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__fact(X)) -> fact(activate(X)) activate(n__p(X)) -> p(activate(X)) activate(n__prod(X1,X2)) -> prod(activate(X1),activate(X2)) activate(n__s(X)) -> s(activate(X)) add(0(),X) -> X add(s(X),Y) -> s(add(X,Y)) fact(X) -> if(zero(X),n__s(n__0()),n__prod(X,n__fact(n__p(X)))) fact(X) -> n__fact(X) if(false(),X,Y) -> activate(Y) if(true(),X,Y) -> activate(X) p(X) -> n__p(X) p(s(X)) -> X prod(X1,X2) -> n__prod(X1,X2) prod(0(),X) -> 0() prod(s(X),Y) -> add(Y,prod(X,Y)) s(X) -> n__s(X) zero(0()) -> true() zero(s(X)) -> false() - Signature: {0/0,activate/1,add/2,fact/1,if/3,p/1,prod/2,s/1,zero/1} / {false/0,n__0/0,n__fact/1,n__p/1,n__prod/2,n__s/1 ,true/0} - Obligation: innermost runtime complexity wrt. defined symbols {0,activate,add,fact,if,p,prod,s ,zero} and constructors {false,n__0,n__fact,n__p,n__prod,n__s,true} + Applied Processor: InnermostRuleRemoval + Details: Arguments of following rules are not normal-forms. add(0(),X) -> X add(s(X),Y) -> s(add(X,Y)) p(s(X)) -> X prod(0(),X) -> 0() prod(s(X),Y) -> add(Y,prod(X,Y)) zero(0()) -> true() zero(s(X)) -> false() 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__fact(X)) -> fact(activate(X)) activate(n__p(X)) -> p(activate(X)) activate(n__prod(X1,X2)) -> prod(activate(X1),activate(X2)) activate(n__s(X)) -> s(activate(X)) fact(X) -> if(zero(X),n__s(n__0()),n__prod(X,n__fact(n__p(X)))) fact(X) -> n__fact(X) if(false(),X,Y) -> activate(Y) if(true(),X,Y) -> activate(X) p(X) -> n__p(X) prod(X1,X2) -> n__prod(X1,X2) s(X) -> n__s(X) - Signature: {0/0,activate/1,add/2,fact/1,if/3,p/1,prod/2,s/1,zero/1} / {false/0,n__0/0,n__fact/1,n__p/1,n__prod/2,n__s/1 ,true/0} - Obligation: innermost runtime complexity wrt. defined symbols {0,activate,add,fact,if,p,prod,s ,zero} and constructors {false,n__0,n__fact,n__p,n__prod,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__fact(X)) -> c_4(fact#(activate(X))) activate#(n__p(X)) -> c_5(p#(activate(X))) activate#(n__prod(X1,X2)) -> c_6(prod#(activate(X1),activate(X2))) activate#(n__s(X)) -> c_7(s#(activate(X))) fact#(X) -> c_8(if#(zero(X),n__s(n__0()),n__prod(X,n__fact(n__p(X))))) fact#(X) -> c_9() if#(false(),X,Y) -> c_10(activate#(Y)) if#(true(),X,Y) -> c_11(activate#(X)) p#(X) -> c_12() prod#(X1,X2) -> c_13() s#(X) -> c_14() 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__fact(X)) -> c_4(fact#(activate(X))) activate#(n__p(X)) -> c_5(p#(activate(X))) activate#(n__prod(X1,X2)) -> c_6(prod#(activate(X1),activate(X2))) activate#(n__s(X)) -> c_7(s#(activate(X))) fact#(X) -> c_8(if#(zero(X),n__s(n__0()),n__prod(X,n__fact(n__p(X))))) fact#(X) -> c_9() if#(false(),X,Y) -> c_10(activate#(Y)) if#(true(),X,Y) -> c_11(activate#(X)) p#(X) -> c_12() prod#(X1,X2) -> c_13() s#(X) -> c_14() - Strict