We are left with following problem, upon which TcT provides the
certificate YES(O(1),O(n^1)).
Strict Trs:
{ prime(0()) -> false()
, prime(s(0())) -> false()
, prime(s(s(x))) -> prime1(s(s(x)), s(x))
, prime1(x, 0()) -> false()
, prime1(x, s(0())) -> true()
, prime1(x, s(s(y))) -> and(not(divp(s(s(y)), x)), prime1(x, s(y)))
, divp(x, y) -> =(rem(x, y), 0()) }
Obligation:
runtime complexity
Answer:
YES(O(1),O(n^1))
We add the following weak dependency pairs:
Strict DPs:
{ prime^#(0()) -> c_1()
, prime^#(s(0())) -> c_2()
, prime^#(s(s(x))) -> c_3(prime1^#(s(s(x)), s(x)))
, prime1^#(x, 0()) -> c_4()
, prime1^#(x, s(0())) -> c_5()
, prime1^#(x, s(s(y))) ->
c_6(divp^#(s(s(y)), x), prime1^#(x, s(y)))
, divp^#(x, y) -> c_7(x, 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:
{ prime^#(0()) -> c_1()
, prime^#(s(0())) -> c_2()
, prime^#(s(s(x))) -> c_3(prime1^#(s(s(x)), s(x)))
, prime1^#(x, 0()) -> c_4()
, prime1^#(x, s(0())) -> c_5()
, prime1^#(x, s(s(y))) ->
c_6(divp^#(s(s(y)), x), prime1^#(x, s(y)))
, divp^#(x, y) -> c_7(x, y) }
Strict Trs:
{ prime(0()) -> false()
, prime(s(0())) -> false()
, prime(s(s(x))) -> prime1(s(s(x)), s(x))
, prime1(x, 0()) -> false()
, prime1(x, s(0())) -> true()
, prime1(x, s(s(y))) -> and(not(divp(s(s(y)), x)), prime1(x, s(y)))
, divp(x, y) -> =(rem(x, y), 0()) }
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:
{ prime^#(0()) -> c_1()
, prime^#(s(0())) -> c_2()
, prime^#(s(s(x))) -> c_3(prime1^#(s(s(x)), s(x)))
, prime1^#(x, 0()) -> c_4()
, prime1^#(x, s(0())) -> c_5()
, prime1^#(x, s(s(y))) ->
c_6(divp^#(s(s(y)), x), prime1^#(x, s(y)))
, divp^#(x, y) -> c_7(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_3) = {1}, Uargs(c_6) = {1, 2}
TcT has computed the following constructor-restricted matrix
interpretation.
[0] = [0]
[0]
[s](x1) = [0]
[0]
[prime^#](x1) = [0]
[0]
[c_1] = [0]
[0]
[c_2] = [0]
[0]
[c_3](x1) = [1 0] x1 + [0]
[0 1] [0]
[prime1^#](x1, x2) = [0]
[0]
[c_4] = [0]
[0]
[c_5] = [0]
[0]
[c_6](x1, x2) = [1 0] x1 + [1 0] x2 + [0]
[0 1] [0 1] [0]
[divp^#](x1, x2) = [1]
[0]
[c_7](x1, x2) = [0]
[0]
The order satisfies the following ordering constraints:
[prime^#(0())] = [0]
[0]
>= [0]
[0]
= [c_1()]
[prime^#(s(0()))] = [0]
[0]
>= [0]
[0]
= [c_2()]
[prime^#(s(s(x)))] = [0]
[0]
>= [0]
[0]
= [c_3(prime1^#(s(s(x)), s(x)))]
[prime1^#(x, 0())] = [0]
[0]
>= [0]
[0]
= [c_4()]
[prime1^#(x, s(0()))] = [0]
[0]
>= [0]
[0]
= [c_5()]
[prime1^#(x, s(s(y)))] = [0]
[0]
? [1]
[0]
= [c_6(divp^#(s(s(y)), x), prime1^#(x, s(y)))]
[divp^#(x, y)] = [1]
[0]
> [0]
[0]
= [c_7(x, 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(n^1)).
