MAYBE * Step 1: DependencyPairs MAYBE + Considered Problem: - Strict TRS: activate(X) -> X activate(n__fib1(X1,X2)) -> fib1(X1,X2) add(0(),X) -> X add(s(X),Y) -> s(add(X,Y)) fib(N) -> sel(N,fib1(s(0()),s(0()))) fib1(X,Y) -> cons(X,n__fib1(Y,add(X,Y))) fib1(X1,X2) -> n__fib1(X1,X2) sel(0(),cons(X,XS)) -> X sel(s(N),cons(X,XS)) -> sel(N,activate(XS)) - Signature: {activate/1,add/2,fib/1,fib1/2,sel/2} / {0/0,cons/2,n__fib1/2,s/1} - Obligation: innermost runtime complexity wrt. defined symbols {activate,add,fib,fib1,sel} and constructors {0,cons ,n__fib1,s} + Applied Processor: DependencyPairs {dpKind_ = DT} + Details: We add the following dependency tuples: Strict DPs activate#(X) -> c_1() activate#(n__fib1(X1,X2)) -> c_2(fib1#(X1,X2)) add#(0(),X) -> c_3() add#(s(X),Y) -> c_4(add#(X,Y)) fib#(N) -> c_5(sel#(N,fib1(s(0()),s(0()))),fib1#(s(0()),s(0()))) fib1#(X,Y) -> c_6(add#(X,Y)) fib1#(X1,X2) -> c_7() sel#(0(),cons(X,XS)) -> c_8() sel#(s(N),cons(X,XS)) -> c_9(sel#(N,activate(XS)),activate#(XS)) Weak DPs and mark the set of starting terms. * Step 2: UsableRules MAYBE + Considered Problem: - Strict DPs: activate#(X) -> c_1() activate#(n__fib1(X1,X2)) -> c_2(fib1#(X1,X2)) add#(0(),X) -> c_3() add#(s(X),Y) -> c_4(add#(X,Y)) fib#(N) -> c_5(sel#(N,fib1(s(0()),s(0()))),fib1#(s(0()),s(0()))) fib1#(X,Y) -> c_6(add#(X,Y)) fib1#(X1,X2) -> c_7() sel#(0(),cons(X,XS)) -> c_8() sel#(s(N),cons(X,XS)) -> c_9(sel#(N,activate(XS)),activate#(XS)) - Weak TRS: activate(X) -> X activate(n__fib1(X1,X2)) -> fib1(X1,X2) add(0(),X) -> X add(s(X),Y) -> s(add(X,Y)) fib(N) -> sel(N,fib1(s(0()),s(0()))) fib1(X,Y) -> cons(X,n__fib1(Y,add(X,Y))) fib1(X1,X2) -> n__fib1(X1,X2) sel(0(),cons(X,XS)) -> X sel(s(N),cons(X,XS)) -> sel(N,activate(XS)) - Signature: {activate/1,add/2,fib/1,fib1/2,sel/2,activate#/1,add#/2,fib#/1,fib1#/2,sel#/2} / {0/0,cons/2,n__fib1/2,s/1 ,c_1/0,c_2/1,c_3/0,c_4/1,c_5/2,c_6/1,c_7/0,c_8/0,c_9/2} - Obligation: innermost runtime complexity wrt. defined symbols {activate#,add#,fib#,fib1#,sel#} and constructors {0,cons ,n__fib1,s} + Applied Processor: UsableRules + Details: We replace rewrite rules by usable rules: activate(X) -> X activate(n__fib1(X1,X2)) -> fib1(X1,X2) add(0(),X) -> X add(s(X),Y) -> s(add(X,Y)) fib1(X,Y) -> cons(X,n__fib1(Y,add(X,Y))) fib1(X1,X2) -> n__fib1(X1,X2) activate#(X) -> c_1() activate#(n__fib1(X1,X2)) -> c_2(fib1#(X1,X2)) add#(0(),X) -> c_3() add#(s(X),Y) -> c_4(add#(X,Y)) fib#(N) -> c_5(sel#(N,fib1(s(0()),s(0()))),fib1#(s(0()),s(0()))) fib1#(X,Y) -> c_6(add#(X,Y)) fib1#(X1,X2) -> c_7() sel#(0(),cons(X,XS)) -> c_8() sel#(s(N),cons(X,XS)) -> c_9(sel#(N,activate(XS)),activate#(XS)) * Step 3: PredecessorEstimation MAYBE + Considered Problem: - Strict DPs: activate#(X) -> c_1() activate#(n__fib1(X1,X2)) -> c_2(fib1#(X1,X2)) add#(0(),X) -> c_3() add#(s(X),Y) -> c_4(add#(X,Y)) fib#(N) -> c_5(sel#(N,fib1(s(0()),s(0()))),fib1#(s(0()),s(0()))) fib1#(X,Y) -> c_6(add#(X,Y)) fib1#(X1,X2) -> c_7() sel#(0(),cons(X,XS)) -> c_8() sel#(s(N),cons(X,XS)) -> c_9(sel#(N,activate(XS)),activate#(XS)) - Weak TRS: activate(X) -> X activate(n__fib1(X1,X2)) -> fib1(X1,X2) add(0(),X) -> X add(s(X),Y) -> s(add(X,Y)) fib1(X,Y) -> cons(X,n__fib1(Y,add(X,Y))) fib1(X1,X2) -> n__fib1(X1,X2) - Signature: {activate/1,add/2,fib/1,fib1/2,sel/2,activate#/1,add#/2,fib#/1,fib1#/2,sel#/2} / {0/0,cons/2,n__fib1/2,s/1 ,c_1/0,c_2/1,c_3/0,c_4/1,c_5/2,c_6/1,c_7/0,c_8/0,c_9/2} - Obligation: innermost runtime complexity wrt. defined symbols {activate#,add#,fib#,fib1#,sel#} and constructors {0,cons ,n__fib1,s} + Applied Processor: PredecessorEstimation {onSelection = all simple predecessor estimation selector} + Details: We estimate the number of