13.52/13.79 YES 13.52/13.79 13.52/13.79 DP problem for innermost termination. 13.52/13.79 P = 13.52/13.79 init#(x1, x2, x3, x4, x5, x6) -> f1#(rnd1, rnd2, rnd3, rnd4, rnd5, rnd6) 13.52/13.79 f5#(I0, I1, I2, I3, I4, I5) -> f5#(I6, I1, I7, I8, I9, I10) [0 <= I6 - 1 /\ 4 <= I0 - 1 /\ 1 <= I1 - 1 /\ I6 + 4 <= I0] 13.52/13.79 f7#(I11, I12, I13, I14, I15, I16) -> f5#(I17, I14, I18, I19, I20, I21) [-1 <= I17 - 1 /\ -1 <= I15 - 1 /\ -1 <= I12 - 1 /\ 4 <= I11 - 1 /\ I17 <= I15 /\ I17 <= I12 /\ 1 <= I14 - 1 /\ I17 + 2 <= I11] 13.52/13.79 f5#(I22, I23, I24, I25, I26, I27) -> f7#(I28, I29, I30, I23, I31, I32) [-1 <= I31 - 1 /\ -1 <= I29 - 1 /\ 7 <= I28 - 1 /\ 0 <= I22 - 1 /\ I31 + 1 <= I22 /\ I29 + 1 <= I22 /\ 1 <= I23 - 1 /\ I28 - 7 <= I22] 13.52/13.79 f8#(I33, I34, I35, I36, I37, I38) -> f7#(I39, I40, I34, I35, I41, I42) [I36 + 6 <= I33 /\ -1 <= I41 - 1 /\ -1 <= I40 - 1 /\ 5 <= I39 - 1 /\ -1 <= I37 - 1 /\ 5 <= I33 - 1 /\ I41 <= I37 /\ I41 + 2 <= I33 /\ I40 <= I37 /\ I40 + 2 <= I33 /\ I39 <= I33] 13.52/13.79 f5#(I43, I44, I45, I46, I47, I48) -> f7#(I49, I50, I51, I44, I52, I53) [-1 <= I52 - 1 /\ -1 <= I50 - 1 /\ 4 <= I49 - 1 /\ 0 <= I43 - 1 /\ I52 + 1 <= I43 /\ 1 <= I44 - 1 /\ I50 + 1 <= I43] 13.52/13.79 f5#(I54, I55, I56, I57, I58, I59) -> f5#(I60, I55, I61, I62, I63, I64) [-1 <= I60 - 1 /\ 1 <= I54 - 1 /\ 1 <= I55 - 1 /\ I60 + 2 <= I54] 13.52/13.79 f6#(I65, I66, I67, I68, I69, I70) -> f6#(I65, I66, I67 + 1, I68, I69, I71) [I66 - 1 <= I66 - 1 /\ -1 <= I66 - 1 /\ I66 - 1 <= I65 - 1 /\ I66 <= I65 - 1 /\ 1 <= I70 - 1 /\ 0 <= I65 - 1 /\ I67 <= I68 - 1 /\ 0 <= I68 - 1] 13.52/13.79 f6#(I72, I73, I74, I75, I76, I77) -> f6#(I72, I73, I74 + 1, I75, I76, I77) [I73 - 1 <= I73 - 1 /\ -1 <= I73 - 1 /\ I73 - 1 <= I72 - 1 /\ I73 <= I72 - 1 /\ 1 <= I77 - 1 /\ 0 <= I72 - 1 /\ I74 <= I75 - 1 /\ 0 <= I75 - 1] 13.52/13.79 f6#(I78, I79, I80, I81, I82, I83) -> f2#(I79 - 1, I82, I83, I84, I85, I86) [I79 - 1 <= I79 - 1 /\ -1 <= I79 - 1 /\ I79 - 1 <= I78 - 1 /\ I79 <= I78 - 1 /\ 1 <= I83 - 1 /\ 0 <= I78 - 1 /\ I80 <= I81 - 1 /\ 0 <= I81 - 1] 13.52/13.79 f2#(I87, I88, I89, I90, I91, I92) -> f6#(I87, I87 - 1, 0, I93, I88, I89 + 1) [-1 <= I93 - 1 /\ I89 <= I88 - 1 /\ 0 <= I89 - 1 /\ 0 <= I87 - 1 /\ -1 <= I88 - 1] 13.52/13.79 f3#(I94, I95, I96, I97, I98, I99) -> f5#(I100, I96, I101, I102, I103, I104) [0 <= y1 - 1 /\ 1 <= I96 - 1 /\ I100 <= I95 /\ 0 <= I94 - 1 /\ -1 <= I95 - 1 /\ -1 <= I100 - 1] 13.52/13.79 f4#(I105, I106, I107, I108, I109, I110) -> f3#(I111, I112, I106, I113, I114, I115) [-1 <= I112 - 1 /\ 0 <= I111 - 1 /\ 0 <= I105 - 1 /\ I112 + 1 <= I105 /\ I111 <= I105] 13.52/13.79 f1#(I116, I117, I118, I119, I120, I121) -> f3#(I122, I123, I124, I125, I126, I127) [-1 <= I128 - 1 /\ 0 <= I117 - 1 /\ I122 <= I116 /\ 0 <= I116 - 1 /\ 0 <= I122 - 1 /\ -1 <= I123 - 1] 13.52/13.79 f1#(I129, I130, I131, I132, I133, I134) -> f2#(I135, I130, 1, I136, I137, I138) [0 <= I129 - 1 /\ 0 <= I130 - 1 /\ -1 <= I135 - 1] 13.52/13.79 R = 13.52/13.79 init(x1, x2, x3, x4, x5, x6) -> f1(rnd1, rnd2, rnd3, rnd4, rnd5, rnd6) 13.52/13.79 f5(I0, I1, I2, I3, I4, I5) -> f5(I6, I1, I7, I8, I9, I10) [0 <= I6 - 1 /\ 4 <= I0 - 1 /\ 1 <= I1 - 1 /\ I6 + 4 <= I0] 13.52/13.79 f7(I11, I12, I13, I14, I15, I16) -> f5(I17, I14, I18, I19, I20, I21) [-1 <= I17 - 1 /\ -1 <= I15 - 1 /\ -1 <= I12 - 1 /\ 4 <= I11 - 1 /\ I17 <= I15 /\ I17 <= I12 /\ 1 <= I14 - 1 /\ I17 + 2 <= I11] 13.52/13.79 f5(I22, I23, I24, I25, I26, I27) -> f7(I28, I29, I30, I23, I31, I32) [-1 <= I31 - 1 /\ -1 <= I29 - 1 /\ 7 <= I28 - 1 /\ 0 <= I22 - 1 /\ I31 + 1 <= I22 /\ I29 + 1 <= I22 /\ 1 <= I23 - 1 /\ I28 - 7 <= I22] 13.52/13.79 f8(I33, I34, I35, I36, I37, I38) -> f7(I39, I40, I34, I35, I41, I42) [I36 + 6 <= I33 /\ -1 <= I41 - 1 /\ -1 <= I40 - 1 /\ 5 <= I39 - 1 /\ -1 <= I37 - 1 /\ 5 <= I33 - 1 /\ I41 <= I37 /\ I41 + 2 <= I33 /\ I40 <= I37 /\ I40 + 2 <= I33 /\ I39 <= I33] 13.52/13.79 f5(I43, I44, I45, I46, I47, I48) -> f7(I49, I50, I51, I44, I52, I53) [-1 <= I52 - 1 /\ -1 <= I50 - 1 /\ 4 <= I49 - 1 /\ 0 <= I43 - 1 /\ I52 + 1 <= I43 /\ 1 <= I44 - 1 /\ I50 + 1 <= I43] 13.52/13.79 f5(I54, I55, I56, I57, I58, I59) -> f5(I60, I55, I61, I62, I63, I64) [-1 <= I60 - 1 /\ 1 <= I54 - 1 /\ 1 <= I55 - 1 /\ I60 + 2 <= I54] 13.52/13.79 f6(I65, I66, I67, I68, I69, I70) -> f6(I65, I66, I67 + 1, I68, I69, I71) [I66 - 1 <= I66 - 1 /\ -1 <= I66 - 1 /\ I66 - 1 <= I65 - 1 /\ I66 <= I65 - 1 /\ 1 <= I70 - 1 /\ 0 <= I65 - 1 /\ I67 <= I68 - 1 /\ 0 <= I68 - 1] 13.52/13.79 f6(I72, I73, I74, I75, I76, I77) -> f6(I72, I73, I74 + 1, I75, I76, I77) [I73 - 1 <= I73 - 1 /\ -1 <= I73 - 1 /\ I73 - 1 <= I72 - 1 /\ I73 <= I72 - 1 /\ 1 <= I77 - 1 /\ 0 <= I72 - 1 /\ I74 <= I75 - 1 /\ 0 <= I75 - 1] 13.52/13.79 f6(I78, I79, I80, I81, I82, I83) -> f2(I79 - 1, I82, I83, I84, I85, I86) [I79 - 1 <= I79 - 1 /\ -1 <= I79 - 1 /\ I79 - 1 <= I78 - 1 /\ I79 <= I78 - 1 /\ 1 <= I83 - 1 /\ 0 <= I78 - 1 /\ I80 <= I81 - 1 /\ 0 <= I81 - 1] 13.52/13.79 f2(I87, I88, I89, I90, I91, I92) -> f6(I87, I87 - 1, 0, I93, I88, I89 + 1) [-1 <= I93 - 1 /\ I89 <= I88 - 1 /\ 0 <= I89 - 1 /\ 0 <= I87 - 1 /\ -1 <= I88 - 1] 13.52/13.79 f3(I94, I95, I96, I97, I98, I99) -> f5(I100, I96, I101, I102, I103, I104) [0 <= y1 - 1 /\ 1 <= I96 - 1 /\ I100 <= I95 /\ 0 <= I94 - 1 /\ -1 <= I95 - 1 /\ -1 <= I100 - 1] 13.52/13.79 f4(I105, I106, I107, I108, I109, I110) -> f3(I111, I112, I106, I113, I114, I115) [-1 <= I112 - 1 /\ 0 <= I111 - 1 /\ 0 <= I105 - 1 /\ I112 + 1 <= I105 /\ I111 <= I105] 13.52/13.79 f1(I116, I117, I118, I119, I120, I121) -> f3(I122, I123, I124, I125, I126, I127) [-1 <= I128 - 1 /\ 0 <= I117 - 1 /\ I122 <= I116 /\ 0 <= I116 - 1 /\ 0 <= I122 - 1 /\ -1 <= I123 - 1] 13.52/13.79 f1(I129, I130, I131, I132, I133, I134) -> f2(I135, I130, 1, I136, I137, I138) [0 <= I129 - 1 /\ 0 <= I130 - 1 /\ -1 <= I135 - 1] 13.52/13.79 13.52/13.79 The dependency graph for this problem is: 13.52/13.79 0 -> 13, 14 13.52/13.79 1 -> 1, 3, 5, 6 13.52/13.79 2 -> 1, 3, 5, 6 13.52/13.79 3 -> 2 13.52/13.79 4 -> 2 13.52/13.79 5 -> 2 13.52/13.79 6 -> 1, 3, 5, 6 13.52/13.79 7 -> 7, 8, 9 13.52/13.79 8 -> 7, 8, 9 13.52/13.79 9 -> 10 13.52/13.79 10 -> 7, 8, 9 13.52/13.79 11 -> 1, 3, 5, 6 13.52/13.79 12 -> 11 13.52/13.79 13 -> 11 13.52/13.79 14 -> 10 13.52/13.79 Where: 13.52/13.79 0) init#(x1, x2, x3, x4, x5, x6) -> f1#(rnd1, rnd2, rnd3, rnd4, rnd5, rnd6) 13.52/13.79 1) f5#(I0, I1, I2, I3, I4, I5) -> f5#(I6, I1, I7, I8, I9, I10) [0 <= I6 - 1 /\ 4 <= I0 - 1 /\ 1 <= I1 - 1 /\ I6 + 4 <= I0] 13.52/13.79 2) f7#(I11, I12, I13, I14, I15, I16) -> f5#(I17, I14, I18, I19, I20, I21) [-1 <= I17 - 1 /\ -1 <= I15 - 1 /\ -1 <= I12 - 1 /\ 4 <= I11 - 1 /\ I17 <= I15 /\ I17 <= I12 /\ 1 <= I14 - 1 /\ I17 + 2 <= I11] 13.52/13.79 3) f5#(I22, I23, I24, I25, I26, I27) -> f7#(I28, I29, I30, I23, I31, I32) [-1 <= I31 - 1 /\ -1 <= I29 - 1 /\ 7 <= I28 - 1 /\ 0 <= I22 - 1 /\ I31 + 1 <= I22 /\ I29 + 1 <= I22 /\ 1 <= I23 - 1 /\ I28 - 7 <= I22] 13.52/13.79 4) f8#(I33, I34, I35, I36, I37, I38) -> f7#(I39, I40, I34, I35, I41, I42) [I36 + 6 <= I33 /\ -1 <= I41 - 1 /\ -1 <= I40 - 1 /\ 5 <= I39 - 1 /\ -1 <= I37 - 1 /\ 5 <= I33 - 1 /\ I41 <= I37 /\ I41 + 2 <= I33 /\ I40 <= I37 /\ I40 + 2 <= I33 /\ I39 <= I33] 13.52/13.79 5) f5#(I43, I44, I45, I46, I47, I48) -> f7#(I49, I50, I51, I44, I52, I53) [-1 <= I52 - 1 /\ -1 <= I50 - 1 /\ 4 <= I49 - 1 /\ 0 <= I43 - 1 /\ I52 + 1 <= I43 /\ 1 <= I44 - 1 /\ I50 + 1 <= I43] 13.52/13.79 6) f5#(I54, I55, I56, I57, I58, I59) -> f5#(I60, I55, I61, I62, I63, I64) [-1 <= I60 - 1 /\ 1 <= I54 - 1 /\ 1 <= I55 - 1 /\ I60 + 2 <= I54] 13.52/13.79 7) f6#(I65, I66, I67, I68, I69, I70) -> f6#(I65, I66, I67 + 1, I68, I69, I71) [I66 - 1 <= I66 - 1 /\ -1 <= I66 - 1 /\ I66 - 1 <= I65 - 1 /\ I66 <= I65 - 1 /\ 1 <= I70 - 1 /\ 0 <= I65 - 1 /\ I67 <= I68 - 1 /\ 0 <= I68 - 1] 13.52/13.79 8) f6#(I72, I73, I74, I75, I76, I77) -> f6#(I72, I73, I74 + 1, I75, I76, I77) [I73 - 1 <= I73 - 1 /\ -1 <= I73 - 1 /\ I73 - 1 <= I72 - 1 /\ I73 <= I72 - 1 /\ 1 <= I77 - 1 /\ 0 <= I72 - 1 /\ I74 <= I75 - 1 /\ 0 <= I75 - 1] 13.52/13.79 9) f6#(I78, I79, I80, I81, I82, I83) -> f2#(I79 - 1, I82, I83, I84, I85, I86) [I79 - 1 <= I79 - 1 /\ -1 <= I79 - 1 /\ I79 - 1 <= I78 - 1 /\ I79 <= I78 - 1 /\ 1 <= I83 - 1 /\ 0 <= I78 - 1 /\ I80 <= I81 - 1 /\ 0 <= I81 - 1] 13.52/13.79 10) f2#(I87, I88, I89, I90, I91, I92) -> f6#(I87, I87 - 1, 0, I93, I88, I89 + 1) [-1 <= I93 - 1 /\ I89 <= I88 - 1 /\ 0 <= I89 - 1 /\ 0 <= I87 - 1 /\ -1 <= I88 - 1] 13.52/13.79 11) f3#(I94, I95, I96, I97, I98, I99) -> f5#(I100, I96, I101, I102, I103, I104) [0 <= y1 - 1 /\ 1 <= I96 - 1 /\ I100 <= I95 /\ 0 <= I94 - 1 /\ -1 <= I95 - 1 /\ -1 <= I100 - 1] 13.52/13.79 12) f4#(I105, I106, I107, I108, I109, I110) -> f3#(I111, I112, I106, I113, I114, I115) [-1 <= I112 - 1 /\ 