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) half(0()) -> 0() half(dbl(X)) -> X half(s(0())) -> 0() half(s(s(X))) -> s(half(X)) 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,half/1,sqr/1,terms/1} / {0/0,cons/1,nil/0,recip/1,s/1} - Obligation: innermost runtime complexity wrt. defined symbols {add,dbl,first,half,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() half#(0()) -> c_7() half#(dbl(X)) -> c_8() half#(s(0())) -> c_9() half#(s(s(X))) -> c_10(half#(X)) sqr#(0()) -> c_11() sqr#(s(X)) -> c_12(add#(sqr(X),dbl(X)),sqr#(X),dbl#(X)) terms#(N) -> c_13(sqr#(N)) Weak DPs and mark the set of starting terms. * Step 2: 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() half#(0()) -> c_7() half#(dbl(X)) -> c_8() half#(s(0())) -> c_9() half#(s(s(X))) -> c_10(half#(X)) sqr#(0()) -> c_11() sqr#(s(X)) -> c_12(add#(sqr(X),dbl(X)),sqr#(X),dbl#(X)) terms#(N) -> c_13(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) half(0()) -> 0() half(dbl(X)) -> X half(s(0())) -> 0() half(s(s(X))) -> s(half(X)) 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,half/1,sqr/1,terms/1,add#/2,dbl#/1,first#/2,half#/1,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_8/0,c_9/0,c_10/1,c_11/0,c_12/3,c_13/1} - Obligation: innermost runtime complexity wrt. defined symbols {add#,dbl#,first#,half#,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 {1,3,5,6,7,8,9,11} by application of Pre({1,3,5,6,7,8,9,11}) = {2,4,10,12,13}. 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: half#(0()) -> c_7() 8: half#(dbl(X)) -> c_8() 9: half#(s(0())) -> c_9() 10: half#(s(s(X))) -> c_10(half#(X)) 11: sqr#(0()) -> c_11() 12: sqr#(s(X)) -> c_12(add#(sqr(X),dbl(X)),sqr#(X),dbl#(X)) 13: terms#(N) -> c_13(sqr#(N)) * Step 3: 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)) half#(s(s(X))) -> c_10(half#(X)) sqr#(s(X)) -> c_12(add#(sqr(X),dbl(X)),sqr#(X),dbl#(X)) terms#(N) -> c_13(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() half#(0()) -> c_7() half#(dbl(X)) -> c_8() half#(s(0())) -> c_9() sqr#(0()) -> c_11() - 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) half(0()) -> 0() half(dbl(X)) -> X half(s(0())) -> 0() half(s(s(X))) -> s(half(X)) 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,half/1,sqr/1,terms/1,add#/2,dbl#/1,first#/2,half#/1,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_8/0,c_9/0,c_10/1,c_11/0,c_12/3,c_13/1} - Obligation: innermost runtime complexity wrt. defined symbols {add#,dbl#,first#,half#,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():6 -->_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():7 -->_1 dbl#(s(X)) -> c_4(dbl#(X)):2 3:S:half#(s(s(X))) -> c_10(half#(X)) -->_1 half#(s(0())) -> c_9():12 -->_1 half#(dbl(X)) -> c_8():11 -->_1 half#(0()) -> c_7():10 -->_1 half#(s(s(X))) -> c_10(half#(X)):3 4:S:sqr#(s(X)) -> c_12(add#(sqr(X),dbl(X)),sqr#(X),dbl#(X)) -->_2 sqr#(0()) -> c_11():13 -->_3 dbl#(0()) -> c_3():7 -->_1 add#(0(),X) -> c_1():6 -->_2 sqr#(s(X)) -> c_12(add#(sqr(X),dbl(X)),sqr#(X),dbl#(X)):4 -->_3 dbl#(s(X)) -> c_4(dbl#(X)):2 -->_1 add#(s(X),Y) -> c_2(add#(X,Y)):1 5:S:terms#(N) -> c_13(sqr#(N)) -->_1 sqr#(0()) -> c_11():13 -->_1 sqr#(s(X)) -> c_12(add#(sqr(X),dbl(X)),sqr#(X),dbl#(X)):4 6:W:add#(0(),X) -> c_1() 7:W:dbl#(0()) -> c_3() 8:W:first#(0(),X) -> c_5() 