0.62/0.69 WORST_CASE(?,O(n^3)) 0.62/0.69 0.62/0.69 Preprocessing Cost Relations 0.62/0.69 ===================================== 0.62/0.69 0.62/0.69 #### Computed strongly connected components 0.62/0.69 0. recursive : [c/8] 0.62/0.69 1. recursive : [b/9,c_loop_cont/10] 0.62/0.69 2. recursive : [b_loop_cont/10,d/9] 0.62/0.69 3. non_recursive : [exit_location/1] 0.62/0.69 4. non_recursive : [halt/9] 0.62/0.69 5. non_recursive : [d_loop_cont/10] 0.62/0.69 6. non_recursive : [a/9] 0.62/0.69 7. non_recursive : [start/9] 0.62/0.69 8. non_recursive : [start0/9] 0.62/0.69 0.62/0.69 #### Obtained direct recursion through partial evaluation 0.62/0.69 0. SCC is partially evaluated into c/8 0.62/0.69 1. SCC is partially evaluated into b/9 0.62/0.69 2. SCC is partially evaluated into d/9 0.62/0.69 3. SCC is completely evaluated into other SCCs 0.62/0.69 4. SCC is completely evaluated into other SCCs 0.62/0.69 5. SCC is partially evaluated into d_loop_cont/10 0.62/0.69 6. SCC is partially evaluated into a/9 0.62/0.69 7. SCC is completely evaluated into other SCCs 0.62/0.69 8. SCC is partially evaluated into start0/9 0.62/0.69 0.62/0.69 Control-Flow Refinement of Cost Relations 0.62/0.69 ===================================== 0.62/0.69 0.62/0.69 ### Specialization of cost equations c/8 0.62/0.69 * CE 15 is refined into CE [16] 0.62/0.69 * CE 14 is refined into CE [17] 0.62/0.69 * CE 13 is refined into CE [18] 0.62/0.69 0.62/0.69 0.62/0.69 ### Cost equations --> "Loop" of c/8 0.62/0.69 * CEs [18] --> Loop 16 0.62/0.69 * CEs [16] --> Loop 17 0.62/0.69 * CEs [17] --> Loop 18 0.62/0.69 0.62/0.69 ### Ranking functions of CR c(A,B,C,E,G,I,J,K) 0.62/0.69 * RF of phase [16]: [-C+E,E-1] 0.62/0.69 0.62/0.69 #### Partial ranking functions of CR c(A,B,C,E,G,I,J,K) 0.62/0.69 * Partial RF of phase [16]: 0.62/0.69 - RF of loop [16:1]: 0.62/0.69 -C+E 0.62/0.69 E-1 0.62/0.69 0.62/0.69 0.62/0.69 ### Specialization of cost equations b/9 0.62/0.69 * CE 11 is refined into CE [19] 0.62/0.69 * CE 9 is refined into CE [20,21] 0.62/0.69 * CE 12 is refined into CE [22] 0.62/0.69 * CE 10 is refined into CE [23] 0.62/0.69 0.62/0.69 0.62/0.69 ### Cost equations --> "Loop" of b/9 0.62/0.69 * CEs [23] --> Loop 19 0.62/0.69 * CEs [19] --> Loop 20 0.62/0.69 * CEs [20,21] --> Loop 21 0.62/0.69 * CEs [22] --> Loop 22 0.62/0.69 0.62/0.69 ### Ranking functions of CR b(A,B,C,E,G,I,J,K,L) 0.62/0.69 * RF of phase [19]: [A-G+1,B-G+1] 0.62/0.69 0.62/0.69 #### Partial ranking