WORST_CASE(?,O(n^1)) * Step 1: InnermostRuleRemoval WORST_CASE(?,O(n^1)) + Considered Problem: - Strict TRS: 0() -> n__0() activate(X) -> X activate(n__0()) -> 0() activate(n__diff(X1,X2)) -> diff(activate(X1),activate(X2)) activate(n__p(X)) -> p(activate(X)) activate(n__s(X)) -> s(activate(X)) diff(X,Y) -> if(leq(X,Y),n__0(),n__s(n__diff(n__p(X),Y))) diff(X1,X2) -> n__diff(X1,X2) if(false(),X,Y) -> activate(Y) if(true(),X,Y) -> activate(X) leq(0(),Y) -> true() leq(s(X),0()) -> false() leq(s(X),s(Y)) -> leq(X,Y) p(X) -> n__p(X) p(0()) -> 0() p(s(X)) -> X s(X) -> n__s(X) - Signature: {0/0,activate/1,diff/2,if/3,leq/2,p/1,s/1} / {false/0,n__0/0,n__diff/2,n__p/1,n__s/1,true/0} - Obligation: innermost runtime complexity wrt. defined symbols {0,activate,diff,if,leq,p,s} and constructors {false,n__0 ,n__diff,n__p,n__s,true} + Applied Processor: InnermostRuleRemoval + Details: Arguments of following rules are not normal-forms. leq(0(),Y) -> true() leq(s(X),0()) -> false() leq(s(X),s(Y)) -> leq(X,Y) p(0()) -> 0() p(s(X)) -> X All above mentioned rules can be savely removed. * Step 2: DependencyPairs WORST_CASE(?,O(n^1)) + Considered Problem: - Strict TRS: 0() -> n__0() activate(X) -> X activate(n__0()) -> 0() activate(n__diff(X1,X2)) -> diff(activate(X1),activate(X2)) activate(n__p(X)) -> p(activate(X)) activate(n__s(X)) -> s(activate(X)) diff(X,Y) -> if(leq(X,Y),n__0(),n__s(n__diff(n__p(X),Y))) diff(X1,X2) -> n__diff(X1,X2) if(false(),X,Y) -> activate(Y) if(true(),X,Y) -> activate(X) p(X) -> n__p(X) s(X) -> n__s(X) - Signature: {0/0,activate/1,diff/2,if/3,leq/2,p/1,s/1} / {false/0,n__0/0,n__diff/2,n__p/1,n__s/1,true/0} - Obligation: innermost runtime complexity wrt. defined symbols {0,activate,diff,if,leq,p,s} and constructors {false,n__0 ,n__diff,n__p,n__s,true} + Applied Processor: DependencyPairs {dpKind_ = WIDP} + Details: We add the following weak innermost dependency pairs: Strict DPs 0#() -> c_1() activate#(X) -> c_2() activate#(n__0()) -> c_3(0#()) activate#(n__diff(X1,X2)) -> c_4(diff#(activate(X1),activate(X2))) activate#(n__p(X)) -> c_5(p#(activate(X))) activate#(n__s(X)) -> c_6(s#(activate(X))) diff#(X,Y) -> c_7(if#(leq(X,Y),n__0(),n__s(n__diff(n__p(X),Y)))) diff#(X1,X2) -> c_8() if#(false(),X,Y) -> c_9(activate#(Y)) if#(true(),X,Y) -> c_10(activate#(X)) p#(X) -> c_11() s#(X) -> c_12() Weak DPs and mark the set of starting terms. * Step 3: UsableRules WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: 