0.06/0.36 WORST_CASE(?,O(n^2)) 0.06/0.36 0.06/0.36 Preprocessing Cost Relations 0.06/0.36 ===================================== 0.06/0.36 0.06/0.36 #### Computed strongly connected components 0.06/0.36 0. recursive : [eval_ex_paper1_3/5,eval_ex_paper1_4/6,eval_ex_paper1_6/7,eval_ex_paper1_7/7,eval_ex_paper1_bb3_in/5,eval_ex_paper1_bb4_in/6] 0.06/0.36 1. recursive : [eval_ex_paper1_bb1_in/4,eval_ex_paper1_bb2_in/4,eval_ex_paper1_bb3_in_loop_cont/5] 0.06/0.36 2. recursive : [eval_ex_paper1_10/4,eval_ex_paper1_11/4,eval_ex_paper1_bb7_in/4,eval_ex_paper1_bb8_in/4] 0.06/0.36 3. recursive : [eval_ex_paper1_bb5_in/3,eval_ex_paper1_bb6_in/3,eval_ex_paper1_bb7_in_loop_cont/4] 0.06/0.36 4. non_recursive : [eval_ex_paper1_stop/1] 0.06/0.36 5. non_recursive : [eval_ex_paper1_bb9_in/1] 0.06/0.36 6. non_recursive : [eval_ex_paper1_bb5_in_loop_cont/2] 0.06/0.36 7. non_recursive : [eval_ex_paper1_bb1_in_loop_cont/4] 0.06/0.36 8. non_recursive : [eval_ex_paper1_bb0_in/4] 0.06/0.36 9. non_recursive : [eval_ex_paper1_start/4] 0.06/0.36 0.06/0.36 #### Obtained direct recursion through partial evaluation 0.06/0.36 0. SCC is partially evaluated into eval_ex_paper1_bb3_in/5 0.06/0.36 1. SCC is partially evaluated into eval_ex_paper1_bb1_in/4 0.06/0.36 2. SCC is partially evaluated into eval_ex_paper1_bb7_in/4 0.06/0.36 3. SCC is partially evaluated into eval_ex_paper1_bb5_in/3 0.06/0.36 4. SCC is completely evaluated into other SCCs 0.06/0.36 5. SCC is completely evaluated into other SCCs 0.06/0.36 6. SCC is completely evaluated into other SCCs 0.06/0.36 7. SCC is partially evaluated into eval_ex_paper1_bb1_in_loop_cont/4 0.06/0.36 8. SCC is partially evaluated into eval_ex_paper1_bb0_in/4 0.06/0.36 9. SCC is partially evaluated into eval_ex_paper1_start/4 0.06/0.36 0.06/0.36 Control-Flow Refinement of Cost Relations 0.06/0.36 ===================================== 0.06/0.36 0.06/0.36 ### Specialization of cost equations eval_ex_paper1_bb3_in/5 0.06/0.36 * CE 9 is refined into CE [14] 0.06/0.36 * CE 8 is refined into CE [15] 0.06/0.36 0.06/0.36 0.06/0.36 ### Cost equations --> "Loop" of eval_ex_paper1_bb3_in/5 0.06/0.36 * CEs [15] --> Loop 14 0.06/0.36 * CEs [14] --> Loop 15 0.06/0.36 0.06/0.36 ### Ranking functions of CR eval_ex_paper1_bb3_in(V__01,V__0,V__1,B,C) 0.06/0.36 * RF of phase [15]: [V__1] 0.06/0.36 0.06/0.36 #### Partial ranking functions of CR eval_ex_paper1_bb3_in(V__01,V__0,V__1,B,C) 0.06/0.36 * Partial RF of phase [15]: 0.06/0.36 - RF of loop [15:1]: 0.06/0.36 V__1 0.06/0.36 0.06/0.36 0.06/0.36 ### Specialization of cost equations eval_ex_paper1_bb1_in/4 0.06/0.36 * CE 6 is refined into CE [16] 0.06/0.36 * CE 5 is refined into CE [17,18] 0.06/0.36 0.06/0.36 0.06/0.36 ### Cost equations --> "Loop" of eval_ex_paper1_bb1_in/4 0.06/0.36 * CEs [18] --> Loop 16 0.06/0.36 * CEs [17] --> Loop 