We are left with following problem, upon which TcT provides the
certificate YES(O(1),O(n^1)).
Strict Trs:
{ from(X) -> cons(X, n__from(s(X)))
, from(X) -> n__from(X)
, after(s(N), cons(X, XS)) -> after(N, activate(XS))
, after(0(), XS) -> XS
, activate(X) -> X
, activate(n__from(X)) -> from(X) }
Obligation:
runtime complexity
Answer:
YES(O(1),O(n^1))
We add the following weak dependency pairs:
Strict DPs:
{ from^#(X) -> c_1(X, X)
, from^#(X) -> c_2(X)
, after^#(s(N), cons(X, XS)) -> c_3(after^#(N, activate(XS)))
, after^#(0(), XS) -> c_4(XS)
, activate^#(X) -> c_5(X)
, activate^#(n__from(X)) -> c_6(from^#(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:
{ from^#(X) -> c_1(X, X)
, from^#(X) -> c_2(X)
, after^#(s(N), cons(X, XS)) -> c_3(after^#(N, activate(XS)))
, after^#(0(), XS) -> c_4(XS)
, activate^#(X) -> c_5(X)
, activate^#(n__from(X)) -> c_6(from^#(X)) }
Strict Trs:
{ from(X) -> cons(X, n__from(s(X)))
, from(X) -> n__from(X)
, after(s(N), cons(X, XS)) -> after(N, activate(XS))
, after(0(), XS) -> XS
, activate(X) -> X
, activate(n__from(X)) -> from(X) }
Obligation:
runtime complexity
Answer:
YES(O(1),O(n^1))
We replace rewrite rules by usable rules:
Strict Usable Rules:
{ from(X) -> cons(X, n__from(s(X)))
, from(X) -> n__from(X)
, activate(X) -> X
, activate(n__from(X)) -> from(X) }
We are left with following problem, upon which TcT provides the
certificate YES(O(1),O(n^1)).
Strict DPs:
{ from^#(X) -> c_1(X, X)
, from^#(X) -> c_2(X)
, after^#(s(N), cons(X, XS)) -> c_3(after^#(N, activate(XS)))
, after^#(0(), XS) -> c_4(XS)
, activate^#(X) -> c_5(X)
, activate^#(n__from(X)) -> c_6(from^#(X)) }
Strict Trs:
{ from(X) -> cons(X, n__from(s(X)))
, from(X) -> n__from(X)
, activate(X) -> X
, activate(n__from(X)) -> 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(after^#) = {2}, Uargs(c_3) = {1}, Uargs(c_4) = {1},
Uargs(c_6) = {1}
TcT has computed the following constructor-restricted matrix
interpretation.
[from](x1) = [1]
[0]
[cons](x1, x2) = [1 0] x2 + [0]
[0 0] [0]
[n__from](x1) = [0]
[0]
[s](x1) = [1 0] x1 + [0]
[0 0] [0]
[0] = [0]
[0]
[activate](x1) = [1 0] x1 + [2]
[0 2] [0]
[from^#](x1) = [1]
[0]
[c_1](x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[1 1] [1 1] [0]
[c_2](x1) = [0]
[0]
[after^#](x1, x2) = [1 0] x2 + [0]
[0 0] [0]
[c_3](x1) = [1 0] x1 + [0]
[0 1] [0]
[c_4](x1) = [1 0] x1 + [0]
[0 1] [0]
[activate^#](x1) = [0]
[0]
[c_5](x1) = [0]
[0]
[c_6](x1) = [1 0] x1 + [1]
[0 1] [0]
The order satisfies the following ordering constraints:
[from(X)] = [1]
[0]
> [0]
[0]
= [cons(X, n__from(s(X)))]
[from(X)] = [1]
[0]
> [0]
[0]
= [n__from(X)]
[activate(X)] = [1 0] X + [2]
[0 2] [0]
> [1 0] X + [0]
[0 1] [0]
= [X]
[activate(n__from(X))] = [2]
[0]
> [1]
[0]
= [from(X)]
[from^#(X)] = [1]
[0]
? [0 0] X + [0]
[2 2] [0]
