WORST_CASE(?,O(n^1)) * Step 1: InnermostRuleRemoval WORST_CASE(?,O(n^1)) + Considered Problem: - Strict TRS: 0() -> n__0() U11(tt(),V2) -> U12(isNat(activate(V2))) U12(tt()) -> tt() U21(tt()) -> tt() U31(tt(),N) -> activate(N) U41(tt(),M,N) -> U42(isNat(activate(N)),activate(M),activate(N)) U42(tt(),M,N) -> s(plus(activate(N),activate(M))) activate(X) -> X activate(n__0()) -> 0() activate(n__plus(X1,X2)) -> plus(X1,X2) activate(n__s(X)) -> s(X) isNat(n__0()) -> tt() isNat(n__plus(V1,V2)) -> U11(isNat(activate(V1)),activate(V2)) isNat(n__s(V1)) -> U21(isNat(activate(V1))) plus(N,0()) -> U31(isNat(N),N) plus(N,s(M)) -> U41(isNat(M),M,N) plus(X1,X2) -> n__plus(X1,X2) s(X) -> n__s(X) - Signature: {0/0,U11/2,U12/1,U21/1,U31/2,U41/3,U42/3,activate/1,isNat/1,plus/2,s/1} / {n__0/0,n__plus/2,n__s/1,tt/0} - Obligation: innermost runtime complexity wrt. defined symbols {0,U11,U12,U21,U31,U41,U42,activate,isNat,plus ,s} and constructors {n__0,n__plus,n__s,tt} + Applied Processor: InnermostRuleRemoval + Details: Arguments of following rules are not normal-forms. plus(N,0()) -> U31(isNat(N),N) plus(N,s(M)) -> U41(isNat(M),M,N) All above mentioned rules can be savely removed. * Step 2: DependencyPairs WORST_CASE(?,O(n^1)) + Considered Problem: - Strict TRS: 0() -> n__0() U11(tt(),V2) -> U12(isNat(activate(V2))) U12(tt()) -> tt() U21(tt()) -> tt() U31(tt(),N) -> activate(N) U41(tt(),M,N) -> U42(isNat(activate(N)),activate(M),activate(N)) U42(tt(),M,N) -> s(plus(activate(N),activate(M))) activate(X) -> X activate(n__0()) -> 0() activate(n__plus(X1,X2)) -> plus(X1,X2) activate(n__s(X)) -> s(X) isNat(n__0()) -> tt() isNat(n__plus(V1,V2)) -> U11(isNat(activate(V1)),activate(V2)) isNat(n__s(V1)) -> U21(isNat(activate(V1))) plus(X1,X2) -> n__plus(X1,X2) s(X) -> n__s(X) - Signature: {0/0,U11/2,U12/1,U21/1,U31/2,U41/3,U42/3,activate/1,isNat/1,plus/2,s/1} / {n__0/0,n__plus/2,n__s/1,tt/0} - Obligation: innermost runtime complexity wrt. defined symbols {0,U11,U12,U21,U31,U41,U42,activate,isNat,plus ,s} and constructors {n__0,n__plus,n__s,tt} + Applied Processor: DependencyPairs {dpKind_ = WIDP} + Details: We add the following weak innermost dependency pairs: Strict DPs 0#() -> c_1() U11#(tt(),V2) -> c_2(U12#(isNat(activate(V2)))) U12#(tt()) -> c_3() U21#(tt()) -> c_4() U31#(tt(),N) -> c_5(activate#(N)) U41#(tt(),M,N) -> c_6(U42#(isNat(activate(N)),activate(M),activate(N))) U42#(tt(),M,N) -> c_7(s#(plus(activate(N),activate(M)))) activate#(X) -> c_8() activate#(n__0()) -> c_9(0#()) activate#(n__plus(X1,X2)) -> c_10(plus#(X1,X2)) activate#(n__s(X)) -> c_11(s#(X)) isNat#(n__0()) -> c_12() isNat#(n__plus(V1,V2)) -> c_13(U11#(isNat(activate(V1)),activate(V2))) isNat#(n__s(V1)) -> c_14(U21#(isNat(activate(V1)))) plus#(X1,X2) -> c_15() s#(X) -> c_16() Weak DPs and mark the set of starting terms. * Step 3: UsableRules WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: 0#() -> c_1() U11#(tt(),V2) -> c_2(U12#(isNat(activate(V2)))) U12#(tt()) -> c_3() U21#(tt()) -> c_4() U31#(tt(),N) -> c_5(activate#(N)) U41#(tt(),M,N) -> c_6(U42#(isNat(activate(N)),activate(M),activate(N))) U42#(tt(),M,N) -> c_7(s#(plus(activate(N),activate(M)))) activate#(X) -> c_8() activate#(n__0()) -> c_9(0#()) activate#(n__plus(X1,X2)) -> c_10(plus#(X1,X2)) activate#(n__s(X)) -> c_11(s#(X)) isNat#(n__0()) -> c_12() isNat#(n__plus(V1,V2)) -> c_13(U11#(isNat(activate(V1)),activate(V2))) isNat#(n__s(V1)) -> c_14(U21#(isNat(activate(V1)))) plus#(X1,X2) -> c_15() s#(X) -> c_16() - Strict TRS: 