0.04/0.30 WORST_CASE(?,O(n^2)) 0.04/0.30 0.04/0.30 Preprocessing Cost Relations 0.04/0.30 ===================================== 0.04/0.30 0.04/0.30 #### Computed strongly connected components 0.04/0.30 0. recursive : [eval_abc_bb2_in/4,eval_abc_bb3_in/4] 0.04/0.30 1. recursive : [eval_abc_8/8,eval_abc_9/8,eval_abc_bb1_in/8,eval_abc_bb2_in_loop_cont/9,eval_abc_bb4_in/8] 0.04/0.30 2. non_recursive : [eval_abc_stop/5] 0.04/0.30 3. non_recursive : [eval_abc_bb5_in/5] 0.04/0.30 4. non_recursive : [exit_location/1] 0.04/0.30 5. non_recursive : [eval_abc_bb1_in_loop_cont/6] 0.04/0.30 6. non_recursive : [eval_abc_4/5] 0.04/0.30 7. non_recursive : [eval_abc_3/5] 0.04/0.30 8. non_recursive : [eval_abc_2/5] 0.04/0.30 9. non_recursive : [eval_abc_1/5] 0.04/0.30 10. non_recursive : [eval_abc_0/5] 0.04/0.30 11. non_recursive : [eval_abc_bb0_in/5] 0.04/0.30 12. non_recursive : [eval_abc_start/5] 0.04/0.30 0.04/0.30 #### Obtained direct recursion through partial evaluation 0.04/0.30 0. SCC is partially evaluated into eval_abc_bb2_in/4 0.04/0.30 1. SCC is partially evaluated into eval_abc_bb1_in/8 0.04/0.30 2. SCC is completely evaluated into other SCCs 0.04/0.30 3. SCC is completely evaluated into other SCCs 0.04/0.30 4. SCC is completely evaluated into other SCCs 0.04/0.30 5. SCC is partially evaluated into eval_abc_bb1_in_loop_cont/6 0.04/0.30 6. SCC is partially evaluated into eval_abc_4/5 0.04/0.30 7. SCC is completely evaluated into other SCCs 0.04/0.30 8. SCC is completely evaluated into other SCCs 0.04/0.30 9. SCC is completely evaluated into other SCCs 0.04/0.30 10. SCC is completely evaluated into other SCCs 0.04/0.30 11. SCC is completely evaluated into other SCCs 0.04/0.30 12. SCC is partially evaluated into eval_abc_start/5 0.04/0.30 0.04/0.30 Control-Flow Refinement of Cost Relations 0.04/0.30 ===================================== 0.04/0.30 0.04/0.30 ### Specialization of cost equations eval_abc_bb2_in/4 0.04/0.30 * CE 11 is refined into CE [12] 0.04/0.30 * CE 10 is refined into CE [13] 0.04/0.30 * CE 9 is refined into CE [14] 0.04/0.30 0.04/0.30 0.04/0.30 ### Cost equations --> "Loop" of eval_abc_bb2_in/4 0.04/0.30 * CEs [14] --> Loop 12 0.04/0.30 * CEs [12] --> Loop 13 0.04/0.30 * CEs [13] --> Loop 14 0.04/0.30 0.04/0.30 ### Ranking functions of CR eval_abc_bb2_in(V_j_0,V_n,B,C) 0.04/0.30 * RF of phase [12]: [-V_j_0+V_n+1] 0.04/0.30 0.04/0.30 #### Partial ranking functions of CR eval_abc_bb2_in(V_j_0,V_n,B,C) 0.04/0.30 * Partial RF of phase [12]: 0.04/0.30 - RF of loop [12:1]: 0.04/0.30 -V_j_0+V_n+1 0.04/0.30 0.04/0.30 0.04/0.30 ### Specialization of cost equations eval_abc_bb1_in/8 0.04/0.30 * CE 5 is refined into CE [15] 0.04/0.30 * CE 3 is refined into CE [16,17] 0.04/0.30 * CE 6 is refined into CE [18] 0.04/0.30 * CE 4 is refined into CE [19] 0.04/0.30 0.04/0.30 0.04/0.30 ### Cost equations --> "Loop" of eval_abc_bb1_in/8 0.04/0.30 * CEs [19] --> Loop 15 0.04/0.30 * CEs [15] --> Loop 16 0.04/0.30 * CEs [16,17] --> Loop 17 0.04/0.30 * CEs [18] --> Loop 18 0.04/0.30 0.04/0.30 ### Ranking functions of CR eval_abc_bb1_in(V_3,V_i_0,V_j_0,V_n,B,C,D,E) 