```* Step 1: DependencyPairs WORST_CASE(?,O(1))
+ Considered Problem:
- Strict TRS:
c(b(a(),a()),b(y,z),x) -> b(a(),b(z,z))
f(b(b(x,f(y)),z)) -> c(z,x,f(b(b(f(a()),y),y)))
f(c(c(a(),y,a()),b(x,z),a())) -> b(y,f(c(f(a()),z,z)))
- Signature:
{c/3,f/1} / {a/0,b/2}
- Obligation:
runtime complexity wrt. defined symbols {c,f} and constructors {a,b}
+ Applied Processor:
DependencyPairs {dpKind_ = DT}
+ Details:
We add the following weak dependency pairs:

Strict DPs
c#(b(a(),a()),b(y,z),x) -> c_1(z,z)
f#(b(b(x,f(y)),z)) -> c_2(c#(z,x,f(b(b(f(a()),y),y))))
f#(c(c(a(),y,a()),b(x,z),a())) -> c_3(y,f#(c(f(a()),z,z)))
Weak DPs

and mark the set of starting terms.
* Step 2: UsableRules WORST_CASE(?,O(1))
+ Considered Problem:
- Strict DPs:
c#(b(a(),a()),b(y,z),x) -> c_1(z,z)
f#(b(b(x,f(y)),z)) -> c_2(c#(z,x,f(b(b(f(a()),y),y))))
f#(c(c(a(),y,a()),b(x,z),a())) -> c_3(y,f#(c(f(a()),z,z)))
- Strict TRS:
c(b(a(),a()),b(y,z),x) -> b(a(),b(z,z))
f(b(b(x,f(y)),z)) -> c(z,x,f(b(b(f(a()),y),y)))
f(c(c(a(),y,a()),b(x,z),a())) -> b(y,f(c(f(a()),z,z)))
- Signature:
{c/3,f/1,c#/3,f#/1} / {a/0,b/2,c_1/2,c_2/1,c_3/2}
- Obligation:
runtime complexity wrt. defined symbols {c#,f#} and constructors {a,b}
+ Applied Processor:
UsableRules
+ Details:
We replace rewrite rules by usable rules:
c#(b(a(),a()),b(y,z),x) -> c_1(z,z)
* Step 3: WeightGap WORST_CASE(?,O(1))
+ Considered Problem:
- Strict DPs:
c#(b(a(),a()),b(y,z),x) -> c_1(z,z)
- Signature:
{c/3,f/1,c#/3,f#/1} / {a/0,b/2,c_1/2,c_2/1,c_3/2}
- Obligation:
runtime complexity wrt. defined symbols {c#,f#} and constructors {a,b}
+ 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:
none

Following symbols are considered usable:
all
TcT has computed the following interpretation:
p(a) = 
p(b) = 
p(c) = 
p(f) = 
p(c#) = 
p(f#) = 
p(c_1) = 
p(c_2) = 
p(c_3) = 

Following rules are strictly oriented:
c#(b(a(),a()),b(y,z),x) = 
> 
= c_1(z,z)

Following rules are (at-least) weakly oriented:

Further, it can be verified that all rules not oriented are covered by the weightgap condition.
* Step 4: EmptyProcessor WORST_CASE(?,O(1))
+ Considered Problem:
- Weak DPs:
c#(b(a(),a()),b(y,z),x) -> c_1(z,z)
- Signature:
{c/3,f/1,c#/3,f#/1} / {a/0,b/2,c_1/2,c_2/1,c_3/2}
- Obligation:
runtime complexity wrt. defined symbols {c#,f#} and constructors {a,b}
+ Applied Processor:
EmptyProcessor
+ Details:
The problem is already closed. The intended complexity is O(1).

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