0 <= I111 - 1 /\ 0 <= I105 - 1 /\ I112 + 1 <= I105 /\ I111 <= I105] 13.52/13.79 13) f1#(I116, I117, I118, I119, I120, I121) -> f3#(I122, I123, I124, I125, I126, I127) [-1 <= I128 - 1 /\ 0 <= I117 - 1 /\ I122 <= I116 /\ 0 <= I116 - 1 /\ 0 <= I122 - 1 /\ -1 <= I123 - 1] 13.52/13.79 14) f1#(I129, I130, I131, I132, I133, I134) -> f2#(I135, I130, 1, I136, I137, I138) [0 <= I129 - 1 /\ 0 <= I130 - 1 /\ -1 <= I135 - 1] 13.52/13.79 13.52/13.79 We have the following SCCs. 13.52/13.79 { 1, 2, 3, 5, 6 } 13.52/13.79 { 7, 8, 9, 10 } 13.52/13.79 13.52/13.79 DP problem for innermost termination. 13.52/13.79 P = 13.52/13.79 f6#(I65, I66, I67, I68, I69, I70) -> f6#(I65, I66, I67 + 1, I68, I69, I71) [I66 - 1 <= I66 - 1 /\ -1 <= I66 - 1 /\ I66 - 1 <= I65 - 1 /\ I66 <= I65 - 1 /\ 1 <= I70 - 1 /\ 0 <= I65 - 1 /\ I67 <= I68 - 1 /\ 0 <= I68 - 1] 13.52/13.79 f6#(I72, I73, I74, I75, I76, I77) -> f6#(I72, I73, I74 + 1, I75, I76, I77) [I73 - 1 <= I73 - 1 /\ -1 <= I73 - 1 /\ I73 - 1 <= I72 - 1 /\ I73 <= I72 - 1 /\ 1 <= I77 - 1 /\ 0 <= I72 - 1 /\ I74 <= I75 - 1 /\ 0 <= I75 - 1] 13.52/13.79 f6#(I78, I79, I80, I81, I82, I83) -> f2#(I79 - 1, I82, I83, I84, I85, I86) [I79 - 1 <= I79 - 1 /\ -1 <= I79 - 1 /\ I79 - 1 <= I78 - 1 /\ I79 <= I78 - 1 /\ 1 <= I83 - 1 /\ 0 <= I78 - 1 /\ I80 <= I81 - 1 /\ 0 <= I81 - 1] 13.52/13.79 f2#(I87, I88, I89, I90, I91, I92) -> f6#(I87, I87 - 1, 0, I93, I88, I89 + 1) [-1 <= I93 - 1 /\ I89 <= I88 - 1 /\ 0 <= I89 - 1 /\ 0 <= I87 - 1 /\ -1 <= I88 - 1] 13.52/13.79 R = 13.52/13.79 init(x1, x2, x3, x4, x5, x6) -> f1(rnd1, rnd2, rnd3, rnd4, rnd5, rnd6) 13.52/13.79 f5(I0, I1, I2, I3, I4, I5) -> f5(I6, I1, I7, I8, I9, I10) [0 <= I6 - 1 /\ 4 <= I0 - 1 /\ 1 <= I1 - 1 /\ I6 + 4 <= I0] 13.52/13.79 f7(I11, I12, I13, I14, I15, I16) -> f5(I17, I14, I18, I19, I20, I21) [-1 <= I17 - 1 /\ -1 <= I15 - 1 /\ -1 <= I12 - 1 /\ 4 <= I11 - 1 /\ I17 <= I15 /\ I17 <= I12 /\ 1 <= I14 - 1 /\ I17 + 2 <= I11] 13.52/13.79 f5(I22, I23, I24, I25, I26, I27) -> f7(I28, I29, I30, I23, I31, I32) [-1 <= I31 - 1 /\ -1 <= I29 - 1 /\ 7 <= I28 - 1 /\ 0 <= I22 - 1 /\ I31 + 1 <= I22 /\ I29 + 1 <= I22 /\ 1 <= I23 - 1 /\ I28 - 7 <= I22] 13.52/13.79 