We are left with following problem, upon which TcT provides the
certificate YES(O(1),O(n^1)).
Strict Trs:
{ del(.(x, .(y, z))) -> f(=(x, y), x, y, z)
, f(true(), x, y, z) -> del(.(y, z))
, f(false(), x, y, z) -> .(x, del(.(y, z)))
, =(.(x, y), .(u(), v())) -> and(=(x, u()), =(y, v()))
, =(.(x, y), nil()) -> false()
, =(nil(), .(y, z)) -> false()
, =(nil(), nil()) -> true() }
Obligation:
innermost runtime complexity
Answer:
YES(O(1),O(n^1))
We add the following weak dependency pairs:
Strict DPs:
{ del^#(.(x, .(y, z))) -> c_1(f^#(=(x, y), x, y, z))
, f^#(true(), x, y, z) -> c_2(del^#(.(y, z)))
, f^#(false(), x, y, z) -> c_3(del^#(.(y, z)))
, =^#(.(x, y), .(u(), v())) -> c_4(=^#(x, u()), =^#(y, v()))
, =^#(.(x, y), nil()) -> c_5()
, =^#(nil(), .(y, z)) -> c_6()
, =^#(nil(), nil()) -> c_7() }
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:
{ del^#(.(x, .(y, z))) -> c_1(f^#(=(x, y), x, y, z))
, f^#(true(), x, y, z) -> c_2(del^#(.(y, z)))
, f^#(false(), x, y, z) -> c_3(del^#(.(y, z)))
, =^#(.(x, y), .(u(), v())) -> c_4(=^#(x, u()), =^#(y, v()))
, =^#(.(x, y), nil()) -> c_5()
, =^#(nil(), .(y, z)) -> c_6()
, =^#(nil(), nil()) -> c_7() }
Strict Trs:
{ del(.(x, .(y, z))) -> f(=(x, y), x, y, z)
, f(true(), x, y, z) -> del(.(y, z))
, f(false(), x, y, z) -> .(x, del(.(y, z)))
, =(.(x, y), .(u(), v())) -> and(=(x, u()), =(y, v()))
, =(.(x, y), nil()) -> false()
, =(nil(), .(y, z)) -> false()
, =(nil(), nil()) -> true() }
Obligation:
innermost runtime complexity
Answer:
YES(O(1),O(n^1))
We replace rewrite rules by usable rules:
Strict Usable Rules:
{ =(.(x, y), .(u(), v())) -> and(=(x, u()), =(y, v()))
, =(.(x, y), nil()) -> false()
, =(nil(), .(y, z)) -> false()
, =(nil(), nil()) -> true() }
We are left with following problem, upon which TcT provides the
certificate YES(O(1),O(n^1)).
Strict DPs:
{ del^#(.(x, .(y, z))) -> c_1(f^#(=(x, y), x, y, z))
, f^#(true(), x, y, z) -> c_2(del^#(.(y, z)))
, f^#(false(), x, y, z) -> c_3(del^#(.(y, z)))
, =^#(.(x, y), .(u(), v())) -> c_4(=^#(x, u()), =^#(y, v()))
, =^#(.(x, y), nil()) -> c_5()
, =^#(nil(), .(y, z)) -> c_6()
, =^#(nil(), nil()) -> c_7() }
Strict Trs:
{ =(.(x, y), .(u(), v())) -> and(=(x, u()), =(y, v()))
, =(.(x, y), nil()) -> false()
, =(nil(), .(y, z)) -> false()
, =(nil(), nil()) -> true() }
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(c_1) = {1}, Uargs(f^#) = {1}, Uargs(c_2) = {1},
Uargs(c_3) = {1}
TcT has computed the following constructor-restricted matrix
interpretation.
