YES(O(1),O(n^2))
We are left with following problem, upon which TcT provides the
certificate YES(O(1),O(n^2)).
Strict Trs:
{ #abs(#0()) -> #0()
, #abs(#neg(@x)) -> #pos(@x)
, #abs(#pos(@x)) -> #pos(@x)
, #abs(#s(@x)) -> #pos(#s(@x))
, #equal(@x, @y) -> #eq(@x, @y)
, #greater(@x, @y) -> #ckgt(#compare(@x, @y))
, +(@x, @y) -> #add(@x, @y)
, firstline(@l) -> firstline#1(@l)
, firstline#1(::(@x, @xs)) -> ::(#abs(#0()), firstline(@xs))
, firstline#1(nil()) -> nil()
, lcs(@l1, @l2) -> lcs#1(lcstable(@l1, @l2))
, lcstable(@l1, @l2) -> lcstable#1(@l1, @l2)
, lcs#1(@m) -> lcs#2(@m)
, lcs#2(::(@l1, @_@2)) -> lcs#3(@l1)
, lcs#2(nil()) -> #abs(#0())
, lcs#3(::(@len, @_@1)) -> @len
, lcs#3(nil()) -> #abs(#0())
, lcstable#1(::(@x, @xs), @l2) ->
lcstable#2(lcstable(@xs, @l2), @l2, @x)
, lcstable#1(nil(), @l2) -> ::(firstline(@l2), nil())
, lcstable#2(@m, @l2, @x) -> lcstable#3(@m, @l2, @x)
, lcstable#3(::(@l, @ls), @l2, @x) ->
::(newline(@x, @l, @l2), ::(@l, @ls))
, lcstable#3(nil(), @l2, @x) -> nil()
, newline(@y, @lastline, @l) -> newline#1(@l, @lastline, @y)
, max(@a, @b) -> max#1(#greater(@a, @b), @a, @b)
, max#1(#false(), @a, @b) -> @b
, max#1(#true(), @a, @b) -> @a
, newline#1(::(@x, @xs), @lastline, @y) ->
newline#2(@lastline, @x, @xs, @y)
, newline#1(nil(), @lastline, @y) -> nil()
, newline#2(::(@belowVal, @lastline'), @x, @xs, @y) ->
newline#3(newline(@y, @lastline', @xs),
@belowVal,
@lastline',
@x,
@y)
, newline#2(nil(), @x, @xs, @y) -> nil()
, newline#3(@nl, @belowVal, @lastline', @x, @y) ->
newline#4(right(@nl), @belowVal, @lastline', @nl, @x, @y)
, right(@l) -> right#1(@l)
, newline#4(@rightVal, @belowVal, @lastline', @nl, @x, @y) ->
newline#5(right(@lastline'), @belowVal, @nl, @rightVal, @x, @y)
, newline#5(@diagVal, @belowVal, @nl, @rightVal, @x, @y) ->
newline#6(newline#7(#equal(@x, @y),
@belowVal,
@diagVal,
@rightVal),
@nl)
, newline#7(#false(), @belowVal, @diagVal, @rightVal) ->
max(@belowVal, @rightVal)
, newline#7(#true(), @belowVal, @diagVal, @rightVal) ->
+(@diagVal, #pos(#s(#0())))
, newline#6(@elem, @nl) -> ::(@elem, @nl)
, right#1(::(@x, @xs)) -> @x
, right#1(nil()) -> #abs(#0()) }
Weak Trs:
{ #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)
, #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()
, #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)
, #ckgt(#EQ()) -> #false()
, #ckgt(#GT()) -> #true()
, #ckgt(#LT()) -> #false()
, #add(#0(), @y) -> @y
, #add(#neg(#s(#0())), @y) -> #pred(@y)
, #add(#neg(#s(#s(@x))), @y) -> #pred(#add(#pos(#s(@x)), @y))
, #add(#pos(#s(#0())), @y) -> #succ(@y)
, #add(#pos(#s(#s(@x))), @y) -> #succ(#add(#pos(#s(@x)), @y))
, #pred(#0()) -> #neg(#s(#0()))
, #pred(#neg(#s(@x))) -> #neg(#s(#s(@x)))
, #pred(#pos(#s(#0()))) -> #0()
, #pred(#pos(#s(#s(@x)))) -> #pos(#s(@x))
, #succ(#0()) -> #pos(#s(#0()))
, #succ(#neg(#s(#0()))) -> #0()
, #succ(#neg(#s(#s(@x)))) -> #neg(#s(@x))
, #succ(#pos(#s(@x))) -> #pos(#s(#s(@x)))
, #and(#false(), #false()) -> #false()
, #and(#false(), #true()) -> #false()
, #and(#true(), #false()) -> #false()
, #and(#true(), #true()) -> #true() }
Obligation:
innermost runtime complexity
Answer:
YES(O(1),O(n^2))
We add following dependency tuples:
Strict DPs:
{ #abs^#(#0()) -> c_1()
, #abs^#(#neg(@x)) -> c_2()
, #abs^#(#pos(@x)) -> c_3()
, #abs^#(#s(@x)) -> c_4()
, #equal^#(@x, @y) -> c_5(#eq^#(@x, @y))
, #greater^#(@x, @y) ->
c_6(#ckgt^#(#compare(@x, @y)), #compare^#(@x, @y))
, +^#(@x, @y) -> c_7(#add^#(@x, @y))
, firstline^#(@l) -> c_8(firstline#1^#(@l))
, firstline#1^#(::(@x, @xs)) -> c_9(#abs^#(#0()), firstline^#(@xs))
, firstline#1^#(nil()) -> c_10()
, lcs^#(@l1, @l2) ->
c_11(lcs#1^#(lcstable(@l1, @l2)), lcstable^#(@l1, @l2))
, lcs#1^#(@m) -> c_13(lcs#2^#(@m))
, lcstable^#(@l1, @l2) -> c_12(lcstable#1^#(@l1, @l2))
, lcstable#1^#(::(@x, @xs), @l2) ->
c_18(lcstable#2^#(lcstable(@xs, @l2), @l2, @x),
lcstable^#(@xs, @l2))
, lcstable#1^#(nil(), @l2) -> c_19(firstline^#(@l2))
, lcs#2^#(::(@l1, @_@2)) -> c_14(lcs#3^#(@l1))
, lcs#2^#(nil()) -> c_15(#abs^#(#0()))
, lcs#3^#(::(@len, @_@1)) -> c_16()
, lcs#3^#(nil()) -> c_17(#abs^#(#0()))
, lcstable#2^#(@m, @l2, @x) -> c_20(lcstable#3^#(@m, @l2, @x))
, lcstable#3^#(::(@l, @ls), @l2, @x) ->
c_21(newline^#(@x, @l, @l2))
, lcstable#3^#(nil(), @l2, @x) -> c_22()
, newline^#(@y, @lastline, @l) ->
c_23(newline#1^#(@l, @lastline, @y))
, newline#1^#(::(@x, @xs), @lastline, @y) ->
c_27(newline#2^#(@lastline, @x, @xs, @y))
, newline#1^#(nil(), @lastline, @y) -> c_28()
, max^#(@a, @b) ->
c_24(max#1^#(#greater(@a, @b), @a, @b), #greater^#(@a, @b))
, max#1^#(#false(), @a, @b) -> c_25()
, max#1^#(#true(), @a, @b) -> c_26()
, newline#2^#(::(@belowVal, @lastline'), @x, @xs, @y) ->
c_29(newline#3^#(newline(@y, @lastline', @xs),
@belowVal,
@lastline',
@x,
@y),
newline^#(@y, @lastline', @xs))
, newline#2^#(nil(), @x, @xs, @y) -> c_30()
, newline#3^#(@nl, @belowVal, @lastline', @x, @y) ->
c_31(newline#4^#(right(@nl), @belowVal, @lastline', @nl, @x, @y),
right^#(@nl))
, newline#4^#(@rightVal, @belowVal, @lastline', @nl, @x, @y) ->
c_33(newline#5^#(right(@lastline'),
@belowVal,
@nl,
@rightVal,
@x,
@y),
right^#(@lastline'))
, right^#(@l) -> c_32(right#1^#(@l))
, right#1^#(::(@x, @xs)) -> c_38()
, right#1^#(nil()) -> c_39(#abs^#(#0()))
, newline#5^#(@diagVal, @belowVal, @nl, @rightVal, @x, @y) ->
c_34(newline#6^#(newline#7(#equal(@x, @y),
@belowVal,
@diagVal,
@rightVal),
@nl),
newline#7^#(#equal(@x, @y), @belowVal, @diagVal, @rightVal),
#equal^#(@x, @y))
, newline#6^#(@elem, @nl) -> c_37()
, newline#7^#(#false(), @belowVal, @diagVal, @rightVal) ->
c_35(max^#(@belowVal, @rightVal))
, newline#7^#(#true(), @belowVal, @diagVal, @rightVal) ->
c_36(+^#(@diagVal, #pos(#s(#0())))) }
Weak DPs:
{ #eq^#(#0(), #0()) -> c_40()
, #eq^#(#0(), #neg(@y)) -> c_41()
, #eq^#(#0(), #pos(@y)) -> c_42()
, #eq^#(#0(), #s(@y)) -> c_43()
, #eq^#(#neg(@x), #0()) -> c_44()
, #eq^#(#neg(@x), #neg(@y)) -> c_45(#eq^#(@x, @y))
, #eq^#(#neg(@x), #pos(@y)) -> c_46()
, #eq^#(#pos(@x), #0()) -> c_47()
, #eq^#(#pos(@x), #neg(@y)) -> c_48()
, #eq^#(#pos(@x), #pos(@y)) -> c_49(#eq^#(@x, @y))
, #eq^#(#s(@x), #0()) -> c_50()
, #eq^#(#s(@x), #s(@y)) -> c_51(#eq^#(@x, @y))
, #eq^#(::(@x_1, @x_2), ::(@y_1, @y_2)) ->
c_52(#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_53()
, #eq^#(nil(), ::(@y_1, @y_2)) -> c_54()
, #eq^#(nil(), nil()) -> c_55()
, #ckgt^#(#EQ()) -> c_68()
, #ckgt^#(#GT()) -> c_69()
, #ckgt^#(#LT()) -> c_70()
, #compare^#(#0(), #0()) -> c_56()
, #compare^#(#0(), #neg(@y)) -> c_57()
, #compare^#(#0(), #pos(@y)) -> c_58()
, #compare^#(#0(), #s(@y)) -> c_59()
, #compare^#(#neg(@x), #0()) -> c_60()
, #compare^#(#neg(@x), #neg(@y)) -> c_61(#compare^#(@y, @x))
, #compare^#(#neg(@x), #pos(@y)) -> c_62()
, #compare^#(#pos(@x), #0()) -> c_63()
, #compare^#(#pos(@x), #neg(@y)) -> c_64()
, #compare^#(#pos(@x), #pos(@y)) -> c_65(#compare^#(@x, @y))
, #compare^#(#s(@x), #0()) -> c_66()
, #compare^#(#s(@x), #s(@y)) -> c_67(#compare^#(@x, @y))
, #add^#(#0(), @y) -> c_71()
, #add^#(#neg(#s(#0())), @y) -> c_72(#pred^#(@y))
, #add^#(#neg(#s(#s(@x))), @y) ->
c_73(#pred^#(#add(#pos(#s(@x)), @y)), #add^#(#pos(#s(@x)), @y))
, #add^#(#pos(#s(#0())), @y) -> c_74(#succ^#(@y))
, #add^#(#pos(#s(#s(@x))), @y) ->
c_75(#succ^#(#add(#pos(#s(@x)), @y)), #add^#(#pos(#s(@x)), @y))
, #and^#(#false(), #false()) -> c_84()
, #and^#(#false(), #true()) -> c_85()
, #and^#(#true(), #false()) -> c_86()
, #and^#(#true(), #true()) -> c_87()
, #pred^#(#0()) -> c_76()
, #pred^#(#neg(#s(@x))) -> c_77()
, #pred^#(#pos(#s(#0()))) -> c_78()
, #pred^#(#pos(#s(#s(@x)))) -> c_79()
, #succ^#(#0()) -> c_80()
, #succ^#(#neg(#s(#0()))) -> c_81()
, #succ^#(#neg(#s(#s(@x)))) -> c_82()
, #succ^#(#pos(#s(@x))) -> c_83() }
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:
{ #abs^#(#0()) -> c_1()
, #abs^#(#neg(@x)) -> c_2()
, #abs^#(#pos(@x)) -> c_3()
, #abs^#(#s(@x)) -> c_4()
, #equal^#(@x, @y) -> c_5(#eq^#(@x, @y))
, #greater^#(@x, @y) ->
c_6(#ckgt^#(#compare(@x, @y)), #compare^#(@x, @y))
, +^#(@x, @y) -> c_7(#add^#(@x, @y))
, firstline^#(@l) -> c_8(firstline#1^#(@l))
, firstline#1^#(::(@x, @xs)) -> c_9(#abs^#(#0()), firstline^#(@xs))
, firstline#1^#(nil()) -> c_10()
, lcs^#(@l1, @l2) ->
c_11(lcs#1^#(lcstable(@l1, @l2)), lcstable^#(@l1, @l2))
, lcs#1^#(@m) -> c_13(lcs#2^#(@m))
, lcstable^#(@l1, @l2) -> c_12(lcstable#1^#(@l1, @l2))
, lcstable#1^#(::(@x, @xs), @l2) ->
c_18(lcstable#2^#(lcstable(@xs, @l2), @l2, @x),
lcstable^#(@xs, @l2))
, lcstable#1^#(nil(), @l2) -> c_19(firstline^#(@l2))
, lcs#2^#(::(@l1, @_@2)) -> c_14(lcs#3^#(@l1))
, lcs#2^#(nil()) -> c_15(#abs^#(#0()))
, lcs#3^#(::(@len, @_@1)) -> c_16()
, lcs#3^#(nil()) -> c_17(#abs^#(#0()))
, lcstable#2^#(@m, @l2, @x) -> c_20(lcstable#3^#(@m, @l2, @x))
, lcstable#3^#(::(@l, @ls), @l2, @x) ->
c_21(newline^#(@x, @l, @l2))
, lcstable#3^#(nil(), @l2, @x) -> c_22()
, newline^#(@y, @lastline, @l) ->
c_23(newline#1^#(@l, @lastline, @y))
, newline#1^#(::(@x, @xs), @lastline, @y) ->
c_27(newline#2^#(@lastline, @x, @xs, @y))
, newline#1^#(nil(), @lastline, @y) -> c_28()
, max^#(@a, @b) ->
c_24(max#1^#(#greater(@a, @b), @a, @b), #greater^#(@a, @b))
, max#1^#(#false(), @a, @b) -> c_25()
, max#1^#(#true(), @a, @b) -> c_26()
, newline#2^#(::(@belowVal, @lastline'), @x, @xs, @y) ->
c_29(newline#3^#(newline(@y, @lastline', @xs),
@belowVal,
@lastline',
@x,
@y),
newline^#(@y, @lastline', @xs))
, newline#2^#(nil(), @x, @xs, @y) -> c_30()
, newline#3^#(@nl, @belowVal, @lastline', @x, @y) ->
c_31(newline#4^#(right(@nl), @belowVal, @lastline', @nl, @x, @y),
right^#(@nl))
, newline#4^#(@rightVal, @belowVal, @lastline', @nl, @x, @y) ->
c_33(newline#5^#(right(@lastline'),
@belowVal,
@nl,
@rightVal,
@x,
@y),
right^#(@lastline'))
, right^#(@l) -> c_32(right#1^#(@l))
, right#1^#(::(@x, @xs)) -> c_38()
, right#1^#(nil()) -> c_39(#abs^#(#0()))
, newline#5^#(@diagVal, @belowVal, @nl, @rightVal, @x, @y) ->
c_34(newline#6^#(newline#7(#equal(@x, @y),
@belowVal,
@diagVal,
@rightVal),
@nl),
newline#7^#(#equal(@x, @y), @belowVal, @diagVal, @rightVal),
#equal^#(@x, @y))
, newline#6^#(@elem, @nl) -> c_37()
, newline#7^#(#false(), @belowVal, @diagVal, @rightVal) ->
c_35(max^#(@belowVal, @rightVal))
, newline#7^#(#true(), @belowVal, @diagVal, @rightVal) ->
c_36(+^#(@diagVal, #pos(#s(#0())))) }
Weak DPs:
{ #eq^#(#0(), #0()) -> c_40()
, #eq^#(#0(), #neg(@y)) -> c_41()
, #eq^#(#0(), #pos(@y)) -> c_42()
, #eq^#(#0(), #s(@y)) -> c_43()
, #eq^#(#neg(@x), #0()) -> c_44()
, #eq^#(#neg(@x), #neg(@y)) -> c_45(#eq^#(@x, @y))
, #eq^#(#neg(@x), #pos(@y)) -> c_46()
, #eq^#(#pos(@x), #0()) -> c_47()
, #eq^#(#pos(@x), #neg(@y)) -> c_48()
, #eq^#(#pos(@x), #pos(@y)) -> c_49(#eq^#(@x, @y))
, #eq^#(#s(@x), #0()) -> c_50()
, #eq^#(#s(@x), #s(@y)) -> c_51(#eq^#(@x, @y))
, #eq^#(::(@x_1, @x_2), ::(@y_1, @y_2)) ->
c_52(#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_53()
, #eq^#(nil(), ::(@y_1, @y_2)) -> c_54()
, #eq^#(nil(), nil()) -> c_55()
, #ckgt^#(#EQ()) -> c_68()
, #ckgt^#(#GT()) -> c_69()
, #ckgt^#(#LT()) -> c_70()
, #compare^#(#0(), #0()) -> c_56()
, #compare^#(#0(), #neg(@y)) -> c_57()
, #compare^#(#0(), #pos(@y)) -> c_58()
, #compare^#(#0(), #s(@y)) -> c_59()
, #compare^#(#neg(@x), #0()) -> c_60()
, #compare^#(#neg(@x), #neg(@y)) -> c_61(#compare^#(@y, @x))
, #compare^#(#neg(@x), #pos(@y)) -> c_62()
, #compare^#(#pos(@x), #0()) -> c_63()
, #compare^#(#pos(@x), #neg(@y)) -> c_64()
, #compare^#(#pos(@x), #pos(@y)) -> c_65(#compare^#(@x, @y))
, #compare^#(#s(@x), #0()) -> c_66()
, #compare^#(#s(@x), #s(@y)) -> c_67(#compare^#(@x, @y))
, #add^#(#0(), @y) -> c_71()
, #add^#(#neg(#s(#0())), @y) -> c_72(#pred^#(@y))
, #add^#(#neg(#s(#s(@x))), @y) ->
c_73(#pred^#(#add(#pos(#s(@x)), @y)), #add^#(#pos(#s(@x)), @y))
, #add^#(#pos(#s(#0())), @y) -> c_74(#succ^#(@y))
, #add^#(#pos(#s(#s(@x))), @y) ->
c_75(#succ^#(#add(#pos(#s(@x)), @y)), #add^#(#pos(#s(@x)), @y))
, #and^#(#false(), #false()) -> c_84()
, #and^#(#false(), #true()) -> c_85()
, #and^#(#true(), #false()) -> c_86()
, #and^#(#true(), #true()) -> c_87()
, #pred^#(#0()) -> c_76()
, #pred^#(#neg(#s(@x))) -> c_77()
, #pred^#(#pos(#s(#0()))) -> c_78()
, #pred^#(#pos(#s(#s(@x)))) -> c_79()
, #succ^#(#0()) -> c_80()
, #succ^#(#neg(#s(#0()))) -> c_81()
, #succ^#(#neg(#s(#s(@x)))) -> c_82()
, #succ^#(#pos(#s(@x))) -> c_83() }
Weak Trs:
{ #abs(#0()) -> #0()
, #abs(#neg(@x)) -> #pos(@x)
, #abs(#pos(@x)) -> #pos(@x)
, #abs(#s(@x)) -> #pos(#s(@x))
, #equal(@x, @y) -> #eq(@x, @y)
, #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)
, #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()
, #greater(@x, @y) -> #ckgt(#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)
, #ckgt(#EQ()) -> #false()
, #ckgt(#GT()) -> #true()
, #ckgt(#LT()) -> #false()
, +(@x, @y) -> #add(@x, @y)
, #add(#0(), @y) -> @y
, #add(#neg(#s(#0())), @y) -> #pred(@y)
, #add(#neg(#s(#s(@x))), @y) -> #pred(#add(#pos(#s(@x)), @y))
, #add(#pos(#s(#0())), @y) -> #succ(@y)
, #add(#pos(#s(#s(@x))), @y) -> #succ(#add(#pos(#s(@x)), @y))
, firstline(@l) -> firstline#1(@l)
, firstline#1(::(@x, @xs)) -> ::(#abs(#0()), firstline(@xs))
, firstline#1(nil()) -> nil()
, lcs(@l1, @l2) -> lcs#1(lcstable(@l1, @l2))
, lcstable(@l1, @l2) -> lcstable#1(@l1, @l2)
, lcs#1(@m) -> lcs#2(@m)
, lcs#2(::(@l1, @_@2)) -> lcs#3(@l1)
, lcs#2(nil()) -> #abs(#0())
, lcs#3(::(@len, @_@1)) -> @len
, lcs#3(nil()) -> #abs(#0())
, lcstable#1(::(@x, @xs), @l2) ->
lcstable#2(lcstable(@xs, @l2), @l2, @x)
, lcstable#1(nil(), @l2) -> ::(firstline(@l2), nil())
, lcstable#2(@m, @l2, @x) -> lcstable#3(@m, @l2, @x)
, lcstable#3(::(@l, @ls), @l2, @x) ->
::(newline(@x, @l, @l2), ::(@l, @ls))
, lcstable#3(nil(), @l2, @x) -> nil()
, newline(@y, @lastline, @l) -> newline#1(@l, @lastline, @y)
, max(@a, @b) -> max#1(#greater(@a, @b), @a, @b)
, max#1(#false(), @a, @b) -> @b
, max#1(#true(), @a, @b) -> @a
, newline#1(::(@x, @xs), @lastline, @y) ->
newline#2(@lastline, @x, @xs, @y)
, newline#1(nil(), @lastline, @y) -> nil()
, newline#2(::(@belowVal, @lastline'), @x, @xs, @y) ->
newline#3(newline(@y, @lastline', @xs),
@belowVal,
@lastline',
@x,
@y)
, newline#2(nil(), @x, @xs, @y) -> nil()
, newline#3(@nl, @belowVal, @lastline', @x, @y) ->
newline#4(right(@nl), @belowVal, @lastline', @nl, @x, @y)
, right(@l) -> right#1(@l)
, newline#4(@rightVal, @belowVal, @lastline', @nl, @x, @y) ->
newline#5(right(@lastline'), @belowVal, @nl, @rightVal, @x, @y)
, newline#5(@diagVal, @belowVal, @nl, @rightVal, @x, @y) ->
newline#6(newline#7(#equal(@x, @y),
@belowVal,
@diagVal,
@rightVal),
@nl)
, newline#7(#false(), @belowVal, @diagVal, @rightVal) ->
max(@belowVal, @rightVal)
, newline#7(#true(), @belowVal, @diagVal, @rightVal) ->
+(@diagVal, #pos(#s(#0())))
, newline#6(@elem, @nl) -> ::(@elem, @nl)
, right#1(::(@x, @xs)) -> @x
, right#1(nil()) -> #abs(#0())
, #pred(#0()) -> #neg(#s(#0()))
, #pred(#neg(#s(@x))) -> #neg(#s(#s(@x)))
, #pred(#pos(#s(#0()))) -> #0()
, #pred(#pos(#s(#s(@x)))) -> #pos(#s(@x))
, #succ(#0()) -> #pos(#s(#0()))
, #succ(#neg(#s(#0()))) -> #0()
, #succ(#neg(#s(#s(@x)))) -> #neg(#s(@x))
, #succ(#pos(#s(@x))) -> #pos(#s(#s(@x)))
, #and(#false(), #false()) -> #false()
, #and(#false(), #true()) -> #false()
, #and(#true(), #false()) -> #false()
, #and(#true(), #true()) -> #true() }
Obligation:
innermost runtime complexity
Answer:
YES(O(1),O(n^2))
Consider the dependency graph
1: #abs^#(#0()) -> c_1()
2: #abs^#(#neg(@x)) -> c_2()
3: #abs^#(#pos(@x)) -> c_3()
4: #abs^#(#s(@x)) -> c_4()
5: #equal^#(@x, @y) -> c_5(#eq^#(@x, @y))
-->_1 #eq^#(::(@x_1, @x_2), ::(@y_1, @y_2)) ->
c_52(#and^#(#eq(@x_1, @y_1), #eq(@x_2, @y_2)),
#eq^#(@x_1, @y_1),
#eq^#(@x_2, @y_2)) :52
-->_1 #eq^#(#s(@x), #s(@y)) -> c_51(#eq^#(@x, @y)) :51
-->_1 #eq^#(#pos(@x), #pos(@y)) -> c_49(#eq^#(@x, @y)) :49
-->_1 #eq^#(#neg(@x), #neg(@y)) -> c_45(#eq^#(@x, @y)) :45
-->_1 #eq^#(nil(), nil()) -> c_55() :55
-->_1 #eq^#(nil(), ::(@y_1, @y_2)) -> c_54() :54
-->_1 #eq^#(::(@x_1, @x_2), nil()) -> c_53() :53
-->_1 #eq^#(#s(@x), #0()) -> c_50() :50
-->_1 #eq^#(#pos(@x), #neg(@y)) -> c_48() :48
-->_1 #eq^#(#pos(@x), #0()) -> c_47() :47
-->_1 #eq^#(#neg(@x), #pos(@y)) -> c_46() :46
-->_1 #eq^#(#neg(@x), #0()) -> c_44() :44
-->_1 #eq^#(#0(), #s(@y)) -> c_43() :43
-->_1 #eq^#(#0(), #pos(@y)) -> c_42() :42
-->_1 #eq^#(#0(), #neg(@y)) -> c_41() :41
-->_1 #eq^#(#0(), #0()) -> c_40() :40
6: #greater^#(@x, @y) ->
c_6(#ckgt^#(#compare(@x, @y)), #compare^#(@x, @y))
-->_2 #compare^#(#s(@x), #s(@y)) -> c_67(#compare^#(@x, @y)) :70
-->_2 #compare^#(#pos(@x), #pos(@y)) ->
c_65(#compare^#(@x, @y)) :68
-->_2 #compare^#(#neg(@x), #neg(@y)) ->
c_61(#compare^#(@y, @x)) :64
-->_2 #compare^#(#s(@x), #0()) -> c_66() :69
-->_2 #compare^#(#pos(@x), #neg(@y)) -> c_64() :67
-->_2 #compare^#(#pos(@x), #0()) -> c_63() :66
-->_2 #compare^#(#neg(@x), #pos(@y)) -> c_62() :65
-->_2 #compare^#(#neg(@x), #0()) -> c_60() :63
-->_2 #compare^#(#0(), #s(@y)) -> c_59() :62
-->_2 #compare^#(#0(), #pos(@y)) -> c_58() :61
-->_2 #compare^#(#0(), #neg(@y)) -> c_57() :60
-->_2 #compare^#(#0(), #0()) -> c_56() :59
-->_1 #ckgt^#(#LT()) -> c_70() :58
-->_1 #ckgt^#(#GT()) -> c_69() :57
-->_1 #ckgt^#(#EQ()) -> c_68() :56
7: +^#(@x, @y) -> c_7(#add^#(@x, @y))
-->_1 #add^#(#pos(#s(#s(@x))), @y) ->
c_75(#succ^#(#add(#pos(#s(@x)), @y)), #add^#(#pos(#s(@x)), @y)) :75
-->_1 #add^#(#pos(#s(#0())), @y) -> c_74(#succ^#(@y)) :74
-->_1 #add^#(#neg(#s(#s(@x))), @y) ->
c_73(#pred^#(#add(#pos(#s(@x)), @y)), #add^#(#pos(#s(@x)), @y)) :73
-->_1 #add^#(#neg(#s(#0())), @y) -> c_72(#pred^#(@y)) :72
-->_1 #add^#(#0(), @y) -> c_71() :71
8: firstline^#(@l) -> c_8(firstline#1^#(@l))
-->_1 firstline#1^#(::(@x, @xs)) ->
c_9(#abs^#(#0()), firstline^#(@xs)) :9
-->_1 firstline#1^#(nil()) -> c_10() :10
9: firstline#1^#(::(@x, @xs)) ->
c_9(#abs^#(#0()), firstline^#(@xs))
-->_2 firstline^#(@l) -> c_8(firstline#1^#(@l)) :8
-->_1 #abs^#(#0()) -> c_1() :1
10: firstline#1^#(nil()) -> c_10()
11: lcs^#(@l1, @l2) ->
c_11(lcs#1^#(lcstable(@l1, @l2)), lcstable^#(@l1, @l2))
-->_2 lcstable^#(@l1, @l2) -> c_12(lcstable#1^#(@l1, @l2)) :13
-->_1 lcs#1^#(@m) -> c_13(lcs#2^#(@m)) :12
12: lcs#1^#(@m) -> c_13(lcs#2^#(@m))
-->_1 lcs#2^#(nil()) -> c_15(#abs^#(#0())) :17
-->_1 lcs#2^#(::(@l1, @_@2)) -> c_14(lcs#3^#(@l1)) :16
13: lcstable^#(@l1, @l2) -> c_12(lcstable#1^#(@l1, @l2))
-->_1 lcstable#1^#(nil(), @l2) -> c_19(firstline^#(@l2)) :15
-->_1 lcstable#1^#(::(@x, @xs), @l2) ->
c_18(lcstable#2^#(lcstable(@xs, @l2), @l2, @x),
lcstable^#(@xs, @l2)) :14
14: lcstable#1^#(::(@x, @xs), @l2) ->
c_18(lcstable#2^#(lcstable(@xs, @l2), @l2, @x),
lcstable^#(@xs, @l2))
-->_1 lcstable#2^#(@m, @l2, @x) ->
c_20(lcstable#3^#(@m, @l2, @x)) :20
-->_2 lcstable^#(@l1, @l2) -> c_12(lcstable#1^#(@l1, @l2)) :13
15: lcstable#1^#(nil(), @l2) -> c_19(firstline^#(@l2))
-->_1 firstline^#(@l) -> c_8(firstline#1^#(@l)) :8
16: lcs#2^#(::(@l1, @_@2)) -> c_14(lcs#3^#(@l1))
-->_1 lcs#3^#(nil()) -> c_17(#abs^#(#0())) :19
-->_1 lcs#3^#(::(@len, @_@1)) -> c_16() :18
17: lcs#2^#(nil()) -> c_15(#abs^#(#0()))
-->_1 #abs^#(#0()) -> c_1() :1
18: lcs#3^#(::(@len, @_@1)) -> c_16()
19: lcs#3^#(nil()) -> c_17(#abs^#(#0()))
-->_1 #abs^#(#0()) -> c_1() :1
20: lcstable#2^#(@m, @l2, @x) -> c_20(lcstable#3^#(@m, @l2, @x))
-->_1 lcstable#3^#(::(@l, @ls), @l2, @x) ->
c_21(newline^#(@x, @l, @l2)) :21
-->_1 lcstable#3^#(nil(), @l2, @x) -> c_22() :22
21: lcstable#3^#(::(@l, @ls), @l2, @x) ->
c_21(newline^#(@x, @l, @l2))
-->_1 newline^#(@y, @lastline, @l) ->
c_23(newline#1^#(@l, @lastline, @y)) :23
22: lcstable#3^#(nil(), @l2, @x) -> c_22()
23: newline^#(@y, @lastline, @l) ->
c_23(newline#1^#(@l, @lastline, @y))
-->_1 newline#1^#(::(@x, @xs), @lastline, @y) ->
c_27(newline#2^#(@lastline, @x, @xs, @y)) :24
-->_1 newline#1^#(nil(), @lastline, @y) -> c_28() :25
24: newline#1^#(::(@x, @xs), @lastline, @y) ->
c_27(newline#2^#(@lastline, @x, @xs, @y))
-->_1 newline#2^#(::(@belowVal, @lastline'), @x, @xs, @y) ->
c_29(newline#3^#(newline(@y, @lastline', @xs),
@belowVal,
@lastline',
@x,
@y),
newline^#(@y, @lastline', @xs)) :29
-->_1 newline#2^#(nil(), @x, @xs, @y) -> c_30() :30
25: newline#1^#(nil(), @lastline, @y) -> c_28()
26: max^#(@a, @b) ->
c_24(max#1^#(#greater(@a, @b), @a, @b), #greater^#(@a, @b))
-->_1 max#1^#(#true(), @a, @b) -> c_26() :28
-->_1 max#1^#(#false(), @a, @b) -> c_25() :27
-->_2 #greater^#(@x, @y) ->
c_6(#ckgt^#(#compare(@x, @y)), #compare^#(@x, @y)) :6
27: max#1^#(#false(), @a, @b) -> c_25()
28: max#1^#(#true(), @a, @b) -> c_26()
29: newline#2^#(::(@belowVal, @lastline'), @x, @xs, @y) ->
c_29(newline#3^#(newline(@y, @lastline', @xs),
@belowVal,
@lastline',
@x,
@y),
newline^#(@y, @lastline', @xs))
-->_1 newline#3^#(@nl, @belowVal, @lastline', @x, @y) ->
c_31(newline#4^#(right(@nl), @belowVal, @lastline', @nl, @x, @y),
right^#(@nl)) :31
-->_2 newline^#(@y, @lastline, @l) ->
c_23(newline#1^#(@l, @lastline, @y)) :23
30: newline#2^#(nil(), @x, @xs, @y) -> c_30()
31: newline#3^#(@nl, @belowVal, @lastline', @x, @y) ->
c_31(newline#4^#(right(@nl), @belowVal, @lastline', @nl, @x, @y),
right^#(@nl))
-->_2 right^#(@l) -> c_32(right#1^#(@l)) :33
-->_1 newline#4^#(@rightVal, @belowVal, @lastline', @nl, @x, @y) ->
c_33(newline#5^#(right(@lastline'),
@belowVal,
@nl,
@rightVal,
@x,
@y),
right^#(@lastline')) :32
32: newline#4^#(@rightVal, @belowVal, @lastline', @nl, @x, @y) ->
c_33(newline#5^#(right(@lastline'),
@belowVal,
@nl,
@rightVal,
@x,
@y),
right^#(@lastline'))
-->_1 newline#5^#(@diagVal, @belowVal, @nl, @rightVal, @x, @y) ->
c_34(newline#6^#(newline#7(#equal(@x, @y),
@belowVal,
@diagVal,
@rightVal),
@nl),
newline#7^#(#equal(@x, @y), @belowVal, @diagVal, @rightVal),
#equal^#(@x, @y)) :36
-->_2 right^#(@l) -> c_32(right#1^#(@l)) :33
33: right^#(@l) -> c_32(right#1^#(@l))
-->_1 right#1^#(nil()) -> c_39(#abs^#(#0())) :35
-->_1 right#1^#(::(@x, @xs)) -> c_38() :34
34: right#1^#(::(@x, @xs)) -> c_38()
35: right#1^#(nil()) -> c_39(#abs^#(#0()))
-->_1 #abs^#(#0()) -> c_1() :1
36: newline#5^#(@diagVal, @belowVal, @nl, @rightVal, @x, @y) ->
c_34(newline#6^#(newline#7(#equal(@x, @y),
@belowVal,
@diagVal,
@rightVal),
@nl),
newline#7^#(#equal(@x, @y), @belowVal, @diagVal, @rightVal),
#equal^#(@x, @y))
-->_2 newline#7^#(#true(), @belowVal, @diagVal, @rightVal) ->
c_36(+^#(@diagVal, #pos(#s(#0())))) :39
-->_2 newline#7^#(#false(), @belowVal, @diagVal, @rightVal) ->
c_35(max^#(@belowVal, @rightVal)) :38
-->_1 newline#6^#(@elem, @nl) -> c_37() :37
-->_3 #equal^#(@x, @y) -> c_5(#eq^#(@x, @y)) :5
37: newline#6^#(@elem, @nl) -> c_37()
38: newline#7^#(#false(), @belowVal, @diagVal, @rightVal) ->
c_35(max^#(@belowVal, @rightVal))
