MAYBE * Step 1: DependencyPairs MAYBE + Considered Problem: - Strict TRS: a__f(X) -> cons(mark(X),f(g(X))) a__f(X) -> f(X) a__g(X) -> g(X) a__g(0()) -> s(0()) a__g(s(X)) -> s(s(a__g(mark(X)))) a__sel(X1,X2) -> sel(X1,X2) a__sel(0(),cons(X,Y)) -> mark(X) a__sel(s(X),cons(Y,Z)) -> a__sel(mark(X),mark(Z)) mark(0()) -> 0() mark(cons(X1,X2)) -> cons(mark(X1),X2) mark(f(X)) -> a__f(mark(X)) mark(g(X)) -> a__g(mark(X)) mark(s(X)) -> s(mark(X)) mark(sel(X1,X2)) -> a__sel(mark(X1),mark(X2)) - Signature: {a__f/1,a__g/1,a__sel/2,mark/1} / {0/0,cons/2,f/1,g/1,s/1,sel/2} - Obligation: innermost runtime complexity wrt. defined symbols {a__f,a__g,a__sel,mark} and constructors {0,cons,f,g,s ,sel} + Applied Processor: DependencyPairs {dpKind_ = DT} + Details: We add the following dependency tuples: Strict DPs a__f#(X) -> c_1(mark#(X)) a__f#(X) -> c_2() a__g#(X) -> c_3() a__g#(0()) -> c_4() a__g#(s(X)) -> c_5(a__g#(mark(X)),mark#(X)) a__sel#(X1,X2) -> c_6() a__sel#(0(),cons(X,Y)) -> c_7(mark#(X)) a__sel#(s(X),cons(Y,Z)) -> c_8(a__sel#(mark(X),mark(Z)),mark#(X),mark#(Z)) mark#(0()) -> c_9() mark#(cons(X1,X2)) -> c_10(mark#(X1)) mark#(f(X)) -> c_11(a__f#(mark(X)),mark#(X)) mark#(g(X)) -> c_12(a__g#(mark(X)),mark#(X)) mark#(s(X)) -> c_13(mark#(X)) mark#(sel(X1,X2)) -> c_14(a__sel#(mark(X1),mark(X2)),mark#(X1),mark#(X2)) Weak DPs and mark the set of starting terms. * Step 2: PredecessorEstimation MAYBE + Considered Problem: - Strict DPs: a__f#(X) -> c_1(mark#(X)) a__f#(X) -> c_2() a__g#(X) -> c_3() a__g#(0()) -> c_4() a__g#(s(X)) -> c_5(a__g#(mark(X)),mark#(X)) a__sel#(X1,X2) -> c_6() a__sel#(0(),cons(X,Y)) -> c_7(mark#(X)) a__sel#(s(X),cons(Y,Z)) -> c_8(a__sel#(mark(X),mark(Z)),mark#(X),mark#(Z)) mark#(0()) -> c_9() mark#(cons(X1,X2)) -> c_10(mark#(X1)) mark#(f(X)) -> c_11(a__f#(mark(X)),mark#(X)) mark#(g(X)) -> c_12(a__g#(mark(X)),mark#(X)) mark#(s(X)) -> c_13(mark#(X)) mark#(sel(X1,X2)) -> c_14(a__sel#(mark(X1),mark(X2)),mark#(X1),mark#(X2)) - Weak TRS: a__f(X) -> cons(mark(X),f(g(X))) a__f(X) -> f(X) a__g(X) -> g(X) a__g(0()) -> s(0()) a__g(s(X)) -> s(s(a__g(mark(X)))) a__sel(X1,X2) -> sel(X1,X2) a__sel(0(),cons(X,Y)) -> mark(X) a__sel(s(X),cons(Y,Z)) -> a__sel(mark(X),mark(Z)) mark(0()) -> 