WORST_CASE(?,O(n^3)) * Step 1: DependencyPairs WORST_CASE(?,O(n^3)) + Considered Problem: - Strict TRS: add(0(),X) -> X add(s(X),Y) -> s(add(X,Y)) dbl(0()) -> 0() dbl(s(X)) -> s(s(dbl(X))) first(0(),X) -> nil() first(s(X),cons(Y)) -> cons(Y) sqr(0()) -> 0() sqr(s(X)) -> s(add(sqr(X),dbl(X))) terms(N) -> cons(recip(sqr(N))) - Signature: {add/2,dbl/1,first/2,sqr/1,terms/1} / {0/0,cons/1,nil/0,recip/1,s/1} - Obligation: innermost runtime complexity wrt. defined symbols {add,dbl,first,sqr,terms} and constructors {0,cons,nil ,recip,s} + Applied Processor: DependencyPairs {dpKind_ = DT} + Details: We add the following dependency tuples: Strict DPs add#(0(),X) -> c_1() add#(s(X),Y) -> c_2(add#(X,Y)) dbl#(0()) -> c_3() dbl#(s(X)) -> c_4(dbl#(X)) first#(0(),X) -> c_5() first#(s(X),cons(Y)) -> c_6() sqr#(0()) -> c_7() sqr#(s(X)) -> c_8(add#(sqr(X),dbl(X)),sqr#(X),dbl#(X)) terms#(N) -> c_9(sqr#(N)) Weak DPs and mark the set of starting terms. * Step 2: UsableRules WORST_CASE(?,O(n^3)) + Considered Problem: - Strict DPs: add#(0(),X) -> c_1() add#(s(X),Y) -> c_2(add#(X,Y)) dbl#(0()) -> c_3() dbl#(s(X)) -> c_4(dbl#(X)) first#(0(),X) -> c_5() first#(s(X),cons(Y)) -> c_6() sqr#(0()) -> c_7() sqr#(s(X)) -> c_8(add#(sqr(X),dbl(X)),sqr#(X),dbl#(X)) terms#(N) -> c_9(sqr#(N)) - Weak TRS: add(0(),X) -> X add(s(X),Y) -> s(add(X,Y)) dbl(0()) -> 0() dbl(s(X)) -> s(s(dbl(X))) first(0(),X) -> nil() first(s(X),cons(Y)) -> cons(Y) sqr(0()) -> 0() sqr(s(X)) -> s(add(sqr(X),dbl(X))) terms(N) -> cons(recip(sqr(N))) - Signature: {add/2,dbl/1,first/2,sqr/1,terms/1,add#/2,dbl#/1,first#/2,sqr#/1,terms#/1} / {0/0,cons/1,nil/0,recip/1,s/1 ,c_1/0,c_2/1,c_3/0,c_4/1,c_5/0,c_6/0,c_7/0,c_8/3,c_9/1} - Obligation: innermost runtime complexity wrt. defined symbols {add#,dbl#,first#,sqr#,terms#} and constructors {0,cons ,nil,recip,s} + Applied Processor: UsableRules + Details: We replace rewrite rules by usable rules: add(0(),X) -> X add(s(X),Y) -> s(add(X,Y)) dbl(0()) -> 0() dbl(s(X)) -> s(s(dbl(X))) sqr(0()) -> 0() sqr(s(X)) -> s(add(sqr(X),dbl(X))) add#(0(),X) -> c_1() add#(s(X),Y) -> c_2(add#(X,Y)) dbl#(0()) -> c_3() dbl#(s(X)) -> c_4(dbl#(X)) first#(0(),X) -> c_5() first#(s(X),cons(Y)) -> c_6() sqr#(0()) -> c_7() sqr#(s(X)) -> c_8(add#(sqr(X),dbl(X)),sqr#(X),dbl#(X)) terms#(N) -> c_9(sqr#(N)) * Step 3: PredecessorEstimation WORST_CASE(?,O(n^3)) + Considered Problem: - Strict DPs: add#(0(),X) -> c_1() add#(s(X),Y) -> c_2(add#(X,Y)) dbl#(0()) -> c_3() dbl#(s(X)) -> c_4(dbl#(X)) first#(0(),X) -> c_5() first#(s(X),cons(Y)) -> c_6() sqr#(0()) -> c_7() sqr#(s(X)) -> c_8(add#(sqr(X),dbl(X)),sqr#(X),dbl#(X)) terms#(N) -> c_9(sqr#(N)) - Weak TRS: add(0(),X) -> X add(s(X),Y) -> s(add(X,Y)) dbl(0()) -> 0() dbl(s(X)) -> s(s(dbl(X))) sqr(0()) -> 0() sqr(s(X)) -> s(add(sqr(X),dbl(X))) - Signature: {add/2,dbl/1,first/2,sqr/1,terms/1,add#/2,dbl#/1,first#/2,sqr#/1,terms#/1} / {0/0,cons/1,nil/0,recip/1,s/1 ,c_1/0,c_2/1,c_3/0,c_4/1,c_5/0,c_6/0,c_7/0,c_8/3,c_9/1} - Obligation: innermost runtime complexity wrt. defined symbols {add#,dbl#,first#,sqr#,terms#} and constructors {0,cons ,nil,recip,s} + Applied Processor: PredecessorEstimation {onSelection = all simple predecessor estimation selector} + Details: We estimate the number of application of {1,3,5,6,7} by application of Pre({1,3,5,6,7}) = {2,4,8,9}. Here rules are labelled as follows: 1: add#(0(),X) -> c_1() 2: add#(s(X),Y) -> c_2(add#(X,Y)) 3: dbl#(0()) -> c_3() 4: dbl#(s(X)) -> c_4(dbl#(X)) 5: first#(0(),X) -> c_5() 6: first#(s(X),cons(Y)) -> c_6() 7: sqr#(0()) -> c_7() 8: sqr#(s(X)) -> c_8(add#(sqr(X),dbl(X)),sqr#(X),dbl#(X)) 9: terms#(N) -> c_9(sqr#(N)) * Step 4: RemoveWeakSuffixes WORST_CASE(?,O(n^3)) + Considered Problem: - Strict DPs: add#(s(X),Y) -> c_2(add#(X,Y)) dbl#(s(X)) -> c_4(dbl#(X)) sqr#(s(X)) -> c_8(add#(sqr(X),dbl(X)),sqr#(X),dbl#(X)) terms#(N) -> c_9(sqr#(N)) - Weak DPs: add#(0(),X) -> c_1() dbl#(0()) -> c_3() first#(0(),X) -> c_5() first#(s(X),cons(Y)) -> c_6() sqr#(0()) -> c_7() - Weak TRS: add(0(),X) -> X add(s(X),Y) -> s(add(X,Y)) dbl(0()) -> 0() dbl(s(X)) -> s(s(dbl(X))) sqr(0()) -> 0() sqr(s(X)) -> s(add(sqr(X),dbl(X))) - Signature: {add/2,dbl/1,first/2,sqr/1,terms/1,add#/2,dbl#/1,first#/2,sqr#/1,terms#/1} / {0/0,cons/1,nil/0,recip/1,s/1 ,c_1/0,c_2/1,c_3/0,c_4/1,c_5/0,c_6/0,c_7/0,c_8/3,c_9/1} - Obligation: innermost runtime complexity wrt. defined symbols {add#,dbl#,first#,sqr#,terms#} and constructors {0,cons ,nil,recip,s} + Applied Processor: RemoveWeakSuffixes + Details: Consider the dependency graph 1:S:add#(s(X),Y) -> c_2(add#(X,Y)) -->_1 