We are left with following problem, upon which TcT provides the
certificate YES(O(1),O(n^1)).
Strict Trs:
{ compS_f#1(compS_f(x2), x1) -> compS_f#1(x2, S(x1))
, compS_f#1(id(), x3) -> S(x3)
, iter#3(S(x6)) -> compS_f(iter#3(x6))
, iter#3(0()) -> id()
, main(S(x9)) -> compS_f#1(iter#3(x9), 0())
, main(0()) -> 0() }
Obligation:
innermost runtime complexity
Answer:
YES(O(1),O(n^1))
We add the following weak dependency pairs:
Strict DPs:
{ compS_f#1^#(compS_f(x2), x1) -> c_1(compS_f#1^#(x2, S(x1)))
, compS_f#1^#(id(), x3) -> c_2()
, iter#3^#(S(x6)) -> c_3(iter#3^#(x6))
, iter#3^#(0()) -> c_4()
, main^#(S(x9)) -> c_5(compS_f#1^#(iter#3(x9), 0()))
, main^#(0()) -> c_6() }
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:
{ compS_f#1^#(compS_f(x2), x1) -> c_1(compS_f#1^#(x2, S(x1)))
, compS_f#1^#(id(), x3) -> c_2()
, iter#3^#(S(x6)) -> c_3(iter#3^#(x6))
, iter#3^#(0()) -> c_4()
, main^#(S(x9)) -> c_5(compS_f#1^#(iter#3(x9), 0()))
, main^#(0()) -> c_6() }
Strict Trs:
{ compS_f#1(compS_f(x2), x1) -> compS_f#1(x2, S(x1))
, compS_f#1(id(), x3) -> S(x3)
, iter#3(S(x6)) -> compS_f(iter#3(x6))
, iter#3(0()) -> id()
, main(S(x9)) -> compS_f#1(iter#3(x9), 0())
, main(0()) -> 0() }
Obligation:
innermost runtime complexity
Answer:
YES(O(1),O(n^1))
We replace rewrite rules by usable rules:
Strict Usable Rules:
{ iter#3(S(x6)) -> compS_f(iter#3(x6))
, iter#3(0()) -> id() }
We are left with following problem, upon which TcT provides the
certificate YES(O(1),O(n^1)).
Strict DPs:
{ compS_f#1^#(compS_f(x2), x1) -> c_1(compS_f#1^#(x2, S(x1)))
, compS_f#1^#(id(), x3) -> c_2()
, iter#3^#(S(x6)) -> c_3(iter#3^#(x6))
, iter#3^#(0()) -> c_4()
, main^#(S(x9)) -> c_5(compS_f#1^#(iter#3(x9), 0()))
, main^#(0()) -> c_6() }
Strict Trs:
{ iter#3(S(x6)) -> compS_f(iter#3(x6))
, iter#3(0()) -> id() }
Obligation:
innermost 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(compS_f) = {1}, Uargs(compS_f#1^#) = {1}, Uargs(c_1) = {1},
Uargs(c_3) = {1}, Uargs(c_5) = {1}
TcT has computed the following constructor-restricted matrix
interpretation.