0.04/0.30 * RF of phase [15]: [-V_i_0+V_n+1] 0.04/0.30 0.04/0.30 #### Partial ranking functions of CR eval_abc_bb1_in(V_3,V_i_0,V_j_0,V_n,B,C,D,E) 0.04/0.30 * Partial RF of phase [15]: 0.04/0.30 - RF of loop [15:1]: 0.04/0.30 -V_i_0+V_n+1 0.04/0.30 0.04/0.30 0.04/0.30 ### Specialization of cost equations eval_abc_bb1_in_loop_cont/6 0.04/0.30 * CE 7 is refined into CE [20] 0.04/0.30 * CE 8 is refined into CE [21] 0.04/0.30 0.04/0.30 0.04/0.30 ### Cost equations --> "Loop" of eval_abc_bb1_in_loop_cont/6 0.04/0.30 * CEs [20] --> Loop 19 0.04/0.30 * CEs [21] --> Loop 20 0.04/0.30 0.04/0.30 ### Ranking functions of CR eval_abc_bb1_in_loop_cont(A,B,C,D,E,F) 0.04/0.30 0.04/0.30 #### Partial ranking functions of CR eval_abc_bb1_in_loop_cont(A,B,C,D,E,F) 0.04/0.30 0.04/0.30 0.04/0.30 ### Specialization of cost equations eval_abc_4/5 0.04/0.30 * CE 2 is refined into CE [22,23,24,25,26] 0.04/0.30 0.04/0.30 0.04/0.30 ### Cost equations --> "Loop" of eval_abc_4/5 0.04/0.30 * CEs [24] --> Loop 21 0.04/0.30 * CEs [23,26] --> Loop 22 0.04/0.30 * CEs [25] --> Loop 23 0.04/0.30 * CEs [22] --> Loop 24 0.04/0.30 0.04/0.30 ### Ranking functions of CR eval_abc_4(V_3,V_i_0,V_j_0,V_n,B) 0.04/0.30 0.04/0.30 #### Partial ranking functions of CR eval_abc_4(V_3,V_i_0,V_j_0,V_n,B) 0.04/0.30 0.04/0.30 0.04/0.30 ### Specialization of cost equations eval_abc_start/5 0.04/0.30 * CE 1 is refined into CE [27,28,29,30] 0.04/0.30 0.04/0.30 0.04/0.30 ### Cost equations --> "Loop" of eval_abc_start/5 0.04/0.31 * CEs [30] --> Loop 25 0.04/0.31 * CEs [29] --> Loop 26 0.04/0.31 * CEs [28] --> Loop 27 0.04/0.31 * CEs [27] --> Loop 28 0.04/0.31 0.04/0.31 ### Ranking functions of CR eval_abc_start(V_3,V_i_0,V_j_0,V_n,B) 0.04/0.31 0.04/0.31 #### Partial ranking functions of CR eval_abc_start(V_3,V_i_0,V_j_0,V_n,B) 0.04/0.31 0.04/0.31 0.04/0.31 Computing Bounds 0.04/0.31 ===================================== 0.04/0.31 0.04/0.31 #### Cost of chains of eval_abc_bb2_in(V_j_0,V_n,B,C): 0.04/0.31 * Chain [[12],14]: 1*it(12)+0 0.04/0.31 Such that:it(12) =< -V_j_0+C 0.04/0.31 0.04/0.31 with precondition: [B=2,V_n+1=C,V_j_0>=1,V_n>=V_j_0] 0.04/0.31 0.04/0.31 * Chain [[12],13]: 1*it(12)+0 0.04/0.31 Such that:it(12) =< -V_j_0+V_n+1 0.04/0.31 0.04/0.31 with precondition: [B=3,V_j_0>=1,V_n>=V_j_0] 0.04/0.31 0.04/0.31 * Chain [13]: 0 0.04/0.31 with precondition: [B=3,V_j_0>=1,V_n>=1,V_n+1>=V_j_0] 0.04/0.31 0.04/0.31 0.04/0.31 #### Cost of chains of eval_abc_bb1_in(V_3,V_i_0,V_j_0,V_n,B,C,D,E): 0.04/0.31 * Chain [[15],18]: 1*it(15)+1*s(3)+0 0.04/0.31 Such that:aux(1) =< V_n 0.04/0.31 aux(3) =< -V_i_0+V_n+1 0.04/0.31 it(15) =< aux(3) 0.04/0.31 aux(1) =< aux(3) 0.04/0.31 s(3) =< it(15)*aux(1) 0.04/0.31 0.04/0.31 with precondition: [B=3,V_i_0>=1,V_n>=V_i_0] 0.04/0.31 0.04/0.31 * Chain [[15],17]: 2*it(15)+1*s(3)+0 0.04/0.31 Such that:aux(1) =< V_n 0.04/0.31 aux(4) =< -V_i_0+V_n 0.04/0.31 aux(5) =< -V_i_0+V_n+1 0.04/0.31 it(15) =< aux(4) 0.04/0.31 it(15) =< aux(5) 0.04/0.31 aux(1) =< aux(5) 0.04/0.31 s(3) =< it(15)*aux(1) 0.04/0.31 0.04/0.31 with precondition: [B=3,V_i_0>=1,V_n>=V_i_0+1] 