17 0.06/0.36 * CEs [16] --> Loop 18 0.06/0.36 0.06/0.36 ### Ranking functions of CR eval_ex_paper1_bb1_in(V__01,V__0,B,C) 0.06/0.36 * RF of phase [16,17]: [V__0] 0.06/0.36 0.06/0.36 #### Partial ranking functions of CR eval_ex_paper1_bb1_in(V__01,V__0,B,C) 0.06/0.36 * Partial RF of phase [16,17]: 0.06/0.36 - RF of loop [16:1,17:1]: 0.06/0.36 V__0 0.06/0.36 0.06/0.36 0.06/0.36 ### Specialization of cost equations eval_ex_paper1_bb7_in/4 0.06/0.36 * CE 13 is refined into CE [19] 0.06/0.36 * CE 12 is refined into CE [20] 0.06/0.36 0.06/0.36 0.06/0.36 ### Cost equations --> "Loop" of eval_ex_paper1_bb7_in/4 0.06/0.36 * CEs [20] --> Loop 19 0.06/0.36 * CEs [19] --> Loop 20 0.06/0.36 0.06/0.36 ### Ranking functions of CR eval_ex_paper1_bb7_in(V_z,V__2,V_i_0,B) 0.06/0.36 * RF of phase [19]: [V_z-V_i_0] 0.06/0.36 0.06/0.36 #### Partial ranking functions of CR eval_ex_paper1_bb7_in(V_z,V__2,V_i_0,B) 0.06/0.36 * Partial RF of phase [19]: 0.06/0.36 - RF of loop [19:1]: 0.06/0.36 V_z-V_i_0 0.06/0.36 0.06/0.36 0.06/0.36 ### Specialization of cost equations eval_ex_paper1_bb5_in/3 0.06/0.36 * CE 11 is refined into CE [21] 0.06/0.36 * CE 10 is refined into CE [22,23] 0.06/0.36 0.06/0.36 0.06/0.36 ### Cost equations --> "Loop" of eval_ex_paper1_bb5_in/3 0.06/0.36 * CEs [23] --> Loop 21 0.06/0.36 * CEs [22] --> Loop 22 0.06/0.36 * CEs [21] --> Loop 23 0.06/0.36 0.06/0.36 ### Ranking functions of CR eval_ex_paper1_bb5_in(V_z,V__2,B) 0.06/0.36 * RF of phase [21]: [V__2] 0.06/0.36 * RF of phase [22]: [V__2] 0.06/0.36 0.06/0.36 #### Partial ranking functions of CR eval_ex_paper1_bb5_in(V_z,V__2,B) 0.06/0.36 * Partial RF of phase [21]: 0.06/0.36 - RF of loop [21:1]: 0.06/0.36 V__2 0.06/0.36 * Partial RF of phase [22]: 0.06/0.36 - RF of loop [22:1]: 0.06/0.36 V__2 0.06/0.36 0.06/0.36 0.06/0.36 ### Specialization of cost equations eval_ex_paper1_bb1_in_loop_cont/4 0.06/0.36 * CE 7 is refined into CE [24,25,26] 0.06/0.36 0.06/0.36 0.06/0.36 ### Cost equations --> "Loop" of eval_ex_paper1_bb1_in_loop_cont/4 0.06/0.36 * CEs [26] --> Loop 24 0.06/0.36 * CEs [25] --> Loop 25 0.06/0.36 * CEs [24] --> Loop 26 0.06/0.36 0.06/0.36 ### Ranking functions of CR eval_ex_paper1_bb1_in_loop_cont(A,B,C,D) 0.06/0.36 0.06/0.36 #### Partial ranking functions of CR eval_ex_paper1_bb1_in_loop_cont(A,B,C,D) 0.06/0.36 0.06/0.36 0.06/0.36 ### Specialization of cost equations eval_ex_paper1_bb0_in/4 0.06/0.36 * CE 4 is refined into CE [27,28,29] 0.06/0.36 * CE 3 is refined into CE [30] 0.06/0.36 * CE 2 is refined into CE [31] 0.06/0.36 0.06/0.36 0.06/0.36 ### Cost equations --> "Loop" of eval_ex_paper1_bb0_in/4 0.06/0.36 * CEs [29] --> Loop 27 0.06/0.36 * CEs [28] --> Loop 28 0.06/0.36 * CEs [27] --> Loop 29 0.06/0.36 * CEs [30] --> Loop 30 0.06/0.36 * CEs [31] --> Loop 31 0.06/0.36 0.06/0.36 ### Ranking functions of CR