4.14/4.13 YES 4.14/4.13 4.14/4.13 DP problem for innermost termination. 4.14/4.13 P = 4.14/4.13 init#(x1, x2, x3, x4) -> f1#(rnd1, rnd2, rnd3, rnd4) 4.14/4.13 f5#(I0, I1, I2, I3) -> f4#(I4, I5, I6, I7) [-1 <= y1 - 1 /\ y2 <= y1 - 1 /\ I4 - 2 <= I0 /\ I4 <= I1 /\ I5 <= I0 /\ 0 <= I0 - 1 /\ 2 <= I1 - 1 /\ 2 <= I4 - 1 /\ 0 <= I5 - 1] 4.14/4.13 f5#(I8, I9, I10, I11) -> f6#(I12, I13, I14, I15) [0 <= I16 - 1 /\ -1 <= I17 - 1 /\ I16 <= I17 - 1 /\ I14 - 2 <= I8 /\ 0 <= I8 - 1 /\ 2 <= I9 - 1 /\ -1 <= I12 - 1 /\ 0 <= I13 - 1 /\ 2 <= I14 - 1 /\ -1 <= I15 - 1] 4.14/4.13 f5#(I18, I19, I20, I21) -> f5#(I22, I23, I20, I24) [0 <= I23 - 1 /\ 2 <= I22 - 1 /\ 2 <= I19 - 1 /\ 0 <= I18 - 1 /\ I22 - 2 <= I18 /\ I24 <= I21 - 1 /\ 0 <= I21 - 1 /\ 0 <= I20 - 1] 4.14/4.13 f6#(I25, I26, I27, I28) -> f4#(I29, I30, I31, I32) [-1 <= I30 - 1 /\ 2 <= I29 - 1 /\ -1 <= I28 - 1 /\ 0 <= I27 - 1 /\ 0 <= I26 - 1 /\ -1 <= I25 - 1 /\ I30 <= I28 /\ I30 + 1 <= I27 /\ I30 + 1 <= I26 /\ I30 <= I25 /\ I29 - 3 <= I28 /\ I29 - 2 <= I27 /\ I29 - 2 <= I26 /\ I29 - 3 <= I25] 4.14/4.13 f6#(I33, I34, I35, I36) -> f4#(I37, I38, I39, I40) [-1 <= I38 - 1 /\ 2 <= I37 - 1 /\ -1 <= I36 - 1 /\ 0 <= I35 - 1 /\ 0 <= I34 - 1 /\ -1 <= I33 - 1 /\ I38 <= I36 /\ I38 + 1 <= I34 /\ I38 <= I33] 4.14/4.13 f4#(I41, I42, I43, I44) -> f6#(I45, I46, I47, I48) [-1 <= I48 - 1 /\ 0 <= I47 - 1 /\ 0 <= I46 - 1 /\ -1 <= I45 - 1 /\ 0 <= I42 - 1 /\ 0 <= I41 - 1 /\ I48 + 1 <= I42 /\ I47 <= I41 /\ I46 <= I42 /\ I45 + 1 <= I42] 4.14/4.13 f3#(I49, I50, I51, I52) -> f5#(I53, I54, I55, I56) [0 <= I54 - 1 /\ 1 <= I53 - 1 /\ 2 <= I49 - 1 /\ I50 <= 0 /\ I53 + 1 <= I49] 4.14/4.13 f3#(I57, I58, I59, I60) -> f4#(I61, I62, I63, I64) [0 <= I62 - 1 /\ 1 <= I61 - 1 /\ 2 <= I57 - 1 /\ I58 <= 0 /\ I61 + 1 <= I57] 4.14/4.13 f3#(I65, I66, I67, I68) -> f4#(I69, I70, I71, I72) [-1 <= I70 - 1 /\ 1 <= I69 - 1 /\ 0 <= I65 - 1 /\ I66 <= 0 /\ I69 - 1 <= I65] 4.14/4.13 f2#(I73, I74, I75, I76) -> f4#(I77, I78, I79, I80) [1 <= I78 - 1 /\ 1 <= I77 - 1 /\ 1 <= I75 - 1 /\ 1 <= I74 - 1 /\ I78 <= I75 /\ I78 <= I74 /\ I77 <= I75 /\ I77 <= I74 /\ I76 <= 0 /\ I73 <= 1] 4.14/4.13 f2#(I81, I82, I83, I84) -> f3#(I85, I81 - 2, I86, I87) [4 <= I85 - 1 /\ 1 <= I83 - 1 /\ 1 <= I82 - 1 /\ I85 - 3 <= I83 /\ I85 - 3 <= I82 /\ I84 <= 0 /\ 1 <= I81 - 1] 4.14/4.13 f3#(I88, I89, I90, I91) -> f3#(I92, I89 - 1, I93, I94) [2 <= I92 - 1 /\ 0 <= I89 - 1 /\ 0 <= I88 - 1] 4.14/4.13 f2#(I95, I96, I97, I98) -> f2#(I95, I99, I100, I98 - 1) [2 <= I100 - 1 /\ 0 <= I99 - 1 /\ 0 <= I97 - 1 /\ 0 <= I96 - 1 /\ I100 - 2 <= I97 /\ 0 <= I98 - 1 /\ I99 <= I96] 4.14/4.13 f2#(I101, I102, I103, I104) -> f3#(I105, I101 - 1, I106, I107) [0 <= I105 - 1 /\ 0 <= I103 - 1 /\ I104 <= 0 /\ 1 <= I102 - 1] 4.14/4.13 f1#(I108, I109, I110, I111) -> f2#(I112, I113, I114, I115) [-1 <= I112 - 1 /\ 1 <= I109 - 1 /\ -1 <= I116 - 1 /\ I113 - 1 <= I108 /\ I114 - 1 <= I108 /\ 0 <= I108 - 1 /\ 1 <= I113 - 1 /\ 1 <= I114 - 1 /\ I116 - 2 = I115] 4.14/4.13 R = 4.14/4.13 init(x1, x2, x3, x4) -> f1(rnd1, rnd2, rnd3, rnd4) 4.14/4.13 f5(I0, I1, I2, I3) -> f4(I4, I5, I6, I7) [-1 <= y1 - 1 /\ y2 <= y1 - 1 /\ I4 - 2 <= I0 /\ I4 <= I1 /\ I5 <= I0 /\ 0 <= I0 - 1 /\ 2 <= I1 - 1 /\ 2 <= I4 - 1 /\ 0 <= I5 - 1] 4.14/4.13 f5(I8, I9, I10, I11) -> f6(I12, I13, I14, I15) [0 <= I16 - 1 /\ -1 <= I17 - 1 /\ I16 <= I17 - 1 /\ I14 - 2 <= I8 /\ 0 <= I8 - 1 /\ 2 <= I9 - 1 /\ -1 <= I12 - 1 /\ 0 <= I13 - 1 /\ 2 <= I14 - 1 /\ -1 <= I15 - 1] 4.14/4.13 f5(I18, I19, I20, I21) -> f5(I22, I23, I20, I24) [0 <= I23 - 1 /\ 2 <= I22 - 1 /\ 2 <= I19 - 1 /\ 0 <= I18 - 1 /\ I22 - 2 <= I18 /\ I24 <= I21 - 1 /\ 0 <= I21 - 1 /\ 0 <= I20 - 1] 4.14/4.13 f6(I25, I26, I27, I28) -> f4(I29, I30, I31, I32) [-1 <= I30 - 1 /\ 2 <= I29 - 1 /\ -1 <= I28 - 1 /\ 0 <= I27 - 1 /\ 0 <= I26 - 1 /\ -1 <= I25 - 1 /\ I30 <= I28 /\ I30 + 1 <= I27 /\ I30 + 1 <= I26 /\ I30 <= I25 /\ I29 - 3 <= I28 /\ I29 - 2 <= I27 /\ I29 - 2 <= I26 /\ I29 - 3 <= I25] 4.14/4.13 f6(I33, I34, I35, I36) -> f4(I37, I38, I39, I40) [-1 <= I38 - 1 /\ 2 <= I37 - 1 /\ -1 <= I36 - 1 /\ 0 <= I35 - 1 /\ 0 <= I34 - 1 /\ -1 <= I33 - 1 /\ I38 <= I36 /\ I38 + 1 <= I34 /\ I38 <= I33] 4.14/4.13 f4(I41, I42, I43, I44) -> f6(I45, I46, I47, I48) [-1 <= I48 - 1 /\ 0 <= I47 - 1 /\ 0 <= I46 - 1 /\ -1 <= I45 - 1 /\ 0 <= I42 - 1 /\ 0 <= I41 - 1 /\ I48 + 1 <= I42 /\ I47 <= I41 /\ I46 <= I42 /\ I45 + 1 <= I42] 4.14/4.13 f3(I49, I50, I51, I52) -> f5(I53, I54, I55, I56) [0 <= I54 - 1 /\ 1 <= I53 - 1 /\ 2 <= I49 - 1 /\ I50 <= 0 /\ I53 + 1 <= I49] 4.14/4.13 f3(I57, I58, I59, I60) -> f4(I61, I62, I63, I64) [0 <= I62 - 1 /\ 1 <= I61 - 1 /\ 2 <= I57 - 1 /\ I58 <= 0 /\ I61 + 1 <= I57] 4.14/4.13 f3(I65, I66, I67, I68) -> f4(I69, I70, I71, I72) [-1 <= I70 - 1 /\ 1 <= I69 - 1 /\ 0 <= I65 - 1 /\ I66 <= 0 /\ I69 - 1 <= I65] 4.14/4.13 f2(I73, I74, I75, I76) -> f4(I77, I78, I79, I80) [1 <= I78 - 1 /\ 1 <= I77 - 1 /\ 1 <= I75 - 1 /\ 1 <= I74 - 1 /\ I78 <= I75 /\ I78 <= I74 /\ I77 <= I75 /\ I77 <= I74 /\ I76 <= 0 /\ I73 <= 1] 4.14/4.13 f2(I81, I82, I83, I84) -> f3(I85, I81 - 2, I86, I87) [4 <= I85 - 1 /\ 1 <= I83 - 1 /\ 1 <= I82 - 1 /\ I85 - 3 <= I83 /\ I85 - 3 <= I82 /\ I84 <= 0 /\ 1 <= I81 - 1] 4.14/4.13 f3(I88, I89, I90, I91) -> f3(I92, I89 - 1, I93, I94) [2 <= I92 - 1 /\ 0 <= I89 - 1 /\ 0 <= I88 - 1] 4.14/4.13 f2(I95, I96, I97, I98) -> f2(I95, I99, I100, I98 - 1) [2 <= I100 - 1 /\ 0 <= I99 - 1 /\ 0 <= I97 - 1 /\ 0 <= I96 - 1 /\ I100 - 2 <= I97 /\ 0 <= I98 - 1 /\ I99 <= I96] 4.14/4.13 f2(I101, I102, I103, I104) -> f3(I105, I101 - 1, I106, I107) [0 <= I105 - 1 /\ 0 <= I103 - 1 /\ I104 <= 0 /\ 1 <= I102 - 1] 4.14/4.13 f1(I108, I109, I110, I111) -> f2(I112, I113, I114, I115) [-1 <= I112 - 1 /\ 1 <= I109 - 1 /\ -1 <= I116 - 1 /\ I113 - 1 <= I108 /\ I114 - 1 <= I108 /\ 0 <= I108 - 1 /\ 1 <= I113 - 1 /\ 1 <= I114 - 1 /\ I116 - 2 = I115] 4.14/4.13 4.14/4.13 The dependency graph for this problem is: 4.14/4.13 0 -> 15 4.14/4.13 1 -> 6 4.14/4.13 2 -> 4, 5 4.14/4.13 3 -> 1, 2, 3 4.14/4.13 4 -> 6 4.14/4.13 5 -> 6 4.14/4.13 6 -> 4, 5 4.14/4.13 7 -> 1, 2, 3 4.14/4.13 8 -> 6 4.14/4.13 9 -> 6 4.14/4.13 10 -> 6 4.14/4.13 11 -> 7, 8, 9, 12 4.14/4.13 12 -> 7, 8, 9, 12 4.14/4.13 13 -> 10, 11, 13, 14 4.14/4.13 14 -> 7, 8, 9, 12 4.14/4.13 15 -> 10, 11, 13, 14 4.14/4.13 Where: 4.14/4.13 0) init#(x1, x2, x3, x4) -> f1#(rnd1, rnd2, rnd3, rnd4) 4.14/4.13 1) f5#(I0, I1, I2, I3) -> f4#(I4, I5, I6, I7) [-1 <= y1 - 1 /\ y2 <= y1 - 1 /\ I4 - 2 <= I0 /\ I4 <= I1 /\ I5 <= I0 /\ 0 <= I0 - 1 /\ 2 <= I1 - 1 /\ 2 <= I4 - 1 /\ 0 <= I5 - 1] 4.14/4.13 2) f5#(I8, I9, I10, I11) -> f6#(I12, I13, I14, I15) [0 <= I16 - 1 /\ -1 <= I17 - 1 /\ I16 <= I17 - 1 /\ I14 - 2 <= I8 /\ 0 <= I8 - 1 /\ 2 <= I9 - 1 /\ -1 <= I12 - 1 /\ 0 <= I13 - 1 /\ 2 <= I14 - 1 /\ -1 <= I15 - 1] 4.14/4.13 3) f5#(I18, I19, I20, I21) -> f5#(I22, I23, I20, I24) [0 <= I23 - 1 /\ 2 <= I22 - 1 /\ 2 <= I19 - 1 /\ 0 <= I18 - 1 /\ I22 - 2 <= I18 /\ I24 <= I21 - 1 /\ 0 <= I21 - 1 /\ 0 <= I20 - 1] 4.14/4.13 4) f6#(I25, I26, I27, I28) -> f4#(I29, I30, I31, I32) [-1 <= I30 - 1 /\ 2 <= I29 - 1 /\ -1 <= I28 - 1 /\ 0 <= I27 - 1 /\ 0 <= I26 - 1 /\ -1 <= I25 - 1 /\ I30 <= I28 /\ I30 + 1 <= I27 /\ I30 + 1 <= I26 /\ I30 <= I25 /\ I29 - 3 <= I28 /\ I29 - 2 <= I27 /\ I29 - 2 <= I26 /\ I29 - 3 <= I25] 4.14/4.13 5) f6#(I33, I34, I35, I36) -> f4#(I37, I38, I39, I40) [-1 <= I38 - 1 /\ 2 <= I37 - 1 /\ -1 <= I36 - 1 /\ 0 <= I35 - 1 /\ 0 <= I34 - 1 /\ -1 <= I33 - 1 /\ I38 <= I36 /\ I38 + 1 <= I34 /\ I38 <= I33] 4.14/4.13 6) f4#(I41, I42, I43, I44) -> f6#(I45, I46, I47, I48) [-1 <= I48 - 1 /\ 0 <= I47 - 1 /\ 0 <= I46 - 1 /\ -1 <= I45 - 1 /\ 0 <= I42 - 1 /\ 0 <= I41 - 1 /\ I48 + 1 <= I42 /\ I47 <= I41 /\ I46 <= I42 /\ I45 + 1 <= I42] 4.14/4.13 7) f3#(I49, I50, I51, I52) -> f5#(I53, I54, I55, I56) [0 <= I54 - 1 /\ 1 <= I53 - 1 /\ 2 <= I49 - 1 /\ I50 <= 0 /\ I53 + 1 <= I49] 4.14/4.13 8) f3#(I57, I58, I59, I60) -> f4#(I61, I62, I63, I64) [0 <= I62 - 1 /\ 1 <= I61 - 1 /\ 2 <= I57 - 1 /\ I58 <= 0 /\ I61 + 1 <= I57] 4.14/4.13 9) f3#(I65, I66, I67, I68) -> f4#(I69, I70, I71, I72) [-1 <= I70 - 1 /\ 1 <= I69 - 1 /\ 0 <= I65 - 1 /\ I66 <= 0 /\ I69 - 1 <= I65] 4.14/4.13 10) f2#(I73, I74, I75, I76) -> f4#(I77, I78, I79, I80) [1 <= I78 - 1 /\ 1 <= I77 - 1 /\ 1 <= I75 - 1 /\ 1 <= I74 - 1 /\ I78 <= I75 /\ I78 <= I74 /\ I77 <= I75 /\ I77 <= I74 /\ I76 <= 0 /\ I73 <= 1] 4.14/4.13 11) f2#(I81, I82, I83, I84) -> f3#(I85, I81 - 2, I86, I87) [4 <= I85 - 1 /\ 1 <= I83 - 1 /\ 1 <= I82 - 1 /\ I85 - 3 <= I83 /\ I85 - 3 <= I82 /\ I84 <= 0 /\ 1 <= I81 - 1] 4.14/4.13 12) f3#(I88, I89, I90, I91) -> f3#(I92, I89 - 1, I93, I94) [2 <= I92 - 1 /\ 0 <= I89 - 1 /\ 0 <= I88 - 1] 4.14/4.13 13) f2#(I95, I96, I97, I98) -> f2#(I95, I99, I100, I98 - 1) [2 <= I100 - 1 /\ 0 <= I99 - 1 /\ 0 <= I97 - 1 /\ 0 <= I96 - 1 /\ I100 - 2 <= I97 /\ 0 <= I98 - 1 /\ I99 <= I96] 4.14/4.13 14) f2#(I101, I102, I103, I104) -> f3#(I105, I101 - 1, I106, I107) [0 <= I105 - 1 /\ 0 <= I103 - 1 /\ I104 <= 0 /\ 1 <= I102 - 1] 4.14/4.13 15) f1#(I108, I109, I110, I111) -> f2#(I112, I113, I114, I115) [-1 <= I112 - 1 /\ 1 <= I109 - 1 /\ -1 <= I116 - 1 /\ I113 - 1 <= I108 /\ I114 - 1 <= I108 /\ 0 <= I108 - 1 /\ 1 <= I113 - 1 /\ 1 <= I114 - 1 /\ I116 - 2 = I115] 4.14/4.13 4.14/4.13 We have the following SCCs. 4.14/4.13 { 13 } 4.14/4.13 { 12 } 4.14/4.13 { 3 } 4.14/4.13 { 4, 5, 6 } 4.14/4.13 4.14/4.13 DP problem for innermost termination. 4.14/4.13 P = 4.14/4.13 f6#(I25, I26, I27, I28) -> f4#(I29, I30, I31, I32) [-1 <= I30 - 1 /\ 2 <= I29 - 1 /\ -1 <= I28 - 1 /\ 0 <= I27 - 1 /\ 0 <= I26 - 1 /\ -1 <= I25 - 1 /\ I30 <= I28 /\ I30 + 1 <= I27 /\ I30 + 1 <= I26 /\ I30 <= I25 /\ I29 - 3 <= I28 /\ I29 - 2 <= I27 /\ I29 - 2 <= I26 /\ I29 - 3 <= I25] 4.14/4.13 f6#(I33, I34, I35, I36) -> f4#(I37, I38, I39, I40) [-1 <= I38 - 1 /\ 2 <= I37 - 1 /\ -1 <= I36 - 1 /\ 0 <= I35 - 1 /\ 0 <= I34 - 1 /\ -1 <= I33 - 1 /\ I38 <= I36 /\ I38 + 1 <= I34 /\ I38 <= I33] 4.14/4.13 f4#(I41, I42, I43, I44) -> f6#(I45, I46, I47, I48) [-1 <= I48 - 1 /\ 0 <= I47 - 1 /\ 0 <= I46 - 1 /\ -1 <= I45 - 1 /\ 0 <= I42 - 1 /\ 0 <= I41 - 1 /\ I48 + 1 <= I42 /\ I47 <= I41 /\ I46 <= I42 /\ I45 + 1 <= I42] 4.14/4.13 R = 4.14/4.13 init(x1, x2, x3, x4) -> f1(rnd1, rnd2, rnd3, rnd4) 4.14/4.13 f5(I0, I1, I2, I3) -> f4(I4, I5, I6, I7) [-1 <= y1 - 1 /\ y2 <= y1 - 1 /\ I4 - 2 <= I0 /\ I4 <= I1 /\ I5 <= I0 /\ 0 <= I0 - 1 /\ 2 <= I1 - 1 /\ 2 <= I4 - 1 /\ 0 <= I5 - 1] 4.14/4.13 f5(I8, I9, I10, I11) -> f6(I12, I13, I14, I15) [0 <= I16 - 1 /\ -1 <= I17 - 1 /\ I16 <= I17 - 1 /\ I14 - 2 <= I8 /\ 