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: 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() 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: 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() 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)) * 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() 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))) 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: 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 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)) 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))) 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#(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 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)) 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))) 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: 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 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)) 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: 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)) 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} Problem (S) - Strict DPs: 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 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,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} ** 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)) 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: 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:half#(s(s(X))) -> c_10(half#(X)) -->_1 half#(s(s(X))) -> c_10(half#(X)):3 4:W:sqr#(s(X)) -> c_12(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_12(add#(sqr(X),dbl(X)),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))) -> c_10(half#(X)) 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_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:add#(s(X),Y) -> c_2(add#(X,Y)) -->_1 add#(s(X),Y) -> c_2(add#(X,Y)):1 4:W:sqr#(s(X)) -> c_12(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_12(add#(sqr(X),dbl(X)),sqr#(X),dbl#(X)):4 Due to missing edges in the depndency graph, the right-hand sides of following rules could be simplified: sqr#(s(X)) -> c_12(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_12(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,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_12/2,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 = Just someStrategy, onLower = Nothing} + Details: We decompose the input problem according to the dependency graph into the upper component sqr#(s(X)) -> c_12(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_12(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,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_12/2,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: 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_12(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_12(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,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_12/2,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 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_12) = {1,2} Following symbols are considered usable: {add#,dbl#,first#,half#,sqr#,terms#} TcT has computed the following interpretation: p(0) = [2] p(add) = [2] p(cons) = [1] x1 + [0] p(dbl) = [2] x1 + [14] p(first) = [0] p(half) = [0] p(nil) = [0] p(recip) = [1] x1 + [0] p(s) = [1] x1 + [1] p(sqr) = [0] p(terms) = [4] x1 + [0] p(add#) = [0] p(dbl#) = [0] p(first#) = [1] x1 + [2] x2 + [0] p(half#) = [0] p(sqr#) = [4] x1 + [0] p(terms#) = [1] x1 + [0] p(c_1) = [0] p(c_2) = [2] x1 + [0] p(c_3) = [0] p(c_4) = [1] x1 + [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 + [0] p(c_11) = [0] p(c_12) = [4] x1 + [1] x2 + [0] p(c_13) = [0] Following rules are strictly oriented: sqr#(s(X)) = [4] X + [4] > [4] X + [0] = c_12(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_12(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,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_12/2,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: 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_12(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,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_12/2,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:W:sqr#(s(X)) -> c_12(add#(sqr(X),dbl(X)),sqr#(X)) -->_2 sqr#(s(X)) -> c_12(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_12(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,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_12/2,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.