We are left with following problem, upon which TcT provides the
certificate YES(O(1),O(n^1)).
Strict Trs:
{ a(Z(), y, z) -> Z()
, a(C(x1, x2), y, z) -> C(a(x1, y, z), a(x2, y, y))
, second(C(x1, x2)) -> x2
, eqZList(Z(), Z()) -> True()
, eqZList(Z(), C(y1, y2)) -> False()
, eqZList(C(x1, x2), Z()) -> False()
, eqZList(C(x1, x2), C(y1, y2)) ->
and(eqZList(x1, y1), eqZList(x2, y2))
, first(C(x1, x2)) -> x1 }
Weak Trs:
{ and(True(), True()) -> True()
, and(True(), False()) -> False()
, and(False(), True()) -> False()
, and(False(), False()) -> False() }
Obligation:
innermost runtime complexity
Answer:
YES(O(1),O(n^1))
We add the following weak dependency pairs:
Strict DPs:
{ a^#(Z(), y, z) -> c_1()
, a^#(C(x1, x2), y, z) -> c_2(a^#(x1, y, z), a^#(x2, y, y))
, second^#(C(x1, x2)) -> c_3()
, eqZList^#(Z(), Z()) -> c_4()
, eqZList^#(Z(), C(y1, y2)) -> c_5()
, eqZList^#(C(x1, x2), Z()) -> c_6()
, eqZList^#(C(x1, x2), C(y1, y2)) ->
c_7(and^#(eqZList(x1, y1), eqZList(x2, y2)))
, first^#(C(x1, x2)) -> c_8() }
Weak DPs:
{ and^#(True(), True()) -> c_9()
, and^#(True(), False()) -> c_10()
, and^#(False(), True()) -> c_11()
, and^#(False(), False()) -> c_12() }
and mark the set of starting terms.
We are left with following problem, upon which TcT provides the
certificate YES(O(1),O(n^1)).
Strict DPs:
{ a^#(Z(), y, z) -> c_1()
, a^#(C(x1, x2), y, z) -> c_2(a^#(x1, y, z), a^#(x2, y, y))
, second^#(C(x1, x2)) -> c_3()
, eqZList^#(Z(), Z()) -> c_4()
, eqZList^#(Z(), C(y1, y2)) -> c_5()
, eqZList^#(C(x1, x2), Z()) -> c_6()
, eqZList^#(C(x1, x2), C(y1, y2)) ->
c_7(and^#(eqZList(x1, y1), eqZList(x2, y2)))
, first^#(C(x1, x2)) -> c_8() }
Strict Trs:
{ a(Z(), y, z) -> Z()
, a(C(x1, x2), y, z) -> C(a(x1, y, z), a(x2, y, y))
, second(C(x1, x2)) -> x2
, eqZList(Z(), Z()) -> True()
, eqZList(Z(), C(y1, y2)) -> False()
, eqZList(C(x1, x2), Z()) -> False()
, eqZList(C(x1, x2), C(y1, y2)) ->
and(eqZList(x1, y1), eqZList(x2, y2))
, first(C(x1, x2)) -> x1 }
Weak DPs:
{ and^#(True(), True()) -> c_9()
, and^#(True(), False()) -> c_10()
, and^#(False(), True()) -> c_11()
, and^#(False(), False()) -> c_12() }
Weak Trs:
{ and(True(), True()) -> True()
, and(True(), False()) -> False()
, and(False(), True()) -> False()
, and(False(), False()) -> False() }
Obligation:
innermost runtime complexity
Answer:
YES(O(1),O(n^1))
We replace rewrite rules by usable rules:
Strict Usable Rules:
{ eqZList(Z(), Z()) -> True()
, eqZList(Z(), C(y1, y2)) -> False()
, eqZList(C(x1, x2), Z()) -> False()
, eqZList(C(x1, x2), C(y1, y2)) ->
and(eqZList(x1, y1), eqZList(x2, y2)) }
Weak Usable Rules:
{ and(True(), True()) -> True()
, and(True(), False()) -> False()
, and(False(), True()) -> False()
, and(False(), False()) -> False() }
We are left with following problem, upon which TcT provides the
certificate YES(O(1),O(n^1)).