0() -> n__0() U11(tt(),V2) -> U12(isNat(activate(V2))) U12(tt()) -> tt() U21(tt()) -> tt() U31(tt(),N) -> activate(N) U41(tt(),M,N) -> U42(isNat(activate(N)),activate(M),activate(N)) U42(tt(),M,N) -> s(plus(activate(N),activate(M))) activate(X) -> X activate(n__0()) -> 0() activate(n__plus(X1,X2)) -> plus(X1,X2) activate(n__s(X)) -> s(X) isNat(n__0()) -> tt() isNat(n__plus(V1,V2)) -> U11(isNat(activate(V1)),activate(V2)) isNat(n__s(V1)) -> U21(isNat(activate(V1))) plus(X1,X2) -> n__plus(X1,X2) s(X) -> n__s(X) - Signature: {0/0,U11/2,U12/1,U21/1,U31/2,U41/3,U42/3,activate/1,isNat/1,plus/2,s/1,0#/0,U11#/2,U12#/1,U21#/1,U31#/2 ,U41#/3,U42#/3,activate#/1,isNat#/1,plus#/2,s#/1} / {n__0/0,n__plus/2,n__s/1,tt/0,c_1/0,c_2/1,c_3/0,c_4/0 ,c_5/1,c_6/1,c_7/1,c_8/0,c_9/1,c_10/1,c_11/1,c_12/0,c_13/1,c_14/1,c_15/0,c_16/0} - Obligation: innermost runtime complexity wrt. defined symbols {0#,U11#,U12#,U21#,U31#,U41#,U42#,activate#,isNat#,plus# ,s#} and constructors {n__0,n__plus,n__s,tt} + Applied Processor: UsableRules + Details: We replace rewrite rules by usable rules: 0() -> n__0() U11(tt(),V2) -> U12(isNat(activate(V2))) U12(tt()) -> tt() U21(tt()) -> tt() activate(X) -> X activate(n__0()) -> 0() activate(n__plus(X1,X2)) -> plus(X1,X2) activate(n__s(X)) -> s(X) isNat(n__0()) -> tt() isNat(n__plus(V1,V2)) -> U11(isNat(activate(V1)),activate(V2)) isNat(n__s(V1)) -> U21(isNat(activate(V1))) plus(X1,X2) -> n__plus(X1,X2) s(X) -> n__s(X) 0#() -> c_1() U11#(tt(),V2) -> c_2(U12#(isNat(activate(V2)))) U12#(tt()) -> c_3() U21#(tt()) -> c_4() U31#(tt(),N) -> c_5(activate#(N)) U41#(tt(),M,N) -> c_6(U42#(isNat(activate(N)),activate(M),activate(N))) U42#(tt(),M,N) -> c_7(s#(plus(activate(N),activate(M)))) activate#(X) -> c_8() activate#(n__0()) -> c_9(0#()) activate#(n__plus(X1,X2)) -> c_10(plus#(X1,X2)) activate#(n__s(X)) -> c_11(s#(X)) isNat#(n__0()) -> c_12() isNat#(n__plus(V1,V2)) -> c_13(U11#(isNat(activate(V1)),activate(V2))) isNat#(n__s(V1)) -> c_14(U21#(isNat(activate(V1)))) plus#(X1,X2) -> c_15() s#(X) -> c_16() * Step 4: WeightGap WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: 0#() -> c_1() U11#(tt(),V2) -> c_2(U12#(isNat(activate(V2)))) U12#(tt()) -> c_3() U21#(tt()) -> c_4() U31#(tt(),N) -> c_5(activate#(N)) U41#(tt(),M,N) -> c_6(U42#(isNat(activate(N)),activate(M),activate(N))) U42#(tt(),M,N) -> c_7(s#(plus(activate(N),activate(M)))) activate#(X) -> c_8() activate#(n__0()) -> c_9(0#()) activate#(n__plus(X1,X2)) -> c_10(plus#(X1,X2)) activate#(n__s(X)) -> c_11(s#(X)) isNat#(n__0()) -> c_12() isNat#(n__plus(V1,V2)) -> c_13(U11#(isNat(activate(V1)),activate(V2))) isNat#(n__s(V1)) -> c_14(U21#(isNat(activate(V1)))) plus#(X1,X2) -> c_15() s#(X) -> c_16() - Strict TRS: 0() -> n__0() U11(tt(),V2) -> U12(isNat(activate(V2))) U12(tt()) -> tt() U21(tt()) -> tt() activate(X) -> X activate(n__0()) -> 0() activate(n__plus(X1,X2)) -> plus(X1,X2) activate(n__s(X)) -> s(X) isNat(n__0()) -> tt() isNat(n__plus(V1,V2)) -> U11(isNat(activate(V1)),activate(V2)) isNat(n__s(V1)) -> U21(isNat(activate(V1))) plus(X1,X2) -> n__plus(X1,X2) s(X) -> n__s(X) - Signature: {0/0,U11/2,U12/1,U21/1,U31/2,U41/3,U42/3,activate/1,isNat/1,plus/2,s/1,0#/0,U11#/2,U12#/1,U21#/1,U31#/2 ,U41#/3,U42#/3,activate#/1,isNat#/1,plus#/2,s#/1} / {n__0/0,n__plus/2,n__s/1,tt/0,c_1/0,c_2/1,c_3/0,c_4/0 ,c_5/1,c_6/1,c_7/1,c_8/0,c_9/1,c_10/1,c_11/1,c_12/0,c_13/1,c_14/1,c_15/0,c_16/0} - Obligation: innermost runtime complexity wrt. defined symbols {0#,U11#,U12#,U21#,U31#,U41#,U42#,activate#,isNat#,plus# ,s#} and constructors {n__0,n__plus,n__s,tt} + Applied Processor: WeightGap {wgDimension = 1, wgDegree = 1, wgKind = Algebraic, wgUArgs = UArgs, wgOn = WgOnTrs} + 