We are left with following problem, upon which TcT provides the
certificate YES(O(1),O(n^1)).
Strict Trs:
{ f(empty(), l) -> l
, f(cons(x, k), l) -> g(k, l, cons(x, k))
, g(a, b, c) -> f(a, cons(b, c)) }
Obligation:
innermost runtime complexity
Answer:
YES(O(1),O(n^1))
We add the following weak dependency pairs:
Strict DPs:
{ f^#(empty(), l) -> c_1()
, f^#(cons(x, k), l) -> c_2(g^#(k, l, cons(x, k)))
, g^#(a, b, c) -> c_3(f^#(a, cons(b, c))) }
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:
{ f^#(empty(), l) -> c_1()
, f^#(cons(x, k), l) -> c_2(g^#(k, l, cons(x, k)))
, g^#(a, b, c) -> c_3(f^#(a, cons(b, c))) }
Strict Trs:
{ f(empty(), l) -> l
, f(cons(x, k), l) -> g(k, l, cons(x, k))
, g(a, b, c) -> f(a, cons(b, c)) }
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^#(empty(), l) -> c_1()
, f^#(cons(x, k), l) -> c_2(g^#(k, l, cons(x, k)))
, g^#(a, b, c) -> c_3(f^#(a, cons(b, c))) }
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_2) = {1}, Uargs(c_3) = {1}
TcT has computed the following constructor-restricted matrix
interpretation.
[empty] = [0]
[0]
[cons](x1, x2) = [0]
[0]
[f^#](x1, x2) = [0]
[0]
[c_1] = [0]
[0]
[c_2](x1) = [1 0] x1 + [0]
[0 1] [2]
[g^#](x1, x2, x3) = [1]
[0]
[c_3](x1) = [1 0] x1 + [0]
[0 1] [0]
The order satisfies the following ordering constraints:
[f^#(empty(), l)] = [0]
[0]
>= [0]
[0]
= [c_1()]
[f^#(cons(x, k), l)] = [0]
[0]
? [1]
[2]
= [c_2(g^#(k, l, cons(x, k)))]
[g^#(a, b, c)] = [1]
[0]
> [0]
[0]
= [c_3(f^#(a, cons(b, c)))]
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:
{ f^#(empty(), l) -> c_1()
, f^#(cons(x, k), l) -> c_2(g^#(k, l, cons(x, k))) }
Weak DPs: { g^#(a, b, c) -> c_3(f^#(a, cons(b, c))) }
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^#(empty(), l) -> c_1() }
Sub-proof:
----------
The following argument positions are usable:
Uargs(c_2) = {1}, Uargs(c_3) = {1}
TcT has computed the following constructor-based matrix
interpretation satisfying not(EDA).
[empty] = [1]
[cons](x1, x2) = [1] x2 + [0]
[f^#](x1, x2) = [1] x1 + [0]
[c_1] = [0]
[c_2](x1) = [1] x1 + [0]
[g^#](x1, x2, x3) = [1] x1 + [0]
[c_3](x1) = [1] x1 + [0]
The order satisfies the following ordering constraints:
[f^#(empty(), l)] = [1]
> [0]
= [c_1()]
[f^#(cons(x, k), l)] = [1] k + [0]
>= [1] k + [0]
= [c_2(g^#(k, l, cons(x, k)))]
[g^#(a, b, c)] = [1] a + [0]
>= [1] a + [0]
= [c_3(f^#(a, cons(b, c)))]
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(n^1)).
Strict DPs: { f^#(cons(x, k), l) -> c_2(g^#(k, l, cons(x, k))) }
Weak DPs:
{ f^#(empty(), l) -> c_1()
, g^#(a, b, c) -> c_3(f^#(a, cons(b, c))) }
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.
{ f^#(empty(), l) -> c_1() }
We are left with following problem, upon which TcT provides the
certificate YES(O(1),O(n^1)).
Strict DPs: { f^#(cons(x, k), l) -> c_2(g^#(k, l, cons(x, k))) }
Weak DPs: { g^#(a, b, c) -> c_3(f^#(a, cons(b, c))) }
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^#(cons(x, k), l) -> c_2(g^#(k, l, cons(x, k)))
, 2: g^#(a, b, c) -> c_3(f^#(a, cons(b, c))) }
Sub-proof:
----------
The following argument positions are usable:
Uargs(c_2) = {1}, Uargs(c_3) = {1}
TcT has computed the following constructor-based matrix
interpretation satisfying not(EDA).
[empty] = [0]
[cons](x1, x2) = [1] x2 + [1]
[f^#](x1, x2) = [4] x1 + [0]
[c_1] = [0]
[c_2](x1) = [1] x1 + [0]
[g^#](x1, x2, x3) = [4] x1 + [1]
[c_3](x1) = [1] x1 + [0]
The order satisfies the following ordering constraints:
[f^#(cons(x, k), l)] = [4] k + [4]
> [4] k + [1]
= [c_2(g^#(k, l, cons(x, k)))]
[g^#(a, b, c)] = [4] a + [1]
> [4] a + [0]
= [c_3(f^#(a, cons(b, c)))]
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^#(cons(x, k), l) -> c_2(g^#(k, l, cons(x, k)))
, g^#(a, b, c) -> c_3(f^#(a, cons(b, c))) }
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^#(cons(x, k), l) -> c_2(g^#(k, l, cons(x, k)))
, g^#(a, b, c) -> c_3(f^#(a, cons(b, c))) }
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))