MAYBE
We are left with following problem, upon which TcT provides the
certificate MAYBE.
Strict Trs:
{ #equal(@x, @y) -> #eq(@x, @y)
, #less(@x, @y) -> #cklt(#compare(@x, @y))
, and(@x, @y) -> #and(@x, @y)
, insert(@x, @l) -> insert#1(@l, @x)
, insert#1(::(@y, @ys), @x) -> insert#2(leq(@x, @y), @x, @y, @ys)
, insert#1(nil(), @x) -> ::(@x, nil())
, leq(@l1, @l2) -> leq#1(@l1, @l2)
, insert#2(#false(), @x, @y, @ys) -> ::(@y, insert(@x, @ys))
, insert#2(#true(), @x, @y, @ys) -> ::(@x, ::(@y, @ys))
, isortlist(@l) -> isortlist#1(@l)
, isortlist#1(::(@x, @xs)) -> insert(@x, isortlist(@xs))
, isortlist#1(nil()) -> nil()
, leq#1(::(@x, @xs), @l2) -> leq#2(@l2, @x, @xs)
, leq#1(nil(), @l2) -> #true()
, leq#2(::(@y, @ys), @x, @xs) ->
or(#less(@x, @y), and(#equal(@x, @y), leq(@xs, @ys)))
, leq#2(nil(), @x, @xs) -> #false()
, or(@x, @y) -> #or(@x, @y) }
Weak Trs:
{ #eq(::(@x_1, @x_2), ::(@y_1, @y_2)) ->
#and(#eq(@x_1, @y_1), #eq(@x_2, @y_2))
, #eq(::(@x_1, @x_2), nil()) -> #false()
, #eq(nil(), ::(@y_1, @y_2)) -> #false()
, #eq(nil(), nil()) -> #true()
, #eq(#0(), #0()) -> #true()
, #eq(#0(), #neg(@y)) -> #false()
, #eq(#0(), #pos(@y)) -> #false()
, #eq(#0(), #s(@y)) -> #false()
, #eq(#neg(@x), #0()) -> #false()
, #eq(#neg(@x), #neg(@y)) -> #eq(@x, @y)
, #eq(#neg(@x), #pos(@y)) -> #false()
, #eq(#pos(@x), #0()) -> #false()
, #eq(#pos(@x), #neg(@y)) -> #false()
, #eq(#pos(@x), #pos(@y)) -> #eq(@x, @y)
, #eq(#s(@x), #0()) -> #false()
, #eq(#s(@x), #s(@y)) -> #eq(@x, @y)
, #compare(#0(), #0()) -> #EQ()
, #compare(#0(), #neg(@y)) -> #GT()
, #compare(#0(), #pos(@y)) -> #LT()
, #compare(#0(), #s(@y)) -> #LT()
, #compare(#neg(@x), #0()) -> #LT()
, #compare(#neg(@x), #neg(@y)) -> #compare(@y, @x)
, #compare(#neg(@x), #pos(@y)) -> #LT()
, #compare(#pos(@x), #0()) -> #GT()
, #compare(#pos(@x), #neg(@y)) -> #GT()
, #compare(#pos(@x), #pos(@y)) -> #compare(@x, @y)
, #compare(#s(@x), #0()) -> #GT()
, #compare(#s(@x), #s(@y)) -> #compare(@x, @y)
, #cklt(#EQ()) -> #false()
, #cklt(#GT()) -> #false()
, #cklt(#LT()) -> #true()
, #and(#false(), #false()) -> #false()
, #and(#false(), #true()) -> #false()
, #and(#true(), #false()) -> #false()
, #and(#true(), #true()) -> #true()
, #or(#false(), #false()) -> #false()
, #or(#false(), #true()) -> #true()
, #or(#true(), #false()) -> #true()
, #or(#true(), #true()) -> #true() }
Obligation:
innermost runtime complexity
Answer:
MAYBE
We add following dependency tuples:
Strict DPs:
{ #equal^#(@x, @y) -> c_1(#eq^#(@x, @y))
, #less^#(@x, @y) ->
c_2(#cklt^#(#compare(@x, @y)), #compare^#(@x, @y))
, and^#(@x, @y) -> c_3(#and^#(@x, @y))
, insert^#(@x, @l) -> c_4(insert#1^#(@l, @x))
, insert#1^#(::(@y, @ys), @x) ->
c_5(insert#2^#(leq(@x, @y), @x, @y, @ys), leq^#(@x, @y))
, insert#1^#(nil(), @x) -> c_6()
, insert#2^#(#false(), @x, @y, @ys) -> c_8(insert^#(@x, @ys))
, insert#2^#(#true(), @x, @y, @ys) -> c_9()
, leq^#(@l1, @l2) -> c_7(leq#1^#(@l1, @l2))
, leq#1^#(::(@x, @xs), @l2) -> c_13(leq#2^#(@l2, @x, @xs))
, leq#1^#(nil(), @l2) -> c_14()
, isortlist^#(@l) -> c_10(isortlist#1^#(@l))
, isortlist#1^#(::(@x, @xs)) ->
c_11(insert^#(@x, isortlist(@xs)), isortlist^#(@xs))
, isortlist#1^#(nil()) -> c_12()
, leq#2^#(::(@y, @ys), @x, @xs) ->
c_15(or^#(#less(@x, @y), and(#equal(@x, @y), leq(@xs, @ys))),
#less^#(@x, @y),
and^#(#equal(@x, @y), leq(@xs, @ys)),
#equal^#(@x, @y),
leq^#(@xs, @ys))
, leq#2^#(nil(), @x, @xs) -> c_16()
, or^#(@x, @y) -> c_17(#or^#(@x, @y)) }
Weak DPs:
{ #eq^#(::(@x_1, @x_2), ::(@y_1, @y_2)) ->
c_18(#and^#(#eq(@x_1, @y_1), #eq(@x_2, @y_2)),
#eq^#(@x_1, @y_1),
#eq^#(@x_2, @y_2))
, #eq^#(::(@x_1, @x_2), nil()) -> c_19()
, #eq^#(nil(), ::(@y_1, @y_2)) -> c_20()
, #eq^#(nil(), nil()) -> c_21()
, #eq^#(#0(), #0()) -> c_22()
, #eq^#(#0(), #neg(@y)) -> c_23()
, #eq^#(#0(), #pos(@y)) -> c_24()
, #eq^#(#0(), #s(@y)) -> c_25()
, #eq^#(#neg(@x), #0()) -> c_26()
, #eq^#(#neg(@x), #neg(@y)) -> c_27(#eq^#(@x, @y))
, #eq^#(#neg(@x), #pos(@y)) -> c_28()
, #eq^#(#pos(@x), #0()) -> c_29()
, #eq^#(#pos(@x), #neg(@y)) -> c_30()
, #eq^#(#pos(@x), #pos(@y)) -> c_31(#eq^#(@x, @y))
, #eq^#(#s(@x), #0()) -> c_32()
, #eq^#(#s(@x), #s(@y)) -> c_33(#eq^#(@x, @y))
, #cklt^#(#EQ()) -> c_46()
, #cklt^#(#GT()) -> c_47()
, #cklt^#(#LT()) -> c_48()
, #compare^#(#0(), #0()) -> c_34()
, #compare^#(#0(), #neg(@y)) -> c_35()
, #compare^#(#0(), #pos(@y)) -> c_36()
, #compare^#(#0(), #s(@y)) -> c_37()
, #compare^#(#neg(@x), #0()) -> c_38()
, #compare^#(#neg(@x), #neg(@y)) -> c_39(#compare^#(@y, @x))
, #compare^#(#neg(@x), #pos(@y)) -> c_40()
, #compare^#(#pos(@x), #0()) -> c_41()
, #compare^#(#pos(@x), #neg(@y)) -> c_42()
, #compare^#(#pos(@x), #pos(@y)) -> c_43(#compare^#(@x, @y))
, #compare^#(#s(@x), #0()) -> c_44()
, #compare^#(#s(@x), #s(@y)) -> c_45(#compare^#(@x, @y))
, #and^#(#false(), #false()) -> c_49()
, #and^#(#false(), #true()) -> c_50()
, #and^#(#true(), #false()) -> c_51()
, #and^#(#true(), #true()) -> c_52()
, #or^#(#false(), #false()) -> c_53()
, #or^#(#false(), #true()) -> c_54()
, #or^#(#true(), #false()) -> c_55()
, #or^#(#true(), #true()) -> c_56() }
and mark the set of starting terms.
We are left with following problem, upon which TcT provides the
certificate MAYBE.