application of {1,3,7,8} by application of Pre({1,3,7,8}) = {2,4,5,6,9}. Here rules are labelled as follows: 1: activate#(X) -> c_1() 2: activate#(n__fib1(X1,X2)) -> c_2(fib1#(X1,X2)) 3: add#(0(),X) -> c_3() 4: add#(s(X),Y) -> c_4(add#(X,Y)) 5: fib#(N) -> c_5(sel#(N,fib1(s(0()),s(0()))),fib1#(s(0()),s(0()))) 6: fib1#(X,Y) -> c_6(add#(X,Y)) 7: fib1#(X1,X2) -> c_7() 8: sel#(0(),cons(X,XS)) -> c_8() 9: sel#(s(N),cons(X,XS)) -> c_9(sel#(N,activate(XS)),activate#(XS)) * Step 4: RemoveWeakSuffixes MAYBE + Considered Problem: - Strict DPs: activate#(n__fib1(X1,X2)) -> c_2(fib1#(X1,X2)) add#(s(X),Y) -> c_4(add#(X,Y)) fib#(N) -> c_5(sel#(N,fib1(s(0()),s(0()))),fib1#(s(0()),s(0()))) fib1#(X,Y) -> c_6(add#(X,Y)) sel#(s(N),cons(X,XS)) -> c_9(sel#(N,activate(XS)),activate#(XS)) - Weak DPs: activate#(X) -> c_1() add#(0(),X) -> c_3() fib1#(X1,X2) -> c_7() sel#(0(),cons(X,XS)) -> c_8() - Weak TRS: activate(X) -> X activate(n__fib1(X1,X2)) -> fib1(X1,X2) add(0(),X) -> X add(s(X),Y) -> s(add(X,Y)) fib1(X,Y) -> cons(X,n__fib1(Y,add(X,Y))) fib1(X1,X2) -> n__fib1(X1,X2) - Signature: {activate/1,add/2,fib/1,fib1/2,sel/2,activate#/1,add#/2,fib#/1,fib1#/2,sel#/2} / {0/0,cons/2,n__fib1/2,s/1 ,c_1/0,c_2/1,c_3/0,c_4/1,c_5/2,c_6/1,c_7/0,c_8/0,c_9/2} - Obligation: innermost runtime complexity wrt. defined symbols {activate#,add#,fib#,fib1#,sel#} and constructors {0,cons ,n__fib1,s} + Applied Processor: RemoveWeakSuffixes + Details: Consider the dependency graph 1:S:activate#(n__fib1(X1,X2)) -> c_2(fib1#(X1,X2)) -->_1 fib1#(X,Y) -> c_6(add#(X,Y)):4 -->_1 fib1#(X1,X2) -> c_7():8 2:S:add#(s(X),Y) -> c_4(add#(X,Y)) -->_1 add#(0(),X) -> c_3():7 -->_1 add#(s(X),Y) -> c_4(add#(X,Y)):2 3:S:fib#(N) -> c_5(sel#(N,fib1(s(0()),s(0()))),fib1#(s(0()),s(0()))) -->_1 sel#(s(N),cons(X,XS)) -> c_9(sel#(N,activate(XS)),activate#(XS)):5 -->_2 fib1#(X,Y) -> c_6(add#(X,Y)):4 -->_1 sel#(0(),cons(X,XS)) -> c_8():9 -->_2 fib1#(X1,X2) -> c_7():8 4:S:fib1#(X,Y) -> c_6(add#(X,Y)) -->_1 add#(0(),X) -> c_3():7 -->_1 add#(s(X),Y) -> c_4(add#(X,Y)):2 5:S:sel#(s(N),cons(X,XS)) -> c_9(sel#(N,activate(XS)),activate#(XS)) -->_1 sel#(0(),cons(X,XS)) -> c_8():9 -->_2 activate#(X) -> c_1():6 -->_1 sel#(s(N),cons(X,XS)) -> c_9(sel#(N,activate(XS)),activate#(XS)):5 -->_2 activate#(n__fib1(X1,X2)) -> c_2(fib1#(X1,X2)):1 6:W:activate#(X) -> c_1() 7:W:add#(0(),X) -> c_3() 8:W:fib1#(X1,X2) -> c_7() 9:W:sel#(0(),cons(X,XS)) -> c_8() The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed. 6: activate#(X) -> c_1() 9: sel#(0(),cons(X,XS)) -> c_8() 8: fib1#(X1,X2) -> c_7() 7: add#(0(),X) -> c_3() * Step 5: Decompose MAYBE + Considered Problem: - Strict DPs: activate#(n__fib1(X1,X2)) -> c_2(fib1#(X1,X2)) add#(s(X),Y) -> c_4(add#(X,Y)) fib#(N) -> c_5(sel#(N,fib1(s(0()),s(0()))),fib1#(s(0()),s(0()))) fib1#(X,Y) -> c_6(add#(X,Y)) sel#(s(N),cons(X,XS)) -> c_9(sel#(N,activate(XS)),activate#(XS)) - Weak TRS: activate(X) -> X activate(n__fib1(X1,X2)) -> fib1(X1,X2) add(0(),X) -> X add(s(X),Y) -> s(add(X,Y)) fib1(X,Y) -> cons(X,n__fib1(Y,add(X,Y))) fib1(X1,X2) -> n__fib1(X1,X2) - Signature: {activate/1,add/2,fib/1,fib1/2,sel/2,activate#/1,add#/2,fib#/1,fib1#/2,sel#/2} / {0/0,cons/2,n__fib1/2,s/1 ,c_1/0,c_2/1,c_3/0,c_4/1,c_5/2,c_6/1,c_7/0,c_8/0,c_9/2} - Obligation: innermost runtime complexity wrt. defined symbols {activate#,add#,fib#,fib1#,sel#} and constructors {0,cons ,n__fib1,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: activate#(n__fib1(X1,X2)) -> c_2(fib1#(X1,X2)) add#(s(X),Y) -> c_4(add#(X,Y)) fib1#(X,Y) -> c_6(add#(X,Y)) - Weak DPs: fib#(N) -> c_5(sel#(N,fib1(s(0()),s(0()))),fib1#(s(0()),s(0()))) sel#(s(N),cons(X,XS)) -> c_9(sel#(N,activate(XS)),activate#(XS)) - Weak TRS: activate(X) -> X activate(n__fib1(X1,X2)) -> fib1(X1,X2) add(0(),X) -> X add(s(X),Y) -> s(add(X,Y)) fib1(X,Y) -> cons(X,n__fib1(Y,add(X,Y))) fib1(X1,X2) -> n__fib1(X1,X2) - Signature: {activate/1,add/2,fib/1,fib1/2,sel/2,activate#/1,add#/2,fib#/1,fib1#/2,sel#/2} / {0/0,cons/2,n__fib1/2,s/1 ,c_1/0,c_2/1,c_3/0,c_4/1,c_5/2,c_6/1,c_7/0,c_8/0,c_9/2} - Obligation: innermost runtime complexity wrt. defined symbols {activate#,add#,fib#,fib1#,sel#} and constructors {0 ,cons,n__fib1,s} Problem (S) - Strict DPs: fib#(N) -> c_5(sel#(N,fib1(s(0()),s(0()))),fib1#(s(0()),s(0()))) sel#(s(N),cons(X,XS)) -> c_9(sel#(N,activate(XS)),activate#(XS)) - Weak DPs: activate#(n__fib1(X1,X2)) -> c_2(fib1#(X1,X2)) add#(s(X),Y) -> c_4(add#(X,Y)) fib1#(X,Y) -> c_6(add#(X,Y)) - Weak TRS: activate(X) -> X activate(n__fib1(X1,X2)) -> fib1(X1,X2) add(0(),X) -> X add(s(X),Y) -> s(add(X,Y)) fib1(X,Y) -> cons(X,n__fib1(Y,add(X,Y))) fib1(X1,X2) -> n__fib1(X1,X2) - Signature: {activate/1,add/2,fib/1,fib1/2,sel/2,activate#/1,add#/2,fib#/1,fib1#/2,sel#/2} / {0/0,cons/2,n__fib1/2,s/1 ,c_1/0,c_2/1,c_3/0,c_4/1,c_5/2,c_6/1,c_7/0,c_8/0,c_9/2} - Obligation: innermost runtime complexity wrt. defined symbols {activate#,add#,fib#,fib1#,sel#} and constructors {0 ,cons,n__fib1,s} ** Step 5.a:1: DecomposeDG MAYBE + Considered Problem: - Strict DPs: activate#(n__fib1(X1,X2)) -> c_2(fib1#(X1,X2)) add#(s(X),Y) -> c_4(add#(X,Y)) fib1#(X,Y) -> c_6(add#(X,Y)) - Weak DPs: fib#(N) -> c_5(sel#(N,fib1(s(0()),s(0()))),fib1#(s(0()),s(0()))) sel#(s(N),cons(X,XS)) -> c_9(sel#(N,activate(XS)),activate#(XS)) - Weak TRS: activate(X) -> X activate(n__fib1(X1,X2)) -> fib1(X1,X2) add(0(),X) -> X add(s(X),Y) -> s(add(X,Y)) fib1(X,Y) -> cons(X,n__fib1(Y,add(X,Y))) fib1(X1,X2) -> n__fib1(X1,X2) - Signature: {activate/1,add/2,fib/1,fib1/2,sel/2,activate#/1,add#/2,fib#/1,fib1#/2,sel#/2} / {0/0,cons/2,n__fib1/2,s/1 ,c_1/0,c_2/1,c_3/0,c_4/1,c_5/2,c_6/1,c_7/0,c_8/0,c_9/2} - Obligation: innermost runtime complexity wrt. defined symbols {activate#,add#,fib#,fib1#,sel#} and constructors {0,cons ,n__fib1,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 fib#(N) -> c_5(sel#(N,fib1(s(0()),s(0()))),fib1#(s(0()),s(0()))) sel#(s(N),cons(X,XS)) -> c_9(sel#(N,activate(XS)),activate#(XS)) and a lower component activate#(n__fib1(X1,X2)) -> c_2(fib1#(X1,X2)) add#(s(X),Y) -> c_4(add#(X,Y)) fib1#(X,Y) -> c_6(add#(X,Y)) Further, following extension rules are added to the lower component. fib#(N) -> fib1#(s(0()),s(0())) fib#(N) -> sel#(N,fib1(s(0()),s(0()))) sel#(s(N),cons(X,XS)) -> activate#(XS) sel#(s(N),cons(X,XS)) -> sel#(N,activate(XS)) *** Step 5.a:1.a:1: PredecessorEstimationCP WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: fib#(N) -> c_5(sel#(N,fib1(s(0()),s(0()))),fib1#(s(0()),s(0()))) sel#(s(N),cons(X,XS)) -> c_9(sel#(N,activate(XS)),activate#(XS)) - Weak TRS: activate(X) -> X activate(n__fib1(X1,X2)) -> fib1(X1,X2) add(0(),X) -> X add(s(X),Y) -> s(add(X,Y)) fib1(X,Y) -> cons(X,n__fib1(Y,add(X,Y))) fib1(X1,X2) -> n__fib1(X1,X2) - Signature: {activate/1,add/2,fib/1,fib1/2,sel/2,activate#/1,add#/2,fib#/1,fib1#/2,sel#/2} / {0/0,cons/2,n__fib1/2,s/1 ,c_1/0,c_2/1,c_3/0,c_4/1,c_5/2,c_6/1,c_7/0,c_8/0,c_9/2} - Obligation: innermost runtime complexity wrt. defined symbols {activate#,add#,fib#,fib1#,sel#} and constructors {0,cons ,n__fib1,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: fib#(N) -> c_5(sel#(N,fib1(s(0()),s(0()))),fib1#(s(0()),s(0()))) 2: sel#(s(N),cons(X,XS)) -> c_9(sel#(N,activate(XS)),activate#(XS)) The strictly oriented rules are moved into the weak component. **** Step 5.a:1.a:1.a:1: NaturalMI WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: fib#(N) -> c_5(sel#(N,fib1(s(0()),s(0()))),fib1#(s(0()),s(0()))) sel#(s(N),cons(X,XS)) -> c_9(sel#(N,activate(XS)),activate#(XS)) - Weak TRS: activate(X) -> X activate(n__fib1(X1,X2)) -> fib1(X1,X2) add(0(),X) -> X add(s(X),Y) -> s(add(X,Y)) fib1(X,Y) -> cons(X,n__fib1(Y,add(X,Y))) fib1(X1,X2) -> n__fib1(X1,X2) - Signature: {activate/1,add/2,fib/1,fib1/2,sel/2,activate#/1,add#/2,fib#/1,fib1#/2,sel#/2} / {0/0,cons/2,n__fib1/2,s/1 ,c_1/0,c_2/1,c_3/0,c_4/1,c_5/2,c_6/1,c_7/0,c_8/0,c_9/2} - Obligation: innermost runtime complexity wrt. defined symbols {activate#,add#,fib#,fib1#,sel#} and constructors {0,cons ,n__fib1,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_5) = {1}, uargs(c_9) = {1} Following symbols are considered usable: {activate#,add#,fib#,fib1#,sel#} TcT has computed the following interpretation: p(0) = [0] p(activate) = [0] p(add) = [4] x1 + [6] p(cons) = [1] x1 + [0] p(fib) = [1] x1 + [2] p(fib1) = [4] x1 + [0] p(n__fib1) = [1] x1 + [1] x2 + [0] p(s) = [1] x1 + [4] p(sel) = [1] x1 + [1] x2 + [1] p(activate#) = [2] p(add#) = [8] x1 + [1] x2 + [1] p(fib#) = [9] x1 + [3] p(fib1#) = [4] x1 + [2] x2 + [4] p(sel#) = [4] x1 + [0] p(c_1) = [2] p(c_2) = [1] p(c_3) = [1] p(c_4) = [4] x1 + [0] p(c_5) = [2] x1 + [0] p(c_6) = [8] x1 + [0] p(c_7) = [1] p(c_8) = [1] p(c_9) = [1] x1 + [1] x2 + [0] Following rules are strictly oriented: fib#(N) = [9] N + [3] > [8] N + [0] = c_5(sel#(N,fib1(s(0()),s(0()))),fib1#(s(0()),s(0()))) sel#(s(N),cons(X,XS)) = [4] N + [16] > [4] N + [2] = c_9(sel#(N,activate(XS)),activate#(XS)) Following rules are (at-least) weakly oriented: **** Step 5.a:1.a:1.a:2: Assumption WORST_CASE(?,O(1)) + Considered Problem: - Weak DPs: fib#(N) -> c_5(sel#(N,fib1(s(0()),s(0()))),fib1#(s(0()),s(0()))) sel#(s(N),cons(X,XS)) -> c_9(sel#(N,activate(XS)),activate#(XS)) - Weak TRS: activate(X) -> X activate(n__fib1(X1,X2)) -> fib1(X1,X2) add(0(),X) -> X add(s(X),Y) -> s(add(X,Y)) fib1(X,Y) -> cons(X,n__fib1(Y,add(X,Y))) fib1(X1,X2) -> n__fib1(X1,X2) - Signature: {activate/1,add/2,fib/1,fib1/2,sel/2,activate#/1,add#/2,fib#/1,fib1#/2,sel#/2} / {0/0,cons/2,n__fib1/2,s/1 ,c_1/0,c_2/1,c_3/0,c_4/1,c_5/2,c_6/1,c_7/0,c_8/0,c_9/2} - Obligation: innermost runtime complexity wrt. defined symbols {activate#,add#,fib#,fib1#,sel#} and constructors {0,cons ,n__fib1,s} + Applied Processor: Assumption {assumed = Certificate {spaceUB = Unknown, spaceLB = Unknown, timeUB = Poly (Just 0), timeLB = Unknown}} + Details: () **** Step 5.a:1.a:1.b:1: RemoveWeakSuffixes WORST_CASE(?,O(1)) + Considered Problem: - Weak DPs: fib#(N) -> c_5(sel#(N,fib1(s(0()),s(0()))),fib1#(s(0()),s(0()))) sel#(s(N),cons(X,XS)) -> c_9(sel#(N,activate(XS)),activate#(XS)) - Weak TRS: activate(X) -> X activate(n__fib1(X1,X2)) -> fib1(X1,X2) add(0(),X) -> X add(s(X),Y) -> s(add(X,Y)) fib1(X,Y) -> cons(X,n__fib1(Y,add(X,Y))) fib1(X1,X2) -> n__fib1(X1,X2) - Signature: {activate/1,add/2,fib/1,fib1/2,sel/2,activate#/1,add#/2,fib#/1,fib1#/2,sel#/2} / {0/0,cons/2,n__fib1/2,s/1 ,c_1/0,c_2/1,c_3/0,c_4/1,c_5/2,c_6/1,c_7/0,c_8/0,c_9/2} - Obligation: innermost runtime complexity wrt. defined symbols {activate#,add#,fib#,fib1#,sel#} and constructors {0,cons ,n__fib1,s} + Applied Processor: RemoveWeakSuffixes + Details: Consider the dependency graph 1:W:fib#(N) -> c_5(sel#(N,fib1(s(0()),s(0()))),fib1#(s(0()),s(0()))) -->_1 sel#(s(N),cons(X,XS)) -> c_9(sel#(N,activate(XS)),activate#(XS)):2 2:W:sel#(s(N),cons(X,XS)) -> c_9(sel#(N,activate(XS)),activate#(XS)) -->_1 sel#(s(N),cons(X,XS)) -> c_9(sel#(N,activate(XS)),activate#(XS)):2 The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed. 1: fib#(N) -> c_5(sel#(N,fib1(s(0()),s(0()))),fib1#(s(0()),s(0()))) 2: sel#(s(N),cons(X,XS)) -> c_9(sel#(N,activate(XS)),activate#(XS)) **** Step 5.a:1.a:1.b:2: EmptyProcessor WORST_CASE(?,O(1)) + Considered Problem: - Weak TRS: activate(X) -> X activate(n__fib1(X1,X2)) -> fib1(X1,X2) add(0(),X) -> X add(s(X),Y) -> s(add(X,Y)) fib1(X,Y) -> cons(X,n__fib1(Y,add(X,Y))) fib1(X1,X2) -> n__fib1(X1,X2) - Signature: {activate/1,add/2,fib/1,fib1/2,sel/2,activate#/1,add#/2,fib#/1,fib1#/2,sel#/2} / {0/0,cons/2,n__fib1/2,s/1 ,c_1/0,c_2/1,c_3/0,c_4/1,c_5/2,c_6/1,c_7/0,c_8/0,c_9/2} - Obligation: innermost runtime complexity wrt. defined symbols {activate#,add#,fib#,fib1#,sel#} and constructors {0,cons ,n__fib1,s} + Applied Processor: EmptyProcessor + Details: The problem is already closed. The intended complexity is O(1). *** Step 5.a:1.b:1: NaturalPI MAYBE + Considered Problem: - Strict DPs: activate#(n__fib1(X1,X2)) -> c_2(fib1#(X1,X2)) add#(s(X),Y) -> c_4(add#(X,Y)) fib1#(X,Y) -> c_6(add#(X,Y)) - Weak DPs: fib#(N) -> fib1#(s(0()),s(0())) fib#(N) -> sel#(N,fib1(s(0()),s(0()))) sel#(s(N),cons(X,XS)) -> activate#(XS) sel#(s(N),cons(X,XS)) -> sel#(N,activate(XS)) - Weak TRS: activate(X) -> X activate(n__fib1(X1,X2)) -> fib1(X1,X2) add(0(),X) -> X add(s(X),Y) -> s(add(X,Y)) fib1(X,Y) -> cons(X,n__fib1(Y,add(X,Y))) fib1(X1,X2) -> n__fib1(X1,X2) - Signature: {activate/1,add/2,fib/1,fib1/2,sel/2,activate#/1,add#/2,fib#/1,fib1#/2,sel#/2} / {0/0,cons/2,n__fib1/2,s/1 ,c_1/0,c_2/1,c_3/0,c_4/1,c_5/2,c_6/1,c_7/0,c_8/0,c_9/2} - Obligation: innermost runtime complexity wrt. defined symbols {activate#,add#,fib#,fib1#,sel#} and constructors {0,cons ,n__fib1,s} + Applied Processor: NaturalPI {shape = Mixed 3, restrict = Restrict, uargs = UArgs, urules = URules, selector = Just any strict-rules} + Details: We apply a polynomial interpretation of kind constructor-based(mixed(3)): The following argument positions are considered usable: uargs(c_2) = {1}, uargs(c_4) = {1}, uargs(c_6) = {1} Following symbols are considered usable: {activate#,add#,fib#,fib1#,sel#} TcT has computed the following interpretation: p(0) = 0 p(activate) = 0 p(add) = 2 + 2*x2 + 2*x2^2 + 2*x2^3 p(cons) = 0 p(fib) = 0 p(fib1) = 0 p(n__fib1) = 0 p(s) = 0 p(sel) = x1^3 + x2 + x2^2 p(activate#) = 3 p(add#) = 1 p(fib#) = 3 p(fib1#) = 3 p(sel#) = 3 p(c_1) = 0 p(c_2) = x1 p(c_3) = 0 p(c_4) = x1 p(c_5) = 2 + x1 p(c_6) = x1 p(c_7) = 0 p(c_8) = 0 p(c_9) = 1 Following rules are strictly oriented: fib1#(X,Y) = 3 > 1 = c_6(add#(X,Y)) Following rules are (at-least) weakly oriented: activate#(n__fib1(X1,X2)) = 3 >= 3 = c_2(fib1#(X1,X2)) add#(s(X),Y) = 1 >= 1 = c_4(add#(X,Y)) fib#(N) = 3 >= 3 = fib1#(s(0()),s(0())) fib#(N) = 3 >= 3 = sel#(N,fib1(s(0()),s(0()))) sel#(s(N),cons(X,XS)) = 3 >= 3 = activate#(XS) sel#(s(N),cons(X,XS)) = 3 >= 3 = sel#(N,activate(XS)) *** Step 5.a:1.b:2: NaturalMI MAYBE + Considered Problem: - Strict DPs: activate#(n__fib1(X1,X2)) -> c_2(fib1#(X1,X2)) add#(s(X),Y) -> c_4(add#(X,Y)) - Weak DPs: fib#(N) -> fib1#(s(0()),s(0())) fib#(N) -> sel#(N,fib1(s(0()),s(0()))) fib1#(X,Y) -> c_6(add#(X,Y)) sel#(s(N),cons(X,XS)) -> activate#(XS) sel#(s(N),cons(X,XS)) -> sel#(N,activate(XS)) - Weak TRS: activate(X) -> X activate(n__fib1(X1,X2)) -> fib1(X1,X2) add(0(),X) -> X add(s(X),Y) -> s(add(X,Y)) fib1(X,Y) -> cons(X,n__fib1(Y,add(X,Y))) fib1(X1,X2) -> n__fib1(X1,X2) - Signature: {activate/1,add/2,fib/1,fib1/2,sel/2,activate#/1,add#/2,fib#/1,fib1#/2,sel#/2} / {0/0,cons/2,n__fib1/2,s/1 ,c_1/0,c_2/1,c_3/0,c_4/1,c_5/2,c_6/1,c_7/0,c_8/0,c_9/2} - Obligation: innermost runtime complexity wrt. defined symbols {activate#,add#,fib#,fib1#,sel#} and constructors {0,cons ,n__fib1,s} + Applied Processor: NaturalMI {miDimension = 2, miDegree = 2, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Just