2.82/2.81 YES 2.82/2.81 2.82/2.81 DP problem for innermost termination. 2.82/2.81 P = 2.82/2.81 init#(x1, x2, x3, x4) -> f3#(rnd1, rnd2, rnd3, rnd4) 2.82/2.81 f7#(I0, I1, I2, I3) -> f7#(I0 - 1, I1 + 1, I4, I5) [0 <= I1 - 1 /\ 0 <= I0 - 1] 2.82/2.81 f3#(I6, I7, I8, I9) -> f7#(I10, 1, I11, I12) [0 <= I6 - 1 /\ -1 <= I10 - 1 /\ -1 <= I7 - 1] 2.82/2.81 f6#(I13, I14, I15, I16) -> f5#(I17, 0, I16, I18) [-1 <= I17 - 1 /\ -1 <= I14 - 1 /\ -1 <= I13 - 1 /\ 1 <= I15 - 1 /\ I17 <= I13] 2.82/2.81 f6#(I19, I20, I21, I22) -> f5#(I23, 0, I22, I24) [-1 <= I23 - 1 /\ -1 <= I20 - 1 /\ -1 <= I19 - 1 /\ I21 <= 1 /\ I23 <= I19] 2.82/2.81 f6#(I25, I26, I27, I28) -> f6#(I29, I30, I27 - 1, I28) [-1 <= I30 - 1 /\ -1 <= I29 - 1 /\ 0 <= I26 - 1 /\ -1 <= I25 - 1 /\ I30 + 1 <= I26 /\ 1 <= I27 - 1 /\ I29 <= I25] 2.82/2.81 f6#(I31, I32, I33, I34) -> f5#(I35, I36, I34, I37) [I36 + 2 <= I32 /\ -1 <= I35 - 1 /\ 0 <= I32 - 1 /\ -1 <= I31 - 1 /\ I33 <= 1 /\ I35 <= I31] 2.82/2.81 f5#(I38, I39, I40, I41) -> f6#(I42, I43, I44, I40 + 1) [-1 <= I43 - 1 /\ -1 <= I42 - 1 /\ 0 <= I38 - 1 /\ I43 + 1 <= I38 /\ I42 + 1 <= I38 /\ 0 <= I39 - 1 /\ 0 <= I40 - 1 /\ -1 <= I44 - 1] 2.82/2.81 f4#(I45, I46, I47, I48) -> f4#(I49, I50, I47 - 1, I48) [-1 <= I50 - 1 /\ -1 <= I49 - 1 /\ 0 <= I46 - 1 /\ -1 <= I45 - 1 /\ I50 + 1 <= I46 /\ 1 <= I47 - 1 /\ I49 <= I45] 2.82/2.81 f4#(I51, I52, I53, I54) -> f5#(I55, I56, I54, I57) [I56 + 2 <= I52 /\ -1 <= I55 - 1 /\ 0 <= I52 - 1 /\ -1 <= I51 - 1 /\ I53 <= 1 /\ I55 <= I51] 2.82/2.81 f2#(I58, I59, I60, I61) -> f4#(I62, I63, I64, I65) [0 <= y1 - 1 /\ -1 <= I64 - 1 /\ I62 <= I58 /\ I63 <= I58 /\ -1 <= I58 - 1 /\ -1 <= I62 - 1 /\ -1 <= I63 - 1 /\ y1 + 1 = I65] 2.82/2.81 f3#(I66, I67, I68, I69) -> f2#(I70, I71, I72, I73) [-1 <= I70 - 1 /\ 0 <= I66 - 1] 2.82/2.81 f1#(I74, I75, I76, I77) -> f2#(I78, I79, I80, I81) [-1 <= I78 - 1 /\ -1 <= I74 - 1 /\ I78 <= I74] 2.82/2.81 R = 2.82/2.81 init(x1, x2, x3, x4) -> f3(rnd1, rnd2, rnd3, rnd4) 2.82/2.81 f7(I0, I1, I2, I3) -> f7(I0 - 1, I1 + 1, I4, I5) [0 <= I1 - 1 /\ 0 <= I0 - 1] 2.82/2.81 f3(I6, I7, I8, I9) -> f7(I10, 1, I11, I12) [0 <= I6 - 1 /\ -1 <= I10 - 1 /\ -1 <= I7 - 1] 2.82/2.81 f6(I13, I14, I15, I16) -> f5(I17, 0, I16, I18) [-1 <= I17 - 1 /\ -1 <= I14 - 1 /\ -1 <= I13 - 1 /\ 1 <= I15 - 1 /\ I17 <= I13] 2.82/2.81 f6(I19, I20, I21, I22) -> f5(I23, 0, I22, I24) [-1 <= I23 - 1 /\ -1 <= I20 - 1 /\ -1 <= I19 - 1 /\ I21 <= 1 /\ I23 <= I19] 2.82/2.81 f6(I25, I26, I27, I28) -> f6(I29, I30, I27 - 1, I28) [-1 <= I30 - 1 /\ -1 <= I29 - 1 /\ 0 <= I26 - 1 /\ -1 <= I25 - 1 /\ I30 + 1 <= I26 /\ 1 <= I27 - 1 /\ I29 <= I25] 2.82/2.81 f6(I31, I32, I33, I34) -> f5(I35, I36, I34, I37) [I36 + 2 <= I32 /\ -1 <= I35 - 1 /\ 0 <= I32 - 1 /\ -1 <= I31 - 1 /\ I33 <= 1 /\ I35 <= I31] 2.82/2.81 f5(I38, I39, I40, I41) -> f6(I42, I43, I44, I40 + 1) [-1 <= I43 - 1 /\ -1 <= I42 - 1 /\ 0 <= I38 - 1 /\ I43 + 1 <= I38 /\ I42 + 1 <= I38 /\ 0 <= I39 - 1 /\ 0 <= I40 - 1 /\ -1 <= I44 - 1] 2.82/2.81 f4(I45, I46, I47, I48) -> f4(I49, I50, I47 - 1, I48) [-1 <= I50 - 1 /\ -1 <= I49 - 1 /\ 0 <= I46 - 1 /\ -1 <= I45 - 1 /\ I50 + 1 <= I46 /\ 1 <= I47 - 1 /\ I49 <= I45] 2.82/2.81 f4(I51, I52, I53, I54) -> f5(I55, I56, I54, I57) [I56 + 2 <= I52 /\ -1 <= I55 - 1 /\ 0 <= I52 - 1 /\ -1 <= I51 - 1 /\ I53 <= 1 /\ I55 <= I51] 2.82/2.81 f2(I58, I59, I60, I61) -> f4(I62, I63, I64, I65) [0 <= y1 - 1 /\ -1 <= I64 - 1 /\ I62 <= I58 /\ I63 <= I58 /\ -1 <= I58 - 1 /\ -1 <= I62 - 1 /\ -1 <= I63 - 1 /\ y1 + 1 = I65] 2.82/2.81 f3(I66, I67, I68, I69) -> f2(I70, I71, I72, I73) [-1 <= I70 - 1 /\ 0 <= I66 - 1] 2.82/2.81 f1(I74, I75, I76, I77) -> f2(I78, I79, I80, I81) [-1 <= I78 - 1 /\ -1 <= I74 - 1 /\ I78 <= I74] 2.82/2.81 2.82/2.81 The dependency graph for this problem is: 2.82/2.81 0 -> 2, 11 2.82/2.81 1 -> 1 2.82/2.81 2 -> 1 2.82/2.81 3 -> 2.82/2.81 4 -> 2.82/2.81 5 -> 3, 4, 5, 6 2.82/2.81 6 -> 7 2.82/2.81 7 -> 3, 4, 5, 6 2.82/2.81 8 -> 8, 9 2.82/2.81 9 -> 7 2.82/2.81 10 -> 8, 9 2.82/2.81 11 -> 10 2.82/2.81 12 -> 10 2.82/2.81 Where: 2.82/2.81 0) init#(x1, x2, x3, x4) -> f3#(rnd1, rnd2, rnd3, rnd4) 2.82/2.81 1) f7#(I0, I1, I2, I3) -> f7#(I0 - 1, I1 + 1, I4, I5) [0 <= I1 - 1 /\ 0 <= I0 - 1] 2.82/2.81 2) f3#(I6, I7, I8, I9) -> f7#(I10, 1, I11, I12) [0 <= I6 - 1 /\ -1 <= I10 - 1 /\ -1 <= I7 - 1] 2.82/2.81 3) f6#(I13, I14, I15, I16) -> f5#(I17, 0, I16, I18) [-1 <= I17 - 1 /\ -1 <= I14 - 1 /\ -1 <= I13 - 1 /\ 1 <= I15 - 1 /\ I17 <= I13] 2.82/2.81 4) f6#(I19, I20, I21, I22) -> f5#(I23, 0, I22, I24) [-1 <= I23 - 1 /\ -1 <= I20 - 1 /\ -1 <= I19 - 1 /\ I21 <= 1 /\ I23 <= I19] 2.82/2.81 5) f6#(I25, I26, I27, I28) -> f6#(I29, I30, I27 - 1, I28) [-1 <= I30 - 1 /\ -1 <= I29 - 1 /\ 0 <= I26 - 1 /\ -1 <= I25 - 1 /\ I30 + 1 <= I26 /\ 1 <= I27 - 1 /\ I29 <= I25] 2.82/2.81 