0#() -> c_1() activate#(X) -> c_2() activate#(n__0()) -> c_3(0#()) activate#(n__diff(X1,X2)) -> c_4(diff#(activate(X1),activate(X2))) activate#(n__p(X)) -> c_5(p#(activate(X))) activate#(n__s(X)) -> c_6(s#(activate(X))) diff#(X,Y) -> c_7(if#(leq(X,Y),n__0(),n__s(n__diff(n__p(X),Y)))) diff#(X1,X2) -> c_8() if#(false(),X,Y) -> c_9(activate#(Y)) if#(true(),X,Y) -> c_10(activate#(X)) p#(X) -> c_11() s#(X) -> c_12() - Strict TRS: 0() -> n__0() activate(X) -> X activate(n__0()) -> 0() activate(n__diff(X1,X2)) -> diff(activate(X1),activate(X2)) activate(n__p(X)) -> p(activate(X)) activate(n__s(X)) -> s(activate(X)) diff(X,Y) -> if(leq(X,Y),n__0(),n__s(n__diff(n__p(X),Y))) diff(X1,X2) -> n__diff(X1,X2) if(false(),X,Y) -> activate(Y) if(true(),X,Y) -> activate(X) p(X) -> n__p(X) s(X) -> n__s(X) - Signature: {0/0,activate/1,diff/2,if/3,leq/2,p/1,s/1,0#/0,activate#/1,diff#/2,if#/3,leq#/2,p#/1,s#/1} / {false/0,n__0/0 ,n__diff/2,n__p/1,n__s/1,true/0,c_1/0,c_2/0,c_3/1,c_4/1,c_5/1,c_6/1,c_7/1,c_8/0,c_9/1,c_10/1,c_11/0,c_12/0} - Obligation: innermost runtime complexity wrt. defined symbols {0#,activate#,diff#,if#,leq#,p# ,s#} and constructors {false,n__0,n__diff,n__p,n__s,true} + Applied Processor: UsableRules + Details: We replace rewrite rules by usable rules: 0() -> n__0() activate(X) -> X activate(n__0()) -> 0() activate(n__diff(X1,X2)) -> diff(activate(X1),activate(X2)) activate(n__p(X)) -> p(activate(X)) activate(n__s(X)) -> s(activate(X)) diff(X,Y) -> if(leq(X,Y),n__0(),n__s(n__diff(n__p(X),Y))) diff(X1,X2) -> n__diff(X1,X2) p(X) -> n__p(X) s(X) -> n__s(X) 0#() -> c_1() activate#(X) -> c_2() activate#(n__0()) -> c_3(0#()) activate#(n__diff(X1,X2)) -> c_4(diff#(activate(X1),activate(X2))) activate#(n__p(X)) -> c_5(p#(activate(X))) activate#(n__s(X)) -> c_6(s#(activate(X))) diff#(X,Y) -> c_7(if#(leq(X,Y),n__0(),n__s(n__diff(n__p(X),Y)))) diff#(X1,X2) -> c_8() if#(false(),X,Y) -> c_9(activate#(Y)) if#(true(),X,Y) -> c_10(activate#(X)) p#(X) -> c_11() s#(X) -> c_12() * Step 4: WeightGap WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: 0#() -> c_1() activate#(X) -> c_2() activate#(n__0()) -> c_3(0#()) activate#(n__diff(X1,X2)) -> c_4(diff#(activate(X1),activate(X2))) activate#(n__p(X)) -> c_5(p#(activate(X))) activate#(n__s(X)) -> c_6(s#(activate(X))) diff#(X,Y) -> c_7(if#(leq(X,Y),n__0(),n__s(n__diff(n__p(X),Y)))) diff#(X1,X2) -> c_8() if#(false(),X,Y) -> c_9(activate#(Y)) if#(true(),X,Y) -> c_10(activate#(X)) p#(X) -> c_11() s#(X) -> c_12() - Strict TRS: 0() -> n__0() activate(X) -> X activate(n__0()) -> 0() activate(n__diff(X1,X2)) -> diff(activate(X1),activate(X2)) activate(n__p(X)) -> p(activate(X)) activate(n__s(X)) -> s(activate(X)) diff(X,Y) -> if(leq(X,Y),n__0(),n__s(n__diff(n__p(X),Y))) diff(X1,X2) -> n__diff(X1,X2) p(X) -> n__p(X) s(X) -> n__s(X) - Signature: {0/0,activate/1,diff/2,if/3,leq/2,p/1,s/1,0#/0,activate#/1,diff#/2,if#/3,leq#/2,p#/1,s#/1} / {false/0,n__0/0 ,n__diff/2,n__p/1,n__s/1,true/0,c_1/0,c_2/0,c_3/1,c_4/1,c_5/1,c_6/1,c_7/1,c_8/0,c_9/1,c_10/1,c_11/0,c_12/0} - Obligation: innermost runtime complexity wrt. defined symbols {0#,activate#,diff#,if#,leq#,p# ,s#} and constructors {false,n__0,n__diff,n__p,n__s,true} + 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(diff) = {1,2}, uargs(p) = {1}, uargs(s) = {1}, uargs(diff#) = {1,2}, uargs(p#) = {1}, uargs(s#) = {1}, uargs(c_3) = {1}, uargs(c_4) = {1}, uargs(c_5) = {1}, uargs(c_6) = {1}, uargs(c_9) = {1}, uargs(c_10) = {1} Following symbols are considered usable: all TcT has computed the following interpretation: p(0) = [2] p(activate) = [3] x1 + [1] p(diff) = [1] x1 + [1] x2 + [3] p(false) = [0] p(if) = [0] p(leq) = [4] p(n__0) = [1] p(n__diff) = [1] x1 + [1] x2 + [2] p(n__p) = [1] x1 + [1] p(n__s) = [1] x1 + [1] p(p) = [1] x1 + [2] p(s) = [1] x1 + [2] p(true) = [1] p(0#) = [3] p(activate#) = [3] x1 + [6] p(diff#) = [1] x1 + [1] x2 + [7] p(if#) = [3] x2 + [3] x3 + [0] p(leq#) = [2] x2 + [0] p(p#) = [1] x1 + [0] p(s#) = [1] x1 + [3] p(c_1) = [0] p(c_2) = [2] p(c_3) = [1] x1 + [6] p(c_4) = [1] x1 + [2] p(c_5) = [1] x1 + [3] p(c_6) = [1] x1 + [2] p(c_7) = [3] p(c_8) = [0] p(c_9) = [1] x1 + [6] p(c_10) = [1] x1 + [3] p(c_11) = [1] p(c_12) = [1] Following rules are strictly oriented: 0#() = [3] > [0] = c_1() activate#(X) = [3] X + [6] > [2] = c_2() activate#(n__diff(X1,X2)) = [3] X1 + [3] X2 + [12] > [3] X1 + [3] X2 + [11] = c_4(diff#(activate(X1),activate(X2))) activate#(n__p(X)) = [3] X + [9] > [3] X + [4] = c_5(p#(activate(X))) activate#(n__s(X)) = [3] X + [9] > [3] X + [6] = c_6(s#(activate(X))) diff#(X,Y) = [1] X + [1] Y + [7] > [3] = c_7(if#(leq(X,Y),n__0(),n__s(n__diff(n__p(X),Y)))) diff#(X1,X2) = [1] X1 + [1] X2 + [7] > [0] = c_8() s#(X) = [1] X + [3] > [1] = c_12() 0() = [2] > [1] = n__0() activate(X) = [3] X + [1] > [1] X + [0] = X activate(n__0()) = [4] > [2] = 0() activate(n__diff(X1,X2)) = [3] X1 + [3] X2 + [7] > [3] X1 + [3] X2 + [5] = diff(activate(X1),activate(X2)) activate(n__p(X)) = [3] X + [4] > [3] X + [3] = p(activate(X)) activate(n__s(X)) = [3] X + [4] > [3] X + [3] = s(activate(X)) diff(X,Y) = [1] X + [1] Y + [3] > [0] = if(leq(X,Y),n__0(),n__s(n__diff(n__p(X),Y))) diff(X1,X2) = [1] X1 + [1] X2 + [3] > [1] X1 + [1] X2 + [2] = n__diff(X1,X2) p(X) = [1] X + [2] > [1] X + [1] = n__p(X) s(X) = [1] X + [2] > [1] X + [1] = n__s(X) Following rules are (at-least) weakly oriented: activate#(n__0()) = [9] >= [9] = c_3(0#()) if#(false(),X,Y) = [3] X + [3] Y + [0] >= [3] Y + [12] = c_9(activate#(Y)) if#(true(),X,Y) = [3] X + [3] Y + [0] >= [3] X + [9] = c_10(activate#(X)) p#(X) = [1] X + [0] >= [1] = c_11() 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: activate#(n__0()) -> c_3(0#()) if#(false(),X,Y) -> c_9(activate#(Y)) if#(true(),X,Y) -> c_10(activate#(X)) p#(X) -> c_11() - Weak DPs: 0#() -> c_1() activate#(X) -> c_2() activate#(n__diff(X1,X2)) -> c_4(diff#(activate(X1),activate(X2))) activate#(n__p(X)) -> c_5(p#(activate(X))) activate#(n__s(X)) -> c_6(s#(activate(X))) diff#(X,Y) -> c_7(if#(leq(X,Y),n__0(),n__s(n__diff(n__p(X),Y)))) diff#(X1,X2) -> c_8() s#(X) -> c_12() - Weak TRS: 0() -> n__0() activate(X) -> X activate(n__0()) -> 0() activate(n__diff(X1,X2)) -> diff(activate(X1),activate(X2)) activate(n__p(X)) -> p(activate(X)) activate(n__s(X)) -> s(activate(X)) diff(X,Y) -> if(leq(X,Y),n__0(),n__s(n__diff(n__p(X),Y))) diff(X1,X2) -> n__diff(X1,X2) p(X) -> n__p(X) s(X) -> n__s(X) - Signature: {0/0,activate/1,diff/2,if/3,leq/2,p/1,s/1,0#/0,activate#/1,diff#/2,if#/3,leq#/2,p#/1,s#/1} / {false/0,n__0/0 ,n__diff/2,n__p/1,n__s/1,true/0,c_1/0,c_2/0,c_3/1,c_4/1,c_5/1,c_6/1,c_7/1,c_8/0,c_9/1,c_10/1,c_11/0,c_12/0} - Obligation: innermost runtime complexity wrt. defined symbols {0#,activate#,diff#,if#,leq#,p# ,s#} and constructors {false,n__0,n__diff,n__p,n__s,true} + Applied Processor: PredecessorEstimation {onSelection = all simple predecessor estimation selector} + Details: We estimate the number of application of {1} by application of Pre({1}) = {2,3}. Here rules are labelled as follows: 1: activate#(n__0()) -> c_3(0#()) 2: if#(false(),X,Y) -> c_9(activate#(Y)) 3: if#(true(),X,Y) -> c_10(activate#(X)) 4: p#(X) -> c_11() 5: 0#() -> c_1() 6: activate#(X) -> c_2() 7: activate#(n__diff(X1,X2)) -> c_4(diff#(activate(X1),activate(X2))) 8: activate#(n__p(X)) -> c_5(p#(activate(X))) 9: activate#(n__s(X)) -> c_6(s#(activate(X))) 10: diff#(X,Y) -> c_7(if#(leq(X,Y),n__0(),n__s(n__diff(n__p(X),Y)))) 11: diff#(X1,X2) -> c_8() 12: s#(X) -> c_12() * Step 6: PredecessorEstimation WORST_CASE(?,O(1)) + Considered Problem: - Strict DPs: if#(false(),X,Y) -> c_9(activate#(Y)) if#(true(),X,Y) -> c_10(activate#(X)) p#(X) -> c_11() - Weak DPs: 0#() -> c_1() activate#(X) -> c_2() activate#(n__0()) -> c_3(0#()) activate#(n__diff(X1,X2)) -> c_4(diff#(activate(X1),activate(X2))) activate#(n__p(X)) -> c_5(p#(activate(X))) activate#(n__s(X)) -> c_6(s#(activate(X))) diff#(X,Y) -> c_7(if#(leq(X,Y),n__0(),n__s(n__diff(n__p(X),Y)))) diff#(X1,X2) -> c_8() s#(X) -> c_12() - Weak