* Step 1: DependencyPairs WORST_CASE(?,O(n^1))
+ Considered Problem:
- Strict TRS:
f(x,0()) -> s(0())
f(s(x),s(y)) -> s(f(x,y))
g(0(),x) -> g(f(x,x),x)
- Signature:
{f/2,g/2} / {0/0,s/1}
- Obligation:
innermost runtime complexity wrt. defined symbols {f,g} and constructors {0,s}
+ Applied Processor:
DependencyPairs {dpKind_ = DT}
+ Details:
We add the following dependency tuples:
Strict DPs
f#(x,0()) -> c_1()
f#(s(x),s(y)) -> c_2(f#(x,y))
g#(0(),x) -> c_3(g#(f(x,x),x),f#(x,x))
Weak DPs
and mark the set of starting terms.
* Step 2: UsableRules WORST_CASE(?,O(n^1))
+ Considered Problem:
- Strict DPs:
f#(x,0()) -> c_1()
f#(s(x),s(y)) -> c_2(f#(x,y))
g#(0(),x) -> c_3(g#(f(x,x),x),f#(x,x))
- Weak TRS:
f(x,0()) -> s(0())
f(s(x),s(y)) -> s(f(x,y))
g(0(),x) -> g(f(x,x),x)
- Signature:
{f/2,g/2,f#/2,g#/2} / {0/0,s/1,c_1/0,c_2/1,c_3/2}
- Obligation:
innermost runtime complexity wrt. defined symbols {f#,g#} and constructors {0,s}
+ Applied Processor:
UsableRules
+ Details:
We replace rewrite rules by usable rules:
f(x,0()) -> s(0())
f(s(x),s(y)) -> s(f(x,y))
f#(x,0()) -> c_1()
f#(s(x),s(y)) -> c_2(f#(x,y))
g#(0(),x) -> c_3(g#(f(x,x),x),f#(x,x))
* Step 3: PredecessorEstimation WORST_CASE(?,O(n^1))
+ Considered Problem:
- Strict DPs:
f#(x,0()) -> c_1()
f#(s(x),s(y)) -> c_2(f#(x,y))
g#(0(),x) -> c_3(g#(f(x,x),x),f#(x,x))
- Weak TRS:
f(x,0()) -> s(0())
f(s(x),s(y)) -> s(f(x,y))
- Signature:
{f/2,g/2,f#/2,g#/2} / {0/0,s/1,c_1/0,c_2/1,c_3/2}
- Obligation:
innermost runtime complexity wrt. defined symbols {f#,g#} and constructors {0,s}
+ Applied Processor:
PredecessorEstimation {onSelection = all simple predecessor estimation selector}
+ Details:
We estimate the number of application of
{1}
by application of
Pre({1}) = {2,3}.
Here rules are labelled as follows:
1: f#(x,0()) -> c_1()
2: f#(s(x),s(y)) -> c_2(f#(x,y))
3: g#(0(),x) -> c_3(g#(f(x,x),x),f#(x,x))
* Step 4: RemoveWeakSuffixes WORST_CASE(?,O(n^1))
+ Considered Problem:
- Strict DPs:
f#(s(x),s(y)) -> c_2(f#(x,y))
g#(0(),x) -> c_3(g#(f(x,x),x),f#(x,x))
- Weak DPs:
f#(x,0()) -> c_1()
- Weak TRS:
f(x,0()) -> s(0())
f(s(x),s(y)) -> s(f(x,y))
- Signature:
{f/2,g/2,f#/2,g#/2} / {0/0,s/1,c_1/0,c_2/1,c_3/2}
- Obligation:
innermost runtime complexity wrt. defined symbols {f#,g#} and constructors {0,s}
+ Applied Processor:
RemoveWeakSuffixes
+ Details:
Consider the dependency graph
1:S:f#(s(x),s(y)) -> c_2(f#(x,y))
-->_1 f#(x,0()) -> c_1():3
-->_1 f#(s(x),s(y)) -> c_2(f#(x,y)):1
2:S:g#(0(),x) -> c_3(g#(f(x,x),x),f#(x,x))
-->_2 f#(x,0()) -> c_1():3
-->_1 g#(0(),x) -> c_3(g#(f(x,x),x),f#(x,x)):2
-->_2 f#(s(x),s(y)) -> c_2(f#(x,y)):1
3:W:f#(x,0()) -> c_1()
The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed.
