WORST_CASE(Omega(n^1),O(n^1)) * Step 1: Sum WORST_CASE(Omega(n^1),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: Sum {left = someStrategy, right = someStrategy} + Details: () ** Step 1.a:1: DecreasingLoops WORST_CASE(Omega(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: DecreasingLoops {bound = AnyLoop, narrow = 10} + Details: The system has following decreasing Loops: circ(y,z){y -> cons(x,y)} = circ(cons(x,y),z) ->^+ cons(msubst(x,z),circ(y,z)) = C[circ(y,z) = circ(y,z){}] ** Step 1.b:1: 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),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: 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),circ(cons(lift(),t),u)) -> c_4(circ#(cons(lift(),circ(s,t)),u),circ#(s,t)) circ#(cons(lift(),s),cons(a,t)) -> c_5(circ#(s,t)) circ#(cons(lift(),s),cons(lift(),t)) -> c_6(circ#(s,t)) circ#(id(),s) -> c_7() msubst#(a,id()) -> c_8() msubst#(msubst(a,s),t) -> c_9(msubst#(a,circ(s,t)),circ#(s,t)) subst#(a,id()) -> c_10() Weak DPs and mark the set of starting terms. ** Step 1.b:2: 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),circ(cons(lift(),t),u)) -> c_4(circ#(cons(lift(),circ(s,t)),u),circ#(s,t)) circ#(cons(lift(),s),cons(a,t)) -> c_5(circ#(s,t)) circ#(cons(lift(),s),cons(lift(),t)) -> c_6(circ#(s,t)) circ#(id(),s) -> c_7() msubst#(a,id()) -> c_8() msubst#(msubst(a,s),t) -> c_9(msubst#(a,circ(s,t)),circ#(s,t)) subst#(a,id()) -> c_10() - 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),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,circ#/2,msubst#/2,subst#/2} / {cons/2,id/0,lift/0,c_1/0,c_2/2,c_3/2,c_4/2,c_5/1 ,c_6/1,c_7/0,c_8/0,c_9/2,c_10/0} - 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 {1,4,7,8,10} by application of Pre({1,4,7,8,10}) = {2,3,5,6,9}. 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),circ(cons(lift(),t),u)) -> c_4(circ#(cons(lift(),circ(s,t)),u),circ#(s,t)) 5: circ#(cons(lift(),s),cons(a,t)) -> c_5(circ#(s,t)) 6: circ#(cons(lift(),s),cons(lift(),t)) -> c_6(circ#(s,t)) 7: circ#(id(),s) -> c_7() 8: msubst#(a,id()) -> c_8() 9: msubst#(msubst(a,s),t) -> c_9(msubst#(a,circ(s,t)),circ#(s,t)) 10: subst#(a,id()) -> c_10() ** Step 1.b:3: 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_5(circ#(s,t)) circ#(cons(lift(),s),cons(lift(),t)) -> c_6(circ#(s,t)) msubst#(msubst(a,s),t) -> c_9(msubst#(a,circ(s,t)),circ#(s,t)) - Weak DPs: circ#(s,id()) -> c_1() circ#(cons(lift(),s),circ(cons(lift(),t),u)) -> c_4(circ#(cons(lift(),circ(s,t)),u),circ#(s,t)) circ#(id(),s) -> c_7() msubst#(a,id()) -> c_8() subst#(a,id()) -> c_10() - 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),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,circ#/2,msubst#/2,subst#/2} / {cons/2,id/0,lift/0,c_1/0,c_2/2,c_3/2,c_4/2,c_5/1 ,c_6/1,c_7/0,c_8/0,c_9/2,c_10/0} - 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_6(circ#(s,t)):4 -->_1 circ#(cons(lift(),s),cons(lift(),t)) -> c_6(circ#(s,t)):4 -->_2 circ#(cons(lift(),s),cons(a,t)) -> c_5(circ#(s,t)):3 -->_1 circ#(cons(lift(),s),cons(a,t)) -> c_5(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_7():8 -->_1 