f8(I33, I34, I35, I36, I37, I38) -> f7(I39, I40, I34, I35, I41, I42) [I36 + 6 <= I33 /\ -1 <= I41 - 1 /\ -1 <= I40 - 1 /\ 5 <= I39 - 1 /\ -1 <= I37 - 1 /\ 5 <= I33 - 1 /\ I41 <= I37 /\ I41 + 2 <= I33 /\ I40 <= I37 /\ I40 + 2 <= I33 /\ I39 <= I33] 13.52/13.79 f5(I43, I44, I45, I46, I47, I48) -> f7(I49, I50, I51, I44, I52, I53) [-1 <= I52 - 1 /\ -1 <= I50 - 1 /\ 4 <= I49 - 1 /\ 0 <= I43 - 1 /\ I52 + 1 <= I43 /\ 1 <= I44 - 1 /\ I50 + 1 <= I43] 13.52/13.79 f5(I54, I55, I56, I57, I58, I59) -> f5(I60, I55, I61, I62, I63, I64) [-1 <= I60 - 1 /\ 1 <= I54 - 1 /\ 1 <= I55 - 1 /\ I60 + 2 <= I54] 13.52/13.79 f6(I65, I66, I67, I68, I69, I70) -> f6(I65, I66, I67 + 1, I68, I69, I71) [I66 - 1 <= I66 - 1 /\ -1 <= I66 - 1 /\ I66 - 1 <= I65 - 1 /\ I66 <= I65 - 1 /\ 1 <= I70 - 1 /\ 0 <= I65 - 1 /\ I67 <= I68 - 1 /\ 0 <= I68 - 1] 13.52/13.79 f6(I72, I73, I74, I75, I76, I77) -> f6(I72, I73, I74 + 1, I75, I76, I77) [I73 - 1 <= I73 - 1 /\ -1 <= I73 - 1 /\ I73 - 1 <= I72 - 1 /\ I73 <= I72 - 1 /\ 1 <= I77 - 1 /\ 0 <= I72 - 1 /\ I74 <= I75 - 1 /\ 0 <= I75 - 1] 13.52/13.79 f6(I78, I79, I80, I81, I82, I83) -> f2(I79 - 1, I82, I83, I84, I85, I86) [I79 - 1 <= I79 - 1 /\ -1 <= I79 - 1 /\ I79 - 1 <= I78 - 1 /\ I79 <= I78 - 1 /\ 1 <= I83 - 1 /\ 0 <= I78 - 1 /\ I80 <= I81 - 1 /\ 0 <= I81 - 1] 13.52/13.79 f2(I87, I88, I89, I90, I91, I92) -> f6(I87, I87 - 1, 0, I93, I88, I89 + 1) [-1 <= I93 - 1 /\ I89 <= I88 - 1 /\ 0 <= I89 - 1 /\ 0 <= I87 - 1 /\ -1 <= I88 - 1] 13.52/13.79 f3(I94, I95, I96, I97, I98, I99) -> f5(I100, I96, I101, I102, I103, I104) [0 <= y1 - 1 /\ 1 <= I96 - 1 /\ I100 <= I95 /\ 0 <= I94 - 1 /\ -1 <= I95 - 1 /\ -1 <= I100 - 1] 13.52/13.79 f4(I105, I106, I107, I108, I109, I110) -> f3(I111, I112, I106, I113, I114, I115) [-1 <= I112 - 1 /\ 0 <= I111 - 1 /\ 0 <= I105 - 1 /\ I112 + 1 <= I105 /\ I111 <= I105] 13.52/13.79 f1(I116, I117, I118, I119, I120, I121) -> f3(I122, I123, I124, I125, I126, I127) [-1 <= I128 - 1 /\ 0 <= I117 - 1 /\ I122 <= I116 /\ 0 <= I116 - 1 /\ 0 <= I122 - 1 /\ -1 <= I123 - 1] 13.52/13.79 f1(I129, I130, I131, I132, I133, I134) -> f2(I135, I130, 1, I136, I137, I138) [0 <= I129 - 1 /\ 0 <= I130 - 1 /\ -1 <= I135 - 1] 13.52/13.79 13.52/13.79 We use the