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

(0) Obligation:

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


The TRS R consists of the following rules:

half(0) → 0
half(s(s(x))) → s(half(x))
log(s(0)) → 0
log(s(s(x))) → s(log(s(half(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^1).


The TRS R consists of the following rules:

half(0) → 0 [1]
half(s(s(x))) → s(half(x)) [1]
log(s(0)) → 0 [1]
log(s(s(x))) → s(log(s(half(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:

half(0) → 0 [1]
half(s(s(x))) → s(half(x)) [1]
log(s(0)) → 0 [1]
log(s(s(x))) → s(log(s(half(x)))) [1]

The TRS has the following type information:

half :: 0:s → 0:s 0 :: 0:s s :: 0:s → 0:s log :: 0:s → 0: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:

half(v0) → null_half [0]
log(v0) → null_log [0]

And the following fresh constants:

null_half, null_log

(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: half(0) → 0 [1] half(s(s(x))) → s(half(x)) [1] log(s(0)) → 0 [1] log(s(s(x))) → s(log(s(half(x)))) [1] half(v0) → null_half [0] log(v0) → null_log [0] The TRS has the following type information: half :: 0:s:null_half:null_log → 0:s:null_half:null_log 0 :: 0:s:null_half:null_log s :: 0:s:null_half:null_log → 0:s:null_half:null_log log :: 0:s:null_half:null_log → 0:s:null_half:null_log null_half :: 0:s:null_half:null_log null_log :: 0:s:null_half:null_log 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_half => 0
null_log => 0

(8) Obligation:

Complexity RNTS consisting of the following rules:

half(z) -{ 1 }→ 0 :|: z = 0
half(z) -{ 0 }→ 0 :|: v0 >= 0, z = v0
half(z) -{ 1 }→ 1 + half(x) :|: x >= 0, z = 1 + (1 + x)
log(z) -{ 1 }→ 0 :|: z = 1 + 0
log(z) -{ 0 }→ 0 :|: v0 >= 0, z = v0
log(z) -{ 1 }→ 1 + log(1 + half(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,[half(V, Out)],[V >= 0]). eq(start(V),0,[log(V, Out)],[V >= 0]). eq(half(V, Out),1,[],[Out = 0,V = 0]). eq(half(V, Out),1,[half(V1, Ret1)],[Out = 1 + Ret1,V1 >= 0,V = 2 + V1]). eq(log(V, Out),1,[],[Out = 0,V = 1]). eq(log(V, Out),1,[half(V2, Ret101),log(1 + Ret101, Ret11)],[Out = 1 + Ret11,V2 >= 0,V = 2 + V2]). eq(half(V, Out),0,[],[Out = 0,V3 >= 0,V = V3]). eq(log(V, Out),0,[],[Out = 0,V4 >= 0,V = V4]). input_output_vars(half(V,Out),[V],[Out]). input_output_vars(log(V,Out),[V],[Out]).

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

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

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

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

### Specialization of cost equations half/2
* CE 4 is refined into CE [10]
* CE 6 is refined into CE [11]
* CE 5 is refined into CE [12]


### Cost equations --> "Loop" of half/2
* CEs [12] --> Loop 6
* CEs [10,11] --> Loop 7

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

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


### Specialization of cost equations log/2
* CE 7 is refined into CE [13]
* CE 9 is refined into CE [14]
* CE 8 is refined into CE [15,16]


### Cost equations --> "Loop" of log/2
* CEs [16] --> Loop 8
* CEs [15] --> Loop 9
* CEs [13,14] --> Loop 10

### Ranking functions of CR log(V,Out)
* RF of phase [8]: [V-3]

#### Partial ranking functions of CR log(V,Out)
* Partial RF of phase [8]:
- RF of loop [8:1]:
V-3


### Specialization of cost equations start/1
* CE 2 is refined into CE [17,18]
* CE 3 is refined into CE [19,20,21,22]


### Cost equations --> "Loop" of start/1
* CEs [17,18,19,20,21,22] --> Loop 11

### Ranking functions of CR start(V)

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


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

#### Cost of chains of half(V,Out):
* Chain [[6],7]: 1*it(6)+1
Such that:it(6) =< 2*Out

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

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


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

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

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

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

* Chain [10]: 1
with precondition: [Out=0,V>=0]

* Chain [9,10]: 3
with precondition: [Out=1,V>=2]


#### Cost of chains of start(V):
* Chain [11]: 5*s(4)+2*s(6)+3
Such that:aux(1) =< V
aux(2) =< 2*V
s(4) =< aux(1)
s(6) =< aux(2)

with precondition: [V>=0]


Closed-form bounds of start(V):
-------------------------------------
* Chain [11] with precondition: [V>=0]
- Upper bound: 9*V+3
- Complexity: n

### Maximum cost of start(V): 9*V+3
Asymptotic class: n
* Total analysis performed in 85 ms.

(10) BOUNDS(1, n^1)