Strict DPs:
{ #equal^#(@x, @y) -> c_1(#eq^#(@x, @y))
, #less^#(@x, @y) ->
c_2(#cklt^#(#compare(@x, @y)), #compare^#(@x, @y))
, and^#(@x, @y) -> c_3(#and^#(@x, @y))
, insert^#(@x, @l) -> c_4(insert#1^#(@l, @x))
, insert#1^#(::(@y, @ys), @x) ->
c_5(insert#2^#(leq(@x, @y), @x, @y, @ys), leq^#(@x, @y))
, insert#1^#(nil(), @x) -> c_6()
, insert#2^#(#false(), @x, @y, @ys) -> c_8(insert^#(@x, @ys))
, insert#2^#(#true(), @x, @y, @ys) -> c_9()
, leq^#(@l1, @l2) -> c_7(leq#1^#(@l1, @l2))
, leq#1^#(::(@x, @xs), @l2) -> c_13(leq#2^#(@l2, @x, @xs))
, leq#1^#(nil(), @l2) -> c_14()
, isortlist^#(@l) -> c_10(isortlist#1^#(@l))
, isortlist#1^#(::(@x, @xs)) ->
c_11(insert^#(@x, isortlist(@xs)), isortlist^#(@xs))
, isortlist#1^#(nil()) -> c_12()
, leq#2^#(::(@y, @ys), @x, @xs) ->
c_15(or^#(#less(@x, @y), and(#equal(@x, @y), leq(@xs, @ys))),
#less^#(@x, @y),
and^#(#equal(@x, @y), leq(@xs, @ys)),
#equal^#(@x, @y),
leq^#(@xs, @ys))
, leq#2^#(nil(), @x, @xs) -> c_16()
, or^#(@x, @y) -> c_17(#or^#(@x, @y)) }
Weak DPs:
{ #eq^#(::(@x_1, @x_2), ::(@y_1, @y_2)) ->
c_18(#and^#(#eq(@x_1, @y_1), #eq(@x_2, @y_2)),
#eq^#(@x_1, @y_1),
#eq^#(@x_2, @y_2))
, #eq^#(::(@x_1, @x_2), nil()) -> c_19()
, #eq^#(nil(), ::(@y_1, @y_2)) -> c_20()
, #eq^#(nil(), nil()) -> c_21()
, #eq^#(#0(), #0()) -> c_22()
, #eq^#(#0(), #neg(@y)) -> c_23()
, #eq^#(#0(), #pos(@y)) -> c_24()
, #eq^#(#0(), #s(@y)) -> c_25()
, #eq^#(#neg(@x), #0()) -> c_26()
, #eq^#(#neg(@x), #neg(@y)) -> c_27(#eq^#(@x, @y))
, #eq^#(#neg(@x), #pos(@y)) -> c_28()
, #eq^#(#pos(@x), #0()) -> c_29()
, #eq^#(#pos(@x), #neg(@y)) -> c_30()
, #eq^#(#pos(@x), #pos(@y)) -> c_31(#eq^#(@x, @y))
, #eq^#(#s(@x), #0()) -> c_32()
, #eq^#(#s(@x), #s(@y)) -> c_33(#eq^#(@x, @y))
, #cklt^#(#EQ()) -> c_46()
, #cklt^#(#GT()) -> c_47()
, #cklt^#(#LT()) -> c_48()
, #compare^#(#0(), #0()) -> c_34()
, #compare^#(#0(), #neg(@y)) -> c_35()
, #compare^#(#0(), #pos(@y)) -> c_36()
, #compare^#(#0(), #s(@y)) -> c_37()
, #compare^#(#neg(@x), #0()) -> c_38()
, #compare^#(#neg(@x), #neg(@y)) -> c_39(#compare^#(@y, @x))
, #compare^#(#neg(@x), #pos(@y)) -> c_40()
, #compare^#(#pos(@x), #0()) -> c_41()
, #compare^#(#pos(@x), #neg(@y)) -> c_42()
, #compare^#(#pos(@x), #pos(@y)) -> c_43(#compare^#(@x, @y))
, #compare^#(#s(@x), #0()) -> c_44()
, #compare^#(#s(@x), #s(@y)) -> c_45(#compare^#(@x, @y))
, #and^#(#false(), #false()) -> c_49()
, #and^#(#false(), #true()) -> c_50()
, #and^#(#true(), #false()) -> c_51()
, #and^#(#true(), #true()) -> c_52()
, #or^#(#false(), #false()) -> c_53()
, #or^#(#false(), #true()) -> c_54()
, #or^#(#true(), #false()) -> c_55()
, #or^#(#true(), #true()) -> c_56() }
Weak Trs:
{ #equal(@x, @y) -> #eq(@x, @y)
, #eq(::(@x_1, @x_2), ::(@y_1, @y_2)) ->
#and(#eq(@x_1, @y_1), #eq(@x_2, @y_2))
, #eq(::(@x_1, @x_2), nil()) -> #false()
, #eq(nil(), ::(@y_1, @y_2)) -> #false()
, #eq(nil(), nil()) -> #true()
, #eq(#0(), #0()) -> #true()
, #eq(#0(), #neg(@y)) -> #false()
, #eq(#0(), #pos(@y)) -> #false()
, #eq(#0(), #s(@y)) -> #false()
, #eq(#neg(@x), #0()) -> #false()
, #eq(#neg(@x), #neg(@y)) -> #eq(@x, @y)
, #eq(#neg(@x), #pos(@y)) -> #false()
, #eq(#pos(@x), #0()) -> #false()
, #eq(#pos(@x), #neg(@y)) -> #false()
, #eq(#pos(@x), #pos(@y)) -> #eq(@x, @y)
, #eq(#s(@x), #0()) -> #false()
, #eq(#s(@x), #s(@y)) -> #eq(@x, @y)
, #less(@x, @y) -> #cklt(#compare(@x, @y))
, #compare(#0(), #0()) -> #EQ()
, #compare(#0(), #neg(@y)) -> #GT()
, #compare(#0(), #pos(@y)) -> #LT()
, #compare(#0(), #s(@y)) -> #LT()
, #compare(#neg(@x), #0()) -> #LT()
, #compare(#neg(@x), #neg(@y)) -> #compare(@y, @x)
, #compare(#neg(@x), #pos(@y)) -> #LT()
, #compare(#pos(@x), #0()) -> #GT()
, #compare(#pos(@x), #neg(@y)) -> #GT()
, #compare(#pos(@x), #pos(@y)) -> #compare(@x, @y)
, #compare(#s(@x), #0()) -> #GT()
, #compare(#s(@x), #s(@y)) -> #compare(@x, @y)
, #cklt(#EQ()) -> #false()
, #cklt(#GT()) -> #false()
, #cklt(#LT()) -> #true()
, and(@x, @y) -> #and(@x, @y)
, #and(#false(), #false()) -> #false()
, #and(#false(), #true()) -> #false()
, #and(#true(), #false()) -> #false()
, #and(#true(), #true()) -> #true()
, insert(@x, @l) -> insert#1(@l, @x)
, insert#1(::(@y, @ys), @x) -> insert#2(leq(@x, @y), @x, @y, @ys)
, insert#1(nil(), @x) -> ::(@x, nil())
, leq(@l1, @l2) -> leq#1(@l1, @l2)
, insert#2(#false(), @x, @y, @ys) -> ::(@y, insert(@x, @ys))
, insert#2(#true(), @x, @y, @ys) -> ::(@x, ::(@y, @ys))
, isortlist(@l) -> isortlist#1(@l)
, isortlist#1(::(@x, @xs)) -> insert(@x, isortlist(@xs))
, isortlist#1(nil()) -> nil()
, leq#1(::(@x, @xs), @l2) -> leq#2(@l2, @x, @xs)
, leq#1(nil(), @l2) -> #true()
, leq#2(::(@y, @ys), @x, @xs) ->
or(#less(@x, @y), and(#equal(@x, @y), leq(@xs, @ys)))
, leq#2(nil(), @x, @xs) -> #false()
, or(@x, @y) -> #or(@x, @y)
, #or(#false(), #false()) -> #false()
, #or(#false(), #true()) -> #true()
, #or(#true(), #false()) -> #true()
, #or(#true(), #true()) -> #true() }
Obligation:
innermost runtime complexity
Answer:
MAYBE
We estimate the number of application of {1,2,3,6,8,11,14,16,17} by
applications of Pre({1,2,3,6,8,11,14,16,17}) = {4,5,9,10,12,15}.
Here rules are labeled as follows:
DPs:
{ 1: #equal^#(@x, @y) -> c_1(#eq^#(@x, @y))
, 2: #less^#(@x, @y) ->
c_2(#cklt^#(#compare(@x, @y)), #compare^#(@x, @y))
, 3: and^#(@x, @y) -> c_3(#and^#(@x, @y))
, 4: insert^#(@x, @l) -> c_4(insert#1^#(@l, @x))
, 5: insert#1^#(::(@y, @ys), @x) ->
c_5(insert#2^#(leq(@x, @y), @x, @y, @ys), leq^#(@x, @y))
, 6: insert#1^#(nil(), @x) -> c_6()
, 7: insert#2^#(#false(), @x, @y, @ys) -> c_8(insert^#(@x, @ys))
, 8: insert#2^#(#true(), @x, @y, @ys) -> c_9()
, 9: leq^#(@l1, @l2) -> c_7(leq#1^#(@l1, @l2))
, 10: leq#1^#(::(@x, @xs), @l2) -> c_13(leq#2^#(@l2, @x, @xs))
, 11: leq#1^#(nil(), @l2) -> c_14()
, 12: isortlist^#(@l) -> c_10(isortlist#1^#(@l))
, 13: isortlist#1^#(::(@x, @xs)) ->
c_11(insert^#(@x, isortlist(@xs)), isortlist^#(@xs))
, 14: isortlist#1^#(nil()) -> c_12()
, 15: leq#2^#(::(@y, @ys), @x, @xs) ->
c_15(or^#(#less(@x, @y), and(#equal(@x, @y), leq(@xs, @ys))),
#less^#(@x, @y),
and^#(#equal(@x, @y), leq(@xs, @ys)),
#equal^#(@x, @y),
leq^#(@xs, @ys))
, 16: leq#2^#(nil(), @x, @xs) -> c_16()
, 17: or^#(@x, @y) -> c_17(#or^#(@x, @y))
, 18: #eq^#(::(@x_1, @x_2), ::(@y_1, @y_2)) ->
c_18(#and^#(#eq(@x_1, @y_1), #eq(@x_2, @y_2)),
#eq^#(@x_1, @y_1),
#eq^#(@x_2, @y_2))
, 19: #eq^#(::(@x_1, @x_2), nil()) -> c_19()
, 20: #eq^#(nil(), ::(@y_1, @y_2)) -> c_20()
, 21: #eq^#(nil(), nil()) -> c_21()
, 22: #eq^#(#0(), #0()) -> c_22()
, 23: #eq^#(#0(), #neg(@y)) -> c_23()
, 24: #eq^#(#0(), #pos(@y)) -> c_24()
, 25: #eq^#(#0(), #s(@y)) -> c_25()
, 26: #eq^#(#neg(@x), #0()) -> c_26()
, 27: #eq^#(#neg(@x), #neg(@y)) -> c_27(#eq^#(@x, @y))
, 28: #eq^#(#neg(@x), #pos(@y)) -> c_28()
, 29: #eq^#(#pos(@x), #0()) -> c_29()
, 30: #eq^#(#pos(@x), #neg(@y)) -> c_30()
, 31: #eq^#(#pos(@x), #pos(@y)) -> c_31(#eq^#(@x, @y))
, 32: #eq^#(#s(@x), #0()) -> c_32()
, 33: #eq^#(#s(@x), #s(@y)) -> c_33(#eq^#(@x, @y))
, 34: #cklt^#(#EQ()) -> c_46()
, 35: #cklt^#(#GT()) -> c_47()
, 36: #cklt^#(#LT()) -> c_48()
, 37: #compare^#(#0(), #0()) -> c_34()
, 38: #compare^#(#0(), #neg(@y)) -> c_35()
, 39: #compare^#(#0(), #pos(@y)) -> c_36()
, 40: #compare^#(#0(), #s(@y)) -> c_37()
, 41: #compare^#(#neg(@x), #0()) -> c_38()
, 42: #compare^#(#neg(@x), #neg(@y)) -> c_39(#compare^#(@y, @x))
, 43: #compare^#(#neg(@x), #pos(@y)) -> c_40()
, 44: #compare^#(#pos(@x), #0()) -> c_41()
, 45: #compare^#(#pos(@x), #neg(@y)) -> c_42()
, 46: #compare^#(#pos(@x), #pos(@y)) -> c_43(#compare^#(@x, @y))
, 47: #compare^#(#s(@x), #0()) -> c_44()
, 48: #compare^#(#s(@x), #s(@y)) -> c_45(#compare^#(@x, @y))
, 49: #and^#(#false(), #false()) -> c_49()
, 50: #and^#(#false(), #true()) -> c_50()
, 51: #and^#(#true(), #false()) -> c_51()
, 52: #and^#(#true(), #true()) -> c_52()
, 53: #or^#(#false(), #false()) -> c_53()
, 54: #or^#(#false(), #true()) -> c_54()
, 55: #or^#(#true(), #false()) -> c_55()
, 56: #or^#(#true(), #true()) -> c_56() }
We are left with following problem, upon which TcT provides the
certificate MAYBE.