6) f6#(I31, I32, I33, I34) -> f5#(I35, I36, I34, I37) [I36 + 2 <= I32 /\ -1 <= I35 - 1 /\ 0 <= I32 - 1 /\ -1 <= I31 - 1 /\ I33 <= 1 /\ I35 <= I31] 2.82/2.81 7) f5#(I38, I39, I40, I41) -> f6#(I42, I43, I44, I40 + 1) [-1 <= I43 - 1 /\ -1 <= I42 - 1 /\ 0 <= I38 - 1 /\ I43 + 1 <= I38 /\ I42 + 1 <= I38 /\ 0 <= I39 - 1 /\ 0 <= I40 - 1 /\ -1 <= I44 - 1] 2.82/2.81 8) f4#(I45, I46, I47, I48) -> f4#(I49, I50, I47 - 1, I48) [-1 <= I50 - 1 /\ -1 <= I49 - 1 /\ 0 <= I46 - 1 /\ -1 <= I45 - 1 /\ I50 + 1 <= I46 /\ 1 <= I47 - 1 /\ I49 <= I45] 2.82/2.81 9) f4#(I51, I52, I53, I54) -> f5#(I55, I56, I54, I57) [I56 + 2 <= I52 /\ -1 <= I55 - 1 /\ 0 <= I52 - 1 /\ -1 <= I51 - 1 /\ I53 <= 1 /\ I55 <= I51] 2.82/2.81 10) f2#(I58, I59, I60, I61) -> f4#(I62, I63, I64, I65) [0 <= y1 - 1 /\ -1 <= I64 - 1 /\ I62 <= I58 /\ I63 <= I58 /\ -1 <= I58 - 1 /\ -1 <= I62 - 1 /\ -1 <= I63 - 1 /\ y1 + 1 = I65] 2.82/2.81 11) f3#(I66, I67, I68, I69) -> f2#(I70, I71, I72, I73) [-1 <= I70 - 1 /\ 0 <= I66 - 1] 2.82/2.81 12) f1#(I74, I75, I76, I77) -> f2#(I78, I79, I80, I81) [-1 <= I78 - 1 /\ -1 <= I74 - 1 /\ I78 <= I74] 2.82/2.81 2.82/2.81 We have the following SCCs. 2.82/2.81 { 1 } 2.82/2.81 { 8 } 2.82/2.81 { 5, 6, 7 } 2.82/2.81 2.82/2.81 DP problem for innermost termination. 2.82/2.81 P = 2.82/2.81 f6#(I25, I26, I27, I28) -> f6#(I29, I30, I27 - 1, I28) [-1 <= I30 - 1 /\ -1 <= I29 - 1 /\ 0 <= I26 - 1 /\ -1 <= I25 - 1 /\ I30 + 1 <= I26 /\ 1 <= I27 - 1 /\ I29 <= I25] 2.82/2.81 f6#(I31, I32, I33, I34) -> f5#(I35, I36, I34, I37) [I36 + 2 <= I32 /\ -1 <= I35 - 1 /\ 0 <= I32 - 1 /\ -1 <= I31 - 1 /\ I33 <= 1 /\ I35 <= I31] 2.82/2.81 f5#(I38, I39, I40, I41) -> f6#(I42, I43, I44, I40 + 1) [-1 <= I43 - 1 /\ -1 <= I42 - 1 /\ 0 <= I38 - 1 /\ I43 + 1 <= I38 /\ I42 + 1 <= I38 /\ 0 <= I39 - 1 /\ 0 <= I40 - 1 /\ -1 <= I44 - 1] 2.82/2.81 R = 2.82/2.81 init(x1, x2, x3, x4) -> f3(rnd1, rnd2, rnd3, rnd4) 2.82/2.81 f7(I0, I1, I2, I3) -> f7(I0 - 1, I1 + 1, I4, I5) [0 <= I1 - 1 /\ 0 <= I0 - 1] 2.82/2.81 f3(I6, I7, I8, I9) -> f7(I10, 1, I11, I12) [0 <= I6 - 1 /\ -1 <= I10 - 1 /\ -1 <= I7 - 1] 2.82/2.81 f6(I13, I14, I15, I16) -> f5(I17, 0, I16, I18) [-1 <= I17 - 1 /\ -1 <= I14 - 1 /\ -1 <= I13 - 1 /\ 1 <= I15 - 1 /\ I17 <= I13] 2.82/2.81 f6(I19, I20, I21, I22) -> f5(I23, 0, I22, I24) [-1 <= I23 - 1 /\ -1 <= I20 - 1 /\ -1 <= I19 - 1 /\ I21 <= 1 /\ I23 <= I19] 2.82/2.81 f6(I25, I26, I27, I28) -> f6(I29, I30, I27 - 