0() mark(cons(X1,X2)) -> cons(mark(X1),X2) mark(f(X)) -> a__f(mark(X)) mark(g(X)) -> a__g(mark(X)) mark(s(X)) -> s(mark(X)) mark(sel(X1,X2)) -> a__sel(mark(X1),mark(X2)) - Signature: {a__f/1,a__g/1,a__sel/2,mark/1,a__f#/1,a__g#/1,a__sel#/2,mark#/1} / {0/0,cons/2,f/1,g/1,s/1,sel/2,c_1/1 ,c_2/0,c_3/0,c_4/0,c_5/2,c_6/0,c_7/1,c_8/3,c_9/0,c_10/1,c_11/2,c_12/2,c_13/1,c_14/3} - Obligation: innermost runtime complexity wrt. defined symbols {a__f#,a__g#,a__sel#,mark#} and constructors {0,cons,f,g,s ,sel} + Applied Processor: PredecessorEstimation {onSelection = all simple predecessor estimation selector} + Details: We estimate the number of application of {2,3,4,6,9} by application of Pre({2,3,4,6,9}) = {1,5,7,8,10,11,12,13,14}. Here rules are labelled as follows: 1: a__f#(X) -> c_1(mark#(X)) 2: a__f#(X) -> c_2() 3: a__g#(X) -> c_3() 4: a__g#(0()) -> c_4() 5: a__g#(s(X)) -> c_5(a__g#(mark(X)),mark#(X)) 6: a__sel#(X1,X2) -> c_6() 7: a__sel#(0(),cons(X,Y)) -> c_7(mark#(X)) 8: a__sel#(s(X),cons(Y,Z)) -> c_8(a__sel#(mark(X),mark(Z)),mark#(X),mark#(Z)) 9: mark#(0()) -> c_9() 10: mark#(cons(X1,X2)) -> c_10(mark#(X1)) 11: mark#(f(X)) -> c_11(a__f#(mark(X)),mark#(X)) 12: mark#(g(X)) -> c_12(a__g#(mark(X)),mark#(X)) 13: mark#(s(X)) -> c_13(mark#(X)) 14: mark#(sel(X1,X2)) -> c_14(a__sel#(mark(X1),mark(X2)),mark#(X1),mark#(X2)) * Step 3: RemoveWeakSuffixes MAYBE + Considered Problem: - Strict DPs: a__f#(X) -> c_1(mark#(X)) a__g#(s(X)) -> c_5(a__g#(mark(X)),mark#(X)) a__sel#(0(),cons(X,Y)) -> c_7(mark#(X)) a__sel#(s(X),cons(Y,Z)) -> c_8(a__sel#(mark(X),mark(Z)),mark#(X),mark#(Z)) mark#(cons(X1,X2)) -> c_10(mark#(X1)) mark#(f(X)) -> c_11(a__f#(mark(X)),mark#(X)) mark#(g(X)) -> c_12(a__g#(mark(X)),mark#(X)) mark#(s(X)) -> c_13(mark#(X)) mark#(sel(X1,X2)) -> c_14(a__sel#(mark(X1),mark(X2)),mark#(X1),mark#(X2)) - Weak DPs: a__f#(X) -> c_2() a__g#(X) -> c_3() a__g#(0()) -> c_4() a__sel#(X1,X2) -> c_6() mark#(0()) -> c_9() - Weak TRS: a__f(X) -> cons(mark(X),f(g(X))) a__f(X) -> f(X) a__g(X) -> g(X) a__g(0()) -> s(0()) a__g(s(X)) -> s(s(a__g(mark(X)))) a__sel(X1,X2) -> sel(X1,X2) a__sel(0(),cons(X,Y)) -> mark(X) a__sel(s(X),cons(Y,Z)) -> a__sel(mark(X),mark(Z)) mark(0()) -> 