MAYBE * Step 1: DependencyPairs MAYBE + 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,Z)) -> cons(Y,first(X,Z)) 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)),terms(s(N))) - Signature: {add/2,dbl/1,first/2,half/1,sqr/1,terms/1} / {0/0,cons/2,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,Z)) -> c_6(first#(X,Z)) 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),terms#(s(N))) Weak DPs and mark the set of starting terms. * Step 2: UsableRules MAYBE + 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,Z)) -> c_6(first#(X,Z)) 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),terms#(s(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,Z)) -> cons(Y,first(X,Z)) 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)),terms(s(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/2 ,nil/0,recip/1,s/1,c_1/0,c_2/1,c_3/0,c_4/1,c_5/0,c_6/1,c_7/0,c_8/0,c_9/0,c_10/1,c_11/0,c_12/3,c_13/2} - 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,Z)) -> c_6(first#(X,Z)) 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),terms#(s(N))) * Step 3: PredecessorEstimation MAYBE + 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,Z)) -> c_6(first#(X,Z)) 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),terms#(s(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/2 ,nil/0,recip/1,s/1,c_1/0,c_2/1,c_3/0,c_4/1,c_5/0,c_6/1,c_7/0,c_8/0,c_9/0,c_10/1,c_11/0,c_12/3,c_13/2} - 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 {1,3,5,7,8,9,11} by application of Pre({1,3,5,7,8,9,11}) = {2,4,6,10,12,13}. 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,Z)) -> c_6(first#(X,Z)) 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),terms#(s(N))) * Step 4: RemoveWeakSuffixes MAYBE + Considered Problem: - Strict DPs: add#(s(X),Y) -> c_2(add#(X,Y)) dbl#(s(X)) -> c_4(dbl#(X)) first#(s(X),cons(Y,Z)) -> c_6(first#(X,Z)) 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),terms#(s(N))) - Weak DPs: add#(0(),X) -> c_1() dbl#(0()) -> c_3() first#(0(),X) -> c_5() 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/2 ,nil/0,recip/1,s/1,c_1/0,c_2/1,c_3/0,c_4/1,c_5/0,c_6/1,c_7/0,c_8/0,c_9/0,c_10/1,c_11/0,c_12/3,c_13/2} - 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():7 -->_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():8 -->_1 dbl#(s(X)) -> c_4(dbl#(X)):2 3:S:first#(s(X),cons(Y,Z)) -> c_6(first#(X,Z)) -->_1 first#(0(),X) -> c_5():9 -->_1 first#(s(X),cons(Y,Z)) -> c_6(first#(X,Z)):3 4: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)):4 5: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():8 -->_1 add#(0(),X) -> c_1():7 -->_2 sqr#(s(X)) -> c_12(add#(sqr(X),dbl(X)),sqr#(X),dbl#(X)):5 -->_3 dbl#(s(X)) -> c_4(dbl#(X)):2 -->_1 add#(s(X),Y) -> c_2(add#(X,Y)):1 6:S:terms#(N) -> c_13(sqr#(N),terms#(s(N))) -->_1 sqr#(0()) -> c_11():13 -->_2 terms#(N) -> c_13(sqr#(N),terms#(s(N))):6 -->_1 sqr#(s(X)) -> c_12(add#(sqr(X),dbl(X)),sqr#(X),dbl#(X)):5 7:W:add#(0(),X) -> c_1() 8:W:dbl#(0()) -> c_3() 9:W:first#(0(),X) -> c_5() 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. 