9:W:first#(s(X),cons(Y)) -> c_6() 10:W:half#(0()) -> c_7() 11:W:half#(dbl(X)) -> c_8() 12:W:half#(s(0())) -> c_9() 13:W:sqr#(0()) -> c_11() The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed. 9: first#(s(X),cons(Y)) -> c_6() 8: first#(0(),X) -> c_5() 13: sqr#(0()) -> c_11() 10: half#(0()) -> c_7() 11: half#(dbl(X)) -> c_8() 12: half#(s(0())) -> c_9() 7: dbl#(0()) -> c_3() 6: add#(0(),X) -> c_1() * Step 4: 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)) half#(s(s(X))) -> c_10(half#(X)) sqr#(s(X)) -> c_12(add#(sqr(X),dbl(X)),sqr#(X),dbl#(X)) terms#(N) -> c_13(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) half(0()) -> 0() half(dbl(X)) -> X half(s(0())) -> 0() half(s(s(X))) -> s(half(X)) 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,half/1,sqr/1,terms/1,add#/2,dbl#/1,first#/2,half#/1,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_8/0,c_9/0,c_10/1,c_11/0,c_12/3,c_13/1} - Obligation: innermost runtime complexity wrt. defined symbols {add#,dbl#,first#,half#,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:half#(s(s(X))) -> c_10(half#(X)) -->_1 half#(s(s(X))) -> c_10(half#(X)):3 4:S:sqr#(s(X)) -> c_12(add#(sqr(X),dbl(X)),sqr#(X),dbl#(X)) -->_2 sqr#(s(X)) -> c_12(add#(sqr(X),dbl(X)),sqr#(X),dbl#(X)):4 -->_3 dbl#(s(X)) -> c_4(dbl#(X)):2 -->_1 add#(s(X),Y) -> c_2(add#(X,Y)):1 5:S:terms#(N) -> c_13(sqr#(N)) -->_1 sqr#(s(X)) -> c_12(add#(sqr(X),dbl(X)),sqr#(X),dbl#(X)):4 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). [(5,terms#(N) -> c_13(sqr#(N)))] * Step 5: UsableRules 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)) half#(s(s(X))) -> c_10(half#(X)) sqr#(s(X)) -> c_12(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))) first(0(),X) -> nil() first(s(X),cons(Y)) -> cons(Y) half(0()) -> 0() half(dbl(X)) -> X half(s(0())) -> 0() half(s(s(X))) -> s(half(X)) 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,half/1,sqr/1,terms/1,add#/2,dbl#/1,first#/2,half#/1,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_8/0,c_9/0,c_10/1,c_11/0,c_12/3,c_13/1} - Obligation: innermost runtime complexity wrt. defined symbols {add#,dbl#,first#,half#,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#(s(X),Y) -> c_2(add#(X,Y)) dbl#(s(X)) -> c_4(dbl#(X)) half#(s(s(X))) -> c_10(half#(X)) sqr#(s(X)) -> c_12(add#(sqr(X),dbl(X)),sqr#(X),dbl#(X)) * Step 6: DecomposeDG 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)) half#(s(s(X))) -> c_10(half#(X)) sqr#(s(X)) -> c_12(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,half/1,sqr/1,terms/1,add#/2,dbl#/1,first#/2,half#/1,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_8/0,c_9/0,c_10/1,c_11/0,c_12/3,c_13/1} - Obligation: innermost runtime complexity wrt. defined symbols {add#,dbl#,first#,half#,sqr#,terms#} and constructors {0 ,cons,nil,recip,s} + 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 half#(s(s(X))) -> c_10(half#(X)) sqr#(s(X)) -> c_12(add#(sqr(X),dbl(X)),sqr#(X),dbl#(X)) and a lower component add#(s(X),Y) -> c_2(add#(X,Y)) dbl#(s(X)) -> c_4(dbl#(X)) Further, following extension rules are added to the lower component. half#(s(s(X))) -> half#(X) sqr#(s(X)) -> add#(sqr(X),dbl(X)) sqr#(s(X)) -> dbl#(X) sqr#(s(X)) -> sqr#(X) ** Step 6.a:1: SimplifyRHS WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: half#(s(s(X))) -> c_10(half#(X)) sqr#(s(X)) -> c_12(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,half/1,sqr/1,terms/1,add#/2,dbl#/1,first#/2,half#/1,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_8/0,c_9/0,c_10/1,c_11/0,c_12/3,c_13/1} - Obligation: innermost runtime complexity wrt. defined symbols {add#,dbl#,first#,half#,sqr#,terms#} and constructors {0 ,cons,nil,recip,s} + Applied Processor: SimplifyRHS + Details: Consider the dependency graph 1:S:half#(s(s(X))) -> c_10(half#(X)) -->_1 half#(s(s(X))) -> c_10(half#(X)):1 2:S:sqr#(s(X)) -> c_12(add#(sqr(X),dbl(X)),sqr#(X),dbl#(X)) -->_2 sqr#(s(X)) -> c_12(add#(sqr(X),dbl(X)),sqr#(X),dbl#(X)):2 Due to missing edges in the depndency graph, the right-hand sides of following rules could be simplified: sqr#(s(X)) -> c_12(sqr#(X)) ** Step 6.a:2: UsableRules WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: half#(s(s(X))) -> c_10(half#(X)) sqr#(s(X)) -> c_12(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,half/1,sqr/1,terms/1,add#/2,dbl#/1,first#/2,half#/1,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_8/0,c_9/0,c_10/1,c_11/0,c_12/1,c_13/1} - Obligation: innermost runtime complexity wrt. defined symbols {add#,dbl#,first#,half#,sqr#,terms#} and constructors {0 ,cons,nil,recip,s} + Applied Processor: UsableRules + Details: We replace rewrite rules by usable rules: half#(s(s(X))) -> c_10(half#(X)) sqr#(s(X)) -> c_12(sqr#(X)) ** Step 6.a:3: WeightGap WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: half#(s(s(X))) -> c_10(half#(X)) sqr#(s(X)) -> c_12(sqr#(X)) - Signature: {add/2,dbl/1,first/2,half/1,sqr/1,terms/1,add#/2,dbl#/1,first#/2,half#/1,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_8/0,c_9/0,c_10/1,c_11/0,c_12/1,c_13/1} - Obligation: innermost runtime complexity wrt. defined symbols {add#,dbl#,first#,half#,sqr#,terms#} and constructors {0 ,cons,nil,recip,s} + Applied Processor: WeightGap {wgDimension = 1, wgDegree = 1, wgKind = Algebraic, wgUArgs = UArgs, wgOn = WgOnAny} + Details: The weightgap principle applies using the following constant growth matrix-interpretation: We apply a matrix interpretation of kind constructor based matrix interpretation: The following argument positions are considered usable: uargs(c_10) = {1}, uargs(c_12) = {1} Following symbols are considered usable: all TcT has computed the following interpretation: p(0) = [0] p(add) = [0] p(cons) = [0] p(dbl) = [0] p(first) = [0] p(half) = [0] p(nil) = [0] p(recip) = [1] x1 + [0] p(s) = [1] x1 + [9] p(sqr) = [0] p(terms) = [0] p(add#) = [0] p(dbl#) = [0] p(first#) = [0] p(half#) = [0] p(sqr#) = [1] x1 + [0] p(terms#) = [0] p(c_1) = [0] p(c_2) = [0] p(c_3) = [0] p(c_4) = [0] p(c_5) = [0] p(c_6) = [0] p(c_7) = [0] p(c_8) = [0] p(c_9) = [0] p(c_10) = [1] x1 + [6] p(c_11) = [0] p(c_12) = [1] x1 + [0] p(c_13) = [0] Following rules are strictly oriented: sqr#(s(X)) = [1] X + [9] > [1] X + [0] = c_12(sqr#(X)) Following rules are (at-least) weakly oriented: half#(s(s(X))) = [0] >= [6] = c_10(half#(X)) Further, it can be verified that all rules not oriented are covered by the weightgap condition. ** Step 6.a:4: WeightGap WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: half#(s(s(X))) -> c_10(half#(X)) - Weak DPs: sqr#(s(X)) -> c_12(sqr#(X)) - Signature: {add/2,dbl/1,first/2,half/1,sqr/1,terms/1,add#/2,dbl#/1,first#/2,half#/1,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_8/0,c_9/0,c_10/1,c_11/0,c_12/1,c_13/1} - Obligation: innermost runtime complexity wrt. defined symbols {add#,dbl#,first#,half#,sqr#,terms#} and constructors {0 ,cons,nil,recip,s} + Applied Processor: WeightGap {wgDimension = 1, wgDegree = 1, wgKind = Algebraic, wgUArgs = UArgs, wgOn = WgOnAny} + Details: The weightgap principle applies using the following constant growth matrix-interpretation: We apply a matrix interpretation of kind constructor based matrix interpretation: The following argument positions are considered usable: uargs(c_10) = {1}, uargs(c_12) = {1} Following symbols are considered usable: all TcT has computed the following interpretation: p(0) = [0] p(add) = [4] x1 + [1] x2 + [8] p(cons) = [1] x1 + [2] p(dbl) = [1] x1 + [0] p(first) = [2] x2 + [0] p(half) = [1] x1 + [0] p(nil) = [4] p(recip) = [8] p(s) = [1] x1 + [1] p(sqr) = [1] x1 + [0] p(terms) = [1] x1 + [1] p(add#) = [4] x1 + [2] x2 + [2] p(dbl#) = [1] x1 + [0] p(first#) = [1] x2 + [1] p(half#) = [2] x1 + [11] p(sqr#) = [3] x1 + [5] p(terms#) = [1] p(c_1) = [0] p(c_2) = [1] x1 + [1] p(c_3) = [0] p(c_4) = [4] x1 + [0] p(c_5) = [4] p(c_6) = [1] p(c_7) = [4] p(c_8) = [0] p(c_9) = [2] p(c_10) = [1] x1 + [1] p(c_11) = [2] p(c_12) = [1] x1 + [1] p(c_13) = [1] x1 + [2] Following rules are strictly oriented: half#(s(s(X))) = [2] X + [15] > [2] X + [12] = c_10(half#(X)) Following rules are (at-least) weakly oriented: sqr#(s(X)) = [3] X + [8] >= [3] X + [6] = c_12(sqr#(X)) Further, it can be verified that all rules not oriented are covered by the weightgap condition. ** Step 6.a:5: EmptyProcessor WORST_CASE(?,O(1)) + Considered Problem: - Weak DPs: half#(s(s(X))) -> c_10(half#(X)) sqr#(s(X)) -> c_12(sqr#(X)) - Signature: {add/2,dbl/1,first/2,half/1,sqr/1,terms/1,add#/2,dbl#/1,first#/2,half#/1,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_8/0,c_9/0,c_10/1,c_11/0,c_12/1,c_13/1} - Obligation: innermost runtime complexity wrt. defined symbols {add#,dbl#,first#,half#,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: add#(s(X),Y) -> c_2(add#(X,Y)) dbl#(s(X)) -> c_4(dbl#(X)) - Weak DPs: half#(s(s(X))) -> half#(X) sqr#(s(X)) -> add#(sqr(X),dbl(X)) sqr#(s(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,half/1,sqr/1,terms/1,add#/2,dbl#/1,first#/2,half#/1,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_8/0,c_9/0,c_10/1,c_11/0,c_12/3,c_13/1} - Obligation: innermost runtime complexity wrt. defined symbols {add#,dbl#,first#,half#,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:S:dbl#(s(X)) -> c_4(dbl#(X)) -->_1 dbl#(s(X)) -> c_4(dbl#(X)):2 3:W:half#(s(s(X))) -> half#(X) -->_1 half#(s(s(X))) -> half#(X):3 4:W:sqr#(s(X)) -> add#(sqr(X),dbl(X)) -->_1 add#(s(X),Y) -> c_2(add#(X,Y)):1 5:W:sqr#(s(X)) -> dbl#(X) -->_1 dbl#(s(X)) -> c_4(dbl#(X)):2 6:W:sqr#(s(X)) -> sqr#(X) -->_1 sqr#(s(X)) -> sqr#(X):6 -->_1 sqr#(s(X)) -> dbl#(X):5 -->_1 sqr#(s(X)) -> add#(sqr(X),dbl(X)):4 The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed. 3: half#(s(s(X))) -> half#(X) ** Step 6.b:2: NaturalMI WORST_CASE(?,O(n^2)) + Considered Problem: - Strict DPs: add#(s(X),Y) -> c_2(add#(X,Y)) dbl#(s(X)) -> c_4(dbl#(X)) - Weak DPs: sqr#(s(X)) -> add#(sqr(X),dbl(X)) sqr#(s(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,half/1,sqr/1,terms/1,add#/2,dbl#/1,first#/2,half#/1,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_8/0,c_9/0,c_10/1,c_11/0,c_12/3,c_13/1} - Obligation: innermost runtime complexity wrt. defined symbols {add#,dbl#,first#,half#,sqr#,terms#} and constructors {0 ,cons,nil,recip,s} + 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}, uargs(c_4) = {1} Following symbols are considered usable: {add#,dbl#,first#,half#,sqr#,terms#} TcT has computed the following interpretation: p(0) = [0] p(add) = [1] x1 + [8] x2 + [7] p(cons) = [1] x1 + [2] p(dbl) = [2] x1 + [1] p(first) = [1] x1 + [1] x2 + [0] p(half) = [1] x1 + [0] p(nil) = [0] p(recip) = [1] p(s) = [1] x1 + [2] p(sqr) = [1] p(terms) = [1] x1 + [0] p(add#) = [0] p(dbl#) = [9] x1 + [0] p(first#) = [1] x1 + [1] x2 + [4] p(half#) = [2] x1 + [2] p(sqr#) = [10] x1 + [2] p(terms#) = [0] p(c_1) = [4] p(c_2) = [2] x1 + [0] p(c_3) = [0] p(c_4) = [1] x1 + [13] p(c_5) = [4] p(c_6) = [2] p(c_7) = [1] p(c_8) = [0] p(c_9) = [1] p(c_10) = [2] x1 + [1] p(c_11) = [0] p(c_12) = [4] p(c_13) = [1] x1 + [0] Following rules are strictly oriented: dbl#(s(X)) = [9] X + [18] > [9] X + [13] = c_4(dbl#(X)) Following rules are (at-least) weakly oriented: add#(s(X),Y) = [0] >= [0] = c_2(add#(X,Y)) sqr#(s(X)) = [10] X + [22] >= [0] = add#(sqr(X),dbl(X)) sqr#(s(X)) = [10] X + [22] >= [9] X + [0] = dbl#(X) sqr#(s(X)) = [10] X + [22] >= [10] X + [2] = sqr#(X) ** Step 6.b:3: Ara WORST_CASE(?,O(n^2)) + 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)) -> add#(sqr(X),dbl(X)) sqr#(s(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,half/1,sqr/1,terms/1,add#/2,dbl#/1,first#/2,half#/1,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_8/0,c_9/0,c_10/1,c_11/0,c_12/3,c_13/1} - Obligation: innermost runtime complexity wrt. defined symbols {add#,dbl#,first#,half#,sqr#,terms#} and constructors {0 ,cons,nil,recip,s} + Applied Processor: Ara {araHeuristics = NoHeuristics, minDegree = 2, maxDegree = 2, araTimeout = 8, araRuleShifting = Just 1} + Details: Signatures used: ---------------- 0 :: [] -(0)-> "A"(2, 0) 0 :: [] -(0)-> "A"(7, 0) 0 :: [] -(0)-> "A"(0, 15) 0 :: [] -(0)-> "A"(5, 7) 0 :: [] -(0)-> "A"(4, 7) add :: ["A"(2, 0) x "A"(2, 0)] -(0)-> "A"(2, 0) dbl :: ["A"(7, 0)] -(9)-> "A"(2, 0) s :: ["A"(2, 0)] -(2)-> "A"(2, 0) s :: ["A"(7, 0)] -(7)-> "A"(7, 0) s :: ["A"(15, 15)] -(15)-> "A"(0, 15) s :: ["A"(9, 0)] -(9)-> "A"(9, 0) sqr :: ["A"(0, 15)] -(0)-> "A"(2, 0) add# :: ["A"(2, 0) x "A"(0, 0)] -(7)-> "A"(0, 1) dbl# :: ["A"(9, 0)] -(8)-> "A"(0, 8) sqr# :: ["A"(0, 15)] -(12)-> "A"(0, 1) c_2 :: ["A"(0, 0)] -(0)-> "A"(4, 4) c_4 :: ["A"(0, 8)] -(0)-> "A"(2, 8) Cost-free Signatures used: -------------------------- Base Constructor Signatures used: --------------------------------- "0_A" :: [] -(0)-> "A"(1, 0) "0_A" :: [] -(0)-> "A"(0, 1) "c_2_A" :: ["A"(0)] -(0)-> "A"(1, 0) "c_2_A" :: ["A"(0)] -(0)-> "A"(0, 1) "c_4_A" :: ["A"(0)] -(0)-> "A"(1, 0) "c_4_A" :: ["A"(0)] -(0)-> "A"(0, 1) "s_A" :: ["A"(1, 0)] -(1)-> "A"(1, 0) "s_A" :: ["A"(1, 1)] -(1)-> "A"(0, 1) Following Still Strict Rules were Typed as: ------------------------------------------- 1. Strict: add#(s(X),Y) -> c_2(add#(X,Y)) 2. Weak: WORST_CASE(?,O(n^3))