```* Step 1: DependencyPairs WORST_CASE(?,O(n^1))
+ Considered Problem:
- Strict TRS:
f(x,0()) -> x
f(0(),y) -> y
f(1(),g(x,y)) -> x
f(2(),g(x,y)) -> y
f(g(x,y),z) -> g(f(x,z),f(y,z))
f(i(x),y) -> i(x)
- Signature:
{f/2} / {0/0,1/0,2/0,g/2,i/1}
- Obligation:
innermost runtime complexity wrt. defined symbols {f} and constructors {0,1,2,g,i}
+ Applied Processor:
DependencyPairs {dpKind_ = WIDP}
+ Details:
We add the following weak innermost dependency pairs:

Strict DPs
f#(x,0()) -> c_1()
f#(0(),y) -> c_2()
f#(1(),g(x,y)) -> c_3()
f#(2(),g(x,y)) -> c_4()
f#(g(x,y),z) -> c_5(f#(x,z),f#(y,z))
f#(i(x),y) -> c_6()
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#(0(),y) -> c_2()
f#(1(),g(x,y)) -> c_3()
f#(2(),g(x,y)) -> c_4()
f#(g(x,y),z) -> c_5(f#(x,z),f#(y,z))
f#(i(x),y) -> c_6()
- Strict TRS:
f(x,0()) -> x
f(0(),y) -> y
f(1(),g(x,y)) -> x
f(2(),g(x,y)) -> y
f(g(x,y),z) -> g(f(x,z),f(y,z))
f(i(x),y) -> i(x)
- Signature:
{f/2,f#/2} / {0/0,1/0,2/0,g/2,i/1,c_1/0,c_2/0,c_3/0,c_4/0,c_5/2,c_6/0}
- Obligation:
innermost runtime complexity wrt. defined symbols {f#} and constructors {0,1,2,g,i}
+ Applied Processor:
UsableRules
+ Details:
We replace rewrite rules by usable rules:
f#(x,0()) -> c_1()
f#(0(),y) -> c_2()
f#(1(),g(x,y)) -> c_3()
f#(2(),g(x,y)) -> c_4()
f#(g(x,y),z) -> c_5(f#(x,z),f#(y,z))
f#(i(x),y) -> c_6()
* Step 3: PredecessorEstimation WORST_CASE(?,O(n^1))
+ Considered Problem:
- Strict DPs:
f#(x,0()) -> c_1()
f#(0(),y) -> c_2()
f#(1(),g(x,y)) -> c_3()
f#(2(),g(x,y)) -> c_4()
f#(g(x,y),z) -> c_5(f#(x,z),f#(y,z))
f#(i(x),y) -> c_6()
- Signature:
{f/2,f#/2} / {0/0,1/0,2/0,g/2,i/1,c_1/0,c_2/0,c_3/0,c_4/0,c_5/2,c_6/0}
- Obligation:
innermost runtime complexity wrt. defined symbols {f#} and constructors {0,1,2,g,i}
+ Applied Processor:
PredecessorEstimation {onSelection = all simple predecessor estimation selector}
+ Details:
We estimate the number of application of
{1,2,3,4,6}
by application of
Pre({1,2,3,4,6}) = {5}.
Here rules are labelled as follows:
1: f#(x,0()) -> c_1()
2: f#(0(),y) -> c_2()
3: f#(1(),g(x,y)) -> c_3()
4: f#(2(),g(x,y)) -> c_4()
5: f#(g(x,y),z) -> c_5(f#(x,z),f#(y,z))
6: f#(i(x),y) -> c_6()
* Step 4: RemoveWeakSuffixes WORST_CASE(?,O(n^1))
+ Considered Problem:
- Strict DPs:
f#(g(x,y),z) -> c_5(f#(x,z),f#(y,z))
- Weak DPs:
f#(x,0()) -> c_1()
f#(0(),y) -> c_2()
f#(1(),g(x,y)) -> c_3()
f#(2(),g(x,y)) -> c_4()
f#(i(x),y) -> c_6()
- Signature:
{f/2,f#/2} / {0/0,1/0,2/0,g/2,i/1,c_1/0,c_2/0,c_3/0,c_4/0,c_5/2,c_6/0}
- Obligation:
innermost runtime complexity wrt. defined symbols {f#} and constructors {0,1,2,g,i}
+ Applied Processor:
RemoveWeakSuffixes
+ Details:
Consider the dependency graph
1:S:f#(g(x,y),z) -> c_5(f#(x,z),f#(y,z))
-->_2 f#(i(x),y) -> c_6():6
-->_1 f#(i(x),y) -> c_6():6
-->_2 f#(2(),g(x,y)) -> c_4():5
-->_1 f#(2(),g(x,y)) -> c_4():5
-->_2 f#(1(),g(x,y)) -> c_3():4
-->_1 f#(1(),g(x,y)) -> c_3():4
-->_2 f#(0(),y) -> c_2():3
-->_1 f#(0(),y) -> c_2():3
-->_2 f#(x,0()) -> c_1():2
-->_1 f#(x,0()) -> c_1():2
-->_2 f#(g(x,y),z) -> c_5(f#(x,z),f#(y,z)):1
-->_1 f#(g(x,y),z) -> c_5(f#(x,z),f#(y,z)):1

2:W:f#(x,0()) -> c_1()

3:W:f#(0(),y) -> c_2()

4:W:f#(1(),g(x,y)) -> c_3()

5:W:f#(2(),g(x,y)) -> c_4()

6:W:f#(i(x),y) -> c_6()

The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed.