any strict-rules} + Details: We apply a matrix interpretation of kind constructor based matrix interpretation: The following argument positions are considered usable: uargs(c_2) = {1}, uargs(c_4) = {1}, uargs(c_6) = {1} Following symbols are considered usable: {activate#,add#,fib#,fib1#,sel#} TcT has computed the following interpretation: p(0) = [0] [2] p(activate) = [0 4] x1 + [0] [2 0] [4] p(add) = [1 4] x1 + [0 1] x2 + [3] [0 2] [0 4] [0] p(cons) = [0 0] x1 + [1 2] x2 + [0] [0 1] [0 0] [1] p(fib) = [1 1] x1 + [0] [2 1] [2] p(fib1) = [0 4] x2 + [0] [6 0] [0] p(n__fib1) = [0 3] x1 + [0 0] x2 + [4] [0 1] [0 1] [0] p(s) = [1] [2] p(sel) = [2 0] x1 + [0] [1 4] [2] p(activate#) = [6] [4] p(add#) = [0] [2] p(fib#) = [1 0] x1 + [7] [0 1] [5] p(fib1#) = [0 0] x2 + [2] [1 0] [4] p(sel#) = [7] [5] p(c_1) = [1] [0] p(c_2) = [2 0] x1 + [0] [2 0] [0] p(c_3) = [1] [0] p(c_4) = [4 0] x1 + [0] [0 0] [2] p(c_5) = [1 0] x1 + [0 1] x2 + [2] [1 2] [4 2] [1] p(c_6) = [1 1] x1 + [0] [0 0] [4] p(c_7) = [0] [0] p(c_8) = [1] [2] p(c_9) = [2 0] x1 + [0] [0 4] [4] Following rules are strictly oriented: activate#(n__fib1(X1,X2)) = [6] [4] > [4] [4] = c_2(fib1#(X1,X2)) Following rules are (at-least) weakly oriented: add#(s(X),Y) = [0] [2] >= [0] [2] = c_4(add#(X,Y)) fib#(N) = [1 0] N + [7] [0 1] [5] >= [2] [5] = fib1#(s(0()),s(0())) fib#(N) = [1 0] N + [7] [0 1] [5] >= [7] [5] = sel#(N,fib1(s(0()),s(0()))) fib1#(X,Y) = [0 0] Y + [2] [1 0] [4] >= [2] [4] = c_6(add#(X,Y)) sel#(s(N),cons(X,XS)) = [7] [5] >= [6] [4] = activate#(XS) sel#(s(N),cons(X,XS)) = [7] [5] >= [7] [5] = sel#(N,activate(XS)) *** Step 5.a:1.b:3: Failure MAYBE + Considered Problem: - Strict DPs: add#(s(X),Y) -> c_4(add#(X,Y)) - Weak DPs: activate#(n__fib1(X1,X2)) -> c_2(fib1#(X1,X2)) fib#(N) -> fib1#(s(0()),s(0())) fib#(N) -> sel#(N,fib1(s(0()),s(0()))) fib1#(X,Y) -> c_6(add#(X,Y)) sel#(s(N),cons(X,XS)) -> activate#(XS) sel#(s(N),cons(X,XS)) -> sel#(N,activate(XS)) - Weak TRS: activate(X) -> X activate(n__fib1(X1,X2)) -> fib1(X1,X2) add(0(),X) -> X add(s(X),Y) -> s(add(X,Y)) fib1(X,Y) -> cons(X,n__fib1(Y,add(X,Y))) fib1(X1,X2) -> n__fib1(X1,X2) - Signature: {activate/1,add/2,fib/1,fib1/2,sel/2,activate#/1,add#/2,fib#/1,fib1#/2,sel#/2} / {0/0,cons/2,n__fib1/2,s/1 ,c_1/0,c_2/1,c_3/0,c_4/1,c_5/2,c_6/1,c_7/0,c_8/0,c_9/2} - Obligation: innermost runtime complexity wrt. defined symbols {activate#,add#,fib#,fib1#,sel#} and constructors {0,cons ,n__fib1,s} + Applied Processor: EmptyProcessor + Details: The problem is still open. ** Step 5.b:1: RemoveWeakSuffixes WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: fib#(N) -> c_5(sel#(N,fib1(s(0()),s(0()))),fib1#(s(0()),s(0()))) sel#(s(N),cons(X,XS)) -> c_9(sel#(N,activate(XS)),activate#(XS)) - Weak DPs: activate#(n__fib1(X1,X2)) -> c_2(fib1#(X1,X2)) add#(s(X),Y) -> c_4(add#(X,Y)) fib1#(X,Y) -> c_6(add#(X,Y)) - Weak TRS: activate(X) -> X activate(n__fib1(X1,X2)) -> fib1(X1,X2) add(0(),X) -> X add(s(X),Y) -> s(add(X,Y)) fib1(X,Y) -> cons(X,n__fib1(Y,add(X,Y))) fib1(X1,X2) -> n__fib1(X1,X2) - Signature: {activate/1,add/2,fib/1,fib1/2,sel/2,activate#/1,add#/2,fib#/1,fib1#/2,sel#/2} / {0/0,cons/2,n__fib1/2,s/1 ,c_1/0,c_2/1,c_3/0,c_4/1,c_5/2,c_6/1,c_7/0,c_8/0,c_9/2} - Obligation: innermost runtime complexity wrt. defined symbols {activate#,add#,fib#,fib1#,sel#} and constructors {0,cons ,n__fib1,s} + Applied Processor: RemoveWeakSuffixes + Details: Consider the dependency graph 1:S:fib#(N) -> c_5(sel#(N,fib1(s(0()),s(0()))),fib1#(s(0()),s(0()))) -->_2 fib1#(X,Y) -> c_6(add#(X,Y)):5 -->_1 sel#(s(N),cons(X,XS)) -> c_9(sel#(N,activate(XS)),activate#(XS)):2 2:S:sel#(s(N),cons(X,XS)) -> c_9(sel#(N,activate(XS)),activate#(XS)) -->_2 