YES(O(1),O(n^1))
We are left with following problem, upon which TcT provides the
certificate YES(O(1),O(n^1)).
Strict Trs:
{ 2nd(cons1(X, cons(Y, Z))) -> Y
, 2nd(cons(X, X1)) -> 2nd(cons1(X, activate(X1)))
, activate(X) -> X
, activate(n__from(X)) -> from(X)
, from(X) -> cons(X, n__from(s(X)))
, from(X) -> n__from(X) }
Obligation:
runtime complexity
Answer:
YES(O(1),O(n^1))
We add the following weak dependency pairs:
Strict DPs:
{ 2nd^#(cons1(X, cons(Y, Z))) -> c_1(Y)
, 2nd^#(cons(X, X1)) -> c_2(2nd^#(cons1(X, activate(X1))))
, activate^#(X) -> c_3(X)
, activate^#(n__from(X)) -> c_4(from^#(X))
, from^#(X) -> c_5(X, X)
, from^#(X) -> c_6(X) }
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:
{ 2nd^#(cons1(X, cons(Y, Z))) -> c_1(Y)
, 2nd^#(cons(X, X1)) -> c_2(2nd^#(cons1(X, activate(X1))))
, activate^#(X) -> c_3(X)
, activate^#(n__from(X)) -> c_4(from^#(X))
, from^#(X) -> c_5(X, X)
, from^#(X) -> c_6(X) }
Strict Trs:
{ 2nd(cons1(X, cons(Y, Z))) -> Y
, 2nd(cons(X, X1)) -> 2nd(cons1(X, activate(X1)))
, activate(X) -> X
, activate(n__from(X)) -> from(X)
, from(X) -> cons(X, n__from(s(X)))
, from(X) -> n__from(X) }
Obligation:
runtime complexity
Answer:
YES(O(1),O(n^1))
We replace rewrite rules by usable rules:
Strict Usable Rules:
{ activate(X) -> X
, activate(n__from(X)) -> from(X)
, from(X) -> cons(X, n__from(s(X)))
, from(X) -> n__from(X) }
We are left with following problem, upon which TcT provides the
certificate YES(O(1),O(n^1)).
Strict DPs:
{ 2nd^#(cons1(X, cons(Y, Z))) -> c_1(Y)
, 2nd^#(cons(X, X1)) -> c_2(2nd^#(cons1(X, activate(X1))))
, activate^#(X) -> c_3(X)
, activate^#(n__from(X)) -> c_4(from^#(X))
, from^#(X) -> c_5(X, X)
, from^#(X) -> c_6(X) }
Strict Trs:
{ activate(X) -> X
, activate(n__from(X)) -> from(X)
, from(X) -> cons(X, n__from(s(X)))
, from(X) -> n__from(X) }
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(cons1) = {2}, Uargs(2nd^#) = {1}, Uargs(c_2) = {1},
Uargs(c_4) = {1}
TcT has computed the following constructor-restricted matrix
interpretation.