Strict DPs:
{ insert^#(@x, @l) -> c_4(insert#1^#(@l, @x))
, insert#1^#(::(@y, @ys), @x) ->
c_5(insert#2^#(leq(@x, @y), @x, @y, @ys), leq^#(@x, @y))
, insert#2^#(#false(), @x, @y, @ys) -> c_8(insert^#(@x, @ys))
, leq^#(@l1, @l2) -> c_7(leq#1^#(@l1, @l2))
, leq#1^#(::(@x, @xs), @l2) -> c_13(leq#2^#(@l2, @x, @xs))
, isortlist^#(@l) -> c_10(isortlist#1^#(@l))
, isortlist#1^#(::(@x, @xs)) ->
c_11(insert^#(@x, isortlist(@xs)), isortlist^#(@xs))
, leq#2^#(::(@y, @ys), @x, @xs) ->
c_15(or^#(#less(@x, @y), and(#equal(@x, @y), leq(@xs, @ys))),
#less^#(@x, @y),
and^#(#equal(@x, @y), leq(@xs, @ys)),
#equal^#(@x, @y),
leq^#(@xs, @ys)) }
Weak DPs:
{ #equal^#(@x, @y) -> c_1(#eq^#(@x, @y))
, #eq^#(::(@x_1, @x_2), ::(@y_1, @y_2)) ->
c_18(#and^#(#eq(@x_1, @y_1), #eq(@x_2, @y_2)),
#eq^#(@x_1, @y_1),
#eq^#(@x_2, @y_2))
, #eq^#(::(@x_1, @x_2), nil()) -> c_19()
, #eq^#(nil(), ::(@y_1, @y_2)) -> c_20()
, #eq^#(nil(), nil()) -> c_21()
, #eq^#(#0(), #0()) -> c_22()
, #eq^#(#0(), #neg(@y)) -> c_23()
, #eq^#(#0(), #pos(@y)) -> c_24()
, #eq^#(#0(), #s(@y)) -> c_25()
, #eq^#(#neg(@x), #0()) -> c_26()
, #eq^#(#neg(@x), #neg(@y)) -> c_27(#eq^#(@x, @y))
, #eq^#(#neg(@x), #pos(@y)) -> c_28()
, #eq^#(#pos(@x), #0()) -> c_29()
, #eq^#(#pos(@x), #neg(@y)) -> c_30()
, #eq^#(#pos(@x), #pos(@y)) -> c_31(#eq^#(@x, @y))
, #eq^#(#s(@x), #0()) -> c_32()
, #eq^#(#s(@x), #s(@y)) -> c_33(#eq^#(@x, @y))
, #less^#(@x, @y) ->
c_2(#cklt^#(#compare(@x, @y)), #compare^#(@x, @y))
, #cklt^#(#EQ()) -> c_46()
, #cklt^#(#GT()) -> c_47()
, #cklt^#(#LT()) -> c_48()
, #compare^#(#0(), #0()) -> c_34()
, #compare^#(#0(), #neg(@y)) -> c_35()
, #compare^#(#0(), #pos(@y)) -> c_36()
, #compare^#(#0(), #s(@y)) -> c_37()
, #compare^#(#neg(@x), #0()) -> c_38()
, #compare^#(#neg(@x), #neg(@y)) -> c_39(#compare^#(@y, @x))
, #compare^#(#neg(@x), #pos(@y)) -> c_40()
, #compare^#(#pos(@x), #0()) -> c_41()
, #compare^#(#pos(@x), #neg(@y)) -> c_42()
, #compare^#(#pos(@x), #pos(@y)) -> c_43(#compare^#(@x, @y))
, #compare^#(#s(@x), #0()) -> c_44()
, #compare^#(#s(@x), #s(@y)) -> c_45(#compare^#(@x, @y))
, and^#(@x, @y) -> c_3(#and^#(@x, @y))
, #and^#(#false(), #false()) -> c_49()
, #and^#(#false(), #true()) -> c_50()
, #and^#(#true(), #false()) -> c_51()
, #and^#(#true(), #true()) -> c_52()
, insert#1^#(nil(), @x) -> c_6()
, insert#2^#(#true(), @x, @y, @ys) -> c_9()
, leq#1^#(nil(), @l2) -> c_14()
, isortlist#1^#(nil()) -> c_12()
, leq#2^#(nil(), @x, @xs) -> c_16()
, or^#(@x, @y) -> c_17(#or^#(@x, @y))
, #or^#(#false(), #false()) -> c_53()
, #or^#(#false(), #true()) -> c_54()
, #or^#(#true(), #false()) -> c_55()
, #or^#(#true(), #true()) -> c_56() }
Weak Trs:
{ #equal(@x, @y) -> #eq(@x, @y)
, #eq(::(@x_1, @x_2), ::(@y_1, @y_2)) ->
#and(#eq(@x_1, @y_1), #eq(@x_2, @y_2))
, #eq(::(@x_1, @x_2), nil()) -> #false()
, #eq(nil(), ::(@y_1, @y_2)) -> #false()
, #eq(nil(), nil()) -> #true()
, #eq(#0(), #0()) -> #true()
, #eq(#0(), #neg(@y)) -> #false()
, #eq(#0(), #pos(@y)) -> #false()
, #eq(#0(), #s(@y)) -> #false()
, #eq(#neg(@x), #0()) -> #false()
, #eq(#neg(@x), #neg(@y)) -> #eq(@x, @y)
, #eq(#neg(@x), #pos(@y)) -> #false()
, #eq(#pos(@x), #0()) -> #false()
, #eq(#pos(@x), #neg(@y)) -> #false()
, #eq(#pos(@x), #pos(@y)) -> #eq(@x, @y)
, #eq(#s(@x), #0()) -> #false()
, #eq(#s(@x), #s(@y)) -> #eq(@x, @y)
, #less(@x, @y) -> #cklt(#compare(@x, @y))
, #compare(#0(), #0()) -> #EQ()
, #compare(#0(), #neg(@y)) -> #GT()
, #compare(#0(), #pos(@y)) -> #LT()
, #compare(#0(), #s(@y)) -> #LT()
, #compare(#neg(@x), #0()) -> #LT()
, #compare(#neg(@x), #neg(@y)) -> #compare(@y, @x)
, #compare(#neg(@x), #pos(@y)) -> #LT()
, #compare(#pos(@x), #0()) -> #GT()
, #compare(#pos(@x), #neg(@y)) -> #GT()
, #compare(#pos(@x), #pos(@y)) -> #compare(@x, @y)
, #compare(#s(@x), #0()) -> #GT()
, #compare(#s(@x), #s(@y)) -> #compare(@x, @y)
, #cklt(#EQ()) -> #false()
, #cklt(#GT()) -> #false()
, #cklt(#LT()) -> #true()
, and(@x, @y) -> #and(@x, @y)
, #and(#false(), #false()) -> #false()
, #and(#false(), #true()) -> #false()
, #and(#true(), #false()) -> #false()
, #and(#true(), #true()) -> #true()
, insert(@x, @l) -> insert#1(@l, @x)
, insert#1(::(@y, @ys), @x) -> insert#2(leq(@x, @y), @x, @y, @ys)
, insert#1(nil(), @x) -> ::(@x, nil())
, leq(@l1, @l2) -> leq#1(@l1, @l2)
, insert#2(#false(), @x, @y, @ys) -> ::(@y, insert(@x, @ys))
, insert#2(#true(), @x, @y, @ys) -> ::(@x, ::(@y, @ys))
, isortlist(@l) -> isortlist#1(@l)
, isortlist#1(::(@x, @xs)) -> insert(@x, isortlist(@xs))
, isortlist#1(nil()) -> nil()
, leq#1(::(@x, @xs), @l2) -> leq#2(@l2, @x, @xs)
, leq#1(nil(), @l2) -> #true()
, leq#2(::(@y, @ys), @x, @xs) ->
or(#less(@x, @y), and(#equal(@x, @y), leq(@xs, @ys)))
, leq#2(nil(), @x, @xs) -> #false()
, or(@x, @y) -> #or(@x, @y)
, #or(#false(), #false()) -> #false()
, #or(#false(), #true()) -> #true()
, #or(#true(), #false()) -> #true()
, #or(#true(), #true()) -> #true() }
Obligation:
innermost runtime complexity
Answer:
MAYBE
The following weak DPs constitute a sub-graph of the DG that is
closed under successors. The DPs are removed.
{ #equal^#(@x, @y) -> c_1(#eq^#(@x, @y))
, #eq^#(::(@x_1, @x_2), ::(@y_1, @y_2)) ->
c_18(#and^#(#eq(@x_1, @y_1), #eq(@x_2, @y_2)),
#eq^#(@x_1, @y_1),
#eq^#(@x_2, @y_2))
, #eq^#(::(@x_1, @x_2), nil()) -> c_19()
, #eq^#(nil(), ::(@y_1, @y_2)) -> c_20()
, #eq^#(nil(), nil()) -> c_21()
, #eq^#(#0(), #0()) -> c_22()
, #eq^#(#0(), #neg(@y)) -> c_23()
, #eq^#(#0(), #pos(@y)) -> c_24()
, #eq^#(#0(), #s(@y)) -> c_25()
, #eq^#(#neg(@x), #0()) -> c_26()
, #eq^#(#neg(@x), #neg(@y)) -> c_27(#eq^#(@x, @y))
, #eq^#(#neg(@x), #pos(@y)) -> c_28()
, #eq^#(#pos(@x), #0()) -> c_29()
, #eq^#(#pos(@x), #neg(@y)) -> c_30()
, #eq^#(#pos(@x), #pos(@y)) -> c_31(#eq^#(@x, @y))
, #eq^#(#s(@x), #0()) -> c_32()
, #eq^#(#s(@x), #s(@y)) -> c_33(#eq^#(@x, @y))
, #less^#(@x, @y) ->
c_2(#cklt^#(#compare(@x, @y)), #compare^#(@x, @y))
, #cklt^#(#EQ()) -> c_46()
, #cklt^#(#GT()) -> c_47()
, #cklt^#(#LT()) -> c_48()
, #compare^#(#0(), #0()) -> c_34()
, #compare^#(#0(), #neg(@y)) -> c_35()
, #compare^#(#0(), #pos(@y)) -> c_36()
, #compare^#(#0(), #s(@y)) -> c_37()
, #compare^#(#neg(@x), #0()) -> c_38()
, #compare^#(#neg(@x), #neg(@y)) -> c_39(#compare^#(@y, @x))
, #compare^#(#neg(@x), #pos(@y)) -> c_40()
, #compare^#(#pos(@x), #0()) -> c_41()
, #compare^#(#pos(@x), #neg(@y)) -> c_42()
, #compare^#(#pos(@x), #pos(@y)) -> c_43(#compare^#(@x, @y))
, #compare^#(#s(@x), #0()) -> c_44()
, #compare^#(#s(@x), #s(@y)) -> c_45(#compare^#(@x, @y))
, and^#(@x, @y) -> c_3(#and^#(@x, @y))
, #and^#(#false(), #false()) -> c_49()
, #and^#(#false(), #true()) -> c_50()
, #and^#(#true(), #false()) -> c_51()
, #and^#(#true(), #true()) -> c_52()
, insert#1^#(nil(), @x) -> c_6()
, insert#2^#(#true(), @x, @y, @ys) -> c_9()
, leq#1^#(nil(), @l2) -> c_14()
, isortlist#1^#(nil()) -> c_12()
, leq#2^#(nil(), @x, @xs) -> c_16()
, or^#(@x, @y) -> c_17(#or^#(@x, @y))
, #or^#(#false(), #false()) -> c_53()
, #or^#(#false(), #true()) -> c_54()
, #or^#(#true(), #false()) -> c_55()
, #or^#(#true(), #true()) -> c_56() }
We are left with following problem, upon which TcT provides the
certificate MAYBE.
