YES(O(1),O(n^2)) We are left with following problem, upon which TcT provides the certificate YES(O(1),O(n^2)). Strict Trs: { #equal(@x, @y) -> #eq(@x, @y) , and(@x, @y) -> #and(@x, @y) , eq(@l1, @l2) -> eq#1(@l1, @l2) , eq#1(::(@x, @xs), @l2) -> eq#3(@l2, @x, @xs) , eq#1(nil(), @l2) -> eq#2(@l2) , eq#3(::(@y, @ys), @x, @xs) -> and(#equal(@x, @y), eq(@xs, @ys)) , eq#3(nil(), @x, @xs) -> #false() , eq#2(::(@y, @ys)) -> #false() , eq#2(nil()) -> #true() , nub(@l) -> nub#1(@l) , nub#1(::(@x, @xs)) -> ::(@x, nub(remove(@x, @xs))) , nub#1(nil()) -> nil() , remove(@x, @l) -> remove#1(@l, @x) , remove#1(::(@y, @ys), @x) -> remove#2(eq(@x, @y), @x, @y, @ys) , remove#1(nil(), @x) -> nil() , remove#2(#false(), @x, @y, @ys) -> ::(@y, remove(@x, @ys)) , remove#2(#true(), @x, @y, @ys) -> remove(@x, @ys) } Weak Trs: { #eq(::(@x_1, @x_2), ::(@y_1, @y_2)) -> #and(#eq(@x_1, @y_1), #eq(@x_2, @y_2)) , #eq(::(@x_1, @x_2), nil()) -> #false() , #eq(nil(), ::(@y_1, @y_2)) -> #false() , #eq(nil(), nil()) -> #true() , #eq(#0(), #0()) -> #true() , #eq(#0(), #neg(@y)) -> #false() , #eq(#0(), #pos(@y)) -> #false() , #eq(#0(), #s(@y)) -> #false() , #eq(#neg(@x), #0()) -> #false() , #eq(#neg(@x), #neg(@y)) -> #eq(@x, @y) , #eq(#neg(@x), #pos(@y)) -> #false() , #eq(#pos(@x), #0()) -> #false() , #eq(#pos(@x), #neg(@y)) -> #false() , #eq(#pos(@x), #pos(@y)) -> #eq(@x, @y) , #eq(#s(@x), #0()) -> #false() , #eq(#s(@x), #s(@y)) -> #eq(@x, @y) , #and(#false(), #false()) -> #false() , #and(#false(), #true()) -> #false() , #and(#true(), #false()) -> #false() , #and(#true(), #true()) -> #true() } Obligation: innermost runtime complexity Answer: YES(O(1),O(n^2)) We add following dependency tuples: Strict DPs: { #equal^#(@x, @y) -> c_1(#eq^#(@x, @y)) , and^#(@x, @y) -> c_2(#and^#(@x, @y)) , eq^#(@l1, @l2) -> c_3(eq#1^#(@l1, @l2)) , eq#1^#(::(@x, @xs), @l2) -> c_4(eq#3^#(@l2, @x, @xs)) , eq#1^#(nil(), @l2) -> c_5(eq#2^#(@l2)) , eq#3^#(::(@y, @ys), @x, @xs) -> c_6(and^#(#equal(@x, @y), eq(@xs, @ys)), #equal^#(@x, @y), eq^#(@xs, @ys)) , eq#3^#(nil(), @x, @xs) -> c_7() , eq#2^#(::(@y, @ys)) -> c_8() , eq#2^#(nil()) -> c_9() , nub^#(@l) -> c_10(nub#1^#(@l)) , nub#1^#(::(@x, @xs)) -> c_11(nub^#(remove(@x, @xs)), remove^#(@x, @xs)) , nub#1^#(nil()) -> c_12() , remove^#(@x, @l) -> c_13(remove#1^#(@l, @x)) , remove#1^#(::(@y, @ys), @x) -> c_14(remove#2^#(eq(@x, @y), @x, @y, @ys), eq^#(@x, @y)) , remove#1^#(nil(), @x) -> c_15() , remove#2^#(#false(), @x, @y, @ys) -> c_16(remove^#(@x, @ys)) , remove#2^#(#true(), @x, @y, @ys) -> c_17(remove^#(@x, @ys)) } Weak DPs: { #eq^#(::(@x_1, @x_2), ::(@y_1, @y_2)) -> c_18(#and^#(#eq(@x_1, @y_1), #eq(@x_2, @y_2)), #eq^#(@x_1, @y_1), #eq^#(@x_2, @y_2)) , #eq^#(::(@x_1, @x_2), nil()) -> c_19() , #eq^#(nil(), ::(@y_1, @y_2)) -> c_20() , #eq^#(nil(), nil()) -> c_21() , #eq^#(#0(), #0()) -> c_22() , #eq^#(#0(), #neg(@y)) -> c_23() , #eq^#(#0(), #pos(@y)) -> c_24() , #eq^#(#0(), #s(@y)) -> c_25() , #eq^#(#neg(@x), #0()) -> c_26() , #eq^#(#neg(@x), #neg(@y)) -> c_27(#eq^#(@x, @y)) , #eq^#(#neg(@x), #pos(@y)) -> c_28() , #eq^#(#pos(@x), #0()) -> c_29() , #eq^#(#pos(@x), #neg(@y)) -> c_30() , #eq^#(#pos(@x), #pos(@y)) -> c_31(#eq^#(@x, @y)) , #eq^#(#s(@x), #0()) -> c_32() , #eq^#(#s(@x), #s(@y)) -> c_33(#eq^#(@x, @y)) , #and^#(#false(), #false()) -> c_34() , #and^#(#false(), #true()) -> c_35() , #and^#(#true(), #false()) -> c_36() , #and^#(#true(), #true()) -> c_37() } and mark the set of starting terms. We are left with following problem, upon which TcT provides the certificate YES(O(1),O(n^2)). Strict DPs: { #equal^#(@x, @y) -> c_1(#eq^#(@x, @y)) , and^#(@x, @y) -> c_2(#and^#(@x, @y)) , eq^#(@l1, @l2) -> c_3(eq#1^#(@l1, @l2)) , eq#1^#(::(@x, @xs), @l2) -> c_4(eq#3^#(@l2, @x, @xs)) , eq#1^#(nil(), @l2) -> c_5(eq#2^#(@l2)) , eq#3^#(::(@y, @ys), @x, @xs) -> c_6(and^#(#equal(@x, @y), eq(@xs, @ys)), #equal^#(@x, @y), eq^#(@xs, @ys)) , eq#3^#(nil(), @x, @xs) -> c_7() , eq#2^#(::(@y, @ys)) -> c_8() , eq#2^#(nil()) -> c_9() , nub^#(@l) -> c_10(nub#1^#(@l)) , nub#1^#(::(@x, @xs)) -> c_11(nub^#(remove(@x, @xs)), remove^#(@x, @xs)) , nub#1^#(nil()) -> c_12() , remove^#(@x, @l) -> c_13(remove#1^#(@l, @x)) , remove#1^#(::(@y, @ys), @x) -> c_14(remove#2^#(eq(@x, @y), @x, @y, @ys), eq^#(@x, @y)) , remove#1^#(nil(), @x) -> c_15() , remove#2^#(#false(), @x, @y, @ys) -> c_16(remove^#(@x, @ys)) , remove#2^#(#true(), @x, @y, @ys) -> c_17(remove^#(@x, @ys)) } Weak DPs: { #eq^#(::(@x_1, @x_2), ::(@y_1, @y_2)) -> c_18(#and^#(#eq(@x_1, @y_1), #eq(@x_2, @y_2)), #eq^#(@x_1, @y_1), #eq^#(@x_2, @y_2)) , #eq^#(::(@x_1, @x_2), nil()) -> c_19() , #eq^#(nil(), ::(@y_1, @y_2)) -> c_20() , #eq^#(nil(), nil()) -> c_21() , #eq^#(#0(), #0()) -> c_22() , #eq^#(#0(), #neg(@y)) -> c_23() , #eq^#(#0(), #pos(@y)) -> c_24() , #eq^#(#0(), #s(@y)) -> c_25() , #eq^#(#neg(@x), #0()) -> c_26() , #eq^#(#neg(@x), #neg(@y)) -> c_27(#eq^#(@x, @y)) , #eq^#(#neg(@x), #pos(@y)) -> c_28() , #eq^#(#pos(@x), #0()) -> c_29() , #eq^#(#pos(@x), #neg(@y)) -> c_30() , #eq^#(#pos(@x), #pos(@y)) -> c_31(#eq^#(@x, @y)) , #eq^#(#s(@x), #0()) -> c_32() , #eq^#(#s(@x), #s(@y)) -> c_33(#eq^#(@x, @y)) , #and^#(#false(), #false()) -> c_34() , #and^#(#false(), #true()) -> c_35() , #and^#(#true(), #false()) -> c_36() , #and^#(#true(), #true()) -> c_37() } Weak Trs: { #equal(@x, @y) -> #eq(@x, @y) , #eq(::(@x_1, @x_2), ::(@y_1, @y_2)) -> #and(#eq(@x_1, @y_1), #eq(@x_2, @y_2)) , #eq(::(@x_1, @x_2), nil()) -> #false() , #eq(nil(), ::(@y_1, @y_2)) -> #false() , #eq(nil(), nil()) -> #true() , #eq(#0(), #0()) -> #true() , #eq(#0(), #neg(@y)) -> #false() , #eq(#0(), #pos(@y)) -> #false() , #eq(#0(), #s(@y)) -> #false() , #eq(#neg(@x), #0()) -> #false() , #eq(#neg(@x), #neg(@y)) -> #eq(@x, @y) , #eq(#neg(@x), #pos(@y)) -> #false() , #eq(#pos(@x), #0()) -> #false() , #eq(#pos(@x), #neg(@y)) -> #false() , #eq(#pos(@x), #pos(@y)) -> #eq(@x, @y) , #eq(#s(@x), #0()) -> #false() , #eq(#s(@x), #s(@y)) -> #eq(@x, @y) , and(@x, @y) -> #and(@x, @y) , #and(#false(), #false()) -> #false() , #and(#false(), #true()) -> #false() , #and(#true(), #false()) -> #false() , #and(#true(), #true()) -> #true() , eq(@l1, @l2) -> eq#1(@l1, @l2) , eq#1(::(@x, @xs), @l2) -> eq#3(@l2, @x, @xs) , eq#1(nil(), @l2) -> eq#2(@l2) , eq#3(::(@y, @ys), @x, @xs) -> and(#equal(@x, @y), eq(@xs, @ys)) , eq#3(nil(), @x, @xs) -> #false() , eq#2(::(@y, @ys)) -> #false() , eq#2(nil()) -> #true() , nub(@l) -> nub#1(@l) , nub#1(::(@x, @xs)) -> ::(@x, nub(remove(@x, @xs))) , nub#1(nil()) -> nil() , remove(@x, @l) -> remove#1(@l, @x) , remove#1(::(@y, @ys), @x) -> remove#2(eq(@x, @y), @x, @y, @ys) , remove#1(nil(), @x) -> nil() , remove#2(#false(), @x, @y, @ys) -> ::(@y, remove(@x, @ys)) , remove#2(#true(), @x, @y, @ys) -> remove(@x, @ys) } Obligation: innermost runtime complexity Answer: YES(O(1),O(n^2)) We estimate the number of application of {1,2,7,8,9,12,15} by applications of Pre({1,2,7,8,9,12,15}) = {4,5,6,10,13}. Here rules are labeled as follows: DPs: { 1: #equal^#(@x, @y) -> c_1(#eq^#(@x, @y)) , 2: and^#(@x, @y) -> c_2(#and^#(@x, @y)) , 3: eq^#(@l1, @l2) -> c_3(eq#1^#(@l1, @l2)) , 4: eq#1^#(::(@x, @xs), @l2) -> c_4(eq#3^#(@l2, @x, @xs)) , 5: eq#1^#(nil(), @l2) -> c_5(eq#2^#(@l2)) , 6: eq#3^#(::(@y, @ys), @x, @xs) -> c_6(and^#(#equal(@x, @y), eq(@xs, @ys)), #equal^#(@x, @y), eq^#(@xs, @ys)) , 7: eq#3^#(nil(), @x, @xs) -> c_7() , 8: eq#2^#(::(@y, @ys)) -> c_8() , 9: eq#2^#(nil()) -> c_9() , 10: nub^#(@l) -> c_10(nub#1^#(@l)) , 11: nub#1^#(::(@x, @xs)) -> c_11(nub^#(remove(@x, @xs)), remove^#(@x, @xs)) , 12: nub#1^#(nil()) -> c_12() , 13: remove^#(@x, @l) -> c_13(remove#1^#(@l, @x)) , 14: remove#1^#(::(@y, @ys), @x) -> c_14(remove#2^#(eq(@x, @y), @x, @y, @ys), eq^#(@x, @y)) , 15: remove#1^#(nil(), @x) -> c_15() , 16: remove#2^#(#false(), @x, @y, @ys) -> c_16(remove^#(@x, @ys)) , 17: remove#2^#(#true(), @x, @y, @ys) -> c_17(remove^#(@x, @ys)) , 18: #eq^#(::(@x_1, @x_2), ::(@y_1, @y_2)) -> c_18(#and^#(#eq(@x_1, @y_1), #eq(@x_2, @y_2)), #eq^#(@x_1, @y_1), #eq^#(@x_2, @y_2)) , 19: #eq^#(::(@x_1, @x_2), nil()) -> c_19() , 20: #eq^#(nil(), ::(@y_1, @y_2)) -> c_20() , 21: #eq^#(nil(), nil()) -> c_21() , 22: #eq^#(#0(), #0()) -> c_22() , 23: #eq^#(#0(), #neg(@y)) -> c_23() , 24: #eq^#(#0(), #pos(@y)) -> c_24() , 25: #eq^#(#0(), #s(@y)) -> c_25() , 26: #eq^#(#neg(@x), #0()) -> c_26() , 27: #eq^#(#neg(@x), #neg(@y)) -> c_27(#eq^#(@x, @y)) , 28: #eq^#(#neg(@x), #pos(@y)) -> c_28() , 29: #eq^#(#pos(@x), #0()) -> c_29() , 30: #eq^#(#pos(@x), #neg(@y)) -> c_30() , 31: #eq^#(#pos(@x), #pos(@y)) -> c_31(#eq^#(@x, @y)) , 32: #eq^#(#s(@x), #0()) -> c_32() , 33: #eq^#(#s(@x), #s(@y)) -> c_33(#eq^#(@x, @y)) , 34: #and^#(#false(), #false()) -> c_34() , 35: #and^#(#false(), #true()) -> c_35() , 36: #and^#(#true(), #false()) -> c_36() , 37: #and^#(#true(), #true()) -> c_37() } We are left with following problem, upon which TcT provides the certificate YES(O(1),O(n^2)). Strict DPs: { eq^#(@l1, @l2) -> c_3(eq#1^#(@l1, @l2)) , eq#1^#(::(@x, @xs), @l2) -> c_4(eq#3^#(@l2, @x, @xs)) , eq#1^#(nil(), @l2) -> c_5(eq#2^#(@l2)) , eq#3^#(::(@y, @ys), @x, @xs) -> c_6(and^#(#equal(@x, @y), eq(@xs, @ys)), #equal^#(@x, @y), eq^#(@xs, @ys)) , nub^#(@l) -> c_10(nub#1^#(@l)) , nub#1^#(::(@x, @xs)) -> c_11(nub^#(remove(@x, @xs)), remove^#(@x, @xs)) , remove^#(@x, @l) -> c_13(remove#1^#(@l, @x)) , remove#1^#(::(@y, @ys), @x) -> c_14(remove#2^#(eq(@x, @y), @x, @y, @ys), eq^#(@x, @y)) , remove#2^#(#false(), @x, @y, @ys) -> c_16(remove^#(@x, @ys)) , remove#2^#(#true(), @x, @y, @ys) -> c_17(remove^#(@x, @ys)) } Weak