[.](x1, x2) = [0]
[0]
[=](x1, x2) = [1]
[0]
[true] = [0]
[0]
[false] = [0]
[0]
[nil] = [0]
[0]
[u] = [0]
[0]
[v] = [0]
[0]
[and](x1, x2) = [0]
[0]
[del^#](x1) = [0]
[0]
[c_1](x1) = [1 0] x1 + [0]
[0 1] [0]
[f^#](x1, x2, x3, x4) = [2 0] x1 + [0]
[0 0] [0]
[c_2](x1) = [1 0] x1 + [0]
[0 1] [0]
[c_3](x1) = [1 0] x1 + [0]
[0 1] [0]
[=^#](x1, x2) = [0]
[0]
[c_4](x1, x2) = [0]
[0]
[c_5] = [0]
[0]
[c_6] = [0]
[0]
[c_7] = [0]
[0]
The order satisfies the following ordering constraints:
[=(.(x, y), .(u(), v()))] = [1]
[0]
> [0]
[0]
= [and(=(x, u()), =(y, v()))]
[=(.(x, y), nil())] = [1]
[0]
> [0]
[0]
= [false()]
[=(nil(), .(y, z))] = [1]
[0]
> [0]
[0]
= [false()]
[=(nil(), nil())] = [1]
[0]
> [0]
[0]
= [true()]
[del^#(.(x, .(y, z)))] = [0]
[0]
? [2]
[0]
= [c_1(f^#(=(x, y), x, y, z))]
[f^#(true(), x, y, z)] = [0]
[0]
>= [0]
[0]
= [c_2(del^#(.(y, z)))]
[f^#(false(), x, y, z)] = [0]
[0]
>= [0]
[0]
= [c_3(del^#(.(y, z)))]
[=^#(.(x, y), .(u(), v()))] = [0]
[0]
>= [0]
[0]
= [c_4(=^#(x, u()), =^#(y, v()))]
[=^#(.(x, y), nil())] = [0]
[0]
>= [0]
[0]
= [c_5()]
[=^#(nil(), .(y, z))] = [0]
[0]
>= [0]
[0]
= [c_6()]
[=^#(nil(), nil())] = [0]
[0]
>= [0]
[0]
= [c_7()]
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:
{ del^#(.(x, .(y, z))) -> c_1(f^#(=(x, y), x, y, z))
, f^#(true(), x, y, z) -> c_2(del^#(.(y, z)))
, f^#(false(), x, y, z) -> c_3(del^#(.(y, z)))
, =^#(.(x, y), .(u(), v())) -> c_4(=^#(x, u()), =^#(y, v()))
, =^#(.(x, y), nil()) -> c_5()
, =^#(nil(), .(y, z)) -> c_6()
, =^#(nil(), nil()) -> c_7() }
Weak Trs:
{ =(.(x, y), .(u(), v())) -> and(=(x, u()), =(y, v()))
, =(.(x, y), nil()) -> false()
, =(nil(), .(y, z)) -> false()
, =(nil(), nil()) -> true() }
Obligation:
innermost runtime complexity
Answer:
YES(O(1),O(n^1))
We estimate the number of application of {4,5,6,7} by applications
of Pre({4,5,6,7}) = {}. Here rules are labeled as follows:
DPs:
{ 1: del^#(.(x, .(y, z))) -> c_1(f^#(=(x, y), x, y, z))
, 2: f^#(true(), x, y, z) -> c_2(del^#(.(y, z)))
, 3: f^#(false(), x, y, z) -> c_3(del^#(.(y, z)))
, 4: =^#(.(x, y), .(u(), v())) -> c_4(=^#(x, u()), =^#(y, v()))
, 5: =^#(.(x, y), nil()) -> c_5()
, 6: =^#(nil(), .(y, z)) -> c_6()
, 7: =^#(nil(), nil()) -> c_7() }
We are left with following problem, upon which TcT provides the
certificate YES(O(1),O(n^1)).
Strict DPs:
{ del^#(.(x, .(y, z))) -> c_1(f^#(=(x, y), x, y, z))
, f^#(true(), x, y, z) -> c_2(del^#(.(y, z)))
, f^#(false(), x, y, z) -> c_3(del^#(.(y, z))) }
Weak DPs:
{ =^#(.(x, y), .(u(), v())) -> c_4(=^#(x, u()), =^#(y, v()))
, =^#(.(x, y), nil()) -> c_5()
, =^#(nil(), .(y, z)) -> c_6()
, =^#(nil(), nil()) -> c_7() }
Weak Trs:
{ =(.(x, y), .(u(), v())) -> and(=(x, u()), =(y, v()))
, =(.(x, y), nil()) -> false()
, =(nil(), .(y, z)) -> false()
, =(nil(), nil()) -> true() }
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.