[cons1](x1, x2) = [1 0] x1 + [1 0] x2 + [0]
[0 1] [0 0] [0]
[cons](x1, x2) = [1 0] x1 + [1 0] x2 + [0]
[0 0] [0 0] [0]
[activate](x1) = [1 0] x1 + [2]
[0 2] [0]
[from](x1) = [1 0] x1 + [1]
[0 0] [0]
[n__from](x1) = [1 0] x1 + [0]
[0 0] [0]
[s](x1) = [0]
[1]
[2nd^#](x1) = [1 0] x1 + [0]
[0 0] [0]
[c_1](x1) = [1 0] x1 + [1]
[1 1] [1]
[c_2](x1) = [1 0] x1 + [2]
[0 1] [2]
[activate^#](x1) = [2 1] x1 + [2]
[2 1] [2]
[c_3](x1) = [1 1] x1 + [1]
[1 1] [1]
[c_4](x1) = [1 0] x1 + [2]
[0 1] [2]
[from^#](x1) = [1 0] x1 + [1]
[2 2] [1]
[c_5](x1, x2) = [0 0] x1 + [1 0] x2 + [2]
[1 2] [2 1] [1]
[c_6](x1) = [1 0] x1 + [2]
[1 1] [1]
The following symbols are considered usable
{activate, from, 2nd^#, activate^#, from^#}
The order satisfies the following ordering constraints:
[activate(X)] = [1 0] X + [2]
[0 2] [0]
> [1 0] X + [0]
[0 1] [0]
= [X]
[activate(n__from(X))] = [1 0] X + [2]
[0 0] [0]
> [1 0] X + [1]
[0 0] [0]
= [from(X)]
[from(X)] = [1 0] X + [1]
[0 0] [0]
> [1 0] X + [0]
[0 0] [0]
= [cons(X, n__from(s(X)))]
[from(X)] = [1 0] X + [1]
[0 0] [0]
> [1 0] X + [0]
[0 0] [0]
= [n__from(X)]
[2nd^#(cons1(X, cons(Y, Z)))] = [1 0] X + [1 0] Y + [1 0] Z + [0]
[0 0] [0 0] [0 0] [0]
? [1 0] Y + [1]
[1 1] [1]
= [c_1(Y)]
[2nd^#(cons(X, X1))] = [1 0] X + [1 0] X1 + [0]
[0 0] [0 0] [0]
? [1 0] X + [1 0] X1 + [4]
[0 0] [0 0] [2]
= [c_2(2nd^#(cons1(X, activate(X1))))]
[activate^#(X)] = [2 1] X + [2]
[2 1] [2]
> [1 1] X + [1]
[1 1] [1]
= [c_3(X)]
[activate^#(n__from(X))] = [2 0] X + [2]
[2 0] [2]
? [1 0] X + [3]
[2 2] [3]
= [c_4(from^#(X))]
[from^#(X)] = [1 0] X + [1]
[2 2] [1]
? [1 0] X + [2]
[3 3] [1]
= [c_5(X, X)]
[from^#(X)] = [1 0] X + [1]
[2 2] [1]
? [1 0] X + [2]
[1 1] [1]
= [c_6(X)]
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:
{ 2nd^#(cons1(X, cons(Y, Z))) -> c_1(Y)
, 2nd^#(cons(X, X1)) -> c_2(2nd^#(cons1(X, activate(X1))))
, activate^#(n__from(X)) -> c_4(from^#(X))
, from^#(X) -> c_5(X, X)
, from^#(X) -> c_6(X) }
Weak DPs: { activate^#(X) -> c_3(X) }
Weak Trs:
{ activate(X) -> X
, activate(n__from(X)) -> from(X)
, from(X) -> cons(X, n__from(s(X)))
, from(X) -> n__from(X) }
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:
{ 3: activate^#(n__from(X)) -> c_4(from^#(X))
, 6: activate^#(X) -> c_3(X) }
Trs:
{ activate(X) -> X
, activate(n__from(X)) -> from(X)
, from(X) -> cons(X, n__from(s(X)))
, from(X) -> n__from(X) }
Sub-proof:
----------
The following argument positions are usable:
Uargs(c_2) = {1}, Uargs(c_4) = {1}
TcT has computed the following constructor-based matrix
interpretation satisfying not(EDA).