add#(0(),X) -> c_1():5 -->_1 add#(s(X),Y) -> c_2(add#(X,Y)):1 2:S:dbl#(s(X)) -> c_4(dbl#(X)) -->_1 dbl#(0()) -> c_3():6 -->_1 dbl#(s(X)) -> c_4(dbl#(X)):2 3:S:sqr#(s(X)) -> c_8(add#(sqr(X),dbl(X)),sqr#(X),dbl#(X)) -->_2 sqr#(0()) -> c_7():9 -->_3 dbl#(0()) -> c_3():6 -->_1 add#(0(),X) -> c_1():5 -->_2 sqr#(s(X)) -> c_8(add#(sqr(X),dbl(X)),sqr#(X),dbl#(X)):3 -->_3 dbl#(s(X)) -> c_4(dbl#(X)):2 -->_1 add#(s(X),Y) -> c_2(add#(X,Y)):1 4:S:terms#(N) -> c_9(sqr#(N)) -->_1 sqr#(0()) -> c_7():9 -->_1 sqr#(s(X)) -> c_8(add#(sqr(X),dbl(X)),sqr#(X),dbl#(X)):3 5:W:add#(0(),X) -> c_1() 6:W:dbl#(0()) -> c_3() 7:W:first#(0(),X) -> c_5() 8:W:first#(s(X),cons(Y)) -> c_6() 9:W:sqr#(0()) -> c_7() The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed. 8: first#(s(X),cons(Y)) -> c_6() 7: first#(0(),X) -> c_5() 9: sqr#(0()) -> c_7() 6: dbl#(0()) -> c_3() 5: add#(0(),X) -> c_1() * Step 5: RemoveHeads WORST_CASE(?,O(n^3)) + Considered Problem: - Strict DPs: add#(s(X),Y) -> c_2(add#(X,Y)) dbl#(s(X)) -> c_4(dbl#(X)) sqr#(s(X)) -> c_8(add#(sqr(X),dbl(X)),sqr#(X),dbl#(X)) terms#(N) -> c_9(sqr#(N)) - Weak TRS: add(0(),X) -> X add(s(X),Y) -> s(add(X,Y)) dbl(0()) -> 0() dbl(s(X)) -> s(s(dbl(X))) sqr(0()) -> 0() sqr(s(X)) -> s(add(sqr(X),dbl(X))) - Signature: {add/2,dbl/1,first/2,sqr/1,terms/1,add#/2,dbl#/1,first#/2,sqr#/1,terms#/1} / {0/0,cons/1,nil/0,recip/1,s/1 ,c_1/0,c_2/1,c_3/0,c_4/1,c_5/0,c_6/0,c_7/0,c_8/3,c_9/1} - Obligation: innermost runtime complexity wrt. defined symbols {add#,dbl#,first#,sqr#,terms#} and constructors {0,cons ,nil,recip,s} + Applied Processor: RemoveHeads + Details: Consider the dependency graph 1:S:add#(s(X),Y) -> c_2(add#(X,Y)) -->_1 add#(s(X),Y) -> c_2(add#(X,Y)):1 2:S:dbl#(s(X)) -> c_4(dbl#(X)) -->_1 dbl#(s(X)) -> c_4(dbl#(X)):2 3:S:sqr#(s(X)) -> c_8(add#(sqr(X),dbl(X)),sqr#(X),dbl#(X)) -->_2 sqr#(s(X)) -> c_8(add#(sqr(X),dbl(X)),sqr#(X),dbl#(X)):3 -->_3 dbl#(s(X)) -> c_4(dbl#(X)):2 -->_1 add#(s(X),Y) -> c_2(add#(X,Y)):1 4:S:terms#(N) -> c_9(sqr#(N)) -->_1 sqr#(s(X)) -> c_8(add#(sqr(X),dbl(X)),sqr#(X),dbl#(X)):3 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). [(4,terms#(N) -> c_9(sqr#(N)))] * Step 6: Decompose WORST_CASE(?,O(n^3)) + Considered Problem: - Strict DPs: add#(s(X),Y) -> c_2(add#(X,Y)) dbl#(s(X)) -> c_4(dbl#(X)) sqr#(s(X)) -> c_8(add#(sqr(X),dbl(X)),sqr#(X),dbl#(X)) - Weak TRS: add(0(),X) -> X add(s(X),Y) -> s(add(X,Y)) dbl(0()) -> 0() dbl(s(X)) -> s(s(dbl(X))) sqr(0()) -> 0() sqr(s(X)) -> s(add(sqr(X),dbl(X))) - Signature: {add/2,dbl/1,first/2,sqr/1,terms/1,add#/2,dbl#/1,first#/2,sqr#/1,terms#/1} / {0/0,cons/1,nil/0,recip/1,s/1 ,c_1/0,c_2/1,c_3/0,c_4/1,c_5/0,c_6/0,c_7/0,c_8/3,c_9/1} - Obligation: innermost runtime complexity wrt. defined symbols {add#,dbl#,first#,sqr#,terms#} and constructors {0,cons ,nil,recip,s} + Applied Processor: Decompose {onSelection = all cycle independent sub-graph, withBound = RelativeAdd} + Details: We analyse the complexity of following sub-problems (R) and (S). Problem (S) is obtained from the input problem by shifting strict rules from (R) into the weak component. Problem (R) - Strict DPs: add#(s(X),Y) -> c_2(add#(X,Y)) - Weak DPs: dbl#(s(X)) -> c_4(dbl#(X)) sqr#(s(X)) -> c_8(add#(sqr(X),dbl(X)),sqr#(X),dbl#(X)) - Weak TRS: add(0(),X) -> X add(s(X),Y) -> s(add(X,Y)) dbl(0()) -> 0() dbl(s(X)) -> s(s(dbl(X))) sqr(0()) -> 0() sqr(s(X)) -> s(add(sqr(X),dbl(X))) - Signature: {add/2,dbl/1,first/2,sqr/1,terms/1,add#/2,dbl#/1,first#/2,sqr#/1,terms#/1} / {0/0,cons/1,nil/0,recip/1,s/1 ,c_1/0,c_2/1,c_3/0,c_4/1,c_5/0,c_6/0,c_7/0,c_8/3,c_9/1} - Obligation: innermost runtime complexity wrt. defined symbols {add#,dbl#,first#,sqr#,terms#} and constructors {0,cons ,nil,recip,s} Problem (S) - Strict DPs: dbl#(s(X)) -> c_4(dbl#(X)) sqr#(s(X)) -> c_8(add#(sqr(X),dbl(X)),sqr#(X),dbl#(X)) - Weak DPs: add#(s(X),Y) -> c_2(add#(X,Y)) - Weak TRS: add(0(),X) -> X add(s(X),Y) -> s(add(X,Y)) dbl(0()) -> 0() dbl(s(X)) -> s(s(dbl(X))) sqr(0()) -> 0() sqr(s(X)) -> s(add(sqr(X),dbl(X))) - Signature: {add/2,dbl/1,first/2,sqr/1,terms/1,add#/2,dbl#/1,first#/2,sqr#/1,terms#/1} / {0/0,cons/1,nil/0,recip/1,s/1 ,c_1/0,c_2/1,c_3/0,c_4/1,c_5/0,c_6/0,c_7/0,c_8/3,c_9/1} - Obligation: innermost runtime complexity wrt. defined symbols {add#,dbl#,first#,sqr#,terms#} and constructors {0,cons ,nil,recip,s} ** Step 6.a:1: RemoveWeakSuffixes WORST_CASE(?,O(n^3)) + Considered Problem: - Strict DPs: add#(s(X),Y) -> c_2(add#(X,Y)) - Weak DPs: dbl#(s(X)) -> c_4(dbl#(X)) sqr#(s(X)) -> c_8(add#(sqr(X),dbl(X)),sqr#(X),dbl#(X)) - Weak TRS: add(0(),X) -> X