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: { remove(@x, @l) -> remove#1(@l, @x) , eq#2(::(@y, @ys)) -> #false() , eq#2(nil()) -> #true() , #equal(@x, @y) -> #eq(@x, @y) , eq(@l1, @l2) -> eq#1(@l1, @l2) , and(@x, @y) -> #and(@x, @y) , remove#1(::(@y, @ys), @x) -> remove#2(eq(@x, @y), @x, @y, @ys) , remove#1(nil(), @x) -> nil() , nub(@l) -> nub#1(@l) , remove#2(#true(), @x, @y, @ys) -> remove(@x, @ys) , remove#2(#false(), @x, @y, @ys) -> ::(@y, remove(@x, @ys)) , 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() , nub#1(::(@x, @xs)) -> ::(@x, nub(remove(@x, @xs))) , nub#1(nil()) -> nil() } 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(#pos(@x), #pos(@y)) -> #eq(@x, @y) , #eq(#pos(@x), #0()) -> #false() , #eq(#pos(@x), #neg(@y)) -> #false() , #eq(#0(), #pos(@y)) -> #false() , #eq(#0(), #0()) -> #true() , #eq(#0(), #neg(@y)) -> #false() , #eq(#0(), #s(@y)) -> #false() , #eq(#neg(@x), #pos(@y)) -> #false() , #eq(#neg(@x), #0()) -> #false() , #eq(#neg(@x), #neg(@y)) -> #eq(@x, @y) , #eq(#s(@x), #0()) -> #false() , #eq(#s(@x), #s(@y)) -> #eq(@x, @y) , #eq(nil(), ::(@y_1, @y_2)) -> #false() , #eq(nil(), nil()) -> #true() , #and(#true(), #true()) -> #true() , #and(#true(), #false()) -> #false() , #and(#false(), #true()) -> #false() , #and(#false(), #false()) -> #false() } Obligation: innermost runtime complexity Answer: YES(O(1),O(n^2)) We add the following dependency tuples: Strict DPs: { remove^#(@x, @l) -> c_1(remove#1^#(@l, @x)) , remove#1^#(::(@y, @ys), @x) -> c_7(remove#2^#(eq(@x, @y), @x, @y, @ys), eq^#(@x, @y)) , remove#1^#(nil(), @x) -> c_8() , eq#2^#(::(@y, @ys)) -> c_2() , eq#2^#(nil()) -> c_3() , #equal^#(@x, @y) -> c_4(#eq^#(@x, @y)) , eq^#(@l1, @l2) -> c_5(eq#1^#(@l1, @l2)) , eq#1^#(::(@x, @xs), @l2) -> c_12(eq#3^#(@l2, @x, @xs)) , eq#1^#(nil(), @l2) -> c_13(eq#2^#(@l2)) , and^#(@x, @y) -> c_6(#and^#(@x, @y)) , remove#2^#(#true(), @x, @y, @ys) -> c_10(remove^#(@x, @ys)) , remove#2^#(#false(), @x, @y, @ys) -> c_11(remove^#(@x, @ys)) , nub^#(@l) -> c_9(nub#1^#(@l)) , nub#1^#(::(@x, @xs)) -> c_16(nub^#(remove(@x, @xs)), remove^#(@x, @xs)) , nub#1^#(nil()) -> c_17() , eq#3^#(::(@y, @ys), @x, @xs) -> c_14(and^#(#equal(@x, @y), eq(@xs, @ys)), #equal^#(@x, @y), eq^#(@xs, @ys)) , eq#3^#(nil(), @x, @xs) -> c_15() } 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^#(#pos(@x), #pos(@y)) -> c_20(#eq^#(@x, @y)) , #eq^#(#pos(@x), #0()) -> c_21() , #eq^#(#pos(@x), #neg(@y)) -> c_22() , #eq^#(#0(), #pos(@y)) -> c_23() , #eq^#(#0(), #0()) -> c_24() , #eq^#(#0(), #neg(@y)) -> c_25() , #eq^#(#0(), #s(@y)) -> c_26() , #eq^#(#neg(@x), #pos(@y)) -> c_27() , #eq^#(#neg(@x), #0()) -> c_28() , #eq^#(#neg(@x), #neg(@y)) -> c_29(#eq^#(@x, @y)) , #eq^#(#s(@x), #0()) -> c_30() , #eq^#(#s(@x), #s(@y)) -> c_31(#eq^#(@x, @y)) , #eq^#(nil(), ::(@y_1, @y_2)) -> c_32() , #eq^#(nil(), nil()) -> c_33() , #and^#(#true(), #true()) -> c_34() , #and^#(#true(), #false()) -> c_35() , #and^#(#false(), #true()) -> c_36() , #and^#(#false(), #false()) -> 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: { remove^#(@x, @l) -> c_1(remove#1^#(@l, @x)) , remove#1^#(::(@y, @ys), @x) -> c_7(remove#2^#(eq(@x, @y), @x, @y, @ys), eq^#(@x, @y)) , remove#1^#(nil(), @x) -> c_8() , eq#2^#(::(@y, @ys)) -> c_2() , eq#2^#(nil()) -> c_3() , #equal^#(@x, @y) -> c_4(#eq^#(@x, @y)) , eq^#(@l1, @l2) -> c_5(eq#1^#(@l1, @l2)) , eq#1^#(::(@x, @xs), @l2) -> c_12(eq#3^#(@l2, @x, @xs)) , eq#1^#(nil(), @l2) -> c_13(eq#2^#(@l2)) , and^#(@x, @y) -> c_6(#and^#(@x, @y)) , remove#2^#(#true(), @x, @y, @ys) -> c_10(remove^#(@x, @ys)) , remove#2^#(#false(), @x, @y, @ys) -> c_11(remove^#(@x, @ys)) , nub^#(@l) -> c_9(nub#1^#(@l)) , nub#1^#(::(@x, @xs)) -> c_16(nub^#(remove(@x, @xs)), remove^#(@x, @xs)) , nub#1^#(nil()) -> c_17() , eq#3^#(::(@y, @ys), @x, @xs) -> c_14(and^#(#equal(@x, @y), eq(@xs, @ys)), #equal^#(@x, @y), eq^#(@xs, @ys)) , eq#3^#(nil(), @x, @xs) -> c_15() } 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^#(#pos(@x), #pos(@y)) -> c_20(#eq^#(@x, @y)) , #eq^#(#pos(@x), #0()) -> c_21() , #eq^#(#pos(@x), #neg(@y)) -> c_22() , #eq^#(#0(), #pos(@y)) -> c_23() , #eq^#(#0(), #0()) -> c_24() , #eq^#(#0(), #neg(@y)) -> c_25() , #eq^#(#0(), #s(@y)) -> c_26() , #eq^#(#neg(@x), #pos(@y)) -> c_27() , #eq^#(#neg(@x), #0()) -> c_28() , #eq^#(#neg(@x), #neg(@y)) -> c_29(#eq^#(@x, @y)) , #eq^#(#s(@x), #0()) -> c_30() , #eq^#(#s(@x), #s(@y)) -> c_31(#eq^#(@x, @y)) , #eq^#(nil(), ::(@y_1, @y_2)) -> c_32() , #eq^#(nil(), nil()) -> c_33() , #and^#(#true(), #true()) -> c_34() , #and^#(#true(), #false()) -> c_35() , #and^#(#false(), #true()) -> c_36() , #and^#(#false(), #false()) -> c_37() } Weak Trs: { remove(@x, @l) -> remove#1(@l, @x) , eq#2(::(@y, @ys)) -> #false() , eq#2(nil()) -> #true() , #equal(@x, @y) -> #eq(@x, @y) , eq(@l1, @l2) -> eq#1(@l1, @l2) , #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(#pos(@x), #pos(@y)) -> #eq(@x, @y) , #eq(#pos(@x), #0()) -> #false() , #eq(#pos(@x), #neg(@y)) -> #false() , #eq(#0(), #pos(@y)) -> #false() , #eq(#0(), #0()) -> #true() , #eq(#0(), #neg(@y)) -> #false() , #eq(#0(), #s(@y)) -> #false() , #eq(#neg(@x), #pos(@y)) -> #false() , #eq(#neg(@x), #0()) -> #false() , #eq(#neg(@x), #neg(@y)) -> #eq(@x, @y) , #eq(#s(@x), #0()) -> #false() , #eq(#s(@x), #s(@y)) -> #eq(@x, @y) , #eq(nil(), ::(@y_1, @y_2)) -> #false() , #eq(nil(), nil()) -> #true() , and(@x, @y) -> #and(@x, @y) , remove#1(::(@y, @ys), @x) -> remove#2(eq(@x, @y), @x, @y, @ys) , remove#1(nil(), @x) -> nil() , nub(@l) -> nub#1(@l) , remove#2(#true(), @x, @y, @ys) -> remove(@x, @ys) , remove#2(#false(), @x, @y, @ys) -> ::(@y, remove(@x, @ys)) , eq#1(::(@x, @xs), @l2) -> eq#3(@l2, @x, @xs) , eq#1(nil(), @l2) -> eq#2(@l2) , #and(#true(), #true()) -> #true() , #and(#true(), #false()) -> #false() , #and(#false(), #true()) -> #false() , #and(#false(), #false()) -> #false() , eq#3(::(@y, @ys), @x, @xs) -> and(#equal(@x, @y), eq(@xs, @ys)) , eq#3(nil(), @x, @xs) -> #false() , nub#1(::(@x, @xs)) -> ::(@x, nub(remove(@x, @xs))) , nub#1(nil()) -> nil() } Obligation: innermost runtime complexity Answer: YES(O(1),O(n^2)) We estimate