/export/starexec/sandbox/solver/bin/starexec_run_its /export/starexec/sandbox/benchmark/theBenchmark.koat /export/starexec/sandbox/output/output_files


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


WORST_CASE(?,O(n^2))  

Preprocessing Cost Relations
=====================================

#### Computed strongly connected components 
0. recursive  : [eval_realheapsort_step1_bb2_in/3,eval_realheapsort_step1_bb3_in/3,eval_realheapsort_step1_bb4_in/3]
1. recursive  : [eval_realheapsort_step1_28/8,eval_realheapsort_step1_29/8,eval_realheapsort_step1__critedge_in/8,eval_realheapsort_step1_bb1_in/8,eval_realheapsort_step1_bb2_in_loop_cont/9]
2. non_recursive  : [eval_realheapsort_step1_stop/5]
3. non_recursive  : [eval_realheapsort_step1_bb5_in/5]
4. non_recursive  : [exit_location/1]
5. non_recursive  : [eval_realheapsort_step1_bb1_in_loop_cont/6]
6. non_recursive  : [eval_realheapsort_step1_2/5]
7. non_recursive  : [eval_realheapsort_step1_1/5]
8. non_recursive  : [eval_realheapsort_step1_0/5]
9. non_recursive  : [eval_realheapsort_step1_bb0_in/5]
10. non_recursive  : [eval_realheapsort_step1_start/5]

#### Obtained direct recursion through partial evaluation 
0. SCC is partially evaluated into eval_realheapsort_step1_bb2_in/3
1. SCC is partially evaluated into eval_realheapsort_step1_bb1_in/8
2. SCC is completely evaluated into other SCCs
3. SCC is completely evaluated into other SCCs
4. SCC is completely evaluated into other SCCs
5. SCC is partially evaluated into eval_realheapsort_step1_bb1_in_loop_cont/6
6. SCC is partially evaluated into eval_realheapsort_step1_2/5
7. SCC is completely evaluated into other SCCs
8. SCC is completely evaluated into other SCCs
9. SCC is completely evaluated into other SCCs
10. SCC is partially evaluated into eval_realheapsort_step1_start/5

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

### Specialization of cost equations eval_realheapsort_step1_bb2_in/3 
* CE 13 is refined into CE [14] 
* CE 10 is refined into CE [15] 
* CE 12 is refined into CE [16] 
* CE 11 is refined into CE [17] 


### Cost equations --> "Loop" of eval_realheapsort_step1_bb2_in/3 
* CEs [17] --> Loop 14 
* CEs [14] --> Loop 15 
* CEs [15] --> Loop 16 
* CEs [16] --> Loop 17 

### Ranking functions of CR eval_realheapsort_step1_bb2_in(V_j_0,B,C) 
* RF of phase [14]: [V_j_0]

#### Partial ranking functions of CR eval_realheapsort_step1_bb2_in(V_j_0,B,C) 
* Partial RF of phase [14]:
  - RF of loop [14:1]:
    V_j_0


### Specialization of cost equations eval_realheapsort_step1_bb1_in/8 
* CE 6 is refined into CE [18] 
* CE 4 is refined into CE [19,20] 
* CE 7 is refined into CE [21] 
* CE 5 is refined into CE [22,23,24] 


### Cost equations --> "Loop" of eval_realheapsort_step1_bb1_in/8 
* CEs [24] --> Loop 18 
* CEs [23] --> Loop 19 
* CEs [22] --> Loop 20 
* CEs [18] --> Loop 21 
* CEs [19,20] --> Loop 22 
* CEs [21] --> Loop 23 

### Ranking functions of CR eval_realheapsort_step1_bb1_in(V_33,V_N,V_j_0,V_k_0,B,C,D,E) 
* RF of phase [18,19,20]: [V_N-V_k_0]

#### Partial ranking functions of CR eval_realheapsort_step1_bb1_in(V_33,V_N,V_j_0,V_k_0,B,C,D,E) 
* Partial RF of phase [18,19,20]:
  - RF of loop [18:1,19:1,20:1]:
    V_N-V_k_0


### Specialization of cost equations eval_realheapsort_step1_bb1_in_loop_cont/6 
* CE 8 is refined into CE [25] 
* CE 9 is refined into CE [26] 


### Cost equations --> "Loop" of eval_realheapsort_step1_bb1_in_loop_cont/6 
* CEs [25] --> Loop 24 
* CEs [26] --> Loop 25 

### Ranking functions of CR eval_realheapsort_step1_bb1_in_loop_cont(A,B,C,D,E,F) 

#### Partial ranking functions of CR eval_realheapsort_step1_bb1_in_loop_cont(A,B,C,D,E,F) 


### Specialization of cost equations eval_realheapsort_step1_2/5 
* CE 3 is refined into CE [27,28,29,30] 
* CE 2 is refined into CE [31] 


