0.05/0.37	WORST_CASE(?,O(n^4))  
0.05/0.37	
0.05/0.37	Preprocessing Cost Relations
0.05/0.37	=====================================
0.05/0.37	
0.05/0.37	#### Computed strongly connected components 
0.05/0.37	0. recursive  : [eval_textbook_ex3_bb4_in/4,eval_textbook_ex3_bb5_in/4]
0.05/0.37	1. recursive  : [eval_textbook_ex3_bb3_in/5,eval_textbook_ex3_bb4_in_loop_cont/6]
0.05/0.37	2. recursive  : [eval_textbook_ex3_bb2_in/5,eval_textbook_ex3_bb3_in_loop_cont/8,eval_textbook_ex3_bb6_in/7]
0.05/0.37	3. recursive  : [eval_textbook_ex3_bb1_in/3,eval_textbook_ex3_bb2_in_loop_cont/5,eval_textbook_ex3_bb7_in/4]
0.05/0.37	4. non_recursive  : [eval_textbook_ex3_stop/1]
0.05/0.37	5. non_recursive  : [eval_textbook_ex3_bb8_in/1]
0.05/0.37	6. non_recursive  : [eval_textbook_ex3_bb1_in_loop_cont/2]
0.05/0.37	7. non_recursive  : [eval_textbook_ex3_bb0_in/2]
0.05/0.37	8. non_recursive  : [eval_textbook_ex3_start/2]
0.05/0.37	
0.05/0.37	#### Obtained direct recursion through partial evaluation 
0.05/0.37	0. SCC is partially evaluated into eval_textbook_ex3_bb4_in/4
0.05/0.37	1. SCC is partially evaluated into eval_textbook_ex3_bb3_in/5
0.05/0.37	2. SCC is partially evaluated into eval_textbook_ex3_bb2_in/5
0.05/0.37	3. SCC is partially evaluated into eval_textbook_ex3_bb1_in/3
0.05/0.37	4. SCC is completely evaluated into other SCCs
0.05/0.37	5. SCC is completely evaluated into other SCCs
0.05/0.37	6. SCC is completely evaluated into other SCCs
0.05/0.37	7. SCC is partially evaluated into eval_textbook_ex3_bb0_in/2
0.05/0.37	8. SCC is partially evaluated into eval_textbook_ex3_start/2
0.05/0.37	
0.05/0.37	Control-Flow Refinement of Cost Relations
0.05/0.37	=====================================
0.05/0.37	
0.05/0.37	### Specialization of cost equations eval_textbook_ex3_bb4_in/4 
0.05/0.37	* CE 10 is refined into CE [11] 
0.05/0.37	* CE 9 is refined into CE [12] 
0.05/0.37	
0.05/0.37	
0.05/0.37	### Cost equations --> "Loop" of eval_textbook_ex3_bb4_in/4 
0.05/0.37	* CEs [12] --> Loop 11 
0.05/0.37	* CEs [11] --> Loop 12 
0.05/0.37	
0.05/0.37	### Ranking functions of CR eval_textbook_ex3_bb4_in(V_i_0_sink,V_2,V_l_0,B) 
0.05/0.37	* RF of phase [11]: [V_2-V_l_0+1,V_i_0_sink-V_l_0+2]
0.05/0.37	
0.05/0.37	#### Partial ranking functions of CR eval_textbook_ex3_bb4_in(V_i_0_sink,V_2,V_l_0,B) 
0.05/0.37	* Partial RF of phase [11]:
0.05/0.37	  - RF of loop [11:1]:
0.05/0.37	    V_2-V_l_0+1
0.05/0.37	    V_i_0_sink-V_l_0+2
0.05/0.37	
0.05/0.37	
0.05/0.37	### Specialization of cost equations eval_textbook_ex3_bb3_in/5 