TRS: 0() -> n__0() activate(X) -> X activate(n__0()) -> 0() activate(n__fact(X)) -> fact(activate(X)) activate(n__p(X)) -> p(activate(X)) activate(n__prod(X1,X2)) -> prod(activate(X1),activate(X2)) activate(n__s(X)) -> s(activate(X)) fact(X) -> if(zero(X),n__s(n__0()),n__prod(X,n__fact(n__p(X)))) fact(X) -> n__fact(X) if(false(),X,Y) -> activate(Y) if(true(),X,Y) -> activate(X) p(X) -> n__p(X) prod(X1,X2) -> n__prod(X1,X2) s(X) -> n__s(X) - Signature: {0/0,activate/1,add/2,fact/1,if/3,p/1,prod/2,s/1,zero/1,0#/0,activate#/1,add#/2,fact#/1,if#/3,p#/1,prod#/2 ,s#/1,zero#/1} / {false/0,n__0/0,n__fact/1,n__p/1,n__prod/2,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_8/1,c_9/0,c_10/1,c_11/1,c_12/0,c_13/0,c_14/0} - Obligation: innermost runtime complexity wrt. defined symbols {0#,activate#,add#,fact#,if#,p#,prod#,s# ,zero#} and constructors {false,n__0,n__fact,n__p,n__prod,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__fact(X)) -> fact(activate(X)) activate(n__p(X)) -> p(activate(X)) activate(n__prod(X1,X2)) -> prod(activate(X1),activate(X2)) activate(n__s(X)) -> s(activate(X)) fact(X) -> if(zero(X),n__s(n__0()),n__prod(X,n__fact(n__p(X)))) fact(X) -> n__fact(X) p(X) -> n__p(X) prod(X1,X2) -> n__prod(X1,X2) s(X) -> n__s(X) 0#() -> c_1() activate#(X) -> c_2() activate#(n__0()) -> c_3(0#()) activate#(n__fact(X)) -> c_4(fact#(activate(X))) activate#(n__p(X)) -> c_5(p#(activate(X))) activate#(n__prod(X1,X2)) -> c_6(prod#(activate(X1),activate(X2))) activate#(n__s(X)) -> c_7(s#(activate(X))) fact#(X) -> c_8(if#(zero(X),n__s(n__0()),n__prod(X,n__fact(n__p(X))))) fact#(X) -> c_9() if#(false(),X,Y) -> c_10(activate#(Y)) if#(true(),X,Y) -> c_11(activate#(X)) p#(X) -> c_12() prod#(X1,X2) -> c_13() s#(X) -> c_14() * 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__fact(X)) -> c_4(fact#(activate(X))) activate#(n__p(X)) -> c_5(p#(activate(X))) activate#(n__prod(X1,X2)) -> c_6(prod#(activate(X1),activate(X2))) activate#(n__s(X)) -> c_7(s#(activate(X))) fact#(X) -> c_8(if#(zero(X),n__s(n__0()),n__prod(X,n__fact(n__p(X))))) fact#(X) -> c_9() if#(false(),X,Y) -> c_10(activate#(Y)) if#(true(),X,Y) -> c_11(activate#(X)) p#(X) -> c_12() prod#(X1,X2) -> c_13() s#(X) -> c_14() - Strict TRS: 0() -> n__0() activate(X) -> X activate(n__0()) -> 0() activate(n__fact(X)) -> fact(activate(X)) activate(n__p(X)) -> p(activate(X)) activate(n__prod(X1,X2)) -> prod(activate(X1),activate(X2)) activate(n__s(X)) -> s(activate(X)) fact(X) -> if(zero(X),n__s(n__0()),n__prod(X,n__fact(n__p(X)))) fact(X) -> n__fact(X) p(X) -> n__p(X) prod(X1,X2) -> n__prod(X1,X2) s(X) -> n__s(X) - Signature: {0/0,activate/1,add/2,fact/1,if/3,p/1,prod/2,s/1,zero/1,0#/0,activate#/1,add#/2,fact#/1,if#/3,p#/1,prod#/2 ,s#/1,zero#/1} / {false/0,n__0/0,n__fact/1,n__p/1,n__prod/2,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_8/1,c_9/0,c_10/1,c_11/1,c_12/0,c_13/0,c_14/0} - Obligation: innermost runtime complexity wrt. defined symbols {0#,activate#,add#,fact#,if#,p#,prod#,s# ,zero#} and constructors {false,n__0,n__fact,n__p,n__prod,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(fact) = {1}, uargs(p) = {1}, uargs(prod) = {1,2}, uargs(s) = {1}, uargs(fact#) = {1}, uargs(p#) = {1}, uargs(prod#) = {1,2}, uargs(s#) = {1}, uargs(c_3) = {1}, uargs(c_4) = {1}, uargs(c_5) = {1}, uargs(c_6) = {1}, uargs(c_7) = {1}, uargs(c_10) = {1}, uargs(c_11) = {1} Following symbols are considered usable: all TcT has computed the following interpretation: p(0) = [1] p(activate) = [3] x1 + [2] p(add) = [4] x1 + [1] x2 + [0] p(fact) = [1] x1 + [2] p(false) = [0] p(if) = [1] p(n__0) = [0] p(n__fact) = [1] x1 + [1] p(n__p) = [1] x1 + [1] p(n__prod) = [1] x1 + [1] x2 + [2] p(n__s) = [1] x1 + [1] p(p) = [1] x1 + [2] p(prod) = [1] x1 + [1] x2 + [3] p(s) = [1] x1 + [2] p(true) = [2] p(zero) = [0] p(0#) = [4] p(activate#) = [3] x1 + [0] p(add#) = [1] x2 + [0] p(fact#) = [1] x1 + [5] p(if#) = [4] x1 + [3] x2 + [3] x3 + [0] p(p#) = [1] x1 + [6] p(prod#) = [1] x1 + [1] x2 + [0] p(s#) = [1] x1 + [0] p(zero#) = [1] x1 + [1] p(c_1) = [0] p(c_2) = [0] p(c_3) = [1] x1 + [0] p(c_4) = [1] x1 + [1] p(c_5) = [1] x1 + [0] p(c_6) = [1] x1 + [4] p(c_7) = [1] x1 + [1] p(c_8) = [1] p(c_9) = [4] p(c_10) = [1] x1 + [0] p(c_11) = [1] x1 + [4] p(c_12) = [0] p(c_13) = [1] p(c_14) = [0] Following rules are strictly oriented: 0#() = [4] > [0] = c_1() fact#(X) = [1] X + [5] > [1] = c_8(if#(zero(X),n__s(n__0()),n__prod(X,n__fact(n__p(X))))) fact#(X) = [1] X + [5] > [4] = c_9() if#(true(),X,Y) = [3] X + [3] Y + [8] > [3] X + [4] = c_11(activate#(X)) p#(X) = [1] X + [6] > [0] = c_12() 0() = [1] > [0] = n__0() activate(X) = [3] X + [2] > [1] X + [0] = X activate(n__0()) = [2] > [1] = 0() activate(n__fact(X)) = [3] X + [5] > [3] X + [4] = fact(activate(X)) activate(n__p(X)) = [3] X + [5] > [3] X + [4] = p(activate(X)) activate(n__prod(X1,X2)) = [3] X1 + [3] X2 + [8] > [3] X1 + [3] X2 + [7] = prod(activate(X1),activate(X2)) activate(n__s(X)) = [3] X + [5] > [3] X + [4] = s(activate(X)) fact(X) = [1] X + [2] > [1] = if(zero(X),n__s(n__0()),n__prod(X,n__fact(n__p(X)))) fact(X) = [1] X + [2] > [1] X + [1] = n__fact(X) p(X) = [1] X + [2] > [1] X + [1] = n__p(X) prod(X1,X2) = [1] X1 + [1] X2 + [3] > [1] X1 + [1] X2 + [2] = n__prod(X1,X2) s(X) = [1] X + [2] > [1] X + [1] = n__s(X) Following rules are (at-least) weakly oriented: activate#(X) = [3] X + [0] >= [0] = c_2() activate#(n__0()) = [0] >= [4] = c_3(0#()) activate#(n__fact(X)) = [3] X + [3] >= [3] X + [8] = c_4(fact#(activate(X))) activate#(n__p(X)) = [3] X + [3] >= [3] X + [8] = c_5(p#(activate(X))) activate#(n__prod(X1,X2)) = [3] X1 + [3] X2 + [6] >= [3] X1 + [3] X2 + [8] = c_6(prod#(activate(X1),activate(X2))) activate#(n__s(X)) = [3] X + [3] >= [3] X + [3] = c_7(s#(activate(X))) if#(false(),X,Y) = [3] X + [3] Y + [0] >= [3] Y + [0] = c_10(activate#(Y)) prod#(X1,X2) = [1] X1 + [1] X2 + [0] >= [1] = c_13() s#(X) = [1] X + [0] >= [0] = c_14() 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#(X) -> c_2() activate#(n__0()) -> c_3(0#()) activate#(n__fact(X)) -> c_4(fact#(activate(X))) activate#(n__p(X)) -> c_5(p#(activate(X))) activate#(n__prod(X1,X2)) -> c_6(prod#(activate(X1),activate(X2))) activate#(n__s(X)) -> c_7(s#(activate(X))) if#(false(),X,Y) -> c_10(activate#(Y)) prod#(X1,X2) -> c_13() s#(X) -> c_14() - Weak DPs: 0#() -> c_1() fact#(X) -> c_8(if#(zero(X),n__s(n__0()),n__prod(X,n__fact(n__p(X))))) fact#(X) -> c_9() if#(true(),X,Y) -> c_11(activate#(X)) p#(X) -> c_12() - Weak TRS: 0() -> n__0() activate(X) -> X activate(n__0()) -> 0() activate(n__fact(X)) -> fact(activate(X)) activate(n__p(X)) -> p(activate(X)) activate(n__prod(X1,X2)) -> prod(activate(X1),activate(X2)) activate(n__s(X)) -> s(activate(X)) fact(X) -> if(zero(X),n__s(n__0()),n__prod(X,n__fact(n__p(X)))) fact(X) -> n__fact(X) p(X) -> n__p(X) prod(X1,X2) -> n__prod(X1,X2) s(X) -> n__s(X) - Signature: {0/0,activate/1,add/2,fact/1,if/3,p/1,prod/2,s/1,zero/1,0#/0,activate#/1,add#/2,fact#/1,if#/3,p#/1,prod#/2 ,s#/1,zero#/1} / {false/0,n__0/0,n__fact/1,n__p/1,n__prod/2,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_8/1,c_9/0,c_10/1,c_11/1,c_12/0,c_13/0,c_14/0} - Obligation: innermost runtime complexity wrt. defined symbols {0#,activate#,add#,fact#,if#,p#,prod#,s# ,zero#} and constructors {false,n__0,n__fact,n__p,n__prod,n__s,true} + Applied Processor: PredecessorEstimation {onSelection = all simple predecessor estimation selector} + Details: We estimate the number of application of {8,9} by application of Pre({8,9}) = {5,6}. Here rules are labelled as follows: 1: activate#(X) -> c_2() 2: activate#(n__0()) -> c_3(0#()) 3: activate#(n__fact(X)) -> c_4(fact#(activate(X))) 4: activate#(n__p(X)) -> c_5(p#(activate(X))) 5: activate#(n__prod(X1,X2)) -> c_6(prod#(activate(X1),activate(X2))) 6: activate#(n__s(X)) -> c_7(s#(activate(X))) 7: if#(false(),X,Y) -> c_10(activate#(Y)) 8: prod#(X1,X2) -> c_13() 9: s#(X) -> c_14() 10: 0#() -> c_1() 11: fact#(X) -> c_8(if#(zero(X),n__s(n__0()),n__prod(X,n__fact(n__p(X))))) 12: fact#(X) -> c_9() 13: if#(true(),X,Y) -> c_11(activate#(X)) 14: p#(X) -> c_12() * Step 6: RemoveWeakSuffixes WORST_CASE(?,O(1)) + Considered Problem: - Strict DPs: activate#(X) -> c_2() activate#(n__0()) -> c_3(0#()) activate#(n__fact(X)) -> c_4(fact#(activate(X))) activate#(n__p(X)) -> c_5(p#(activate(X))) activate#(n__prod(X1,X2)) -> c_6(prod#(activate(X1),activate(X2))) activate#(n__s(X)) -> c_7(s#(activate(X))) if#(false(),X,Y) -> c_10(activate#(Y)) - Weak DPs: 0#() -> c_1() fact#(X) -> c_8(if#(zero(X),n__s(n__0()),n__prod(X,n__fact(n__p(X))))) fact#(X) -> c_9() if#(true(),X,Y) -> c_11(activate#(X)) p#(X) -> c_12() prod#(X1,X2) -> c_13() s#(X) -> c_14() - Weak TRS: 0() -> n__0() activate(X) -> X activate(n__0()) -> 0() activate(n__fact(X)) -> fact(activate(X)) activate(n__p(X)) -> p(activate(X)) activate(n__prod(X1,X2)) -> prod(activate(X1),activate(X2)) activate(n__s(X)) -> s(activate(X)) fact(X) -> if(zero(X),n__s(n__0()),n__prod(X,n__fact(n__p(X)))) fact(X) -> n__fact(X) p(X) -> n__p(X) prod(X1,X2) -> n__prod(X1,X2) s(X) -> n__s(X) - Signature: {0/0,activate/1,add/2,fact/1,if/3,p/1,prod/2,s/1,zero/1,0#/0,activate#/1,add#/2,fact#/1,if#/3,p#/1,prod#/2 ,s#/1,zero#/1} / {false/0,n__0/0,n__fact/1,n__p/1,n__prod/2,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_8/1,c_9/0,c_10/1,c_11/1,c_12/0,c_13/0,c_14/0} - Obligation: innermost runtime complexity wrt. defined symbols {0#,activate#,add#,fact#,if#,p#,prod#,s# ,zero#} and constructors {false,n__0,n__fact,n__p,n__prod,n__s,true} + Applied Processor: RemoveWeakSuffixes + Details: Consider the dependency graph 1:S:activate#(X) -> c_2() 2:S:activate#(n__0()) -> c_3(0#()) -->_1 0#() -> c_1():8 3:S:activate#(n__fact(X)) -> c_4(fact#(activate(X))) -->_1 fact#(X) -> c_9():10 -->_1 fact#(X) -> c_8(if#(zero(X),n__s(n__0()),n__prod(X,n__fact(n__p(X))))):9 4:S:activate#(n__p(X)) -> c_5(p#(activate(X))) -->_1 p#(X) -> c_12():12 5:S:activate#(n__prod(X1,X2)) -> c_6(prod#(activate(X1),activate(X2))) -->_1 prod#(X1,X2) -> c_13():13 6:S:activate#(n__s(X)) -> c_7(s#(activate(X))) -->_1 s#(X) -> c_14():14 7:S:if#(false(),X,Y) -> c_10(activate#(Y)) -->_1 activate#(n__s(X)) -> c_7(s#(activate(X))):6 -->_1 activate#(n__prod(X1,X2)) -> c_6(prod#(activate(X1),activate(X2))):5 -->_1 activate#(n__p(X)) -> c_5(p#(activate(X))):4 -->_1 activate#(n__fact(X)) -> c_4(fact#(activate(X))):3 -->_1 activate#(n__0()) -> c_3(0#()):2 -->_1 activate#(X) -> c_2():1 8:W:0#() -> c_1() 9:W:fact#(X) -> c_8(if#(zero(X),n__s(n__0()),n__prod(X,n__fact(n__p(X))))) 10:W:fact#(X) -> c_9() 11:W:if#(true(),X,Y) -> c_11(activate#(X)) -->_1 activate#(n__s(X)) -> c_7(s#(activate(X))):6 -->_1 activate#(n__prod(X1,X2)) -> c_6(prod#(activate(X1),activate(X2))):5 -->_1 activate#(n__p(X)) -> c_5(p#(activate(X))):4 -->_1 