13: sqr#(0()) -> c_11() 10: half#(0()) -> c_7() 11: half#(dbl(X)) -> c_8() 12: half#(s(0())) -> c_9() 9: first#(0(),X) -> c_5() 8: dbl#(0()) -> c_3() 7: add#(0(),X) -> c_1() * Step 5: Decompose MAYBE + Considered Problem: - Strict DPs: add#(s(X),Y) -> c_2(add#(X,Y)) dbl#(s(X)) -> c_4(dbl#(X)) first#(s(X),cons(Y,Z)) -> c_6(first#(X,Z)) 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),terms#(s(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/2 ,nil/0,recip/1,s/1,c_1/0,c_2/1,c_3/0,c_4/1,c_5/0,c_6/1,c_7/0,c_8/0,c_9/0,c_10/1,c_11/0,c_12/3,c_13/2} - 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)) first#(s(X),cons(Y,Z)) -> c_6(first#(X,Z)) 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),terms#(s(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/2 ,nil/0,recip/1,s/1,c_1/0,c_2/1,c_3/0,c_4/1,c_5/0,c_6/1,c_7/0,c_8/0,c_9/0,c_10/1,c_11/0,c_12/3,c_13/2} - 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)) first#(s(X),cons(Y,Z)) -> c_6(first#(X,Z)) 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),terms#(s(N))) - 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/2 ,nil/0,recip/1,s/1,c_1/0,c_2/1,c_3/0,c_4/1,c_5/0,c_6/1,c_7/0,c_8/0,c_9/0,c_10/1,c_11/0,c_12/3,c_13/2} - Obligation: innermost runtime complexity wrt. defined symbols {add#,dbl#,first#,half#,sqr#,terms#} and constructors {0 ,cons,nil,recip,s} ** Step 5.a:1: RemoveWeakSuffixes MAYBE + Considered Problem: - Strict DPs: add#(s(X),Y) -> c_2(add#(X,Y)) - Weak DPs: dbl#(s(X)) -> c_4(dbl#(X)) first#(s(X),cons(Y,Z)) -> c_6(first#(X,Z)) 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),terms#(s(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/2 ,nil/0,recip/1,s/1,c_1/0,c_2/1,c_3/0,c_4/1,c_5/0,c_6/1,c_7/0,c_8/0,c_9/0,c_10/1,c_11/0,c_12/3,c_13/2} - 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:first#(s(X),cons(Y,Z)) -> c_6(first#(X,Z)) -->_1 first#(s(X),cons(Y,Z)) -> c_6(first#(X,Z)):3 4:W:half#(s(s(X))) -> c_10(half#(X)) -->_1 half#(s(s(X))) -> c_10(half#(X)):4 5: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)):5 6:W:terms#(N) -> c_13(sqr#(N),terms#(s(N))) -->_1 sqr#(s(X)) -> c_12(add#(sqr(X),dbl(X)),sqr#(X),dbl#(X)):5 -->_2 terms#(N) -> c_13(sqr#(N),terms#(s(N))):6 The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed. 4: half#(s(s(X))) -> c_10(half#(X)) 3: first#(s(X),cons(Y,Z)) -> c_6(first#(X,Z)) 2: dbl#(s(X)) -> c_4(dbl#(X)) ** Step 5.a:2: SimplifyRHS MAYBE + 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)) terms#(N) -> c_13(sqr#(N),terms#(s(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/2 ,nil/0,recip/1,s/1,c_1/0,c_2/1,c_3/0,c_4/1,c_5/0,c_6/1,c_7/0,c_8/0,c_9/0,c_10/1,c_11/0,c_12/3,c_13/2} - 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 5: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)):5 6:W:terms#(N) -> c_13(sqr#(N),terms#(s(N))) -->_1 sqr#(s(X)) -> c_12(add#(sqr(X),dbl(X)),sqr#(X),dbl#(X)):5 -->_2 terms#(N) -> c_13(sqr#(N),terms#(s(N))):6 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 5.a:3: Failure MAYBE + 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)) terms#(N) -> c_13(sqr#(N),terms#(s(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/2 ,nil/0,recip/1,s/1,c_1/0,c_2/1,c_3/0,c_4/1,c_5/0,c_6/1,c_7/0,c_8/0,c_9/0,c_10/1,c_11/0,c_12/2,c_13/2} - 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 still open. ** Step 5.b:1: RemoveWeakSuffixes MAYBE + Considered Problem: - Strict DPs: dbl#(s(X)) -> c_4(dbl#(X)) first#(s(X),cons(Y,Z)) -> c_6(first#(X,Z)) 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),terms#(s(N))) - 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/2 ,nil/0,recip/1,s/1,c_1/0,c_2/1,c_3/0,c_4/1,c_5/0,c_6/1,c_7/0,c_8/0,c_9/0,c_10/1,c_11/0,c_12/3,c_13/2} - 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:first#(s(X),cons(Y,Z)) -> c_6(first#(X,Z)) -->_1 first#(s(X),cons(Y,Z)) -> c_6(first#(X,Z)):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)) -->_1 add#(s(X),Y) -> c_2(add#(X,Y)):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)):1 5:S:terms#(N) -> c_13(sqr#(N),terms#(s(N))) -->_2 terms#(N) -> c_13(sqr#(N),terms#(s(N))):5 -->_1 sqr#(s(X)) -> c_12(add#(sqr(X),dbl(X)),sqr#(X),dbl#(X)):4 6:W:add#(s(X),Y) -> c_2(add#(X,Y)) -->_1 add#(s(X),Y) -> c_2(add#(X,Y)):6 The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed. 6: add#(s(X),Y) -> c_2(add#(X,Y)) ** Step 5.b:2: SimplifyRHS MAYBE + Considered Problem: - Strict DPs: dbl#(s(X)) -> c_4(dbl#(X)) first#(s(X),cons(Y,Z)) -> c_6(first#(X,Z)) 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),terms#(s(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/2 ,nil/0,recip/1,s/1,c_1/0,c_2/1,c_3/0,c_4/1,c_5/0,c_6/1,c_7/0,c_8/0,c_9/0,c_10/1,c_11/0,c_12/3,c_13/2} - 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:first#(s(X),cons(Y,Z)) -> c_6(first#(X,Z)) -->_1 first#(s(X),cons(Y,Z)) -> c_6(first#(X,Z)):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)):1 5:S:terms#(N) -> c_13(sqr#(N),terms#(s(N))) -->_2 terms#(N) -> c_13(sqr#(N),terms#(s(N))):5 -->_1 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(sqr#(X),dbl#(X)) ** Step 5.b:3: UsableRules MAYBE + Considered Problem: - Strict DPs: dbl#(s(X)) -> c_4(dbl#(X)) first#(s(X),cons(Y,Z)) -> c_6(first#(X,Z)) half#(s(s(X))) -> c_10(half#(X)) sqr#(s(X)) -> c_12(sqr#(X),dbl#(X)) terms#(N) -> c_13(sqr#(N),terms#(s(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/2 ,nil/0,recip/1,s/1,c_1/0,c_2/1,c_3/0,c_4/1,c_5/0,c_6/1,c_7/0,c_8/0,c_9/0,c_10/1,c_11/0,c_12/2,c_13/2} - 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)) first#(s(X),cons(Y,Z)) -> c_6(first#(X,Z)) half#(s(s(X))) -> c_10(half#(X)) sqr#(s(X)) -> c_12(sqr#(X),dbl#(X)) terms#(N) -> c_13(sqr#(N),terms#(s(N))) ** Step 5.b:4: Decompose MAYBE + Considered Problem: - Strict DPs: dbl#(s(X)) -> c_4(dbl#(X)) first#(s(X),cons(Y,Z)) -> c_6(first#(X,Z)) half#(s(s(X))) -> c_10(half#(X)) sqr#(s(X)) -> c_12(sqr#(X),dbl#(X)) terms#(N) -> c_13(sqr#(N),terms#(s(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/2 ,nil/0,recip/1,s/1,c_1/0,c_2/1,c_3/0,c_4/1,c_5/0,c_6/1,c_7/0,c_8/0,c_9/0,c_10/1,c_11/0,c_12/2,c_13/2} - 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: first#(s(X),cons(Y,Z)) -> c_6(first#(X,Z)) half#(s(s(X))) -> c_10(half#(X)) sqr#(s(X)) -> c_12(sqr#(X),dbl#(X)) terms#(N) -> c_13(sqr#(N),terms#(s(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/2 ,nil/0,recip/1,s/1,c_1/0,c_2/1,c_3/0,c_4/1,c_5/0,c_6/1,c_7/0,c_8/0,c_9/0,c_10/1,c_11/0,c_12/2,c_13/2} - Obligation: innermost runtime complexity wrt. defined symbols {add#,dbl#,first#,half#,sqr#,terms#} and constructors {0 ,cons,nil,recip,s} Problem (S) - Strict DPs: first#(s(X),cons(Y,Z)) -> c_6(first#(X,Z)) half#(s(s(X))) -> c_10(half#(X)) sqr#(s(X)) -> c_12(sqr#(X),dbl#(X)) terms#(N) -> c_13(sqr#(N),terms#(s(N))) - 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/2 ,nil/0,recip/1,s/1,c_1/0,c_2/1,c_3/0,c_4/1,c_5/0,c_6/1,c_7/0,c_8/0,c_9/0,c_10/1,c_11/0,c_12/2,c_13/2} - Obligation: innermost runtime complexity wrt. defined symbols {add#,dbl#,first#,half#,sqr#,terms#} and constructors {0 ,cons,nil,recip,s} *** Step 5.b:4.a:1: RemoveWeakSuffixes MAYBE + Considered Problem: - Strict DPs: dbl#(s(X)) -> c_4(dbl#(X)) - Weak DPs: first#(s(X),cons(Y,Z)) -> c_6(first#(X,Z)) half#(s(s(X))) -> c_10(half#(X)) sqr#(s(X)) -> c_12(sqr#(X),dbl#(X)) terms#(N) -> c_13(sqr#(N),terms#(s(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/2 ,nil/0,recip/1,s/1,c_1/0,c_2/1,c_3/0,c_4/1,c_5/0,c_6/1,c_7/0,c_8/0,c_9/0,c_10/1,c_11/0,c_12/2,c_13/2} - 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:first#(s(X),cons(Y,Z)) -> c_6(first#(X,Z)) -->_1 first#(s(X),cons(Y,Z)) -> c_6(first#(X,Z)):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(sqr#(X),dbl#(X)) -->_2 dbl#(s(X)) -> c_4(dbl#(X)):1 -->_1 sqr#(s(X)) -> c_12(sqr#(X),dbl#(X)):4 5:W:terms#(N) -> c_13(sqr#(N),terms#(s(N))) -->_1 sqr#(s(X)) -> c_12(sqr#(X),dbl#(X)):4 -->_2 terms#(N) -> c_13(sqr#(N),terms#(s(N))):5 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: first#(s(X),cons(Y,Z)) -> c_6(first#(X,Z)) *** Step 5.b:4.a:2: Failure MAYBE + Considered Problem: - Strict DPs: dbl#(s(X)) -> c_4(dbl#(X)) - Weak DPs: sqr#(s(X)) -> c_12(sqr#(X),dbl#(X)) terms#(N) -> c_13(sqr#(N),terms#(s(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/2 ,nil/0,recip/1,s/1,c_1/0,c_2/1,c_3/0,c_4/1,c_5/0,c_6/1,c_7/0,c_8/0,c_9/0,c_10/1,c_11/0,c_12/2,c_13/2} - 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 still open. *** Step 5.b:4.b:1: RemoveWeakSuffixes MAYBE + Considered Problem: - Strict DPs: first#(s(X),cons(Y,Z)) -> c_6(first#(X,Z)) half#(s(s(X))) -> c_10(half#(X)) sqr#(s(X)) -> c_12(sqr#(X),dbl#(X)) terms#(N) -> c_13(sqr#(N),terms#(s(N))) - 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/2 ,nil/0,recip/1,s/1,c_1/0,c_2/1,c_3/0,c_4/1,c_5/0,c_6/1,c_7/0,c_8/0,c_9/0,c_10/1,c_11/0,c_12/2,c_13/2} - 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:first#(s(X),cons(Y,Z)) -> c_6(first#(X,Z)) -->_1 first#(s(X),cons(Y,Z)) -> c_6(first#(X,Z)):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(sqr#(X),dbl#(X)) -->_2 dbl#(s(X)) -> c_4(dbl#(X)):5 -->_1 sqr#(s(X)) -> c_12(sqr#(X),dbl#(X)):3 4:S:terms#(N) -> c_13(sqr#(N),terms#(s(N))) -->_2 terms#(N) -> c_13(sqr#(N),terms#(s(N))):4 -->_1 sqr#(s(X)) -> c_12(sqr#(X),dbl#(X)):3 5:W:dbl#(s(X)) -> c_4(dbl#(X)) -->_1 dbl#(s(X)) -> c_4(dbl#(X)):5 The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed. 5: dbl#(s(X)) -> c_4(dbl#(X)) *** Step 5.b:4.b:2: SimplifyRHS MAYBE + Considered Problem: - Strict DPs: first#(s(X),cons(Y,Z)) -> c_6(first#(X,Z)) half#(s(s(X))) -> c_10(half#(X)) sqr#(s(X)) -> c_12(sqr#(X),dbl#(X)) terms#(N) -> c_13(sqr#(N),terms#(s(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/2 ,nil/0,recip/1,s/1,c_1/0,c_2/1,c_3/0,c_4/1,c_5/0,c_6/1,c_7/0,c_8/0,c_9/0,c_10/1,c_11/0,c_12/2,c_13/2} - 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:first#(s(X),cons(Y,Z)) -> c_6(first#(X,Z)) -->_1 