* Step 1: Sum WORST_CASE(?,O(1))
+ Considered Problem:
- Strict TRS:
activate(X) -> X
activate(n__f(X)) -> f(X)
f(X) -> n__f(X)
f(0()) -> cons(0(),n__f(s(0())))
f(s(0())) -> f(p(s(0())))
p(s(0())) -> 0()
- Signature:
{activate/1,f/1,p/1} / {0/0,cons/2,n__f/1,s/1}
- Obligation:
innermost runtime complexity wrt. defined symbols {activate,f,p} and constructors {0,cons,n__f,s}
+ Applied Processor:
Sum {left = someStrategy, right = someStrategy}
+ Details:
()
* Step 2: DependencyPairs WORST_CASE(?,O(1))
+ Considered Problem:
- Strict TRS:
activate(X) -> X
activate(n__f(X)) -> f(X)
f(X) -> n__f(X)
f(0()) -> cons(0(),n__f(s(0())))
f(s(0())) -> f(p(s(0())))
p(s(0())) -> 0()
- Signature:
{activate/1,f/1,p/1} / {0/0,cons/2,n__f/1,s/1}
- Obligation:
innermost runtime complexity wrt. defined symbols {activate,f,p} and constructors {0,cons,n__f,s}
+ Applied Processor:
DependencyPairs {dpKind_ = DT}
+ Details:
We add the following dependency tuples:
Strict DPs
activate#(X) -> c_1()
activate#(n__f(X)) -> c_2(f#(X))
f#(X) -> c_3()
f#(0()) -> c_4()
f#(s(0())) -> c_5(f#(p(s(0()))),p#(s(0())))
p#(s(0())) -> c_6()
Weak DPs
and mark the set of starting terms.
* Step 3: UsableRules WORST_CASE(?,O(1))
+ Considered Problem:
- Strict DPs:
activate#(X) -> c_1()
activate#(n__f(X)) -> c_2(f#(X))
f#(X) -> c_3()
f#(0()) -> c_4()
f#(s(0())) -> c_5(f#(p(s(0()))),p#(s(0())))
p#(s(0())) -> c_6()
- Weak TRS:
activate(X) -> X
activate(n__f(X)) -> f(X)
f(X) -> n__f(X)
f(0()) -> cons(0(),n__f(s(0())))
f(s(0())) -> f(p(s(0())))
p(s(0())) -> 0()
- Signature:
{activate/1,f/1,p/1,activate#/1,f#/1,p#/1} / {0/0,cons/2,n__f/1,s/1,c_1/0,c_2/1,c_3/0,c_4/0,c_5/2,c_6/0}
- Obligation:
innermost runtime complexity wrt. defined symbols {activate#,f#,p#} and constructors {0,cons,n__f,s}
+ Applied Processor:
UsableRules
+ Details:
We replace rewrite rules by usable rules:
p(s(0())) -> 0()
activate#(X) -> c_1()
activate#(n__f(X)) -> c_2(f#(X))
f#(X) -> c_3()
f#(0()) -> c_4()
f#(s(0())) -> c_5(f#(p(s(0()))),p#(s(0())))
p#(s(0())) -> c_6()
* Step 4: PredecessorEstimation WORST_CASE(?,O(1))
+ Considered Problem:
- Strict DPs:
activate#(X) -> c_1()
activate#(n__f(X)) -> c_2(f#(X))
f#(X) -> c_3()
f#(0()) -> c_4()
f#(s(0())) -> c_5(f#(p(s(0()))),p#(s(0())))
p#(s(0())) -> c_6()
- Weak TRS:
p(s(0())) -> 0()
- Signature:
{activate/1,f/1,p/1,activate#/1,f#/1,p#/1} / {0/0,cons/2,n__f/1,s/1,c_1/0,c_2/1,c_3/0,c_4/0,c_5/2,c_6/0}
- Obligation:
innermost runtime complexity wrt. defined symbols {activate#,f#,p#} and constructors {0,cons,n__f,s}
+ Applied Processor:
PredecessorEstimation {onSelection = all simple predecessor estimation selector}
+ Details:
We estimate the number of application of
{1,3,4,6}
by application of
Pre({1,3,4,6}) = {2,5}.