Strict DPs:
{ prime^#(0()) -> c_1()
, prime^#(s(0())) -> c_2()
, prime^#(s(s(x))) -> c_3(prime1^#(s(s(x)), s(x)))
, prime1^#(x, 0()) -> c_4()
, prime1^#(x, s(0())) -> c_5()
, prime1^#(x, s(s(y))) ->
c_6(divp^#(s(s(y)), x), prime1^#(x, s(y))) }
Weak DPs: { divp^#(x, y) -> c_7(x, y) }
Obligation:
runtime complexity
Answer:
YES(?,O(n^1))
We employ 'linear path analysis' using the following approximated
dependency graph:
->{3,7,6} [ ? ]
|
|->{1} [ YES(O(1),O(n^1)) ]
|
|->{2} [ YES(O(1),O(n^1)) ]
|
|->{4} [ YES(O(1),O(n^1)) ]
|
`->{5} [ YES(O(1),O(n^1)) ]
Here dependency-pairs are as follows:
Strict DPs:
{ 1: prime^#(0()) -> c_1()
, 2: prime^#(s(0())) -> c_2()
, 3: prime^#(s(s(x))) -> c_3(prime1^#(s(s(x)), s(x)))
, 4: prime1^#(x, 0()) -> c_4()
, 5: prime1^#(x, s(0())) -> c_5()
, 6: prime1^#(x, s(s(y))) ->
c_6(divp^#(s(s(y)), x), prime1^#(x, s(y))) }
Weak DPs:
{ 7: divp^#(x, y) -> c_7(x, y) }
* Path {3,7,6}->{1}: 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 DPs:
{ prime^#(0()) -> c_1()
, prime^#(s(s(x))) -> c_3(prime1^#(s(s(x)), s(x)))
, prime1^#(x, s(s(y))) ->
c_6(divp^#(s(s(y)), x), prime1^#(x, s(y))) }
Weak DPs: { divp^#(x, y) -> c_7(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: prime^#(0()) -> c_1() }
Sub-proof:
----------
The following argument positions are usable:
Uargs(c_3) = {1}, Uargs(c_6) = {1, 2}
TcT has computed the following constructor-based matrix
interpretation satisfying not(EDA).
[0] = [2]
[s](x1) = [1] x1 + [0]
[prime^#](x1) = [4] x1 + [0]
[c_1] = [0]
[c_3](x1) = [1] x1 + [0]
[prime1^#](x1, x2) = [4] x2 + [0]
[c_6](x1, x2) = [4] x1 + [1] x2 + [0]
[divp^#](x1, x2) = [0]
[c_7](x1, x2) = [0]
The order satisfies the following ordering constraints:
[prime^#(0())] = [8]
> [0]
= [c_1()]
[prime^#(s(s(x)))] = [4] x + [0]
>= [4] x + [0]
= [c_3(prime1^#(s(s(x)), s(x)))]
[prime1^#(x, s(s(y)))] = [4] y + [0]
>= [4] y + [0]
= [c_6(divp^#(s(s(y)), x), prime1^#(x, s(y)))]
[divp^#(x, y)] = [0]
>= [0]
= [c_7(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:
{ prime^#(s(s(x))) -> c_3(prime1^#(s(s(x)), s(x)))
, prime1^#(x, s(s(y))) ->
c_6(divp^#(s(s(y)), x), prime1^#(x, s(y))) }
Weak DPs:
{ prime^#(0()) -> c_1()
, divp^#(x, y) -> c_7(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.
{ prime^#(0()) -> c_1() }
We are left with following problem, upon which TcT provides the
certificate YES(O(1),O(n^1)).
Strict DPs:
{ prime^#(s(s(x))) -> c_3(prime1^#(s(s(x)), s(x)))
, prime1^#(x, s(s(y))) ->
c_6(divp^#(s(s(y)), x), prime1^#(x, s(y))) }
Weak DPs: { divp^#(x, y) -> c_7(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: prime^#(s(s(x))) -> c_3(prime1^#(s(s(x)), s(x))) }
Sub-proof:
----------
The following argument positions are usable:
Uargs(c_3) = {1}, Uargs(c_6) = {1, 2}
TcT has computed the following constructor-based matrix
interpretation satisfying not(EDA).
