The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(1, 1).

(0) Obligation:

The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(1, 1).


The TRS R consists of the following rules:

f(f(X)) → f(a(b(f(X))))
f(a(g(X))) → b(X)
b(X) → a(X)

Rewrite Strategy: INNERMOST

(1) DependencyGraphProof (BOTH BOUNDS(ID, ID) transformation)

The following rules are not reachable from basic terms in the dependency graph and can be removed:
f(f(X)) → f(a(b(f(X))))

(2) Obligation:

The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(1, 1).


The TRS R consists of the following rules:

f(a(g(X))) → b(X)
b(X) → a(X)

Rewrite Strategy: INNERMOST

(3) TrsToWeightedTrsProof (BOTH BOUNDS(ID, ID) transformation)

Transformed TRS to weighted TRS

(4) Obligation:

The Runtime Complexity (innermost) of the given CpxWeightedTrs could be proven to be BOUNDS(1, 1).


The TRS R consists of the following rules:

f(a(g(X))) → b(X) [1]
b(X) → a(X) [1]

Rewrite Strategy: INNERMOST

(5) TypeInferenceProof (BOTH BOUNDS(ID, ID) transformation)

Infered types.

(6) Obligation:

Runtime Complexity Weighted TRS with Types.
The TRS R consists of the following rules:

f(a(g(X))) → b(X) [1]
b(X) → a(X) [1]

The TRS has the following type information:

f :: a → a a :: g → a g :: g → g b :: g → a

Rewrite Strategy: INNERMOST

(7) CompletionProof (UPPER BOUND(ID) transformation)

The TRS is a completely defined constructor system, as every type has a constant constructor and the following rules were added:

f(v0) → null_f [0]

And the following fresh constants:

null_f, const

(8) Obligation:

Runtime Complexity Weighted TRS where all functions are completely defined. The underlying TRS is:

Runtime Complexity Weighted TRS with Types. The TRS R consists of the following rules: f(a(g(X))) → b(X) [1] b(X) → a(X) [1] f(v0) → null_f [0] The TRS has the following type information: f :: a:null_f → a:null_f a :: g → a:null_f g :: g → g b :: g → a:null_f null_f :: a:null_f const :: g Rewrite Strategy: INNERMOST

(9) CpxTypedWeightedTrsToRntsProof (UPPER BOUND(ID) transformation)

Transformed the TRS into an over-approximating RNTS by (improved) Size Abstraction.
The constant constructors are abstracted as follows:

null_f => 0
const => 0

(10) Obligation:

Complexity RNTS consisting of the following rules:

b(z) -{ 1 }→ 1 + X :|: X >= 0, z = X
f(z) -{ 1 }→ b(X) :|: z = 1 + (1 + X), X >= 0
f(z) -{ 0 }→ 0 :|: v0 >= 0, z = v0

Only complete derivations are relevant for the runtime complexity.

(11) CompleteCoflocoProof (EQUIVALENT transformation)

Transformed the RNTS (where only complete derivations are relevant) into cost relations for CoFloCo:

eq(start(V),0,[f(V, Out)],[V >= 0]). eq(start(V),0,[b(V, Out)],[V >= 0]). eq(f(V, Out),1,[b(X1, Ret)],[Out = Ret,V = 2 + X1,X1 >= 0]). eq(b(V, Out),1,[],[Out = 1 + X2,X2 >= 0,V = X2]). eq(f(V, Out),0,[],[Out = 0,V1 >= 0,V = V1]). input_output_vars(f(V,Out),[V],[Out]). input_output_vars(b(V,Out),[V],[Out]).

CoFloCo proof output:
Preprocessing Cost Relations
=====================================

#### Computed strongly connected components
0. non_recursive : [b/2]
1. non_recursive : [f/2]
2. non_recursive : [start/1]

#### Obtained direct recursion through partial evaluation
0. SCC is completely evaluated into other SCCs
1. SCC is partially evaluated into f/2
2. SCC is partially evaluated into start/1

Control-Flow Refinement of Cost Relations
=====================================

### Specialization of cost equations f/2
* CE 4 is refined into CE [6]
* CE 5 is refined into CE [7]


### Cost equations --> "Loop" of f/2
* CEs [6] --> Loop 4
* CEs [7] --> Loop 5

### Ranking functions of CR f(V,Out)

#### Partial ranking functions of CR f(V,Out)


### Specialization of cost equations start/1
* CE 2 is refined into CE [8,9]
* CE 3 is refined into CE [10]


### Cost equations --> "Loop" of start/1
* CEs [8,9,10] --> Loop 6

### Ranking functions of CR start(V)

#### Partial ranking functions of CR start(V)


Computing Bounds
=====================================

#### Cost of chains of f(V,Out):
* Chain [5]: 0
with precondition: [Out=0,V>=0]

* Chain [4]: 2
with precondition: [V=Out+1,V>=2]


#### Cost of chains of start(V):
* Chain [6]: 2
with precondition: [V>=0]


Closed-form bounds of start(V):
-------------------------------------
* Chain [6] with precondition: [V>=0]
- Upper bound: 2
- Complexity: constant

### Maximum cost of start(V): 2
Asymptotic class: constant
* Total analysis performed in 22 ms.

(12) BOUNDS(1, 1)