Here rules are labelled as follows:
1: activate#(X) -> c_1()
2: activate#(n__f(X)) -> c_2(f#(X))
3: f#(X) -> c_3()
4: f#(0()) -> c_4()
5: f#(s(0())) -> c_5(f#(p(s(0()))),p#(s(0())))
6: p#(s(0())) -> c_6()
* Step 5: RemoveWeakSuffixes WORST_CASE(?,O(1))
+ Considered Problem:
- Strict DPs:
activate#(n__f(X)) -> c_2(f#(X))
f#(s(0())) -> c_5(f#(p(s(0()))),p#(s(0())))
- Weak DPs:
activate#(X) -> c_1()
f#(X) -> c_3()
f#(0()) -> c_4()
p#(s(0())) -> c_6()
- Weak TRS:
p(s(0())) -> 0()
- Signature:
{activate/1,f/1,p/1,activate#/1,f#/1,p#/1} / {0/0,cons/2,n__f/1,s/1,c_1/0,c_2/1,c_3/0,c_4/0,c_5/2,c_6/0}
- Obligation:
innermost runtime complexity wrt. defined symbols {activate#,f#,p#} and constructors {0,cons,n__f,s}
+ Applied Processor:
RemoveWeakSuffixes
+ Details:
Consider the dependency graph
1:S:activate#(n__f(X)) -> c_2(f#(X))
-->_1 f#(s(0())) -> c_5(f#(p(s(0()))),p#(s(0()))):2
-->_1 f#(0()) -> c_4():5
-->_1 f#(X) -> c_3():4
2:S:f#(s(0())) -> c_5(f#(p(s(0()))),p#(s(0())))
-->_2 p#(s(0())) -> c_6():6
-->_1 f#(0()) -> c_4():5
-->_1 f#(X) -> c_3():4
-->_1 f#(s(0())) -> c_5(f#(p(s(0()))),p#(s(0()))):2
3:W:activate#(X) -> c_1()
4:W:f#(X) -> c_3()
5:W:f#(0()) -> c_4()
6:W:p#(s(0())) -> c_6()
The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed.
3: activate#(X) -> c_1()
4: f#(X) -> c_3()
5: f#(0()) -> c_4()
6: p#(s(0())) -> c_6()
* Step 6: SimplifyRHS WORST_CASE(?,O(1))
+ Considered Problem:
- Strict DPs:
activate#(n__f(X)) -> c_2(f#(X))
f#(s(0())) -> c_5(f#(p(s(0()))),p#(s(0())))
- Weak TRS:
p(s(0())) -> 0()
- Signature:
{activate/1,f/1,p/1,activate#/1,f#/1,p#/1} / {0/0,cons/2,n__f/1,s/1,c_1/0,c_2/1,c_3/0,c_4/0,c_5/2,c_6/0}
- Obligation:
innermost runtime complexity wrt. defined symbols {activate#,f#,p#} and constructors {0,cons,n__f,s}
+ Applied Processor:
SimplifyRHS
+ Details:
Consider the dependency graph
1:S:activate#(n__f(X)) -> c_2(f#(X))
-->_1 f#(s(0())) -> c_5(f#(p(s(0()))),p#(s(0()))):2
2:S:f#(s(0())) -> c_5(f#(p(s(0()))),p#(s(0())))
-->_1 f#(s(0())) -> c_5(f#(p(s(0()))),p#(s(0()))):2
Due to missing edges in the depndency graph, the right-hand sides of following rules could be simplified:
f#(s(0())) -> c_5(f#(p(s(0()))))
* Step 7: RemoveHeads WORST_CASE(?,O(1))
+ Considered Problem:
- Strict DPs:
activate#(n__f(X)) -> c_2(f#(X))
f#(s(0())) -> c_5(f#(p(s(0()))))
- Weak TRS:
p(s(0())) -> 0()
- Signature:
{activate/1,f/1,p/1,activate#/1,f#/1,p#/1} / {0/0,cons/2,n__f/1,s/1,c_1/0,c_2/1,c_3/0,c_4/0,c_5/1,c_6/0}
- Obligation:
innermost runtime complexity wrt. defined symbols {activate#,f#,p#} and constructors {0,cons,n__f,s}
+ Applied Processor:
RemoveHeads
+ Details:
Consider the dependency graph
1:S:activate#(n__f(X)) -> c_2(f#(X))
-->_1 f#(s(0())) -> c_5(f#(p(s(0())))):2
2:S:f#(s(0())) -> c_5(f#(p(s(0()))))
-->_1 f#(s(0())) -> c_5(f#(p(s(0())))):2
Following roots of the dependency graph are removed, as the considered set of starting terms is closed under reduction with respect to these rules (modulo compound contexts).
