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


The TRS R consists of the following rules:

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

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

The TRS has the following type information:

half :: 0:s → 0:s 0 :: 0:s s :: 0:s → 0:s bits :: 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:
none

And the following fresh constants: none

(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(0)) → 0 [1] half(s(s(x))) → s(half(x)) [1] bits(0) → 0 [1] bits(s(x)) → s(bits(half(s(x)))) [1] The TRS has the following type information: half :: 0:s → 0:s 0 :: 0:s s :: 0:s → 0:s bits :: 0:s → 0: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:

0 => 0

(8) Obligation:

Complexity RNTS consisting of the following rules:

bits(z) -{ 1 }→ 0 :|: z = 0
bits(z) -{ 1 }→ 1 + bits(half(1 + x)) :|: x >= 0, z = 1 + x
half(z) -{ 1 }→ 0 :|: z = 0
half(z) -{ 1 }→ 0 :|: z = 1 + 0
half(z) -{ 1 }→ 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,[bits(V, Out)],[V >= 0]). eq(half(V, Out),1,[],[Out = 0,V = 0]). eq(half(V, Out),1,[],[Out = 0,V = 1]). eq(half(V, Out),1,[half(V1, Ret1)],[Out = 1 + Ret1,V1 >= 0,V = 2 + V1]). eq(bits(V, Out),1,[],[Out = 0,V = 0]). eq(bits(V, Out),1,[half(1 + V2, Ret10),bits(Ret10, Ret11)],[Out = 1 + Ret11,V2 >= 0,V = 1 + V2]). input_output_vars(half(V,Out),[V],[Out]). input_output_vars(bits(V,Out),[V],[Out]).

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

#### Computed strongly connected components
0. recursive : [half/2]
1. recursive : [bits/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 bits/2
2. SCC is partially evaluated into start/1

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

### Specialization of cost equations half/2
* CE 6 is refined into CE [9]
* CE 5 is refined into CE [10]
* CE 4 is refined into CE [11]


### Cost equations --> "Loop" of half/2
* CEs [10] --> Loop 7
* CEs [11] --> Loop 8
* CEs [9] --> Loop 9

### Ranking functions of CR half(V,Out)
* RF of phase [9]: [V-1]

#### Partial ranking functions of CR half(V,Out)
* Partial RF of phase [9]:
- RF of loop [9:1]:
V-1


### Specialization of cost equations bits/2
* CE 8 is refined into CE [12,13,14]
* CE 7 is refined into CE [15]


### Cost equations --> "Loop" of bits/2
* CEs [15] --> Loop 10
* CEs [14] --> Loop 11
* CEs [13] --> Loop 12
* CEs [12] --> Loop 13

### Ranking functions of CR bits(V,Out)
* RF of phase [11,12]: [V-1]

#### Partial ranking functions of CR bits(V,Out)
* Partial RF of phase [11,12]:
- RF of loop [11:1]:
V/2-1
- RF of loop [12:1]:
V-1


### Specialization of cost equations start/1
* CE 2 is refined into CE [16,17,18,19]
* CE 3 is refined into CE [20,21,22]


### Cost equations --> "Loop" of start/1
* CEs [18,19,22] --> Loop 14
* CEs [17,21] --> Loop 15
* CEs [16,20] --> Loop 16

### Ranking functions of CR start(V)

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


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

#### Cost of chains of half(V,Out):
* Chain [[9],8]: 1*it(9)+1
Such that:it(9) =< 2*Out

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

* Chain [[9],7]: 1*it(9)+1
Such that:it(9) =< 2*Out

with precondition: [V=2*Out+1,V>=3]

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

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


#### Cost of chains of bits(V,Out):
* Chain [[11,12],13,10]: 2*it(11)+2*it(12)+2*s(5)+3
Such that:it(11) =< V/2
aux(5) =< V
aux(6) =< 2*V
it(11) =< aux(5)
it(12) =< aux(5)
it(12) =< aux(6)
s(5) =< aux(6)

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

* Chain [13,10]: 3
with precondition: [V=1,Out=1]

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


#### Cost of chains of start(V):
* Chain [16]: 1
with precondition: [V=0]

* Chain [15]: 3
with precondition: [V=1]

* Chain [14]: 2*s(7)+2*s(9)+2*s(12)+2*s(13)+3
Such that:s(11) =< 2*V
s(9) =< V/2
aux(7) =< V
s(7) =< aux(7)
s(9) =< aux(7)
s(12) =< aux(7)
s(12) =< s(11)
s(13) =< s(11)

with precondition: [V>=2]


Closed-form bounds of start(V):
-------------------------------------
* Chain [16] with precondition: [V=0]
- Upper bound: 1
- Complexity: constant
* Chain [15] with precondition: [V=1]
- Upper bound: 3
- Complexity: constant
* Chain [14] with precondition: [V>=2]
- Upper bound: 9*V+3
- Complexity: n

### Maximum cost of start(V): 9*V+3
Asymptotic class: n
* Total analysis performed in 128 ms.

(10) BOUNDS(1, n^1)