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,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_12/2,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: 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,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_12/2,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: 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#,half#,sqr#,terms#} TcT has computed the following interpretation: p(0) = 0 p(add) = 1 + x1 + x2 p(cons) = 1 p(dbl) = 3*x1 p(first) = 1 + x1 + 4*x1^2 p(half) = 0 p(nil) = 0 p(recip) = x1 p(s) = 1 + x1 p(sqr) = x1 + 2*x1^2 p(terms) = 1 p(add#) = 2*x1 p(dbl#) = x1^2 p(first#) = 1 + x2 + x2^2 p(half#) = 0 p(sqr#) = 2 + 6*x1 + 4*x1^2 p(terms#) = 0 p(c_1) = 1 p(c_2) = x1 p(c_3) = 2 p(c_4) = 1 + x1 p(c_5) = 0 p(c_6) = 2 p(c_7) = 0 p(c_8) = 4 p(c_9) = 1 p(c_10) = 1 + x1 p(c_11) = 4 p(c_12) = 4 p(c_13) = 4 + x1 Following rules are strictly oriented: add#(s(X),Y) = 2 + 2*X > 2*X = c_2(add#(X,Y)) Following rules are (at-least) weakly oriented: sqr#(s(X)) = 12 + 14*X + 4*X^2 >= 2*X + 4*X^2 = add#(sqr(X),dbl(X)) sqr#(s(X)) = 12 + 14*X + 4*X^2 >= 2 + 6*X + 4*X^2 = sqr#(X) add(0(),X) = 1 + X >= X = X add(s(X),Y) = 2 + X + Y >= 2 + X + Y = s(add(X,Y)) dbl(0()) = 0 >= 0 = 0() dbl(s(X)) = 3 + 3*X >= 2 + 3*X = s(s(dbl(X))) sqr(0()) = 0 >= 0 = 0() sqr(s(X)) = 3 + 5*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,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_12/2,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: 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,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_12/2,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: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,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_12/2,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: 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 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,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:dbl#(s(X)) -> c_4(dbl#(X)) -->_1 dbl#(s(X)) -> c_4(dbl#(X)):1 2:S:half#(s(s(X))) -> c_10(half#(X)) -->_1 half#(s(s(X))) -> c_10(half#(X)):2 3:S:sqr#(s(X)) -> c_12(add#(sqr(X),dbl(X)),sqr#(X),dbl#(X)) -->_1 add#(s(X),Y) -> c_2(add#(X,Y)):4 -->_2 sqr#(s(X)) -> c_12(add#(sqr(X),dbl(X)),sqr#(X),dbl#(X)):3 -->_3 dbl#(s(X)) -> c_4(dbl#(X)):1 4:W:add#(s(X),Y) -> c_2(add#(X,Y)) -->_1 add#(s(X),Y) -> c_2(add#(X,Y)):4 The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed. 4: 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)) 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:dbl#(s(X)) -> c_4(dbl#(X)) -->_1 dbl#(s(X)) -> c_4(dbl#(X)):1 2:S:half#(s(s(X))) -> c_10(half#(X)) -->_1 half#(s(s(X))) -> c_10(half#(X)):2 3: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)):3 -->_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_12(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)) half#(s(s(X))) -> c_10(half#(X)) sqr#(s(X)) -> c_12(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_12/2,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: dbl#(s(X)) -> c_4(dbl#(X)) half#(s(s(X))) -> c_10(half#(X)) sqr#(s(X)) -> c_12(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)) half#(s(s(X))) -> c_10(half#(X)) sqr#(s(X)) -> c_12(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_12/2,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: 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: half#(s(s(X))) -> c_10(half#(X)) sqr#(s(X)) -> c_12(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_12/2,c_13/1} - Obligation: innermost runtime complexity wrt. defined symbols {add#,dbl#,first#,half#,sqr#,terms#} and constructors {0 ,cons,nil,recip,s} Problem (S) - Strict DPs: half#(s(s(X))) -> c_10(half#(X)) sqr#(s(X)) -> c_12(sqr#(X),dbl#(X)) - Weak DPs: dbl#(s(X)) -> c_4(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_12/2,c_13/1} - Obligation: innermost runtime complexity wrt. defined symbols {add#,dbl#,first#,half#,sqr#,terms#} and constructors {0 ,cons,nil,recip,s} *** Step 6.b:4.a:1: RemoveWeakSuffixes WORST_CASE(?,O(n^2)) + Considered Problem: - Strict DPs: dbl#(s(X)) -> c_4(dbl#(X)) - Weak DPs: half#(s(s(X))) -> c_10(half#(X)) sqr#(s(X)) -> c_12(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_12/2,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:dbl#(s(X)) -> c_4(dbl#(X)) -->_1 dbl#(s(X)) -> c_4(dbl#(X)):1 2:W:half#(s(s(X))) -> c_10(half#(X)) -->_1 half#(s(s(X))) -> c_10(half#(X)):2 3:W:sqr#(s(X)) -> c_12(sqr#(X),dbl#(X)) -->_2 dbl#(s(X)) -> c_4(dbl#(X)):1 -->_1 sqr#(s(X)) -> c_12(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: half#(s(s(X))) -> c_10(half#(X)) *** Step 6.b:4.a:2: PredecessorEstimationCP WORST_CASE(?,O(n^2)) + Considered Problem: - Strict DPs: dbl#(s(X)) -> c_4(dbl#(X)) - Weak DPs: sqr#(s(X)) -> c_12(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_12/2,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: 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:2.