0.05/0.37	* CE 8 is refined into CE [13] 
0.05/0.37	* CE 7 is refined into CE [14] 
0.05/0.37	
0.05/0.37	
0.05/0.37	### Cost equations --> "Loop" of eval_textbook_ex3_bb3_in/5 
0.05/0.37	* CEs [14] --> Loop 13 
0.05/0.37	* CEs [13] --> Loop 14 
0.05/0.37	
0.05/0.37	### Ranking functions of CR eval_textbook_ex3_bb3_in(V_m,V_i_0_sink,B,C,D) 
0.05/0.37	* RF of phase [13]: [V_m-V_i_0_sink]
0.05/0.37	
0.05/0.37	#### Partial ranking functions of CR eval_textbook_ex3_bb3_in(V_m,V_i_0_sink,B,C,D) 
0.05/0.37	* Partial RF of phase [13]:
0.05/0.37	  - RF of loop [13:1]:
0.05/0.37	    V_m-V_i_0_sink
0.05/0.37	
0.05/0.37	
0.05/0.37	### Specialization of cost equations eval_textbook_ex3_bb2_in/5 
0.05/0.37	* CE 6 is refined into CE [15] 
0.05/0.37	* CE 5 is refined into CE [16,17] 
0.05/0.37	
0.05/0.37	
0.05/0.37	### Cost equations --> "Loop" of eval_textbook_ex3_bb2_in/5 
0.05/0.37	* CEs [17] --> Loop 15 
0.05/0.37	* CEs [16] --> Loop 16 
0.05/0.37	* CEs [15] --> Loop 17 
0.05/0.37	
0.05/0.37	### Ranking functions of CR eval_textbook_ex3_bb2_in(V_m,V_i_0,V_j_0,B,C) 
0.05/0.37	* RF of phase [15]: [V_i_0-V_j_0+1,V_m-V_j_0]
0.05/0.37	* RF of phase [16]: [V_i_0-V_j_0+1,V_m-V_j_0+1]
0.05/0.37	
0.05/0.37	#### Partial ranking functions of CR eval_textbook_ex3_bb2_in(V_m,V_i_0,V_j_0,B,C) 
0.05/0.37	* Partial RF of phase [15]:
0.05/0.37	  - RF of loop [15:1]:
0.05/0.37	    V_i_0-V_j_0+1
0.05/0.37	    V_m-V_j_0
0.05/0.37	* Partial RF of phase [16]:
0.05/0.37	  - RF of loop [16:1]:
0.05/0.37	    V_i_0-V_j_0+1
0.05/0.37	    V_m-V_j_0+1
0.05/0.37	
0.05/0.37	
0.05/0.37	### Specialization of cost equations eval_textbook_ex3_bb1_in/3 
0.05/0.37	* CE 4 is refined into CE [18] 
0.05/0.37	* CE 3 is refined into CE [19,20] 
0.05/0.37	
0.05/0.37	
0.05/0.37	### Cost equations --> "Loop" of eval_textbook_ex3_bb1_in/3 
0.05/0.37	* CEs [20] --> Loop 18 
0.05/0.37	* CEs [19] --> Loop 19 
0.05/0.37	* CEs [18] --> Loop 20 
0.05/0.37	
0.05/0.37	### Ranking functions of CR eval_textbook_ex3_bb1_in(V_m,V_i_0,B) 
0.05/0.37	* RF of phase [18]: [V_m-V_i_0]
0.05/0.37	
0.05/0.37	#### Partial ranking functions of CR eval_textbook_ex3_bb1_in(V_m,V_i_0,B) 
0.05/0.37	* Partial RF of phase [18]:
0.05/0.37	  - RF of loop [18:1]:
0.05/0.37	    V_m-V_i_0
0.05/0.37	
0.05/0.37	
0.05/0.37	### Specialization of cost equations eval_textbook_ex3_bb0_in/2 
0.05/0.37	* CE 2 is refined into CE [21,22,23] 
0.05/0.37	
0.05/0.37	
0.05/0.37	### Cost equations --> "Loop" of eval_textbook_ex3_bb0_in/2 
0.05/0.37	* CEs [23] --> Loop 21 
0.05/0.37	* CEs [22] --> Loop 22 
0.05/0.37	* CEs [21] --> Loop 23 