-->_1 max^#(@a, @b) ->
c_24(max#1^#(#greater(@a, @b), @a, @b), #greater^#(@a, @b)) :26
39: newline#7^#(#true(), @belowVal, @diagVal, @rightVal) ->
c_36(+^#(@diagVal, #pos(#s(#0()))))
-->_1 +^#(@x, @y) -> c_7(#add^#(@x, @y)) :7
40: #eq^#(#0(), #0()) -> c_40()
41: #eq^#(#0(), #neg(@y)) -> c_41()
42: #eq^#(#0(), #pos(@y)) -> c_42()
43: #eq^#(#0(), #s(@y)) -> c_43()
44: #eq^#(#neg(@x), #0()) -> c_44()
45: #eq^#(#neg(@x), #neg(@y)) -> c_45(#eq^#(@x, @y))
-->_1 #eq^#(::(@x_1, @x_2), ::(@y_1, @y_2)) ->
c_52(#and^#(#eq(@x_1, @y_1), #eq(@x_2, @y_2)),
#eq^#(@x_1, @y_1),
#eq^#(@x_2, @y_2)) :52
-->_1 #eq^#(#s(@x), #s(@y)) -> c_51(#eq^#(@x, @y)) :51
-->_1 #eq^#(#pos(@x), #pos(@y)) -> c_49(#eq^#(@x, @y)) :49
-->_1 #eq^#(nil(), nil()) -> c_55() :55
-->_1 #eq^#(nil(), ::(@y_1, @y_2)) -> c_54() :54
-->_1 #eq^#(::(@x_1, @x_2), nil()) -> c_53() :53
-->_1 #eq^#(#s(@x), #0()) -> c_50() :50
-->_1 #eq^#(#pos(@x), #neg(@y)) -> c_48() :48
-->_1 #eq^#(#pos(@x), #0()) -> c_47() :47
-->_1 #eq^#(#neg(@x), #pos(@y)) -> c_46() :46
-->_1 #eq^#(#neg(@x), #neg(@y)) -> c_45(#eq^#(@x, @y)) :45
-->_1 #eq^#(#neg(@x), #0()) -> c_44() :44
-->_1 #eq^#(#0(), #s(@y)) -> c_43() :43
-->_1 #eq^#(#0(), #pos(@y)) -> c_42() :42
-->_1 #eq^#(#0(), #neg(@y)) -> c_41() :41
-->_1 #eq^#(#0(), #0()) -> c_40() :40
46: #eq^#(#neg(@x), #pos(@y)) -> c_46()
47: #eq^#(#pos(@x), #0()) -> c_47()
48: #eq^#(#pos(@x), #neg(@y)) -> c_48()
49: #eq^#(#pos(@x), #pos(@y)) -> c_49(#eq^#(@x, @y))
-->_1 #eq^#(::(@x_1, @x_2), ::(@y_1, @y_2)) ->
c_52(#and^#(#eq(@x_1, @y_1), #eq(@x_2, @y_2)),
#eq^#(@x_1, @y_1),
#eq^#(@x_2, @y_2)) :52
-->_1 #eq^#(#s(@x), #s(@y)) -> c_51(#eq^#(@x, @y)) :51
-->_1 #eq^#(nil(), nil()) -> c_55() :55
-->_1 #eq^#(nil(), ::(@y_1, @y_2)) -> c_54() :54
-->_1 #eq^#(::(@x_1, @x_2), nil()) -> c_53() :53
-->_1 #eq^#(#s(@x), #0()) -> c_50() :50
-->_1 #eq^#(#pos(@x), #pos(@y)) -> c_49(#eq^#(@x, @y)) :49
-->_1 #eq^#(#pos(@x), #neg(@y)) -> c_48() :48
-->_1 #eq^#(#pos(@x), #0()) -> c_47() :47
-->_1 #eq^#(#neg(@x), #pos(@y)) -> c_46() :46
-->_1 #eq^#(#neg(@x), #neg(@y)) -> c_45(#eq^#(@x, @y)) :45
-->_1 #eq^#(#neg(@x), #0()) -> c_44() :44
-->_1 #eq^#(#0(), #s(@y)) -> c_43() :43
-->_1 #eq^#(#0(), #pos(@y)) -> c_42() :42
-->_1 #eq^#(#0(), #neg(@y)) -> c_41() :41
-->_1 #eq^#(#0(), #0()) -> c_40() :40
50: #eq^#(#s(@x), #0()) -> c_50()
51: #eq^#(#s(@x), #s(@y)) -> c_51(#eq^#(@x, @y))
-->_1 #eq^#(::(@x_1, @x_2), ::(@y_1, @y_2)) ->
c_52(#and^#(#eq(@x_1, @y_1), #eq(@x_2, @y_2)),
#eq^#(@x_1, @y_1),
#eq^#(@x_2, @y_2)) :52
-->_1 #eq^#(nil(), nil()) -> c_55() :55
-->_1 #eq^#(nil(), ::(@y_1, @y_2)) -> c_54() :54
-->_1 #eq^#(::(@x_1, @x_2), nil()) -> c_53() :53
-->_1 #eq^#(#s(@x), #s(@y)) -> c_51(#eq^#(@x, @y)) :51
-->_1 #eq^#(#s(@x), #0()) -> c_50() :50
-->_1 #eq^#(#pos(@x), #pos(@y)) -> c_49(#eq^#(@x, @y)) :49
-->_1 #eq^#(#pos(@x), #neg(@y)) -> c_48() :48
-->_1 #eq^#(#pos(@x), #0()) -> c_47() :47
-->_1 #eq^#(#neg(@x), #pos(@y)) -> c_46() :46
-->_1 #eq^#(#neg(@x), #neg(@y)) -> c_45(#eq^#(@x, @y)) :45
-->_1 #eq^#(#neg(@x), #0()) -> c_44() :44
-->_1 #eq^#(#0(), #s(@y)) -> c_43() :43
-->_1 #eq^#(#0(), #pos(@y)) -> c_42() :42
-->_1 #eq^#(#0(), #neg(@y)) -> c_41() :41
-->_1 #eq^#(#0(), #0()) -> c_40() :40
52: #eq^#(::(@x_1, @x_2), ::(@y_1, @y_2)) ->
c_52(#and^#(#eq(@x_1, @y_1), #eq(@x_2, @y_2)),
#eq^#(@x_1, @y_1),
#eq^#(@x_2, @y_2))
-->_1 #and^#(#true(), #true()) -> c_87() :79
-->_1 #and^#(#true(), #false()) -> c_86() :78
-->_1 #and^#(#false(), #true()) -> c_85() :77
-->_1 #and^#(#false(), #false()) -> c_84() :76
-->_3 #eq^#(nil(), nil()) -> c_55() :55
-->_2 #eq^#(nil(), nil()) -> c_55() :55
-->_3 #eq^#(nil(), ::(@y_1, @y_2)) -> c_54() :54
-->_2 #eq^#(nil(), ::(@y_1, @y_2)) -> c_54() :54
-->_3 #eq^#(::(@x_1, @x_2), nil()) -> c_53() :53
-->_2 #eq^#(::(@x_1, @x_2), nil()) -> c_53() :53
-->_3 #eq^#(::(@x_1, @x_2), ::(@y_1, @y_2)) ->
c_52(#and^#(#eq(@x_1, @y_1), #eq(@x_2, @y_2)),
#eq^#(@x_1, @y_1),
#eq^#(@x_2, @y_2)) :52
-->_2 #eq^#(::(@x_1, @x_2), ::(@y_1, @y_2)) ->
c_52(#and^#(#eq(@x_1, @y_1), #eq(@x_2, @y_2)),
#eq^#(@x_1, @y_1),
#eq^#(@x_2, @y_2)) :52
-->_3 #eq^#(#s(@x), #s(@y)) -> c_51(#eq^#(@x, @y)) :51
-->_2 #eq^#(#s(@x), #s(@y)) -> c_51(#eq^#(@x, @y)) :51
-->_3 #eq^#(#s(@x), #0()) -> c_50() :50
-->_2 #eq^#(#s(@x), #0()) -> c_50() :50
-->_3 #eq^#(#pos(@x), #pos(@y)) -> c_49(#eq^#(@x, @y)) :49
-->_2 #eq^#(#pos(@x), #pos(@y)) -> c_49(#eq^#(@x, @y)) :49
-->_3 #eq^#(#pos(@x), #neg(@y)) -> c_48() :48
-->_2 #eq^#(#pos(@x), #neg(@y)) -> c_48() :48
-->_3 #eq^#(#pos(@x), #0()) -> c_47() :47
-->_2 #eq^#(#pos(@x), #0()) -> c_47() :47
-->_3 #eq^#(#neg(@x), #pos(@y)) -> c_46() :46
-->_2 #eq^#(#neg(@x), #pos(@y)) -> c_46() :46
-->_3 #eq^#(#neg(@x), #neg(@y)) -> c_45(#eq^#(@x, @y)) :45
-->_2 #eq^#(#neg(@x), #neg(@y)) -> c_45(#eq^#(@x, @y)) :45
-->_3 #eq^#(#neg(@x), #0()) -> c_44() :44
-->_2 #eq^#(#neg(@x), #0()) -> c_44() :44
-->_3 #eq^#(#0(), #s(@y)) -> c_43() :43
-->_2 #eq^#(#0(), #s(@y)) -> c_43() :43
-->_3 #eq^#(#0(), #pos(@y)) -> c_42() :42
-->_2 #eq^#(#0(), #pos(@y)) -> c_42() :42
-->_3 #eq^#(#0(), #neg(@y)) -> c_41() :41
-->_2 #eq^#(#0(), #neg(@y)) -> c_41() :41
-->_3 #eq^#(#0(), #0()) -> c_40() :40
-->_2 #eq^#(#0(), #0()) -> c_40() :40
53: #eq^#(::(@x_1, @x_2), nil()) -> c_53()
54: #eq^#(nil(), ::(@y_1, @y_2)) -> c_54()
55: #eq^#(nil(), nil()) -> c_55()
56: #ckgt^#(#EQ()) -> c_68()
57: #ckgt^#(#GT()) -> c_69()
58: #ckgt^#(#LT()) -> c_70()
59: #compare^#(#0(), #0()) -> c_56()
60: #compare^#(#0(), #neg(@y)) -> c_57()
61: #compare^#(#0(), #pos(@y)) -> c_58()
62: #compare^#(#0(), #s(@y)) -> c_59()
63: #compare^#(#neg(@x), #0()) -> c_60()
64: #compare^#(#neg(@x), #neg(@y)) -> c_61(#compare^#(@y, @x))
-->_1 #compare^#(#s(@x), #s(@y)) -> c_67(#compare^#(@x, @y)) :70
-->_1 #compare^#(#pos(@x), #pos(@y)) ->
c_65(#compare^#(@x, @y)) :68
-->_1 #compare^#(#s(@x), #0()) -> c_66() :69
-->_1 #compare^#(#pos(@x), #neg(@y)) -> c_64() :67
-->_1 #compare^#(#pos(@x), #0()) -> c_63() :66
-->_1 #compare^#(#neg(@x), #pos(@y)) -> c_62() :65
-->_1 #compare^#(#neg(@x), #neg(@y)) ->
c_61(#compare^#(@y, @x)) :64
-->_1 #compare^#(#neg(@x), #0()) -> c_60() :63
-->_1 #compare^#(#0(), #s(@y)) -> c_59() :62
-->_1 #compare^#(#0(), #pos(@y)) -> c_58() :61
-->_1 #compare^#(#0(), #neg(@y)) -> c_57() :60
-->_1 #compare^#(#0(), #0()) -> c_56() :59
65: #compare^#(#neg(@x), #pos(@y)) -> c_62()
66: #compare^#(#pos(@x), #0()) -> c_63()
67: #compare^#(#pos(@x), #neg(@y)) -> c_64()
68: #compare^#(#pos(@x), #pos(@y)) -> c_65(#compare^#(@x, @y))
-->_1 #compare^#(#s(@x), #s(@y)) -> c_67(#compare^#(@x, @y)) :70
-->_1 #compare^#(#s(@x), #0()) -> c_66() :69
-->_1 #compare^#(#pos(@x), #pos(@y)) ->
c_65(#compare^#(@x, @y)) :68
-->_1 #compare^#(#pos(@x), #neg(@y)) -> c_64() :67
-->_1 #compare^#(#pos(@x), #0()) -> c_63() :66
-->_1 #compare^#(#neg(@x), #pos(@y)) -> c_62() :65
-->_1 #compare^#(#neg(@x), #neg(@y)) ->
c_61(#compare^#(@y, @x)) :64
-->_1 #compare^#(#neg(@x), #0()) -> c_60() :63
-->_1 #compare^#(#0(), #s(@y)) -> c_59() :62
-->_1 #compare^#(#0(), #pos(@y)) -> c_58() :61
-->_1 #compare^#(#0(), #neg(@y)) -> c_57() :60
-->_1 #compare^#(#0(), #0()) -> c_56() :59
69: #compare^#(#s(@x), #0()) -> c_66()
70: #compare^#(#s(@x), #s(@y)) -> c_67(#compare^#(@x, @y))
-->_1 #compare^#(#s(@x), #s(@y)) -> c_67(#compare^#(@x, @y)) :70
-->_1 #compare^#(#s(@x), #0()) -> c_66() :69
-->_1 #compare^#(#pos(@x), #pos(@y)) ->
c_65(#compare^#(@x, @y)) :68
-->_1 #compare^#(#pos(@x), #neg(@y)) -> c_64() :67
-->_1 #compare^#(#pos(@x), #0()) -> c_63() :66
-->_1 #compare^#(#neg(@x), #pos(@y)) -> c_62() :65
-->_1 #compare^#(#neg(@x), #neg(@y)) ->
c_61(#compare^#(@y, @x)) :64
-->_1 #compare^#(#neg(@x), #0()) -> c_60() :63
-->_1 #compare^#(#0(), #s(@y)) -> c_59() :62
-->_1 #compare^#(#0(), #pos(@y)) -> c_58() :61
-->_1 #compare^#(#0(), #neg(@y)) -> c_57() :60
-->_1 #compare^#(#0(), #0()) -> c_56() :59
71: #add^#(#0(), @y) -> c_71()
72: #add^#(#neg(#s(#0())), @y) -> c_72(#pred^#(@y))
-->_1 #pred^#(#pos(#s(#s(@x)))) -> c_79() :83
-->_1 #pred^#(#pos(#s(#0()))) -> c_78() :82
-->_1 #pred^#(#neg(#s(@x))) -> c_77() :81
-->_1 #pred^#(#0()) -> c_76() :80
73: #add^#(#neg(#s(#s(@x))), @y) ->
c_73(#pred^#(#add(#pos(#s(@x)), @y)), #add^#(#pos(#s(@x)), @y))
-->_2 #add^#(#pos(#s(#s(@x))), @y) ->
c_75(#succ^#(#add(#pos(#s(@x)), @y)), #add^#(#pos(#s(@x)), @y)) :75
-->_2 #add^#(#pos(#s(#0())), @y) -> c_74(#succ^#(@y)) :74
-->_1 #pred^#(#pos(#s(#s(@x)))) -> c_79() :83
-->_1 #pred^#(#pos(#s(#0()))) -> c_78() :82
-->_1 #pred^#(#neg(#s(@x))) -> c_77() :81
-->_1 #pred^#(#0()) -> c_76() :80
74: #add^#(#pos(#s(#0())), @y) -> c_74(#succ^#(@y))
-->_1 #succ^#(#pos(#s(@x))) -> c_83() :87
-->_1 #succ^#(#neg(#s(#s(@x)))) -> c_82() :86
-->_1 #succ^#(#neg(#s(#0()))) -> c_81() :85
-->_1 #succ^#(#0()) -> c_80() :84
75: #add^#(#pos(#s(#s(@x))), @y) ->
c_75(#succ^#(#add(#pos(#s(@x)), @y)), #add^#(#pos(#s(@x)), @y))
-->_1 #succ^#(#pos(#s(@x))) -> c_83() :87
-->_1 #succ^#(#neg(#s(#s(@x)))) -> c_82() :86
-->_1 #succ^#(#neg(#s(#0()))) -> c_81() :85
-->_1 #succ^#(#0()) -> c_80() :84
-->_2 #add^#(#pos(#s(#s(@x))), @y) ->
c_75(#succ^#(#add(#pos(#s(@x)), @y)), #add^#(#pos(#s(@x)), @y)) :75
-->_2 #add^#(#pos(#s(#0())), @y) -> c_74(#succ^#(@y)) :74
76: #and^#(#false(), #false()) -> c_84()
77: #and^#(#false(), #true()) -> c_85()
78: #and^#(#true(), #false()) -> c_86()
79: #and^#(#true(), #true()) -> c_87()
80: #pred^#(#0()) -> c_76()
81: #pred^#(#neg(#s(@x))) -> c_77()
82: #pred^#(#pos(#s(#0()))) -> c_78()
83: #pred^#(#pos(#s(#s(@x)))) -> c_79()
84: #succ^#(#0()) -> c_80()
85: #succ^#(#neg(#s(#0()))) -> c_81()
86: #succ^#(#neg(#s(#s(@x)))) -> c_82()
87: #succ^#(#pos(#s(@x))) -> c_83()
Following roots of the dependency graph are removed, as the
considered set of starting terms is closed under reduction with
respect to these rules (modulo compound contexts).
{ #abs^#(#neg(@x)) -> c_2()
, #abs^#(#pos(@x)) -> c_3()
, #abs^#(#s(@x)) -> c_4() }
We are left with following problem, upon which TcT provides the
certificate YES(O(1),O(n^2)).
Strict DPs:
{ #abs^#(#0()) -> c_1()
, #equal^#(@x, @y) -> c_5(#eq^#(@x, @y))
, #greater^#(@x, @y) ->
c_6(#ckgt^#(#compare(@x, @y)), #compare^#(@x, @y))
, +^#(@x, @y) -> c_7(#add^#(@x, @y))
, firstline^#(@l) -> c_8(firstline#1^#(@l))
, firstline#1^#(::(@x, @xs)) -> c_9(#abs^#(#0()), firstline^#(@xs))
, firstline#1^#(nil()) -> c_10()
, lcs^#(@l1, @l2) ->
c_11(lcs#1^#(lcstable(@l1, @l2)), lcstable^#(@l1, @l2))
, lcs#1^#(@m) -> c_13(lcs#2^#(@m))
, lcstable^#(@l1, @l2) -> c_12(lcstable#1^#(@l1, @l2))
, lcstable#1^#(::(@x, @xs), @l2) ->
c_18(lcstable#2^#(lcstable(@xs, @l2), @l2, @x),
lcstable^#(@xs, @l2))
, lcstable#1^#(nil(), @l2) -> c_19(firstline^#(@l2))
, lcs#2^#(::(@l1, @_@2)) -> c_14(lcs#3^#(@l1))
, lcs#2^#(nil()) -> c_15(#abs^#(#0()))
, lcs#3^#(::(@len, @_@1)) -> c_16()
, lcs#3^#(nil()) -> c_17(#abs^#(#0()))
, lcstable#2^#(@m, @l2, @x) -> c_20(lcstable#3^#(@m, @l2, @x))
, lcstable#3^#(::(@l, @ls), @l2, @x) ->
c_21(newline^#(@x, @l, @l2))
, lcstable#3^#(nil(), @l2, @x) -> c_22()
, newline^#(@y, @lastline, @l) ->
c_23(newline#1^#(@l, @lastline, @y))
, newline#1^#(::(@x, @xs), @lastline, @y) ->
c_27(newline#2^#(@lastline, @x, @xs, @y))
, newline#1^#(nil(), @lastline, @y) -> c_28()
, max^#(@a, @b) ->
c_24(max#1^#(#greater(@a, @b), @a, @b), #greater^#(@a, @b))
, max#1^#(#false(), @a, @b) -> c_25()
, max#1^#(#true(), @a, @b) -> c_26()
, newline#2^#(::(@belowVal, @lastline'), @x, @xs, @y) ->
c_29(newline#3^#(newline(@y, @lastline', @xs),
@belowVal,
@lastline',
@x,
@y),
newline^#(@y, @lastline', @xs))
, newline#2^#(nil(), @x, @xs, @y) -> c_30()
, newline#3^#(@nl, @belowVal, @lastline', @x, @y) ->
c_31(newline#4^#(right(@nl), @belowVal, @lastline', @nl, @x, @y),
right^#(@nl))
, newline#4^#(@rightVal, @belowVal, @lastline', @nl, @x, @y) ->
c_33(newline#5^#(right(@lastline'),
@belowVal,
@nl,
@rightVal,
@x,
@y),
right^#(@lastline'))
, right^#(@l) -> c_32(right#1^#(@l))
, right#1^#(::(@x, @xs)) -> c_38()
, right#1^#(nil()) -> c_39(#abs^#(#0()))
, newline#5^#(@diagVal, @belowVal, @nl, @rightVal, @x, @y) ->
c_34(newline#6^#(newline#7(#equal(@x, @y),
@belowVal,
@diagVal,
@rightVal),
@nl),
newline#7^#(#equal(@x, @y), @belowVal, @diagVal, @rightVal),
#equal^#(@x, @y))
, newline#6^#(@elem, @nl) -> c_37()
, newline#7^#(#false(), @belowVal, @diagVal, @rightVal) ->
c_35(max^#(@belowVal, @rightVal))
, newline#7^#(#true(), @belowVal, @diagVal, @rightVal) ->
c_36(+^#(@diagVal, #pos(#s(#0())))) }
Weak DPs:
{ #eq^#(#0(), #0()) -> c_40()
, #eq^#(#0(), #neg(@y)) -> c_41()
, #eq^#(#0(), #pos(@y)) -> c_42()
, #eq^#(#0(), #s(@y)) -> c_43()
, #eq^#(#neg(@x), #0()) -> c_44()
, #eq^#(#neg(@x), #neg(@y)) -> c_45(#eq^#(@x, @y))
, #eq^#(#neg(@x), #pos(@y)) -> c_46()
, #eq^#(#pos(@x), #0()) -> c_47()
, #eq^#(#pos(@x), #neg(@y)) -> c_48()
, #eq^#(#pos(@x), #pos(@y)) -> c_49(#eq^#(@x, @y))
, #eq^#(#s(@x), #0()) -> c_50()
, #eq^#(#s(@x), #s(@y)) -> c_51(#eq^#(@x, @y))
, #eq^#(::(@x_1, @x_2), ::(@y_1, @y_2)) ->
c_52(#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_53()
, #eq^#(nil(), ::(@y_1, @y_2)) -> c_54()
, #eq^#(nil(), nil()) -> c_55()
, #ckgt^#(#EQ()) -> c_68()
, #ckgt^#(#GT()) -> c_69()
, #ckgt^#(#LT()) -> c_70()
, #compare^#(#0(), #0()) -> c_56()
, #compare^#(#0(), #neg(@y)) -> c_57()
, #compare^#(#0(), #pos(@y)) -> c_58()
, #compare^#(#0(), #s(@y)) -> c_59()
, #compare^#(#neg(@x), #0()) -> c_60()
, #compare^#(#neg(@x), #neg(@y)) -> c_61(#compare^#(@y, @x))
, #compare^#(#neg(@x), #pos(@y)) -> c_62()
, #compare^#(#pos(@x), #0()) -> c_63()
, #compare^#(#pos(@x), #neg(@y)) -> c_64()
, #compare^#(#pos(@x), #pos(@y)) -> c_65(#compare^#(@x, @y))
, #compare^#(#s(@x), #0()) -> c_66()
, #compare^#(#s(@x), #s(@y)) -> c_67(#compare^#(@x, @y))
, #add^#(#0(), @y) -> c_71()
, #add^#(#neg(#s(#0())), @y) -> c_72(#pred^#(@y))
, #add^#(#neg(#s(#s(@x))), @y) ->
c_73(#pred^#(#add(#pos(#s(@x)), @y)), #add^#(#pos(#s(@x)), @y))
, #add^#(#pos(#s(#0())), @y) -> c_74(#succ^#(@y))
, #add^#(#pos(#s(#s(@x))), @y) ->
c_75(#succ^#(#add(#pos(#s(@x)), @y)), #add^#(#pos(#s(@x)), @y))
, #and^#(#false(), #false()) -> c_84()
, #and^#(#false(), #true()) -> c_85()
, #and^#(#true(), #false()) -> c_86()
, #and^#(#true(), #true()) -> c_87()
, #pred^#(#0()) -> c_76()
, #pred^#(#neg(#s(@x))) -> c_77()
, #pred^#(#pos(#s(#0()))) -> c_78()
, #pred^#(#pos(#s(#s(@x)))) -> c_79()
, #succ^#(#0()) -> c_80()
, #succ^#(#neg(#s(#0()))) -> c_81()
, #succ^#(#neg(#s(#s(@x)))) -> c_82()
, #succ^#(#pos(#s(@x))) -> c_83() }
Weak Trs:
{ #abs(#0()) -> #0()
, #abs(#neg(@x)) -> #pos(@x)
, #abs(#pos(@x)) -> #pos(@x)
, #abs(#s(@x)) -> #pos(#s(@x))
, #equal(@x, @y) -> #eq(@x, @y)
, #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)
, #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()
, #greater(@x, @y) -> #ckgt(#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)
, #ckgt(#EQ()) -> #false()
, #ckgt(#GT()) -> #true()
, #ckgt(#LT()) -> #false()
, +(@x, @y) -> #add(@x, @y)
, #add(#0(), @y) -> @y
, #add(#neg(#s(#0())), @y) -> #pred(@y)
, #add(#neg(#s(#s(@x))), @y) -> #pred(#add(#pos(#s(@x)), @y))
, #add(#pos(#s(#0())), @y) -> #succ(@y)
, #add(#pos(#s(#s(@x))), @y) -> #succ(#add(#pos(#s(@x)), @y))
, firstline(@l) -> firstline#1(@l)
, firstline#1(::(@x, @xs)) -> ::(#abs(#0()), firstline(@xs))
, firstline#1(nil()) -> nil()
, lcs(@l1, @l2) -> lcs#1(lcstable(@l1, @l2))
, lcstable(@l1, @l2) -> lcstable#1(@l1, @l2)
, lcs#1(@m) -> lcs#2(@m)
, lcs#2(::(@l1, @_@2)) -> lcs#3(@l1)
, lcs#2(nil()) -> #abs(#0())
, lcs#3(::(@len, @_@1)) -> @len
, lcs#3(nil()) -> #abs(#0())
, lcstable#1(::(@x, @xs), @l2) ->
lcstable#2(lcstable(@xs, @l2), @l2, @x)
, lcstable#1(nil(), @l2) -> ::(firstline(@l2), nil())
, lcstable#2(@m, @l2, @x) -> lcstable#3(@m, @l2, @x)
, lcstable#3(::(@l, @ls), @l2, @x) ->
::(newline(@x, @l, @l2), ::(@l, @ls))
, lcstable#3(nil(), @l2, @x) -> nil()
, newline(@y, @lastline, @l) -> newline#1(@l, @lastline, @y)
, max(@a, @b) -> max#1(#greater(@a, @b), @a, @b)
, max#1(#false(), @a, @b) -> @b
, max#1(#true(), @a, @b) -> @a
, newline#1(::(@x, @xs), @lastline, @y) ->
newline#2(@lastline, @x, @xs, @y)
, newline#1(nil(), @lastline, @y) -> nil()
, newline#2(::(@belowVal, @lastline'), @x, @xs, @y) ->
newline#3(newline(@y, @lastline', @xs),
@belowVal,
@lastline',
@x,
@y)
, newline#2(nil(), @x, @xs, @y) -> nil()
, newline#3(@nl, @belowVal, @lastline', @x, @y) ->
newline#4(right(@nl), @belowVal, @lastline', @nl, @x, @y)
, right(@l) -> right#1(@l)
, newline#4(@rightVal, @belowVal, @lastline', @nl, @x, @y) ->
newline#5(right(@lastline'), @belowVal, @nl, @rightVal, @x, @y)
, newline#5(@diagVal, @belowVal, @nl, @rightVal, @x, @y) ->
newline#6(newline#7(#equal(@x, @y),
@belowVal,
@diagVal,
@rightVal),
@nl)
, newline#7(#false(), @belowVal, @diagVal, @rightVal) ->
max(@belowVal, @rightVal)
, newline#7(#true(), @belowVal, @diagVal, @rightVal) ->
+(@diagVal, #pos(#s(#0())))
, newline#6(@elem, @nl) -> ::(@elem, @nl)
, right#1(::(@x, @xs)) -> @x
, right#1(nil()) -> #abs(#0())
, #pred(#0()) -> #neg(#s(#0()))
, #pred(#neg(#s(@x))) -> #neg(#s(#s(@x)))
, #pred(#pos(#s(#0()))) -> #0()
, #pred(#pos(#s(#s(@x)))) -> #pos(#s(@x))
, #succ(#0()) -> #pos(#s(#0()))
, #succ(#neg(#s(#0()))) -> #0()
, #succ(#neg(#s(#s(@x)))) -> #neg(#s(@x))
, #succ(#pos(#s(@x))) -> #pos(#s(#s(@x)))
, #and(#false(), #false()) -> #false()
, #and(#false(), #true()) -> #false()
, #and(#true(), #false()) -> #false()
, #and(#true(), #true()) -> #true() }
Obligation:
innermost runtime complexity
Answer:
YES(O(1),O(n^2))
We estimate the number of application of
{1,2,3,4,7,15,19,22,24,25,27,31,34} by applications of
Pre({1,2,3,4,7,15,19,22,24,25,27,31,34}) =
{5,6,13,14,16,17,20,21,23,30,32,33,36}. Here rules are labeled as
follows:
DPs:
{ 1: #abs^#(#0()) -> c_1()
, 2: #equal^#(@x, @y) -> c_5(#eq^#(@x, @y))
, 3: #greater^#(@x, @y) ->
c_6(#ckgt^#(#compare(@x, @y)), #compare^#(@x, @y))
, 4: +^#(@x, @y) -> c_7(#add^#(@x, @y))
, 5: firstline^#(@l) -> c_8(firstline#1^#(@l))
, 6: firstline#1^#(::(@x, @xs)) ->
c_9(#abs^#(#0()), firstline^#(@xs))
, 7: firstline#1^#(nil()) -> c_10()
, 8: lcs^#(@l1, @l2) ->
c_11(lcs#1^#(lcstable(@l1, @l2)), lcstable^#(@l1, @l2))
, 9: lcs#1^#(@m) -> c_13(lcs#2^#(@m))
, 10: lcstable^#(@l1, @l2) -> c_12(lcstable#1^#(@l1, @l2))
, 11: lcstable#1^#(::(@x, @xs), @l2) ->
c_18(lcstable#2^#(lcstable(@xs, @l2), @l2, @x),
lcstable^#(@xs, @l2))
, 12: lcstable#1^#(nil(), @l2) -> c_19(firstline^#(@l2))
, 13: lcs#2^#(::(@l1, @_@2)) -> c_14(lcs#3^#(@l1))
, 14: lcs#2^#(nil()) -> c_15(#abs^#(#0()))
, 15: lcs#3^#(::(@len, @_@1)) -> c_16()
, 16: lcs#3^#(nil()) -> c_17(#abs^#(#0()))
, 17: lcstable#2^#(@m, @l2, @x) -> c_20(lcstable#3^#(@m, @l2, @x))
, 18: lcstable#3^#(::(@l, @ls), @l2, @x) ->
c_21(newline^#(@x, @l, @l2))
, 19: lcstable#3^#(nil(), @l2, @x) -> c_22()
, 20: newline^#(@y, @lastline, @l) ->
c_23(newline#1^#(@l, @lastline, @y))
, 21: newline#1^#(::(@x, @xs), @lastline, @y) ->
c_27(newline#2^#(@lastline, @x, @xs, @y))
, 22: newline#1^#(nil(), @lastline, @y) -> c_28()
, 23: max^#(@a, @b) ->
c_24(max#1^#(#greater(@a, @b), @a, @b), #greater^#(@a, @b))
, 24: max#1^#(#false(), @a, @b) -> c_25()
, 25: max#1^#(#true(), @a, @b) -> c_26()
, 26: newline#2^#(::(@belowVal, @lastline'), @x, @xs, @y) ->
c_29(newline#3^#(newline(@y, @lastline', @xs),
@belowVal,
@lastline',
@x,
@y),
newline^#(@y, @lastline', @xs))
, 27: newline#2^#(nil(), @x, @xs, @y) -> c_30()
, 28: newline#3^#(@nl, @belowVal, @lastline', @x, @y) ->
c_31(newline#4^#(right(@nl), @belowVal, @lastline', @nl, @x, @y),
right^#(@nl))
, 29: newline#4^#(@rightVal, @belowVal, @lastline', @nl, @x, @y) ->
c_33(newline#5^#(right(@lastline'),
@belowVal,
@nl,
@rightVal,
@x,
@y),
right^#(@lastline'))
, 30: right^#(@l) -> c_32(right#1^#(@l))
, 31: right#1^#(::(@x, @xs)) -> c_38()
, 32: right#1^#(nil()) -> c_39(#abs^#(#0()))
, 33: newline#5^#(@diagVal, @belowVal, @nl, @rightVal, @x, @y) ->
c_34(newline#6^#(newline#7(#equal(@x, @y),
@belowVal,
@diagVal,
@rightVal),
@nl),
newline#7^#(#equal(@x, @y), @belowVal, @diagVal, @rightVal),
#equal^#(@x, @y))
, 34: newline#6^#(@elem, @nl) -> c_37()
, 35: newline#7^#(#false(), @belowVal, @diagVal, @rightVal) ->
c_35(max^#(@belowVal, @rightVal))
, 36: newline#7^#(#true(), @belowVal, @diagVal, @rightVal) ->
c_36(+^#(@diagVal, #pos(#s(#0()))))
, 37: #eq^#(#0(), #0()) -> c_40()
, 38: #eq^#(#0(), #neg(@y)) -> c_41()
, 39: #eq^#(#0(), #pos(@y)) -> c_42()
, 40: #eq^#(#0(), #s(@y)) -> c_43()
, 41: #eq^#(#neg(@x), #0()) -> c_44()
, 42: #eq^#(#neg(@x), #neg(@y)) -> c_45(#eq^#(@x, @y))
, 43: #eq^#(#neg(@x), #pos(@y)) -> c_46()
, 44: #eq^#(#pos(@x), #0()) -> c_47()
, 45: #eq^#(#pos(@x), #neg(@y)) -> c_48()
, 46: #eq^#(#pos(@x), #pos(@y)) -> c_49(#eq^#(@x, @y))
, 47: #eq^#(#s(@x), #0()) -> c_50()
, 48: #eq^#(#s(@x), #s(@y)) -> c_51(#eq^#(@x, @y))
, 49: #eq^#(::(@x_1, @x_2), ::(@y_1, @y_2)) ->
c_52(#and^#(#eq(@x_1, @y_1), #eq(@x_2, @y_2)),
#eq^#(@x_1, @y_1),
#eq^#(@x_2, @y_2))
, 50: #eq^#(::(@x_1, @x_2), nil()) -> c_53()
, 51: #eq^#(nil(), ::(@y_1, @y_2)) -> c_54()
, 52: #eq^#(nil(), nil()) -> c_55()
, 53: #ckgt^#(#EQ()) -> c_68()
, 54: #ckgt^#(#GT()) -> c_69()
, 55: #ckgt^#(#LT()) -> c_70()
, 56: #compare^#(#0(), #0()) -> c_56()
, 57: #compare^#(#0(), #neg(@y)) -> c_57()
, 58: #compare^#(#0(), #pos(@y)) -> c_58()
, 59: #compare^#(#0(), #s(@y)) -> c_59()
, 60: #compare^#(#neg(@x), #0()) -> c_60()
, 61: #compare^#(#neg(@x), #neg(@y)) -> c_61(#compare^#(@y, @x))
, 62: #compare^#(#neg(@x), #pos(@y)) -> c_62()
, 63: #compare^#(#pos(@x), #0()) -> c_63()
, 64: #compare^#(#pos(@x), #neg(@y)) -> c_64()
, 65: #compare^#(#pos(@x), #pos(@y)) -> c_65(#compare^#(@x, @y))
, 66: #compare^#(#s(@x), #0()) -> c_66()
, 67: #compare^#(#s(@x), #s(@y)) -> c_67(#compare^#(@x, @y))
, 68: #add^#(#0(), @y) -> c_71()
, 69: #add^#(#neg(#s(#0())), @y) -> c_72(#pred^#(@y))
, 70: #add^#(#neg(#s(#s(@x))), @y) ->
c_73(#pred^#(#add(#pos(#s(@x)), @y)), #add^#(#pos(#s(@x)), @y))
, 71: #add^#(#pos(#s(#0())), @y) -> c_74(#succ^#(@y))
, 72: #add^#(#pos(#s(#s(@x))), @y) ->
c_75(#succ^#(#add(#pos(#s(@x)), @y)), #add^#(#pos(#s(@x)), @y))
, 73: #and^#(#false(), #false()) -> c_84()
, 74: #and^#(#false(), #true()) -> c_85()
, 75: #and^#(#true(), #false()) -> c_86()
, 76: #and^#(#true(), #true()) -> c_87()
, 77: #pred^#(#0()) -> c_76()
, 78: #pred^#(#neg(#s(@x))) -> c_77()
, 79: #pred^#(#pos(#s(#0()))) -> c_78()
, 80: #pred^#(#pos(#s(#s(@x)))) -> c_79()
, 81: #succ^#(#0()) -> c_80()
, 82: #succ^#(#neg(#s(#0()))) -> c_81()
, 83: #succ^#(#neg(#s(#s(@x)))) -> c_82()
, 84: #succ^#(#pos(#s(@x))) -> c_83() }
We are left with following problem, upon which TcT provides the
certificate YES(O(1),O(n^2)).
