WORST_CASE(?,O(n^1)) * Step 1: InnermostRuleRemoval WORST_CASE(?,O(n^1)) + Considered Problem: - Strict TRS: circ(s,id()) -> s circ(circ(s,t),u) -> circ(s,circ(t,u)) circ(cons(a,s),t) -> cons(msubst(a,t),circ(s,t)) circ(cons(lift(),s),circ(cons(lift(),t),u)) -> circ(cons(lift(),circ(s,t)),u) circ(cons(lift(),s),cons(a,t)) -> cons(a,circ(s,t)) circ(cons(lift(),s),cons(lift(),t)) -> cons(lift(),circ(s,t)) circ(id(),s) -> s msubst(a,id()) -> a msubst(msubst(a,s),t) -> msubst(a,circ(s,t)) subst(a,id()) -> a - Signature: {circ/2,msubst/2,subst/2} / {cons/2,id/0,lift/0} - Obligation: innermost runtime complexity wrt. defined symbols {circ,msubst,subst} and constructors {cons,id,lift} + Applied Processor: InnermostRuleRemoval + Details: Arguments of following rules are not normal-forms. circ(cons(lift(),s),circ(cons(lift(),t),u)) -> circ(cons(lift(),circ(s,t)),u) All above mentioned rules can be savely removed. * Step 2: DependencyPairs WORST_CASE(?,O(n^1)) + Considered Problem: - Strict TRS: circ(s,id()) -> s circ(circ(s,t),u) -> circ(s,circ(t,u)) circ(cons(a,s),t) -> cons(msubst(a,t),circ(s,t)) circ(cons(lift(),s),cons(a,t)) -> cons(a,circ(s,t)) circ(cons(lift(),s),cons(lift(),t)) -> cons(lift(),circ(s,t)) circ(id(),s) -> s msubst(a,id()) -> a msubst(msubst(a,s),t) -> msubst(a,circ(s,t)) subst(a,id()) -> a - Signature: {circ/2,msubst/2,subst/2} / {cons/2,id/0,lift/0} - Obligation: innermost runtime complexity wrt. defined symbols {circ,msubst,subst} and constructors {cons,id,lift} + Applied Processor: DependencyPairs {dpKind_ = DT} + Details: We add the following dependency tuples: Strict DPs circ#(s,id()) -> c_1() circ#(circ(s,t),u) -> c_2(circ#(s,circ(t,u)),circ#(t,u)) circ#(cons(a,s),t) -> c_3(msubst#(a,t),circ#(s,t)) circ#(cons(lift(),s),cons(a,t)) -> c_4(circ#(s,t)) circ#(cons(lift(),s),cons(lift(),t)) -> c_5(circ#(s,t)) circ#(id(),s) -> c_6() msubst#(a,id()) -> c_7() msubst#(msubst(a,s),t) -> c_8(msubst#(a,circ(s,t)),circ#(s,t)) subst#(a,id()) -> c_9() Weak DPs and mark the set of starting terms. * Step 3: UsableRules WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: circ#(s,id()) -> c_1() circ#(circ(s,t),u) -> c_2(circ#(s,circ(t,u)),circ#(t,u)) circ#(cons(a,s),t) -> c_3(msubst#(a,t),circ#(s,t)) circ#(cons(lift(),s),cons(a,t)) -> c_4(circ#(s,t)) circ#(cons(lift(),s),cons(lift(),t)) -> c_5(circ#(s,t)) circ#(id(),s) -> c_6() msubst#(a,id()) -> c_7() msubst#(msubst(a,s),t) -> c_8(msubst#(a,circ(s,t)),circ#(s,t)) subst#(a,id()) -> c_9() - Weak TRS: circ(s,id()) -> s circ(circ(s,t),u) -> circ(s,circ(t,u)) circ(cons(a,s),t) -> cons(msubst(a,t),circ(s,t)) circ(cons(lift(),s),cons(a,t)) -> cons(a,circ(s,t)) circ(cons(lift(),s),cons(lift(),t)) -> cons(lift(),circ(s,t)) circ(id(),s) -> s