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(U11) = {1,2}, uargs(U12) = {1}, uargs(U21) = {1}, uargs(isNat) = {1}, uargs(plus) = {1,2}, uargs(U11#) = {1,2}, uargs(U12#) = {1}, uargs(U21#) = {1}, uargs(U42#) = {1,2,3}, uargs(s#) = {1}, uargs(c_2) = {1}, uargs(c_5) = {1}, uargs(c_6) = {1}, uargs(c_7) = {1}, uargs(c_9) = {1}, uargs(c_10) = {1}, uargs(c_11) = {1}, uargs(c_13) = {1}, uargs(c_14) = {1} Following symbols are considered usable: all TcT has computed the following interpretation: p(0) = [6] p(U11) = [1] x1 + [1] x2 + [0] p(U12) = [1] x1 + [1] p(U21) = [1] x1 + [2] p(U31) = [0] p(U41) = [0] p(U42) = [0] p(activate) = [1] x1 + [2] p(isNat) = [1] x1 + [0] p(n__0) = [5] p(n__plus) = [1] x1 + [1] x2 + [5] p(n__s) = [1] x1 + [5] p(plus) = [1] x1 + [1] x2 + [6] p(s) = [1] x1 + [6] p(tt) = [4] p(0#) = [5] p(U11#) = [1] x1 + [1] x2 + [0] p(U12#) = [1] x1 + [0] p(U21#) = [1] x1 + [0] p(U31#) = [4] x2 + [0] p(U41#) = [1] x2 + [4] x3 + [0] p(U42#) = [1] x1 + [1] x2 + [1] x3 + [6] p(activate#) = [2] x1 + [3] p(isNat#) = [2] x1 + [1] p(plus#) = [2] x2 + [0] p(s#) = [1] x1 + [0] p(c_1) = [0] p(c_2) = [1] x1 + [0] p(c_3) = [0] p(c_4) = [0] p(c_5) = [1] x1 + [0] p(c_6) = [1] x1 + [0] p(c_7) = [1] x1 + [0] p(c_8) = [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) = [1] x1 + [0] p(c_14) = [1] x1 + [0] p(c_15) = [0] p(c_16) = [0] Following rules are strictly oriented: 0#() = [5] > [0] = c_1() U11#(tt(),V2) = [1] V2 + [4] > [1] V2 + [2] = c_2(U12#(isNat(activate(V2)))) U12#(tt()) = [4] > [0] = c_3() U21#(tt()) = [4] > [0] = c_4() activate#(X) = [2] X + [3] > [0] = c_8() activate#(n__0()) = [13] > [5] = c_9(0#()) activate#(n__plus(X1,X2)) = [2] X1 + [2] X2 + [13] > [2] X2 + [0] = c_10(plus#(X1,X2)) activate#(n__s(X)) = [2] X + [13] > [1] X + [0] = c_11(s#(X)) isNat#(n__0()) = [11] > [0] = c_12() isNat#(n__plus(V1,V2)) = [2] V1 + [2] V2 + [11] > [1] V1 + [1] V2 + [4] = c_13(U11#(isNat(activate(V1)),activate(V2))) isNat#(n__s(V1)) = [2] V1 + [11] > [1] V1 + [2] = c_14(U21#(isNat(activate(V1)))) 0() = [6] > [5] = n__0() U11(tt(),V2) = [1] V2 + [4] > [1] V2 + [3] = U12(isNat(activate(V2))) U12(tt()) = [5] > [4] = tt() U21(tt()) = [6] > [4] = tt() activate(X) = [1] X + [2] > [1] X + [0] = X activate(n__0()) = [7] > [6] = 0() activate(n__plus(X1,X2)) = [1] X1 + [1] X2 + [7] > [1] X1 + [1] X2 + [6] = plus(X1,X2) activate(n__s(X)) = [1] X + [7] > [1] X + [6] = s(X) isNat(n__0()) = [5] > [4] = tt() isNat(n__plus(V1,V2)) = [1] V1 + [1] V2 + [5] > [1] V1 + [1] V2 + [4] = U11(isNat(activate(V1)),activate(V2)) isNat(n__s(V1)) = [1] V1 + [5] > [1] V1 + [4] = U21(isNat(activate(V1))) plus(X1,X2) = [1] X1 + [1] X2 + [6] > [1] X1 + [1] X2 + [5] = n__plus(X1,X2) s(X) = [1] X + [6] > [1] X + [5] = n__s(X) Following rules are (at-least) weakly oriented: U31#(tt(),N) = [4] N + [0] >= [2] N + [3] = c_5(activate#(N)) U41#(tt(),M,N) = [1] M + [4] N + [0] >= [1] M + [2] N + [12] = c_6(U42#(isNat(activate(N)),activate(M),activate(N))) U42#(tt(),M,N) = [1] M + [1] N + [10] >= [1] M + [1] N + [10] = c_7(s#(plus(activate(N),activate(M)))) plus#(X1,X2) = [2] X2 + [0] >= [0] = c_15() s#(X) = [1] X + [0] >= [0] = c_16() Further, it can be verified that all rules not oriented are covered by the weightgap condition. * Step 5: PredecessorEstimation WORST_CASE(?,O(1)) + Considered Problem: - Strict DPs: U31#(tt(),N) -> c_5(activate#(N)) U41#(tt(),M,N) -> c_6(U42#(isNat(activate(N)),activate(M),activate(N))) U42#(tt(),M,N) -> c_7(s#(plus(activate(N),activate(M)))) plus#(X1,X2) -> c_15() s#(X) -> c_16() - Weak DPs: 