activate#(n__fact(X)) -> c_4(fact#(activate(X))):3 -->_1 activate#(n__0()) -> c_3(0#()):2 -->_1 activate#(X) -> c_2():1 12:W:p#(X) -> c_12() 13:W:prod#(X1,X2) -> c_13() 14:W:s#(X) -> c_14() The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed. 14: s#(X) -> c_14() 13: prod#(X1,X2) -> c_13() 12: p#(X) -> c_12() 9: fact#(X) -> c_8(if#(zero(X),n__s(n__0()),n__prod(X,n__fact(n__p(X))))) 10: fact#(X) -> c_9() 8: 0#() -> c_1() * Step 7: SimplifyRHS WORST_CASE(?,O(1)) + Considered Problem: - Strict DPs: activate#(X) -> c_2() activate#(n__0()) -> c_3(0#()) activate#(n__fact(X)) -> c_4(fact#(activate(X))) activate#(n__p(X)) -> c_5(p#(activate(X))) activate#(n__prod(X1,X2)) -> c_6(prod#(activate(X1),activate(X2))) activate#(n__s(X)) -> c_7(s#(activate(X))) if#(false(),X,Y) -> c_10(activate#(Y)) - Weak DPs: if#(true(),X,Y) -> c_11(activate#(X)) - Weak TRS: 0() -> n__0() activate(X) -> X activate(n__0()) -> 0() activate(n__fact(X)) -> fact(activate(X)) activate(n__p(X)) -> p(activate(X)) activate(n__prod(X1,X2)) -> prod(activate(X1),activate(X2)) activate(n__s(X)) -> s(activate(X)) fact(X) -> if(zero(X),n__s(n__0()),n__prod(X,n__fact(n__p(X)))) fact(X) -> n__fact(X) p(X) -> n__p(X) prod(X1,X2) -> n__prod(X1,X2) s(X) -> n__s(X) - Signature: {0/0,activate/1,add/2,fact/1,if/3,p/1,prod/2,s/1,zero/1,0#/0,activate#/1,add#/2,fact#/1,if#/3,p#/1,prod#/2 ,s#/1,zero#/1} / {false/0,n__0/0,n__fact/1,n__p/1,n__prod/2,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_8/1,c_9/0,c_10/1,c_11/1,c_12/0,c_13/0,c_14/0} - Obligation: innermost runtime complexity wrt. defined symbols {0#,activate#,add#,fact#,if#,p#,prod#,s# ,zero#} and constructors {false,n__0,n__fact,n__p,n__prod,n__s,true} + Applied Processor: SimplifyRHS + Details: Consider the dependency graph 1:S:activate#(X) -> c_2() 2:S:activate#(n__0()) -> c_3(0#()) 3:S:activate#(n__fact(X)) -> c_4(fact#(activate(X))) 4:S:activate#(n__p(X)) -> c_5(p#(activate(X))) 5:S:activate#(n__prod(X1,X2)) -> c_6(prod#(activate(X1),activate(X2))) 6:S:activate#(n__s(X)) -> c_7(s#(activate(X))) 7:S:if#(false(),X,Y) -> c_10(activate#(Y)) -->_1 activate#(n__s(X)) -> c_7(s#(activate(X))):6 -->_1 activate#(n__prod(X1,X2)) -> c_6(prod#(activate(X1),activate(X2))):5 -->_1 activate#(n__p(X)) -> c_5(p#(activate(X))):4 -->_1 activate#(n__fact(X)) -> c_4(fact#(activate(X))):3 -->_1 activate#(n__0()) -> c_3(0#()):2 -->_1 activate#(X) -> c_2():1 11:W:if#(true(),X,Y) -> c_11(activate#(X)) -->_1 activate#(n__s(X)) -> c_7(s#(activate(X))):6 -->_1 activate#(n__prod(X1,X2)) -> c_6(prod#(activate(X1),activate(X2))):5 -->_1 activate#(n__p(X)) -> c_5(p#(activate(X))):4 -->_1 activate#(n__fact(X)) -> c_4(fact#(activate(X))):3 -->_1 activate#(n__0()) -> c_3(0#()):2 -->_1 