msubst(a,id()) -> a msubst(msubst(a,s),t) -> msubst(a,circ(s,t)) subst(a,id()) -> a - Signature: {circ/2,msubst/2,subst/2,circ#/2,msubst#/2,subst#/2} / {cons/2,id/0,lift/0,c_1/0,c_2/2,c_3/2,c_4/1,c_5/1 ,c_6/0,c_7/0,c_8/2,c_9/0} - Obligation: innermost runtime complexity wrt. defined symbols {circ#,msubst#,subst#} and constructors {cons,id,lift} + Applied Processor: UsableRules + Details: We replace rewrite rules by usable rules: circ(s,id()) -> s circ(circ(s,t),u) -> circ(s,circ(t,u)) circ(cons(a,s),t) -> cons(msubst(a,t),circ(s,t)) circ(cons(lift(),s),cons(a,t)) -> cons(a,circ(s,t)) circ(cons(lift(),s),cons(lift(),t)) -> cons(lift(),circ(s,t)) circ(id(),s) -> s msubst(a,id()) -> a msubst(msubst(a,s),t) -> msubst(a,circ(s,t)) circ#(s,id()) -> c_1() circ#(circ(s,t),u) -> c_2(circ#(s,circ(t,u)),circ#(t,u)) circ#(cons(a,s),t) -> c_3(msubst#(a,t),circ#(s,t)) circ#(cons(lift(),s),cons(a,t)) -> c_4(circ#(s,t)) circ#(cons(lift(),s),cons(lift(),t)) -> c_5(circ#(s,t)) circ#(id(),s) -> c_6() msubst#(a,id()) -> c_7() msubst#(msubst(a,s),t) -> c_8(msubst#(a,circ(s,t)),circ#(s,t)) subst#(a,id()) -> c_9() * Step 4: PredecessorEstimation WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: circ#(s,id()) -> c_1() circ#(circ(s,t),u) -> c_2(circ#(s,circ(t,u)),circ#(t,u)) circ#(cons(a,s),t) -> c_3(msubst#(a,t),circ#(s,t)) circ#(cons(lift(),s),cons(a,t)) -> c_4(circ#(s,t)) circ#(cons(lift(),s),cons(lift(),t)) -> c_5(circ#(s,t)) circ#(id(),s) -> c_6() msubst#(a,id()) -> c_7() msubst#(msubst(a,s),t) -> c_8(msubst#(a,circ(s,t)),circ#(s,t)) subst#(a,id()) -> c_9() - Weak TRS: circ(s,id()) -> s circ(circ(s,t),u) -> circ(s,circ(t,u)) circ(cons(a,s),t) -> cons(msubst(a,t),circ(s,t)) circ(cons(lift(),s),cons(a,t)) -> cons(a,circ(s,t)) circ(cons(lift(),s),cons(lift(),t)) -> cons(lift(),circ(s,t)) circ(id(),s) -> s msubst(a,id()) -> a msubst(msubst(a,s),t) -> msubst(a,circ(s,t)) - Signature: {circ/2,msubst/2,subst/2,circ#/2,msubst#/2,subst#/2} / {cons/2,id/0,lift/0,c_1/0,c_2/2,c_3/2,c_4/1,c_5/1 ,c_6/0,c_7/0,c_8/2,c_9/0} - Obligation: innermost runtime complexity wrt. defined symbols {circ#,msubst#,subst#} and constructors {cons,id,lift} + Applied Processor: PredecessorEstimation {onSelection = all simple predecessor estimation selector} + Details: We estimate the number of application of {1,6,7,9} by application of Pre({1,6,7,9}) = {2,3,4,5,8}. Here rules are labelled as follows: 1: circ#(s,id()) -> c_1() 2: circ#(circ(s,t),u) -> c_2(circ#(s,circ(t,u)),circ#(t,u)) 3: circ#(cons(a,s),t) -> c_3(msubst#(a,t),circ#(s,t)) 4: circ#(cons(lift(),s),cons(a,t)) -> c_4(circ#(s,t)) 5: circ#(cons(lift(),s),cons(lift(),t)) -> c_5(circ#(s,t)) 6: circ#(id(),s) -> c_6() 7: msubst#(a,id()) -> c_7() 8: msubst#(msubst(a,s),t) -> c_8(msubst#(a,circ(s,t)),circ#(s,t)) 9: subst#(a,id()) -> c_9() * Step 5: RemoveWeakSuffixes WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: circ#(circ(s,t),u) -> c_2(circ#(s,circ(t,u)),circ#(t,u)) circ#(cons(a,s),t) -> c_3(msubst#(a,t),circ#(s,t)) circ#(cons(lift(),s),cons(a,t)) -> c_4(circ#(s,t)) circ#(cons(lift(),s),cons(lift(),t)) -> c_5(circ#(s,t)) msubst#(msubst(a,s),t) -> c_8(msubst#(a,circ(s,t)),circ#(s,t)) - Weak DPs: circ#(s,id()) -> c_1() circ#(id(),s) -> c_6() msubst#(a,id()) -> c_7() subst#(a,id()) -> c_9() - Weak TRS: circ(s,id()) -> s circ(circ(s,t),u) -> circ(s,circ(t,u)) circ(cons(a,s),t) -> cons(msubst(a,t),circ(s,t)) circ(cons(lift(),s),cons(a,t)) -> cons(a,circ(s,t)) circ(cons(lift(),s),cons(lift(),t)) -> cons(lift(),circ(s,t)) circ(id(),s) -> s msubst(a,id()) -> a msubst(msubst(a,s),t) -> msubst(a,circ(s,t)) - Signature: {circ/2,msubst/2,subst/2,circ#/2,msubst#/2,subst#/2} / {cons/2,id/0,lift/0,c_1/0,c_2/2,c_3/2,c_4/1,c_5/1 ,c_6/0,c_7/0,c_8/2,c_9/0} - Obligation: innermost runtime complexity wrt. defined symbols {circ#,msubst#,subst#} and constructors {cons,id,lift} + Applied Processor: RemoveWeakSuffixes + Details: Consider the dependency graph 1:S:circ#(circ(s,t),u) -> c_2(circ#(s,circ(t,u)),circ#(t,u)) -->_2 circ#(cons(lift(),s),cons(lift(),t)) -> c_5(circ#(s,t)):4 -->_1 circ#(cons(lift(),s),cons(lift(),t)) -> c_5(circ#(s,t)):4 -->_2 circ#(cons(lift(),s),cons(a,t)) -> c_4(circ#(s,t)):3 -->_1 circ#(cons(lift(),s),cons(a,t)) -> c_4(circ#(s,t)):3 -->_2 circ#(cons(a,s),t) -> c_3(msubst#(a,t),circ#(s,t)):2 -->_1 circ#(cons(a,s),t) -> c_3(msubst#(a,t),circ#(s,t)):2 -->_2 circ#(id(),s) -> c_6():7 -->_1 circ#(id(),s) -> c_6():7 -->_2 circ#(s,id()) -> c_1():6 -->_1 circ#(s,id()) -> c_1():6 -->_2 circ#(circ(s,t),u) -> c_2(circ#(s,circ(t,u)),circ#(t,u)):1 -->_1 circ#(circ(s,t),u) -> c_2(circ#(s,circ(t,u)),circ#(t,u)):1 2:S:circ#(cons(a,s),t) -> c_3(msubst#(a,t),circ#(s,t)) -->_1 msubst#(msubst(a,s),t) -> c_8(msubst#(a,circ(s,t)),circ#(s,t)):5 -->_2 circ#(cons(lift(),s),cons(lift(),t)) -> c_5(circ#(s,t)):4 -->_2 circ#(cons(lift(),s),cons(a,t)) -> c_4(circ#(s,t)):3 -->_1 msubst#(a,id()) -> c_7():8 -->_2 circ#(id(),s) -> c_6():7 -->_2 circ#(s,id()) -> c_1():6 -->_2 circ#(cons(a,s),t) -> c_3(msubst#(a,t),circ#(s,t)):2 -->_2 circ#(circ(s,t),u) -> c_2(circ#(s,circ(t,u)),circ#(t,u)):1 3:S:circ#(cons(lift(),s),cons(a,t)) -> c_4(circ#(s,t)) -->_1 circ#(cons(lift(),s),cons(lift(),t)) -> c_5(circ#(s,t)):4 -->_1 circ#(id(),s) -> c_6():7 -->_1 circ#(s,id()) -> c_1():6 -->_1 