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:

g(x, y) → x
g(x, y) → y
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:

g(x, y) → x [1]
g(x, y) → y [1]
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:

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

The TRS has the following type information:

g :: g → g → g 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, const1

(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: g(x, y) → x [1] g(x, y) → y [1] 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: g :: g → g → g f :: s → s → s → null_f s :: s → s null_f :: null_f const :: g const1 :: 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
const1 => 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
g(z, z') -{ 1 }→ x :|: x >= 0, y >= 0, z = x, z' = y
g(z, z') -{ 1 }→ y :|: x >= 0, y >= 0, z = x, z' = y

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, V6),0,[g(V, V1, Out)],[V >= 0,V1 >= 0]). eq(start(V, V1, V6),0,[f(V, V1, V6, Out)],[V >= 0,V1 >= 0,V6 >= 0]). eq(g(V, V1, Out),1,[],[Out = V2,V2 >= 0,V3 >= 0,V = V2,V1 = V3]). eq(g(V, V1, Out),1,[],[Out = V4,V5 >= 0,V4 >= 0,V = V5,V1 = V4]). eq(f(V, V1, V6, Out),1,[f(V7, V8, 1 + V8, Ret)],[Out = Ret,V6 = V7,V8 >= 0,V7 >= 0,V = 1 + V8,V1 = V7]). eq(f(V, V1, V6, Out),0,[],[Out = 0,V9 >= 0,V6 = V10,V11 >= 0,V = V9,V1 = V11,V10 >= 0]). input_output_vars(g(V,V1,Out),[V,V1],[Out]). input_output_vars(f(V,V1,V6,Out),[V,V1,V6],[Out]).

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

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

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

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

### Specialization of cost equations f/4
* CE 7 is refined into CE [8]
* CE 6 is refined into CE [9]


### Cost equations --> "Loop" of f/4
* CEs [9] --> Loop 6
* CEs [8] --> Loop 7

### Ranking functions of CR f(V,V1,V6,Out)

#### Partial ranking functions of CR f(V,V1,V6,Out)


### Specialization of cost equations g/3
* CE 5 is refined into CE [10]
* CE 4 is refined into CE [11]


### Cost equations --> "Loop" of g/3
* CEs [10] --> Loop 8
* CEs [11] --> Loop 9

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

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


### Specialization of cost equations start/3
* CE 2 is refined into CE [12,13]
* CE 3 is refined into CE [14]


### Cost equations --> "Loop" of start/3
* CEs [12,13,14] --> Loop 10

### Ranking functions of CR start(V,V1,V6)

#### Partial ranking functions of CR start(V,V1,V6)


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

#### Cost of chains of f(V,V1,V6,Out):
* Chain [7]: 0
with precondition: [Out=0,V>=0,V1>=0,V6>=0]

* Chain [6,7]: 1
with precondition: [Out=0,V1=V6,V>=1,V1>=0]


#### Cost of chains of g(V,V1,Out):
* Chain [9]: 1
with precondition: [V=Out,V>=0,V1>=0]

* Chain [8]: 1
with precondition: [V1=Out,V>=0,V1>=0]


#### Cost of chains of start(V,V1,V6):
* Chain [10]: 1
with precondition: [V>=0,V1>=0]


Closed-form bounds of start(V,V1,V6):
-------------------------------------
* Chain [10] with precondition: [V>=0,V1>=0]
- Upper bound: 1
- Complexity: constant

### Maximum cost of start(V,V1,V6): 1
Asymptotic class: constant
* Total analysis performed in 54 ms.

(10) BOUNDS(1, 1)