0.63/0.65 WORST_CASE(?,O(n^2)) 0.63/0.65 0.63/0.65 Preprocessing Cost Relations 0.63/0.65 ===================================== 0.63/0.65 0.63/0.65 #### Computed strongly connected components 0.63/0.65 0. non_recursive : [eval_realheapsort_step2_stop/1] 0.63/0.65 1. non_recursive : [eval_realheapsort_step2_bb11_in/1] 0.63/0.65 2. recursive : [eval_realheapsort_step2_14/5,eval_realheapsort_step2_15/6,eval_realheapsort_step2_16/7,eval_realheapsort_step2_23/6,eval_realheapsort_step2_24/7,eval_realheapsort_step2_25/8,eval_realheapsort_step2_26/8,eval_realheapsort_step2_27/8,eval_realheapsort_step2_bb3_in/5,eval_realheapsort_step2_bb4_in/5,eval_realheapsort_step2_bb5_in/5,eval_realheapsort_step2_bb6_in/5,eval_realheapsort_step2_bb7_in/7,eval_realheapsort_step2_bb8_in/6,eval_realheapsort_step2_bb9_in/8] 0.63/0.65 3. recursive : [eval_realheapsort_step2_2/3,eval_realheapsort_step2_3/3,eval_realheapsort_step2_bb10_in/4,eval_realheapsort_step2_bb1_in/3,eval_realheapsort_step2_bb2_in/3,eval_realheapsort_step2_bb3_in_loop_cont/5] 0.63/0.65 4. non_recursive : [eval_realheapsort_step2_bb1_in_loop_cont/2] 0.63/0.65 5. non_recursive : [eval_realheapsort_step2_bb0_in/2] 0.63/0.65 6. non_recursive : [eval_realheapsort_step2_start/2] 0.63/0.65 0.63/0.65 #### Obtained direct recursion through partial evaluation 0.63/0.65 0. SCC is completely evaluated into other SCCs 0.63/0.65 1. SCC is completely evaluated into other SCCs 0.63/0.65 2. SCC is partially evaluated into eval_realheapsort_step2_bb3_in/5 0.63/0.65 3. SCC is partially evaluated into eval_realheapsort_step2_bb1_in/3 0.63/0.65 4. SCC is completely evaluated into other SCCs 0.63/0.65 5. SCC is partially evaluated into eval_realheapsort_step2_bb0_in/2 0.63/0.65 6. SCC is partially evaluated into eval_realheapsort_step2_start/2 0.63/0.65 0.63/0.65 Control-Flow Refinement of Cost Relations 0.63/0.65 ===================================== 0.63/0.65 0.63/0.65 ### Specialization of cost equations eval_realheapsort_step2_bb3_in/5 0.63/0.65 * CE 11 is refined into CE [12] 0.63/0.65 * CE 9 is refined into CE [13] 0.63/0.65 * CE 6 is refined into CE [14] 0.63/0.65 * CE 7 is refined into CE [15] 0.63/0.65 * CE 8 is refined into CE [16] 0.63/0.65 * CE 10 is refined into CE [17] 0.63/0.65 0.63/0.65 0.63/0.65 ### Cost equations --> "Loop" of eval_realheapsort_step2_bb3_in/5 0.63/0.65 * CEs [13] --> Loop 12 0.63/0.65 * CEs [14] --> Loop 13 0.63/0.65 * CEs [15] --> Loop 14 0.63/0.65 * CEs [16] --> Loop 15 0.63/0.65 * CEs [17] --> Loop 16 0.63/0.65 * CEs [12] --> Loop 17 0.63/0.65 0.63/0.65 ### Ranking functions of CR