0 <= I8 - 1 /\ 2 <= I9 - 1 /\ -1 <= I12 - 1 /\ 0 <= I13 - 1 /\ 2 <= I14 - 1 /\ -1 <= I15 - 1] 4.14/4.13 f5(I18, I19, I20, I21) -> f5(I22, I23, I20, I24) [0 <= I23 - 1 /\ 2 <= I22 - 1 /\ 2 <= I19 - 1 /\ 0 <= I18 - 1 /\ I22 - 2 <= I18 /\ I24 <= I21 - 1 /\ 0 <= I21 - 1 /\ 0 <= I20 - 1] 4.14/4.13 f6(I25, I26, I27, I28) -> f4(I29, I30, I31, I32) [-1 <= I30 - 1 /\ 2 <= I29 - 1 /\ -1 <= I28 - 1 /\ 0 <= I27 - 1 /\ 0 <= I26 - 1 /\ -1 <= I25 - 1 /\ I30 <= I28 /\ I30 + 1 <= I27 /\ I30 + 1 <= I26 /\ I30 <= I25 /\ I29 - 3 <= I28 /\ I29 - 2 <= I27 /\ I29 - 2 <= I26 /\ I29 - 3 <= I25] 4.14/4.13 f6(I33, I34, I35, I36) -> f4(I37, I38, I39, I40) [-1 <= I38 - 1 /\ 2 <= I37 - 1 /\ -1 <= I36 - 1 /\ 0 <= I35 - 1 /\ 0 <= I34 - 1 /\ -1 <= I33 - 1 /\ I38 <= I36 /\ I38 + 1 <= I34 /\ I38 <= I33] 4.14/4.13 f4(I41, I42, I43, I44) -> f6(I45, I46, I47, I48) [-1 <= I48 - 1 /\ 0 <= I47 - 1 /\ 0 <= I46 - 1 /\ -1 <= I45 - 1 /\ 0 <= I42 - 1 /\ 0 <= I41 - 1 /\ I48 + 1 <= I42 /\ I47 <= I41 /\ I46 <= I42 /\ I45 + 1 <= I42] 4.14/4.13 f3(I49, I50, I51, I52) -> f5(I53, I54, I55, I56) [0 <= I54 - 1 /\ 1 <= I53 - 1 /\ 2 <= I49 - 1 /\ I50 <= 0 /\ I53 + 1 <= I49] 4.14/4.13 f3(I57, I58, I59, I60) -> f4(I61, I62, I63, I64) [0 <= I62 - 1 /\ 1 <= I61 - 1 /\ 2 <= I57 - 1 /\ I58 <= 0 /\ I61 + 1 <= I57] 4.14/4.13 f3(I65, I66, I67, I68) -> f4(I69, I70, I71, I72) [-1 <= I70 - 1 /\ 1 <= I69 - 1 /\ 0 <= I65 - 1 /\ I66 <= 0 /\ I69 - 1 <= I65] 4.14/4.13 f2(I73, I74, I75, I76) -> f4(I77, I78, I79, I80) [1 <= I78 - 1 /\ 1 <= I77 - 1 /\ 1 <= I75 - 1 /\ 1 <= I74 - 1 /\ I78 <= I75 /\ I78 <= I74 /\ I77 <= I75 /\ I77 <= I74 /\ I76 <= 0 /\ I73 <= 1] 4.14/4.13 f2(I81, I82, I83, I84) -> f3(I85, I81 - 2, I86, I87) [4 <= I85 - 1 /\ 1 <= I83 - 1 /\ 1 <= I82 - 1 /\ I85 - 3 <= I83 /\ I85 - 3 <= I82 /\ I84 <= 0 /\ 1 <= I81 - 1] 4.14/4.13 f3(I88, I89, I90, I91) -> f3(I92, I89 - 1, I93, I94) [2 <= I92 - 1 /\ 0 <= I89 - 1 /\ 0 <= I88 - 1] 4.14/4.13 f2(I95, I96, I97, I98) -> f2(I95, I99, I100, I98 - 1) [2 <= I100 - 1 /\ 0 <= I99 - 1 /\ 0 <= I97 - 1 /\ 0 <= I96 - 1 /\ I100 - 2 <= I97 /\ 0 <= I98 - 1 /\ I99 <= I96] 4.14/4.13 f2(I101, I102, I103, I104) -> f3(I105, I101 - 1, I106, I107) [0 <= I105 - 1 /\ 0 <= I103 - 1 /\ I104 <= 0 /\ 1 <= I102 - 1] 4.14/4.13 f1(I108, I109, I110, I111) -> f2(I112, I113, I114, I115) [-1 <= I112 - 1 /\ 1 <= I109 - 1 /\ -1 <= I116 - 1 /\ I113 - 1 <= I108 /\ I114 - 1 <= I108 /\ 0 <= I108 - 1 /\ 1 <= I113 - 1 /\ 1 <= I114 - 1 /\ I116 - 2 = I115] 4.14/4.13 4.14/4.13 We use the basic value criterion with the projection function NU: 4.14/4.13 NU[f4#(z1,z2,z3,z4)] = z2 4.14/4.13 NU[f6#(z1,z2,z3,z4)] = z2 4.14/4.13 4.14/4.13 This gives the following inequalities: 4.14/4.13 -1 <= I30 - 1 /\ 2 <= I29 - 1 /\ -1 <= I28 - 1 /\ 0 <= I27 - 1 /\ 0 <= I26 - 1 /\ -1 <= I25 - 1 /\ I30 <= I28 /\ I30 + 1 <= I27 /\ I30 + 1 <= I26 /\ I30 <= I25 /\ I29 - 3 <= I28 /\ I29 - 2 <= I27 /\ I29 - 2 <= I26 /\ I29 - 3 <= I25 ==> I26 >! I30 4.14/4.13 -1 <= I38 - 1 /\ 2 <= I37 - 1 /\ -1 <= I36 - 1 /\ 0 <= I35 - 1 /\ 0 <= I34 - 1 /\ -1 <= I33 - 1 /\ I38 <= I36 /\ I38 + 1 <= I34 /\ I38 <= I33 ==> I34 >! I38 4.14/4.13 -1 <= I48 - 1 /\ 0 <= I47 - 1 /\ 0 <= I46 - 1 /\ -1 <= I45 - 1 /\ 0 <= I42 - 1 /\ 0 <= I41 - 1 /\ I48 + 1 <= I42 /\ I47 <= I41 /\ I46 <= I42 /\ I45 + 1 <= I42 ==> I42 (>! \union =) I46 4.14/4.13 4.14/4.13 We remove all the strictly oriented dependency pairs. 4.14/4.13 4.14/4.13 DP problem for innermost termination. 