Strict DPs:
{ a^#(Z(), y, z) -> c_1()
, a^#(C(x1, x2), y, z) -> c_2(a^#(x1, y, z), a^#(x2, y, y))
, second^#(C(x1, x2)) -> c_3()
, eqZList^#(Z(), Z()) -> c_4()
, eqZList^#(Z(), C(y1, y2)) -> c_5()
, eqZList^#(C(x1, x2), Z()) -> c_6()
, eqZList^#(C(x1, x2), C(y1, y2)) ->
c_7(and^#(eqZList(x1, y1), eqZList(x2, y2)))
, first^#(C(x1, x2)) -> c_8() }
Strict Trs:
{ eqZList(Z(), Z()) -> True()
, eqZList(Z(), C(y1, y2)) -> False()
, eqZList(C(x1, x2), Z()) -> False()
, eqZList(C(x1, x2), C(y1, y2)) ->
and(eqZList(x1, y1), eqZList(x2, y2)) }
Weak DPs:
{ and^#(True(), True()) -> c_9()
, and^#(True(), False()) -> c_10()
, and^#(False(), True()) -> c_11()
, and^#(False(), False()) -> c_12() }
Weak Trs:
{ and(True(), True()) -> True()
, and(True(), False()) -> False()
, and(False(), True()) -> False()
, and(False(), False()) -> False() }
Obligation:
innermost runtime complexity
Answer:
YES(O(1),O(n^1))
The weightgap principle applies (using the following constant
growth matrix-interpretation)
The following argument positions are usable:
Uargs(and) = {1, 2}, Uargs(c_2) = {1, 2}, Uargs(c_7) = {1},
Uargs(and^#) = {1, 2}
TcT has computed the following constructor-restricted matrix
interpretation.
[eqZList](x1, x2) = [1 1] x2 + [1]
[0 0] [1]
[Z] = [0]
[0]
[True] = [0]
[0]
[C](x1, x2) = [1 0] x1 + [1 1] x2 + [2]
[0 1] [0 0] [2]
[and](x1, x2) = [1 1] x1 + [1 0] x2 + [1]
[0 0] [0 0] [0]
[False] = [0]
[0]
[a^#](x1, x2, x3) = [0 0] x2 + [0 0] x3 + [0]
[1 1] [1 1] [0]
[c_1] = [0]
[0]
[c_2](x1, x2) = [1 0] x1 + [1 0] x2 + [2]
[0 1] [0 1] [2]
[second^#](x1) = [0]
[0]
[c_3] = [0]
[0]
[eqZList^#](x1, x2) = [2 1] x2 + [0]
[0 0] [0]
[c_4] = [0]
[0]
[c_5] = [0]
[0]
[c_6] = [0]
[0]
[c_7](x1) = [1 0] x1 + [0]
[0 1] [0]
[and^#](x1, x2) = [1 2] x1 + [2 0] x2 + [0]
[0 0] [0 0] [0]
[first^#](x1) = [0]
[0]
[c_8] = [0]
[0]
[c_9] = [0]
[0]
[c_10] = [0]
[0]
[c_11] = [0]
[0]
[c_12] = [0]
[0]
The order satisfies the following ordering constraints:
[eqZList(Z(), Z())] = [1]
[1]
> [0]
[0]
= [True()]
[eqZList(Z(), C(y1, y2))] = [1 1] y1 + [1 1] y2 + [5]
[0 0] [0 0] [1]
> [0]
[0]
= [False()]
[eqZList(C(x1, x2), Z())] = [1]
[1]
> [0]
[0]
= [False()]
[eqZList(C(x1, x2), C(y1, y2))] = [1 1] y1 + [1 1] y2 + [5]
[0 0] [0 0] [1]
> [1 1] y1 + [1 1] y2 + [4]
[0 0] [0 0] [0]
= [and(eqZList(x1, y1), eqZList(x2, y2))]
[and(True(), True())] = [1]
[0]
> [0]
[0]
= [True()]
[and(True(), False())] = [1]
[0]
> [0]
[0]
= [False()]
[and(False(), True())] = [1]
[0]
> [0]
[0]
= [False()]
[and(False(), False())] = [1]
[0]
> [0]
[0]
= [False()]
[a^#(Z(), y, z)] = [0 0] y + [0 0] z + [0]
[1 1] [1 1] [0]
>= [0]
[0]
= [c_1()]
[a^#(C(x1, x2), y, z)] = [0 0] y + [0 0] z + [0]
