/export/starexec/sandbox/solver/bin/starexec_run_complexity /export/starexec/sandbox/benchmark/theBenchmark.xml /export/starexec/sandbox/output/output_files


--------------------------------------------------------------------------------


WORST_CASE(Omega(n^1), O(n^1))
proof of /export/starexec/sandbox/benchmark/theBenchmark.xml
# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty


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

(0) CpxTRS
(1) RelTrsToWeightedTrsProof [BOTH BOUNDS(ID, ID), 0 ms]
(2) CpxWeightedTrs
    (3) TypeInferenceProof [BOTH BOUNDS(ID, ID), 0 ms]
    (4) CpxTypedWeightedTrs
    (5) CompletionProof [UPPER BOUND(ID), 0 ms]
    (6) CpxTypedWeightedCompleteTrs
    (7) CpxTypedWeightedTrsToRntsProof [UPPER BOUND(ID), 0 ms]
    (8) CpxRNTS
    (9) CompleteCoflocoProof [FINISHED, 329 ms]
    (10) BOUNDS(1, n^1)
(11) RelTrsToDecreasingLoopProblemProof [LOWER BOUND(ID), 0 ms]
(12) TRS for Loop Detection
    (13) DecreasingLoopProof [LOWER BOUND(ID), 0 ms]
    (14) BEST
        (15) proven lower bound
            (16) LowerBoundPropagationProof [FINISHED, 0 ms]
            (17) BOUNDS(n^1, INF)
        (18) TRS for Loop Detection


----------------------------------------

(0)
Obligation:
The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(n^1, n^1).


The TRS R consists of the following rules:

   minus(0, Y) -> 0
   minus(s(X), s(Y)) -> minus(X, Y)
   geq(X, 0) -> true
   geq(0, s(Y)) -> false
   geq(s(X), s(Y)) -> geq(X, Y)
   div(0, s(Y)) -> 0
   div(s(X), s(Y)) -> if(geq(X, Y), s(div(minus(X, Y), s(Y))), 0)
   if(true, X, Y) -> X
   if(false, X, Y) -> Y

S is empty.
Rewrite Strategy: INNERMOST
----------------------------------------

(1) RelTrsToWeightedTrsProof (BOTH BOUNDS(ID, ID))
Transformed relative 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:

   minus(0, Y) -> 0 [1]
   minus(s(X), s(Y)) -> minus(X, Y) [1]
   geq(X, 0) -> true [1]
   geq(0, s(Y)) -> false [1]
   geq(s(X), s(Y)) -> geq(X, Y) [1]
   div(0, s(Y)) -> 0 [1]
   div(s(X), s(Y)) -> if(geq(X, Y), s(div(minus(X, Y), s(Y))), 0) [1]
   if(true, X, Y) -> X [1]
   if(false, X, Y) -> Y [1]

Rewrite Strategy: INNERMOST
----------------------------------------

(3) TypeInferenceProof (BOTH BOUNDS(ID, ID))
Infered types.
----------------------------------------

(4)
Obligation:
Runtime Complexity Weighted TRS with Types.
The TRS R consists of the following rules:

   minus(0, Y) -> 0 [1]
   minus(s(X), s(Y)) -> minus(X, Y) [1]
   geq(X, 0) -> true [1]
   geq(0, s(Y)) -> false [1]
   geq(s(X), s(Y)) -> geq(X, Y) [1]
   div(0, s(Y)) -> 0 [1]
   div(s(X), s(Y)) -> if(geq(X, Y), s(div(minus(X, Y), s(Y))), 0) [1]
   if(true, X, Y) -> X [1]
   if(false, X, Y) -> Y [1]

The TRS has the following type information:
minus :: 0:s -> 0:s -> 0:s
0 :: 0:s
s :: 0:s -> 0:s
geq :: 0:s -> 0:s -> true:false
true :: true:false
false :: true:false
div :: 0:s -> 0:s -> 0:s
if :: true:false -> 0:s -> 0:s -> 0:s

Rewrite Strategy: INNERMOST
----------------------------------------

(5) CompletionProof (UPPER BOUND(ID))
The TRS is a completely defined constructor system, as every type has a constant constructor and the following rules were added:

   minus(v0, v1) -> null_minus [0]
   div(v0, v1) -> null_div [0]
   geq(v0, v1) -> null_geq [0]
   if(v0, v1, v2) -> null_if [0]