4.14/4.13 P = 4.14/4.13 f4#(I41, I42, I43, I44) -> f6#(I45, I46, I47, I48) [-1 <= I48 - 1 /\ 0 <= I47 - 1 /\ 0 <= I46 - 1 /\ -1 <= I45 - 1 /\ 0 <= I42 - 1 /\ 0 <= I41 - 1 /\ I48 + 1 <= I42 /\ I47 <= I41 /\ I46 <= I42 /\ I45 + 1 <= I42] 4.14/4.13 R = 4.14/4.13 init(x1, x2, x3, x4) -> f1(rnd1, rnd2, rnd3, rnd4) 4.14/4.13 f5(I0, I1, I2, I3) -> f4(I4, I5, I6, I7) [-1 <= y1 - 1 /\ y2 <= y1 - 1 /\ I4 - 2 <= I0 /\ I4 <= I1 /\ I5 <= I0 /\ 0 <= I0 - 1 /\ 2 <= I1 - 1 /\ 2 <= I4 - 1 /\ 0 <= I5 - 1] 4.14/4.13 f5(I8, I9, I10, I11) -> f6(I12, I13, I14, I15) [0 <= I16 - 1 /\ -1 <= I17 - 1 /\ I16 <= I17 - 1 /\ I14 - 2 <= I8 /\ 0 <= I8 - 1 /\ 2 <= I9 - 1 /\ -1 <= I12 - 1 /\ 0 <= I13 - 1 /\ 2 <= I14 - 1 /\ -1 <= I15 - 1] 4.14/4.13 f5(I18, I19, I20, I21) -> f5(I22, I23, I20, I24) [0 <= I23 - 1 /\ 2 <= I22 - 1 /\ 2 <= I19 - 1 /\ 0 <= I18 - 1 /\ I22 - 2 <= I18 /\ I24 <= I21 - 1 /\ 0 <= I21 - 1 /\ 0 <= I20 - 1] 4.14/4.13 f6(I25, I26, I27, I28) -> f4(I29, I30, I31, I32) [-1 <= I30 - 1 /\ 2 <= I29 - 1 /\ -1 <= I28 - 1 /\ 0 <= I27 - 1 /\ 0 <= I26 - 1 /\ -1 <= I25 - 1 /\ I30 <= I28 /\ I30 + 1 <= I27 /\ I30 + 1 <= I26 /\ I30 <= I25 /\ I29 - 3 <= I28 /\ I29 - 2 <= I27 /\ I29 - 2 <= I26 /\ I29 - 3 <= I25] 4.14/4.13 f6(I33, I34, I35, I36) -> f4(I37, I38, I39, I40) [-1 <= I38 - 1 /\ 2 <= I37 - 1 /\ -1 <= I36 - 1 /\ 0 <= I35 - 1 /\ 0 <= I34 - 1 /\ -1 <= I33 - 1 /\ I38 <= I36 /\ I38 + 1 <= I34 /\ I38 <= I33] 4.14/4.13 f4(I41, I42, I43, I44) -> f6(I45, I46, I47, I48) [-1 <= I48 - 1 /\ 0 <= I47 - 1 /\ 0 <= I46 - 1 /\ -1 <= I45 - 1 /\ 0 <= I42 - 1 /\ 0 <= I41 - 1 /\ I48 + 1 <= I42 /\ I47 <= I41 /\ I46 <= I42 /\ I45 + 1 <= I42] 4.14/4.13 f3(I49, I50, I51, I52) -> f5(I53, I54, I55, I56) [0 <= I54 - 1 /\ 1 <= I53 - 1 /\ 2 <= I49 - 1 /\ I50 <= 0 /\ I53 + 1 <= I49] 4.14/4.13 f3(I57, I58, I59, I60) -> f4(I61, I62, I63, I64) [0 <= I62 - 1 /\ 1 <= I61 - 1 /\ 2 <= I57 - 1 /\ I58 <= 0 /\ I61 + 1 <= I57] 4.14/4.13 f3(I65, I66, I67, I68) -> f4(I69, I70, I71, I72) [-1 <= I70 - 1 /\ 1 <= I69 - 1 /\ 0 <= I65 - 1 /\ I66 <= 0 /\ I69 - 1 <= I65] 4.14/4.13 f2(I73, I74, I75, I76) -> f4(I77, I78, I79, I80) [1 <= I78 - 1 /\ 1 <= I77 - 1 /\ 1 <= I75 - 1 /\ 1 <= I74 - 1 /\ I78 <= I75 /\ I78 <= I74 /\ I77 <= I75 /\ I77 <= I74 /\ I76 <= 0 /\ I73 <= 1] 4.14/4.13 f2(I81, I82, I83, I84) -> f3(I85, I81 - 2, I86, I87) [4 <= I85 - 1 /\ 1 <= I83 - 1 /\ 1 <= I82 - 1 /\ I85 - 3 <= I83 /\ I85 - 3 <= I82 /\ I84 <= 0 /\ 1 <= I81 - 1] 4.14/4.13 f3(I88, I89, I90, I91) -> f3(I92, I89 - 1, I93, I94) [2 <= I92 - 1 /\ 0 <= I89 - 1 /\ 0 <= I88 - 1] 4.14/4.13 f2(I95, I96, I97, I98) -> f2(I95, I99, I100, I98 - 1) [2 <= I100 - 1 /\ 0 <= I99 - 1 /\ 0 <= I97 - 1 /\ 0 <= I96 - 1 /\ I100 - 2 <= I97 /\ 0 <= I98 - 1 /\ I99 <= I96] 4.14/4.13 f2(I101, I102, I103, I104) -> f3(I105, I101 - 1, I106, I107) [0 <= I105 - 1 /\ 0 <= I103 - 1 /\ I104 <= 0 /\ 1 <= I102 - 1] 4.14/4.13 f1(I108, I109, I110, I111) -> f2(I112, I113, I114, I115) [-1 <= I112 - 1 /\ 1 <= I109 - 1 /\ -1 <= I116 - 1 /\ I113 - 1 <= I108 /\ I114 - 1 <= I108 /\ 0 <= I108 - 1 /\ 1 <= I113 - 1 /\ 1 <= I114 - 1 /\ I116 - 2 = I115] 4.14/4.13 4.14/4.13 The dependency graph for this problem is: 4.14/4.13 6 -> 4.14/4.13 Where: 4.14/4.13 6) f4#(I41, I42, I43, I44) -> f6#(I45, I46, I47, I48) [-1 <= I48 - 1 /\ 0 <= I47 - 1 /\ 0 <= I46 - 1 /\ -1 <= I45 - 1 /\ 0 <= I42 - 1 /\ 0 <= I41 - 1 /\ I48 + 1 <= I42 /\ I47 <= I41 /\ I46 <= I42 /\ I45 + 1 <= I42] 4.14/4.13 4.14/4.13 We have the following SCCs. 4.14/4.13 4.14/4.13 4.14/4.13 DP problem for innermost termination. 