0.05/0.37	
0.05/0.37	### Ranking functions of CR eval_textbook_ex3_bb0_in(V_m,B) 
0.05/0.37	
0.05/0.37	#### Partial ranking functions of CR eval_textbook_ex3_bb0_in(V_m,B) 
0.05/0.37	
0.05/0.37	
0.05/0.37	### Specialization of cost equations eval_textbook_ex3_start/2 
0.05/0.37	* CE 1 is refined into CE [24,25,26] 
0.05/0.37	
0.05/0.37	
0.05/0.37	### Cost equations --> "Loop" of eval_textbook_ex3_start/2 
0.05/0.37	* CEs [26] --> Loop 24 
0.05/0.37	* CEs [25] --> Loop 25 
0.05/0.37	* CEs [24] --> Loop 26 
0.05/0.37	
0.05/0.37	### Ranking functions of CR eval_textbook_ex3_start(V_m,B) 
0.05/0.37	
0.05/0.37	#### Partial ranking functions of CR eval_textbook_ex3_start(V_m,B) 
0.05/0.37	
0.05/0.37	
0.05/0.37	Computing Bounds
0.05/0.37	=====================================
0.05/0.37	
0.05/0.37	#### Cost of chains of eval_textbook_ex3_bb4_in(V_i_0_sink,V_2,V_l_0,B):
0.05/0.37	* Chain [[11],12]: 1*it(11)+0
0.05/0.37	  Such that:it(11) =< V_i_0_sink-V_l_0+2
0.05/0.37	
0.05/0.37	  with precondition: [B=2,V_i_0_sink+1=V_2,V_i_0_sink>=1,V_l_0>=1,V_i_0_sink+1>=V_l_0] 
0.05/0.37	
0.05/0.37	
0.05/0.37	#### Cost of chains of eval_textbook_ex3_bb3_in(V_m,V_i_0_sink,B,C,D):
0.05/0.37	* Chain [[13],14]: 1*it(13)+1*s(3)+0
0.05/0.37	  Such that:it(13) =< -V_i_0_sink+D
0.05/0.37	aux(1) =< D
0.05/0.37	s(3) =< it(13)*aux(1)
0.05/0.37	
0.05/0.37	  with precondition: [B=3,V_m=C,V_m+1=D,V_i_0_sink>=1,V_m>=V_i_0_sink+1] 
0.05/0.37	
0.05/0.37	* Chain [14]: 0
0.05/0.37	  with precondition: [B=3,V_i_0_sink=V_m,V_i_0_sink=C,V_i_0_sink+1=D,V_i_0_sink>=1] 
0.05/0.37	
0.05/0.37	
0.05/0.37	#### Cost of chains of eval_textbook_ex3_bb2_in(V_m,V_i_0,V_j_0,B,C):
0.05/0.37	* Chain [[16],17]: 1*it(16)+0
0.05/0.37	  Such that:it(16) =< -V_j_0+C
0.05/0.37	
0.05/0.37	  with precondition: [B=4,V_m=V_i_0,V_m+1=C,V_j_0>=1,V_m>=V_j_0] 
0.05/0.37	
0.05/0.37	* Chain [[15],17]: 1*it(15)+1*s(10)+1*s(11)+0
0.05/0.37	  Such that:s(7) =< V_m+1
0.05/0.37	aux(2) =< V_m-C+2
0.05/0.37	it(15) =< -V_j_0+C
0.05/0.37	s(10) =< it(15)*aux(2)
0.05/0.37	s(11) =< s(10)*s(7)
0.05/0.37	
0.05/0.37	  with precondition: [B=4,V_i_0+1=C,V_j_0>=1,V_m>=V_i_0+1,V_i_0>=V_j_0] 
0.05/0.37	
0.05/0.37	
0.05/0.37	#### Cost of chains of eval_textbook_ex3_bb1_in(V_m,V_i_0,B):
0.05/0.37	* Chain [[18],19,20]: 1*it(18)+1*s(12)+1*s(23)+1*s(24)+1*s(25)+1
0.05/0.37	  Such that:aux(3) =< V_m
0.05/0.37	it(18) =< V_m-V_i_0
0.05/0.37	s(19) =< V_m-V_i_0+1
0.05/0.37	aux(4) =< V_m+1
0.05/0.37	s(12) =< aux(4)
0.05/0.37	aux(3) =< aux(4)-1
0.05/0.37	s(23) =< it(18)*aux(3)
