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