0#() -> c_1() U11#(tt(),V2) -> c_2(U12#(isNat(activate(V2)))) U12#(tt()) -> c_3() U21#(tt()) -> c_4() activate#(X) -> c_8() activate#(n__0()) -> c_9(0#()) activate#(n__plus(X1,X2)) -> c_10(plus#(X1,X2)) activate#(n__s(X)) -> c_11(s#(X)) isNat#(n__0()) -> c_12() isNat#(n__plus(V1,V2)) -> c_13(U11#(isNat(activate(V1)),activate(V2))) isNat#(n__s(V1)) -> c_14(U21#(isNat(activate(V1)))) - Weak TRS: 0() -> n__0() U11(tt(),V2) -> U12(isNat(activate(V2))) U12(tt()) -> tt() U21(tt()) -> tt() activate(X) -> X activate(n__0()) -> 0() activate(n__plus(X1,X2)) -> plus(X1,X2) activate(n__s(X)) -> s(X) isNat(n__0()) -> tt() isNat(n__plus(V1,V2)) -> U11(isNat(activate(V1)),activate(V2)) isNat(n__s(V1)) -> U21(isNat(activate(V1))) plus(X1,X2) -> n__plus(X1,X2) s(X) -> n__s(X) - Signature: {0/0,U11/2,U12/1,U21/1,U31/2,U41/3,U42/3,activate/1,isNat/1,plus/2,s/1,0#/0,U11#/2,U12#/1,U21#/1,U31#/2 ,U41#/3,U42#/3,activate#/1,isNat#/1,plus#/2,s#/1} / {n__0/0,n__plus/2,n__s/1,tt/0,c_1/0,c_2/1,c_3/0,c_4/0 ,c_5/1,c_6/1,c_7/1,c_8/0,c_9/1,c_10/1,c_11/1,c_12/0,c_13/1,c_14/1,c_15/0,c_16/0} - Obligation: innermost runtime complexity wrt. defined symbols {0#,U11#,U12#,U21#,U31#,U41#,U42#,activate#,isNat#,plus# ,s#} and constructors {n__0,n__plus,n__s,tt} + Applied Processor: PredecessorEstimation {onSelection = all simple predecessor estimation selector} + Details: We estimate the number of application of {1} by application of Pre({1}) = {}. Here rules are labelled as follows: 1: U31#(tt(),N) -> c_5(activate#(N)) 2: U41#(tt(),M,N) -> c_6(U42#(isNat(activate(N)),activate(M),activate(N))) 3: U42#(tt(),M,N) -> c_7(s#(plus(activate(N),activate(M)))) 4: plus#(X1,X2) -> c_15() 5: s#(X) -> c_16() 6: 0#() -> c_1() 7: U11#(tt(),V2) -> c_2(U12#(isNat(activate(V2)))) 8: U12#(tt()) -> c_3() 9: U21#(tt()) -> c_4() 10: activate#(X) -> c_8() 11: activate#(n__0()) -> c_9(0#()) 12: activate#(n__plus(X1,X2)) -> c_10(plus#(X1,X2)) 13: activate#(n__s(X)) -> c_11(s#(X)) 14: isNat#(n__0()) -> c_12() 15: isNat#(n__plus(V1,V2)) -> c_13(U11#(isNat(activate(V1)),activate(V2))) 16: isNat#(n__s(V1)) -> c_14(U21#(isNat(activate(V1)))) * Step 6: RemoveWeakSuffixes WORST_CASE(?,O(1)) + Considered Problem: - Strict DPs: U41#(tt(),M,N) -> c_6(U42#(isNat(activate(N)),activate(M),activate(N))) U42#(tt(),M,N) -> c_7(s#(plus(activate(N),activate(M)))) plus#(X1,X2) -> c_15() s#(X) -> c_16() - Weak DPs: 0#() -> c_1() U11#(tt(),V2) -> c_2(U12#(isNat(activate(V2)))) U12#(tt()) -> c_3() U21#(tt()) -> c_4() U31#(tt(),N) -> c_5(activate#(N)) activate#(X) -> c_8() activate#(n__0()) -> c_9(0#()) activate#(n__plus(X1,X2)) -> c_10(plus#(X1,X2)) activate#(n__s(X)) -> c_11(s#(X)) isNat#(n__0()) -> c_12() isNat#(n__plus(V1,V2)) -> c_13(U11#(isNat(activate(V1)),activate(V2))) isNat#(n__s(V1)) -> c_14(U21#(isNat(activate(V1)))) - Weak TRS: 0() -> n__0() U11(tt(),V2) -> U12(isNat(activate(V2))) U12(tt()) -> tt() U21(tt()) -> tt() activate(X) -> X activate(n__0()) -> 0() activate(n__plus(X1,X2)) -> plus(X1,X2) activate(n__s(X)) -> s(X) isNat(n__0()) -> tt() isNat(n__plus(V1,V2)) -> U11(isNat(activate(V1)),activate(V2)) isNat(n__s(V1)) -> U21(isNat(activate(V1))) plus(X1,X2) -> n__plus(X1,X2) s(X) -> n__s(X) - Signature: {0/0,U11/2,U12/1,U21/1,U31/2,U41/3,U42/3,activate/1,isNat/1,plus/2,s/1,0#/0,U11#/2,U12#/1,U21#/1,U31#/2 ,U41#/3,U42#/3,activate#/1,isNat#/1,plus#/2,s#/1} / {n__0/0,n__plus/2,n__s/1,tt/0,c_1/0,c_2/1,c_3/0,c_4/0 ,c_5/1,c_6/1,c_7/1,c_8/0,c_9/1,c_10/1,c_11/1,c_12/0,c_13/1,c_14/1,c_15/0,c_16/0} - Obligation: innermost runtime complexity wrt. defined symbols {0#,U11#,U12#,U21#,U31#,U41#,U42#,activate#,isNat#,plus# ,s#} and constructors {n__0,n__plus,n__s,tt} + Applied