Strict DPs:
{ insert^#(@x, @l) -> c_4(insert#1^#(@l, @x))
, insert#1^#(::(@y, @ys), @x) ->
c_5(insert#2^#(leq(@x, @y), @x, @y, @ys), leq^#(@x, @y))
, insert#2^#(#false(), @x, @y, @ys) -> c_8(insert^#(@x, @ys))
, leq^#(@l1, @l2) -> c_7(leq#1^#(@l1, @l2))
, leq#1^#(::(@x, @xs), @l2) -> c_13(leq#2^#(@l2, @x, @xs))
, isortlist^#(@l) -> c_10(isortlist#1^#(@l))
, isortlist#1^#(::(@x, @xs)) ->
c_11(insert^#(@x, isortlist(@xs)), isortlist^#(@xs))
, leq#2^#(::(@y, @ys), @x, @xs) ->
c_15(or^#(#less(@x, @y), and(#equal(@x, @y), leq(@xs, @ys))),
#less^#(@x, @y),
and^#(#equal(@x, @y), leq(@xs, @ys)),
#equal^#(@x, @y),
leq^#(@xs, @ys)) }
Weak Trs:
{ #equal(@x, @y) -> #eq(@x, @y)
, #eq(::(@x_1, @x_2), ::(@y_1, @y_2)) ->
#and(#eq(@x_1, @y_1), #eq(@x_2, @y_2))
, #eq(::(@x_1, @x_2), nil()) -> #false()
, #eq(nil(), ::(@y_1, @y_2)) -> #false()
, #eq(nil(), nil()) -> #true()
, #eq(#0(), #0()) -> #true()
, #eq(#0(), #neg(@y)) -> #false()
, #eq(#0(), #pos(@y)) -> #false()
, #eq(#0(), #s(@y)) -> #false()
, #eq(#neg(@x), #0()) -> #false()
, #eq(#neg(@x), #neg(@y)) -> #eq(@x, @y)
, #eq(#neg(@x), #pos(@y)) -> #false()
, #eq(#pos(@x), #0()) -> #false()
, #eq(#pos(@x), #neg(@y)) -> #false()
, #eq(#pos(@x), #pos(@y)) -> #eq(@x, @y)
, #eq(#s(@x), #0()) -> #false()
, #eq(#s(@x), #s(@y)) -> #eq(@x, @y)
, #less(@x, @y) -> #cklt(#compare(@x, @y))
, #compare(#0(), #0()) -> #EQ()
, #compare(#0(), #neg(@y)) -> #GT()
, #compare(#0(), #pos(@y)) -> #LT()
, #compare(#0(), #s(@y)) -> #LT()
, #compare(#neg(@x), #0()) -> #LT()
, #compare(#neg(@x), #neg(@y)) -> #compare(@y, @x)
, #compare(#neg(@x), #pos(@y)) -> #LT()
, #compare(#pos(@x), #0()) -> #GT()
, #compare(#pos(@x), #neg(@y)) -> #GT()
, #compare(#pos(@x), #pos(@y)) -> #compare(@x, @y)
, #compare(#s(@x), #0()) -> #GT()
, #compare(#s(@x), #s(@y)) -> #compare(@x, @y)
, #cklt(#EQ()) -> #false()
, #cklt(#GT()) -> #false()
, #cklt(#LT()) -> #true()
, and(@x, @y) -> #and(@x, @y)
, #and(#false(), #false()) -> #false()
, #and(#false(), #true()) -> #false()
, #and(#true(), #false()) -> #false()
, #and(#true(), #true()) -> #true()
, insert(@x, @l) -> insert#1(@l, @x)
, insert#1(::(@y, @ys), @x) -> insert#2(leq(@x, @y), @x, @y, @ys)
, insert#1(nil(), @x) -> ::(@x, nil())
, leq(@l1, @l2) -> leq#1(@l1, @l2)
, insert#2(#false(), @x, @y, @ys) -> ::(@y, insert(@x, @ys))
, insert#2(#true(), @x, @y, @ys) -> ::(@x, ::(@y, @ys))
, isortlist(@l) -> isortlist#1(@l)
, isortlist#1(::(@x, @xs)) -> insert(@x, isortlist(@xs))
, isortlist#1(nil()) -> nil()
, leq#1(::(@x, @xs), @l2) -> leq#2(@l2, @x, @xs)
, leq#1(nil(), @l2) -> #true()
, leq#2(::(@y, @ys), @x, @xs) ->
or(#less(@x, @y), and(#equal(@x, @y), leq(@xs, @ys)))
, leq#2(nil(), @x, @xs) -> #false()
, or(@x, @y) -> #or(@x, @y)
, #or(#false(), #false()) -> #false()
, #or(#false(), #true()) -> #true()
, #or(#true(), #false()) -> #true()
, #or(#true(), #true()) -> #true() }
Obligation:
innermost runtime complexity
Answer:
MAYBE
Due to missing edges in the dependency-graph, the right-hand sides
of following rules could be simplified:
{ leq#2^#(::(@y, @ys), @x, @xs) ->
c_15(or^#(#less(@x, @y), and(#equal(@x, @y), leq(@xs, @ys))),
#less^#(@x, @y),
and^#(#equal(@x, @y), leq(@xs, @ys)),
#equal^#(@x, @y),
leq^#(@xs, @ys)) }
We are left with following problem, upon which TcT provides the
certificate MAYBE.
Strict DPs:
{ insert^#(@x, @l) -> c_1(insert#1^#(@l, @x))
, insert#1^#(::(@y, @ys), @x) ->
c_2(insert#2^#(leq(@x, @y), @x, @y, @ys), leq^#(@x, @y))
, insert#2^#(#false(), @x, @y, @ys) -> c_3(insert^#(@x, @ys))
, leq^#(@l1, @l2) -> c_4(leq#1^#(@l1, @l2))
, leq#1^#(::(@x, @xs), @l2) -> c_5(leq#2^#(@l2, @x, @xs))
, isortlist^#(@l) -> c_6(isortlist#1^#(@l))
, isortlist#1^#(::(@x, @xs)) ->
c_7(insert^#(@x, isortlist(@xs)), isortlist^#(@xs))
, leq#2^#(::(@y, @ys), @x, @xs) -> c_8(leq^#(@xs, @ys)) }
Weak Trs:
{ #equal(@x, @y) -> #eq(@x, @y)
, #eq(::(@x_1, @x_2), ::(@y_1, @y_2)) ->
#and(#eq(@x_1, @y_1), #eq(@x_2, @y_2))
, #eq(::(@x_1, @x_2), nil()) -> #false()
, #eq(nil(), ::(@y_1, @y_2)) -> #false()
, #eq(nil(), nil()) -> #true()
, #eq(#0(), #0()) -> #true()
, #eq(#0(), #neg(@y)) -> #false()
, #eq(#0(), #pos(@y)) -> #false()
, #eq(#0(), #s(@y)) -> #false()
, #eq(#neg(@x), #0()) -> #false()
, #eq(#neg(@x), #neg(@y)) -> #eq(@x, @y)
, #eq(#neg(@x), #pos(@y)) -> #false()
, #eq(#pos(@x), #0()) -> #false()
, #eq(#pos(@x), #neg(@y)) -> #false()
, #eq(#pos(@x), #pos(@y)) -> #eq(@x, @y)
, #eq(#s(@x), #0()) -> #false()
, #eq(#s(@x), #s(@y)) -> #eq(@x, @y)
, #less(@x, @y) -> #cklt(#compare(@x, @y))
, #compare(#0(), #0()) -> #EQ()
, #compare(#0(), #neg(@y)) -> #GT()
, #compare(#0(), #pos(@y)) -> #LT()
, #compare(#0(), #s(@y)) -> #LT()
, #compare(#neg(@x), #0()) -> #LT()
, #compare(#neg(@x), #neg(@y)) -> #compare(@y, @x)
, #compare(#neg(@x), #pos(@y)) -> #LT()
, #compare(#pos(@x), #0()) -> #GT()
, #compare(#pos(@x), #neg(@y)) -> #GT()
, #compare(#pos(@x), #pos(@y)) -> #compare(@x, @y)
, #compare(#s(@x), #0()) -> #GT()
, #compare(#s(@x), #s(@y)) -> #compare(@x, @y)
, #cklt(#EQ()) -> #false()
, #cklt(#GT()) -> #false()
, #cklt(#LT()) -> #true()
, and(@x, @y) -> #and(@x, @y)
, #and(#false(), #false()) -> #false()
, #and(#false(), #true()) -> #false()
, #and(#true(), #false()) -> #false()
, #and(#true(), #true()) -> #true()
, insert(@x, @l) -> insert#1(@l, @x)
, insert#1(::(@y, @ys), @x) -> insert#2(leq(@x, @y), @x, @y, @ys)
, insert#1(nil(), @x) -> ::(@x, nil())
, leq(@l1, @l2) -> leq#1(@l1, @l2)
, insert#2(#false(), @x, @y, @ys) -> ::(@y, insert(@x, @ys))
, insert#2(#true(), @x, @y, @ys) -> ::(@x, ::(@y, @ys))
, isortlist(@l) -> isortlist#1(@l)
, isortlist#1(::(@x, @xs)) -> insert(@x, isortlist(@xs))
, isortlist#1(nil()) -> nil()
, leq#1(::(@x, @xs), @l2) -> leq#2(@l2, @x, @xs)
, leq#1(nil(), @l2) -> #true()
, leq#2(::(@y, @ys), @x, @xs) ->
or(#less(@x, @y), and(#equal(@x, @y), leq(@xs, @ys)))
, leq#2(nil(), @x, @xs) -> #false()
, or(@x, @y) -> #or(@x, @y)
, #or(#false(), #false()) -> #false()
, #or(#false(), #true()) -> #true()
, #or(#true(), #false()) -> #true()
, #or(#true(), #true()) -> #true() }
Obligation:
innermost runtime complexity
Answer:
MAYBE
None of the processors succeeded.
Details of failed attempt(s):
-----------------------------
1) 'empty' failed due to the following reason:
Empty strict component of the problem is NOT empty.
2) 'Fastest' failed due to the following reason:
None of the processors succeeded.
Details of failed attempt(s):
-----------------------------
1) 'trivial' failed due to the following reason:
We use the processor 'matrix interpretation of dimension 3' to
orient following rules strictly.
DPs:
{ 6: isortlist^#(@l) -> c_6(isortlist#1^#(@l)) }
Trs:
{ #eq(::(@x_1, @x_2), ::(@y_1, @y_2)) ->
#and(#eq(@x_1, @y_1), #eq(@x_2, @y_2))
, #eq(::(@x_1, @x_2), nil()) -> #false()
, #eq(nil(), ::(@y_1, @y_2)) -> #false()
, #eq(nil(), nil()) -> #true()
, insert#1(::(@y, @ys), @x) -> insert#2(leq(@x, @y), @x, @y, @ys)
, isortlist#1(nil()) -> nil()
, leq#2(::(@y, @ys), @x, @xs) ->
or(#less(@x, @y), and(#equal(@x, @y), leq(@xs, @ys)))
, leq#2(nil(), @x, @xs) -> #false() }
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_7) = {1, 2}, Uargs(c_8) = {1}
TcT has computed following constructor-based matrix interpretation
satisfying not(EDA).