4.14/4.13 P = 4.14/4.13 f5#(I18, I19, I20, I21) -> f5#(I22, I23, I20, I24) [0 <= I23 - 1 /\ 2 <= I22 - 1 /\ 2 <= I19 - 1 /\ 0 <= I18 - 1 /\ I22 - 2 <= I18 /\ I24 <= I21 - 1 /\ 0 <= I21 - 1 /\ 0 <= I20 - 1] 4.14/4.13 R = 4.14/4.13 init(x1, x2, x3, x4) -> f1(rnd1, rnd2, rnd3, rnd4) 4.14/4.13 f5(I0, I1, I2, I3) -> f4(I4, I5, I6, I7) [-1 <= y1 - 1 /\ y2 <= y1 - 1 /\ I4 - 2 <= I0 /\ I4 <= I1 /\ I5 <= I0 /\ 0 <= I0 - 1 /\ 2 <= I1 - 1 /\ 2 <= I4 - 1 /\ 0 <= I5 - 1] 4.14/4.13 f5(I8, I9, I10, I11) -> f6(I12, I13, I14, I15) [0 <= I16 - 1 /\ -1 <= I17 - 1 /\ I16 <= I17 - 1 /\ I14 - 2 <= I8 /\ 0 <= I8 - 1 /\ 2 <= I9 - 1 /\ -1 <= I12 - 1 /\ 0 <= I13 - 1 /\ 2 <= I14 - 1 /\ -1 <= I15 - 1] 4.14/4.13 f5(I18, I19, I20, I21) -> f5(I22, I23, I20, I24) [0 <= I23 - 1 /\ 2 <= I22 - 1 /\ 2 <= I19 - 1 /\ 0 <= I18 - 1 /\ I22 - 2 <= I18 /\ I24 <= I21 - 1 /\ 0 <= I21 - 1 /\ 0 <= I20 - 1] 4.14/4.13 f6(I25, I26, I27, I28) -> f4(I29, I30, I31, I32) [-1 <= I30 - 1 /\ 2 <= I29 - 1 /\ -1 <= I28 - 1 /\ 0 <= I27 - 1 /\ 0 <= I26 - 1 /\ -1 <= I25 - 1 /\ I30 <= I28 /\ I30 + 1 <= I27 /\ I30 + 1 <= I26 /\ I30 <= I25 /\ I29 - 3 <= I28 /\ I29 - 2 <= I27 /\ I29 - 2 <= I26 /\ I29 - 3 <= I25] 4.14/4.13 f6(I33, I34, I35, I36) -> f4(I37, I38, I39, I40) [-1 <= I38 - 1 /\ 2 <= I37 - 1 /\ -1 <= I36 - 1 /\ 0 <= I35 - 1 /\ 0 <= I34 - 1 /\ -1 <= I33 - 1 /\ I38 <= I36 /\ I38 + 1 <= I34 /\ I38 <= I33] 4.14/4.13 f4(I41, I42, I43, I44) -> f6(I45, I46, I47, I48) [-1 <= I48 - 1 /\ 0 <= I47 - 1 /\ 0 <= I46 - 1 /\ -1 <= I45 - 1 /\ 0 <= I42 - 1 /\ 0 <= I41 - 1 /\ I48 + 1 <= I42 /\ I47 <= I41 /\ I46 <= I42 /\ I45 + 1 <= I42] 4.14/4.13 f3(I49, I50, I51, I52) -> f5(I53, I54, I55, I56) [0 <= I54 - 1 /\ 1 <= I53 - 1 /\ 2 <= I49 - 1 /\ I50 <= 0 /\ I53 + 1 <= I49] 4.14/4.13 f3(I57, I58, I59, I60) -> f4(I61, I62, I63, I64) [0 <= I62 - 1 /\ 1 <= I61 - 1 /\ 2 <= I57 - 1 /\ I58 <= 0 /\ I61 + 1 <= I57] 4.14/4.13 f3(I65, I66, I67, I68) -> f4(I69, I70, I71, I72) [-1 <= I70 - 1 /\ 1 <= I69 - 1 /\ 0 <= I65 - 1 /\ I66 <= 0 /\ I69 - 1 <= I65] 4.14/4.13 f2(I73, I74, I75, I76) -> f4(I77, I78, I79, I80) [1 <= I78 - 1 /\ 1 <= I77 - 1 /\ 1 <= I75 - 1 /\ 1 <= I74 - 1 /\ I78 <= I75 /\ I78 <= I74 /\ I77 <= I75 /\ I77 <= I74 /\ I76 <= 0 /\ I73 <= 1] 4.14/4.13 f2(I81, I82, I83, I84) -> f3(I85, I81 - 2, I86, I87) [4 <= I85 - 1 /\ 1 <= I83 - 1 /\ 1 <= I82 - 1 /\ I85 - 3 <= I83 /\ I85 - 3 <= I82 /\ I84 <= 0 /\ 1 <= I81 - 1] 4.14/4.13 f3(I88, I89, I90, I91) -> f3(I92, I89 - 1, I93, I94) [2 <= I92 - 1 /\ 0 <= I89 - 1 /\ 0 <= I88 - 1] 4.14/4.13 f2(I95, I96, I97, I98) -> f2(I95, I99, I100, I98 - 1) [2 <= I100 - 1 /\ 0 <= I99 - 1 /\ 0 <= I97 - 1 /\ 0 <= I96 - 1 /\ I100 - 2 <= I97 /\ 0 <= I98 - 1 /\ I99 <= I96] 4.14/4.13 f2(I101, I102, I103, I104) -> f3(I105, I101 - 1, I106, I107) [0 <= I105 - 1 /\ 0 <= I103 - 1 /\ I104 <= 0 /\ 1 <= I102 - 1] 4.14/4.13 f1(I108, I109, I110, I111) -> f2(I112, I113, I114, I115) [-1 <= I112 - 1 /\ 1 <= I109 - 1 /\ -1 <= I116 - 1 /\ I113 - 1 <= I108 /\ I114 - 1 <= I108 /\ 0 <= I108 - 1 /\ 1 <= I113 - 1 /\ 1 <= I114 - 1 /\ I116 - 2 = I115] 4.14/4.13 4.14/4.13 We use the basic value criterion with the projection function NU: 4.14/4.13 NU[f5#(z1,z2,z3,z4)] = z4 4.14/4.13 4.14/4.13 This gives the following inequalities: 4.14/4.13 0 <= I23 - 1 /\ 2 <= I22 - 1 /\ 2 <= I19 - 1 /\ 0 <= I18 - 1 /\ I22 - 2 <= I18 /\ I24 <= I21 - 1 /\ 0 <= I21 - 1 /\ 0 <= I20 - 1 ==> I21 >! I24 4.14/4.13 4.14/4.13 All dependency pairs are strictly oriented, so the entire dependency pair problem may be removed. 