Strict DPs:
{ firstline^#(@l) -> c_8(firstline#1^#(@l))
, firstline#1^#(::(@x, @xs)) -> c_9(#abs^#(#0()), firstline^#(@xs))
, lcs^#(@l1, @l2) ->
c_11(lcs#1^#(lcstable(@l1, @l2)), lcstable^#(@l1, @l2))
, lcs#1^#(@m) -> c_13(lcs#2^#(@m))
, lcstable^#(@l1, @l2) -> c_12(lcstable#1^#(@l1, @l2))
, lcstable#1^#(::(@x, @xs), @l2) ->
c_18(lcstable#2^#(lcstable(@xs, @l2), @l2, @x),
lcstable^#(@xs, @l2))
, lcstable#1^#(nil(), @l2) -> c_19(firstline^#(@l2))
, lcs#2^#(::(@l1, @_@2)) -> c_14(lcs#3^#(@l1))
, lcs#2^#(nil()) -> c_15(#abs^#(#0()))
, lcs#3^#(nil()) -> c_17(#abs^#(#0()))
, lcstable#2^#(@m, @l2, @x) -> c_20(lcstable#3^#(@m, @l2, @x))
, lcstable#3^#(::(@l, @ls), @l2, @x) ->
c_21(newline^#(@x, @l, @l2))
, newline^#(@y, @lastline, @l) ->
c_23(newline#1^#(@l, @lastline, @y))
, newline#1^#(::(@x, @xs), @lastline, @y) ->
c_27(newline#2^#(@lastline, @x, @xs, @y))
, max^#(@a, @b) ->
c_24(max#1^#(#greater(@a, @b), @a, @b), #greater^#(@a, @b))
, newline#2^#(::(@belowVal, @lastline'), @x, @xs, @y) ->
c_29(newline#3^#(newline(@y, @lastline', @xs),
@belowVal,
@lastline',
@x,
@y),
newline^#(@y, @lastline', @xs))
, newline#3^#(@nl, @belowVal, @lastline', @x, @y) ->
c_31(newline#4^#(right(@nl), @belowVal, @lastline', @nl, @x, @y),
right^#(@nl))
, newline#4^#(@rightVal, @belowVal, @lastline', @nl, @x, @y) ->
c_33(newline#5^#(right(@lastline'),
@belowVal,
@nl,
@rightVal,
@x,
@y),
right^#(@lastline'))
, right^#(@l) -> c_32(right#1^#(@l))
, right#1^#(nil()) -> c_39(#abs^#(#0()))
, newline#5^#(@diagVal, @belowVal, @nl, @rightVal, @x, @y) ->
c_34(newline#6^#(newline#7(#equal(@x, @y),
@belowVal,
@diagVal,
@rightVal),
@nl),
newline#7^#(#equal(@x, @y), @belowVal, @diagVal, @rightVal),
#equal^#(@x, @y))
, newline#7^#(#false(), @belowVal, @diagVal, @rightVal) ->
c_35(max^#(@belowVal, @rightVal))
, newline#7^#(#true(), @belowVal, @diagVal, @rightVal) ->
c_36(+^#(@diagVal, #pos(#s(#0())))) }
Weak DPs:
{ #abs^#(#0()) -> c_1()
, #equal^#(@x, @y) -> c_5(#eq^#(@x, @y))
, #eq^#(#0(), #0()) -> c_40()
, #eq^#(#0(), #neg(@y)) -> c_41()
, #eq^#(#0(), #pos(@y)) -> c_42()
, #eq^#(#0(), #s(@y)) -> c_43()
, #eq^#(#neg(@x), #0()) -> c_44()
, #eq^#(#neg(@x), #neg(@y)) -> c_45(#eq^#(@x, @y))
, #eq^#(#neg(@x), #pos(@y)) -> c_46()
, #eq^#(#pos(@x), #0()) -> c_47()
, #eq^#(#pos(@x), #neg(@y)) -> c_48()
, #eq^#(#pos(@x), #pos(@y)) -> c_49(#eq^#(@x, @y))
, #eq^#(#s(@x), #0()) -> c_50()
, #eq^#(#s(@x), #s(@y)) -> c_51(#eq^#(@x, @y))
, #eq^#(::(@x_1, @x_2), ::(@y_1, @y_2)) ->
c_52(#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_53()
, #eq^#(nil(), ::(@y_1, @y_2)) -> c_54()
, #eq^#(nil(), nil()) -> c_55()
, #greater^#(@x, @y) ->
c_6(#ckgt^#(#compare(@x, @y)), #compare^#(@x, @y))
, #ckgt^#(#EQ()) -> c_68()
, #ckgt^#(#GT()) -> c_69()
, #ckgt^#(#LT()) -> c_70()
, #compare^#(#0(), #0()) -> c_56()
, #compare^#(#0(), #neg(@y)) -> c_57()
, #compare^#(#0(), #pos(@y)) -> c_58()
, #compare^#(#0(), #s(@y)) -> c_59()
, #compare^#(#neg(@x), #0()) -> c_60()
, #compare^#(#neg(@x), #neg(@y)) -> c_61(#compare^#(@y, @x))
, #compare^#(#neg(@x), #pos(@y)) -> c_62()
, #compare^#(#pos(@x), #0()) -> c_63()
, #compare^#(#pos(@x), #neg(@y)) -> c_64()
, #compare^#(#pos(@x), #pos(@y)) -> c_65(#compare^#(@x, @y))
, #compare^#(#s(@x), #0()) -> c_66()
, #compare^#(#s(@x), #s(@y)) -> c_67(#compare^#(@x, @y))
, +^#(@x, @y) -> c_7(#add^#(@x, @y))
, #add^#(#0(), @y) -> c_71()
, #add^#(#neg(#s(#0())), @y) -> c_72(#pred^#(@y))
, #add^#(#neg(#s(#s(@x))), @y) ->
c_73(#pred^#(#add(#pos(#s(@x)), @y)), #add^#(#pos(#s(@x)), @y))
, #add^#(#pos(#s(#0())), @y) -> c_74(#succ^#(@y))
, #add^#(#pos(#s(#s(@x))), @y) ->
c_75(#succ^#(#add(#pos(#s(@x)), @y)), #add^#(#pos(#s(@x)), @y))
, firstline#1^#(nil()) -> c_10()
, lcs#3^#(::(@len, @_@1)) -> c_16()
, lcstable#3^#(nil(), @l2, @x) -> c_22()
, newline#1^#(nil(), @lastline, @y) -> c_28()
, max#1^#(#false(), @a, @b) -> c_25()
, max#1^#(#true(), @a, @b) -> c_26()
, newline#2^#(nil(), @x, @xs, @y) -> c_30()
, right#1^#(::(@x, @xs)) -> c_38()
, newline#6^#(@elem, @nl) -> c_37()
, #and^#(#false(), #false()) -> c_84()
, #and^#(#false(), #true()) -> c_85()
, #and^#(#true(), #false()) -> c_86()
, #and^#(#true(), #true()) -> c_87()
, #pred^#(#0()) -> c_76()
, #pred^#(#neg(#s(@x))) -> c_77()
, #pred^#(#pos(#s(#0()))) -> c_78()
, #pred^#(#pos(#s(#s(@x)))) -> c_79()
, #succ^#(#0()) -> c_80()
, #succ^#(#neg(#s(#0()))) -> c_81()
, #succ^#(#neg(#s(#s(@x)))) -> c_82()
, #succ^#(#pos(#s(@x))) -> c_83() }
Weak Trs:
{ #abs(#0()) -> #0()
, #abs(#neg(@x)) -> #pos(@x)
, #abs(#pos(@x)) -> #pos(@x)
, #abs(#s(@x)) -> #pos(#s(@x))
, #equal(@x, @y) -> #eq(@x, @y)
, #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)
, #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()
, #greater(@x, @y) -> #ckgt(#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)
, #ckgt(#EQ()) -> #false()
, #ckgt(#GT()) -> #true()
, #ckgt(#LT()) -> #false()
, +(@x, @y) -> #add(@x, @y)
, #add(#0(), @y) -> @y
, #add(#neg(#s(#0())), @y) -> #pred(@y)
, #add(#neg(#s(#s(@x))), @y) -> #pred(#add(#pos(#s(@x)), @y))
, #add(#pos(#s(#0())), @y) -> #succ(@y)
, #add(#pos(#s(#s(@x))), @y) -> #succ(#add(#pos(#s(@x)), @y))
, firstline(@l) -> firstline#1(@l)
, firstline#1(::(@x, @xs)) -> ::(#abs(#0()), firstline(@xs))
, firstline#1(nil()) -> nil()
, lcs(@l1, @l2) -> lcs#1(lcstable(@l1, @l2))
, lcstable(@l1, @l2) -> lcstable#1(@l1, @l2)
, lcs#1(@m) -> lcs#2(@m)
, lcs#2(::(@l1, @_@2)) -> lcs#3(@l1)
, lcs#2(nil()) -> #abs(#0())
, lcs#3(::(@len, @_@1)) -> @len
, lcs#3(nil()) -> #abs(#0())
, lcstable#1(::(@x, @xs), @l2) ->
lcstable#2(lcstable(@xs, @l2), @l2, @x)
, lcstable#1(nil(), @l2) -> ::(firstline(@l2), nil())
, lcstable#2(@m, @l2, @x) -> lcstable#3(@m, @l2, @x)
, lcstable#3(::(@l, @ls), @l2, @x) ->
::(newline(@x, @l, @l2), ::(@l, @ls))
, lcstable#3(nil(), @l2, @x) -> nil()
, newline(@y, @lastline, @l) -> newline#1(@l, @lastline, @y)
, max(@a, @b) -> max#1(#greater(@a, @b), @a, @b)
, max#1(#false(), @a, @b) -> @b
, max#1(#true(), @a, @b) -> @a
, newline#1(::(@x, @xs), @lastline, @y) ->
newline#2(@lastline, @x, @xs, @y)
, newline#1(nil(), @lastline, @y) -> nil()
, newline#2(::(@belowVal, @lastline'), @x, @xs, @y) ->
newline#3(newline(@y, @lastline', @xs),
@belowVal,
@lastline',
@x,
@y)
, newline#2(nil(), @x, @xs, @y) -> nil()
, newline#3(@nl, @belowVal, @lastline', @x, @y) ->
newline#4(right(@nl), @belowVal, @lastline', @nl, @x, @y)
, right(@l) -> right#1(@l)
, newline#4(@rightVal, @belowVal, @lastline', @nl, @x, @y) ->
newline#5(right(@lastline'), @belowVal, @nl, @rightVal, @x, @y)
, newline#5(@diagVal, @belowVal, @nl, @rightVal, @x, @y) ->
newline#6(newline#7(#equal(@x, @y),
@belowVal,
@diagVal,
@rightVal),
@nl)
, newline#7(#false(), @belowVal, @diagVal, @rightVal) ->
max(@belowVal, @rightVal)
, newline#7(#true(), @belowVal, @diagVal, @rightVal) ->
+(@diagVal, #pos(#s(#0())))
, newline#6(@elem, @nl) -> ::(@elem, @nl)
, right#1(::(@x, @xs)) -> @x
, right#1(nil()) -> #abs(#0())
, #pred(#0()) -> #neg(#s(#0()))
, #pred(#neg(#s(@x))) -> #neg(#s(#s(@x)))
, #pred(#pos(#s(#0()))) -> #0()
, #pred(#pos(#s(#s(@x)))) -> #pos(#s(@x))
, #succ(#0()) -> #pos(#s(#0()))
, #succ(#neg(#s(#0()))) -> #0()
, #succ(#neg(#s(#s(@x)))) -> #neg(#s(@x))
, #succ(#pos(#s(@x))) -> #pos(#s(#s(@x)))
, #and(#false(), #false()) -> #false()
, #and(#false(), #true()) -> #false()
, #and(#true(), #false()) -> #false()
, #and(#true(), #true()) -> #true() }
Obligation:
innermost runtime complexity
Answer:
YES(O(1),O(n^2))
We estimate the number of application of {9,10,15,20,23} by
applications of Pre({9,10,15,20,23}) = {4,8,19,21,22}. Here rules
are labeled as follows:
DPs:
{ 1: firstline^#(@l) -> c_8(firstline#1^#(@l))
, 2: firstline#1^#(::(@x, @xs)) ->
c_9(#abs^#(#0()), firstline^#(@xs))
, 3: lcs^#(@l1, @l2) ->
c_11(lcs#1^#(lcstable(@l1, @l2)), lcstable^#(@l1, @l2))
, 4: lcs#1^#(@m) -> c_13(lcs#2^#(@m))
, 5: lcstable^#(@l1, @l2) -> c_12(lcstable#1^#(@l1, @l2))
, 6: lcstable#1^#(::(@x, @xs), @l2) ->
c_18(lcstable#2^#(lcstable(@xs, @l2), @l2, @x),
lcstable^#(@xs, @l2))
, 7: lcstable#1^#(nil(), @l2) -> c_19(firstline^#(@l2))
, 8: lcs#2^#(::(@l1, @_@2)) -> c_14(lcs#3^#(@l1))
, 9: lcs#2^#(nil()) -> c_15(#abs^#(#0()))
, 10: lcs#3^#(nil()) -> c_17(#abs^#(#0()))
, 11: lcstable#2^#(@m, @l2, @x) -> c_20(lcstable#3^#(@m, @l2, @x))
, 12: lcstable#3^#(::(@l, @ls), @l2, @x) ->
c_21(newline^#(@x, @l, @l2))
, 13: newline^#(@y, @lastline, @l) ->
c_23(newline#1^#(@l, @lastline, @y))
, 14: newline#1^#(::(@x, @xs), @lastline, @y) ->
c_27(newline#2^#(@lastline, @x, @xs, @y))
, 15: max^#(@a, @b) ->
c_24(max#1^#(#greater(@a, @b), @a, @b), #greater^#(@a, @b))
, 16: newline#2^#(::(@belowVal, @lastline'), @x, @xs, @y) ->
c_29(newline#3^#(newline(@y, @lastline', @xs),
@belowVal,
@lastline',
@x,
@y),
newline^#(@y, @lastline', @xs))
, 17: newline#3^#(@nl, @belowVal, @lastline', @x, @y) ->
c_31(newline#4^#(right(@nl), @belowVal, @lastline', @nl, @x, @y),
right^#(@nl))
, 18: newline#4^#(@rightVal, @belowVal, @lastline', @nl, @x, @y) ->
c_33(newline#5^#(right(@lastline'),
@belowVal,
@nl,
@rightVal,
@x,
@y),
right^#(@lastline'))
, 19: right^#(@l) -> c_32(right#1^#(@l))
, 20: right#1^#(nil()) -> c_39(#abs^#(#0()))
, 21: newline#5^#(@diagVal, @belowVal, @nl, @rightVal, @x, @y) ->
c_34(newline#6^#(newline#7(#equal(@x, @y),
@belowVal,
@diagVal,
@rightVal),
@nl),
newline#7^#(#equal(@x, @y), @belowVal, @diagVal, @rightVal),
#equal^#(@x, @y))
, 22: newline#7^#(#false(), @belowVal, @diagVal, @rightVal) ->
c_35(max^#(@belowVal, @rightVal))
, 23: newline#7^#(#true(), @belowVal, @diagVal, @rightVal) ->
c_36(+^#(@diagVal, #pos(#s(#0()))))
, 24: #abs^#(#0()) -> c_1()
, 25: #equal^#(@x, @y) -> c_5(#eq^#(@x, @y))
, 26: #eq^#(#0(), #0()) -> c_40()
, 27: #eq^#(#0(), #neg(@y)) -> c_41()
, 28: #eq^#(#0(), #pos(@y)) -> c_42()
, 29: #eq^#(#0(), #s(@y)) -> c_43()
, 30: #eq^#(#neg(@x), #0()) -> c_44()
, 31: #eq^#(#neg(@x), #neg(@y)) -> c_45(#eq^#(@x, @y))
, 32: #eq^#(#neg(@x), #pos(@y)) -> c_46()
, 33: #eq^#(#pos(@x), #0()) -> c_47()
, 34: #eq^#(#pos(@x), #neg(@y)) -> c_48()
, 35: #eq^#(#pos(@x), #pos(@y)) -> c_49(#eq^#(@x, @y))
, 36: #eq^#(#s(@x), #0()) -> c_50()
, 37: #eq^#(#s(@x), #s(@y)) -> c_51(#eq^#(@x, @y))
, 38: #eq^#(::(@x_1, @x_2), ::(@y_1, @y_2)) ->
c_52(#and^#(#eq(@x_1, @y_1), #eq(@x_2, @y_2)),
#eq^#(@x_1, @y_1),
#eq^#(@x_2, @y_2))
, 39: #eq^#(::(@x_1, @x_2), nil()) -> c_53()
, 40: #eq^#(nil(), ::(@y_1, @y_2)) -> c_54()
, 41: #eq^#(nil(), nil()) -> c_55()
, 42: #greater^#(@x, @y) ->
c_6(#ckgt^#(#compare(@x, @y)), #compare^#(@x, @y))
, 43: #ckgt^#(#EQ()) -> c_68()
, 44: #ckgt^#(#GT()) -> c_69()
, 45: #ckgt^#(#LT()) -> c_70()
, 46: #compare^#(#0(), #0()) -> c_56()
, 47: #compare^#(#0(), #neg(@y)) -> c_57()
, 48: #compare^#(#0(), #pos(@y)) -> c_58()
, 49: #compare^#(#0(), #s(@y)) -> c_59()
, 50: #compare^#(#neg(@x), #0()) -> c_60()
, 51: #compare^#(#neg(@x), #neg(@y)) -> c_61(#compare^#(@y, @x))
, 52: #compare^#(#neg(@x), #pos(@y)) -> c_62()
, 53: #compare^#(#pos(@x), #0()) -> c_63()
, 54: #compare^#(#pos(@x), #neg(@y)) -> c_64()
, 55: #compare^#(#pos(@x), #pos(@y)) -> c_65(#compare^#(@x, @y))
, 56: #compare^#(#s(@x), #0()) -> c_66()
, 57: #compare^#(#s(@x), #s(@y)) -> c_67(#compare^#(@x, @y))
, 58: +^#(@x, @y) -> c_7(#add^#(@x, @y))
, 59: #add^#(#0(), @y) -> c_71()
, 60: #add^#(#neg(#s(#0())), @y) -> c_72(#pred^#(@y))
, 61: #add^#(#neg(#s(#s(@x))), @y) ->
c_73(#pred^#(#add(#pos(#s(@x)), @y)), #add^#(#pos(#s(@x)), @y))
, 62: #add^#(#pos(#s(#0())), @y) -> c_74(#succ^#(@y))
, 63: #add^#(#pos(#s(#s(@x))), @y) ->
c_75(#succ^#(#add(#pos(#s(@x)), @y)), #add^#(#pos(#s(@x)), @y))
, 64: firstline#1^#(nil()) -> c_10()
, 65: lcs#3^#(::(@len, @_@1)) -> c_16()
, 66: lcstable#3^#(nil(), @l2, @x) -> c_22()
, 67: newline#1^#(nil(), @lastline, @y) -> c_28()
, 68: max#1^#(#false(), @a, @b) -> c_25()
, 69: max#1^#(#true(), @a, @b) -> c_26()
, 70: newline#2^#(nil(), @x, @xs, @y) -> c_30()
, 71: right#1^#(::(@x, @xs)) -> c_38()
, 72: newline#6^#(@elem, @nl) -> c_37()
, 73: #and^#(#false(), #false()) -> c_84()
, 74: #and^#(#false(), #true()) -> c_85()
, 75: #and^#(#true(), #false()) -> c_86()
, 76: #and^#(#true(), #true()) -> c_87()
, 77: #pred^#(#0()) -> c_76()
, 78: #pred^#(#neg(#s(@x))) -> c_77()
, 79: #pred^#(#pos(#s(#0()))) -> c_78()
, 80: #pred^#(#pos(#s(#s(@x)))) -> c_79()
, 81: #succ^#(#0()) -> c_80()
, 82: #succ^#(#neg(#s(#0()))) -> c_81()
, 83: #succ^#(#neg(#s(#s(@x)))) -> c_82()
, 84: #succ^#(#pos(#s(@x))) -> c_83() }
We are left with following problem, upon which TcT provides the
certificate YES(O(1),O(n^2)).
Strict DPs:
{ firstline^#(@l) -> c_8(firstline#1^#(@l))
, firstline#1^#(::(@x, @xs)) -> c_9(#abs^#(#0()), firstline^#(@xs))
, lcs^#(@l1, @l2) ->
c_11(lcs#1^#(lcstable(@l1, @l2)), lcstable^#(@l1, @l2))
, lcs#1^#(@m) -> c_13(lcs#2^#(@m))
, lcstable^#(@l1, @l2) -> c_12(lcstable#1^#(@l1, @l2))
, lcstable#1^#(::(@x, @xs), @l2) ->
c_18(lcstable#2^#(lcstable(@xs, @l2), @l2, @x),
lcstable^#(@xs, @l2))
, lcstable#1^#(nil(), @l2) -> c_19(firstline^#(@l2))
, lcs#2^#(::(@l1, @_@2)) -> c_14(lcs#3^#(@l1))
, lcstable#2^#(@m, @l2, @x) -> c_20(lcstable#3^#(@m, @l2, @x))
, lcstable#3^#(::(@l, @ls), @l2, @x) ->
c_21(newline^#(@x, @l, @l2))
, newline^#(@y, @lastline, @l) ->
c_23(newline#1^#(@l, @lastline, @y))
, newline#1^#(::(@x, @xs), @lastline, @y) ->
c_27(newline#2^#(@lastline, @x, @xs, @y))
, newline#2^#(::(@belowVal, @lastline'), @x, @xs, @y) ->
c_29(newline#3^#(newline(@y, @lastline', @xs),
@belowVal,
@lastline',
@x,
@y),
newline^#(@y, @lastline', @xs))
, newline#3^#(@nl, @belowVal, @lastline', @x, @y) ->
c_31(newline#4^#(right(@nl), @belowVal, @lastline', @nl, @x, @y),
right^#(@nl))
, newline#4^#(@rightVal, @belowVal, @lastline', @nl, @x, @y) ->
c_33(newline#5^#(right(@lastline'),
@belowVal,
@nl,
@rightVal,
@x,
@y),
right^#(@lastline'))
, right^#(@l) -> c_32(right#1^#(@l))
, newline#5^#(@diagVal, @belowVal, @nl, @rightVal, @x, @y) ->
c_34(newline#6^#(newline#7(#equal(@x, @y),
@belowVal,
@diagVal,
@rightVal),
@nl),
newline#7^#(#equal(@x, @y), @belowVal, @diagVal, @rightVal),
#equal^#(@x, @y))
, newline#7^#(#false(), @belowVal, @diagVal, @rightVal) ->
c_35(max^#(@belowVal, @rightVal)) }
Weak DPs:
{ #abs^#(#0()) -> c_1()
, #equal^#(@x, @y) -> c_5(#eq^#(@x, @y))
, #eq^#(#0(), #0()) -> c_40()
, #eq^#(#0(), #neg(@y)) -> c_41()
, #eq^#(#0(), #pos(@y)) -> c_42()
, #eq^#(#0(), #s(@y)) -> c_43()
, #eq^#(#neg(@x), #0()) -> c_44()
, #eq^#(#neg(@x), #neg(@y)) -> c_45(#eq^#(@x, @y))
, #eq^#(#neg(@x), #pos(@y)) -> c_46()
, #eq^#(#pos(@x), #0()) -> c_47()
, #eq^#(#pos(@x), #neg(@y)) -> c_48()
, #eq^#(#pos(@x), #pos(@y)) -> c_49(#eq^#(@x, @y))
, #eq^#(#s(@x), #0()) -> c_50()
, #eq^#(#s(@x), #s(@y)) -> c_51(#eq^#(@x, @y))
, #eq^#(::(@x_1, @x_2), ::(@y_1, @y_2)) ->
c_52(#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_53()
, #eq^#(nil(), ::(@y_1, @y_2)) -> c_54()
, #eq^#(nil(), nil()) -> c_55()
, #greater^#(@x, @y) ->
c_6(#ckgt^#(#compare(@x, @y)), #compare^#(@x, @y))
, #ckgt^#(#EQ()) -> c_68()
, #ckgt^#(#GT()) -> c_69()
, #ckgt^#(#LT()) -> c_70()
, #compare^#(#0(), #0()) -> c_56()
, #compare^#(#0(), #neg(@y)) -> c_57()
, #compare^#(#0(), #pos(@y)) -> c_58()
, #compare^#(#0(), #s(@y)) -> c_59()
, #compare^#(#neg(@x), #0()) -> c_60()
, #compare^#(#neg(@x), #neg(@y)) -> c_61(#compare^#(@y, @x))
, #compare^#(#neg(@x), #pos(@y)) -> c_62()
, #compare^#(#pos(@x), #0()) -> c_63()
, #compare^#(#pos(@x), #neg(@y)) -> c_64()
, #compare^#(#pos(@x), #pos(@y)) -> c_65(#compare^#(@x, @y))
, #compare^#(#s(@x), #0()) -> c_66()
, #compare^#(#s(@x), #s(@y)) -> c_67(#compare^#(@x, @y))
, +^#(@x, @y) -> c_7(#add^#(@x, @y))
, #add^#(#0(), @y) -> c_71()
, #add^#(#neg(#s(#0())), @y) -> c_72(#pred^#(@y))
, #add^#(#neg(#s(#s(@x))), @y) ->
c_73(#pred^#(#add(#pos(#s(@x)), @y)), #add^#(#pos(#s(@x)), @y))
, #add^#(#pos(#s(#0())), @y) -> c_74(#succ^#(@y))
, #add^#(#pos(#s(#s(@x))), @y) ->
c_75(#succ^#(#add(#pos(#s(@x)), @y)), #add^#(#pos(#s(@x)), @y))
, firstline#1^#(nil()) -> c_10()
, lcs#2^#(nil()) -> c_15(#abs^#(#0()))
, lcs#3^#(::(@len, @_@1)) -> c_16()
, lcs#3^#(nil()) -> c_17(#abs^#(#0()))
, lcstable#3^#(nil(), @l2, @x) -> c_22()
, newline#1^#(nil(), @lastline, @y) -> c_28()
, max^#(@a, @b) ->
c_24(max#1^#(#greater(@a, @b), @a, @b), #greater^#(@a, @b))
, max#1^#(#false(), @a, @b) -> c_25()
, max#1^#(#true(), @a, @b) -> c_26()
, newline#2^#(nil(), @x, @xs, @y) -> c_30()
, right#1^#(::(@x, @xs)) -> c_38()
, right#1^#(nil()) -> c_39(#abs^#(#0()))
, newline#6^#(@elem, @nl) -> c_37()
, newline#7^#(#true(), @belowVal, @diagVal, @rightVal) ->
c_36(+^#(@diagVal, #pos(#s(#0()))))
, #and^#(#false(), #false()) -> c_84()
, #and^#(#false(), #true()) -> c_85()
, #and^#(#true(), #false()) -> c_86()
, #and^#(#true(), #true()) -> c_87()
, #pred^#(#0()) -> c_76()
, #pred^#(#neg(#s(@x))) -> c_77()
, #pred^#(#pos(#s(#0()))) -> c_78()
, #pred^#(#pos(#s(#s(@x)))) -> c_79()
, #succ^#(#0()) -> c_80()
, #succ^#(#neg(#s(#0()))) -> c_81()
, #succ^#(#neg(#s(#s(@x)))) -> c_82()
, #succ^#(#pos(#s(@x))) -> c_83() }
Weak Trs:
{ #abs(#0()) -> #0()
, #abs(#neg(@x)) -> #pos(@x)
, #abs(#pos(@x)) -> #pos(@x)
, #abs(#s(@x)) -> #pos(#s(@x))
, #equal(@x, @y) -> #eq(@x, @y)
, #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)
, #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()
, #greater(@x, @y) -> #ckgt(#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)
, #ckgt(#EQ()) -> #false()
, #ckgt(#GT()) -> #true()
, #ckgt(#LT()) -> #false()
, +(@x, @y) -> #add(@x, @y)
, #add(#0(), @y) -> @y
, #add(#neg(#s(#0())), @y) -> #pred(@y)
, #add(#neg(#s(#s(@x))), @y) -> #pred(#add(#pos(#s(@x)), @y))
, #add(#pos(#s(#0())), @y) -> #succ(@y)
, #add(#pos(#s(#s(@x))), @y) -> #succ(#add(#pos(#s(@x)), @y))
, firstline(@l) -> firstline#1(@l)
, firstline#1(::(@x, @xs)) -> ::(#abs(#0()), firstline(@xs))
, firstline#1(nil()) -> nil()
, lcs(@l1, @l2) -> lcs#1(lcstable(@l1, @l2))
, lcstable(@l1, @l2) -> lcstable#1(@l1, @l2)
, lcs#1(@m) -> lcs#2(@m)
, lcs#2(::(@l1, @_@2)) -> lcs#3(@l1)
, lcs#2(nil()) -> #abs(#0())
, lcs#3(::(@len, @_@1)) -> @len
, lcs#3(nil()) -> #abs(#0())
, lcstable#1(::(@x, @xs), @l2) ->
lcstable#2(lcstable(@xs, @l2), @l2, @x)
, lcstable#1(nil(), @l2) -> ::(firstline(@l2), nil())
, lcstable#2(@m, @l2, @x) -> lcstable#3(@m, @l2, @x)
, lcstable#3(::(@l, @ls), @l2, @x) ->
::(newline(@x, @l, @l2), ::(@l, @ls))
, lcstable#3(nil(), @l2, @x) -> nil()
, newline(@y, @lastline, @l) -> newline#1(@l, @lastline, @y)
, max(@a, @b) -> max#1(#greater(@a, @b), @a, @b)
, max#1(#false(), @a, @b) -> @b
, max#1(#true(), @a, @b) -> @a
, newline#1(::(@x, @xs), @lastline, @y) ->
newline#2(@lastline, @x, @xs, @y)
, newline#1(nil(), @lastline, @y) -> nil()
, newline#2(::(@belowVal, @lastline'), @x, @xs, @y) ->
newline#3(newline(@y, @lastline', @xs),
@belowVal,
@lastline',
@x,
@y)
, newline#2(nil(), @x, @xs, @y) -> nil()
, newline#3(@nl, @belowVal, @lastline', @x, @y) ->
newline#4(right(@nl), @belowVal, @lastline', @nl, @x, @y)
, right(@l) -> right#1(@l)
, newline#4(@rightVal, @belowVal, @lastline', @nl, @x, @y) ->
newline#5(right(@lastline'), @belowVal, @nl, @rightVal, @x, @y)
, newline#5(@diagVal, @belowVal, @nl, @rightVal, @x, @y) ->
newline#6(newline#7(#equal(@x, @y),
@belowVal,
@diagVal,
@rightVal),
@nl)
, newline#7(#false(), @belowVal, @diagVal, @rightVal) ->
max(@belowVal, @rightVal)
, newline#7(#true(), @belowVal, @diagVal, @rightVal) ->
+(@diagVal, #pos(#s(#0())))
, newline#6(@elem, @nl) -> ::(@elem, @nl)
, right#1(::(@x, @xs)) -> @x
, right#1(nil()) -> #abs(#0())
, #pred(#0()) -> #neg(#s(#0()))
, #pred(#neg(#s(@x))) -> #neg(#s(#s(@x)))
, #pred(#pos(#s(#0()))) -> #0()
, #pred(#pos(#s(#s(@x)))) -> #pos(#s(@x))
, #succ(#0()) -> #pos(#s(#0()))
, #succ(#neg(#s(#0()))) -> #0()
, #succ(#neg(#s(#s(@x)))) -> #neg(#s(@x))
, #succ(#pos(#s(@x))) -> #pos(#s(#s(@x)))
, #and(#false(), #false()) -> #false()
, #and(#false(), #true()) -> #false()
, #and(#true(), #false()) -> #false()
, #and(#true(), #true()) -> #true() }
Obligation:
innermost runtime complexity
Answer:
YES(O(1),O(n^2))
We estimate the number of application of {8,16,18} by applications
of Pre({8,16,18}) = {4,14,15,17}. Here rules are labeled as
follows:
DPs:
{ 1: firstline^#(@l) -> c_8(firstline#1^#(@l))
, 2: firstline#1^#(::(@x, @xs)) ->
c_9(#abs^#(#0()), firstline^#(@xs))
, 3: lcs^#(@l1, @l2) ->
c_11(lcs#1^#(lcstable(@l1, @l2)), lcstable^#(@l1, @l2))
, 4: lcs#1^#(@m) -> c_13(lcs#2^#(@m))
, 5: lcstable^#(@l1, @l2) -> c_12(lcstable#1^#(@l1, @l2))
, 6: lcstable#1^#(::(@x, @xs), @l2) ->
c_18(lcstable#2^#(lcstable(@xs, @l2), @l2, @x),
lcstable^#(@xs, @l2))
, 7: lcstable#1^#(nil(), @l2) -> c_19(firstline^#(@l2))
, 8: lcs#2^#(::(@l1, @_@2)) -> c_14(lcs#3^#(@l1))
, 9: lcstable#2^#(@m, @l2, @x) -> c_20(lcstable#3^#(@m, @l2, @x))
, 10: lcstable#3^#(::(@l, @ls), @l2, @x) ->
c_21(newline^#(@x, @l, @l2))
, 11: newline^#(@y, @lastline, @l) ->
c_23(newline#1^#(@l, @lastline, @y))
, 12: newline#1^#(::(@x, @xs), @lastline, @y) ->
c_27(newline#2^#(@lastline, @x, @xs, @y))
, 13: newline#2^#(::(@belowVal, @lastline'), @x, @xs, @y) ->
c_29(newline#3^#(newline(@y, @lastline', @xs),
@belowVal,
@lastline',
@x,
@y),
newline^#(@y, @lastline', @xs))
, 14: newline#3^#(@nl, @belowVal, @lastline', @x, @y) ->
c_31(newline#4^#(right(@nl), @belowVal, @lastline', @nl, @x, @y),
right^#(@nl))
, 15: newline#4^#(@rightVal, @belowVal, @lastline', @nl, @x, @y) ->
c_33(newline#5^#(right(@lastline'),
@belowVal,
@nl,
@rightVal,
@x,
@y),
right^#(@lastline'))
, 16: right^#(@l) -> c_32(right#1^#(@l))
, 17: newline#5^#(@diagVal, @belowVal, @nl, @rightVal, @x, @y) ->
c_34(newline#6^#(newline#7(#equal(@x, @y),
@belowVal,
@diagVal,
@rightVal),
@nl),
newline#7^#(#equal(@x, @y), @belowVal, @diagVal, @rightVal),
#equal^#(@x, @y))
, 18: newline#7^#(#false(), @belowVal, @diagVal, @rightVal) ->
c_35(max^#(@belowVal, @rightVal))
, 19: #abs^#(#0()) -> c_1()
, 20: #equal^#(@x, @y) -> c_5(#eq^#(@x, @y))
, 21: #eq^#(#0(), #0()) -> c_40()
, 22: #eq^#(#0(), #neg(@y)) -> c_41()
, 23: #eq^#(#0(), #pos(@y)) -> c_42()
, 24: #eq^#(#0(), #s(@y)) -> c_43()
, 25: #eq^#(#neg(@x), #0()) -> c_44()
, 26: #eq^#(#neg(@x), #neg(@y)) -> c_45(#eq^#(@x, @y))
, 27: #eq^#(#neg(@x), #pos(@y)) -> c_46()
, 28: #eq^#(#pos(@x), #0()) -> c_47()
, 29: #eq^#(#pos(@x), #neg(@y)) -> c_48()
, 30: #eq^#(#pos(@x), #pos(@y)) -> c_49(#eq^#(@x, @y))
, 31: #eq^#(#s(@x), #0()) -> c_50()
, 32: #eq^#(#s(@x), #s(@y)) -> c_51(#eq^#(@x, @y))
, 33: #eq^#(::(@x_1, @x_2), ::(@y_1, @y_2)) ->
c_52(#and^#(#eq(@x_1, @y_1), #eq(@x_2, @y_2)),
#eq^#(@x_1, @y_1),
#eq^#(@x_2, @y_2))
, 34: #eq^#(::(@x_1, @x_2), nil()) -> c_53()
, 35: #eq^#(nil(), ::(@y_1, @y_2)) -> c_54()
, 36: #eq^#(nil(), nil()) -> c_55()
, 37: #greater^#(@x, @y) ->
c_6(#ckgt^#(#compare(@x, @y)), #compare^#(@x, @y))
, 38: #ckgt^#(#EQ()) -> c_68()
, 39: #ckgt^#(#GT()) -> c_69()
, 40: #ckgt^#(#LT()) -> c_70()
, 41: #compare^#(#0(), #0()) -> c_56()
, 42: #compare^#(#0(), #neg(@y)) -> c_57()
, 43: #compare^#(#0(), #pos(@y)) -> c_58()
, 44: #compare^#(#0(), #s(@y)) -> c_59()
, 45: #compare^#(#neg(@x), #0()) -> c_60()
, 46: #compare^#(#neg(@x), #neg(@y)) -> c_61(#compare^#(@y, @x))
, 47: #compare^#(#neg(@x), #pos(@y)) -> c_62()
, 48: #compare^#(#pos(@x), #0()) -> c_63()
, 49: #compare^#(#pos(@x), #neg(@y)) -> c_64()
, 50: #compare^#(#pos(@x), #pos(@y)) -> c_65(#compare^#(@x, @y))
, 51: #compare^#(#s(@x), #0()) -> c_66()
, 52: #compare^#(#s(@x), #s(@y)) -> c_67(#compare^#(@x, @y))
, 53: +^#(@x, @y) -> c_7(#add^#(@x, @y))
, 54: #add^#(#0(), @y) -> c_71()
, 55: #add^#(#neg(#s(#0())), @y) -> c_72(#pred^#(@y))
, 56: #add^#(#neg(#s(#s(@x))), @y) ->
c_73(#pred^#(#add(#pos(#s(@x)), @y)), #add^#(#pos(#s(@x)), @y))
, 57: #add^#(#pos(#s(#0())), @y) -> c_74(#succ^#(@y))
, 58: #add^#(#pos(#s(#s(@x))), @y) ->
c_75(#succ^#(#add(#pos(#s(@x)), @y)), #add^#(#pos(#s(@x)), @y))
, 59: firstline#1^#(nil()) -> c_10()
, 60: lcs#2^#(nil()) -> c_15(#abs^#(#0()))
, 61: lcs#3^#(::(@len, @_@1)) -> c_16()
, 62: lcs#3^#(nil()) -> c_17(#abs^#(#0()))
, 63: lcstable#3^#(nil(), @l2, @x) -> c_22()
, 64: newline#1^#(nil(), @lastline, @y) -> c_28()
, 65: max^#(@a, @b) ->
c_24(max#1^#(#greater(@a, @b), @a, @b), #greater^#(@a, @b))
, 66: max#1^#(#false(), @a, @b) -> c_25()
, 67: max#1^#(#true(), @a, @b) -> c_26()
, 68: newline#2^#(nil(), @x, @xs, @y) -> c_30()
, 69: right#1^#(::(@x, @xs)) -> c_38()
, 70: right#1^#(nil()) -> c_39(#abs^#(#0()))
, 71: newline#6^#(@elem, @nl) -> c_37()
, 72: newline#7^#(#true(), @belowVal, @diagVal, @rightVal) ->
c_36(+^#(@diagVal, #pos(#s(#0()))))
, 73: #and^#(#false(), #false()) -> c_84()
, 74: #and^#(#false(), #true()) -> c_85()
, 75: #and^#(#true(), #false()) -> c_86()
, 76: #and^#(#true(), #true()) -> c_87()
, 77: #pred^#(#0()) -> c_76()
, 78: #pred^#(#neg(#s(@x))) -> c_77()
, 79: #pred^#(#pos(#s(#0()))) -> c_78()
, 80: #pred^#(#pos(#s(#s(@x)))) -> c_79()
, 81: #succ^#(#0()) -> c_80()
, 82: #succ^#(#neg(#s(#0()))) -> c_81()
, 83: #succ^#(#neg(#s(#s(@x)))) -> c_82()
, 84: #succ^#(#pos(#s(@x))) -> c_83() }
We are left with following problem, upon which TcT provides the
certificate YES(O(1),O(n^2)).
