WORST_CASE(?,O(n^3)) * Step 1: DependencyPairs WORST_CASE(?,O(n^3)) + Considered Problem: - Strict TRS: append(cons(x,xs),ys) -> cons(x,append(xs,ys)) append(nil(),ys) -> ys attach(x,cons(y,ys)) -> cons(pair(x,y),attach(x,ys)) attach(x,nil()) -> nil() pairsp(cons(x,xs)) -> append(pairsp(xs),attach(x,xs)) pairsp(nil()) -> nil() - Signature: {append/2,attach/2,pairsp/1} / {cons/2,nil/0,pair/2} - Obligation: innermost runtime complexity wrt. defined symbols {append,attach,pairsp} and constructors {cons,nil,pair} + Applied Processor: DependencyPairs {dpKind_ = DT} + Details: We add the following dependency tuples: Strict DPs append#(cons(x,xs),ys) -> c_1(append#(xs,ys)) append#(nil(),ys) -> c_2() attach#(x,cons(y,ys)) -> c_3(attach#(x,ys)) attach#(x,nil()) -> c_4() pairsp#(cons(x,xs)) -> c_5(append#(pairsp(xs),attach(x,xs)),pairsp#(xs),attach#(x,xs)) pairsp#(nil()) -> c_6() Weak DPs and mark the set of starting terms. * Step 2: PredecessorEstimation WORST_CASE(?,O(n^3)) + Considered Problem: - Strict DPs: append#(cons(x,xs),ys) -> c_1(append#(xs,ys)) append#(nil(),ys) -> c_2() attach#(x,cons(y,ys)) -> c_3(attach#(x,ys)) attach#(x,nil()) -> c_4() pairsp#(cons(x,xs)) -> c_5(append#(pairsp(xs),attach(x,xs)),pairsp#(xs),attach#(x,xs)) pairsp#(nil()) -> c_6() - Weak TRS: append(cons(x,xs),ys) -> cons(x,append(xs,ys)) append(nil(),ys) -> ys attach(x,cons(y,ys)) -> cons(pair(x,y),attach(x,ys)) attach(x,nil()) -> nil() pairsp(cons(x,xs)) -> append(pairsp(xs),attach(x,xs)) pairsp(nil()) -> nil() - Signature: {append/2,attach/2,pairsp/1,append#/2,attach#/2,pairsp#/1} / {cons/2,nil/0,pair/2,c_1/1,c_2/0,c_3/1,c_4/0 ,c_5/3,c_6/0} - Obligation: innermost runtime complexity wrt. defined symbols {append#,attach#,pairsp#} and constructors {cons,nil,pair} + Applied Processor: PredecessorEstimation {onSelection = all simple predecessor estimation selector} + Details: We estimate the number of application of {2,4,6} by application of Pre({2,4,6}) = {1,3,5}. Here rules are labelled as follows: 1: append#(cons(x,xs),ys) -> c_1(append#(xs,ys)) 2: append#(nil(),ys) -> c_2() 3: attach#(x,cons(y,ys)) -> c_3(attach#(x,ys)) 4: attach#(x,nil()) -> c_4() 5: pairsp#(cons(x,xs)) -> c_5(append#(pairsp(xs),attach(x,xs)),pairsp#(xs),attach#(x,xs)) 6: pairsp#(nil()) -> c_6() * Step 3: RemoveWeakSuffixes WORST_CASE(?,O(n^3)) + Considered Problem: - Strict DPs: append#(cons(x,xs),ys) -> c_1(append#(xs,ys)) attach#(x,cons(y,ys)) -> c_3(attach#(x,ys)) pairsp#(cons(x,xs)) -> c_5(append#(pairsp(xs),attach(x,xs)),pairsp#(xs),attach#(x,xs)) - Weak DPs: append#(nil(),ys) -> c_2() attach#(x,nil()) -> c_4() pairsp#(nil()) -> c_6() - Weak TRS: append(cons(x,xs),ys) -> cons(x,append(xs,ys)) append(nil(),ys) -> ys