TRS: 0() -> n__0() activate(X) -> X activate(n__0()) -> 0() activate(n__diff(X1,X2)) -> diff(activate(X1),activate(X2)) activate(n__p(X)) -> p(activate(X)) activate(n__s(X)) -> s(activate(X)) diff(X,Y) -> if(leq(X,Y),n__0(),n__s(n__diff(n__p(X),Y))) diff(X1,X2) -> n__diff(X1,X2) p(X) -> n__p(X) s(X) -> n__s(X) - Signature: {0/0,activate/1,diff/2,if/3,leq/2,p/1,s/1,0#/0,activate#/1,diff#/2,if#/3,leq#/2,p#/1,s#/1} / {false/0,n__0/0 ,n__diff/2,n__p/1,n__s/1,true/0,c_1/0,c_2/0,c_3/1,c_4/1,c_5/1,c_6/1,c_7/1,c_8/0,c_9/1,c_10/1,c_11/0,c_12/0} - Obligation: innermost runtime complexity wrt. defined symbols {0#,activate#,diff#,if#,leq#,p# ,s#} and constructors {false,n__0,n__diff,n__p,n__s,true} + Applied Processor: PredecessorEstimation {onSelection = all simple predecessor estimation selector} + Details: We estimate the number of application of {1,2} by application of Pre({1,2}) = {}. Here rules are labelled as follows: 1: if#(false(),X,Y) -> c_9(activate#(Y)) 2: if#(true(),X,Y) -> c_10(activate#(X)) 3: p#(X) -> c_11() 4: 0#() -> c_1() 5: activate#(X) -> c_2() 6: activate#(n__0()) -> c_3(0#()) 7: activate#(n__diff(X1,X2)) -> c_4(diff#(activate(X1),activate(X2))) 8: activate#(n__p(X)) -> c_5(p#(activate(X))) 9: activate#(n__s(X)) -> c_6(s#(activate(X))) 10: diff#(X,Y) -> c_7(if#(leq(X,Y),n__0(),n__s(n__diff(n__p(X),Y)))) 11: diff#(X1,X2) -> c_8() 12: s#(X) -> c_12() * Step 7: RemoveWeakSuffixes WORST_CASE(?,O(1)) + Considered Problem: - Strict DPs: p#(X) -> c_11() - Weak DPs: 0#() -> c_1() activate#(X) -> c_2() activate#(n__0()) -> c_3(0#()) activate#(n__diff(X1,X2)) -> c_4(diff#(activate(X1),activate(X2))) activate#(n__p(X)) -> c_5(p#(activate(X))) activate#(n__s(X)) -> c_6(s#(activate(X))) diff#(X,Y) -> c_7(if#(leq(X,Y),n__0(),n__s(n__diff(n__p(X),Y)))) diff#(X1,X2) -> c_8() if#(false(),X,Y) -> c_9(activate#(Y)) if#(true(),X,Y) -> c_10(activate#(X)) s#(X) -> c_12() - Weak TRS: 0() -> n__0() activate(X) -> X activate(n__0()) -> 0() activate(n__diff(X1,X2)) -> diff(activate(X1),activate(X2)) activate(n__p(X)) -> p(activate(X)) activate(n__s(X)) -> s(activate(X)) diff(X,Y) -> if(leq(X,Y),n__0(),n__s(n__diff(n__p(X),Y))) diff(X1,X2) -> n__diff(X1,X2) p(X) -> n__p(X) s(X) -> n__s(X) - Signature: {0/0,activate/1,diff/2,if/3,leq/2,p/1,s/1,0#/0,activate#/1,diff#/2,if#/3,leq#/2,p#/1,s#/1} / {false/0,n__0/0 ,n__diff/2,n__p/1,n__s/1,true/0,c_1/0,c_2/0,c_3/1,c_4/1,c_5/1,c_6/1,c_7/1,c_8/0,c_9/1,c_10/1,c_11/0,c_12/0} - Obligation: innermost runtime complexity wrt. defined symbols {0#,activate#,diff#,if#,leq#,p# ,s#} and constructors {false,n__0,n__diff,n__p,n__s,true} + Applied Processor: RemoveWeakSuffixes + Details: Consider the dependency graph 1:S:p#(X) -> c_11() 2:W:0#() -> c_1() 3:W:activate#(X) -> c_2() 4:W:activate#(n__0()) -> c_3(0#()) -->_1 0#() -> c_1():2 5:W:activate#(n__diff(X1,X2)) -> c_4(diff#(activate(X1),activate(X2))) -->_1 diff#(X1,X2) -> c_8():9 -->_1 diff#(X,Y) -> c_7(if#(leq(X,Y),n__0(),n__s(n__diff(n__p(X),Y)))):8 6:W:activate#(n__p(X)) -> c_5(p#(activate(X))) -->_1 p#(X) -> c_11():1 7:W:activate#(n__s(X)) -> c_6(s#(activate(X))) -->_1 s#(X) -> c_12():12 8:W:diff#(X,Y) -> c_7(if#(leq(X,Y),n__0(),n__s(n__diff(n__p(X),Y)))) 9:W:diff#(X1,X2) -> c_8() 10:W:if#(false(),X,Y) -> c_9(activate#(Y)) -->_1 activate#(n__s(X)) -> c_6(s#(activate(X))):7 -->_1 activate#(n__p(X)) -> c_5(p#(activate(X))):6 -->_1 activate#(n__diff(X1,X2)) -> c_4(diff#(activate(X1),activate(X2))):5 -->_1 activate#(n__0()) -> c_3(0#()):4 -->_1 activate#(X) -> c_2():3 11:W:if#(true(),X,Y) -> c_10(activate#(X)) -->_1 activate#(n__s(X)) -> c_6(s#(activate(X))):7 -->_1 activate#(n__p(X)) -> c_5(p#(activate(X))):6 -->_1 activate#(n__diff(X1,X2)) -> c_4(diff#(activate(X1),activate(X2))):5 -->_1 activate#(n__0()) -> c_3(0#()):4 -->_1 activate#(X) -> c_2():3 12:W:s#(X) -> c_12() The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed. 7: activate#(n__s(X)) -> c_6(s#(activate(X))) 12: s#(X) -> c_12() 5: activate#(n__diff(X1,X2)) -> c_4(diff#(activate(X1),activate(X2))) 8: diff#(X,Y) -> c_7(if#(leq(X,Y),n__0(),n__s(n__diff(n__p(X),Y)))) 9: diff#(X1,X2) -> c_8() 4: activate#(n__0()) -> c_3(0#()) 3: activate#(X) -> c_2() 2: 0#() -> c_1() * Step 8: RemoveHeads WORST_CASE(?,O(1)) + Considered Problem: - Strict DPs: p#(X) -> c_11() - Weak DPs: activate#(n__p(X)) -> c_5(p#(activate(X))) if#(false(),X,Y) -> c_9(activate#(Y)) if#(true(),X,Y) -> c_10(activate#(X)) - Weak TRS: 0() -> n__0() activate(X) -> X activate(n__0()) -> 0() activate(n__diff(X1,X2)) -> diff(activate(X1),activate(X2)) activate(n__p(X)) -> p(activate(X)) activate(n__s(X)) -> s(activate(X)) diff(X,Y) -> if(leq(X,Y),n__0(),n__s(n__diff(n__p(X),Y))) diff(X1,X2) -> n__diff(X1,X2) p(X) -> n__p(X) s(X) -> n__s(X) - Signature: {0/0,activate/1,diff/2,if/3,leq/2,p/1,s/1,0#/0,activate#/1,diff#/2,if#/3,leq#/2,p#/1,s#/1} / {false/0,n__0/0 ,n__diff/2,n__p/1,n__s/1,true/0,c_1/0,c_2/0,c_3/1,c_4/1,c_5/1,c_6/1,c_7/1,c_8/0,c_9/1,c_10/1,c_11/0,c_12/0} - Obligation: innermost runtime complexity wrt. defined