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_12(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_12/2,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: 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_12) = {1,2} Following symbols are considered usable: {add#,dbl#,first#,half#,sqr#,terms#} TcT has computed the following interpretation: p(0) = 0 p(add) = x1 + 8*x1*x2 + 2*x1^2 + 2*x2 + 8*x2^2 p(cons) = 1 p(dbl) = 1 + x1^2 p(first) = 1 + x1*x2 + x2 + 2*x2^2 p(half) = 2 + x1^2 p(nil) = 0 p(recip) = 1 + x1 p(s) = 2 + x1 p(sqr) = 1 + x1 + x1^2 p(terms) = 0 p(add#) = 1 + 2*x1 + x1*x2 + 2*x1^2 + 2*x2 + 4*x2^2 p(dbl#) = x1 p(first#) = 2 + 4*x1 + x1*x2 + x2^2 p(half#) = 1 + x1 + x1^2 p(sqr#) = 2 + 2*x1 + 3*x1^2 p(terms#) = 8 p(c_1) = 0 p(c_2) = 2 p(c_3) = 2 p(c_4) = x1 p(c_5) = 1 p(c_6) = 2 p(c_7) = 2 p(c_8) = 0 p(c_9) = 0 p(c_10) = x1 p(c_11) = 8 p(c_12) = 1 + x1 + x2 p(c_13) = 2 Following rules are strictly oriented: dbl#(s(X)) = 2 + X > X = c_4(dbl#(X)) Following rules are (at-least) weakly oriented: sqr#(s(X)) = 18 + 14*X + 3*X^2 >= 3 + 3*X + 3*X^2 = c_12(sqr#(X),dbl#(X)) **** Step 6.b:4.a:2.a:2: Assumption WORST_CASE(?,O(1)) + Considered Problem: - Weak DPs: dbl#(s(X)) -> c_4(dbl#(X)) sqr#(s(X)) -> c_12(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_12/2,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: Assumption {assumed = Certificate {spaceUB = Unknown, spaceLB = Unknown, timeUB = Poly (Just 0), timeLB = Unknown}} + Details: () **** Step 6.b:4.a:2.b:1: RemoveWeakSuffixes WORST_CASE(?,O(1)) + Considered Problem: - Weak DPs: dbl#(s(X)) -> c_4(dbl#(X)) sqr#(s(X)) -> c_12(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_12/2,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:W:dbl#(s(X)) -> c_4(dbl#(X)) -->_1 dbl#(s(X)) -> c_4(dbl#(X)):1 2:W:sqr#(s(X)) -> c_12(sqr#(X),dbl#(X)) -->_1 sqr#(s(X)) -> c_12(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_12(sqr#(X),dbl#(X)) 1: dbl#(s(X)) -> c_4(dbl#(X)) **** Step 6.b:4.a:2.b:2: EmptyProcessor WORST_CASE(?,O(1)) + Considered Problem: - 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_12/2,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:4.b:1: RemoveWeakSuffixes WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: half#(s(s(X))) -> c_10(half#(X)) sqr#(s(X)) -> c_12(sqr#(X),dbl#(X)) - Weak DPs: dbl#(s(X)) -> c_4(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_12/2,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: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(sqr#(X),dbl#(X)) -->_2 dbl#(s(X)) -> c_4(dbl#(X)):3 -->_1 sqr#(s(X)) -> c_12(sqr#(X),dbl#(X)):2 3:W:dbl#(s(X)) -> c_4(dbl#(X)) -->_1 dbl#(s(X)) -> c_4(dbl#(X)):3 The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed. 3: dbl#(s(X)) -> c_4(dbl#(X)) *** Step 6.b:4.b:2: SimplifyRHS WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: half#(s(s(X))) -> c_10(half#(X)) sqr#(s(X)) -> c_12(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_12/2,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(sqr#(X),dbl#(X)) -->_1 sqr#(s(X)) -> c_12(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.b:4.b:3: Decompose 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: 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: 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} Problem (S) - Strict DPs: sqr#(s(X)) -> c_12(sqr#(X)) - Weak DPs: half#(s(s(X))) -> c_10(half#(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} **** Step 6.b:4.b:3.a:1: RemoveWeakSuffixes 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: RemoveWeakSuffixes + 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:W:sqr#(s(X)) -> c_12(sqr#(X)) -->_1 sqr#(s(X)) -> c_12(sqr#(X)):2 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_12(sqr#(X)) **** Step 6.b:4.b:3.a:2: PredecessorEstimationCP WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: half#(s(s(X))) -> c_10(half#(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: 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: half#(s(s(X))) -> c_10(half#(X)) The strictly