first#(s(X),cons(Y,Z)) -> c_6(first#(X,Z)):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(sqr#(X),dbl#(X)) -->_1 sqr#(s(X)) -> c_12(sqr#(X),dbl#(X)):3 4:S:terms#(N) -> c_13(sqr#(N),terms#(s(N))) -->_2 terms#(N) -> c_13(sqr#(N),terms#(s(N))):4 -->_1 sqr#(s(X)) -> c_12(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_12(sqr#(X)) *** Step 5.b:4.b:3: Decompose MAYBE + Considered Problem: - Strict DPs: first#(s(X),cons(Y,Z)) -> c_6(first#(X,Z)) half#(s(s(X))) -> c_10(half#(X)) sqr#(s(X)) -> c_12(sqr#(X)) terms#(N) -> c_13(sqr#(N),terms#(s(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/2 ,nil/0,recip/1,s/1,c_1/0,c_2/1,c_3/0,c_4/1,c_5/0,c_6/1,c_7/0,c_8/0,c_9/0,c_10/1,c_11/0,c_12/1,c_13/2} - 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: first#(s(X),cons(Y,Z)) -> c_6(first#(X,Z)) - Weak DPs: half#(s(s(X))) -> c_10(half#(X)) sqr#(s(X)) -> c_12(sqr#(X)) terms#(N) -> c_13(sqr#(N),terms#(s(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/2 ,nil/0,recip/1,s/1,c_1/0,c_2/1,c_3/0,c_4/1,c_5/0,c_6/1,c_7/0,c_8/0,c_9/0,c_10/1,c_11/0,c_12/1,c_13/2} - 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)) terms#(N) -> c_13(sqr#(N),terms#(s(N))) - Weak DPs: first#(s(X),cons(Y,Z)) -> c_6(first#(X,Z)) - 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/2 ,nil/0,recip/1,s/1,c_1/0,c_2/1,c_3/0,c_4/1,c_5/0,c_6/1,c_7/0,c_8/0,c_9/0,c_10/1,c_11/0,c_12/1,c_13/2} - Obligation: innermost runtime complexity wrt. defined symbols {add#,dbl#,first#,half#,sqr#,terms#} and constructors {0 ,cons,nil,recip,s} **** Step 5.b:4.b:3.a:1: RemoveWeakSuffixes WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: first#(s(X),cons(Y,Z)) -> c_6(first#(X,Z)) - Weak DPs: half#(s(s(X))) -> c_10(half#(X)) sqr#(s(X)) -> c_12(sqr#(X)) terms#(N) -> c_13(sqr#(N),terms#(s(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/2 ,nil/0,recip/1,s/1,c_1/0,c_2/1,c_3/0,c_4/1,c_5/0,c_6/1,c_7/0,c_8/0,c_9/0,c_10/1,c_11/0,c_12/1,c_13/2} - 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:first#(s(X),cons(Y,Z)) -> c_6(first#(X,Z)) -->_1 first#(s(X),cons(Y,Z)) -> c_6(first#(X,Z)):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)) -->_1 sqr#(s(X)) -> c_12(sqr#(X)):3 4:W:terms#(N) -> c_13(sqr#(N),terms#(s(N))) -->_1 sqr#(s(X)) -> c_12(sqr#(X)):3 -->_2 terms#(N) -> c_13(sqr#(N),terms#(s(N))):4 The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed. 4: terms#(N) -> c_13(sqr#(N),terms#(s(N))) 3: sqr#(s(X)) -> c_12(sqr#(X)) 2: half#(s(s(X))) -> c_10(half#(X)) **** Step 5.b:4.b:3.a:2: PredecessorEstimationCP WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: first#(s(X),cons(Y,Z)) -> c_6(first#(X,Z)) - 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/2 ,nil/0,recip/1,s/1,c_1/0,c_2/1,c_3/0,c_4/1,c_5/0,c_6/1,c_7/0,c_8/0,c_9/0,c_10/1,c_11/0,c_12/1,c_13/2} - 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: first#(s(X),cons(Y,Z)) -> c_6(first#(X,Z)) The strictly oriented rules are moved into the weak component. ***** Step 5.b:4.b:3.a:2.a:1: NaturalMI WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: first#(s(X),cons(Y,Z)) -> c_6(first#(X,Z)) - 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/2 ,nil/0,recip/1,s/1,c_1/0,c_2/1,c_3/0,c_4/1,c_5/0,c_6/1,c_7/0,c_8/0,c_9/0,c_10/1,c_11/0,c_12/1,c_13/2} - 