And the following fresh constants:    null_minus, null_div, null_geq, null_if

----------------------------------------

(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:

   minus(0, Y) -> 0 [1]
   minus(s(X), s(Y)) -> minus(X, Y) [1]
   geq(X, 0) -> true [1]
   geq(0, s(Y)) -> false [1]
   geq(s(X), s(Y)) -> geq(X, Y) [1]
   div(0, s(Y)) -> 0 [1]
   div(s(X), s(Y)) -> if(geq(X, Y), s(div(minus(X, Y), s(Y))), 0) [1]
   if(true, X, Y) -> X [1]
   if(false, X, Y) -> Y [1]
   minus(v0, v1) -> null_minus [0]
   div(v0, v1) -> null_div [0]
   geq(v0, v1) -> null_geq [0]
   if(v0, v1, v2) -> null_if [0]

The TRS has the following type information:
minus :: 0:s:null_minus:null_div:null_if -> 0:s:null_minus:null_div:null_if -> 0:s:null_minus:null_div:null_if
0 :: 0:s:null_minus:null_div:null_if
s :: 0:s:null_minus:null_div:null_if -> 0:s:null_minus:null_div:null_if
geq :: 0:s:null_minus:null_div:null_if -> 0:s:null_minus:null_div:null_if -> true:false:null_geq
true :: true:false:null_geq
false :: true:false:null_geq
div :: 0:s:null_minus:null_div:null_if -> 0:s:null_minus:null_div:null_if -> 0:s:null_minus:null_div:null_if
if :: true:false:null_geq -> 0:s:null_minus:null_div:null_if -> 0:s:null_minus:null_div:null_if -> 0:s:null_minus:null_div:null_if
null_minus :: 0:s:null_minus:null_div:null_if
null_div :: 0:s:null_minus:null_div:null_if
null_geq :: true:false:null_geq
null_if :: 0:s:null_minus:null_div:null_if

Rewrite Strategy: INNERMOST
----------------------------------------

(7) CpxTypedWeightedTrsToRntsProof (UPPER BOUND(ID))
Transformed the TRS into an over-approximating RNTS by (improved) Size Abstraction.
The constant constructors are abstracted as follows:

   0 => 0
   true => 2
   false => 1
   null_minus => 0
   null_div => 0
   null_geq => 0
   null_if => 0

----------------------------------------

(8)
Obligation:
Complexity RNTS consisting of the following rules:

   div(z, z') -{ 1 }-> if(geq(X, Y), 1 + div(minus(X, Y), 1 + Y), 0) :|: z = 1 + X, Y >= 0, z' = 1 + Y, X >= 0
   div(z, z') -{ 1 }-> 0 :|: Y >= 0, z' = 1 + Y, z = 0
   div(z, z') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z = v0, z' = v1
   geq(z, z') -{ 1 }-> geq(X, Y) :|: z = 1 + X, Y >= 0, z' = 1 + Y, X >= 0
   geq(z, z') -{ 1 }-> 2 :|: X >= 0, z = X, z' = 0
   geq(z, z') -{ 1 }-> 1 :|: Y >= 0, z' = 1 + Y, z = 0
   geq(z, z') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z = v0, z' = v1
   if(z, z', z'') -{ 1 }-> X :|: z = 2, z' = X, Y >= 0, z'' = Y, X >= 0
   if(z, z', z'') -{ 1 }-> Y :|: z' = X, Y >= 0, z = 1, z'' = Y, X >= 0
   if(z, z', z'') -{ 0 }-> 0 :|: v0 >= 0, z'' = v2, v1 >= 0, z = v0, z' = v1, v2 >= 0
   minus(z, z') -{ 1 }-> minus(X, Y) :|: z = 1 + X, Y >= 0, z' = 1 + Y, X >= 0
   minus(z, z') -{ 1 }-> 0 :|: z' = Y, Y >= 0, z = 0
   minus(z, z') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z = v0, z' = v1

Only complete derivations are relevant for the runtime complexity.