DPs: { #equal^#(@x, @y) -> c_1(#eq^#(@x, @y)) , #eq^#(::(@x_1, @x_2), ::(@y_1, @y_2)) -> c_18(#and^#(#eq(@x_1, @y_1), #eq(@x_2, @y_2)), #eq^#(@x_1, @y_1), #eq^#(@x_2, @y_2)) , #eq^#(::(@x_1, @x_2), nil()) -> c_19() , #eq^#(nil(), ::(@y_1, @y_2)) -> c_20() , #eq^#(nil(), nil()) -> c_21() , #eq^#(#0(), #0()) -> c_22() , #eq^#(#0(), #neg(@y)) -> c_23() , #eq^#(#0(), #pos(@y)) -> c_24() , #eq^#(#0(), #s(@y)) -> c_25() , #eq^#(#neg(@x), #0()) -> c_26() , #eq^#(#neg(@x), #neg(@y)) -> c_27(#eq^#(@x, @y)) , #eq^#(#neg(@x), #pos(@y)) -> c_28() , #eq^#(#pos(@x), #0()) -> c_29() , #eq^#(#pos(@x), #neg(@y)) -> c_30() , #eq^#(#pos(@x), #pos(@y)) -> c_31(#eq^#(@x, @y)) , #eq^#(#s(@x), #0()) -> c_32() , #eq^#(#s(@x), #s(@y)) -> c_33(#eq^#(@x, @y)) , and^#(@x, @y) -> c_2(#and^#(@x, @y)) , #and^#(#false(), #false()) -> c_34() , #and^#(#false(), #true()) -> c_35() , #and^#(#true(), #false()) -> c_36() , #and^#(#true(), #true()) -> c_37() , eq#3^#(nil(), @x, @xs) -> c_7() , eq#2^#(::(@y, @ys)) -> c_8() , eq#2^#(nil()) -> c_9() , nub#1^#(nil()) -> c_12() , remove#1^#(nil(), @x) -> c_15() } Weak Trs: { #equal(@x, @y) -> #eq(@x, @y) , #eq(::(@x_1, @x_2), ::(@y_1, @y_2)) -> #and(#eq(@x_1, @y_1), #eq(@x_2, @y_2)) , #eq(::(@x_1, @x_2), nil()) -> #false() , #eq(nil(), ::(@y_1, @y_2)) -> #false() , #eq(nil(), nil()) -> #true() , #eq(#0(), #0()) -> #true() , #eq(#0(), #neg(@y)) -> #false() , #eq(#0(), #pos(@y)) -> #false() , #eq(#0(), #s(@y)) -> #false() , #eq(#neg(@x), #0()) -> #false() , #eq(#neg(@x), #neg(@y)) -> #eq(@x, @y) , #eq(#neg(@x), #pos(@y)) -> #false() , #eq(#pos(@x), #0()) -> #false() , #eq(#pos(@x), #neg(@y)) -> #false() , #eq(#pos(@x), #pos(@y)) -> #eq(@x, @y) , #eq(#s(@x), #0()) -> #false() , #eq(#s(@x), #s(@y)) -> #eq(@x, @y) , and(@x, @y) -> #and(@x, @y) , #and(#false(), #false()) -> #false() , #and(#false(), #true()) -> #false() , #and(#true(), #false()) -> #false() , #and(#true(), #true()) -> #true() , eq(@l1, @l2) -> eq#1(@l1, @l2) , eq#1(::(@x, @xs), @l2) -> eq#3(@l2, @x, @xs) , eq#1(nil(), @l2) -> eq#2(@l2) , eq#3(::(@y, @ys), @x, @xs) -> and(#equal(@x, @y), eq(@xs, @ys)) , eq#3(nil(), @x, @xs) -> #false() , eq#2(::(@y, @ys)) -> #false() , eq#2(nil()) -> #true() , nub(@l) -> nub#1(@l) , nub#1(::(@x, @xs)) -> ::(@x, nub(remove(@x, @xs))) , nub#1(nil()) -> nil() , remove(@x, @l) -> remove#1(@l, @x) , remove#1(::(@y, @ys), @x) -> remove#2(eq(@x, @y), @x, @y, @ys) , remove#1(nil(), @x) -> nil() , remove#2(#false(), @x, @y, @ys) -> ::(@y, remove(@x, @ys)) , remove#2(#true(), @x, @y, @ys) -> remove(@x, @ys) } Obligation: innermost runtime complexity Answer: YES(O(1),O(n^2)) We estimate the number of application of {3} by applications of Pre({3}) = {1}. Here rules are labeled as follows: DPs: { 1: eq^#(@l1, @l2) -> c_3(eq#1^#(@l1, @l2)) , 2: eq#1^#(::(@x, @xs), @l2) -> c_4(eq#3^#(@l2, @x, @xs)) , 3: eq#1^#(nil(), @l2) -> c_5(eq#2^#(@l2)) , 4: eq#3^#(::(@y, @ys), @x, @xs) -> c_6(and^#(#equal(@x, @y), eq(@xs, @ys)), #equal^#(@x, @y), eq^#(@xs, @ys)) , 5: nub^#(@l) -> c_10(nub#1^#(@l)) , 6: nub#1^#(::(@x, @xs)) -> c_11(nub^#(remove(@x, @xs)), remove^#(@x, @xs)) , 7: remove^#(@x, @l) -> c_13(remove#1^#(@l, @x)) , 8: remove#1^#(::(@y, @ys), @x) -> c_14(remove#2^#(eq(@x, @y), @x, @y, @ys), eq^#(@x, @y)) , 9: remove#2^#(#false(), @x, @y, @ys) -> c_16(remove^#(@x, @ys)) , 10: remove#2^#(#true(), @x, @y, @ys) -> c_17(remove^#(@x, @ys)) , 11: #equal^#(@x, @y) -> c_1(#eq^#(@x, @y)) , 12: #eq^#(::(@x_1, @x_2), ::(@y_1, @y_2)) -> c_18(#and^#(#eq(@x_1, @y_1), #eq(@x_2, @y_2)), #eq^#(@x_1, @y_1), #eq^#(@x_2, @y_2)) , 13: #eq^#(::(@x_1, @x_2), nil()) -> c_19() , 14: #eq^#(nil(), ::(@y_1, @y_2)) -> c_20() , 15: #eq^#(nil(), nil()) -> c_21() , 16: #eq^#(#0(), #0()) -> c_22() , 17: #eq^#(#0(), #neg(@y)) -> c_23() , 18: #eq^#(#0(), #pos(@y)) -> c_24() , 19: #eq^#(#0(), #s(@y)) -> c_25() , 20: #eq^#(#neg(@x), #0()) -> c_26() , 21: #eq^#(#neg(@x), #neg(@y)) -> c_27(#eq^#(@x, @y)) , 22: #eq^#(#neg(@x), #pos(@y)) -> c_28() , 23: #eq^#(#pos(@x), #0()) -> c_29() , 24: #eq^#(#pos(@x), #neg(@y)) -> c_30() , 25: #eq^#(#pos(@x), #pos(@y)) -> c_31(#eq^#(@x, @y)) , 26: #eq^#(#s(@x), #0()) -> c_32() , 27: #eq^#(#s(@x), #s(@y)) -> c_33(#eq^#(@x, @y)) , 28: and^#(@x, @y) -> c_2(#and^#(@x, @y)) , 29: #and^#(#false(), #false()) -> c_34() , 30: #and^#(#false(), #true()) -> c_35() , 31: #and^#(#true(), #false()) -> c_36() , 32: #and^#(#true(), #true()) -> c_37() , 33: eq#3^#(nil(), @x, @xs) -> c_7() , 34: eq#2^#(::(@y, @ys)) -> c_8() , 35: eq#2^#(nil()) -> c_9() , 36: nub#1^#(nil()) -> c_12() , 37: remove#1^#(nil(), @x) -> c_15() } We are left with following problem, upon which TcT provides the certificate