We are left with following problem, upon which TcT provides the
certificate YES(O(1),O(n^1)).
Strict Trs:
{ 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))
, f(Z()) -> Z()
, f(C(x1, x2)) -> C(f(x1), f(x2))
, first(C(x1, x2)) -> x1
, g(x) -> x }
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:
{ second^#(C(x1, x2)) -> c_1()
, eqZList^#(Z(), Z()) -> c_2()
, eqZList^#(Z(), C(y1, y2)) -> c_3()
, eqZList^#(C(x1, x2), Z()) -> c_4()
, eqZList^#(C(x1, x2), C(y1, y2)) ->
c_5(and^#(eqZList(x1, y1), eqZList(x2, y2)))
, f^#(Z()) -> c_6()
, f^#(C(x1, x2)) -> c_7(f^#(x1), f^#(x2))
, first^#(C(x1, x2)) -> c_8()
, g^#(x) -> c_9() }
Weak DPs:
{ and^#(True(), True()) -> c_10()
, and^#(True(), False()) -> c_11()
, and^#(False(), True()) -> c_12()
, and^#(False(), False()) -> c_13() }
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:
{ second^#(C(x1, x2)) -> c_1()
, eqZList^#(Z(), Z()) -> c_2()
, eqZList^#(Z(), C(y1, y2)) -> c_3()
, eqZList^#(C(x1, x2), Z()) -> c_4()
, eqZList^#(C(x1, x2), C(y1, y2)) ->
c_5(and^#(eqZList(x1, y1), eqZList(x2, y2)))
, f^#(Z()) -> c_6()
, f^#(C(x1, x2)) -> c_7(f^#(x1), f^#(x2))
, first^#(C(x1, x2)) -> c_8()
, g^#(x) -> c_9() }
Strict Trs:
{ 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))
, f(Z()) -> Z()
, f(C(x1, x2)) -> C(f(x1), f(x2))
, first(C(x1, x2)) -> x1
, g(x) -> x }
Weak DPs:
{ and^#(True(), True()) -> c_10()
, and^#(True(), False()) -> c_11()
, and^#(False(), True()) -> c_12()
, and^#(False(), False()) -> c_13() }
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:
{ second^#(C(x1, x2)) -> c_1()
, eqZList^#(Z(), Z()) -> c_2()
, eqZList^#(Z(), C(y1, y2)) -> c_3()
, eqZList^#(C(x1, x2), Z()) -> c_4()
, eqZList^#(C(x1, x2), C(y1, y2)) ->
c_5(and^#(eqZList(x1, y1), eqZList(x2, y2)))
, f^#(Z()) -> c_6()
, f^#(C(x1, x2)) -> c_7(f^#(x1), f^#(x2))
, first^#(C(x1, x2)) -> c_8()
, g^#(x) -> c_9() }
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_10()
, and^#(True(), False()) -> c_11()
, and^#(False(), True()) -> c_12()
, and^#(False(), False()) -> c_13() }
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_5) = {1}, Uargs(and^#) = {1, 2},
Uargs(c_7) = {1, 2}
TcT has computed the following constructor-restricted matrix
interpretation.