Strict DPs:
{ firstline^#(@l) -> c_8(firstline#1^#(@l))
, firstline#1^#(::(@x, @xs)) -> c_9(#abs^#(#0()), firstline^#(@xs))
, lcs^#(@l1, @l2) ->
c_11(lcs#1^#(lcstable(@l1, @l2)), lcstable^#(@l1, @l2))
, lcs#1^#(@m) -> c_13(lcs#2^#(@m))
, lcstable^#(@l1, @l2) -> c_12(lcstable#1^#(@l1, @l2))
, lcstable#1^#(::(@x, @xs), @l2) ->
c_18(lcstable#2^#(lcstable(@xs, @l2), @l2, @x),
lcstable^#(@xs, @l2))
, lcstable#1^#(nil(), @l2) -> c_19(firstline^#(@l2))
, lcstable#2^#(@m, @l2, @x) -> c_20(lcstable#3^#(@m, @l2, @x))
, lcstable#3^#(::(@l, @ls), @l2, @x) ->
c_21(newline^#(@x, @l, @l2))
, newline^#(@y, @lastline, @l) ->
c_23(newline#1^#(@l, @lastline, @y))
, newline#1^#(::(@x, @xs), @lastline, @y) ->
c_27(newline#2^#(@lastline, @x, @xs, @y))
, newline#2^#(::(@belowVal, @lastline'), @x, @xs, @y) ->
c_29(newline#3^#(newline(@y, @lastline', @xs),
@belowVal,
@lastline',
@x,
@y),
newline^#(@y, @lastline', @xs))
, newline#3^#(@nl, @belowVal, @lastline', @x, @y) ->
c_31(newline#4^#(right(@nl), @belowVal, @lastline', @nl, @x, @y),
right^#(@nl))
, newline#4^#(@rightVal, @belowVal, @lastline', @nl, @x, @y) ->
c_33(newline#5^#(right(@lastline'),
@belowVal,
@nl,
@rightVal,
@x,
@y),
right^#(@lastline'))
, newline#5^#(@diagVal, @belowVal, @nl, @rightVal, @x, @y) ->
c_34(newline#6^#(newline#7(#equal(@x, @y),
@belowVal,
@diagVal,
@rightVal),
@nl),
newline#7^#(#equal(@x, @y), @belowVal, @diagVal, @rightVal),
#equal^#(@x, @y)) }
Weak DPs:
{ #abs^#(#0()) -> c_1()
, #equal^#(@x, @y) -> c_5(#eq^#(@x, @y))
, #eq^#(#0(), #0()) -> c_40()
, #eq^#(#0(), #neg(@y)) -> c_41()
, #eq^#(#0(), #pos(@y)) -> c_42()
, #eq^#(#0(), #s(@y)) -> c_43()
, #eq^#(#neg(@x), #0()) -> c_44()
, #eq^#(#neg(@x), #neg(@y)) -> c_45(#eq^#(@x, @y))
, #eq^#(#neg(@x), #pos(@y)) -> c_46()
, #eq^#(#pos(@x), #0()) -> c_47()
, #eq^#(#pos(@x), #neg(@y)) -> c_48()
, #eq^#(#pos(@x), #pos(@y)) -> c_49(#eq^#(@x, @y))
, #eq^#(#s(@x), #0()) -> c_50()
, #eq^#(#s(@x), #s(@y)) -> c_51(#eq^#(@x, @y))
, #eq^#(::(@x_1, @x_2), ::(@y_1, @y_2)) ->
c_52(#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_53()
, #eq^#(nil(), ::(@y_1, @y_2)) -> c_54()
, #eq^#(nil(), nil()) -> c_55()
, #greater^#(@x, @y) ->
c_6(#ckgt^#(#compare(@x, @y)), #compare^#(@x, @y))
, #ckgt^#(#EQ()) -> c_68()
, #ckgt^#(#GT()) -> c_69()
, #ckgt^#(#LT()) -> c_70()
, #compare^#(#0(), #0()) -> c_56()
, #compare^#(#0(), #neg(@y)) -> c_57()
, #compare^#(#0(), #pos(@y)) -> c_58()
, #compare^#(#0(), #s(@y)) -> c_59()
, #compare^#(#neg(@x), #0()) -> c_60()
, #compare^#(#neg(@x), #neg(@y)) -> c_61(#compare^#(@y, @x))
, #compare^#(#neg(@x), #pos(@y)) -> c_62()
, #compare^#(#pos(@x), #0()) -> c_63()
, #compare^#(#pos(@x), #neg(@y)) -> c_64()
, #compare^#(#pos(@x), #pos(@y)) -> c_65(#compare^#(@x, @y))
, #compare^#(#s(@x), #0()) -> c_66()
, #compare^#(#s(@x), #s(@y)) -> c_67(#compare^#(@x, @y))
, +^#(@x, @y) -> c_7(#add^#(@x, @y))
, #add^#(#0(), @y) -> c_71()
, #add^#(#neg(#s(#0())), @y) -> c_72(#pred^#(@y))
, #add^#(#neg(#s(#s(@x))), @y) ->
c_73(#pred^#(#add(#pos(#s(@x)), @y)), #add^#(#pos(#s(@x)), @y))
, #add^#(#pos(#s(#0())), @y) -> c_74(#succ^#(@y))
, #add^#(#pos(#s(#s(@x))), @y) ->
c_75(#succ^#(#add(#pos(#s(@x)), @y)), #add^#(#pos(#s(@x)), @y))
, firstline#1^#(nil()) -> c_10()
, lcs#2^#(::(@l1, @_@2)) -> c_14(lcs#3^#(@l1))
, lcs#2^#(nil()) -> c_15(#abs^#(#0()))
, lcs#3^#(::(@len, @_@1)) -> c_16()
, lcs#3^#(nil()) -> c_17(#abs^#(#0()))
, lcstable#3^#(nil(), @l2, @x) -> c_22()
, newline#1^#(nil(), @lastline, @y) -> c_28()
, max^#(@a, @b) ->
c_24(max#1^#(#greater(@a, @b), @a, @b), #greater^#(@a, @b))
, max#1^#(#false(), @a, @b) -> c_25()
, max#1^#(#true(), @a, @b) -> c_26()
, newline#2^#(nil(), @x, @xs, @y) -> c_30()
, right^#(@l) -> c_32(right#1^#(@l))
, right#1^#(::(@x, @xs)) -> c_38()
, right#1^#(nil()) -> c_39(#abs^#(#0()))
, newline#6^#(@elem, @nl) -> c_37()
, newline#7^#(#false(), @belowVal, @diagVal, @rightVal) ->
c_35(max^#(@belowVal, @rightVal))
, newline#7^#(#true(), @belowVal, @diagVal, @rightVal) ->
c_36(+^#(@diagVal, #pos(#s(#0()))))
, #and^#(#false(), #false()) -> c_84()
, #and^#(#false(), #true()) -> c_85()
, #and^#(#true(), #false()) -> c_86()
, #and^#(#true(), #true()) -> c_87()
, #pred^#(#0()) -> c_76()
, #pred^#(#neg(#s(@x))) -> c_77()
, #pred^#(#pos(#s(#0()))) -> c_78()
, #pred^#(#pos(#s(#s(@x)))) -> c_79()
, #succ^#(#0()) -> c_80()
, #succ^#(#neg(#s(#0()))) -> c_81()
, #succ^#(#neg(#s(#s(@x)))) -> c_82()
, #succ^#(#pos(#s(@x))) -> c_83() }
Weak Trs:
{ #abs(#0()) -> #0()
, #abs(#neg(@x)) -> #pos(@x)
, #abs(#pos(@x)) -> #pos(@x)
, #abs(#s(@x)) -> #pos(#s(@x))
, #equal(@x, @y) -> #eq(@x, @y)
, #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)
, #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()
, #greater(@x, @y) -> #ckgt(#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)
, #ckgt(#EQ()) -> #false()
, #ckgt(#GT()) -> #true()
, #ckgt(#LT()) -> #false()
, +(@x, @y) -> #add(@x, @y)
, #add(#0(), @y) -> @y
, #add(#neg(#s(#0())), @y) -> #pred(@y)
, #add(#neg(#s(#s(@x))), @y) -> #pred(#add(#pos(#s(@x)), @y))
, #add(#pos(#s(#0())), @y) -> #succ(@y)
, #add(#pos(#s(#s(@x))), @y) -> #succ(#add(#pos(#s(@x)), @y))
, firstline(@l) -> firstline#1(@l)
, firstline#1(::(@x, @xs)) -> ::(#abs(#0()), firstline(@xs))
, firstline#1(nil()) -> nil()
, lcs(@l1, @l2) -> lcs#1(lcstable(@l1, @l2))
, lcstable(@l1, @l2) -> lcstable#1(@l1, @l2)
, lcs#1(@m) -> lcs#2(@m)
, lcs#2(::(@l1, @_@2)) -> lcs#3(@l1)
, lcs#2(nil()) -> #abs(#0())
, lcs#3(::(@len, @_@1)) -> @len
, lcs#3(nil()) -> #abs(#0())
, lcstable#1(::(@x, @xs), @l2) ->
lcstable#2(lcstable(@xs, @l2), @l2, @x)
, lcstable#1(nil(), @l2) -> ::(firstline(@l2), nil())
, lcstable#2(@m, @l2, @x) -> lcstable#3(@m, @l2, @x)
, lcstable#3(::(@l, @ls), @l2, @x) ->
::(newline(@x, @l, @l2), ::(@l, @ls))
, lcstable#3(nil(), @l2, @x) -> nil()
, newline(@y, @lastline, @l) -> newline#1(@l, @lastline, @y)
, max(@a, @b) -> max#1(#greater(@a, @b), @a, @b)
, max#1(#false(), @a, @b) -> @b
, max#1(#true(), @a, @b) -> @a
, newline#1(::(@x, @xs), @lastline, @y) ->
newline#2(@lastline, @x, @xs, @y)
, newline#1(nil(), @lastline, @y) -> nil()
, newline#2(::(@belowVal, @lastline'), @x, @xs, @y) ->
newline#3(newline(@y, @lastline', @xs),
@belowVal,
@lastline',
@x,
@y)
, newline#2(nil(), @x, @xs, @y) -> nil()
, newline#3(@nl, @belowVal, @lastline', @x, @y) ->
newline#4(right(@nl), @belowVal, @lastline', @nl, @x, @y)
, right(@l) -> right#1(@l)
, newline#4(@rightVal, @belowVal, @lastline', @nl, @x, @y) ->
newline#5(right(@lastline'), @belowVal, @nl, @rightVal, @x, @y)
, newline#5(@diagVal, @belowVal, @nl, @rightVal, @x, @y) ->
newline#6(newline#7(#equal(@x, @y),
@belowVal,
@diagVal,
@rightVal),
@nl)
, newline#7(#false(), @belowVal, @diagVal, @rightVal) ->
max(@belowVal, @rightVal)
, newline#7(#true(), @belowVal, @diagVal, @rightVal) ->
+(@diagVal, #pos(#s(#0())))
, newline#6(@elem, @nl) -> ::(@elem, @nl)
, right#1(::(@x, @xs)) -> @x
, right#1(nil()) -> #abs(#0())
, #pred(#0()) -> #neg(#s(#0()))
, #pred(#neg(#s(@x))) -> #neg(#s(#s(@x)))
, #pred(#pos(#s(#0()))) -> #0()
, #pred(#pos(#s(#s(@x)))) -> #pos(#s(@x))
, #succ(#0()) -> #pos(#s(#0()))
, #succ(#neg(#s(#0()))) -> #0()
, #succ(#neg(#s(#s(@x)))) -> #neg(#s(@x))
, #succ(#pos(#s(@x))) -> #pos(#s(#s(@x)))
, #and(#false(), #false()) -> #false()
, #and(#false(), #true()) -> #false()
, #and(#true(), #false()) -> #false()
, #and(#true(), #true()) -> #true() }
Obligation:
innermost runtime complexity
Answer:
YES(O(1),O(n^2))
We estimate the number of application of {4,15} by applications of
Pre({4,15}) = {3,14}. Here rules are labeled as follows:
DPs:
{ 1: firstline^#(@l) -> c_8(firstline#1^#(@l))
, 2: firstline#1^#(::(@x, @xs)) ->
c_9(#abs^#(#0()), firstline^#(@xs))
, 3: lcs^#(@l1, @l2) ->
c_11(lcs#1^#(lcstable(@l1, @l2)), lcstable^#(@l1, @l2))
, 4: lcs#1^#(@m) -> c_13(lcs#2^#(@m))
, 5: lcstable^#(@l1, @l2) -> c_12(lcstable#1^#(@l1, @l2))
, 6: lcstable#1^#(::(@x, @xs), @l2) ->
c_18(lcstable#2^#(lcstable(@xs, @l2), @l2, @x),
lcstable^#(@xs, @l2))
, 7: lcstable#1^#(nil(), @l2) -> c_19(firstline^#(@l2))
, 8: lcstable#2^#(@m, @l2, @x) -> c_20(lcstable#3^#(@m, @l2, @x))
, 9: lcstable#3^#(::(@l, @ls), @l2, @x) ->
c_21(newline^#(@x, @l, @l2))
, 10: newline^#(@y, @lastline, @l) ->
c_23(newline#1^#(@l, @lastline, @y))
, 11: newline#1^#(::(@x, @xs), @lastline, @y) ->
c_27(newline#2^#(@lastline, @x, @xs, @y))
, 12: newline#2^#(::(@belowVal, @lastline'), @x, @xs, @y) ->
c_29(newline#3^#(newline(@y, @lastline', @xs),
@belowVal,
@lastline',
@x,
@y),
newline^#(@y, @lastline', @xs))
, 13: newline#3^#(@nl, @belowVal, @lastline', @x, @y) ->
c_31(newline#4^#(right(@nl), @belowVal, @lastline', @nl, @x, @y),
right^#(@nl))
, 14: newline#4^#(@rightVal, @belowVal, @lastline', @nl, @x, @y) ->
c_33(newline#5^#(right(@lastline'),
@belowVal,
@nl,
@rightVal,
@x,
@y),
right^#(@lastline'))
, 15: newline#5^#(@diagVal, @belowVal, @nl, @rightVal, @x, @y) ->
c_34(newline#6^#(newline#7(#equal(@x, @y),
@belowVal,
@diagVal,
@rightVal),
@nl),
newline#7^#(#equal(@x, @y), @belowVal, @diagVal, @rightVal),
#equal^#(@x, @y))
, 16: #abs^#(#0()) -> c_1()
, 17: #equal^#(@x, @y) -> c_5(#eq^#(@x, @y))
, 18: #eq^#(#0(), #0()) -> c_40()
, 19: #eq^#(#0(), #neg(@y)) -> c_41()
, 20: #eq^#(#0(), #pos(@y)) -> c_42()
, 21: #eq^#(#0(), #s(@y)) -> c_43()
, 22: #eq^#(#neg(@x), #0()) -> c_44()
, 23: #eq^#(#neg(@x), #neg(@y)) -> c_45(#eq^#(@x, @y))
, 24: #eq^#(#neg(@x), #pos(@y)) -> c_46()
, 25: #eq^#(#pos(@x), #0()) -> c_47()
, 26: #eq^#(#pos(@x), #neg(@y)) -> c_48()
, 27: #eq^#(#pos(@x), #pos(@y)) -> c_49(#eq^#(@x, @y))
, 28: #eq^#(#s(@x), #0()) -> c_50()
, 29: #eq^#(#s(@x), #s(@y)) -> c_51(#eq^#(@x, @y))
, 30: #eq^#(::(@x_1, @x_2), ::(@y_1, @y_2)) ->
c_52(#and^#(#eq(@x_1, @y_1), #eq(@x_2, @y_2)),
#eq^#(@x_1, @y_1),
#eq^#(@x_2, @y_2))
, 31: #eq^#(::(@x_1, @x_2), nil()) -> c_53()
, 32: #eq^#(nil(), ::(@y_1, @y_2)) -> c_54()
, 33: #eq^#(nil(), nil()) -> c_55()
, 34: #greater^#(@x, @y) ->
c_6(#ckgt^#(#compare(@x, @y)), #compare^#(@x, @y))
, 35: #ckgt^#(#EQ()) -> c_68()
, 36: #ckgt^#(#GT()) -> c_69()
, 37: #ckgt^#(#LT()) -> c_70()
, 38: #compare^#(#0(), #0()) -> c_56()
, 39: #compare^#(#0(), #neg(@y)) -> c_57()
, 40: #compare^#(#0(), #pos(@y)) -> c_58()
, 41: #compare^#(#0(), #s(@y)) -> c_59()
, 42: #compare^#(#neg(@x), #0()) -> c_60()
, 43: #compare^#(#neg(@x), #neg(@y)) -> c_61(#compare^#(@y, @x))
, 44: #compare^#(#neg(@x), #pos(@y)) -> c_62()
, 45: #compare^#(#pos(@x), #0()) -> c_63()
, 46: #compare^#(#pos(@x), #neg(@y)) -> c_64()
, 47: #compare^#(#pos(@x), #pos(@y)) -> c_65(#compare^#(@x, @y))
, 48: #compare^#(#s(@x), #0()) -> c_66()
, 49: #compare^#(#s(@x), #s(@y)) -> c_67(#compare^#(@x, @y))
, 50: +^#(@x, @y) -> c_7(#add^#(@x, @y))
, 51: #add^#(#0(), @y) -> c_71()
, 52: #add^#(#neg(#s(#0())), @y) -> c_72(#pred^#(@y))
, 53: #add^#(#neg(#s(#s(@x))), @y) ->
c_73(#pred^#(#add(#pos(#s(@x)), @y)), #add^#(#pos(#s(@x)), @y))
, 54: #add^#(#pos(#s(#0())), @y) -> c_74(#succ^#(@y))
, 55: #add^#(#pos(#s(#s(@x))), @y) ->
c_75(#succ^#(#add(#pos(#s(@x)), @y)), #add^#(#pos(#s(@x)), @y))
, 56: firstline#1^#(nil()) -> c_10()
, 57: lcs#2^#(::(@l1, @_@2)) -> c_14(lcs#3^#(@l1))
, 58: lcs#2^#(nil()) -> c_15(#abs^#(#0()))
, 59: lcs#3^#(::(@len, @_@1)) -> c_16()
, 60: lcs#3^#(nil()) -> c_17(#abs^#(#0()))
, 61: lcstable#3^#(nil(), @l2, @x) -> c_22()
, 62: newline#1^#(nil(), @lastline, @y) -> c_28()
, 63: max^#(@a, @b) ->
c_24(max#1^#(#greater(@a, @b), @a, @b), #greater^#(@a, @b))
, 64: max#1^#(#false(), @a, @b) -> c_25()
, 65: max#1^#(#true(), @a, @b) -> c_26()
, 66: newline#2^#(nil(), @x, @xs, @y) -> c_30()
, 67: right^#(@l) -> c_32(right#1^#(@l))
, 68: right#1^#(::(@x, @xs)) -> c_38()
, 69: right#1^#(nil()) -> c_39(#abs^#(#0()))
, 70: newline#6^#(@elem, @nl) -> c_37()
, 71: newline#7^#(#false(), @belowVal, @diagVal, @rightVal) ->
c_35(max^#(@belowVal, @rightVal))
, 72: newline#7^#(#true(), @belowVal, @diagVal, @rightVal) ->
c_36(+^#(@diagVal, #pos(#s(#0()))))
, 73: #and^#(#false(), #false()) -> c_84()
, 74: #and^#(#false(), #true()) -> c_85()
, 75: #and^#(#true(), #false()) -> c_86()
, 76: #and^#(#true(), #true()) -> c_87()
, 77: #pred^#(#0()) -> c_76()
, 78: #pred^#(#neg(#s(@x))) -> c_77()
, 79: #pred^#(#pos(#s(#0()))) -> c_78()
, 80: #pred^#(#pos(#s(#s(@x)))) -> c_79()
, 81: #succ^#(#0()) -> c_80()
, 82: #succ^#(#neg(#s(#0()))) -> c_81()
, 83: #succ^#(#neg(#s(#s(@x)))) -> c_82()
, 84: #succ^#(#pos(#s(@x))) -> c_83() }
We are left with following problem, upon which TcT provides the
certificate YES(O(1),O(n^2)).
