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(x, x) → f(g(x), x)
g(x) → s(x)

Rewrite Strategy: INNERMOST

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

Transformed TRS to weighted TRS

(2) 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(x, x) → f(g(x), x) [1]
g(x) → s(x) [1]

Rewrite Strategy: INNERMOST

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

Infered types.

(4) Obligation:

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

f(x, x) → f(g(x), x) [1]
g(x) → s(x) [1]

The TRS has the following type information:

f :: s → s → f g :: s → s s :: s → s

Rewrite Strategy: INNERMOST

(5) 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, v1) → null_f [0]

And the following fresh constants:

null_f, const

(6) 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(x, x) → f(g(x), x) [1] g(x) → s(x) [1] f(v0, v1) → null_f [0] The TRS has the following type information: f :: s → s → null_f g :: s → s s :: s → s null_f :: null_f const :: s Rewrite Strategy: INNERMOST

(7) 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

(8) Obligation:

Complexity RNTS consisting of the following rules:

f(z, z') -{ 1 }→ f(g(x), x) :|: z' = x, x >= 0, z = x
f(z, z') -{ 0 }→ 0 :|: v0 >= 0, v1 >= 0, z = v0, z' = v1
g(z) -{ 1 }→ 1 + x :|: x >= 0, z = x

Only complete derivations are relevant for the runtime complexity.

(9) CompleteCoflocoProof (EQUIVALENT transformation)

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

eq(start(V, V1),0,[f(V, V1, Out)],[V >= 0,V1 >= 0]). eq(start(V, V1),0,[g(V, Out)],[V >= 0]). eq(f(V, V1, Out),1,[g(V2, Ret0),f(Ret0, V2, Ret)],[Out = Ret,V1 = V2,V2 >= 0,V = V2]). eq(g(V, Out),1,[],[Out = 1 + V3,V3 >= 0,V = V3]). eq(f(V, V1, Out),0,[],[Out = 0,V4 >= 0,V5 >= 0,V = V4,V1 = V5]). input_output_vars(f(V,V1,Out),[V,V1],[Out]). input_output_vars(g(V,Out),[V],[Out]).

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

#### Computed strongly connected components
0. non_recursive : [g/2]
1. recursive : [f/3]
2. non_recursive : [start/2]

#### Obtained direct recursion through partial evaluation
0. SCC is completely evaluated into other SCCs
1. SCC is partially evaluated into f/3
2. SCC is partially evaluated into start/2

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

### Specialization of cost equations f/3
* CE 5 is refined into CE [6]
* CE 4 is refined into CE [7]


### Cost equations --> "Loop" of f/3
* CEs [7] --> Loop 4
* CEs [6] --> Loop 5

### Ranking functions of CR f(V,V1,Out)

#### Partial ranking functions of CR f(V,V1,Out)


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


### Cost equations --> "Loop" of start/2
* CEs [8,9] --> Loop 6

### Ranking functions of CR start(V,V1)

#### Partial ranking functions of CR start(V,V1)


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

#### Cost of chains of f(V,V1,Out):
* Chain [5]: 0
with precondition: [Out=0,V>=0,V1>=0]

* Chain [4,5]: 2
with precondition: [Out=0,V=V1,V>=0]


#### Cost of chains of start(V,V1):
* Chain [6]: 2
with precondition: [V>=0]


Closed-form bounds of start(V,V1):
-------------------------------------
* Chain [6] with precondition: [V>=0]
- Upper bound: 2
- Complexity: constant

### Maximum cost of start(V,V1): 2
Asymptotic class: constant
* Total analysis performed in 34 ms.

(10) BOUNDS(1, 1)