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:
{ 1: nub^#(@l) -> c_7(nub#1^#(@l))
, 2: nub#1^#(::(@x, @xs)) ->
c_8(nub^#(remove(@x, @xs)), remove^#(@x, @xs)) }
Trs:
{ remove#1(nil(), @x) -> nil()
, 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 + [3]
[eq#2](x1) = [0]
[#equal](x1, x2) = [0]
[eq](x1, x2) = [0]
[#eq](x1, x2) = [0]
[#true] = [0]
[#false] = [0]
[::](x1, x2) = [1] x2 + [4]
[and](x1, x2) = [0]
[remove#1](x1, x2) = [1] x1 + [3]
[#pos](x1) = [1] x1 + [0]
[#0] = [0]
[#neg](x1) = [1] x1 + [0]
[remove#2](x1, x2, x3, x4) = [3] x1 + [1] x4 + [7]
[eq#1](x1, x2) = [0]
[#and](x1, x2) = [0]
[eq#3](x1, x2, x3) = [0]
[#s](x1) = [1] x1 + [0]
[nil] = [0]
[remove^#](x1, x2) = [0]
[nub^#](x1) = [3] x1 + [2]
[nub#1^#](x1) = [3] x1 + [1]
[c_7](x1) = [1] x1 + [0]
[c_8](x1, x2) = [1] x1 + [1] x2 + [0]
The order satisfies the following ordering constraints:
[remove(@x, @l)] = [1] @l + [3]
>= [1] @l + [3]
= [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)] = [0]
>= [0]
= [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] @ys + [7]
>= [1] @ys + [7]
= [remove#2(eq(@x, @y), @x, @y, @ys)]
[remove#1(nil(), @x)] = [3]
> [0]
= [nil()]
[remove#2(#true(), @x, @y, @ys)] = [1] @ys + [7]
> [1] @ys + [3]
= [remove(@x, @ys)]
[remove#2(#false(), @x, @y, @ys)] = [1] @ys + [7]
>= [1] @ys + [7]
= [::(@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)] = [3] @l + [2]
> [3] @l + [1]
= [c_7(nub#1^#(@l))]
[nub#1^#(::(@x, @xs))] = [3] @xs + [13]
> [3] @xs + [11]
= [c_8(nub^#(remove(@x, @xs)), remove^#(@x, @xs))]
The strictly oriented rules are moved 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))
, 7: eq#3^#(::(@y, @ys), @x, @xs) -> c_9(eq^#(@xs, @ys))
, 9: nub#1^#(::(@x, @xs)) -> remove^#(@x, @xs)
, 10: nub#1^#(::(@x, @xs)) -> nub^#(remove(@x, @xs)) }
Trs:
{ eq(@l1, @l2) -> eq#1(@l1, @l2)
, 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, 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 + [0]
[eq#2](x1) = [0]
[#equal](x1, x2) = [0]
[eq](x1, x2) = [1]
[#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 + [0]
[#pos](x1) = [1] x1 + [0]
[#0] = [0]
[#neg](x1) = [1] x1 + [0]
[remove#2](x1, x2, x3, x4) = [1] x3 + [1] x4 + [2]
[eq#1](x1, x2) = [0]
[#and](x1, x2) = [0]
[eq#3](x1, x2, x3) = [0]
[#s](x1) = [1] x1 + [0]
[nil] = [0]
[remove^#](x1, x2) = [4] x1 + [4] x2 + [1]
[remove#1^#](x1, x2) = [4] x1 + [4] x2 + [1]
[eq^#](x1, x2) = [2] x2 + [1]
[eq#1^#](x1, x2) = [2] x2 + [1]
[remove#2^#](x1, x2, x3, x4) = [2] x1 + [4] x2 + [4] x4 + [1]
[nub^#](x1) = [4] x1 + [4]
[nub#1^#](x1) = [4] x1 + [4]
[eq#3^#](x1, x2, x3) = [2] x1 + [1]
[c_1](x1) = [1] x1 + [0]
[c_2](x1, x2) = [1] x1 + [1] x2 + [3]
[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_9](x1) = [1] x1 + [0]
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)] = [0]
>= [0]
= [#eq(@x, @y)]
[eq(@l1, @l2)] = [1]
> [0]
= [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 + [2]
>= [1] @y + [1] @ys + [2]
= [remove#2(eq(@x, @y), @x, @y, @ys)]
[remove#1(nil(), @x)] = [0]
>= [0]
= [nil()]
[remove#2(#true(), @x, @y, @ys)] = [1] @y + [1] @ys + [2]
> [1] @ys + [0]
= [remove(@x, @ys)]
[remove#2(#false(), @x, @y, @ys)] = [1] @y + [1] @ys + [2]
>= [1] @y + [1] @ys + [2]
= [::(@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()]
[remove^#(@x, @l)] = [4] @x + [4] @l + [1]
>= [4] @x + [4] @l + [1]
= [c_1(remove#1^#(@l, @x))]
[remove#1^#(::(@y, @ys), @x)] = [4] @x + [4] @y + [4] @ys + [9]
> [4] @x + [2] @y + [4] @ys + [7]
= [c_2(remove#2^#(eq(@x, @y), @x, @y, @ys), eq^#(@x, @y))]
[eq^#(@l1, @l2)] = [2] @l2 + [1]
>= [2] @l2 + [1]
= [c_3(eq#1^#(@l1, @l2))]
[eq#1^#(::(@x, @xs), @l2)] = [2] @l2 + [1]
>= [2] @l2 + [1]
= [c_4(eq#3^#(@l2, @x, @xs))]
[remove#2^#(#true(), @x, @y, @ys)] = [4] @x + [4] @ys + [1]
>= [4] @x + [4] @ys + [1]
= [c_5(remove^#(@x, @ys))]
[remove#2^#(#false(), @x, @y, @ys)] = [4] @x + [4] @ys + [1]
>= [4] @x + [4] @ys + [1]
= [c_6(remove^#(@x, @ys))]
[nub^#(@l)] = [4] @l + [4]
>= [4] @l + [4]
= [nub#1^#(@l)]
[nub#1^#(::(@x, @xs))] = [4] @x + [4] @xs + [12]
> [4] @x + [4] @xs + [1]
= [remove^#(@x, @xs)]
[nub#1^#(::(@x, @xs))] = [4] @x + [4] @xs + [12]
> [4] @xs + [4]
= [nub^#(remove(@x, @xs))]
[eq#3^#(::(@y, @ys), @x, @xs)] = [2] @y + [2] @ys + [5]
> [2] @ys + [1]
= [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 {2,7,9,10}. These cover all (indirect) predecessors of
dependency pairs {1,2,3,4,5,6,7,8,9,10}, 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))