4.14/4.13 4.14/4.13 DP problem for innermost termination. 4.14/4.13 P = 4.14/4.13 f3#(I88, I89, I90, I91) -> f3#(I92, I89 - 1, I93, I94) [2 <= I92 - 1 /\ 0 <= I89 - 1 /\ 0 <= I88 - 1] 4.14/4.13 R = 4.14/4.13 init(x1, x2, x3, x4) -> f1(rnd1, rnd2, rnd3, rnd4) 4.14/4.13 f5(I0, I1, I2, I3) -> f4(I4, I5, I6, I7) [-1 <= y1 - 1 /\ y2 <= y1 - 1 /\ I4 - 2 <= I0 /\ I4 <= I1 /\ I5 <= I0 /\ 0 <= I0 - 1 /\ 2 <= I1 - 1 /\ 2 <= I4 - 1 /\ 0 <= I5 - 1] 4.14/4.13 f5(I8, I9, I10, I11) -> f6(I12, I13, I14, I15) [0 <= I16 - 1 /\ -1 <= I17 - 1 /\ I16 <= I17 - 1 /\ I14 - 2 <= I8 /\ 0 <= I8 - 1 /\ 2 <= I9 - 1 /\ -1 <= I12 - 1 /\ 0 <= I13 - 1 /\ 2 <= I14 - 1 /\ -1 <= I15 - 1] 4.14/4.13 f5(I18, I19, I20, I21) -> f5(I22, I23, I20, I24) [0 <= I23 - 1 /\ 2 <= I22 - 1 /\ 2 <= I19 - 1 /\ 0 <= I18 - 1 /\ I22 - 2 <= I18 /\ I24 <= I21 - 1 /\ 0 <= I21 - 1 /\ 0 <= I20 - 1] 4.14/4.13 f6(I25, I26, I27, I28) -> f4(I29, I30, I31, I32) [-1 <= I30 - 1 /\ 2 <= I29 - 1 /\ -1 <= I28 - 1 /\ 0 <= I27 - 1 /\ 0 <= I26 - 1 /\ -1 <= I25 - 1 /\ I30 <= I28 /\ I30 + 1 <= I27 /\ I30 + 1 <= I26 /\ I30 <= I25 /\ I29 - 3 <= I28 /\ I29 - 2 <= I27 /\ I29 - 2 <= I26 /\ I29 - 3 <= I25] 4.14/4.13 f6(I33, I34, I35, I36) -> f4(I37, I38, I39, I40) [-1 <= I38 - 1 /\ 2 <= I37 - 1 /\ -1 <= I36 - 1 /\ 0 <= I35 - 1 /\ 0 <= I34 - 1 /\ -1 <= I33 - 1 /\ I38 <= I36 /\ I38 + 1 <= I34 /\ I38 <= I33] 4.14/4.13 f4(I41, I42, I43, I44) -> f6(I45, I46, I47, I48) [-1 <= I48 - 1 /\ 0 <= I47 - 1 /\ 0 <= I46 - 1 /\ -1 <= I45 - 1 /\ 0 <= I42 - 1 /\ 0 <= I41 - 1 /\ I48 + 1 <= I42 /\ I47 <= I41 /\ I46 <= I42 /\ I45 + 1 <= I42] 4.14/4.13 f3(I49, I50, I51, I52) -> f5(I53, I54, I55, I56) [0 <= I54 - 1 /\ 1 <= I53 - 1 /\ 2 <= I49 - 1 /\ I50 <= 0 /\ I53 + 1 <= I49] 4.14/4.13 f3(I57, I58, I59, I60) -> f4(I61, I62, I63, I64) [0 <= I62 - 1 /\ 1 <= I61 - 1 /\ 2 <= I57 - 1 /\ I58 <= 0 /\ I61 + 1 <= I57] 4.14/4.13 f3(I65, I66, I67, I68) -> f4(I69, I70, I71, I72) [-1 <= I70 - 1 /\ 1 <= I69 - 1 /\ 0 <= I65 - 1 /\ I66 <= 0 /\ I69 - 1 <= I65] 4.14/4.13 f2(I73, I74, I75, I76) -> f4(I77, I78, I79, I80) [1 <= I78 - 1 /\ 1 <= I77 - 1 /\ 1 <= I75 - 1 /\ 1 <= I74 - 1 /\ I78 <= I75 /\ I78 <= I74 /\ I77 <= I75 /\ I77 <= I74 /\ I76 <= 0 /\ I73 <= 1] 4.14/4.13 f2(I81, I82, I83, I84) -> f3(I85, I81 - 2, I86, I87) [4 <= I85 - 1 /\ 1 <= I83 - 1 /\ 1 <= I82 - 1 /\ I85 - 3 <= I83 /\ I85 - 3 <= I82 /\ I84 <= 0 /\ 1 <= I81 - 1] 4.14/4.13 f3(I88, I89, I90, I91) -> f3(I92, I89 - 1, I93, I94) [2 <= I92 - 1 /\ 0 <= I89 - 1 /\ 0 <= I88 - 1] 4.14/4.13 f2(I95, I96, I97, I98) -> f2(I95, I99, I100, I98 - 1) [2 <= I100 - 1 /\ 0 <= I99 - 1 /\ 0 <= I97 - 1 /\ 0 <= I96 - 1 /\ I100 - 2 <= I97 /\ 0 <= I98 - 1 /\ I99 <= I96] 4.14/4.13 f2(I101, I102, I103, I104) -> f3(I105, I101 - 1, I106, I107) [0 <= I105 - 1 /\ 0 <= I103 - 1 /\ I104 <= 0 /\ 1 <= I102 - 1] 4.14/4.13 f1(I108, I109, I110, I111) -> f2(I112, I113, I114, I115) [-1 <= I112 - 1 /\ 1 <= I109 - 1 /\ -1 <= I116 - 1 /\ I113 - 1 <= I108 /\ I114 - 1 <= I108 /\ 0 <= I108 - 1 /\ 1 <= I113 - 1 /\ 1 <= I114 - 1 /\ I116 - 2 = I115] 4.14/4.13 4.14/4.13 We use the basic value criterion with the projection function NU: 4.14/4.13 NU[f3#(z1,z2,z3,z4)] = z2 4.14/4.13 4.14/4.13 This gives the following inequalities: 4.14/4.13 2 <= I92 - 1 /\ 0 <= I89 - 1 /\ 0 <= I88 - 1 ==> I89 >! I89 - 1 4.14/4.13 4.14/4.13 All dependency pairs are strictly oriented, so the entire dependency pair problem may be removed. 4.14/4.13 4.14/4.13 DP problem for innermost termination. 