0() mark(cons(X1,X2)) -> cons(mark(X1),X2) mark(f(X)) -> a__f(mark(X)) mark(g(X)) -> a__g(mark(X)) mark(s(X)) -> s(mark(X)) mark(sel(X1,X2)) -> a__sel(mark(X1),mark(X2)) - Signature: {a__f/1,a__g/1,a__sel/2,mark/1,a__f#/1,a__g#/1,a__sel#/2,mark#/1} / {0/0,cons/2,f/1,g/1,s/1,sel/2,c_1/1 ,c_2/0,c_3/0,c_4/0,c_5/2,c_6/0,c_7/1,c_8/3,c_9/0,c_10/1,c_11/2,c_12/2,c_13/1,c_14/3} - Obligation: innermost runtime complexity wrt. defined symbols {a__f#,a__g#,a__sel#,mark#} and constructors {0,cons,f,g,s ,sel} + Applied Processor: RemoveWeakSuffixes + Details: Consider the dependency graph 1:S:a__f#(X) -> c_1(mark#(X)) -->_1 mark#(sel(X1,X2)) -> c_14(a__sel#(mark(X1),mark(X2)),mark#(X1),mark#(X2)):9 -->_1 mark#(s(X)) -> c_13(mark#(X)):8 -->_1 mark#(g(X)) -> c_12(a__g#(mark(X)),mark#(X)):7 -->_1 mark#(f(X)) -> c_11(a__f#(mark(X)),mark#(X)):6 -->_1 mark#(cons(X1,X2)) -> c_10(mark#(X1)):5 -->_1 mark#(0()) -> c_9():14 2:S:a__g#(s(X)) -> c_5(a__g#(mark(X)),mark#(X)) -->_2 mark#(sel(X1,X2)) -> c_14(a__sel#(mark(X1),mark(X2)),mark#(X1),mark#(X2)):9 -->_2 mark#(s(X)) -> c_13(mark#(X)):8 -->_2 mark#(g(X)) -> c_12(a__g#(mark(X)),mark#(X)):7 -->_2 mark#(f(X)) -> c_11(a__f#(mark(X)),mark#(X)):6 -->_2 mark#(cons(X1,X2)) -> c_10(mark#(X1)):5 -->_2 mark#(0()) -> c_9():14 -->_1 a__g#(0()) -> c_4():12 -->_1 a__g#(X) -> c_3():11 -->_1 a__g#(s(X)) -> c_5(a__g#(mark(X)),mark#(X)):2 3:S:a__sel#(0(),cons(X,Y)) -> c_7(mark#(X)) -->_1 mark#(sel(X1,X2)) -> c_14(a__sel#(mark(X1),mark(X2)),mark#(X1),mark#(X2)):9 -->_1 mark#(s(X)) -> c_13(mark#(X)):8 -->_1 mark#(g(X)) -> c_12(a__g#(mark(X)),mark#(X)):7 -->_1 mark#(f(X)) -> c_11(a__f#(mark(X)),mark#(X)):6 -->_1 mark#(cons(X1,X2)) -> c_10(mark#(X1)):5 -->_1 mark#(0()) -> c_9():14 4:S:a__sel#(s(X),cons(Y,Z)) -> c_8(a__sel#(mark(X),mark(Z)),mark#(X),mark#(Z)) -->_3 mark#(sel(X1,X2)) -> c_14(a__sel#(mark(X1),mark(X2)),mark#(X1),mark#(X2)):9 -->_2 mark#(sel(X1,X2)) -> c_14(a__sel#(mark(X1),mark(X2)),mark#(X1),mark#(X2)):9 -->_3 mark#(s(X)) -> c_13(mark#(X)):8 -->_2 mark#(s(X)) -> c_13(mark#(X)):8 -->_3 mark#(g(X)) -> c_12(a__g#(mark(X)),mark#(X)):7 -->_2 mark#(g(X)) -> c_12(a__g#(mark(X)),mark#(X)):7 -->_3 mark#(f(X)) -> c_11(a__f#(mark(X)),mark#(X)):6 -->_2 mark#(f(X)) -> c_11(a__f#(mark(X)),mark#(X)):6 -->_3 mark#(cons(X1,X2)) -> c_10(mark#(X1)):5 -->_2 mark#(cons(X1,X2)) -> c_10(mark#(X1)):5 -->_3 mark#(0()) -> c_9():14 -->_2 mark#(0()) -> c_9():14 -->_1 a__sel#(X1,X2) -> c_6():13 -->_1 a__sel#(s(X),cons(Y,Z)) -> c_8(a__sel#(mark(X),mark(Z)),mark#(X),mark#(Z)):4 -->_1 a__sel#(0(),cons(X,Y)) -> c_7(mark#(X)):3 5:S:mark#(cons(X1,X2)) -> c_10(mark#(X1)) -->_1 mark#(sel(X1,X2)) -> c_14(a__sel#(mark(X1),mark(X2)),mark#(X1),mark#(X2)):9 -->_1 mark#(s(X)) -> c_13(mark#(X)):8 -->_1 mark#(g(X)) -> c_12(a__g#(mark(X)),mark#(X)):7 -->_1 mark#(f(X)) -> c_11(a__f#(mark(X)),mark#(X)):6 -->_1 mark#(0()) -> c_9():14 -->_1 mark#(cons(X1,X2)) -> c_10(mark#(X1)):5 6:S:mark#(f(X)) -> c_11(a__f#(mark(X)),mark#(X)) -->_2 mark#(sel(X1,X2)) -> c_14(a__sel#(mark(X1),mark(X2)),mark#(X1),mark#(X2)):9 -->_2 mark#(s(X)) -> c_13(mark#(X)):8 -->_2 mark#(g(X)) -> c_12(a__g#(mark(X)),mark#(X)):7 -->_2 mark#(0()) -> c_9():14 -->_1 a__f#(X) -> c_2():10 -->_2 mark#(f(X)) -> c_11(a__f#(mark(X)),mark#(X)):6 -->_2 mark#(cons(X1,X2)) -> c_10(mark#(X1)):5 -->_1 a__f#(X) -> c_1(mark#(X)):1 7:S:mark#(g(X)) -> c_12(a__g#(mark(X)),mark#(X)) -->_2 mark#(sel(X1,X2)) -> c_14(a__sel#(mark(X1),mark(X2)),mark#(X1),mark#(X2)):9 -->_2 mark#(s(X)) -> c_13(mark#(X)):8 -->_2 mark#(0()) -> c_9():14 -->_1 a__g#(0()) -> c_4():12 -->_1 a__g#(X) -> c_3():11 -->_2 mark#(g(X)) -> c_12(a__g#(mark(X)),mark#(X)):7 -->_2 mark#(f(X)) -> c_11(a__f#(mark(X)),mark#(X)):6 -->_2 mark#(cons(X1,X2)) -> c_10(mark#(X1)):5 -->_1 a__g#(s(X)) -> c_5(a__g#(mark(X)),mark#(X)):2 8:S:mark#(s(X)) -> c_13(mark#(X)) -->_1 mark#(sel(X1,X2)) -> c_14(a__sel#(mark(X1),mark(X2)),mark#(X1),mark#(X2)):9 -->_1 mark#(0()) -> c_9():14 -->_1 mark#(s(X)) -> c_13(mark#(X)):8 -->_1 mark#(g(X)) -> c_12(a__g#(mark(X)),mark#(X)):7 -->_1 mark#(f(X)) -> c_11(a__f#(mark(X)),mark#(X)):6 -->_1 mark#(cons(X1,X2)) -> c_10(mark#(X1)):5 9:S:mark#(sel(X1,X2)) -> c_14(a__sel#(mark(X1),mark(X2)),mark#(X1),mark#(X2)) -->_3 mark#(0()) -> c_9():14 -->_2 mark#(0()) -> c_9():14 -->_1 a__sel#(X1,X2) -> c_6():13 -->_3 mark#(sel(X1,X2)) -> c_14(a__sel#(mark(X1),mark(X2)),mark#(X1),mark#(X2)):9 -->_2 