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_6) = {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 + [2] x2 + [0] p(cons) = [1] x2 + [1] p(dbl) = [1] p(first) = [2] p(half) = [0] p(nil) = [0] p(recip) = [1] p(s) = [1] x1 + [9] p(sqr) = [2] p(terms) = [1] p(add#) = [1] x1 + [4] x2 + [0] p(dbl#) = [2] p(first#) = [1] x1 + [12] x2 + [8] p(half#) = [2] p(sqr#) = [2] x1 + [1] p(terms#) = [2] x1 + [1] p(c_1) = [1] p(c_2) = [8] x1 + [1] p(c_3) = [8] p(c_4) = [1] x1 + [0] p(c_5) = [2] p(c_6) = [1] x1 + [13] p(c_7) = [4] p(c_8) = [0] p(c_9) = [2] p(c_10) = [1] x1 + [1] p(c_11) = [0] p(c_12) = [4] x1 + [2] p(c_13) = [4] x1 + [2] Following rules are strictly oriented: first#(s(X),cons(Y,Z)) = [1] X + [12] Z + [29] > [1] X + [12] Z + [21] = c_6(first#(X,Z)) Following rules are (at-least) weakly oriented: ***** Step 5.b:4.b:3.a:2.a:2: Assumption WORST_CASE(?,O(1)) + Considered Problem: - Weak DPs: first#(s(X),cons(Y,Z)) -> c_6(first#(X,Z)) - 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/2 ,nil/0,recip/1,s/1,c_1/0,c_2/1,c_3/0,c_4/1,c_5/0,c_6/1,c_7/0,c_8/0,c_9/0,c_10/1,c_11/0,c_12/1,c_13/2} - 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 5.b:4.b:3.a:2.b:1: RemoveWeakSuffixes WORST_CASE(?,O(1)) + Considered Problem: - Weak DPs: first#(s(X),cons(Y,Z)) -> c_6(first#(X,Z)) - 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/2 ,nil/0,recip/1,s/1,c_1/0,c_2/1,c_3/0,c_4/1,c_5/0,c_6/1,c_7/0,c_8/0,c_9/0,c_10/1,c_11/0,c_12/1,c_13/2} - 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:first#(s(X),cons(Y,Z)) -> c_6(first#(X,Z)) -->_1 first#(s(X),cons(Y,Z)) -> c_6(first#(X,Z)):1 The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed. 1: first#(s(X),cons(Y,Z)) -> c_6(first#(X,Z)) ***** Step 5.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/2 ,nil/0,recip/1,s/1,c_1/0,c_2/1,c_3/0,c_4/1,c_5/0,c_6/1,c_7/0,c_8/0,c_9/0,c_10/1,c_11/0,c_12/1,c_13/2} - 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 5.b:4.b:3.b:1: RemoveWeakSuffixes MAYBE + Considered Problem: - Strict DPs: half#(s(s(X))) -> c_10(half#(X)) sqr#(s(X)) -> c_12(sqr#(X)) terms#(N) -> c_13(sqr#(N),terms#(s(N))) - Weak DPs: first#(s(X),cons(Y,Z)) -> c_6(first#(X,Z)) - 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/2 ,nil/0,recip/1,s/1,c_1/0,c_2/1,c_3/0,c_4/1,c_5/0,c_6/1,c_7/0,c_8/0,c_9/0,c_10/1,c_11/0,c_12/1,c_13/2} - 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)) -->_1 sqr#(s(X)) -> c_12(sqr#(X)):2 3:S:terms#(N) -> c_13(sqr#(N),terms#(s(N))) -->_2 terms#(N) -> c_13(sqr#(N),terms#(s(N))):3 -->_1 sqr#(s(X)) -> c_12(sqr#(X)):2 4:W:first#(s(X),cons(Y,Z)) -> c_6(first#(X,Z)) -->_1 first#(s(X),cons(Y,Z)) -> c_6(first#(X,Z)):4 The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed. 4: first#(s(X),cons(Y,Z)) -> c_6(first#(X,Z)) **** Step 5.b:4.b:3.b:2: Decompose MAYBE + Considered Problem: - Strict DPs: half#(s(s(X))) -> c_10(half#(X)) sqr#(s(X)) -> c_12(sqr#(X)) terms#(N) -> c_13(sqr#(N),terms#(s(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/2 ,nil/0,recip/1,s/1,c_1/0,c_2/1,c_3/0,c_4/1,c_5/0,c_6/1,c_7/0,c_8/0,c_9/0,c_10/1,c_11/0,c_12/1,c_13/2} - 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)) terms#(N) -> c_13(sqr#(N),terms#(s(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/2 ,nil/0,recip/1,s/1,c_1/0,c_2/1,c_3/0,c_4/1,c_5/0,c_6/1,c_7/0,c_8/0,c_9/0,c_10/1,c_11/0,c_12/1,c_13/2} - 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)) terms#(N) -> c_13(sqr#(N),terms#(s(N))) - 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/2 ,nil/0,recip/1,s/1,c_1/0,c_2/1,c_3/0,c_4/1,c_5/0,c_6/1,c_7/0,c_8/0,c_9/0,c_10/1,c_11/0,c_12/1,c_13/2} - Obligation: innermost runtime complexity wrt. defined symbols {add#,dbl#,first#,half#,sqr#,terms#} and constructors {0 ,cons,nil,recip,s} ***** Step 5.b:4.b:3.b:2.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)) terms#(N) -> c_13(sqr#(N),terms#(s(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/2 ,nil/0,recip/1,s/1,c_1/0,c_2/1,c_3/0,c_4/1,c_5/0,c_6/1,c_7/0,c_8/0,c_9/0,c_10/1,c_11/0,c_12/1,c_13/2} - 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 3:W:terms#(N) -> c_13(sqr#(N),terms#(s(N))) -->_1 sqr#(s(X)) -> c_12(sqr#(X)):2 -->_2 terms#(N) -> c_13(sqr#(N),terms#(s(N))):3 The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed. 3: terms#(N) -> c_13(sqr#(N),terms#(s(N))) 2: sqr#(s(X)) -> c_12(sqr#(X)) ***** Step 5.b:4.b:3.b:2.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/2 ,nil/0,recip/1,s/1,c_1/0,c_2/1,c_3/0,c_4/1,c_5/0,c_6/1,c_7/0,c_8/0,c_9/0,c_10/1,c_11/0,c_12/1,c_13/2} - 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 5.b:4.b:3.b:2.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/2 ,nil/0,recip/1,s/1,c_1/0,c_2/1,c_3/0,c_4/1,c_5/0,c_6/1,c_7/0,c_8/0,c_9/0,c_10/1,c_11/0,c_12/1,c_13/2} - 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) = [1] p(add) = [2] p(cons) = [0] p(dbl) = [2] x1 + [8] p(first) = [1] x1 + [0] p(half) = [1] x1 + [2] p(nil) = [1] p(recip) = [1] x1 + [0] p(s) = [1] x1 + [1] p(sqr) = [2] p(terms) = [1] p(add#) = [1] x1 + [1] p(dbl#) = [8] p(first#) = [2] x1 + [2] x2 + [2] p(half#) = [8] x1 + [3] p(sqr#) = [2] p(terms#) = [1] x1 + [1] p(c_1) = [0] p(c_2) = [1] x1 + [1] p(c_3) = [4] p(c_4) = [1] p(c_5) = [1] p(c_6) = [2] x1 + [0] p(c_7) = [1] p(c_8) = [0] p(c_9) = [1] p(c_10) = [1] x1 + [11] p(c_11) = [2] p(c_12) = [4] x1 + [0] p(c_13) = [0] Following rules are strictly oriented: half#(s(s(X))) = [8] X + [19] > [8] X + [14] = c_10(half#(X)) Following rules are (at-least) weakly oriented: ****** Step 5.b:4.b:3.b:2.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/2 ,nil/0,recip/1,s/1,c_1/0,c_2/1,c_3/0,c_4/1,c_5/0,c_6/1,c_7/0,c_8/0,c_9/0,c_10/1,c_11/0,c_12/1,c_13/2} - 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 5.b:4.b:3.b:2.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/2 ,nil/0,recip/1,s/1,c_1/0,c_2/1,c_3/0,c_4/1,c_5/0,c_6/1,c_7/0,c_8/0,c_9/0,c_10/1,c_11/0,c_12/1,c_13/2} - 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 5.b:4.b:3.b:2.