Strict DPs:
{ firstline^#(@l) -> c_8(firstline#1^#(@l))
, firstline#1^#(::(@x, @xs)) -> c_9(#abs^#(#0()), firstline^#(@xs))
, lcs^#(@l1, @l2) ->
c_11(lcs#1^#(lcstable(@l1, @l2)), lcstable^#(@l1, @l2))
, lcstable^#(@l1, @l2) -> c_12(lcstable#1^#(@l1, @l2))
, lcstable#1^#(::(@x, @xs), @l2) ->
c_18(lcstable#2^#(lcstable(@xs, @l2), @l2, @x),
lcstable^#(@xs, @l2))
, lcstable#1^#(nil(), @l2) -> c_19(firstline^#(@l2))
, lcstable#2^#(@m, @l2, @x) -> c_20(lcstable#3^#(@m, @l2, @x))
, lcstable#3^#(::(@l, @ls), @l2, @x) ->
c_21(newline^#(@x, @l, @l2))
, newline^#(@y, @lastline, @l) ->
c_23(newline#1^#(@l, @lastline, @y))
, newline#1^#(::(@x, @xs), @lastline, @y) ->
c_27(newline#2^#(@lastline, @x, @xs, @y))
, newline#2^#(::(@belowVal, @lastline'), @x, @xs, @y) ->
c_29(newline#3^#(newline(@y, @lastline', @xs),
@belowVal,
@lastline',
@x,
@y),
newline^#(@y, @lastline', @xs))
, newline#3^#(@nl, @belowVal, @lastline', @x, @y) ->
c_31(newline#4^#(right(@nl), @belowVal, @lastline', @nl, @x, @y),
right^#(@nl))
, newline#4^#(@rightVal, @belowVal, @lastline', @nl, @x, @y) ->
c_33(newline#5^#(right(@lastline'),
@belowVal,
@nl,
@rightVal,
@x,
@y),
right^#(@lastline')) }
Weak DPs:
{ #abs^#(#0()) -> c_1()
, #equal^#(@x, @y) -> c_5(#eq^#(@x, @y))
, #eq^#(#0(), #0()) -> c_40()
, #eq^#(#0(), #neg(@y)) -> c_41()
, #eq^#(#0(), #pos(@y)) -> c_42()
, #eq^#(#0(), #s(@y)) -> c_43()
, #eq^#(#neg(@x), #0()) -> c_44()
, #eq^#(#neg(@x), #neg(@y)) -> c_45(#eq^#(@x, @y))
, #eq^#(#neg(@x), #pos(@y)) -> c_46()
, #eq^#(#pos(@x), #0()) -> c_47()
, #eq^#(#pos(@x), #neg(@y)) -> c_48()
, #eq^#(#pos(@x), #pos(@y)) -> c_49(#eq^#(@x, @y))
, #eq^#(#s(@x), #0()) -> c_50()
, #eq^#(#s(@x), #s(@y)) -> c_51(#eq^#(@x, @y))
, #eq^#(::(@x_1, @x_2), ::(@y_1, @y_2)) ->
c_52(#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_53()
, #eq^#(nil(), ::(@y_1, @y_2)) -> c_54()
, #eq^#(nil(), nil()) -> c_55()
, #greater^#(@x, @y) ->
c_6(#ckgt^#(#compare(@x, @y)), #compare^#(@x, @y))
, #ckgt^#(#EQ()) -> c_68()
, #ckgt^#(#GT()) -> c_69()
, #ckgt^#(#LT()) -> c_70()
, #compare^#(#0(), #0()) -> c_56()
, #compare^#(#0(), #neg(@y)) -> c_57()
, #compare^#(#0(), #pos(@y)) -> c_58()
, #compare^#(#0(), #s(@y)) -> c_59()
, #compare^#(#neg(@x), #0()) -> c_60()
, #compare^#(#neg(@x), #neg(@y)) -> c_61(#compare^#(@y, @x))
, #compare^#(#neg(@x), #pos(@y)) -> c_62()
, #compare^#(#pos(@x), #0()) -> c_63()
, #compare^#(#pos(@x), #neg(@y)) -> c_64()
, #compare^#(#pos(@x), #pos(@y)) -> c_65(#compare^#(@x, @y))
, #compare^#(#s(@x), #0()) -> c_66()
, #compare^#(#s(@x), #s(@y)) -> c_67(#compare^#(@x, @y))
, +^#(@x, @y) -> c_7(#add^#(@x, @y))
, #add^#(#0(), @y) -> c_71()
, #add^#(#neg(#s(#0())), @y) -> c_72(#pred^#(@y))
, #add^#(#neg(#s(#s(@x))), @y) ->
c_73(#pred^#(#add(#pos(#s(@x)), @y)), #add^#(#pos(#s(@x)), @y))
, #add^#(#pos(#s(#0())), @y) -> c_74(#succ^#(@y))
, #add^#(#pos(#s(#s(@x))), @y) ->
c_75(#succ^#(#add(#pos(#s(@x)), @y)), #add^#(#pos(#s(@x)), @y))
, firstline#1^#(nil()) -> c_10()
, lcs#1^#(@m) -> c_13(lcs#2^#(@m))
, lcs#2^#(::(@l1, @_@2)) -> c_14(lcs#3^#(@l1))
, lcs#2^#(nil()) -> c_15(#abs^#(#0()))
, lcs#3^#(::(@len, @_@1)) -> c_16()
, lcs#3^#(nil()) -> c_17(#abs^#(#0()))
, lcstable#3^#(nil(), @l2, @x) -> c_22()
, newline#1^#(nil(), @lastline, @y) -> c_28()
, max^#(@a, @b) ->
c_24(max#1^#(#greater(@a, @b), @a, @b), #greater^#(@a, @b))
, max#1^#(#false(), @a, @b) -> c_25()
, max#1^#(#true(), @a, @b) -> c_26()
, newline#2^#(nil(), @x, @xs, @y) -> c_30()
, right^#(@l) -> c_32(right#1^#(@l))
, right#1^#(::(@x, @xs)) -> c_38()
, right#1^#(nil()) -> c_39(#abs^#(#0()))
, newline#5^#(@diagVal, @belowVal, @nl, @rightVal, @x, @y) ->
c_34(newline#6^#(newline#7(#equal(@x, @y),
@belowVal,
@diagVal,
@rightVal),
@nl),
newline#7^#(#equal(@x, @y), @belowVal, @diagVal, @rightVal),
#equal^#(@x, @y))
, newline#6^#(@elem, @nl) -> c_37()
, newline#7^#(#false(), @belowVal, @diagVal, @rightVal) ->
c_35(max^#(@belowVal, @rightVal))
, newline#7^#(#true(), @belowVal, @diagVal, @rightVal) ->
c_36(+^#(@diagVal, #pos(#s(#0()))))
, #and^#(#false(), #false()) -> c_84()
, #and^#(#false(), #true()) -> c_85()
, #and^#(#true(), #false()) -> c_86()
, #and^#(#true(), #true()) -> c_87()
, #pred^#(#0()) -> c_76()
, #pred^#(#neg(#s(@x))) -> c_77()
, #pred^#(#pos(#s(#0()))) -> c_78()
, #pred^#(#pos(#s(#s(@x)))) -> c_79()
, #succ^#(#0()) -> c_80()
, #succ^#(#neg(#s(#0()))) -> c_81()
, #succ^#(#neg(#s(#s(@x)))) -> c_82()
, #succ^#(#pos(#s(@x))) -> c_83() }
Weak Trs:
{ #abs(#0()) -> #0()
, #abs(#neg(@x)) -> #pos(@x)
, #abs(#pos(@x)) -> #pos(@x)
, #abs(#s(@x)) -> #pos(#s(@x))
, #equal(@x, @y) -> #eq(@x, @y)
, #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)
, #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()
, #greater(@x, @y) -> #ckgt(#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)
, #ckgt(#EQ()) -> #false()
, #ckgt(#GT()) -> #true()
, #ckgt(#LT()) -> #false()
, +(@x, @y) -> #add(@x, @y)
, #add(#0(), @y) -> @y
, #add(#neg(#s(#0())), @y) -> #pred(@y)
, #add(#neg(#s(#s(@x))), @y) -> #pred(#add(#pos(#s(@x)), @y))
, #add(#pos(#s(#0())), @y) -> #succ(@y)
, #add(#pos(#s(#s(@x))), @y) -> #succ(#add(#pos(#s(@x)), @y))
, firstline(@l) -> firstline#1(@l)
, firstline#1(::(@x, @xs)) -> ::(#abs(#0()), firstline(@xs))
, firstline#1(nil()) -> nil()
, lcs(@l1, @l2) -> lcs#1(lcstable(@l1, @l2))
, lcstable(@l1, @l2) -> lcstable#1(@l1, @l2)
, lcs#1(@m) -> lcs#2(@m)
, lcs#2(::(@l1, @_@2)) -> lcs#3(@l1)
, lcs#2(nil()) -> #abs(#0())
, lcs#3(::(@len, @_@1)) -> @len
, lcs#3(nil()) -> #abs(#0())
, lcstable#1(::(@x, @xs), @l2) ->
lcstable#2(lcstable(@xs, @l2), @l2, @x)
, lcstable#1(nil(), @l2) -> ::(firstline(@l2), nil())
, lcstable#2(@m, @l2, @x) -> lcstable#3(@m, @l2, @x)
, lcstable#3(::(@l, @ls), @l2, @x) ->
::(newline(@x, @l, @l2), ::(@l, @ls))
, lcstable#3(nil(), @l2, @x) -> nil()
, newline(@y, @lastline, @l) -> newline#1(@l, @lastline, @y)
, max(@a, @b) -> max#1(#greater(@a, @b), @a, @b)
, max#1(#false(), @a, @b) -> @b
, max#1(#true(), @a, @b) -> @a
, newline#1(::(@x, @xs), @lastline, @y) ->
newline#2(@lastline, @x, @xs, @y)
, newline#1(nil(), @lastline, @y) -> nil()
, newline#2(::(@belowVal, @lastline'), @x, @xs, @y) ->
newline#3(newline(@y, @lastline', @xs),
@belowVal,
@lastline',
@x,
@y)
, newline#2(nil(), @x, @xs, @y) -> nil()
, newline#3(@nl, @belowVal, @lastline', @x, @y) ->
newline#4(right(@nl), @belowVal, @lastline', @nl, @x, @y)
, right(@l) -> right#1(@l)
, newline#4(@rightVal, @belowVal, @lastline', @nl, @x, @y) ->
newline#5(right(@lastline'), @belowVal, @nl, @rightVal, @x, @y)
, newline#5(@diagVal, @belowVal, @nl, @rightVal, @x, @y) ->
newline#6(newline#7(#equal(@x, @y),
@belowVal,
@diagVal,
@rightVal),
@nl)
, newline#7(#false(), @belowVal, @diagVal, @rightVal) ->
max(@belowVal, @rightVal)
, newline#7(#true(), @belowVal, @diagVal, @rightVal) ->
+(@diagVal, #pos(#s(#0())))
, newline#6(@elem, @nl) -> ::(@elem, @nl)
, right#1(::(@x, @xs)) -> @x
, right#1(nil()) -> #abs(#0())
, #pred(#0()) -> #neg(#s(#0()))
, #pred(#neg(#s(@x))) -> #neg(#s(#s(@x)))
, #pred(#pos(#s(#0()))) -> #0()
, #pred(#pos(#s(#s(@x)))) -> #pos(#s(@x))
, #succ(#0()) -> #pos(#s(#0()))
, #succ(#neg(#s(#0()))) -> #0()
, #succ(#neg(#s(#s(@x)))) -> #neg(#s(@x))
, #succ(#pos(#s(@x))) -> #pos(#s(#s(@x)))
, #and(#false(), #false()) -> #false()
, #and(#false(), #true()) -> #false()
, #and(#true(), #false()) -> #false()
, #and(#true(), #true()) -> #true() }
Obligation:
innermost runtime complexity
Answer:
YES(O(1),O(n^2))
We estimate the number of application of {13} by applications of
Pre({13}) = {12}. Here rules are labeled as follows:
DPs:
{ 1: firstline^#(@l) -> c_8(firstline#1^#(@l))
, 2: firstline#1^#(::(@x, @xs)) ->
c_9(#abs^#(#0()), firstline^#(@xs))
, 3: lcs^#(@l1, @l2) ->
c_11(lcs#1^#(lcstable(@l1, @l2)), lcstable^#(@l1, @l2))
, 4: lcstable^#(@l1, @l2) -> c_12(lcstable#1^#(@l1, @l2))
, 5: lcstable#1^#(::(@x, @xs), @l2) ->
c_18(lcstable#2^#(lcstable(@xs, @l2), @l2, @x),
lcstable^#(@xs, @l2))
, 6: lcstable#1^#(nil(), @l2) -> c_19(firstline^#(@l2))
, 7: lcstable#2^#(@m, @l2, @x) -> c_20(lcstable#3^#(@m, @l2, @x))
, 8: lcstable#3^#(::(@l, @ls), @l2, @x) ->
c_21(newline^#(@x, @l, @l2))
, 9: newline^#(@y, @lastline, @l) ->
c_23(newline#1^#(@l, @lastline, @y))
, 10: newline#1^#(::(@x, @xs), @lastline, @y) ->
c_27(newline#2^#(@lastline, @x, @xs, @y))
, 11: newline#2^#(::(@belowVal, @lastline'), @x, @xs, @y) ->
c_29(newline#3^#(newline(@y, @lastline', @xs),
@belowVal,
@lastline',
@x,
@y),
newline^#(@y, @lastline', @xs))
, 12: newline#3^#(@nl, @belowVal, @lastline', @x, @y) ->
c_31(newline#4^#(right(@nl), @belowVal, @lastline', @nl, @x, @y),
right^#(@nl))
, 13: newline#4^#(@rightVal, @belowVal, @lastline', @nl, @x, @y) ->
c_33(newline#5^#(right(@lastline'),
@belowVal,
@nl,
@rightVal,
@x,
@y),
right^#(@lastline'))
, 14: #abs^#(#0()) -> c_1()
, 15: #equal^#(@x, @y) -> c_5(#eq^#(@x, @y))
, 16: #eq^#(#0(), #0()) -> c_40()
, 17: #eq^#(#0(), #neg(@y)) -> c_41()
, 18: #eq^#(#0(), #pos(@y)) -> c_42()
, 19: #eq^#(#0(), #s(@y)) -> c_43()
, 20: #eq^#(#neg(@x), #0()) -> c_44()
, 21: #eq^#(#neg(@x), #neg(@y)) -> c_45(#eq^#(@x, @y))
, 22: #eq^#(#neg(@x), #pos(@y)) -> c_46()
, 23: #eq^#(#pos(@x), #0()) -> c_47()
, 24: #eq^#(#pos(@x), #neg(@y)) -> c_48()
, 25: #eq^#(#pos(@x), #pos(@y)) -> c_49(#eq^#(@x, @y))
, 26: #eq^#(#s(@x), #0()) -> c_50()
, 27: #eq^#(#s(@x), #s(@y)) -> c_51(#eq^#(@x, @y))
, 28: #eq^#(::(@x_1, @x_2), ::(@y_1, @y_2)) ->
c_52(#and^#(#eq(@x_1, @y_1), #eq(@x_2, @y_2)),
#eq^#(@x_1, @y_1),
#eq^#(@x_2, @y_2))
, 29: #eq^#(::(@x_1, @x_2), nil()) -> c_53()
, 30: #eq^#(nil(), ::(@y_1, @y_2)) -> c_54()
, 31: #eq^#(nil(), nil()) -> c_55()
, 32: #greater^#(@x, @y) ->
c_6(#ckgt^#(#compare(@x, @y)), #compare^#(@x, @y))
, 33: #ckgt^#(#EQ()) -> c_68()
, 34: #ckgt^#(#GT()) -> c_69()
, 35: #ckgt^#(#LT()) -> c_70()
, 36: #compare^#(#0(), #0()) -> c_56()
, 37: #compare^#(#0(), #neg(@y)) -> c_57()
, 38: #compare^#(#0(), #pos(@y)) -> c_58()
, 39: #compare^#(#0(), #s(@y)) -> c_59()
, 40: #compare^#(#neg(@x), #0()) -> c_60()
, 41: #compare^#(#neg(@x), #neg(@y)) -> c_61(#compare^#(@y, @x))
, 42: #compare^#(#neg(@x), #pos(@y)) -> c_62()
, 43: #compare^#(#pos(@x), #0()) -> c_63()
, 44: #compare^#(#pos(@x), #neg(@y)) -> c_64()
, 45: #compare^#(#pos(@x), #pos(@y)) -> c_65(#compare^#(@x, @y))
, 46: #compare^#(#s(@x), #0()) -> c_66()
, 47: #compare^#(#s(@x), #s(@y)) -> c_67(#compare^#(@x, @y))
, 48: +^#(@x, @y) -> c_7(#add^#(@x, @y))
, 49: #add^#(#0(), @y) -> c_71()
, 50: #add^#(#neg(#s(#0())), @y) -> c_72(#pred^#(@y))
, 51: #add^#(#neg(#s(#s(@x))), @y) ->
c_73(#pred^#(#add(#pos(#s(@x)), @y)), #add^#(#pos(#s(@x)), @y))
, 52: #add^#(#pos(#s(#0())), @y) -> c_74(#succ^#(@y))
, 53: #add^#(#pos(#s(#s(@x))), @y) ->
c_75(#succ^#(#add(#pos(#s(@x)), @y)), #add^#(#pos(#s(@x)), @y))
, 54: firstline#1^#(nil()) -> c_10()
, 55: lcs#1^#(@m) -> c_13(lcs#2^#(@m))
, 56: lcs#2^#(::(@l1, @_@2)) -> c_14(lcs#3^#(@l1))
, 57: lcs#2^#(nil()) -> c_15(#abs^#(#0()))
, 58: lcs#3^#(::(@len, @_@1)) -> c_16()
, 59: lcs#3^#(nil()) -> c_17(#abs^#(#0()))
, 60: lcstable#3^#(nil(), @l2, @x) -> c_22()
, 61: newline#1^#(nil(), @lastline, @y) -> c_28()
, 62: max^#(@a, @b) ->
c_24(max#1^#(#greater(@a, @b), @a, @b), #greater^#(@a, @b))
, 63: max#1^#(#false(), @a, @b) -> c_25()
, 64: max#1^#(#true(), @a, @b) -> c_26()
, 65: newline#2^#(nil(), @x, @xs, @y) -> c_30()
, 66: right^#(@l) -> c_32(right#1^#(@l))
, 67: right#1^#(::(@x, @xs)) -> c_38()
, 68: right#1^#(nil()) -> c_39(#abs^#(#0()))
, 69: newline#5^#(@diagVal, @belowVal, @nl, @rightVal, @x, @y) ->
c_34(newline#6^#(newline#7(#equal(@x, @y),
@belowVal,
@diagVal,
@rightVal),
@nl),
newline#7^#(#equal(@x, @y), @belowVal, @diagVal, @rightVal),
#equal^#(@x, @y))
, 70: newline#6^#(@elem, @nl) -> c_37()
, 71: newline#7^#(#false(), @belowVal, @diagVal, @rightVal) ->
c_35(max^#(@belowVal, @rightVal))
, 72: newline#7^#(#true(), @belowVal, @diagVal, @rightVal) ->
c_36(+^#(@diagVal, #pos(#s(#0()))))
, 73: #and^#(#false(), #false()) -> c_84()
, 74: #and^#(#false(), #true()) -> c_85()
, 75: #and^#(#true(), #false()) -> c_86()
, 76: #and^#(#true(), #true()) -> c_87()
, 77: #pred^#(#0()) -> c_76()
, 78: #pred^#(#neg(#s(@x))) -> c_77()
, 79: #pred^#(#pos(#s(#0()))) -> c_78()
, 80: #pred^#(#pos(#s(#s(@x)))) -> c_79()
, 81: #succ^#(#0()) -> c_80()
, 82: #succ^#(#neg(#s(#0()))) -> c_81()
, 83: #succ^#(#neg(#s(#s(@x)))) -> c_82()
, 84: #succ^#(#pos(#s(@x))) -> c_83() }
We are left with following problem, upon which TcT provides the
certificate YES(O(1),O(n^2)).
Strict DPs:
{ firstline^#(@l) -> c_8(firstline#1^#(@l))
, firstline#1^#(::(@x, @xs)) -> c_9(#abs^#(#0()), firstline^#(@xs))
, lcs^#(@l1, @l2) ->
c_11(lcs#1^#(lcstable(@l1, @l2)), lcstable^#(@l1, @l2))
, lcstable^#(@l1, @l2) -> c_12(lcstable#1^#(@l1, @l2))
, lcstable#1^#(::(@x, @xs), @l2) ->
c_18(lcstable#2^#(lcstable(@xs, @l2), @l2, @x),
lcstable^#(@xs, @l2))
, lcstable#1^#(nil(), @l2) -> c_19(firstline^#(@l2))
, lcstable#2^#(@m, @l2, @x) -> c_20(lcstable#3^#(@m, @l2, @x))
, lcstable#3^#(::(@l, @ls), @l2, @x) ->
c_21(newline^#(@x, @l, @l2))
, newline^#(@y, @lastline, @l) ->
c_23(newline#1^#(@l, @lastline, @y))
, newline#1^#(::(@x, @xs), @lastline, @y) ->
c_27(newline#2^#(@lastline, @x, @xs, @y))
, newline#2^#(::(@belowVal, @lastline'), @x, @xs, @y) ->
c_29(newline#3^#(newline(@y, @lastline', @xs),
@belowVal,
@lastline',
@x,
@y),
newline^#(@y, @lastline', @xs))
, newline#3^#(@nl, @belowVal, @lastline', @x, @y) ->
c_31(newline#4^#(right(@nl), @belowVal, @lastline', @nl, @x, @y),
right^#(@nl)) }
Weak DPs:
{ #abs^#(#0()) -> c_1()
, #equal^#(@x, @y) -> c_5(#eq^#(@x, @y))
, #eq^#(#0(), #0()) -> c_40()
, #eq^#(#0(), #neg(@y)) -> c_41()
, #eq^#(#0(), #pos(@y)) -> c_42()
, #eq^#(#0(), #s(@y)) -> c_43()
, #eq^#(#neg(@x), #0()) -> c_44()
, #eq^#(#neg(@x), #neg(@y)) -> c_45(#eq^#(@x, @y))
, #eq^#(#neg(@x), #pos(@y)) -> c_46()
, #eq^#(#pos(@x), #0()) -> c_47()
, #eq^#(#pos(@x), #neg(@y)) -> c_48()
, #eq^#(#pos(@x), #pos(@y)) -> c_49(#eq^#(@x, @y))
, #eq^#(#s(@x), #0()) -> c_50()
, #eq^#(#s(@x), #s(@y)) -> c_51(#eq^#(@x, @y))
, #eq^#(::(@x_1, @x_2), ::(@y_1, @y_2)) ->
c_52(#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_53()
, #eq^#(nil(), ::(@y_1, @y_2)) -> c_54()
, #eq^#(nil(), nil()) -> c_55()
, #greater^#(@x, @y) ->
c_6(#ckgt^#(#compare(@x, @y)), #compare^#(@x, @y))
, #ckgt^#(#EQ()) -> c_68()
, #ckgt^#(#GT()) -> c_69()
, #ckgt^#(#LT()) -> c_70()
, #compare^#(#0(), #0()) -> c_56()
, #compare^#(#0(), #neg(@y)) -> c_57()
, #compare^#(#0(), #pos(@y)) -> c_58()
, #compare^#(#0(), #s(@y)) -> c_59()
, #compare^#(#neg(@x), #0()) -> c_60()
, #compare^#(#neg(@x), #neg(@y)) -> c_61(#compare^#(@y, @x))
, #compare^#(#neg(@x), #pos(@y)) -> c_62()
, #compare^#(#pos(@x), #0()) -> c_63()
, #compare^#(#pos(@x), #neg(@y)) -> c_64()
, #compare^#(#pos(@x), #pos(@y)) -> c_65(#compare^#(@x, @y))
, #compare^#(#s(@x), #0()) -> c_66()
, #compare^#(#s(@x), #s(@y)) -> c_67(#compare^#(@x, @y))
, +^#(@x, @y) -> c_7(#add^#(@x, @y))
, #add^#(#0(), @y) -> c_71()
, #add^#(#neg(#s(#0())), @y) -> c_72(#pred^#(@y))
, #add^#(#neg(#s(#s(@x))), @y) ->
c_73(#pred^#(#add(#pos(#s(@x)), @y)), #add^#(#pos(#s(@x)), @y))
, #add^#(#pos(#s(#0())), @y) -> c_74(#succ^#(@y))
, #add^#(#pos(#s(#s(@x))), @y) ->
c_75(#succ^#(#add(#pos(#s(@x)), @y)), #add^#(#pos(#s(@x)), @y))
, firstline#1^#(nil()) -> c_10()
, lcs#1^#(@m) -> c_13(lcs#2^#(@m))
, lcs#2^#(::(@l1, @_@2)) -> c_14(lcs#3^#(@l1))
, lcs#2^#(nil()) -> c_15(#abs^#(#0()))
, lcs#3^#(::(@len, @_@1)) -> c_16()
, lcs#3^#(nil()) -> c_17(#abs^#(#0()))
, lcstable#3^#(nil(), @l2, @x) -> c_22()
, newline#1^#(nil(), @lastline, @y) -> c_28()
, max^#(@a, @b) ->
c_24(max#1^#(#greater(@a, @b), @a, @b), #greater^#(@a, @b))
, max#1^#(#false(), @a, @b) -> c_25()
, max#1^#(#true(), @a, @b) -> c_26()
, newline#2^#(nil(), @x, @xs, @y) -> c_30()
, newline#4^#(@rightVal, @belowVal, @lastline', @nl, @x, @y) ->
c_33(newline#5^#(right(@lastline'),
@belowVal,
@nl,
@rightVal,
@x,
@y),
right^#(@lastline'))
, right^#(@l) -> c_32(right#1^#(@l))
, right#1^#(::(@x, @xs)) -> c_38()
, right#1^#(nil()) -> c_39(#abs^#(#0()))
, newline#5^#(@diagVal, @belowVal, @nl, @rightVal, @x, @y) ->
c_34(newline#6^#(newline#7(#equal(@x, @y),
@belowVal,
@diagVal,
@rightVal),
@nl),
newline#7^#(#equal(@x, @y), @belowVal, @diagVal, @rightVal),
#equal^#(@x, @y))
, newline#6^#(@elem, @nl) -> c_37()
, newline#7^#(#false(), @belowVal, @diagVal, @rightVal) ->
c_35(max^#(@belowVal, @rightVal))
, newline#7^#(#true(), @belowVal, @diagVal, @rightVal) ->
c_36(+^#(@diagVal, #pos(#s(#0()))))
, #and^#(#false(), #false()) -> c_84()
, #and^#(#false(), #true()) -> c_85()
, #and^#(#true(), #false()) -> c_86()
, #and^#(#true(), #true()) -> c_87()
, #pred^#(#0()) -> c_76()
, #pred^#(#neg(#s(@x))) -> c_77()
, #pred^#(#pos(#s(#0()))) -> c_78()
, #pred^#(#pos(#s(#s(@x)))) -> c_79()
, #succ^#(#0()) -> c_80()
, #succ^#(#neg(#s(#0()))) -> c_81()
, #succ^#(#neg(#s(#s(@x)))) -> c_82()
, #succ^#(#pos(#s(@x))) -> c_83() }
Weak Trs:
{ #abs(#0()) -> #0()
, #abs(#neg(@x)) -> #pos(@x)
, #abs(#pos(@x)) -> #pos(@x)
, #abs(#s(@x)) -> #pos(#s(@x))
, #equal(@x, @y) -> #eq(@x, @y)
, #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)
, #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()
, #greater(@x, @y) -> #ckgt(#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)
, #ckgt(#EQ()) -> #false()
, #ckgt(#GT()) -> #true()
, #ckgt(#LT()) -> #false()
, +(@x, @y) -> #add(@x, @y)
, #add(#0(), @y) -> @y
, #add(#neg(#s(#0())), @y) -> #pred(@y)
, #add(#neg(#s(#s(@x))), @y) -> #pred(#add(#pos(#s(@x)), @y))
, #add(#pos(#s(#0())), @y) -> #succ(@y)
, #add(#pos(#s(#s(@x))), @y) -> #succ(#add(#pos(#s(@x)), @y))
, firstline(@l) -> firstline#1(@l)
, firstline#1(::(@x, @xs)) -> ::(#abs(#0()), firstline(@xs))
, firstline#1(nil()) -> nil()
, lcs(@l1, @l2) -> lcs#1(lcstable(@l1, @l2))
, lcstable(@l1, @l2) -> lcstable#1(@l1, @l2)
, lcs#1(@m) -> lcs#2(@m)
, lcs#2(::(@l1, @_@2)) -> lcs#3(@l1)
, lcs#2(nil()) -> #abs(#0())
, lcs#3(::(@len, @_@1)) -> @len
, lcs#3(nil()) -> #abs(#0())
, lcstable#1(::(@x, @xs), @l2) ->
lcstable#2(lcstable(@xs, @l2), @l2, @x)
, lcstable#1(nil(), @l2) -> ::(firstline(@l2), nil())
, lcstable#2(@m, @l2, @x) -> lcstable#3(@m, @l2, @x)
, lcstable#3(::(@l, @ls), @l2, @x) ->
::(newline(@x, @l, @l2), ::(@l, @ls))
, lcstable#3(nil(), @l2, @x) -> nil()
, newline(@y, @lastline, @l) -> newline#1(@l, @lastline, @y)
, max(@a, @b) -> max#1(#greater(@a, @b), @a, @b)
, max#1(#false(), @a, @b) -> @b
, max#1(#true(), @a, @b) -> @a
, newline#1(::(@x, @xs), @lastline, @y) ->
newline#2(@lastline, @x, @xs, @y)
, newline#1(nil(), @lastline, @y) -> nil()
, newline#2(::(@belowVal, @lastline'), @x, @xs, @y) ->
newline#3(newline(@y, @lastline', @xs),
@belowVal,
@lastline',
@x,
@y)
, newline#2(nil(), @x, @xs, @y) -> nil()
, newline#3(@nl, @belowVal, @lastline', @x, @y) ->
newline#4(right(@nl), @belowVal, @lastline', @nl, @x, @y)
, right(@l) -> right#1(@l)
, newline#4(@rightVal, @belowVal, @lastline', @nl, @x, @y) ->
newline#5(right(@lastline'), @belowVal, @nl, @rightVal, @x, @y)
, newline#5(@diagVal, @belowVal, @nl, @rightVal, @x, @y) ->
newline#6(newline#7(#equal(@x, @y),
@belowVal,
@diagVal,
@rightVal),
@nl)
, newline#7(#false(), @belowVal, @diagVal, @rightVal) ->
max(@belowVal, @rightVal)
, newline#7(#true(), @belowVal, @diagVal, @rightVal) ->
+(@diagVal, #pos(#s(#0())))
, newline#6(@elem, @nl) -> ::(@elem, @nl)
, right#1(::(@x, @xs)) -> @x
, right#1(nil()) -> #abs(#0())
, #pred(#0()) -> #neg(#s(#0()))
, #pred(#neg(#s(@x))) -> #neg(#s(#s(@x)))
, #pred(#pos(#s(#0()))) -> #0()
, #pred(#pos(#s(#s(@x)))) -> #pos(#s(@x))
, #succ(#0()) -> #pos(#s(#0()))
, #succ(#neg(#s(#0()))) -> #0()
, #succ(#neg(#s(#s(@x)))) -> #neg(#s(@x))
, #succ(#pos(#s(@x))) -> #pos(#s(#s(@x)))
, #and(#false(), #false()) -> #false()
, #and(#false(), #true()) -> #false()
, #and(#true(), #false()) -> #false()
, #and(#true(), #true()) -> #true() }
Obligation:
innermost runtime complexity
Answer:
YES(O(1),O(n^2))
We estimate the number of application of {12} by applications of
Pre({12}) = {11}. Here rules are labeled as follows:
DPs:
{ 1: firstline^#(@l) -> c_8(firstline#1^#(@l))
, 2: firstline#1^#(::(@x, @xs)) ->
c_9(#abs^#(#0()), firstline^#(@xs))
, 3: lcs^#(@l1, @l2) ->
c_11(lcs#1^#(lcstable(@l1, @l2)), lcstable^#(@l1, @l2))
, 4: lcstable^#(@l1, @l2) -> c_12(lcstable#1^#(@l1, @l2))
, 5: lcstable#1^#(::(@x, @xs), @l2) ->
c_18(lcstable#2^#(lcstable(@xs, @l2), @l2, @x),
lcstable^#(@xs, @l2))
, 6: lcstable#1^#(nil(), @l2) -> c_19(firstline^#(@l2))
, 7: lcstable#2^#(@m, @l2, @x) -> c_20(lcstable#3^#(@m, @l2, @x))
, 8: lcstable#3^#(::(@l, @ls), @l2, @x) ->
c_21(newline^#(@x, @l, @l2))
, 9: newline^#(@y, @lastline, @l) ->
c_23(newline#1^#(@l, @lastline, @y))
, 10: newline#1^#(::(@x, @xs), @lastline, @y) ->
c_27(newline#2^#(@lastline, @x, @xs, @y))
, 11: newline#2^#(::(@belowVal, @lastline'), @x, @xs, @y) ->
c_29(newline#3^#(newline(@y, @lastline', @xs),
@belowVal,
@lastline',
@x,
@y),
newline^#(@y, @lastline', @xs))
, 12: newline#3^#(@nl, @belowVal, @lastline', @x, @y) ->
c_31(newline#4^#(right(@nl), @belowVal, @lastline', @nl, @x, @y),
right^#(@nl))
, 13: #abs^#(#0()) -> c_1()
, 14: #equal^#(@x, @y) -> c_5(#eq^#(@x, @y))
, 15: #eq^#(#0(), #0()) -> c_40()
, 16: #eq^#(#0(), #neg(@y)) -> c_41()
, 17: #eq^#(#0(), #pos(@y)) -> c_42()
, 18: #eq^#(#0(), #s(@y)) -> c_43()
, 19: #eq^#(#neg(@x), #0()) -> c_44()
, 20: #eq^#(#neg(@x), #neg(@y)) -> c_45(#eq^#(@x, @y))
, 21: #eq^#(#neg(@x), #pos(@y)) -> c_46()
, 22: #eq^#(#pos(@x), #0()) -> c_47()
, 23: #eq^#(#pos(@x), #neg(@y)) -> c_48()
, 24: #eq^#(#pos(@x), #pos(@y)) -> c_49(#eq^#(@x, @y))
, 25: #eq^#(#s(@x), #0()) -> c_50()
, 26: #eq^#(#s(@x), #s(@y)) -> c_51(#eq^#(@x, @y))
, 27: #eq^#(::(@x_1, @x_2), ::(@y_1, @y_2)) ->
c_52(#and^#(#eq(@x_1, @y_1), #eq(@x_2, @y_2)),
#eq^#(@x_1, @y_1),
#eq^#(@x_2, @y_2))
, 28: #eq^#(::(@x_1, @x_2), nil()) -> c_53()
, 29: #eq^#(nil(), ::(@y_1, @y_2)) -> c_54()
, 30: #eq^#(nil(), nil()) -> c_55()
, 31: #greater^#(@x, @y) ->
c_6(#ckgt^#(#compare(@x, @y)), #compare^#(@x, @y))
, 32: #ckgt^#(#EQ()) -> c_68()
, 33: #ckgt^#(#GT()) -> c_69()
, 34: #ckgt^#(#LT()) -> c_70()
, 35: #compare^#(#0(), #0()) -> c_56()
, 36: #compare^#(#0(), #neg(@y)) -> c_57()
, 37: #compare^#(#0(), #pos(@y)) -> c_58()
, 38: #compare^#(#0(), #s(@y)) -> c_59()
, 39: #compare^#(#neg(@x), #0()) -> c_60()
, 40: #compare^#(#neg(@x), #neg(@y)) -> c_61(#compare^#(@y, @x))
, 41: #compare^#(#neg(@x), #pos(@y)) -> c_62()
, 42: #compare^#(#pos(@x), #0()) -> c_63()
, 43: #compare^#(#pos(@x), #neg(@y)) -> c_64()
, 44: #compare^#(#pos(@x), #pos(@y)) -> c_65(#compare^#(@x, @y))
, 45: #compare^#(#s(@x), #0()) -> c_66()
, 46: #compare^#(#s(@x), #s(@y)) -> c_67(#compare^#(@x, @y))
, 47: +^#(@x, @y) -> c_7(#add^#(@x, @y))
, 48: #add^#(#0(), @y) -> c_71()
, 49: #add^#(#neg(#s(#0())), @y) -> c_72(#pred^#(@y))
, 50: #add^#(#neg(#s(#s(@x))), @y) ->
c_73(#pred^#(#add(#pos(#s(@x)), @y)), #add^#(#pos(#s(@x)), @y))
, 51: #add^#(#pos(#s(#0())), @y) -> c_74(#succ^#(@y))
, 52: #add^#(#pos(#s(#s(@x))), @y) ->
c_75(#succ^#(#add(#pos(#s(@x)), @y)), #add^#(#pos(#s(@x)), @y))
, 53: firstline#1^#(nil()) -> c_10()
, 54: lcs#1^#(@m) -> c_13(lcs#2^#(@m))
, 55: lcs#2^#(::(@l1, @_@2)) -> c_14(lcs#3^#(@l1))
, 56: lcs#2^#(nil()) -> c_15(#abs^#(#0()))
, 57: lcs#3^#(::(@len, @_@1)) -> c_16()
, 58: lcs#3^#(nil()) -> c_17(#abs^#(#0()))
, 59: lcstable#3^#(nil(), @l2, @x) -> c_22()
, 60: newline#1^#(nil(), @lastline, @y) -> c_28()
, 61: max^#(@a, @b) ->
c_24(max#1^#(#greater(@a, @b), @a, @b), #greater^#(@a, @b))
, 62: max#1^#(#false(), @a, @b) -> c_25()
, 63: max#1^#(#true(), @a, @b) -> c_26()
, 64: newline#2^#(nil(), @x, @xs, @y) -> c_30()
, 65: newline#4^#(@rightVal, @belowVal, @lastline', @nl, @x, @y) ->
c_33(newline#5^#(right(@lastline'),
@belowVal,
@nl,
@rightVal,
@x,
@y),
right^#(@lastline'))
, 66: right^#(@l) -> c_32(right#1^#(@l))
, 67: right#1^#(::(@x, @xs)) -> c_38()
, 68: right#1^#(nil()) -> c_39(#abs^#(#0()))
, 69: newline#5^#(@diagVal, @belowVal, @nl, @rightVal, @x, @y) ->
c_34(newline#6^#(newline#7(#equal(@x, @y),
@belowVal,
@diagVal,
@rightVal),
@nl),
newline#7^#(#equal(@x, @y), @belowVal, @diagVal, @rightVal),
#equal^#(@x, @y))
, 70: newline#6^#(@elem, @nl) -> c_37()
, 71: newline#7^#(#false(), @belowVal, @diagVal, @rightVal) ->
c_35(max^#(@belowVal, @rightVal))
, 72: newline#7^#(#true(), @belowVal, @diagVal, @rightVal) ->
c_36(+^#(@diagVal, #pos(#s(#0()))))
, 73: #and^#(#false(), #false()) -> c_84()
, 74: #and^#(#false(), #true()) -> c_85()
, 75: #and^#(#true(), #false()) -> c_86()
, 76: #and^#(#true(), #true()) -> c_87()
, 77: #pred^#(#0()) -> c_76()
, 78: #pred^#(#neg(#s(@x))) -> c_77()
, 79: #pred^#(#pos(#s(#0()))) -> c_78()
, 80: #pred^#(#pos(#s(#s(@x)))) -> c_79()
, 81: #succ^#(#0()) -> c_80()
, 82: #succ^#(#neg(#s(#0()))) -> c_81()
, 83: #succ^#(#neg(#s(#s(@x)))) -> c_82()
, 84: #succ^#(#pos(#s(@x))) -> c_83() }
We are left with following problem, upon which TcT provides the
certificate YES(O(1),O(n^2)).