mark#(sel(X1,X2)) -> c_14(a__sel#(mark(X1),mark(X2)),mark#(X1),mark#(X2)):9 -->_3 mark#(s(X)) -> c_13(mark#(X)):8 -->_2 mark#(s(X)) -> c_13(mark#(X)):8 -->_3 mark#(g(X)) -> c_12(a__g#(mark(X)),mark#(X)):7 -->_2 mark#(g(X)) -> c_12(a__g#(mark(X)),mark#(X)):7 -->_3 mark#(f(X)) -> c_11(a__f#(mark(X)),mark#(X)):6 -->_2 mark#(f(X)) -> c_11(a__f#(mark(X)),mark#(X)):6 -->_3 mark#(cons(X1,X2)) -> c_10(mark#(X1)):5 -->_2 mark#(cons(X1,X2)) -> c_10(mark#(X1)):5 -->_1 a__sel#(s(X),cons(Y,Z)) -> c_8(a__sel#(mark(X),mark(Z)),mark#(X),mark#(Z)):4 -->_1 a__sel#(0(),cons(X,Y)) -> c_7(mark#(X)):3 10:W:a__f#(X) -> c_2() 11:W:a__g#(X) -> c_3() 12:W:a__g#(0()) -> c_4() 13:W:a__sel#(X1,X2) -> c_6() 14:W:mark#(0()) -> c_9() The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed. 10: a__f#(X) -> c_2() 11: a__g#(X) -> c_3() 12: a__g#(0()) -> c_4() 13: a__sel#(X1,X2) -> c_6() 14: mark#(0()) -> c_9() * Step 4: Failure MAYBE + Considered Problem: - Strict DPs: a__f#(X) -> c_1(mark#(X)) a__g#(s(X)) -> c_5(a__g#(mark(X)),mark#(X)) a__sel#(0(),cons(X,Y)) -> c_7(mark#(X)) a__sel#(s(X),cons(Y,Z)) -> c_8(a__sel#(mark(X),mark(Z)),mark#(X),mark#(Z)) mark#(cons(X1,X2)) -> c_10(mark#(X1)) mark#(f(X)) -> c_11(a__f#(mark(X)),mark#(X)) mark#(g(X)) -> c_12(a__g#(mark(X)),mark#(X)) mark#(s(X)) -> c_13(mark#(X)) mark#(sel(X1,X2)) -> c_14(a__sel#(mark(X1),mark(X2)),mark#(X1),mark#(X2)) - Weak TRS: a__f(X) -> cons(mark(X),f(g(X))) a__f(X) -> f(X) a__g(X) -> g(X) a__g(0()) -> s(0()) a__g(s(X)) -> s(s(a__g(mark(X)))) a__sel(X1,X2) -> sel(X1,X2) a__sel(0(),cons(X,Y)) -> mark(X) a__sel(s(X),cons(Y,Z)) -> a__sel(mark(X),mark(Z)) mark(0()) -> 0() mark(cons(X1,X2)) -> cons(mark(X1),X2) mark(f(X)) -> a__f(mark(X)) mark(g(X)) -> a__g(mark(X)) mark(s(X)) -> s(mark(X)) mark(sel(X1,X2)) -> a__sel(mark(X1),mark(X2)) - Signature: {a__f/1,a__g/1,a__sel/2,mark/1,a__f#/1,a__g#/1,a__sel#/2,mark#/1} / {0/0,cons/2,f/1,g/1,s/1,sel/2,c_1/1 ,c_2/0,c_3/0,c_4/0,c_5/2,c_6/0,c_7/1,c_8/3,c_9/0,c_10/1,c_11/2,c_12/2,c_13/1,c_14/3} - Obligation: innermost runtime complexity wrt. defined symbols {a__f#,a__g#,a__sel#,mark#} and constructors {0,cons,f,g,s ,sel} + Applied Processor: EmptyProcessor + Details: The problem is still open. MAYBE