1, I28) [-1 <= I30 - 1 /\ -1 <= I29 - 1 /\ 0 <= I26 - 1 /\ -1 <= I25 - 1 /\ I30 + 1 <= I26 /\ 1 <= I27 - 1 /\ I29 <= I25] 2.82/2.81 f6(I31, I32, I33, I34) -> f5(I35, I36, I34, I37) [I36 + 2 <= I32 /\ -1 <= I35 - 1 /\ 0 <= I32 - 1 /\ -1 <= I31 - 1 /\ I33 <= 1 /\ I35 <= I31] 2.82/2.81 f5(I38, I39, I40, I41) -> f6(I42, I43, I44, I40 + 1) [-1 <= I43 - 1 /\ -1 <= I42 - 1 /\ 0 <= I38 - 1 /\ I43 + 1 <= I38 /\ I42 + 1 <= I38 /\ 0 <= I39 - 1 /\ 0 <= I40 - 1 /\ -1 <= I44 - 1] 2.82/2.81 f4(I45, I46, I47, I48) -> f4(I49, I50, I47 - 1, I48) [-1 <= I50 - 1 /\ -1 <= I49 - 1 /\ 0 <= I46 - 1 /\ -1 <= I45 - 1 /\ I50 + 1 <= I46 /\ 1 <= I47 - 1 /\ I49 <= I45] 2.82/2.81 f4(I51, I52, I53, I54) -> f5(I55, I56, I54, I57) [I56 + 2 <= I52 /\ -1 <= I55 - 1 /\ 0 <= I52 - 1 /\ -1 <= I51 - 1 /\ I53 <= 1 /\ I55 <= I51] 2.82/2.81 f2(I58, I59, I60, I61) -> f4(I62, I63, I64, I65) [0 <= y1 - 1 /\ -1 <= I64 - 1 /\ I62 <= I58 /\ I63 <= I58 /\ -1 <= I58 - 1 /\ -1 <= I62 - 1 /\ -1 <= I63 - 1 /\ y1 + 1 = I65] 2.82/2.81 f3(I66, I67, I68, I69) -> f2(I70, I71, I72, I73) [-1 <= I70 - 1 /\ 0 <= I66 - 1] 2.82/2.81 f1(I74, I75, I76, I77) -> f2(I78, I79, I80, I81) [-1 <= I78 - 1 /\ -1 <= I74 - 1 /\ I78 <= I74] 2.82/2.81 2.82/2.81 We use the basic value criterion with the projection function NU: 2.82/2.81 NU[f5#(z1,z2,z3,z4)] = z1 2.82/2.81 NU[f6#(z1,z2,z3,z4)] = z1 2.82/2.81 2.82/2.81 This gives the following inequalities: 2.82/2.81 -1 <= I30 - 1 /\ -1 <= I29 - 1 /\ 0 <= I26 - 1 /\ -1 <= I25 - 1 /\ I30 + 1 <= I26 /\ 1 <= I27 - 1 /\ I29 <= I25 ==> I25 (>! \union =) I29 2.82/2.81 I36 + 2 <= I32 /\ -1 <= I35 - 1 /\ 0 <= I32 - 1 /\ -1 <= I31 - 1 /\ I33 <= 1 /\ I35 <= I31 ==> I31 (>! \union =) I35 2.82/2.81 -1 <= I43 - 1 /\ -1 <= I42 - 1 /\ 0 <= I38 - 1 /\ I43 + 1 <= I38 /\ I42 + 1 <= I38 /\ 0 <= I39 - 1 /\ 0 <= I40 - 1 /\ -1 <= I44 - 1 ==> I38 >! I42 2.82/2.81 2.82/2.81 We remove all the strictly oriented dependency pairs. 2.82/2.81 2.82/2.81 DP problem for innermost termination. 2.82/2.81 P = 2.82/2.81 f6#(I25, I26, I27, I28) -> f6#(I29, I30, I27 - 1, I28) [-1 <= I30 - 1 /\ -1 <= I29 - 1 /\ 0 <= I26 - 1 /\ -1 <= I25 - 1 /\ I30 + 1 <= I26 /\ 1 <= I27 - 1 /\ I29 <= I25] 2.82/2.81 f6#(I31, I32, I33, I34) -> f5#(I35, I36, I34, I37) [I36 + 2 <= I32 /\ -1 <= I35 - 1 /\ 0 <= I32 - 1 /\ -1 <= I31 - 1 /\ I33 <= 1 /\ I35 <= I31] 2.82/2.81 R = 2.82/2.81 init(x1, x2, x3, x4) -> f3(rnd1, rnd2, rnd3, rnd4) 