attach(x,cons(y,ys)) -> cons(pair(x,y),attach(x,ys)) attach(x,nil()) -> nil() pairsp(cons(x,xs)) -> append(pairsp(xs),attach(x,xs)) pairsp(nil()) -> nil() - Signature: {append/2,attach/2,pairsp/1,append#/2,attach#/2,pairsp#/1} / {cons/2,nil/0,pair/2,c_1/1,c_2/0,c_3/1,c_4/0 ,c_5/3,c_6/0} - Obligation: innermost runtime complexity wrt. defined symbols {append#,attach#,pairsp#} and constructors {cons,nil,pair} + Applied Processor: RemoveWeakSuffixes + Details: Consider the dependency graph 1:S:append#(cons(x,xs),ys) -> c_1(append#(xs,ys)) -->_1 append#(nil(),ys) -> c_2():4 -->_1 append#(cons(x,xs),ys) -> c_1(append#(xs,ys)):1 2:S:attach#(x,cons(y,ys)) -> c_3(attach#(x,ys)) -->_1 attach#(x,nil()) -> c_4():5 -->_1 attach#(x,cons(y,ys)) -> c_3(attach#(x,ys)):2 3:S:pairsp#(cons(x,xs)) -> c_5(append#(pairsp(xs),attach(x,xs)),pairsp#(xs),attach#(x,xs)) -->_2 pairsp#(nil()) -> c_6():6 -->_3 attach#(x,nil()) -> c_4():5 -->_1 append#(nil(),ys) -> c_2():4 -->_2 pairsp#(cons(x,xs)) -> c_5(append#(pairsp(xs),attach(x,xs)),pairsp#(xs),attach#(x,xs)):3 -->_3 attach#(x,cons(y,ys)) -> c_3(attach#(x,ys)):2 -->_1 append#(cons(x,xs),ys) -> c_1(append#(xs,ys)):1 4:W:append#(nil(),ys) -> c_2() 5:W:attach#(x,nil()) -> c_4() 6:W:pairsp#(nil()) -> c_6() The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed. 6: pairsp#(nil()) -> c_6() 5: attach#(x,nil()) -> c_4() 4: append#(nil(),ys) -> c_2() * Step 4: DecomposeDG WORST_CASE(?,O(n^3)) + Considered Problem: - Strict DPs: append#(cons(x,xs),ys) -> c_1(append#(xs,ys)) attach#(x,cons(y,ys)) -> c_3(attach#(x,ys)) pairsp#(cons(x,xs)) -> c_5(append#(pairsp(xs),attach(x,xs)),pairsp#(xs),attach#(x,xs)) - Weak TRS: append(cons(x,xs),ys) -> cons(x,append(xs,ys)) append(nil(),ys) -> ys attach(x,cons(y,ys)) -> cons(pair(x,y),attach(x,ys)) attach(x,nil()) -> nil() pairsp(cons(x,xs)) -> append(pairsp(xs),attach(x,xs)) pairsp(nil()) -> nil() - Signature: {append/2,attach/2,pairsp/1,append#/2,attach#/2,pairsp#/1} / {cons/2,nil/0,pair/2,c_1/1,c_2/0,c_3/1,c_4/0 ,c_5/3,c_6/0} - Obligation: innermost runtime complexity wrt. defined symbols {append#,attach#,pairsp#} and constructors {cons,nil,pair} + Applied Processor: DecomposeDG {onSelection = all below first cut in WDG, onUpper = Nothing, onLower = Nothing} + Details: We decompose the input problem according to the dependency graph into the upper component pairsp#(cons(x,xs)) -> c_5(append#(pairsp(xs),attach(x,xs)),pairsp#(xs),attach#(x,xs)) and a lower component append#(cons(x,xs),ys) -> c_1(append#(xs,ys)) attach#(x,cons(y,ys)) -> c_3(attach#(x,ys)) Further, following extension rules are added to the lower component. pairsp#(cons(x,xs)) -> append#(pairsp(xs),attach(x,xs)) pairsp#(cons(x,xs)) -> attach#(x,xs) pairsp#(cons(x,xs)) -> pairsp#(xs) ** Step 4.a:1: SimplifyRHS WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: pairsp#(cons(x,xs)) -> c_5(append#(pairsp(xs),attach(x,xs)),pairsp#(xs),attach#(x,xs)) - Weak TRS: append(cons(x,xs),ys) -> cons(x,append(xs,ys)) append(nil(),ys) -> ys attach(x,cons(y,ys)) -> cons(pair(x,y),attach(x,ys)) attach(x,nil()) -> nil() pairsp(cons(x,xs)) -> append(pairsp(xs),attach(x,xs)) pairsp(nil()) -> nil() - Signature: {append/2,attach/2,pairsp/1,append#/2,attach#/2,pairsp#/1} / {cons/2,nil/0,pair/2,c_1/1,c_2/0,c_3/1,c_4/0 ,c_5/3,c_6/0} - Obligation: innermost runtime complexity wrt. defined symbols {append#,attach#,pairsp#} and constructors {cons,nil,pair} + Applied Processor: SimplifyRHS + Details: Consider the dependency graph 1:S:pairsp#(cons(x,xs)) -> c_5(append#(pairsp(xs),attach(x,xs)),pairsp#(xs),attach#(x,xs)) -->_2 pairsp#(cons(x,xs)) -> c_5(append#(pairsp(xs),attach(x,xs)),pairsp#(xs),attach#(x,xs)):1 Due to missing edges in the depndency graph, the right-hand sides of following rules could be simplified: pairsp#(cons(x,xs)) -> c_5(pairsp#(xs)) ** Step 4.a:2: UsableRules WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: pairsp#(cons(x,xs)) -> c_5(pairsp#(xs)) - Weak TRS: append(cons(x,xs),ys) -> cons(x,append(xs,ys)) append(nil(),ys) -> ys attach(x,cons(y,ys)) -> cons(pair(x,y),attach(x,ys)) attach(x,nil()) -> nil() pairsp(cons(x,xs)) -> append(pairsp(xs),attach(x,xs)) pairsp(nil()) -> nil() - Signature: {append/2,attach/2,pairsp/1,append#/2,attach#/2,pairsp#/1} / {cons/2,nil/0,pair/2,c_1/1,c_2/0,c_3/1,c_4/0 ,c_5/1,c_6/0} - Obligation: innermost runtime complexity wrt. defined symbols {append#,attach#,pairsp#} and constructors {cons,nil,pair} + Applied Processor: UsableRules + Details: We replace rewrite rules by usable rules: pairsp#(cons(x,xs)) -> c_5(pairsp#(xs)) ** Step 4.a:3: NaturalMI WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: pairsp#(cons(x,xs)) -> c_5(pairsp#(xs)) - Signature: {append/2,attach/2,pairsp/1,append#/2,attach#/2,pairsp#/1} / {cons/2,nil/0,pair/2,c_1/1,c_2/0,c_3/1,c_4/0 ,c_5/1,c_6/0} - Obligation: innermost runtime complexity wrt. defined symbols {append#,attach#,pairsp#} and constructors {cons,nil,pair} + 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_5) = {1} Following symbols are considered usable: {append#,attach#,pairsp#} TcT has computed the following interpretation: p(append) = [1] x1 + [4] x2 + [8] p(attach) = [2] x1 + [8] x2 + [1] p(cons) = [1] x2 + [1] p(nil) = [0] p(pair) = [1] x2 + [0] p(pairsp) = [1] x1 + [8] p(append#) = [1] x1 + [1] p(attach#) = [8] p(pairsp#) = [8] x1 + [0] p(c_1) = [1] p(c_2) = [0] p(c_3) = [2] x1 + [1] p(c_4) = [2] p(c_5) = [1] x1 + [7] p(c_6) = [0] Following rules are strictly oriented: pairsp#(cons(x,xs)) = [8] xs + [8] > [8] xs + [7] = c_5(pairsp#(xs)) Following rules are (at-least) weakly