[0]
[#equal](x1, x2) = [0]
[0]
[0 0 0] [1 1 1] [0]
[#eq](x1, x2) = [1 1 0] x1 + [0 1 1] x2 + [0]
[0 0 0] [0 0 1] [0]
[0 0 0] [0]
[#less](x1, x2) = [1 0 0] x1 + [0]
[0 0 0] [0]
[1 1 0] [1 1 0] [0]
[#compare](x1, x2) = [1 1 1] x1 + [0 0 0] x2 + [0]
[0 0 0] [1 1 0] [0]
[0]
[#cklt](x1) = [0]
[0]
[0]
[and](x1, x2) = [0]
[0]
[0]
[#and](x1, x2) = [0]
[0]
[0 1 1] [1 1 1] [0]
[insert](x1, x2) = [1 1 1] x1 + [1 0 0] x2 + [1]
[1 1 1] [1 1 1] [0]
[1 1 1] [1 1 1] [0]
[insert#1](x1, x2) = [1 1 1] x1 + [1 1 1] x2 + [0]
[1 1 1] [1 1 1] [0]
[0 1 0] [0 1 0] [1]
[::](x1, x2) = [0 0 1] x1 + [1 0 1] x2 + [1]
[1 0 0] [0 0 0] [0]
[0]
[leq](x1, x2) = [0]
[0]
[0 1 1] [1 1 1] [1 1
1] [0]
[insert#2](x1, x2, x3, x4) = [0 1 1] x2 + [1 1 1] x3 + [1 1
1] x4 + [0]
[1 1 1] [1 1 1] [1 1
1] [0]
[0]
[nil] = [0]
[1]
[0]
[#false] = [0]
[0]
[0]
[#true] = [0]
[0]
[1 1 1] [0]
[isortlist](x1) = [1 1 1] x1 + [0]
[0 0 1] [0]
[1 1 1] [0]
[isortlist#1](x1) = [1 1 1] x1 + [0]
[1 1 1] [0]
[1 0 0] [0]
[leq#1](x1, x2) = [1 0 1] x1 + [0]
[0 0 0] [0]
[1 1 1] [1 0 0] [0 0
0] [0]
[leq#2](x1, x2, x3) = [1 1 1] x1 + [0 0 0] x2 + [0 0
0] x3 + [0]
[0 0 1] [0 0 0] [0 1
0] [0]
[0 1 0] [0]
[or](x1, x2) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0]
[#or](x1, x2) = [0]
[0]
[0]
[#EQ] = [0]
[0]
[0]
[#GT] = [0]
[0]
[0]
[#LT] = [0]
[0]
[0]
[#0] = [0]
[0]
[0 0 1] [0]
[#neg](x1) = [0 0 0] x1 + [0]
[1 1 0] [0]
[0 0 1] [0]
[#pos](x1) = [1 1 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
[#s](x1) = [0 0 0] x1 + [0]
[1 0 1] [0]
[0]
[#equal^#](x1, x2) = [0]
[0]
[0]
[#eq^#](x1, x2) = [0]
[0]
[0]
[#less^#](x1, x2) = [0]
[0]
[0]
[#cklt^#](x1) = [0]
[0]
[0]
[#compare^#](x1, x2) = [0]
[0]
[0]
[and^#](x1, x2) = [0]
[0]
[0]
[#and^#](x1, x2) = [0]
[0]
[1]
[insert^#](x1, x2) = [1]
[0]
[0 0 0] [1]
[insert#1^#](x1, x2) = [0 0 0] x1 + [0]
[0 0 1] [0]
[1]
[insert#2^#](x1, x2, x3, x4) = [0]
[0]
[0]
[leq^#](x1, x2) = [0]
[0]
[0]
[leq#1^#](x1, x2) = [0]
[0]
[1 1 1] [1]
[isortlist^#](x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[1 1 1] [0]
[isortlist#1^#](x1) = [1 1 1] x1 + [0]
[0 0 0] [0]
[0]
[leq#2^#](x1, x2, x3) = [0]
[0]
[0]
[or^#](x1, x2) = [0]
[0]
[0]
[#or^#](x1, x2) = [0]
[0]
[1 0 0] [0]
[c_1](x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[1 0 1] [1 0 0] [0]
[c_2](x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
[0 0 0] [0 0 0] [0]
[1 0 0] [0]
[c_3](x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[1 0 0] [0]
[c_4](x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[1 0 0] [0]
[c_5](x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[1 0 1] [0]
[c_6](x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[1 0 1] [1 0 0] [0]
[c_7](x1, x2) = [1 0 0] x1 + [0 0 0] x2 + [0]
[0 0 0] [0 0 0] [0]
[1 0 0] [0]
[c_8](x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
This order satisfies following ordering constraints
[insert^#(@x, @l)] = [1]
[1]
[0]
>= [1]
[0]
[0]
= [c_1(insert#1^#(@l, @x))]
[insert#1^#(::(@y, @ys), @x)] = [0 0 0] [1]
[0 0 0] @y + [0]
[1 0 0] [0]
>= [1]
[0]
[0]
= [c_2(insert#2^#(leq(@x, @y), @x, @y, @ys), leq^#(@x, @y))]
[insert#2^#(#false(), @x, @y, @ys)] = [1]
[0]
[0]
>= [1]
[0]
[0]
= [c_3(insert^#(@x, @ys))]
[leq^#(@l1, @l2)] = [0]
[0]
[0]
>= [0]
[0]
[0]
= [c_4(leq#1^#(@l1, @l2))]
[leq#1^#(::(@x, @xs), @l2)] = [0]
[0]
[0]
>= [0]
[0]
[0]
= [c_5(leq#2^#(@l2, @x, @xs))]
[isortlist^#(@l)] = [1 1 1] [1]
[0 0 0] @l + [0]
[0 0 0] [0]
> [1 1 1] [0]
[0 0 0] @l + [0]
[0 0 0] [0]
= [c_6(isortlist#1^#(@l))]
[isortlist#1^#(::(@x, @xs))] = [1 1 1] [1 1 1] [2]
[1 1 1] @x + [1 1 1] @xs + [2]
[0 0 0] [0 0 0] [0]
>= [1 1 1] [2]
[0 0 0] @xs + [1]
[0 0 0] [0]
= [c_7(insert^#(@x, isortlist(@xs)), isortlist^#(@xs))]
[leq#2^#(::(@y, @ys), @x, @xs)] = [0]
[0]
[0]
>= [0]
[0]
[0]
= [c_8(leq^#(@xs, @ys))]
Consider the set of all dependency pairs
DPs:
{ 1: insert^#(@x, @l) -> c_1(insert#1^#(@l, @x))
, 2: insert#1^#(::(@y, @ys), @x) ->
c_2(insert#2^#(leq(@x, @y), @x, @y, @ys), leq^#(@x, @y))
, 3: insert#2^#(#false(), @x, @y, @ys) -> c_3(insert^#(@x, @ys))
, 4: leq^#(@l1, @l2) -> c_4(leq#1^#(@l1, @l2))
, 5: leq#1^#(::(@x, @xs), @l2) -> c_5(leq#2^#(@l2, @x, @xs))
, 6: isortlist^#(@l) -> c_6(isortlist#1^#(@l))
, 7: isortlist#1^#(::(@x, @xs)) ->
c_7(insert^#(@x, isortlist(@xs)), isortlist^#(@xs))
, 8: leq#2^#(::(@y, @ys), @x, @xs) -> c_8(leq^#(@xs, @ys)) }
Processor 'matrix interpretation of dimension 3' induces the
complexity certificate YES(?,O(n^3)) on application of dependency
pairs {6}. These cover all (indirect) predecessors of dependency
pairs {6,7}, their number of application is equally bounded. The
dependency pairs are shifted into the corresponding weak
component(s).
We are left with following problem, upon which TcT provides the
certificate MAYBE.
Strict DPs:
{ insert^#(@x, @l) -> c_1(insert#1^#(@l, @x))
, insert#1^#(::(@y, @ys), @x) ->
c_2(insert#2^#(leq(@x, @y), @x, @y, @ys), leq^#(@x, @y))
, insert#2^#(#false(), @x, @y, @ys) -> c_3(insert^#(@x, @ys))
, leq^#(@l1, @l2) -> c_4(leq#1^#(@l1, @l2))
, leq#1^#(::(@x, @xs), @l2) -> c_5(leq#2^#(@l2, @x, @xs))
, leq#2^#(::(@y, @ys), @x, @xs) -> c_8(leq^#(@xs, @ys)) }
Weak DPs:
{ isortlist^#(@l) -> c_6(isortlist#1^#(@l))
, isortlist#1^#(::(@x, @xs)) ->
c_7(insert^#(@x, isortlist(@xs)), isortlist^#(@xs)) }
Weak Trs:
{ #equal(@x, @y) -> #eq(@x, @y)
, #eq(::(@x_1, @x_2), ::(@y_1, @y_2)) ->
#and(#eq(@x_1, @y_1), #eq(@x_2, @y_2))
, #eq(::(@x_1, @x_2), nil()) -> #false()
, #eq(nil(), ::(@y_1, @y_2)) -> #false()
, #eq(nil(), nil()) -> #true()
, #eq(#0(), #0()) -> #true()
, #eq(#0(), #neg(@y)) -> #false()
, #eq(#0(), #pos(@y)) -> #false()
, #eq(#0(), #s(@y)) -> #false()
, #eq(#neg(@x), #0()) -> #false()
, #eq(#neg(@x), #neg(@y)) -> #eq(@x, @y)
, #eq(#neg(@x), #pos(@y)) -> #false()
, #eq(#pos(@x), #0()) -> #false()
, #eq(#pos(@x), #neg(@y)) -> #false()
, #eq(#pos(@x), #pos(@y)) -> #eq(@x, @y)
, #eq(#s(@x), #0()) -> #false()
, #eq(#s(@x), #s(@y)) -> #eq(@x, @y)
, #less(@x, @y) -> #cklt(#compare(@x, @y))
, #compare(#0(), #0()) -> #EQ()
, #compare(#0(), #neg(@y)) -> #GT()
, #compare(#0(), #pos(@y)) -> #LT()
, #compare(#0(), #s(@y)) -> #LT()
, #compare(#neg(@x), #0()) -> #LT()
, #compare(#neg(@x), #neg(@y)) -> #compare(@y, @x)
, #compare(#neg(@x), #pos(@y)) -> #LT()
, #compare(#pos(@x), #0()) -> #GT()
, #compare(#pos(@x), #neg(@y)) -> #GT()
, #compare(#pos(@x), #pos(@y)) -> #compare(@x, @y)
, #compare(#s(@x), #0()) -> #GT()
, #compare(#s(@x), #s(@y)) -> #compare(@x, @y)
, #cklt(#EQ()) -> #false()
, #cklt(#GT()) -> #false()
, #cklt(#LT()) -> #true()
, and(@x, @y) -> #and(@x, @y)
, #and(#false(), #false()) -> #false()
, #and(#false(), #true()) -> #false()
, #and(#true(), #false()) -> #false()
, #and(#true(), #true()) -> #true()
, insert(@x, @l) -> insert#1(@l, @x)
, insert#1(::(@y, @ys), @x) -> insert#2(leq(@x, @y), @x, @y, @ys)
, insert#1(nil(), @x) -> ::(@x, nil())
, leq(@l1, @l2) -> leq#1(@l1, @l2)
, insert#2(#false(), @x, @y, @ys) -> ::(@y, insert(@x, @ys))
, insert#2(#true(), @x, @y, @ys) -> ::(@x, ::(@y, @ys))
, isortlist(@l) -> isortlist#1(@l)
, isortlist#1(::(@x, @xs)) -> insert(@x, isortlist(@xs))
, isortlist#1(nil()) -> nil()
, leq#1(::(@x, @xs), @l2) -> leq#2(@l2, @x, @xs)
, leq#1(nil(), @l2) -> #true()
, leq#2(::(@y, @ys), @x, @xs) ->
or(#less(@x, @y), and(#equal(@x, @y), leq(@xs, @ys)))
, leq#2(nil(), @x, @xs) -> #false()
, or(@x, @y) -> #or(@x, @y)
, #or(#false(), #false()) -> #false()
, #or(#false(), #true()) -> #true()
, #or(#true(), #false()) -> #true()
, #or(#true(), #true()) -> #true() }
Obligation:
innermost runtime complexity
Answer:
MAYBE
We use the processor 'matrix interpretation of dimension 3' to
orient following rules strictly.
DPs:
{ 3: insert#2^#(#false(), @x, @y, @ys) -> c_3(insert^#(@x, @ys)) }
Trs: { isortlist#1(nil()) -> nil() }
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_7) = {1, 2}, Uargs(c_8) = {1}
TcT has computed following constructor-based matrix interpretation
satisfying not(EDA).