0.05/0.37	s(24) =< s(23)*s(19)
0.05/0.37	s(25) =< s(24)*aux(4)
0.05/0.37	
0.05/0.37	  with precondition: [B=5,V_i_0>=1,V_m>=V_i_0+1] 
0.05/0.37	
0.05/0.37	* Chain [20]: 0
0.05/0.37	  with precondition: [B=5,V_i_0>=1,V_i_0>=V_m+1] 
0.05/0.37	
0.05/0.37	* Chain [19,20]: 1*s(12)+1
0.05/0.37	  Such that:s(12) =< V_m+1
0.05/0.37	
0.05/0.37	  with precondition: [B=5,V_m=V_i_0,V_m>=1] 
0.05/0.37	
0.05/0.37	
0.05/0.37	#### Cost of chains of eval_textbook_ex3_bb0_in(V_m,B):
0.05/0.37	* Chain [23]: 1*s(26)+1
0.05/0.37	  Such that:s(26) =< 2
0.05/0.37	
0.05/0.37	  with precondition: [V_m=1] 
0.05/0.37	
0.05/0.37	* Chain [22]: 0
0.05/0.37	  with precondition: [0>=V_m] 
0.05/0.37	
0.05/0.37	* Chain [21]: 1*s(28)+1*s(31)+1*s(32)+1*s(33)+1*s(34)+1
0.05/0.37	  Such that:s(30) =< V_m+1
0.05/0.37	aux(5) =< V_m
0.05/0.37	s(27) =< aux(5)
0.05/0.37	s(28) =< aux(5)
0.05/0.37	s(31) =< s(30)
0.05/0.37	s(27) =< s(30)-1
0.05/0.37	s(32) =< s(28)*s(27)
0.05/0.37	s(33) =< s(32)*aux(5)
0.05/0.37	s(34) =< s(33)*s(30)
0.05/0.37	
0.05/0.37	  with precondition: [V_m>=2] 
0.05/0.37	
0.05/0.37	
0.05/0.37	#### Cost of chains of eval_textbook_ex3_start(V_m,B):
0.05/0.37	* Chain [26]: 1*s(35)+1
0.05/0.37	  Such that:s(35) =< 2
0.05/0.37	
0.05/0.37	  with precondition: [V_m=1] 
0.05/0.37	
0.05/0.37	* Chain [25]: 0
0.05/0.37	  with precondition: [0>=V_m] 
0.05/0.37	
0.05/0.37	* Chain [24]: 1*s(39)+1*s(40)+1*s(41)+1*s(42)+1*s(43)+1
0.05/0.37	  Such that:s(37) =< V_m
0.05/0.37	s(36) =< V_m+1
0.05/0.37	s(38) =< s(37)
0.05/0.37	s(39) =< s(37)
0.05/0.37	s(40) =< s(36)
0.05/0.37	s(38) =< s(36)-1
0.05/0.37	s(41) =< s(39)*s(38)
0.05/0.37	s(42) =< s(41)*s(37)
0.05/0.37	s(43) =< s(42)*s(36)
0.05/0.37	
0.05/0.37	  with precondition: [V_m>=2] 
0.05/0.37	
0.05/0.37	
0.05/0.37	Closed-form bounds of eval_textbook_ex3_start(V_m,B): 
0.05/0.37	-------------------------------------
0.05/0.37	* Chain [26] with precondition: [V_m=1] 
0.05/0.37	    - Upper bound: 3 
0.05/0.37	    - Complexity: constant 
0.05/0.37	* Chain [25] with precondition: [0>=V_m] 
0.05/0.37	    - Upper bound: 0 
0.05/0.37	    - Complexity: constant 
0.05/0.37	* Chain [24] with precondition: [V_m>=2] 
0.05/0.37	    - Upper bound: V_m+1+V_m*V_m+V_m*V_m*V_m+(V_m+1)*(V_m*V_m*V_m)+(V_m+1) 
0.05/0.37	    - Complexity: n^4 
0.05/0.37	
0.05/0.37	### Maximum cost of eval_textbook_ex3_start(V_m,B): max([3,nat(V_m)+1+nat(V_m)*nat(V_m)+nat(V_m)*nat(V_m)*nat(V_m)+nat(V_m)*nat(V_m)*nat(V_m)*nat(V_m+1)+nat(V_m+1)]) 
0.05/0.37	Asymptotic class: n^4 
0.05/0.37	* Total analysis performed in 283 ms.
0.05/0.37	
0.05/0.47	EOF