[2nd](x1) = [7] x1 + [0]
[cons1](x1, x2) = [0]
[cons](x1, x2) = [1] x1 + [1] x2 + [0]
[activate](x1) = [2] x1 + [6]
[from](x1) = [2] x1 + [4]
[n__from](x1) = [1] x1 + [1]
[s](x1) = [0]
[2nd^#](x1) = [4] x1 + [0]
[c_1](x1) = [0]
[c_2](x1) = [2] x1 + [0]
[activate^#](x1) = [7]
[c_3](x1) = [5]
[c_4](x1) = [4] x1 + [1]
[from^#](x1) = [0]
[c_5](x1, x2) = [0]
[c_6](x1) = [0]
The following symbols are considered usable
{activate, from, 2nd^#, activate^#, from^#}
The order satisfies the following ordering constraints:
[activate(X)] = [2] X + [6]
> [1] X + [0]
= [X]
[activate(n__from(X))] = [2] X + [8]
> [2] X + [4]
= [from(X)]
[from(X)] = [2] X + [4]
> [1] X + [1]
= [cons(X, n__from(s(X)))]
[from(X)] = [2] X + [4]
> [1] X + [1]
= [n__from(X)]
[2nd^#(cons1(X, cons(Y, Z)))] = [0]
>= [0]
= [c_1(Y)]
[2nd^#(cons(X, X1))] = [4] X + [4] X1 + [0]
>= [0]
= [c_2(2nd^#(cons1(X, activate(X1))))]
[activate^#(X)] = [7]
> [5]
= [c_3(X)]
[activate^#(n__from(X))] = [7]
> [1]
= [c_4(from^#(X))]
[from^#(X)] = [0]
>= [0]
= [c_5(X, X)]
[from^#(X)] = [0]
>= [0]
= [c_6(X)]
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:
{ 2nd^#(cons1(X, cons(Y, Z))) -> c_1(Y)
, 2nd^#(cons(X, X1)) -> c_2(2nd^#(cons1(X, activate(X1))))
, from^#(X) -> c_5(X, X)
, from^#(X) -> c_6(X) }
Weak DPs:
{ activate^#(X) -> c_3(X)
, activate^#(n__from(X)) -> c_4(from^#(X)) }
Weak Trs:
{ activate(X) -> X
, activate(n__from(X)) -> from(X)
, from(X) -> cons(X, n__from(s(X)))
, from(X) -> n__from(X) }
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:
{ 3: from^#(X) -> c_5(X, X)
, 4: from^#(X) -> c_6(X)
, 5: activate^#(X) -> c_3(X)
, 6: activate^#(n__from(X)) -> c_4(from^#(X)) }
Sub-proof:
----------
The following argument positions are usable:
Uargs(c_2) = {1}, Uargs(c_4) = {1}
TcT has computed the following constructor-based matrix
interpretation satisfying not(EDA).
[2nd](x1) = [7] x1 + [0]
[cons1](x1, x2) = [1] x1 + [0]
[cons](x1, x2) = [1] x1 + [0]
[activate](x1) = [4] x1 + [0]
[from](x1) = [4] x1 + [0]
[n__from](x1) = [1] x1 + [0]
[s](x1) = [1] x1 + [7]
[2nd^#](x1) = [0]
[c_1](x1) = [0]
[c_2](x1) = [4] x1 + [0]
[activate^#](x1) = [5] x1 + [7]
[c_3](x1) = [5] x1 + [6]
[c_4](x1) = [1] x1 + [1]
[from^#](x1) = [3] x1 + [4]
[c_5](x1, x2) = [3] x2 + [2]
[c_6](x1) = [3] x1 + [2]
The following symbols are considered usable
{activate, from, 2nd^#, activate^#, from^#}
The order satisfies the following ordering constraints:
[activate(X)] = [4] X + [0]
>= [1] X + [0]
= [X]
[activate(n__from(X))] = [4] X + [0]
>= [4] X + [0]
= [from(X)]
[from(X)] = [4] X + [0]
>= [1] X + [0]
= [cons(X, n__from(s(X)))]
[from(X)] = [4] X + [0]
>= [1] X + [0]
= [n__from(X)]
[2nd^#(cons1(X, cons(Y, Z)))] = [0]
>= [0]
= [c_1(Y)]
[2nd^#(cons(X, X1))] = [0]
>= [0]