eval_realheapsort_step2_bb3_in(V_N,V_k_0,V_j_0,B,C) 0.63/0.65 * RF of phase [13,14]: [V_N/2-V_j_0-3/2,V_N/2-V_k_0/2-V_j_0-3/2] 0.63/0.65 0.63/0.65 #### Partial ranking functions of CR eval_realheapsort_step2_bb3_in(V_N,V_k_0,V_j_0,B,C) 0.63/0.65 * Partial RF of phase [13,14]: 0.63/0.65 - RF of loop [13:1,14:1]: 0.63/0.65 V_N/2-V_j_0-3/2 0.63/0.65 V_N/2-V_k_0/2-V_j_0-3/2 0.63/0.65 0.63/0.65 0.63/0.65 ### Specialization of cost equations eval_realheapsort_step2_bb1_in/3 0.63/0.65 * CE 5 is refined into CE [18] 0.63/0.65 * CE 4 is refined into CE [19,20,21,22,23,24,25,26] 0.63/0.65 0.63/0.65 0.63/0.65 ### Cost equations --> "Loop" of eval_realheapsort_step2_bb1_in/3 0.63/0.65 * CEs [21] --> Loop 18 0.63/0.65 * CEs [20] --> Loop 19 0.63/0.65 * CEs [22] --> Loop 20 0.63/0.65 * CEs [25] --> Loop 21 0.63/0.65 * CEs [26] --> Loop 22 0.63/0.65 * CEs [19,24] --> Loop 23 0.63/0.65 * CEs [23] --> Loop 24 0.63/0.65 * CEs [18] --> Loop 25 0.63/0.65 0.63/0.65 ### Ranking functions of CR eval_realheapsort_step2_bb1_in(V_N,V_k_0,B) 0.63/0.65 * RF of phase [18,19,20,21,22]: [V_N-V_k_0-3] 0.63/0.65 0.63/0.65 #### Partial ranking functions of CR eval_realheapsort_step2_bb1_in(V_N,V_k_0,B) 0.63/0.65 * Partial RF of phase [18,19,20,21,22]: 0.63/0.65 - RF of loop [18:1]: 0.63/0.65 V_N-V_k_0-5 0.63/0.65 - RF of loop [19:1,21:1]: 0.63/0.65 V_N-V_k_0-4 0.63/0.65 - RF of loop [20:1,22:1]: 0.63/0.65 V_N-V_k_0-3 0.63/0.65 0.63/0.65 0.63/0.65 ### Specialization of cost equations eval_realheapsort_step2_bb0_in/2 0.63/0.65 * CE 3 is refined into CE [27,28] 0.63/0.65 * CE 2 is refined into CE [29] 0.63/0.65 0.63/0.65 0.63/0.65 ### Cost equations --> "Loop" of eval_realheapsort_step2_bb0_in/2 0.63/0.65 * CEs [28] --> Loop 26 0.63/0.65 * CEs [29] --> Loop 27 0.63/0.65 * CEs [27] --> Loop 28 0.63/0.65 0.63/0.65 ### Ranking functions of CR eval_realheapsort_step2_bb0_in(V_N,B) 0.63/0.65 0.63/0.65 #### Partial ranking functions of CR eval_realheapsort_step2_bb0_in(V_N,B) 0.63/0.65 0.63/0.65 0.63/0.65 ### Specialization of cost equations eval_realheapsort_step2_start/2 0.63/0.65 * CE 1 is refined into CE [30,31,32] 0.63/0.65 0.63/0.65 0.63/0.65 ### Cost equations --> "Loop" of eval_realheapsort_step2_start/2 0.63/0.65 * CEs [32] --> Loop 29 0.63/0.65 * CEs [31] --> Loop 30 0.63/0.65 * CEs [30] --> Loop 31 0.63/0.65 0.63/0.65 ### Ranking functions of CR eval_realheapsort_step2_start(V_N,B) 0.63/0.65 0.63/0.65 #### Partial ranking functions of CR eval_realheapsort_step2_start(V_N,B) 0.63/0.65 0.63/0.65 0.63/0.65 Computing Bounds 0.63/0.65 ===================================== 0.63/0.65 0.63/0.65 #### Cost of chains of