[1 1] [1 1] [0]
? [0 0] y + [0 0] z + [2]
[3 3] [1 1] [2]
= [c_2(a^#(x1, y, z), a^#(x2, y, y))]
[second^#(C(x1, x2))] = [0]
[0]
>= [0]
[0]
= [c_3()]
[eqZList^#(Z(), Z())] = [0]
[0]
>= [0]
[0]
= [c_4()]
[eqZList^#(Z(), C(y1, y2))] = [2 1] y1 + [2 2] y2 + [6]
[0 0] [0 0] [0]
> [0]
[0]
= [c_5()]
[eqZList^#(C(x1, x2), Z())] = [0]
[0]
>= [0]
[0]
= [c_6()]
[eqZList^#(C(x1, x2), C(y1, y2))] = [2 1] y1 + [2 2] y2 + [6]
[0 0] [0 0] [0]
> [1 1] y1 + [2 2] y2 + [5]
[0 0] [0 0] [0]
= [c_7(and^#(eqZList(x1, y1), eqZList(x2, y2)))]
[and^#(True(), True())] = [0]
[0]
>= [0]
[0]
= [c_9()]
[and^#(True(), False())] = [0]
[0]
>= [0]
[0]
= [c_10()]
[and^#(False(), True())] = [0]
[0]
>= [0]
[0]
= [c_11()]
[and^#(False(), False())] = [0]
[0]
>= [0]
[0]
= [c_12()]
[first^#(C(x1, x2))] = [0]
[0]
>= [0]
[0]
= [c_8()]
Further, it can be verified that all rules not oriented are covered by the weightgap condition.
We are left with following problem, upon which TcT provides the
certificate YES(O(1),O(n^1)).
Strict DPs:
{ a^#(Z(), y, z) -> c_1()
, a^#(C(x1, x2), y, z) -> c_2(a^#(x1, y, z), a^#(x2, y, y))
, second^#(C(x1, x2)) -> c_3()
, eqZList^#(Z(), Z()) -> c_4()
, eqZList^#(C(x1, x2), Z()) -> c_6()
, first^#(C(x1, x2)) -> c_8() }
Weak DPs:
{ eqZList^#(Z(), C(y1, y2)) -> c_5()
, eqZList^#(C(x1, x2), C(y1, y2)) ->
c_7(and^#(eqZList(x1, y1), eqZList(x2, y2)))
, and^#(True(), True()) -> c_9()
, and^#(True(), False()) -> c_10()
, and^#(False(), True()) -> c_11()
, and^#(False(), False()) -> c_12() }
Weak Trs:
{ eqZList(Z(), Z()) -> True()
, eqZList(Z(), C(y1, y2)) -> False()
, eqZList(C(x1, x2), Z()) -> False()
, eqZList(C(x1, x2), C(y1, y2)) ->
and(eqZList(x1, y1), eqZList(x2, y2))
, and(True(), True()) -> True()
, and(True(), False()) -> False()
, and(False(), True()) -> False()
, and(False(), False()) -> False() }
Obligation:
innermost runtime complexity
Answer:
YES(O(1),O(n^1))
We estimate the number of application of {1,3,4,5,6} by
applications of Pre({1,3,4,5,6}) = {2}. Here rules are labeled as
follows:
DPs:
{ 1: a^#(Z(), y, z) -> c_1()
, 2: a^#(C(x1, x2), y, z) -> c_2(a^#(x1, y, z), a^#(x2, y, y))
, 3: second^#(C(x1, x2)) -> c_3()
, 4: eqZList^#(Z(), Z()) -> c_4()
, 5: eqZList^#(C(x1, x2), Z()) -> c_6()
, 6: first^#(C(x1, x2)) -> c_8()
, 7: eqZList^#(Z(), C(y1, y2)) -> c_5()
, 8: eqZList^#(C(x1, x2), C(y1, y2)) ->
c_7(and^#(eqZList(x1, y1), eqZList(x2, y2)))
, 9: and^#(True(), True()) -> c_9()
, 10: and^#(True(), False()) -> c_10()
, 11: and^#(False(), True()) -> c_11()
, 12: and^#(False(), False()) -> c_12() }
We are left with following problem, upon which TcT provides the
certificate YES(O(1),O(n^1)).