[(1,activate#(n__f(X)) -> c_2(f#(X)))]
* Step 8: WeightGap WORST_CASE(?,O(1))
+ Considered Problem:
- Strict DPs:
f#(s(0())) -> c_5(f#(p(s(0()))))
- Weak TRS:
p(s(0())) -> 0()
- Signature:
{activate/1,f/1,p/1,activate#/1,f#/1,p#/1} / {0/0,cons/2,n__f/1,s/1,c_1/0,c_2/1,c_3/0,c_4/0,c_5/1,c_6/0}
- Obligation:
innermost runtime complexity wrt. defined symbols {activate#,f#,p#} and constructors {0,cons,n__f,s}
+ Applied Processor:
WeightGap {wgDimension = 1, wgDegree = 0, wgKind = Algebraic, wgUArgs = UArgs, wgOn = WgOnAny}
+ Details:
The weightgap principle applies using the following constant growth matrix-interpretation:
We apply a matrix interpretation of kind constructor based matrix interpretation (containing no more than 0 non-zero interpretation-entries in the diagonal of the component-wise maxima):
The following argument positions are considered usable:
uargs(f#) = {1},
uargs(c_5) = {1}
Following symbols are considered usable:
all
TcT has computed the following interpretation:
p(0) = [0]
p(activate) = [0]
p(cons) = [0]
p(f) = [0]
p(n__f) = [0]
p(p) = [1]
p(s) = [11]
p(activate#) = [0]
p(f#) = [1] x1 + [6]
p(p#) = [1] x1 + [1]
p(c_1) = [2]
p(c_2) = [1]
p(c_3) = [0]
p(c_4) = [2]
p(c_5) = [1] x1 + [5]
p(c_6) = [1]
Following rules are strictly oriented:
f#(s(0())) = [17]
> [12]
= c_5(f#(p(s(0()))))
Following rules are (at-least) weakly oriented:
p(s(0())) = [1]
>= [0]
= 0()
Further, it can be verified that all rules not oriented are covered by the weightgap condition.
* Step 9: EmptyProcessor WORST_CASE(?,O(1))
+ Considered Problem:
- Weak DPs:
f#(s(0())) -> c_5(f#(p(s(0()))))
- Weak TRS:
p(s(0())) -> 0()
- Signature:
{activate/1,f/1,p/1,activate#/1,f#/1,p#/1} / {0/0,cons/2,n__f/1,s/1,c_1/0,c_2/1,c_3/0,c_4/0,c_5/1,c_6/0}
- Obligation:
innermost runtime complexity wrt. defined symbols {activate#,f#,p#} and constructors {0,cons,n__f,s}
+ Applied Processor:
EmptyProcessor
+ Details:
The problem is already closed. The intended complexity is O(1).
WORST_CASE(?,O(1))