circ#(cons(lift(),s),cons(a,t)) -> c_4(circ#(s,t)):3 -->_1 circ#(cons(a,s),t) -> c_3(msubst#(a,t),circ#(s,t)):2 -->_1 circ#(circ(s,t),u) -> c_2(circ#(s,circ(t,u)),circ#(t,u)):1 4:S:circ#(cons(lift(),s),cons(lift(),t)) -> c_5(circ#(s,t)) -->_1 circ#(id(),s) -> c_6():7 -->_1 circ#(s,id()) -> c_1():6 -->_1 circ#(cons(lift(),s),cons(lift(),t)) -> c_5(circ#(s,t)):4 -->_1 circ#(cons(lift(),s),cons(a,t)) -> c_4(circ#(s,t)):3 -->_1 circ#(cons(a,s),t) -> c_3(msubst#(a,t),circ#(s,t)):2 -->_1 circ#(circ(s,t),u) -> c_2(circ#(s,circ(t,u)),circ#(t,u)):1 5:S:msubst#(msubst(a,s),t) -> c_8(msubst#(a,circ(s,t)),circ#(s,t)) -->_1 msubst#(a,id()) -> c_7():8 -->_2 circ#(id(),s) -> c_6():7 -->_2 circ#(s,id()) -> c_1():6 -->_1 msubst#(msubst(a,s),t) -> c_8(msubst#(a,circ(s,t)),circ#(s,t)):5 -->_2 circ#(cons(lift(),s),cons(lift(),t)) -> c_5(circ#(s,t)):4 -->_2 circ#(cons(lift(),s),cons(a,t)) -> c_4(circ#(s,t)):3 -->_2 circ#(cons(a,s),t) -> c_3(msubst#(a,t),circ#(s,t)):2 -->_2 circ#(circ(s,t),u) -> c_2(circ#(s,circ(t,u)),circ#(t,u)):1 6:W:circ#(s,id()) -> c_1() 7:W:circ#(id(),s) -> c_6() 8:W:msubst#(a,id()) -> c_7() 9:W:subst#(a,id()) -> c_9() The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed. 9: subst#(a,id()) -> c_9() 8: msubst#(a,id()) -> c_7() 6: circ#(s,id()) -> c_1() 7: circ#(id(),s) -> c_6() * Step 6: PredecessorEstimationCP WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: circ#(circ(s,t),u) -> c_2(circ#(s,circ(t,u)),circ#(t,u)) circ#(cons(a,s),t) -> c_3(msubst#(a,t),circ#(s,t)) circ#(cons(lift(),s),cons(a,t)) -> c_4(circ#(s,t)) circ#(cons(lift(),s),cons(lift(),t)) -> c_5(circ#(s,t)) msubst#(msubst(a,s),t) -> c_8(msubst#(a,circ(s,t)),circ#(s,t)) - Weak TRS: circ(s,id()) -> s circ(circ(s,t),u) -> circ(s,circ(t,u)) circ(cons(a,s),t) -> cons(msubst(a,t),circ(s,t)) circ(cons(lift(),s),cons(a,t)) -> cons(a,circ(s,t)) circ(cons(lift(),s),cons(lift(),t)) -> cons(lift(),circ(s,t)) circ(id(),s) -> s msubst(a,id()) -> a msubst(msubst(a,s),t) -> msubst(a,circ(s,t)) - Signature: {circ/2,msubst/2,subst/2,circ#/2,msubst#/2,subst#/2} / {cons/2,id/0,lift/0,c_1/0,c_2/2,c_3/2,c_4/1,c_5/1 ,c_6/0,c_7/0,c_8/2,c_9/0} - Obligation: innermost runtime complexity wrt. defined symbols {circ#,msubst#,subst#} and constructors {cons,id,lift} + Applied Processor: PredecessorEstimationCP {onSelectionCP = any intersect of rules of CDG leaf and strict-rules, withComplexityPair = NaturalMI {miDimension = 1, miDegree = 1, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Nothing}} + Details: We first use the processor NaturalMI {miDimension = 1, miDegree = 1, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Nothing} to orient following rules strictly: 