[compS_f](x1) = [1 0] x1 + [0]
[0 0] [0]
[S](x1) = [1 0] x1 + [2]
[0 1] [1]
[id] = [0]
[0]
[iter#3](x1) = [0 1] x1 + [0]
[2 0] [0]
[0] = [0]
[1]
[compS_f#1^#](x1, x2) = [1 0] x1 + [0]
[0 0] [0]
[c_1](x1) = [1 0] x1 + [0]
[0 1] [0]
[c_2] = [0]
[0]
[iter#3^#](x1) = [0]
[0]
[c_3](x1) = [1 0] x1 + [0]
[0 1] [0]
[c_4] = [0]
[0]
[main^#](x1) = [0 1] x1 + [0]
[0 0] [0]
[c_5](x1) = [1 0] x1 + [0]
[0 1] [0]
[c_6] = [0]
[0]
The order satisfies the following ordering constraints:
[iter#3(S(x6))] = [0 1] x6 + [1]
[2 0] [4]
> [0 1] x6 + [0]
[0 0] [0]
= [compS_f(iter#3(x6))]
[iter#3(0())] = [1]
[0]
> [0]
[0]
= [id()]
[compS_f#1^#(compS_f(x2), x1)] = [1 0] x2 + [0]
[0 0] [0]
>= [1 0] x2 + [0]
[0 0] [0]
= [c_1(compS_f#1^#(x2, S(x1)))]
[compS_f#1^#(id(), x3)] = [0]
[0]
>= [0]
[0]
= [c_2()]
[iter#3^#(S(x6))] = [0]
[0]
>= [0]
[0]
= [c_3(iter#3^#(x6))]
[iter#3^#(0())] = [0]
[0]
>= [0]
[0]
= [c_4()]
[main^#(S(x9))] = [0 1] x9 + [1]
[0 0] [0]
> [0 1] x9 + [0]
[0 0] [0]
= [c_5(compS_f#1^#(iter#3(x9), 0()))]
[main^#(0())] = [1]
[0]
> [0]
[0]
= [c_6()]
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:
{ compS_f#1^#(compS_f(x2), x1) -> c_1(compS_f#1^#(x2, S(x1)))
, compS_f#1^#(id(), x3) -> c_2()
, iter#3^#(S(x6)) -> c_3(iter#3^#(x6))
, iter#3^#(0()) -> c_4() }
Weak DPs:
{ main^#(S(x9)) -> c_5(compS_f#1^#(iter#3(x9), 0()))
, main^#(0()) -> c_6() }
Weak Trs:
{ iter#3(S(x6)) -> compS_f(iter#3(x6))
, iter#3(0()) -> id() }
Obligation:
innermost runtime complexity
Answer:
YES(O(1),O(n^1))
We estimate the number of application of {4} by applications of
Pre({4}) = {3}. Here rules are labeled as follows:
DPs:
{ 1: compS_f#1^#(compS_f(x2), x1) -> c_1(compS_f#1^#(x2, S(x1)))
, 2: compS_f#1^#(id(), x3) -> c_2()
, 3: iter#3^#(S(x6)) -> c_3(iter#3^#(x6))
, 4: iter#3^#(0()) -> c_4()
, 5: main^#(S(x9)) -> c_5(compS_f#1^#(iter#3(x9), 0()))
, 6: main^#(0()) -> c_6() }
We are left with following problem, upon which TcT provides the
certificate YES(O(1),O(n^1)).
Strict DPs:
{ compS_f#1^#(compS_f(x2), x1) -> c_1(compS_f#1^#(x2, S(x1)))
, compS_f#1^#(id(), x3) -> c_2()
, iter#3^#(S(x6)) -> c_3(iter#3^#(x6)) }
Weak DPs:
{ iter#3^#(0()) -> c_4()
, main^#(S(x9)) -> c_5(compS_f#1^#(iter#3(x9), 0()))
, main^#(0()) -> c_6() }
Weak Trs:
{ iter#3(S(x6)) -> compS_f(iter#3(x6))
, iter#3(0()) -> id() }
Obligation:
innermost 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.
{ iter#3^#(0()) -> c_4()
, main^#(0()) -> c_6() }
We are left with following problem, upon which TcT provides the
certificate YES(O(1),O(n^1)).
Strict DPs:
{ compS_f#1^#(compS_f(x2), x1) -> c_1(compS_f#1^#(x2, S(x1)))
, compS_f#1^#(id(), x3) -> c_2()
, iter#3^#(S(x6)) -> c_3(iter#3^#(x6)) }
Weak DPs: { main^#(S(x9)) -> c_5(compS_f#1^#(iter#3(x9), 0())) }
Weak Trs:
{ iter#3(S(x6)) -> compS_f(iter#3(x6))
, iter#3(0()) -> id() }
Obligation:
innermost runtime complexity
Answer:
YES(O(1),O(n^1))
We analyse the complexity of following sub-problems (R) and (S).