----------------------------------------

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

eq(start(V1, V, V2),0,[minus(V1, V, Out)],[V1 >= 0,V >= 0]).
eq(start(V1, V, V2),0,[geq(V1, V, Out)],[V1 >= 0,V >= 0]).
eq(start(V1, V, V2),0,[div(V1, V, Out)],[V1 >= 0,V >= 0]).
eq(start(V1, V, V2),0,[if(V1, V, V2, Out)],[V1 >= 0,V >= 0,V2 >= 0]).
eq(minus(V1, V, Out),1,[],[Out = 0,V = Y1,Y1 >= 0,V1 = 0]).
eq(minus(V1, V, Out),1,[minus(X1, Y2, Ret)],[Out = Ret,V1 = 1 + X1,Y2 >= 0,V = 1 + Y2,X1 >= 0]).
eq(geq(V1, V, Out),1,[],[Out = 2,X2 >= 0,V1 = X2,V = 0]).
eq(geq(V1, V, Out),1,[],[Out = 1,Y3 >= 0,V = 1 + Y3,V1 = 0]).
eq(geq(V1, V, Out),1,[geq(X3, Y4, Ret1)],[Out = Ret1,V1 = 1 + X3,Y4 >= 0,V = 1 + Y4,X3 >= 0]).
eq(div(V1, V, Out),1,[],[Out = 0,Y5 >= 0,V = 1 + Y5,V1 = 0]).
eq(div(V1, V, Out),1,[geq(X4, Y6, Ret0),minus(X4, Y6, Ret110),div(Ret110, 1 + Y6, Ret11),if(Ret0, 1 + Ret11, 0, Ret2)],[Out = Ret2,V1 = 1 + X4,Y6 >= 0,V = 1 + Y6,X4 >= 0]).
eq(if(V1, V, V2, Out),1,[],[Out = X5,V1 = 2,V = X5,Y7 >= 0,V2 = Y7,X5 >= 0]).
eq(if(V1, V, V2, Out),1,[],[Out = Y8,V = X6,Y8 >= 0,V1 = 1,V2 = Y8,X6 >= 0]).
eq(minus(V1, V, Out),0,[],[Out = 0,V4 >= 0,V3 >= 0,V1 = V4,V = V3]).
eq(div(V1, V, Out),0,[],[Out = 0,V6 >= 0,V5 >= 0,V1 = V6,V = V5]).
eq(geq(V1, V, Out),0,[],[Out = 0,V8 >= 0,V7 >= 0,V1 = V8,V = V7]).
eq(if(V1, V, V2, Out),0,[],[Out = 0,V9 >= 0,V2 = V11,V10 >= 0,V1 = V9,V = V10,V11 >= 0]).
input_output_vars(minus(V1,V,Out),[V1,V],[Out]).
input_output_vars(geq(V1,V,Out),[V1,V],[Out]).
input_output_vars(div(V1,V,Out),[V1,V],[Out]).
input_output_vars(if(V1,V,V2,Out),[V1,V,V2],[Out]).


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

#### Computed strongly connected components 
0. recursive  : [geq/3]
1. non_recursive  : [if/4]
2. recursive  : [minus/3]
3. recursive [non_tail] : [(div)/3]
4. non_recursive  : [start/3]

#### Obtained direct recursion through partial evaluation 
0. SCC is partially evaluated into geq/3
1. SCC is partially evaluated into if/4
2. SCC is partially evaluated into minus/3
3. SCC is partially evaluated into (div)/3
4. SCC is partially evaluated into start/3

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

### Specialization of cost equations geq/3 
* CE 11 is refined into CE [18] 
* CE 8 is refined into CE [19] 
* CE 9 is refined into CE [20] 
* CE 10 is refined into CE [21] 


### Cost equations --> "Loop" of geq/3 
* CEs [21] --> Loop 13 
* CEs [18] --> Loop 14 
* CEs [19] --> Loop 15 
* CEs [20] --> Loop 16 

### Ranking functions of CR geq(V1,V,Out) 
* RF of phase [13]: [V,V1]

#### Partial ranking functions of CR geq(V1,V,Out) 
* Partial RF of phase [13]:
  - RF of loop [13:1]:
    V
    V1


### Specialization of cost equations if/4 
* CE 17 is refined into CE [22] 
* CE 15 is refined into CE [23] 
* CE 16 is refined into CE [24] 