= [c_1(X, X)]
[from^#(X)] = [1]
[0]
> [0]
[0]
= [c_2(X)]
[after^#(s(N), cons(X, XS))] = [1 0] XS + [0]
[0 0] [0]
? [1 0] XS + [2]
[0 0] [0]
= [c_3(after^#(N, activate(XS)))]
[after^#(0(), XS)] = [1 0] XS + [0]
[0 0] [0]
? [1 0] XS + [0]
[0 1] [0]
= [c_4(XS)]
[activate^#(X)] = [0]
[0]
>= [0]
[0]
= [c_5(X)]
[activate^#(n__from(X))] = [0]
[0]
? [2]
[0]
= [c_6(from^#(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:
{ from^#(X) -> c_1(X, X)
, after^#(s(N), cons(X, XS)) -> c_3(after^#(N, activate(XS)))
, after^#(0(), XS) -> c_4(XS)
, activate^#(X) -> c_5(X)
, activate^#(n__from(X)) -> c_6(from^#(X)) }
Weak DPs: { from^#(X) -> c_2(X) }
Weak Trs:
{ from(X) -> cons(X, n__from(s(X)))
, from(X) -> n__from(X)
, activate(X) -> X
, activate(n__from(X)) -> 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:
{ 4: activate^#(X) -> c_5(X)
, 5: activate^#(n__from(X)) -> c_6(from^#(X)) }
Sub-proof:
----------
The following argument positions are usable:
Uargs(c_3) = {1}, Uargs(c_4) = {1}, Uargs(c_6) = {1}
TcT has computed the following constructor-based matrix
interpretation satisfying not(EDA).
[from](x1) = [0]
[cons](x1, x2) = [1] x2 + [0]
[n__from](x1) = [0]
[s](x1) = [1] x1 + [0]
[0] = [0]
[activate](x1) = [1] x1 + [0]
[from^#](x1) = [0]
[c_1](x1, x2) = [0]
[c_2](x1) = [0]
[after^#](x1, x2) = [4] x2 + [0]
[c_3](x1) = [1] x1 + [0]
[c_4](x1) = [1] x1 + [0]
[activate^#](x1) = [5]
[c_5](x1) = [0]
[c_6](x1) = [4] x1 + [0]
The order satisfies the following ordering constraints:
[from(X)] = [0]
>= [0]
= [cons(X, n__from(s(X)))]
[from(X)] = [0]
>= [0]
= [n__from(X)]
[activate(X)] = [1] X + [0]
>= [1] X + [0]
= [X]
[activate(n__from(X))] = [0]
>= [0]
= [from(X)]
[from^#(X)] = [0]
>= [0]
= [c_1(X, X)]
[from^#(X)] = [0]
>= [0]
= [c_2(X)]
[after^#(s(N), cons(X, XS))] = [4] XS + [0]
>= [4] XS + [0]
= [c_3(after^#(N, activate(XS)))]
[after^#(0(), XS)] = [4] XS + [0]
>= [1] XS + [0]
= [c_4(XS)]
[activate^#(X)] = [5]
> [0]
= [c_5(X)]
[activate^#(n__from(X))] = [5]
> [0]
= [c_6(from^#(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:
{ from^#(X) -> c_1(X, X)
, after^#(s(N), cons(X, XS)) -> c_3(after^#(N, activate(XS)))
, after^#(0(), XS) -> c_4(XS) }
Weak DPs:
{ from^#(X) -> c_2(X)
, activate^#(X) -> c_5(X)
, activate^#(n__from(X)) -> c_6(from^#(X)) }
Weak Trs:
{ from(X) -> cons(X, n__from(s(X)))
, from(X) -> n__from(X)
, activate(X) -> X
, activate(n__from(X)) -> 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: from^#(X) -> c_1(X, X)
, 4: from^#(X) -> c_2(X)
, 5: activate^#(X) -> c_5(X)
, 6: activate^#(n__from(X)) -> c_6(from^#(X)) }
Sub-proof:
----------
The following argument positions are usable:
Uargs(c_3) = {1}, Uargs(c_4) = {1}, Uargs(c_6) = {1}
TcT has computed the following constructor-based matrix
interpretation satisfying not(EDA).