3: f#(x,0()) -> c_1()
* Step 5: Decompose WORST_CASE(?,O(n^1))
+ Considered Problem:
- Strict DPs:
f#(s(x),s(y)) -> c_2(f#(x,y))
g#(0(),x) -> c_3(g#(f(x,x),x),f#(x,x))
- Weak TRS:
f(x,0()) -> s(0())
f(s(x),s(y)) -> s(f(x,y))
- Signature:
{f/2,g/2,f#/2,g#/2} / {0/0,s/1,c_1/0,c_2/1,c_3/2}
- Obligation:
innermost runtime complexity wrt. defined symbols {f#,g#} and constructors {0,s}
+ Applied Processor:
Decompose {onSelection = all cycle independent sub-graph, withBound = RelativeAdd}
+ Details:
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:
f#(s(x),s(y)) -> c_2(f#(x,y))
- Weak DPs:
g#(0(),x) -> c_3(g#(f(x,x),x),f#(x,x))
- Weak TRS:
f(x,0()) -> s(0())
f(s(x),s(y)) -> s(f(x,y))
- Signature:
{f/2,g/2,f#/2,g#/2} / {0/0,s/1,c_1/0,c_2/1,c_3/2}
- Obligation:
innermost runtime complexity wrt. defined symbols {f#,g#} and constructors {0,s}
Problem (S)
- Strict DPs:
g#(0(),x) -> c_3(g#(f(x,x),x),f#(x,x))
- Weak DPs:
f#(s(x),s(y)) -> c_2(f#(x,y))
- Weak TRS:
f(x,0()) -> s(0())
f(s(x),s(y)) -> s(f(x,y))
- Signature:
{f/2,g/2,f#/2,g#/2} / {0/0,s/1,c_1/0,c_2/1,c_3/2}
- Obligation:
innermost runtime complexity wrt. defined symbols {f#,g#} and constructors {0,s}
** Step 5.a:1: DecomposeDG WORST_CASE(?,O(n^1))
+ Considered Problem:
- Strict DPs:
f#(s(x),s(y)) -> c_2(f#(x,y))
- Weak DPs:
g#(0(),x) -> c_3(g#(f(x,x),x),f#(x,x))
- Weak TRS:
f(x,0()) -> s(0())
f(s(x),s(y)) -> s(f(x,y))
- Signature:
{f/2,g/2,f#/2,g#/2} / {0/0,s/1,c_1/0,c_2/1,c_3/2}
- Obligation:
innermost runtime complexity wrt. defined symbols {f#,g#} and constructors {0,s}
+ Applied Processor:
DecomposeDG {onSelection = all below first cut in WDG, onUpper = Just someStrategy, onLower = Nothing}
+ Details:
We decompose the input problem according to the dependency graph into the upper component
g#(0(),x) -> c_3(g#(f(x,x),x),f#(x,x))
and a lower component
f#(s(x),s(y)) -> c_2(f#(x,y))
Further, following extension rules are added to the lower component.
g#(0(),x) -> f#(x,x)
g#(0(),x) -> g#(f(x,x),x)
*** Step 5.a:1.a:1: PredecessorEstimationCP WORST_CASE(?,O(1))
+ Considered Problem:
- Strict DPs:
g#(0(),x) -> c_3(g#(f(x,x),x),f#(x,x))
- Weak TRS:
f(x,0()) -> s(0())
f(s(x),s(y)) -> s(f(x,y))
- Signature:
{f/2,g/2,f#/2,g#/2} / {0/0,s/1,c_1/0,c_2/1,c_3/2}
- Obligation:
innermost runtime complexity wrt. defined symbols {f#,g#} and constructors {0,s}
+ Applied Processor:
PredecessorEstimationCP {onSelectionCP = any intersect of rules of CDG leaf and strict-rules, withComplexityPair = NaturalMI {miDimension = 1, miDegree = 0, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Nothing}}
+ Details:
We first use the processor NaturalMI {miDimension = 1, miDegree = 0, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Nothing} to orient following rules strictly:
1: g#(0(),x) -> c_3(g#(f(x,x),x),f#(x,x))
The strictly oriented rules are moved into the weak component.