[0]
[#equal](x1, x2) = [0]
[0]
[0]
[#eq](x1, x2) = [0]
[0]
[1]
[#less](x1, x2) = [0]
[0]
[0]
[#compare](x1, x2) = [0]
[0]
[1]
[#cklt](x1) = [0]
[0]
[0]
[and](x1, x2) = [0]
[0]
[0]
[#and](x1, x2) = [0]
[0]
[1 0 1] [1 0 1] [1]
[insert](x1, x2) = [0 0 0] x1 + [0 1 0] x2 + [1]
[0 0 0] [0 1 0] [0]
[1 0 1] [1 0 1] [1]
[insert#1](x1, x2) = [0 1 0] x1 + [0 0 0] x2 + [1]
[0 1 0] [0 0 0] [0]
[1 0 1] [1 0 1] [1]
[::](x1, x2) = [0 0 0] x1 + [0 1 0] x2 + [1]
[0 0 0] [0 1 0] [0]
[1]
[leq](x1, x2) = [0]
[0]
[1 1 1] [1 0 1] [1 0
1] [1 1 1] [1]
[insert#2](x1, x2, x3, x4) = [1 0 0] x1 + [0 0 0] x2 + [0 0
0] x3 + [0 1 0] x4 + [1]
[1 0 0] [0 0 0] [0 0
0] [0 1 0] [0]
[1]
[nil] = [1]
[1]
[1]
[#false] = [0]
[0]
[1]
[#true] = [0]
[0]
[1 0 1] [0]
[isortlist](x1) = [0 1 0] x1 + [1]
[0 1 0] [0]
[1 0 1] [0]
[isortlist#1](x1) = [0 1 0] x1 + [1]
[0 1 0] [0]
[1]
[leq#1](x1, x2) = [0]
[0]
[1]
[leq#2](x1, x2, x3) = [0]
[0]
[0 0 0] [1]
[or](x1, x2) = [0 1 0] x1 + [0]
[0 1 0] [0]
[0 0 0] [1]
[#or](x1, x2) = [0 1 0] x1 + [0]
[0 1 0] [0]
[0]
[#EQ] = [0]
[0]
[0]
[#GT] = [0]
[0]
[0]
[#LT] = [0]
[0]
[0]
[#0] = [0]
[0]
[1 1 0] [0]
[#neg](x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[1 0 1] [0]
[#pos](x1) = [0 1 0] x1 + [0]
[0 1 0] [0]
[1 0 0] [0]
[#s](x1) = [0 0 0] x1 + [0]
[0 0 1] [0]
[0]
[#equal^#](x1, x2) = [0]
[0]
[0]
[#eq^#](x1, x2) = [0]
[0]
[0]
[#less^#](x1, x2) = [0]
[0]
[0]
[#cklt^#](x1) = [0]
[0]
[0]
[#compare^#](x1, x2) = [0]
[0]
[0]
[and^#](x1, x2) = [0]
[0]
[0]
[#and^#](x1, x2) = [0]
[0]
[1 0 0] [0 1 0] [0]
[insert^#](x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
[0 0 0] [0 0 0] [0]
[0 1 0] [1 0 0] [0]
[insert#1^#](x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [1]
[0 0 0] [0 1 1] [1]
[1 1 0] [1 0 0] [0 1
0] [0]
[insert#2^#](x1, x2, x3, x4) = [0 0 0] x1 + [0 1 1] x2 + [0 0
0] x4 + [0]
[0 1 0] [0 0 0] [0 0
0] [0]
[0]
[leq^#](x1, x2) = [1]
[1]
[0]
[leq#1^#](x1, x2) = [1]
[1]
[1 0 1] [0]
[isortlist^#](x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[1 0 1] [0]
[isortlist#1^#](x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0]
[leq#2^#](x1, x2, x3) = [0]
[1]
[0]
[or^#](x1, x2) = [0]
[0]
[0]
[#or^#](x1, x2) = [0]
[0]
[1 0 0] [0]
[c_1](x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[1 0 1] [1 0 0] [0]
[c_2](x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
[0 1 1] [0 0 0] [0]
[1 0 0] [0]
[c_3](x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[1 0 0] [0]
[c_4](x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[1 0 0] [0]
[c_5](x1) = [0 0 0] x1 + [0]
[1 0 0] [0]
[1 0 0] [0]
[c_6](x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[1 1 0] [1 0 0] [0]
[c_7](x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
[0 0 0] [0 0 0] [0]
[1 0 0] [0]
[c_8](x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
This order satisfies following ordering constraints
[#less(@x, @y)] = [1]
[0]
[0]
>= [1]
[0]
[0]
= [#cklt(#compare(@x, @y))]
[#cklt(#EQ())] = [1]
[0]
[0]
>= [1]
[0]
[0]
= [#false()]
[#cklt(#GT())] = [1]
[0]
[0]
>= [1]
[0]
[0]
= [#false()]
[#cklt(#LT())] = [1]
[0]
[0]
>= [1]
[0]
[0]
= [#true()]
[insert(@x, @l)] = [1 0 1] [1 0 1] [1]
[0 1 0] @l + [0 0 0] @x + [1]
[0 1 0] [0 0 0] [0]
>= [1 0 1] [1 0 1] [1]
[0 1 0] @l + [0 0 0] @x + [1]
[0 1 0] [0 0 0] [0]
= [insert#1(@l, @x)]
[insert#1(::(@y, @ys), @x)] = [1 0 1] [1 0 1] [1 1 1] [2]
[0 0 0] @x + [0 0 0] @y + [0 1 0] @ys + [2]
[0 0 0] [0 0 0] [0 1 0] [1]
>= [1 0 1] [1 0 1] [1 1 1] [2]
[0 0 0] @x + [0 0 0] @y + [0 1 0] @ys + [2]
[0 0 0] [0 0 0] [0 1 0] [1]
= [insert#2(leq(@x, @y), @x, @y, @ys)]
[insert#1(nil(), @x)] = [1 0 1] [3]
[0 0 0] @x + [2]
[0 0 0] [1]
>= [1 0 1] [3]
[0 0 0] @x + [2]
[0 0 0] [1]
= [::(@x, nil())]
[leq(@l1, @l2)] = [1]
[0]
[0]
>= [1]
[0]
[0]
= [leq#1(@l1, @l2)]
[insert#2(#false(), @x, @y, @ys)] = [1 0 1] [1 0 1] [1 1 1] [2]
[0 0 0] @x + [0 0 0] @y + [0 1 0] @ys + [2]
[0 0 0] [0 0 0] [0 1 0] [1]
>= [1 0 1] [1 0 1] [1 1 1] [2]
[0 0 0] @x + [0 0 0] @y + [0 1 0] @ys + [2]
[0 0 0] [0 0 0] [0 1 0] [1]
= [::(@y, insert(@x, @ys))]
[insert#2(#true(), @x, @y, @ys)] = [1 0 1] [1 0 1] [1 1 1] [2]
[0 0 0] @x + [0 0 0] @y + [0 1 0] @ys + [2]
[0 0 0] [0 0 0] [0 1 0] [1]
>= [1 0 1] [1 0 1] [1 1 1] [2]
[0 0 0] @x + [0 0 0] @y + [0 1 0] @ys + [2]
[0 0 0] [0 0 0] [0 1 0] [1]
= [::(@x, ::(@y, @ys))]
[isortlist(@l)] = [1 0 1] [0]
[0 1 0] @l + [1]
[0 1 0] [0]
>= [1 0 1] [0]
[0 1 0] @l + [1]
[0 1 0] [0]
= [isortlist#1(@l)]
[isortlist#1(::(@x, @xs))] = [1 0 1] [1 1 1] [1]
[0 0 0] @x + [0 1 0] @xs + [2]
[0 0 0] [0 1 0] [1]
>= [1 0 1] [1 1 1] [1]
[0 0 0] @x + [0 1 0] @xs + [2]
[0 0 0] [0 1 0] [1]
= [insert(@x, isortlist(@xs))]
[isortlist#1(nil())] = [2]
[2]
[1]
> [1]
[1]
[1]
= [nil()]
[leq#1(::(@x, @xs), @l2)] = [1]
[0]
[0]
>= [1]
[0]
[0]
= [leq#2(@l2, @x, @xs)]
[leq#1(nil(), @l2)] = [1]
[0]
[0]
>= [1]
[0]
[0]
= [#true()]
[leq#2(::(@y, @ys), @x, @xs)] = [1]
[0]
[0]
>= [1]
[0]
[0]
= [or(#less(@x, @y), and(#equal(@x, @y), leq(@xs, @ys)))]
[leq#2(nil(), @x, @xs)] = [1]
[0]
[0]
>= [1]
[0]
[0]
= [#false()]
[or(@x, @y)] = [0 0 0] [1]
[0 1 0] @x + [0]
[0 1 0] [0]
>= [0 0 0] [1]
[0 1 0] @x + [0]
[0 1 0] [0]
= [#or(@x, @y)]
[#or(#false(), #false())] = [1]
[0]
[0]
>= [1]
[0]
[0]
= [#false()]
[#or(#false(), #true())] = [1]
[0]
[0]
>= [1]
[0]
[0]
= [#true()]
[#or(#true(), #false())] = [1]
[0]
[0]
>= [1]
[0]
[0]
= [#true()]
[#or(#true(), #true())] = [1]
[0]
[0]
>= [1]
[0]
[0]
= [#true()]
[insert^#(@x, @l)] = [0 1 0] [1 0 0] [0]
[0 0 0] @l + [0 0 0] @x + [0]
[0 0 0] [0 0 0] [0]
>= [0 1 0] [1 0 0] [0]
[0 0 0] @l + [0 0 0] @x + [0]
[0 0 0] [0 0 0] [0]
= [c_1(insert#1^#(@l, @x))]
[insert#1^#(::(@y, @ys), @x)] = [1 0 0] [0 1 0] [1]
[0 0 0] @x + [0 0 0] @ys + [1]
[0 1 1] [0 0 0] [1]
>= [1 0 0] [0 1 0] [1]
[0 0 0] @x + [0 0 0] @ys + [0]
[0 1 1] [0 0 0] [0]
= [c_2(insert#2^#(leq(@x, @y), @x, @y, @ys), leq^#(@x, @y))]
[insert#2^#(#false(), @x, @y, @ys)] = [1 0 0] [0 1 0] [1]
[0 1 1] @x + [0 0 0] @ys + [0]
[0 0 0] [0 0 0] [0]
> [1 0 0] [0 1 0] [0]
[0 0 0] @x + [0 0 0] @ys + [0]
[0 0 0] [0 0 0] [0]
= [c_3(insert^#(@x, @ys))]
[leq^#(@l1, @l2)] = [0]
[1]
[1]
>= [0]
[0]
[0]
= [c_4(leq#1^#(@l1, @l2))]
[leq#1^#(::(@x, @xs), @l2)] = [0]
[1]
[1]
>= [0]
[0]
[0]
= [c_5(leq#2^#(@l2, @x, @xs))]
[isortlist^#(@l)] = [1 0 1] [0]
[0 0 0] @l + [0]
[0 0 0] [0]
>= [1 0 1] [0]
[0 0 0] @l + [0]
[0 0 0] [0]
= [c_6(isortlist#1^#(@l))]
[isortlist#1^#(::(@x, @xs))] = [1 0 1] [1 1 1] [1]
[0 0 0] @x + [0 0 0] @xs + [0]
[0 0 0] [0 0 0] [0]
>= [1 0 0] [1 1 1] [1]
[0 0 0] @x + [0 0 0] @xs + [0]
[0 0 0] [0 0 0] [0]
= [c_7(insert^#(@x, isortlist(@xs)), isortlist^#(@xs))]
[leq#2^#(::(@y, @ys), @x, @xs)] = [0]
[0]
[1]
>= [0]
[0]
[0]
= [c_8(leq^#(@xs, @ys))]
The strictly oriented rules are moved into the corresponding weak
component(s).
We are left with following problem, upon which TcT provides the
certificate MAYBE.