oriented: ** Step 4.a:4: EmptyProcessor WORST_CASE(?,O(1)) + Considered Problem: - Weak DPs: pairsp#(cons(x,xs)) -> c_5(pairsp#(xs)) - Signature: {append/2,attach/2,pairsp/1,append#/2,attach#/2,pairsp#/1} / {cons/2,nil/0,pair/2,c_1/1,c_2/0,c_3/1,c_4/0 ,c_5/1,c_6/0} - Obligation: innermost runtime complexity wrt. defined symbols {append#,attach#,pairsp#} and constructors {cons,nil,pair} + Applied Processor: EmptyProcessor + Details: The problem is already closed. The intended complexity is O(1). ** Step 4.b:1: NaturalMI WORST_CASE(?,O(n^2)) + Considered Problem: - Strict DPs: append#(cons(x,xs),ys) -> c_1(append#(xs,ys)) attach#(x,cons(y,ys)) -> c_3(attach#(x,ys)) - Weak DPs: pairsp#(cons(x,xs)) -> append#(pairsp(xs),attach(x,xs)) pairsp#(cons(x,xs)) -> attach#(x,xs) pairsp#(cons(x,xs)) -> pairsp#(xs) - Weak TRS: append(cons(x,xs),ys) -> cons(x,append(xs,ys)) append(nil(),ys) -> ys attach(x,cons(y,ys)) -> cons(pair(x,y),attach(x,ys)) attach(x,nil()) -> nil() pairsp(cons(x,xs)) -> append(pairsp(xs),attach(x,xs)) pairsp(nil()) -> nil() - Signature: {append/2,attach/2,pairsp/1,append#/2,attach#/2,pairsp#/1} / {cons/2,nil/0,pair/2,c_1/1,c_2/0,c_3/1,c_4/0 ,c_5/3,c_6/0} - Obligation: innermost runtime complexity wrt. defined symbols {append#,attach#,pairsp#} and constructors {cons,nil,pair} + 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_1) = {1}, uargs(c_3) = {1} Following symbols are considered usable: {append#,attach#,pairsp#} TcT has computed the following interpretation: p(append) = [4] x1 + [0] p(attach) = [6] x1 + [0] p(cons) = [1] x1 + [1] x2 + [4] p(nil) = [5] p(pair) = [1] x2 + [10] p(pairsp) = [2] p(append#) = [0] p(attach#) = [4] x2 + [0] p(pairsp#) = [7] x1 + [2] p(c_1) = [1] x1 + [0] p(c_2) = [2] p(c_3) = [1] x1 + [14] p(c_4) = [1] p(c_5) = [1] x3 + [4] p(c_6) = [4] Following rules are strictly oriented: attach#(x,cons(y,ys)) = [4] y + [4] ys + [16] > [4] ys + [14] = c_3(attach#(x,ys)) Following rules are (at-least) weakly oriented: append#(cons(x,xs),ys) = [0] >= [0] = c_1(append#(xs,ys)) pairsp#(cons(x,xs)) = [7] x + [7] xs + [30] >= [0] = append#(pairsp(xs),attach(x,xs)) pairsp#(cons(x,xs)) = [7] x + [7] xs + [30] >= [4] xs + [0] = attach#(x,xs) pairsp#(cons(x,xs)) = [7] x + [7] xs + [30] >= [7] xs + [2] = pairsp#(xs) ** Step 4.b:2: NaturalPI WORST_CASE(?,O(n^2)) + Considered Problem: - Strict DPs: append#(cons(x,xs),ys) -> c_1(append#(xs,ys)) - Weak DPs: attach#(x,cons(y,ys)) -> c_3(attach#(x,ys)) pairsp#(cons(x,xs)) -> append#(pairsp(xs),attach(x,xs)) pairsp#(cons(x,xs)) -> attach#(x,xs) pairsp#(cons(x,xs)) -> pairsp#(xs) - Weak TRS: append(cons(x,xs),ys) -> cons(x,append(xs,ys)) append(nil(),ys) -> ys attach(x,cons(y,ys)) -> cons(pair(x,y),attach(x,ys)) attach(x,nil()) -> nil() pairsp(cons(x,xs)) -> append(pairsp(xs),attach(x,xs)) pairsp(nil()) -> nil() - Signature: {append/2,attach/2,pairsp/1,append#/2,attach#/2,pairsp#/1} / {cons/2,nil/0,pair/2,c_1/1,c_2/0,c_3/1,c_4/0 ,c_5/3,c_6/0} - Obligation: innermost runtime complexity wrt. defined symbols {append#,attach#,pairsp#} and constructors {cons,nil,pair} + Applied Processor: NaturalPI {shape = Mixed 2, restrict = Restrict, uargs = UArgs, urules = URules, selector = Just any strict-rules} + Details: We apply a polynomial interpretation of kind constructor-based(mixed(2)): The following argument positions are considered usable: uargs(c_1) = {1}, uargs(c_3) = {1} Following symbols are considered usable: {append,attach,pairsp,append#,attach#,pairsp#} TcT has computed the following interpretation: p(append) = 4 + x1 + 4*x2 p(attach) = 2*x2 p(cons) = 2 + x2 p(nil) = 0 p(pair) = 0 p(pairsp) = 2*x1 + 2*x1^2 p(append#) = 2 + x1 + x2 p(attach#) = x2 + x2^2 p(pairsp#) = 2*x1^2 p(c_1) = x1 p(c_2) = 0 p(c_3) = x1 p(c_4) = 1 p(c_5) = x1 + x2 p(c_6) = 1 Following rules are strictly oriented: append#(cons(x,xs),ys) = 4 + xs + ys > 2 + xs + ys = c_1(append#(xs,ys)) Following rules are (at-least) weakly oriented: attach#(x,cons(y,ys)) = 6 + 5*ys + ys^2 >= ys + ys^2 = c_3(attach#(x,ys)) pairsp#(cons(x,xs)) = 8 + 8*xs + 2*xs^2 >= 2 + 4*xs + 2*xs^2 = append#(pairsp(xs),attach(x,xs)) pairsp#(cons(x,xs)) = 8 + 8*xs + 2*xs^2 >= xs + xs^2 = attach#(x,xs) pairsp#(cons(x,xs)) = 8 + 8*xs + 2*xs^2 >= 2*xs^2 = pairsp#(xs) append(cons(x,xs),ys) = 6 + xs + 4*ys >= 6 + xs + 4*ys = cons(x,append(xs,ys)) append(nil(),ys) = 4 + 4*ys >= ys = ys attach(x,cons(y,ys)) = 4 + 2*ys >= 2 + 2*ys = cons(pair(x,y),attach(x,ys)) attach(x,nil()) = 0 >= 0 = nil() pairsp(cons(x,xs)) = 12 + 10*xs + 2*xs^2 >= 4 + 10*xs + 2*xs^2 = append(pairsp(xs),attach(x,xs)) pairsp(nil()) = 0 >= 0 = nil() ** Step 4.b:3: EmptyProcessor WORST_CASE(?,O(1)) + Considered Problem: - Weak DPs: append#(cons(x,xs),ys) -> c_1(append#(xs,ys)) attach#(x,cons(y,ys)) -> c_3(attach#(x,ys)) pairsp#(cons(x,xs)) -> append#(pairsp(xs),attach(x,xs)) pairsp#(cons(x,xs)) -> attach#(x,xs) pairsp#(cons(x,xs)) -> pairsp#(xs) - Weak TRS: append(cons(x,xs),ys) -> cons(x,append(xs,ys)) append(nil(),ys) -> ys attach(x,cons(y,ys)) -> cons(pair(x,y),attach(x,ys)) attach(x,nil()) -> nil() pairsp(cons(x,xs)) -> append(pairsp(xs),attach(x,xs)) pairsp(nil()) -> nil() - Signature: {append/2,attach/2,pairsp/1,append#/2,attach#/2,pairsp#/1} / {cons/2,nil/0,pair/2,c_1/1,c_2/0,c_3/1,c_4/0 ,c_5/3,c_6/0} - Obligation: innermost runtime complexity wrt. defined symbols {append#,attach#,pairsp#} and constructors {cons,nil,pair} + Applied Processor: EmptyProcessor + Details: The problem is already closed. The intended complexity is O(1). WORST_CASE(?,O(n^3))