eval_realheapsort_step2_bb3_in(V_N,V_k_0,V_j_0,B,C): 0.63/0.65 * Chain [[13,14],17]: 2*it(13)+0 0.63/0.65 Such that:aux(1) =< V_N/2-V_k_0/2-V_j_0 0.63/0.65 aux(3) =< V_N/2-V_j_0 0.63/0.65 aux(5) =< -V_j_0+C 0.63/0.65 it(13) =< aux(1) 0.63/0.65 it(13) =< aux(5) 0.63/0.65 it(13) =< aux(3) 0.63/0.65 0.63/0.65 with precondition: [B=2,V_k_0>=0,V_j_0>=0,C>=2*V_j_0+1,V_N>=2*V_j_0+V_k_0+4,V_k_0+2*C+2>=V_N,V_N>=V_k_0+C+2] 0.63/0.65 0.63/0.65 * Chain [[13,14],16,17]: 2*it(13)+1 0.63/0.65 Such that:aux(3) =< -V_j_0+C/2 0.63/0.65 aux(6) =< -V_k_0/2-V_j_0+C/2 0.63/0.65 it(13) =< aux(6) 0.63/0.65 it(13) =< aux(3) 0.63/0.65 0.63/0.65 with precondition: [B=2,V_N=C,V_k_0>=0,V_j_0>=0,V_N>=4*V_j_0+V_k_0+5] 0.63/0.65 0.63/0.65 * Chain [[13,14],15,17]: 2*it(13)+1 0.63/0.65 Such that:aux(3) =< -V_j_0+C/2 0.63/0.65 aux(7) =< -V_k_0/2-V_j_0+C/2 0.63/0.65 it(13) =< aux(7) 0.63/0.65 it(13) =< aux(3) 0.63/0.65 0.63/0.65 with precondition: [B=2,V_N=C,V_k_0>=0,V_j_0>=0,V_N>=4*V_j_0+V_k_0+6] 0.63/0.65 0.63/0.65 * Chain [[13,14],12,17]: 2*it(13)+1 0.63/0.65 Such that:aux(3) =< V_k_0/2-V_j_0+C/2+1 0.63/0.65 aux(1) =< -V_j_0+C/2+1 0.63/0.65 aux(8) =< -V_j_0+C/2 0.63/0.65 it(13) =< aux(1) 0.63/0.65 it(13) =< aux(8) 0.63/0.65 it(13) =< aux(3) 0.63/0.65 0.63/0.65 with precondition: [B=2,V_N=V_k_0+C+2,V_k_0>=0,V_j_0>=0,V_N>=4*V_j_0+V_k_0+5] 0.63/0.65 0.63/0.65 * Chain [17]: 0 0.63/0.65 with precondition: [B=2,V_j_0=C,V_N>=3,V_k_0>=0,V_N>=V_k_0+2,V_N>=V_j_0,4*V_N>=3*V_k_0+V_j_0+9,V_k_0+2*V_j_0+2>=V_N] 0.63/0.65 0.63/0.65 * Chain [16,17]: 1 0.63/0.65 with precondition: [B=2,V_N=C,V_N=2*V_j_0+V_k_0+3,V_k_0>=0,V_N>=V_k_0+3] 0.63/0.65 0.63/0.65 * Chain [15,17]: 1 0.63/0.65 with precondition: [B=2,V_N=C,V_k_0>=0,V_j_0>=0,V_N>=2*V_j_0+V_k_0+4] 0.63/0.65 0.63/0.65 * Chain [12,17]: 1 0.63/0.65 with precondition: [B=2,V_N=2*V_j_0+V_k_0+3,V_N=V_k_0+C+2,V_k_0>=0,V_N>=V_k_0+3] 0.63/0.65 0.63/0.65 0.63/0.65 #### Cost of chains of eval_realheapsort_step2_bb1_in(V_N,V_k_0,B): 0.63/0.65 * Chain [[18,19,20,21,22],23,24,25]: 9*it(18)+2*s(29)+2*s(32)+2*s(35)+2*s(39)+3 0.63/0.65 Such that:aux(9) =< V_N/2 0.63/0.65 aux(10) =< V_N/2-V_k_0/2 0.63/0.65 aux(21) =< V_N-V_k_0 0.63/0.65 it(18) =< aux(21) 0.63/0.65 aux(12) =< aux(10) 0.63/0.65 aux(11) =< aux(9) 0.63/0.65 aux(16) =< aux(10)*2-1 0.63/0.65 aux(14) =< aux(10)-1/2 0.63/0.65 s(31) =< it(18)*aux(10) 0.63/0.65 s(30) =< it(18)*aux(9) 0.63/0.65 s(34) =< it(18)*aux(12) 0.63/0.65 s(33) =< it(18)*aux(11) 0.63/0.65 s(41) =< it(18)*aux(16) 0.63/0.65 s(37) =< it(18)*aux(14) 0.63/0.65 s(39) =< s(34) 0.63/0.65 s(39) =< s(41) 0.63/0.65 s(39) =< s(33) 0.63/0.65 s(35) =< s(34) 0.63/0.65 