/1,c11/0,c12/1,c13/0,c2/2,c3/2,c4/2,c5/2,c6/3,c7/3,c8/2,c9/0,s/1,tt/0,c_1/0,c_2/0,c_3/1,c_4/4 ,c_5/4,c_6/4,c_7/4,c_8/4,c_9/4,c_10/9,c_11/9,c_12/6,c_13/0,c_14/1,c_15/3,c_16/3,c_17/3,c_18/5,c_19/0,c_20/0 ,c_21/1,c_22/0,c_23/1} - Obligation: innermost runtime complexity wrt. defined symbols {ACTIVATE#,PLUS#,U11#,U11'#,U12#,U12'#,U21#,U21'#,U22# ,U22'#,X#,activate#,plus#,x#} and constructors {0,c,c1,c10,c11,c12,c13,c2,c3,c4,c5,c6,c7,c8,c9,s,tt} + Applied Processor: PredecessorEstimation {onSelection = all simple predecessor estimation selector} + Details: We estimate the number of application of {1,2,13} by application of Pre({1,2,13}) = {4,5,6,7,8,9,10,11,12}. Here rules are labelled as follows: 1: ACTIVATE#(z0) -> c_1() 2: PLUS#(z0,0()) -> c_2() 3: PLUS#(z0,s(z1)) -> c_3(U11'#(tt(),z1,z0)) 4: U11'#(tt(),z0,z1) -> c_4(U12'#(tt(),activate(z0),activate(z1)),activate#(z0),activate#(z1),ACTIVATE#(z0)) 5: U11'#(tt(),z0,z1) -> c_5(U12'#(tt(),activate(z0),activate(z1)),activate#(z0),activate#(z1),ACTIVATE#(z1)) 6: U12'#(tt(),z0,z1) -> c_6(PLUS#(activate(z1),activate(z0)),activate#(z1),activate#(z0),ACTIVATE#(z1)) 7: U12'#(tt(),z0,z1) -> c_7(PLUS#(activate(z1),activate(z0)),activate#(z1),activate#(z0),ACTIVATE#(z0)) 8: U21'#(tt(),z0,z1) -> c_8(U22'#(tt(),activate(z0),activate(z1)),activate#(z0),activate#(z1),ACTIVATE#(z0)) 9: U21'#(tt(),z0,z1) -> c_9(U22'#(tt(),activate(z0),activate(z1)),activate#(z0),activate#(z1),ACTIVATE#(z1)) 10: U22'#(tt(),z0,z1) -> c_10(PLUS#(x(activate(z1),activate(z0)),activate(z1)) ,x#(activate(z1),activate(z0)) ,activate#(z1) ,activate#(z0) ,activate#(z1) ,X#(activate(z1),activate(z0)) ,activate#(z1) ,activate#(z0) ,ACTIVATE#(z1)) 11: U22'#(tt(),z0,z1) -> c_11(PLUS#(x(activate(z1),activate(z0)),activate(z1)) ,x#(activate(z1),activate(z0)) ,activate#(z1) ,activate#(z0) ,activate#(z1) ,X#(activate(z1),activate(z0)) ,activate#(z1) ,activate#(z0) ,ACTIVATE#(z0)) 12: U22'#(tt(),z0,z1) -> c_12(PLUS#(x(activate(z1),activate(z0)),activate(z1)) ,x#(activate(z1),activate(z0)) ,activate#(z1) ,activate#(z0) ,activate#(z1) ,ACTIVATE#(z1)) 13: X#(z0,0()) -> c_13() 14: X#(z0,s(z1)) -> c_14(U21'#(tt(),z1,z0)) 15: U11#(tt(),z0,z1) -> c_15(U12#(tt(),activate(z0),activate(z1)),activate#(z0),activate#(z1)) 16: U12#(tt(),z0,z1) -> c_16(plus#(activate(z1),activate(z0)),activate#(z1),activate#(z0)) 17: U21#(tt(),z0,z1) -> c_17(U22#(tt(),activate(z0),activate(z1)),activate#(z0),activate#(z1)) 18: U22#(tt(),z0,z1) -> c_18(plus#(x(activate(z1),activate(z0)),activate(z1)) ,x#(activate(z1),activate(z0)) ,activate#(z1) ,activate#(z0) ,activate#(z1)) 19: activate#(z0) -> c_19() 20: plus#(z0,0()) -> c_20() 21: plus#(z0,s(z1)) -> c_21(U11#(tt(),z1,z0)) 22: x#(z0,0()) -> c_22() 23: x#(z0,s(z1)) -> c_23(U21#(tt(),z1,z0)) * Step 4: RemoveWeakSuffixes. WORST_CASE(?,O(n^2)) + Considered Problem: - Strict DPs: PLUS#(z0,s(z1)) -> c_3(U11'#(tt(),z1,z0)) U11'#(tt(),z0,z1) -> c_4(U12'#(tt(),activate(z0),activate(z1)),activate#(z0),activate#(z1),ACTIVATE#(z0)) U11'#(tt(),z0,z1) -> c_5(U12'#(tt(),activate(z0),activate(z1)),activate#(z0),activate#(z1),ACTIVATE#(z1)) U12'#(tt(),z0,z1) -> c_6(PLUS#(activate(z1),activate(z0)),activate#(z1),activate#(z0),ACTIVATE#(z1)) U12'#(tt(),z0,z1) -> c_7(PLUS#(activate(z1),activate(z0)),activate#(z1),activate#(z0),ACTIVATE#(z0)) U21'#(tt(),z0,z1) -> c_8(U22'#(tt(),activate(z0),activate(z1)),activate#(z0),activate#(z1),ACTIVATE#(z0)) U21'#(tt(),z0,z1) -> c_9(U22'#(tt(),activate(z0),activate(z1)),activate#(z0),activate#(z1),ACTIVATE#(z1)) U22'#(tt(),z0,z1) -> c_10(PLUS#(x(activate(z1),activate(z0)),activate(z1)) ,x#(activate(z1),activate(z0)) ,activate#(z1) ,activate#(z0) ,activate#(z1) ,X#(activate(z1),activate(z0)) ,activate#(z1) ,activate#(z0) ,ACTIVATE#(z1)) U22'#(tt(),z0,z1) -> c_11(PLUS#(x(activate(z1),activate(z0)),activate(z1)) ,x#(activate(z1),activate(z0)) ,activate#(z1) ,activate#(z0) ,activate#(z1) ,X#(activate(z1),activate(z0)) ,activate#(z1) ,activate#(z0) ,ACTIVATE#(z0)) U22'#(tt(),z0,z1) -> c_12(PLUS#(x(activate(z1),activate(z0)),activate(z1)) ,x#(activate(z1),activate(z0)) ,activate#(z1) ,activate#(z0) ,activate#(z1) ,ACTIVATE#(z1)) X#(z0,s(z1)) -> c_14(U21'#(tt(),z1,z0)) - Weak DPs: ACTIVATE#(z0) -> c_1() PLUS#(z0,0()) -> c_2() U11#(tt(),z0,z1) -> c_15(U12#(tt(),activate(z0),activate(z1)),activate#(z0),activate#(z1)) U12#(tt(),z0,z1) -> c_16(plus#(activate(z1),activate(z0)),activate#(z1),activate#(z0)) U21#(tt(),z0,z1) -> c_17(U22#(tt(),activate(z0),activate(z1)),activate#(z0),activate#(z1)) U22#(tt(),z0,z1) -> c_18(plus#(x(activate(z1),activate(z0)),activate(z1)) ,x#(activate(z1),activate(z0)) ,activate#(z1) ,activate#(z0) ,activate#(z1)) X#(z0,0()) -> c_13() activate#(z0) -> c_19() plus#(z0,0()) -> c_20() plus#(z0,s(z1)) -> c_21(U11#(tt(),z1,z0)) x#(z0,0()) -> c_22() x#(z0,s(z1)) -> c_23(U21#(tt(),z1,z0)) - Weak TRS: ACTIVATE(z0) -> c13() PLUS(z0,0()) -> c9() PLUS(z0,s(z1)) -> c10(U11'(tt(),z1,z0)) U11(tt(),z0,z1) -> U12(tt(),activate(z0),activate(z1)) U11'(tt(),z0,z1) -> c(U12'(tt(),activate(z0),activate(z1)),ACTIVATE(z0)) U11'(tt(),z0,z1) -> c1(U12'(tt(),activate(z0),activate(z1)),ACTIVATE(z1)) U12(tt(),z0,z1) -> s(plus(activate(z1),activate(z0))) U12'(tt(),z0,z1) -> c2(PLUS(activate(z1),activate(z0)),ACTIVATE(z1)) U12'(tt(),z0,z1) -> c3(PLUS(activate(z1),activate(z0)),ACTIVATE(z0)) U21(tt(),z0,z1) -> U22(tt(),activate(z0),activate(z1)) U21'(tt(),z0,z1) -> c4(U22'(tt(),activate(z0),activate(z1)),ACTIVATE(z0)) U21'(tt(),z0,z1) -> c5(U22'(tt(),activate(z0),activate(z1)),ACTIVATE(z1)) U22(tt(),z0,z1) -> plus(x(activate(z1),activate(z0)),activate(z1)) U22'(tt(),z0,z1) -> c6(PLUS(x(activate(z1),activate(z0)),activate(z1)) ,X(activate(z1),activate(z0)) ,ACTIVATE(z1)) U22'(tt(),z0,z1) -> c7(PLUS(x(activate(z1),activate(z0)),activate(z1)) ,X(activate(z1),activate(z0)) ,ACTIVATE(z0)) U22'(tt(),z0,z1) -> c8(PLUS(x(activate(z1),activate(z0)),activate(z1)),ACTIVATE(z1)) X(z0,0()) -> c11() X(z0,s(z1)) -> c12(U21'(tt(),z1,z0)) activate(z0) -> z0 plus(z0,0()) -> z0 plus(z0,s(z1)) -> U11(tt(),z1,z0) x(z0,0()) -> 0() x(z0,s(z1)) -> U21(tt(),z1,z0) - Signature: {ACTIVATE/1,PLUS/2,U11/3,U11'/3,U12/3,U12'/3,U21/3,U21'/3,U22/3,U22'/3,X/2,activate/1,plus/2,x/2,ACTIVATE#/1 ,PLUS#/2,U11#/3,U11'#/3,U12#/3,U12'#/3,U21#/3,U21'#/3,U22#/3,U22'#/3,X#/2,activate#/1,plus#/2,x#/2} / {0/0 ,c/2,c1/2,c10/1,c11/0,c12/1,c13/0,c2/2,c3/2,c4/2,c5/2,c6/3,c7/3,c8/2,c9/0,s/1,tt/0,c_1/0,c_2/0,c_3/1,c_4/4 ,c_5/4,c_6/4,c_7/4,c_8/4,c_9/4,c_10/9,c_11/9,c_12/6,c_13/0,c_14/1,c_15/3,c_16/3,c_17/3,c_18/5,c_19/0,c_20/0 ,c_21/1,c_22/0,c_23/1} - Obligation: innermost runtime complexity wrt. defined symbols {ACTIVATE#,PLUS#,U11#,U11'#,U12#,U12'#,U21#,U21'#,U22# ,U22'#,X#,activate#,plus#,x#} and constructors {0,c,c1,c10,c11,c12,c13,c2,c3,c4,c5,c6,c7,c8,c9,s,tt} + Applied Processor: RemoveWeakSuffixes + Details: Consider the dependency graph 1:S:PLUS#(z0,s(z1)) -> c_3(U11'#(tt(),z1,z0)) -->_1 U11'#(tt(),z0,z1) -> c_5(U12'#(tt(),activate(z0),activate(z1)) ,activate#(z0) ,activate#(z1) ,ACTIVATE#(z1)):3 -->_1 U11'#(tt(),z0,z1) -> c_4(U12'#(tt(),activate(z0),activate(z1)) ,activate#(z0) ,activate#(z1) ,ACTIVATE#(z0)):2 2:S:U11'#(tt(),z0,z1) -> c_4(U12'#(tt(),activate(z0),activate(z1)) ,activate#(z0) ,activate#(z1) ,ACTIVATE#(z0)) -->_1 U12'#(tt(),z0,z1) -> c_7(PLUS#(activate(z1),activate(z0)) ,activate#(z1) ,activate#(z0) ,ACTIVATE#(z0)):5 -->_1 U12'#(tt(),z0,z1) -> c_6(PLUS#(activate(z1),activate(z0)) ,activate#(z1) ,activate#(z0) ,ACTIVATE#(z1)):4 -->_3 activate#(z0) -> c_19():19 -->_2 activate#(z0) -> c_19():19 -->_4 ACTIVATE#(z0) -> c_1():12 3:S:U11'#(tt(),z0,z1) -> c_5(U12'#(tt(),activate(z0),activate(z1)) ,activate#(z0) ,activate#(z1) ,ACTIVATE#(z1)) -->_1 U12'#(tt(),z0,z1) -> c_7(PLUS#(activate(z1),activate(z0)) ,activate#(z1) ,activate#(z0) ,ACTIVATE#(z0)):5 -->_1 U12'#(tt(),z0,z1) -> c_6(PLUS#(activate(z1),activate(z0)) ,activate#(z1) ,activate#(z0) ,ACTIVATE#(z1)):4 -->_3 activate#(z0) -> c_19():19 -->_2 activate#(z0) -> c_19():19 -->_4 ACTIVATE#(z0) -> c_1():12 4:S:U12'#(tt(),z0,z1) -> c_6(PLUS#(activate(z1),activate(z0)),activate#(z1),activate#(z0),ACTIVATE#(z1)) -->_3 activate#(z0) -> c_19():19 -->_2 activate#(z0) -> c_19():19 -->_1 PLUS#(z0,0()) -> c_2():13 -->_4 ACTIVATE#(z0) -> c_1():12 -->_1 PLUS#(z0,s(z1)) -> c_3(U11'#(tt(),z1,z0)):1 5:S:U12'#(tt(),z0,z1) -> c_7(PLUS#(activate(z1),activate(z0)),activate#(z1),activate#(z0),ACTIVATE#(z0)) -->_3 activate#(z0) -> c_19():19 -->_2 activate#(z0) -> c_19():19 -->_1 PLUS#(z0,0()) -> c_2():13 -->_4 ACTIVATE#(z0) -> c_1():12 -->_1 PLUS#(z0,s(z1)) -> c_3(U11'#(tt(),z1,z0)):1 6:S:U21'#(tt(),z0,z1) -> c_8(U22'#(tt(),activate(z0),activate(z1)) ,activate#(z0) ,activate#(z1) ,ACTIVATE#(z0)) -->_1 U22'#(tt(),z0,z1) -> c_12(PLUS#(x(activate(z1),activate(z0)),activate(z1)) ,x#(activate(z1),activate(z0)) ,activate#(z1) ,activate#(z0) ,activate#(z1) ,ACTIVATE#(z1)):10 -->_1 U22'#(tt(),z0,z1) -> c_11(PLUS#(x(activate(z1),activate(z0)),activate(z1)) ,x#(activate(z1),activate(z0)) ,activate#(z1) ,activate#(z0) ,activate#(z1) ,X#(activate(z1),activate(z0)) ,activate#(z1) ,activate#(z0) ,ACTIVATE#(z0)):9 -->_1 