[eqZList](x1, x2) = [0 1] x2 + [0]
[0 0] [0]
[Z] = [0]
[1]
[True] = [0]
[0]
[C](x1, x2) = [1 0] x1 + [1 0] x2 + [1]
[0 1] [0 1] [1]
[and](x1, x2) = [1 0] x1 + [1 0] x2 + [0]
[0 0] [0 0] [0]
[False] = [0]
[0]
[second^#](x1) = [0]
[0]
[c_1] = [0]
[0]
[eqZList^#](x1, x2) = [0 1] x2 + [0]
[0 0] [0]
[c_2] = [0]
[0]
[c_3] = [0]
[0]
[c_4] = [0]
[0]
[c_5](x1) = [1 0] x1 + [0]
[0 1] [0]
[and^#](x1, x2) = [1 0] x1 + [1 1] x2 + [0]
[0 0] [0 0] [0]
[f^#](x1) = [0]
[0]
[c_6] = [0]
[0]
[c_7](x1, x2) = [1 0] x1 + [1 0] x2 + [2]
[0 1] [0 1] [2]
[first^#](x1) = [0]
[0]
[c_8] = [0]
[0]
[g^#](x1) = [0]
[0]
[c_9] = [0]
[0]
[c_10] = [0]
[0]
[c_11] = [0]
[0]
[c_12] = [0]
[0]
[c_13] = [0]
[0]
The order satisfies the following ordering constraints:
[eqZList(Z(), Z())] = [1]
[0]
> [0]
[0]
= [True()]
[eqZList(Z(), C(y1, y2))] = [0 1] y1 + [0 1] y2 + [1]
[0 0] [0 0] [0]
> [0]
[0]
= [False()]
[eqZList(C(x1, x2), Z())] = [1]
[0]
> [0]
[0]
= [False()]
[eqZList(C(x1, x2), C(y1, y2))] = [0 1] y1 + [0 1] y2 + [1]
[0 0] [0 0] [0]
> [0 1] y1 + [0 1] y2 + [0]
[0 0] [0 0] [0]
= [and(eqZList(x1, y1), eqZList(x2, y2))]
[and(True(), True())] = [0]
[0]
>= [0]
[0]
= [True()]
[and(True(), False())] = [0]
[0]
>= [0]
[0]
= [False()]
[and(False(), True())] = [0]
[0]
>= [0]
[0]
= [False()]
[and(False(), False())] = [0]
[0]
>= [0]
[0]
= [False()]
[second^#(C(x1, x2))] = [0]
[0]
>= [0]
[0]
= [c_1()]
[eqZList^#(Z(), Z())] = [1]
[0]
> [0]
[0]
= [c_2()]
[eqZList^#(Z(), C(y1, y2))] = [0 1] y1 + [0 1] y2 + [1]
[0 0] [0 0] [0]
> [0]
[0]
= [c_3()]
[eqZList^#(C(x1, x2), Z())] = [1]
[0]
> [0]
[0]
= [c_4()]
[eqZList^#(C(x1, x2), C(y1, y2))] = [0 1] y1 + [0 1] y2 + [1]
[0 0] [0 0] [0]
> [0 1] y1 + [0 1] y2 + [0]
[0 0] [0 0] [0]
= [c_5(and^#(eqZList(x1, y1), eqZList(x2, y2)))]
[and^#(True(), True())] = [0]
[0]
>= [0]
[0]
= [c_10()]
[and^#(True(), False())] = [0]
[0]
>= [0]
[0]
= [c_11()]
[and^#(False(), True())] = [0]
[0]
>= [0]
[0]
= [c_12()]
[and^#(False(), False())] = [0]
[0]
>= [0]
[0]
= [c_13()]
[f^#(Z())] = [0]
[0]
>= [0]
[0]
= [c_6()]
[f^#(C(x1, x2))] = [0]
[0]
? [2]
[2]
= [c_7(f^#(x1), f^#(x2))]
[first^#(C(x1, x2))] = [0]
[0]
>= [0]
[0]
= [c_8()]
[g^#(x)] = [0]
[0]
>= [0]
[0]
= [c_9()]
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:
{ second^#(C(x1, x2)) -> c_1()
, f^#(Z()) -> c_6()
, f^#(C(x1, x2)) -> c_7(f^#(x1), f^#(x2))
, first^#(C(x1, x2)) -> c_8()
, g^#(x) -> c_9() }
Weak DPs:
{ eqZList^#(Z(), Z()) -> c_2()
, eqZList^#(Z(), C(y1, y2)) -> c_3()
, eqZList^#(C(x1, x2), Z()) -> c_4()
, eqZList^#(C(x1, x2), C(y1, y2)) ->
c_5(and^#(eqZList(x1, y1), eqZList(x2, y2)))
, and^#(True(), True()) -> c_10()
, and^#(True(), False()) -> c_11()
, and^#(False(), True()) -> c_12()
, and^#(False(), False()) -> c_13() }