2.82/2.81 f7(I0, I1, I2, I3) -> f7(I0 - 1, I1 + 1, I4, I5) [0 <= I1 - 1 /\ 0 <= I0 - 1] 2.82/2.81 f3(I6, I7, I8, I9) -> f7(I10, 1, I11, I12) [0 <= I6 - 1 /\ -1 <= I10 - 1 /\ -1 <= I7 - 1] 2.82/2.81 f6(I13, I14, I15, I16) -> f5(I17, 0, I16, I18) [-1 <= I17 - 1 /\ -1 <= I14 - 1 /\ -1 <= I13 - 1 /\ 1 <= I15 - 1 /\ I17 <= I13] 2.82/2.81 f6(I19, I20, I21, I22) -> f5(I23, 0, I22, I24) [-1 <= I23 - 1 /\ -1 <= I20 - 1 /\ -1 <= I19 - 1 /\ I21 <= 1 /\ I23 <= I19] 2.82/2.81 f6(I25, I26, I27, I28) -> f6(I29, I30, I27 - 1, I28) [-1 <= I30 - 1 /\ -1 <= I29 - 1 /\ 0 <= I26 - 1 /\ -1 <= I25 - 1 /\ I30 + 1 <= I26 /\ 1 <= I27 - 1 /\ I29 <= I25] 2.82/2.81 f6(I31, I32, I33, I34) -> f5(I35, I36, I34, I37) [I36 + 2 <= I32 /\ -1 <= I35 - 1 /\ 0 <= I32 - 1 /\ -1 <= I31 - 1 /\ I33 <= 1 /\ I35 <= I31] 2.82/2.81 f5(I38, I39, I40, I41) -> f6(I42, I43, I44, I40 + 1) [-1 <= I43 - 1 /\ -1 <= I42 - 1 /\ 0 <= I38 - 1 /\ I43 + 1 <= I38 /\ I42 + 1 <= I38 /\ 0 <= I39 - 1 /\ 0 <= I40 - 1 /\ -1 <= I44 - 1] 2.82/2.81 f4(I45, I46, I47, I48) -> f4(I49, I50, I47 - 1, I48) [-1 <= I50 - 1 /\ -1 <= I49 - 1 /\ 0 <= I46 - 1 /\ -1 <= I45 - 1 /\ I50 + 1 <= I46 /\ 1 <= I47 - 1 /\ I49 <= I45] 2.82/2.81 f4(I51, I52, I53, I54) -> f5(I55, I56, I54, I57) [I56 + 2 <= I52 /\ -1 <= I55 - 1 /\ 0 <= I52 - 1 /\ -1 <= I51 - 1 /\ I53 <= 1 /\ I55 <= I51] 2.82/2.81 f2(I58, I59, I60, I61) -> f4(I62, I63, I64, I65) [0 <= y1 - 1 /\ -1 <= I64 - 1 /\ I62 <= I58 /\ I63 <= I58 /\ -1 <= I58 - 1 /\ -1 <= I62 - 1 /\ -1 <= I63 - 1 /\ y1 + 1 = I65] 2.82/2.81 f3(I66, I67, I68, I69) -> f2(I70, I71, I72, I73) [-1 <= I70 - 1 /\ 0 <= I66 - 1] 2.82/2.81 f1(I74, I75, I76, I77) -> f2(I78, I79, I80, I81) [-1 <= I78 - 1 /\ -1 <= I74 - 1 /\ I78 <= I74] 2.82/2.81 2.82/2.81 The dependency graph for this problem is: 2.82/2.81 5 -> 5, 6 2.82/2.81 6 -> 2.82/2.81 Where: 2.82/2.81 5) f6#(I25, I26, I27, I28) -> f6#(I29, I30, I27 - 1, I28) [-1 <= I30 - 1 /\ -1 <= I29 - 1 /\ 0 <= I26 - 1 /\ -1 <= I25 - 1 /\ I30 + 1 <= I26 /\ 1 <= I27 - 1 /\ I29 <= I25] 2.82/2.81 6) f6#(I31, I32, I33, I34) -> f5#(I35, I36, I34, I37) [I36 + 2 <= I32 /\ -1 <= I35 - 1 /\ 0 <= I32 - 1 /\ -1 <= I31 - 1 /\ I33 <= 1 /\ I35 <= I31] 2.82/2.81 2.82/2.81 We have the following SCCs. 2.82/2.81 { 5 } 2.82/2.81 2.82/2.81 DP problem for innermost termination. 