### Cost equations --> "Loop" of eval_realheapsort_step1_2/5 
* CEs [27,28,29,30] --> Loop 26 
* CEs [31] --> Loop 27 

### Ranking functions of CR eval_realheapsort_step1_2(V_33,V_N,V_j_0,V_k_0,B) 

#### Partial ranking functions of CR eval_realheapsort_step1_2(V_33,V_N,V_j_0,V_k_0,B) 


### Specialization of cost equations eval_realheapsort_step1_start/5 
* CE 1 is refined into CE [32,33] 


### Cost equations --> "Loop" of eval_realheapsort_step1_start/5 
* CEs [33] --> Loop 28 
* CEs [32] --> Loop 29 

### Ranking functions of CR eval_realheapsort_step1_start(V_33,V_N,V_j_0,V_k_0,B) 

#### Partial ranking functions of CR eval_realheapsort_step1_start(V_33,V_N,V_j_0,V_k_0,B) 


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

#### Cost of chains of eval_realheapsort_step1_bb2_in(V_j_0,B,C):
* Chain [[14],17]: 1*it(14)+0
  Such that:it(14) =< V_j_0

  with precondition: [B=2,0>=C,V_j_0>=1,2*C+1>=0] 

* Chain [[14],16]: 1*it(14)+0
  Such that:it(14) =< V_j_0-C

  with precondition: [B=2,C>=1,V_j_0>=2*C+1] 

* Chain [[14],15]: 1*it(14)+0
  Such that:it(14) =< V_j_0

  with precondition: [B=3,V_j_0>=1] 

* Chain [16]: 0
  with precondition: [B=2,V_j_0=C,V_j_0>=1] 

* Chain [15]: 0
  with precondition: [B=3,2*V_j_0+1>=0] 


#### Cost of chains of eval_realheapsort_step1_bb1_in(V_33,V_N,V_j_0,V_k_0,B,C,D,E):
* Chain [[18,19,20],23]: 3*it(18)+1*s(5)+1*s(6)+0
  Such that:aux(1) =< V_N
aux(5) =< V_N-V_k_0
it(18) =< aux(5)
aux(2) =< aux(1)+1
s(5) =< it(18)*aux(1)
s(6) =< it(18)*aux(2)

  with precondition: [B=3,V_N>=3,V_k_0>=1,V_N>=V_k_0+1] 

* Chain [[18,19,20],22]: 3*it(18)+1*s(5)+1*s(6)+1*s(7)+0
  Such that:aux(6) =< V_N
aux(7) =< V_N-V_k_0
s(7) =< aux(6)
it(18) =< aux(7)
aux(2) =< aux(6)+1
s(5) =< it(18)*aux(6)
s(6) =< it(18)*aux(2)

  with precondition: [B=3,V_k_0>=1,V_N>=V_k_0+2] 

* Chain [[18,19,20],21]: 3*it(18)+1*s(5)+1*s(6)+0
  Such that:aux(1) =< C
aux(8) =< -V_k_0+C
it(18) =< aux(8)
aux(2) =< aux(1)+1
s(5) =< it(18)*aux(1)
s(6) =< it(18)*aux(2)

  with precondition: [B=4,V_N=C,V_N=E,V_N>=3,V_k_0>=1,2*D+1>=0,V_N>=V_k_0+1,V_N>=D+1] 

* Chain [23]: 0
  with precondition: [B=3,V_N>=3,V_k_0>=1] 

* Chain [22]: 1*s(7)+0
  Such that:s(7) =< V_k_0

  with precondition: [B=3,V_N>=3,V_k_0>=1,V_N>=V_k_0+1] 


#### Cost of chains of eval_realheapsort_step1_bb1_in_loop_cont(A,B,C,D,E,F):
* Chain [25]: 0
  with precondition: [A=3,C>=3] 

* Chain [24]: 0
  with precondition: [A=4,C>=3] 


#### Cost of chains of eval_realheapsort_step1_2(V_33,V_N,V_j_0,V_k_0,B):
* Chain [27]: 0
  with precondition: [2>=V_N] 

* Chain [26]: 1*s(17)+10*s(18)+3*s(20)+3*s(21)+0
  Such that:s(17) =< 1
aux(12) =< V_N
s(18) =< aux(12)
s(19) =< aux(12)+1
s(20) =< s(18)*aux(12)
s(21) =< s(18)*s(19)

  with precondition: [V_N>=3] 


#### Cost of chains of eval_realheapsort_step1_start(V_33,V_N,V_j_0,V_k_0,B):
* Chain [29]: 0
  with precondition: [2>=V_N] 

* Chain [28]: 1*s(35)+10*s(37)+3*s(39)+3*s(40)+0
  Such that:s(35) =< 1
s(36) =< V_N
s(37) =< s(36)
s(38) =< s(36)+1
s(39) =< s(37)*s(36)
s(40) =< s(37)*s(38)

  with precondition: [V_N>=3] 


Closed-form bounds of eval_realheapsort_step1_start(V_33,V_N,V_j_0,V_k_0,B): 
-------------------------------------
* Chain [29] with precondition: [2>=V_N] 
    - Upper bound: 0 
    - Complexity: constant 
* Chain [28] with precondition: [V_N>=3] 
    - Upper bound: 13*V_N+1+6*V_N*V_N 
    - Complexity: n^2 

### Maximum cost of eval_realheapsort_step1_start(V_33,V_N,V_j_0,V_k_0,B): nat(V_N)*13+1+nat(V_N)*6*nat(V_N) 
Asymptotic class: n^2 
* Total analysis performed in 335 ms.