= [c_2(2nd^#(cons1(X, activate(X1))))]
[activate^#(X)] = [5] X + [7]
> [5] X + [6]
= [c_3(X)]
[activate^#(n__from(X))] = [5] X + [7]
> [3] X + [5]
= [c_4(from^#(X))]
[from^#(X)] = [3] X + [4]
> [3] X + [2]
= [c_5(X, X)]
[from^#(X)] = [3] X + [4]
> [3] X + [2]
= [c_6(X)]
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:
{ 2nd^#(cons1(X, cons(Y, Z))) -> c_1(Y)
, 2nd^#(cons(X, X1)) -> c_2(2nd^#(cons1(X, activate(X1)))) }
Weak DPs:
{ activate^#(X) -> c_3(X)
, activate^#(n__from(X)) -> c_4(from^#(X))
, from^#(X) -> c_5(X, X)
, from^#(X) -> c_6(X) }
Weak Trs:
{ activate(X) -> X
, activate(n__from(X)) -> from(X)
, from(X) -> cons(X, n__from(s(X)))
, from(X) -> n__from(X) }
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: 2nd^#(cons1(X, cons(Y, Z))) -> c_1(Y)
, 3: activate^#(X) -> c_3(X)
, 4: activate^#(n__from(X)) -> c_4(from^#(X)) }
Trs:
{ activate(n__from(X)) -> from(X)
, from(X) -> cons(X, n__from(s(X)))
, from(X) -> n__from(X) }
Sub-proof:
----------
The following argument positions are usable:
Uargs(c_2) = {1}, Uargs(c_4) = {1}
TcT has computed the following constructor-based matrix
interpretation satisfying not(EDA).
[2nd](x1) = [7] x1 + [0]
[cons1](x1, x2) = [1] x1 + [0]
[cons](x1, x2) = [1] x1 + [0]
[activate](x1) = [4] x1 + [0]
[from](x1) = [4] x1 + [4]
[n__from](x1) = [1] x1 + [2]
[s](x1) = [1] x1 + [7]
[2nd^#](x1) = [1]
[c_1](x1) = [0]
[c_2](x1) = [1] x1 + [0]
[activate^#](x1) = [7]
[c_3](x1) = [6]
[c_4](x1) = [1] x1 + [1]
[from^#](x1) = [0]
[c_5](x1, x2) = [0]
[c_6](x1) = [0]
The following symbols are considered usable
{activate, from, 2nd^#, activate^#, from^#}
The order satisfies the following ordering constraints:
[activate(X)] = [4] X + [0]
>= [1] X + [0]
= [X]
[activate(n__from(X))] = [4] X + [8]
> [4] X + [4]
= [from(X)]
[from(X)] = [4] X + [4]
> [1] X + [0]
= [cons(X, n__from(s(X)))]
[from(X)] = [4] X + [4]
> [1] X + [2]
= [n__from(X)]
[2nd^#(cons1(X, cons(Y, Z)))] = [1]
> [0]
= [c_1(Y)]
[2nd^#(cons(X, X1))] = [1]
>= [1]
= [c_2(2nd^#(cons1(X, activate(X1))))]
[activate^#(X)] = [7]
> [6]
= [c_3(X)]
[activate^#(n__from(X))] = [7]
> [1]
= [c_4(from^#(X))]
[from^#(X)] = [0]
>= [0]
= [c_5(X, X)]
[from^#(X)] = [0]
>= [0]
= [c_6(X)]
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)).
Strict DPs:
{ 2nd^#(cons(X, X1)) -> c_2(2nd^#(cons1(X, activate(X1)))) }
Weak DPs:
{ 2nd^#(cons1(X, cons(Y, Z))) -> c_1(Y)
, activate^#(X) -> c_3(X)
, activate^#(n__from(X)) -> c_4(from^#(X))
, from^#(X) -> c_5(X, X)
, from^#(X) -> c_6(X) }
Weak Trs:
{ activate(X) -> X
, activate(n__from(X)) -> from(X)
, from(X) -> cons(X, n__from(s(X)))
, from(X) -> n__from(X) }
Obligation:
runtime complexity
Answer:
YES(O(1),O(1))
We use the processor 'matrix interpretation of dimension 1' to
orient following rules strictly.