Strict DPs:
{ firstline^#(@l) -> c_8(firstline#1^#(@l))
, firstline#1^#(::(@x, @xs)) -> c_9(#abs^#(#0()), firstline^#(@xs))
, lcs^#(@l1, @l2) ->
c_11(lcs#1^#(lcstable(@l1, @l2)), lcstable^#(@l1, @l2))
, lcstable^#(@l1, @l2) -> c_12(lcstable#1^#(@l1, @l2))
, lcstable#1^#(::(@x, @xs), @l2) ->
c_18(lcstable#2^#(lcstable(@xs, @l2), @l2, @x),
lcstable^#(@xs, @l2))
, lcstable#1^#(nil(), @l2) -> c_19(firstline^#(@l2))
, lcstable#2^#(@m, @l2, @x) -> c_20(lcstable#3^#(@m, @l2, @x))
, lcstable#3^#(::(@l, @ls), @l2, @x) ->
c_21(newline^#(@x, @l, @l2))
, newline^#(@y, @lastline, @l) ->
c_23(newline#1^#(@l, @lastline, @y))
, newline#1^#(::(@x, @xs), @lastline, @y) ->
c_27(newline#2^#(@lastline, @x, @xs, @y))
, newline#2^#(::(@belowVal, @lastline'), @x, @xs, @y) ->
c_29(newline#3^#(newline(@y, @lastline', @xs),
@belowVal,
@lastline',
@x,
@y),
newline^#(@y, @lastline', @xs)) }
Weak DPs:
{ #abs^#(#0()) -> c_1()
, #equal^#(@x, @y) -> c_5(#eq^#(@x, @y))
, #eq^#(#0(), #0()) -> c_40()
, #eq^#(#0(), #neg(@y)) -> c_41()
, #eq^#(#0(), #pos(@y)) -> c_42()
, #eq^#(#0(), #s(@y)) -> c_43()
, #eq^#(#neg(@x), #0()) -> c_44()
, #eq^#(#neg(@x), #neg(@y)) -> c_45(#eq^#(@x, @y))
, #eq^#(#neg(@x), #pos(@y)) -> c_46()
, #eq^#(#pos(@x), #0()) -> c_47()
, #eq^#(#pos(@x), #neg(@y)) -> c_48()
, #eq^#(#pos(@x), #pos(@y)) -> c_49(#eq^#(@x, @y))
, #eq^#(#s(@x), #0()) -> c_50()
, #eq^#(#s(@x), #s(@y)) -> c_51(#eq^#(@x, @y))
, #eq^#(::(@x_1, @x_2), ::(@y_1, @y_2)) ->
c_52(#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_53()
, #eq^#(nil(), ::(@y_1, @y_2)) -> c_54()
, #eq^#(nil(), nil()) -> c_55()
, #greater^#(@x, @y) ->
c_6(#ckgt^#(#compare(@x, @y)), #compare^#(@x, @y))
, #ckgt^#(#EQ()) -> c_68()
, #ckgt^#(#GT()) -> c_69()
, #ckgt^#(#LT()) -> c_70()
, #compare^#(#0(), #0()) -> c_56()
, #compare^#(#0(), #neg(@y)) -> c_57()
, #compare^#(#0(), #pos(@y)) -> c_58()
, #compare^#(#0(), #s(@y)) -> c_59()
, #compare^#(#neg(@x), #0()) -> c_60()
, #compare^#(#neg(@x), #neg(@y)) -> c_61(#compare^#(@y, @x))
, #compare^#(#neg(@x), #pos(@y)) -> c_62()
, #compare^#(#pos(@x), #0()) -> c_63()
, #compare^#(#pos(@x), #neg(@y)) -> c_64()
, #compare^#(#pos(@x), #pos(@y)) -> c_65(#compare^#(@x, @y))
, #compare^#(#s(@x), #0()) -> c_66()
, #compare^#(#s(@x), #s(@y)) -> c_67(#compare^#(@x, @y))
, +^#(@x, @y) -> c_7(#add^#(@x, @y))
, #add^#(#0(), @y) -> c_71()
, #add^#(#neg(#s(#0())), @y) -> c_72(#pred^#(@y))
, #add^#(#neg(#s(#s(@x))), @y) ->
c_73(#pred^#(#add(#pos(#s(@x)), @y)), #add^#(#pos(#s(@x)), @y))
, #add^#(#pos(#s(#0())), @y) -> c_74(#succ^#(@y))
, #add^#(#pos(#s(#s(@x))), @y) ->
c_75(#succ^#(#add(#pos(#s(@x)), @y)), #add^#(#pos(#s(@x)), @y))
, firstline#1^#(nil()) -> c_10()
, lcs#1^#(@m) -> c_13(lcs#2^#(@m))
, lcs#2^#(::(@l1, @_@2)) -> c_14(lcs#3^#(@l1))
, lcs#2^#(nil()) -> c_15(#abs^#(#0()))
, lcs#3^#(::(@len, @_@1)) -> c_16()
, lcs#3^#(nil()) -> c_17(#abs^#(#0()))
, lcstable#3^#(nil(), @l2, @x) -> c_22()
, newline#1^#(nil(), @lastline, @y) -> c_28()
, max^#(@a, @b) ->
c_24(max#1^#(#greater(@a, @b), @a, @b), #greater^#(@a, @b))
, max#1^#(#false(), @a, @b) -> c_25()
, max#1^#(#true(), @a, @b) -> c_26()
, newline#2^#(nil(), @x, @xs, @y) -> c_30()
, newline#3^#(@nl, @belowVal, @lastline', @x, @y) ->
c_31(newline#4^#(right(@nl), @belowVal, @lastline', @nl, @x, @y),
right^#(@nl))
, newline#4^#(@rightVal, @belowVal, @lastline', @nl, @x, @y) ->
c_33(newline#5^#(right(@lastline'),
@belowVal,
@nl,
@rightVal,
@x,
@y),
right^#(@lastline'))
, right^#(@l) -> c_32(right#1^#(@l))
, right#1^#(::(@x, @xs)) -> c_38()
, right#1^#(nil()) -> c_39(#abs^#(#0()))
, newline#5^#(@diagVal, @belowVal, @nl, @rightVal, @x, @y) ->
c_34(newline#6^#(newline#7(#equal(@x, @y),
@belowVal,
@diagVal,
@rightVal),
@nl),
newline#7^#(#equal(@x, @y), @belowVal, @diagVal, @rightVal),
#equal^#(@x, @y))
, newline#6^#(@elem, @nl) -> c_37()
, newline#7^#(#false(), @belowVal, @diagVal, @rightVal) ->
c_35(max^#(@belowVal, @rightVal))
, newline#7^#(#true(), @belowVal, @diagVal, @rightVal) ->
c_36(+^#(@diagVal, #pos(#s(#0()))))
, #and^#(#false(), #false()) -> c_84()
, #and^#(#false(), #true()) -> c_85()
, #and^#(#true(), #false()) -> c_86()
, #and^#(#true(), #true()) -> c_87()
, #pred^#(#0()) -> c_76()
, #pred^#(#neg(#s(@x))) -> c_77()
, #pred^#(#pos(#s(#0()))) -> c_78()
, #pred^#(#pos(#s(#s(@x)))) -> c_79()
, #succ^#(#0()) -> c_80()
, #succ^#(#neg(#s(#0()))) -> c_81()
, #succ^#(#neg(#s(#s(@x)))) -> c_82()
, #succ^#(#pos(#s(@x))) -> c_83() }
Weak Trs:
{ #abs(#0()) -> #0()
, #abs(#neg(@x)) -> #pos(@x)
, #abs(#pos(@x)) -> #pos(@x)
, #abs(#s(@x)) -> #pos(#s(@x))
, #equal(@x, @y) -> #eq(@x, @y)
, #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)
, #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()
, #greater(@x, @y) -> #ckgt(#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)
, #ckgt(#EQ()) -> #false()
, #ckgt(#GT()) -> #true()
, #ckgt(#LT()) -> #false()
, +(@x, @y) -> #add(@x, @y)
, #add(#0(), @y) -> @y
, #add(#neg(#s(#0())), @y) -> #pred(@y)
, #add(#neg(#s(#s(@x))), @y) -> #pred(#add(#pos(#s(@x)), @y))
, #add(#pos(#s(#0())), @y) -> #succ(@y)
, #add(#pos(#s(#s(@x))), @y) -> #succ(#add(#pos(#s(@x)), @y))
, firstline(@l) -> firstline#1(@l)
, firstline#1(::(@x, @xs)) -> ::(#abs(#0()), firstline(@xs))
, firstline#1(nil()) -> nil()
, lcs(@l1, @l2) -> lcs#1(lcstable(@l1, @l2))
, lcstable(@l1, @l2) -> lcstable#1(@l1, @l2)
, lcs#1(@m) -> lcs#2(@m)
, lcs#2(::(@l1, @_@2)) -> lcs#3(@l1)
, lcs#2(nil()) -> #abs(#0())
, lcs#3(::(@len, @_@1)) -> @len
, lcs#3(nil()) -> #abs(#0())
, lcstable#1(::(@x, @xs), @l2) ->
lcstable#2(lcstable(@xs, @l2), @l2, @x)
, lcstable#1(nil(), @l2) -> ::(firstline(@l2), nil())
, lcstable#2(@m, @l2, @x) -> lcstable#3(@m, @l2, @x)
, lcstable#3(::(@l, @ls), @l2, @x) ->
::(newline(@x, @l, @l2), ::(@l, @ls))
, lcstable#3(nil(), @l2, @x) -> nil()
, newline(@y, @lastline, @l) -> newline#1(@l, @lastline, @y)
, max(@a, @b) -> max#1(#greater(@a, @b), @a, @b)
, max#1(#false(), @a, @b) -> @b
, max#1(#true(), @a, @b) -> @a
, newline#1(::(@x, @xs), @lastline, @y) ->
newline#2(@lastline, @x, @xs, @y)
, newline#1(nil(), @lastline, @y) -> nil()
, newline#2(::(@belowVal, @lastline'), @x, @xs, @y) ->
newline#3(newline(@y, @lastline', @xs),
@belowVal,
@lastline',
@x,
@y)
, newline#2(nil(), @x, @xs, @y) -> nil()
, newline#3(@nl, @belowVal, @lastline', @x, @y) ->
newline#4(right(@nl), @belowVal, @lastline', @nl, @x, @y)
, right(@l) -> right#1(@l)
, newline#4(@rightVal, @belowVal, @lastline', @nl, @x, @y) ->
newline#5(right(@lastline'), @belowVal, @nl, @rightVal, @x, @y)
, newline#5(@diagVal, @belowVal, @nl, @rightVal, @x, @y) ->
newline#6(newline#7(#equal(@x, @y),
@belowVal,
@diagVal,
@rightVal),
@nl)
, newline#7(#false(), @belowVal, @diagVal, @rightVal) ->
max(@belowVal, @rightVal)
, newline#7(#true(), @belowVal, @diagVal, @rightVal) ->
+(@diagVal, #pos(#s(#0())))
, newline#6(@elem, @nl) -> ::(@elem, @nl)
, right#1(::(@x, @xs)) -> @x
, right#1(nil()) -> #abs(#0())
, #pred(#0()) -> #neg(#s(#0()))
, #pred(#neg(#s(@x))) -> #neg(#s(#s(@x)))
, #pred(#pos(#s(#0()))) -> #0()
, #pred(#pos(#s(#s(@x)))) -> #pos(#s(@x))
, #succ(#0()) -> #pos(#s(#0()))
, #succ(#neg(#s(#0()))) -> #0()
, #succ(#neg(#s(#s(@x)))) -> #neg(#s(@x))
, #succ(#pos(#s(@x))) -> #pos(#s(#s(@x)))
, #and(#false(), #false()) -> #false()
, #and(#false(), #true()) -> #false()
, #and(#true(), #false()) -> #false()
, #and(#true(), #true()) -> #true() }
Obligation:
innermost runtime complexity
Answer:
YES(O(1),O(n^2))
The following weak DPs constitute a sub-graph of the DG that is
closed under successors. The DPs are removed.
{ #abs^#(#0()) -> c_1()
, #equal^#(@x, @y) -> c_5(#eq^#(@x, @y))
, #eq^#(#0(), #0()) -> c_40()
, #eq^#(#0(), #neg(@y)) -> c_41()
, #eq^#(#0(), #pos(@y)) -> c_42()
, #eq^#(#0(), #s(@y)) -> c_43()
, #eq^#(#neg(@x), #0()) -> c_44()
, #eq^#(#neg(@x), #neg(@y)) -> c_45(#eq^#(@x, @y))
, #eq^#(#neg(@x), #pos(@y)) -> c_46()
, #eq^#(#pos(@x), #0()) -> c_47()
, #eq^#(#pos(@x), #neg(@y)) -> c_48()
, #eq^#(#pos(@x), #pos(@y)) -> c_49(#eq^#(@x, @y))
, #eq^#(#s(@x), #0()) -> c_50()
, #eq^#(#s(@x), #s(@y)) -> c_51(#eq^#(@x, @y))
, #eq^#(::(@x_1, @x_2), ::(@y_1, @y_2)) ->
c_52(#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_53()
, #eq^#(nil(), ::(@y_1, @y_2)) -> c_54()
, #eq^#(nil(), nil()) -> c_55()
, #greater^#(@x, @y) ->
c_6(#ckgt^#(#compare(@x, @y)), #compare^#(@x, @y))
, #ckgt^#(#EQ()) -> c_68()
, #ckgt^#(#GT()) -> c_69()
, #ckgt^#(#LT()) -> c_70()
, #compare^#(#0(), #0()) -> c_56()
, #compare^#(#0(), #neg(@y)) -> c_57()
, #compare^#(#0(), #pos(@y)) -> c_58()
, #compare^#(#0(), #s(@y)) -> c_59()
, #compare^#(#neg(@x), #0()) -> c_60()
, #compare^#(#neg(@x), #neg(@y)) -> c_61(#compare^#(@y, @x))
, #compare^#(#neg(@x), #pos(@y)) -> c_62()
, #compare^#(#pos(@x), #0()) -> c_63()
, #compare^#(#pos(@x), #neg(@y)) -> c_64()
, #compare^#(#pos(@x), #pos(@y)) -> c_65(#compare^#(@x, @y))
, #compare^#(#s(@x), #0()) -> c_66()
, #compare^#(#s(@x), #s(@y)) -> c_67(#compare^#(@x, @y))
, +^#(@x, @y) -> c_7(#add^#(@x, @y))
, #add^#(#0(), @y) -> c_71()
, #add^#(#neg(#s(#0())), @y) -> c_72(#pred^#(@y))
, #add^#(#neg(#s(#s(@x))), @y) ->
c_73(#pred^#(#add(#pos(#s(@x)), @y)), #add^#(#pos(#s(@x)), @y))
, #add^#(#pos(#s(#0())), @y) -> c_74(#succ^#(@y))
, #add^#(#pos(#s(#s(@x))), @y) ->
c_75(#succ^#(#add(#pos(#s(@x)), @y)), #add^#(#pos(#s(@x)), @y))
, firstline#1^#(nil()) -> c_10()
, lcs#1^#(@m) -> c_13(lcs#2^#(@m))
, lcs#2^#(::(@l1, @_@2)) -> c_14(lcs#3^#(@l1))
, lcs#2^#(nil()) -> c_15(#abs^#(#0()))
, lcs#3^#(::(@len, @_@1)) -> c_16()
, lcs#3^#(nil()) -> c_17(#abs^#(#0()))
, lcstable#3^#(nil(), @l2, @x) -> c_22()
, newline#1^#(nil(), @lastline, @y) -> c_28()
, max^#(@a, @b) ->
c_24(max#1^#(#greater(@a, @b), @a, @b), #greater^#(@a, @b))
, max#1^#(#false(), @a, @b) -> c_25()
, max#1^#(#true(), @a, @b) -> c_26()
, newline#2^#(nil(), @x, @xs, @y) -> c_30()
, newline#3^#(@nl, @belowVal, @lastline', @x, @y) ->
c_31(newline#4^#(right(@nl), @belowVal, @lastline', @nl, @x, @y),
right^#(@nl))
, newline#4^#(@rightVal, @belowVal, @lastline', @nl, @x, @y) ->
c_33(newline#5^#(right(@lastline'),
@belowVal,
@nl,
@rightVal,
@x,
@y),
right^#(@lastline'))
, right^#(@l) -> c_32(right#1^#(@l))
, right#1^#(::(@x, @xs)) -> c_38()
, right#1^#(nil()) -> c_39(#abs^#(#0()))
, newline#5^#(@diagVal, @belowVal, @nl, @rightVal, @x, @y) ->
c_34(newline#6^#(newline#7(#equal(@x, @y),
@belowVal,
@diagVal,
@rightVal),
@nl),
newline#7^#(#equal(@x, @y), @belowVal, @diagVal, @rightVal),
#equal^#(@x, @y))
, newline#6^#(@elem, @nl) -> c_37()
, newline#7^#(#false(), @belowVal, @diagVal, @rightVal) ->
c_35(max^#(@belowVal, @rightVal))
, newline#7^#(#true(), @belowVal, @diagVal, @rightVal) ->
c_36(+^#(@diagVal, #pos(#s(#0()))))
, #and^#(#false(), #false()) -> c_84()
, #and^#(#false(), #true()) -> c_85()
, #and^#(#true(), #false()) -> c_86()
, #and^#(#true(), #true()) -> c_87()
, #pred^#(#0()) -> c_76()
, #pred^#(#neg(#s(@x))) -> c_77()
, #pred^#(#pos(#s(#0()))) -> c_78()
, #pred^#(#pos(#s(#s(@x)))) -> c_79()
, #succ^#(#0()) -> c_80()
, #succ^#(#neg(#s(#0()))) -> c_81()
, #succ^#(#neg(#s(#s(@x)))) -> c_82()
, #succ^#(#pos(#s(@x))) -> c_83() }
We are left with following problem, upon which TcT provides the
certificate YES(O(1),O(n^2)).
Strict DPs:
{ firstline^#(@l) -> c_8(firstline#1^#(@l))
, firstline#1^#(::(@x, @xs)) -> c_9(#abs^#(#0()), firstline^#(@xs))
, lcs^#(@l1, @l2) ->
c_11(lcs#1^#(lcstable(@l1, @l2)), lcstable^#(@l1, @l2))
, lcstable^#(@l1, @l2) -> c_12(lcstable#1^#(@l1, @l2))
, lcstable#1^#(::(@x, @xs), @l2) ->
c_18(lcstable#2^#(lcstable(@xs, @l2), @l2, @x),
lcstable^#(@xs, @l2))
, lcstable#1^#(nil(), @l2) -> c_19(firstline^#(@l2))
, lcstable#2^#(@m, @l2, @x) -> c_20(lcstable#3^#(@m, @l2, @x))
, lcstable#3^#(::(@l, @ls), @l2, @x) ->
c_21(newline^#(@x, @l, @l2))
, newline^#(@y, @lastline, @l) ->
c_23(newline#1^#(@l, @lastline, @y))
, newline#1^#(::(@x, @xs), @lastline, @y) ->
c_27(newline#2^#(@lastline, @x, @xs, @y))
, newline#2^#(::(@belowVal, @lastline'), @x, @xs, @y) ->
c_29(newline#3^#(newline(@y, @lastline', @xs),
@belowVal,
@lastline',
@x,
@y),
newline^#(@y, @lastline', @xs)) }
Weak Trs:
{ #abs(#0()) -> #0()
, #abs(#neg(@x)) -> #pos(@x)
, #abs(#pos(@x)) -> #pos(@x)
, #abs(#s(@x)) -> #pos(#s(@x))
, #equal(@x, @y) -> #eq(@x, @y)
, #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)
, #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()
, #greater(@x, @y) -> #ckgt(#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)
, #ckgt(#EQ()) -> #false()
, #ckgt(#GT()) -> #true()
, #ckgt(#LT()) -> #false()
, +(@x, @y) -> #add(@x, @y)
, #add(#0(), @y) -> @y
, #add(#neg(#s(#0())), @y) -> #pred(@y)
, #add(#neg(#s(#s(@x))), @y) -> #pred(#add(#pos(#s(@x)), @y))
, #add(#pos(#s(#0())), @y) -> #succ(@y)
, #add(#pos(#s(#s(@x))), @y) -> #succ(#add(#pos(#s(@x)), @y))
, firstline(@l) -> firstline#1(@l)
, firstline#1(::(@x, @xs)) -> ::(#abs(#0()), firstline(@xs))
, firstline#1(nil()) -> nil()
, lcs(@l1, @l2) -> lcs#1(lcstable(@l1, @l2))
, lcstable(@l1, @l2) -> lcstable#1(@l1, @l2)
, lcs#1(@m) -> lcs#2(@m)
, lcs#2(::(@l1, @_@2)) -> lcs#3(@l1)
, lcs#2(nil()) -> #abs(#0())
, lcs#3(::(@len, @_@1)) -> @len
, lcs#3(nil()) -> #abs(#0())
, lcstable#1(::(@x, @xs), @l2) ->
lcstable#2(lcstable(@xs, @l2), @l2, @x)
, lcstable#1(nil(), @l2) -> ::(firstline(@l2), nil())
, lcstable#2(@m, @l2, @x) -> lcstable#3(@m, @l2, @x)
, lcstable#3(::(@l, @ls), @l2, @x) ->
::(newline(@x, @l, @l2), ::(@l, @ls))
, lcstable#3(nil(), @l2, @x) -> nil()
, newline(@y, @lastline, @l) -> newline#1(@l, @lastline, @y)
, max(@a, @b) -> max#1(#greater(@a, @b), @a, @b)
, max#1(#false(), @a, @b) -> @b
, max#1(#true(), @a, @b) -> @a
, newline#1(::(@x, @xs), @lastline, @y) ->
newline#2(@lastline, @x, @xs, @y)
, newline#1(nil(), @lastline, @y) -> nil()
, newline#2(::(@belowVal, @lastline'), @x, @xs, @y) ->
newline#3(newline(@y, @lastline', @xs),
@belowVal,
@lastline',
@x,
@y)
, newline#2(nil(), @x, @xs, @y) -> nil()
, newline#3(@nl, @belowVal, @lastline', @x, @y) ->
newline#4(right(@nl), @belowVal, @lastline', @nl, @x, @y)
, right(@l) -> right#1(@l)
, newline#4(@rightVal, @belowVal, @lastline', @nl, @x, @y) ->
newline#5(right(@lastline'), @belowVal, @nl, @rightVal, @x, @y)
, newline#5(@diagVal, @belowVal, @nl, @rightVal, @x, @y) ->
newline#6(newline#7(#equal(@x, @y),
@belowVal,
@diagVal,
@rightVal),
@nl)
, newline#7(#false(), @belowVal, @diagVal, @rightVal) ->
max(@belowVal, @rightVal)
, newline#7(#true(), @belowVal, @diagVal, @rightVal) ->
+(@diagVal, #pos(#s(#0())))
, newline#6(@elem, @nl) -> ::(@elem, @nl)
, right#1(::(@x, @xs)) -> @x
, right#1(nil()) -> #abs(#0())
, #pred(#0()) -> #neg(#s(#0()))
, #pred(#neg(#s(@x))) -> #neg(#s(#s(@x)))
, #pred(#pos(#s(#0()))) -> #0()
, #pred(#pos(#s(#s(@x)))) -> #pos(#s(@x))
, #succ(#0()) -> #pos(#s(#0()))
, #succ(#neg(#s(#0()))) -> #0()
, #succ(#neg(#s(#s(@x)))) -> #neg(#s(@x))
, #succ(#pos(#s(@x))) -> #pos(#s(#s(@x)))
, #and(#false(), #false()) -> #false()
, #and(#false(), #true()) -> #false()
, #and(#true(), #false()) -> #false()
, #and(#true(), #true()) -> #true() }
Obligation:
innermost runtime complexity
Answer:
YES(O(1),O(n^2))
Due to missing edges in the dependency-graph, the right-hand sides
of following rules could be simplified:
{ firstline#1^#(::(@x, @xs)) -> c_9(#abs^#(#0()), firstline^#(@xs))
, lcs^#(@l1, @l2) ->
c_11(lcs#1^#(lcstable(@l1, @l2)), lcstable^#(@l1, @l2))
, newline#2^#(::(@belowVal, @lastline'), @x, @xs, @y) ->
c_29(newline#3^#(newline(@y, @lastline', @xs),
@belowVal,
@lastline',
@x,
@y),
newline^#(@y, @lastline', @xs)) }
We are left with following problem, upon which TcT provides the
certificate YES(O(1),O(n^2)).
Strict DPs:
{ firstline^#(@l) -> c_1(firstline#1^#(@l))
, firstline#1^#(::(@x, @xs)) -> c_2(firstline^#(@xs))
, lcs^#(@l1, @l2) -> c_3(lcstable^#(@l1, @l2))
, lcstable^#(@l1, @l2) -> c_4(lcstable#1^#(@l1, @l2))
, lcstable#1^#(::(@x, @xs), @l2) ->
c_5(lcstable#2^#(lcstable(@xs, @l2), @l2, @x),
lcstable^#(@xs, @l2))
, lcstable#1^#(nil(), @l2) -> c_6(firstline^#(@l2))
, lcstable#2^#(@m, @l2, @x) -> c_7(lcstable#3^#(@m, @l2, @x))
, lcstable#3^#(::(@l, @ls), @l2, @x) -> c_8(newline^#(@x, @l, @l2))
, newline^#(@y, @lastline, @l) ->
c_9(newline#1^#(@l, @lastline, @y))
, newline#1^#(::(@x, @xs), @lastline, @y) ->
c_10(newline#2^#(@lastline, @x, @xs, @y))
, newline#2^#(::(@belowVal, @lastline'), @x, @xs, @y) ->
c_11(newline^#(@y, @lastline', @xs)) }
Weak Trs:
{ #abs(#0()) -> #0()
, #abs(#neg(@x)) -> #pos(@x)
, #abs(#pos(@x)) -> #pos(@x)
, #abs(#s(@x)) -> #pos(#s(@x))
, #equal(@x, @y) -> #eq(@x, @y)
, #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)
, #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()
, #greater(@x, @y) -> #ckgt(#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)
, #ckgt(#EQ()) -> #false()
, #ckgt(#GT()) -> #true()
, #ckgt(#LT()) -> #false()
, +(@x, @y) -> #add(@x, @y)
, #add(#0(), @y) -> @y
, #add(#neg(#s(#0())), @y) -> #pred(@y)
, #add(#neg(#s(#s(@x))), @y) -> #pred(#add(#pos(#s(@x)), @y))
, #add(#pos(#s(#0())), @y) -> #succ(@y)
, #add(#pos(#s(#s(@x))), @y) -> #succ(#add(#pos(#s(@x)), @y))
, firstline(@l) -> firstline#1(@l)
, firstline#1(::(@x, @xs)) -> ::(#abs(#0()), firstline(@xs))
, firstline#1(nil()) -> nil()
, lcs(@l1, @l2) -> lcs#1(lcstable(@l1, @l2))
, lcstable(@l1, @l2) -> lcstable#1(@l1, @l2)
, lcs#1(@m) -> lcs#2(@m)
, lcs#2(::(@l1, @_@2)) -> lcs#3(@l1)
, lcs#2(nil()) -> #abs(#0())
, lcs#3(::(@len, @_@1)) -> @len
, lcs#3(nil()) -> #abs(#0())
, lcstable#1(::(@x, @xs), @l2) ->
lcstable#2(lcstable(@xs, @l2), @l2, @x)
, lcstable#1(nil(), @l2) -> ::(firstline(@l2), nil())
, lcstable#2(@m, @l2, @x) -> lcstable#3(@m, @l2, @x)
, lcstable#3(::(@l, @ls), @l2, @x) ->
::(newline(@x, @l, @l2), ::(@l, @ls))
, lcstable#3(nil(), @l2, @x) -> nil()
, newline(@y, @lastline, @l) -> newline#1(@l, @lastline, @y)
, max(@a, @b) -> max#1(#greater(@a, @b), @a, @b)
, max#1(#false(), @a, @b) -> @b
, max#1(#true(), @a, @b) -> @a
, newline#1(::(@x, @xs), @lastline, @y) ->
newline#2(@lastline, @x, @xs, @y)
, newline#1(nil(), @lastline, @y) -> nil()
, newline#2(::(@belowVal, @lastline'), @x, @xs, @y) ->
newline#3(newline(@y, @lastline', @xs),
@belowVal,
@lastline',
@x,
@y)
, newline#2(nil(), @x, @xs, @y) -> nil()
, newline#3(@nl, @belowVal, @lastline', @x, @y) ->
newline#4(right(@nl), @belowVal, @lastline', @nl, @x, @y)
, right(@l) -> right#1(@l)
, newline#4(@rightVal, @belowVal, @lastline', @nl, @x, @y) ->
newline#5(right(@lastline'), @belowVal, @nl, @rightVal, @x, @y)
, newline#5(@diagVal, @belowVal, @nl, @rightVal, @x, @y) ->
newline#6(newline#7(#equal(@x, @y),
@belowVal,
@diagVal,
@rightVal),
@nl)
, newline#7(#false(), @belowVal, @diagVal, @rightVal) ->
max(@belowVal, @rightVal)
, newline#7(#true(), @belowVal, @diagVal, @rightVal) ->
+(@diagVal, #pos(#s(#0())))
, newline#6(@elem, @nl) -> ::(@elem, @nl)
, right#1(::(@x, @xs)) -> @x
, right#1(nil()) -> #abs(#0())
, #pred(#0()) -> #neg(#s(#0()))
, #pred(#neg(#s(@x))) -> #neg(#s(#s(@x)))
, #pred(#pos(#s(#0()))) -> #0()
, #pred(#pos(#s(#s(@x)))) -> #pos(#s(@x))
, #succ(#0()) -> #pos(#s(#0()))
, #succ(#neg(#s(#0()))) -> #0()
, #succ(#neg(#s(#s(@x)))) -> #neg(#s(@x))
, #succ(#pos(#s(@x))) -> #pos(#s(#s(@x)))
, #and(#false(), #false()) -> #false()
, #and(#false(), #true()) -> #false()
, #and(#true(), #false()) -> #false()
, #and(#true(), #true()) -> #true() }
Obligation:
innermost runtime complexity
Answer:
YES(O(1),O(n^2))
We replace rewrite rules by usable rules:
Weak Usable Rules:
{ #abs(#0()) -> #0()
, #abs(#neg(@x)) -> #pos(@x)
, #abs(#pos(@x)) -> #pos(@x)
, #abs(#s(@x)) -> #pos(#s(@x))
, #equal(@x, @y) -> #eq(@x, @y)
, #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)
, #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()
, #greater(@x, @y) -> #ckgt(#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)
, #ckgt(#EQ()) -> #false()
, #ckgt(#GT()) -> #true()
, #ckgt(#LT()) -> #false()
, +(@x, @y) -> #add(@x, @y)
, #add(#0(), @y) -> @y
, #add(#neg(#s(#0())), @y) -> #pred(@y)
, #add(#neg(#s(#s(@x))), @y) -> #pred(#add(#pos(#s(@x)), @y))
, #add(#pos(#s(#0())), @y) -> #succ(@y)
, #add(#pos(#s(#s(@x))), @y) -> #succ(#add(#pos(#s(@x)), @y))
, firstline(@l) -> firstline#1(@l)
, firstline#1(::(@x, @xs)) -> ::(#abs(#0()), firstline(@xs))
, firstline#1(nil()) -> nil()
, lcstable(@l1, @l2) -> lcstable#1(@l1, @l2)
, lcstable#1(::(@x, @xs), @l2) ->
lcstable#2(lcstable(@xs, @l2), @l2, @x)
, lcstable#1(nil(), @l2) -> ::(firstline(@l2), nil())
, lcstable#2(@m, @l2, @x) -> lcstable#3(@m, @l2, @x)
, lcstable#3(::(@l, @ls), @l2, @x) ->
::(newline(@x, @l, @l2), ::(@l, @ls))
, lcstable#3(nil(), @l2, @x) -> nil()
, newline(@y, @lastline, @l) -> newline#1(@l, @lastline, @y)
, max(@a, @b) -> max#1(#greater(@a, @b), @a, @b)
, max#1(#false(), @a, @b) -> @b
, max#1(#true(), @a, @b) -> @a
, newline#1(::(@x, @xs), @lastline, @y) ->
newline#2(@lastline, @x, @xs, @y)
, newline#1(nil(), @lastline, @y) -> nil()
, newline#2(::(@belowVal, @lastline'), @x, @xs, @y) ->
newline#3(newline(@y, @lastline', @xs),
@belowVal,
@lastline',
@x,
@y)
, newline#2(nil(), @x, @xs, @y) -> nil()
, newline#3(@nl, @belowVal, @lastline', @x, @y) ->
newline#4(right(@nl), @belowVal, @lastline', @nl, @x, @y)
, right(@l) -> right#1(@l)
, newline#4(@rightVal, @belowVal, @lastline', @nl, @x, @y) ->
newline#5(right(@lastline'), @belowVal, @nl, @rightVal, @x, @y)
, newline#5(@diagVal, @belowVal, @nl, @rightVal, @x, @y) ->
newline#6(newline#7(#equal(@x, @y),
@belowVal,
@diagVal,
@rightVal),
@nl)
, newline#7(#false(), @belowVal, @diagVal, @rightVal) ->
max(@belowVal, @rightVal)
, newline#7(#true(), @belowVal, @diagVal, @rightVal) ->
+(@diagVal, #pos(#s(#0())))
, newline#6(@elem, @nl) -> ::(@elem, @nl)
, right#1(::(@x, @xs)) -> @x
, right#1(nil()) -> #abs(#0())
, #pred(#0()) -> #neg(#s(#0()))
, #pred(#neg(#s(@x))) -> #neg(#s(#s(@x)))
, #pred(#pos(#s(#0()))) -> #0()
, #pred(#pos(#s(#s(@x)))) -> #pos(#s(@x))
, #succ(#0()) -> #pos(#s(#0()))
, #succ(#neg(#s(#0()))) -> #0()
, #succ(#neg(#s(#s(@x)))) -> #neg(#s(@x))
, #succ(#pos(#s(@x))) -> #pos(#s(#s(@x)))
, #and(#false(), #false()) -> #false()
, #and(#false(), #true()) -> #false()
, #and(#true(), #false()) -> #false()
, #and(#true(), #true()) -> #true() }
We are left with following problem, upon which TcT provides the
certificate YES(O(1),O(n^2)).