### Cost equations --> "Loop" of if/4 
* CEs [22] --> Loop 17 
* CEs [23] --> Loop 18 
* CEs [24] --> Loop 19 

### Ranking functions of CR if(V1,V,V2,Out) 

#### Partial ranking functions of CR if(V1,V,V2,Out) 


### Specialization of cost equations minus/3 
* CE 5 is refined into CE [25] 
* CE 7 is refined into CE [26] 
* CE 6 is refined into CE [27] 


### Cost equations --> "Loop" of minus/3 
* CEs [27] --> Loop 20 
* CEs [25,26] --> Loop 21 

### Ranking functions of CR minus(V1,V,Out) 
* RF of phase [20]: [V,V1]

#### Partial ranking functions of CR minus(V1,V,Out) 
* Partial RF of phase [20]:
  - RF of loop [20:1]:
    V
    V1


### Specialization of cost equations (div)/3 
* CE 12 is refined into CE [28] 
* CE 14 is refined into CE [29] 
* CE 13 is refined into CE [30,31,32,33,34,35,36,37,38] 


### Cost equations --> "Loop" of (div)/3 
* CEs [37] --> Loop 22 
* CEs [38] --> Loop 23 
* CEs [32] --> Loop 24 
* CEs [33] --> Loop 25 
* CEs [30,31,34,35,36] --> Loop 26 
* CEs [28,29] --> Loop 27 

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

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


### Specialization of cost equations start/3 
* CE 1 is refined into CE [39] 
* CE 2 is refined into CE [40,41,42,43,44] 
* CE 3 is refined into CE [45,46,47] 
* CE 4 is refined into CE [48,49,50] 


### Cost equations --> "Loop" of start/3 
* CEs [46] --> Loop 28 
* CEs [41] --> Loop 29 
* CEs [49] --> Loop 30 
* CEs [48] --> Loop 31 
* CEs [39,40,42,43,44,45,47,50] --> Loop 32 

### Ranking functions of CR start(V1,V,V2) 

#### Partial ranking functions of CR start(V1,V,V2) 


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

#### Cost of chains of geq(V1,V,Out):
* Chain [[13],16]: 1*it(13)+1
  Such that:it(13) =< V1

  with precondition: [Out=1,V1>=1,V>=V1+1] 

* Chain [[13],15]: 1*it(13)+1
  Such that:it(13) =< V

  with precondition: [Out=2,V>=1,V1>=V] 

* Chain [[13],14]: 1*it(13)+0
  Such that:it(13) =< V

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

* Chain [16]: 1
  with precondition: [V1=0,Out=1,V>=1] 

* Chain [15]: 1
  with precondition: [V=0,Out=2,V1>=0] 

* Chain [14]: 0
  with precondition: [Out=0,V1>=0,V>=0] 


#### Cost of chains of if(V1,V,V2,Out):
* Chain [19]: 1
  with precondition: [V1=1,V2=Out,V>=0,V2>=0] 

* Chain [18]: 1
  with precondition: [V1=2,V=Out,V>=0,V2>=0] 

* Chain [17]: 0
  with precondition: [Out=0,V1>=0,V>=0,V2>=0] 


#### Cost of chains of minus(V1,V,Out):
* Chain [[20],21]: 1*it(20)+1
  Such that:it(20) =< V

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

* Chain [21]: 1
  with precondition: [Out=0,V1>=0,V>=0] 


#### Cost of chains of div(V1,V,Out):
* Chain [27]: 1
  with precondition: [Out=0,V1>=0,V>=0] 

* Chain [26,27]: 6*s(3)+2*s(7)+5
  Such that:aux(2) =< V1
aux(4) =< V
s(7) =< aux(2)
s(3) =< aux(4)

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

* Chain [25,27]: 4
  with precondition: [V=1,Out=0,V1>=1] 

* Chain [24,27]: 5
  with precondition: [V=1,Out=1,V1>=1] 

* Chain [23,27]: 2*s(13)+4
  Such that:aux(5) =< V
s(13) =< aux(5)

  with precondition: [Out=0,V>=2,V1>=V] 

* Chain [22,27]: 2*s(15)+5
  Such that:aux(6) =< V
s(15) =< aux(6)

  with precondition: [Out=1,V>=2,V1>=V] 