functions of CR b(A,B,C,E,G,I,J,K,L) 0.62/0.69 * Partial RF of phase [19]: 0.62/0.69 - RF of loop [19:1]: 0.62/0.69 A-G+1 0.62/0.69 B-G+1 0.62/0.69 0.62/0.69 0.62/0.69 ### Specialization of cost equations d/9 0.62/0.69 * CE 5 is refined into CE [24] 0.62/0.69 * CE 3 is refined into CE [25,26,27,28] 0.62/0.69 * CE 6 is refined into CE [29] 0.62/0.69 * CE 4 is refined into CE [30] 0.62/0.69 0.62/0.69 0.62/0.69 ### Cost equations --> "Loop" of d/9 0.62/0.69 * CEs [30] --> Loop 23 0.62/0.69 * CEs [24] --> Loop 24 0.62/0.69 * CEs [28] --> Loop 25 0.62/0.69 * CEs [25,26,27] --> Loop 26 0.62/0.69 * CEs [29] --> Loop 27 0.62/0.69 0.62/0.69 ### Ranking functions of CR d(A,B,C,E,G,I,J,K,L) 0.62/0.69 * RF of phase [23]: [A-C,B-C] 0.62/0.69 0.62/0.69 #### Partial ranking functions of CR d(A,B,C,E,G,I,J,K,L) 0.62/0.69 * Partial RF of phase [23]: 0.62/0.69 - RF of loop [23:1]: 0.62/0.69 A-C 0.62/0.69 B-C 0.62/0.69 0.62/0.69 0.62/0.69 ### Specialization of cost equations d_loop_cont/10 0.62/0.69 * CE 7 is refined into CE [31] 0.62/0.69 * CE 8 is refined into CE [32] 0.62/0.69 0.62/0.69 0.62/0.69 ### Cost equations --> "Loop" of d_loop_cont/10 0.62/0.69 * CEs [31] --> Loop 28 0.62/0.69 * CEs [32] --> Loop 29 0.62/0.69 0.62/0.69 ### Ranking functions of CR d_loop_cont(A,B,C,D,E,F,G,H,I,J) 0.62/0.69 0.62/0.69 #### Partial ranking functions of CR d_loop_cont(A,B,C,D,E,F,G,H,I,J) 0.62/0.69 0.62/0.69 0.62/0.69 ### Specialization of cost equations a/9 0.62/0.69 * CE 2 is refined into CE [33,34,35,36,37,38,39,40] 0.62/0.69 0.62/0.69 0.62/0.69 ### Cost equations --> "Loop" of a/9 0.62/0.69 * CEs [38] --> Loop 30 0.62/0.69 * CEs [35,37] --> Loop 31 0.62/0.69 * CEs [34,36,40] --> Loop 32 0.62/0.69 * CEs [33] --> Loop 33 0.62/0.69 * CEs [39] --> Loop 34 0.62/0.69 0.62/0.69 ### Ranking functions of CR a(A,B,C,D,E,F,G,H,I) 0.62/0.69 0.62/0.69 #### Partial ranking functions of CR a(A,B,C,D,E,F,G,H,I) 0.62/0.69 0.62/0.69 0.62/0.69 ### Specialization of cost equations start0/9 0.62/0.69 * CE 1 is refined into CE [41,42,43,44,45] 0.62/0.69 0.62/0.69 0.62/0.69 ### Cost equations --> "Loop" of start0/9 0.62/0.69 * CEs [45] --> Loop 35 0.62/0.69 * CEs [44] --> Loop 36 0.62/0.69 * CEs [43] --> Loop 37 0.62/0.69 * CEs [42] --> Loop 38 0.62/0.69 * CEs [41] --> Loop 39 0.62/0.69 0.62/0.69 ### Ranking functions of CR start0(A,B,C,D,E,F,G,H,I) 0.62/0.69 0.62/0.69 #### Partial ranking functions of CR start0(A,B,C,D,E,F,G,H,I) 0.62/0.69 0.62/0.69 0.62/0.69 