activate#(n__fib1(X1,X2)) -> c_2(fib1#(X1,X2)):3 -->_1 sel#(s(N),cons(X,XS)) -> c_9(sel#(N,activate(XS)),activate#(XS)):2 3:W:activate#(n__fib1(X1,X2)) -> c_2(fib1#(X1,X2)) -->_1 fib1#(X,Y) -> c_6(add#(X,Y)):5 4:W:add#(s(X),Y) -> c_4(add#(X,Y)) -->_1 add#(s(X),Y) -> c_4(add#(X,Y)):4 5:W:fib1#(X,Y) -> c_6(add#(X,Y)) -->_1 add#(s(X),Y) -> c_4(add#(X,Y)):4 The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed. 3: activate#(n__fib1(X1,X2)) -> c_2(fib1#(X1,X2)) 5: fib1#(X,Y) -> c_6(add#(X,Y)) 4: add#(s(X),Y) -> c_4(add#(X,Y)) ** Step 5.b:2: SimplifyRHS WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: fib#(N) -> c_5(sel#(N,fib1(s(0()),s(0()))),fib1#(s(0()),s(0()))) sel#(s(N),cons(X,XS)) -> c_9(sel#(N,activate(XS)),activate#(XS)) - Weak TRS: activate(X) -> X activate(n__fib1(X1,X2)) -> fib1(X1,X2) add(0(),X) -> X add(s(X),Y) -> s(add(X,Y)) fib1(X,Y) -> cons(X,n__fib1(Y,add(X,Y))) fib1(X1,X2) -> n__fib1(X1,X2) - Signature: {activate/1,add/2,fib/1,fib1/2,sel/2,activate#/1,add#/2,fib#/1,fib1#/2,sel#/2} / {0/0,cons/2,n__fib1/2,s/1 ,c_1/0,c_2/1,c_3/0,c_4/1,c_5/2,c_6/1,c_7/0,c_8/0,c_9/2} - Obligation: innermost runtime complexity wrt. defined symbols {activate#,add#,fib#,fib1#,sel#} and constructors {0,cons ,n__fib1,s} + Applied Processor: SimplifyRHS + Details: Consider the dependency graph 1:S:fib#(N) -> c_5(sel#(N,fib1(s(0()),s(0()))),fib1#(s(0()),s(0()))) -->_1 sel#(s(N),cons(X,XS)) -> c_9(sel#(N,activate(XS)),activate#(XS)):2 2:S:sel#(s(N),cons(X,XS)) -> c_9(sel#(N,activate(XS)),activate#(XS)) -->_1 sel#(s(N),cons(X,XS)) -> c_9(sel#(N,activate(XS)),activate#(XS)):2 Due to missing edges in the depndency graph, the right-hand sides of following rules could be simplified: fib#(N) -> c_5(sel#(N,fib1(s(0()),s(0())))) sel#(s(N),cons(X,XS)) -> c_9(sel#(N,activate(XS))) ** Step 5.b:3: PredecessorEstimationCP WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: fib#(N) -> c_5(sel#(N,fib1(s(0()),s(0())))) sel#(s(N),cons(X,XS)) -> c_9(sel#(N,activate(XS))) - Weak TRS: activate(X) -> X activate(n__fib1(X1,X2)) -> fib1(X1,X2) add(0(),X) -> X add(s(X),Y) -> s(add(X,Y)) fib1(X,Y) -> cons(X,n__fib1(Y,add(X,Y))) fib1(X1,X2) -> n__fib1(X1,X2) - Signature: {activate/1,add/2,fib/1,fib1/2,sel/2,activate#/1,add#/2,fib#/1,fib1#/2,sel#/2} / {0/0,cons/2,n__fib1/2,s/1 ,c_1/0,c_2/1,c_3/0,c_4/1,c_5/1,c_6/1,c_7/0,c_8/0,c_9/1} - Obligation: innermost runtime complexity wrt. defined symbols {activate#,add#,fib#,fib1#,sel#} and constructors {0,cons ,n__fib1,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: 2: sel#(s(N),cons(X,XS)) -> c_9(sel#(N,activate(XS))) Consider the set of all dependency pairs 1: fib#(N) -> c_5(sel#(N,fib1(s(0()),s(0())))) 2: sel#(s(N),cons(X,XS)) -> c_9(sel#(N,activate(XS))) Processor NaturalMI {miDimension = 1, miDegree = 1, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Nothing}induces the complexity certificateTIME (?,O(n^1)) SPACE(?,?)on application of the dependency pairs {2} These cover all (indirect) predecessors of dependency pairs {1,2} their number of applications is equally bounded. The dependency pairs are shifted into the weak component. *** Step 5.b:3.a:1: NaturalMI WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: fib#(N) -> c_5(sel#(N,fib1(s(0()),s(0())))) sel#(s(N),cons(X,XS)) -> c_9(sel#(N,activate(XS))) - Weak TRS: activate(X) -> X activate(n__fib1(X1,X2)) -> fib1(X1,X2) add(0(),X) -> X add(s(X),Y) -> s(add(X,Y)) fib1(X,Y) -> cons(X,n__fib1(Y,add(X,Y))) fib1(X1,X2) -> n__fib1(X1,X2) - Signature: {activate/1,add/2,fib/1,fib1/2,sel/2,activate#/1,add#/2,fib#/1,fib1#/2,sel#/2} / {0/0,cons/2,n__fib1/2,s/1 ,c_1/0,c_2/1,c_3/0,c_4/1,c_5/1,c_6/1,c_7/0,c_8/0,c_9/1} - Obligation: innermost runtime complexity wrt. defined symbols {activate#,add#,fib#,fib1#,sel#} and constructors {0,cons ,n__fib1,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_5) = {1}, uargs(c_9) = {1} Following symbols are considered usable: {activate#,add#,fib#,fib1#,sel#} TcT has computed the following interpretation: p(0) = [0] p(activate) = [2] p(add) = [0] p(cons) = [1] x1 + [1] x2 + [0] p(fib) = [0] p(fib1) = [0] p(n__fib1) = [1] x1 + [1] x2 + [0] p(s) = [1] x1 + [9] p(sel) = [0] p(activate#) = [0] p(add#) = [1] x2 + [0] p(fib#) = [2] x1 + [0] p(fib1#) = [1] x1 + [1] x2 + [0] p(sel#) = [1] x1 + [0] p(c_1) = [0] p(c_2) = [1] x1 + [1] p(c_3) = [1] p(c_4) = [1] p(c_5) = [1] x1 + [0] p(c_6) = [0] p(c_7) = [1] p(c_8) = [1] p(c_9) = [1] x1 + [8] Following rules are strictly oriented: sel#(s(N),cons(X,XS)) = [1] N + [9] > [1] N + [8] = c_9(sel#(N,activate(XS))) Following rules are (at-least) weakly oriented: fib#(N) = [2] N + [0] >= [1] N + [0] = c_5(sel#(N,fib1(s(0()),s(0())))) *** Step 5.b:3.a:2: Assumption WORST_CASE(?,O(1)) + Considered Problem: - Strict DPs: fib#(N) -> c_5(sel#(N,fib1(s(0()),s(0())))) - Weak DPs: sel#(s(N),cons(X,XS)) -> c_9(sel#(N,activate(XS))) - Weak TRS: activate(X) -> X activate(n__fib1(X1,X2)) -> fib1(X1,X2) add(0(),X) -> X add(s(X),Y) -> s(add(X,Y)) fib1(X,Y) -> cons(X,n__fib1(Y,add(X,Y))) fib1(X1,X2) -> n__fib1(X1,X2) - Signature: {activate/1,add/2,fib/1,fib1/2,sel/2,activate#/1,add#/2,fib#/1,fib1#/2,sel#/2} / {0/0,cons/2,n__fib1/2,s/1 ,c_1/0,c_2/1,c_3/0,c_4/1,c_5/1,c_6/1,c_7/0,c_8/0,c_9/1} - Obligation: innermost runtime complexity wrt. defined symbols {activate#,add#,fib#,fib1#,sel#} and constructors {0,cons ,n__fib1,s} + Applied Processor: Assumption {assumed = Certificate {spaceUB = Unknown, spaceLB = Unknown, timeUB = Poly (Just 0), timeLB = Unknown}} + Details: () *** Step 5.b:3.b:1: RemoveWeakSuffixes WORST_CASE(?,O(1)) + Considered Problem: - Weak DPs: fib#(N) -> c_5(sel#(N,fib1(s(0()),s(0())))) sel#(s(N),cons(X,XS)) -> c_9(sel#(N,activate(XS))) - Weak TRS: activate(X) -> X activate(n__fib1(X1,X2)) -> fib1(X1,X2) add(0(),X) -> X add(s(X),Y) -> s(add(X,Y)) fib1(X,Y) -> cons(X,n__fib1(Y,add(X,Y))) fib1(X1,X2) -> n__fib1(X1,X2) - Signature: {activate/1,add/2,fib/1,fib1/2,sel/2,activate#/1,add#/2,fib#/1,fib1#/2,sel#/2} / {0/0,cons/2,n__fib1/2,s/1 ,c_1/0,c_2/1,c_3/0,c_4/1,c_5/1,c_6/1,c_7/0,c_8/0,c_9/1} - Obligation: innermost runtime complexity wrt. defined symbols {activate#,add#,fib#,fib1#,sel#} and constructors {0,cons ,n__fib1,s} + Applied Processor: RemoveWeakSuffixes + Details: Consider the dependency graph 1:W:fib#(N) -> c_5(sel#(N,fib1(s(0()),s(0())))) -->_1 sel#(s(N),cons(X,XS)) -> c_9(sel#(N,activate(XS))):2 2:W:sel#(s(N),cons(X,XS)) -> c_9(sel#(N,activate(XS))) -->_1 sel#(s(N),cons(X,XS)) -> c_9(sel#(N,activate(XS))):2 The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed. 1: fib#(N) -> c_5(sel#(N,fib1(s(0()),s(0())))) 2: sel#(s(N),cons(X,XS)) -> c_9(sel#(N,activate(XS))) *** Step 5.b:3.b:2: EmptyProcessor WORST_CASE(?,O(1)) + Considered Problem: - Weak TRS: activate(X) -> X activate(n__fib1(X1,X2)) -> fib1(X1,X2) add(0(),X) -> X add(s(X),Y) -> s(add(X,Y)) fib1(X,Y) -> cons(X,n__fib1(Y,add(X,Y))) fib1(X1,X2) -> n__fib1(X1,X2) - Signature: {activate/1,add/2,fib/1,fib1/2,sel/2,activate#/1,add#/2,fib#/1,fib1#/2,sel#/2} / {0/0,cons/2,n__fib1/2,s/1 ,c_1/0,c_2/1,c_3/0,c_4/1,c_5/1,c_6/1,c_7/0,c_8/0,c_9/1} - Obligation: innermost runtime complexity wrt. defined symbols {activate#,add#,fib#,fib1#,sel#} and constructors {0,cons ,n__fib1,s} + Applied Processor: EmptyProcessor + Details: The problem is already closed. The intended complexity is O(1). MAYBE