The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(1, n^2).

(0) Obligation:

The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(1, n^2).


The TRS R consists of the following rules:

fac(s(x)) → *(fac(p(s(x))), s(x))
p(s(0)) → 0
p(s(s(x))) → s(p(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, n^2).


The TRS R consists of the following rules:

fac(s(x)) → *(fac(p(s(x))), s(x)) [1]
p(s(0)) → 0 [1]
p(s(s(x))) → s(p(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:

fac(s(x)) → *(fac(p(s(x))), s(x)) [1]
p(s(0)) → 0 [1]
p(s(s(x))) → s(p(s(x))) [1]

The TRS has the following type information:

fac :: s:0 → * s :: s:0 → s:0 * :: * → s:0 → * p :: s:0 → s:0 0 :: s:0

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:

fac(v0) → null_fac [0]
p(v0) → null_p [0]

And the following fresh constants:

null_fac, null_p

(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: fac(s(x)) → *(fac(p(s(x))), s(x)) [1] p(s(0)) → 0 [1] p(s(s(x))) → s(p(s(x))) [1] fac(v0) → null_fac [0] p(v0) → null_p [0] The TRS has the following type information: fac :: s:0:null_p → *:null_fac s :: s:0:null_p → s:0:null_p * :: *:null_fac → s:0:null_p → *:null_fac p :: s:0:null_p → s:0:null_p 0 :: s:0:null_p null_fac :: *:null_fac null_p :: s:0:null_p 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:

0 => 0
null_fac => 0
null_p => 0

(8) Obligation:

Complexity RNTS consisting of the following rules:

fac(z) -{ 0 }→ 0 :|: v0 >= 0, z = v0
fac(z) -{ 1 }→ 1 + fac(p(1 + x)) + (1 + x) :|: x >= 0, z = 1 + x
p(z) -{ 1 }→ 0 :|: z = 1 + 0
p(z) -{ 0 }→ 0 :|: v0 >= 0, z = v0
p(z) -{ 1 }→ 1 + p(1 + x) :|: x >= 0, z = 1 + (1 + 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),0,[fac(V, Out)],[V >= 0]). eq(start(V),0,[p(V, Out)],[V >= 0]). eq(fac(V, Out),1,[p(1 + V1, Ret010),fac(Ret010, Ret01)],[Out = 2 + Ret01 + V1,V1 >= 0,V = 1 + V1]). eq(p(V, Out),1,[],[Out = 0,V = 1]). eq(p(V, Out),1,[p(1 + V2, Ret1)],[Out = 1 + Ret1,V2 >= 0,V = 2 + V2]). eq(fac(V, Out),0,[],[Out = 0,V3 >= 0,V = V3]). eq(p(V, Out),0,[],[Out = 0,V4 >= 0,V = V4]). input_output_vars(fac(V,Out),[V],[Out]). input_output_vars(p(V,Out),[V],[Out]).

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

#### Computed strongly connected components
0. recursive : [p/2]
1. recursive : [fac/2]
2. non_recursive : [start/1]

#### Obtained direct recursion through partial evaluation
0. SCC is partially evaluated into p/2
1. SCC is partially evaluated into fac/2
2. SCC is partially evaluated into start/1

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

### Specialization of cost equations p/2
* CE 6 is refined into CE [9]
* CE 8 is refined into CE [10]
* CE 7 is refined into CE [11]


### Cost equations --> "Loop" of p/2
* CEs [11] --> Loop 6
* CEs [9,10] --> Loop 7

### Ranking functions of CR p(V,Out)
* RF of phase [6]: [V-1]

#### Partial ranking functions of CR p(V,Out)
* Partial RF of phase [6]:
- RF of loop [6:1]:
V-1


### Specialization of cost equations fac/2
* CE 5 is refined into CE [12]
* CE 4 is refined into CE [13,14]


### Cost equations --> "Loop" of fac/2
* CEs [14] --> Loop 8
* CEs [13] --> Loop 9
* CEs [12] --> Loop 10

### Ranking functions of CR fac(V,Out)
* RF of phase [8]: [V-1]

#### Partial ranking functions of CR fac(V,Out)
* Partial RF of phase [8]:
- RF of loop [8:1]:
V-1


### Specialization of cost equations start/1
* CE 2 is refined into CE [15,16,17]
* CE 3 is refined into CE [18,19]


### Cost equations --> "Loop" of start/1
* CEs [15,16,17,18,19] --> Loop 11

### Ranking functions of CR start(V)

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


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

#### Cost of chains of p(V,Out):
* Chain [[6],7]: 1*it(6)+1
Such that:it(6) =< Out

with precondition: [Out>=1,V>=Out+1]

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


#### Cost of chains of fac(V,Out):
* Chain [[8],10]: 2*it(8)+1*s(3)+0
Such that:aux(3) =< V
it(8) =< aux(3)
s(3) =< it(8)*aux(3)

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

* Chain [[8],9,10]: 2*it(8)+1*s(3)+2
Such that:aux(4) =< V
it(8) =< aux(4)
s(3) =< it(8)*aux(4)

with precondition: [V>=2,Out>=V+3]

* Chain [10]: 0
with precondition: [Out=0,V>=0]

* Chain [9,10]: 2
with precondition: [V+1=Out,V>=1]


#### Cost of chains of start(V):
* Chain [11]: 5*s(11)+2*s(12)+2
Such that:aux(6) =< V
s(11) =< aux(6)
s(12) =< s(11)*aux(6)

with precondition: [V>=0]


Closed-form bounds of start(V):
-------------------------------------
* Chain [11] with precondition: [V>=0]
- Upper bound: 5*V+2+2*V*V
- Complexity: n^2

### Maximum cost of start(V): 5*V+2+2*V*V
Asymptotic class: n^2
* Total analysis performed in 87 ms.

(10) BOUNDS(1, n^2)