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

#### Computed strongly connected components 
0. recursive  : [lZZ1/9]
1. recursive  : [lM1/17,lZZ1_loop_cont/18]
2. non_recursive  : [exit_location/1]
3. non_recursive  : [stop/9]
4. non_recursive  : [lM1_loop_cont/10]
5. non_recursive  : [start/9]
6. non_recursive  : [start0/9]

#### Obtained direct recursion through partial evaluation 
0. SCC is partially evaluated into lZZ1/9
1. SCC is partially evaluated into lM1/17
2. SCC is completely evaluated into other SCCs
3. SCC is completely evaluated into other SCCs
4. SCC is partially evaluated into lM1_loop_cont/10
5. SCC is partially evaluated into start/9
6. SCC is partially evaluated into start0/9

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

### Specialization of cost equations lZZ1/9 
* CE 14 is refined into CE [15] 
* CE 13 is refined into CE [16] 


### Cost equations --> "Loop" of lZZ1/9 
* CEs [15] --> Loop 15 
* CEs [16] --> Loop 16 

### Ranking functions of CR lZZ1(A,B,D,F,G,H,I,J,K) 

#### Partial ranking functions of CR lZZ1(A,B,D,F,G,H,I,J,K) 


### Specialization of cost equations lM1/17 
* CE 6 is refined into CE [17] 
* CE 10 is refined into CE [18] 
* CE 8 is refined into CE [19] 
* CE 9 is refined into CE [20] 
* CE 7 is refined into CE [21] 


### Cost equations --> "Loop" of lM1/17 
* CEs [20] --> Loop 17 
* CEs [21] --> Loop 18 
* CEs [18] --> Loop 19 
* CEs [17] --> Loop 20 
* CEs [19] --> Loop 21 

### Ranking functions of CR lM1(A,B,C,D,E,F,G,H,I,J,K,L,M,N,O,P,Q) 
* RF of phase [17,18]: [D]

#### Partial ranking functions of CR lM1(A,B,C,D,E,F,G,H,I,J,K,L,M,N,O,P,Q) 
* Partial RF of phase [17,18]:
  - RF of loop [17:1]:
    A-B depends on loops [18:1] 
    -B+F depends on loops [18:1] 
    -B+G depends on loops [18:1] 
    -B+H depends on loops [18:1] 
  - RF of loop [17:1,18:1]:
    D
  - RF of loop [18:1]:
    -A+B+1 depends on loops [17:1] 
    B-1 depends on loops [17:1] 
    B-H+1 depends on loops [17:1] 


### Specialization of cost equations lM1_loop_cont/10 
* CE 12 is refined into CE [22] 
* CE 11 is refined into CE [23] 


### Cost equations --> "Loop" of lM1_loop_cont/10 
* CEs [22] --> Loop 22 
* CEs [23] --> Loop 23 

### Ranking functions of CR lM1_loop_cont(A,B,C,D,E,F,G,H,I,J) 

#### Partial ranking functions of CR lM1_loop_cont(A,B,C,D,E,F,G,H,I,J) 


### Specialization of cost equations start/9 
* CE 5 is refined into CE [24,25,26,27,28,29] 
* CE 3 is refined into CE [30] 
* CE 2 is refined into CE [31] 
* CE 4 is refined into CE [32] 


### Cost equations --> "Loop" of start/9 
* CEs [29] --> Loop 24 
* CEs [25,28] --> Loop 25 
* CEs [26] --> Loop 26 
* CEs [30] --> Loop 27 
* CEs [31] --> Loop 28 
* CEs [24] --> Loop 29 
* CEs [32] --> Loop 30 
* CEs [27] --> Loop 31 

### Ranking functions of CR start(A,B,C,D,E,F,G,H,I) 

#### Partial ranking functions of CR start(A,B,C,D,E,F,G,H,I) 


### Specialization of cost equations start0/9 
* CE 1 is refined into CE [33,34,35,36,37,38,39,40] 


### Cost equations --> "Loop" of start0/9 
* CEs [40] --> Loop 32 
* CEs [39] --> Loop 33 
* CEs [38] --> Loop 34 
* CEs [36] --> Loop 35 
* CEs [37] --> Loop 36 
* CEs [35] --> Loop 37 
* CEs [34] --> Loop 38 
* CEs [33] --> Loop 39 

### Ranking functions of CR start0(A,B,C,D,E,F,G,H,I) 

#### Partial ranking functions of CR start0(A,B,C,D,E,F,G,H,I) 


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

#### Cost of chains of lZZ1(A,B,D,F,G,H,I,J,K):
* Chain [16]: 0
  with precondition: [B=0,I=3,J=1,H=A,G=F,D=K+1,D>=1,H>=2,G>=D+H] 

* Chain [15]: 0
  with precondition: [B=0,I=4,H=A,G=F,D>=1,H>=1,G>=D+H] 


