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:

a__f(f(a)) → a__f(g(f(a)))
mark(f(X)) → a__f(mark(X))
mark(a) → a
mark(g(X)) → g(X)
a__f(X) → f(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:

a__f(f(a)) → a__f(g(f(a))) [1]
mark(f(X)) → a__f(mark(X)) [1]
mark(a) → a [1]
mark(g(X)) → g(X) [1]
a__f(X) → f(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:

a__f(f(a)) → a__f(g(f(a))) [1]
mark(f(X)) → a__f(mark(X)) [1]
mark(a) → a [1]
mark(g(X)) → g(X) [1]
a__f(X) → f(X) [1]

The TRS has the following type information:

a__f :: a:f:g → a:f:g f :: a:f:g → a:f:g a :: a:f:g g :: a:f:g → a:f:g mark :: a:f:g → a:f:g

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: a__f(f(a)) → a__f(g(f(a))) [1] mark(f(X)) → a__f(mark(X)) [1] mark(a) → a [1] mark(g(X)) → g(X) [1] a__f(X) → f(X) [1] The TRS has the following type information: a__f :: a:f:g → a:f:g f :: a:f:g → a:f:g a :: a:f:g g :: a:f:g → a:f:g mark :: a:f:g → a:f:g 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:

a => 0

(8) Obligation:

Complexity RNTS consisting of the following rules:

a__f(z) -{ 1 }→ a__f(1 + (1 + 0)) :|: z = 1 + 0
a__f(z) -{ 1 }→ 1 + X :|: X >= 0, z = X
mark(z) -{ 1 }→ a__f(mark(X)) :|: z = 1 + X, X >= 0
mark(z) -{ 1 }→ 0 :|: z = 0
mark(z) -{ 1 }→ 1 + X :|: z = 1 + X, X >= 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),0,[fun(V, Out)],[V >= 0]). eq(start(V),0,[mark(V, Out)],[V >= 0]). eq(fun(V, Out),1,[fun(1 + (1 + 0), Ret)],[Out = Ret,V = 1]). eq(mark(V, Out),1,[mark(X1, Ret0),fun(Ret0, Ret1)],[Out = Ret1,V = 1 + X1,X1 >= 0]). eq(mark(V, Out),1,[],[Out = 0,V = 0]). eq(mark(V, Out),1,[],[Out = 1 + X2,V = 1 + X2,X2 >= 0]). eq(fun(V, Out),1,[],[Out = 1 + X3,X3 >= 0,V = X3]). input_output_vars(fun(V,Out),[V],[Out]). input_output_vars(mark(V,Out),[V],[Out]).

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

#### Computed strongly connected components
0. recursive : [fun/2]
1. recursive [non_tail] : [mark/2]
2. non_recursive : [start/1]

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

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

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


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

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

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


### Specialization of cost equations mark/2
* CE 8 is refined into CE [11]
* CE 7 is refined into CE [12]
* CE 6 is refined into CE [13,14]


### Cost equations --> "Loop" of mark/2
* CEs [14] --> Loop 9
* CEs [13] --> Loop 10
* CEs [11] --> Loop 11
* CEs [12] --> Loop 12

### Ranking functions of CR mark(V,Out)
* RF of phase [9,10]: [V]

#### Partial ranking functions of CR mark(V,Out)
* Partial RF of phase [9,10]:
- RF of loop [9:1,10:1]:
V


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


### Cost equations --> "Loop" of start/1
* CEs [15] --> Loop 13
* CEs [16,17,18,19] --> Loop 14

### Ranking functions of CR start(V)

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


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

#### Cost of chains of fun(V,Out):
* Chain [8]: 1
with precondition: [V+1=Out,V>=0]

* Chain [7,8]: 2
with precondition: [V=1,Out=3]


#### Cost of chains of mark(V,Out):
* Chain [[9,10],12]: 5*it(9)+1
Such that:aux(3) =< V
it(9) =< aux(3)

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

* Chain [[9,10],11]: 5*it(9)+1
Such that:aux(4) =< V
it(9) =< aux(4)

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

* Chain [12]: 1
with precondition: [V=0,Out=0]

* Chain [11]: 1
with precondition: [V=Out,V>=1]


#### Cost of chains of start(V):
* Chain [14]: 10*s(4)+1
Such that:aux(5) =< V
s(4) =< aux(5)

with precondition: [V>=0]

* Chain [13]: 2
with precondition: [V=1]


Closed-form bounds of start(V):
-------------------------------------
* Chain [14] with precondition: [V>=0]
- Upper bound: 10*V+1
- Complexity: n
* Chain [13] with precondition: [V=1]
- Upper bound: 2
- Complexity: constant

### Maximum cost of start(V): max([10*V,1])+1
Asymptotic class: n
* Total analysis performed in 84 ms.

(10) BOUNDS(1, n^1)