[0] = [0]
[s](x1) = [1] x1 + [0]
[prime^#](x1) = [5]
[c_1] = [0]
[c_3](x1) = [4] x1 + [0]
[prime1^#](x1, x2) = [0]
[c_6](x1, x2) = [2] x1 + [4] x2 + [0]
[divp^#](x1, x2) = [0]
[c_7](x1, x2) = [0]
The order satisfies the following ordering constraints:
[prime^#(s(s(x)))] = [5]
> [0]
= [c_3(prime1^#(s(s(x)), s(x)))]
[prime1^#(x, s(s(y)))] = [0]
>= [0]
= [c_6(divp^#(s(s(y)), x), prime1^#(x, s(y)))]
[divp^#(x, y)] = [0]
>= [0]
= [c_7(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:
{ prime1^#(x, s(s(y))) ->
c_6(divp^#(s(s(y)), x), prime1^#(x, s(y))) }
Weak DPs:
{ prime^#(s(s(x))) -> c_3(prime1^#(s(s(x)), s(x)))
, divp^#(x, y) -> c_7(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: prime1^#(x, s(s(y))) ->
c_6(divp^#(s(s(y)), x), prime1^#(x, s(y)))
, 2: prime^#(s(s(x))) -> c_3(prime1^#(s(s(x)), s(x)))
, 3: divp^#(x, y) -> c_7(x, y) }
Sub-proof:
----------
The following argument positions are usable:
Uargs(c_3) = {1}, Uargs(c_6) = {1, 2}
TcT has computed the following constructor-based matrix
interpretation satisfying not(EDA).
[0] = [0]
[s](x1) = [1] x1 + [2]
[prime^#](x1) = [2] x1 + [5]
[c_1] = [0]
[c_3](x1) = [1] x1 + [0]
[prime1^#](x1, x2) = [1] x2 + [0]
[c_6](x1, x2) = [1] x1 + [1] x2 + [0]
[divp^#](x1, x2) = [1]
[c_7](x1, x2) = [0]
The order satisfies the following ordering constraints:
[prime^#(s(s(x)))] = [2] x + [13]
> [1] x + [2]
= [c_3(prime1^#(s(s(x)), s(x)))]
[prime1^#(x, s(s(y)))] = [1] y + [4]
> [1] y + [3]
= [c_6(divp^#(s(s(y)), x), prime1^#(x, s(y)))]
[divp^#(x, y)] = [1]
> [0]
= [c_7(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:
{ prime^#(s(s(x))) -> c_3(prime1^#(s(s(x)), s(x)))
, prime1^#(x, s(s(y))) ->
c_6(divp^#(s(s(y)), x), prime1^#(x, s(y)))
, divp^#(x, y) -> c_7(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.
{ prime^#(s(s(x))) -> c_3(prime1^#(s(s(x)), s(x)))
, prime1^#(x, s(s(y))) ->
c_6(divp^#(s(s(y)), x), prime1^#(x, s(y)))
, divp^#(x, y) -> c_7(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
* Path {3,7,6}->{2}: 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 DPs:
{ prime^#(s(0())) -> c_2()
, prime^#(s(s(x))) -> c_3(prime1^#(s(s(x)), s(x)))
, prime1^#(x, s(s(y))) ->
c_6(divp^#(s(s(y)), x), prime1^#(x, s(y))) }
Weak DPs: { divp^#(x, y) -> c_7(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: prime^#(s(0())) -> c_2()
, 2: prime^#(s(s(x))) -> c_3(prime1^#(s(s(x)), s(x))) }
Sub-proof:
----------
The following argument positions are usable:
Uargs(c_3) = {1}, Uargs(c_6) = {1, 2}
TcT has computed the following constructor-based matrix
interpretation satisfying not(EDA).