**** Step 5.a:1.a:1.a:1: NaturalMI WORST_CASE(?,O(1))
+ Considered Problem:
- Strict DPs:
g#(0(),x) -> c_3(g#(f(x,x),x),f#(x,x))
- Weak TRS:
f(x,0()) -> s(0())
f(s(x),s(y)) -> s(f(x,y))
- Signature:
{f/2,g/2,f#/2,g#/2} / {0/0,s/1,c_1/0,c_2/1,c_3/2}
- Obligation:
innermost runtime complexity wrt. defined symbols {f#,g#} and constructors {0,s}
+ Applied Processor:
NaturalMI {miDimension = 1, miDegree = 0, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Just first alternative for predecessorEstimation on any intersect of rules of CDG leaf and strict-rules}
+ Details:
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(c_3) = {1}
Following symbols are considered usable:
{f,f#,g#}
TcT has computed the following interpretation:
p(0) = [9]
p(f) = [0]
p(g) = [1] x2 + [0]
p(s) = [0]
p(f#) = [0]
p(g#) = [2] x1 + [0]
p(c_1) = [2]
p(c_2) = [2]
p(c_3) = [8] x1 + [8] x2 + [12]
Following rules are strictly oriented:
g#(0(),x) = [18]
> [12]
= c_3(g#(f(x,x),x),f#(x,x))
Following rules are (at-least) weakly oriented:
f(x,0()) = [0]
>= [0]
= s(0())
f(s(x),s(y)) = [0]
>= [0]
= s(f(x,y))
**** Step 5.a:1.a:1.a:2: Assumption WORST_CASE(?,O(1))
+ Considered Problem:
- Weak DPs:
g#(0(),x) -> c_3(g#(f(x,x),x),f#(x,x))
- Weak TRS:
f(x,0()) -> s(0())
f(s(x),s(y)) -> s(f(x,y))
- Signature:
{f/2,g/2,f#/2,g#/2} / {0/0,s/1,c_1/0,c_2/1,c_3/2}
- Obligation:
innermost runtime complexity wrt. defined symbols {f#,g#} and constructors {0,s}
+ Applied Processor:
Assumption {assumed = Certificate {spaceUB = Unknown, spaceLB = Unknown, timeUB = Poly (Just 0), timeLB = Unknown}}
+ Details:
()
**** Step 5.a:1.a:1.b:1: RemoveWeakSuffixes WORST_CASE(?,O(1))
+ Considered Problem:
- Weak DPs:
g#(0(),x) -> c_3(g#(f(x,x),x),f#(x,x))
- Weak TRS:
f(x,0()) -> s(0())
f(s(x),s(y)) -> s(f(x,y))
- Signature:
{f/2,g/2,f#/2,g#/2} / {0/0,s/1,c_1/0,c_2/1,c_3/2}
- Obligation:
innermost runtime complexity wrt. defined symbols {f#,g#} and constructors {0,s}
+ Applied Processor:
RemoveWeakSuffixes
+ Details:
Consider the dependency graph
1:W:g#(0(),x) -> c_3(g#(f(x,x),x),f#(x,x))
-->_1 g#(0(),x) -> c_3(g#(f(x,x),x),f#(x,x)):1
The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed.
1: g#(0(),x) -> c_3(g#(f(x,x),x),f#(x,x))
**** Step 5.a:1.a:1.b:2: EmptyProcessor WORST_CASE(?,O(1))
+ Considered Problem:
- Weak TRS:
f(x,0()) -> s(0())
f(s(x),s(y)) -> s(f(x,y))
- Signature:
{f/2,g/2,f#/2,g#/2} / {0/0,s/1,c_1/0,c_2/1,c_3/2}
- Obligation:
innermost runtime complexity wrt. defined symbols {f#,g#} and constructors {0,s}
+ Applied Processor:
EmptyProcessor
+ Details:
The problem is already closed. The intended complexity is O(1).