DPs:
{ 1: 2nd^#(cons(X, X1)) -> c_2(2nd^#(cons1(X, activate(X1))))
, 3: activate^#(X) -> c_3(X)
, 4: activate^#(n__from(X)) -> c_4(from^#(X)) }
Trs:
{ activate(X) -> X
, activate(n__from(X)) -> from(X)
, from(X) -> cons(X, n__from(s(X)))
, from(X) -> n__from(X) }
Sub-proof:
----------
The following argument positions are usable:
Uargs(c_2) = {1}, Uargs(c_4) = {1}
TcT has computed the following constructor-restricted matrix
interpretation. Note that the diagonal of the component-wise maxima
of interpretation-entries (of constructors) contains no more than 0
non-zero entries.
[2nd](x1) = [0]
[cons1](x1, x2) = [0]
[cons](x1, x2) = [1]
[activate](x1) = [3] x1 + [3]
[from](x1) = [7]
[n__from](x1) = [4]
[s](x1) = [7]
[2nd^#](x1) = [4] x1 + [0]
[c_1](x1) = [0]
[c_2](x1) = [2] x1 + [1]
[activate^#](x1) = [1]
[c_3](x1) = [0]
[c_4](x1) = [4] x1 + [0]
[from^#](x1) = [0]
[c_5](x1, x2) = [0]
[c_6](x1) = [0]
The following symbols are considered usable
{activate, from, 2nd^#, activate^#, from^#}
The order satisfies the following ordering constraints:
[activate(X)] = [3] X + [3]
> [1] X + [0]
= [X]
[activate(n__from(X))] = [15]
> [7]
= [from(X)]
[from(X)] = [7]
> [1]
= [cons(X, n__from(s(X)))]
[from(X)] = [7]
> [4]
= [n__from(X)]
[2nd^#(cons1(X, cons(Y, Z)))] = [0]
>= [0]
= [c_1(Y)]
[2nd^#(cons(X, X1))] = [4]
> [1]
= [c_2(2nd^#(cons1(X, activate(X1))))]
[activate^#(X)] = [1]
> [0]
= [c_3(X)]
[activate^#(n__from(X))] = [1]
> [0]
= [c_4(from^#(X))]
[from^#(X)] = [0]
>= [0]
= [c_5(X, X)]
[from^#(X)] = [0]
>= [0]
= [c_6(X)]
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:
{ 2nd^#(cons1(X, cons(Y, Z))) -> c_1(Y)
, 2nd^#(cons(X, X1)) -> c_2(2nd^#(cons1(X, activate(X1))))
, activate^#(X) -> c_3(X)
, activate^#(n__from(X)) -> c_4(from^#(X))
, from^#(X) -> c_5(X, X)
, from^#(X) -> c_6(X) }
Weak Trs:
{ activate(X) -> X
, activate(n__from(X)) -> from(X)
, from(X) -> cons(X, n__from(s(X)))
, from(X) -> n__from(X) }
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.
{ 2nd^#(cons1(X, cons(Y, Z))) -> c_1(Y)
, 2nd^#(cons(X, X1)) -> c_2(2nd^#(cons1(X, activate(X1))))
, activate^#(X) -> c_3(X)
, activate^#(n__from(X)) -> c_4(from^#(X))
, from^#(X) -> c_5(X, X)
, from^#(X) -> c_6(X) }
We are left with following problem, upon which TcT provides the
certificate YES(O(1),O(1)).
Weak Trs:
{ activate(X) -> X
, activate(n__from(X)) -> from(X)
, from(X) -> cons(X, n__from(s(X)))
, from(X) -> n__from(X) }
Obligation:
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:
runtime complexity
Answer:
YES(O(1),O(1))
Empty rules are trivially bounded
Hurray, we answered YES(O(1),O(n^1))