U22'#(tt(),z0,z1) -> c_10(PLUS#(x(activate(z1),activate(z0)),activate(z1)) ,x#(activate(z1),activate(z0)) ,activate#(z1) ,activate#(z0) ,activate#(z1) ,X#(activate(z1),activate(z0)) ,activate#(z1) ,activate#(z0) ,ACTIVATE#(z1)):8 -->_3 activate#(z0) -> c_19():19 -->_2 activate#(z0) -> c_19():19 -->_4 ACTIVATE#(z0) -> c_1():12 7:S:U21'#(tt(),z0,z1) -> c_9(U22'#(tt(),activate(z0),activate(z1)) ,activate#(z0) ,activate#(z1) ,ACTIVATE#(z1)) -->_1 U22'#(tt(),z0,z1) -> c_12(PLUS#(x(activate(z1),activate(z0)),activate(z1)) ,x#(activate(z1),activate(z0)) ,activate#(z1) ,activate#(z0) ,activate#(z1) ,ACTIVATE#(z1)):10 -->_1 U22'#(tt(),z0,z1) -> c_11(PLUS#(x(activate(z1),activate(z0)),activate(z1)) ,x#(activate(z1),activate(z0)) ,activate#(z1) ,activate#(z0) ,activate#(z1) ,X#(activate(z1),activate(z0)) ,activate#(z1) ,activate#(z0) ,ACTIVATE#(z0)):9 -->_1 U22'#(tt(),z0,z1) -> c_10(PLUS#(x(activate(z1),activate(z0)),activate(z1)) ,x#(activate(z1),activate(z0)) ,activate#(z1) ,activate#(z0) ,activate#(z1) ,X#(activate(z1),activate(z0)) ,activate#(z1) ,activate#(z0) ,ACTIVATE#(z1)):8 -->_3 activate#(z0) -> c_19():19 -->_2 activate#(z0) -> c_19():19 -->_4 ACTIVATE#(z0) -> c_1():12 8:S:U22'#(tt(),z0,z1) -> c_10(PLUS#(x(activate(z1),activate(z0)),activate(z1)) ,x#(activate(z1),activate(z0)) ,activate#(z1) ,activate#(z0) ,activate#(z1) ,X#(activate(z1),activate(z0)) ,activate#(z1) ,activate#(z0) ,ACTIVATE#(z1)) -->_2 x#(z0,s(z1)) -> c_23(U21#(tt(),z1,z0)):23 -->_6 X#(z0,s(z1)) -> c_14(U21'#(tt(),z1,z0)):11 -->_2 x#(z0,0()) -> c_22():22 -->_8 activate#(z0) -> c_19():19 -->_7 activate#(z0) -> c_19():19 -->_5 activate#(z0) -> c_19():19 -->_4 activate#(z0) -> c_19():19 -->_3 activate#(z0) -> c_19():19 -->_6 X#(z0,0()) -> c_13():18 -->_1 PLUS#(z0,0()) -> c_2():13 -->_9 ACTIVATE#(z0) -> c_1():12 -->_1 PLUS#(z0,s(z1)) -> c_3(U11'#(tt(),z1,z0)):1 9:S:U22'#(tt(),z0,z1) -> c_11(PLUS#(x(activate(z1),activate(z0)),activate(z1)) ,x#(activate(z1),activate(z0)) ,activate#(z1) ,activate#(z0) ,activate#(z1) ,X#(activate(z1),activate(z0)) ,activate#(z1) ,activate#(z0) ,ACTIVATE#(z0)) -->_2 x#(z0,s(z1)) -> c_23(U21#(tt(),z1,z0)):23 -->_6 X#(z0,s(z1)) -> c_14(U21'#(tt(),z1,z0)):11 -->_2 x#(z0,0()) -> c_22():22 -->_8 activate#(z0) -> c_19():19 -->_7 activate#(z0) -> c_19():19 -->_5 activate#(z0) -> c_19():19 -->_4 activate#(z0) -> c_19():19 -->_3 activate#(z0) -> c_19():19 -->_6 X#(z0,0()) -> c_13():18 -->_1 PLUS#(z0,0()) -> c_2():13 -->_9 ACTIVATE#(z0) -> c_1():12 -->_1 PLUS#(z0,s(z1)) -> c_3(U11'#(tt(),z1,z0)):1 10:S:U22'#(tt(),z0,z1) -> c_12(PLUS#(x(activate(z1),activate(z0)),activate(z1)) ,x#(activate(z1),activate(z0)) ,activate#(z1) ,activate#(z0) ,activate#(z1) ,ACTIVATE#(z1)) -->_2 x#(z0,s(z1)) -> c_23(U21#(tt(),z1,z0)):23 -->_2 x#(z0,0()) -> c_22():22 -->_5 activate#(z0) -> c_19():19 -->_4 activate#(z0) -> c_19():19 -->_3 activate#(z0) -> c_19():19 -->_1 PLUS#(z0,0()) -> c_2():13 -->_6 ACTIVATE#(z0) -> c_1():12 -->_1 PLUS#(z0,s(z1)) -> c_3(U11'#(tt(),z1,z0)):1 11:S:X#(z0,s(z1)) -> c_14(U21'#(tt(),z1,z0)) -->_1 U21'#(tt(),z0,z1) -> c_9(U22'#(tt(),activate(z0),activate(z1)) ,activate#(z0) ,activate#(z1) ,ACTIVATE#(z1)):7 -->_1 U21'#(tt(),z0,z1) -> c_8(U22'#(tt(),activate(z0),activate(z1)) ,activate#(z0) ,activate#(z1) ,ACTIVATE#(z0)):6 12:W:ACTIVATE#(z0) -> c_1() 13:W:PLUS#(z0,0()) -> c_2() 14:W:U11#(tt(),z0,z1) -> c_15(U12#(tt(),activate(z0),activate(z1)),activate#(z0),activate#(z1)) -->_1 U12#(tt(),z0,z1) -> c_16(plus#(activate(z1),activate(z0)),activate#(z1),activate#(z0)):15 -->_3 activate#(z0) -> c_19():19 -->_2 activate#(z0) -> c_19():19 15:W:U12#(tt(),z0,z1) -> c_16(plus#(activate(z1),activate(z0)),activate#(z1),activate#(z0)) -->_1 plus#(z0,s(z1)) -> c_21(U11#(tt(),z1,z0)):21 -->_1 plus#(z0,0()) -> c_20():20 -->_3 activate#(z0) -> c_19():19 -->_2 activate#(z0) -> c_19():19 16:W:U21#(tt(),z0,z1) -> c_17(U22#(tt(),activate(z0),activate(z1)),activate#(z0),activate#(z1)) -->_1 U22#(tt(),z0,z1) -> c_18(plus#(x(activate(z1),activate(z0)),activate(z1)) ,x#(activate(z1),activate(z0)) ,activate#(z1) ,activate#(z0) ,activate#(z1)):17 -->_3 activate#(z0) -> c_19():19 -->_2 activate#(z0) -> c_19():19 17:W:U22#(tt(),z0,z1) -> c_18(plus#(x(activate(z1),activate(z0)),activate(z1)) ,x#(activate(z1),activate(z0)) ,activate#(z1) ,activate#(z0) ,activate#(z1)) -->_2 x#(z0,s(z1)) -> c_23(U21#(tt(),z1,z0)):23 -->_1 plus#(z0,s(z1)) -> c_21(U11#(tt(),z1,z0)):21 -->_2 x#(z0,0()) -> c_22():22 -->_1 plus#(z0,0()) -> c_20():20 -->_5 activate#(z0) -> c_19():19 -->_4 activate#(z0) -> c_19():19 -->_3 activate#(z0) -> c_19():19 18:W:X#(z0,0()) -> c_13() 19:W:activate#(z0) -> c_19() 20:W:plus#(z0,0()) -> c_20() 21:W:plus#(z0,s(z1)) -> c_21(U11#(tt(),z1,z0)) -->_1 U11#(tt(),z0,z1) -> c_15(U12#(tt(),activate(z0),activate(z1)),activate#(z0),activate#(z1)):14 22:W:x#(z0,0()) -> c_22() 23:W:x#(z0,s(z1)) -> c_23(U21#(tt(),z1,z0)) -->_1 U21#(tt(),z0,z1) -> c_17(U22#(tt(),activate(z0),activate(z1)),activate#(z0),activate#(z1)):16 The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed. 18: X#(z0,0()) -> c_13() 23: x#(z0,s(z1)) -> c_23(U21#(tt(),z1,z0)) 17: U22#(tt(),z0,z1) -> c_18(plus#(x(activate(z1),activate(z0)),activate(z1)) ,x#(activate(z1),activate(z0)) ,activate#(z1) ,activate#(z0) ,activate#(z1)) 16: U21#(tt(),z0,z1) -> c_17(U22#(tt(),activate(z0),activate(z1)),activate#(z0),activate#(z1)) 22: x#(z0,0()) -> c_22() 21: plus#(z0,s(z1)) -> c_21(U11#(tt(),z1,z0)) 15: U12#(tt(),z0,z1) -> c_16(plus#(activate(z1),activate(z0)),activate#(z1),activate#(z0)) 14: U11#(tt(),z0,z1) -> c_15(U12#(tt(),activate(z0),activate(z1)),activate#(z0),activate#(z1)) 20: plus#(z0,0()) -> c_20() 12: ACTIVATE#(z0) -> c_1() 13: PLUS#(z0,0()) -> c_2() 19: activate#(z0) -> c_19() * Step 5: SimplifyRHS. WORST_CASE(?,O(n^2)) + Considered Problem: - Strict DPs: PLUS#(z0,s(z1)) -> c_3(U11'#(tt(),z1,z0)) U11'#(tt(),z0,z1) -> c_4(U12'#(tt(),activate(z0),activate(z1)),activate#(z0),activate#(z1),ACTIVATE#(z0)) U11'#(tt(),z0,z1) -> c_5(U12'#(tt(),activate(z0),activate(z1)),activate#(z0),activate#(z1),ACTIVATE#(z1)) U12'#(tt(),z0,z1) -> c_6(PLUS#(activate(z1),activate(z0)),activate#(z1),activate#(z0),ACTIVATE#(z1)) U12'#(tt(),z0,z1) -> c_7(PLUS#(activate(z1),activate(z0)),activate#(z1),activate#(z0),ACTIVATE#(z0)) U21'#(tt(),z0,z1) -> c_8(U22'#(tt(),activate(z0),activate(z1)),activate#(z0),activate#(z1),ACTIVATE#(z0)) U21'#(tt(),z0,z1) -> c_9(U22'#(tt(),activate(z0),activate(z1)),activate#(z0),activate#(z1),ACTIVATE#(z1)) U22'#(tt(),z0,z1) -> c_10(PLUS#(x(activate(z1),activate(z0)),activate(z1)) ,x#(activate(z1),activate(z0)) ,activate#(z1) ,activate#(z0) ,activate#(z1) ,X#(activate(z1),activate(z0)) ,activate#(z1) ,activate#(z0) ,ACTIVATE#(z1)) U22'#(tt(),z0,z1) -> c_11(PLUS#(x(activate(z1),activate(z0)),activate(z1)) ,x#(activate(z1),activate(z0)) ,activate#(z1) ,activate#(z0) ,activate#(z1) ,X#(activate(z1),activate(z0)) ,activate#(z1) ,activate#(z0) ,ACTIVATE#(z0)) U22'#(tt(),z0,z1) -> c_12(PLUS#(x(activate(z1),activate(z0)),activate(z1)) ,x#(activate(z1),activate(z0)) ,activate#(z1) ,activate#(z0) ,activate#(z1) ,ACTIVATE#(z1)) X#(z0,s(z1)) -> c_14(U21'#(tt(),z1,z0)) - Weak TRS: ACTIVATE(z0) -> c13() PLUS(z0,0()) -> c9() PLUS(z0,s(z1)) -> c10(U11'(tt(),z1,z0)) U11(tt(),z0,z1) -> U12(tt(),activate(z0),activate(z1)) U11'(tt(),z0,z1) -> c(U12'(tt(),activate(z0),activate(z1)),ACTIVATE(z0)) U11'(tt(),z0,z1) -> c1(U12'(tt(),activate(z0),activate(z1)),ACTIVATE(z1)) U12(tt(),z0,z1) -> s(plus(activate(z1),activate(z0))) U12'(tt(),z0,z1) -> c2(PLUS(activate(z1),activate(z0)),ACTIVATE(z1)) U12'(tt(),z0,z1) -> c3(PLUS(activate(z1),activate(z0)),ACTIVATE(z0)) U21(tt(),z0,z1) -> U22(tt(),activate(z0),activate(z1)) U21'(tt(),z0,z1) -> c4(U22'(tt(),activate(z0),activate(z1)),ACTIVATE(z0)) U21'(tt(),z0,z1) -> c5(U22'(tt(),activate(z0),activate(z1)),ACTIVATE(z1)) U22(tt(),z0,z1) -> plus(x(activate(z1),activate(z0)),activate(z1)) U22'(tt(),z0,z1) -> c6(PLUS(x(activate(z1),activate(z0)),activate(z1)) ,X(activate(z1),activate(z0)) ,ACTIVATE(z1)) U22'(tt(),z0,z1) -> c7(PLUS(x(activate(z1),activate(z0)),activate(z1)) ,X(activate(z1),activate(z0)) ,ACTIVATE(z0)) U22'(tt(),z0,z1) -> c8(PLUS(x(activate(z1),activate(z0)),activate(z1)),ACTIVATE(z1)) X(z0,0()) -> c11() X(z0,s(z1)) -> c12(U21'(tt(),z1,z0)) activate(z0) -> z0 plus(z0,0()) -> z0 plus(z0,s(z1)) -> U11(tt(),z1,z0) x(z0,0()) -> 0() x(z0,s(z1)) -> U21(tt(),z1,z0) - Signature: {ACTIVATE/1,PLUS/2,U11/3,U11'/3,U12/3,U12'/3,U21/3,U21'/3,U22/3,U22'/3,X/2,activate/1,plus/2,x/2,ACTIVATE#/1 ,PLUS#/2,U11#/3,U11'#/3,U12#/3,U12'#/3,U21#/3,U21'#/3,U22#/3,U22'#/3,X#/2,activate#/1,plus#/2,x#/2} / {0/0 ,c/2,c1/2,c10/1,c11/0,c12/1,c13/0,c2/2,c3/2,c4/2,c5/2,c6/3,c7/3,c8/2,c9/0,s/1,tt/0,c_1/0,c_2/0,c_3/1,c_4/4 ,c_5/4,c_6/4,c_7/4,c_8/4,c_9/4,c_10/9,c_11/9,c_12/6,c_13/0,c_14/1,c_15/3,c_16/3,c_17/3,c_18/5,c_19/0,c_20/0 ,c_21/1,c_22/0,c_23/1} - Obligation: innermost runtime complexity wrt. defined symbols {ACTIVATE#,PLUS#,U11#,U11'#,U12#,U12'#,U21#,U21'#,U22# ,U22'#,X#,activate#,plus#,x#} and constructors {0,c,c1,c10,c11,c12,c13,c2,c3,c4,c5,c6,c7,c8,c9,s,tt} + Applied Processor: SimplifyRHS + Details: Consider the dependency