[from](x1) = [3]
[cons](x1, x2) = [1] x2 + [0]
[n__from](x1) = [3]
[s](x1) = [1] x1 + [0]
[0] = [0]
[activate](x1) = [1] x1 + [0]
[from^#](x1) = [1]
[c_1](x1, x2) = [0]
[c_2](x1) = [0]
[after^#](x1, x2) = [1] x2 + [0]
[c_3](x1) = [1] x1 + [0]
[c_4](x1) = [1] x1 + [0]
[activate^#](x1) = [5]
[c_5](x1) = [0]
[c_6](x1) = [1] x1 + [1]
The order satisfies the following ordering constraints:
[from(X)] = [3]
>= [3]
= [cons(X, n__from(s(X)))]
[from(X)] = [3]
>= [3]
= [n__from(X)]
[activate(X)] = [1] X + [0]
>= [1] X + [0]
= [X]
[activate(n__from(X))] = [3]
>= [3]
= [from(X)]
[from^#(X)] = [1]
> [0]
= [c_1(X, X)]
[from^#(X)] = [1]
> [0]
= [c_2(X)]
[after^#(s(N), cons(X, XS))] = [1] XS + [0]
>= [1] XS + [0]
= [c_3(after^#(N, activate(XS)))]
[after^#(0(), XS)] = [1] XS + [0]
>= [1] XS + [0]
= [c_4(XS)]
[activate^#(X)] = [5]
> [0]
= [c_5(X)]
[activate^#(n__from(X))] = [5]
> [2]
= [c_6(from^#(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:
{ after^#(s(N), cons(X, XS)) -> c_3(after^#(N, activate(XS)))
, after^#(0(), XS) -> c_4(XS) }
Weak DPs:
{ from^#(X) -> c_1(X, X)
, from^#(X) -> c_2(X)
, activate^#(X) -> c_5(X)
, activate^#(n__from(X)) -> c_6(from^#(X)) }
Weak Trs:
{ from(X) -> cons(X, n__from(s(X)))
, from(X) -> n__from(X)
, activate(X) -> X
, activate(n__from(X)) -> 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: after^#(s(N), cons(X, XS)) -> c_3(after^#(N, activate(XS)))
, 5: activate^#(X) -> c_5(X)
, 6: activate^#(n__from(X)) -> c_6(from^#(X)) }
Trs:
{ from(X) -> cons(X, n__from(s(X)))
, from(X) -> n__from(X)
, activate(X) -> X }
Sub-proof:
----------
The following argument positions are usable:
Uargs(c_3) = {1}, Uargs(c_4) = {1}, Uargs(c_6) = {1}
TcT has computed the following constructor-based matrix
interpretation satisfying not(EDA).
[from](x1) = [2]
[cons](x1, x2) = [1] x2 + [1]
[n__from](x1) = [0]
[s](x1) = [1] x1 + [2]
[0] = [0]
[activate](x1) = [1] x1 + [2]
[from^#](x1) = [0]
[c_1](x1, x2) = [0]
[c_2](x1) = [0]
[after^#](x1, x2) = [4] x1 + [4] x2 + [0]
[c_3](x1) = [1] x1 + [3]
[c_4](x1) = [2] x1 + [0]
[activate^#](x1) = [5]
[c_5](x1) = [0]
[c_6](x1) = [4] x1 + [0]
The order satisfies the following ordering constraints:
[from(X)] = [2]
> [1]
= [cons(X, n__from(s(X)))]
[from(X)] = [2]
> [0]
= [n__from(X)]
[activate(X)] = [1] X + [2]
> [1] X + [0]
= [X]
[activate(n__from(X))] = [2]
>= [2]
= [from(X)]
[from^#(X)] = [0]
>= [0]
= [c_1(X, X)]
[from^#(X)] = [0]
>= [0]
= [c_2(X)]
[after^#(s(N), cons(X, XS))] = [4] XS + [4] N + [12]
> [4] XS + [4] N + [11]
= [c_3(after^#(N, activate(XS)))]
[after^#(0(), XS)] = [4] XS + [0]
>= [2] XS + [0]
= [c_4(XS)]
[activate^#(X)] = [5]
> [0]
= [c_5(X)]
[activate^#(n__from(X))] = [5]