circ#(id(),s) -> c_7():8 -->_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_9(msubst#(a,circ(s,t)),circ#(s,t)):5 -->_2 circ#(cons(lift(),s),cons(lift(),t)) -> c_6(circ#(s,t)):4 -->_2 circ#(cons(lift(),s),cons(a,t)) -> c_5(circ#(s,t)):3 -->_1 msubst#(a,id()) -> c_8():9 -->_2 circ#(id(),s) -> c_7():8 -->_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_5(circ#(s,t)) -->_1 circ#(cons(lift(),s),cons(lift(),t)) -> c_6(circ#(s,t)):4 -->_1 circ#(id(),s) -> c_7():8 -->_1 circ#(s,id()) -> c_1():6 -->_1 circ#(cons(lift(),s),cons(a,t)) -> c_5(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_6(circ#(s,t)) -->_1 circ#(id(),s) -> c_7():8 -->_1 circ#(s,id()) -> c_1():6 -->_1 circ#(cons(lift(),s),cons(lift(),t)) -> c_6(circ#(s,t)):4 -->_1 circ#(cons(lift(),s),cons(a,t)) -> c_5(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_9(msubst#(a,circ(s,t)),circ#(s,t)) -->_1 msubst#(a,id()) -> c_8():9 -->_2 circ#(id(),s) -> c_7():8 -->_2 circ#(s,id()) -> c_1():6 -->_1 msubst#(msubst(a,s),t) -> c_9(msubst#(a,circ(s,t)),circ#(s,t)):5 -->_2 circ#(cons(lift(),s),cons(lift(),t)) -> c_6(circ#(s,t)):4 -->_2 circ#(cons(lift(),s),cons(a,t)) -> c_5(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#(cons(lift(),s),circ(cons(lift(),t),u)) -> c_4(circ#(cons(lift(),circ(s,t)),u),circ#(s,t)) 8:W:circ#(id(),s) -> c_7() 9:W:msubst#(a,id()) -> c_8() 10:W:subst#(a,id()) -> c_10() The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed. 10: subst#(a,id()) -> c_10() 7: circ#(cons(lift(),s),circ(cons(lift(),t),u)) -> c_4(circ#(cons(lift(),circ(s,t)),u),circ#(s,t)) 9: msubst#(a,id()) -> c_8() 6: circ#(s,id()) -> c_1() 8: circ#(id(),s) -> c_7() ** Step 1.b:4: UsableRules 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_5(circ#(s,t)) circ#(cons(lift(),s),cons(lift(),t)) -> c_6(circ#(s,t)) msubst#(msubst(a,s),t) -> c_9(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),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,circ#/2,msubst#/2,subst#/2} / {cons/2,id/0,lift/0,c_1/0,c_2/2,c_3/2,c_4/2,c_5/1 ,c_6/1,c_7/0,c_8/0,c_9/2,c_10/0} - 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),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)) 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_5(circ#(s,t)) circ#(cons(lift(),s),cons(lift(),t)) -> c_6(circ#(s,t)) msubst#(msubst(a,s),t) -> c_9(msubst#(a,circ(s,t)),circ#(s,t)) ** Step 1.b:5: 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_5(circ#(s,t)) circ#(cons(lift(),s),cons(lift(),t)) -> c_6(circ#(s,t)) msubst#(msubst(a,s),t) -> c_9(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),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)) - 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_4/2,c_5/1 ,c_6/1,c_7/0,c_8/0,c_9/2,c_10/0} - 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 any 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_5) = {1}, uargs(c_6) = {1}, uargs(c_9) = {1,2} Following symbols are considered usable: {circ#,msubst#,subst#} TcT has computed the following interpretation: p(circ) = [2] x1 + [2] x2 + [1] p(cons) = [1] x1 + [1] x2 + [2] p(id) = [2] p(lift) = [0] p(msubst) = [1] x1 + [6] x2 + [2] p(subst) = [0] p(circ#) = [4] x1 + [0] p(msubst#) = [1] x1 + [0] p(subst#) = [1] x1 + [4] x2 + [0] p(c_1) = [4] p(c_2) = [2] x1 + [2] x2 + [2] p(c_3) = [4] x1 + [1] x2 + [7] p(c_4) = [0] p(c_5) = [1] x1 + [7] p(c_6) = [1] x1 + [7] p(c_7) = [4] p(c_8) = [1] p(c_9) = [1] x1 + [1] x2 + [2] p(c_10) = [4] Following rules are strictly oriented: circ#(circ(s,t),u) = [8] s + [8] t + [4] > [8] s + [8] t + [2] = c_2(circ#(s,circ(t,u)),circ#(t,u)) circ#(cons(a,s),t) = [4] a + [4] s + [8] > [4] a + [4] s + [7] = c_3(msubst#(a,t),circ#(s,t)) circ#(cons(lift(),s),cons(a,t)) = [4] s + [8] > [4] s + [7] = c_5(circ#(s,t)) circ#(cons(lift(),s),cons(lift(),t)) = [4] s + [8] > [4] s + [7] = c_6(circ#(s,t)) Following rules are (at-least) weakly oriented: msubst#(msubst(a,s),t) = [1] a + [6] s + [2] >= [1] a + [4] s + [2] = c_9(msubst#(a,circ(s,t)),circ#(s,t)) ** Step 1.b:6: NaturalMI WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: msubst#(msubst(a,s),t) -> c_9(msubst#(a,circ(s,t)),circ#(s,t)) - 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_5(circ#(s,t)) circ#(cons(lift(),s),cons(lift(),t)) -> c_6(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),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)) - 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_4/2,c_5/1 ,c_6/1,c_7/0,c_8/0,c_9/2,c_10/0} - 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 any 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_5) = {1}, uargs(c_6) = {1}, uargs(c_9) = {1,2} Following symbols are considered usable: {circ#,msubst#,subst#} TcT has computed the following interpretation: p(circ) = [1] x1 + [2] x2 + [0] p(cons) = [1] x1 + [1] x2 + [0] p(id) = [0] p(lift) = [2] p(msubst) = [2] x1 + [4] x2 + [4] p(subst) = [4] x1 + [4] x2 + [1] p(circ#) = [4] x1 + [0] p(msubst#) = [1] x1 + [0] p(subst#) = [2] x1 + [2] x2 + [4] p(c_1) = [0] p(c_2) = [1] x1 + [2] x2 + [0] p(c_3) = [2] x1 + [1] x2 + [0] p(c_4) = [4] x1 + [0] p(c_5) = [1] x1 + [4] p(c_6) = [1] x1 + [0] p(c_7) = [1] p(c_8) = [1] p(c_9) = [2] x1 + [1] x2 + [0] p(c_10) = [4] Following rules are strictly oriented: msubst#(msubst(a,s),t) = [2] a + [4] s + [4] > [2] a + [4] s + [0] = c_9(msubst#(a,circ(s,t)),circ#(s,t)) Following rules are (at-least) weakly oriented: circ#(circ(s,t),u) = [4] s + [8] t + [0] >= [4] s + [8] t + [0] = c_2(circ#(s,circ(t,u)),circ#(t,u)) circ#(cons(a,s),t) = [4] a + [4] s + [0] >= [2] a + [4] s + [0] = c_3(msubst#(a,t),circ#(s,t)) circ#(cons(lift(),s),cons(a,t)) = [4] s + [8] >= [4] s + [4] = c_5(circ#(s,t)) circ#(cons(lift(),s),cons(lift(),t)) = [4] s + [8] >= [4] s + [0] = c_6(circ#(s,t)) ** Step 1.b:7: EmptyProcessor 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_5(circ#(s,t)) circ#(cons(lift(),s),cons(lift(),t)) -> c_6(circ#(s,t)) msubst#(msubst(a,s),t) -> c_9(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),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)) - 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_4/2,c_5/1 ,c_6/1,c_7/0,c_8/0,c_9/2,c_10/0} - 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(Omega(n^1),O(n^1))