add(s(X),Y) -> s(add(X,Y)) dbl(0()) -> 0() dbl(s(X)) -> s(s(dbl(X))) sqr(0()) -> 0() sqr(s(X)) -> s(add(sqr(X),dbl(X))) - Signature: {add/2,dbl/1,first/2,sqr/1,terms/1,add#/2,dbl#/1,first#/2,sqr#/1,terms#/1} / {0/0,cons/1,nil/0,recip/1,s/1 ,c_1/0,c_2/1,c_3/0,c_4/1,c_5/0,c_6/0,c_7/0,c_8/3,c_9/1} - Obligation: innermost runtime complexity wrt. defined symbols {add#,dbl#,first#,sqr#,terms#} and constructors {0,cons ,nil,recip,s} + Applied Processor: RemoveWeakSuffixes + Details: Consider the dependency graph 1:S:add#(s(X),Y) -> c_2(add#(X,Y)) -->_1 add#(s(X),Y) -> c_2(add#(X,Y)):1 2:W:dbl#(s(X)) -> c_4(dbl#(X)) -->_1 dbl#(s(X)) -> c_4(dbl#(X)):2 3:W:sqr#(s(X)) -> c_8(add#(sqr(X),dbl(X)),sqr#(X),dbl#(X)) -->_1 add#(s(X),Y) -> c_2(add#(X,Y)):1 -->_3 dbl#(s(X)) -> c_4(dbl#(X)):2 -->_2 sqr#(s(X)) -> c_8(add#(sqr(X),dbl(X)),sqr#(X),dbl#(X)):3 The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed. 2: dbl#(s(X)) -> c_4(dbl#(X)) ** Step 6.a:2: SimplifyRHS WORST_CASE(?,O(n^3)) + Considered Problem: - Strict DPs: add#(s(X),Y) -> c_2(add#(X,Y)) - Weak DPs: sqr#(s(X)) -> c_8(add#(sqr(X),dbl(X)),sqr#(X),dbl#(X)) - Weak TRS: add(0(),X) -> X add(s(X),Y) -> s(add(X,Y)) dbl(0()) -> 0() dbl(s(X)) -> s(s(dbl(X))) sqr(0()) -> 0() sqr(s(X)) -> s(add(sqr(X),dbl(X))) - Signature: {add/2,dbl/1,first/2,sqr/1,terms/1,add#/2,dbl#/1,first#/2,sqr#/1,terms#/1} / {0/0,cons/1,nil/0,recip/1,s/1 ,c_1/0,c_2/1,c_3/0,c_4/1,c_5/0,c_6/0,c_7/0,c_8/3,c_9/1} - Obligation: innermost runtime complexity wrt. defined symbols {add#,dbl#,first#,sqr#,terms#} and constructors {0,cons ,nil,recip,s} + Applied Processor: SimplifyRHS + Details: Consider the dependency graph 1:S:add#(s(X),Y) -> c_2(add#(X,Y)) -->_1 add#(s(X),Y) -> c_2(add#(X,Y)):1 3:W:sqr#(s(X)) -> c_8(add#(sqr(X),dbl(X)),sqr#(X),dbl#(X)) -->_1 add#(s(X),Y) -> c_2(add#(X,Y)):1 -->_2 sqr#(s(X)) -> c_8(add#(sqr(X),dbl(X)),sqr#(X),dbl#(X)):3 Due to missing edges in the depndency graph, the right-hand sides of following rules could be simplified: sqr#(s(X)) -> c_8(add#(sqr(X),dbl(X)),sqr#(X)) ** Step 6.a:3: DecomposeDG WORST_CASE(?,O(n^3)) + Considered Problem: - Strict DPs: add#(s(X),Y) -> c_2(add#(X,Y)) - Weak DPs: sqr#(s(X)) -> c_8(add#(sqr(X),dbl(X)),sqr#(X)) - Weak TRS: add(0(),X) -> X add(s(X),Y) -> s(add(X,Y)) dbl(0()) -> 0() dbl(s(X)) -> s(s(dbl(X))) sqr(0()) -> 0() sqr(s(X)) -> s(add(sqr(X),dbl(X))) - Signature: {add/2,dbl/1,first/2,sqr/1,terms/1,add#/2,dbl#/1,first#/2,sqr#/1,terms#/1} / {0/0,cons/1,nil/0,recip/1,s/1 ,c_1/0,c_2/1,c_3/0,c_4/1,c_5/0,c_6/0,c_7/0,c_8/2,c_9/1} - Obligation: innermost runtime complexity wrt. defined symbols {add#,dbl#,first#,sqr#,terms#} and constructors {0,cons ,nil,recip,s} + Applied Processor: DecomposeDG {onSelection = all below first cut in WDG, onUpper = Just someStrategy, onLower = Nothing} + Details: We decompose the input problem according to the dependency graph into the upper component sqr#(s(X)) -> c_8(add#(sqr(X),dbl(X)),sqr#(X)) and a lower component add#(s(X),Y) -> c_2(add#(X,Y)) Further, following extension rules are added to the lower component. sqr#(s(X)) -> add#(sqr(X),dbl(X)) sqr#(s(X)) -> sqr#(X) *** Step 6.a:3.a:1: PredecessorEstimationCP WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: sqr#(s(X)) -> c_8(add#(sqr(X),dbl(X)),sqr#(X)) - Weak TRS: add(0(),X) -> X add(s(X),Y) -> s(add(X,Y)) dbl(0()) -> 0() dbl(s(X)) -> s(s(dbl(X))) sqr(0()) -> 0() sqr(s(X)) -> s(add(sqr(X),dbl(X))) - Signature: {add/2,dbl/1,first/2,sqr/1,terms/1,add#/2,dbl#/1,first#/2,sqr#/1,terms#/1} / {0/0,cons/1,nil/0,recip/1,s/1 ,c_1/0,c_2/1,c_3/0,c_4/1,c_5/0,c_6/0,c_7/0,c_8/2,c_9/1} - Obligation: innermost runtime complexity wrt. defined symbols {add#,dbl#,first#,sqr#,terms#} and constructors {0,cons ,nil,recip,s} + 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: sqr#(s(X)) -> c_8(add#(sqr(X),dbl(X)),sqr#(X)) The strictly oriented rules are moved into the weak component. **** Step 6.a:3.a:1.a:1: NaturalMI WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: sqr#(s(X)) -> c_8(add#(sqr(X),dbl(X)),sqr#(X)) - Weak TRS: add(0(),X) -> X add(s(X),Y) -> s(add(X,Y)) dbl(0()) -> 0() dbl(s(X)) -> s(s(dbl(X))) sqr(0()) -> 0() sqr(s(X)) -> s(add(sqr(X),dbl(X))) - Signature: {add/2,dbl/1,first/2,sqr/1,terms/1,add#/2,dbl#/1,first#/2,sqr#/1,terms#/1} / {0/0,cons/1,nil/0,recip/1,s/1 ,c_1/0,c_2/1,c_3/0,c_4/1,c_5/0,c_6/0,c_7/0,c_8/2,c_9/1} - Obligation: innermost runtime complexity wrt. defined symbols {add#,dbl#,first#,sqr#,terms#} and constructors {0,cons ,nil,recip,s} + 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_8) = {1,2} Following