the number of application of {3,4,5,6,10,15,17} by applications of Pre({3,4,5,6,10,15,17}) = {1,8,9,13,16}. Here rules are labeled as follows: DPs: { 1: remove^#(@x, @l) -> c_1(remove#1^#(@l, @x)) , 2: remove#1^#(::(@y, @ys), @x) -> c_7(remove#2^#(eq(@x, @y), @x, @y, @ys), eq^#(@x, @y)) , 3: remove#1^#(nil(), @x) -> c_8() , 4: eq#2^#(::(@y, @ys)) -> c_2() , 5: eq#2^#(nil()) -> c_3() , 6: #equal^#(@x, @y) -> c_4(#eq^#(@x, @y)) , 7: eq^#(@l1, @l2) -> c_5(eq#1^#(@l1, @l2)) , 8: eq#1^#(::(@x, @xs), @l2) -> c_12(eq#3^#(@l2, @x, @xs)) , 9: eq#1^#(nil(), @l2) -> c_13(eq#2^#(@l2)) , 10: and^#(@x, @y) -> c_6(#and^#(@x, @y)) , 11: remove#2^#(#true(), @x, @y, @ys) -> c_10(remove^#(@x, @ys)) , 12: remove#2^#(#false(), @x, @y, @ys) -> c_11(remove^#(@x, @ys)) , 13: nub^#(@l) -> c_9(nub#1^#(@l)) , 14: nub#1^#(::(@x, @xs)) -> c_16(nub^#(remove(@x, @xs)), remove^#(@x, @xs)) , 15: nub#1^#(nil()) -> c_17() , 16: eq#3^#(::(@y, @ys), @x, @xs) -> c_14(and^#(#equal(@x, @y), eq(@xs, @ys)), #equal^#(@x, @y), eq^#(@xs, @ys)) , 17: eq#3^#(nil(), @x, @xs) -> c_15() , 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^#(#pos(@x), #pos(@y)) -> c_20(#eq^#(@x, @y)) , 21: #eq^#(#pos(@x), #0()) -> c_21() , 22: #eq^#(#pos(@x), #neg(@y)) -> c_22() , 23: #eq^#(#0(), #pos(@y)) -> c_23() , 24: #eq^#(#0(), #0()) -> c_24() , 25: #eq^#(#0(), #neg(@y)) -> c_25() , 26: #eq^#(#0(), #s(@y)) -> c_26() , 27: #eq^#(#neg(@x), #pos(@y)) -> c_27() , 28: #eq^#(#neg(@x), #0()) -> c_28() , 29: #eq^#(#neg(@x), #neg(@y)) -> c_29(#eq^#(@x, @y)) , 30: #eq^#(#s(@x), #0()) -> c_30() , 31: #eq^#(#s(@x), #s(@y)) -> c_31(#eq^#(@x, @y)) , 32: #eq^#(nil(), ::(@y_1, @y_2)) -> c_32() , 33: #eq^#(nil(), nil()) -> c_33() , 34: #and^#(#true(), #true()) -> c_34() , 35: #and^#(#true(), #false()) -> c_35() , 36: #and^#(#false(), #true()) -> c_36() , 37: #and^#(#false(), #false()) -> c_37() } We are left with following problem, upon which TcT provides the certificate YES(O(1),O(n^2)). Strict DPs: { remove^#(@x, @l) -> c_1(remove#1^#(@l, @x)) , remove#1^#(::(@y, @ys), @x) -> c_7(remove#2^#(eq(@x, @y), @x, @y, @ys), eq^#(@x, @y)) , eq^#(@l1, @l2) -> c_5(eq#1^#(@l1, @l2)) , eq#1^#(::(@x, @xs), @l2) -> c_12(eq#3^#(@l2, @x, @xs)) , eq#1^#(nil(), @l2) -> c_13(eq#2^#(@l2)) , remove#2^#(#true(), @x, @y, @ys) -> c_10(remove^#(@x, @ys)) , remove#2^#(#false(), @x, @y, @ys) -> c_11(remove^#(@x, @ys)) , nub^#(@l) -> c_9(nub#1^#(@l)) , nub#1^#(::(@x, @xs)) -> c_16(nub^#(remove(@x, @xs)), remove^#(@x, @xs)) , eq#3^#(::(@y, @ys), @x, @xs) -> c_14(and^#(#equal(@x, @y), eq(@xs, @ys)), #equal^#(@x, @y), eq^#(@xs, @ys)) } Weak DPs: { remove#1^#(nil(), @x) -> c_8() , eq#2^#(::(@y, @ys)) -> c_2() , eq#2^#(nil()) -> c_3() , #equal^#(@x, @y) -> c_4(#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^#(#pos(@x), #pos(@y)) -> c_20(#eq^#(@x, @y)) , #eq^#(#pos(@x), #0()) -> c_21() , #eq^#(#pos(@x), #neg(@y)) -> c_22() , #eq^#(#0(), #pos(@y)) -> c_23() , #eq^#(#0(), #0()) -> c_24() , #eq^#(#0(), #neg(@y)) -> c_25() , #eq^#(#0(), #s(@y)) -> c_26() , #eq^#(#neg(@x), #pos(@y)) -> c_27() , #eq^#(#neg(@x), #0()) -> c_28() , #eq^#(#neg(@x), #neg(@y)) -> c_29(#eq^#(@x, @y)) , #eq^#(#s(@x), #0()) -> c_30() , #eq^#(#s(@x), #s(@y)) -> c_31(#eq^#(@x, @y)) , #eq^#(nil(), ::(@y_1, @y_2)) -> c_32() , #eq^#(nil(), nil()) -> c_33() , and^#(@x, @y) -> c_6(#and^#(@x, @y)) , #and^#(#true(), #true()) -> c_34() , #and^#(#true(), #false()) -> c_35() , #and^#(#false(), #true()) -> c_36() , #and^#(#false(), #false()) -> c_37() , nub#1^#(nil()) -> c_17() , eq#3^#(nil(), @x, @xs) -> c_15() } Weak Trs: { remove(@x, @l) -> remove#1(@l, @x) , eq#2(::(@y, @ys)) -> #false() , eq#2(nil()) -> #true() , #equal(@x, @y) -> #eq(@x, @y) , eq(@l1, @l2) -> eq#1(@l1, @l2) , #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(#pos(@x), #pos(@y)) -> #eq(@x, @y) , #eq(#pos(@x), #0()) -> #false() , #eq(#pos(@x), #neg(@y)) -> #false() , #eq(#0(), #pos(@y)) -> #false() , #eq(#0(), #0()) -> #true() , #eq(#0(), #neg(@y)) -> #false() , #eq(#0(), #s(@y)) -> #false() , #eq(#neg(@x), #pos(@y)) -> #false() , #eq(#neg(@x), #0()) -> #false() , #eq(#neg(@x), #neg(@y)) -> #eq(@x, @y) , #eq(#s(@x), #0()) -> #false() , #eq(#s(@x), #s(@y)) -> #eq(@x, @y) , #eq(nil(), ::(@y_1, @y_2)) -> #false() , #eq(nil(), nil()) -> #true() , and(@x, @y) -> #and(@x, @y) , remove#1(::(@y, @ys), @x) -> remove#2(eq(@x, @y), @x, @y, @ys) , remove#1(nil(), @x) -> nil() , nub(@l) -> nub#1(@l) , remove#2(#true(), @x, @y, @ys) -> remove(@x, @ys) , remove#2(#false(), @x, @y, @ys) -> ::(@y, remove(@x, @ys)) , eq#1(::(@x, @xs), @l2) -> eq#3(@l2, @x, @xs) , eq#1(nil(), @l2) -> eq#2(@l2) , #and(#true(), #true()) -> #true() , #and(#true(), #false()) -> #false() , #and(#false(), #true()) -> #false() , #and(#false(), #false()) -> #false() , eq#3(::(@y, @ys), @x, @xs) -> and(#equal(@x, @y), eq(@xs, @ys)) , eq#3(nil(), @x, @xs) -> #false() , nub#1(::(@x, @xs)) -> ::(@x, nub(remove(@x, @xs))) , nub#1(nil()) -> nil() } Obligation: innermost runtime complexity Answer: YES(O(1),O(n^2)) We estimate the number of application of {5} by applications of Pre({5}) = {3}. Here rules are labeled as follows: DPs: { 1: remove^#(@x, @l) -> c_1(remove#1^#(@l, @x)) , 2: remove#1^#(::(@y, @ys), @x) -> c_7(remove#2^#(eq(@x, @y), @x, @y, @ys), eq^#(@x, @y)) , 3: eq^#(@l1, @l2) -> c_5(eq#1^#(@l1, @l2)) , 4: eq#1^#(::(@x, @xs), @l2) -> c_12(eq#3^#(@l2, @x, @xs)) , 5: eq#1^#(nil(), @l2) -> c_13(eq#2^#(@l2)) , 6: remove#2^#(#true(), @x, @y, @ys) -> c_10(remove^#(@x, @ys)) , 7: remove#2^#(#false(), @x, @y, @ys) -> c_11(remove^#(@x, @ys)) , 8: nub^#(@l) -> c_9(nub#1^#(@l)) , 9: nub#1^#(::(@x, @xs)) -> c_16(nub^#(remove(@x, @xs)), remove^#(@x, @xs)) , 10: eq#3^#(::(@y, @ys), @x, @xs) -> c_14(and^#(#equal(@x, @y), eq(@xs, @ys)), #equal^#(@x, @y), eq^#(@xs, @ys)) , 11: remove#1^#(nil(), @x) -> c_8() , 12: eq#2^#(::(@y, @ys)) -> c_2() , 13: eq#2^#(nil()) -> c_3() , 14: #equal^#(@x, @y) -> c_4(#eq^#(@x, @y)) , 15: #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)) , 16: #eq^#(::(@x_1, @x_2), nil()) -> c_19() , 17: #eq^#(#pos(@x), #pos(@y)) -> c_20(#eq^#(@x, @y)) , 18: #eq^#(#pos(@x), #0()) -> c_21() , 