#### Cost of chains of start(V1,V,V2):
* Chain [32]: 13*s(23)+3*s(25)+5
  Such that:aux(8) =< V1
aux(9) =< V
s(25) =< aux(8)
s(23) =< aux(9)

  with precondition: [V1>=0,V>=0] 

* Chain [31]: 1
  with precondition: [V1=1,V>=0,V2>=0] 

* Chain [30]: 1
  with precondition: [V1=2,V>=0,V2>=0] 

* Chain [29]: 1
  with precondition: [V=0,V1>=0] 

* Chain [28]: 5
  with precondition: [V=1,V1>=1] 


Closed-form bounds of start(V1,V,V2): 
-------------------------------------
* Chain [32] with precondition: [V1>=0,V>=0] 
    - Upper bound: 3*V1+13*V+5 
    - Complexity: n 
* Chain [31] with precondition: [V1=1,V>=0,V2>=0] 
    - Upper bound: 1 
    - Complexity: constant 
* Chain [30] with precondition: [V1=2,V>=0,V2>=0] 
    - Upper bound: 1 
    - Complexity: constant 
* Chain [29] with precondition: [V=0,V1>=0] 
    - Upper bound: 1 
    - Complexity: constant 
* Chain [28] with precondition: [V=1,V1>=1] 
    - Upper bound: 5 
    - Complexity: constant 

### Maximum cost of start(V1,V,V2): 3*V1+13*V+5 
Asymptotic class: n 
* Total analysis performed in 257 ms.


----------------------------------------

(10)
BOUNDS(1, n^1)

----------------------------------------

(11) RelTrsToDecreasingLoopProblemProof (LOWER BOUND(ID))
Transformed a relative TRS into a decreasing-loop problem.
----------------------------------------

(12)
Obligation:
Analyzing the following TRS for decreasing loops:

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


The TRS R consists of the following rules:

   minus(0, Y) -> 0
   minus(s(X), s(Y)) -> minus(X, Y)
   geq(X, 0) -> true
   geq(0, s(Y)) -> false
   geq(s(X), s(Y)) -> geq(X, Y)
   div(0, s(Y)) -> 0
   div(s(X), s(Y)) -> if(geq(X, Y), s(div(minus(X, Y), s(Y))), 0)
   if(true, X, Y) -> X
   if(false, X, Y) -> Y

S is empty.
Rewrite Strategy: INNERMOST
----------------------------------------

(13) DecreasingLoopProof (LOWER BOUND(ID))
The following loop(s) give(s) rise to the lower bound Omega(n^1):

The rewrite sequence

minus(s(X), s(Y)) ->^+ minus(X, Y)

gives rise to a decreasing loop by considering the right hand sides subterm at position [].

The pumping substitution is [X / s(X), Y / s(Y)].

The result substitution is [ ].




----------------------------------------

(14)
Complex Obligation (BEST)

----------------------------------------

(15)
Obligation:
Proved the lower bound n^1 for the following obligation:

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


The TRS R consists of the following rules:

   minus(0, Y) -> 0
   minus(s(X), s(Y)) -> minus(X, Y)
   geq(X, 0) -> true
   geq(0, s(Y)) -> false
   geq(s(X), s(Y)) -> geq(X, Y)
   div(0, s(Y)) -> 0
   div(s(X), s(Y)) -> if(geq(X, Y), s(div(minus(X, Y), s(Y))), 0)
   if(true, X, Y) -> X
   if(false, X, Y) -> Y

S is empty.
Rewrite Strategy: INNERMOST
----------------------------------------

(16) LowerBoundPropagationProof (FINISHED)
Propagated lower bound.
----------------------------------------

(17)
BOUNDS(n^1, INF)

----------------------------------------

(18)
Obligation:
Analyzing the following TRS for decreasing loops:

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


The TRS R consists of the following rules:

   minus(0, Y) -> 0
   minus(s(X), s(Y)) -> minus(X, Y)
   geq(X, 0) -> true
   geq(0, s(Y)) -> false
   geq(s(X), s(Y)) -> geq(X, Y)
   div(0, s(Y)) -> 0
   div(s(X), s(Y)) -> if(geq(X, Y), s(div(minus(X, Y), s(Y))), 0)
   if(true, X, Y) -> X
   if(false, X, Y) -> Y

S is empty.
Rewrite Strategy: INNERMOST