symbols are considered usable: {add#,dbl#,first#,sqr#,terms#} TcT has computed the following interpretation: p(0) = [5] p(add) = [4] x1 + [0] p(cons) = [8] p(dbl) = [3] x1 + [0] p(first) = [1] p(nil) = [0] p(recip) = [2] p(s) = [1] x1 + [4] p(sqr) = [1] x1 + [3] p(terms) = [1] p(add#) = [0] p(dbl#) = [0] p(first#) = [2] p(sqr#) = [4] x1 + [8] p(terms#) = [2] p(c_1) = [1] p(c_2) = [1] p(c_3) = [1] p(c_4) = [1] p(c_5) = [1] p(c_6) = [1] p(c_7) = [0] p(c_8) = [8] x1 + [1] x2 + [12] p(c_9) = [1] x1 + [0] Following rules are strictly oriented: sqr#(s(X)) = [4] X + [24] > [4] X + [20] = c_8(add#(sqr(X),dbl(X)),sqr#(X)) Following rules are (at-least) weakly oriented: **** Step 6.a:3.a:1.a:2: Assumption WORST_CASE(?,O(1)) + Considered Problem: - Weak DPs: sqr#(s(X)) -> c_8(add#(sqr(X),dbl(X)),sqr#(X)) - Weak TRS: add(0(),X) -> X add(s(X),Y) -> s(add(X,Y)) dbl(0()) -> 0() dbl(s(X)) -> s(s(dbl(X))) sqr(0()) -> 0() sqr(s(X)) -> s(add(sqr(X),dbl(X))) - Signature: {add/2,dbl/1,first/2,sqr/1,terms/1,add#/2,dbl#/1,first#/2,sqr#/1,terms#/1} / {0/0,cons/1,nil/0,recip/1,s/1 ,c_1/0,c_2/1,c_3/0,c_4/1,c_5/0,c_6/0,c_7/0,c_8/2,c_9/1} - Obligation: innermost runtime complexity wrt. defined symbols {add#,dbl#,first#,sqr#,terms#} and constructors {0,cons ,nil,recip,s} + Applied Processor: Assumption {assumed = Certificate {spaceUB = Unknown, spaceLB = Unknown, timeUB = Poly (Just 0), timeLB = Unknown}} + Details: () **** Step 6.a:3.a:1.b:1: RemoveWeakSuffixes WORST_CASE(?,O(1)) + Considered Problem: - Weak DPs: sqr#(s(X)) -> c_8(add#(sqr(X),dbl(X)),sqr#(X)) - Weak TRS: add(0(),X) -> X add(s(X),Y) -> s(add(X,Y)) dbl(0()) -> 0() dbl(s(X)) -> s(s(dbl(X))) sqr(0()) -> 0() sqr(s(X)) -> s(add(sqr(X),dbl(X))) - Signature: {add/2,dbl/1,first/2,sqr/1,terms/1,add#/2,dbl#/1,first#/2,sqr#/1,terms#/1} / {0/0,cons/1,nil/0,recip/1,s/1 ,c_1/0,c_2/1,c_3/0,c_4/1,c_5/0,c_6/0,c_7/0,c_8/2,c_9/1} - Obligation: innermost runtime complexity wrt. defined symbols {add#,dbl#,first#,sqr#,terms#} and constructors {0,cons ,nil,recip,s} + Applied Processor: RemoveWeakSuffixes + Details: Consider the dependency graph 1:W:sqr#(s(X)) -> c_8(add#(sqr(X),dbl(X)),sqr#(X)) -->_2 sqr#(s(X)) -> c_8(add#(sqr(X),dbl(X)),sqr#(X)):1 The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed. 1: sqr#(s(X)) -> c_8(add#(sqr(X),dbl(X)),sqr#(X)) **** Step 6.a:3.a:1.b:2: EmptyProcessor WORST_CASE(?,O(1)) + Considered Problem: - Weak TRS: add(0(),X) -> X add(s(X),Y) -> s(add(X,Y)) dbl(0()) -> 0() dbl(s(X)) -> s(s(dbl(X))) sqr(0()) -> 0() sqr(s(X)) -> s(add(sqr(X),dbl(X))) - Signature: {add/2,dbl/1,first/2,sqr/1,terms/1,add#/2,dbl#/1,first#/2,sqr#/1,terms#/1} / {0/0,cons/1,nil/0,recip/1,s/1 ,c_1/0,c_2/1,c_3/0,c_4/1,c_5/0,c_6/0,c_7/0,c_8/2,c_9/1} - Obligation: innermost runtime complexity wrt. defined symbols {add#,dbl#,first#,sqr#,terms#} and constructors {0,cons ,nil,recip,s} + Applied Processor: EmptyProcessor + Details: The problem is already closed. The intended complexity is O(1). *** Step 6.a:3.b:1: PredecessorEstimationCP WORST_CASE(?,O(n^2)) + Considered Problem: - Strict DPs: add#(s(X),Y) -> c_2(add#(X,Y)) - Weak DPs: sqr#(s(X)) -> add#(sqr(X),dbl(X)) sqr#(s(X)) -> sqr#(X) - Weak TRS: add(0(),X) -> X add(s(X),Y) -> s(add(X,Y)) dbl(0()) -> 0() dbl(s(X)) -> s(s(dbl(X))) sqr(0()) -> 0() sqr(s(X)) -> s(add(sqr(X),dbl(X))) - Signature: {add/2,dbl/1,first/2,sqr/1,terms/1,add#/2,dbl#/1,first#/2,sqr#/1,terms#/1} / {0/0,cons/1,nil/0,recip/1,s/1 ,c_1/0,c_2/1,c_3/0,c_4/1,c_5/0,c_6/0,c_7/0,c_8/2,c_9/1} - Obligation: innermost runtime complexity wrt. defined symbols {add#,dbl#,first#,sqr#,terms#} and constructors {0,cons ,nil,recip,s} + Applied Processor: PredecessorEstimationCP {onSelectionCP = any intersect of rules of CDG leaf and strict-rules, withComplexityPair = NaturalPI {shape = Mixed 2, restrict = Restrict, uargs = UArgs, urules = URules, selector = Nothing}} + Details: We first use the processor NaturalPI {shape = Mixed 2, restrict = Restrict, uargs = UArgs, urules = URules, selector = Nothing} to orient following rules strictly: 1: add#(s(X),Y) -> c_2(add#(X,Y)) The strictly oriented rules are moved into the weak component. **** Step 6.a:3.b:1.a:1: NaturalPI WORST_CASE(?,O(n^2)) + Considered Problem: - Strict DPs: add#(s(X),Y) -> c_2(add#(X,Y)) - Weak DPs: sqr#(s(X)) -> add#(sqr(X),dbl(X)) sqr#(s(X)) -> sqr#(X) - Weak TRS: add(0(),X) -> X add(s(X),Y) -> s(add(X,Y)) dbl(0()) -> 0() dbl(s(X)) -> s(s(dbl(X))) sqr(0()) -> 0() sqr(s(X)) -> s(add(sqr(X),dbl(X))) - Signature: {add/2,dbl/1,first/2,sqr/1,terms/1,add#/2,dbl#/1,first#/2,sqr#/1,terms#/1} / {0/0,cons/1,nil/0,recip/1,s/1 ,c_1/0,c_2/1,c_3/0,c_4/1,c_5/0,c_6/0,c_7/0,c_8/2,c_9/1} - Obligation: innermost runtime complexity wrt. defined symbols {add#,dbl#,first#,sqr#,terms#} and constructors {0,cons ,nil,recip,s} + Applied Processor: NaturalPI {shape = Mixed 2, restrict = Restrict, 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 polynomial interpretation of kind constructor-based(mixed(2)): The following argument positions are considered usable: uargs(c_2) = {1} Following symbols are considered usable: {add,dbl,sqr,add#,dbl#,first#,sqr#,terms#} TcT has computed the following interpretation: p(0) = 0 p(add) = x1 + 2*x2 p(cons) = 1 + x1 p(dbl) = 2*x1 p(first) = 1 + x1 + x1^2 p(nil) = 1 p(recip) = x1 p(s) = 2 + x1 p(sqr) = 2*x1^2 p(terms) = 2 + x1 p(add#) = x1 + 5*x2 p(dbl#) = 1 p(first#) = 4 + x1 + 2*x1^2 + x2 p(sqr#) = 1 + x1 + 3*x1^2 p(terms#) = 1 p(c_1) = 1 p(c_2) = x1 p(c_3) = 0 p(c_4) = 4 + x1 p(c_5) = 2 p(c_6) = 1 p(c_7) = 1 p(c_8) = 1 + x1 p(c_9) = 2 Following rules are strictly oriented: add#(s(X),Y) = 2 + X + 5*Y > X + 5*Y = c_2(add#(X,Y)) Following rules are (at-least) weakly oriented: sqr#(s(X)) = 15 + 13*X + 3*X^2 >= 10*X + 2*X^2 = add#(sqr(X),dbl(X)) sqr#(s(X)) = 15 + 13*X + 3*X^2 >= 1 + X + 3*X^2 = sqr#(X) add(0(),X) = 2*X >= X = X add(s(X),Y) = 2 + X + 2*Y >= 2 + X + 2*Y = s(add(X,Y)) dbl(0()) = 0 >= 0 = 0() dbl(s(X)) = 4 + 2*X >= 4 + 2*X = s(s(dbl(X))) sqr(0()) = 0 >= 0 = 0() sqr(s(X)) = 8 + 8*X + 2*X^2 >= 2 + 4*X + 2*X^2 = s(add(sqr(X),dbl(X))) **** Step 6.a:3.b:1.a:2: Assumption WORST_CASE(?,O(1)) + Considered Problem: - Weak DPs: add#(s(X),Y) -> c_2(add#(X,Y)) sqr#(s(X)) -> add#(sqr(X),dbl(X)) sqr#(s(X)) -> sqr#(X) - Weak TRS: add(0(),X) -> X add(s(X),Y) -> s(add(X,Y)) dbl(0()) -> 0() dbl(s(X)) -> s(s(dbl(X))) sqr(0()) -> 0() sqr(s(X)) -> s(add(sqr(X),dbl(X))) - Signature: {add/2,dbl/1,first/2,sqr/1,terms/1,add#/2,dbl#/1,first#/2,sqr#/1,terms#/1} / {0/0,cons/1,nil/0,recip/1,s/1 ,c_1/0,c_2/1,c_3/0,c_4/1,c_5/0,c_6/0,c_7/0,c_8/2,c_9/1} - Obligation: innermost runtime complexity wrt. defined symbols {add#,dbl#,first#,sqr#,terms#} and constructors {0,cons ,nil,recip,s} + Applied Processor: Assumption {assumed = Certificate {spaceUB = Unknown, spaceLB = Unknown, timeUB = Poly (Just 0), timeLB = Unknown}} + Details: () **** Step 6.a:3.b:1.b:1: RemoveWeakSuffixes WORST_CASE(?,O(1)) + Considered Problem: - Weak DPs: add#(s(X),Y) -> c_2(add#(X,Y)) sqr#(s(X)) -> add#(sqr(X),dbl(X)) sqr#(s(X)) -> sqr#(X) - Weak TRS: add(0(),X) -> X add(s(X),Y) -> s(add(X,Y)) dbl(0()) -> 0() dbl(s(X)) -> s(s(dbl(X))) sqr(0()) -> 0() sqr(s(X)) -> s(add(sqr(X),dbl(X))) - Signature: {add/2,dbl/1,first/2,sqr/1,terms/1,add#/2,dbl#/1,first#/2,sqr#/1,terms#/1} / {0/0,cons/1,nil/0,recip/1,s/1 ,c_1/0,c_2/1,c_3/0,c_4/1,c_5/0,c_6/0,c_7/0,c_8/2,c_9/1} - Obligation: innermost runtime complexity wrt. defined symbols {add#,dbl#,first#,sqr#,terms#} and constructors {0,cons ,nil,recip,s} + Applied Processor: RemoveWeakSuffixes + Details: Consider the dependency graph 1:W:add#(s(X),Y) -> c_2(add#(X,Y)) -->_1 add#(s(X),Y) -> c_2(add#(X,Y)):1 2:W:sqr#(s(X)) -> add#(sqr(X),dbl(X)) -->_1 add#(s(X),Y) -> c_2(add#(X,Y)):1 3:W:sqr#(s(X)) -> sqr#(X) -->_1 sqr#(s(X)) -> sqr#(X):3 -->_1 sqr#(s(X)) -> add#(sqr(X),dbl(X)):2 The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed. 3: sqr#(s(X)) -> sqr#(X) 2: sqr#(s(X)) -> add#(sqr(X),dbl(X)) 1: add#(s(X),Y) -> c_2(add#(X,Y)) **** Step 6.a:3.b:1.b:2: EmptyProcessor WORST_CASE(?,O(1)) + Considered Problem: - Weak TRS: add(0(),X) -> X add(s(X),Y) -> s(add(X,Y)) dbl(0()) -> 0() dbl(s(X)) -> s(s(dbl(X))) sqr(0()) -> 0() sqr(s(X)) -> s(add(sqr(X),dbl(X))) - Signature: {add/2,dbl/1,first/2,sqr/1,terms/1,add#/2,dbl#/1,first#/2,sqr#/1,terms#/1} / {0/0,cons/1,nil/0,recip/1,s/1 ,c_1/0,c_2/1,c_3/0,c_4/1,c_5/0,c_6/0,c_7/0,c_8/2,c_9/1} - Obligation: innermost runtime complexity wrt. defined symbols {add#,dbl#,first#,sqr#,terms#} and constructors {0,cons ,nil,recip,s} + Applied Processor: EmptyProcessor + Details: The problem is already closed. The intended complexity is O(1). ** Step 6.b:1: RemoveWeakSuffixes WORST_CASE(?,O(n^2)) + Considered Problem: - Strict DPs: dbl#(s(X)) -> c_4(dbl#(X)) sqr#(s(X)) -> c_8(add#(sqr(X),dbl(X)),sqr#(X),dbl#(X)) - Weak DPs: add#(s(X),Y) -> c_2(add#(X,Y)) - Weak TRS: add(0(),X) -> X add(s(X),Y) -> s(add(X,Y)) dbl(0()) -> 0() dbl(s(X)) -> s(s(dbl(X))) sqr(0()) -> 0() sqr(s(X)) -> s(add(sqr(X),dbl(X))) - Signature: {add/2,dbl/1,first/2,sqr/1,terms/1,add#/2,dbl#/1,first#/2,sqr#/1,terms#/1} / {0/0,cons/1,nil/0,recip/1,s/1 ,c_1/0,c_2/1,c_3/0,c_4/1,c_5/0,c_6/0,c_7/0,c_8/3,c_9/1} - Obligation: innermost runtime complexity wrt. defined symbols {add#,dbl#,first#,sqr#,terms#} and constructors {0,cons ,nil,recip,s} + Applied Processor: RemoveWeakSuffixes + Details: Consider the dependency graph 1:S:dbl#(s(X)) -> c_4(dbl#(X)) -->_1 dbl#(s(X)) -> c_4(dbl#(X)):1 2:S:sqr#(s(X)) -> c_8(add#(sqr(X),dbl(X)),sqr#(X),dbl#(X)) -->_1 add#(s(X),Y) -> c_2(add#(X,Y)):3 -->_2 sqr#(s(X)) -> c_8(add#(sqr(X),dbl(X)),sqr#(X),dbl#(X)):2 -->_3 dbl#(s(X)) -> c_4(dbl#(X)):1 3:W:add#(s(X),Y) -> c_2(add#(X,Y)) -->_1 add#(s(X),Y) -> c_2(add#(X,Y)):3 The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed. 3: add#(s(X),Y) -> c_2(add#(X,Y)) ** Step 6.b:2: SimplifyRHS WORST_CASE(?,O(n^2)) + Considered Problem: - Strict DPs: dbl#(s(X)) -> c_4(dbl#(X)) sqr#(s(X)) -> c_8(add#(sqr(X),dbl(X)),sqr#(X),dbl#(X)) - Weak TRS: add(0(),X) -> X add(s(X),Y) -> s(add(X,Y)) dbl(0()) -> 0() dbl(s(X)) -> s(s(dbl(X))) sqr(0()) -> 0() sqr(s(X)) -> s(add(sqr(X),dbl(X))) - Signature: {add/2,dbl/1,first/2,sqr/1,terms/1,add#/2,dbl#/1,first#/2,sqr#/1,terms#/1} / {0/0,cons/1,nil/0,recip/1,s/1 ,c_1/0,c_2/1,c_3/0,c_4/1,c_5/0,c_6/0,c_7/0,c_8/3,c_9/1} - Obligation: innermost runtime complexity wrt. defined symbols {add#,dbl#,first#,sqr#,terms#} and constructors {0,cons ,nil,recip,s} + Applied Processor: SimplifyRHS + Details: Consider the dependency graph 1:S:dbl#(s(X)) -> c_4(dbl#(X)) -->_1 dbl#(s(X)) -> c_4(dbl#(X)):1 2:S:sqr#(s(X)) -> c_8(add#(sqr(X),dbl(X)),sqr#(X),dbl#(X)) -->_2 sqr#(s(X)) -> c_8(add#(sqr(X),dbl(X)),sqr#(X),dbl#(X)):2 -->_3 dbl#(s(X)) -> c_4(dbl#(X)):1 Due to missing edges in the depndency graph, the right-hand sides of following rules could be simplified: sqr#(s(X)) -> c_8(sqr#(X),dbl#(X)) ** Step 6.b:3: UsableRules WORST_CASE(?,O(n^2)) + Considered Problem: - Strict DPs: dbl#(s(X)) -> c_4(dbl#(X)) sqr#(s(X)) -> c_8(sqr#(X),dbl#(X)) - Weak TRS: add(0(),X) -> X add(s(X),Y) -> s(add(X,Y)) dbl(0()) -> 0() dbl(s(X)) -> s(s(dbl(X))) sqr(0()) -> 0() sqr(s(X)) -> s(add(sqr(X),dbl(X))) - Signature: {add/2,dbl/1,first/2,sqr/1,terms/1,add#/2,dbl#/1,first#/2,sqr#/1,terms#/1} / {0/0,cons/1,nil/0,recip/1,s/1 ,c_1/0,c_2/1,c_3/0,c_4/1,c_5/0,c_6/0,c_7/0,c_8/2,c_9/1} - Obligation: innermost runtime complexity wrt. defined symbols {add#,dbl#,first#,sqr#,terms#} and constructors {0,cons ,nil,recip,s} + Applied Processor: UsableRules + Details: We replace rewrite rules by usable rules: dbl#(s(X)) -> c_4(dbl#(X)) sqr#(s(X)) -> c_8(sqr#(X),dbl#(X)) ** Step 6.b:4: Decompose WORST_CASE(?,O(n^2)) + Considered Problem: - Strict DPs: dbl#(s(X)) -> c_4(dbl#(X)) sqr#(s(X)) -> c_8(sqr#(X),dbl#(X)) - Signature: {add/2,dbl/1,first/2,sqr/1,terms/1,add#/2,dbl#/1,first#/2,sqr#/1,terms#/1} / {0/0,cons/1,nil/0,recip/1,s/1 ,c_1/0,c_2/1,c_3/0,c_4/1,c_5/0,c_6/0,c_7/0,c_8/2,c_9/1} - Obligation: innermost runtime complexity wrt. defined symbols {add#,dbl#,first#,sqr#,terms#} and constructors {0,cons ,nil,recip,s} + Applied Processor: Decompose {onSelection = all cycle independent sub-graph, withBound = RelativeAdd} + Details: We analyse the complexity of following sub-problems (R) and (S). Problem (S) is obtained from the input problem by shifting strict rules from (R) into the weak component. Problem (R) - Strict DPs: dbl#(s(X)) -> c_4(dbl#(X)) - Weak DPs: sqr#(s(X)) -> c_8(sqr#(X),dbl#(X)) - Signature: {add/2,dbl/1,first/2,sqr/1,terms/1,add#/2,dbl#/1,first#/2,sqr#/1,terms#/1} / {0/0,cons/1,nil/0,recip/1,s/1 ,c_1/0,c_2/1,c_3/0,c_4/1,c_5/0,c_6/0,c_7/0,c_8/2,c_9/1} - Obligation: innermost runtime complexity wrt. defined symbols {add#,dbl#,first#,sqr#,terms#} and constructors {0,cons ,nil,recip,s} Problem (S) - Strict DPs: sqr#(s(X)) -> c_8(sqr#(X),dbl#(X)) - Weak DPs: dbl#(s(X)) -> c_4(dbl#(X)) - Signature: {add/2,dbl/1,first/2,sqr/1,terms/1,add#/2,dbl#/1,first#/2,sqr#/1,terms#/1} / {0/0,cons/1,nil/0,recip/1,s/1 ,c_1/0,c_2/1,c_3/0,c_4/1,c_5/0,c_6/0,c_7/0,c_8/2,c_9/1} - Obligation: innermost runtime complexity wrt. defined symbols {add#,dbl#,first#,sqr#,terms#} and constructors {0,cons ,nil,recip,s} *** Step 6.b:4.a:1: PredecessorEstimationCP WORST_CASE(?,O(n^2)) + Considered Problem: - Strict DPs: dbl#(s(X)) -> c_4(dbl#(X)) - Weak DPs: sqr#(s(X)) -> c_8(sqr#(X),dbl#(X)) - Signature: {add/2,dbl/1,first/2,sqr/1,terms/1,add#/2,dbl#/1,first#/2,sqr#/1,terms#/1} / {0/0,cons/1,nil/0,recip/1,s/1 ,c_1/0,c_2/1,c_3/0,c_4/1,c_5/0,c_6/0,c_7/0,c_8/2,c_9/1} - Obligation: innermost runtime complexity wrt. defined symbols {add#,dbl#,first#,sqr#,terms#} and