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 {1,2,4,5} by applications
of Pre({1,2,4,5}) = {3}. Here rules are labeled as follows:
DPs:
{ 1: second^#(C(x1, x2)) -> c_1()
, 2: f^#(Z()) -> c_6()
, 3: f^#(C(x1, x2)) -> c_7(f^#(x1), f^#(x2))
, 4: first^#(C(x1, x2)) -> c_8()
, 5: g^#(x) -> c_9()
, 6: eqZList^#(Z(), Z()) -> c_2()
, 7: eqZList^#(Z(), C(y1, y2)) -> c_3()
, 8: eqZList^#(C(x1, x2), Z()) -> c_4()
, 9: eqZList^#(C(x1, x2), C(y1, y2)) ->
c_5(and^#(eqZList(x1, y1), eqZList(x2, y2)))
, 10: and^#(True(), True()) -> c_10()
, 11: and^#(True(), False()) -> c_11()
, 12: and^#(False(), True()) -> c_12()
, 13: and^#(False(), False()) -> c_13() }
We are left with following problem, upon which TcT provides the
certificate YES(O(1),O(n^1)).
Strict DPs: { f^#(C(x1, x2)) -> c_7(f^#(x1), f^#(x2)) }
Weak DPs:
{ second^#(C(x1, x2)) -> c_1()
, eqZList^#(Z(), Z()) -> c_2()
, eqZList^#(Z(), C(y1, y2)) -> c_3()
, eqZList^#(C(x1, x2), Z()) -> c_4()
, eqZList^#(C(x1, x2), C(y1, y2)) ->
c_5(and^#(eqZList(x1, y1), eqZList(x2, y2)))
, and^#(True(), True()) -> c_10()
, and^#(True(), False()) -> c_11()
, and^#(False(), True()) -> c_12()
, and^#(False(), False()) -> c_13()
, f^#(Z()) -> c_6()
, first^#(C(x1, x2)) -> c_8()
, g^#(x) -> c_9() }
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.
{ second^#(C(x1, x2)) -> c_1()
, eqZList^#(Z(), Z()) -> c_2()
, eqZList^#(Z(), C(y1, y2)) -> c_3()
, eqZList^#(C(x1, x2), Z()) -> c_4()
, eqZList^#(C(x1, x2), C(y1, y2)) ->
c_5(and^#(eqZList(x1, y1), eqZList(x2, y2)))
, and^#(True(), True()) -> c_10()
, and^#(True(), False()) -> c_11()
, and^#(False(), True()) -> c_12()
, and^#(False(), False()) -> c_13()
, f^#(Z()) -> c_6()
, first^#(C(x1, x2)) -> c_8()
, g^#(x) -> c_9() }
We are left with following problem, upon which TcT provides the
certificate YES(O(1),O(n^1)).
Strict DPs: { f^#(C(x1, x2)) -> c_7(f^#(x1), f^#(x2)) }
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: { f^#(C(x1, x2)) -> c_7(f^#(x1), f^#(x2)) }
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: f^#(C(x1, x2)) -> c_7(f^#(x1), f^#(x2)) }
Sub-proof:
----------
The following argument positions are usable:
Uargs(c_7) = {1, 2}
TcT has computed the following constructor-based matrix
interpretation satisfying not(EDA).
[C](x1, x2) = [1] x1 + [1] x2 + [4]
[f^#](x1) = [2] x1 + [0]
[c_7](x1, x2) = [1] x1 + [1] x2 + [1]
The order satisfies the following ordering constraints:
[f^#(C(x1, x2))] = [2] x1 + [2] x2 + [8]
> [2] x1 + [2] x2 + [1]
= [c_7(f^#(x1), f^#(x2))]
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: { f^#(C(x1, x2)) -> c_7(f^#(x1), f^#(x2)) }
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.
{ f^#(C(x1, x2)) -> c_7(f^#(x1), f^#(x2)) }
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))