Strict DPs:
{ firstline^#(@l) -> c_1(firstline#1^#(@l))
, firstline#1^#(::(@x, @xs)) -> c_2(firstline^#(@xs))
, lcs^#(@l1, @l2) -> c_3(lcstable^#(@l1, @l2))
, lcstable^#(@l1, @l2) -> c_4(lcstable#1^#(@l1, @l2))
, lcstable#1^#(::(@x, @xs), @l2) ->
c_5(lcstable#2^#(lcstable(@xs, @l2), @l2, @x),
lcstable^#(@xs, @l2))
, lcstable#1^#(nil(), @l2) -> c_6(firstline^#(@l2))
, lcstable#2^#(@m, @l2, @x) -> c_7(lcstable#3^#(@m, @l2, @x))
, lcstable#3^#(::(@l, @ls), @l2, @x) -> c_8(newline^#(@x, @l, @l2))
, newline^#(@y, @lastline, @l) ->
c_9(newline#1^#(@l, @lastline, @y))
, newline#1^#(::(@x, @xs), @lastline, @y) ->
c_10(newline#2^#(@lastline, @x, @xs, @y))
, newline#2^#(::(@belowVal, @lastline'), @x, @xs, @y) ->
c_11(newline^#(@y, @lastline', @xs)) }
Weak Trs:
{ #abs(#0()) -> #0()
, #abs(#neg(@x)) -> #pos(@x)
, #abs(#pos(@x)) -> #pos(@x)
, #abs(#s(@x)) -> #pos(#s(@x))
, #equal(@x, @y) -> #eq(@x, @y)
, #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)
, #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()
, #greater(@x, @y) -> #ckgt(#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)
, #ckgt(#EQ()) -> #false()
, #ckgt(#GT()) -> #true()
, #ckgt(#LT()) -> #false()
, +(@x, @y) -> #add(@x, @y)
, #add(#0(), @y) -> @y
, #add(#neg(#s(#0())), @y) -> #pred(@y)
, #add(#neg(#s(#s(@x))), @y) -> #pred(#add(#pos(#s(@x)), @y))
, #add(#pos(#s(#0())), @y) -> #succ(@y)
, #add(#pos(#s(#s(@x))), @y) -> #succ(#add(#pos(#s(@x)), @y))
, firstline(@l) -> firstline#1(@l)
, firstline#1(::(@x, @xs)) -> ::(#abs(#0()), firstline(@xs))
, firstline#1(nil()) -> nil()
, lcstable(@l1, @l2) -> lcstable#1(@l1, @l2)
, lcstable#1(::(@x, @xs), @l2) ->
lcstable#2(lcstable(@xs, @l2), @l2, @x)
, lcstable#1(nil(), @l2) -> ::(firstline(@l2), nil())
, lcstable#2(@m, @l2, @x) -> lcstable#3(@m, @l2, @x)
, lcstable#3(::(@l, @ls), @l2, @x) ->
::(newline(@x, @l, @l2), ::(@l, @ls))
, lcstable#3(nil(), @l2, @x) -> nil()
, newline(@y, @lastline, @l) -> newline#1(@l, @lastline, @y)
, max(@a, @b) -> max#1(#greater(@a, @b), @a, @b)
, max#1(#false(), @a, @b) -> @b
, max#1(#true(), @a, @b) -> @a
, newline#1(::(@x, @xs), @lastline, @y) ->
newline#2(@lastline, @x, @xs, @y)
, newline#1(nil(), @lastline, @y) -> nil()
, newline#2(::(@belowVal, @lastline'), @x, @xs, @y) ->
newline#3(newline(@y, @lastline', @xs),
@belowVal,
@lastline',
@x,
@y)
, newline#2(nil(), @x, @xs, @y) -> nil()
, newline#3(@nl, @belowVal, @lastline', @x, @y) ->
newline#4(right(@nl), @belowVal, @lastline', @nl, @x, @y)
, right(@l) -> right#1(@l)
, newline#4(@rightVal, @belowVal, @lastline', @nl, @x, @y) ->
newline#5(right(@lastline'), @belowVal, @nl, @rightVal, @x, @y)
, newline#5(@diagVal, @belowVal, @nl, @rightVal, @x, @y) ->
newline#6(newline#7(#equal(@x, @y),
@belowVal,
@diagVal,
@rightVal),
@nl)
, newline#7(#false(), @belowVal, @diagVal, @rightVal) ->
max(@belowVal, @rightVal)
, newline#7(#true(), @belowVal, @diagVal, @rightVal) ->
+(@diagVal, #pos(#s(#0())))
, newline#6(@elem, @nl) -> ::(@elem, @nl)
, right#1(::(@x, @xs)) -> @x
, right#1(nil()) -> #abs(#0())
, #pred(#0()) -> #neg(#s(#0()))
, #pred(#neg(#s(@x))) -> #neg(#s(#s(@x)))
, #pred(#pos(#s(#0()))) -> #0()
, #pred(#pos(#s(#s(@x)))) -> #pos(#s(@x))
, #succ(#0()) -> #pos(#s(#0()))
, #succ(#neg(#s(#0()))) -> #0()
, #succ(#neg(#s(#s(@x)))) -> #neg(#s(@x))
, #succ(#pos(#s(@x))) -> #pos(#s(#s(@x)))
, #and(#false(), #false()) -> #false()
, #and(#false(), #true()) -> #false()
, #and(#true(), #false()) -> #false()
, #and(#true(), #true()) -> #true() }
Obligation:
innermost runtime complexity
Answer:
YES(O(1),O(n^2))
We decompose the input problem according to the dependency graph
into the upper component
{ lcs^#(@l1, @l2) -> c_3(lcstable^#(@l1, @l2))
, lcstable^#(@l1, @l2) -> c_4(lcstable#1^#(@l1, @l2))
, lcstable#1^#(::(@x, @xs), @l2) ->
c_5(lcstable#2^#(lcstable(@xs, @l2), @l2, @x),
lcstable^#(@xs, @l2)) }
and lower component
{ firstline^#(@l) -> c_1(firstline#1^#(@l))
, firstline#1^#(::(@x, @xs)) -> c_2(firstline^#(@xs))
, lcstable#1^#(nil(), @l2) -> c_6(firstline^#(@l2))
, lcstable#2^#(@m, @l2, @x) -> c_7(lcstable#3^#(@m, @l2, @x))
, lcstable#3^#(::(@l, @ls), @l2, @x) -> c_8(newline^#(@x, @l, @l2))
, newline^#(@y, @lastline, @l) ->
c_9(newline#1^#(@l, @lastline, @y))
, newline#1^#(::(@x, @xs), @lastline, @y) ->
c_10(newline#2^#(@lastline, @x, @xs, @y))
, newline#2^#(::(@belowVal, @lastline'), @x, @xs, @y) ->
c_11(newline^#(@y, @lastline', @xs)) }
Further, following extension rules are added to the lower
component.
{ lcs^#(@l1, @l2) -> lcstable^#(@l1, @l2)
, lcstable^#(@l1, @l2) -> lcstable#1^#(@l1, @l2)
, lcstable#1^#(::(@x, @xs), @l2) -> lcstable^#(@xs, @l2)
, lcstable#1^#(::(@x, @xs), @l2) ->
lcstable#2^#(lcstable(@xs, @l2), @l2, @x) }
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:
{ lcs^#(@l1, @l2) -> c_3(lcstable^#(@l1, @l2))
, lcstable^#(@l1, @l2) -> c_4(lcstable#1^#(@l1, @l2))
, lcstable#1^#(::(@x, @xs), @l2) ->
c_5(lcstable#2^#(lcstable(@xs, @l2), @l2, @x),
lcstable^#(@xs, @l2)) }
Weak Trs:
{ #abs(#0()) -> #0()
, #abs(#neg(@x)) -> #pos(@x)
, #abs(#pos(@x)) -> #pos(@x)
, #abs(#s(@x)) -> #pos(#s(@x))
, #equal(@x, @y) -> #eq(@x, @y)
, #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)
, #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()
, #greater(@x, @y) -> #ckgt(#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)
, #ckgt(#EQ()) -> #false()
, #ckgt(#GT()) -> #true()
, #ckgt(#LT()) -> #false()
, +(@x, @y) -> #add(@x, @y)
, #add(#0(), @y) -> @y
, #add(#neg(#s(#0())), @y) -> #pred(@y)
, #add(#neg(#s(#s(@x))), @y) -> #pred(#add(#pos(#s(@x)), @y))
, #add(#pos(#s(#0())), @y) -> #succ(@y)
, #add(#pos(#s(#s(@x))), @y) -> #succ(#add(#pos(#s(@x)), @y))
, firstline(@l) -> firstline#1(@l)
, firstline#1(::(@x, @xs)) -> ::(#abs(#0()), firstline(@xs))
, firstline#1(nil()) -> nil()
, lcstable(@l1, @l2) -> lcstable#1(@l1, @l2)
, lcstable#1(::(@x, @xs), @l2) ->
lcstable#2(lcstable(@xs, @l2), @l2, @x)
, lcstable#1(nil(), @l2) -> ::(firstline(@l2), nil())
, lcstable#2(@m, @l2, @x) -> lcstable#3(@m, @l2, @x)
, lcstable#3(::(@l, @ls), @l2, @x) ->
::(newline(@x, @l, @l2), ::(@l, @ls))
, lcstable#3(nil(), @l2, @x) -> nil()
, newline(@y, @lastline, @l) -> newline#1(@l, @lastline, @y)
, max(@a, @b) -> max#1(#greater(@a, @b), @a, @b)
, max#1(#false(), @a, @b) -> @b
, max#1(#true(), @a, @b) -> @a
, newline#1(::(@x, @xs), @lastline, @y) ->
newline#2(@lastline, @x, @xs, @y)
, newline#1(nil(), @lastline, @y) -> nil()
, newline#2(::(@belowVal, @lastline'), @x, @xs, @y) ->
newline#3(newline(@y, @lastline', @xs),
@belowVal,
@lastline',
@x,
@y)
, newline#2(nil(), @x, @xs, @y) -> nil()
, newline#3(@nl, @belowVal, @lastline', @x, @y) ->
newline#4(right(@nl), @belowVal, @lastline', @nl, @x, @y)
, right(@l) -> right#1(@l)
, newline#4(@rightVal, @belowVal, @lastline', @nl, @x, @y) ->
newline#5(right(@lastline'), @belowVal, @nl, @rightVal, @x, @y)
, newline#5(@diagVal, @belowVal, @nl, @rightVal, @x, @y) ->
newline#6(newline#7(#equal(@x, @y),
@belowVal,
@diagVal,
@rightVal),
@nl)
, newline#7(#false(), @belowVal, @diagVal, @rightVal) ->
max(@belowVal, @rightVal)
, newline#7(#true(), @belowVal, @diagVal, @rightVal) ->
+(@diagVal, #pos(#s(#0())))
, newline#6(@elem, @nl) -> ::(@elem, @nl)
, right#1(::(@x, @xs)) -> @x
, right#1(nil()) -> #abs(#0())
, #pred(#0()) -> #neg(#s(#0()))
, #pred(#neg(#s(@x))) -> #neg(#s(#s(@x)))
, #pred(#pos(#s(#0()))) -> #0()
, #pred(#pos(#s(#s(@x)))) -> #pos(#s(@x))
, #succ(#0()) -> #pos(#s(#0()))
, #succ(#neg(#s(#0()))) -> #0()
, #succ(#neg(#s(#s(@x)))) -> #neg(#s(@x))
, #succ(#pos(#s(@x))) -> #pos(#s(#s(@x)))
, #and(#false(), #false()) -> #false()
, #and(#false(), #true()) -> #false()
, #and(#true(), #false()) -> #false()
, #and(#true(), #true()) -> #true() }
Obligation:
innermost runtime complexity
Answer:
YES(O(1),O(n^1))
Due to missing edges in the dependency-graph, the right-hand sides
of following rules could be simplified:
{ lcstable#1^#(::(@x, @xs), @l2) ->
c_5(lcstable#2^#(lcstable(@xs, @l2), @l2, @x),
lcstable^#(@xs, @l2)) }
We are left with following problem, upon which TcT provides the
certificate YES(O(1),O(n^1)).
Strict DPs:
{ lcs^#(@l1, @l2) -> c_1(lcstable^#(@l1, @l2))
, lcstable^#(@l1, @l2) -> c_2(lcstable#1^#(@l1, @l2))
, lcstable#1^#(::(@x, @xs), @l2) -> c_3(lcstable^#(@xs, @l2)) }
Weak Trs:
{ #abs(#0()) -> #0()
, #abs(#neg(@x)) -> #pos(@x)
, #abs(#pos(@x)) -> #pos(@x)
, #abs(#s(@x)) -> #pos(#s(@x))
, #equal(@x, @y) -> #eq(@x, @y)
, #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)
, #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()
, #greater(@x, @y) -> #ckgt(#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)
, #ckgt(#EQ()) -> #false()
, #ckgt(#GT()) -> #true()
, #ckgt(#LT()) -> #false()
, +(@x, @y) -> #add(@x, @y)
, #add(#0(), @y) -> @y
, #add(#neg(#s(#0())), @y) -> #pred(@y)
, #add(#neg(#s(#s(@x))), @y) -> #pred(#add(#pos(#s(@x)), @y))
, #add(#pos(#s(#0())), @y) -> #succ(@y)
, #add(#pos(#s(#s(@x))), @y) -> #succ(#add(#pos(#s(@x)), @y))
, firstline(@l) -> firstline#1(@l)
, firstline#1(::(@x, @xs)) -> ::(#abs(#0()), firstline(@xs))
, firstline#1(nil()) -> nil()
, lcstable(@l1, @l2) -> lcstable#1(@l1, @l2)
, lcstable#1(::(@x, @xs), @l2) ->
lcstable#2(lcstable(@xs, @l2), @l2, @x)
, lcstable#1(nil(), @l2) -> ::(firstline(@l2), nil())
, lcstable#2(@m, @l2, @x) -> lcstable#3(@m, @l2, @x)
, lcstable#3(::(@l, @ls), @l2, @x) ->
::(newline(@x, @l, @l2), ::(@l, @ls))
, lcstable#3(nil(), @l2, @x) -> nil()
, newline(@y, @lastline, @l) -> newline#1(@l, @lastline, @y)
, max(@a, @b) -> max#1(#greater(@a, @b), @a, @b)
, max#1(#false(), @a, @b) -> @b
, max#1(#true(), @a, @b) -> @a
, newline#1(::(@x, @xs), @lastline, @y) ->
newline#2(@lastline, @x, @xs, @y)
, newline#1(nil(), @lastline, @y) -> nil()
, newline#2(::(@belowVal, @lastline'), @x, @xs, @y) ->
newline#3(newline(@y, @lastline', @xs),
@belowVal,
@lastline',
@x,
@y)
, newline#2(nil(), @x, @xs, @y) -> nil()
, newline#3(@nl, @belowVal, @lastline', @x, @y) ->
newline#4(right(@nl), @belowVal, @lastline', @nl, @x, @y)
, right(@l) -> right#1(@l)
, newline#4(@rightVal, @belowVal, @lastline', @nl, @x, @y) ->
newline#5(right(@lastline'), @belowVal, @nl, @rightVal, @x, @y)
, newline#5(@diagVal, @belowVal, @nl, @rightVal, @x, @y) ->
newline#6(newline#7(#equal(@x, @y),
@belowVal,
@diagVal,
@rightVal),
@nl)
, newline#7(#false(), @belowVal, @diagVal, @rightVal) ->
max(@belowVal, @rightVal)
, newline#7(#true(), @belowVal, @diagVal, @rightVal) ->
+(@diagVal, #pos(#s(#0())))
, newline#6(@elem, @nl) -> ::(@elem, @nl)
, right#1(::(@x, @xs)) -> @x
, right#1(nil()) -> #abs(#0())
, #pred(#0()) -> #neg(#s(#0()))
, #pred(#neg(#s(@x))) -> #neg(#s(#s(@x)))
, #pred(#pos(#s(#0()))) -> #0()
, #pred(#pos(#s(#s(@x)))) -> #pos(#s(@x))
, #succ(#0()) -> #pos(#s(#0()))
, #succ(#neg(#s(#0()))) -> #0()
, #succ(#neg(#s(#s(@x)))) -> #neg(#s(@x))
, #succ(#pos(#s(@x))) -> #pos(#s(#s(@x)))
, #and(#false(), #false()) -> #false()
, #and(#false(), #true()) -> #false()
, #and(#true(), #false()) -> #false()
, #and(#true(), #true()) -> #true() }
Obligation:
innermost runtime complexity
Answer:
YES(O(1),O(n^1))
Consider the dependency graph
1: lcs^#(@l1, @l2) -> c_1(lcstable^#(@l1, @l2))
-->_1 lcstable^#(@l1, @l2) -> c_2(lcstable#1^#(@l1, @l2)) :2
2: lcstable^#(@l1, @l2) -> c_2(lcstable#1^#(@l1, @l2))
-->_1 lcstable#1^#(::(@x, @xs), @l2) ->
c_3(lcstable^#(@xs, @l2)) :3
3: lcstable#1^#(::(@x, @xs), @l2) -> c_3(lcstable^#(@xs, @l2))
-->_1 lcstable^#(@l1, @l2) -> c_2(lcstable#1^#(@l1, @l2)) :2
Following roots of the dependency graph are removed, as the
considered set of starting terms is closed under reduction with
respect to these rules (modulo compound contexts).
{ lcs^#(@l1, @l2) -> c_1(lcstable^#(@l1, @l2)) }
We are left with following problem, upon which TcT provides the
certificate YES(O(1),O(n^1)).
Strict DPs:
{ lcstable^#(@l1, @l2) -> c_2(lcstable#1^#(@l1, @l2))
, lcstable#1^#(::(@x, @xs), @l2) -> c_3(lcstable^#(@xs, @l2)) }
Weak Trs:
{ #abs(#0()) -> #0()
, #abs(#neg(@x)) -> #pos(@x)
, #abs(#pos(@x)) -> #pos(@x)
, #abs(#s(@x)) -> #pos(#s(@x))
, #equal(@x, @y) -> #eq(@x, @y)
, #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)
, #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()
, #greater(@x, @y) -> #ckgt(#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)
, #ckgt(#EQ()) -> #false()
, #ckgt(#GT()) -> #true()
, #ckgt(#LT()) -> #false()
, +(@x, @y) -> #add(@x, @y)
, #add(#0(), @y) -> @y
, #add(#neg(#s(#0())), @y) -> #pred(@y)
, #add(#neg(#s(#s(@x))), @y) -> #pred(#add(#pos(#s(@x)), @y))
, #add(#pos(#s(#0())), @y) -> #succ(@y)
, #add(#pos(#s(#s(@x))), @y) -> #succ(#add(#pos(#s(@x)), @y))
, firstline(@l) -> firstline#1(@l)
, firstline#1(::(@x, @xs)) -> ::(#abs(#0()), firstline(@xs))
, firstline#1(nil()) -> nil()
, lcstable(@l1, @l2) -> lcstable#1(@l1, @l2)
, lcstable#1(::(@x, @xs), @l2) ->
lcstable#2(lcstable(@xs, @l2), @l2, @x)
, lcstable#1(nil(), @l2) -> ::(firstline(@l2), nil())
, lcstable#2(@m, @l2, @x) -> lcstable#3(@m, @l2, @x)
, lcstable#3(::(@l, @ls), @l2, @x) ->
::(newline(@x, @l, @l2), ::(@l, @ls))
, lcstable#3(nil(), @l2, @x) -> nil()
, newline(@y, @lastline, @l) -> newline#1(@l, @lastline, @y)
, max(@a, @b) -> max#1(#greater(@a, @b), @a, @b)
, max#1(#false(), @a, @b) -> @b
, max#1(#true(), @a, @b) -> @a
, newline#1(::(@x, @xs), @lastline, @y) ->
newline#2(@lastline, @x, @xs, @y)
, newline#1(nil(), @lastline, @y) -> nil()
, newline#2(::(@belowVal, @lastline'), @x, @xs, @y) ->
newline#3(newline(@y, @lastline', @xs),
@belowVal,
@lastline',
@x,
@y)
, newline#2(nil(), @x, @xs, @y) -> nil()
, newline#3(@nl, @belowVal, @lastline', @x, @y) ->
newline#4(right(@nl), @belowVal, @lastline', @nl, @x, @y)
, right(@l) -> right#1(@l)
, newline#4(@rightVal, @belowVal, @lastline', @nl, @x, @y) ->
newline#5(right(@lastline'), @belowVal, @nl, @rightVal, @x, @y)
, newline#5(@diagVal, @belowVal, @nl, @rightVal, @x, @y) ->
newline#6(newline#7(#equal(@x, @y),
@belowVal,
@diagVal,
@rightVal),
@nl)
, newline#7(#false(), @belowVal, @diagVal, @rightVal) ->
max(@belowVal, @rightVal)
, newline#7(#true(), @belowVal, @diagVal, @rightVal) ->
+(@diagVal, #pos(#s(#0())))
, newline#6(@elem, @nl) -> ::(@elem, @nl)
, right#1(::(@x, @xs)) -> @x
, right#1(nil()) -> #abs(#0())
, #pred(#0()) -> #neg(#s(#0()))
, #pred(#neg(#s(@x))) -> #neg(#s(#s(@x)))
, #pred(#pos(#s(#0()))) -> #0()
, #pred(#pos(#s(#s(@x)))) -> #pos(#s(@x))
, #succ(#0()) -> #pos(#s(#0()))
, #succ(#neg(#s(#0()))) -> #0()
, #succ(#neg(#s(#s(@x)))) -> #neg(#s(@x))
, #succ(#pos(#s(@x))) -> #pos(#s(#s(@x)))
, #and(#false(), #false()) -> #false()
, #and(#false(), #true()) -> #false()
, #and(#true(), #false()) -> #false()
, #and(#true(), #true()) -> #true() }
Obligation:
innermost runtime complexity
Answer:
YES(O(1),O(n^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(n^1)).
Strict DPs:
{ lcstable^#(@l1, @l2) -> c_2(lcstable#1^#(@l1, @l2))
, lcstable#1^#(::(@x, @xs), @l2) -> c_3(lcstable^#(@xs, @l2)) }
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: lcstable^#(@l1, @l2) -> c_2(lcstable#1^#(@l1, @l2)) }
Sub-proof:
----------
The following argument positions are usable:
Uargs(c_2) = {1}, Uargs(c_3) = {1}
TcT has computed following constructor-based matrix interpretation
satisfying not(EDA).
[#0] = [0]
[#abs](x1) = [0]
[#neg](x1) = [0]
[#pos](x1) = [0]
[#s](x1) = [0]
[#equal](x1, x2) = [0]
[#eq](x1, x2) = [0]
[#greater](x1, x2) = [0]
[#compare](x1, x2) = [0]
[#ckgt](x1) = [0]
[+](x1, x2) = [0]
[#add](x1, x2) = [0]
[firstline](x1) = [0]
[firstline#1](x1) = [0]
[::](x1, x2) = [1] x2 + [1]
[nil] = [0]
[lcstable](x1, x2) = [0]
[lcstable#1](x1, x2) = [0]
[lcstable#2](x1, x2, x3) = [0]
[lcstable#3](x1, x2, x3) = [0]
[newline](x1, x2, x3) = [0]
[max](x1, x2) = [0]
[max#1](x1, x2, x3) = [0]
[#false] = [0]
[#true] = [0]
[newline#1](x1, x2, x3) = [0]
[newline#2](x1, x2, x3, x4) = [0]
[newline#3](x1, x2, x3, x4, x5) = [0]
[right](x1) = [0]
[newline#4](x1, x2, x3, x4, x5, x6) = [0]
[newline#5](x1, x2, x3, x4, x5, x6) = [0]
[newline#7](x1, x2, x3, x4) = [0]
[newline#6](x1, x2) = [0]
[right#1](x1) = [0]
[#pred](x1) = [0]
[#succ](x1) = [0]
[#and](x1, x2) = [0]
[#EQ] = [0]
[#GT] = [0]
[#LT] = [0]
[#abs^#](x1) = [0]
[#equal^#](x1, x2) = [0]
[#eq^#](x1, x2) = [0]
[#greater^#](x1, x2) = [0]
[#ckgt^#](x1) = [0]
[#compare^#](x1, x2) = [0]
[+^#](x1, x2) = [0]
[#add^#](x1, x2) = [0]
[firstline^#](x1) = [0]
[firstline#1^#](x1) = [0]
[lcs^#](x1, x2) = [0]
[lcs#1^#](x1) = [0]
[lcstable^#](x1, x2) = [1] x1 + [1]
[lcstable#1^#](x1, x2) = [1] x1 + [0]
[lcs#2^#](x1) = [0]
[lcs#3^#](x1) = [0]
[lcstable#2^#](x1, x2, x3) = [0]
[lcstable#3^#](x1, x2, x3) = [0]
[newline^#](x1, x2, x3) = [0]
[newline#1^#](x1, x2, x3) = [0]
[max^#](x1, x2) = [0]
[max#1^#](x1, x2, x3) = [0]
[newline#2^#](x1, x2, x3, x4) = [0]
[newline#3^#](x1, x2, x3, x4, x5) = [0]
[newline#4^#](x1, x2, x3, x4, x5, x6) = [0]
[right^#](x1) = [0]
[right#1^#](x1) = [0]
[newline#5^#](x1, x2, x3, x4, x5, x6) = [0]
[newline#6^#](x1, x2) = [0]
[newline#7^#](x1, x2, x3, x4) = [0]
[#and^#](x1, x2) = [0]
[#pred^#](x1) = [0]
[#succ^#](x1) = [0]
[c_3](x1) = [0]
[c_4](x1) = [0]
[c_5](x1, x2) = [0]
[c] = [0]
[c_1](x1) = [0]
[c_2](x1) = [1] x1 + [0]
[c_3](x1) = [1] x1 + [0]
This order satisfies following ordering constraints
[lcstable^#(@l1, @l2)] = [1] @l1 + [1]
> [1] @l1 + [0]
= [c_2(lcstable#1^#(@l1, @l2))]
[lcstable#1^#(::(@x, @xs), @l2)] = [1] @xs + [1]
>= [1] @xs + [1]
= [c_3(lcstable^#(@xs, @l2))]
Consider the set of all dependency pairs
DPs:
{ 1: lcstable^#(@l1, @l2) -> c_2(lcstable#1^#(@l1, @l2))
, 2: lcstable#1^#(::(@x, @xs), @l2) -> c_3(lcstable^#(@xs, @l2)) }
Processor 'matrix interpretation of dimension 1' induces the
complexity certificate YES(?,O(n^1)) on application of dependency
pairs {1}. These cover all (indirect) predecessors of dependency
pairs {1,2}, their number of application is equally bounded. The
dependency pairs are shifted into the corresponding weak
component(s).
We apply the transformation 'removetails' on the sub-problem:
Weak DPs:
{ lcstable^#(@l1, @l2) -> c_2(lcstable#1^#(@l1, @l2))
, lcstable#1^#(::(@x, @xs), @l2) -> c_3(lcstable^#(@xs, @l2)) }
StartTerms: basic terms
Strategy: innermost
The following weak DPs constitute a sub-graph of the DG that is
closed under successors. The DPs are removed.
{ lcstable^#(@l1, @l2) -> c_2(lcstable#1^#(@l1, @l2))
, lcstable#1^#(::(@x, @xs), @l2) -> c_3(lcstable^#(@xs, @l2)) }
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:
{ firstline^#(@l) -> c_1(firstline#1^#(@l))
, firstline#1^#(::(@x, @xs)) -> c_2(firstline^#(@xs))
, lcstable#1^#(nil(), @l2) -> c_6(firstline^#(@l2))
, lcstable#2^#(@m, @l2, @x) -> c_7(lcstable#3^#(@m, @l2, @x))
, lcstable#3^#(::(@l, @ls), @l2, @x) -> c_8(newline^#(@x, @l, @l2))
, newline^#(@y, @lastline, @l) ->
c_9(newline#1^#(@l, @lastline, @y))
, newline#1^#(::(@x, @xs), @lastline, @y) ->
c_10(newline#2^#(@lastline, @x, @xs, @y))
, newline#2^#(::(@belowVal, @lastline'), @x, @xs, @y) ->
c_11(newline^#(@y, @lastline', @xs)) }
Weak DPs:
{ lcs^#(@l1, @l2) -> lcstable^#(@l1, @l2)
, lcstable^#(@l1, @l2) -> lcstable#1^#(@l1, @l2)
, lcstable#1^#(::(@x, @xs), @l2) -> lcstable^#(@xs, @l2)
, lcstable#1^#(::(@x, @xs), @l2) ->
lcstable#2^#(lcstable(@xs, @l2), @l2, @x) }
Weak Trs:
{ #abs(#0()) -> #0()
, #abs(#neg(@x)) -> #pos(@x)
, #abs(#pos(@x)) -> #pos(@x)
, #abs(#s(@x)) -> #pos(#s(@x))
, #equal(@x, @y) -> #eq(@x, @y)
, #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)
, #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()
, #greater(@x, @y) -> #ckgt(#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)
, #ckgt(#EQ()) -> #false()
, #ckgt(#GT()) -> #true()
, #ckgt(#LT()) -> #false()
, +(@x, @y) -> #add(@x, @y)
, #add(#0(), @y) -> @y
, #add(#neg(#s(#0())), @y) -> #pred(@y)
, #add(#neg(#s(#s(@x))), @y) -> #pred(#add(#pos(#s(@x)), @y))
, #add(#pos(#s(#0())), @y) -> #succ(@y)
, #add(#pos(#s(#s(@x))), @y) -> #succ(#add(#pos(#s(@x)), @y))
, firstline(@l) -> firstline#1(@l)
, firstline#1(::(@x, @xs)) -> ::(#abs(#0()), firstline(@xs))
, firstline#1(nil()) -> nil()
, lcstable(@l1, @l2) -> lcstable#1(@l1, @l2)
, lcstable#1(::(@x, @xs), @l2) ->
lcstable#2(lcstable(@xs, @l2), @l2, @x)
, lcstable#1(nil(), @l2) -> ::(firstline(@l2), nil())
, lcstable#2(@m, @l2, @x) -> lcstable#3(@m, @l2, @x)
, lcstable#3(::(@l, @ls), @l2, @x) ->
::(newline(@x, @l, @l2), ::(@l, @ls))
, lcstable#3(nil(), @l2, @x) -> nil()
, newline(@y, @lastline, @l) -> newline#1(@l, @lastline, @y)
, max(@a, @b) -> max#1(#greater(@a, @b), @a, @b)
, max#1(#false(), @a, @b) -> @b
, max#1(#true(), @a, @b) -> @a
, newline#1(::(@x, @xs), @lastline, @y) ->
newline#2(@lastline, @x, @xs, @y)
, newline#1(nil(), @lastline, @y) -> nil()
, newline#2(::(@belowVal, @lastline'), @x, @xs, @y) ->
newline#3(newline(@y, @lastline', @xs),
@belowVal,
@lastline',
@x,
@y)
, newline#2(nil(), @x, @xs, @y) -> nil()
, newline#3(@nl, @belowVal, @lastline', @x, @y) ->
newline#4(right(@nl), @belowVal, @lastline', @nl, @x, @y)
, right(@l) -> right#1(@l)
, newline#4(@rightVal, @belowVal, @lastline', @nl, @x, @y) ->
newline#5(right(@lastline'), @belowVal, @nl, @rightVal, @x, @y)
, newline#5(@diagVal, @belowVal, @nl, @rightVal, @x, @y) ->
newline#6(newline#7(#equal(@x, @y),
@belowVal,
@diagVal,
@rightVal),
@nl)
, newline#7(#false(), @belowVal, @diagVal, @rightVal) ->
max(@belowVal, @rightVal)
, newline#7(#true(), @belowVal, @diagVal, @rightVal) ->
+(@diagVal, #pos(#s(#0())))
, newline#6(@elem, @nl) -> ::(@elem, @nl)
, right#1(::(@x, @xs)) -> @x
, right#1(nil()) -> #abs(#0())
, #pred(#0()) -> #neg(#s(#0()))
, #pred(#neg(#s(@x))) -> #neg(#s(#s(@x)))
, #pred(#pos(#s(#0()))) -> #0()
, #pred(#pos(#s(#s(@x)))) -> #pos(#s(@x))
, #succ(#0()) -> #pos(#s(#0()))
, #succ(#neg(#s(#0()))) -> #0()
, #succ(#neg(#s(#s(@x)))) -> #neg(#s(@x))
, #succ(#pos(#s(@x))) -> #pos(#s(#s(@x)))
, #and(#false(), #false()) -> #false()
, #and(#false(), #true()) -> #false()
, #and(#true(), #false()) -> #false()
, #and(#true(), #true()) -> #true() }
Obligation:
innermost runtime complexity
Answer:
YES(O(1),O(n^1))
Consider the dependency graph
1: firstline^#(@l) -> c_1(firstline#1^#(@l))
-->_1 firstline#1^#(::(@x, @xs)) -> c_2(firstline^#(@xs)) :2
2: firstline#1^#(::(@x, @xs)) -> c_2(firstline^#(@xs))
-->_1 firstline^#(@l) -> c_1(firstline#1^#(@l)) :1
3: lcstable#1^#(nil(), @l2) -> c_6(firstline^#(@l2))
-->_1 firstline^#(@l) -> c_1(firstline#1^#(@l)) :1
4: lcstable#2^#(@m, @l2, @x) -> c_7(lcstable#3^#(@m, @l2, @x))
-->_1 lcstable#3^#(::(@l, @ls), @l2, @x) ->
c_8(newline^#(@x, @l, @l2)) :5
5: lcstable#3^#(::(@l, @ls), @l2, @x) ->
c_8(newline^#(@x, @l, @l2))
-->_1 newline^#(@y, @lastline, @l) ->
c_9(newline#1^#(@l, @lastline, @y)) :6
6: newline^#(@y, @lastline, @l) ->
c_9(newline#1^#(@l, @lastline, @y))
-->_1 newline#1^#(::(@x, @xs), @lastline, @y) ->
c_10(newline#2^#(@lastline, @x, @xs, @y)) :7
7: newline#1^#(::(@x, @xs), @lastline, @y) ->
c_10(newline#2^#(@lastline, @x, @xs, @y))
-->_1 newline#2^#(::(@belowVal, @lastline'), @x, @xs, @y) ->
c_11(newline^#(@y, @lastline', @xs)) :8
8: newline#2^#(::(@belowVal, @lastline'), @x, @xs, @y) ->
c_11(newline^#(@y, @lastline', @xs))
-->_1 newline^#(@y, @lastline, @l) ->
c_9(newline#1^#(@l, @lastline, @y)) :6
9: lcs^#(@l1, @l2) -> lcstable^#(@l1, @l2)
-->_1 lcstable^#(@l1, @l2) -> lcstable#1^#(@l1, @l2) :10
10: lcstable^#(@l1, @l2) -> lcstable#1^#(@l1, @l2)
-->_1 lcstable#1^#(::(@x, @xs), @l2) ->
lcstable#2^#(lcstable(@xs, @l2), @l2, @x) :12
-->_1 lcstable#1^#(::(@x, @xs), @l2) -> lcstable^#(@xs, @l2) :11
-->_1 lcstable#1^#(nil(), @l2) -> c_6(firstline^#(@l2)) :3
11: lcstable#1^#(::(@x, @xs), @l2) -> lcstable^#(@xs, @l2)
-->_1 lcstable^#(@l1, @l2) -> lcstable#1^#(@l1, @l2) :10
12: lcstable#1^#(::(@x, @xs), @l2) ->
lcstable#2^#(lcstable(@xs, @l2), @l2, @x)
-->_1 lcstable#2^#(@m, @l2, @x) ->
c_7(lcstable#3^#(@m, @l2, @x)) :4
Following roots of the dependency graph are removed, as the
considered set of starting terms is closed under reduction with
respect to these rules (modulo compound contexts).
{ lcs^#(@l1, @l2) -> lcstable^#(@l1, @l2) }
We are left with following problem, upon which TcT provides the
certificate YES(O(1),O(n^1)).
