We are left with following problem, upon which TcT provides the
certificate YES(O(1),O(n^1)).
Strict Trs:
{ or(x, true()) -> true()
, or(true(), y) -> true()
, or(false(), false()) -> false()
, mem(x, nil()) -> false()
, mem(x, set(y)) -> =(x, y)
, mem(x, union(y, z)) -> or(mem(x, y), mem(x, z)) }
Obligation:
runtime complexity
Answer:
YES(O(1),O(n^1))
We add the following weak dependency pairs:
Strict DPs:
{ or^#(x, true()) -> c_1()
, or^#(true(), y) -> c_2()
, or^#(false(), false()) -> c_3()
, mem^#(x, nil()) -> c_4()
, mem^#(x, set(y)) -> c_5(x, y)
, mem^#(x, union(y, z)) -> c_6(or^#(mem(x, y), mem(x, z))) }
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:
{ or^#(x, true()) -> c_1()
, or^#(true(), y) -> c_2()
, or^#(false(), false()) -> c_3()
, mem^#(x, nil()) -> c_4()
, mem^#(x, set(y)) -> c_5(x, y)
, mem^#(x, union(y, z)) -> c_6(or^#(mem(x, y), mem(x, z))) }
Strict Trs:
{ or(x, true()) -> true()
, or(true(), y) -> true()
, or(false(), false()) -> false()
, mem(x, nil()) -> false()
, mem(x, set(y)) -> =(x, y)
, mem(x, union(y, z)) -> or(mem(x, y), mem(x, z)) }
Obligation:
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(or) = {1, 2}, Uargs(or^#) = {1, 2}, Uargs(c_6) = {1}
TcT has computed the following constructor-restricted matrix
interpretation.
[or](x1, x2) = [1 0] x1 + [1 0] x2 + [2]
[0 0] [0 0] [1]
[true] = [2]
[0]
[false] = [1]
[1]
[mem](x1, x2) = [1 1] x2 + [1]
[0 0] [1]
[nil] = [2]
[1]
[set](x1) = [0]
[0]
[=](x1, x2) = [0]
[0]
[union](x1, x2) = [1 0] x1 + [1 0] x2 + [2]
[0 1] [0 1] [2]
[or^#](x1, x2) = [1 0] x1 + [1 1] x2 + [0]
[0 0] [0 0] [0]
[c_1] = [0]
[0]
[c_2] = [0]
[0]
[c_3] = [0]
[0]
[mem^#](x1, x2) = [1 1] x1 + [2 1] x2 + [0]
[0 0] [0 0] [0]
[c_4] = [0]
[0]
[c_5](x1, x2) = [0]
[0]
[c_6](x1) = [1 0] x1 + [0]
[0 1] [0]
The order satisfies the following ordering constraints:
[or(x, true())] = [1 0] x + [4]
[0 0] [1]
> [2]
[0]
= [true()]
[or(true(), y)] = [1 0] y + [4]
[0 0] [1]
> [2]
[0]
= [true()]
[or(false(), false())] = [4]
[1]
> [1]
[1]
= [false()]
[mem(x, nil())] = [4]
[1]
> [1]
[1]
= [false()]
[mem(x, set(y))] = [1]
[1]
> [0]
[0]
= [=(x, y)]
[mem(x, union(y, z))] = [1 1] y + [1 1] z + [5]
[0 0] [0 0] [1]
> [1 1] y + [1 1] z + [4]
[0 0] [0 0] [1]
= [or(mem(x, y), mem(x, z))]
[or^#(x, true())] = [1 0] x + [2]
[0 0] [0]
> [0]
[0]
= [c_1()]
[or^#(true(), y)] = [1 1] y + [2]
[0 0] [0]
> [0]
[0]
= [c_2()]
[or^#(false(), false())] = [3]
[0]
> [0]
[0]
= [c_3()]
[mem^#(x, nil())] = [1 1] x + [5]
[0 0] [0]
> [0]
[0]
= [c_4()]
[mem^#(x, set(y))] = [1 1] x + [0]
[0 0] [0]
>= [0]
[0]
= [c_5(x, y)]
[mem^#(x, union(y, z))] = [2 1] y + [1 1] x + [2 1] z + [6]
[0 0] [0 0] [0 0] [0]
> [1 1] y + [1 1] z + [3]
[0 0] [0 0] [0]
= [c_6(or^#(mem(x, y), mem(x, z)))]
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: { mem^#(x, set(y)) -> c_5(x, y) }
Weak DPs:
{ or^#(x, true()) -> c_1()
, or^#(true(), y) -> c_2()
, or^#(false(), false()) -> c_3()
, mem^#(x, nil()) -> c_4()
, mem^#(x, union(y, z)) -> c_6(or^#(mem(x, y), mem(x, z))) }
Weak Trs:
{ or(x, true()) -> true()
, or(true(), y) -> true()
, or(false(), false()) -> false()
, mem(x, nil()) -> false()
, mem(x, set(y)) -> =(x, y)
, mem(x, union(y, z)) -> or(mem(x, y), mem(x, z)) }
Obligation:
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.
{ or^#(x, true()) -> c_1()
, or^#(true(), y) -> c_2()
, or^#(false(), false()) -> c_3()
, mem^#(x, nil()) -> c_4()
, mem^#(x, union(y, z)) -> c_6(or^#(mem(x, y), mem(x, z))) }
We are left with following problem, upon which TcT provides the
certificate YES(O(1),O(n^1)).
Strict DPs: { mem^#(x, set(y)) -> c_5(x, y) }
Weak Trs:
{ or(x, true()) -> true()
, or(true(), y) -> true()
, or(false(), false()) -> false()
, mem(x, nil()) -> false()
, mem(x, set(y)) -> =(x, y)
, mem(x, union(y, z)) -> or(mem(x, y), mem(x, z)) }
Obligation:
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: { mem^#(x, set(y)) -> c_5(x, y) }
Obligation:
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: mem^#(x, set(y)) -> c_5(x, y) }
Sub-proof:
----------
The following argument positions are usable:
none
TcT has computed the following constructor-based matrix
interpretation satisfying not(EDA).
[set](x1) = [1]
[mem^#](x1, x2) = [7] x1 + [1] x2 + [4]
[c_5](x1, x2) = [1] x1 + [1]
The order satisfies the following ordering constraints:
[mem^#(x, set(y))] = [7] x + [5]
> [1] x + [1]
= [c_5(x, 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: { mem^#(x, set(y)) -> c_5(x, y) }
Obligation:
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.
{ mem^#(x, set(y)) -> c_5(x, y) }
We are left with following problem, upon which TcT provides the
certificate YES(O(1),O(1)).
Rules: Empty
Obligation:
runtime complexity
Answer:
YES(O(1),O(1))
Empty rules are trivially bounded
Hurray, we answered YES(O(1),O(n^1))