Strict DPs:
{ insert^#(@x, @l) -> c_1(insert#1^#(@l, @x))
, insert#1^#(::(@y, @ys), @x) ->
c_2(insert#2^#(leq(@x, @y), @x, @y, @ys), leq^#(@x, @y))
, leq^#(@l1, @l2) -> c_4(leq#1^#(@l1, @l2))
, leq#1^#(::(@x, @xs), @l2) -> c_5(leq#2^#(@l2, @x, @xs))
, leq#2^#(::(@y, @ys), @x, @xs) -> c_8(leq^#(@xs, @ys)) }
Weak DPs:
{ insert#2^#(#false(), @x, @y, @ys) -> c_3(insert^#(@x, @ys))
, isortlist^#(@l) -> c_6(isortlist#1^#(@l))
, isortlist#1^#(::(@x, @xs)) ->
c_7(insert^#(@x, isortlist(@xs)), isortlist^#(@xs)) }
Weak Trs:
{ #equal(@x, @y) -> #eq(@x, @y)
, #eq(::(@x_1, @x_2), ::(@y_1, @y_2)) ->
#and(#eq(@x_1, @y_1), #eq(@x_2, @y_2))
, #eq(::(@x_1, @x_2), nil()) -> #false()
, #eq(nil(), ::(@y_1, @y_2)) -> #false()
, #eq(nil(), nil()) -> #true()
, #eq(#0(), #0()) -> #true()
, #eq(#0(), #neg(@y)) -> #false()
, #eq(#0(), #pos(@y)) -> #false()
, #eq(#0(), #s(@y)) -> #false()
, #eq(#neg(@x), #0()) -> #false()
, #eq(#neg(@x), #neg(@y)) -> #eq(@x, @y)
, #eq(#neg(@x), #pos(@y)) -> #false()
, #eq(#pos(@x), #0()) -> #false()
, #eq(#pos(@x), #neg(@y)) -> #false()
, #eq(#pos(@x), #pos(@y)) -> #eq(@x, @y)
, #eq(#s(@x), #0()) -> #false()
, #eq(#s(@x), #s(@y)) -> #eq(@x, @y)
, #less(@x, @y) -> #cklt(#compare(@x, @y))
, #compare(#0(), #0()) -> #EQ()
, #compare(#0(), #neg(@y)) -> #GT()
, #compare(#0(), #pos(@y)) -> #LT()
, #compare(#0(), #s(@y)) -> #LT()
, #compare(#neg(@x), #0()) -> #LT()
, #compare(#neg(@x), #neg(@y)) -> #compare(@y, @x)
, #compare(#neg(@x), #pos(@y)) -> #LT()
, #compare(#pos(@x), #0()) -> #GT()
, #compare(#pos(@x), #neg(@y)) -> #GT()
, #compare(#pos(@x), #pos(@y)) -> #compare(@x, @y)
, #compare(#s(@x), #0()) -> #GT()
, #compare(#s(@x), #s(@y)) -> #compare(@x, @y)
, #cklt(#EQ()) -> #false()
, #cklt(#GT()) -> #false()
, #cklt(#LT()) -> #true()
, and(@x, @y) -> #and(@x, @y)
, #and(#false(), #false()) -> #false()
, #and(#false(), #true()) -> #false()
, #and(#true(), #false()) -> #false()
, #and(#true(), #true()) -> #true()
, insert(@x, @l) -> insert#1(@l, @x)
, insert#1(::(@y, @ys), @x) -> insert#2(leq(@x, @y), @x, @y, @ys)
, insert#1(nil(), @x) -> ::(@x, nil())
, leq(@l1, @l2) -> leq#1(@l1, @l2)
, insert#2(#false(), @x, @y, @ys) -> ::(@y, insert(@x, @ys))
, insert#2(#true(), @x, @y, @ys) -> ::(@x, ::(@y, @ys))
, isortlist(@l) -> isortlist#1(@l)
, isortlist#1(::(@x, @xs)) -> insert(@x, isortlist(@xs))
, isortlist#1(nil()) -> nil()
, leq#1(::(@x, @xs), @l2) -> leq#2(@l2, @x, @xs)
, leq#1(nil(), @l2) -> #true()
, leq#2(::(@y, @ys), @x, @xs) ->
or(#less(@x, @y), and(#equal(@x, @y), leq(@xs, @ys)))
, leq#2(nil(), @x, @xs) -> #false()
, or(@x, @y) -> #or(@x, @y)
, #or(#false(), #false()) -> #false()
, #or(#false(), #true()) -> #true()
, #or(#true(), #false()) -> #true()
, #or(#true(), #true()) -> #true() }
Obligation:
innermost runtime complexity
Answer:
MAYBE
We use the processor 'matrix interpretation of dimension 3' to
orient following rules strictly.
DPs:
{ 1: insert^#(@x, @l) -> c_1(insert#1^#(@l, @x)) }
Trs:
{ #eq(::(@x_1, @x_2), ::(@y_1, @y_2)) ->
#and(#eq(@x_1, @y_1), #eq(@x_2, @y_2))
, #eq(#s(@x), #s(@y)) -> #eq(@x, @y)
, isortlist#1(nil()) -> nil() }
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_7) = {1, 2}, Uargs(c_8) = {1}
TcT has computed following constructor-based matrix interpretation
satisfying not(EDA).
[0]
[#equal](x1, x2) = [0]
[0]
[0 0 1] [0 0 0] [0]
[#eq](x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
[0 0 0] [0 1 0] [0]
[1]
[#less](x1, x2) = [0]
[1]
[1]
[#compare](x1, x2) = [0]
[0]
[1 1 1] [0]
[#cklt](x1) = [0 0 0] x1 + [0]
[0 0 0] [1]
[0]
[and](x1, x2) = [0]
[0]
[0]
[#and](x1, x2) = [0]
[0]
[0 0 0] [1 0 0] [1]
[insert](x1, x2) = [1 0 0] x1 + [0 1 1] x2 + [0]
[0 0 0] [0 0 1] [1]
[1 0 0] [0 0 0] [1]
[insert#1](x1, x2) = [0 1 1] x1 + [1 0 0] x2 + [0]
[0 0 1] [0 0 0] [1]
[0 0 0] [0 0 1] [0]
[::](x1, x2) = [1 0 0] x1 + [1 1 0] x2 + [1]
[0 0 0] [0 0 1] [1]
[0 0 0] [0 0 0] [1]
[leq](x1, x2) = [1 0 1] x1 + [0 0 0] x2 + [0]
[1 1 0] [0 1 0] [1]
[0 0 0] [0 0 0] [0 0
0] [0 0 1] [1]
[insert#2](x1, x2, x3, x4) = [1 0 0] x1 + [1 0 0] x2 + [1 0
0] x3 + [1 1 1] x4 + [1]
[1 0 0] [0 0 0] [0 0
0] [0 0 1] [1]
[0]
[nil] = [1]
[1]
[1]
[#false] = [0]
[1]
[1]
[#true] = [0]
[0]
[1 1 0] [1]
[isortlist](x1) = [1 1 0] x1 + [0]
[0 0 1] [0]
[1 1 0] [1]
[isortlist#1](x1) = [1 1 0] x1 + [0]
[0 0 1] [0]
[0 0 0] [0 0 0] [1]
[leq#1](x1, x2) = [0 0 1] x1 + [0 0 0] x2 + [0]
[0 0 0] [0 1 0] [1]
[1]
[leq#2](x1, x2, x3) = [1]
[1]
[1 0 0] [0]
[or](x1, x2) = [1 1 0] x1 + [0]
[1 0 0] [0]
[1 0 0] [0]
[#or](x1, x2) = [0 0 0] x1 + [0]
[1 0 0] [0]
[1]
[#EQ] = [0]
[0]
[1]
[#GT] = [0]
[0]
[1]
[#LT] = [0]
[0]
[0]
[#0] = [0]
[0]
[1 0 0] [1]
[#neg](x1) = [1 1 0] x1 + [0]
[1 0 0] [0]
[1 0 0] [0]
[#pos](x1) = [1 1 0] x1 + [0]
[0 0 1] [0]
[1 0 0] [0]
[#s](x1) = [0 1 0] x1 + [0]
[0 0 1] [1]
[0]
[#equal^#](x1, x2) = [0]
[0]
[0]
[#eq^#](x1, x2) = [0]
[0]
[0]
[#less^#](x1, x2) = [0]
[0]
[0]
[#cklt^#](x1) = [0]
[0]
[0]
[#compare^#](x1, x2) = [0]
[0]
[0]
[and^#](x1, x2) = [0]
[0]
[0]
[#and^#](x1, x2) = [0]
[0]
[0 0 1] [1]
[insert^#](x1, x2) = [0 0 0] x2 + [0]
[0 0 0] [0]
[0 0 1] [0]
[insert#1^#](x1, x2) = [0 0 0] x1 + [0]
[0 0 0] [1]
[1 0 0] [0 0 1] [0]
[insert#2^#](x1, x2, x3, x4) = [0 0 0] x1 + [0 0 0] x4 + [0]
[0 0 0] [0 0 0] [0]
[0]
[leq^#](x1, x2) = [1]
[1]
[0]
[leq#1^#](x1, x2) = [0]
[0]
[1 1 0] [0]
[isortlist^#](x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[1 1 0] [0]
[isortlist#1^#](x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0]
[leq#2^#](x1, x2, x3) = [0]
[1]
[0]
[or^#](x1, x2) = [0]
[0]
[0]
[#or^#](x1, x2) = [0]
[0]
[1 1 0] [0]
[c_1](x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[1 1 1] [1 0 0] [0]
[c_2](x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
[0 0 0] [0 0 0] [0]
[1 0 0] [0]
[c_3](x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[1 1 1] [0]
[c_4](x1) = [0 0 0] x1 + [1]
[0 0 0] [1]
[1 1 0] [0]
[c_5](x1) = [0 0 0] x1 + [0]
[1 0 0] [0]
[1 0 0] [0]
[c_6](x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[1 1 1] [1 0 0] [0]
[c_7](x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
[0 0 0] [0 0 0] [0]
[1 0 0] [0]
[c_8](x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
This order satisfies following ordering constraints
[#less(@x, @y)] = [1]
[0]
[1]
>= [1]
[0]
[1]
= [#cklt(#compare(@x, @y))]
[#compare(#0(), #0())] = [1]
[0]
[0]
>= [1]
[0]
[0]
= [#EQ()]
[#compare(#0(), #neg(@y))] = [1]
[0]
[0]
>= [1]
[0]
[0]
= [#GT()]
[#compare(#0(), #pos(@y))] = [1]
[0]
[0]
>= [1]
[0]
[0]
= [#LT()]
[#compare(#0(), #s(@y))] = [1]
[0]
[0]
>= [1]
[0]
[0]
= [#LT()]
[#compare(#neg(@x), #0())] = [1]
[0]
[0]
>= [1]
[0]
[0]
= [#LT()]
[#compare(#neg(@x), #neg(@y))] = [1]
[0]
[0]
>= [1]
[0]
[0]
= [#compare(@y, @x)]
[#compare(#neg(@x), #pos(@y))] = [1]
[0]
[0]
>= [1]
[0]
[0]
= [#LT()]
[#compare(#pos(@x), #0())] = [1]
[0]
[0]
>= [1]
[0]
[0]
= [#GT()]
[#compare(#pos(@x), #neg(@y))] = [1]
[0]
[0]
>= [1]
[0]
[0]
= [#GT()]
[#compare(#pos(@x), #pos(@y))] = [1]
[0]
[0]
>= [1]
[0]
[0]
= [#compare(@x, @y)]
[#compare(#s(@x), #0())] = [1]
[0]
[0]
>= [1]
[0]
[0]
= [#GT()]
[#compare(#s(@x), #s(@y))] = [1]
[0]
[0]
>= [1]
[0]
[0]
= [#compare(@x, @y)]
[#cklt(#EQ())] = [1]
[0]
[1]
>= [1]
[0]
[1]
= [#false()]
[#cklt(#GT())] = [1]
[0]
[1]
>= [1]
[0]
[1]
= [#false()]
[#cklt(#LT())] = [1]
[0]
[1]
>= [1]
[0]
[0]
= [#true()]
[insert(@x, @l)] = [1 0 0] [0 0 0] [1]
[0 1 1] @l + [1 0 0] @x + [0]
[0 0 1] [0 0 0] [1]
>= [1 0 0] [0 0 0] [1]
[0 1 1] @l + [1 0 0] @x + [0]
[0 0 1] [0 0 0] [1]
= [insert#1(@l, @x)]
[insert#1(::(@y, @ys), @x)] = [0 0 0] [0 0 0] [0 0 1] [1]
[1 0 0] @x + [1 0 0] @y + [1 1 1] @ys + [2]
[0 0 0] [0 0 0] [0 0 1] [2]
>= [0 0 0] [0 0 0] [0 0 1] [1]
[1 0 0] @x + [1 0 0] @y + [1 1 1] @ys + [2]
[0 0 0] [0 0 0] [0 0 1] [2]
= [insert#2(leq(@x, @y), @x, @y, @ys)]
[insert#1(nil(), @x)] = [0 0 0] [1]
[1 0 0] @x + [2]
[0 0 0] [2]
>= [0 0 0] [1]