0.04/0.31 0.04/0.31 * Chain [[15],16]: 1*it(15)+1*s(3)+0 0.04/0.31 Such that:aux(1) =< C 0.04/0.31 aux(6) =< -V_i_0+C 0.04/0.31 it(15) =< aux(6) 0.04/0.31 aux(1) =< aux(6) 0.04/0.31 s(3) =< it(15)*aux(1) 0.04/0.31 0.04/0.31 with precondition: [B=4,V_n+1=C,V_n+1=D,V_n+1=E,V_i_0>=1,V_n>=V_i_0] 0.04/0.31 0.04/0.31 * Chain [18]: 0 0.04/0.31 with precondition: [B=3,V_i_0>=1] 0.04/0.31 0.04/0.31 * Chain [17]: 1*s(4)+0 0.04/0.31 Such that:s(4) =< -V_i_0+V_n+1 0.04/0.31 0.04/0.31 with precondition: [B=3,V_i_0>=1,V_n>=V_i_0] 0.04/0.31 0.04/0.31 * Chain [16]: 0 0.04/0.31 with precondition: [B=4,C=V_3,E=V_j_0,V_i_0=D,V_i_0>=1,V_i_0>=V_n+1] 0.04/0.31 0.04/0.31 0.04/0.31 #### Cost of chains of eval_abc_bb1_in_loop_cont(A,B,C,D,E,F): 0.04/0.31 * Chain [20]: 0 0.04/0.31 with precondition: [A=3] 0.04/0.31 0.04/0.31 * Chain [19]: 0 0.04/0.31 with precondition: [A=4] 0.04/0.31 0.04/0.31 0.04/0.31 #### Cost of chains of eval_abc_4(V_3,V_i_0,V_j_0,V_n,B): 0.04/0.31 * Chain [24]: 0 0.04/0.31 with precondition: [] 0.04/0.31 0.04/0.31 * Chain [23]: 0 0.04/0.31 with precondition: [0>=V_n] 0.04/0.31 0.04/0.31 * Chain [22]: 3*s(12)+1*s(13)+1*s(17)+0 0.04/0.31 Such that:s(14) =< V_n+1 0.04/0.31 aux(9) =< V_n 0.04/0.31 s(12) =< aux(9) 0.04/0.31 s(13) =< s(12)*aux(9) 0.04/0.31 s(14) =< aux(9) 0.04/0.31 s(17) =< s(12)*s(14) 0.04/0.31 0.04/0.31 with precondition: [V_n>=1] 0.04/0.31 0.04/0.31 * Chain [21]: 2*s(21)+1*s(22)+0 0.04/0.31 Such that:aux(10) =< V_n 0.04/0.31 s(21) =< aux(10) 0.04/0.31 s(22) =< s(21)*aux(10) 0.04/0.31 0.04/0.31 with precondition: [V_n>=2] 0.04/0.31 0.04/0.31 0.04/0.31 #### Cost of chains of eval_abc_start(V_3,V_i_0,V_j_0,V_n,B): 0.04/0.31 * Chain [28]: 0 0.04/0.31 with precondition: [] 0.04/0.31 0.04/0.31 * Chain [27]: 0 0.04/0.31 with precondition: [0>=V_n] 0.04/0.31 0.04/0.31 * Chain [26]: 3*s(25)+1*s(26)+1*s(27)+0 0.04/0.31 Such that:s(24) =< V_n 0.04/0.31 s(23) =< V_n+1 0.04/0.31 s(25) =< s(24) 0.04/0.31 s(26) =< s(25)*s(24) 0.04/0.31 s(23) =< s(24) 0.04/0.31 s(27) =< s(25)*s(23) 0.04/0.31 0.04/0.31 with precondition: [V_n>=1] 0.04/0.31 0.04/0.31 * Chain [25]: 2*s(29)+1*s(30)+0 0.04/0.31 Such that:s(28) =< V_n 0.04/0.31 s(29) =< s(28) 0.04/0.31 s(30) =< s(29)*s(28) 0.04/0.31 0.04/0.31 with precondition: [V_n>=2] 0.04/0.31 0.04/0.31 0.04/0.31 Closed-form bounds of eval_abc_start(V_3,V_i_0,V_j_0,V_n,B): 0.04/0.31 ------------------------------------- 0.04/0.31 * Chain [28] with precondition: [] 0.04/0.31 - Upper bound: 0 0.04/0.31 - Complexity: constant 0.04/0.31 * Chain [27] with precondition: [0>=V_n] 0.04/0.31 - Upper bound: 0 0.04/0.31 - Complexity: constant 0.04/0.31 * Chain [26] with precondition: [V_n>=1] 0.04/0.31 - Upper bound: 3*V_n+V_n*V_n+(V_n+1)*V_n 0.04/0.31 - Complexity: n^2 0.04/0.31 * Chain [25] with precondition: [V_n>=2] 0.04/0.31 - Upper bound: 2*V_n+V_n*V_n 0.04/0.31 - Complexity: n^2 0.04/0.31 0.04/0.31 ### Maximum cost of eval_abc_start(V_3,V_i_0,V_j_0,V_n,B): nat(V_n+1)*nat(V_n)+nat(V_n)+(nat(V_n)*nat(V_n)+nat(V_n)*2) 0.04/0.31 Asymptotic class: n^2 0.04/0.31 * Total analysis performed in 230 ms. 0.04/0.31 0.04/0.41 EOF