Computing Bounds 0.62/0.69 ===================================== 0.62/0.69 0.62/0.69 #### Cost of chains of c(A,B,C,E,G,I,J,K): 0.62/0.69 * Chain [[16],18]: 1*it(16)+0 0.62/0.69 Such that:it(16) =< E-J 0.62/0.69 0.62/0.69 with precondition: [I=2,A=B,C=J,G+1=K,C>=1,E>=C+1,G>=C+1,A>=E,A>=G] 0.62/0.69 0.62/0.69 * Chain [[16],17]: 1*it(16)+0 0.62/0.69 Such that:it(16) =< -C+E 0.62/0.69 0.62/0.69 with precondition: [I=3,A=B,C>=1,E>=C+1,G>=C+1,A>=E,A>=G] 0.62/0.69 0.62/0.69 * Chain [17]: 0 0.62/0.69 with precondition: [I=3,B=A,C>=1,G>=C+1,B>=E,B>=G] 0.62/0.69 0.62/0.69 0.62/0.69 #### Cost of chains of b(A,B,C,E,G,I,J,K,L): 0.62/0.69 * Chain [[19],22]: 1*it(19)+1*s(3)+0 0.62/0.69 Such that:aux(1) =< A-C 0.62/0.69 it(19) =< A-G+1 0.62/0.69 s(3) =< it(19)*aux(1) 0.62/0.69 0.62/0.69 with precondition: [I=3,A=B,C>=1,G>=C+1,A>=G] 0.62/0.69 0.62/0.69 * Chain [[19],21]: 1*it(19)+1*s(3)+1*s(4)+0 0.62/0.69 Such that:it(19) =< B-G 0.62/0.69 aux(2) =< B-C 0.62/0.69 s(4) =< aux(2) 0.62/0.69 s(3) =< it(19)*aux(2) 0.62/0.69 0.62/0.69 with precondition: [I=3,A=B,C>=1,G>=C+1,A>=G+1] 0.62/0.69 0.62/0.69 * Chain [[19],20]: 1*it(19)+1*s(3)+0 0.62/0.69 Such that:it(19) =< A-G+1 0.62/0.69 aux(1) =< A-K 0.62/0.69 s(3) =< it(19)*aux(1) 0.62/0.69 0.62/0.69 with precondition: [I=4,A=B,C+1=J,C=K,A+1=L,C>=1,G>=C+1,A>=G] 0.62/0.69 0.62/0.69 * Chain [22]: 0 0.62/0.69 with precondition: [I=3,B=A,C>=1,B>=C+1,G>=C+1] 0.62/0.69 0.62/0.69 * Chain [21]: 1*s(4)+0 0.62/0.69 Such that:s(4) =< A-C 0.62/0.69 0.62/0.69 with precondition: [I=3,B=A,C>=1,G>=C+1,B>=G] 0.62/0.69 0.62/0.69 0.62/0.69 #### Cost of chains of d(A,B,C,E,G,I,J,K,L): 0.62/0.69 * Chain [[23],27]: 1*it(23)+1*s(11)+1*s(12)+0 0.62/0.69 Such that:aux(7) =< A-C 0.62/0.69 it(23) =< aux(7) 0.62/0.69 aux(4) =< aux(7) 0.62/0.69 aux(4) =< aux(7) 0.62/0.69 s(13) =< it(23)*aux(4) 0.62/0.69 s(11) =< s(13) 0.62/0.69 s(12) =< s(11)*aux(7) 0.62/0.69 0.62/0.69 with precondition: [I=3,A=B,C>=1,A>=C+1] 0.62/0.69 0.62/0.69 * Chain [[23],26]: 3*it(23)+1*s(11)+1*s(12)+1*s(17)+0 0.62/0.69 Such that:aux(9) =< A-C 0.62/0.69 it(23) =< aux(9) 0.62/0.69 s(17) =< it(23)*aux(9) 0.62/0.69 aux(4) =< aux(9) 0.62/0.69 aux(4) =< aux(9) 0.62/0.69 s(13) =< it(23)*aux(4) 0.62/0.69 s(11) =< s(13) 0.62/0.69 s(12) =< s(11)*aux(9) 0.62/0.69 0.62/0.69 with precondition: [I=3,A=B,C>=1,A>=C+2] 0.62/0.69 0.62/0.69 * Chain [[23],25]: 3*it(23)+1*s(11)+1*s(12)+1*s(21)+0 0.62/0.69 Such that:aux(11) =< A-C 0.62/0.69 it(23) =< aux(11) 