Problem (S) is obtained from the input problem by shifting strict
rules from (R) into the weak component:
Problem (R):
------------
Strict DPs: { compS_f#1^#(id(), x3) -> c_2() }
Weak DPs:
{ compS_f#1^#(compS_f(x2), x1) -> c_1(compS_f#1^#(x2, S(x1)))
, iter#3^#(S(x6)) -> c_3(iter#3^#(x6))
, main^#(S(x9)) -> c_5(compS_f#1^#(iter#3(x9), 0())) }
Weak Trs:
{ iter#3(S(x6)) -> compS_f(iter#3(x6))
, iter#3(0()) -> id() }
StartTerms: basic terms
Strategy: innermost
Problem (S):
------------
Strict DPs:
{ compS_f#1^#(compS_f(x2), x1) -> c_1(compS_f#1^#(x2, S(x1)))
, iter#3^#(S(x6)) -> c_3(iter#3^#(x6)) }
Weak DPs:
{ compS_f#1^#(id(), x3) -> c_2()
, main^#(S(x9)) -> c_5(compS_f#1^#(iter#3(x9), 0())) }
Weak Trs:
{ iter#3(S(x6)) -> compS_f(iter#3(x6))
, iter#3(0()) -> id() }
StartTerms: basic terms
Strategy: innermost
Overall, the transformation results in the following sub-problem(s):
Generated new problems:
-----------------------
R) Strict DPs: { compS_f#1^#(id(), x3) -> c_2() }
Weak DPs:
{ compS_f#1^#(compS_f(x2), x1) -> c_1(compS_f#1^#(x2, S(x1)))
, iter#3^#(S(x6)) -> c_3(iter#3^#(x6))
, main^#(S(x9)) -> c_5(compS_f#1^#(iter#3(x9), 0())) }
Weak Trs:
{ iter#3(S(x6)) -> compS_f(iter#3(x6))
, iter#3(0()) -> id() }
StartTerms: basic terms
Strategy: innermost
This problem was proven YES(O(1),O(n^1)).
S) Strict DPs:
{ compS_f#1^#(compS_f(x2), x1) -> c_1(compS_f#1^#(x2, S(x1)))
, iter#3^#(S(x6)) -> c_3(iter#3^#(x6)) }
Weak DPs:
{ compS_f#1^#(id(), x3) -> c_2()
, main^#(S(x9)) -> c_5(compS_f#1^#(iter#3(x9), 0())) }
Weak Trs:
{ iter#3(S(x6)) -> compS_f(iter#3(x6))
, iter#3(0()) -> id() }
StartTerms: basic terms
Strategy: innermost
This problem was proven YES(O(1),O(n^1)).
Proofs for generated problems:
------------------------------
R) We are left with following problem, upon which TcT provides the
certificate YES(O(1),O(n^1)).
Strict DPs: { compS_f#1^#(id(), x3) -> c_2() }
Weak DPs:
{ compS_f#1^#(compS_f(x2), x1) -> c_1(compS_f#1^#(x2, S(x1)))
, iter#3^#(S(x6)) -> c_3(iter#3^#(x6))
, main^#(S(x9)) -> c_5(compS_f#1^#(iter#3(x9), 0())) }
Weak Trs:
{ iter#3(S(x6)) -> compS_f(iter#3(x6))
, iter#3(0()) -> id() }
Obligation:
innermost 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.
{ iter#3^#(S(x6)) -> c_3(iter#3^#(x6)) }
We are left with following problem, upon which TcT provides the
certificate YES(O(1),O(n^1)).
Strict DPs: { compS_f#1^#(id(), x3) -> c_2() }
Weak DPs:
{ compS_f#1^#(compS_f(x2), x1) -> c_1(compS_f#1^#(x2, S(x1)))
, main^#(S(x9)) -> c_5(compS_f#1^#(iter#3(x9), 0())) }
Weak Trs:
{ iter#3(S(x6)) -> compS_f(iter#3(x6))
, iter#3(0()) -> id() }
Obligation:
innermost runtime complexity
Answer:
YES(O(1),O(n^1))
We use the processor 'Small Polynomial Path Order (PS,1-bounded)'
to orient following rules strictly.
DPs:
{ 1: compS_f#1^#(id(), x3) -> c_2()
, 2: compS_f#1^#(compS_f(x2), x1) -> c_1(compS_f#1^#(x2, S(x1)))
, 3: main^#(S(x9)) -> c_5(compS_f#1^#(iter#3(x9), 0())) }
Sub-proof:
----------
The input was oriented with the instance of 'Small Polynomial Path
Order (PS,1-bounded)' as induced by the safe mapping
safe(compS_f) = {1}, safe(S) = {1}, safe(id) = {},
safe(iter#3) = {}, safe(0) = {}, safe(compS_f#1^#) = {2},
safe(c_1) = {}, safe(c_2) = {}, safe(main^#) = {}, safe(c_5) = {}
and precedence
compS_f#1^# ~ main^# .
Following symbols are considered recursive:
{compS_f#1^#, main^#}
The recursion depth is 1.