2: f#(x,0()) -> c_1()
3: f#(0(),y) -> c_2()
4: f#(1(),g(x,y)) -> c_3()
5: f#(2(),g(x,y)) -> c_4()
6: f#(i(x),y) -> c_6()
* Step 5: PredecessorEstimationCP WORST_CASE(?,O(n^1))
+ Considered Problem:
- Strict DPs:
f#(g(x,y),z) -> c_5(f#(x,z),f#(y,z))
- Signature:
{f/2,f#/2} / {0/0,1/0,2/0,g/2,i/1,c_1/0,c_2/0,c_3/0,c_4/0,c_5/2,c_6/0}
- Obligation:
innermost runtime complexity wrt. defined symbols {f#} and constructors {0,1,2,g,i}
+ 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#(g(x,y),z) -> c_5(f#(x,z),f#(y,z))

The strictly oriented rules are moved into the weak component.
** Step 5.a:1: NaturalMI WORST_CASE(?,O(n^1))
+ Considered Problem:
- Strict DPs:
f#(g(x,y),z) -> c_5(f#(x,z),f#(y,z))
- Signature:
{f/2,f#/2} / {0/0,1/0,2/0,g/2,i/1,c_1/0,c_2/0,c_3/0,c_4/0,c_5/2,c_6/0}
- Obligation:
innermost runtime complexity wrt. defined symbols {f#} and constructors {0,1,2,g,i}
+ 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_5) = {1,2}

Following symbols are considered usable:
{f#}
TcT has computed the following interpretation:
p(0) = 
p(1) = 
p(2) = 
p(f) =  x2 + 
p(g) =  x1 +  x2 + 
p(i) = 
p(f#) =  x1 + 
p(c_1) = 
p(c_2) = 
p(c_3) = 
p(c_4) = 
p(c_5) =  x1 +  x2 + 
p(c_6) = 

Following rules are strictly oriented:
f#(g(x,y),z) =  x +  y + 
>  x +  y + 
= c_5(f#(x,z),f#(y,z))

Following rules are (at-least) weakly oriented:

** Step 5.a:2: Assumption WORST_CASE(?,O(1))
+ Considered Problem:
- Weak DPs:
f#(g(x,y),z) -> c_5(f#(x,z),f#(y,z))
- Signature:
{f/2,f#/2} / {0/0,1/0,2/0,g/2,i/1,c_1/0,c_2/0,c_3/0,c_4/0,c_5/2,c_6/0}
- Obligation:
innermost runtime complexity wrt. defined symbols {f#} and constructors {0,1,2,g,i}
+ Applied Processor:
Assumption {assumed = Certificate {spaceUB = Unknown, spaceLB = Unknown, timeUB = Poly (Just 0), timeLB = Unknown}}
+ Details:
()

** Step 5.b:1: RemoveWeakSuffixes WORST_CASE(?,O(1))
+ Considered Problem:
- Weak DPs:
f#(g(x,y),z) -> c_5(f#(x,z),f#(y,z))
- Signature:
{f/2,f#/2} / {0/0,1/0,2/0,g/2,i/1,c_1/0,c_2/0,c_3/0,c_4/0,c_5/2,c_6/0}
- Obligation:
innermost runtime complexity wrt. defined symbols {f#} and constructors {0,1,2,g,i}
+ Applied Processor:
RemoveWeakSuffixes
+ Details:
Consider the dependency graph
1:W:f#(g(x,y),z) -> c_5(f#(x,z),f#(y,z))
-->_2 f#(g(x,y),z) -> c_5(f#(x,z),f#(y,z)):1
-->_1 f#(g(x,y),z) -> c_5(f#(x,z),f#(y,z)):1

The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed.
1: f#(g(x,y),z) -> c_5(f#(x,z),f#(y,z))
** Step 5.b:2: EmptyProcessor WORST_CASE(?,O(1))
+ Considered Problem:

- Signature:
{f/2,f#/2} / {0/0,1/0,2/0,g/2,i/1,c_1/0,c_2/0,c_3/0,c_4/0,c_5/2,c_6/0}
- Obligation:
innermost runtime complexity wrt. defined symbols {f#} and constructors {0,1,2,g,i}
+ Applied Processor:
EmptyProcessor
+ Details:
The problem is already closed. The intended complexity is O(1).

WORST_CASE(?,O(n^1))
```