Processor: RemoveWeakSuffixes + Details: Consider the dependency graph 1:S:U41#(tt(),M,N) -> c_6(U42#(isNat(activate(N)),activate(M),activate(N))) -->_1 U42#(tt(),M,N) -> c_7(s#(plus(activate(N),activate(M)))):2 2:S:U42#(tt(),M,N) -> c_7(s#(plus(activate(N),activate(M)))) -->_1 s#(X) -> c_16():4 3:S:plus#(X1,X2) -> c_15() 4:S:s#(X) -> c_16() 5:W:0#() -> c_1() 6:W:U11#(tt(),V2) -> c_2(U12#(isNat(activate(V2)))) -->_1 U12#(tt()) -> c_3():7 7:W:U12#(tt()) -> c_3() 8:W:U21#(tt()) -> c_4() 9:W:U31#(tt(),N) -> c_5(activate#(N)) -->_1 activate#(n__s(X)) -> c_11(s#(X)):13 -->_1 activate#(n__plus(X1,X2)) -> c_10(plus#(X1,X2)):12 -->_1 activate#(n__0()) -> c_9(0#()):11 -->_1 activate#(X) -> c_8():10 10:W:activate#(X) -> c_8() 11:W:activate#(n__0()) -> c_9(0#()) -->_1 0#() -> c_1():5 12:W:activate#(n__plus(X1,X2)) -> c_10(plus#(X1,X2)) -->_1 plus#(X1,X2) -> c_15():3 13:W:activate#(n__s(X)) -> c_11(s#(X)) -->_1 s#(X) -> c_16():4 14:W:isNat#(n__0()) -> c_12() 15:W:isNat#(n__plus(V1,V2)) -> c_13(U11#(isNat(activate(V1)),activate(V2))) -->_1 U11#(tt(),V2) -> c_2(U12#(isNat(activate(V2)))):6 16:W:isNat#(n__s(V1)) -> c_14(U21#(isNat(activate(V1)))) -->_1 U21#(tt()) -> c_4():8 The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed. 16: isNat#(n__s(V1)) -> c_14(U21#(isNat(activate(V1)))) 15: isNat#(n__plus(V1,V2)) -> c_13(U11#(isNat(activate(V1)),activate(V2))) 14: isNat#(n__0()) -> c_12() 10: activate#(X) -> c_8() 11: activate#(n__0()) -> c_9(0#()) 8: U21#(tt()) -> c_4() 6: U11#(tt(),V2) -> c_2(U12#(isNat(activate(V2)))) 7: U12#(tt()) -> c_3() 5: 0#() -> c_1() * Step 7: RemoveHeads WORST_CASE(?,O(1)) + Considered Problem: - Strict DPs: U41#(tt(),M,N) -> c_6(U42#(isNat(activate(N)),activate(M),activate(N))) U42#(tt(),M,N) -> c_7(s#(plus(activate(N),activate(M)))) plus#(X1,X2) -> c_15() s#(X) -> c_16() - Weak DPs: U31#(tt(),N) -> c_5(activate#(N)) activate#(n__plus(X1,X2)) -> c_10(plus#(X1,X2)) activate#(n__s(X)) -> c_11(s#(X)) - Weak TRS: 0() -> n__0() U11(tt(),V2) -> U12(isNat(activate(V2))) U12(tt()) -> tt() U21(tt()) -> tt() activate(X) -> X activate(n__0()) -> 0() activate(n__plus(X1,X2)) -> plus(X1,X2) activate(n__s(X)) -> s(X) isNat(n__0()) -> tt() isNat(n__plus(V1,V2)) -> U11(isNat(activate(V1)),activate(V2)) isNat(n__s(V1)) -> U21(isNat(activate(V1))) plus(X1,X2) -> n__plus(X1,X2) s(X) -> n__s(X) - Signature: {0/0,U11/2,U12/1,U21/1,U31/2,U41/3,U42/3,activate/1,isNat/1,plus/2,s/1,0#/0,U11#/2,U12#/1,U21#/1,U31#/2 ,U41#/3,U42#/3,activate#/1,isNat#/1,plus#/2,s#/1} / {n__0/0,n__plus/2,n__s/1,tt/0,c_1/0,c_2/1,c_3/0,c_4/0 ,c_5/1,c_6/1,c_7/1,c_8/0,c_9/1,c_10/1,c_11/1,c_12/0,c_13/1,c_14/1,c_15/0,c_16/0} - Obligation: innermost runtime complexity wrt. defined symbols {0#,U11#,U12#,U21#,U31#,U41#,U42#,activate#,isNat#,plus# ,s#} and constructors {n__0,n__plus,n__s,tt} + Applied Processor: RemoveHeads + Details: Consider the dependency graph 1:S:U41#(tt(),M,N) -> c_6(U42#(isNat(activate(N)),activate(M),activate(N))) -->_1 U42#(tt(),M,N) -> c_7(s#(plus(activate(N),activate(M)))):2 2:S:U42#(tt(),M,N) -> c_7(s#(plus(activate(N),activate(M)))) -->_1 s#(X) -> c_16():4 3:S:plus#(X1,X2) -> c_15() 4:S:s#(X) -> c_16() 9:W:U31#(tt(),N) -> c_5(activate#(N)) -->_1 activate#(n__s(X)) -> c_11(s#(X)):13 -->_1 activate#(n__plus(X1,X2)) -> c_10(plus#(X1,X2)):12 12:W:activate#(n__plus(X1,X2)) -> c_10(plus#(X1,X2)) -->_1 plus#(X1,X2) -> c_15():3 13:W:activate#(n__s(X)) -> c_11(s#(X)) -->_1 s#(X) -> c_16():4 Following roots of the dependency graph are removed, as the considered set of starting terms is closed under reduction with respect to these rules (modulo compound