YES(O(1),O(n^2)). Strict DPs: { eq^#(@l1, @l2) -> c_3(eq#1^#(@l1, @l2)) , eq#1^#(::(@x, @xs), @l2) -> c_4(eq#3^#(@l2, @x, @xs)) , eq#3^#(::(@y, @ys), @x, @xs) -> c_6(and^#(#equal(@x, @y), eq(@xs, @ys)), #equal^#(@x, @y), eq^#(@xs, @ys)) , nub^#(@l) -> c_10(nub#1^#(@l)) , nub#1^#(::(@x, @xs)) -> c_11(nub^#(remove(@x, @xs)), remove^#(@x, @xs)) , remove^#(@x, @l) -> c_13(remove#1^#(@l, @x)) , remove#1^#(::(@y, @ys), @x) -> c_14(remove#2^#(eq(@x, @y), @x, @y, @ys), eq^#(@x, @y)) , remove#2^#(#false(), @x, @y, @ys) -> c_16(remove^#(@x, @ys)) , remove#2^#(#true(), @x, @y, @ys) -> c_17(remove^#(@x, @ys)) } Weak DPs: { #equal^#(@x, @y) -> c_1(#eq^#(@x, @y)) , #eq^#(::(@x_1, @x_2), ::(@y_1, @y_2)) -> c_18(#and^#(#eq(@x_1, @y_1), #eq(@x_2, @y_2)), #eq^#(@x_1, @y_1), #eq^#(@x_2, @y_2)) , #eq^#(::(@x_1, @x_2), nil()) -> c_19() , #eq^#(nil(), ::(@y_1, @y_2)) -> c_20() , #eq^#(nil(), nil()) -> c_21() , #eq^#(#0(), #0()) -> c_22() , #eq^#(#0(), #neg(@y)) -> c_23() , #eq^#(#0(), #pos(@y)) -> c_24() , #eq^#(#0(), #s(@y)) -> c_25() , #eq^#(#neg(@x), #0()) -> c_26() , #eq^#(#neg(@x), #neg(@y)) -> c_27(#eq^#(@x, @y)) , #eq^#(#neg(@x), #pos(@y)) -> c_28() , #eq^#(#pos(@x), #0()) -> c_29() , #eq^#(#pos(@x), #neg(@y)) -> c_30() , #eq^#(#pos(@x), #pos(@y)) -> c_31(#eq^#(@x, @y)) , #eq^#(#s(@x), #0()) -> c_32() , #eq^#(#s(@x), #s(@y)) -> c_33(#eq^#(@x, @y)) , and^#(@x, @y) -> c_2(#and^#(@x, @y)) , #and^#(#false(), #false()) -> c_34() , #and^#(#false(), #true()) -> c_35() , #and^#(#true(), #false()) -> c_36() , #and^#(#true(), #true()) -> c_37() , eq#1^#(nil(), @l2) -> c_5(eq#2^#(@l2)) , eq#3^#(nil(), @x, @xs) -> c_7() , eq#2^#(::(@y, @ys)) -> c_8() , eq#2^#(nil()) -> c_9() , nub#1^#(nil()) -> c_12() , remove#1^#(nil(), @x) -> c_15() } Weak Trs: { #equal(@x, @y) -> #eq(@x, @y) , #eq(::(@x_1, @x_2), ::(@y_1, @y_2)) -> #and(#eq(@x_1, @y_1), #eq(@x_2, @y_2)) , #eq(::(@x_1, @x_2), nil()) -> #false() , #eq(nil(), ::(@y_1, @y_2)) -> #false() , #eq(nil(), nil()) -> #true() , #eq(#0(), #0()) -> #true() , #eq(#0(), #neg(@y)) -> #false() , #eq(#0(), #pos(@y)) -> #false() , #eq(#0(), #s(@y)) -> #false() , #eq(#neg(@x), #0()) -> #false() , #eq(#neg(@x), #neg(@y)) -> #eq(@x, @y) , #eq(#neg(@x), #pos(@y)) -> #false() , #eq(#pos(@x), #0()) -> #false() , #eq(#pos(@x), #neg(@y)) -> #false() , #eq(#pos(@x), #pos(@y)) -> #eq(@x, @y) , #eq(#s(@x), #0()) -> #false() , #eq(#s(@x), #s(@y)) -> #eq(@x, @y) , and(@x, @y) -> #and(@x, @y) , #and(#false(), #false()) -> #false() , #and(#false(), #true()) -> #false() , #and(#true(), #false()) -> #false() , #and(#true(), #true()) -> #true() , eq(@l1, @l2) -> eq#1(@l1, @l2) , eq#1(::(@x, @xs), @l2) -> eq#3(@l2, @x, @xs) , eq#1(nil(), @l2) -> eq#2(@l2) , eq#3(::(@y, @ys), @x, @xs) -> and(#equal(@x, @y), eq(@xs, @ys)) , eq#3(nil(), @x, @xs) -> #false() , eq#2(::(@y, @ys)) -> #false() , eq#2(nil()) -> #true() , nub(@l) -> nub#1(@l) , nub#1(::(@x, @xs)) -> ::(@x, nub(remove(@x, @xs))) , nub#1(nil()) -> nil() , remove(@x, @l) -> remove#1(@l, @x) , remove#1(::(@y, @ys), @x) -> remove#2(eq(@x, @y), @x, @y, @ys) , remove#1(nil(), @x) -> nil() , remove#2(#false(), @x, @y, @ys) -> ::(@y, remove(@x, @ys)) , remove#2(#true(), @x, @y, @ys) -> remove(@x, @ys) } Obligation: innermost runtime complexity Answer: YES(O(1),O(n^2)) The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed. { #equal^#(@x, @y) -> c_1(#eq^#(@x, @y)) , #eq^#(::(@x_1, @x_2), ::(@y_1, @y_2)) -> c_18(#and^#(#eq(@x_1, @y_1), #eq(@x_2, @y_2)), #eq^#(@x_1, @y_1), #eq^#(@x_2, @y_2)) , #eq^#(::(@x_1, @x_2), nil()) -> c_19() , #eq^#(nil(), ::(@y_1, @y_2)) -> c_20() , #eq^#(nil(), nil()) -> c_21() , #eq^#(#0(), #0()) -> c_22() , #eq^#(#0(), #neg(@y)) -> c_23() , #eq^#(#0(), #pos(@y)) -> c_24() , #eq^#(#0(), #s(@y)) -> c_25() , #eq^#(#neg(@x), #0()) -> c_26() , #eq^#(#neg(@x), #neg(@y)) -> c_27(#eq^#(@x, @y)) , #eq^#(#neg(@x), #pos(@y)) -> c_28() , #eq^#(#pos(@x), #0()) -> c_29() , #eq^#(#pos(@x), #neg(@y)) -> c_30() , #eq^#(#pos(@x), #pos(@y)) -> c_31(#eq^#(@x, @y)) , #eq^#(#s(@x), #0()) -> c_32() , #eq^#(#s(@x), #s(@y)) -> c_33(#eq^#(@x, @y)) , and^#(@x, @y) -> c_2(#and^#(@x, @y)) , #and^#(#false(), #false()) -> c_34() , #and^#(#false(), #true()) -> c_35() , #and^#(#true(), #false()) -> c_36() , #and^#(#true(), #true()) -> c_37() , eq#1^#(nil(), @l2) -> c_5(eq#2^#(@l2)) , eq#3^#(nil(), @x, @xs) -> c_7() , eq#2^#(::(@y, @ys)) -> c_8() , eq#2^#(nil()) -> c_9() , nub#1^#(nil()) -> c_12() , remove#1^#(nil(), @x) -> c_15() } We are left with following problem, upon which TcT provides the certificate YES(O(1),O(n^2)). Strict DPs: { eq^#(@l1, @l2) -> c_3(eq#1^#(@l1, @l2)) , eq#1^#(::(@x, @xs), @l2) -> c_4(eq#3^#(@l2, @x, @xs)) , eq#3^#(::(@y, @ys), @x, @xs) -> c_6(and^#(#equal(@x, @y), eq(@xs, @ys)), #equal^#(@x, @y), eq^#(@xs, @ys)) , nub^#(@l) -> c_10(nub#1^#(@l)) , nub#1^#(::(@x, @xs)) -> c_11(nub^#(remove(@x, @xs)), remove^#(@x, @xs)) , remove^#(@x, @l) -> c_13(remove#1^#(@l, @x)) , remove#1^#(::(@y, @ys), @x) -> c_14(remove#2^#(eq(@x, @y), @x, @y, @ys), eq^#(@x, @y)) , remove#2^#(#false(), @x, @y, @ys) -> c_16(remove^#(@x, @ys)) , remove#2^#(#true(), @x, @y, @ys) -> c_17(remove^#(@x, @ys)) } Weak Trs: { #equal(@x, @y) -> #eq(@x, @y) , #eq(::(@x_1, @x_2), ::(@y_1, @y_2)) -> #and(#eq(@x_1, @y_1), #eq(@x_2, @y_2)) , #eq(::(@x_1, @x_2), nil()) -> #false() , #eq(nil(), ::(@y_1, @y_2)) -> #false() , #eq(nil(), nil()) -> #true() , #eq(#0(), #0()) -> #true() , #eq(#0(), #neg(@y)) -> #false() , #eq(#0(), #pos(@y)) -> #false() , #eq(#0(), #s(@y)) -> #false() , #eq(#neg(@x), #0()) -> #false() , #eq(#neg(@x), #neg(@y)) -> #eq(@x, @y) , #eq(#neg(@x), #pos(@y)) -> #false() , #eq(#pos(@x), #0()) -> #false() , #eq(#pos(@x), #neg(@y)) -> #false() , #eq(#pos(@x), #pos(@y)) -> #eq(@x, @y) , #eq(#s(@x), #0()) -> #false() , #eq(#s(@x), #s(@y)) -> #eq(@x, @y) , and(@x, @y) -> #and(@x, @y) , #and(#false(), #false()) -> #false() , #and(#false(), #true()) -> #false() , #and(#true(), #false()) -> #false() , #and(#true(), #true()) -> #true() , eq(@l1, @l2) -> eq#1(@l1, @l2) , eq#1(::(@x, @xs), @l2) -> eq#3(@l2, @x, @xs) , eq#1(nil(), @l2) -> eq#2(@l2) , eq#3(::(@y, @ys), @x, @xs) -> and(#equal(@x, @y), eq(@xs, @ys)) , eq#3(nil(), @x, @xs) -> #false() , eq#2(::(@y, @ys)) -> #false() , eq#2(nil()) -> #true() , nub(@l) -> nub#1(@l) , nub#1(::(@x, @xs)) -> ::(@x, nub(remove(@x, @xs))) , nub#1(nil()) -> nil() , remove(@x, @l) -> remove#1(@l, @x) , remove#1(::(@y, @ys), @x) -> remove#2(eq(@x, @y), @x, @y, @ys) , remove#1(nil(), @x) -> nil() , remove#2(#false(), @x, @y, @ys) -> ::(@y, remove(@x, @ys)) , remove#2(#true(), @x, @y, @ys) -> remove(@x, @ys) } Obligation: innermost runtime complexity Answer: YES(O(1),O(n^2)) Due to missing edges in the dependency-graph, the right-hand sides of following rules could be simplified: { eq#3^#(::(@y, @ys), @x, @xs) -> c_6(and^#(#equal(@x, @y), eq(@xs, @ys)), #equal^#(@x, @y), eq^#(@xs, @ys)) } We are left with following problem, upon which TcT provides the certificate YES(O(1),O(n^2)). Strict DPs: { eq^#(@l1, @l2) -> c_1(eq#1^#(@l1, @l2)) , eq#1^#(::(@x, @xs), @l2) -> c_2(eq#3^#(@l2, @x, @xs)) , eq#3^#(::(@y, @ys), @x, @xs) -> c_3(eq^#(@xs, @ys)) , nub^#(@l) -> c_4(nub#1^#(@l)) , nub#1^#(::(@x, @xs)) -> c_5(nub^#(remove(@x, @xs)), remove^#(@x, @xs)) , remove^#(@x, @l) -> c_6(remove#1^#(@l, @x)) , remove#1^#(::(@y, @ys), @x) -> c_7(remove#2^#(eq(@x, @y), @x, @y, @ys), eq^#(@x, @y)) , remove#2^#(#false(), @x, @y, @ys) -> c_8(remove^#(@x, @ys)) , remove#2^#(#true(), @x, @y, @ys) -> c_9(remove^#(@x, @ys)) } Weak Trs: { #equal(@x, @y) -> #eq(@x, @y) , #eq(::(@x_1, @x_2), ::(@y_1, @y_2)) -> #and(#eq(@x_1, @y_1), #eq(@x_2, @y_2)) , #eq(::(@x_1, @x_2), nil()) -> #false() , #eq(nil(), ::(@y_1, @y_2)) -> #false() , #eq(nil(), nil()) -> #true() , #eq(#0(), #0()) -> #true() , #eq(#0(), #neg(@y)) -> #false() , #eq(#0(), #pos(@y)) -> #false() , #eq(#0(), #s(@y)) -> #false() , #eq(#neg(@x), #0()) -> #false() , #eq(#neg(@x), #neg(@y)) -> #eq(@x, @y) , #eq(#neg(@x), #pos(@y)) -> #false() , #eq(#pos(@x), #0()) -> #false() , #eq(#pos(@x), #neg(@y)) -> #false() , #eq(#pos(@x), #pos(@y)) -> #eq(@x, @y) , #eq(#s(@x), #0()) -> #false() , #eq(#s(@x), #s(@y)) -> #eq(@x, @y) , and(@x, @y) -> #and(@x, @y) , #and(#false(), #false()) -> #false() , #and(#false(), #true()) -> #false() , #and(#true(), #false()) -> #false() , #and(#true(), #true()) -> #true() , eq(@l1, @l2) -> eq#1(@l1, @l2) , eq#1(::(@x, @xs), @l2) -> eq#3(@l2, @x, @xs) , eq#1(nil(), @l2) -> eq#2(@l2) , eq#3(::(@y, @ys), @x, @xs) -> and(#equal(@x, @y), eq(@xs, @ys)) , eq#3(nil(), @x, @xs) -> #false() , eq#2(::(@y, @ys)) -> #false() , eq#2(nil()) -> #true() , nub(@l) -> nub#1(@l) , nub#1(::(@x, @xs)) -> ::(@x, nub(remove(@x, @xs))) , nub#1(nil()) -> nil() , remove(@x, @l) -> remove#1(@l, @x) , remove#1(::(@y, @ys), @x) -> remove#2(eq(@x, @y), @x, @y, @ys) , remove#1(nil(), @x) -> nil() , remove#2(#false(), @x, @y, @ys) -> ::(@y, remove(@x, @ys)) , remove#2(#true(), @x, @y, @ys) -> remove(@x, @ys) } Obligation: innermost runtime complexity Answer: YES(O(1),O(n^2)) We replace rewrite rules by usable rules: Weak Usable Rules: { #equal(@x, @y) -> #eq(@x, @y) , #eq(::(@x_1, @x_2), ::(@y_1, @y_2)) -> #and(#eq(@x_1, @y_1), #eq(@x_2, @y_2)) , #eq(::(@x_1, @x_2), nil()) -> #false() , #eq(nil(), ::(@y_1, @y_2)) -> #false() , #eq(nil(), nil()) -> #true() , #eq(#0(), #0()) -> #true() , #eq(#0(), #neg(@y)) -> #false() , #eq(#0(), #pos(@y)) -> #false() , #eq(#0(), #s(@y)) -> #false() , #eq(#neg(@x), #0()) -> #false() , #eq(#neg(@x), #neg(@y)) -> #eq(@x, @y) , #eq(#neg(@x), #pos(@y)) -> #false() , #eq(#pos(@x), #0()) -> #false() , #eq(#pos(@x), #neg(@y)) -> #false() , #eq(#pos(@x), #pos(@y)) -> #eq(@x, @y) , #eq(#s(@x), #0()) -> #false() , #eq(#s(@x), #s(@y)) -> #eq(@x, @y) , and(@x, @y) -> #and(@x, @y) , #and(#false(), #false()) -> #false() , #and(#false(), #true()) -> #false() , #and(#true(), #false()) -> #false() , #and(#true(), #true()) -> #true() , eq(@l1, @l2) -> eq#1(@l1, @l2) , eq#1(::(@x, @xs), @l2) -> eq#3(@l2, @x, @xs) , eq#1(nil(), @l2) -> eq#2(@l2) , eq#3(::(@y, @ys), @x, @xs) -> and(#equal(@x, @y), eq(@xs, @ys)) , eq#3(nil(), @x, @xs) -> #false() , eq#2(::(@y, @ys)) -> #false() , eq#2(nil()) -> #true() , remove(@x, @l) -> remove#1(@l, @x) , remove#1(::(@y, @ys), @x) -> remove#2(eq(@x, @y), @x, @y, @ys) , remove#1(nil(), @x) -> nil() , remove#2(#false(), @x, @y, @ys) -> ::(@y, remove(@x, @ys)) , remove#2(#true(), @x, @y, @ys) -> remove(@x, @ys) } We are left with following problem, upon which TcT provides the certificate YES(O(1),O(n^2)). Strict DPs: { eq^#(@l1, @l2) -> c_1(eq#1^#(@l1, @l2)) , eq#1^#(::(@x, @xs), @l2) -> c_2(eq#3^#(@l2, @x, @xs)) , eq#3^#(::(@y, @ys), @x, @xs) -> c_3(eq^#(@xs, @ys)) , nub^#(@l) -> c_4(nub#1^#(@l)) , nub#1^#(::(@x, @xs)) -> c_5(nub^#(remove(@x, @xs)), remove^#(@x, @xs)) , remove^#(@x, @l) -> c_6(remove#1^#(@l, @x)) , remove#1^#(::(@y, @ys), @x) -> c_7(remove#2^#(eq(@x, @y), @x, @y, @ys), eq^#(@x, @y)) , remove#2^#(#false(), @x, @y, @ys) -> c_8(remove^#(@x, @ys)) , remove#2^#(#true(), @x, @y, @ys) -> c_9(remove^#(@x, @ys)) } Weak Trs: { #equal(@x, @y) -> #eq(@x, @y) , #eq(::(@x_1, @x_2), ::(@y_1, @y_2)) -> #and(#eq(@x_1, @y_1), #eq(@x_2, @y_2)) , #eq(::(@x_1, @x_2), nil()) -> #false() , #eq(nil(), ::(@y_1, @y_2)) -> #false() , #eq(nil(), nil()) -> #true() , #eq(#0(), #0()) -> #true() , #eq(#0(), #neg(@y)) -> #false() , #eq(#0(), #pos(@y)) -> #false() , #eq(#0(), #s(@y)) -> #false() , #eq(#neg(@x), #0()) -> #false() , #eq(#neg(@x), #neg(@y)) -> #eq(@x, @y) , #eq(#neg(@x), #pos(@y)) -> #false() , #eq(#pos(@x), #0()) -> #false() , #eq(#pos(@x), #neg(@y)) -> #false() , #eq(#pos(@x), #pos(@y)) -> #eq(@x, @y) , #eq(#s(@x), #0()) -> #false() , #eq(#s(@x), #s(@y)) -> #eq(@x, @y) , and(@x, @y) -> #and(@x, @y) , #and(#false(), #false()) -> #false() , #and(#false(), #true()) -> #false() , #and(#true(), #false()) -> #false() , #and(#true(), #true()) -> #true() , eq(@l1, @l2) -> eq#1(@l1, @l2) , eq#1(::(@x, @xs), @l2) -> eq#3(@l2, @x, @xs) , eq#1(nil(), @l2) -> eq#2(@l2) , eq#3(::(@y, @ys), @x, @xs) -> and(#equal(@x, @y), eq(@xs, @ys)) , eq#3(nil(), @x, @xs) -> #false() , eq#2(::(@y, @ys)) -> #false() , eq#2(nil()) -> #true() , remove(@x, @l) -> remove#1(@l, @x) , remove#1(::(@y, @ys), @x) -> remove#2(eq(@x, @y), @x, @y, @ys) , remove#1(nil(), @x) -> nil() , remove#2(#false(), @x, @y, @ys) -> ::(@y, remove(@x, @ys)) , remove#2(#true(), @x, @y, @ys) -> remove(@x, @ys) } Obligation: innermost runtime complexity Answer: YES(O(1),O(n^2)) We use the processor 'matrix interpretation of dimension 2' to orient following rules strictly. DPs: { 3: eq#3^#(::(@y, @ys), @x, @xs) -> c_3(eq^#(@xs, @ys)) , 5: nub#1^#(::(@x, @xs)) -> c_5(nub^#(remove(@x, @xs)), remove^#(@x, @xs)) , 7: remove#1^#(::(@y, @ys), @x) -> c_7(remove#2^#(eq(@x, @y), @x, @y, @ys), eq^#(@x, @y)) } Trs: { remove#2(#true(), @x, @y, @ys) -> remove(@x, @ys) } Sub-proof: ---------- The following argument positions are usable: Uargs(c_1) = {1}, Uargs(c_2) = {1}, Uargs(c_3) = {1}, Uargs(c_4) = {1}, Uargs(c_5) = {1, 2}, Uargs(c_6) = {1}, Uargs(c_7) = {1, 2}, Uargs(c_8) = {1}, Uargs(c_9) = {1} TcT has computed following constructor-based matrix interpretation satisfying not(EDA). [#equal](x1, x2) = [0] [0] [#eq](x1, x2) = [0] [0] [and](x1, x2) = [0 0] x1 + [0] [0 1] [0] [#and](x1, x2) = [0] [0] [eq](x1, x2) = [0 0] x1 + [0 0] x2 + [0] [0 1] [0 1] [1] [eq#1](x1, x2) = [0 0] x2 + [0] [0 1] [0] [::](x1, x2) = [0 1] x1 + [1 1] x2 + [1] [0 1] [0 1] [1] [eq#3](x1, x2, x3) = [0] [0] [nil] = [0] [0] [eq#2](x1) = [0] [0] [#false] = [0] [0] [#true] = [0] [0] [remove](x1, x2) = [1 0] x2 + [0] [0 1] [0] [remove#1](x1, x2) = [1 0] x1 + [0] [0 1] [0] [remove#2](x1, x2, x3, x4) = [1 0] x1 + [0 1] x3 + [1 1] x4 + [1] [0 0] [0 1] [0 1] [1] [#0] = [0] [0] [#neg](x1) = [1 1] x1 + [0] [0 0] [0] [#pos](x1) = [1 1] x1 + [0] [0 1] [0] [#s](x1) = [0] [0] [#equal^#](x1, x2) = [0] [0] [#eq^#](x1, x2) = [0] [0] [and^#](x1, x2) = [0] [0] [#and^#](x1, x2) = [0] [0] [eq^#](x1, x2) = [0 1] x2 + [0] [0 0] [0] [eq#1^#](x1, x2) = [0 1] x2 + [0] [0 0] [0] [eq#3^#](x1, x2, x3) = [0 1] x1 + [0] [0 0] [0] [eq#2^#](x1) = [0] [0] [nub^#](x1) = [1 1] x1 + [0] [0 0] [0] [nub#1^#](x1) = [1 1] x1 + [0] [0 0] [0] [remove^#](x1, x2) = [0 1] x2 + [0] [0 0] [0] [remove#1^#](x1, x2) = [0 1] x1 + [0] [0 0] [0] [remove#2^#](x1, x2, x3, x4) = [1 0] x1 + [0 1] x4 + [0] [1 0] [0 0] [0] [c_1](x1) = [1 1] x1 + [0] [0 0] [0] [c_2](x1) = [1 1] x1 + [0] [0 0] [0] [c_3](x1) = [1 1] x1 + [0] [0 0] [0] [c_4](x1) = [1 1] x1 + [0] [0 0] [0] [c_5](x1, x2) = [1 1] x1 + [1 1] x2 + [0] [0 0] [0 0] [0] [c_6](x1) = [1 1] x1 + [0] [0 0] [0] [c_7](x1, x2) = [1 1] x1 + [1 1] x2 + [0] [0 0] [0 0] [0] [c_8](x1) = [1 0] x1 + [0] [0 0] [0] [c_9](x1) = [1 0] x1 + [0] [0 0] [0] This order satisfies following ordering constraints [#equal(@x, @y)] = [0] [0] >= [0] [0] = [#eq(@x, @y)] [#eq(::(@x_1, @x_2), ::(@y_1, @y_2))] = [0] [0] >= [0] [0] = [#and(#eq(@x_1, @y_1), #eq(@x_2, @y_2))] [#eq(::(@x_1, @x_2), nil())] = [0] [0] >= [0] [0] = [#false()] [#eq(nil(), ::(@y_1, @y_2))] = [0] [0] >= [0] [0] = [#false()] [#eq(nil(), nil())] = [0] [0] >= [0] [0] = [#true()] [#eq(#0(), #0())] = [0] [0] >= [0] [0] = [#true()] [#eq(#0(), #neg(@y))] = [0] [0] >= [0] [0] = [#false()] [#eq(#0(), #pos(@y))] = [0] [0] >= [0] [0] = [#false()] [#eq(#0(), #s(@y))] = [0] [0] >= [0] [0] = [#false()] [#eq(#neg(@x), #0())] = [0] [0] >= [0] [0] = [#false()] [#eq(#neg(@x), #neg(@y))] = [0] [0] >= [0] [0] = [#eq(@x, @y)] [#eq(#neg(@x), #pos(@y))] = [0] [0] >= [0] [0] = [#false()] [#eq(#pos(@x), #0())] = [0] [0] >= [0] [0] = [#false()] [#eq(#pos(@x), #neg(@y))] = [0] [0] >= [0] [0] = [#false()] [#eq(#pos(@x), #pos(@y))] = [0] [0] >= [0] [0] = [#eq(@x, @y)] [#eq(#s(@x), #0())] = [0] [0] >= [0] [0] = [#false()] [#eq(#s(@x), #s(@y))] = [0] [0] >= [0] [0] = [#eq(@x, @y)] [and(@x, @y)] = [0 0] @x + [0] [0 1] [0] >= [0] [0] = [#and(@x, @y)] [#and(#false(), #false())] = [0] [0] >= [0] [0] = [#false()] [#and(#false(), #true())] = [0] [0] >= [0] [0] = [#false()] [#and(#true(), #false())] = [0] [0] >= [0] [0] = [#false()] [#and(#true(), #true())] = [0] [0] >= [0] [0] = [#true()] [eq(@l1, @l2)] = [0 0] @l1 + [0 0] @l2 + [0] [0 1] [0 1] [1] >= [0 0] @l2 + [0] [0 1] [0] = [eq#1(@l1, @l2)] [eq#1(::(@x, @xs), @l2)] = [0 0] @l2 + [0] [0 1] [0] >= [0] [0] = [eq#3(@l2, @x, @xs)] [eq#1(nil(), @l2)] = [0 0] @l2 + [0] [0 1] [0] >= [0] [0] = [eq#2(@l2)] [eq#3(::(@y, @ys), @x, @xs)] = [0] [0] >= [0] [0] = [and(#equal(@x, @y), eq(@xs, @ys))] [eq#3(nil(), @x, @xs)] = [0] [0] >= [0] [0] = [#false()] [eq#2(::(@y, @ys))] = [0] [0] >= [0] [0] = [#false()] [eq#2(nil())] = [0] [0] >= [0] [0] = [#true()] [remove(@x, @l)] = [1 0] @l + [0] [0 1] [0] >= [1 0] @l + [0] [0 1] [0] = [remove#1(@l, @x)] [remove#1(::(@y, @ys), @x)] = [0 1] @y + [1 1] @ys + [1] [0 1] [0 1] [1] >= [0 1] @y + [1 1] @ys + [1] [0 1] [0 1] [1] = [remove#2(eq(@x, @y), @x, @y, @ys)] [remove#1(nil(), @x)] = [0] [0] >= [0] [0] = [nil()] [remove#2(#false(), @x, @y, @ys)] = [0 1] @y + [1 1] @ys + [1] [0 1] [0 1] [1] >= [0 1] @y + [1 1] @ys + [1] [0 1] [0 1] [1] = [::(@y, remove(@x, @ys))] [remove#2(#true(), @x, @y, @ys)] = [0 1] @y + [1 1] @ys + [1] [0 1] [0 1] [1] > [1 0] @ys + [0] [0 1] [0] = [remove(@x, @ys)] [eq^#(@l1, @l2)] = [0 1] @l2 + [0] [0 0] [0] >= [0 1] @l2 + [0] [0 0] [0] = [c_1(eq#1^#(@l1, @l2))] [eq#1^#(::(@x, @xs), @l2)] = [0 1] @l2 + [0] [0 0] [0] >= [0 1] @l2 + [0] [0 0] [0] = [c_2(eq#3^#(@l2, @x, @xs))] [eq#3^#(::(@y, @ys), @x, @xs)] = [0 1] @y + [0 1] @ys + [1] [0 0] [0 0] [0] > [0 1] @ys + [0] [0 0] [0] = [c_3(eq^#(@xs, @ys))] [nub^#(@l)] = [1 1] @l + [0] [0 0] [0] >= [1 1] @l + [0] [0 0] [0] = [c_4(nub#1^#(@l))] [nub#1^#(::(@x, @xs))] = [0 2] @x + [1 2] @xs + [2] [0 0] [0 0] [0] > [1 2] @xs + [0] [0 0] [0] = [c_5(nub^#(remove(@x, @xs)), remove^#(@x, @xs))] [remove^#(@x, @l)] = [0 1] @l + [0] [0 0] [0] >= [0 1] @l + [0] [0 0] [0] = [c_6(remove#1^#(@l, @x))] [remove#1^#(::(@y, @ys), @x)] = [0 1] @y + [0 1] @ys + [1] [0 0] [0 0] [0] > [0 1] @y + [0 1] @ys + [0] [0 0] [0 0] [0] = [c_7(remove#2^#(eq(@x, @y), @x, @y, @ys), eq^#(@x, @y))] [remove#2^#(#false(), @x, @y, @ys)] = [0 1] @ys + [0] [0 0] [0] >= [0 1] @ys + [0] [0 0] [0] = [c_8(remove^#(@x, @ys))] [remove#2^#(#true(), @x, @y, @ys)] = [0 1] @ys + [0] [0 0] [0] >= [0 1] @ys + [0] [0 0] [0] = [c_9(remove^#(@x, @ys))] Consider the set of all dependency pairs DPs: { 1: eq^#(@l1, @l2) -> c_1(eq#1^#(@l1, @l2)) , 2: eq#1^#(::(@x, @xs), @l2) -> c_2(eq#3^#(@l2, @x, @xs)) , 3: eq#3^#(::(@y, @ys), @x, @xs) -> c_3(eq^#(@xs, @ys)) , 4: nub^#(@l) -> c_4(nub#1^#(@l)) , 5: nub#1^#(::(@x, @xs)) -> c_5(nub^#(remove(@x, @xs)), remove^#(@x, @xs)) , 6: remove^#(@x, @l) -> c_6(remove#1^#(@l, @x)) , 7: remove#1^#(::(@y, @ys), @x) -> c_7(remove#2^#(eq(@x, @y), @x, @y, @ys), eq^#(@x, @y)) , 8: remove#2^#(#false(), @x, @y, @ys) -> c_8(remove^#(@x, @ys)) , 9: remove#2^#(#true(), @x, @y, @ys) -> c_9(remove^#(@x, @ys)) } Processor 