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

Strict DPs
:#(z,+(x,f(y))) -> c_1(:#(g(z,y),+(x,a())))
:#(+(x,y),z) -> c_2(:#(x,z),:#(y,z))
Weak DPs

and mark the set of starting terms.
* Step 2: UsableRules WORST_CASE(?,O(n^1))
+ Considered Problem:
- Strict DPs:
:#(z,+(x,f(y))) -> c_1(:#(g(z,y),+(x,a())))
:#(+(x,y),z) -> c_2(:#(x,z),:#(y,z))
- Strict TRS:
:(z,+(x,f(y))) -> :(g(z,y),+(x,a()))
:(+(x,y),z) -> +(:(x,z),:(y,z))
- Signature:
{:/2,:#/2} / {+/2,a/0,f/1,g/2,c_1/1,c_2/2}
- Obligation:
innermost runtime complexity wrt. defined symbols {:#} and constructors {+,a,f,g}
+ Applied Processor:
UsableRules
+ Details:
We replace rewrite rules by usable rules:
:#(z,+(x,f(y))) -> c_1(:#(g(z,y),+(x,a())))
:#(+(x,y),z) -> c_2(:#(x,z),:#(y,z))
* Step 3: PredecessorEstimation WORST_CASE(?,O(n^1))
+ Considered Problem:
- Strict DPs:
:#(z,+(x,f(y))) -> c_1(:#(g(z,y),+(x,a())))
:#(+(x,y),z) -> c_2(:#(x,z),:#(y,z))
- Signature:
{:/2,:#/2} / {+/2,a/0,f/1,g/2,c_1/1,c_2/2}
- Obligation:
innermost runtime complexity wrt. defined symbols {:#} and constructors {+,a,f,g}
+ 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: :#(z,+(x,f(y))) -> c_1(:#(g(z,y),+(x,a())))
2: :#(+(x,y),z) -> c_2(:#(x,z),:#(y,z))
* Step 4: RemoveWeakSuffixes WORST_CASE(?,O(n^1))
+ Considered Problem:
- Strict DPs:
:#(+(x,y),z) -> c_2(:#(x,z),:#(y,z))
- Weak DPs:
:#(z,+(x,f(y))) -> c_1(:#(g(z,y),+(x,a())))
- Signature:
{:/2,:#/2} / {+/2,a/0,f/1,g/2,c_1/1,c_2/2}
- Obligation:
innermost runtime complexity wrt. defined symbols {:#} and constructors {+,a,f,g}
+ Applied Processor:
RemoveWeakSuffixes
+ Details:
Consider the dependency graph
1:S::#(+(x,y),z) -> c_2(:#(x,z),:#(y,z))
-->_2 :#(z,+(x,f(y))) -> c_1(:#(g(z,y),+(x,a()))):2
-->_1 :#(z,+(x,f(y))) -> c_1(:#(g(z,y),+(x,a()))):2
-->_2 :#(+(x,y),z) -> c_2(:#(x,z),:#(y,z)):1
-->_1 :#(+(x,y),z) -> c_2(:#(x,z),:#(y,z)):1

2:W::#(z,+(x,f(y))) -> c_1(:#(g(z,y),+(x,a())))

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

Following symbols are considered usable:
{:#}
TcT has computed the following interpretation:
p(+) = [1] x1 + [1] x2 + [9]
p(:) = [1] x1 + [8] x2 + [4]
p(a) = [0]
p(f) = [1]
p(g) = [1] x1 + [1]
p(:#) = [2] x1 + [5]
p(c_1) = [1] x1 + [1]
p(c_2) = [1] x1 + [1] x2 + [1]

Following rules are strictly oriented:
:#(+(x,y),z) = [2] x + [2] y + [23]
> [2] x + [2] y + [11]
= c_2(:#(x,z),:#(y,z))

Following rules are (at-least) weakly oriented:

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

The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed.
1: :#(+(x,y),z) -> c_2(:#(x,z),:#(y,z))
** Step 5.b:2: EmptyProcessor WORST_CASE(?,O(1))
+ Considered Problem:

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

WORST_CASE(?,O(n^1))