#### Cost of chains of lM1(A,B,C,D,E,F,G,H,I,J,K,L,M,N,O,P,Q):
* Chain [[17,18],21]: 2*it(17)+0
  Such that:aux(59) =< D
it(17) =< aux(59)

  with precondition: [I=2,M=0,F=G,A=H,A=J,C=L,E=N,F=O,F=P,A=Q,A>=2,B>=1,D>=1,K>=1,A>=B,A>=K,F>=B+D,B+D>=K,B+D+K>=4,D+2*A>=B+K+2] 

* Chain [[17,18],20]: 2*it(17)+0
  Such that:aux(60) =< D
it(17) =< aux(60)

  with precondition: [I=4,F=G,A=H,A>=2,B>=1,A>=B,B+D>=A+1,F>=B+D,A+D>=B+3] 

* Chain [[17,18],19]: 2*it(17)+0
  Such that:aux(61) =< D
it(17) =< aux(61)

  with precondition: [I=4,F=G,A=H,A>=2,B>=1,D>=1,A>=B,F>=B+D] 

* Chain [21]: 0
  with precondition: [D=0,I=2,M=0,H=A,L=C,N=E,G=F,H=J,B=K,G=O,G=P,H=Q,B>=1,G>=B,H>=B] 

* Chain [20]: 0
  with precondition: [I=4,A=B,G=F,A=H,A>=1,D>=1,G>=A+D] 

* Chain [19]: 0
  with precondition: [I=4,H=A,G=F,B>=1,D>=0,H>=B,G>=B+D] 


#### Cost of chains of lM1_loop_cont(A,B,C,D,E,F,G,H,I,J):
* Chain [23]: 0
  with precondition: [A=2] 

* Chain [22]: 0
  with precondition: [A=4] 


#### Cost of chains of start(A,B,C,D,E,F,G,H,I):
* Chain [31]: 0
  with precondition: [A=1,H=1,C=B,E=D,G=F,G>=2] 

* Chain [30]: 0
  with precondition: [F=0,G=0,H=A,C=B,E=D,H>=1] 

* Chain [29]: 0
  with precondition: [F=1,G=1,H=A,C=B,E=D,H>=1] 

* Chain [28]: 0
  with precondition: [H=A,C=B,E=D,G=F,0>=H] 

* Chain [27]: 0
  with precondition: [H=A,C=B,E=D,G=F,0>=G+1] 

* Chain [26]: 0
  with precondition: [H=A,C=B,E=D,G=F,G>=1,H>=1] 

* Chain [25]: 4*s(2)+0
  Such that:aux(62) =< F
s(2) =< aux(62)

  with precondition: [H=A,C=B,E=D,G=F,G>=2,H>=2] 

* Chain [24]: 2*s(6)+0
  Such that:s(5) =< F
s(6) =< s(5)

  with precondition: [H=A,C=B,E=D,G=F,H>=2,G>=H+1] 


#### Cost of chains of start0(A,B,C,D,E,F,G,H,I):
* Chain [39]: 0
  with precondition: [A=1,G>=2] 

* Chain [38]: 0
  with precondition: [G=0,A>=1] 

* Chain [37]: 0
  with precondition: [G=1,A>=1] 

* Chain [36]: 0
  with precondition: [0>=A] 

* Chain [35]: 0
  with precondition: [0>=G+1] 

* Chain [34]: 0
  with precondition: [A>=1,G>=1] 

* Chain [33]: 4*s(8)+0
  Such that:s(7) =< G
s(8) =< s(7)

  with precondition: [A>=2,G>=2] 

* Chain [32]: 2*s(10)+0
  Such that:s(9) =< G
s(10) =< s(9)

  with precondition: [A>=2,G>=A+1] 


Closed-form bounds of start0(A,B,C,D,E,F,G,H,I): 
-------------------------------------
* Chain [39] with precondition: [A=1,G>=2] 
    - Upper bound: 0 
    - Complexity: constant 
* Chain [38] with precondition: [G=0,A>=1] 
    - Upper bound: 0 
    - Complexity: constant 
* Chain [37] with precondition: [G=1,A>=1] 
    - Upper bound: 0 
    - Complexity: constant 
* Chain [36] with precondition: [0>=A] 
    - Upper bound: 0 
    - Complexity: constant 
* Chain [35] with precondition: [0>=G+1] 
    - Upper bound: 0 
    - Complexity: constant 
* Chain [34] with precondition: [A>=1,G>=1] 
    - Upper bound: 0 
    - Complexity: constant 
* Chain [33] with precondition: [A>=2,G>=2] 
    - Upper bound: 4*G 
    - Complexity: n 
* Chain [32] with precondition: [A>=2,G>=A+1] 
    - Upper bound: 2*G 
    - Complexity: n 

### Maximum cost of start0(A,B,C,D,E,F,G,H,I): nat(G)*4 
Asymptotic class: n 
* Total analysis performed in 339 ms.