Strict DPs:
{ firstline^#(@l) -> c_1(firstline#1^#(@l))
, firstline#1^#(::(@x, @xs)) -> c_2(firstline^#(@xs))
, lcstable#1^#(nil(), @l2) -> c_6(firstline^#(@l2))
, lcstable#2^#(@m, @l2, @x) -> c_7(lcstable#3^#(@m, @l2, @x))
, lcstable#3^#(::(@l, @ls), @l2, @x) -> c_8(newline^#(@x, @l, @l2))
, newline^#(@y, @lastline, @l) ->
c_9(newline#1^#(@l, @lastline, @y))
, newline#1^#(::(@x, @xs), @lastline, @y) ->
c_10(newline#2^#(@lastline, @x, @xs, @y))
, newline#2^#(::(@belowVal, @lastline'), @x, @xs, @y) ->
c_11(newline^#(@y, @lastline', @xs)) }
Weak DPs:
{ lcstable^#(@l1, @l2) -> lcstable#1^#(@l1, @l2)
, lcstable#1^#(::(@x, @xs), @l2) -> lcstable^#(@xs, @l2)
, lcstable#1^#(::(@x, @xs), @l2) ->
lcstable#2^#(lcstable(@xs, @l2), @l2, @x) }
Weak Trs:
{ #abs(#0()) -> #0()
, #abs(#neg(@x)) -> #pos(@x)
, #abs(#pos(@x)) -> #pos(@x)
, #abs(#s(@x)) -> #pos(#s(@x))
, #equal(@x, @y) -> #eq(@x, @y)
, #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)
, #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()
, #greater(@x, @y) -> #ckgt(#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)
, #ckgt(#EQ()) -> #false()
, #ckgt(#GT()) -> #true()
, #ckgt(#LT()) -> #false()
, +(@x, @y) -> #add(@x, @y)
, #add(#0(), @y) -> @y
, #add(#neg(#s(#0())), @y) -> #pred(@y)
, #add(#neg(#s(#s(@x))), @y) -> #pred(#add(#pos(#s(@x)), @y))
, #add(#pos(#s(#0())), @y) -> #succ(@y)
, #add(#pos(#s(#s(@x))), @y) -> #succ(#add(#pos(#s(@x)), @y))
, firstline(@l) -> firstline#1(@l)
, firstline#1(::(@x, @xs)) -> ::(#abs(#0()), firstline(@xs))
, firstline#1(nil()) -> nil()
, lcstable(@l1, @l2) -> lcstable#1(@l1, @l2)
, lcstable#1(::(@x, @xs), @l2) ->
lcstable#2(lcstable(@xs, @l2), @l2, @x)
, lcstable#1(nil(), @l2) -> ::(firstline(@l2), nil())
, lcstable#2(@m, @l2, @x) -> lcstable#3(@m, @l2, @x)
, lcstable#3(::(@l, @ls), @l2, @x) ->
::(newline(@x, @l, @l2), ::(@l, @ls))
, lcstable#3(nil(), @l2, @x) -> nil()
, newline(@y, @lastline, @l) -> newline#1(@l, @lastline, @y)
, max(@a, @b) -> max#1(#greater(@a, @b), @a, @b)
, max#1(#false(), @a, @b) -> @b
, max#1(#true(), @a, @b) -> @a
, newline#1(::(@x, @xs), @lastline, @y) ->
newline#2(@lastline, @x, @xs, @y)
, newline#1(nil(), @lastline, @y) -> nil()
, newline#2(::(@belowVal, @lastline'), @x, @xs, @y) ->
newline#3(newline(@y, @lastline', @xs),
@belowVal,
@lastline',
@x,
@y)
, newline#2(nil(), @x, @xs, @y) -> nil()
, newline#3(@nl, @belowVal, @lastline', @x, @y) ->
newline#4(right(@nl), @belowVal, @lastline', @nl, @x, @y)
, right(@l) -> right#1(@l)
, newline#4(@rightVal, @belowVal, @lastline', @nl, @x, @y) ->
newline#5(right(@lastline'), @belowVal, @nl, @rightVal, @x, @y)
, newline#5(@diagVal, @belowVal, @nl, @rightVal, @x, @y) ->
newline#6(newline#7(#equal(@x, @y),
@belowVal,
@diagVal,
@rightVal),
@nl)
, newline#7(#false(), @belowVal, @diagVal, @rightVal) ->
max(@belowVal, @rightVal)
, newline#7(#true(), @belowVal, @diagVal, @rightVal) ->
+(@diagVal, #pos(#s(#0())))
, newline#6(@elem, @nl) -> ::(@elem, @nl)
, right#1(::(@x, @xs)) -> @x
, right#1(nil()) -> #abs(#0())
, #pred(#0()) -> #neg(#s(#0()))
, #pred(#neg(#s(@x))) -> #neg(#s(#s(@x)))
, #pred(#pos(#s(#0()))) -> #0()
, #pred(#pos(#s(#s(@x)))) -> #pos(#s(@x))
, #succ(#0()) -> #pos(#s(#0()))
, #succ(#neg(#s(#0()))) -> #0()
, #succ(#neg(#s(#s(@x)))) -> #neg(#s(@x))
, #succ(#pos(#s(@x))) -> #pos(#s(#s(@x)))
, #and(#false(), #false()) -> #false()
, #and(#false(), #true()) -> #false()
, #and(#true(), #false()) -> #false()
, #and(#true(), #true()) -> #true() }
Obligation:
innermost runtime complexity
Answer:
YES(O(1),O(n^1))
We use the processor 'matrix interpretation of dimension 1' to
orient following rules strictly.
DPs:
{ 2: firstline#1^#(::(@x, @xs)) -> c_2(firstline^#(@xs)) }
Trs:
{ #greater(@x, @y) -> #ckgt(#compare(@x, @y))
, max(@a, @b) -> max#1(#greater(@a, @b), @a, @b)
, newline#2(::(@belowVal, @lastline'), @x, @xs, @y) ->
newline#3(newline(@y, @lastline', @xs),
@belowVal,
@lastline',
@x,
@y)
, right#1(::(@x, @xs)) -> @x
, right#1(nil()) -> #abs(#0()) }
Sub-proof:
----------
The following argument positions are usable:
Uargs(c_1) = {1}, Uargs(c_2) = {1}, Uargs(c_6) = {1},
Uargs(c_7) = {1}, Uargs(c_8) = {1}, Uargs(c_9) = {1},
Uargs(c_10) = {1}, Uargs(c_11) = {1}
TcT has computed following constructor-based matrix interpretation
satisfying not(EDA).
[#0] = [0]
[#abs](x1) = [0]
[#neg](x1) = [0]
[#pos](x1) = [0]
[#s](x1) = [1]
[#equal](x1, x2) = [0]
[#eq](x1, x2) = [0]
[#greater](x1, x2) = [1]
[#compare](x1, x2) = [0]
[#ckgt](x1) = [1] x1 + [0]
[+](x1, x2) = [0]
[#add](x1, x2) = [1] x2 + [0]
[firstline](x1) = [0]
[firstline#1](x1) = [0]
[::](x1, x2) = [1] x1 + [1] x2 + [1]
[nil] = [1]
[lcstable](x1, x2) = [0]
[lcstable#1](x1, x2) = [0]
[lcstable#2](x1, x2, x3) = [0]
[lcstable#3](x1, x2, x3) = [1] x1 + [0]
[newline](x1, x2, x3) = [0]
[max](x1, x2) = [1]
[max#1](x1, x2, x3) = [0]
[#false] = [0]
[#true] = [1]
[newline#1](x1, x2, x3) = [1]
[newline#2](x1, x2, x3, x4) = [1] x1 + [0]
[newline#3](x1, x2, x3, x4, x5) = [0]
[right](x1) = [0]
[newline#4](x1, x2, x3, x4, x5, x6) = [0]
[newline#5](x1, x2, x3, x4, x5, x6) = [0]
[newline#7](x1, x2, x3, x4) = [0]
[newline#6](x1, x2) = [0]
[right#1](x1) = [1] x1 + [0]
[#pred](x1) = [0]
[#succ](x1) = [1] x1 + [0]
[#and](x1, x2) = [0]
[#EQ] = [0]
[#GT] = [0]
[#LT] = [0]
[#abs^#](x1) = [0]
[#equal^#](x1, x2) = [0]
[#eq^#](x1, x2) = [0]
[#greater^#](x1, x2) = [0]
[#ckgt^#](x1) = [0]
[#compare^#](x1, x2) = [0]
[+^#](x1, x2) = [0]
[#add^#](x1, x2) = [0]
[firstline^#](x1) = [1] x1 + [1]
[firstline#1^#](x1) = [1] x1 + [1]
[lcs^#](x1, x2) = [0]
[lcs#1^#](x1) = [0]
[lcstable^#](x1, x2) = [1] x2 + [1]
[lcstable#1^#](x1, x2) = [1] x2 + [1]
[lcs#2^#](x1) = [0]
[lcs#3^#](x1) = [0]
[lcstable#2^#](x1, x2, x3) = [1]
[lcstable#3^#](x1, x2, x3) = [1]
[newline^#](x1, x2, x3) = [1]
[newline#1^#](x1, x2, x3) = [1]
[max^#](x1, x2) = [0]
[max#1^#](x1, x2, x3) = [0]
[newline#2^#](x1, x2, x3, x4) = [1]
[newline#3^#](x1, x2, x3, x4, x5) = [0]
[newline#4^#](x1, x2, x3, x4, x5, x6) = [0]
[right^#](x1) = [0]
[right#1^#](x1) = [0]
[newline#5^#](x1, x2, x3, x4, x5, x6) = [0]
[newline#6^#](x1, x2) = [0]
[newline#7^#](x1, x2, x3, x4) = [0]
[#and^#](x1, x2) = [0]
[#pred^#](x1) = [0]
[#succ^#](x1) = [0]
[c_1](x1) = [1] x1 + [0]
[c_2](x1) = [1] x1 + [0]
[c_6](x1) = [1] x1 + [0]
[c_7](x1) = [1] x1 + [0]
[c_8](x1) = [1] x1 + [0]
[c_9](x1) = [1] x1 + [0]
[c_10](x1) = [1] x1 + [0]
[c_11](x1) = [1] x1 + [0]
This order satisfies following ordering constraints
[firstline^#(@l)] = [1] @l + [1]
>= [1] @l + [1]
= [c_1(firstline#1^#(@l))]
[firstline#1^#(::(@x, @xs))] = [1] @x + [1] @xs + [2]
> [1] @xs + [1]
= [c_2(firstline^#(@xs))]
[lcstable^#(@l1, @l2)] = [1] @l2 + [1]
>= [1] @l2 + [1]
= [lcstable#1^#(@l1, @l2)]
[lcstable#1^#(::(@x, @xs), @l2)] = [1] @l2 + [1]
>= [1] @l2 + [1]
= [lcstable^#(@xs, @l2)]
[lcstable#1^#(::(@x, @xs), @l2)] = [1] @l2 + [1]
>= [1]
= [lcstable#2^#(lcstable(@xs, @l2), @l2, @x)]
[lcstable#1^#(nil(), @l2)] = [1] @l2 + [1]
>= [1] @l2 + [1]
= [c_6(firstline^#(@l2))]
[lcstable#2^#(@m, @l2, @x)] = [1]
>= [1]
= [c_7(lcstable#3^#(@m, @l2, @x))]
[lcstable#3^#(::(@l, @ls), @l2, @x)] = [1]
>= [1]
= [c_8(newline^#(@x, @l, @l2))]
[newline^#(@y, @lastline, @l)] = [1]
>= [1]
= [c_9(newline#1^#(@l, @lastline, @y))]
[newline#1^#(::(@x, @xs), @lastline, @y)] = [1]
>= [1]
= [c_10(newline#2^#(@lastline, @x, @xs, @y))]
[newline#2^#(::(@belowVal, @lastline'), @x, @xs, @y)] = [1]
>= [1]
= [c_11(newline^#(@y, @lastline', @xs))]
The strictly oriented rules are moved into the corresponding weak
component(s).
We are left with following problem, upon which TcT provides the
certificate YES(O(1),O(n^1)).
Strict DPs:
{ firstline^#(@l) -> c_1(firstline#1^#(@l))
, lcstable#1^#(nil(), @l2) -> c_6(firstline^#(@l2))
, lcstable#2^#(@m, @l2, @x) -> c_7(lcstable#3^#(@m, @l2, @x))
, lcstable#3^#(::(@l, @ls), @l2, @x) -> c_8(newline^#(@x, @l, @l2))
, newline^#(@y, @lastline, @l) ->
c_9(newline#1^#(@l, @lastline, @y))
, newline#1^#(::(@x, @xs), @lastline, @y) ->
c_10(newline#2^#(@lastline, @x, @xs, @y))
, newline#2^#(::(@belowVal, @lastline'), @x, @xs, @y) ->
c_11(newline^#(@y, @lastline', @xs)) }
Weak DPs:
{ firstline#1^#(::(@x, @xs)) -> c_2(firstline^#(@xs))
, lcstable^#(@l1, @l2) -> lcstable#1^#(@l1, @l2)
, lcstable#1^#(::(@x, @xs), @l2) -> lcstable^#(@xs, @l2)
, lcstable#1^#(::(@x, @xs), @l2) ->
lcstable#2^#(lcstable(@xs, @l2), @l2, @x) }
Weak Trs:
{ #abs(#0()) -> #0()
, #abs(#neg(@x)) -> #pos(@x)
, #abs(#pos(@x)) -> #pos(@x)
, #abs(#s(@x)) -> #pos(#s(@x))
, #equal(@x, @y) -> #eq(@x, @y)
, #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)
, #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()
, #greater(@x, @y) -> #ckgt(#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)
, #ckgt(#EQ()) -> #false()
, #ckgt(#GT()) -> #true()
, #ckgt(#LT()) -> #false()
, +(@x, @y) -> #add(@x, @y)
, #add(#0(), @y) -> @y
, #add(#neg(#s(#0())), @y) -> #pred(@y)
, #add(#neg(#s(#s(@x))), @y) -> #pred(#add(#pos(#s(@x)), @y))
, #add(#pos(#s(#0())), @y) -> #succ(@y)
, #add(#pos(#s(#s(@x))), @y) -> #succ(#add(#pos(#s(@x)), @y))
, firstline(@l) -> firstline#1(@l)
, firstline#1(::(@x, @xs)) -> ::(#abs(#0()), firstline(@xs))
, firstline#1(nil()) -> nil()
, lcstable(@l1, @l2) -> lcstable#1(@l1, @l2)
, lcstable#1(::(@x, @xs), @l2) ->
lcstable#2(lcstable(@xs, @l2), @l2, @x)
, lcstable#1(nil(), @l2) -> ::(firstline(@l2), nil())
, lcstable#2(@m, @l2, @x) -> lcstable#3(@m, @l2, @x)
, lcstable#3(::(@l, @ls), @l2, @x) ->
::(newline(@x, @l, @l2), ::(@l, @ls))
, lcstable#3(nil(), @l2, @x) -> nil()
, newline(@y, @lastline, @l) -> newline#1(@l, @lastline, @y)
, max(@a, @b) -> max#1(#greater(@a, @b), @a, @b)
, max#1(#false(), @a, @b) -> @b
, max#1(#true(), @a, @b) -> @a
, newline#1(::(@x, @xs), @lastline, @y) ->
newline#2(@lastline, @x, @xs, @y)
, newline#1(nil(), @lastline, @y) -> nil()
, newline#2(::(@belowVal, @lastline'), @x, @xs, @y) ->
newline#3(newline(@y, @lastline', @xs),
@belowVal,
@lastline',
@x,
@y)
, newline#2(nil(), @x, @xs, @y) -> nil()
, newline#3(@nl, @belowVal, @lastline', @x, @y) ->
newline#4(right(@nl), @belowVal, @lastline', @nl, @x, @y)
, right(@l) -> right#1(@l)
, newline#4(@rightVal, @belowVal, @lastline', @nl, @x, @y) ->
newline#5(right(@lastline'), @belowVal, @nl, @rightVal, @x, @y)
, newline#5(@diagVal, @belowVal, @nl, @rightVal, @x, @y) ->
newline#6(newline#7(#equal(@x, @y),
@belowVal,
@diagVal,
@rightVal),
@nl)
, newline#7(#false(), @belowVal, @diagVal, @rightVal) ->
max(@belowVal, @rightVal)
, newline#7(#true(), @belowVal, @diagVal, @rightVal) ->
+(@diagVal, #pos(#s(#0())))
, newline#6(@elem, @nl) -> ::(@elem, @nl)
, right#1(::(@x, @xs)) -> @x
, right#1(nil()) -> #abs(#0())
, #pred(#0()) -> #neg(#s(#0()))
, #pred(#neg(#s(@x))) -> #neg(#s(#s(@x)))
, #pred(#pos(#s(#0()))) -> #0()
, #pred(#pos(#s(#s(@x)))) -> #pos(#s(@x))
, #succ(#0()) -> #pos(#s(#0()))
, #succ(#neg(#s(#0()))) -> #0()
, #succ(#neg(#s(#s(@x)))) -> #neg(#s(@x))
, #succ(#pos(#s(@x))) -> #pos(#s(#s(@x)))
, #and(#false(), #false()) -> #false()
, #and(#false(), #true()) -> #false()
, #and(#true(), #false()) -> #false()
, #and(#true(), #true()) -> #true() }
Obligation:
innermost runtime complexity
Answer:
YES(O(1),O(n^1))
We use the processor 'matrix interpretation of dimension 1' to
orient following rules strictly.
DPs:
{ 2: lcstable#1^#(nil(), @l2) -> c_6(firstline^#(@l2))
, 7: newline#2^#(::(@belowVal, @lastline'), @x, @xs, @y) ->
c_11(newline^#(@y, @lastline', @xs))
, 8: firstline#1^#(::(@x, @xs)) -> c_2(firstline^#(@xs))
, 11: lcstable#1^#(::(@x, @xs), @l2) ->
lcstable#2^#(lcstable(@xs, @l2), @l2, @x) }
Trs:
{ #greater(@x, @y) -> #ckgt(#compare(@x, @y))
, #add(#pos(#s(#0())), @y) -> #succ(@y)
, #add(#pos(#s(#s(@x))), @y) -> #succ(#add(#pos(#s(@x)), @y))
, lcstable#1(::(@x, @xs), @l2) ->
lcstable#2(lcstable(@xs, @l2), @l2, @x)
, max(@a, @b) -> max#1(#greater(@a, @b), @a, @b)
, newline#1(nil(), @lastline, @y) -> nil()
, newline#2(::(@belowVal, @lastline'), @x, @xs, @y) ->
newline#3(newline(@y, @lastline', @xs),
@belowVal,
@lastline',
@x,
@y) }
Sub-proof:
----------
The following argument positions are usable:
Uargs(c_1) = {1}, Uargs(c_2) = {1}, Uargs(c_6) = {1},
Uargs(c_7) = {1}, Uargs(c_8) = {1}, Uargs(c_9) = {1},
Uargs(c_10) = {1}, Uargs(c_11) = {1}
TcT has computed following constructor-based matrix interpretation
satisfying not(EDA).
[#0] = [0]
[#abs](x1) = [0]
[#neg](x1) = [0]
[#pos](x1) = [1]
[#s](x1) = [0]
[#equal](x1, x2) = [0]
[#eq](x1, x2) = [0]
[#greater](x1, x2) = [1]
[#compare](x1, x2) = [0]
[#ckgt](x1) = [1] x1 + [0]
[+](x1, x2) = [0]
[#add](x1, x2) = [1] x1 + [1] x2 + [0]
[firstline](x1) = [0]
[firstline#1](x1) = [0]
[::](x1, x2) = [1] x2 + [1]
[nil] = [0]
[lcstable](x1, x2) = [0]
[lcstable#1](x1, x2) = [1] x2 + [1]
[lcstable#2](x1, x2, x3) = [0]
[lcstable#3](x1, x2, x3) = [1] x1 + [0]
[newline](x1, x2, x3) = [0]
[max](x1, x2) = [1]
[max#1](x1, x2, x3) = [0]
[#false] = [0]
[#true] = [1]
[newline#1](x1, x2, x3) = [1]
[newline#2](x1, x2, x3, x4) = [1] x1 + [0]
[newline#3](x1, x2, x3, x4, x5) = [0]
[right](x1) = [0]
[newline#4](x1, x2, x3, x4, x5, x6) = [0]
[newline#5](x1, x2, x3, x4, x5, x6) = [0]
[newline#7](x1, x2, x3, x4) = [0]
[newline#6](x1, x2) = [0]
[right#1](x1) = [1] x1 + [0]
[#pred](x1) = [0]
[#succ](x1) = [0]
[#and](x1, x2) = [0]
[#EQ] = [0]
[#GT] = [1]
[#LT] = [0]
[#abs^#](x1) = [0]
[#equal^#](x1, x2) = [0]
[#eq^#](x1, x2) = [0]
[#greater^#](x1, x2) = [0]
[#ckgt^#](x1) = [0]
[#compare^#](x1, x2) = [0]
[+^#](x1, x2) = [0]
[#add^#](x1, x2) = [0]
[firstline^#](x1) = [1] x1 + [0]
[firstline#1^#](x1) = [1] x1 + [0]
[lcs^#](x1, x2) = [0]
[lcs#1^#](x1) = [0]
[lcstable^#](x1, x2) = [1] x2 + [1]
[lcstable#1^#](x1, x2) = [1] x2 + [1]
[lcs#2^#](x1) = [0]
[lcs#3^#](x1) = [0]
[lcstable#2^#](x1, x2, x3) = [1] x2 + [0]
[lcstable#3^#](x1, x2, x3) = [1] x2 + [0]
[newline^#](x1, x2, x3) = [1] x3 + [0]
[newline#1^#](x1, x2, x3) = [1] x1 + [0]
[max^#](x1, x2) = [0]
[max#1^#](x1, x2, x3) = [0]
[newline#2^#](x1, x2, x3, x4) = [1] x3 + [1]
[newline#3^#](x1, x2, x3, x4, x5) = [0]
[newline#4^#](x1, x2, x3, x4, x5, x6) = [0]
[right^#](x1) = [0]
[right#1^#](x1) = [0]
[newline#5^#](x1, x2, x3, x4, x5, x6) = [0]
[newline#6^#](x1, x2) = [0]
[newline#7^#](x1, x2, x3, x4) = [0]
[#and^#](x1, x2) = [0]
[#pred^#](x1) = [0]
[#succ^#](x1) = [0]
[c_1](x1) = [1] x1 + [0]
[c_2](x1) = [1] x1 + [0]
[c_6](x1) = [1] x1 + [0]
[c_7](x1) = [1] x1 + [0]
[c_8](x1) = [1] x1 + [0]
[c_9](x1) = [1] x1 + [0]
[c_10](x1) = [1] x1 + [0]
[c_11](x1) = [1] x1 + [0]
This order satisfies following ordering constraints
[firstline^#(@l)] = [1] @l + [0]
>= [1] @l + [0]
= [c_1(firstline#1^#(@l))]
[firstline#1^#(::(@x, @xs))] = [1] @xs + [1]
> [1] @xs + [0]
= [c_2(firstline^#(@xs))]
[lcstable^#(@l1, @l2)] = [1] @l2 + [1]
>= [1] @l2 + [1]
= [lcstable#1^#(@l1, @l2)]
[lcstable#1^#(::(@x, @xs), @l2)] = [1] @l2 + [1]
>= [1] @l2 + [1]
= [lcstable^#(@xs, @l2)]
[lcstable#1^#(::(@x, @xs), @l2)] = [1] @l2 + [1]
> [1] @l2 + [0]
= [lcstable#2^#(lcstable(@xs, @l2), @l2, @x)]
[lcstable#1^#(nil(), @l2)] = [1] @l2 + [1]
> [1] @l2 + [0]
= [c_6(firstline^#(@l2))]
[lcstable#2^#(@m, @l2, @x)] = [1] @l2 + [0]
>= [1] @l2 + [0]
= [c_7(lcstable#3^#(@m, @l2, @x))]
[lcstable#3^#(::(@l, @ls), @l2, @x)] = [1] @l2 + [0]
>= [1] @l2 + [0]
= [c_8(newline^#(@x, @l, @l2))]
[newline^#(@y, @lastline, @l)] = [1] @l + [0]
>= [1] @l + [0]
= [c_9(newline#1^#(@l, @lastline, @y))]
[newline#1^#(::(@x, @xs), @lastline, @y)] = [1] @xs + [1]
>= [1] @xs + [1]
= [c_10(newline#2^#(@lastline, @x, @xs, @y))]
[newline#2^#(::(@belowVal, @lastline'), @x, @xs, @y)] = [1] @xs + [1]
> [1] @xs + [0]
= [c_11(newline^#(@y, @lastline', @xs))]
Consider the set of all dependency pairs
DPs:
{ 1: firstline^#(@l) -> c_1(firstline#1^#(@l))
, 2: lcstable#1^#(nil(), @l2) -> c_6(firstline^#(@l2))
, 3: lcstable#2^#(@m, @l2, @x) -> c_7(lcstable#3^#(@m, @l2, @x))
, 4: lcstable#3^#(::(@l, @ls), @l2, @x) ->
c_8(newline^#(@x, @l, @l2))
, 5: newline^#(@y, @lastline, @l) ->
c_9(newline#1^#(@l, @lastline, @y))
, 6: newline#1^#(::(@x, @xs), @lastline, @y) ->
c_10(newline#2^#(@lastline, @x, @xs, @y))
, 7: newline#2^#(::(@belowVal, @lastline'), @x, @xs, @y) ->
c_11(newline^#(@y, @lastline', @xs))
, 8: firstline#1^#(::(@x, @xs)) -> c_2(firstline^#(@xs))
, 9: lcstable^#(@l1, @l2) -> lcstable#1^#(@l1, @l2)
, 10: lcstable#1^#(::(@x, @xs), @l2) -> lcstable^#(@xs, @l2)
, 11: lcstable#1^#(::(@x, @xs), @l2) ->
lcstable#2^#(lcstable(@xs, @l2), @l2, @x) }
Processor 'matrix interpretation of dimension 1' induces the
complexity certificate YES(?,O(n^1)) on application of dependency
pairs {2,7,8,11}. These cover all (indirect) predecessors of
dependency pairs {1,2,3,4,5,6,7,8,11}, their number of application
is equally bounded. The dependency pairs are shifted into the
corresponding weak component(s).
We apply the transformation 'removetails' on the sub-problem:
Weak DPs:
{ firstline^#(@l) -> c_1(firstline#1^#(@l))
, firstline#1^#(::(@x, @xs)) -> c_2(firstline^#(@xs))
, lcstable^#(@l1, @l2) -> lcstable#1^#(@l1, @l2)
, lcstable#1^#(::(@x, @xs), @l2) -> lcstable^#(@xs, @l2)
, lcstable#1^#(::(@x, @xs), @l2) ->
lcstable#2^#(lcstable(@xs, @l2), @l2, @x)
, lcstable#1^#(nil(), @l2) -> c_6(firstline^#(@l2))
, lcstable#2^#(@m, @l2, @x) -> c_7(lcstable#3^#(@m, @l2, @x))
, lcstable#3^#(::(@l, @ls), @l2, @x) -> c_8(newline^#(@x, @l, @l2))
, newline^#(@y, @lastline, @l) ->
c_9(newline#1^#(@l, @lastline, @y))
, newline#1^#(::(@x, @xs), @lastline, @y) ->
c_10(newline#2^#(@lastline, @x, @xs, @y))
, newline#2^#(::(@belowVal, @lastline'), @x, @xs, @y) ->
c_11(newline^#(@y, @lastline', @xs)) }
Weak Trs:
{ #abs(#0()) -> #0()
, #abs(#neg(@x)) -> #pos(@x)
, #abs(#pos(@x)) -> #pos(@x)
, #abs(#s(@x)) -> #pos(#s(@x))
, #equal(@x, @y) -> #eq(@x, @y)
, #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)
, #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()
, #greater(@x, @y) -> #ckgt(#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)
, #ckgt(#EQ()) -> #false()
, #ckgt(#GT()) -> #true()
, #ckgt(#LT()) -> #false()
, +(@x, @y) -> #add(@x, @y)
, #add(#0(), @y) -> @y
, #add(#neg(#s(#0())), @y) -> #pred(@y)
, #add(#neg(#s(#s(@x))), @y) -> #pred(#add(#pos(#s(@x)), @y))
, #add(#pos(#s(#0())), @y) -> #succ(@y)
, #add(#pos(#s(#s(@x))), @y) -> #succ(#add(#pos(#s(@x)), @y))
, firstline(@l) -> firstline#1(@l)
, firstline#1(::(@x, @xs)) -> ::(#abs(#0()), firstline(@xs))
, firstline#1(nil()) -> nil()
, lcstable(@l1, @l2) -> lcstable#1(@l1, @l2)
, lcstable#1(::(@x, @xs), @l2) ->
lcstable#2(lcstable(@xs, @l2), @l2, @x)
, lcstable#1(nil(), @l2) -> ::(firstline(@l2), nil())
, lcstable#2(@m, @l2, @x) -> lcstable#3(@m, @l2, @x)
, lcstable#3(::(@l, @ls), @l2, @x) ->
::(newline(@x, @l, @l2), ::(@l, @ls))
, lcstable#3(nil(), @l2, @x) -> nil()
, newline(@y, @lastline, @l) -> newline#1(@l, @lastline, @y)
, max(@a, @b) -> max#1(#greater(@a, @b), @a, @b)
, max#1(#false(), @a, @b) -> @b
, max#1(#true(), @a, @b) -> @a
, newline#1(::(@x, @xs), @lastline, @y) ->
newline#2(@lastline, @x, @xs, @y)
, newline#1(nil(), @lastline, @y) -> nil()
, newline#2(::(@belowVal, @lastline'), @x, @xs, @y) ->
newline#3(newline(@y, @lastline', @xs),
@belowVal,
@lastline',
@x,
@y)
, newline#2(nil(), @x, @xs, @y) -> nil()
, newline#3(@nl, @belowVal, @lastline', @x, @y) ->
newline#4(right(@nl), @belowVal, @lastline', @nl, @x, @y)
, right(@l) -> right#1(@l)
, newline#4(@rightVal, @belowVal, @lastline', @nl, @x, @y) ->
newline#5(right(@lastline'), @belowVal, @nl, @rightVal, @x, @y)
, newline#5(@diagVal, @belowVal, @nl, @rightVal, @x, @y) ->
newline#6(newline#7(#equal(@x, @y),
@belowVal,
@diagVal,
@rightVal),
@nl)
, newline#7(#false(), @belowVal, @diagVal, @rightVal) ->
max(@belowVal, @rightVal)
, newline#7(#true(), @belowVal, @diagVal, @rightVal) ->
+(@diagVal, #pos(#s(#0())))
, newline#6(@elem, @nl) -> ::(@elem, @nl)
, right#1(::(@x, @xs)) -> @x
, right#1(nil()) -> #abs(#0())
, #pred(#0()) -> #neg(#s(#0()))
, #pred(#neg(#s(@x))) -> #neg(#s(#s(@x)))
, #pred(#pos(#s(#0()))) -> #0()
, #pred(#pos(#s(#s(@x)))) -> #pos(#s(@x))
, #succ(#0()) -> #pos(#s(#0()))
, #succ(#neg(#s(#0()))) -> #0()
, #succ(#neg(#s(#s(@x)))) -> #neg(#s(@x))
, #succ(#pos(#s(@x))) -> #pos(#s(#s(@x)))
, #and(#false(), #false()) -> #false()
, #and(#false(), #true()) -> #false()
, #and(#true(), #false()) -> #false()
, #and(#true(), #true()) -> #true() }
StartTerms: basic terms
Strategy: innermost
The following weak DPs constitute a sub-graph of the DG that is
closed under successors. The DPs are removed.
{ firstline^#(@l) -> c_1(firstline#1^#(@l))
, firstline#1^#(::(@x, @xs)) -> c_2(firstline^#(@xs))
, lcstable^#(@l1, @l2) -> lcstable#1^#(@l1, @l2)
, lcstable#1^#(::(@x, @xs), @l2) -> lcstable^#(@xs, @l2)
, lcstable#1^#(::(@x, @xs), @l2) ->
lcstable#2^#(lcstable(@xs, @l2), @l2, @x)
, lcstable#1^#(nil(), @l2) -> c_6(firstline^#(@l2))
, lcstable#2^#(@m, @l2, @x) -> c_7(lcstable#3^#(@m, @l2, @x))
, lcstable#3^#(::(@l, @ls), @l2, @x) -> c_8(newline^#(@x, @l, @l2))
, newline^#(@y, @lastline, @l) ->
c_9(newline#1^#(@l, @lastline, @y))
, newline#1^#(::(@x, @xs), @lastline, @y) ->
c_10(newline#2^#(@lastline, @x, @xs, @y))
, newline#2^#(::(@belowVal, @lastline'), @x, @xs, @y) ->
c_11(newline^#(@y, @lastline', @xs)) }
We apply the transformation 'usablerules' on the sub-problem:
Weak Trs:
{ #abs(#0()) -> #0()
, #abs(#neg(@x)) -> #pos(@x)
, #abs(#pos(@x)) -> #pos(@x)
, #abs(#s(@x)) -> #pos(#s(@x))
, #equal(@x, @y) -> #eq(@x, @y)
, #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)
, #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()
, #greater(@x, @y) -> #ckgt(#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)
, #ckgt(#EQ()) -> #false()
, #ckgt(#GT()) -> #true()
, #ckgt(#LT()) -> #false()
, +(@x, @y) -> #add(@x, @y)
, #add(#0(), @y) -> @y
, #add(#neg(#s(#0())), @y) -> #pred(@y)
, #add(#neg(#s(#s(@x))), @y) -> #pred(#add(#pos(#s(@x)), @y))
, #add(#pos(#s(#0())), @y) -> #succ(@y)
, #add(#pos(#s(#s(@x))), @y) -> #succ(#add(#pos(#s(@x)), @y))
, firstline(@l) -> firstline#1(@l)
, firstline#1(::(@x, @xs)) -> ::(#abs(#0()), firstline(@xs))
, firstline#1(nil()) -> nil()
, lcstable(@l1, @l2) -> lcstable#1(@l1, @l2)
, lcstable#1(::(@x, @xs), @l2) ->
lcstable#2(lcstable(@xs, @l2), @l2, @x)
, lcstable#1(nil(), @l2) -> ::(firstline(@l2), nil())
, lcstable#2(@m, @l2, @x) -> lcstable#3(@m, @l2, @x)
, lcstable#3(::(@l, @ls), @l2, @x) ->
::(newline(@x, @l, @l2), ::(@l, @ls))
, lcstable#3(nil(), @l2, @x) -> nil()
, newline(@y, @lastline, @l) -> newline#1(@l, @lastline, @y)
, max(@a, @b) -> max#1(#greater(@a, @b), @a, @b)
, max#1(#false(), @a, @b) -> @b
, max#1(#true(), @a, @b) -> @a
, newline#1(::(@x, @xs), @lastline, @y) ->
newline#2(@lastline, @x, @xs, @y)
, newline#1(nil(), @lastline, @y) -> nil()
, newline#2(::(@belowVal, @lastline'), @x, @xs, @y) ->
newline#3(newline(@y, @lastline', @xs),
@belowVal,
@lastline',
@x,
@y)
, newline#2(nil(), @x, @xs, @y) -> nil()
, newline#3(@nl, @belowVal, @lastline', @x, @y) ->
newline#4(right(@nl), @belowVal, @lastline', @nl, @x, @y)
, right(@l) -> right#1(@l)
, newline#4(@rightVal, @belowVal, @lastline', @nl, @x, @y) ->
newline#5(right(@lastline'), @belowVal, @nl, @rightVal, @x, @y)
, newline#5(@diagVal, @belowVal, @nl, @rightVal, @x, @y) ->
newline#6(newline#7(#equal(@x, @y),
@belowVal,
@diagVal,
@rightVal),
@nl)
, newline#7(#false(), @belowVal, @diagVal, @rightVal) ->
max(@belowVal, @rightVal)
, newline#7(#true(), @belowVal, @diagVal, @rightVal) ->
+(@diagVal, #pos(#s(#0())))
, newline#6(@elem, @nl) -> ::(@elem, @nl)
, right#1(::(@x, @xs)) -> @x
, right#1(nil()) -> #abs(#0())
, #pred(#0()) -> #neg(#s(#0()))
, #pred(#neg(#s(@x))) -> #neg(#s(#s(@x)))
, #pred(#pos(#s(#0()))) -> #0()
, #pred(#pos(#s(#s(@x)))) -> #pos(#s(@x))
, #succ(#0()) -> #pos(#s(#0()))
, #succ(#neg(#s(#0()))) -> #0()
, #succ(#neg(#s(#s(@x)))) -> #neg(#s(@x))
, #succ(#pos(#s(@x))) -> #pos(#s(#s(@x)))
, #and(#false(), #false()) -> #false()
, #and(#false(), #true()) -> #false()
, #and(#true(), #false()) -> #false()
, #and(#true(), #true()) -> #true() }
StartTerms: basic terms
Strategy: innermost
No rule is usable, rules are removed from the input problem.
We are left with following problem, upon which TcT provides the
certificate YES(O(1),O(1)).
Rules: Empty
Obligation:
innermost runtime complexity
Answer:
YES(O(1),O(1))
Empty rules are trivially bounded
Wall-time: 0.109137s
CPU-time: 0.784s
Wall-time: 2.9685s
CPU-time: 21.298s
Hurray, we answered YES(O(1),O(n^2))