Further, following argument filtering is employed:
pi(compS_f) = [1], pi(S) = [1], pi(id) = [], pi(iter#3) = 1,
pi(0) = [], pi(compS_f#1^#) = [1, 2], pi(c_1) = [1], pi(c_2) = [],
pi(main^#) = [1], pi(c_5) = [1]
Usable defined function symbols are a subset of:
{iter#3, compS_f#1^#, main^#}
For your convenience, here are the satisfied ordering constraints:
pi(compS_f#1^#(compS_f(x2), x1)) = compS_f#1^#(compS_f(; x2); x1)
> c_1(compS_f#1^#(x2; S(; x1));)
= pi(c_1(compS_f#1^#(x2, S(x1))))
pi(compS_f#1^#(id(), x3)) = compS_f#1^#(id(); x3)
> c_2()
= pi(c_2())
pi(main^#(S(x9))) = main^#(S(; x9);)
> c_5(compS_f#1^#(x9; 0());)
= pi(c_5(compS_f#1^#(iter#3(x9), 0())))
pi(iter#3(S(x6))) = S(; x6)
>= compS_f(; x6)
= pi(compS_f(iter#3(x6)))
pi(iter#3(0())) = 0()
>= id()
= pi(id())
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:
{ compS_f#1^#(compS_f(x2), x1) -> c_1(compS_f#1^#(x2, S(x1)))
, compS_f#1^#(id(), x3) -> c_2()
, main^#(S(x9)) -> c_5(compS_f#1^#(iter#3(x9), 0())) }
Weak Trs:
{ iter#3(S(x6)) -> compS_f(iter#3(x6))
, iter#3(0()) -> id() }
Obligation:
innermost 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.
{ compS_f#1^#(compS_f(x2), x1) -> c_1(compS_f#1^#(x2, S(x1)))
, compS_f#1^#(id(), x3) -> c_2()
, main^#(S(x9)) -> c_5(compS_f#1^#(iter#3(x9), 0())) }
We are left with following problem, upon which TcT provides the
certificate YES(O(1),O(1)).
Weak Trs:
{ iter#3(S(x6)) -> compS_f(iter#3(x6))
, iter#3(0()) -> id() }
Obligation:
innermost 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:
innermost runtime complexity
Answer:
YES(O(1),O(1))
Empty rules are trivially bounded
S) We are left with following problem, upon which TcT provides the
certificate YES(O(1),O(n^1)).
Strict DPs:
{ compS_f#1^#(compS_f(x2), x1) -> c_1(compS_f#1^#(x2, S(x1)))
, iter#3^#(S(x6)) -> c_3(iter#3^#(x6)) }
Weak DPs:
{ compS_f#1^#(id(), x3) -> c_2()
, main^#(S(x9)) -> c_5(compS_f#1^#(iter#3(x9), 0())) }
Weak Trs:
{ iter#3(S(x6)) -> compS_f(iter#3(x6))
, iter#3(0()) -> id() }
Obligation:
innermost 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.
{ compS_f#1^#(id(), x3) -> c_2() }
We are left with following problem, upon which TcT provides the
certificate YES(O(1),O(n^1)).
Strict DPs:
{ compS_f#1^#(compS_f(x2), x1) -> c_1(compS_f#1^#(x2, S(x1)))
, iter#3^#(S(x6)) -> c_3(iter#3^#(x6)) }
Weak DPs: { main^#(S(x9)) -> c_5(compS_f#1^#(iter#3(x9), 0())) }
Weak Trs:
{ iter#3(S(x6)) -> compS_f(iter#3(x6))
, iter#3(0()) -> id() }
Obligation:
innermost runtime complexity
Answer:
YES(O(1),O(n^1))
We analyse the complexity of following sub-problems (R) and (S).