1: circ#(circ(s,t),u) -> c_2(circ#(s,circ(t,u)),circ#(t,u)) 2: circ#(cons(a,s),t) -> c_3(msubst#(a,t),circ#(s,t)) 3: circ#(cons(lift(),s),cons(a,t)) -> c_4(circ#(s,t)) 4: circ#(cons(lift(),s),cons(lift(),t)) -> c_5(circ#(s,t)) 5: msubst#(msubst(a,s),t) -> c_8(msubst#(a,circ(s,t)),circ#(s,t)) The strictly oriented rules are moved into the weak component. ** Step 6.a:1: NaturalMI WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: circ#(circ(s,t),u) -> c_2(circ#(s,circ(t,u)),circ#(t,u)) circ#(cons(a,s),t) -> c_3(msubst#(a,t),circ#(s,t)) circ#(cons(lift(),s),cons(a,t)) -> c_4(circ#(s,t)) circ#(cons(lift(),s),cons(lift(),t)) -> c_5(circ#(s,t)) msubst#(msubst(a,s),t) -> c_8(msubst#(a,circ(s,t)),circ#(s,t)) - Weak TRS: circ(s,id()) -> s circ(circ(s,t),u) -> circ(s,circ(t,u)) circ(cons(a,s),t) -> cons(msubst(a,t),circ(s,t)) circ(cons(lift(),s),cons(a,t)) -> cons(a,circ(s,t)) circ(cons(lift(),s),cons(lift(),t)) -> cons(lift(),circ(s,t)) circ(id(),s) -> s msubst(a,id()) -> a msubst(msubst(a,s),t) -> msubst(a,circ(s,t)) - Signature: {circ/2,msubst/2,subst/2,circ#/2,msubst#/2,subst#/2} / {cons/2,id/0,lift/0,c_1/0,c_2/2,c_3/2,c_4/1,c_5/1 ,c_6/0,c_7/0,c_8/2,c_9/0} - Obligation: innermost runtime complexity wrt. defined symbols {circ#,msubst#,subst#} and constructors {cons,id,lift} + Applied Processor: NaturalMI {miDimension = 1, miDegree = 1, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Just first alternative for predecessorEstimation on any intersect of rules of CDG leaf and strict-rules} + Details: We apply a matrix interpretation of kind constructor based matrix interpretation: The following argument positions are considered usable: uargs(c_2) = {1,2}, uargs(c_3) = {1,2}, uargs(c_4) = {1}, uargs(c_5) = {1}, uargs(c_8) = {1,2} Following symbols are considered usable: {circ#,msubst#,subst#} TcT has computed the following interpretation: p(circ) = [1] x1 + [3] x2 + [2] p(cons) = [1] x1 + [1] x2 + [2] p(id) = [3] p(lift) = [0] p(msubst) = [2] x1 + [4] x2 + [3] p(subst) = [1] x1 + [0] p(circ#) = [4] x1 + [2] p(msubst#) = [2] x1 + [0] p(subst#) = [2] p(c_1) = [1] p(c_2) = [1] x1 + [2] x2 + [1] p(c_3) = [2] x1 + [1] x2 + [0] p(c_4) = [1] x1 + [1] p(c_5) = [1] x1 + [4] p(c_6) = [1] p(c_7) = [1] p(c_8) = [2] x1 + [2] x2 + [0] p(c_9) = [1] Following rules are strictly oriented: circ#(circ(s,t),u) = [4] s + [12] t + [10] > [4] s + [8] t + [7] = c_2(circ#(s,circ(t,u)),circ#(t,u)) circ#(cons(a,s),t) = [4] a + [4] s + [10] > [4] a + [4] s + [2] = c_3(msubst#(a,t),circ#(s,t)) circ#(cons(lift(),s),cons(a,t)) = [4] s + [10] > [4] s + [3] = c_4(circ#(s,t)) circ#(cons(lift(),s),cons(lift(),t)) = [4] s + [10] > [4] s + [6] = c_5(circ#(s,t)) msubst#(msubst(a,s),t) = [4] a + [8] s + [6] > [4] a + [8] s + [4] = c_8(msubst#(a,circ(s,t)),circ#(s,t)) Following rules are (at-least) weakly oriented: ** Step 6.a:2: Assumption WORST_CASE(?,O(1)) + Considered Problem: - Weak DPs: circ#(circ(s,t),u) -> c_2(circ#(s,circ(t,u)),circ#(t,u)) circ#(cons(a,s),t) -> c_3(msubst#(a,t),circ#(s,t)) circ#(cons(lift(),s),cons(a,t)) -> c_4(circ#(s,t)) circ#(cons(lift(),s),cons(lift(),t)) -> c_5(circ#(s,t)) msubst#(msubst(a,s),t) -> c_8(msubst#(a,circ(s,t)),circ#(s,t)) - Weak TRS: circ(s,id()) -> s circ(circ(s,t),u) -> circ(s,circ(t,u)) circ(cons(a,s),t) -> cons(msubst(a,t),circ(s,t)) circ(cons(lift(),s),cons(a,t)) -> cons(a,circ(s,t)) circ(cons(lift(),s),cons(lift(),t)) -> cons(lift(),circ(s,t)) circ(id(),s) -> s msubst(a,id()) -> a msubst(msubst(a,s),t) -> msubst(a,circ(s,t)) - Signature: {circ/2,msubst/2,subst/2,circ#/2,msubst#/2,subst#/2} / {cons/2,id/0,lift/0,c_1/0,c_2/2,c_3/2,c_4/1,c_5/1 ,c_6/0,c_7/0,c_8/2,c_9/0} - Obligation: innermost runtime complexity wrt. defined symbols {circ#,msubst#,subst#} and constructors {cons,id,lift} + Applied Processor: Assumption {assumed = Certificate {spaceUB = Unknown, spaceLB = Unknown, timeUB = Poly (Just 0), timeLB = Unknown}} + Details: () ** Step 6.b:1: RemoveWeakSuffixes WORST_CASE(?,O(1)) + Considered Problem: - Weak DPs: circ#(circ(s,t),u) -> c_2(circ#(s,circ(t,u)),circ#(t,u)) circ#(cons(a,s),t) -> c_3(msubst#(a,t),circ#(s,t)) circ#(cons(lift(),s),cons(a,t)) -> c_4(circ#(s,t)) circ#(cons(lift(),s),cons(lift(),t)) -> c_5(circ#(s,t)) msubst#(msubst(a,s),t) -> c_8(msubst#(a,circ(s,t)),circ#(s,t)) - Weak TRS: circ(s,id()) -> s circ(circ(s,t),u) -> circ(s,circ(t,u)) circ(cons(a,s),t) -> cons(msubst(a,t),circ(s,t)) circ(cons(lift(),s),cons(a,t)) -> cons(a,circ(s,t)) circ(cons(lift(),s),cons(lift(),t)) -> cons(lift(),circ(s,t)) circ(id(),s) -> s msubst(a,id()) -> a msubst(msubst(a,s),t) -> msubst(a,circ(s,t)) - Signature: {circ/2,msubst/2,subst/2,circ#/2,msubst#/2,subst#/2} / {cons/2,id/0,lift/0,c_1/0,c_2/2,c_3/2,c_4/1,c_5/1 ,c_6/0,c_7/0,c_8/2,c_9/0} - Obligation: innermost runtime complexity wrt. defined symbols {circ#,msubst#,subst#} and constructors {cons,id,lift} + Applied Processor: RemoveWeakSuffixes + Details: Consider the dependency graph 1:W:circ#(circ(s,t),u) -> c_2(circ#(s,circ(t,u)),circ#(t,u)) -->_2 circ#(cons(lift(),s),cons(lift(),t)) -> c_5(circ#(s,t)):4 -->_1 circ#(cons(lift(),s),cons(lift(),t)) -> c_5(circ#(s,t)):4 -->_2 circ#(cons(lift(),s),cons(a,t)) -> c_4(circ#(s,t)):3 -->_1 circ#(cons(lift(),s),cons(a,t)) -> c_4(circ#(s,t)):3 -->_2 circ#(cons(a,s),t) -> c_3(msubst#(a,t),circ#(s,t)):2 -->_1 