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:

terms(N) → cons(recip(sqr(N)))
sqr(0) → 0
sqr(s) → s
dbl(0) → 0
dbl(s) → s
add(0, X) → X
add(s, Y) → s
first(0, X) → nil
first(s, cons(Y)) → cons(Y)

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:

terms(N) → cons(recip(sqr(N))) [1]
sqr(0) → 0 [1]
sqr(s) → s [1]
dbl(0) → 0 [1]
dbl(s) → s [1]
add(0, X) → X [1]
add(s, Y) → s [1]
first(0, X) → nil [1]
first(s, cons(Y)) → cons(Y) [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:

terms(N) → cons(recip(sqr(N))) [1]
sqr(0) → 0 [1]
sqr(s) → s [1]
dbl(0) → 0 [1]
dbl(s) → s [1]
add(0, X) → X [1]
add(s, Y) → s [1]
first(0, X) → nil [1]
first(s, cons(Y)) → cons(Y) [1]

The TRS has the following type information:

terms :: 0:s → cons:nil cons :: recip → cons:nil recip :: 0:s → recip sqr :: 0:s → 0:s 0 :: 0:s s :: 0:s dbl :: 0:s → 0:s add :: 0:s → 0:s → 0:s first :: 0:s → cons:nil → cons:nil nil :: cons:nil

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:

first(v0, v1) → null_first [0]

And the following fresh constants:

null_first, 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: terms(N) → cons(recip(sqr(N))) [1] sqr(0) → 0 [1] sqr(s) → s [1] dbl(0) → 0 [1] dbl(s) → s [1] add(0, X) → X [1] add(s, Y) → s [1] first(0, X) → nil [1] first(s, cons(Y)) → cons(Y) [1] first(v0, v1) → null_first [0] The TRS has the following type information: terms :: 0:s → cons:nil:null_first cons :: recip → cons:nil:null_first recip :: 0:s → recip sqr :: 0:s → 0:s 0 :: 0:s s :: 0:s dbl :: 0:s → 0:s add :: 0:s → 0:s → 0:s first :: 0:s → cons:nil:null_first → cons:nil:null_first nil :: cons:nil:null_first null_first :: cons:nil:null_first const :: recip 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
s => 1
nil => 0
null_first => 0
const => 0

(8) Obligation:

Complexity RNTS consisting of the following rules:

add(z, z') -{ 1 }→ X :|: z' = X, X >= 0, z = 0
add(z, z') -{ 1 }→ 1 :|: z' = Y, Y >= 0, z = 1
dbl(z) -{ 1 }→ 1 :|: z = 1
dbl(z) -{ 1 }→ 0 :|: z = 0
first(z, z') -{ 1 }→ 0 :|: z' = X, X >= 0, z = 0
first(z, z') -{ 0 }→ 0 :|: v0 >= 0, v1 >= 0, z = v0, z' = v1
first(z, z') -{ 1 }→ 1 + Y :|: Y >= 0, z = 1, z' = 1 + Y
sqr(z) -{ 1 }→ 1 :|: z = 1
sqr(z) -{ 1 }→ 0 :|: z = 0
terms(z) -{ 1 }→ 1 + (1 + sqr(N)) :|: z = N, N >= 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),0,[terms(V, Out)],[V >= 0]). eq(start(V, V1),0,[sqr(V, Out)],[V >= 0]). eq(start(V, V1),0,[dbl(V, Out)],[V >= 0]). eq(start(V, V1),0,[add(V, V1, Out)],[V >= 0,V1 >= 0]). eq(start(V, V1),0,[first(V, V1, Out)],[V >= 0,V1 >= 0]). eq(terms(V, Out),1,[sqr(N1, Ret11)],[Out = 2 + Ret11,V = N1,N1 >= 0]). eq(sqr(V, Out),1,[],[Out = 0,V = 0]). eq(sqr(V, Out),1,[],[Out = 1,V = 1]). eq(dbl(V, Out),1,[],[Out = 0,V = 0]). eq(dbl(V, Out),1,[],[Out = 1,V = 1]). eq(add(V, V1, Out),1,[],[Out = X1,V1 = X1,X1 >= 0,V = 0]). eq(add(V, V1, Out),1,[],[Out = 1,V1 = Y1,Y1 >= 0,V = 1]). eq(first(V, V1, Out),1,[],[Out = 0,V1 = X2,X2 >= 0,V = 0]). eq(first(V, V1, Out),1,[],[Out = 1 + Y2,Y2 >= 0,V = 1,V1 = 1 + Y2]). eq(first(V, V1, Out),0,[],[Out = 0,V2 >= 0,V3 >= 0,V = V2,V1 = V3]). input_output_vars(terms(V,Out),[V],[Out]). input_output_vars(sqr(V,Out),[V],[Out]). input_output_vars(dbl(V,Out),[V],[Out]). input_output_vars(add(V,V1,Out),[V,V1],[Out]). input_output_vars(first(V,V1,Out),[V,V1],[Out]).