constructors {0,cons ,nil,recip,s} + Applied Processor: PredecessorEstimationCP {onSelectionCP = any intersect of rules of CDG leaf and strict-rules, withComplexityPair = NaturalPI {shape = Mixed 2, restrict = Restrict, uargs = UArgs, urules = URules, selector = Nothing}} + Details: We first use the processor NaturalPI {shape = Mixed 2, restrict = Restrict, uargs = UArgs, urules = URules, selector = Nothing} to orient following rules strictly: 1: dbl#(s(X)) -> c_4(dbl#(X)) The strictly oriented rules are moved into the weak component. **** Step 6.b:4.a:1.a:1: NaturalPI WORST_CASE(?,O(n^2)) + Considered Problem: - Strict DPs: dbl#(s(X)) -> c_4(dbl#(X)) - Weak DPs: sqr#(s(X)) -> c_8(sqr#(X),dbl#(X)) - Signature: {add/2,dbl/1,first/2,sqr/1,terms/1,add#/2,dbl#/1,first#/2,sqr#/1,terms#/1} / {0/0,cons/1,nil/0,recip/1,s/1 ,c_1/0,c_2/1,c_3/0,c_4/1,c_5/0,c_6/0,c_7/0,c_8/2,c_9/1} - Obligation: innermost runtime complexity wrt. defined symbols {add#,dbl#,first#,sqr#,terms#} and constructors {0,cons ,nil,recip,s} + Applied Processor: NaturalPI {shape = Mixed 2, restrict = Restrict, 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 polynomial interpretation of kind constructor-based(mixed(2)): The following argument positions are considered usable: uargs(c_4) = {1}, uargs(c_8) = {1,2} Following symbols are considered usable: {add#,dbl#,first#,sqr#,terms#} TcT has computed the following interpretation: p(0) = 2 p(add) = 1 + 4*x1 + x1^2 p(cons) = 1 + x1 p(dbl) = 1 + 2*x1 p(first) = x1 + x1^2 + 8*x2 + x2^2 p(nil) = 2 p(recip) = 1 + x1 p(s) = 1 + x1 p(sqr) = 2 + 8*x1 + 2*x1^2 p(terms) = 4 + x1 + x1^2 p(add#) = 8*x1 + x1^2 + 8*x2 + x2^2 p(dbl#) = x1 p(first#) = 1 + x2 + 8*x2^2 p(sqr#) = 2 + 8*x1^2 p(terms#) = x1 + x1^2 p(c_1) = 1 p(c_2) = 1 p(c_3) = 1 p(c_4) = x1 p(c_5) = 0 p(c_6) = 1 p(c_7) = 0 p(c_8) = 5 + x1 + x2 p(c_9) = 1 Following rules are strictly oriented: dbl#(s(X)) = 1 + X > X = c_4(dbl#(X)) Following rules are (at-least) weakly oriented: sqr#(s(X)) = 10 + 16*X + 8*X^2 >= 7 + X + 8*X^2 = c_8(sqr#(X),dbl#(X)) **** Step 6.b:4.a:1.a:2: Assumption WORST_CASE(?,O(1)) + Considered Problem: - Weak DPs: dbl#(s(X)) -> c_4(dbl#(X)) sqr#(s(X)) -> c_8(sqr#(X),dbl#(X)) - Signature: {add/2,dbl/1,first/2,sqr/1,terms/1,add#/2,dbl#/1,first#/2,sqr#/1,terms#/1} / {0/0,cons/1,nil/0,recip/1,s/1 ,c_1/0,c_2/1,c_3/0,c_4/1,c_5/0,c_6/0,c_7/0,c_8/2,c_9/1} - Obligation: innermost runtime complexity wrt. defined symbols {add#,dbl#,first#,sqr#,terms#} and constructors {0,cons ,nil,recip,s} + Applied Processor: Assumption {assumed = Certificate {spaceUB = Unknown, spaceLB = Unknown, timeUB = Poly (Just 0), timeLB = Unknown}} + Details: () **** Step 6.b:4.a:1.b:1: RemoveWeakSuffixes WORST_CASE(?,O(1)) + Considered Problem: - Weak DPs: dbl#(s(X)) -> c_4(dbl#(X)) sqr#(s(X)) -> c_8(sqr#(X),dbl#(X)) - Signature: {add/2,dbl/1,first/2,sqr/1,terms/1,add#/2,dbl#/1,first#/2,sqr#/1,terms#/1} / {0/0,cons/1,nil/0,recip/1,s/1 ,c_1/0,c_2/1,c_3/0,c_4/1,c_5/0,c_6/0,c_7/0,c_8/2,c_9/1} - Obligation: innermost runtime complexity wrt. defined symbols {add#,dbl#,first#,sqr#,terms#} and constructors {0,cons ,nil,recip,s} + Applied Processor: RemoveWeakSuffixes + Details: Consider the dependency graph 1:W:dbl#(s(X)) -> c_4(dbl#(X)) -->_1 dbl#(s(X)) -> c_4(dbl#(X)):1 2:W:sqr#(s(X)) -> c_8(sqr#(X),dbl#(X)) -->_1 sqr#(s(X)) -> c_8(sqr#(X),dbl#(X)):2 -->_2 dbl#(s(X)) -> c_4(dbl#(X)):1 The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed. 2: sqr#(s(X)) -> c_8(sqr#(X),dbl#(X)) 1: dbl#(s(X)) -> c_4(dbl#(X)) **** Step 6.b:4.a:1.b:2: EmptyProcessor WORST_CASE(?,O(1)) + Considered Problem: - Signature: {add/2,dbl/1,first/2,sqr/1,terms/1,add#/2,dbl#/1,first#/2,sqr#/1,terms#/1} / {0/0,cons/1,nil/0,recip/1,s/1 ,c_1/0,c_2/1,c_3/0,c_4/1,c_5/0,c_6/0,c_7/0,c_8/2,c_9/1} - Obligation: innermost runtime complexity wrt. defined symbols {add#,dbl#,first#,sqr#,terms#} and constructors {0,cons ,nil,recip,s} + Applied Processor: EmptyProcessor + Details: The problem is already closed. The intended complexity is O(1). *** Step 6.b:4.b:1: RemoveWeakSuffixes WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: sqr#(s(X)) -> c_8(sqr#(X),dbl#(X)) - Weak DPs: dbl#(s(X)) -> c_4(dbl#(X)) - Signature: {add/2,dbl/1,first/2,sqr/1,terms/1,add#/2,dbl#/1,first#/2,sqr#/1,terms#/1} / {0/0,cons/1,nil/0,recip/1,s/1 ,c_1/0,c_2/1,c_3/0,c_4/1,c_5/0,c_6/0,c_7/0,c_8/2,c_9/1} - Obligation: innermost runtime complexity wrt. defined symbols {add#,dbl#,first#,sqr#,terms#} and constructors {0,cons ,nil,recip,s} + Applied Processor: RemoveWeakSuffixes + Details: Consider the dependency graph 1:S:sqr#(s(X)) -> c_8(sqr#(X),dbl#(X)) -->_2 dbl#(s(X)) -> c_4(dbl#(X)):2 -->_1 sqr#(s(X)) -> c_8(sqr#(X),dbl#(X)):1 2:W:dbl#(s(X)) -> c_4(dbl#(X)) -->_1 dbl#(s(X)) -> c_4(dbl#(X)):2 The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed. 2: dbl#(s(X)) -> c_4(dbl#(X)) *** Step 6.b:4.b:2: SimplifyRHS WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: sqr#(s(X)) -> c_8(sqr#(X),dbl#(X)) - Signature: {add/2,dbl/1,first/2,sqr/1,terms/1,add#/2,dbl#/1,first#/2,sqr#/1,terms#/1} / {0/0,cons/1,nil/0,recip/1,s/1 ,c_1/0,c_2/1,c_3/0,c_4/1,c_5/0,c_6/0,c_7/0,c_8/2,c_9/1} - Obligation: innermost runtime complexity wrt. defined symbols {add#,dbl#,first#,sqr#,terms#} and constructors {0,cons ,nil,recip,s} + Applied Processor: SimplifyRHS + Details: Consider the dependency graph 1:S:sqr#(s(X)) -> c_8(sqr#(X),dbl#(X)) -->_1 sqr#(s(X)) -> c_8(sqr#(X),dbl#(X)):1 Due to missing edges in the depndency graph, the right-hand sides of following rules could be simplified: sqr#(s(X)) -> c_8(sqr#(X)) *** Step 6.b:4.b:3: PredecessorEstimationCP WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: sqr#(s(X)) -> c_8(sqr#(X)) - Signature: {add/2,dbl/1,first/2,sqr/1,terms/1,add#/2,dbl#/1,first#/2,sqr#/1,terms#/1} / {0/0,cons/1,nil/0,recip/1,s/1 ,c_1/0,c_2/1,c_3/0,c_4/1,c_5/0,c_6/0,c_7/0,c_8/1,c_9/1} - Obligation: innermost runtime complexity wrt. defined symbols {add#,dbl#,first#,sqr#,terms#} and constructors {0,cons ,nil,recip,s} + 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: sqr#(s(X)) -> c_8(sqr#(X)) The strictly oriented rules are moved into the weak component. **** Step 6.b:4.b:3.a:1: NaturalMI WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: sqr#(s(X)) -> c_8(sqr#(X)) - Signature: {add/2,dbl/1,first/2,sqr/1,terms/1,add#/2,dbl#/1,first#/2,sqr#/1,terms#/1} / {0/0,cons/1,nil/0,recip/1,s/1 ,c_1/0,c_2/1,c_3/0,c_4/1,c_5/0,c_6/0,c_7/0,c_8/1,c_9/1} - Obligation: innermost runtime complexity wrt. defined symbols {add#,dbl#,first#,sqr#,terms#} and constructors {0,cons ,nil,recip,s} + 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_8) = {1} Following symbols are considered usable: {add#,dbl#,first#,sqr#,terms#} TcT has computed the following interpretation: p(0) = [0] p(add) = [0] p(cons) = [2] p(dbl) = [8] p(first) = [1] p(nil) = [4] p(recip) = [8] p(s) = [1] x1 + [2] p(sqr) = [8] p(terms) = [8] x1 + [0] p(add#) = [1] x1 + [1] x2 + [2] p(dbl#) = [1] p(first#) = [1] x1 + [2] x2 + [2] p(sqr#) = [1] x1 + [1] p(terms#) = [2] p(c_1) = [1] p(c_2) = [1] p(c_3) = [1] p(c_4) = [1] x1 + [1] p(c_5) = [0] p(c_6) = [1] p(c_7) = [4] p(c_8) = [1] x1 + [0] p(c_9) = [8] Following rules are strictly oriented: sqr#(s(X)) = [1] X + [3] > [1] X + [1] = c_8(sqr#(X)) Following rules are (at-least) weakly oriented: **** Step 6.b:4.b:3.a:2: Assumption WORST_CASE(?,O(1)) + Considered Problem: - Weak DPs: sqr#(s(X)) -> c_8(sqr#(X)) - Signature: {add/2,dbl/1,first/2,sqr/1,terms/1,add#/2,dbl#/1,first#/2,sqr#/1,terms#/1} / {0/0,cons/1,nil/0,recip/1,s/1 ,c_1/0,c_2/1,c_3/0,c_4/1,c_5/0,c_6/0,c_7/0,c_8/1,c_9/1} - Obligation: innermost runtime complexity wrt. defined symbols {add#,dbl#,first#,sqr#,terms#} and constructors {0,cons ,nil,recip,s} + Applied Processor: Assumption {assumed = Certificate {spaceUB = Unknown, spaceLB = Unknown, timeUB = Poly (Just 0), timeLB = Unknown}} + Details: () **** Step 6.b:4.b:3.b:1: RemoveWeakSuffixes WORST_CASE(?,O(1)) + Considered Problem: - Weak DPs: sqr#(s(X)) -> c_8(sqr#(X)) - Signature: {add/2,dbl/1,first/2,sqr/1,terms/1,add#/2,dbl#/1,first#/2,sqr#/1,terms#/1} / {0/0,cons/1,nil/0,recip/1,s/1 ,c_1/0,c_2/1,c_3/0,c_4/1,c_5/0,c_6/0,c_7/0,c_8/1,c_9/1} - Obligation: innermost runtime complexity wrt. defined symbols {add#,dbl#,first#,sqr#,terms#} and constructors {0,cons ,nil,recip,s} + Applied Processor: RemoveWeakSuffixes + Details: Consider the dependency graph 1:W:sqr#(s(X)) -> c_8(sqr#(X)) -->_1 sqr#(s(X)) -> c_8(sqr#(X)):1 The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed. 1: sqr#(s(X)) -> c_8(sqr#(X)) **** Step 6.b:4.b:3.b:2: EmptyProcessor WORST_CASE(?,O(1)) + Considered Problem: - Signature: {add/2,dbl/1,first/2,sqr/1,terms/1,add#/2,dbl#/1,first#/2,sqr#/1,terms#/1} / {0/0,cons/1,nil/0,recip/1,s/1 ,c_1/0,c_2/1,c_3/0,c_4/1,c_5/0,c_6/0,c_7/0,c_8/1,c_9/1} - Obligation: innermost runtime complexity wrt. defined symbols {add#,dbl#,first#,sqr#,terms#} and constructors {0,cons ,nil,recip,s} + Applied Processor: EmptyProcessor + Details: The problem is already closed. The intended complexity is O(1). WORST_CASE(?,O(n^3))