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)) 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,sqr/1,terms/1} / {0/0,cons/2,nil/0,recip/1,s/1} - Obligation: innermost runtime complexity wrt. defined symbols {add,dbl,first,sqr,terms} and constructors {0,cons,nil ,recip,s} + Applied Processor: DependencyPairs {dpKind_ = DT} + Details: We add the following dependency tuples: Strict DPs add#(0(),X) -> c_1() add#(s(X),Y) -> c_2(add#(X,Y)) dbl#(0()) -> c_3() dbl#(s(X)) -> c_4(dbl#(X)) first#(0(),X) -> c_5() first#(s(X),cons(Y,Z)) -> c_6(first#(X,Z)) sqr#(0()) -> c_7() sqr#(s(X)) -> c_8(add#(sqr(X),dbl(X)),sqr#(X),dbl#(X)) terms#(N) -> c_9(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)) sqr#(0()) -> c_7() sqr#(s(X)) -> c_8(add#(sqr(X),dbl(X)),sqr#(X),dbl#(X)) terms#(N) -> c_9(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)) 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,sqr/1,terms/1,add#/2,dbl#/1,first#/2,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_8/3,c_9/2} - Obligation: innermost runtime complexity wrt. defined symbols {add#,dbl#,first#,sqr#,terms#} and constructors {0,cons ,nil,recip,s} + Applied Processor: UsableRules + Details: We replace rewrite rules by usable rules: add(0(),X) -> X add(s(X),Y) -> s(add(X,Y)) dbl(0()) -> 0() dbl(s(X)) -> s(s(dbl(X))) sqr(0()) -> 0() sqr(s(X)) -> s(add(sqr(X),dbl(X))) add#(0(),X) -> c_1() add#(s(X),Y) -> c_2(add#(X,Y)) dbl#(0()) -> c_3() dbl#(s(X)) -> c_4(dbl#(X)) first#(0(),X) -> c_5() first#(s(X),cons(Y,Z)) -> c_6(first#(X,Z)) sqr#(0()) -> c_7() sqr#(s(X)) -> c_8(add#(sqr(X),dbl(X)),sqr#(X),dbl#(X)) terms#(N) -> c_9(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)) sqr#(0()) -> c_7() sqr#(s(X)) -> c_8(add#(sqr(X),dbl(X)),sqr#(X),dbl#(X)) terms#(N) -> c_9(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,sqr/1,terms/1,add#/2,dbl#/1,first#/2,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_8/3,c_9/2} - Obligation: innermost runtime complexity wrt. defined symbols {add#,dbl#,first#,sqr#,terms#} and constructors {0,cons ,nil,recip,s} + Applied Processor: PredecessorEstimation {onSelection = all simple predecessor estimation selector} + Details: We estimate the number of application of {1,3,5,7} by application of Pre({1,3,5,7}) = {2,4,6,8,9}. Here rules are labelled as follows: 1: add#(0(),X) -> c_1() 2: add#(s(X),Y) -> c_2(add#(X,Y)) 3: dbl#(0()) -> c_3() 4: dbl#(s(X)) -> c_4(dbl#(X)) 5: first#(0(),X) -> c_5() 6: first#(s(X),cons(Y,Z)) -> c_6(first#(X,Z)) 7: sqr#(0()) -> c_7() 8: sqr#(s(X)) -> c_8(add#(sqr(X),dbl(X)),sqr#(X),dbl#(X)) 9: terms#(N) -> c_9(sqr#(N),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)) sqr#(s(X)) -> c_8(add#(sqr(X),dbl(X)),sqr#(X),dbl#(X)) terms#(N) -> c_9(sqr#(N),terms#(s(N))) - Weak DPs: add#(0(),X) -> c_1() dbl#(0()) -> c_3() first#(0(),X) -> c_5() sqr#(0()) -> c_7() - Weak TRS: add(0(),X) -> X add(s(X),Y) -> s(add(X,Y)) dbl(0()) -> 0() dbl(s(X)) -> s(s(dbl(X))) sqr(0()) -> 0() sqr(s(X)) -> s(add(sqr(X),dbl(X))) - Signature: {add/2,dbl/1,first/2,sqr/1,terms/1,add#/2,dbl#/1,first#/2,sqr#/1,terms#/1} / {0/0,cons/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_8/3,c_9/2} - Obligation: innermost runtime complexity wrt. defined symbols {add#,dbl#,first#,sqr#,terms#} and constructors {0,cons ,nil,recip,s} + Applied Processor: RemoveWeakSuffixes + Details: Consider the dependency graph 1:S:add#(s(X),Y) -> c_2(add#(X,Y)) -->_1 add#(0(),X) -> c_1():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:first#(s(X),cons(Y,Z)) -> c_6(first#(X,Z)) -->_1 first#(0(),X) -> c_5():8 -->_1 first#(s(X),cons(Y,Z)) -> c_6(first#(X,Z)):3 4:S:sqr#(s(X)) -> c_8(add#(sqr(X),dbl(X)),sqr#(X),dbl#(X)) -->_2 sqr#(0()) -> c_7():9 -->_3 dbl#(0()) -> c_3():7 -->_1 add#(0(),X) -> c_1():6 -->_2 sqr#(s(X)) -> c_8(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_9(sqr#(N),terms#(s(N))) -->_1 sqr#(0()) -> c_7():9 -->_2 terms#(N) -> c_9(sqr#(N),terms#(s(N))):5 -->_1 sqr#(s(X)) -> c_8(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:sqr#(0()) -> c_7() The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed. 9: sqr#(0()) -> c_7() 8: first#(0(),X) -> c_5() 7: dbl#(0()) -> c_3() 6: add#(0(),X) -> c_1() * Step 5: Failure 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)) sqr#(s(X)) -> c_8(add#(sqr(X),dbl(X)),sqr#(X),dbl#(X)) terms#(N) -> c_9(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,sqr/1,terms/1,add#/2,dbl#/1,first#/2,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_8/3,c_9/2} - Obligation: innermost runtime complexity wrt. defined symbols {add#,dbl#,first#,sqr#,terms#} and constructors {0,cons ,nil,recip,s} + Applied Processor: EmptyProcessor + Details: The problem is still open. MAYBE