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

#### Computed strongly connected components
0. non_recursive : [add/3]
1. non_recursive : [dbl/2]
2. non_recursive : [first/3]
3. non_recursive : [sqr/2]
4. non_recursive : [terms/2]
5. non_recursive : [start/2]

#### Obtained direct recursion through partial evaluation
0. SCC is partially evaluated into add/3
1. SCC is partially evaluated into dbl/2
2. SCC is partially evaluated into first/3
3. SCC is partially evaluated into sqr/2
4. SCC is completely evaluated into other SCCs
5. SCC is partially evaluated into start/2

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

### Specialization of cost equations add/3
* CE 12 is refined into CE [16]
* CE 11 is refined into CE [17]


### Cost equations --> "Loop" of add/3
* CEs [16] --> Loop 10
* CEs [17] --> Loop 11

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

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


### Specialization of cost equations dbl/2
* CE 10 is refined into CE [18]
* CE 9 is refined into CE [19]


### Cost equations --> "Loop" of dbl/2
* CEs [18] --> Loop 12
* CEs [19] --> Loop 13

### Ranking functions of CR dbl(V,Out)

#### Partial ranking functions of CR dbl(V,Out)


### Specialization of cost equations first/3
* CE 14 is refined into CE [20]
* CE 13 is refined into CE [21]
* CE 15 is refined into CE [22]


### Cost equations --> "Loop" of first/3
* CEs [20] --> Loop 14
* CEs [21,22] --> Loop 15

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

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


### Specialization of cost equations sqr/2
* CE 8 is refined into CE [23]
* CE 7 is refined into CE [24]


### Cost equations --> "Loop" of sqr/2
* CEs [23] --> Loop 16
* CEs [24] --> Loop 17

### Ranking functions of CR sqr(V,Out)

#### Partial ranking functions of CR sqr(V,Out)


### Specialization of cost equations start/2
* CE 2 is refined into CE [25,26]
* CE 3 is refined into CE [27,28]
* CE 4 is refined into CE [29,30]
* CE 5 is refined into CE [31,32]
* CE 6 is refined into CE [33,34]


### Cost equations --> "Loop" of start/2
* CEs [34] --> Loop 18
* CEs [26,28,30,32,33] --> Loop 19
* CEs [25,27,29,31] --> Loop 20

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

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


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

#### Cost of chains of add(V,V1,Out):
* Chain [11]: 1
with precondition: [V=0,V1=Out,V1>=0]

* Chain [10]: 1
with precondition: [V=1,Out=1,V1>=0]


#### Cost of chains of dbl(V,Out):
* Chain [13]: 1
with precondition: [V=0,Out=0]

* Chain [12]: 1
with precondition: [V=1,Out=1]


#### Cost of chains of first(V,V1,Out):
* Chain [15]: 1
with precondition: [Out=0,V>=0,V1>=0]

* Chain [14]: 1
with precondition: [V=1,V1=Out,V1>=1]


#### Cost of chains of sqr(V,Out):
* Chain [17]: 1
with precondition: [V=0,Out=0]

* Chain [16]: 1
with precondition: [V=1,Out=1]


#### Cost of chains of start(V,V1):
* Chain [20]: 2
with precondition: [V=0]

* Chain [19]: 2
with precondition: [V=1]

* Chain [18]: 1
with precondition: [V>=0,V1>=0]


Closed-form bounds of start(V,V1):
-------------------------------------
* Chain [20] with precondition: [V=0]
- Upper bound: 2
- Complexity: constant
* Chain [19] with precondition: [V=1]
- Upper bound: 2
- Complexity: constant
* Chain [18] with precondition: [V>=0,V1>=0]
- Upper bound: 1
- Complexity: constant

### Maximum cost of start(V,V1): 2
Asymptotic class: constant
* Total analysis performed in 80 ms.

(10) BOUNDS(1, 1)