WORST_CASE(?,O(n^1)) * Step 1: DependencyPairs WORST_CASE(?,O(n^1)) + Considered Problem: - Strict TRS: append(l1,l2) -> ifappend(l1,l2,l1) hd(cons(x,l)) -> x ifappend(l1,l2,cons(x,l)) -> cons(x,append(l,l2)) ifappend(l1,l2,nil()) -> l2 is_empty(cons(x,l)) -> false() is_empty(nil()) -> true() tl(cons(x,l)) -> l - Signature: {append/2,hd/1,ifappend/3,is_empty/1,tl/1} / {cons/2,false/0,nil/0,true/0} - Obligation: innermost runtime complexity wrt. defined symbols {append,hd,ifappend,is_empty,tl} and constructors {cons ,false,nil,true} + Applied Processor: DependencyPairs {dpKind_ = WIDP} + Details: We add the following weak innermost dependency pairs: Strict DPs append#(l1,l2) -> c_1(ifappend#(l1,l2,l1)) hd#(cons(x,l)) -> c_2() ifappend#(l1,l2,cons(x,l)) -> c_3(append#(l,l2)) ifappend#(l1,l2,nil()) -> c_4() is_empty#(cons(x,l)) -> c_5() is_empty#(nil()) -> c_6() tl#(cons(x,l)) -> c_7() Weak DPs and mark the set of starting terms. * Step 2: UsableRules WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: append#(l1,l2) -> c_1(ifappend#(l1,l2,l1)) hd#(cons(x,l)) -> c_2() ifappend#(l1,l2,cons(x,l)) -> c_3(append#(l,l2)) ifappend#(l1,l2,nil()) -> c_4() is_empty#(cons(x,l)) -> c_5() is_empty#(nil()) -> c_6() tl#(cons(x,l)) -> c_7() - Strict TRS: append(l1,l2) -> ifappend(l1,l2,l1) hd(cons(x,l)) -> x ifappend(l1,l2,cons(x,l)) -> cons(x,append(l,l2)) ifappend(l1,l2,nil()) -> l2 is_empty(cons(x,l)) -> false() is_empty(nil()) -> true() tl(cons(x,l)) -> l - Signature: {append/2,hd/1,ifappend/3,is_empty/1,tl/1,append#/2,hd#/1,ifappend#/3,is_empty#/1,tl#/1} / {cons/2,false/0 ,nil/0,true/0,c_1/1,c_2/0,c_3/1,c_4/0,c_5/0,c_6/0,c_7/0} - Obligation: innermost runtime complexity wrt. defined symbols {append#,hd#,ifappend#,is_empty# ,tl#} and constructors {cons,false,nil,true} + Applied Processor: UsableRules + Details: We replace rewrite rules by usable rules: append#(l1,l2) -> c_1(ifappend#(l1,l2,l1)) hd#(cons(x,l)) -> c_2() ifappend#(l1,l2,cons(x,l)) -> c_3(append#(l,l2)) ifappend#(l1,l2,nil()) -> c_4() is_empty#(cons(x,l)) -> c_5() is_empty#(nil()) -> c_6() tl#(cons(x,l)) -> c_7() * Step 3: PredecessorEstimation WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: append#(l1,l2) -> c_1(ifappend#(l1,l2,l1)) hd#(cons(x,l)) -> c_2() ifappend#(l1,l2,cons(x,l)) -> c_3(append#(l,l2)) ifappend#(l1,l2,nil()) -> c_4() is_empty#(cons(x,l)) -> c_5() is_empty#(nil()) -> c_6() tl#(cons(x,l)) -> c_7() - Signature: {append/2,hd/1,ifappend/3,is_empty/1,tl/1,append#/2,hd#/1,ifappend#/3,is_empty#/1,tl#/1} / {cons/2,false/0 ,nil/0,true/0,c_1/1,c_2/0,c_3/1,c_4/0,c_5/0,c_6/0,c_7/0} - Obligation: innermost runtime complexity wrt. defined symbols {append#,hd#,ifappend#,is_empty# ,tl#} and constructors {cons,false,nil,true} + Applied Processor: PredecessorEstimation {onSelection = all simple predecessor estimation selector} + Details: We estimate the number of application of {2,4,5,6,7} by application of Pre({2,4,5,6,7}) = {1}. Here rules are labelled as follows: 1: append#(l1,l2) -> c_1(ifappend#(l1,l2,l1)) 2: hd#(cons(x,l)) -> c_2() 3: ifappend#(l1,l2,cons(x,l)) -> c_3(append#(l,l2)) 4: ifappend#(l1,l2,nil()) -> c_4() 5: is_empty#(cons(x,l)) -> c_5() 6: is_empty#(nil()) -> c_6() 7: tl#(cons(x,l)) -> c_7() * Step 4: RemoveWeakSuffixes WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: append#(l1,l2) -> c_1(ifappend#(l1,l2,l1)) ifappend#(l1,l2,cons(x,l)) -> c_3(append#(l,l2)) - Weak DPs: hd#(cons(x,l)) -> c_2() ifappend#(l1,l2,nil()) -> c_4() is_empty#(cons(x,l)) -> c_5() is_empty#(nil()) -> c_6() tl#(cons(x,l)) -> c_7() - Signature: {append/2,hd/1,ifappend/3,is_empty/1,tl/1,append#/2,hd#/1,ifappend#/3,is_empty#/1,tl#/1} / {cons/2,false/0 ,nil/0,true/0,c_1/1,c_2/0,c_3/1,c_4/0,c_5/0,c_6/0,c_7/0} - Obligation: innermost runtime complexity wrt. defined symbols {append#,hd#,ifappend#,is_empty# ,tl#} and constructors {cons,false,nil,true} + Applied Processor: RemoveWeakSuffixes + Details: Consider the dependency graph 1:S:append#(l1,l2) -> c_1(ifappend#(l1,l2,l1)) -->_1 ifappend#(l1,l2,cons(x,l)) -> c_3(append#(l,l2)):2 -->_1 ifappend#(l1,l2,nil()) -> c_4():4 2:S:ifappend#(l1,l2,cons(x,l)) -> c_3(append#(l,l2)) -->_1 append#(l1,l2) -> c_1(ifappend#(l1,l2,l1)):1 3:W:hd#(cons(x,l)) -> c_2() 4:W:ifappend#(l1,l2,nil()) -> c_4() 5:W:is_empty#(cons(x,l)) -> c_5() 6:W:is_empty#(nil()) -> c_6() 7:W:tl#(cons(x,l)) -> c_7() The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed. 7: tl#(cons(x,l)) -> c_7() 6: is_empty#(nil()) -> c_6() 5: is_empty#(cons(x,l)) -> c_5() 3: hd#(cons(x,l)) -> c_2() 4: ifappend#(l1,l2,nil()) -> c_4() * Step 5: PredecessorEstimationCP WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: append#(l1,l2) -> c_1(ifappend#(l1,l2,l1)) ifappend#(l1,l2,cons(x,l)) -> c_3(append#(l,l2)) - Signature: {append/2,hd/1,ifappend/3,is_empty/1,tl/1,append#/2,hd#/1,ifappend#/3,is_empty#/1,tl#/1} / {cons/2,false/0 ,nil/0,true/0,c_1/1,c_2/0,c_3/1,c_4/0,c_5/0,c_6/0,c_7/0} - Obligation: innermost runtime complexity wrt. defined symbols {append#,hd#,ifappend#,is_empty# ,tl#} and constructors {cons,false,nil,true} + 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: 2: ifappend#(l1,l2,cons(x,l)) -> c_3(append#(l,l2)) Consider the set of all dependency pairs 1: append#(l1,l2) -> c_1(ifappend#(l1,l2,l1)) 2: ifappend#(l1,l2,cons(x,l)) -> c_3(append#(l,l2)) Processor NaturalMI {miDimension = 1, miDegree = 1, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Nothing}induces the complexity certificateTIME (?,O(n^1)) SPACE(?,?)on application of the dependency pairs {2} These cover all (indirect) predecessors of dependency pairs {1,2} their number of applications is equally bounded. The dependency pairs are shifted into the weak component. ** Step 5.a:1: NaturalMI WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: append#(l1,l2) -> c_1(ifappend#(l1,l2,l1)) ifappend#(l1,l2,cons(x,l)) -> c_3(append#(l,l2)) - Signature: {append/2,hd/1,ifappend/3,is_empty/1,tl/1,append#/2,hd#/1,ifappend#/3,is_empty#/1,tl#/1} / {cons/2,false/0 ,nil/0,true/0,c_1/1,c_2/0,c_3/1,c_4/0,c_5/0,c_6/0,c_7/0} - Obligation: innermost runtime complexity wrt. defined symbols {append#,hd#,ifappend#,is_empty# ,tl#} and constructors {cons,false,nil,true} + 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_1) = {1}, uargs(c_3) = {1} Following symbols are considered usable: {append#,hd#,ifappend#,is_empty#,tl#} TcT has computed the following interpretation: p(append) = [1] p(cons) = [1] x2 + [2] p(false) = [1] p(hd) = [0] p(ifappend) = [1] x1 + [2] x2 + [4] x3 + [8] p(is_empty) = [1] p(nil) = [2] p(tl) = [0] p(true) = [2] p(append#) = [15] x1 + [4] x2 + [0] p(hd#) = [1] p(ifappend#) = [4] x2 + [15] x3 + [0] p(is_empty#) = [1] p(tl#) = [1] x1 + [2] p(c_1) = [1] x1 + [0] p(c_2) = [2] p(c_3) = [1] x1 + [12] p(c_4) = [0] p(c_5) = [1] p(c_6) = [1] p(c_7) = [1] Following rules are strictly oriented: ifappend#(l1,l2,cons(x,l)) = [15] l + [4] l2 + [30] > [15] l + [4] l2 + [12] = c_3(append#(l,l2)) Following rules are (at-least) weakly oriented: append#(l1,l2) = [15] l1 + [4] l2 + [0] >= [15] l1 + [4] l2 + [0] = c_1(ifappend#(l1,l2,l1)) ** Step 5.a:2: Assumption WORST_CASE(?,O(1)) + Considered Problem: - Strict DPs: append#(l1,l2) -> c_1(ifappend#(l1,l2,l1)) - Weak DPs: ifappend#(l1,l2,cons(x,l)) -> c_3(append#(l,l2)) - Signature: {append/2,hd/1,ifappend/3,is_empty/1,tl/1,append#/2,hd#/1,ifappend#/3,is_empty#/1,tl#/1} / {cons/2,false/0 ,nil/0,true/0,c_1/1,c_2/0,c_3/1,c_4/0,c_5/0,c_6/0,c_7/0} - Obligation: innermost runtime complexity wrt. defined symbols {append#,hd#,ifappend#,is_empty# ,tl#} and constructors {cons,false,nil,true} + Applied Processor: Assumption {assumed = Certificate {spaceUB = Unknown, spaceLB = Unknown, timeUB = Poly (Just 0), timeLB = Unknown}} + Details: () ** Step 5.b:1: RemoveWeakSuffixes WORST_CASE(?,O(1)) + Considered Problem: - Weak DPs: append#(l1,l2) -> c_1(ifappend#(l1,l2,l1)) ifappend#(l1,l2,cons(x,l)) -> c_3(append#(l,l2)) - Signature: {append/2,hd/1,ifappend/3,is_empty/1,tl/1,append#/2,hd#/1,ifappend#/3,is_empty#/1,tl#/1} / {cons/2,false/0 ,nil/0,true/0,c_1/1,c_2/0,c_3/1,c_4/0,c_5/0,c_6/0,c_7/0} - Obligation: innermost runtime complexity wrt. defined symbols {append#,hd#,ifappend#,is_empty# ,tl#} and constructors {cons,false,nil,true} + Applied Processor: RemoveWeakSuffixes + Details: Consider the dependency graph 1:W:append#(l1,l2) -> c_1(ifappend#(l1,l2,l1)) -->_1 ifappend#(l1,l2,cons(x,l)) -> c_3(append#(l,l2)):2 2:W:ifappend#(l1,l2,cons(x,l)) -> c_3(append#(l,l2)) -->_1 append#(l1,l2) -> c_1(ifappend#(l1,l2,l1)):1 The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed. 1: append#(l1,l2) -> c_1(ifappend#(l1,l2,l1)) 2: ifappend#(l1,l2,cons(x,l)) -> c_3(append#(l,l2)) ** Step 5.b:2: EmptyProcessor WORST_CASE(?,O(1)) + Considered Problem: - Signature: {append/2,hd/1,ifappend/3,is_empty/1,tl/1,append#/2,hd#/1,ifappend#/3,is_empty#/1,tl#/1} / {cons/2,false/0 ,nil/0,true/0,c_1/1,c_2/0,c_3/1,c_4/0,c_5/0,c_6/0,c_7/0} - Obligation: innermost runtime complexity wrt. defined symbols {append#,hd#,ifappend#,is_empty# ,tl#} and constructors {cons,false,nil,true} + Applied Processor: EmptyProcessor + Details: The problem is already closed. The intended complexity is O(1). WORST_CASE(?,O(n^1))