[0] = [0]
[s](x1) = [0]
[prime^#](x1) = [1]
[c_2] = [0]
[c_3](x1) = [5] x1 + [0]
[prime1^#](x1, x2) = [0]
[c_6](x1, x2) = [4] x1 + [1] x2 + [0]
[divp^#](x1, x2) = [0]
[c_7](x1, x2) = [0]
The order satisfies the following ordering constraints:
[prime^#(s(0()))] = [1]
> [0]
= [c_2()]
[prime^#(s(s(x)))] = [1]
> [0]
= [c_3(prime1^#(s(s(x)), s(x)))]
[prime1^#(x, s(s(y)))] = [0]
>= [0]
= [c_6(divp^#(s(s(y)), x), prime1^#(x, s(y)))]
[divp^#(x, y)] = [0]
>= [0]
= [c_7(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:
{ prime1^#(x, s(s(y))) ->
c_6(divp^#(s(s(y)), x), prime1^#(x, s(y))) }
Weak DPs:
{ prime^#(s(0())) -> c_2()
, prime^#(s(s(x))) -> c_3(prime1^#(s(s(x)), s(x)))
, divp^#(x, y) -> c_7(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.
{ prime^#(s(0())) -> c_2() }
We are left with following problem, upon which TcT provides the
certificate YES(O(1),O(n^1)).
Strict DPs:
{ prime1^#(x, s(s(y))) ->
c_6(divp^#(s(s(y)), x), prime1^#(x, s(y))) }
Weak DPs:
{ prime^#(s(s(x))) -> c_3(prime1^#(s(s(x)), s(x)))
, divp^#(x, y) -> c_7(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: prime1^#(x, s(s(y))) ->
c_6(divp^#(s(s(y)), x), prime1^#(x, s(y)))
, 2: prime^#(s(s(x))) -> c_3(prime1^#(s(s(x)), s(x)))
, 3: divp^#(x, y) -> c_7(x, y) }
Sub-proof:
----------
The following argument positions are usable:
Uargs(c_3) = {1}, Uargs(c_6) = {1, 2}
TcT has computed the following constructor-based matrix
interpretation satisfying not(EDA).
[0] = [0]
[s](x1) = [1] x1 + [2]
[prime^#](x1) = [2] x1 + [5]
[c_2] = [0]
[c_3](x1) = [1] x1 + [0]
[prime1^#](x1, x2) = [1] x2 + [0]
[c_6](x1, x2) = [1] x1 + [1] x2 + [0]
[divp^#](x1, x2) = [1]
[c_7](x1, x2) = [0]
The order satisfies the following ordering constraints:
[prime^#(s(s(x)))] = [2] x + [13]
> [1] x + [2]
= [c_3(prime1^#(s(s(x)), s(x)))]
[prime1^#(x, s(s(y)))] = [1] y + [4]
> [1] y + [3]
= [c_6(divp^#(s(s(y)), x), prime1^#(x, s(y)))]
[divp^#(x, y)] = [1]
> [0]
= [c_7(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:
{ prime^#(s(s(x))) -> c_3(prime1^#(s(s(x)), s(x)))
, prime1^#(x, s(s(y))) ->
c_6(divp^#(s(s(y)), x), prime1^#(x, s(y)))
, divp^#(x, y) -> c_7(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.
{ prime^#(s(s(x))) -> c_3(prime1^#(s(s(x)), s(x)))
, prime1^#(x, s(s(y))) ->
c_6(divp^#(s(s(y)), x), prime1^#(x, s(y)))
, divp^#(x, y) -> c_7(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
* Path {3,7,6}->{4}: 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 DPs:
{ prime^#(s(s(x))) -> c_3(prime1^#(s(s(x)), s(x)))
, prime1^#(x, 0()) -> c_4()
, prime1^#(x, s(s(y))) ->
c_6(divp^#(s(s(y)), x), prime1^#(x, s(y))) }
Weak DPs: { divp^#(x, y) -> c_7(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: prime^#(s(s(x))) -> c_3(prime1^#(s(s(x)), s(x)))
, 2: prime1^#(x, 0()) -> c_4() }
Sub-proof:
----------
The following argument positions are usable:
Uargs(c_3) = {1}, Uargs(c_6) = {1, 2}
TcT has computed the following constructor-based matrix
interpretation satisfying not(EDA).