oriented rules are moved into the weak component. ***** Step 6.b:4.b:3.a:2.a:1: NaturalMI WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: half#(s(s(X))) -> c_10(half#(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: 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_10) = {1} Following symbols are considered usable: {add#,dbl#,first#,half#,sqr#,terms#} TcT has computed the following interpretation: p(0) = [2] p(add) = [2] x2 + [1] p(cons) = [1] p(dbl) = [1] p(first) = [2] x1 + [1] x2 + [2] p(half) = [4] x1 + [2] p(nil) = [1] p(recip) = [0] p(s) = [1] x1 + [4] p(sqr) = [0] p(terms) = [2] x1 + [8] p(add#) = [1] x1 + [1] x2 + [0] p(dbl#) = [2] x1 + [2] p(first#) = [8] x1 + [1] x2 + [1] p(half#) = [1] x1 + [4] p(sqr#) = [2] x1 + [0] p(terms#) = [4] x1 + [2] p(c_1) = [2] p(c_2) = [1] p(c_3) = [0] p(c_4) = [2] x1 + [0] p(c_5) = [0] p(c_6) = [0] p(c_7) = [0] p(c_8) = [0] p(c_9) = [1] p(c_10) = [1] x1 + [0] p(c_11) = [0] p(c_12) = [1] x1 + [2] p(c_13) = [2] Following rules are strictly oriented: half#(s(s(X))) = [1] X + [12] > [1] X + [4] = c_10(half#(X)) Following rules are (at-least) weakly oriented: ***** Step 6.b:4.b:3.a:2.a:2: Assumption WORST_CASE(?,O(1)) + Considered Problem: - Weak DPs: half#(s(s(X))) -> c_10(half#(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: Assumption {assumed = Certificate {spaceUB = Unknown, spaceLB = Unknown, timeUB = Poly (Just 0), timeLB = Unknown}} + Details: () ***** Step 6.b:4.b:3.a:2.b:1: RemoveWeakSuffixes WORST_CASE(?,O(1)) + Considered Problem: - Weak DPs: half#(s(s(X))) -> c_10(half#(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: RemoveWeakSuffixes + Details: Consider the dependency graph 1:W:half#(s(s(X))) -> c_10(half#(X)) -->_1 half#(s(s(X))) -> c_10(half#(X)):1 The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed. 1: half#(s(s(X))) -> c_10(half#(X)) ***** Step 6.b:4.b:3.a:2.b:2: EmptyProcessor WORST_CASE(?,O(1)) + Considered Problem: - 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:4.b:3.b:1: RemoveWeakSuffixes WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: sqr#(s(X)) -> c_12(sqr#(X)) - Weak DPs: half#(s(s(X))) -> c_10(half#(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: RemoveWeakSuffixes + Details: Consider the dependency graph 1:S:sqr#(s(X)) -> c_12(sqr#(X)) -->_1 sqr#(s(X)) -> c_12(sqr#(X)):1 2:W:half#(s(s(X))) -> c_10(half#(X)) -->_1 half#(s(s(X))) -> c_10(half#(X)):2 The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed. 2: half#(s(s(X))) -> c_10(half#(X)) **** Step 6.b:4.b:3.b:2: PredecessorEstimationCP WORST_CASE(?,O(n^1)) + Considered Problem: - Strict 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: 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_12(sqr#(X)) The strictly oriented rules are moved into the weak component. ***** Step 6.b:4.b:3.b:2.a:1: NaturalMI WORST_CASE(?,O(n^1)) + Considered Problem: - Strict 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: 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_12) = {1} Following symbols are considered usable: {add#,dbl#,first#,half#,sqr#,terms#} TcT has computed the following interpretation: p(0) = [0] p(add) = [0] p(cons) = [1] x1 + [0] p(dbl) = [0] p(first) = [0] p(half) = [0] p(nil) = [0] p(recip) = [1] x1 + [0] p(s) = [1] x1 + [4] p(sqr) = [0] p(terms) = [0] p(add#) = [0] p(dbl#) = [1] x1 + [0] p(first#) = [8] x2 + [1] p(half#) = [8] x1 + [8] p(sqr#) = [4] x1 + [8] p(terms#) = [1] x1 + [1] p(c_1) = [4] p(c_2) = [1] p(c_3) = [2] p(c_4) = [2] p(c_5) = [4] p(c_6) = [8] p(c_7) = [1] p(c_8) = [1] p(c_9) = [1] p(c_10) = [1] x1 + [8] p(c_11) = [1] p(c_12) = [1] x1 + [8] p(c_13) = [2] x1 + [0] Following rules are strictly oriented: sqr#(s(X)) = [4] X + [24] > [4] X + [16] = c_12(sqr#(X)) Following rules are (at-least) weakly oriented: ***** Step 6.b:4.b:3.b:2.a:2: Assumption WORST_CASE(?,O(1)) + Considered Problem: - 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: Assumption {assumed = Certificate {spaceUB = Unknown, spaceLB = Unknown, timeUB = Poly (Just 0), timeLB = Unknown}} + Details: () ***** Step 6.b:4.b:3.b:2.b:1: RemoveWeakSuffixes WORST_CASE(?,O(1)) + Considered Problem: - 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: RemoveWeakSuffixes + Details: Consider the dependency graph 1:W:sqr#(s(X)) -> c_12(sqr#(X)) -->_1 sqr#(s(X)) -> c_12(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_12(sqr#(X)) ***** Step 6.b:4.b:3.b:2.b:2: EmptyProcessor WORST_CASE(?,O(1)) + Considered Problem: - 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). WORST_CASE(?,O(n^3))