'matrix interpretation of dimension 2' induces the complexity certificate YES(?,O(n^2)) on application of dependency pairs {3,5,7}. These cover all (indirect) predecessors of dependency pairs {1,2,3,4,5,6,7,8,9}, their number of application is equally bounded. The dependency pairs are shifted into the corresponding weak component(s). We apply the transformation 'removetails' on the sub-problem: Weak DPs: { eq^#(@l1, @l2) -> c_1(eq#1^#(@l1, @l2)) , eq#1^#(::(@x, @xs), @l2) -> c_2(eq#3^#(@l2, @x, @xs)) , eq#3^#(::(@y, @ys), @x, @xs) -> c_3(eq^#(@xs, @ys)) , nub^#(@l) -> c_4(nub#1^#(@l)) , nub#1^#(::(@x, @xs)) -> c_5(nub^#(remove(@x, @xs)), remove^#(@x, @xs)) , remove^#(@x, @l) -> c_6(remove#1^#(@l, @x)) , remove#1^#(::(@y, @ys), @x) -> c_7(remove#2^#(eq(@x, @y), @x, @y, @ys), eq^#(@x, @y)) , remove#2^#(#false(), @x, @y, @ys) -> c_8(remove^#(@x, @ys)) , remove#2^#(#true(), @x, @y, @ys) -> c_9(remove^#(@x, @ys)) } Weak Trs: { #equal(@x, @y) -> #eq(@x, @y) , #eq(::(@x_1, @x_2), ::(@y_1, @y_2)) -> #and(#eq(@x_1, @y_1), #eq(@x_2, @y_2)) , #eq(::(@x_1, @x_2), nil()) -> #false() , #eq(nil(), ::(@y_1, @y_2)) -> #false() , #eq(nil(), nil()) -> #true() , #eq(#0(), #0()) -> #true() , #eq(#0(), #neg(@y)) -> #false() , #eq(#0(), #pos(@y)) -> #false() , #eq(#0(), #s(@y)) -> #false() , #eq(#neg(@x), #0()) -> #false() , #eq(#neg(@x), #neg(@y)) -> #eq(@x, @y) , #eq(#neg(@x), #pos(@y)) -> #false() , #eq(#pos(@x), #0()) -> #false() , #eq(#pos(@x), #neg(@y)) -> #false() , #eq(#pos(@x), #pos(@y)) -> #eq(@x, @y) , #eq(#s(@x), #0()) -> #false() , #eq(#s(@x), #s(@y)) -> #eq(@x, @y) , and(@x, @y) -> #and(@x, @y) , #and(#false(), #false()) -> #false() , #and(#false(), #true()) -> #false() , #and(#true(), #false()) -> #false() , #and(#true(), #true()) -> #true() , eq(@l1, @l2) -> eq#1(@l1, @l2) , eq#1(::(@x, @xs), @l2) -> eq#3(@l2, @x, @xs) , eq#1(nil(), @l2) -> eq#2(@l2) , eq#3(::(@y, @ys), @x, @xs) -> and(#equal(@x, @y), eq(@xs, @ys)) , eq#3(nil(), @x, @xs) -> #false() , eq#2(::(@y, @ys)) -> #false() , eq#2(nil()) -> #true() , remove(@x, @l) -> remove#1(@l, @x) , remove#1(::(@y, @ys), @x) -> remove#2(eq(@x, @y), @x, @y, @ys) , remove#1(nil(), @x) -> nil() , remove#2(#false(), @x, @y, @ys) -> ::(@y, remove(@x, @ys)) , remove#2(#true(), @x, @y, @ys) -> remove(@x, @ys) } StartTerms: basic terms Strategy: innermost The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed. { eq^#(@l1, @l2) -> c_1(eq#1^#(@l1, @l2)) , eq#1^#(::(@x, @xs), @l2) -> c_2(eq#3^#(@l2, @x, @xs)) , eq#3^#(::(@y, @ys), @x, @xs) -> c_3(eq^#(@xs, @ys)) , nub^#(@l) -> c_4(nub#1^#(@l)) , nub#1^#(::(@x, @xs)) -> c_5(nub^#(remove(@x, @xs)), remove^#(@x, @xs)) , remove^#(@x, @l) -> c_6(remove#1^#(@l, @x)) , remove#1^#(::(@y, @ys), @x) -> c_7(remove#2^#(eq(@x, @y), @x, @y, @ys), eq^#(@x, @y)) , remove#2^#(#false(), @x, @y, @ys) -> c_8(remove^#(@x, @ys)) , remove#2^#(#true(), @x, @y, @ys) -> c_9(remove^#(@x, @ys)) } We apply the transformation 'usablerules' on the sub-problem: Weak Trs: { #equal(@x, @y) -> #eq(@x, @y) , #eq(::(@x_1, @x_2), ::(@y_1, @y_2)) -> #and(#eq(@x_1, @y_1), #eq(@x_2, @y_2)) , #eq(::(@x_1, @x_2), nil()) -> #false() , #eq(nil(), ::(@y_1, @y_2)) -> #false() , #eq(nil(), nil()) -> #true() , #eq(#0(), #0()) -> #true() , #eq(#0(), #neg(@y)) -> #false() , #eq(#0(), #pos(@y)) -> #false() , #eq(#0(), #s(@y)) -> #false() , #eq(#neg(@x), #0()) -> #false() , #eq(#neg(@x), #neg(@y)) -> #eq(@x, @y) , #eq(#neg(@x), #pos(@y)) -> #false() , #eq(#pos(@x), #0()) -> #false() , #eq(#pos(@x), #neg(@y)) -> #false() , #eq(#pos(@x), #pos(@y)) -> #eq(@x, @y) , #eq(#s(@x), #0()) -> #false() , #eq(#s(@x), #s(@y)) -> #eq(@x, @y) , and(@x, @y) -> #and(@x, @y) , #and(#false(), #false()) -> #false() , #and(#false(), #true()) -> #false() , #and(#true(), #false()) -> #false() , #and(#true(), #true()) -> #true() , eq(@l1, @l2) -> eq#1(@l1, @l2) , eq#1(::(@x, @xs), @l2) -> eq#3(@l2, @x, @xs) , eq#1(nil(), @l2) -> eq#2(@l2) , eq#3(::(@y, @ys), @x, @xs) -> and(#equal(@x, @y), eq(@xs, @ys)) , eq#3(nil(), @x, @xs) -> #false() , eq#2(::(@y, @ys)) -> #false() , eq#2(nil()) -> #true() , remove(@x, @l) -> remove#1(@l, @x) , remove#1(::(@y, @ys), @x) -> remove#2(eq(@x, @y), @x, @y, @ys) , remove#1(nil(), @x) -> nil() , remove#2(#false(), @x, @y, @ys) -> ::(@y, remove(@x, @ys)) , remove#2(#true(), @x, @y, @ys) -> remove(@x, @ys) } StartTerms: basic terms Strategy: innermost No rule is usable, rules are removed from the input problem. We are left with following problem, upon which TcT provides the certificate YES(O(1),O(1)). Rules: Empty Obligation: innermost runtime complexity Answer: YES(O(1),O(1)) Empty rules are trivially bounded Wall-time: 3.261926s CPU-time: 23.657s Hurray, we answered YES(O(1),O(n^2))