* Step 1: Sum WORST_CASE(Omega(n^1),O(n^1))
+ Considered Problem:
- Strict TRS:
f(g(x),s(0()),y) -> f(y,y,g(x))
g(0()) -> 0()
g(s(x)) -> s(g(x))
- Signature:
{f/3,g/1} / {0/0,s/1}
- Obligation:
innermost runtime complexity wrt. defined symbols {f,g} and constructors {0,s}
+ Applied Processor:
Sum {left = someStrategy, right = someStrategy}
+ Details:
()
** Step 1.a:1: DecreasingLoops WORST_CASE(Omega(n^1),?)
+ Considered Problem:
- Strict TRS:
f(g(x),s(0()),y) -> f(y,y,g(x))
g(0()) -> 0()
g(s(x)) -> s(g(x))
- Signature:
{f/3,g/1} / {0/0,s/1}
- Obligation:
innermost runtime complexity wrt. defined symbols {f,g} and constructors {0,s}
+ Applied Processor:
DecreasingLoops {bound = AnyLoop, narrow = 10}
+ Details:
The system has following decreasing Loops:
g(x){x -> s(x)} =
g(s(x)) ->^+ s(g(x))
= C[g(x) = g(x){}]
** Step 1.b:1: DependencyPairs WORST_CASE(?,O(n^1))
+ Considered Problem:
- Strict TRS:
f(g(x),s(0()),y) -> f(y,y,g(x))
g(0()) -> 0()
g(s(x)) -> s(g(x))
- Signature:
{f/3,g/1} / {0/0,s/1}
- Obligation:
innermost runtime complexity wrt. defined symbols {f,g} and constructors {0,s}
+ Applied Processor:
DependencyPairs {dpKind_ = WIDP}
+ Details:
We add the following weak innermost dependency pairs:
Strict DPs
f#(g(x),s(0()),y) -> c_1(f#(y,y,g(x)))
g#(0()) -> c_2()
g#(s(x)) -> c_3(g#(x))
Weak DPs
and mark the set of starting terms.
** Step 1.b:2: UsableRules WORST_CASE(?,O(n^1))
+ Considered Problem:
- Strict DPs:
f#(g(x),s(0()),y) -> c_1(f#(y,y,g(x)))
g#(0()) -> c_2()
g#(s(x)) -> c_3(g#(x))
- Strict TRS:
f(g(x),s(0()),y) -> f(y,y,g(x))
g(0()) -> 0()
g(s(x)) -> s(g(x))
- Signature:
{f/3,g/1,f#/3,g#/1} / {0/0,s/1,c_1/1,c_2/0,c_3/1}
- Obligation:
innermost runtime complexity wrt. defined symbols {f#,g#} and constructors {0,s}
+ Applied Processor:
UsableRules
+ Details:
We replace rewrite rules by usable rules:
g#(0()) -> c_2()
g#(s(x)) -> c_3(g#(x))
** Step 1.b:3: PredecessorEstimation WORST_CASE(?,O(n^1))
+ Considered Problem:
- Strict DPs:
g#(0()) -> c_2()
g#(s(x)) -> c_3(g#(x))
- Signature:
{f/3,g/1,f#/3,g#/1} / {0/0,s/1,c_1/1,c_2/0,c_3/1}
- 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}.
Here rules are labelled as follows:
1: g#(0()) -> c_2()
2: g#(s(x)) -> c_3(g#(x))
** Step 1.b:4: RemoveWeakSuffixes WORST_CASE(?,O(n^1))
+ Considered Problem:
- Strict DPs:
g#(s(x)) -> c_3(g#(x))
- Weak DPs:
g#(0()) -> c_2()
- Signature:
{f/3,g/1,f#/3,g#/1} / {0/0,s/1,c_1/1,c_2/0,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:S:g#(s(x)) -> c_3(g#(x))
-->_1 g#(0()) -> c_2():2
-->_1 g#(s(x)) -> c_3(g#(x)):1
2:W:g#(0()) -> c_2()
The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed.
2: g#(0()) -> c_2()
** Step 1.b:5: PredecessorEstimationCP WORST_CASE(?,O(n^1))
+ Considered Problem:
- Strict DPs:
g#(s(x)) -> c_3(g#(x))
- Signature:
{f/3,g/1,f#/3,g#/1} / {0/0,s/1,c_1/1,c_2/0,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#(s(x)) -> c_3(g#(x))
The strictly oriented rules are moved into the weak component.
*** Step 1.b:5.a:1: NaturalMI WORST_CASE(?,O(n^1))
+ Considered Problem:
- Strict DPs:
g#(s(x)) -> c_3(g#(x))
- Signature:
{f/3,g/1,f#/3,g#/1} / {0/0,s/1,c_1/1,c_2/0,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#,g#}
TcT has computed the following interpretation:
p(0) = [0]
p(f) = [1] x1 + [8] x2 + [2] x3 + [8]
p(g) = [1]
p(s) = [1] x1 + [9]
p(f#) = [4] x1 + [1] x3 + [8]
p(g#) = [2] x1 + [2]
p(c_1) = [2] x1 + [1]
p(c_2) = [0]
p(c_3) = [1] x1 + [14]
Following rules are strictly oriented:
g#(s(x)) = [2] x + [20]
> [2] x + [16]
= c_3(g#(x))
Following rules are (at-least) weakly oriented:
*** Step 1.b:5.a:2: Assumption WORST_CASE(?,O(1))
+ Considered Problem:
- Weak DPs:
g#(s(x)) -> c_3(g#(x))
- Signature:
{f/3,g/1,f#/3,g#/1} / {0/0,s/1,c_1/1,c_2/0,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 1.b:5.b:1: RemoveWeakSuffixes WORST_CASE(?,O(1))
+ Considered Problem:
- Weak DPs:
g#(s(x)) -> c_3(g#(x))
- Signature:
{f/3,g/1,f#/3,g#/1} / {0/0,s/1,c_1/1,c_2/0,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#(s(x)) -> c_3(g#(x))
-->_1 g#(s(x)) -> c_3(g#(x)):1
The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed.
1: g#(s(x)) -> c_3(g#(x))
*** Step 1.b:5.b:2: EmptyProcessor WORST_CASE(?,O(1))
+ Considered Problem:
- Signature:
{f/3,g/1,f#/3,g#/1} / {0/0,s/1,c_1/1,c_2/0,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(Omega(n^1),O(n^1))