We are left with following problem, upon which TcT provides the
certificate YES(O(1),O(n^1)).
Strict Trs:
{ admit(x, nil()) -> nil()
, admit(x, .(u, .(v, .(w(), z)))) ->
cond(=(sum(x, u, v), w()),
.(u, .(v, .(w(), admit(carry(x, u, v), z)))))
, cond(true(), y) -> y }
Obligation:
runtime complexity
Answer:
YES(O(1),O(n^1))
We add the following weak dependency pairs:
Strict DPs:
{ admit^#(x, nil()) -> c_1()
, admit^#(x, .(u, .(v, .(w(), z)))) ->
c_2(cond^#(=(sum(x, u, v), w()),
.(u, .(v, .(w(), admit(carry(x, u, v), z))))))
, cond^#(true(), y) -> c_3(y) }
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:
{ admit^#(x, nil()) -> c_1()
, admit^#(x, .(u, .(v, .(w(), z)))) ->
c_2(cond^#(=(sum(x, u, v), w()),
.(u, .(v, .(w(), admit(carry(x, u, v), z))))))
, cond^#(true(), y) -> c_3(y) }
Strict Trs:
{ admit(x, nil()) -> nil()
, admit(x, .(u, .(v, .(w(), z)))) ->
cond(=(sum(x, u, v), w()),
.(u, .(v, .(w(), admit(carry(x, u, v), z)))))
, cond(true(), y) -> y }
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(.) = {2}, Uargs(cond) = {2}, Uargs(c_2) = {1},
Uargs(cond^#) = {2}, Uargs(c_3) = {1}
TcT has computed the following constructor-restricted matrix
interpretation.
[admit](x1, x2) = [0 2] x1 + [0 1] x2 + [0]
[0 0] [0 1] [0]
[nil] = [0]
[1]
[.](x1, x2) = [0 1] x1 + [1 0] x2 + [0]
[0 1] [0 1] [1]
[w] = [0]
[2]
[cond](x1, x2) = [1 0] x2 + [2]
[0 1] [0]
[=](x1, x2) = [0]
[0]
[sum](x1, x2, x3) = [0]
[0]
[carry](x1, x2, x3) = [1]
[0]
[true] = [0]
[0]
[admit^#](x1, x2) = [1 2] x1 + [0 1] x2 + [0]
[0 0] [0 0] [0]
[c_1] = [0]
[0]
[c_2](x1) = [1 0] x1 + [0]
[0 1] [0]
[cond^#](x1, x2) = [1 0] x2 + [0]
[0 0] [0]
[c_3](x1) = [1 0] x1 + [0]
[0 1] [0]
The order satisfies the following ordering constraints:
[admit(x, nil())] = [0 2] x + [1]
[0 0] [1]
> [0]
[1]
= [nil()]
[admit(x, .(u, .(v, .(w(), z))))] = [0 2] x + [0 1] u + [0 1] v + [0 1] z + [5]
[0 0] [0 1] [0 1] [0 1] [5]
> [0 1] u + [0 1] v + [0 1] z + [4]
[0 1] [0 1] [0 1] [5]
= [cond(=(sum(x, u, v), w()),
.(u, .(v, .(w(), admit(carry(x, u, v), z)))))]
[cond(true(), y)] = [1 0] y + [2]
[0 1] [0]
> [1 0] y + [0]
[0 1] [0]
= [y]
[admit^#(x, nil())] = [1 2] x + [1]
[0 0] [0]
> [0]
[0]
= [c_1()]
[admit^#(x, .(u, .(v, .(w(), z))))] = [1 2] x + [0 1] u + [0 1] v + [0 1] z + [5]
[0 0] [0 0] [0 0] [0 0] [0]
> [0 1] u + [0 1] v + [0 1] z + [2]
[0 0] [0 0] [0 0] [0]
= [c_2(cond^#(=(sum(x, u, v), w()),
.(u, .(v, .(w(), admit(carry(x, u, v), z))))))]
[cond^#(true(), y)] = [1 0] y + [0]
[0 0] [0]
? [1 0] y + [0]
[0 1] [0]
= [c_3(y)]
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: { cond^#(true(), y) -> c_3(y) }
Weak DPs:
{ admit^#(x, nil()) -> c_1()
, admit^#(x, .(u, .(v, .(w(), z)))) ->
c_2(cond^#(=(sum(x, u, v), w()),
.(u, .(v, .(w(), admit(carry(x, u, v), z)))))) }
Weak Trs:
{ admit(x, nil()) -> nil()
, admit(x, .(u, .(v, .(w(), z)))) ->
cond(=(sum(x, u, v), w()),
.(u, .(v, .(w(), admit(carry(x, u, v), z)))))
, cond(true(), y) -> y }
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.
{ admit^#(x, nil()) -> c_1()
, admit^#(x, .(u, .(v, .(w(), z)))) ->
c_2(cond^#(=(sum(x, u, v), w()),
.(u, .(v, .(w(), admit(carry(x, u, v), z)))))) }
We are left with following problem, upon which TcT provides the
certificate YES(O(1),O(n^1)).
Strict DPs: { cond^#(true(), y) -> c_3(y) }
Weak Trs:
{ admit(x, nil()) -> nil()
, admit(x, .(u, .(v, .(w(), z)))) ->
cond(=(sum(x, u, v), w()),
.(u, .(v, .(w(), admit(carry(x, u, v), z)))))
, cond(true(), y) -> y }
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: { cond^#(true(), y) -> c_3(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: cond^#(true(), y) -> c_3(y) }
Sub-proof:
----------
The following argument positions are usable:
none
TcT has computed the following constructor-based matrix
interpretation satisfying not(EDA).
[true] = [1]
[cond^#](x1, x2) = [1] x1 + [4]
[c_3](x1) = [0]
The order satisfies the following ordering constraints:
[cond^#(true(), y)] = [5]
> [0]
= [c_3(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: { cond^#(true(), y) -> c_3(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.
{ cond^#(true(), y) -> c_3(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))