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:

a(c(d(x))) → c(x)
u(b(d(d(x)))) → b(x)
v(a(a(x))) → u(v(x))
v(a(c(x))) → u(b(d(x)))
v(c(x)) → b(x)
w(a(a(x))) → u(w(x))
w(a(c(x))) → u(b(d(x)))
w(c(x)) → b(x)

Rewrite Strategy: INNERMOST

(1) DependencyGraphProof (BOTH BOUNDS(ID, ID) transformation)

The following rules are not reachable from basic terms in the dependency graph and can be removed:
v(a(a(x))) → u(v(x))
v(a(c(x))) → u(b(d(x)))
w(a(a(x))) → u(w(x))
w(a(c(x))) → u(b(d(x)))

(2) 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:

v(c(x)) → b(x)
a(c(d(x))) → c(x)
u(b(d(d(x)))) → b(x)
w(c(x)) → b(x)

Rewrite Strategy: INNERMOST

(3) TrsToWeightedTrsProof (BOTH BOUNDS(ID, ID) transformation)

Transformed TRS to weighted TRS

(4) 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:

v(c(x)) → b(x) [1]
a(c(d(x))) → c(x) [1]
u(b(d(d(x)))) → b(x) [1]
w(c(x)) → b(x) [1]

Rewrite Strategy: INNERMOST

(5) TypeInferenceProof (BOTH BOUNDS(ID, ID) transformation)

Infered types.

(6) Obligation:

Runtime Complexity Weighted TRS with Types.
The TRS R consists of the following rules:

v(c(x)) → b(x) [1]
a(c(d(x))) → c(x) [1]
u(b(d(d(x)))) → b(x) [1]
w(c(x)) → b(x) [1]

The TRS has the following type information:

v :: c → b c :: d → c b :: d → b a :: c → c d :: d → d u :: b → b w :: c → b

Rewrite Strategy: INNERMOST

(7) 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:

v(v0) → null_v [0]
a(v0) → null_a [0]
u(v0) → null_u [0]
w(v0) → null_w [0]

And the following fresh constants:

null_v, null_a, null_u, null_w, const

(8) 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: v(c(x)) → b(x) [1] a(c(d(x))) → c(x) [1] u(b(d(d(x)))) → b(x) [1] w(c(x)) → b(x) [1] v(v0) → null_v [0] a(v0) → null_a [0] u(v0) → null_u [0] w(v0) → null_w [0] The TRS has the following type information: v :: c:null_a → b:null_v:null_u:null_w c :: d → c:null_a b :: d → b:null_v:null_u:null_w a :: c:null_a → c:null_a d :: d → d u :: b:null_v:null_u:null_w → b:null_v:null_u:null_w w :: c:null_a → b:null_v:null_u:null_w null_v :: b:null_v:null_u:null_w null_a :: c:null_a null_u :: b:null_v:null_u:null_w null_w :: b:null_v:null_u:null_w const :: d Rewrite Strategy: INNERMOST

(9) 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_v => 0
null_a => 0
null_u => 0
null_w => 0
const => 0

(10) Obligation:

Complexity RNTS consisting of the following rules:

a(z) -{ 0 }→ 0 :|: v0 >= 0, z = v0
a(z) -{ 1 }→ 1 + x :|: x >= 0, z = 1 + (1 + x)
u(z) -{ 0 }→ 0 :|: v0 >= 0, z = v0
u(z) -{ 1 }→ 1 + x :|: z = 1 + (1 + (1 + x)), x >= 0
v(z) -{ 0 }→ 0 :|: v0 >= 0, z = v0
v(z) -{ 1 }→ 1 + x :|: x >= 0, z = 1 + x
w(z) -{ 0 }→ 0 :|: v0 >= 0, z = v0
w(z) -{ 1 }→ 1 + x :|: x >= 0, z = 1 + x

Only complete derivations are relevant for the runtime complexity.

(11) CompleteCoflocoProof (EQUIVALENT transformation)

Transformed the RNTS (where only complete derivations are relevant) into cost relations for CoFloCo:

eq(start(V),0,[v(V, Out)],[V >= 0]). eq(start(V),0,[a(V, Out)],[V >= 0]). eq(start(V),0,[u(V, Out)],[V >= 0]). eq(start(V),0,[w(V, Out)],[V >= 0]). eq(v(V, Out),1,[],[Out = 1 + V1,V1 >= 0,V = 1 + V1]). eq(a(V, Out),1,[],[Out = 1 + V2,V2 >= 0,V = 2 + V2]). eq(u(V, Out),1,[],[Out = 1 + V3,V = 3 + V3,V3 >= 0]). eq(w(V, Out),1,[],[Out = 1 + V4,V4 >= 0,V = 1 + V4]). eq(v(V, Out),0,[],[Out = 0,V5 >= 0,V = V5]). eq(a(V, Out),0,[],[Out = 0,V6 >= 0,V = V6]). eq(u(V, Out),0,[],[Out = 0,V7 >= 0,V = V7]). eq(w(V, Out),0,[],[Out = 0,V8 >= 0,V = V8]). input_output_vars(v(V,Out),[V],[Out]). input_output_vars(a(V,Out),[V],[Out]). input_output_vars(u(V,Out),[V],[Out]). input_output_vars(w(V,Out),[V],[Out]).

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

#### Computed strongly connected components
0. non_recursive : [a/2]
1. non_recursive : [u/2]
2. non_recursive : [v/2]
3. non_recursive : [w/2]
4. non_recursive : [start/1]

#### Obtained direct recursion through partial evaluation
0. SCC is partially evaluated into a/2
1. SCC is partially evaluated into u/2
2. SCC is partially evaluated into v/2
3. SCC is partially evaluated into w/2
4. SCC is partially evaluated into start/1

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

### Specialization of cost equations a/2
* CE 8 is refined into CE [14]
* CE 9 is refined into CE [15]


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

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

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


### Specialization of cost equations u/2
* CE 10 is refined into CE [16]
* CE 11 is refined into CE [17]


### Cost equations --> "Loop" of u/2
* CEs [16] --> Loop 12
* CEs [17] --> Loop 13

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

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


### Specialization of cost equations v/2
* CE 6 is refined into CE [18]
* CE 7 is refined into CE [19]


### Cost equations --> "Loop" of v/2
* CEs [18] --> Loop 14
* CEs [19] --> Loop 15

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

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


### Specialization of cost equations w/2
* CE 12 is refined into CE [20]
* CE 13 is refined into CE [21]


### Cost equations --> "Loop" of w/2
* CEs [20] --> Loop 16
* CEs [21] --> Loop 17

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

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


### Specialization of cost equations start/1
* CE 2 is refined into CE [22,23]
* CE 3 is refined into CE [24,25]
* CE 4 is refined into CE [26,27]
* CE 5 is refined into CE [28,29]


### Cost equations --> "Loop" of start/1
* CEs [22,23,24,25,26,27,28,29] --> Loop 18

### Ranking functions of CR start(V)

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


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

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

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


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

* Chain [12]: 1
with precondition: [V=Out+2,V>=3]


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

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


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

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


#### Cost of chains of start(V):
* Chain [18]: 1
with precondition: [V>=0]


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

### Maximum cost of start(V): 1
Asymptotic class: constant
* Total analysis performed in 58 ms.

(12) BOUNDS(1, 1)