contexts). [(9,U31#(tt(),N) -> c_5(activate#(N)))] * Step 8: RemoveHeads WORST_CASE(?,O(1)) + Considered Problem: - Strict DPs: U41#(tt(),M,N) -> c_6(U42#(isNat(activate(N)),activate(M),activate(N))) U42#(tt(),M,N) -> c_7(s#(plus(activate(N),activate(M)))) plus#(X1,X2) -> c_15() s#(X) -> c_16() - Weak DPs: activate#(n__plus(X1,X2)) -> c_10(plus#(X1,X2)) activate#(n__s(X)) -> c_11(s#(X)) - Weak TRS: 0() -> n__0() U11(tt(),V2) -> U12(isNat(activate(V2))) U12(tt()) -> tt() U21(tt()) -> tt() activate(X) -> X activate(n__0()) -> 0() activate(n__plus(X1,X2)) -> plus(X1,X2) activate(n__s(X)) -> s(X) isNat(n__0()) -> tt() isNat(n__plus(V1,V2)) -> U11(isNat(activate(V1)),activate(V2)) isNat(n__s(V1)) -> U21(isNat(activate(V1))) plus(X1,X2) -> n__plus(X1,X2) s(X) -> n__s(X) - Signature: {0/0,U11/2,U12/1,U21/1,U31/2,U41/3,U42/3,activate/1,isNat/1,plus/2,s/1,0#/0,U11#/2,U12#/1,U21#/1,U31#/2 ,U41#/3,U42#/3,activate#/1,isNat#/1,plus#/2,s#/1} / {n__0/0,n__plus/2,n__s/1,tt/0,c_1/0,c_2/1,c_3/0,c_4/0 ,c_5/1,c_6/1,c_7/1,c_8/0,c_9/1,c_10/1,c_11/1,c_12/0,c_13/1,c_14/1,c_15/0,c_16/0} - Obligation: innermost runtime complexity wrt. defined symbols {0#,U11#,U12#,U21#,U31#,U41#,U42#,activate#,isNat#,plus# ,s#} and constructors {n__0,n__plus,n__s,tt} + Applied Processor: RemoveHeads + Details: Consider the dependency graph 1:S:U41#(tt(),M,N) -> c_6(U42#(isNat(activate(N)),activate(M),activate(N))) -->_1 U42#(tt(),M,N) -> c_7(s#(plus(activate(N),activate(M)))):2 2:S:U42#(tt(),M,N) -> c_7(s#(plus(activate(N),activate(M)))) -->_1 s#(X) -> c_16():4 3:S:plus#(X1,X2) -> c_15() 4:S:s#(X) -> c_16() 12:W:activate#(n__plus(X1,X2)) -> c_10(plus#(X1,X2)) -->_1 plus#(X1,X2) -> c_15():3 13:W:activate#(n__s(X)) -> c_11(s#(X)) -->_1 s#(X) -> c_16():4 Following roots of the dependency graph are removed, as the considered set of starting terms is closed under reduction with respect to these rules (modulo compound contexts). [(12,activate#(n__plus(X1,X2)) -> c_10(plus#(X1,X2))),(13,activate#(n__s(X)) -> c_11(s#(X)))] * Step 9: RemoveHeads WORST_CASE(?,O(1)) + Considered Problem: - Strict DPs: U41#(tt(),M,N) -> c_6(U42#(isNat(activate(N)),activate(M),activate(N))) U42#(tt(),M,N) -> c_7(s#(plus(activate(N),activate(M)))) plus#(X1,X2) -> c_15() s#(X) -> c_16() - Weak TRS: 0() -> n__0() U11(tt(),V2) -> U12(isNat(activate(V2))) U12(tt()) -> tt() U21(tt()) -> tt() activate(X) -> X activate(n__0()) -> 0() activate(n__plus(X1,X2)) -> plus(X1,X2) activate(n__s(X)) -> s(X) isNat(n__0()) -> tt() isNat(n__plus(V1,V2)) -> U11(isNat(activate(V1)),activate(V2)) isNat(n__s(V1)) -> U21(isNat(activate(V1))) plus(X1,X2) -> n__plus(X1,X2) s(X) -> n__s(X) - Signature: {0/0,U11/2,U12/1,U21/1,U31/2,U41/3,U42/3,activate/1,isNat/1,plus/2,s/1,0#/0,U11#/2,U12#/1,U21#/1,U31#/2 ,U41#/3,U42#/3,activate#/1,isNat#/1,plus#/2,s#/1} / {n__0/0,n__plus/2,n__s/1,tt/0,c_1/0,c_2/1,c_3/0,c_4/0 ,c_5/1,c_6/1,c_7/1,c_8/0,c_9/1,c_10/1,c_11/1,c_12/0,c_13/1,c_14/1,c_15/0,c_16/0} - Obligation: innermost runtime complexity wrt. defined symbols {0#,U11#,U12#,U21#,U31#,U41#,U42#,activate#,isNat#,plus# ,s#} and constructors {n__0,n__plus,n__s,tt} + Applied Processor: RemoveHeads + Details: Consider the dependency graph 1:S:U41#(tt(),M,N) -> c_6(U42#(isNat(activate(N)),activate(M),activate(N))) -->_1 U42#(tt(),M,N) -> c_7(s#(plus(activate(N),activate(M)))):2 2:S:U42#(tt(),M,N) -> c_7(s#(plus(activate(N),activate(M)))) -->_1 s#(X) -> c_16():4 3:S:plus#(X1,X2) -> c_15() 4:S:s#(X) -> c_16() Following roots of the dependency graph are removed, as the considered set of starting terms is closed under reduction with respect