eval_ex_paper1_bb0_in(V_x,V_y,V_z,B) 0.06/0.36 0.06/0.36 #### Partial ranking functions of CR eval_ex_paper1_bb0_in(V_x,V_y,V_z,B) 0.06/0.36 0.06/0.36 0.06/0.36 ### Specialization of cost equations eval_ex_paper1_start/4 0.06/0.36 * CE 1 is refined into CE [32,33,34,35,36] 0.06/0.36 0.06/0.36 0.06/0.36 ### Cost equations --> "Loop" of eval_ex_paper1_start/4 0.06/0.36 * CEs [36] --> Loop 32 0.06/0.36 * CEs [35] --> Loop 33 0.06/0.36 * CEs [34] --> Loop 34 0.06/0.36 * CEs [33] --> Loop 35 0.06/0.36 * CEs [32] --> Loop 36 0.06/0.36 0.06/0.36 ### Ranking functions of CR eval_ex_paper1_start(V_x,V_y,V_z,B) 0.06/0.36 0.06/0.36 #### Partial ranking functions of CR eval_ex_paper1_start(V_x,V_y,V_z,B) 0.06/0.36 0.06/0.36 0.06/0.36 Computing Bounds 0.06/0.36 ===================================== 0.06/0.36 0.06/0.36 #### Cost of chains of eval_ex_paper1_bb3_in(V__01,V__0,V__1,B,C): 0.06/0.36 * Chain [[15],14]: 1*it(15)+0 0.06/0.36 Such that:it(15) =< V__1-C 0.06/0.36 0.06/0.36 with precondition: [B=3,V__0>=1,C>=0,V__01+1>=V__1,V__1>=C+1] 0.06/0.36 0.06/0.36 * Chain [14]: 0 0.06/0.36 with precondition: [B=3,V__1=C,V__0>=1,V__01+1>=V__1] 0.06/0.36 0.06/0.36 0.06/0.36 #### Cost of chains of eval_ex_paper1_bb1_in(V__01,V__0,B,C): 0.06/0.36 * Chain [[16,17],18]: 2*it(16)+1*s(3)+0 0.06/0.36 Such that:s(3) =< V__01+V__0-C 0.06/0.36 aux(3) =< V__0 0.06/0.36 it(16) =< aux(3) 0.06/0.36 0.06/0.36 with precondition: [B=2,V__01>=0,V__0>=1,C>=0,V__0+V__01>=C] 0.06/0.36 0.06/0.36 0.06/0.36 #### Cost of chains of eval_ex_paper1_bb7_in(V_z,V__2,V_i_0,B): 0.06/0.36 * Chain [[19],20]: 1*it(19)+0 0.06/0.36 Such that:it(19) =< V_z-V_i_0 0.06/0.36 0.06/0.36 with precondition: [B=2,V__2>=1,V_i_0>=0,V_z>=V_i_0+1] 0.06/0.36 0.06/0.36 * Chain [20]: 0 0.06/0.36 with precondition: [B=2,V__2>=1,V_i_0>=0,V_i_0>=V_z] 0.06/0.36 0.06/0.36 0.06/0.36 #### Cost of chains of eval_ex_paper1_bb5_in(V_z,V__2,B): 0.06/0.36 * Chain [[22],23]: 1*it(22)+0 0.06/0.36 Such that:it(22) =< V__2 0.06/0.36 0.06/0.36 with precondition: [B=4,0>=V_z,V__2>=1] 0.06/0.36 0.06/0.36 * Chain [[21],23]: 1*it(21)+1*s(6)+0 0.06/0.36 Such that:aux(4) =< V_z 0.06/0.36 it(21) =< V__2 0.06/0.36 s(6) =< it(21)*aux(4) 0.06/0.36 0.06/0.36 with precondition: [B=4,V_z>=1,V__2>=1] 0.06/0.36 0.06/0.36 * Chain [23]: 0 0.06/0.36 with precondition: [B=4,0>=V__2] 0.06/0.36 0.06/0.36 0.06/0.36 #### Cost of chains of eval_ex_paper1_bb1_in_loop_cont(A,B,C,D): 0.06/0.36 * Chain [26]: 1*s(7)+0 0.06/0.36 Such that:s(7) =< C 0.06/0.36 0.06/0.36 with precondition: [A=2,0>=B,C>=1] 0.06/0.36 0.06/0.36 * Chain [25]: 0 0.06/0.36 with precondition: [A=2,0>=C] 0.06/0.36 0.06/0.36 * Chain [24]: 1*s(9)+1*s(10)+0 0.06/0.36 Such that:s(8) =< B 0.06/0.36 s(9) =< C 0.06/0.36 s(10) =< s(9)*s(8) 0.06/0.36 0.06/0.36 with precondition: [A=2,B>=1,C>=1] 0.06/0.36 0.06/0.36 0.06/0.36 #### Cost