2.82/2.81 P = 2.82/2.81 f6#(I25, I26, I27, I28) -> f6#(I29, I30, I27 - 1, I28) [-1 <= I30 - 1 /\ -1 <= I29 - 1 /\ 0 <= I26 - 1 /\ -1 <= I25 - 1 /\ I30 + 1 <= I26 /\ 1 <= I27 - 1 /\ I29 <= I25] 2.82/2.81 R = 2.82/2.81 init(x1, x2, x3, x4) -> f3(rnd1, rnd2, rnd3, rnd4) 2.82/2.81 f7(I0, I1, I2, I3) -> f7(I0 - 1, I1 + 1, I4, I5) [0 <= I1 - 1 /\ 0 <= I0 - 1] 2.82/2.81 f3(I6, I7, I8, I9) -> f7(I10, 1, I11, I12) [0 <= I6 - 1 /\ -1 <= I10 - 1 /\ -1 <= I7 - 1] 2.82/2.81 f6(I13, I14, I15, I16) -> f5(I17, 0, I16, I18) [-1 <= I17 - 1 /\ -1 <= I14 - 1 /\ -1 <= I13 - 1 /\ 1 <= I15 - 1 /\ I17 <= I13] 2.82/2.81 f6(I19, I20, I21, I22) -> f5(I23, 0, I22, I24) [-1 <= I23 - 1 /\ -1 <= I20 - 1 /\ -1 <= I19 - 1 /\ I21 <= 1 /\ I23 <= I19] 2.82/2.81 f6(I25, I26, I27, I28) -> f6(I29, I30, I27 - 1, I28) [-1 <= I30 - 1 /\ -1 <= I29 - 1 /\ 0 <= I26 - 1 /\ -1 <= I25 - 1 /\ I30 + 1 <= I26 /\ 1 <= I27 - 1 /\ I29 <= I25] 2.82/2.81 f6(I31, I32, I33, I34) -> f5(I35, I36, I34, I37) [I36 + 2 <= I32 /\ -1 <= I35 - 1 /\ 0 <= I32 - 1 /\ -1 <= I31 - 1 /\ I33 <= 1 /\ I35 <= I31] 2.82/2.81 f5(I38, I39, I40, I41) -> f6(I42, I43, I44, I40 + 1) [-1 <= I43 - 1 /\ -1 <= I42 - 1 /\ 0 <= I38 - 1 /\ I43 + 1 <= I38 /\ I42 + 1 <= I38 /\ 0 <= I39 - 1 /\ 0 <= I40 - 1 /\ -1 <= I44 - 1] 2.82/2.81 f4(I45, I46, I47, I48) -> f4(I49, I50, I47 - 1, I48) [-1 <= I50 - 1 /\ -1 <= I49 - 1 /\ 0 <= I46 - 1 /\ -1 <= I45 - 1 /\ I50 + 1 <= I46 /\ 1 <= I47 - 1 /\ I49 <= I45] 2.82/2.81 f4(I51, I52, I53, I54) -> f5(I55, I56, I54, I57) [I56 + 2 <= I52 /\ -1 <= I55 - 1 /\ 0 <= I52 - 1 /\ -1 <= I51 - 1 /\ I53 <= 1 /\ I55 <= I51] 2.82/2.81 f2(I58, I59, I60, I61) -> f4(I62, I63, I64, I65) [0 <= y1 - 1 /\ -1 <= I64 - 1 /\ I62 <= I58 /\ I63 <= I58 /\ -1 <= I58 - 1 /\ -1 <= I62 - 1 /\ -1 <= I63 - 1 /\ y1 + 1 = I65] 2.82/2.81 f3(I66, I67, I68, I69) -> f2(I70, I71, I72, I73) [-1 <= I70 - 1 /\ 0 <= I66 - 1] 2.82/2.81 f1(I74, I75, I76, I77) -> f2(I78, I79, I80, I81) [-1 <= I78 - 1 /\ -1 <= I74 - 1 /\ I78 <= I74] 2.82/2.81 2.82/2.81 We use the basic value criterion with the projection function NU: 2.82/2.81 NU[f6#(z1,z2,z3,z4)] = z3 2.82/2.81 2.82/2.81 This gives the following inequalities: 2.82/2.81 -1 <= I30 - 1 /\ -1 <= I29 - 1 /\ 0 <= I26 - 1 /\ -1 <= I25 - 1 /\ I30 + 1 <= I26 /\ 1 <= I27 - 1 /\ I29 <= I25 ==> I27 >! I27 - 1 2.82/2.81 2.82/2.81 All dependency pairs are strictly oriented, so the entire dependency pair problem may be removed. 2.82/2.81 2.82/2.81 DP problem for innermost termination. 