Problem (S) is obtained from the input problem by shifting strict
rules from (R) into the weak component:
Problem (R):
------------
Strict DPs:
{ compS_f#1^#(compS_f(x2), x1) -> c_1(compS_f#1^#(x2, S(x1))) }
Weak DPs:
{ iter#3^#(S(x6)) -> c_3(iter#3^#(x6))
, main^#(S(x9)) -> c_5(compS_f#1^#(iter#3(x9), 0())) }
Weak Trs:
{ iter#3(S(x6)) -> compS_f(iter#3(x6))
, iter#3(0()) -> id() }
StartTerms: basic terms
Strategy: innermost
Problem (S):
------------
Strict DPs: { iter#3^#(S(x6)) -> c_3(iter#3^#(x6)) }
Weak DPs:
{ compS_f#1^#(compS_f(x2), x1) -> c_1(compS_f#1^#(x2, S(x1)))
, main^#(S(x9)) -> c_5(compS_f#1^#(iter#3(x9), 0())) }
Weak Trs:
{ iter#3(S(x6)) -> compS_f(iter#3(x6))
, iter#3(0()) -> id() }
StartTerms: basic terms
Strategy: innermost
Overall, the transformation results in the following sub-problem(s):
Generated new problems:
-----------------------
R) Strict DPs:
{ compS_f#1^#(compS_f(x2), x1) -> c_1(compS_f#1^#(x2, S(x1))) }
Weak DPs:
{ iter#3^#(S(x6)) -> c_3(iter#3^#(x6))
, main^#(S(x9)) -> c_5(compS_f#1^#(iter#3(x9), 0())) }
Weak Trs:
{ iter#3(S(x6)) -> compS_f(iter#3(x6))
, iter#3(0()) -> id() }
StartTerms: basic terms
Strategy: innermost
This problem was proven YES(O(1),O(n^1)).
S) Strict DPs: { iter#3^#(S(x6)) -> c_3(iter#3^#(x6)) }
Weak DPs:
{ compS_f#1^#(compS_f(x2), x1) -> c_1(compS_f#1^#(x2, S(x1)))
, main^#(S(x9)) -> c_5(compS_f#1^#(iter#3(x9), 0())) }
Weak Trs:
{ iter#3(S(x6)) -> compS_f(iter#3(x6))
, iter#3(0()) -> id() }
StartTerms: basic terms
Strategy: innermost
This problem was proven YES(O(1),O(n^1)).
Proofs for generated problems:
------------------------------
R) We are left with following problem, upon which TcT provides the
certificate YES(O(1),O(n^1)).
Strict DPs:
{ compS_f#1^#(compS_f(x2), x1) -> c_1(compS_f#1^#(x2, S(x1))) }
Weak DPs:
{ iter#3^#(S(x6)) -> c_3(iter#3^#(x6))
, main^#(S(x9)) -> c_5(compS_f#1^#(iter#3(x9), 0())) }
Weak Trs:
{ iter#3(S(x6)) -> compS_f(iter#3(x6))
, iter#3(0()) -> id() }
Obligation:
innermost 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.
{ iter#3^#(S(x6)) -> c_3(iter#3^#(x6)) }
We are left with following problem, upon which TcT provides the
certificate YES(O(1),O(n^1)).
Strict DPs:
{ compS_f#1^#(compS_f(x2), x1) -> c_1(compS_f#1^#(x2, S(x1))) }
Weak DPs: { main^#(S(x9)) -> c_5(compS_f#1^#(iter#3(x9), 0())) }
Weak Trs:
{ iter#3(S(x6)) -> compS_f(iter#3(x6))
, iter#3(0()) -> id() }
Obligation:
innermost runtime complexity
Answer:
YES(O(1),O(n^1))
We use the processor 'Small Polynomial Path Order (PS,1-bounded)'
to orient following rules strictly.
DPs:
{ 1: compS_f#1^#(compS_f(x2), x1) -> c_1(compS_f#1^#(x2, S(x1)))
, 2: main^#(S(x9)) -> c_5(compS_f#1^#(iter#3(x9), 0())) }
Sub-proof:
----------
The input was oriented with the instance of 'Small Polynomial Path
Order (PS,1-bounded)' as induced by the safe mapping
safe(compS_f) = {1}, safe(S) = {1}, safe(id) = {},
safe(iter#3) = {}, safe(0) = {}, safe(compS_f#1^#) = {2},
safe(c_1) = {}, safe(main^#) = {}, safe(c_5) = {}
and precedence
compS_f#1^# ~ main^# .
Following symbols are considered recursive:
{compS_f#1^#, main^#}
The recursion depth is 1.