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/2 ,nil/0,recip/1,s/1,c_1/0,c_2/1,c_3/0,c_4/1,c_5/0,c_6/1,c_7/0,c_8/0,c_9/0,c_10/1,c_11/0,c_12/1,c_13/2} - 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 5.b:4.b:3.b:2.b:1: RemoveWeakSuffixes MAYBE + Considered Problem: - Strict DPs: sqr#(s(X)) -> c_12(sqr#(X)) terms#(N) -> c_13(sqr#(N),terms#(s(N))) - 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/2 ,nil/0,recip/1,s/1,c_1/0,c_2/1,c_3/0,c_4/1,c_5/0,c_6/1,c_7/0,c_8/0,c_9/0,c_10/1,c_11/0,c_12/1,c_13/2} - 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:S:terms#(N) -> c_13(sqr#(N),terms#(s(N))) -->_2 terms#(N) -> c_13(sqr#(N),terms#(s(N))):2 -->_1 sqr#(s(X)) -> c_12(sqr#(X)):1 3:W:half#(s(s(X))) -> c_10(half#(X)) -->_1 half#(s(s(X))) -> c_10(half#(X)):3 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)) ***** Step 5.b:4.b:3.b:2.b:2: Decompose MAYBE + Considered Problem: - Strict DPs: sqr#(s(X)) -> c_12(sqr#(X)) terms#(N) -> c_13(sqr#(N),terms#(s(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/2 ,nil/0,recip/1,s/1,c_1/0,c_2/1,c_3/0,c_4/1,c_5/0,c_6/1,c_7/0,c_8/0,c_9/0,c_10/1,c_11/0,c_12/1,c_13/2} - 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: sqr#(s(X)) -> c_12(sqr#(X)) - Weak DPs: terms#(N) -> c_13(sqr#(N),terms#(s(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/2 ,nil/0,recip/1,s/1,c_1/0,c_2/1,c_3/0,c_4/1,c_5/0,c_6/1,c_7/0,c_8/0,c_9/0,c_10/1,c_11/0,c_12/1,c_13/2} - Obligation: innermost runtime complexity wrt. defined symbols {add#,dbl#,first#,half#,sqr#,terms#} and constructors {0 ,cons,nil,recip,s} Problem (S) - Strict DPs: terms#(N) -> c_13(sqr#(N),terms#(s(N))) - 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/2 ,nil/0,recip/1,s/1,c_1/0,c_2/1,c_3/0,c_4/1,c_5/0,c_6/1,c_7/0,c_8/0,c_9/0,c_10/1,c_11/0,c_12/1,c_13/2} - Obligation: innermost runtime complexity wrt. defined symbols {add#,dbl#,first#,half#,sqr#,terms#} and constructors {0 ,cons,nil,recip,s} ****** Step 5.b:4.b:3.b:2.b:2.a:1: Failure MAYBE + Considered Problem: - Strict DPs: sqr#(s(X)) -> c_12(sqr#(X)) - Weak DPs: terms#(N) -> c_13(sqr#(N),terms#(s(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/2 ,nil/0,recip/1,s/1,c_1/0,c_2/1,c_3/0,c_4/1,c_5/0,c_6/1,c_7/0,c_8/0,c_9/0,c_10/1,c_11/0,c_12/1,c_13/2} - 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 still open. ****** Step 5.b:4.b:3.b:2.b:2.b:1: RemoveWeakSuffixes MAYBE + Considered Problem: - Strict DPs: terms#(N) -> c_13(sqr#(N),terms#(s(N))) - 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/2 ,nil/0,recip/1,s/1,c_1/0,c_2/1,c_3/0,c_4/1,c_5/0,c_6/1,c_7/0,c_8/0,c_9/0,c_10/1,c_11/0,c_12/1,c_13/2} - 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:terms#(N) -> c_13(sqr#(N),terms#(s(N))) -->_1 sqr#(s(X)) -> c_12(sqr#(X)):2 -->_2 terms#(N) -> c_13(sqr#(N),terms#(s(N))):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 5.b:4.b:3.b:2.b:2.b:2: SimplifyRHS MAYBE + Considered Problem: - Strict DPs: terms#(N) -> c_13(sqr#(N),terms#(s(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/2 ,nil/0,recip/1,s/1,c_1/0,c_2/1,c_3/0,c_4/1,c_5/0,c_6/1,c_7/0,c_8/0,c_9/0,c_10/1,c_11/0,c_12/1,c_13/2} - 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:terms#(N) -> c_13(sqr#(N),terms#(s(N))) -->_2 terms#(N) -> c_13(sqr#(N),terms#(s(N))):1 Due to missing edges in the depndency graph, the right-hand sides of following rules could be simplified: terms#(N) -> c_13(terms#(s(N))) ****** Step 5.b:4.b:3.b:2.b:2.b:3: Failure MAYBE + Considered Problem: - Strict DPs: terms#(N) -> c_13(terms#(s(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/2 ,nil/0,recip/1,s/1,c_1/0,c_2/1,c_3/0,c_4/1,c_5/0,c_6/1,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 still open. MAYBE