of chains of eval_ex_paper1_bb0_in(V_x,V_y,V_z,B): 0.06/0.36 * Chain [31]: 0 0.06/0.36 with precondition: [0>=V_x] 0.06/0.36 0.06/0.36 * Chain [30]: 0 0.06/0.36 with precondition: [0>=V_y] 0.06/0.36 0.06/0.36 * Chain [29]: 2*s(11)+2*s(13)+0 0.06/0.36 Such that:s(12) =< V_x 0.06/0.36 aux(5) =< V_x+V_y 0.06/0.36 s(11) =< aux(5) 0.06/0.36 s(13) =< s(12) 0.06/0.36 0.06/0.36 with precondition: [0>=V_z,V_x>=1,V_y>=1] 0.06/0.36 0.06/0.36 * Chain [28]: 1*s(15)+2*s(17)+0 0.06/0.36 Such that:s(16) =< V_x 0.06/0.36 s(15) =< V_x+V_y 0.06/0.36 s(17) =< s(16) 0.06/0.36 0.06/0.36 with precondition: [V_x>=1,V_y>=1] 0.06/0.36 0.06/0.36 * Chain [27]: 2*s(18)+2*s(20)+1*s(23)+0 0.06/0.36 Such that:s(19) =< V_x 0.06/0.36 s(21) =< V_z 0.06/0.36 aux(6) =< V_x+V_y 0.06/0.36 s(18) =< aux(6) 0.06/0.36 s(23) =< s(18)*s(21) 0.06/0.36 s(20) =< s(19) 0.06/0.36 0.06/0.36 with precondition: [V_x>=1,V_y>=1,V_z>=1] 0.06/0.36 0.06/0.36 0.06/0.36 #### Cost of chains of eval_ex_paper1_start(V_x,V_y,V_z,B): 0.06/0.36 * Chain [36]: 0 0.06/0.36 with precondition: [0>=V_x] 0.06/0.36 0.06/0.36 * Chain [35]: 0 0.06/0.36 with precondition: [0>=V_y] 0.06/0.36 0.06/0.36 * Chain [34]: 2*s(26)+2*s(27)+0 0.06/0.36 Such that:s(24) =< V_x 0.06/0.36 s(25) =< V_x+V_y 0.06/0.36 s(26) =< s(25) 0.06/0.36 s(27) =< s(24) 0.06/0.36 0.06/0.36 with precondition: [0>=V_z,V_x>=1,V_y>=1] 0.06/0.36 0.06/0.36 * Chain [33]: 1*s(29)+2*s(30)+0 0.06/0.36 Such that:s(28) =< V_x 0.06/0.36 s(29) =< V_x+V_y 0.06/0.36 s(30) =< s(28) 0.06/0.36 0.06/0.36 with precondition: [V_x>=1,V_y>=1] 0.06/0.36 0.06/0.36 * Chain [32]: 2*s(34)+1*s(35)+2*s(36)+0 0.06/0.36 Such that:s(31) =< V_x 0.06/0.36 s(33) =< V_x+V_y 0.06/0.36 s(32) =< V_z 0.06/0.36 s(34) =< s(33) 0.06/0.36 s(35) =< s(34)*s(32) 0.06/0.36 s(36) =< s(31) 0.06/0.36 0.06/0.36 with precondition: [V_x>=1,V_y>=1,V_z>=1] 0.06/0.36 0.06/0.36 0.06/0.36 Closed-form bounds of eval_ex_paper1_start(V_x,V_y,V_z,B): 0.06/0.36 ------------------------------------- 0.06/0.36 * Chain [36] with precondition: [0>=V_x] 0.06/0.36 - Upper bound: 0 0.06/0.36 - Complexity: constant 0.06/0.36 * Chain [35] with precondition: [0>=V_y] 0.06/0.36 - Upper bound: 0 0.06/0.36 - Complexity: constant 0.06/0.36 * Chain [34] with precondition: [0>=V_z,V_x>=1,V_y>=1] 0.06/0.36 - Upper bound: 4*V_x+2*V_y 0.06/0.36 - Complexity: n 0.06/0.36 * Chain [33] with precondition: [V_x>=1,V_y>=1] 0.06/0.36 - Upper bound: 3*V_x+V_y 0.06/0.36 - Complexity: n 0.06/0.36 * Chain [32] with precondition: [V_x>=1,V_y>=1,V_z>=1] 0.06/0.36 - Upper bound: (V_x+V_y)*V_z+2*V_x+(2*V_x+2*V_y) 0.06/0.36 - Complexity: n^2 0.06/0.36 0.06/0.36 ### Maximum cost of eval_ex_paper1_start(V_x,V_y,V_z,B): nat(V_x+V_y)*nat(V_z)+nat(V_x+V_y)+(nat(V_x)*2+nat(V_x+V_y)) 0.06/0.36 Asymptotic class: n^2 0.06/0.36 * Total analysis performed in 267 ms. 0.06/0.36 0.06/0.46 EOF