[0] = [2]
[s](x1) = [1] x1 + [0]
[prime^#](x1) = [4] x1 + [5]
[c_3](x1) = [1] x1 + [0]
[prime1^#](x1, x2) = [4] x2 + [0]
[c_4] = [3]
[c_6](x1, x2) = [4] x1 + [1] x2 + [0]
[divp^#](x1, x2) = [0]
[c_7](x1, x2) = [0]
The order satisfies the following ordering constraints:
[prime^#(s(s(x)))] = [4] x + [5]
> [4] x + [0]
= [c_3(prime1^#(s(s(x)), s(x)))]
[prime1^#(x, 0())] = [8]
> [3]
= [c_4()]
[prime1^#(x, s(s(y)))] = [4] y + [0]
>= [4] y + [0]
= [c_6(divp^#(s(s(y)), x), prime1^#(x, s(y)))]
[divp^#(x, y)] = [0]
>= [0]
= [c_7(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:
{ prime1^#(x, s(s(y))) ->
c_6(divp^#(s(s(y)), x), prime1^#(x, s(y))) }
Weak DPs:
{ prime^#(s(s(x))) -> c_3(prime1^#(s(s(x)), s(x)))
, prime1^#(x, 0()) -> c_4()
, divp^#(x, y) -> c_7(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.
{ prime1^#(x, 0()) -> c_4() }
We are left with following problem, upon which TcT provides the
certificate YES(O(1),O(n^1)).
Strict DPs:
{ prime1^#(x, s(s(y))) ->
c_6(divp^#(s(s(y)), x), prime1^#(x, s(y))) }
Weak DPs:
{ prime^#(s(s(x))) -> c_3(prime1^#(s(s(x)), s(x)))
, divp^#(x, y) -> c_7(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: prime1^#(x, s(s(y))) ->
c_6(divp^#(s(s(y)), x), prime1^#(x, s(y)))
, 2: prime^#(s(s(x))) -> c_3(prime1^#(s(s(x)), s(x)))
, 3: divp^#(x, y) -> c_7(x, y) }
Sub-proof:
----------
The following argument positions are usable:
Uargs(c_3) = {1}, Uargs(c_6) = {1, 2}
TcT has computed the following constructor-based matrix
interpretation satisfying not(EDA).
[0] = [0]
[s](x1) = [1] x1 + [2]
[prime^#](x1) = [2] x1 + [5]
[c_3](x1) = [1] x1 + [0]
[prime1^#](x1, x2) = [1] x2 + [0]
[c_4] = [0]
[c_6](x1, x2) = [1] x1 + [1] x2 + [0]
[divp^#](x1, x2) = [1]
[c_7](x1, x2) = [0]
The order satisfies the following ordering constraints:
[prime^#(s(s(x)))] = [2] x + [13]
> [1] x + [2]
= [c_3(prime1^#(s(s(x)), s(x)))]
[prime1^#(x, s(s(y)))] = [1] y + [4]
> [1] y + [3]
= [c_6(divp^#(s(s(y)), x), prime1^#(x, s(y)))]
[divp^#(x, y)] = [1]
> [0]
= [c_7(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:
{ prime^#(s(s(x))) -> c_3(prime1^#(s(s(x)), s(x)))
, prime1^#(x, s(s(y))) ->
c_6(divp^#(s(s(y)), x), prime1^#(x, s(y)))
, divp^#(x, y) -> c_7(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.
{ prime^#(s(s(x))) -> c_3(prime1^#(s(s(x)), s(x)))
, prime1^#(x, s(s(y))) ->
c_6(divp^#(s(s(y)), x), prime1^#(x, s(y)))
, divp^#(x, y) -> c_7(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
* Path {3,7,6}->{5}: 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 DPs:
{ prime^#(s(s(x))) -> c_3(prime1^#(s(s(x)), s(x)))
, prime1^#(x, s(0())) -> c_5()
, prime1^#(x, s(s(y))) ->
c_6(divp^#(s(s(y)), x), prime1^#(x, s(y))) }
Weak DPs: { divp^#(x, y) -> c_7(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: prime^#(s(s(x))) -> c_3(prime1^#(s(s(x)), s(x)))
, 2: prime1^#(x, s(0())) -> c_5() }
Sub-proof:
----------
The following argument positions are usable:
Uargs(c_3) = {1}, Uargs(c_6) = {1, 2}
TcT has computed the following constructor-based matrix
interpretation satisfying not(EDA).