*** Step 5.a:1.b:1: PredecessorEstimationCP WORST_CASE(?,O(n^1))
+ Considered Problem:
- Strict DPs:
f#(s(x),s(y)) -> c_2(f#(x,y))
- Weak DPs:
g#(0(),x) -> f#(x,x)
g#(0(),x) -> g#(f(x,x),x)
- Weak TRS:
f(x,0()) -> s(0())
f(s(x),s(y)) -> s(f(x,y))
- Signature:
{f/2,g/2,f#/2,g#/2} / {0/0,s/1,c_1/0,c_2/1,c_3/2}
- Obligation:
innermost runtime complexity wrt. defined symbols {f#,g#} and constructors {0,s}
+ Applied Processor:
PredecessorEstimationCP {onSelectionCP = any intersect of rules of CDG leaf and strict-rules, withComplexityPair = NaturalMI {miDimension = 1, miDegree = 1, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Nothing}}
+ Details:
We first use the processor NaturalMI {miDimension = 1, miDegree = 1, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Nothing} to orient following rules strictly:
1: f#(s(x),s(y)) -> c_2(f#(x,y))
The strictly oriented rules are moved into the weak component.
**** Step 5.a:1.b:1.a:1: NaturalMI WORST_CASE(?,O(n^1))
+ Considered Problem:
- Strict DPs:
f#(s(x),s(y)) -> c_2(f#(x,y))
- Weak DPs:
g#(0(),x) -> f#(x,x)
g#(0(),x) -> g#(f(x,x),x)
- Weak TRS:
f(x,0()) -> s(0())
f(s(x),s(y)) -> s(f(x,y))
- Signature:
{f/2,g/2,f#/2,g#/2} / {0/0,s/1,c_1/0,c_2/1,c_3/2}
- Obligation:
innermost runtime complexity wrt. defined symbols {f#,g#} and constructors {0,s}
+ Applied Processor:
NaturalMI {miDimension = 1, miDegree = 1, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Just first alternative for predecessorEstimation on any intersect of rules of CDG leaf and strict-rules}
+ Details:
We apply a matrix interpretation of kind constructor based matrix interpretation:
The following argument positions are considered usable:
uargs(c_2) = {1}
Following symbols are considered usable:
{f#,g#}
TcT has computed the following interpretation:
p(0) = [3]
p(f) = [0]
p(g) = [1] x1 + [2] x2 + [2]
p(s) = [1] x1 + [1]
p(f#) = [2] x1 + [0]
p(g#) = [6] x2 + [0]
p(c_1) = [4]
p(c_2) = [1] x1 + [0]
p(c_3) = [1] x2 + [1]
Following rules are strictly oriented:
f#(s(x),s(y)) = [2] x + [2]
> [2] x + [0]
= c_2(f#(x,y))
Following rules are (at-least) weakly oriented:
g#(0(),x) = [6] x + [0]
>= [2] x + [0]
= f#(x,x)
g#(0(),x) = [6] x + [0]
>= [6] x + [0]
= g#(f(x,x),x)
**** Step 5.a:1.b:1.a:2: Assumption WORST_CASE(?,O(1))
+ Considered Problem:
- Weak DPs:
f#(s(x),s(y)) -> c_2(f#(x,y))
g#(0(),x) -> f#(x,x)
g#(0(),x) -> g#(f(x,x),x)
- Weak TRS:
f(x,0()) -> s(0())
f(s(x),s(y)) -> s(f(x,y))
- Signature:
{f/2,g/2,f#/2,g#/2} / {0/0,s/1,c_1/0,c_2/1,c_3/2}
- Obligation:
innermost runtime complexity wrt. defined symbols {f#,g#} and constructors {0,s}
+ Applied Processor:
Assumption {assumed = Certificate {spaceUB = Unknown, spaceLB = Unknown, timeUB = Poly (Just 0), timeLB = Unknown}}
+ Details:
()
**** Step 5.a:1.b:1.b:1: RemoveWeakSuffixes WORST_CASE(?,O(1))
+ Considered Problem:
- Weak DPs:
f#(s(x),s(y)) -> c_2(f#(x,y))
g#(0(),x) -> f#(x,x)
g#(0(),x) -> g#(f(x,x),x)
- Weak TRS:
f(x,0()) -> s(0())
f(s(x),s(y)) -> s(f(x,y))
- Signature:
{f/2,g/2,f#/2,g#/2} / {0/0,s/1,c_1/0,c_2/1,c_3/2}
- Obligation:
innermost runtime complexity wrt. defined symbols {f#,g#} and constructors {0,s}
+ Applied Processor:
RemoveWeakSuffixes
+ Details:
Consider the dependency graph
1:W:f#(s(x),s(y)) -> c_2(f#(x,y))
-->_1 f#(s(x),s(y)) -> c_2(f#(x,y)):1
2:W:g#(0(),x) -> f#(x,x)
-->_1 f#(s(x),s(y)) -> c_2(f#(x,y)):1
3:W:g#(0(),x) -> g#(f(x,x),x)
-->_1 g#(0(),x) -> g#(f(x,x),x):3
-->_1 g#(0(),x) -> f#(x,x):2
The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed.