19: #eq^#(#pos(@x), #neg(@y)) -> c_22() , 20: #eq^#(#0(), #pos(@y)) -> c_23() , 21: #eq^#(#0(), #0()) -> c_24() , 22: #eq^#(#0(), #neg(@y)) -> c_25() , 23: #eq^#(#0(), #s(@y)) -> c_26() , 24: #eq^#(#neg(@x), #pos(@y)) -> c_27() , 25: #eq^#(#neg(@x), #0()) -> c_28() , 26: #eq^#(#neg(@x), #neg(@y)) -> c_29(#eq^#(@x, @y)) , 27: #eq^#(#s(@x), #0()) -> c_30() , 28: #eq^#(#s(@x), #s(@y)) -> c_31(#eq^#(@x, @y)) , 29: #eq^#(nil(), ::(@y_1, @y_2)) -> c_32() , 30: #eq^#(nil(), nil()) -> c_33() , 31: and^#(@x, @y) -> c_6(#and^#(@x, @y)) , 32: #and^#(#true(), #true()) -> c_34() , 33: #and^#(#true(), #false()) -> c_35() , 34: #and^#(#false(), #true()) -> c_36() , 35: #and^#(#false(), #false()) -> c_37() , 36: nub#1^#(nil()) -> c_17() , 37: eq#3^#(nil(), @x, @xs) -> c_15() } We are left with following problem, upon which TcT provides the certificate YES(O(1),O(n^2)). Strict DPs: { remove^#(@x, @l) -> c_1(remove#1^#(@l, @x)) , remove#1^#(::(@y, @ys), @x) -> c_7(remove#2^#(eq(@x, @y), @x, @y, @ys), eq^#(@x, @y)) , eq^#(@l1, @l2) -> c_5(eq#1^#(@l1, @l2)) , eq#1^#(::(@x, @xs), @l2) -> c_12(eq#3^#(@l2, @x, @xs)) , remove#2^#(#true(), @x, @y, @ys) -> c_10(remove^#(@x, @ys)) , remove#2^#(#false(), @x, @y, @ys) -> c_11(remove^#(@x, @ys)) , nub^#(@l) -> c_9(nub#1^#(@l)) , nub#1^#(::(@x, @xs)) -> c_16(nub^#(remove(@x, @xs)), remove^#(@x, @xs)) , eq#3^#(::(@y, @ys), @x, @xs) -> c_14(and^#(#equal(@x, @y), eq(@xs, @ys)), #equal^#(@x, @y), eq^#(@xs, @ys)) } Weak DPs: { remove#1^#(nil(), @x) -> c_8() , eq#2^#(::(@y, @ys)) -> c_2() , eq#2^#(nil()) -> c_3() , #equal^#(@x, @y) -> c_4(#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^#(#pos(@x), #pos(@y)) -> c_20(#eq^#(@x, @y)) , #eq^#(#pos(@x), #0()) -> c_21() , #eq^#(#pos(@x), #neg(@y)) -> c_22() , #eq^#(#0(), #pos(@y)) -> c_23() , #eq^#(#0(), #0()) -> c_24() , #eq^#(#0(), #neg(@y)) -> c_25() , #eq^#(#0(), #s(@y)) -> c_26() , #eq^#(#neg(@x), #pos(@y)) -> c_27() , #eq^#(#neg(@x), #0()) -> c_28() , #eq^#(#neg(@x), #neg(@y)) -> c_29(#eq^#(@x, @y)) , #eq^#(#s(@x), #0()) -> c_30() , #eq^#(#s(@x), #s(@y)) -> c_31(#eq^#(@x, @y)) , #eq^#(nil(), ::(@y_1, @y_2)) -> c_32() , #eq^#(nil(), nil()) -> c_33() , eq#1^#(nil(), @l2) -> c_13(eq#2^#(@l2)) , and^#(@x, @y) -> c_6(#and^#(@x, @y)) , #and^#(#true(), #true()) -> c_34() , #and^#(#true(), #false()) -> c_35() , #and^#(#false(), #true()) -> c_36() , #and^#(#false(), #false()) -> c_37() , nub#1^#(nil()) -> c_17() , eq#3^#(nil(), @x, @xs) -> c_15() } Weak Trs: { remove(@x, @l) -> remove#1(@l, @x) , eq#2(::(@y, @ys)) -> #false() , eq#2(nil()) -> #true() , #equal(@x, @y) -> #eq(@x, @y) , eq(@l1, @l2) -> eq#1(@l1, @l2) , #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(#pos(@x), #pos(@y)) -> #eq(@x, @y) , #eq(#pos(@x), #0()) -> #false() , #eq(#pos(@x), #neg(@y)) -> #false() , #eq(#0(), #pos(@y)) -> #false() , #eq(#0(), #0()) -> #true() , #eq(#0(), #neg(@y)) -> #false() , #eq(#0(), #s(@y)) -> #false() , #eq(#neg(@x), #pos(@y)) -> #false() , #eq(#neg(@x), #0()) -> #false() , #eq(#neg(@x), #neg(@y)) -> #eq(@x, @y) , #eq(#s(@x), #0()) -> #false() , #eq(#s(@x), #s(@y)) -> #eq(@x, @y) , #eq(nil(), ::(@y_1, @y_2)) -> #false() , #eq(nil(), nil()) -> #true() , and(@x, @y) -> #and(@x, @y) , remove#1(::(@y, @ys), @x) -> remove#2(eq(@x, @y), @x, @y, @ys) , remove#1(nil(), @x) -> nil() , nub(@l) -> nub#1(@l) , remove#2(#true(), @x, @y, @ys) -> remove(@x, @ys) , remove#2(#false(), @x, @y, @ys) -> ::(@y, remove(@x, @ys)) , eq#1(::(@x, @xs), @l2) -> eq#3(@l2, @x, @xs) , eq#1(nil(), @l2) -> eq#2(@l2) , #and(#true(), #true()) -> #true() , #and(#true(), #false()) -> #false() , #and(#false(), #true()) -> #false() , #and(#false(), #false()) -> #false() , eq#3(::(@y, @ys), @x, @xs) -> and(#equal(@x, @y), eq(@xs, @ys)) , eq#3(nil(), @x, @xs) -> #false() , nub#1(::(@x, @xs)) -> ::(@x, nub(remove(@x, @xs))) , nub#1(nil()) -> nil() } 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. { remove#1^#(nil(), @x) -> c_8() , eq#2^#(::(@y, @ys)) -> c_2() , eq#2^#(nil()) -> c_3() , #equal^#(@x, @y) -> c_4(#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^#(#pos(@x), #pos(@y)) -> c_20(#eq^#(@x, @y)) , #eq^#(#pos(@x), #0()) -> c_21() , #eq^#(#pos(@x), #neg(@y)) -> c_22() , #eq^#(#0(), #pos(@y)) -> c_23() , #eq^#(#0(), #0()) -> c_24() , #eq^#(#0(), #neg(@y)) -> c_25() , #eq^#(#0(), #s(@y)) -> c_26() , #eq^#(#neg(@x), #pos(@y)) -> c_27() , #eq^#(#neg(@x), #0()) -> c_28() , #eq^#(#neg(@x), #neg(@y)) -> c_29(#eq^#(@x, @y)) , #eq^#(#s(@x), #0()) -> c_30() , #eq^#(#s(@x), #s(@y)) -> c_31(#eq^#(@x, @y)) , #eq^#(nil(), ::(@y_1, @y_2)) -> c_32() , #eq^#(nil(), nil()) -> c_33() , eq#1^#(nil(), @l2) -> c_13(eq#2^#(@l2)) , and^#(@x, @y) -> c_6(#and^#(@x, @y)) , #and^#(#true(), #true()) -> c_34() , #and^#(#true(), #false()) -> c_35() , #and^#(#false(), #true()) -> c_36() , #and^#(#false(), #false()) -> c_37() , nub#1^#(nil()) -> c_17() , eq#3^#(nil(), @x, @xs) -> c_15() } We are left with following problem, upon which TcT provides the certificate YES(O(1),O(n^2)). Strict DPs: { remove^#(@x, @l) -> c_1(remove#1^#(@l, @x)) , remove#1^#(::(@y, @ys), @x) -> c_7(remove#2^#(eq(@x, @y), @x, @y, @ys), eq^#(@x, @y)) , eq^#(@l1, @l2) -> c_5(eq#1^#(@l1, @l2)) , eq#1^#(::(@x, @xs), @l2) -> c_12(eq#3^#(@l2, @x, @xs)) , remove#2^#(#true(), @x, @y, @ys) -> c_10(remove^#(@x, @ys)) , remove#2^#(#false(), @x, @y, @ys) -> c_11(remove^#(@x, @ys)) , nub^#(@l) -> c_9(nub#1^#(@l)) , nub#1^#(::(@x, @xs)) -> c_16(nub^#(remove(@x, @xs)), remove^#(@x, @xs)) , eq#3^#(::(@y, @ys), @x, @xs) -> c_14(and^#(#equal(@x, @y), eq(@xs, @ys)), #equal^#(@x, @y), eq^#(@xs, @ys)) } Weak Trs: { remove(@x, @l) -> remove#1(@l, @x) , eq#2(::(@y, @ys)) -> #false() , eq#2(nil()) -> #true() , #equal(@x, @y) -> #eq(@x, @y) , eq(@l1, @l2) -> eq#1(@l1, @l2) , #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(#pos(@x), #pos(@y)) -> #eq(@x, @y) , #eq(#pos(@x), #0()) -> #false() , #eq(#pos(@x), #neg(@y)) -> #false() , #eq(#0(), #pos(@y)) -> #false() , #eq(#0(), #0()) -> #true() , #eq(#0(), #neg(@y)) -> #false() , #eq(#0(), #s(@y)) -> #false() , #eq(#neg(@x), #pos(@y)) -> #false() , #eq(#neg(@x), #0()) -> #false() , #eq(#neg(@x), #neg(@y)) -> #eq(@x, @y) , #eq(#s(@x), #0()) -> #false() , #eq(#s(@x), #s(@y)) -> #eq(@x, @y) , #eq(nil(), ::(@y_1, @y_2)) -> #false() , #eq(nil(), nil()) -> #true() , and(@x, @y) -> #and(@x, @y) , remove#1(::(@y, @ys), @x) -> remove#2(eq(@x, @y), @x, @y, @ys) , remove#1(nil(), @x) -> nil() , nub(@l) -> nub#1(@l) , remove#2(#true(), @x, @y, @ys) -> remove(@x, @ys) , remove#2(#false(), @x, @y, @ys) -> ::(@y, remove(@x, @ys)) , eq#1(::(@x, @xs), @l2) -> eq#3(@l2, @x, @xs) , eq#1(nil(), @l2) -> eq#2(@l2) , #and(#true(), #true()) -> #true() , #and(#true(), #false()) -> #false() , #and(#false(), #true()) -> #false() , #and(#false(), #false()) -> #false() , eq#3(::(@y, @ys), @x, @xs) -> and(#equal(@x, @y), eq(@xs, @ys)) , eq#3(nil(), @x, @xs) -> #false() , nub#1(::(@x, @xs)) -> ::(@x, nub(remove(@x, @xs))) , nub#1(nil()) -> nil() } 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_14(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: { remove^#(@x, @l) -> c_1(remove#1^#(@l, @x)) , remove#1^#(::(@y, @ys), @x) -> c_2(remove#2^#(eq(@x, @y), @x, @y, @ys), eq^#(@x, @y)) , eq^#(@l1, @l2) -> c_3(eq#1^#(@l1, @l2)) , eq#1^#(::(@x, @xs), @l2) -> c_4(eq#3^#(@l2, @x, @xs)) , remove#2^#(#true(), @x, @y, @ys) -> c_5(remove^#(@x, @ys)) , remove#2^#(#false(), @x, @y, @ys) -> c_6(remove^#(@x, @ys)) , nub^#(@l) -> c_7(nub#1^#(@l)) , nub#1^#(::(@x, @xs)) -> c_8(nub^#(remove(@x, @xs)), remove^#(@x, @xs)) , eq#3^#(::(@y, @ys), @x, @xs) -> c_9(eq^#(@xs, @ys)) } Weak Trs: { remove(@x, @l) -> remove#1(@l, @x) , eq#2(::(@y, @ys)) -> #false() , eq#2(nil()) -> #true() , #equal(@x, @y) -> #eq(@x, @y) , eq(@l1, @l2) -> eq#1(@l1, @l2) , #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(#pos(@x), #pos(@y)) -> #eq(@x, @y) , #eq(#pos(@x), #0()) -> #false() , #eq(#pos(@x), #neg(@y)) -> #false() , #eq(#0(), #pos(@y)) -> #false() , #eq(#0(), #0()) -> #true() , #eq(#0(), #neg(@y)) -> #false() , #eq(#0(), #s(@y)) -> #false() , #eq(#neg(@x), #pos(@y)) -> #false() , #eq(#neg(@x), #0()) -> #false() , #eq(#neg(@x), #neg(@y)) -> #eq(@x, @y) , #eq(#s(@x), #0()) -> #false() , #eq(#s(@x), #s(@y)) -> #eq(@x, @y) , #eq(nil(), ::(@y_1, @y_2)) -> #false() , #eq(nil(), nil()) -> #true() , and(@x, @y) -> #and(@x, @y) , remove#1(::(@y, @ys), @x) -> remove#2(eq(@x, @y), @x, @y, @ys) , remove#1(nil(), @x) -> nil() , nub(@l) -> nub#1(@l) , remove#2(#true(), @x, @y, @ys) -> remove(@x, @ys) , remove#2(#false(), @x, @y, @ys) -> ::(@y, remove(@x, @ys)) , eq#1(::(@x, @xs), @l2) -> eq#3(@l2, @x, @xs) , eq#1(nil(), @l2) -> eq#2(@l2) , #and(#true(), #true()) -> #true() , #and(#true(), #false()) -> #false() , #and(#false(), #true()) -> #false() , #and(#false(), #false()) -> #false() , eq#3(::(@y, @ys), @x, @xs) -> and(#equal(@x, @y), eq(@xs, @ys)) , eq#3(nil(), @x, @xs) -> #false() , nub#1(::(@x, @xs)) -> ::(@x, nub(remove(@x, @xs))) , nub#1(nil()) -> nil() } Obligation: innermost runtime complexity Answer: YES(O(1),O(n^2)) We replace rewrite rules by usable rules: Weak Usable Rules: { remove(@x, @l) -> remove#1(@l, @x) , eq#2(::(@y, @ys)) -> #false() , eq#2(nil()) -> #true() , #equal(@x, @y) -> #eq(@x, @y) , eq(@l1, @l2) -> eq#1(@l1, @l2) , #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(#pos(@x), #pos(@y)) -> #eq(@x, @y) , #eq(#pos(@x), #0()) -> #false() , #eq(#pos(@x), #neg(@y)) -> #false() , #eq(#0(), #pos(@y)) -> #false() , #eq(#0(), #0()) -> #true() , #eq(#0(), #neg(@y)) -> #false() , #eq(#0(), #s(@y)) -> #false() , #eq(#neg(@x), #pos(@y)) -> #false() , #eq(#neg(@x), #0()) -> #false() , #eq(#neg(@x), #neg(@y)) -> #eq(@x, @y) , #eq(#s(@x), #0()) -> #false() , #eq(#s(@x), #s(@y)) -> #eq(@x, @y) , #eq(nil(), ::(@y_1, @y_2)) -> #false() , #eq(nil(), nil()) -> #true() , and(@x, @y) -> #and(@x, @y) , remove#1(::(@y, @ys), @x) -> remove#2(eq(@x, @y), @x, @y, @ys) , remove#1(nil(), @x) -> nil() , remove#2(#true(), @x, @y, @ys) -> remove(@x, @ys) , remove#2(#false(), @x, @y, @ys) -> ::(@y, remove(@x, @ys)) , eq#1(::(@x, @xs), @l2) -> eq#3(@l2, @x, @xs) , eq#1(nil(), @l2) -> eq#2(@l2) , #and(#true(), #true()) -> #true() , #and(#true(), #false()) -> #false() , #and(#false(), #true()) -> #false() , #and(#false(), #false()) -> #false() , eq#3(::(@y, @ys), @x, @xs) -> and(#equal(@x, @y), eq(@xs, @ys)) , eq#3(nil(), @x, @xs) -> #false() } We are left with following problem, upon which TcT provides the certificate YES(O(1),O(n^2)). Strict DPs: { remove^#(@x, @l) -> c_1(remove#1^#(@l, @x)) , remove#1^#(::(@y, @ys), @x) -> c_2(remove#2^#(eq(@x, @y), @x, @y, @ys), eq^#(@x, @y)) , eq^#(@l1, @l2) -> c_3(eq#1^#(@l1, @l2)) , eq#1^#(::(@x, @xs), @l2) -> c_4(eq#3^#(@l2, @x, @xs)) , remove#2^#(#true(), @x, @y, @ys) -> c_5(remove^#(@x, @ys)) , remove#2^#(#false(), @x, @y, @ys) -> c_6(remove^#(@x, @ys)) , nub^#(@l) -> c_7(nub#1^#(@l)) , nub#1^#(::(@x, @xs)) -> c_8(nub^#(remove(@x, @xs)), remove^#(@x, @xs)) , eq#3^#(::(@y, @ys), @x, @xs) -> c_9(eq^#(@xs, @ys)) } Weak Trs: { remove(@x, @l) -> remove#1(@l, @x) , eq#2(::(@y, @ys)) -> #false() , eq#2(nil()) -> #true() , #equal(@x, @y) -> #eq(@x, @y) , eq(@l1, @l2) -> eq#1(@l1, @l2) , #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(#pos(@x), #pos(@y)) -> #eq(@x, @y) , #eq(#pos(@x), #0()) -> #false() , #eq(#pos(@x), #neg(@y)) -> #false() , #eq(#0(), #pos(@y)) -> #false() , #eq(#0(), #0()) -> #true() , #eq(#0(), #neg(@y)) -> #false() , #eq(#0(), #s(@y)) -> #false() , #eq(#neg(@x), #pos(@y)) -> #false() , #eq(#neg(@x), #0()) -> #false() , #eq(#neg(@x), #neg(@y)) -> #eq(@x, @y) , #eq(#s(@x), #0()) -> #false() , #eq(#s(@x), #s(@y)) -> #eq(@x, @y) , #eq(nil(), ::(@y_1, @y_2)) -> #false() , #eq(nil(), nil()) -> #true() , and(@x, @y) -> #and(@x, @y) , remove#1(::(@y, @ys), @x) -> remove#2(eq(@x, @y), @x, @y, @ys) , remove#1(nil(), @x) -> nil() , remove#2(#true(), @x, @y, @ys) -> remove(@x, @ys) , remove#2(#false(), @x, @y, @ys) -> ::(@y, remove(@x, @ys)) , eq#1(::(@x, @xs), @l2) -> eq#3(@l2, @x, @xs) , eq#1(nil(), @l2) -> eq#2(@l2) , #and(#true(), #true()) -> #true() , #and(#true(), #false()) -> #false() , #and(#false(), #true()) -> #false() , #and(#false(), #false()) -> #false() , eq#3(::(@y, @ys), @x, @xs) -> and(#equal(@x, @y), eq(@xs, @ys)) , eq#3(nil(), @x, @xs) -> #false() } Obligation: innermost runtime complexity Answer: YES(O(1),O(n^2)) We decompose the input problem according to the dependency graph into the upper component { nub^#(@l) -> c_7(nub#1^#(@l)) , nub#1^#(::(@x, @xs)) -> c_8(nub^#(remove(@x, @xs)), remove^#(@x, @xs)) } and lower component { remove^#(@x, @l) -> c_1(remove#1^#(@l, @x)) , remove#1^#(::(@y, @ys), @x) -> c_2(remove#2^#(eq(@x, @y), @x, @y, @ys), eq^#(@x, @y)) , eq^#(@l1, @l2) -> c_3(eq#1^#(@l1, @l2)) , eq#1^#(::(@x, @xs), @l2) -> c_4(eq#3^#(@l2, @x, @xs)) , remove#2^#(#true(), @x, @y, @ys) -> c_5(remove^#(@x, @ys)) , remove#2^#(#false(), @x, @y, @ys) -> c_6(remove^#(@x, @ys)) , eq#3^#(::(@y, @ys), @x, @xs) -> c_9(eq^#(@xs, @ys)) } Further, following extension rules are added to the lower component. { nub^#(@l) -> nub#1^#(@l) , nub#1^#(::(@x, @xs)) -> remove^#(@x, @xs) , nub#1^#(::(@x, @xs)) -> nub^#(remove(@x, @xs)) } TcT solves the upper component with certificate YES(O(1),O(n^1)). Sub-proof: ---------- We are left with following problem, upon which TcT provides the certificate YES(O(1),O(n^1)). Strict DPs: { nub^#(@l) -> c_7(nub#1^#(@l)) , nub#1^#(::(@x, @xs)) -> c_8(nub^#(remove(@x, @xs)), remove^#(@x, @xs)) } Weak Trs: { remove(@x, @l) -> remove#1(@l, @x) , eq#2(::(@y, @ys)) -> #false() , eq#2(nil()) -> #true() , #equal(@x, @y) -> #eq(@x, @y) , eq(@l1, @l2) -> eq#1(@l1, @l2) , #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(#pos(@x), #pos(@y)) -> #eq(@x, @y) , #eq(#pos(@x), #0()) -> #false() , #eq(#pos(@x), #neg(@y)) -> #false() , #eq(#0(), #pos(@y)) -> #false() , #eq(#0(), #0()) -> #true() , #eq(#0(), #neg(@y)) -> #false() , #eq(#0(), #s(@y)) -> #false() , #eq(#neg(@x), #pos(@y)) -> #false() , #eq(#neg(@x), #0()) -> #false() , #eq(#neg(@x), #neg(@y)) -> #eq(@x, @y) , #eq(#s(@x), #0()) -> #false() , #eq(#s(@x), #s(@y)) -> #eq(@x, @y) , #eq(nil(), ::(@y_1, @y_2)) -> #false() , #eq(nil(), nil()) -> #true() , and(@x, @y) -> #and(@x, @y) , remove#1(::(@y, @ys), @x) -> remove#2(eq(@x, @y), @x, @y, @ys) , remove#1(nil(), @x) -> nil() , remove#2(#true(), @x, @y, @ys) -> remove(@x, @ys) , remove#2(#false(), @x, @y, @ys) -> ::(@y, remove(@x, @ys)) , eq#1(::(@x, @xs), @l2) -> eq#3(@l2, @x, @xs) , eq#1(nil(), @l2) -> eq#2(@l2) , #and(#true(), #true()) -> #true() , #and(#true(), #false()) -> #false() , #and(#false(), #true()) -> #false() , #and(#false(), #false()) -> #false() , eq#3(::(@y, @ys), @x, @xs) -> and(#equal(@x, @y), eq(@xs, @ys)) , eq#3(nil(), @x, @xs) -> #false() } Obligation: innermost runtime complexity Answer: YES(O(1),O(n^1)) We use the processor 'matrix interpretation of dimension 1' to orient following rules strictly. DPs: { 2: nub#1^#(::(@x, @xs)) -> c_8(nub^#(remove(@x, @xs)), remove^#(@x, @xs)) } Trs: { #eq(::(@x_1, @x_2), ::(@y_1, @y_2)) -> #and(#eq(@x_1, @y_1), #eq(@x_2, @y_2)) , #eq(#pos(@x), #neg(@y)) -> #false() , #eq(#0(), #neg(@y)) -> #false() , #eq(#neg(@x), #neg(@y)) -> #eq(@x, @y) , #eq(nil(), ::(@y_1, @y_2)) -> #false() , remove#2(#true(), @x, @y, @ys) -> remove(@x, @ys) } Sub-proof: ---------- The following argument positions are usable: Uargs(c_7) = {1}, Uargs(c_8) = {1} TcT has computed the following constructor-based matrix interpretation satisfying not(EDA). [remove](x1, x2) = [1] x2 + [0] [eq#2](x1) = [0] [#equal](x1, x2) = [4] x2 + [0] [eq](x1, x2) = [0] [#eq](x1, x2) = [4] x2 + [0] [#true] = [0] [#false] = [0] [::](x1, x2) = [1] x2 + [1] [and](x1, x2) = [4] x2 + [0] [remove#1](x1, x2) = [1] x1 + [0] [nub](x1) = [7] x1 + [0] [#pos](x1) = [1] x1 + [0] [#0] = [0] [#neg](x1) = [1] x1 + [2] [remove#2](x1, x2, x3, x4) = [1] x4 + [1] [eq#1](x1, x2) = [0] [#and](x1, x2) = [0] [eq#3](x1, x2, x3) = [0] [#s](x1) = [1] x1 + [0] [nub#1](x1) = [7] x1 + [0] [nil] = [0] [remove^#](x1, x2) = [7] x1 + [7] x2 + [7] [c_1](x1) = [7] x1 + [0] [remove#1^#](x1, x2) = [7] x1 + [7] x2 + [0] [eq#2^#](x1) = [7] x1 + [0] [c_2] = [0] [c_3] = [0] [#equal^#](x1, x2) = [7] x1 + [7] x2 + [0] [c_4](x1) = [7] x1 + [0] [#eq^#](x1, x2) = [7] x1 + [7] x2 + [0] [eq^#](x1, x2) = [7] x1 + [7] x2 + [0] [c_5](x1) = [7] x1 + [0] [eq#1^#](x1, x2) = [7] x1 + [7] x2 + [0] [and^#](x1, x2) = [7] x1 + [7] x2 + [0] [c_6](x1) = [7] x1 + [0] [#and^#](x1, x2) = [7] x1 + [7] x2 + [0] [c_7](x1, x2) = [7] x1 + [7] x2 + [0] [remove#2^#](x1, x2, x3, x4) = [7] x1 + [7] x2 + [7] x3 + [7] x4 + [0] [c_8] = [0] [nub^#](x1) = [1] x1 + [0] [c_9](x1) = [7] x1 + [0] [nub#1^#](x1) = [1] x1 + [0] [c_10](x1) = [7] x1 + [0] [c_11](x1) = [7] x1 + [0] [c_12](x1) = [7] x1 + [0] [eq#3^#](x1, x2, x3) = [7] x1 + [7] x2 + [7] x3 + [0] [c_13](x1) = [7] x1 + [0] [c_14](x1, x2, x3) = [7] x1 + [7] x2 + [7] x3 + [0] [c_15] = [0] [c_16](x1, x2) = [7] x1 + [7] x2 + [0] [c_17] = [0] [c_18](x1, x2, x3) = [7] x1 + [7] x2 + [7] x3 + [0] [c_19] = [0] [c_20](x1) = [7] x1 + [0] [c_21] = [0] [c_22] = [0] [c_23] = [0] [c_24] = [0] [c_25] = [0] [c_26] = [0] [c_27] = [0] [c_28] = [0] [c_29](x1) = [7] x1 + [0] [c_30] = [0] [c_31](x1) = [7] x1 + [0] [c_32] = [0] [c_33] = [0] [c_34] = [0] [c_35] = [0] [c_36] = [0] [c_37] = [0] [c] = [0] [c_1](x1) = [7] x1 + [0] [c_2](x1, x2) = [7] x1 + [7] x2 + [0] [c_3](x1) = [7] x1 + [0] [c_4](x1) = [7] x1 + [0] [c_5](x1) = [7] x1 + [0] [c_6](x1) = [7] x1 + [0] [c_7](x1) = [1] x1 + [0] [c_8](x1, x2) = [1] x1 + [0] [c_9](x1) = [7] x1 + [0] The following symbols are considered usable {remove, eq#2, #equal, eq, #eq, and, remove#1, remove#2, eq#1, #and, eq#3, nub^#, nub#1^#} The order satisfies the following ordering constraints: [remove(@x, @l)] = [1] @l + [0] >= [1] @l + [0] = [remove#1(@l, @x)] [eq#2(::(@y, @ys))] = [0] >= [0] = [#false()] [eq#2(nil())] = [0] >= [0] = [#true()] [#equal(@x, @y)] = [4] @y + [0] >= [4] @y + [0] = [#eq(@x, @y)] [eq(@l1, @l2)] = [0] >= [0] = [eq#1(@l1, @l2)] [#eq(::(@x_1, @x_2), ::(@y_1, @y_2))] = [4] @y_2 + [4] > [0] = [#and(#eq(@x_1, @y_1), #eq(@x_2, @y_2))] [#eq(::(@x_1, @x_2), nil())] = [0] >= [0] = [#false()] [#eq(#pos(@x), #pos(@y))] = [4] @y + [0] >= [4] @y + [0] = [#eq(@x, @y)] [#eq(#pos(@x), #0())] = [0] >= [0] = [#false()] [#eq(#pos(@x), #neg(@y))] = [4] @y + [8] > [0] = [#false()] [#eq(#0(), #pos(@y))] = [4] @y + [0] >= [0] = [#false()] [#eq(#0(), #0())] = [0] >= [0] = [#true()] [#eq(#0(), #neg(@y))] = [4] @y + [8] > [0] = [#false()] [#eq(#0(), #s(@y))] = [4] @y + [0] >= [0] = [#false()] [#eq(#neg(@x), #pos(@y))] = [4] @y + [0] >= [0] = [#false()] [#eq(#neg(@x), #0())] = [0] >= [0] = [#false()] [#eq(#neg(@x), #neg(@y))] = [4] @y + [8] > [4] @y + [0] = [#eq(@x, @y)] [#eq(#s(@x), #0())] = [0] >= [0] = [#false()] [#eq(#s(@x), #s(@y))] = [4] @y + [0] >= [4] @y + [0] = [#eq(@x, @y)] [#eq(nil(), ::(@y_1, @y_2))] = [4] @y_2 + [4] > [0] = [#false()] [#eq(nil(), nil())] = [0] >= [0] = [#true()] [and(@x, @y)] = [4] @y + [0] >= [0] = [#and(@x, @y)] [remove#1(::(@y, @ys), @x)] = [1] @ys + [1] >= [1] @ys + [1] = [remove#2(eq(@x, @y), @x, @y, @ys)] [remove#1(nil(), @x)] = [0] >= [0] = [nil()] [remove#2(#true(), @x, @y, @ys)] = [1] @ys + [1] > [1] @ys + [0] = [remove(@x, @ys)] [remove#2(#false(), @x, @y, @ys)] = [1] @ys + [1] >= [1] @ys + [1] = [::(@y, remove(@x, @ys))] [eq#1(::(@x, @xs), @l2)] = [0] >= [0] = [eq#3(@l2, @x, @xs)] [eq#1(nil(), @l2)] = [0] >= [0] = [eq#2(@l2)] [#and(#true(), #true())] = [0] >= [0] = [#true()] [#and(#true(), #false())] = [0] >= [0] = [#false()] [#and(#false(), #true())] = [0] >= [0] = [#false()] [#and(#false(), #false())] = [0] >= [0] = [#false()] [eq#3(::(@y, @ys), @x, @xs)] = [0] >= [0] = [and(#equal(@x, @y), eq(@xs, @ys))] [eq#3(nil(), @x, @xs)] = [0] >= [0] = [#false()] [nub^#(@l)] = [1] @l + [0] >= [1] @l + [0] = [c_7(nub#1^#(@l))] [nub#1^#(::(@x, @xs))] = [1] @xs + [1] > [1] @xs + [0] = [c_8(nub^#(remove(@x, @xs)), remove^#(@x, @xs))] We return to the main proof. Consider the set of all dependency pairs : { 1: nub^#(@l) -> c_7(nub#1^#(@l)) , 2: nub#1^#(::(@x, @xs)) -> c_8(nub^#(remove(@x, @xs)), remove^#(@x, @xs)) } Processor 'matrix interpretation of dimension 1' induces the complexity certificate YES(?,O(n^1)) on application of dependency pairs {2}. These cover all (indirect) predecessors of dependency pairs {1,2}, their number of application is equally bounded. The dependency pairs are shifted into the weak component. We are left with following problem, upon which TcT provides the certificate YES(O(1),O(1)). Weak DPs: { nub^#(@l) -> c_7(nub#1^#(@l)) , nub#1^#(::(@x, @xs)) -> c_8(nub^#(remove(@x, @xs)), remove^#(@x, @xs)) } Weak Trs: { remove(@x, @l) -> remove#1(@l, @x) , eq#2(::(@y, @ys)) -> #false() , eq#2(nil()) -> #true() , #equal(@x, @y) -> #eq(@x, @y) , eq(@l1, @l2) -> eq#1(@l1, @l2) , #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(#pos(@x), #pos(@y)) -> #eq(@x, @y) , #eq(#pos(@x), #0()) -> #false() , #eq(#pos(@x), #neg(@y)) -> #false() , #eq(#0(), #pos(@y)) -> #false() , #eq(#0(), #0()) -> #true() , #eq(#0(), #neg(@y)) -> #false() , #eq(#0(), #s(@y)) -> #false() , #eq(#neg(@x), #pos(@y)) -> #false() , #eq(#neg(@x), #0()) -> #false() , #eq(#neg(@x), #neg(@y)) -> #eq(@x, @y) , #eq(#s(@x), #0()) -> #false() , #eq(#s(@x), #s(@y)) -> #eq(@x, @y) , #eq(nil(), ::(@y_1, @y_2)) -> #false() , #eq(nil(), nil()) -> #true() , and(@x, @y) -> #and(@x, @y) , remove#1(::(@y, @ys), @x) -> remove#2(eq(@x, @y), @x, @y, @ys) , remove#1(nil(), @x) -> nil() , remove#2(#true(), @x, @y, @ys) -> remove(@x, @ys) , remove#2(#false(), @x, @y, @ys) -> ::(@y, remove(@x, @ys)) , eq#1(::(@x, @xs), @l2) -> eq#3(@l2, @x, @xs) , eq#1(nil(), @l2) -> eq#2(@l2) , #and(#true(), #true()) -> #true() , #and(#true(), #false()) -> #false() , #and(#false(), #true()) -> #false() , #and(#false(), #false()) -> #false() , eq#3(::(@y, @ys), @x, @xs) -> and(#equal(@x, @y), eq(@xs, @ys)) , eq#3(nil(), @x, @xs) -> #false() } Obligation: innermost runtime complexity Answer: YES(O(1),O(1)) The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed. { nub^#(@l) -> c_7(nub#1^#(@l)) , nub#1^#(::(@x, @xs)) -> c_8(nub^#(remove(@x, @xs)), remove^#(@x, @xs)) } We are left with following problem, upon which TcT provides the certificate YES(O(1),O(1)). Weak Trs: { remove(@x, @l) -> remove#1(@l, @x) , eq#2(::(@y, @ys)) -> #false() , eq#2(nil()) -> #true() , #equal(@x, @y) -> #eq(@x, @y) , eq(@l1, @l2) -> eq#1(@l1, @l2) , #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(#pos(@x), #pos(@y)) -> #eq(@x, @y) , #eq(#pos(@x), #0()) -> #false() , #eq(#pos(@x), #neg(@y)) -> #false() , #eq(#0(), #pos(@y)) -> #false() , #eq(#0(), #0()) -> #true() , #eq(#0(), #neg(@y)) -> #false() , #eq(#0(), #s(@y)) -> #false() , #eq(#neg(@x), #pos(@y)) -> #false() , #eq(#neg(@x), #0()) -> #false() , #eq(#neg(@x), #neg(@y)) -> #eq(@x, @y) , #eq(#s(@x), #0()) -> #false() , #eq(#s(@x), #s(@y)) -> #eq(@x, @y) , #eq(nil(), ::(@y_1, @y_2)) -> #false() , #eq(nil(), nil()) -> #true() , and(@x, @y) -> #and(@x, @y) , remove#1(::(@y, @ys), @x) -> remove#2(eq(@x, @y), @x, @y, @ys) , remove#1(nil(), @x) -> nil() , remove#2(#true(), @x, @y, @ys) -> remove(@x, @ys) , remove#2(#false(), @x, @y, @ys) -> ::(@y, remove(@x, @ys)) , eq#1(::(@x, @xs), @l2) -> eq#3(@l2, @x, @xs) , eq#1(nil(), @l2) -> eq#2(@l2) , #and(#true(), #true()) -> #true() , #and(#true(), #false()) -> #false() , #and(#false(), #true()) -> #false() , #and(#false(), #false()) -> #false() , eq#3(::(@y, @ys), @x, @xs) -> and(#equal(@x, @y), eq(@xs, @ys)) , eq#3(nil(), @x, @xs) -> #false() } Obligation: innermost runtime complexity Answer: YES(O(1),O(1)) 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 We return to the main proof. We are left with following problem, upon which TcT provides the certificate YES(O(1),O(n^1)). Strict DPs: { remove^#(@x, @l) -> c_1(remove#1^#(@l, @x)) , remove#1^#(::(@y, @ys), @x) -> c_2(remove#2^#(eq(@x, @y), @x, @y, @ys), eq^#(@x, @y)) , eq^#(@l1, @l2) -> c_3(eq#1^#(@l1, @l2)) , eq#1^#(::(@x, @xs), @l2) -> c_4(eq#3^#(@l2, @x, @xs)) , remove#2^#(#true(), @x, @y, @ys) -> c_5(remove^#(@x, @ys)) , remove#2^#(#false(), @x, @y, @ys) -> c_6(remove^#(@x, @ys)) , eq#3^#(::(@y, @ys), @x, @xs) -> c_9(eq^#(@xs, @ys)) } Weak DPs: { nub^#(@l) -> nub#1^#(@l) , nub#1^#(::(@x, @xs)) -> remove^#(@x, @xs) , nub#1^#(::(@x, @xs)) -> nub^#(remove(@x, @xs)) } Weak Trs: { remove(@x, @l) -> remove#1(@l, @x) , eq#2(::(@y, @ys)) -> #false() , eq#2(nil()) -> #true() , #equal(@x, @y) -> #eq(@x, @y) , eq(@l1, @l2) -> eq#1(@l1, @l2) , #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(#pos(@x), #pos(@y)) -> #eq(@x, @y) , #eq(#pos(@x), #0()) -> #false() , #eq(#pos(@x), #neg(@y)) -> #false() , #eq(#0(), #pos(@y)) -> #false() , #eq(#0(), #0()) -> #true() , #eq(#0(), #neg(@y)) -> #false() , #eq(#0(), #s(@y)) -> #false() , #eq(#neg(@x), #pos(@y)) -> #false() , #eq(#neg(@x), #0()) -> #false() , #eq(#neg(@x), #neg(@y)) -> #eq(@x, @y) , #eq(#s(@x), #0()) -> #false() , #eq(#s(@x), #s(@y)) -> #eq(@x, @y) , #eq(nil(), ::(@y_1, @y_2)) -> #false() , #eq(nil(), nil()) -> #true() , and(@x, @y) -> #and(@x, @y) , remove#1(::(@y, @ys), @x) -> remove#2(eq(@x, @y), @x, @y, @ys) , remove#1(nil(), @x) -> nil() , remove#2(#true(), @x, @y, @ys) -> remove(@x, @ys) , remove#2(#false(), @x, @y, @ys) -> ::(@y, remove(@x, @ys)) , eq#1(::(@x, @xs), @l2) -> eq#3(@l2, @x, @xs) , eq#1(nil(), @l2) -> eq#2(@l2) , #and(#true(), #true()) -> #true() , #and(#true(), #false()) -> #false() , #and(#false(), #true()) -> #false() , #and(#false(), #false()) -> #false() , eq#3(::(@y, @ys), @x, @xs) -> and(#equal(@x, @y), eq(@xs, @ys)) , eq#3(nil(), @x, @xs) -> #false() } Obligation: innermost runtime complexity Answer: YES(O(1),O(n^1)) We use the processor 'matrix interpretation of dimension 1' to orient following rules strictly. DPs: { 2: remove#1^#(::(@y, @ys), @x) -> c_2(remove#2^#(eq(@x, @y), @x, @y, @ys), eq^#(@x, @y)) , 3: eq^#(@l1, @l2) -> c_3(eq#1^#(@l1, @l2)) , 7: eq#3^#(::(@y, @ys), @x, @xs) -> c_9(eq^#(@xs, @ys)) , 9: nub#1^#(::(@x, @xs)) -> remove^#(@x, @xs) } Trs: { remove#1(nil(), @x) -> nil() , remove#2(#true(), @x, @y, @ys) -> remove(@x, @ys) , eq#1(::(@x, @xs), @l2) -> eq#3(@l2, @x, @xs) , eq#1(nil(), @l2) -> eq#2(@l2) } Sub-proof: ---------- The following argument positions are usable: Uargs(c_1) = {1}, Uargs(c_2) = {1, 2}, Uargs(c_3) = {1}, Uargs(c_4) = {1}, Uargs(c_5) = {1}, Uargs(c_6) = {1}, Uargs(c_9) = {1} TcT has computed the following constructor-based matrix interpretation satisfying not(EDA). [remove](x1, x2) = [1] x2 + [2] [eq#2](x1) = [0] [#equal](x1, x2) = [0] [eq](x1, x2) = [4] [#eq](x1, x2) = [0] [#true] = [0] [#false] = [0] [::](x1, x2) = [1] x1 + [1] x2 + [2] [and](x1, x2) = [0] [remove#1](x1, x2) = [1] x1 + [2] [nub](x1) = [7] x1 + [0] [#pos](x1) = [1] x1 + [0] [#0] = [0] [#neg](x1) = [1] x1 + [0] [remove#2](x1, x2, x3, x4) = [1] x3 + [1] x4 + [4] [eq#1](x1, x2) = [4] [#and](x1, x2) = [0] [eq#3](x1, x2, x3) = [0] [#s](x1) = [1] x1 + [0] [nub#1](x1) = [7] x1 + [0] [nil] = [0] [remove^#](x1, x2) = [4] x2 + [0] [c_1](x1) = [7] x1 + [0] [remove#1^#](x1, x2) = [4] x1 + [0] [eq#2^#](x1) = [7] x1 + [0] [c_2] = [0] [c_3] = [0] [#equal^#](x1, x2) = [7] x1 + [7] x2 + [0] [c_4](x1) = [7] x1 + [0] [#eq^#](x1, x2) = [7] x1 + [7] x2 + [0] [eq^#](x1, x2) = [4] x2 + [1] [c_5](x1) = [7] x1 + [0] [eq#1^#](x1, x2) = [4] x2 + [0] [and^#](x1, x2) = [7] x1 + [7] x2 + [0] [c_6](x1) = [7] x1 + [0] [#and^#](x1, x2) = [7] x1 + [7] x2 + [0] [c_7](x1, x2) = [7] x1 + [7] x2 + [0] [remove#2^#](x1, x2, x3, x4) = [4] x4 + [0] [c_8] = [0] [nub^#](x1) = [4] x1 + [0] [c_9](x1) = [7] x1 + [0] [nub#1^#](x1) = [4] x1 + [0] [c_10](x1) = [7] x1 + [0] [c_11](x1) = [7] x1 + [0] [c_12](x1) = [7] x1 + [0] [eq#3^#](x1, x2, x3) = [4] x1 + [0] [c_13](x1) = [7] x1 + [0] [c_14](x1, x2, x3) = [7] x1 + [7] x2 + [7] x3 + [0] [c_15] = [0] [c_16](x1, x2) = [7] x1 + [7] x2 + [0] [c_17] = [0] [c_18](x1, x2, x3) = [7] x1 + [7] x2 + [7] x3 + [0] [c_19] = [0] [c_20](x1) = [7] x1 + [0] [c_21] = [0] [c_22] = [0] [c_23] = [0] [c_24] = [0] [c_25] = [0] [c_26] = [0] [c_27] = [0] [c_28] = [0] [c_29](x1) = [7] x1 + [0] [c_30] = [0] [c_31](x1) = [7] x1 + [0] [c_32] = [0] [c_33] = [0] [c_34] = [0] [c_35] = [0] [c_36] = [0] [c_37] = [0] [c] = [0] [c_1](x1) = [1] x1 + [0] [c_2](x1, x2) = [1] x1 + [1] x2 + [5] [c_3](x1) = [1] x1 + [0] [c_4](x1) = [1] x1 + [0] [c_5](x1) = [1] x1 + [0] [c_6](x1) = [1] x1 + [0] [c_7](x1) = [7] x1 + [0] [c_8](x1, x2) = [7] x1 + [7] x2 + [0] [c_9](x1) = [1] x1 + [5] The following symbols are considered usable {remove, eq#2, #equal, eq, #eq, and, remove#1, remove#2, eq#1, #and, eq#3, remove^#, remove#1^#, eq^#, eq#1^#, remove#2^#, nub^#, nub#1^#, eq#3^#} The order satisfies the following ordering constraints: [remove(@x, @l)] = [1] @l + [2] >= [1] @l + [2] = [remove#1(@l, @x)] [eq#2(::(@y, @ys))] = [0] >= [0] = [#false()] [eq#2(nil())] = [0] >= [0] = [#true()] [#equal(@x, @y)] = [0] >= [0] = [#eq(@x, @y)] [eq(@l1, @l2)] = [4] >= [4] = [eq#1(@l1, @l2)] [#eq(::(@x_1, @x_2), ::(@y_1, @y_2))] = [0] >= [0] = [#and(#eq(@x_1, @y_1), #eq(@x_2, @y_2))] [#eq(::(@x_1, @x_2), nil())] = [0] >= [0] = [#false()] [#eq(#pos(@x), #pos(@y))] = [0] >= [0] = [#eq(@x, @y)] [#eq(#pos(@x), #0())] = [0] >= [0] = [#false()] [#eq(#pos(@x), #neg(@y))] = [0] >= [0] = [#false()] [#eq(#0(), #pos(@y))] = [0] >= [0] = [#false()] [#eq(#0(), #0())] = [0] >= [0] = [#true()] [#eq(#0(), #neg(@y))] = [0] >= [0] = [#false()] [#eq(#0(), #s(@y))] = [0] >= [0] = [#false()] [#eq(#neg(@x), #pos(@y))] = [0] >= [0] = [#false()] [#eq(#neg(@x), #0())] = [0] >= [0] = [#false()] [#eq(#neg(@x), #neg(@y))] = [0] >= [0] = [#eq(@x, @y)] [#eq(#s(@x), #0())] = [0] >= [0] = [#false()] [#eq(#s(@x), #s(@y))] = [0] >= [0] = [#eq(@x, @y)] [#eq(nil(), ::(@y_1, @y_2))] = [0] >= [0] = [#false()] [#eq(nil(), nil())] = [0] >= [0] = [#true()] [and(@x, @y)] = [0] >= [0] = [#and(@x, @y)] [remove#1(::(@y, @ys), @x)] = [1] @y + [1] @ys + [4] >= [1] @y + [1] @ys + [4] = [remove#2(eq(@x, @y), @x, @y, @ys)] [remove#1(nil(), @x)] = [2] > [0] = [nil()] [remove#2(#true(), @x, @y, @ys)] = [1] @y + [1] @ys + [4] > [1] @ys + [2] = [remove(@x, @ys)] [remove#2(#false(), @x, @y, @ys)] = [1] @y + [1] @ys + [4] >= [1] @y + [1] @ys + [4] = [::(@y, remove(@x, @ys))] [eq#1(::(@x, @xs), @l2)] = [4] > [0] = [eq#3(@l2, @x, @xs)] [eq#1(nil(), @l2)] = [4] > [0] = [eq#2(@l2)] [#and(#true(), #true())] = [0] >= [0] = [#true()] [#and(#true(), #false())] = [0] >= [0] = [#false()] [#and(#false(), #true())] = [0] >= [0] = [#false()] [#and(#false(), #false())] = [0] >= [0] = [#false()] [eq#3(::(@y, @ys), @x, @xs)] = [0] >= [0] = [and(#equal(@x, @y), eq(@xs, @ys))] [eq#3(nil(), @x, @xs)] = [0] >= [0] = [#false()] [remove^#(@x, @l)] = [4] @l + [0] >= [4] @l + [0] = [c_1(remove#1^#(@l, @x))] [remove#1^#(::(@y, @ys), @x)] = [4] @y + [4] @ys + [8] > [4] @y + [4] @ys + [6] = [c_2(remove#2^#(eq(@x, @y), @x, @y, @ys), eq^#(@x, @y))] [eq^#(@l1, @l2)] = [4] @l2 + [1] > [4] @l2 + [0] = [c_3(eq#1^#(@l1, @l2))] [eq#1^#(::(@x, @xs), @l2)] = [4] @l2 + [0] >= [4] @l2 + [0] = [c_4(eq#3^#(@l2, @x, @xs))] [remove#2^#(#true(), @x, @y, @ys)] = [4] @ys + [0] >= [4] @ys + [0] = [c_5(remove^#(@x, @ys))] [remove#2^#(#false(), @x, @y, @ys)] = [4] @ys + [0] >= [4] @ys + [0] = [c_6(remove^#(@x, @ys))] [nub^#(@l)] = [4] @l + [0] >= [4] @l + [0] = [nub#1^#(@l)] [nub#1^#(::(@x, @xs))] = [4] @x + [4] @xs + [8] > [4] @xs + [0] = [remove^#(@x, @xs)] [nub#1^#(::(@x, @xs))] = [4] @x + [4] @xs + [8] >= [4] @xs + [8] = [nub^#(remove(@x, @xs))] [eq#3^#(::(@y, @ys), @x, @xs)] = [4] @y + [4] @ys + [8] > [4] @ys + [6] = [c_9(eq^#(@xs, @ys))] We return to the main proof. Consider the set of all dependency pairs : { 1: remove^#(@x, @l) -> c_1(remove#1^#(@l, @x)) , 2: remove#1^#(::(@y, @ys), @x) -> c_2(remove#2^#(eq(@x, @y), @x, @y, @ys), eq^#(@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: remove#2^#(#true(), @x, @y, @ys) -> c_5(remove^#(@x, @ys)) , 6: remove#2^#(#false(), @x, @y, @ys) -> c_6(remove^#(@x, @ys)) , 7: eq#3^#(::(@y, @ys), @x, @xs) -> c_9(eq^#(@xs, @ys)) , 8: nub^#(@l) -> nub#1^#(@l) , 9: nub#1^#(::(@x, @xs)) -> remove^#(@x, @xs) , 10: nub#1^#(::(@x, @xs)) -> nub^#(remove(@x, @xs)) } Processor 'matrix interpretation of dimension 1' induces the complexity certificate YES(?,O(n^1)) on application of dependency pairs {2,3,7,9}. These cover all (indirect) predecessors of dependency pairs {1,2,3,4,5,6,7,9}, their number of application is equally bounded. The dependency pairs are shifted into the weak component. We are left with following problem, upon which TcT provides the certificate YES(O(1),O(1)). Weak DPs: { remove^#(@x, @l) -> c_1(remove#1^#(@l, @x)) , remove#1^#(::(@y, @ys), @x) -> c_2(remove#2^#(eq(@x, @y), @x, @y, @ys), eq^#(@x, @y)) , eq^#(@l1, @l2) -> c_3(eq#1^#(@l1, @l2)) , eq#1^#(::(@x, @xs), @l2) -> c_4(eq#3^#(@l2, @x, @xs)) , remove#2^#(#true(), @x, @y, @ys) -> c_5(remove^#(@x, @ys)) , remove#2^#(#false(), @x, @y, @ys) -> c_6(remove^#(@x, @ys)) , nub^#(@l) -> nub#1^#(@l) , nub#1^#(::(@x, @xs)) -> remove^#(@x, @xs) , nub#1^#(::(@x, @xs)) -> nub^#(remove(@x, @xs)) , eq#3^#(::(@y, @ys), @x, @xs) -> c_9(eq^#(@xs, @ys)) } Weak Trs: { remove(@x, @l) -> remove#1(@l, @x) , eq#2(::(@y, @ys)) -> #false() , eq#2(nil()) -> #true() , #equal(@x, @y) -> #eq(@x, @y) , eq(@l1, @l2) -> eq#1(@l1, @l2) , #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(#pos(@x), #pos(@y)) -> #eq(@x, @y) , #eq(#pos(@x), #0()) -> #false() , #eq(#pos(@x), #neg(@y)) -> #false() , #eq(#0(), #pos(@y)) -> #false() , #eq(#0(), #0()) -> #true() , #eq(#0(), #neg(@y)) -> #false() , #eq(#0(), #s(@y)) -> #false() , #eq(#neg(@x), #pos(@y)) -> #false() , #eq(#neg(@x), #0()) -> #false() , #eq(#neg(@x), #neg(@y)) -> #eq(@x, @y) , #eq(#s(@x), #0()) -> #false() , #eq(#s(@x), #s(@y)) -> #eq(@x, @y) , #eq(nil(), ::(@y_1, @y_2)) -> #false() , #eq(nil(), nil()) -> #true() , and(@x, @y) -> #and(@x, @y) , remove#1(::(@y, @ys), @x) -> remove#2(eq(@x, @y), @x, @y, @ys) , remove#1(nil(), @x) -> nil() , remove#2(#true(), @x, @y, @ys) -> remove(@x, @ys) , remove#2(#false(), @x, @y, @ys) -> ::(@y, remove(@x, @ys)) , eq#1(::(@x, @xs), @l2) -> eq#3(@l2, @x, @xs) , eq#1(nil(), @l2) -> eq#2(@l2) , #and(#true(), #true()) -> #true() , #and(#true(), #false()) -> #false() , #and(#false(), #true()) -> #false() , #and(#false(), #false()) -> #false() , eq#3(::(@y, @ys), @x, @xs) -> and(#equal(@x, @y), eq(@xs, @ys)) , eq#3(nil(), @x, @xs) -> #false() } Obligation: innermost runtime complexity Answer: YES(O(1),O(1)) The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed. { remove^#(@x, @l) -> c_1(remove#1^#(@l, @x)) , remove#1^#(::(@y, @ys), @x) -> c_2(remove#2^#(eq(@x, @y), @x, @y, @ys), eq^#(@x, @y)) , eq^#(@l1, @l2) -> c_3(eq#1^#(@l1, @l2)) , eq#1^#(::(@x, @xs), @l2) -> c_4(eq#3^#(@l2, @x, @xs)) , remove#2^#(#true(), @x, @y, @ys) -> c_5(remove^#(@x, @ys)) , remove#2^#(#false(), @x, @y, @ys) -> c_6(remove^#(@x, @ys)) , nub^#(@l) -> nub#1^#(@l) , nub#1^#(::(@x, @xs)) -> remove^#(@x, @xs) , nub#1^#(::(@x, @xs)) -> nub^#(remove(@x, @xs)) , eq#3^#(::(@y, @ys), @x, @xs) -> c_9(eq^#(@xs, @ys)) } We are left with following problem, upon which TcT provides the certificate YES(O(1),O(1)). Weak Trs: { remove(@x, @l) -> remove#1(@l, @x) , eq#2(::(@y, @ys)) -> #false() , eq#2(nil()) -> #true() , #equal(@x, @y) -> #eq(@x, @y) , eq(@l1, @l2) -> eq#1(@l1, @l2) , #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(#pos(@x), #pos(@y)) -> #eq(@x, @y) , #eq(#pos(@x), #0()) -> #false() , #eq(#pos(@x), #neg(@y)) -> #false() , #eq(#0(), #pos(@y)) -> #false() , #eq(#0(), #0()) -> #true() , #eq(#0(), #neg(@y)) -> #false() , #eq(#0(), #s(@y)) -> #false() , #eq(#neg(@x), #pos(@y)) -> #false() , #eq(#neg(@x), #0()) -> #false() , #eq(#neg(@x), #neg(@y)) -> #eq(@x, @y) , #eq(#s(@x), #0()) -> #false() , #eq(#s(@x), #s(@y)) -> #eq(@x, @y) , #eq(nil(), ::(@y_1, @y_2)) -> #false() , #eq(nil(), nil()) -> #true() , and(@x, @y) -> #and(@x, @y) , remove#1(::(@y, @ys), @x) -> remove#2(eq(@x, @y), @x, @y, @ys) , remove#1(nil(), @x) -> nil() , remove#2(#true(), @x, @y, @ys) -> remove(@x, @ys) , remove#2(#false(), @x, @y, @ys) -> ::(@y, remove(@x, @ys)) , eq#1(::(@x, @xs), @l2) -> eq#3(@l2, @x, @xs) , eq#1(nil(), @l2) -> eq#2(@l2) , #and(#true(), #true()) -> #true() , #and(#true(), #false()) -> #false() , #and(#false(), #true()) -> #false() , #and(#false(), #false()) -> #false() , eq#3(::(@y, @ys), @x, @xs) -> and(#equal(@x, @y), eq(@xs, @ys)) , eq#3(nil(), @x, @xs) -> #false() } Obligation: innermost runtime complexity Answer: YES(O(1),O(1)) 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 Hurray, we answered YES(O(1),O(n^2))