2.82/2.81 P = 2.82/2.81 f4#(I45, I46, I47, I48) -> f4#(I49, I50, I47 - 1, I48) [-1 <= I50 - 1 /\ -1 <= I49 - 1 /\ 0 <= I46 - 1 /\ -1 <= I45 - 1 /\ I50 + 1 <= I46 /\ 1 <= I47 - 1 /\ I49 <= I45] 2.82/2.81 R = 2.82/2.81 init(x1, x2, x3, x4) -> f3(rnd1, rnd2, rnd3, rnd4) 2.82/2.81 f7(I0, I1, I2, I3) -> f7(I0 - 1, I1 + 1, I4, I5) [0 <= I1 - 1 /\ 0 <= I0 - 1] 2.82/2.81 f3(I6, I7, I8, I9) -> f7(I10, 1, I11, I12) [0 <= I6 - 1 /\ -1 <= I10 - 1 /\ -1 <= I7 - 1] 2.82/2.81 f6(I13, I14, I15, I16) -> f5(I17, 0, I16, I18) [-1 <= I17 - 1 /\ -1 <= I14 - 1 /\ -1 <= I13 - 1 /\ 1 <= I15 - 1 /\ I17 <= I13] 2.82/2.81 f6(I19, I20, I21, I22) -> f5(I23, 0, I22, I24) [-1 <= I23 - 1 /\ -1 <= I20 - 1 /\ -1 <= I19 - 1 /\ I21 <= 1 /\ I23 <= I19] 2.82/2.81 f6(I25, I26, I27, I28) -> f6(I29, I30, I27 - 1, I28) [-1 <= I30 - 1 /\ -1 <= I29 - 1 /\ 0 <= I26 - 1 /\ -1 <= I25 - 1 /\ I30 + 1 <= I26 /\ 1 <= I27 - 1 /\ I29 <= I25] 2.82/2.81 f6(I31, I32, I33, I34) -> f5(I35, I36, I34, I37) [I36 + 2 <= I32 /\ -1 <= I35 - 1 /\ 0 <= I32 - 1 /\ -1 <= I31 - 1 /\ I33 <= 1 /\ I35 <= I31] 2.82/2.81 f5(I38, I39, I40, I41) -> f6(I42, I43, I44, I40 + 1) [-1 <= I43 - 1 /\ -1 <= I42 - 1 /\ 0 <= I38 - 1 /\ I43 + 1 <= I38 /\ I42 + 1 <= I38 /\ 0 <= I39 - 1 /\ 0 <= I40 - 1 /\ -1 <= I44 - 1] 2.82/2.81 f4(I45, I46, I47, I48) -> f4(I49, I50, I47 - 1, I48) [-1 <= I50 - 1 /\ -1 <= I49 - 1 /\ 0 <= I46 - 1 /\ -1 <= I45 - 1 /\ I50 + 1 <= I46 /\ 1 <= I47 - 1 /\ I49 <= I45] 2.82/2.81 f4(I51, I52, I53, I54) -> f5(I55, I56, I54, I57) [I56 + 2 <= I52 /\ -1 <= I55 - 1 /\ 0 <= I52 - 1 /\ -1 <= I51 - 1 /\ I53 <= 1 /\ I55 <= I51] 2.82/2.81 f2(I58, I59, I60, I61) -> f4(I62, I63, I64, I65) [0 <= y1 - 1 /\ -1 <= I64 - 1 /\ I62 <= I58 /\ I63 <= I58 /\ -1 <= I58 - 1 /\ -1 <= I62 - 1 /\ -1 <= I63 - 1 /\ y1 + 1 = I65] 2.82/2.81 f3(I66, I67, I68, I69) -> f2(I70, I71, I72, I73) [-1 <= I70 - 1 /\ 0 <= I66 - 1] 2.82/2.81 f1(I74, I75, I76, I77) -> f2(I78, I79, I80, I81) [-1 <= I78 - 1 /\ -1 <= I74 - 1 /\ I78 <= I74] 2.82/2.81 2.82/2.81 We use the basic value criterion with the projection function NU: 2.82/2.81 NU[f4#(z1,z2,z3,z4)] = z3 2.82/2.81 2.82/2.81 This gives the following inequalities: 2.82/2.81 -1 <= I50 - 1 /\ -1 <= I49 - 1 /\ 0 <= I46 - 1 /\ -1 <= I45 - 1 /\ I50 + 1 <= I46 /\ 1 <= I47 - 1 /\ I49 <= I45 ==> I47 >! I47 - 1 2.82/2.81 2.82/2.81 All dependency pairs are strictly oriented, so the entire dependency pair problem may be removed. 