3: g#(0(),x) -> g#(f(x,x),x)
2: g#(0(),x) -> f#(x,x)
1: f#(s(x),s(y)) -> c_2(f#(x,y))
**** Step 5.a:1.b:1.b:2: EmptyProcessor WORST_CASE(?,O(1))
+ Considered Problem:
- Weak TRS:
f(x,0()) -> s(0())
f(s(x),s(y)) -> s(f(x,y))
- Signature:
{f/2,g/2,f#/2,g#/2} / {0/0,s/1,c_1/0,c_2/1,c_3/2}
- Obligation:
innermost runtime complexity wrt. defined symbols {f#,g#} and constructors {0,s}
+ Applied Processor:
EmptyProcessor
+ Details:
The problem is already closed. The intended complexity is O(1).
** Step 5.b:1: RemoveWeakSuffixes WORST_CASE(?,O(1))
+ Considered Problem:
- Strict DPs:
g#(0(),x) -> c_3(g#(f(x,x),x),f#(x,x))
- Weak DPs:
f#(s(x),s(y)) -> c_2(f#(x,y))
- Weak TRS:
f(x,0()) -> s(0())
f(s(x),s(y)) -> s(f(x,y))
- Signature:
{f/2,g/2,f#/2,g#/2} / {0/0,s/1,c_1/0,c_2/1,c_3/2}
- Obligation:
innermost runtime complexity wrt. defined symbols {f#,g#} and constructors {0,s}
+ Applied Processor:
RemoveWeakSuffixes
+ Details:
Consider the dependency graph
1:S:g#(0(),x) -> c_3(g#(f(x,x),x),f#(x,x))
-->_2 f#(s(x),s(y)) -> c_2(f#(x,y)):2
-->_1 g#(0(),x) -> c_3(g#(f(x,x),x),f#(x,x)):1
2:W:f#(s(x),s(y)) -> c_2(f#(x,y))
-->_1 f#(s(x),s(y)) -> c_2(f#(x,y)):2
The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed.
2: f#(s(x),s(y)) -> c_2(f#(x,y))
** Step 5.b:2: SimplifyRHS WORST_CASE(?,O(1))
+ Considered Problem:
- Strict DPs:
g#(0(),x) -> c_3(g#(f(x,x),x),f#(x,x))
- Weak TRS:
f(x,0()) -> s(0())
f(s(x),s(y)) -> s(f(x,y))
- Signature:
{f/2,g/2,f#/2,g#/2} / {0/0,s/1,c_1/0,c_2/1,c_3/2}
- Obligation:
innermost runtime complexity wrt. defined symbols {f#,g#} and constructors {0,s}
+ Applied Processor:
SimplifyRHS
+ Details:
Consider the dependency graph
1:S:g#(0(),x) -> c_3(g#(f(x,x),x),f#(x,x))
-->_1 g#(0(),x) -> c_3(g#(f(x,x),x),f#(x,x)):1
Due to missing edges in the depndency graph, the right-hand sides of following rules could be simplified:
g#(0(),x) -> c_3(g#(f(x,x),x))
** Step 5.b:3: PredecessorEstimationCP WORST_CASE(?,O(1))
+ Considered Problem:
- Strict DPs:
g#(0(),x) -> c_3(g#(f(x,x),x))
- Weak TRS:
f(x,0()) -> s(0())
f(s(x),s(y)) -> s(f(x,y))
- Signature:
{f/2,g/2,f#/2,g#/2} / {0/0,s/1,c_1/0,c_2/1,c_3/1}
- Obligation:
innermost runtime complexity wrt. defined symbols {f#,g#} and constructors {0,s}
+ Applied Processor:
PredecessorEstimationCP {onSelectionCP = any intersect of rules of CDG leaf and strict-rules, withComplexityPair = NaturalMI {miDimension = 1, miDegree = 1, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Nothing}}
+ Details:
We first use the processor NaturalMI {miDimension = 1, miDegree = 1, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Nothing} to orient following rules strictly:
1: g#(0(),x) -> c_3(g#(f(x,x),x))
The strictly oriented rules are moved into the weak component.