[0] = [2]
[s](x1) = [1] x1 + [0]
[prime^#](x1) = [4] x1 + [4]
[c_3](x1) = [1] x1 + [0]
[prime1^#](x1, x2) = [4] x2 + [0]
[c_5] = [0]
[c_6](x1, x2) = [4] x1 + [1] x2 + [0]
[divp^#](x1, x2) = [0]
[c_7](x1, x2) = [0]
The order satisfies the following ordering constraints:
[prime^#(s(s(x)))] = [4] x + [4]
> [4] x + [0]
= [c_3(prime1^#(s(s(x)), s(x)))]
[prime1^#(x, s(0()))] = [8]
> [0]
= [c_5()]
[prime1^#(x, s(s(y)))] = [4] y + [0]
>= [4] y + [0]
= [c_6(divp^#(s(s(y)), x), prime1^#(x, s(y)))]
[divp^#(x, y)] = [0]
>= [0]
= [c_7(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:
{ prime1^#(x, s(s(y))) ->
c_6(divp^#(s(s(y)), x), prime1^#(x, s(y))) }
Weak DPs:
{ prime^#(s(s(x))) -> c_3(prime1^#(s(s(x)), s(x)))
, prime1^#(x, s(0())) -> c_5()
, divp^#(x, y) -> c_7(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.
{ prime1^#(x, s(0())) -> c_5() }
We are left with following problem, upon which TcT provides the
certificate YES(O(1),O(n^1)).
Strict DPs:
{ prime1^#(x, s(s(y))) ->
c_6(divp^#(s(s(y)), x), prime1^#(x, s(y))) }
Weak DPs:
{ prime^#(s(s(x))) -> c_3(prime1^#(s(s(x)), s(x)))
, divp^#(x, y) -> c_7(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: prime1^#(x, s(s(y))) ->
c_6(divp^#(s(s(y)), x), prime1^#(x, s(y)))
, 2: prime^#(s(s(x))) -> c_3(prime1^#(s(s(x)), s(x)))
, 3: divp^#(x, y) -> c_7(x, y) }
Sub-proof:
----------
The following argument positions are usable:
Uargs(c_3) = {1}, Uargs(c_6) = {1, 2}
TcT has computed the following constructor-based matrix
interpretation satisfying not(EDA).
[0] = [0]
[s](x1) = [1] x1 + [2]
[prime^#](x1) = [2] x1 + [5]
[c_3](x1) = [1] x1 + [0]
[prime1^#](x1, x2) = [1] x2 + [0]
[c_5] = [0]
[c_6](x1, x2) = [1] x1 + [1] x2 + [0]
[divp^#](x1, x2) = [1]
[c_7](x1, x2) = [0]
The order satisfies the following ordering constraints:
[prime^#(s(s(x)))] = [2] x + [13]
> [1] x + [2]
= [c_3(prime1^#(s(s(x)), s(x)))]
[prime1^#(x, s(s(y)))] = [1] y + [4]
> [1] y + [3]
= [c_6(divp^#(s(s(y)), x), prime1^#(x, s(y)))]
[divp^#(x, y)] = [1]
> [0]
= [c_7(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:
{ prime^#(s(s(x))) -> c_3(prime1^#(s(s(x)), s(x)))
, prime1^#(x, s(s(y))) ->
c_6(divp^#(s(s(y)), x), prime1^#(x, s(y)))
, divp^#(x, y) -> c_7(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.
{ prime^#(s(s(x))) -> c_3(prime1^#(s(s(x)), s(x)))
, prime1^#(x, s(s(y))) ->
c_6(divp^#(s(s(y)), x), prime1^#(x, s(y)))
, divp^#(x, y) -> c_7(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))