0.62/0.69 s(21) =< it(23)*aux(11) 0.62/0.69 aux(4) =< aux(11) 0.62/0.69 aux(4) =< aux(11) 0.62/0.69 s(13) =< it(23)*aux(4) 0.62/0.69 s(11) =< s(13) 0.62/0.69 s(12) =< s(11)*aux(11) 0.62/0.69 0.62/0.69 with precondition: [I=3,A=B,C>=1,A>=C+3] 0.62/0.69 0.62/0.69 * Chain [[23],24]: 1*it(23)+1*s(11)+1*s(12)+0 0.62/0.69 Such that:aux(12) =< -C+J 0.62/0.69 it(23) =< aux(12) 0.62/0.69 aux(4) =< aux(12) 0.62/0.69 aux(4) =< aux(12) 0.62/0.69 s(13) =< it(23)*aux(4) 0.62/0.69 s(11) =< s(13) 0.62/0.69 s(12) =< s(11)*aux(12) 0.62/0.69 0.62/0.69 with precondition: [I=5,A=B,A=J,A=K+1,A+1=L,C>=1,A>=C+1] 0.62/0.69 0.62/0.69 * Chain [27]: 0 0.62/0.69 with precondition: [I=3,B=A,C>=1,B>=C] 0.62/0.69 0.62/0.69 * Chain [26]: 1*s(14)+1*s(16)+1*s(17)+0 0.62/0.69 Such that:aux(8) =< A-C 0.62/0.69 s(14) =< B-C 0.62/0.69 s(16) =< aux(8) 0.62/0.69 s(17) =< s(16)*aux(8) 0.62/0.69 0.62/0.69 with precondition: [I=3,B=A,C>=1,B>=C+1] 0.62/0.69 0.62/0.69 * Chain [25]: 2*s(18)+1*s(21)+0 0.62/0.69 Such that:aux(10) =< A-C 0.62/0.69 s(18) =< aux(10) 0.62/0.69 s(21) =< s(18)*aux(10) 0.62/0.69 0.62/0.69 with precondition: [I=3,B=A,C>=1,B>=C+2] 0.62/0.69 0.62/0.69 * Chain [24]: 0 0.62/0.69 with precondition: [I=5,B=A,B=C,K=E,L=G,B=J,B>=1] 0.62/0.69 0.62/0.69 0.62/0.69 #### Cost of chains of d_loop_cont(A,B,C,D,E,F,G,H,I,J): 0.62/0.69 * Chain [29]: 0 0.62/0.69 with precondition: [A=3,C=B,C>=1] 0.62/0.69 0.62/0.69 * Chain [28]: 0 0.62/0.69 with precondition: [A=5,C=B,C>=1] 0.62/0.69 0.62/0.69 0.62/0.69 #### Cost of chains of a(A,B,C,D,E,F,G,H,I): 0.62/0.69 * Chain [34]: 0 0.62/0.69 with precondition: [A=1,B=1,D=C,F=E,H=G] 0.62/0.69 0.62/0.69 * Chain [33]: 0 0.62/0.69 with precondition: [B=A,D=C,F=E,H=G,B>=1] 0.62/0.69 0.62/0.69 * Chain [32]: 4*s(23)+1*s(25)+2*s(30)+2*s(31)+0 0.62/0.69 Such that:aux(14) =< A 0.62/0.69 s(23) =< aux(14) 0.62/0.69 s(25) =< s(23)*aux(14) 0.62/0.69 s(28) =< aux(14) 0.62/0.69 s(28) =< aux(14) 0.62/0.69 s(29) =< s(23)*s(28) 0.62/0.69 s(30) =< s(29) 0.62/0.69 s(31) =< s(30)*aux(14) 0.62/0.69 0.62/0.69 with precondition: [B=A,D=C,F=E,H=G,B>=2] 0.62/0.69 0.62/0.69 * Chain [31]: 5*s(39)+2*s(40)+1*s(46)+1*s(47)+0 0.62/0.69 Such that:aux(15) =< A 0.62/0.69 s(39) =< aux(15) 0.62/0.69 s(40) =< s(39)*aux(15) 0.62/0.69 s(44) =< aux(15) 0.62/0.69 s(44) =< aux(15) 0.62/0.69 s(45) =< s(39)*s(44) 0.62/0.69 s(46) =< s(45) 0.62/0.69 s(47) =< s(46)*aux(15) 0.62/0.69 0.62/0.69 with precondition: [B=A,D=C,F=E,H=G,B>=3] 0.62/0.69 0.62/0.69 * Chain [30]: 