activate#(X) -> c_2():1 Due to missing edges in the depndency graph, the right-hand sides of following rules could be simplified: activate#(n__0()) -> c_3() activate#(n__fact(X)) -> c_4() activate#(n__p(X)) -> c_5() activate#(n__prod(X1,X2)) -> c_6() activate#(n__s(X)) -> c_7() * Step 8: UsableRules WORST_CASE(?,O(1)) + Considered Problem: - Strict DPs: activate#(X) -> c_2() activate#(n__0()) -> c_3() activate#(n__fact(X)) -> c_4() activate#(n__p(X)) -> c_5() activate#(n__prod(X1,X2)) -> c_6() activate#(n__s(X)) -> c_7() if#(false(),X,Y) -> c_10(activate#(Y)) - Weak DPs: if#(true(),X,Y) -> c_11(activate#(X)) - Weak TRS: 0() -> n__0() activate(X) -> X activate(n__0()) -> 0() activate(n__fact(X)) -> fact(activate(X)) activate(n__p(X)) -> p(activate(X)) activate(n__prod(X1,X2)) -> prod(activate(X1),activate(X2)) activate(n__s(X)) -> s(activate(X)) fact(X) -> if(zero(X),n__s(n__0()),n__prod(X,n__fact(n__p(X)))) fact(X) -> n__fact(X) p(X) -> n__p(X) prod(X1,X2) -> n__prod(X1,X2) s(X) -> n__s(X) - Signature: {0/0,activate/1,add/2,fact/1,if/3,p/1,prod/2,s/1,zero/1,0#/0,activate#/1,add#/2,fact#/1,if#/3,p#/1,prod#/2 ,s#/1,zero#/1} / {false/0,n__0/0,n__fact/1,n__p/1,n__prod/2,n__s/1,true/0,c_1/0,c_2/0,c_3/0,c_4/0,c_5/0 ,c_6/0,c_7/0,c_8/1,c_9/0,c_10/1,c_11/1,c_12/0,c_13/0,c_14/0} - Obligation: innermost runtime complexity wrt. defined symbols {0#,activate#,add#,fact#,if#,p#,prod#,s# ,zero#} and constructors {false,n__0,n__fact,n__p,n__prod,n__s,true} + Applied Processor: UsableRules + Details: We replace rewrite rules by usable rules: activate#(X) -> c_2() activate#(n__0()) -> c_3() activate#(n__fact(X)) -> c_4() activate#(n__p(X)) -> c_5() activate#(n__prod(X1,X2)) -> c_6() activate#(n__s(X)) -> c_7() if#(false(),X,Y) -> c_10(activate#(Y)) if#(true(),X,Y) -> c_11(activate#(X)) * Step 9: RemoveHeads WORST_CASE(?,O(1)) + Considered Problem: - Strict DPs: activate#(X) -> c_2() activate#(n__0()) -> c_3() activate#(n__fact(X)) -> c_4() activate#(n__p(X)) -> c_5() activate#(n__prod(X1,X2)) -> c_6() activate#(n__s(X)) -> c_7() if#(false(),X,Y) -> c_10(activate#(Y)) - Weak DPs: if#(true(),X,Y) -> c_11(activate#(X)) - Signature: {0/0,activate/1,add/2,fact/1,if/3,p/1,prod/2,s/1,zero/1,0#/0,activate#/1,add#/2,fact#/1,if#/3,p#/1,prod#/2 ,s#/1,zero#/1} / {false/0,n__0/0,n__fact/1,n__p/1,n__prod/2,n__s/1,true/0,c_1/0,c_2/0,c_3/0,c_4/0,c_5/0 ,c_6/0,c_7/0,c_8/1,c_9/0,c_10/1,c_11/1,c_12/0,c_13/0,c_14/0} - Obligation: innermost runtime complexity wrt. defined symbols {0#,activate#,add#,fact#,if#,p#,prod#,s# ,zero#} and constructors {false,n__0,n__fact,n__p,n__prod,n__s,true} + Applied Processor: RemoveHeads + Details: Consider the dependency graph 1:S:activate#(X) -> c_2() 2:S:activate#(n__0()) -> c_3() 