> [0]
= [c_6(from^#(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: { after^#(0(), XS) -> c_4(XS) }
Weak DPs:
{ from^#(X) -> c_1(X, X)
, from^#(X) -> c_2(X)
, after^#(s(N), cons(X, XS)) -> c_3(after^#(N, activate(XS)))
, activate^#(X) -> c_5(X)
, activate^#(n__from(X)) -> c_6(from^#(X)) }
Weak Trs:
{ from(X) -> cons(X, n__from(s(X)))
, from(X) -> n__from(X)
, activate(X) -> X
, activate(n__from(X)) -> 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: after^#(0(), XS) -> c_4(XS)
, 5: activate^#(X) -> c_5(X)
, 6: activate^#(n__from(X)) -> c_6(from^#(X)) }
Trs:
{ activate(X) -> X
, activate(n__from(X)) -> from(X) }
Sub-proof:
----------
The following argument positions are usable:
Uargs(c_3) = {1}, Uargs(c_4) = {1}, Uargs(c_6) = {1}
TcT has computed the following constructor-based matrix
interpretation satisfying not(EDA).
[from](x1) = [7]
[cons](x1, x2) = [1] x2 + [0]
[n__from](x1) = [7]
[s](x1) = [1] x1 + [1]
[0] = [1]
[activate](x1) = [1] x1 + [1]
[from^#](x1) = [0]
[c_1](x1, x2) = [0]
[c_2](x1) = [0]
[after^#](x1, x2) = [1] x1 + [1] x2 + [0]
[c_3](x1) = [1] x1 + [0]
[c_4](x1) = [1] x1 + [0]
[activate^#](x1) = [6]
[c_5](x1) = [0]
[c_6](x1) = [1] x1 + [1]
The order satisfies the following ordering constraints:
[from(X)] = [7]
>= [7]
= [cons(X, n__from(s(X)))]
[from(X)] = [7]
>= [7]
= [n__from(X)]
[activate(X)] = [1] X + [1]
> [1] X + [0]
= [X]
[activate(n__from(X))] = [8]
> [7]
= [from(X)]
[from^#(X)] = [0]
>= [0]
= [c_1(X, X)]
[from^#(X)] = [0]
>= [0]
= [c_2(X)]
[after^#(s(N), cons(X, XS))] = [1] XS + [1] N + [1]
>= [1] XS + [1] N + [1]
= [c_3(after^#(N, activate(XS)))]
[after^#(0(), XS)] = [1] XS + [1]
> [1] XS + [0]
= [c_4(XS)]
[activate^#(X)] = [6]
> [0]
= [c_5(X)]
[activate^#(n__from(X))] = [6]
> [1]
= [c_6(from^#(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:
{ from^#(X) -> c_1(X, X)
, from^#(X) -> c_2(X)
, after^#(s(N), cons(X, XS)) -> c_3(after^#(N, activate(XS)))
, after^#(0(), XS) -> c_4(XS)
, activate^#(X) -> c_5(X)
, activate^#(n__from(X)) -> c_6(from^#(X)) }
Weak Trs:
{ from(X) -> cons(X, n__from(s(X)))
, from(X) -> n__from(X)
, activate(X) -> X
, activate(n__from(X)) -> 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.
{ from^#(X) -> c_1(X, X)
, from^#(X) -> c_2(X)
, after^#(s(N), cons(X, XS)) -> c_3(after^#(N, activate(XS)))
, after^#(0(), XS) -> c_4(XS)
, activate^#(X) -> c_5(X)
, activate^#(n__from(X)) -> c_6(from^#(X)) }
We are left with following problem, upon which TcT provides the
certificate YES(O(1),O(1)).
Weak Trs:
{ from(X) -> cons(X, n__from(s(X)))
, from(X) -> n__from(X)
, activate(X) -> X
, activate(n__from(X)) -> 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))