Strict DPs:
{ a^#(C(x1, x2), y, z) -> c_2(a^#(x1, y, z), a^#(x2, y, y)) }
Weak DPs:
{ a^#(Z(), y, z) -> c_1()
, second^#(C(x1, x2)) -> c_3()
, eqZList^#(Z(), Z()) -> c_4()
, eqZList^#(Z(), C(y1, y2)) -> c_5()
, eqZList^#(C(x1, x2), Z()) -> c_6()
, eqZList^#(C(x1, x2), C(y1, y2)) ->
c_7(and^#(eqZList(x1, y1), eqZList(x2, y2)))
, and^#(True(), True()) -> c_9()
, and^#(True(), False()) -> c_10()
, and^#(False(), True()) -> c_11()
, and^#(False(), False()) -> c_12()
, first^#(C(x1, x2)) -> c_8() }
Weak Trs:
{ eqZList(Z(), Z()) -> True()
, eqZList(Z(), C(y1, y2)) -> False()
, eqZList(C(x1, x2), Z()) -> False()
, eqZList(C(x1, x2), C(y1, y2)) ->
and(eqZList(x1, y1), eqZList(x2, y2))
, and(True(), True()) -> True()
, and(True(), False()) -> False()
, and(False(), True()) -> False()
, and(False(), False()) -> False() }
Obligation:
innermost runtime complexity
Answer:
YES(O(1),O(n^1))
The following weak DPs constitute a sub-graph of the DG that is
closed under successors. The DPs are removed.
{ a^#(Z(), y, z) -> c_1()
, second^#(C(x1, x2)) -> c_3()
, eqZList^#(Z(), Z()) -> c_4()
, eqZList^#(Z(), C(y1, y2)) -> c_5()
, eqZList^#(C(x1, x2), Z()) -> c_6()
, eqZList^#(C(x1, x2), C(y1, y2)) ->
c_7(and^#(eqZList(x1, y1), eqZList(x2, y2)))
, and^#(True(), True()) -> c_9()
, and^#(True(), False()) -> c_10()
, and^#(False(), True()) -> c_11()
, and^#(False(), False()) -> c_12()
, first^#(C(x1, x2)) -> c_8() }
We are left with following problem, upon which TcT provides the
certificate YES(O(1),O(n^1)).
Strict DPs:
{ a^#(C(x1, x2), y, z) -> c_2(a^#(x1, y, z), a^#(x2, y, y)) }
Weak Trs:
{ eqZList(Z(), Z()) -> True()
, eqZList(Z(), C(y1, y2)) -> False()
, eqZList(C(x1, x2), Z()) -> False()
, eqZList(C(x1, x2), C(y1, y2)) ->
and(eqZList(x1, y1), eqZList(x2, y2))
, and(True(), True()) -> True()
, and(True(), False()) -> False()
, and(False(), True()) -> False()
, and(False(), False()) -> False() }
Obligation:
innermost runtime complexity
Answer:
YES(O(1),O(n^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:
{ a^#(C(x1, x2), y, z) -> c_2(a^#(x1, y, z), a^#(x2, y, y)) }
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: a^#(C(x1, x2), y, z) -> c_2(a^#(x1, y, z), a^#(x2, y, y)) }
Sub-proof:
----------
The following argument positions are usable:
Uargs(c_2) = {1, 2}
TcT has computed the following constructor-based matrix
interpretation satisfying not(EDA).
[C](x1, x2) = [1] x1 + [1] x2 + [2]
[a^#](x1, x2, x3) = [4] x1 + [7]
[c_2](x1, x2) = [1] x1 + [1] x2 + [0]
The order satisfies the following ordering constraints:
[a^#(C(x1, x2), y, z)] = [4] x1 + [4] x2 + [15]
> [4] x1 + [4] x2 + [14]
= [c_2(a^#(x1, y, z), a^#(x2, y, y))]
The strictly oriented rules are moved into the weak component.
We are left with following problem, upon which TcT provides the
certificate YES(O(1),O(1)).
Weak DPs:
{ a^#(C(x1, x2), y, z) -> c_2(a^#(x1, y, z), a^#(x2, y, y)) }
Obligation:
innermost runtime complexity
Answer:
YES(O(1),O(1))
The following weak DPs constitute a sub-graph of the DG that is
closed under successors. The DPs are removed.
{ a^#(C(x1, x2), y, z) -> c_2(a^#(x1, y, z), a^#(x2, y, y)) }
We are left with following problem, upon which TcT provides the
certificate YES(O(1),O(1)).
Rules: Empty
Obligation:
innermost runtime complexity
Answer:
YES(O(1),O(1))
Empty rules are trivially bounded
Hurray, we answered YES(O(1),O(n^1))