3:S:activate#(n__fact(X)) -> c_4() 4:S:activate#(n__p(X)) -> c_5() 5:S:activate#(n__prod(X1,X2)) -> c_6() 6:S:activate#(n__s(X)) -> c_7() 7:S:if#(false(),X,Y) -> c_10(activate#(Y)) -->_1 activate#(n__s(X)) -> c_7():6 -->_1 activate#(n__prod(X1,X2)) -> c_6():5 -->_1 activate#(n__p(X)) -> c_5():4 -->_1 activate#(n__fact(X)) -> c_4():3 -->_1 activate#(n__0()) -> c_3():2 -->_1 activate#(X) -> c_2():1 8:W:if#(true(),X,Y) -> c_11(activate#(X)) -->_1 activate#(n__s(X)) -> c_7():6 -->_1 activate#(n__prod(X1,X2)) -> c_6():5 -->_1 activate#(n__p(X)) -> c_5():4 -->_1 activate#(n__fact(X)) -> c_4():3 -->_1 activate#(n__0()) -> c_3():2 -->_1 activate#(X) -> c_2():1 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). [(7,if#(false(),X,Y) -> c_10(activate#(Y))),(8,if#(true(),X,Y) -> c_11(activate#(X)))] * Step 10: RemoveHeads WORST_CASE(?,O(1)) + Considered Problem: - Strict DPs: activate#(X) -> c_2() activate#(n__0()) -> c_3() activate#(n__fact(X)) -> c_4() activate#(n__p(X)) -> c_5() activate#(n__prod(X1,X2)) -> c_6() activate#(n__s(X)) -> c_7() - Signature: {0/0,activate/1,add/2,fact/1,if/3,p/1,prod/2,s/1,zero/1,0#/0,activate#/1,add#/2,fact#/1,if#/3,p#/1,prod#/2 ,s#/1,zero#/1} / {false/0,n__0/0,n__fact/1,n__p/1,n__prod/2,n__s/1,true/0,c_1/0,c_2/0,c_3/0,c_4/0,c_5/0 ,c_6/0,c_7/0,c_8/1,c_9/0,c_10/1,c_11/1,c_12/0,c_13/0,c_14/0} - Obligation: innermost runtime complexity wrt. defined symbols {0#,activate#,add#,fact#,if#,p#,prod#,s# ,zero#} and constructors {false,n__0,n__fact,n__p,n__prod,n__s,true} + Applied Processor: RemoveHeads + Details: Consider the dependency graph 1:S:activate#(X) -> c_2() 2:S:activate#(n__0()) -> c_3() 3:S:activate#(n__fact(X)) -> c_4() 4:S:activate#(n__p(X)) -> c_5() 5:S:activate#(n__prod(X1,X2)) -> c_6() 6:S:activate#(n__s(X)) -> c_7() 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). [(1,activate#(X) -> c_2()) ,(2,activate#(n__0()) -> c_3()) ,(3,activate#(n__fact(X)) -> c_4()) ,(4,activate#(n__p(X)) -> c_5()) ,(5,activate#(n__prod(X1,X2)) -> c_6()) ,(6,activate#(n__s(X)) -> c_7())] * Step 11: EmptyProcessor WORST_CASE(?,O(1)) + Considered Problem: - Signature: {0/0,activate/1,add/2,fact/1,if/3,p/1,prod/2,s/1,zero/1,0#/0,activate#/1,add#/2,fact#/1,if#/3,p#/1,prod#/2 ,s#/1,zero#/1} / {false/0,n__0/0,n__fact/1,n__p/1,n__prod/2,n__s/1,true/0,c_1/0,c_2/0,c_3/0,c_4/0,c_5/0 ,c_6/0,c_7/0,c_8/1,c_9/0,c_10/1,c_11/1,c_12/0,c_13/0,c_14/0} - Obligation: innermost runtime complexity wrt. defined symbols {0#,activate#,add#,fact#,if#,p#,prod#,s# ,zero#} and constructors {false,n__0,n__fact,n__p,n__prod,n__s,true} + Applied Processor: EmptyProcessor + Details: The problem is already closed. The intended complexity is O(1). WORST_CASE(?,O(n^1))