symbols {0#,activate#,diff#,if#,leq#,p# ,s#} and constructors {false,n__0,n__diff,n__p,n__s,true} + Applied Processor: RemoveHeads + Details: Consider the dependency graph 1:S:p#(X) -> c_11() 6:W:activate#(n__p(X)) -> c_5(p#(activate(X))) -->_1 p#(X) -> c_11():1 10:W:if#(false(),X,Y) -> c_9(activate#(Y)) -->_1 activate#(n__p(X)) -> c_5(p#(activate(X))):6 11:W:if#(true(),X,Y) -> c_10(activate#(X)) -->_1 activate#(n__p(X)) -> c_5(p#(activate(X))):6 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). [(10,if#(false(),X,Y) -> c_9(activate#(Y))),(11,if#(true(),X,Y) -> c_10(activate#(X)))] * Step 9: Trivial WORST_CASE(?,O(1)) + Considered Problem: - Strict DPs: p#(X) -> c_11() - Weak DPs: activate#(n__p(X)) -> c_5(p#(activate(X))) - Weak TRS: 0() -> n__0() activate(X) -> X activate(n__0()) -> 0() activate(n__diff(X1,X2)) -> diff(activate(X1),activate(X2)) activate(n__p(X)) -> p(activate(X)) activate(n__s(X)) -> s(activate(X)) diff(X,Y) -> if(leq(X,Y),n__0(),n__s(n__diff(n__p(X),Y))) diff(X1,X2) -> n__diff(X1,X2) p(X) -> n__p(X) s(X) -> n__s(X) - Signature: {0/0,activate/1,diff/2,if/3,leq/2,p/1,s/1,0#/0,activate#/1,diff#/2,if#/3,leq#/2,p#/1,s#/1} / {false/0,n__0/0 ,n__diff/2,n__p/1,n__s/1,true/0,c_1/0,c_2/0,c_3/1,c_4/1,c_5/1,c_6/1,c_7/1,c_8/0,c_9/1,c_10/1,c_11/0,c_12/0} - Obligation: innermost runtime complexity wrt. defined symbols {0#,activate#,diff#,if#,leq#,p# ,s#} and constructors {false,n__0,n__diff,n__p,n__s,true} + Applied Processor: Trivial + Details: Consider the dependency graph 1:S:p#(X) -> c_11() 6:W:activate#(n__p(X)) -> c_5(p#(activate(X))) -->_1 p#(X) -> c_11():1 The dependency graph contains no loops, we remove all dependency pairs. * Step 10: EmptyProcessor WORST_CASE(?,O(1)) + Considered Problem: - Weak TRS: 0() -> n__0() activate(X) -> X activate(n__0()) -> 0() activate(n__diff(X1,X2)) -> diff(activate(X1),activate(X2)) activate(n__p(X)) -> p(activate(X)) activate(n__s(X)) -> s(activate(X)) diff(X,Y) -> if(leq(X,Y),n__0(),n__s(n__diff(n__p(X),Y))) diff(X1,X2) -> n__diff(X1,X2) p(X) -> n__p(X) s(X) -> n__s(X) - Signature: {0/0,activate/1,diff/2,if/3,leq/2,p/1,s/1,0#/0,activate#/1,diff#/2,if#/3,leq#/2,p#/1,s#/1} / {false/0,n__0/0 ,n__diff/2,n__p/1,n__s/1,true/0,c_1/0,c_2/0,c_3/1,c_4/1,c_5/1,c_6/1,c_7/1,c_8/0,c_9/1,c_10/1,c_11/0,c_12/0} - Obligation: innermost runtime complexity wrt. defined symbols {0#,activate#,diff#,if#,leq#,p# ,s#} and constructors {false,n__0,n__diff,n__p,n__s,true} + Applied Processor: EmptyProcessor + Details: The problem is already closed. The intended complexity is O(1). WORST_CASE(?,O(n^1))