* Step 1: DependencyPairs WORST_CASE(?,O(1))
+ Considered Problem:
- Strict TRS:
f(x,x,y) -> x
f(x,y,y) -> y
f(x,y,g(y)) -> x
f(g(x),x,y) -> y
- Signature:
{f/3} / {g/1}
- Obligation:
innermost runtime complexity wrt. defined symbols {f} and constructors {g}
+ Applied Processor:
DependencyPairs {dpKind_ = DT}
+ Details:
We add the following dependency tuples:

Strict DPs
f#(x,x,y) -> c_1()
f#(x,y,y) -> c_2()
f#(x,y,g(y)) -> c_3()
f#(g(x),x,y) -> c_4()
Weak DPs

and mark the set of starting terms.
* Step 2: UsableRules WORST_CASE(?,O(1))
+ Considered Problem:
- Strict DPs:
f#(x,x,y) -> c_1()
f#(x,y,y) -> c_2()
f#(x,y,g(y)) -> c_3()
f#(g(x),x,y) -> c_4()
- Weak TRS:
f(x,x,y) -> x
f(x,y,y) -> y
f(x,y,g(y)) -> x
f(g(x),x,y) -> y
- Signature:
{f/3,f#/3} / {g/1,c_1/0,c_2/0,c_3/0,c_4/0}
- Obligation:
innermost runtime complexity wrt. defined symbols {f#} and constructors {g}
+ Applied Processor:
UsableRules
+ Details:
We replace rewrite rules by usable rules:
f#(x,x,y) -> c_1()
f#(x,y,y) -> c_2()
f#(x,y,g(y)) -> c_3()
f#(g(x),x,y) -> c_4()
* Step 3: Trivial WORST_CASE(?,O(1))
+ Considered Problem:
- Strict DPs:
f#(x,x,y) -> c_1()
f#(x,y,y) -> c_2()
f#(x,y,g(y)) -> c_3()
f#(g(x),x,y) -> c_4()
- Signature:
{f/3,f#/3} / {g/1,c_1/0,c_2/0,c_3/0,c_4/0}
- Obligation:
innermost runtime complexity wrt. defined symbols {f#} and constructors {g}
+ Applied Processor:
Trivial
+ Details:
Consider the dependency graph
1:S:f#(x,x,y) -> c_1()

2:S:f#(x,y,y) -> c_2()

3:S:f#(x,y,g(y)) -> c_3()

4:S:f#(g(x),x,y) -> c_4()

The dependency graph contains no loops, we remove all dependency pairs.
* Step 4: EmptyProcessor WORST_CASE(?,O(1))
+ Considered Problem:

- Signature:
{f/3,f#/3} / {g/1,c_1/0,c_2/0,c_3/0,c_4/0}
- Obligation:
innermost runtime complexity wrt. defined symbols {f#} and constructors {g}
+ Applied Processor:
EmptyProcessor
+ Details:
The problem is already closed. The intended complexity is O(1).

WORST_CASE(?,O(1))