3*s(49)+1*s(50)+1*s(53)+1*s(54)+0 0.62/0.69 Such that:s(48) =< A 0.62/0.69 s(49) =< s(48) 0.62/0.69 s(50) =< s(49)*s(48) 0.62/0.69 s(51) =< s(48) 0.62/0.69 s(51) =< s(48) 0.62/0.69 s(52) =< s(49)*s(51) 0.62/0.69 s(53) =< s(52) 0.62/0.69 s(54) =< s(53)*s(48) 0.62/0.69 0.62/0.69 with precondition: [B=A,D=C,F=E,H=G,B>=4] 0.62/0.69 0.62/0.69 0.62/0.69 #### Cost of chains of start0(A,B,C,D,E,F,G,H,I): 0.62/0.69 * Chain [39]: 0 0.62/0.69 with precondition: [A=1] 0.62/0.69 0.62/0.69 * Chain [38]: 0 0.62/0.69 with precondition: [A>=1] 0.62/0.69 0.62/0.69 * Chain [37]: 4*s(56)+1*s(57)+2*s(60)+2*s(61)+0 0.62/0.69 Such that:s(55) =< A 0.62/0.69 s(56) =< s(55) 0.62/0.69 s(57) =< s(56)*s(55) 0.62/0.69 s(58) =< s(55) 0.62/0.69 s(58) =< s(55) 0.62/0.69 s(59) =< s(56)*s(58) 0.62/0.69 s(60) =< s(59) 0.62/0.69 s(61) =< s(60)*s(55) 0.62/0.69 0.62/0.69 with precondition: [A>=2] 0.62/0.69 0.62/0.69 * Chain [36]: 5*s(63)+2*s(64)+1*s(67)+1*s(68)+0 0.62/0.69 Such that:s(62) =< A 0.62/0.69 s(63) =< s(62) 0.62/0.69 s(64) =< s(63)*s(62) 0.62/0.69 s(65) =< s(62) 0.62/0.69 s(65) =< s(62) 0.62/0.69 s(66) =< s(63)*s(65) 0.62/0.69 s(67) =< s(66) 0.62/0.69 s(68) =< s(67)*s(62) 0.62/0.69 0.62/0.69 with precondition: [A>=3] 0.62/0.69 0.62/0.69 * Chain [35]: 3*s(70)+1*s(71)+1*s(74)+1*s(75)+0 0.62/0.69 Such that:s(69) =< A 0.62/0.69 s(70) =< s(69) 0.62/0.69 s(71) =< s(70)*s(69) 0.62/0.69 s(72) =< s(69) 0.62/0.69 s(72) =< s(69) 0.62/0.69 s(73) =< s(70)*s(72) 0.62/0.69 s(74) =< s(73) 0.62/0.69 s(75) =< s(74)*s(69) 0.62/0.69 0.62/0.69 with precondition: [A>=4] 0.62/0.69 0.62/0.69 0.62/0.69 Closed-form bounds of start0(A,B,C,D,E,F,G,H,I): 0.62/0.69 ------------------------------------- 0.62/0.69 * Chain [39] with precondition: [A=1] 0.62/0.69 - Upper bound: 0 0.62/0.69 - Complexity: constant 0.62/0.69 * Chain [38] with precondition: [A>=1] 0.62/0.69 - Upper bound: 0 0.62/0.69 - Complexity: constant 0.62/0.69 * Chain [37] with precondition: [A>=2] 0.62/0.69 - Upper bound: 3*A*A+4*A+2*A*A*A 0.62/0.69 - Complexity: n^3 0.62/0.69 * Chain [36] with precondition: [A>=3] 0.62/0.69 - Upper bound: 3*A*A+5*A+A*A*A 0.62/0.69 - Complexity: n^3 0.62/0.69 * Chain [35] with precondition: [A>=4] 0.62/0.69 - Upper bound: 2*A*A+3*A+A*A*A 0.62/0.69 - Complexity: n^3 0.62/0.69 0.62/0.69 ### Maximum cost of start0(A,B,C,D,E,F,G,H,I): 2*A*A+3*A+A*A*A+(A*A+A+max([A,A*A*A])) 0.62/0.69 Asymptotic class: n^3 0.62/0.69 * Total analysis performed in 599 ms. 0.62/0.69 0.69/0.80 EOF