[1 0 0] @x + [2]
[0 0 0] [2]
= [::(@x, nil())]
[leq(@l1, @l2)] = [0 0 0] [0 0 0] [1]
[1 0 1] @l1 + [0 0 0] @l2 + [0]
[1 1 0] [0 1 0] [1]
>= [0 0 0] [0 0 0] [1]
[0 0 1] @l1 + [0 0 0] @l2 + [0]
[0 0 0] [0 1 0] [1]
= [leq#1(@l1, @l2)]
[insert#2(#false(), @x, @y, @ys)] = [0 0 0] [0 0 0] [0 0 1] [1]
[1 0 0] @x + [1 0 0] @y + [1 1 1] @ys + [2]
[0 0 0] [0 0 0] [0 0 1] [2]
>= [0 0 0] [0 0 0] [0 0 1] [1]
[1 0 0] @x + [1 0 0] @y + [1 1 1] @ys + [2]
[0 0 0] [0 0 0] [0 0 1] [2]
= [::(@y, insert(@x, @ys))]
[insert#2(#true(), @x, @y, @ys)] = [0 0 0] [0 0 0] [0 0 1] [1]
[1 0 0] @x + [1 0 0] @y + [1 1 1] @ys + [2]
[0 0 0] [0 0 0] [0 0 1] [2]
>= [0 0 0] [0 0 0] [0 0 1] [1]
[1 0 0] @x + [1 0 0] @y + [1 1 1] @ys + [2]
[0 0 0] [0 0 0] [0 0 1] [2]
= [::(@x, ::(@y, @ys))]
[isortlist(@l)] = [1 1 0] [1]
[1 1 0] @l + [0]
[0 0 1] [0]
>= [1 1 0] [1]
[1 1 0] @l + [0]
[0 0 1] [0]
= [isortlist#1(@l)]
[isortlist#1(::(@x, @xs))] = [1 0 0] [1 1 1] [2]
[1 0 0] @x + [1 1 1] @xs + [1]
[0 0 0] [0 0 1] [1]
>= [0 0 0] [1 1 0] [2]
[1 0 0] @x + [1 1 1] @xs + [0]
[0 0 0] [0 0 1] [1]
= [insert(@x, isortlist(@xs))]
[isortlist#1(nil())] = [2]
[1]
[1]
> [0]
[1]
[1]
= [nil()]
[leq#1(::(@x, @xs), @l2)] = [0 0 0] [0 0 0] [1]
[0 0 0] @l2 + [0 0 1] @xs + [1]
[0 1 0] [0 0 0] [1]
>= [1]
[1]
[1]
= [leq#2(@l2, @x, @xs)]
[leq#1(nil(), @l2)] = [0 0 0] [1]
[0 0 0] @l2 + [1]
[0 1 0] [1]
>= [1]
[0]
[0]
= [#true()]
[leq#2(::(@y, @ys), @x, @xs)] = [1]
[1]
[1]
>= [1]
[1]
[1]
= [or(#less(@x, @y), and(#equal(@x, @y), leq(@xs, @ys)))]
[leq#2(nil(), @x, @xs)] = [1]
[1]
[1]
>= [1]
[0]
[1]
= [#false()]
[or(@x, @y)] = [1 0 0] [0]
[1 1 0] @x + [0]
[1 0 0] [0]
>= [1 0 0] [0]
[0 0 0] @x + [0]
[1 0 0] [0]
= [#or(@x, @y)]
[#or(#false(), #false())] = [1]
[0]
[1]
>= [1]
[0]
[1]
= [#false()]
[#or(#false(), #true())] = [1]
[0]
[1]
>= [1]
[0]
[0]
= [#true()]
[#or(#true(), #false())] = [1]
[0]
[1]
>= [1]
[0]
[0]
= [#true()]
[#or(#true(), #true())] = [1]
[0]
[1]
>= [1]
[0]
[0]
= [#true()]
[insert^#(@x, @l)] = [0 0 1] [1]
[0 0 0] @l + [0]
[0 0 0] [0]
> [0 0 1] [0]
[0 0 0] @l + [0]
[0 0 0] [0]
= [c_1(insert#1^#(@l, @x))]
[insert#1^#(::(@y, @ys), @x)] = [0 0 1] [1]
[0 0 0] @ys + [0]
[0 0 0] [1]
>= [0 0 1] [1]
[0 0 0] @ys + [0]
[0 0 0] [0]
= [c_2(insert#2^#(leq(@x, @y), @x, @y, @ys), leq^#(@x, @y))]
[insert#2^#(#false(), @x, @y, @ys)] = [0 0 1] [1]
[0 0 0] @ys + [0]
[0 0 0] [0]
>= [0 0 1] [1]
[0 0 0] @ys + [0]
[0 0 0] [0]
= [c_3(insert^#(@x, @ys))]
[leq^#(@l1, @l2)] = [0]
[1]
[1]
>= [0]
[1]
[1]
= [c_4(leq#1^#(@l1, @l2))]
[leq#1^#(::(@x, @xs), @l2)] = [0]
[0]
[0]
>= [0]
[0]
[0]
= [c_5(leq#2^#(@l2, @x, @xs))]
[isortlist^#(@l)] = [1 1 0] [0]
[0 0 0] @l + [0]
[0 0 0] [0]
>= [1 1 0] [0]
[0 0 0] @l + [0]
[0 0 0] [0]
= [c_6(isortlist#1^#(@l))]
[isortlist#1^#(::(@x, @xs))] = [1 0 0] [1 1 1] [1]
[0 0 0] @x + [0 0 0] @xs + [0]
[0 0 0] [0 0 0] [0]
>= [1 1 1] [1]
[0 0 0] @xs + [0]
[0 0 0] [0]
= [c_7(insert^#(@x, isortlist(@xs)), isortlist^#(@xs))]
[leq#2^#(::(@y, @ys), @x, @xs)] = [0]
[0]
[1]
>= [0]
[0]
[0]
= [c_8(leq^#(@xs, @ys))]
Consider the set of all dependency pairs
DPs:
{ 1: insert^#(@x, @l) -> c_1(insert#1^#(@l, @x))
, 2: insert#1^#(::(@y, @ys), @x) ->
c_2(insert#2^#(leq(@x, @y), @x, @y, @ys), leq^#(@x, @y))
, 3: leq^#(@l1, @l2) -> c_4(leq#1^#(@l1, @l2))
, 4: leq#1^#(::(@x, @xs), @l2) -> c_5(leq#2^#(@l2, @x, @xs))
, 5: leq#2^#(::(@y, @ys), @x, @xs) -> c_8(leq^#(@xs, @ys))
, 6: insert#2^#(#false(), @x, @y, @ys) -> c_3(insert^#(@x, @ys))
, 7: isortlist^#(@l) -> c_6(isortlist#1^#(@l))
, 8: isortlist#1^#(::(@x, @xs)) ->
c_7(insert^#(@x, isortlist(@xs)), isortlist^#(@xs)) }
Processor 'matrix interpretation of dimension 3' induces the
complexity certificate YES(?,O(n^3)) on application of dependency
pairs {1}. These cover all (indirect) predecessors of dependency
pairs {1,2,6}, their number of application is equally bounded. The
dependency pairs are shifted into the corresponding weak
component(s).
We are left with following problem, upon which TcT provides the
certificate MAYBE.
Strict DPs:
{ leq^#(@l1, @l2) -> c_4(leq#1^#(@l1, @l2))
, leq#1^#(::(@x, @xs), @l2) -> c_5(leq#2^#(@l2, @x, @xs))
, leq#2^#(::(@y, @ys), @x, @xs) -> c_8(leq^#(@xs, @ys)) }
Weak DPs:
{ insert^#(@x, @l) -> c_1(insert#1^#(@l, @x))
, insert#1^#(::(@y, @ys), @x) ->
c_2(insert#2^#(leq(@x, @y), @x, @y, @ys), leq^#(@x, @y))
, insert#2^#(#false(), @x, @y, @ys) -> c_3(insert^#(@x, @ys))
, isortlist^#(@l) -> c_6(isortlist#1^#(@l))
, isortlist#1^#(::(@x, @xs)) ->
c_7(insert^#(@x, isortlist(@xs)), isortlist^#(@xs)) }
Weak Trs:
{ #equal(@x, @y) -> #eq(@x, @y)
, #eq(::(@x_1, @x_2), ::(@y_1, @y_2)) ->
#and(#eq(@x_1, @y_1), #eq(@x_2, @y_2))
, #eq(::(@x_1, @x_2), nil()) -> #false()
, #eq(nil(), ::(@y_1, @y_2)) -> #false()
, #eq(nil(), nil()) -> #true()
, #eq(#0(), #0()) -> #true()
, #eq(#0(), #neg(@y)) -> #false()
, #eq(#0(), #pos(@y)) -> #false()
, #eq(#0(), #s(@y)) -> #false()
, #eq(#neg(@x), #0()) -> #false()
, #eq(#neg(@x), #neg(@y)) -> #eq(@x, @y)
, #eq(#neg(@x), #pos(@y)) -> #false()
, #eq(#pos(@x), #0()) -> #false()
, #eq(#pos(@x), #neg(@y)) -> #false()
, #eq(#pos(@x), #pos(@y)) -> #eq(@x, @y)
, #eq(#s(@x), #0()) -> #false()
, #eq(#s(@x), #s(@y)) -> #eq(@x, @y)
, #less(@x, @y) -> #cklt(#compare(@x, @y))
, #compare(#0(), #0()) -> #EQ()
, #compare(#0(), #neg(@y)) -> #GT()
, #compare(#0(), #pos(@y)) -> #LT()
, #compare(#0(), #s(@y)) -> #LT()
, #compare(#neg(@x), #0()) -> #LT()
, #compare(#neg(@x), #neg(@y)) -> #compare(@y, @x)
, #compare(#neg(@x), #pos(@y)) -> #LT()
, #compare(#pos(@x), #0()) -> #GT()
, #compare(#pos(@x), #neg(@y)) -> #GT()
, #compare(#pos(@x), #pos(@y)) -> #compare(@x, @y)
, #compare(#s(@x), #0()) -> #GT()
, #compare(#s(@x), #s(@y)) -> #compare(@x, @y)
, #cklt(#EQ()) -> #false()
, #cklt(#GT()) -> #false()
, #cklt(#LT()) -> #true()
, and(@x, @y) -> #and(@x, @y)
, #and(#false(), #false()) -> #false()
, #and(#false(), #true()) -> #false()
, #and(#true(), #false()) -> #false()
, #and(#true(), #true()) -> #true()
, insert(@x, @l) -> insert#1(@l, @x)
, insert#1(::(@y, @ys), @x) -> insert#2(leq(@x, @y), @x, @y, @ys)
, insert#1(nil(), @x) -> ::(@x, nil())
, leq(@l1, @l2) -> leq#1(@l1, @l2)
, insert#2(#false(), @x, @y, @ys) -> ::(@y, insert(@x, @ys))
, insert#2(#true(), @x, @y, @ys) -> ::(@x, ::(@y, @ys))
, isortlist(@l) -> isortlist#1(@l)
, isortlist#1(::(@x, @xs)) -> insert(@x, isortlist(@xs))
, isortlist#1(nil()) -> nil()
, leq#1(::(@x, @xs), @l2) -> leq#2(@l2, @x, @xs)
, leq#1(nil(), @l2) -> #true()
, leq#2(::(@y, @ys), @x, @xs) ->
or(#less(@x, @y), and(#equal(@x, @y), leq(@xs, @ys)))
, leq#2(nil(), @x, @xs) -> #false()
, or(@x, @y) -> #or(@x, @y)
, #or(#false(), #false()) -> #false()
, #or(#false(), #true()) -> #true()
, #or(#true(), #false()) -> #true()
, #or(#true(), #true()) -> #true() }
Obligation:
innermost runtime complexity
Answer:
MAYBE
None of the processors succeeded.
Details of failed attempt(s):
-----------------------------
1) 'empty' failed due to the following reason:
Empty strict component of the problem is NOT empty.
2) 'Sequentially' failed due to the following reason:
None of the processors succeeded.
Details of failed attempt(s):
-----------------------------
1) 'Polynomial Path Order (PS)' failed due to the following reason:
The input cannot be shown compatible
2) 'Polynomial Path Order' failed due to the following reason:
The input cannot be shown compatible
2) 'Fastest (timeout of 5 seconds)' failed due to the following
reason:
None of the processors succeeded.
Details of failed attempt(s):
-----------------------------
1) 'Bounds with perSymbol-enrichment and initial automaton 'match''
failed due to the following reason:
match-boundness of the problem could not be verified.
2) 'Bounds with minimal-enrichment and initial automaton 'match''
failed due to the following reason:
match-boundness of the problem could not be verified.
Arrrr..