circ#(cons(a,s),t) -> c_3(msubst#(a,t),circ#(s,t)):2 -->_2 circ#(circ(s,t),u) -> c_2(circ#(s,circ(t,u)),circ#(t,u)):1 -->_1 circ#(circ(s,t),u) -> c_2(circ#(s,circ(t,u)),circ#(t,u)):1 2:W:circ#(cons(a,s),t) -> c_3(msubst#(a,t),circ#(s,t)) -->_1 msubst#(msubst(a,s),t) -> c_8(msubst#(a,circ(s,t)),circ#(s,t)):5 -->_2 circ#(cons(lift(),s),cons(lift(),t)) -> c_5(circ#(s,t)):4 -->_2 circ#(cons(lift(),s),cons(a,t)) -> c_4(circ#(s,t)):3 -->_2 circ#(cons(a,s),t) -> c_3(msubst#(a,t),circ#(s,t)):2 -->_2 circ#(circ(s,t),u) -> c_2(circ#(s,circ(t,u)),circ#(t,u)):1 3:W:circ#(cons(lift(),s),cons(a,t)) -> c_4(circ#(s,t)) -->_1 circ#(cons(lift(),s),cons(lift(),t)) -> c_5(circ#(s,t)):4 -->_1 circ#(cons(lift(),s),cons(a,t)) -> c_4(circ#(s,t)):3 -->_1 circ#(cons(a,s),t) -> c_3(msubst#(a,t),circ#(s,t)):2 -->_1 circ#(circ(s,t),u) -> c_2(circ#(s,circ(t,u)),circ#(t,u)):1 4:W:circ#(cons(lift(),s),cons(lift(),t)) -> c_5(circ#(s,t)) -->_1 circ#(cons(lift(),s),cons(lift(),t)) -> c_5(circ#(s,t)):4 -->_1 circ#(cons(lift(),s),cons(a,t)) -> c_4(circ#(s,t)):3 -->_1 circ#(cons(a,s),t) -> c_3(msubst#(a,t),circ#(s,t)):2 -->_1 circ#(circ(s,t),u) -> c_2(circ#(s,circ(t,u)),circ#(t,u)):1 5:W:msubst#(msubst(a,s),t) -> c_8(msubst#(a,circ(s,t)),circ#(s,t)) -->_1 msubst#(msubst(a,s),t) -> c_8(msubst#(a,circ(s,t)),circ#(s,t)):5 -->_2 circ#(cons(lift(),s),cons(lift(),t)) -> c_5(circ#(s,t)):4 -->_2 circ#(cons(lift(),s),cons(a,t)) -> c_4(circ#(s,t)):3 -->_2 circ#(cons(a,s),t) -> c_3(msubst#(a,t),circ#(s,t)):2 -->_2 circ#(circ(s,t),u) -> c_2(circ#(s,circ(t,u)),circ#(t,u)):1 The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed. 1: circ#(circ(s,t),u) -> c_2(circ#(s,circ(t,u)),circ#(t,u)) 5: msubst#(msubst(a,s),t) -> c_8(msubst#(a,circ(s,t)),circ#(s,t)) 2: circ#(cons(a,s),t) -> c_3(msubst#(a,t),circ#(s,t)) 4: circ#(cons(lift(),s),cons(lift(),t)) -> c_5(circ#(s,t)) 3: circ#(cons(lift(),s),cons(a,t)) -> c_4(circ#(s,t)) ** Step 6.b:2: EmptyProcessor WORST_CASE(?,O(1)) + Considered Problem: - Weak TRS: circ(s,id()) -> s circ(circ(s,t),u) -> circ(s,circ(t,u)) circ(cons(a,s),t) -> cons(msubst(a,t),circ(s,t)) circ(cons(lift(),s),cons(a,t)) -> cons(a,circ(s,t)) circ(cons(lift(),s),cons(lift(),t)) -> cons(lift(),circ(s,t)) circ(id(),s) -> s msubst(a,id()) -> a msubst(msubst(a,s),t) -> msubst(a,circ(s,t)) - Signature: {circ/2,msubst/2,subst/2,circ#/2,msubst#/2,subst#/2} / {cons/2,id/0,lift/0,c_1/0,c_2/2,c_3/2,c_4/1,c_5/1 ,c_6/0,c_7/0,c_8/2,c_9/0} - Obligation: innermost runtime complexity wrt. defined symbols {circ#,msubst#,subst#} and constructors {cons,id,lift} + Applied Processor: EmptyProcessor + Details: The problem is already closed. The intended complexity is O(1). WORST_CASE(?,O(n^1))