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

The TRS has the following type information:

f :: s → s → s → f 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, v2) → 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(s(x), y, y) → f(y, x, s(x)) [1] f(v0, v1, v2) → null_f [0] The TRS has the following type information: f :: s → s → s → null_f 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', z'') -{ 1 }→ f(y, x, 1 + x) :|: z'' = y, x >= 0, y >= 0, z = 1 + x, z' = y
f(z, z', z'') -{ 0 }→ 0 :|: v0 >= 0, z'' = v2, v1 >= 0, z = v0, z' = v1, v2 >= 0

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, V2),0,[f(V, V1, V2, Out)],[V >= 0,V1 >= 0,V2 >= 0]). eq(f(V, V1, V2, Out),1,[f(V3, V4, 1 + V4, Ret)],[Out = Ret,V2 = V3,V4 >= 0,V3 >= 0,V = 1 + V4,V1 = V3]). eq(f(V, V1, V2, Out),0,[],[Out = 0,V5 >= 0,V2 = V6,V7 >= 0,V = V5,V1 = V7,V6 >= 0]). input_output_vars(f(V,V1,V2,Out),[V,V1,V2],[Out]).

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

#### Computed strongly connected components
0. recursive : [f/4]
1. non_recursive : [start/3]

#### Obtained direct recursion through partial evaluation
0. SCC is partially evaluated into f/4
1. SCC is partially evaluated into start/3

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

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


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

### Ranking functions of CR f(V,V1,V2,Out)

#### Partial ranking functions of CR f(V,V1,V2,Out)


### Specialization of cost equations start/3
* CE 2 is refined into CE [7]


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

### Ranking functions of CR start(V,V1,V2)

#### Partial ranking functions of CR start(V,V1,V2)


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

#### Cost of chains of f(V,V1,V2,Out):
* Chain [5]: 0
with precondition: [Out=0,V>=0,V1>=0,V2>=0]

* Chain [4,5]: 1
with precondition: [Out=0,V1=V2,V>=1,V1>=0]


#### Cost of chains of start(V,V1,V2):
* Chain [6]: 1
with precondition: [V>=0,V1>=0,V2>=0]


Closed-form bounds of start(V,V1,V2):
-------------------------------------
* Chain [6] with precondition: [V>=0,V1>=0,V2>=0]
- Upper bound: 1
- Complexity: constant

### Maximum cost of start(V,V1,V2): 1
Asymptotic class: constant
* Total analysis performed in 40 ms.

(10) BOUNDS(1, 1)