*** Step 5.b:3.a:1: NaturalMI WORST_CASE(?,O(1))
+ Considered Problem:
- Strict DPs:
g#(0(),x) -> c_3(g#(f(x,x),x))
- Weak TRS:
f(x,0()) -> s(0())
f(s(x),s(y)) -> s(f(x,y))
- Signature:
{f/2,g/2,f#/2,g#/2} / {0/0,s/1,c_1/0,c_2/1,c_3/1}
- Obligation:
innermost runtime complexity wrt. defined symbols {f#,g#} and constructors {0,s}
+ Applied Processor:
NaturalMI {miDimension = 1, miDegree = 1, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Just first alternative for predecessorEstimation on any intersect of rules of CDG leaf and strict-rules}
+ Details:
We apply a matrix interpretation of kind constructor based matrix interpretation:
The following argument positions are considered usable:
uargs(c_3) = {1}
Following symbols are considered usable:
{f,f#,g#}
TcT has computed the following interpretation:
p(0) = [7]
p(f) = [0]
p(g) = [4] x2 + [2]
p(s) = [0]
p(f#) = [2] x2 + [1]
p(g#) = [3] x1 + [4]
p(c_1) = [1]
p(c_2) = [2] x1 + [1]
p(c_3) = [2] x1 + [4]
Following rules are strictly oriented:
g#(0(),x) = [25]
> [12]
= c_3(g#(f(x,x),x))
Following rules are (at-least) weakly oriented:
f(x,0()) = [0]
>= [0]
= s(0())
f(s(x),s(y)) = [0]
>= [0]
= s(f(x,y))
*** Step 5.b:3.a:2: Assumption WORST_CASE(?,O(1))
+ Considered Problem:
- Weak DPs:
g#(0(),x) -> c_3(g#(f(x,x),x))
- Weak TRS:
f(x,0()) -> s(0())
f(s(x),s(y)) -> s(f(x,y))
- Signature:
{f/2,g/2,f#/2,g#/2} / {0/0,s/1,c_1/0,c_2/1,c_3/1}
- Obligation:
innermost runtime complexity wrt. defined symbols {f#,g#} and constructors {0,s}
+ Applied Processor:
Assumption {assumed = Certificate {spaceUB = Unknown, spaceLB = Unknown, timeUB = Poly (Just 0), timeLB = Unknown}}
+ Details:
()
*** Step 5.b:3.b:1: RemoveWeakSuffixes WORST_CASE(?,O(1))
+ Considered Problem:
- Weak DPs:
g#(0(),x) -> c_3(g#(f(x,x),x))
- Weak TRS:
f(x,0()) -> s(0())
f(s(x),s(y)) -> s(f(x,y))
- Signature:
{f/2,g/2,f#/2,g#/2} / {0/0,s/1,c_1/0,c_2/1,c_3/1}
- Obligation:
innermost runtime complexity wrt. defined symbols {f#,g#} and constructors {0,s}
+ Applied Processor:
RemoveWeakSuffixes
+ Details:
Consider the dependency graph
1:W:g#(0(),x) -> c_3(g#(f(x,x),x))
-->_1 g#(0(),x) -> c_3(g#(f(x,x),x)):1
The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed.
1: g#(0(),x) -> c_3(g#(f(x,x),x))
*** Step 5.b:3.b:2: EmptyProcessor WORST_CASE(?,O(1))
+ Considered Problem:
- Weak TRS:
f(x,0()) -> s(0())
f(s(x),s(y)) -> s(f(x,y))
- Signature:
{f/2,g/2,f#/2,g#/2} / {0/0,s/1,c_1/0,c_2/1,c_3/1}
- Obligation:
innermost runtime complexity wrt. defined symbols {f#,g#} and constructors {0,s}
+ Applied Processor:
EmptyProcessor
+ Details:
The problem is already closed. The intended complexity is O(1).
WORST_CASE(?,O(n^1))