s(35) =< s(37) 0.63/0.65 s(35) =< s(33) 0.63/0.65 s(32) =< s(34) 0.63/0.65 s(32) =< s(33) 0.63/0.65 s(29) =< s(31) 0.63/0.65 s(29) =< s(30) 0.63/0.65 0.63/0.65 with precondition: [B=3,V_k_0>=0,V_N>=V_k_0+4] 0.63/0.65 0.63/0.65 * Chain [23,24,25]: 3 0.63/0.65 with precondition: [B=3,V_N=V_k_0+3,V_N>=3] 0.63/0.65 0.63/0.65 0.63/0.65 #### Cost of chains of eval_realheapsort_step2_bb0_in(V_N,B): 0.63/0.65 * Chain [28]: 3 0.63/0.65 with precondition: [V_N=3] 0.63/0.65 0.63/0.65 * Chain [27]: 0 0.63/0.65 with precondition: [2>=V_N] 0.63/0.65 0.63/0.65 * Chain [26]: 9*s(46)+2*s(57)+2*s(58)+2*s(59)+2*s(60)+3 0.63/0.65 Such that:s(45) =< V_N 0.63/0.65 aux(22) =< V_N/2 0.63/0.65 s(46) =< s(45) 0.63/0.65 s(47) =< aux(22) 0.63/0.65 s(49) =< aux(22)*2-1 0.63/0.65 s(50) =< aux(22)-1/2 0.63/0.65 s(51) =< s(46)*aux(22) 0.63/0.65 s(53) =< s(46)*s(47) 0.63/0.65 s(55) =< s(46)*s(49) 0.63/0.65 s(56) =< s(46)*s(50) 0.63/0.65 s(57) =< s(53) 0.63/0.65 s(57) =< s(55) 0.63/0.65 s(58) =< s(53) 0.63/0.65 s(58) =< s(56) 0.63/0.65 s(59) =< s(53) 0.63/0.65 s(60) =< s(51) 0.63/0.65 0.63/0.65 with precondition: [V_N>=4] 0.63/0.65 0.63/0.65 0.63/0.65 #### Cost of chains of eval_realheapsort_step2_start(V_N,B): 0.63/0.65 * Chain [31]: 3 0.63/0.65 with precondition: [V_N=3] 0.63/0.65 0.63/0.65 * Chain [30]: 0 0.63/0.65 with precondition: [2>=V_N] 0.63/0.65 0.63/0.65 * Chain [29]: 9*s(63)+2*s(71)+2*s(72)+2*s(73)+2*s(74)+3 0.63/0.65 Such that:s(61) =< V_N 0.63/0.65 s(62) =< V_N/2 0.63/0.65 s(63) =< s(61) 0.63/0.65 s(64) =< s(62) 0.63/0.65 s(65) =< s(62)*2-1 0.63/0.65 s(66) =< s(62)-1/2 0.63/0.65 s(67) =< s(63)*s(62) 0.63/0.65 s(68) =< s(63)*s(64) 0.63/0.65 s(69) =< s(63)*s(65) 0.63/0.65 s(70) =< s(63)*s(66) 0.63/0.65 s(71) =< s(68) 0.63/0.65 s(71) =< s(69) 0.63/0.65 s(72) =< s(68) 0.63/0.65 s(72) =< s(70) 0.63/0.65 s(73) =< s(68) 0.63/0.65 s(74) =< s(67) 0.63/0.65 0.63/0.65 with precondition: [V_N>=4] 0.63/0.65 0.63/0.65 0.63/0.65 Closed-form bounds of eval_realheapsort_step2_start(V_N,B): 0.63/0.65 ------------------------------------- 0.63/0.65 * Chain [31] with precondition: [V_N=3] 0.63/0.65 - Upper bound: 3 0.63/0.65 - Complexity: constant 0.63/0.65 * Chain [30] with precondition: [2>=V_N] 0.63/0.65 - Upper bound: 0 0.63/0.65 - Complexity: constant 0.63/0.65 * Chain [29] with precondition: [V_N>=4] 0.63/0.65 - Upper bound: 9*V_N+3+V_N/2*(8*V_N) 0.63/0.65 - Complexity: n^2 0.63/0.65 0.63/0.65 ### Maximum cost of eval_realheapsort_step2_start(V_N,B): max([3,nat(V_N)*9+3+nat(V_N)*8*nat(V_N/2)]) 0.63/0.65 Asymptotic class: n^2 0.63/0.65 * Total analysis performed in 513 ms. 0.63/0.65 0.66/0.75 EOF