to these rules (modulo compound contexts). [(3,plus#(X1,X2) -> c_15())] * Step 10: PredecessorEstimation WORST_CASE(?,O(1)) + Considered Problem: - Strict DPs: U41#(tt(),M,N) -> c_6(U42#(isNat(activate(N)),activate(M),activate(N))) U42#(tt(),M,N) -> c_7(s#(plus(activate(N),activate(M)))) s#(X) -> c_16() - Weak TRS: 0() -> n__0() U11(tt(),V2) -> U12(isNat(activate(V2))) U12(tt()) -> tt() U21(tt()) -> tt() activate(X) -> X activate(n__0()) -> 0() activate(n__plus(X1,X2)) -> plus(X1,X2) activate(n__s(X)) -> s(X) isNat(n__0()) -> tt() isNat(n__plus(V1,V2)) -> U11(isNat(activate(V1)),activate(V2)) isNat(n__s(V1)) -> U21(isNat(activate(V1))) plus(X1,X2) -> n__plus(X1,X2) s(X) -> n__s(X) - Signature: {0/0,U11/2,U12/1,U21/1,U31/2,U41/3,U42/3,activate/1,isNat/1,plus/2,s/1,0#/0,U11#/2,U12#/1,U21#/1,U31#/2 ,U41#/3,U42#/3,activate#/1,isNat#/1,plus#/2,s#/1} / {n__0/0,n__plus/2,n__s/1,tt/0,c_1/0,c_2/1,c_3/0,c_4/0 ,c_5/1,c_6/1,c_7/1,c_8/0,c_9/1,c_10/1,c_11/1,c_12/0,c_13/1,c_14/1,c_15/0,c_16/0} - Obligation: innermost runtime complexity wrt. defined symbols {0#,U11#,U12#,U21#,U31#,U41#,U42#,activate#,isNat#,plus# ,s#} and constructors {n__0,n__plus,n__s,tt} + Applied Processor: PredecessorEstimation {onSelection = all simple predecessor estimation selector} + Details: We estimate the number of application of {4} by application of Pre({4}) = {2}. Here rules are labelled as follows: 1: U41#(tt(),M,N) -> c_6(U42#(isNat(activate(N)),activate(M),activate(N))) 2: U42#(tt(),M,N) -> c_7(s#(plus(activate(N),activate(M)))) 4: s#(X) -> c_16() * Step 11: PredecessorEstimation WORST_CASE(?,O(1)) + Considered Problem: - Strict DPs: U41#(tt(),M,N) -> c_6(U42#(isNat(activate(N)),activate(M),activate(N))) U42#(tt(),M,N) -> c_7(s#(plus(activate(N),activate(M)))) - Weak DPs: s#(X) -> c_16() - Weak TRS: 0() -> n__0() U11(tt(),V2) -> U12(isNat(activate(V2))) U12(tt()) -> tt() U21(tt()) -> tt() activate(X) -> X activate(n__0()) -> 0() activate(n__plus(X1,X2)) -> plus(X1,X2) activate(n__s(X)) -> s(X) isNat(n__0()) -> tt() isNat(n__plus(V1,V2)) -> U11(isNat(activate(V1)),activate(V2)) isNat(n__s(V1)) -> U21(isNat(activate(V1))) plus(X1,X2) -> n__plus(X1,X2) s(X) -> n__s(X) - Signature: {0/0,U11/2,U12/1,U21/1,U31/2,U41/3,U42/3,activate/1,isNat/1,plus/2,s/1,0#/0,U11#/2,U12#/1,U21#/1,U31#/2 ,U41#/3,U42#/3,activate#/1,isNat#/1,plus#/2,s#/1} / {n__0/0,n__plus/2,n__s/1,tt/0,c_1/0,c_2/1,c_3/0,c_4/0 ,c_5/1,c_6/1,c_7/1,c_8/0,c_9/1,c_10/1,c_11/1,c_12/0,c_13/1,c_14/1,c_15/0,c_16/0} - Obligation: innermost runtime complexity wrt. defined symbols {0#,U11#,U12#,U21#,U31#,U41#,U42#,activate#,isNat#,plus# ,s#} and constructors {n__0,n__plus,n__s,tt} + Applied Processor: PredecessorEstimation {onSelection = all simple predecessor estimation selector} + Details: We estimate the number of application of {2} by application of Pre({2}) = {1}. Here rules are labelled as follows: 1: U41#(tt(),M,N) -> c_6(U42#(isNat(activate(N)),activate(M),activate(N))) 2: U42#(tt(),M,N) -> c_7(s#(plus(activate(N),activate(M)))) 3: s#(X) -> c_16() * Step 12: PredecessorEstimation WORST_CASE(?,O(1)) + Considered Problem: - Strict DPs: U41#(tt(),M,N) -> c_6(U42#(isNat(activate(N)),activate(M),activate(N))) - Weak DPs: U42#(tt(),M,N) -> c_7(s#(plus(activate(N),activate(M)))) s#(X) -> c_16() - Weak TRS: 0() -> n__0() U11(tt(),V2) -> U12(isNat(activate(V2))) U12(tt()) -> tt() U21(tt()) -> tt() activate(X) -> X activate(n__0()) -> 0() activate(n__plus(X1,X2)) -> plus(X1,X2) activate(n__s(X)) -> s(X) isNat(n__0()) -> tt() isNat(n__plus(V1,V2)) -> U11(isNat(activate(V1)),activate(V2)) isNat(n__s(V1)) -> U21(isNat(activate(V1))) plus(X1,X2) -> n__plus(X1,X2) s(X) -> n__s(X) - Signature: {0/0,U11/2,U12/1,U21/1,U31/2,U41/3,U42/3,activate/1,isNat/1,plus/2,s/1,0#/0,U11#/2,U12#/1,U21#/1,U31#/2 ,U41#/3,U42#/3,activate#/1,isNat#/1,plus#/2,s#/1} / {n__0/0,n__plus/2,n__s/1,tt/0,c_1/0,c_2/1,c_3/0,c_4/0 ,c_5/1,c_6/1,c_7/1,c_8/0,c_9/1,c_10/1,c_11/1,c_12/0,c_13/1,c_14/1,c_15/0,c_16/0} - Obligation: innermost runtime complexity wrt. defined symbols {0#,U11#,U12#,U21#,U31#,U41#,U42#,activate#,isNat#,plus# ,s#} and constructors {n__0,n__plus,n__s,tt} + Applied Processor: PredecessorEstimation {onSelection = all simple predecessor estimation selector} + Details: We estimate the number of application of {1} by application of Pre({1}) = {}. Here rules are labelled as follows: 1: U41#(tt(),M,N) -> c_6(U42#(isNat(activate(N)),activate(M),activate(N))) 2: U42#(tt(),M,N) -> c_7(s#(plus(activate(N),activate(M)))) 3: s#(X) -> c_16() * Step 13: RemoveWeakSuffixes WORST_CASE(?,O(1)) + Considered Problem: - Weak DPs: U41#(tt(),M,N) -> c_6(U42#(isNat(activate(N)),activate(M),activate(N))) U42#(tt(),M,N) -> c_7(s#(plus(activate(N),activate(M)))) s#(X) -> c_16() - Weak TRS: 0() -> n__0() U11(tt(),V2) -> U12(isNat(activate(V2))) U12(tt()) -> tt() U21(tt()) -> tt() activate(X) -> X activate(n__0()) -> 0() activate(n__plus(X1,X2)) -> plus(X1,X2) activate(n__s(X)) -> s(X) isNat(n__0()) -> tt() isNat(n__plus(V1,V2)) -> U11(isNat(activate(V1)),activate(V2)) isNat(n__s(V1)) -> U21(isNat(activate(V1))) plus(X1,X2) -> n__plus(X1,X2) s(X) -> n__s(X) - Signature: {0/0,U11/2,U12/1,U21/1,U31/2,U41/3,U42/3,activate/1,isNat/1,plus/2,s/1,0#/0,U11#/2,U12#/1,U21#/1,U31#/2 ,U41#/3,U42#/3,activate#/1,isNat#/1,plus#/2,s#/1} / {n__0/0,n__plus/2,n__s/1,tt/0,c_1/0,c_2/1,c_3/0,c_4/0 ,c_5/1,c_6/1,c_7/1,c_8/0,c_9/1,c_10/1,c_11/1,c_12/0,c_13/1,c_14/1,c_15/0,c_16/0} - Obligation: innermost runtime complexity wrt. defined symbols {0#,U11#,U12#,U21#,U31#,U41#,U42#,activate#,isNat#,plus# ,s#} and constructors {n__0,n__plus,n__s,tt} + Applied Processor: RemoveWeakSuffixes + Details: Consider the dependency graph 1:W:U41#(tt(),M,N) -> c_6(U42#(isNat(activate(N)),activate(M),activate(N))) -->_1 U42#(tt(),M,N) -> c_7(s#(plus(activate(N),activate(M)))):2 2:W:U42#(tt(),M,N) -> c_7(s#(plus(activate(N),activate(M)))) -->_1 s#(X) -> c_16():3 3:W:s#(X) -> c_16() The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed. 1: U41#(tt(),M,N) -> c_6(U42#(isNat(activate(N)),activate(M),activate(N))) 2: U42#(tt(),M,N) -> c_7(s#(plus(activate(N),activate(M)))) 3: s#(X) -> c_16() * Step 14: EmptyProcessor WORST_CASE(?,O(1)) + Considered Problem: - Weak TRS: 0() -> n__0() U11(tt(),V2) -> U12(isNat(activate(V2))) U12(tt()) -> tt() U21(tt()) -> tt() activate(X) -> X activate(n__0()) -> 0() activate(n__plus(X1,X2)) -> plus(X1,X2) activate(n__s(X)) -> s(X) isNat(n__0()) -> tt() isNat(n__plus(V1,V2)) -> U11(isNat(activate(V1)),activate(V2)) isNat(n__s(V1)) -> U21(isNat(activate(V1))) plus(X1,X2) -> n__plus(X1,X2) s(X) -> n__s(X) - Signature: {0/0,U11/2,U12/1,U21/1,U31/2,U41/3,U42/3,activate/1,isNat/1,plus/2,s/1,0#/0,U11#/2,U12#/1,U21#/1,U31#/2 ,U41#/3,U42#/3,activate#/1,isNat#/1,plus#/2,s#/1} / {n__0/0,n__plus/2,n__s/1,tt/0,c_1/0,c_2/1,c_3/0,c_4/0 ,c_5/1,c_6/1,c_7/1,c_8/0,c_9/1,c_10/1,c_11/1,c_12/0,c_13/1,c_14/1,c_15/0,c_16/0} - Obligation: innermost runtime complexity wrt. defined symbols {0#,U11#,U12#,U21#,U31#,U41#,U42#,activate#,isNat#,plus# ,s#} and constructors {n__0,n__plus,n__s,tt} + Applied Processor: EmptyProcessor + Details: The problem is already closed. The intended complexity is O(1). WORST_CASE(?,O(n^1))