4.14/4.13 P = 4.14/4.13 f2#(I95, I96, I97, I98) -> f2#(I95, I99, I100, I98 - 1) [2 <= I100 - 1 /\ 0 <= I99 - 1 /\ 0 <= I97 - 1 /\ 0 <= I96 - 1 /\ I100 - 2 <= I97 /\ 0 <= I98 - 1 /\ I99 <= I96] 4.14/4.13 R = 4.14/4.13 init(x1, x2, x3, x4) -> f1(rnd1, rnd2, rnd3, rnd4) 4.14/4.13 f5(I0, I1, I2, I3) -> f4(I4, I5, I6, I7) [-1 <= y1 - 1 /\ y2 <= y1 - 1 /\ I4 - 2 <= I0 /\ I4 <= I1 /\ I5 <= I0 /\ 0 <= I0 - 1 /\ 2 <= I1 - 1 /\ 2 <= I4 - 1 /\ 0 <= I5 - 1] 4.14/4.13 f5(I8, I9, I10, I11) -> f6(I12, I13, I14, I15) [0 <= I16 - 1 /\ -1 <= I17 - 1 /\ I16 <= I17 - 1 /\ I14 - 2 <= I8 /\ 0 <= I8 - 1 /\ 2 <= I9 - 1 /\ -1 <= I12 - 1 /\ 0 <= I13 - 1 /\ 2 <= I14 - 1 /\ -1 <= I15 - 1] 4.14/4.13 f5(I18, I19, I20, I21) -> f5(I22, I23, I20, I24) [0 <= I23 - 1 /\ 2 <= I22 - 1 /\ 2 <= I19 - 1 /\ 0 <= I18 - 1 /\ I22 - 2 <= I18 /\ I24 <= I21 - 1 /\ 0 <= I21 - 1 /\ 0 <= I20 - 1] 4.14/4.13 f6(I25, I26, I27, I28) -> f4(I29, I30, I31, I32) [-1 <= I30 - 1 /\ 2 <= I29 - 1 /\ -1 <= I28 - 1 /\ 0 <= I27 - 1 /\ 0 <= I26 - 1 /\ -1 <= I25 - 1 /\ I30 <= I28 /\ I30 + 1 <= I27 /\ I30 + 1 <= I26 /\ I30 <= I25 /\ I29 - 3 <= I28 /\ I29 - 2 <= I27 /\ I29 - 2 <= I26 /\ I29 - 3 <= I25] 4.14/4.13 f6(I33, I34, I35, I36) -> f4(I37, I38, I39, I40) [-1 <= I38 - 1 /\ 2 <= I37 - 1 /\ -1 <= I36 - 1 /\ 0 <= I35 - 1 /\ 0 <= I34 - 1 /\ -1 <= I33 - 1 /\ I38 <= I36 /\ I38 + 1 <= I34 /\ I38 <= I33] 4.14/4.13 f4(I41, I42, I43, I44) -> f6(I45, I46, I47, I48) [-1 <= I48 - 1 /\ 0 <= I47 - 1 /\ 0 <= I46 - 1 /\ -1 <= I45 - 1 /\ 0 <= I42 - 1 /\ 0 <= I41 - 1 /\ I48 + 1 <= I42 /\ I47 <= I41 /\ I46 <= I42 /\ I45 + 1 <= I42] 4.14/4.13 f3(I49, I50, I51, I52) -> f5(I53, I54, I55, I56) [0 <= I54 - 1 /\ 1 <= I53 - 1 /\ 2 <= I49 - 1 /\ I50 <= 0 /\ I53 + 1 <= I49] 4.14/4.13 f3(I57, I58, I59, I60) -> f4(I61, I62, I63, I64) [0 <= I62 - 1 /\ 1 <= I61 - 1 /\ 2 <= I57 - 1 /\ I58 <= 0 /\ I61 + 1 <= I57] 4.14/4.13 f3(I65, I66, I67, I68) -> f4(I69, I70, I71, I72) [-1 <= I70 - 1 /\ 1 <= I69 - 1 /\ 0 <= I65 - 1 /\ I66 <= 0 /\ I69 - 1 <= I65] 4.14/4.13 f2(I73, I74, I75, I76) -> f4(I77, I78, I79, I80) [1 <= I78 - 1 /\ 1 <= I77 - 1 /\ 1 <= I75 - 1 /\ 1 <= I74 - 1 /\ I78 <= I75 /\ I78 <= I74 /\ I77 <= I75 /\ I77 <= I74 /\ I76 <= 0 /\ I73 <= 1] 4.14/4.13 f2(I81, I82, I83, I84) -> f3(I85, I81 - 2, I86, I87) [4 <= I85 - 1 /\ 1 <= I83 - 1 /\ 1 <= I82 - 1 /\ I85 - 3 <= I83 /\ I85 - 3 <= I82 /\ I84 <= 0 /\ 1 <= I81 - 1] 4.14/4.13 f3(I88, I89, I90, I91) -> f3(I92, I89 - 1, I93, I94) [2 <= I92 - 1 /\ 0 <= I89 - 1 /\ 0 <= I88 - 1] 4.14/4.13 f2(I95, I96, I97, I98) -> f2(I95, I99, I100, I98 - 1) [2 <= I100 - 1 /\ 0 <= I99 - 1 /\ 0 <= I97 - 1 /\ 0 <= I96 - 1 /\ I100 - 2 <= I97 /\ 0 <= I98 - 1 /\ I99 <= I96] 4.14/4.13 f2(I101, I102, I103, I104) -> f3(I105, I101 - 1, I106, I107) [0 <= I105 - 1 /\ 0 <= I103 - 1 /\ I104 <= 0 /\ 1 <= I102 - 1] 4.14/4.13 f1(I108, I109, I110, I111) -> f2(I112, I113, I114, I115) [-1 <= I112 - 1 /\ 1 <= I109 - 1 /\ -1 <= I116 - 1 /\ I113 - 1 <= I108 /\ I114 - 1 <= I108 /\ 0 <= I108 - 1 /\ 1 <= I113 - 1 /\ 1 <= I114 - 1 /\ I116 - 2 = I115] 4.14/4.13 4.14/4.13 We use the basic value criterion with the projection function NU: 4.14/4.13 NU[f2#(z1,z2,z3,z4)] = z4 4.14/4.13 4.14/4.13 This gives the following inequalities: 4.14/4.13 2 <= I100 - 1 /\ 0 <= I99 - 1 /\ 0 <= I97 - 1 /\ 0 <= I96 - 1 /\ I100 - 2 <= I97 /\ 0 <= I98 - 1 /\ I99 <= I96 ==> I98 >! I98 - 1 4.14/4.13 4.14/4.13 All dependency pairs are strictly oriented, so the entire dependency pair problem may be removed. 4.14/7.11 EOF