graph 1:S:PLUS#(z0,s(z1)) -> c_3(U11'#(tt(),z1,z0)) -->_1 U11'#(tt(),z0,z1) -> c_5(U12'#(tt(),activate(z0),activate(z1)) ,activate#(z0) ,activate#(z1) ,ACTIVATE#(z1)):3 -->_1 U11'#(tt(),z0,z1) -> c_4(U12'#(tt(),activate(z0),activate(z1)) ,activate#(z0) ,activate#(z1) ,ACTIVATE#(z0)):2 2:S:U11'#(tt(),z0,z1) -> c_4(U12'#(tt(),activate(z0),activate(z1)) ,activate#(z0) ,activate#(z1) ,ACTIVATE#(z0)) -->_1 U12'#(tt(),z0,z1) -> c_7(PLUS#(activate(z1),activate(z0)) ,activate#(z1) ,activate#(z0) ,ACTIVATE#(z0)):5 -->_1 U12'#(tt(),z0,z1) -> c_6(PLUS#(activate(z1),activate(z0)) ,activate#(z1) ,activate#(z0) ,ACTIVATE#(z1)):4 3:S:U11'#(tt(),z0,z1) -> c_5(U12'#(tt(),activate(z0),activate(z1)) ,activate#(z0) ,activate#(z1) ,ACTIVATE#(z1)) -->_1 U12'#(tt(),z0,z1) -> c_7(PLUS#(activate(z1),activate(z0)) ,activate#(z1) ,activate#(z0) ,ACTIVATE#(z0)):5 -->_1 U12'#(tt(),z0,z1) -> c_6(PLUS#(activate(z1),activate(z0)) ,activate#(z1) ,activate#(z0) ,ACTIVATE#(z1)):4 4:S:U12'#(tt(),z0,z1) -> c_6(PLUS#(activate(z1),activate(z0)),activate#(z1),activate#(z0),ACTIVATE#(z1)) -->_1 PLUS#(z0,s(z1)) -> c_3(U11'#(tt(),z1,z0)):1 5:S:U12'#(tt(),z0,z1) -> c_7(PLUS#(activate(z1),activate(z0)),activate#(z1),activate#(z0),ACTIVATE#(z0)) -->_1 PLUS#(z0,s(z1)) -> c_3(U11'#(tt(),z1,z0)):1 6:S:U21'#(tt(),z0,z1) -> c_8(U22'#(tt(),activate(z0),activate(z1)) ,activate#(z0) ,activate#(z1) ,ACTIVATE#(z0)) -->_1 U22'#(tt(),z0,z1) -> c_12(PLUS#(x(activate(z1),activate(z0)),activate(z1)) ,x#(activate(z1),activate(z0)) ,activate#(z1) ,activate#(z0) ,activate#(z1) ,ACTIVATE#(z1)):10 -->_1 U22'#(tt(),z0,z1) -> c_11(PLUS#(x(activate(z1),activate(z0)),activate(z1)) ,x#(activate(z1),activate(z0)) ,activate#(z1) ,activate#(z0) ,activate#(z1) ,X#(activate(z1),activate(z0)) ,activate#(z1) ,activate#(z0) ,ACTIVATE#(z0)):9 -->_1 U22'#(tt(),z0,z1) -> c_10(PLUS#(x(activate(z1),activate(z0)),activate(z1)) ,x#(activate(z1),activate(z0)) ,activate#(z1) ,activate#(z0) ,activate#(z1) ,X#(activate(z1),activate(z0)) ,activate#(z1) ,activate#(z0) ,ACTIVATE#(z1)):8 7:S:U21'#(tt(),z0,z1) -> c_9(U22'#(tt(),activate(z0),activate(z1)) ,activate#(z0) ,activate#(z1) ,ACTIVATE#(z1)) -->_1 U22'#(tt(),z0,z1) -> c_12(PLUS#(x(activate(z1),activate(z0)),activate(z1)) ,x#(activate(z1),activate(z0)) ,activate#(z1) ,activate#(z0) ,activate#(z1) ,ACTIVATE#(z1)):10 -->_1 U22'#(tt(),z0,z1) -> c_11(PLUS#(x(activate(z1),activate(z0)),activate(z1)) ,x#(activate(z1),activate(z0)) ,activate#(z1) ,activate#(z0) ,activate#(z1) ,X#(activate(z1),activate(z0)) ,activate#(z1) ,activate#(z0) ,ACTIVATE#(z0)):9 -->_1 U22'#(tt(),z0,z1) -> c_10(PLUS#(x(activate(z1),activate(z0)),activate(z1)) ,x#(activate(z1),activate(z0)) ,activate#(z1) ,activate#(z0) ,activate#(z1) ,X#(activate(z1),activate(z0)) ,activate#(z1) ,activate#(z0) ,ACTIVATE#(z1)):8 8:S:U22'#(tt(),z0,z1) -> c_10(PLUS#(x(activate(z1),activate(z0)),activate(z1)) ,x#(activate(z1),activate(z0)) ,activate#(z1) ,activate#(z0) ,activate#(z1) ,X#(activate(z1),activate(z0)) ,activate#(z1) ,activate#(z0) ,ACTIVATE#(z1)) -->_6 X#(z0,s(z1)) -> c_14(U21'#(tt(),z1,z0)):11 -->_1 PLUS#(z0,s(z1)) -> c_3(U11'#(tt(),z1,z0)):1 9:S:U22'#(tt(),z0,z1) -> c_11(PLUS#(x(activate(z1),activate(z0)),activate(z1)) ,x#(activate(z1),activate(z0)) ,activate#(z1) ,activate#(z0) ,activate#(z1) ,X#(activate(z1),activate(z0)) ,activate#(z1) ,activate#(z0) ,ACTIVATE#(z0)) -->_6 X#(z0,s(z1)) -> c_14(U21'#(tt(),z1,z0)):11 -->_1 PLUS#(z0,s(z1)) -> c_3(U11'#(tt(),z1,z0)):1 10:S:U22'#(tt(),z0,z1) -> c_12(PLUS#(x(activate(z1),activate(z0)),activate(z1)) ,x#(activate(z1),activate(z0)) ,activate#(z1) ,activate#(z0) ,activate#(z1) ,ACTIVATE#(z1)) -->_1 PLUS#(z0,s(z1)) -> c_3(U11'#(tt(),z1,z0)):1 11:S:X#(z0,s(z1)) -> c_14(U21'#(tt(),z1,z0)) -->_1 U21'#(tt(),z0,z1) -> c_9(U22'#(tt(),activate(z0),activate(z1)) ,activate#(z0) ,activate#(z1) ,ACTIVATE#(z1)):7 -->_1 U21'#(tt(),z0,z1) -> c_8(U22'#(tt(),activate(z0),activate(z1)) ,activate#(z0) ,activate#(z1) ,ACTIVATE#(z0)):6 Due to missing edges in the depndency graph, the right-hand sides of following rules could be simplified: U11'#(tt(),z0,z1) -> c_4(U12'#(tt(),activate(z0),activate(z1))) U11'#(tt(),z0,z1) -> c_5(U12'#(tt(),activate(z0),activate(z1))) U12'#(tt(),z0,z1) -> c_6(PLUS#(activate(z1),activate(z0))) U12'#(tt(),z0,z1) -> c_7(PLUS#(activate(z1),activate(z0))) U21'#(tt(),z0,z1) -> c_8(U22'#(tt(),activate(z0),activate(z1))) U21'#(tt(),z0,z1) -> c_9(U22'#(tt(),activate(z0),activate(z1))) U22'#(tt(),z0,z1) -> c_10(PLUS#(x(activate(z1),activate(z0)),activate(z1)),X#(activate(z1),activate(z0))) U22'#(tt(),z0,z1) -> c_11(PLUS#(x(activate(z1),activate(z0)),activate(z1)),X#(activate(z1),activate(z0))) U22'#(tt(),z0,z1) -> c_12(PLUS#(x(activate(z1),activate(z0)),activate(z1))) * Step 6: UsableRules. WORST_CASE(?,O(n^2)) + Considered Problem: - Strict DPs: PLUS#(z0,s(z1)) -> c_3(U11'#(tt(),z1,z0)) U11'#(tt(),z0,z1) -> c_4(U12'#(tt(),activate(z0),activate(z1))) U11'#(tt(),z0,z1) -> c_5(U12'#(tt(),activate(z0),activate(z1))) U12'#(tt(),z0,z1) -> c_6(PLUS#(activate(z1),activate(z0))) U12'#(tt(),z0,z1) -> c_7(PLUS#(activate(z1),activate(z0))) U21'#(tt(),z0,z1) -> c_8(U22'#(tt(),activate(z0),activate(z1))) U21'#(tt(),z0,z1) -> c_9(U22'#(tt(),activate(z0),activate(z1))) U22'#(tt(),z0,z1) -> c_10(PLUS#(x(activate(z1),activate(z0)),activate(z1)),X#(activate(z1),activate(z0))) U22'#(tt(),z0,z1) -> c_11(PLUS#(x(activate(z1),activate(z0)),activate(z1)),X#(activate(z1),activate(z0))) U22'#(tt(),z0,z1) -> c_12(PLUS#(x(activate(z1),activate(z0)),activate(z1))) X#(z0,s(z1)) -> c_14(U21'#(tt(),z1,z0)) - Weak TRS: ACTIVATE(z0) -> c13() PLUS(z0,0()) -> c9() PLUS(z0,s(z1)) -> c10(U11'(tt(),z1,z0)) U11(tt(),z0,z1) -> U12(tt(),activate(z0),activate(z1)) U11'(tt(),z0,z1) -> c(U12'(tt(),activate(z0),activate(z1)),ACTIVATE(z0)) U11'(tt(),z0,z1) -> c1(U12'(tt(),activate(z0),activate(z1)),ACTIVATE(z1)) U12(tt(),z0,z1) -> s(plus(activate(z1),activate(z0))) U12'(tt(),z0,z1) -> c2(PLUS(activate(z1),activate(z0)),ACTIVATE(z1)) U12'(tt(),z0,z1) -> c3(PLUS(activate(z1),activate(z0)),ACTIVATE(z0)) U21(tt(),z0,z1) -> U22(tt(),activate(z0),activate(z1)) U21'(tt(),z0,z1) -> c4(U22'(tt(),activate(z0),activate(z1)),ACTIVATE(z0)) U21'(tt(),z0,z1) -> c5(U22'(tt(),activate(z0),activate(z1)),ACTIVATE(z1)) U22(tt(),z0,z1) -> plus(x(activate(z1),activate(z0)),activate(z1)) U22'(tt(),z0,z1) -> c6(PLUS(x(activate(z1),activate(z0)),activate(z1)) ,X(activate(z1),activate(z0)) ,ACTIVATE(z1)) U22'(tt(),z0,z1) -> c7(PLUS(x(activate(z1),activate(z0)),activate(z1)) ,X(activate(z1),activate(z0)) ,ACTIVATE(z0)) U22'(tt(),z0,z1) -> c8(PLUS(x(activate(z1),activate(z0)),activate(z1)),ACTIVATE(z1)) X(z0,0()) -> c11() X(z0,s(z1)) -> c12(U21'(tt(),z1,z0)) activate(z0) -> z0 plus(z0,0()) -> z0 plus(z0,s(z1)) -> U11(tt(),z1,z0) x(z0,0()) -> 0() x(z0,s(z1)) -> U21(tt(),z1,z0) - Signature: {ACTIVATE/1,PLUS/2,U11/3,U11'/3,U12/3,U12'/3,U21/3,U21'/3,U22/3,U22'/3,X/2,activate/1,plus/2,x/2,ACTIVATE#/1 ,PLUS#/2,U11#/3,U11'#/3,U12#/3,U12'#/3,U21#/3,U21'#/3,U22#/3,U22'#/3,X#/2,activate#/1,plus#/2,x#/2} / {0/0 ,c/2,c1/2,c10/1,c11/0,c12/1,c13/0,c2/2,c3/2,c4/2,c5/2,c6/3,c7/3,c8/2,c9/0,s/1,tt/0,c_1/0,c_2/0,c_3/1,c_4/1 ,c_5/1,c_6/1,c_7/1,c_8/1,c_9/1,c_10/2,c_11/2,c_12/1,c_13/0,c_14/1,c_15/3,c_16/3,c_17/3,c_18/5,c_19/0,c_20/0 ,c_21/1,c_22/0,c_23/1} - Obligation: innermost runtime complexity wrt. defined symbols {ACTIVATE#,PLUS#,U11#,U11'#,U12#,U12'#,U21#,U21'#,U22# ,U22'#,X#,activate#,plus#,x#} and constructors {0,c,c1,c10,c11,c12,c13,c2,c3,c4,c5,c6,c7,c8,c9,s,tt} + Applied Processor: UsableRules + Details: We replace rewrite rules by usable rules: U11(tt(),z0,z1) -> U12(tt(),activate(z0),activate(z1)) U12(tt(),z0,z1) -> s(plus(activate(z1),activate(z0))) U21(tt(),z0,z1) -> U22(tt(),activate(z0),activate(z1)) U22(tt(),z0,z1) -> plus(x(activate(z1),activate(z0)),activate(z1)) activate(z0) -> z0 plus(z0,0()) -> z0 plus(z0,s(z1)) -> U11(tt(),z1,z0) x(z0,0()) -> 0() x(z0,s(z1)) -> U21(tt(),z1,z0) PLUS#(z0,s(z1)) -> c_3(U11'#(tt(),z1,z0)) U11'#(tt(),z0,z1) -> c_4(U12'#(tt(),activate(z0),activate(z1))) U11'#(tt(),z0,z1) -> c_5(U12'#(tt(),activate(z0),activate(z1))) U12'#(tt(),z0,z1) -> c_6(PLUS#(activate(z1),activate(z0))) U12'#(tt(),z0,z1) -> c_7(PLUS#(activate(z1),activate(z0))) U21'#(tt(),z0,z1) -> c_8(U22'#(tt(),activate(z0),activate(z1))) U21'#(tt(),z0,z1) -> c_9(U22'#(tt(),activate(z0),activate(z1))) U22'#(tt(),z0,z1) -> c_10(PLUS#(x(activate(z1),activate(z0)),activate(z1)),X#(activate(z1),activate(z0))) U22'#(tt(),z0,z1) -> c_11(PLUS#(x(activate(z1),activate(z0)),activate(z1)),X#(activate(z1),activate(z0))) U22'#(tt(),z0,z1) -> c_12(PLUS#(x(activate(z1),activate(z0)),activate(z1))) X#(z0,s(z1)) -> c_14(U21'#(tt(),z1,z0)) * Step 7: DecomposeDG. WORST_CASE(?,O(n^2)) + Considered Problem: - Strict DPs: PLUS#(z0,s(z1)) -> c_3(U11'#(tt(),z1,z0)) U11'#(tt(),z0,z1) -> c_4(U12'#(tt(),activate(z0),activate(z1))) U11'#(tt(),z0,z1) -> c_5(U12'#(tt(),activate(z0),activate(z1))) U12'#(tt(),z0,z1) -> c_6(PLUS#(activate(z1),activate(z0))) U12'#(tt(),z0,z1) -> c_7(PLUS#(activate(z1),activate(z0))) U21'#(tt(),z0,z1) -> c_8(U22'#(tt(),activate(z0),activate(z1))) U21'#(tt(),z0,z1) -> c_9(U22'#(tt(),activate(z0),activate(z1))) U22'#(tt(),z0,z1) -> c_10(PLUS#(x(activate(z1),activate(z0)),activate(z1)),X#(activate(z1),activate(z0))) U22'#(tt(),z0,z1) -> c_11(PLUS#(x(activate(z1),activate(z0)),activate(z1)),X#(activate(z1),activate(z0))) U22'#(tt(),z0,z1) -> c_12(PLUS#(x(activate(z1),activate(z0)),activate(z1))) X#(z0,s(z1)) -> c_14(U21'#(tt(),z1,z0)) - Weak TRS: U11(tt(),z0,z1) -> U12(tt(),activate(z0),activate(z1)) U12(tt(),z0,z1) -> s(plus(activate(z1),activate(z0))) U21(tt(),z0,z1) -> U22(tt(),activate(z0),activate(z1)) U22(tt(),z0,z1) -> plus(x(activate(z1),activate(z0)),activate(z1)) activate(z0) -> z0 plus(z0,0()) -> z0 plus(z0,s(z1)) -> U11(tt(),z1,z0) x(z0,0()) -> 0() x(z0,s(z1)) -> U21(tt(),z1,z0) - Signature: {ACTIVATE/1,PLUS/2,U11/3,U11'/3,U12/3,U12'/3,U21/3,U21'/3,U22/3,U22'/3,X/2,activate/1,plus/2,x/2,ACTIVATE#/1 ,PLUS#/2,U11#/3,U11'#/3,U12#/3,U12'#/3,U21#/3,U21'#/3,U22#/3,U22'#/3,X#/2,activate#/1,plus#/2,x#/2} / {0/0 ,c/2,c1/2,c10/1,c11/0,c12/1,c13/0,c2/2,c3/2,c4/2,c5/2,c6/3,c7/3,c8/2,c9/0,s/1,tt/0,c_1/0,c_2/0,c_3/1,c_4/1 ,c_5/1,c_6/1,c_7/1,c_8/1,c_9/1,c_10/2,c_11/2,c_12/1,c_13/0,c_14/1,c_15/3,c_16/3,c_17/3,c_18/5,c_19/0,c_20/0 ,c_21/1,c_22/0,c_23/1} - Obligation: innermost runtime complexity wrt. defined symbols {ACTIVATE#,PLUS#,U11#,U11'#,U12#,U12'#,U21#,U21'#,U22# ,U22'#,X#,activate#,plus#,x#} and constructors {0,c,c1,c10,c11,c12,c13,c2,c3,c4,c5,c6,c7,c8,c9,s,tt} + Applied Processor: DecomposeDG {onSelection = all below first cut in WDG, onUpper = Nothing, onLower = Nothing} + Details: We decompose the input problem according to the dependency graph into the upper component U21'#(tt(),z0,z1) -> c_8(U22'#(tt(),activate(z0),activate(z1))) U21'#(tt(),z0,z1) -> c_9(U22'#(tt(),activate(z0),activate(z1))) U22'#(tt(),z0,z1) -> c_10(PLUS#(x(activate(z1),activate(z0)),activate(z1)),X#(activate(z1),activate(z0))) U22'#(tt(),z0,z1) -> c_11(PLUS#(x(activate(z1),activate(z0)),activate(z1)),X#(activate(z1),activate(z0))) X#(z0,s(z1)) -> c_14(U21'#(tt(),z1,z0)) and a lower component PLUS#(z0,s(z1)) -> c_3(U11'#(tt(),z1,z0)) U11'#(tt(),z0,z1) -> c_4(U12'#(tt(),activate(z0),activate(z1))) U11'#(tt(),z0,z1) -> c_5(U12'#(tt(),activate(z0),activate(z1))) U12'#(tt(),z0,z1) -> c_6(PLUS#(activate(z1),activate(z0))) U12'#(tt(),z0,z1) -> c_7(PLUS#(activate(z1),activate(z0))) U22'#(tt(),z0,z1) -> c_12(PLUS#(x(activate(z1),activate(z0)),activate(z1))) Further, following extension rules are added to the lower component. U21'#(tt(),z0,z1) -> U22'#(tt(),activate(z0),activate(z1)) U22'#(tt(),z0,z1) -> PLUS#(x(activate(z1),activate(z0)),activate(z1)) U22'#(tt(),z0,z1) -> X#(activate(z1),activate(z0)) X#(z0,s(z1)) -> U21'#(tt(),z1,z0) ** Step 7.a:1: SimplifyRHS. WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: U21'#(tt(),z0,z1) -> c_8(U22'#(tt(),activate(z0),activate(z1))) U21'#(tt(),z0,z1) -> c_9(U22'#(tt(),activate(z0),activate(z1))) U22'#(tt(),z0,z1) -> c_10(PLUS#(x(activate(z1),activate(z0)),activate(z1)),X#(activate(z1),activate(z0))) U22'#(tt(),z0,z1) -> c_11(PLUS#(x(activate(z1),activate(z0)),activate(z1)),X#(activate(z1),activate(z0))) X#(z0,s(z1)) -> c_14(U21'#(tt(),z1,z0)) - Weak TRS: U11(tt(),z0,z1) -> U12(tt(),activate(z0),activate(z1)) U12(tt(),z0,z1) -> s(plus(activate(z1),activate(z0))) U21(tt(),z0,z1) -> U22(tt(),activate(z0),activate(z1)) U22(tt(),z0,z1) -> plus(x(activate(z1),activate(z0)),activate(z1)) activate(z0) -> z0 plus(z0,0()) -> z0 plus(z0,s(z1)) -> U11(tt(),z1,z0) x(z0,0()) -> 0() x(z0,s(z1)) -> U21(tt(),z1,z0) - Signature: {ACTIVATE/1,PLUS/2,U11/3,U11'/3,U12/3,U12'/3,U21/3,U21'/3,U22/3,U22'/3,X/2,activate/1,plus/2,x/2,ACTIVATE#/1 ,PLUS#/2,U11#/3,U11'#/3,U12#/3,U12'#/3,U21#/3,U21'#/3,U22#/3,U22'#/3,X#/2,activate#/1,plus#/2,x#/2} / {0/0 ,c/2,c1/2,c10/1,c11/0,c12/1,c13/0,c2/2,c3/2,c4/2,c5/2,c6/3,c7/3,c8/2,c9/0,s/1,tt/0,c_1/0,c_2/0,c_3/1,c_4/1 ,c_5/1,c_6/1,c_7/1,c_8/1,c_9/1,c_10/2,c_11/2,c_12/1,c_13/0,c_14/1,c_15/3,c_16/3,c_17/3,c_18/5,c_19/0,c_20/0 ,c_21/1,c_22/0,c_23/1} - Obligation: innermost runtime complexity wrt. defined symbols {ACTIVATE#,PLUS#,U11#,U11'#,U12#,U12'#,U21#,U21'#,U22# ,U22'#,X#,activate#,plus#,x#} and constructors {0,c,c1,c10,c11,c12,c13,c2,c3,c4,c5,c6,c7,c8,c9,s,tt} + Applied Processor: SimplifyRHS + Details: Consider the dependency graph 1:S:U21'#(tt(),z0,z1) -> c_8(U22'#(tt(),activate(z0),activate(z1))) -->_1 U22'#(tt(),z0,z1) -> c_11(PLUS#(x(activate(z1),activate(z0)),activate(z1)) ,X#(activate(z1),activate(z0))):4 -->_1 U22'#(tt(),z0,z1) -> c_10(PLUS#(x(activate(z1),activate(z0)),activate(z1)) ,X#(activate(z1),activate(z0))):3 2:S:U21'#(tt(),z0,z1) -> c_9(U22'#(tt(),activate(z0),activate(z1))) -->_1 U22'#(tt(),z0,z1) -> c_11(PLUS#(x(activate(z1),activate(z0)),activate(z1)) ,X#(activate(z1),activate(z0))):4 -->_1 U22'#(tt(),z0,z1) -> c_10(PLUS#(x(activate(z1),activate(z0)),activate(z1)) ,X#(activate(z1),activate(z0))):3 3:S:U22'#(tt(),z0,z1) -> c_10(PLUS#(x(activate(z1),activate(z0)),activate(z1)) ,X#(activate(z1),activate(z0))) -->_2 X#(z0,s(z1)) -> c_14(U21'#(tt(),z1,z0)):5 4:S:U22'#(tt(),z0,z1) -> c_11(PLUS#(x(activate(z1),activate(z0)),activate(z1)) ,X#(activate(z1),activate(z0))) -->_2 X#(z0,s(z1)) -> c_14(U21'#(tt(),z1,z0)):5 5:S:X#(z0,s(z1)) -> c_14(U21'#(tt(),z1,z0)) -->_1 U21'#(tt(),z0,z1) -> c_9(U22'#(tt(),activate(z0),activate(z1))):2 -->_1 U21'#(tt(),z0,z1) -> c_8(U22'#(tt(),activate(z0),activate(z1))):1 Due to missing edges in the depndency graph, the right-hand sides of following rules could be simplified: U22'#(tt(),z0,z1) -> c_10(X#(activate(z1),activate(z0))) U22'#(tt(),z0,z1) -> c_11(X#(activate(z1),activate(z0))) ** Step 7.a:2: UsableRules. WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: U21'#(tt(),z0,z1) -> c_8(U22'#(tt(),activate(z0),activate(z1))) U21'#(tt(),z0,z1) -> c_9(U22'#(tt(),activate(z0),activate(z1))) U22'#(tt(),z0,z1) -> c_10(X#(activate(z1),activate(z0))) U22'#(tt(),z0,z1) -> c_11(X#(activate(z1),activate(z0))) X#(z0,s(z1)) -> c_14(U21'#(tt(),z1,z0)) - Weak TRS: U11(tt(),z0,z1) -> U12(tt(),activate(z0),activate(z1)) U12(tt(),z0,z1) -> s(plus(activate(z1),activate(z0))) U21(tt(),z0,z1) -> U22(tt(),activate(z0),activate(z1)) U22(tt(),z0,z1) -> plus(x(activate(z1),activate(z0)),activate(z1)) activate(z0) -> z0 plus(z0,0()) -> z0 plus(z0,s(z1)) -> U11(tt(),z1,z0) x(z0,0()) -> 0() x(z0,s(z1)) -> U21(tt(),z1,z0) - Signature: {ACTIVATE/1,PLUS/2,U11/3,U11'/3,U12/3,U12'/3,U21/3,U21'/3,U22/3,U22'/3,X/2,activate/1,plus/2,x/2,ACTIVATE#/1 ,PLUS#/2,U11#/3,U11'#/3,U12#/3,U12'#/3,U21#/3,U21'#/3,U22#/3,U22'#/3,X#/2,activate#/1,plus#/2,x#/2} / {0/0 ,c/2,c1/2,c10/1,c11/0,c12/1,c13/0,c2/2,c3/2,c4/2,c5/2,c6/3,c7/3,c8/2,c9/0,s/1,tt/0,c_1/0,c_2/0,c_3/1,c_4/1 ,c_5/1,c_6/1,c_7/1,c_8/1,c_9/1,c_10/1,c_11/1,c_12/1,c_13/0,c_14/1,c_15/3,c_16/3,c_17/3,c_18/5,c_19/0,c_20/0 ,c_21/1,c_22/0,c_23/1} - Obligation: innermost runtime complexity wrt. defined symbols {ACTIVATE#,PLUS#,U11#,U11'#,U12#,U12'#,U21#,U21'#,U22# ,U22'#,X#,activate#,plus#,x#} and constructors {0,c,c1,c10,c11,c12,c13,c2,c3,c4,c5,c6,c7,c8,c9,s,tt} + Applied Processor: UsableRules + Details: We replace rewrite rules by usable rules: activate(z0) -> z0 U21'#(tt(),z0,z1) -> c_8(U22'#(tt(),activate(z0),activate(z1))) U21'#(tt(),z0,z1) -> c_9(U22'#(tt(),activate(z0),activate(z1))) U22'#(tt(),z0,z1) -> c_10(X#(activate(z1),activate(z0))) U22'#(tt(),z0,z1) -> c_11(X#(activate(z1),activate(z0))) X#(z0,s(z1)) -> c_14(U21'#(tt(),z1,z0)) ** Step 7.a:3: WeightGap. WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: U21'#(tt(),z0,z1) -> c_8(U22'#(tt(),activate(z0),activate(z1))) U21'#(tt(),z0,z1) -> c_9(U22'#(tt(),activate(z0),activate(z1))) U22'#(tt(),z0,z1) -> c_10(X#(activate(z1),activate(z0))) U22'#(tt(),z0,z1) -> c_11(X#(activate(z1),activate(z0))) X#(z0,s(z1)) -> c_14(U21'#(tt(),z1,z0)) - Weak TRS: activate(z0) -> z0 - Signature: {ACTIVATE/1,PLUS/2,U11/3,U11'/3,U12/3,U12'/3,U21/3,U21'/3,U22/3,U22'/3,X/2,activate/1,plus/2,x/2,ACTIVATE#/1 ,PLUS#/2,U11#/3,U11'#/3,U12#/3,U12'#/3,U21#/3,U21'#/3,U22#/3,U22'#/3,X#/2,activate#/1,plus#/2,x#/2} / {0/0 ,c/2,c1/2,c10/1,c11/0,c12/1,c13/0,c2/2,c3/2,c4/2,c5/2,c6/3,c7/3,c8/2,c9/0,s/1,tt/0,c_1/0,c_2/0,c_3/1,c_4/1 ,c_5/1,c_6/1,c_7/1,c_8/1,c_9/1,c_10/1,c_11/1,c_12/1,c_13/0,c_14/1,c_15/3,c_16/3,c_17/3,c_18/5,c_19/0,c_20/0 ,c_21/1,c_22/0,c_23/1} - Obligation: innermost runtime complexity wrt. defined symbols {ACTIVATE#,PLUS#,U11#,U11'#,U12#,U12'#,U21#,U21'#,U22# ,U22'#,X#,activate#,plus#,x#} and constructors {0,c,c1,c10,c11,c12,c13,c2,c3,c4,c5,c6,c7,c8,c9,s,tt} + Applied Processor: WeightGap {wgDimension = 1, wgDegree = 1, wgKind = Algebraic, wgUArgs = UArgs, wgOn = WgOnAny} + Details: The weightgap principle applies using the following constant growth matrix-interpretation: We apply a matrix interpretation of kind constructor based matrix interpretation: The following argument positions are considered usable: uargs(U22'#) = {2,3}, uargs(X#) = {1,2}, uargs(c_8) = {1}, uargs(c_9) = {1}, uargs(c_10) = {1}, uargs(c_11) = {1}, uargs(c_14) = {1} Following symbols are considered usable: all TcT has computed the following interpretation: p(0) = [2] p(ACTIVATE) = [1] p(PLUS) = [0] p(U11) = [0] p(U11') = [0] p(U12) = [0] p(U12') = [0] p(U21) = [0] p(U21') = [0] p(U22) = [0] p(U22') = [0] p(X) = [1] x2 + [0] p(activate) = [1] x1 + [2] p(c) = [1] x1 + [1] x2 + [0] p(c1) = [1] x1 + [1] x2 + [0] p(c10) = [1] x1 + [0] p(c11) = [0] p(c12) = [1] x1 + [0] p(c13) = [0] p(c2) = [1] x1 + [1] x2 + [0] p(c3) = [1] x1 + [1] x2 + [0] p(c4) = [1] x1 + [1] x2 + [0] p(c5) = [1] x1 + [1] x2 + [0] p(c6) = [1] x2 + [1] x3 + [0] p(c7) = [1] x1 + [0] p(c8) = [1] x1 + [1] x2 + [0] p(c9) = [0] p(plus) = [4] x2 + [0] p(s) = [1] x1 + [1] p(tt) = [6] p(x) = [0] p(ACTIVATE#) = [0] p(PLUS#) = [0] p(U11#) = [0] p(U11'#) = [2] p(U12#) = [0] p(U12'#) = [0] p(U21#) = [0] p(U21'#) = [2] x1 + [1] x2 + [1] x3 + [12] p(U22#) = [4] x1 + [1] x2 + [1] x3 + [2] p(U22'#) = [3] x1 + [1] x2 + [1] x3 + [0] p(X#) = [1] x1 + [1] x2 + [4] p(activate#) = [0] p(plus#) = [0] p(x#) = [0] p(c_1) = [0] p(c_2) = [0] p(c_3) = [0] p(c_4) = [0] p(c_5) = [0] p(c_6) = [0] p(c_7) = [0] p(c_8) = [1] x1 + [0] p(c_9) = [1] x1 + [0] p(c_10) = [1] x1 + [0] p(c_11) = [1] x1 + [0] p(c_12) = [0] p(c_13) = [0] p(c_14) = [1] x1 + [0] p(c_15) = [0] p(c_16) = [0] p(c_17) = [0] p(c_18) = [0] p(c_19) = [0] p(c_20) = [0] p(c_21) = [0] p(c_22) = [0] p(c_23) = [0] Following rules are strictly oriented: U21'#(tt(),z0,z1) = [1] z0 + [1] z1 + [24] > [1] z0 + [1] z1 + [22] = c_8(U22'#(tt(),activate(z0),activate(z1))) U21'#(tt(),z0,z1) = [1] z0 + [1] z1 + [24] > [1] z0 + [1] z1 + [22] = c_9(U22'#(tt(),activate(z0),activate(z1))) U22'#(tt(),z0,z1) = [1] z0 + [1] z1 + [18] > [1] z0 + [1] z1 + [8] = c_10(X#(activate(z1),activate(z0))) U22'#(tt(),z0,z1) = [1] z0 + [1] z1 + [18] > [1] z0 + [1] z1 + [8] = c_11(X#(activate(z1),activate(z0))) Following rules are (at-least) weakly oriented: X#(z0,s(z1)) = [1] z0 + [1] z1 + [5] >= [1] z0 + [1] z1 + [24] = c_14(U21'#(tt(),z1,z0)) activate(z0) = [1] z0 + [2] >= [1] z0 + [0] = z0 Further, it can be verified that all rules not oriented are covered by the weightgap condition. ** Step 7.a:4: WeightGap. WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: X#(z0,s(z1)) -> c_14(U21'#(tt(),z1,z0)) - Weak DPs: U21'#(tt(),z0,z1) -> c_8(U22'#(tt(),activate(z0),activate(z1))) U21'#(tt(),z0,z1) -> c_9(U22'#(tt(),activate(z0),activate(z1))) U22'#(tt(),z0,z1) -> c_10(X#(activate(z1),activate(z0))) U22'#(tt(),z0,z1) -> c_11(X#(activate(z1),activate(z0))) - Weak TRS: activate(z0) -> z0 - Signature: {ACTIVATE/1,PLUS/2,U11/3,U11'/3,U12/3,U12'/3,U21/3,U21'/3,U22/3,U22'/3,X/2,activate/1,plus/2,x/2,ACTIVATE#/1 ,PLUS#/2,U11#/3,U11'#/3,U12#/3,U12'#/3,U21#/3,U21'#/3,U22#/3,U22'#/3,X#/2,activate#/1,plus#/2,x#/2} / {0/0 ,c/2,c1/2,c10/1,c11/0,c12/1,c13/0,c2/2,c3/2,c4/2,c5/2,c6/3,c7/3,c8/2,c9/0,s/1,tt/0,c_1/0,c_2/0,c_3/1,c_4/1 ,c_5/1,c_6/1,c_7/1,c_8/1,c_9/1,c_10/1,c_11/1,c_12/1,c_13/0,c_14/1,c_15/3,c_16/3,c_17/3,c_18/5,c_19/0,c_20/0 ,c_21/1,c_22/0,c_23/1} - Obligation: innermost runtime complexity wrt. defined symbols {ACTIVATE#,PLUS#,U11#,U11'#,U12#,U12'#,U21#,U21'#,U22# ,U22'#,X#,activate#,plus#,x#} and constructors {0,c,c1,c10,c11,c12,c13,c2,c3,c4,c5,c6,c7,c8,c9,s,tt} + Applied Processor: WeightGap {wgDimension = 1, wgDegree = 1, wgKind = Algebraic, wgUArgs = UArgs, wgOn = WgOnAny} + Details: The weightgap principle applies using the following constant growth matrix-interpretation: We apply a matrix interpretation of kind constructor based matrix interpretation: The following argument positions are considered usable: uargs(U22'#) = {2,3}, uargs(X#) = {1,2}, uargs(c_8) = {1}, uargs(c_9) = {1}, uargs(c_10) = {1}, uargs(c_11) = {1}, uargs(c_14) = {1} Following symbols are considered usable: all TcT has computed the following interpretation: p(0) = [0] p(ACTIVATE) = [0] p(PLUS) = [0] p(U11) = [0] p(U11') = [0] p(U12) = [4] x3 + [0] p(U12') = [2] x3 + [0] p(U21) = [0] p(U21') = [0] p(U22) = [0] p(U22') = [0] p(X) = [1] x1 + [0] p(activate) = [1] x1 + [0] p(c) = [1] x1 + [1] x2 + [0] p(c1) = [1] x1 + [0] p(c10) = [1] x1 + [0] p(c11) = [1] p(c12) = [2] p(c13) = [0] p(c2) = [1] x1 + [1] x2 + [1] p(c3) = [1] x1 + [1] p(c4) = [0] p(c5) = [1] p(c6) = [1] x1 + [1] x3 + [1] p(c7) = [1] x1 + [1] x2 + [1] x3 + [0] p(c8) = [1] x1 + [1] x2 + [0] p(c9) = [0] p(plus) = [0] p(s) = [1] x1 + [10] p(tt) = [4] p(x) = [0] p(ACTIVATE#) = [0] p(PLUS#) = [0] p(U11#) = [0] p(U11'#) = [0] p(U12#) = [0] p(U12'#) = [0] p(U21#) = [0] p(U21'#) = [1] x2 + [1] x3 + [8] p(U22#) = [1] x2 + [2] x3 + [0] p(U22'#) = [1] x2 + [1] x3 + [8] p(X#) = [1] x1 + [1] x2 + [5] p(activate#) = [0] p(plus#) = [0] p(x#) = [0] p(c_1) = [0] p(c_2) = [0] p(c_3) = [0] p(c_4) = [2] x1 + [0] p(c_5) = [1] p(c_6) = [0] p(c_7) = [0] p(c_8) = [1] x1 + [0] p(c_9) = [1] x1 + [0] p(c_10) = [1] x1 + [1] p(c_11) = [1] x1 + [2] p(c_12) = [0] p(c_13) = [0] p(c_14) = [1] x1 + [0] p(c_15) = [0] p(c_16) = [0] p(c_17) = [1] x3 + [0] p(c_18) = [2] x1 + [1] x2 + [1] x3 + [0] p(c_19) = [2] p(c_20) = [0] p(c_21) = [2] x1 + [1] p(c_22) = [1] p(c_23) = [2] x1 + [1] Following rules are strictly oriented: X#(z0,s(z1)) = [1] z0 + [1] z1 + [15] > [1] z0 + [1] z1 + [8] = c_14(U21'#(tt(),z1,z0)) Following rules are (at-least) weakly oriented: U21'#(tt(),z0,z1) = [1] z0 + [1] z1 + [8] >= [1] z0 + [1] z1 + [8] = c_8(U22'#(tt(),activate(z0),activate(z1))) U21'#(tt(),z0,z1) = [1] z0 + [1] z1 + [8] >= [1] z0 + [1] z1 + [8] = c_9(U22'#(tt(),activate(z0),activate(z1))) U22'#(tt(),z0,z1) = [1] z0 + [1] z1 + [8] >= [1] z0 + [1] z1 + [6] = c_10(X#(activate(z1),activate(z0))) U22'#(tt(),z0,z1) = [1] z0 + [1] z1 + [8] >= [1] z0 + [1] z1 + [7] = c_11(X#(activate(z1),activate(z0))) activate(z0) = [1] z0 + [0] >= [1] z0 + [0] = z0 Further, it can be verified that all rules not oriented are covered by the weightgap condition. ** Step 7.a:5: EmptyProcessor. WORST_CASE(?,O(1)) + Considered Problem: - Weak DPs: U21'#(tt(),z0,z1) -> c_8(U22'#(tt(),activate(z0),activate(z1))) U21'#(tt(),z0,z1) -> c_9(U22'#(tt(),activate(z0),activate(z1))) U22'#(tt(),z0,z1) -> c_10(X#(activate(z1),activate(z0))) U22'#(tt(),z0,z1) -> c_11(X#(activate(z1),activate(z0))) X#(z0,s(z1)) -> c_14(U21'#(tt(),z1,z0)) - Weak TRS: activate(z0) -> z0 - Signature: {ACTIVATE/1,PLUS/2,U11/3,U11'/3,U12/3,U12'/3,U21/3,U21'/3,U22/3,U22'/3,X/2,activate/1,plus/2,x/2,ACTIVATE#/1 ,PLUS#/2,U11#/3,U11'#/3,U12#/3,U12'#/3,U21#/3,U21'#/3,U22#/3,U22'#/3,X#/2,activate#/1,plus#/2,x#/2} / {0/0 ,c/2,c1/2,c10/1,c11/0,c12/1,c13/0,c2/2,c3/2,c4/2,c5/2,c6/3,c7/3,c8/2,c9/0,s/1,tt/0,c_1/0,c_2/0,c_3/1,c_4/1 ,c_5/1,c_6/1,c_7/1,c_8/1,c_9/1,c_10/1,c_11/1,c_12/1,c_13/0,c_14/1,c_15/3,c_16/3,c_17/3,c_18/5,c_19/0,c_20/0 ,c_21/1,c_22/0,c_23/1} - Obligation: innermost runtime complexity wrt. defined symbols {ACTIVATE#,PLUS#,U11#,U11'#,U12#,U12'#,U21#,U21'#,U22# ,U22'#,X#,activate#,plus#,x#} and constructors {0,c,c1,c10,c11,c12,c13,c2,c3,c4,c5,c6,c7,c8,c9,s,tt} + Applied Processor: EmptyProcessor + Details: The problem is already closed. The intended complexity is O(1). ** Step 7.b:1: Ara. WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: PLUS#(z0,s(z1)) -> c_3(U11'#(tt(),z1,z0)) U11'#(tt(),z0,z1) -> c_4(U12'#(tt(),activate(z0),activate(z1))) U11'#(tt(),z0,z1) -> c_5(U12'#(tt(),activate(z0),activate(z1))) U12'#(tt(),z0,z1) -> c_6(PLUS#(activate(z1),activate(z0))) U12'#(tt(),z0,z1) -> c_7(PLUS#(activate(z1),activate(z0))) U22'#(tt(),z0,z1) -> c_12(PLUS#(x(activate(z1),activate(z0)),activate(z1))) - Weak DPs: U21'#(tt(),z0,z1) -> U22'#(tt(),activate(z0),activate(z1)) U22'#(tt(),z0,z1) -> PLUS#(x(activate(z1),activate(z0)),activate(z1)) U22'#(tt(),z0,z1) -> X#(activate(z1),activate(z0)) X#(z0,s(z1)) -> U21'#(tt(),z1,z0) - Weak TRS: U11(tt(),z0,z1) -> U12(tt(),activate(z0),activate(z1)) U12(tt(),z0,z1) -> s(plus(activate(z1),activate(z0))) U21(tt(),z0,z1) -> U22(tt(),activate(z0),activate(z1)) U22(tt(),z0,z1) -> plus(x(activate(z1),activate(z0)),activate(z1)) activate(z0) -> z0 plus(z0,0()) -> z0 plus(z0,s(z1)) -> U11(tt(),z1,z0) x(z0,0()) -> 0() x(z0,s(z1)) -> U21(tt(),z1,z0) - Signature: {ACTIVATE/1,PLUS/2,U11/3,U11'/3,U12/3,U12'/3,U21/3,U21'/3,U22/3,U22'/3,X/2,activate/1,plus/2,x/2,ACTIVATE#/1 ,PLUS#/2,U11#/3,U11'#/3,U12#/3,U12'#/3,U21#/3,U21'#/3,U22#/3,U22'#/3,X#/2,activate#/1,plus#/2,x#/2} / {0/0 ,c/2,c1/2,c10/1,c11/0,c12/1,c13/0,c2/2,c3/2,c4/2,c5/2,c6/3,c7/3,c8/2,c9/0,s/1,tt/0,c_1/0,c_2/0,c_3/1,c_4/1 ,c_5/1,c_6/1,c_7/1,c_8/1,c_9/1,c_10/2,c_11/2,c_12/1,c_13/0,c_14/1,c_15/3,c_16/3,c_17/3,c_18/5,c_19/0,c_20/0 ,c_21/1,c_22/0,c_23/1} - Obligation: innermost runtime complexity wrt. defined symbols {ACTIVATE#,PLUS#,U11#,U11'#,U12#,U12'#,U21#,U21'#,U22# ,U22'#,X#,activate#,plus#,x#} and constructors {0,c,c1,c10,c11,c12,c13,c2,c3,c4,c5,c6,c7,c8,c9,s,tt} + Applied Processor: Ara {minDegree = 1, maxDegree = 2, araTimeout = 8, araRuleShifting = Just 1, isBestCase = False, mkCompletelyDefined = False, verboseOutput = False} + Details: Signatures used: ---------------- F (TrsFun "0") :: [] -(0)-> "A"(0) F (TrsFun "U11") :: ["A"(0) x "A"(0) x "A"(0)] -(0)-> "A"(0) F (TrsFun "U12") :: ["A"(0) x "A"(0) x "A"(0)] -(0)-> "A"(0) F (TrsFun "U21") :: ["A"(0) x "A"(0) x "A"(0)] -(0)-> "A"(0) F (TrsFun "U22") :: ["A"(0) x "A"(0) x "A"(0)] -(0)-> "A"(0) F (TrsFun "activate") :: ["A"(0)] -(0)-> "A"(0) F (TrsFun "plus") :: ["A"(0) x "A"(0)] -(0)-> "A"(0) F (TrsFun "s") :: ["A"(0)] -(0)-> "A"(0) F (TrsFun "tt") :: [] -(0)-> "A"(0) F (TrsFun "x") :: ["A"(0) x "A"(0)] -(0)-> "A"(0) F (DpFun "PLUS") :: ["A"(0) x "A"(0)] -(0)-> "A"(0) F (DpFun "U11'") :: ["A"(0) x "A"(0) x "A"(0)] -(0)-> "A"(0) F (DpFun "U12'") :: ["A"(0) x "A"(0) x "A"(0)] -(0)-> "A"(0) F (DpFun "U21'") :: ["A"(0) x "A"(0) x "A"(0)] -(1)-> "A"(0) F (DpFun "U22'") :: ["A"(0) x "A"(0) x "A"(0)] -(1)-> "A"(0) F (DpFun "X") :: ["A"(0) x "A"(0)] -(1)-> "A"(0) F (ComFun 3) :: ["A"(0)] -(0)-> "A"(0) F (ComFun 4) :: ["A"(0)] -(0)-> "A"(0) F (ComFun 5) :: ["A"(0)] -(0)-> "A"(0) F (ComFun 6) :: ["A"(0)] -(0)-> "A"(0) F (ComFun 7) :: ["A"(0)] -(0)-> "A"(0) F (ComFun 12) :: ["A"(0)] -(0)-> "A"(0) Cost-free Signatures used: -------------------------- Base Constructor Signatures used: --------------------------------- Following Still Strict Rules were Typed as: ------------------------------------------- 1. Strict: U22'#(tt(),z0,z1) -> c_12(PLUS#(x(activate(z1),activate(z0)),activate(z1))) 2. Weak: PLUS#(z0,s(z1)) -> c_3(U11'#(tt(),z1,z0)) U11'#(tt(),z0,z1) -> c_4(U12'#(tt(),activate(z0),activate(z1))) U11'#(tt(),z0,z1) -> c_5(U12'#(tt(),activate(z0),activate(z1))) U12'#(tt(),z0,z1) -> c_6(PLUS#(activate(z1),activate(z0))) U12'#(tt(),z0,z1) -> c_7(PLUS#(activate(z1),activate(z0))) ** Step 7.b:2: NaturalMI. WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: PLUS#(z0,s(z1)) -> c_3(U11'#(tt(),z1,z0)) U11'#(tt(),z0,z1) -> c_4(U12'#(tt(),activate(z0),activate(z1))) U11'#(tt(),z0,z1) -> c_5(U12'#(tt(),activate(z0),activate(z1))) U12'#(tt(),z0,z1) -> c_6(PLUS#(activate(z1),activate(z0))) U12'#(tt(),z0,z1) -> c_7(PLUS#(activate(z1),activate(z0))) - Weak DPs: U21'#(tt(),z0,z1) -> U22'#(tt(),activate(z0),activate(z1)) U22'#(tt(),z0,z1) -> PLUS#(x(activate(z1),activate(z0)),activate(z1)) U22'#(tt(),z0,z1) -> X#(activate(z1),activate(z0)) U22'#(tt(),z0,z1) -> c_12(PLUS#(x(activate(z1),activate(z0)),activate(z1))) X#(z0,s(z1)) -> U21'#(tt(),z1,z0) - Weak TRS: U11(tt(),z0,z1) -> U12(tt(),activate(z0),activate(z1)) U12(tt(),z0,z1) -> s(plus(activate(z1),activate(z0))) U21(tt(),z0,z1) -> U22(tt(),activate(z0),activate(z1)) U22(tt(),z0,z1) -> plus(x(activate(z1),activate(z0)),activate(z1)) activate(z0) -> z0 plus(z0,0()) -> z0 plus(z0,s(z1)) -> U11(tt(),z1,z0) x(z0,0()) -> 0() x(z0,s(z1)) -> U21(tt(),z1,z0) - Signature: {ACTIVATE/1,PLUS/2,U11/3,U11'/3,U12/3,U12'/3,U21/3,U21'/3,U22/3,U22'/3,X/2,activate/1,plus/2,x/2,ACTIVATE#/1 ,PLUS#/2,U11#/3,U11'#/3,U12#/3,U12'#/3,U21#/3,U21'#/3,U22#/3,U22'#/3,X#/2,activate#/1,plus#/2,x#/2} / {0/0 ,c/2,c1/2,c10/1,c11/0,c12/1,c13/0,c2/2,c3/2,c4/2,c5/2,c6/3,c7/3,c8/2,c9/0,s/1,tt/0,c_1/0,c_2/0,c_3/1,c_4/1 ,c_5/1,c_6/1,c_7/1,c_8/1,c_9/1,c_10/2,c_11/2,c_12/1,c_13/0,c_14/1,c_15/3,c_16/3,c_17/3,c_18/5,c_19/0,c_20/0 ,c_21/1,c_22/0,c_23/1} - Obligation: innermost runtime complexity wrt. defined symbols {ACTIVATE#,PLUS#,U11#,U11'#,U12#,U12'#,U21#,U21'#,U22# ,U22'#,X#,activate#,plus#,x#} and constructors {0,c,c1,c10,c11,c12,c13,c2,c3,c4,c5,c6,c7,c8,c9,s,tt} + Applied Processor: NaturalMI {miDimension = 1, miDegree = 1, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Just any strict-rules} + Details: We apply a matrix interpretation of kind constructor based matrix interpretation: The following argument positions are considered usable: uargs(c_3) = {1}, uargs(c_4) = {1}, uargs(c_5) = {1}, uargs(c_6) = {1}, uargs(c_7) = {1}, uargs(c_12) = {1} Following symbols are considered usable: {activate,ACTIVATE#,PLUS#,U11#,U11'#,U12#,U12'#,U21#,U21'#,U22#,U22'#,X#,activate#,plus#,x#} TcT has computed the following interpretation: p(0) = [0] p(ACTIVATE) = [4] x1 + [4] p(PLUS) = [1] p(U11) = [4] x2 + [4] p(U11') = [1] x3 + [2] p(U12) = [1] p(U12') = [1] p(U21) = [0] p(U21') = [2] x1 + [4] p(U22) = [4] x2 + [1] x3 + [1] p(U22') = [2] x1 + [1] x2 + [2] p(X) = [1] p(activate) = [1] x1 + [0] p(c) = [1] x2 + [0] p(c1) = [0] p(c10) = [2] p(c11) = [1] p(c12) = [2] p(c13) = [0] p(c2) = [0] p(c3) = [2] p(c4) = [1] x1 + [1] p(c5) = [1] x1 + [1] x2 + [1] p(c6) = [0] p(c7) = [1] x1 + [1] p(c8) = [1] p(c9) = [2] p(plus) = [3] x1 + [0] p(s) = [1] x1 + [3] p(tt) = [4] p(x) = [3] x1 + [0] p(ACTIVATE#) = [0] p(PLUS#) = [4] x2 + [0] p(U11#) = [1] x1 + [4] x2 + [1] p(U11'#) = [4] x2 + [1] p(U12#) = [1] x2 + [1] x3 + [1] p(U12'#) = [4] x2 + [1] p(U21#) = [1] x3 + [1] p(U21'#) = [1] x1 + [1] x2 + [4] x3 + [4] p(U22#) = [1] p(U22'#) = [1] x1 + [1] x2 + [4] x3 + [4] p(X#) = [4] x1 + [1] x2 + [5] p(activate#) = [2] p(plus#) = [4] x1 + [2] p(x#) = [2] x1 + [1] x2 + [1] p(c_1) = [2] p(c_2) = [0] p(c_3) = [1] x1 + [7] p(c_4) = [1] x1 + [0] p(c_5) = [1] x1 + [0] p(c_6) = [1] x1 + [1] p(c_7) = [1] x1 + [1] p(c_8) = [1] x1 + [2] p(c_9) = [1] p(c_10) = [1] x1 + [2] x2 + [0] p(c_11) = [1] x1 + [1] p(c_12) = [1] x1 + [5] p(c_13) = [2] p(c_14) = [0] p(c_15) = [4] x2 + [1] p(c_16) = [1] x2 + [1] p(c_17) = [2] x1 + [2] x2 + [1] p(c_18) = [2] x2 + [4] x5 + [4] p(c_19) = [4] p(c_20) = [0] p(c_21) = [1] x1 + [1] p(c_22) = [1] p(c_23) = [4] x1 + [0] Following rules are strictly oriented: PLUS#(z0,s(z1)) = [4] z1 + [12] > [4] z1 + [8] = c_3(U11'#(tt(),z1,z0)) Following rules are (at-least) weakly oriented: U11'#(tt(),z0,z1) = [4] z0 + [1] >= [4] z0 + [1] = c_4(U12'#(tt(),activate(z0),activate(z1))) U11'#(tt(),z0,z1) = [4] z0 + [1] >= [4] z0 + [1] = c_5(U12'#(tt(),activate(z0),activate(z1))) U12'#(tt(),z0,z1) = [4] z0 + [1] >= [4] z0 + [1] = c_6(PLUS#(activate(z1),activate(z0))) U12'#(tt(),z0,z1) = [4] z0 + [1] >= [4] z0 + [1] = c_7(PLUS#(activate(z1),activate(z0))) U21'#(tt(),z0,z1) = [1] z0 + [4] z1 + [8] >= [1] z0 + [4] z1 + [8] = U22'#(tt(),activate(z0),activate(z1)) U22'#(tt(),z0,z1) = [1] z0 + [4] z1 + [8] >= [4] z1 + [0] = PLUS#(x(activate(z1),activate(z0)),activate(z1)) U22'#(tt(),z0,z1) = [1] z0 + [4] z1 + [8] >= [1] z0 + [4] z1 + [5] = X#(activate(z1),activate(z0)) U22'#(tt(),z0,z1) = [1] z0 + [4] z1 + [8] >= [4] z1 + [5] = c_12(PLUS#(x(activate(z1),activate(z0)),activate(z1))) X#(z0,s(z1)) = [4] z0 + [1] z1 + [8] >= [4] z0 + [1] z1 + [8] = U21'#(tt(),z1,z0) activate(z0) = [1] z0 + [0] >= [1] z0 + [0] = z0 ** Step 7.b:3: NaturalMI. WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: U11'#(tt(),z0,z1) -> c_4(U12'#(tt(),activate(z0),activate(z1))) U11'#(tt(),z0,z1) -> c_5(U12'#(tt(),activate(z0),activate(z1))) U12'#(tt(),z0,z1) -> c_6(PLUS#(activate(z1),activate(z0))) U12'#(tt(),z0,z1) -> c_7(PLUS#(activate(z1),activate(z0))) - Weak DPs: PLUS#(z0,s(z1)) -> c_3(U11'#(tt(),z1,z0)) U21'#(tt(),z0,z1) -> U22'#(tt(),activate(z0),activate(z1)) U22'#(tt(),z0,z1) -> PLUS#(x(activate(z1),activate(z0)),activate(z1)) U22'#(tt(),z0,z1) -> X#(activate(z1),activate(z0)) U22'#(tt(),z0,z1) -> c_12(PLUS#(x(activate(z1),activate(z0)),activate(z1))) X#(z0,s(z1)) -> U21'#(tt(),z1,z0) - Weak TRS: U11(tt(),z0,z1) -> U12(tt(),activate(z0),activate(z1)) U12(tt(),z0,z1) -> s(plus(activate(z1),activate(z0))) U21(tt(),z0,z1) -> U22(tt(),activate(z0),activate(z1)) U22(tt(),z0,z1) -> plus(x(activate(z1),activate(z0)),activate(z1)) activate(z0) -> z0 plus(z0,0()) -> z0 plus(z0,s(z1)) -> U11(tt(),z1,z0) x(z0,0()) -> 0() x(z0,s(z1)) -> U21(tt(),z1,z0) - Signature: {ACTIVATE/1,PLUS/2,U11/3,U11'/3,U12/3,U12'/3,U21/3,U21'/3,U22/3,U22'/3,X/2,activate/1,plus/2,x/2,ACTIVATE#/1 ,PLUS#/2,U11#/3,U11'#/3,U12#/3,U12'#/3,U21#/3,U21'#/3,U22#/3,U22'#/3,X#/2,activate#/1,plus#/2,x#/2} / {0/0 ,c/2,c1/2,c10/1,c11/0,c12/1,c13/0,c2/2,c3/2,c4/2,c5/2,c6/3,c7/3,c8/2,c9/0,s/1,tt/0,c_1/0,c_2/0,c_3/1,c_4/1 ,c_5/1,c_6/1,c_7/1,c_8/1,c_9/1,c_10/2,c_11/2,c_12/1,c_13/0,c_14/1,c_15/3,c_16/3,c_17/3,c_18/5,c_19/0,c_20/0 ,c_21/1,c_22/0,c_23/1} - Obligation: innermost runtime complexity wrt. defined symbols {ACTIVATE#,PLUS#,U11#,U11'#,U12#,U12'#,U21#,U21'#,U22# ,U22'#,X#,activate#,plus#,x#} and constructors {0,c,c1,c10,c11,c12,c13,c2,c3,c4,c5,c6,c7,c8,c9,s,tt} + Applied Processor: NaturalMI {miDimension = 1, miDegree = 1, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Just any strict-rules} + Details: We apply a matrix interpretation of kind constructor based matrix interpretation: The following argument positions are considered usable: uargs(c_3) = {1}, uargs(c_4) = {1}, uargs(c_5) = {1}, uargs(c_6) = {1}, uargs(c_7) = {1}, uargs(c_12) = {1} Following symbols are considered usable: {activate,ACTIVATE#,PLUS#,U11#,U11'#,U12#,U12'#,U21#,U21'#,U22#,U22'#,X#,activate#,plus#,x#} TcT has computed the following interpretation: p(0) = [2] p(ACTIVATE) = [0] p(PLUS) = [1] x1 + [2] p(U11) = [2] x1 + [0] p(U11') = [1] x2 + [2] x3 + [0] p(U12) = [3] x1 + [4] x3 + [2] p(U12') = [4] x3 + [1] p(U21) = [4] x1 + [2] x3 + [0] p(U21') = [0] p(U22) = [1] x3 + [1] p(U22') = [1] x2 + [1] x3 + [0] p(X) = [2] x1 + [1] x2 + [0] p(activate) = [1] x1 + [0] p(c) = [1] x1 + [1] p(c1) = [1] x1 + [1] x2 + [1] p(c10) = [1] x1 + [2] p(c11) = [0] p(c12) = [4] p(c13) = [1] p(c2) = [0] p(c3) = [1] p(c4) = [0] p(c5) = [4] p(c6) = [1] p(c7) = [1] x2 + [1] p(c8) = [1] x2 + [1] p(c9) = [0] p(plus) = [2] x1 + [1] x2 + [4] p(s) = [1] x1 + [7] p(tt) = [1] p(x) = [2] x1 + [1] x2 + [0] p(ACTIVATE#) = [1] x1 + [0] p(PLUS#) = [1] x2 + [1] p(U11#) = [0] p(U11'#) = [1] x2 + [2] p(U12#) = [1] x2 + [1] x3 + [2] p(U12'#) = [1] x2 + [1] p(U21#) = [0] p(U21'#) = [4] x1 + [4] x3 + [0] p(U22#) = [1] x2 + [2] p(U22'#) = [4] x3 + [4] p(X#) = [4] x1 + [4] p(activate#) = [0] p(plus#) = [4] x1 + [1] x2 + [4] p(x#) = [1] x1 + [0] p(c_1) = [1] p(c_2) = [1] p(c_3) = [1] x1 + [6] p(c_4) = [1] x1 + [0] p(c_5) = [1] x1 + [1] p(c_6) = [1] x1 + [0] p(c_7) = [1] x1 + [0] p(c_8) = [1] x1 + [2] p(c_9) = [2] x1 + [0] p(c_10) = [0] p(c_11) = [1] x1 + [1] p(c_12) = [2] x1 + [0] p(c_13) = [1] p(c_14) = [1] p(c_15) = [1] x3 + [1] p(c_16) = [1] x3 + [2] p(c_17) = [4] x1 + [0] p(c_18) = [4] x1 + [1] x2 + [4] x3 + [0] p(c_19) = [0] p(c_20) = [0] p(c_21) = [0] p(c_22) = [4] p(c_23) = [2] x1 + [2] Following rules are strictly oriented: U11'#(tt(),z0,z1) = [1] z0 + [2] > [1] z0 + [1] = c_4(U12'#(tt(),activate(z0),activate(z1))) Following rules are (at-least) weakly oriented: PLUS#(z0,s(z1)) = [1] z1 + [8] >= [1] z1 + [8] = c_3(U11'#(tt(),z1,z0)) U11'#(tt(),z0,z1) = [1] z0 + [2] >= [1] z0 + [2] = c_5(U12'#(tt(),activate(z0),activate(z1))) U12'#(tt(),z0,z1) = [1] z0 + [1] >= [1] z0 + [1] = c_6(PLUS#(activate(z1),activate(z0))) U12'#(tt(),z0,z1) = [1] z0 + [1] >= [1] z0 + [1] = c_7(PLUS#(activate(z1),activate(z0))) U21'#(tt(),z0,z1) = [4] z1 + [4] >= [4] z1 + [4] = U22'#(tt(),activate(z0),activate(z1)) U22'#(tt(),z0,z1) = [4] z1 + [4] >= [1] z1 + [1] = PLUS#(x(activate(z1),activate(z0)),activate(z1)) U22'#(tt(),z0,z1) = [4] z1 + [4] >= [4] z1 + [4] = X#(activate(z1),activate(z0)) U22'#(tt(),z0,z1) = [4] z1 + [4] >= [2] z1 + [2] = c_12(PLUS#(x(activate(z1),activate(z0)),activate(z1))) X#(z0,s(z1)) = [4] z0 + [4] >= [4] z0 + [4] = U21'#(tt(),z1,z0) activate(z0) = [1] z0 + [0] >= [1] z0 + [0] = z0 ** Step 7.b:4: NaturalMI. WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: U11'#(tt(),z0,z1) -> c_5(U12'#(tt(),activate(z0),activate(z1))) U12'#(tt(),z0,z1) -> c_6(PLUS#(activate(z1),activate(z0))) U12'#(tt(),z0,z1) -> c_7(PLUS#(activate(z1),activate(z0))) - Weak DPs: PLUS#(z0,s(z1)) -> c_3(U11'#(tt(),z1,z0)) U11'#(tt(),z0,z1) -> c_4(U12'#(tt(),activate(z0),activate(z1))) U21'#(tt(),z0,z1) -> U22'#(tt(),activate(z0),activate(z1)) U22'#(tt(),z0,z1) -> PLUS#(x(activate(z1),activate(z0)),activate(z1)) U22'#(tt(),z0,z1) -> X#(activate(z1),activate(z0)) U22'#(tt(),z0,z1) -> c_12(PLUS#(x(activate(z1),activate(z0)),activate(z1))) X#(z0,s(z1)) -> U21'#(tt(),z1,z0) - Weak TRS: U11(tt(),z0,z1) -> U12(tt(),activate(z0),activate(z1)) U12(tt(),z0,z1) -> s(plus(activate(z1),activate(z0))) U21(tt(),z0,z1) -> U22(tt(),activate(z0),activate(z1)) U22(tt(),z0,z1) -> plus(x(activate(z1),activate(z0)),activate(z1)) activate(z0) -> z0 plus(z0,0()) -> z0 plus(z0,s(z1)) -> U11(tt(),z1,z0) x(z0,0()) -> 0() x(z0,s(z1)) -> U21(tt(),z1,z0) - Signature: {ACTIVATE/1,PLUS/2,U11/3,U11'/3,U12/3,U12'/3,U21/3,U21'/3,U22/3,U22'/3,X/2,activate/1,plus/2,x/2,ACTIVATE#/1 ,PLUS#/2,U11#/3,U11'#/3,U12#/3,U12'#/3,U21#/3,U21'#/3,U22#/3,U22'#/3,X#/2,activate#/1,plus#/2,x#/2} / {0/0 ,c/2,c1/2,c10/1,c11/0,c12/1,c13/0,c2/2,c3/2,c4/2,c5/2,c6/3,c7/3,c8/2,c9/0,s/1,tt/0,c_1/0,c_2/0,c_3/1,c_4/1 ,c_5/1,c_6/1,c_7/1,c_8/1,c_9/1,c_10/2,c_11/2,c_12/1,c_13/0,c_14/1,c_15/3,c_16/3,c_17/3,c_18/5,c_19/0,c_20/0 ,c_21/1,c_22/0,c_23/1} - Obligation: innermost runtime complexity wrt. defined symbols {ACTIVATE#,PLUS#,U11#,U11'#,U12#,U12'#,U21#,U21'#,U22# ,U22'#,X#,activate#,plus#,x#} and constructors {0,c,c1,c10,c11,c12,c13,c2,c3,c4,c5,c6,c7,c8,c9,s,tt} + Applied Processor: NaturalMI {miDimension = 1, miDegree = 1, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Just any strict-rules} + Details: We apply a matrix interpretation of kind constructor based matrix interpretation: The following argument positions are considered usable: uargs(c_3) = {1}, uargs(c_4) = {1}, uargs(c_5) = {1}, uargs(c_6) = {1}, uargs(c_7) = {1}, uargs(c_12) = {1} Following symbols are considered usable: {activate,ACTIVATE#,PLUS#,U11#,U11'#,U12#,U12'#,U21#,U21'#,U22#,U22'#,X#,activate#,plus#,x#} TcT has computed the following interpretation: p(0) = [0] p(ACTIVATE) = [0] p(PLUS) = [0] p(U11) = [0] p(U11') = [0] p(U12) = [0] p(U12') = [0] p(U21) = [4] x3 + [0] p(U21') = [0] p(U22) = [1] x3 + [0] p(U22') = [0] p(X) = [0] p(activate) = [1] x1 + [0] p(c) = [1] x1 + [1] x2 + [0] p(c1) = [1] x1 + [1] x2 + [0] p(c10) = [1] x1 + [0] p(c11) = [0] p(c12) = [1] x1 + [0] p(c13) = [1] p(c2) = [0] p(c3) = [0] p(c4) = [0] p(c5) = [1] x2 + [1] p(c6) = [1] x2 + [1] p(c7) = [1] x1 + [1] p(c8) = [2] p(c9) = [0] p(plus) = [4] x1 + [1] x2 + [1] p(s) = [1] x1 + [2] p(tt) = [0] p(x) = [2] p(ACTIVATE#) = [0] p(PLUS#) = [4] x2 + [0] p(U11#) = [1] x1 + [2] p(U11'#) = [4] x1 + [4] x2 + [4] p(U12#) = [4] x1 + [1] p(U12'#) = [4] x2 + [2] p(U21#) = [4] x1 + [1] x3 + [0] p(U21'#) = [1] x1 + [2] x2 + [4] x3 + [4] p(U22#) = [1] x2 + [4] p(U22'#) = [1] x1 + [2] x2 + [4] x3 + [1] p(X#) = [4] x1 + [2] x2 + [1] p(activate#) = [0] p(plus#) = [2] x1 + [1] x2 + [0] p(x#) = [1] p(c_1) = [1] p(c_2) = [0] p(c_3) = [1] x1 + [4] p(c_4) = [1] x1 + [0] p(c_5) = [1] x1 + [2] p(c_6) = [1] x1 + [2] p(c_7) = [1] x1 + [0] p(c_8) = [1] x1 + [2] p(c_9) = [4] x1 + [1] p(c_10) = [2] x2 + [4] p(c_11) = [0] p(c_12) = [1] x1 + [1] p(c_13) = [1] p(c_14) = [1] x1 + [0] p(c_15) = [1] x2 + [0] p(c_16) = [1] x1 + [1] p(c_17) = [1] p(c_18) = [1] x1 + [1] x2 + [4] x3 + [4] x4 + [1] p(c_19) = [0] p(c_20) = [2] p(c_21) = [1] x1 + [1] p(c_22) = [0] p(c_23) = [0] Following rules are strictly oriented: U12'#(tt(),z0,z1) = [4] z0 + [2] > [4] z0 + [0] = c_7(PLUS#(activate(z1),activate(z0))) Following rules are (at-least) weakly oriented: PLUS#(z0,s(z1)) = [4] z1 + [8] >= [4] z1 + [8] = c_3(U11'#(tt(),z1,z0)) U11'#(tt(),z0,z1) = [4] z0 + [4] >= [4] z0 + [2] = c_4(U12'#(tt(),activate(z0),activate(z1))) U11'#(tt(),z0,z1) = [4] z0 + [4] >= [4] z0 + [4] = c_5(U12'#(tt(),activate(z0),activate(z1))) U12'#(tt(),z0,z1) = [4] z0 + [2] >= [4] z0 + [2] = c_6(PLUS#(activate(z1),activate(z0))) U21'#(tt(),z0,z1) = [2] z0 + [4] z1 + [4] >= [2] z0 + [4] z1 + [1] = U22'#(tt(),activate(z0),activate(z1)) U22'#(tt(),z0,z1) = [2] z0 + [4] z1 + [1] >= [4] z1 + [0] = PLUS#(x(activate(z1),activate(z0)),activate(z1)) U22'#(tt(),z0,z1) = [2] z0 + [4] z1 + [1] >= [2] z0 + [4] z1 + [1] = X#(activate(z1),activate(z0)) U22'#(tt(),z0,z1) = [2] z0 + [4] z1 + [1] >= [4] z1 + [1] = c_12(PLUS#(x(activate(z1),activate(z0)),activate(z1))) X#(z0,s(z1)) = [4] z0 + [2] z1 + [5] >= [4] z0 + [2] z1 + [4] = U21'#(tt(),z1,z0) activate(z0) = [1] z0 + [0] >= [1] z0 + [0] = z0 ** Step 7.b:5: NaturalMI. WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: U11'#(tt(),z0,z1) -> c_5(U12'#(tt(),activate(z0),activate(z1))) U12'#(tt(),z0,z1) -> c_6(PLUS#(activate(z1),activate(z0))) - Weak DPs: PLUS#(z0,s(z1)) -> c_3(U11'#(tt(),z1,z0)) U11'#(tt(),z0,z1) -> c_4(U12'#(tt(),activate(z0),activate(z1))) U12'#(tt(),z0,z1) -> c_7(PLUS#(activate(z1),activate(z0))) U21'#(tt(),z0,z1) -> U22'#(tt(),activate(z0),activate(z1)) U22'#(tt(),z0,z1) -> PLUS#(x(activate(z1),activate(z0)),activate(z1)) U22'#(tt(),z0,z1) -> X#(activate(z1),activate(z0)) U22'#(tt(),z0,z1) -> c_12(PLUS#(x(activate(z1),activate(z0)),activate(z1))) X#(z0,s(z1)) -> U21'#(tt(),z1,z0) - Weak TRS: U11(tt(),z0,z1) -> U12(tt(),activate(z0),activate(z1)) U12(tt(),z0,z1) -> s(plus(activate(z1),activate(z0))) U21(tt(),z0,z1) -> U22(tt(),activate(z0),activate(z1)) U22(tt(),z0,z1) -> plus(x(activate(z1),activate(z0)),activate(z1)) activate(z0) -> z0 plus(z0,0()) -> z0 plus(z0,s(z1)) -> U11(tt(),z1,z0) x(z0,0()) -> 0() x(z0,s(z1)) -> U21(tt(),z1,z0) - Signature: {ACTIVATE/1,PLUS/2,U11/3,U11'/3,U12/3,U12'/3,U21/3,U21'/3,U22/3,U22'/3,X/2,activate/1,plus/2,x/2,ACTIVATE#/1 ,PLUS#/2,U11#/3,U11'#/3,U12#/3,U12'#/3,U21#/3,U21'#/3,U22#/3,U22'#/3,X#/2,activate#/1,plus#/2,x#/2} / {0/0 ,c/2,c1/2,c10/1,c11/0,c12/1,c13/0,c2/2,c3/2,c4/2,c5/2,c6/3,c7/3,c8/2,c9/0,s/1,tt/0,c_1/0,c_2/0,c_3/1,c_4/1 ,c_5/1,c_6/1,c_7/1,c_8/1,c_9/1,c_10/2,c_11/2,c_12/1,c_13/0,c_14/1,c_15/3,c_16/3,c_17/3,c_18/5,c_19/0,c_20/0 ,c_21/1,c_22/0,c_23/1} - Obligation: innermost runtime complexity wrt. defined symbols {ACTIVATE#,PLUS#,U11#,U11'#,U12#,U12'#,U21#,U21'#,U22# ,U22'#,X#,activate#,plus#,x#} and constructors {0,c,c1,c10,c11,c12,c13,c2,c3,c4,c5,c6,c7,c8,c9,s,tt} + Applied Processor: NaturalMI {miDimension = 1, miDegree = 1, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Just any strict-rules} + Details: We apply a matrix interpretation of kind constructor based matrix interpretation: The following argument positions are considered usable: uargs(c_3) = {1}, uargs(c_4) = {1}, uargs(c_5) = {1}, uargs(c_6) = {1}, uargs(c_7) = {1}, uargs(c_12) = {1} Following symbols are considered usable: {activate,ACTIVATE#,PLUS#,U11#,U11'#,U12#,U12'#,U21#,U21'#,U22#,U22'#,X#,activate#,plus#,x#} TcT has computed the following interpretation: p(0) = [0] p(ACTIVATE) = [0] p(PLUS) = [0] p(U11) = [0] p(U11') = [0] p(U12) = [1] p(U12') = [0] p(U21) = [0] p(U21') = [0] p(U22) = [0] p(U22') = [0] p(X) = [0] p(activate) = [1] x1 + [0] p(c) = [1] x1 + [1] x2 + [0] p(c1) = [1] x1 + [1] x2 + [0] p(c10) = [1] x1 + [0] p(c11) = [0] p(c12) = [1] x1 + [0] p(c13) = [0] p(c2) = [1] x1 + [1] x2 + [0] p(c3) = [1] x1 + [1] x2 + [0] p(c4) = [1] x1 + [1] x2 + [0] p(c5) = [1] x1 + [1] x2 + [0] p(c6) = [1] x1 + [1] x2 + [1] x3 + [0] p(c7) = [1] x2 + [1] x3 + [0] p(c8) = [1] x1 + [1] x2 + [0] p(c9) = [4] p(plus) = [3] x1 + [4] p(s) = [1] x1 + [2] p(tt) = [2] p(x) = [3] x1 + [2] x2 + [1] p(ACTIVATE#) = [0] p(PLUS#) = [4] x2 + [0] p(U11#) = [1] x1 + [1] x2 + [1] x3 + [2] p(U11'#) = [4] x1 + [4] x2 + [0] p(U12#) = [4] x2 + [0] p(U12'#) = [4] x2 + [0] p(U21#) = [1] x1 + [1] x2 + [1] p(U21'#) = [4] x2 + [4] x3 + [3] p(U22#) = [1] x3 + [1] p(U22'#) = [4] x2 + [4] x3 + [0] p(X#) = [4] x1 + [4] x2 + [0] p(activate#) = [1] p(plus#) = [1] x1 + [4] p(x#) = [2] x1 + [1] x2 + [1] p(c_1) = [0] p(c_2) = [0] p(c_3) = [1] x1 + [0] p(c_4) = [1] x1 + [3] p(c_5) = [1] x1 + [7] p(c_6) = [1] x1 + [0] p(c_7) = [1] x1 + [0] p(c_8) = [4] x1 + [1] p(c_9) = [0] p(c_10) = [1] x2 + [0] p(c_11) = [1] p(c_12) = [1] x1 + [0] p(c_13) = [0] p(c_14) = [1] x1 + [0] p(c_15) = [1] x2 + [1] p(c_16) = [1] x1 + [1] x2 + [0] p(c_17) = [1] x1 + [4] x2 + [2] p(c_18) = [1] x1 + [1] x3 + [2] x4 + [4] x5 + [2] p(c_19) = [0] p(c_20) = [1] p(c_21) = [4] p(c_22) = [1] p(c_23) = [1] x1 + [0] Following rules are strictly oriented: U11'#(tt(),z0,z1) = [4] z0 + [8] > [4] z0 + [7] = c_5(U12'#(tt(),activate(z0),activate(z1))) Following rules are (at-least) weakly oriented: PLUS#(z0,s(z1)) = [4] z1 + [8] >= [4] z1 + [8] = c_3(U11'#(tt(),z1,z0)) U11'#(tt(),z0,z1) = [4] z0 + [8] >= [4] z0 + [3] = c_4(U12'#(tt(),activate(z0),activate(z1))) U12'#(tt(),z0,z1) = [4] z0 + [0] >= [4] z0 + [0] = c_6(PLUS#(activate(z1),activate(z0))) U12'#(tt(),z0,z1) = [4] z0 + [0] >= [4] z0 + [0] = c_7(PLUS#(activate(z1),activate(z0))) U21'#(tt(),z0,z1) = [4] z0 + [4] z1 + [3] >= [4] z0 + [4] z1 + [0] = U22'#(tt(),activate(z0),activate(z1)) U22'#(tt(),z0,z1) = [4] z0 + [4] z1 + [0] >= [4] z1 + [0] = PLUS#(x(activate(z1),activate(z0)),activate(z1)) U22'#(tt(),z0,z1) = [4] z0 + [4] z1 + [0] >= [4] z0 + [4] z1 + [0] = X#(activate(z1),activate(z0)) U22'#(tt(),z0,z1) = [4] z0 + [4] z1 + [0] >= [4] z1 + [0] = c_12(PLUS#(x(activate(z1),activate(z0)),activate(z1))) X#(z0,s(z1)) = [4] z0 + [4] z1 + [8] >= [4] z0 + [4] z1 + [3] = U21'#(tt(),z1,z0) activate(z0) = [1] z0 + [0] >= [1] z0 + [0] = z0 ** Step 7.b:6: NaturalMI. WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: U12'#(tt(),z0,z1) -> c_6(PLUS#(activate(z1),activate(z0))) - Weak DPs: PLUS#(z0,s(z1)) -> c_3(U11'#(tt(),z1,z0)) U11'#(tt(),z0,z1) -> c_4(U12'#(tt(),activate(z0),activate(z1))) U11'#(tt(),z0,z1) -> c_5(U12'#(tt(),activate(z0),activate(z1))) U12'#(tt(),z0,z1) -> c_7(PLUS#(activate(z1),activate(z0))) U21'#(tt(),z0,z1) -> U22'#(tt(),activate(z0),activate(z1)) U22'#(tt(),z0,z1) -> PLUS#(x(activate(z1),activate(z0)),activate(z1)) U22'#(tt(),z0,z1) -> X#(activate(z1),activate(z0)) U22'#(tt(),z0,z1) -> c_12(PLUS#(x(activate(z1),activate(z0)),activate(z1))) X#(z0,s(z1)) -> U21'#(tt(),z1,z0) - Weak TRS: U11(tt(),z0,z1) -> U12(tt(),activate(z0),activate(z1)) U12(tt(),z0,z1) -> s(plus(activate(z1),activate(z0))) U21(tt(),z0,z1) -> U22(tt(),activate(z0),activate(z1)) U22(tt(),z0,z1) -> plus(x(activate(z1),activate(z0)),activate(z1)) activate(z0) -> z0 plus(z0,0()) -> z0 plus(z0,s(z1)) -> U11(tt(),z1,z0) x(z0,0()) -> 0() x(z0,s(z1)) -> U21(tt(),z1,z0) - Signature: {ACTIVATE/1,PLUS/2,U11/3,U11'/3,U12/3,U12'/3,U21/3,U21'/3,U22/3,U22'/3,X/2,activate/1,plus/2,x/2,ACTIVATE#/1 ,PLUS#/2,U11#/3,U11'#/3,U12#/3,U12'#/3,U21#/3,U21'#/3,U22#/3,U22'#/3,X#/2,activate#/1,plus#/2,x#/2} / {0/0 ,c/2,c1/2,c10/1,c11/0,c12/1,c13/0,c2/2,c3/2,c4/2,c5/2,c6/3,c7/3,c8/2,c9/0,s/1,tt/0,c_1/0,c_2/0,c_3/1,c_4/1 ,c_5/1,c_6/1,c_7/1,c_8/1,c_9/1,c_10/2,c_11/2,c_12/1,c_13/0,c_14/1,c_15/3,c_16/3,c_17/3,c_18/5,c_19/0,c_20/0 ,c_21/1,c_22/0,c_23/1} - Obligation: innermost runtime complexity wrt. defined symbols {ACTIVATE#,PLUS#,U11#,U11'#,U12#,U12'#,U21#,U21'#,U22# ,U22'#,X#,activate#,plus#,x#} and constructors {0,c,c1,c10,c11,c12,c13,c2,c3,c4,c5,c6,c7,c8,c9,s,tt} + Applied Processor: NaturalMI {miDimension = 1, miDegree = 1, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Just any strict-rules} + Details: We apply a matrix interpretation of kind constructor based matrix interpretation: The following argument positions are considered usable: uargs(c_3) = {1}, uargs(c_4) = {1}, uargs(c_5) = {1}, uargs(c_6) = {1}, uargs(c_7) = {1}, uargs(c_12) = {1} Following symbols are considered usable: {activate,ACTIVATE#,PLUS#,U11#,U11'#,U12#,U12'#,U21#,U21'#,U22#,U22'#,X#,activate#,plus#,x#} TcT has computed the following interpretation: p(0) = [1] p(ACTIVATE) = [2] p(PLUS) = [1] x1 + [1] x2 + [4] p(U11) = [0] p(U11') = [2] x2 + [0] p(U12) = [1] p(U12') = [1] x1 + [4] x3 + [0] p(U21) = [4] x1 + [1] x2 + [0] p(U21') = [1] x1 + [1] x2 + [1] p(U22) = [4] x1 + [6] p(U22') = [1] x1 + [0] p(X) = [2] p(activate) = [1] x1 + [0] p(c) = [1] x1 + [1] x2 + [0] p(c1) = [1] p(c10) = [0] p(c11) = [1] p(c12) = [1] x1 + [0] p(c13) = [0] p(c2) = [1] x2 + [1] p(c3) = [1] x1 + [1] x2 + [4] p(c4) = [1] x1 + [1] x2 + [1] p(c5) = [1] x1 + [1] p(c6) = [1] x2 + [0] p(c7) = [1] p(c8) = [1] p(c9) = [1] p(plus) = [5] x1 + [1] x2 + [0] p(s) = [1] x1 + [4] p(tt) = [2] p(x) = [1] x1 + [0] p(ACTIVATE#) = [1] x1 + [2] p(PLUS#) = [2] x2 + [6] p(U11#) = [4] x2 + [1] x3 + [2] p(U11'#) = [4] x1 + [2] x2 + [6] p(U12#) = [1] x1 + [1] p(U12'#) = [5] x1 + [2] x2 + [4] p(U21#) = [2] x1 + [1] x2 + [4] x3 + [0] p(U21'#) = [4] x1 + [2] x2 + [4] x3 + [0] p(U22#) = [1] x1 + [4] x3 + [4] p(U22'#) = [1] x1 + [2] x2 + [4] x3 + [6] p(X#) = [4] x1 + [2] x2 + [2] p(activate#) = [1] x1 + [1] p(plus#) = [4] x1 + [0] p(x#) = [0] p(c_1) = [1] p(c_2) = [1] p(c_3) = [1] x1 + [0] p(c_4) = [1] x1 + [0] p(c_5) = [1] x1 + [0] p(c_6) = [1] x1 + [7] p(c_7) = [1] x1 + [0] p(c_8) = [1] x1 + [4] p(c_9) = [4] p(c_10) = [2] x1 + [0] p(c_11) = [1] p(c_12) = [1] x1 + [0] p(c_13) = [0] p(c_14) = [4] p(c_15) = [1] x1 + [4] x2 + [4] x3 + [0] p(c_16) = [4] x3 + [1] p(c_17) = [1] x1 + [0] p(c_18) = [4] x1 + [2] x4 + [4] x5 + [2] p(c_19) = [4] p(c_20) = [1] p(c_21) = [1] p(c_22) = [4] p(c_23) = [1] x1 + [1] Following rules are strictly oriented: U12'#(tt(),z0,z1) = [2] z0 + [14] > [2] z0 + [13] = c_6(PLUS#(activate(z1),activate(z0))) Following rules are (at-least) weakly oriented: PLUS#(z0,s(z1)) = [2] z1 + [14] >= [2] z1 + [14] = c_3(U11'#(tt(),z1,z0)) U11'#(tt(),z0,z1) = [2] z0 + [14] >= [2] z0 + [14] = c_4(U12'#(tt(),activate(z0),activate(z1))) U11'#(tt(),z0,z1) = [2] z0 + [14] >= [2] z0 + [14] = c_5(U12'#(tt(),activate(z0),activate(z1))) U12'#(tt(),z0,z1) = [2] z0 + [14] >= [2] z0 + [6] = c_7(PLUS#(activate(z1),activate(z0))) U21'#(tt(),z0,z1) = [2] z0 + [4] z1 + [8] >= [2] z0 + [4] z1 + [8] = U22'#(tt(),activate(z0),activate(z1)) U22'#(tt(),z0,z1) = [2] z0 + [4] z1 + [8] >= [2] z1 + [6] = PLUS#(x(activate(z1),activate(z0)),activate(z1)) U22'#(tt(),z0,z1) = [2] z0 + [4] z1 + [8] >= [2] z0 + [4] z1 + [2] = X#(activate(z1),activate(z0)) U22'#(tt(),z0,z1) = [2] z0 + [4] z1 + [8] >= [2] z1 + [6] = c_12(PLUS#(x(activate(z1),activate(z0)),activate(z1))) X#(z0,s(z1)) = [4] z0 + [2] z1 + [10] >= [4] z0 + [2] z1 + [8] = U21'#(tt(),z1,z0) activate(z0) = [1] z0 + [0] >= [1] z0 + [0] = z0 ** Step 7.b:7: EmptyProcessor. WORST_CASE(?,O(1)) + Considered Problem: - Weak DPs: PLUS#(z0,s(z1)) -> c_3(U11'#(tt(),z1,z0)) U11'#(tt(),z0,z1) -> c_4(U12'#(tt(),activate(z0),activate(z1))) U11'#(tt(),z0,z1) -> c_5(U12'#(tt(),activate(z0),activate(z1))) U12'#(tt(),z0,z1) -> c_6(PLUS#(activate(z1),activate(z0))) U12'#(tt(),z0,z1) -> c_7(PLUS#(activate(z1),activate(z0))) U21'#(tt(),z0,z1) -> U22'#(tt(),activate(z0),activate(z1)) U22'#(tt(),z0,z1) -> PLUS#(x(activate(z1),activate(z0)),activate(z1)) U22'#(tt(),z0,z1) -> X#(activate(z1),activate(z0)) U22'#(tt(),z0,z1) -> c_12(PLUS#(x(activate(z1),activate(z0)),activate(z1))) X#(z0,s(z1)) -> U21'#(tt(),z1,z0) - Weak TRS: U11(tt(),z0,z1) -> U12(tt(),activate(z0),activate(z1)) U12(tt(),z0,z1) -> s(plus(activate(z1),activate(z0))) U21(tt(),z0,z1) -> U22(tt(),activate(z0),activate(z1)) U22(tt(),z0,z1) -> plus(x(activate(z1),activate(z0)),activate(z1)) activate(z0) -> z0 plus(z0,0()) -> z0 plus(z0,s(z1)) -> U11(tt(),z1,z0) x(z0,0()) -> 0() x(z0,s(z1)) -> U21(tt(),z1,z0) - Signature: {ACTIVATE/1,PLUS/2,U11/3,U11'/3,U12/3,U12'/3,U21/3,U21'/3,U22/3,U22'/3,X/2,activate/1,plus/2,x/2,ACTIVATE#/1 ,PLUS#/2,U11#/3,U11'#/3,U12#/3,U12'#/3,U21#/3,U21'#/3,U22#/3,U22'#/3,X#/2,activate#/1,plus#/2,x#/2} / {0/0 ,c/2,c1/2,c10/1,c11/0,c12/1,c13/0,c2/2,c3/2,c4/2,c5/2,c6/3,c7/3,c8/2,c9/0,s/1,tt/0,c_1/0,c_2/0,c_3/1,c_4/1 ,c_5/1,c_6/1,c_7/1,c_8/1,c_9/1,c_10/2,c_11/2,c_12/1,c_13/0,c_14/1,c_15/3,c_16/3,c_17/3,c_18/5,c_19/0,c_20/0 ,c_21/1,c_22/0,c_23/1} - Obligation: innermost runtime complexity wrt. defined symbols {ACTIVATE#,PLUS#,U11#,U11'#,U12#,U12'#,U21#,U21'#,U22# ,U22'#,X#,activate#,plus#,x#} and constructors {0,c,c1,c10,c11,c12,c13,c2,c3,c4,c5,c6,c7,c8,c9,s,tt} + Applied Processor: EmptyProcessor + Details: The problem is already closed. The intended complexity is O(1). WORST_CASE(?,O(n^2))