{ =^#(.(x, y), .(u(), v())) -> c_4(=^#(x, u()), =^#(y, v()))
, =^#(.(x, y), nil()) -> c_5()
, =^#(nil(), .(y, z)) -> c_6()
, =^#(nil(), nil()) -> c_7() }
We are left with following problem, upon which TcT provides the
certificate YES(O(1),O(n^1)).
Strict DPs:
{ del^#(.(x, .(y, z))) -> c_1(f^#(=(x, y), x, y, z))
, f^#(true(), x, y, z) -> c_2(del^#(.(y, z)))
, f^#(false(), x, y, z) -> c_3(del^#(.(y, z))) }
Weak Trs:
{ =(.(x, y), .(u(), v())) -> and(=(x, u()), =(y, v()))
, =(.(x, y), nil()) -> false()
, =(nil(), .(y, z)) -> false()
, =(nil(), nil()) -> 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:
{ 1: del^#(.(x, .(y, z))) -> c_1(f^#(=(x, y), x, y, z))
, 2: f^#(true(), x, y, z) -> c_2(del^#(.(y, z)))
, 3: f^#(false(), x, y, z) -> c_3(del^#(.(y, z))) }
Trs:
{ =(.(x, y), .(u(), v())) -> and(=(x, u()), =(y, v()))
, =(nil(), .(y, z)) -> false() }
Sub-proof:
----------
The following argument positions are usable:
Uargs(c_1) = {1}, Uargs(c_2) = {1}, Uargs(c_3) = {1}
TcT has computed the following constructor-based matrix
interpretation satisfying not(EDA).
[.](x1, x2) = [1] x1 + [1] x2 + [1]
[=](x1, x2) = [1] x1 + [0]
[true] = [4]
[false] = [1]
[nil] = [4]
[u] = [0]
[v] = [0]
[and](x1, x2) = [0]
[del^#](x1) = [3] x1 + [2]
[c_1](x1) = [1] x1 + [2]
[f^#](x1, x2, x3, x4) = [2] x1 + [3] x3 + [3] x4 + [5]
[c_2](x1) = [1] x1 + [4]
[c_3](x1) = [1] x1 + [0]
The order satisfies the following ordering constraints:
[=(.(x, y), .(u(), v()))] = [1] x + [1] y + [1]
> [0]
= [and(=(x, u()), =(y, v()))]
[=(.(x, y), nil())] = [1] x + [1] y + [1]
>= [1]
= [false()]
[=(nil(), .(y, z))] = [4]
> [1]
= [false()]
[=(nil(), nil())] = [4]
>= [4]
= [true()]
[del^#(.(x, .(y, z)))] = [3] x + [3] y + [3] z + [8]
> [2] x + [3] y + [3] z + [7]
= [c_1(f^#(=(x, y), x, y, z))]
[f^#(true(), x, y, z)] = [3] y + [3] z + [13]
> [3] y + [3] z + [9]
= [c_2(del^#(.(y, z)))]
[f^#(false(), x, y, z)] = [3] y + [3] z + [7]
> [3] y + [3] z + [5]
= [c_3(del^#(.(y, z)))]
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:
{ del^#(.(x, .(y, z))) -> c_1(f^#(=(x, y), x, y, z))
, f^#(true(), x, y, z) -> c_2(del^#(.(y, z)))
, f^#(false(), x, y, z) -> c_3(del^#(.(y, z))) }
Weak Trs:
{ =(.(x, y), .(u(), v())) -> and(=(x, u()), =(y, v()))
, =(.(x, y), nil()) -> false()
, =(nil(), .(y, z)) -> false()
, =(nil(), nil()) -> true() }
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.
{ del^#(.(x, .(y, z))) -> c_1(f^#(=(x, y), x, y, z))
, f^#(true(), x, y, z) -> c_2(del^#(.(y, z)))
, f^#(false(), x, y, z) -> c_3(del^#(.(y, z))) }
We are left with following problem, upon which TcT provides the
certificate YES(O(1),O(1)).
Weak Trs:
{ =(.(x, y), .(u(), v())) -> and(=(x, u()), =(y, v()))
, =(.(x, y), nil()) -> false()
, =(nil(), .(y, z)) -> false()
, =(nil(), nil()) -> true() }
Obligation:
innermost runtime complexity
Answer:
YES(O(1),O(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(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))