2.82/2.81 2.82/2.81 DP problem for innermost termination. 2.82/2.81 P = 2.82/2.81 f7#(I0, I1, I2, I3) -> f7#(I0 - 1, I1 + 1, I4, I5) [0 <= I1 - 1 /\ 0 <= I0 - 1] 2.82/2.81 R = 2.82/2.81 init(x1, x2, x3, x4) -> f3(rnd1, rnd2, rnd3, rnd4) 2.82/2.81 f7(I0, I1, I2, I3) -> f7(I0 - 1, I1 + 1, I4, I5) [0 <= I1 - 1 /\ 0 <= I0 - 1] 2.82/2.81 f3(I6, I7, I8, I9) -> f7(I10, 1, I11, I12) [0 <= I6 - 1 /\ -1 <= I10 - 1 /\ -1 <= I7 - 1] 2.82/2.81 f6(I13, I14, I15, I16) -> f5(I17, 0, I16, I18) [-1 <= I17 - 1 /\ -1 <= I14 - 1 /\ -1 <= I13 - 1 /\ 1 <= I15 - 1 /\ I17 <= I13] 2.82/2.81 f6(I19, I20, I21, I22) -> f5(I23, 0, I22, I24) [-1 <= I23 - 1 /\ -1 <= I20 - 1 /\ -1 <= I19 - 1 /\ I21 <= 1 /\ I23 <= I19] 2.82/2.81 f6(I25, I26, I27, I28) -> f6(I29, I30, I27 - 1, I28) [-1 <= I30 - 1 /\ -1 <= I29 - 1 /\ 0 <= I26 - 1 /\ -1 <= I25 - 1 /\ I30 + 1 <= I26 /\ 1 <= I27 - 1 /\ I29 <= I25] 2.82/2.81 f6(I31, I32, I33, I34) -> f5(I35, I36, I34, I37) [I36 + 2 <= I32 /\ -1 <= I35 - 1 /\ 0 <= I32 - 1 /\ -1 <= I31 - 1 /\ I33 <= 1 /\ I35 <= I31] 2.82/2.81 f5(I38, I39, I40, I41) -> f6(I42, I43, I44, I40 + 1) [-1 <= I43 - 1 /\ -1 <= I42 - 1 /\ 0 <= I38 - 1 /\ I43 + 1 <= I38 /\ I42 + 1 <= I38 /\ 0 <= I39 - 1 /\ 0 <= I40 - 1 /\ -1 <= I44 - 1] 2.82/2.81 f4(I45, I46, I47, I48) -> f4(I49, I50, I47 - 1, I48) [-1 <= I50 - 1 /\ -1 <= I49 - 1 /\ 0 <= I46 - 1 /\ -1 <= I45 - 1 /\ I50 + 1 <= I46 /\ 1 <= I47 - 1 /\ I49 <= I45] 2.82/2.81 f4(I51, I52, I53, I54) -> f5(I55, I56, I54, I57) [I56 + 2 <= I52 /\ -1 <= I55 - 1 /\ 0 <= I52 - 1 /\ -1 <= I51 - 1 /\ I53 <= 1 /\ I55 <= I51] 2.82/2.81 f2(I58, I59, I60, I61) -> f4(I62, I63, I64, I65) [0 <= y1 - 1 /\ -1 <= I64 - 1 /\ I62 <= I58 /\ I63 <= I58 /\ -1 <= I58 - 1 /\ -1 <= I62 - 1 /\ -1 <= I63 - 1 /\ y1 + 1 = I65] 2.82/2.81 f3(I66, I67, I68, I69) -> f2(I70, I71, I72, I73) [-1 <= I70 - 1 /\ 0 <= I66 - 1] 2.82/2.81 f1(I74, I75, I76, I77) -> f2(I78, I79, I80, I81) [-1 <= I78 - 1 /\ -1 <= I74 - 1 /\ I78 <= I74] 2.82/2.81 2.82/2.81 We use the basic value criterion with the projection function NU: 2.82/2.81 NU[f7#(z1,z2,z3,z4)] = z1 2.82/2.81 2.82/2.81 This gives the following inequalities: 2.82/2.81 0 <= I1 - 1 /\ 0 <= I0 - 1 ==> I0 >! I0 - 1 2.82/2.81 2.82/2.81 All dependency pairs are strictly oriented, so the entire dependency pair problem may be removed. 2.82/5.80 EOF