Further, following argument filtering is employed:
pi(compS_f) = [1], pi(S) = [1], pi(id) = [], pi(iter#3) = 1,
pi(0) = [], pi(compS_f#1^#) = [1], pi(c_1) = [1], pi(main^#) = [1],
pi(c_5) = [1]
Usable defined function symbols are a subset of:
{iter#3, compS_f#1^#, main^#}
For your convenience, here are the satisfied ordering constraints:
pi(compS_f#1^#(compS_f(x2), x1)) = compS_f#1^#(compS_f(; x2);)
> c_1(compS_f#1^#(x2;);)
= pi(c_1(compS_f#1^#(x2, S(x1))))
pi(main^#(S(x9))) = main^#(S(; x9);)
> c_5(compS_f#1^#(x9;);)
= pi(c_5(compS_f#1^#(iter#3(x9), 0())))
pi(iter#3(S(x6))) = S(; x6)
>= compS_f(; x6)
= pi(compS_f(iter#3(x6)))
pi(iter#3(0())) = 0()
>= id()
= pi(id())
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:
{ compS_f#1^#(compS_f(x2), x1) -> c_1(compS_f#1^#(x2, S(x1)))
, main^#(S(x9)) -> c_5(compS_f#1^#(iter#3(x9), 0())) }
Weak Trs:
{ iter#3(S(x6)) -> compS_f(iter#3(x6))
, iter#3(0()) -> id() }
Obligation:
innermost 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.
{ compS_f#1^#(compS_f(x2), x1) -> c_1(compS_f#1^#(x2, S(x1)))
, main^#(S(x9)) -> c_5(compS_f#1^#(iter#3(x9), 0())) }
We are left with following problem, upon which TcT provides the
certificate YES(O(1),O(1)).
Weak Trs:
{ iter#3(S(x6)) -> compS_f(iter#3(x6))
, iter#3(0()) -> id() }
Obligation:
innermost 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:
innermost runtime complexity
Answer:
YES(O(1),O(1))
Empty rules are trivially bounded
S) We are left with following problem, upon which TcT provides the
certificate YES(O(1),O(n^1)).
Strict DPs: { iter#3^#(S(x6)) -> c_3(iter#3^#(x6)) }
Weak DPs:
{ compS_f#1^#(compS_f(x2), x1) -> c_1(compS_f#1^#(x2, S(x1)))
, main^#(S(x9)) -> c_5(compS_f#1^#(iter#3(x9), 0())) }
Weak Trs:
{ iter#3(S(x6)) -> compS_f(iter#3(x6))
, iter#3(0()) -> id() }
Obligation:
innermost 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.
{ compS_f#1^#(compS_f(x2), x1) -> c_1(compS_f#1^#(x2, S(x1)))
, main^#(S(x9)) -> c_5(compS_f#1^#(iter#3(x9), 0())) }
We are left with following problem, upon which TcT provides the
certificate YES(O(1),O(n^1)).
Strict DPs: { iter#3^#(S(x6)) -> c_3(iter#3^#(x6)) }
Weak Trs:
{ iter#3(S(x6)) -> compS_f(iter#3(x6))
, iter#3(0()) -> id() }
Obligation:
innermost 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: { iter#3^#(S(x6)) -> c_3(iter#3^#(x6)) }
Obligation:
innermost runtime complexity
Answer:
YES(O(1),O(n^1))
We use the processor 'Small Polynomial Path Order (PS,1-bounded)'
to orient following rules strictly.
DPs:
{ 1: iter#3^#(S(x6)) -> c_3(iter#3^#(x6)) }
Sub-proof:
----------
The input was oriented with the instance of 'Small Polynomial Path
Order (PS,1-bounded)' as induced by the safe mapping
safe(S) = {1}, safe(iter#3^#) = {}, safe(c_3) = {}
and precedence
empty .
Following symbols are considered recursive:
{iter#3^#}
The recursion depth is 1.
Further, following argument filtering is employed:
pi(S) = [1], pi(iter#3^#) = [1], pi(c_3) = [1]
Usable defined function symbols are a subset of:
{iter#3^#}
For your convenience, here are the satisfied ordering constraints:
pi(iter#3^#(S(x6))) = iter#3^#(S(; x6);)
> c_3(iter#3^#(x6;);)
= pi(c_3(iter#3^#(x6)))
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: { iter#3^#(S(x6)) -> c_3(iter#3^#(x6)) }
Obligation:
innermost 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.
{ iter#3^#(S(x6)) -> c_3(iter#3^#(x6)) }
We are left with following problem, upon which TcT provides the
certificate YES(O(1),O(1)).
Rules: Empty
Obligation:
innermost runtime complexity
Answer:
YES(O(1),O(1))
Empty rules are trivially bounded
Hurray, we answered YES(O(1),O(n^1))