basic value criterion with the projection function NU: 13.52/13.79 NU[f2#(z1,z2,z3,z4,z5,z6)] = z1 13.52/13.79 NU[f6#(z1,z2,z3,z4,z5,z6)] = z2 13.52/13.79 13.52/13.79 This gives the following inequalities: 13.52/13.79 I66 - 1 <= I66 - 1 /\ -1 <= I66 - 1 /\ I66 - 1 <= I65 - 1 /\ I66 <= I65 - 1 /\ 1 <= I70 - 1 /\ 0 <= I65 - 1 /\ I67 <= I68 - 1 /\ 0 <= I68 - 1 ==> I66 (>! \union =) I66 13.52/13.79 I73 - 1 <= I73 - 1 /\ -1 <= I73 - 1 /\ I73 - 1 <= I72 - 1 /\ I73 <= I72 - 1 /\ 1 <= I77 - 1 /\ 0 <= I72 - 1 /\ I74 <= I75 - 1 /\ 0 <= I75 - 1 ==> I73 (>! \union =) I73 13.52/13.79 I79 - 1 <= I79 - 1 /\ -1 <= I79 - 1 /\ I79 - 1 <= I78 - 1 /\ I79 <= I78 - 1 /\ 1 <= I83 - 1 /\ 0 <= I78 - 1 /\ I80 <= I81 - 1 /\ 0 <= I81 - 1 ==> I79 >! I79 - 1 13.52/13.79 -1 <= I93 - 1 /\ I89 <= I88 - 1 /\ 0 <= I89 - 1 /\ 0 <= I87 - 1 /\ -1 <= I88 - 1 ==> I87 >! I87 - 1 13.52/13.79 13.52/13.79 We remove all the strictly oriented dependency pairs. 13.52/13.79 13.52/13.79 DP problem for innermost termination. 13.52/13.79 P = 13.52/13.79 f6#(I65, I66, I67, I68, I69, I70) -> f6#(I65, I66, I67 + 1, I68, I69, I71) [I66 - 1 <= I66 - 1 /\ -1 <= I66 - 1 /\ I66 - 1 <= I65 - 1 /\ I66 <= I65 - 1 /\ 1 <= I70 - 1 /\ 0 <= I65 - 1 /\ I67 <= I68 - 1 /\ 0 <= I68 - 1] 13.52/13.79 f6#(I72, I73, I74, I75, I76, I77) -> f6#(I72, I73, I74 + 1, I75, I76, I77) [I73 - 1 <= I73 - 1 /\ -1 <= I73 - 1 /\ I73 - 1 <= I72 - 1 /\ I73 <= I72 - 1 /\ 1 <= I77 - 1 /\ 0 <= I72 - 1 /\ I74 <= I75 - 1 /\ 0 <= I75 - 1] 13.52/13.79 R = 13.52/13.79 init(x1, x2, x3, x4, x5, x6) -> f1(rnd1, rnd2, rnd3, rnd4, rnd5, rnd6) 13.52/13.79 f5(I0, I1, I2, I3, I4, I5) -> f5(I6, I1, I7, I8, I9, I10) [0 <= I6 - 1 /\ 4 <= I0 - 1 /\ 1 <= I1 - 1 /\ I6 + 4 <= I0] 13.52/13.79 f7(I11, I12, I13, I14, I15, I16) -> f5(I17, I14, I18, I19, I20, I21) [-1 <= I17 - 1 /\ -1 <= I15 - 1 /\ -1 <= I12 - 1 /\ 4 <= I11 - 1 /\ I17 <= I15 /\ I17 <= I12 /\ 1 <= I14 - 1 /\ I17 + 2 <= I11] 13.52/13.79 f5(I22, I23, I24, I25, I26, I27) -> f7(I28, I29, I30, I23, I31, I32) [-1 <= I31 - 1 /\ -1 <= I29 - 1 /\ 7 <= I28 - 1 /\ 0 <= I22 - 1 /\ I31 + 1 <= I22 /\ I29 + 1 <= I22 /\ 1 <= I23 - 1 /\ I28 - 7 <= I22] 13.52/13.79 f8(I33, I34, I35, I36, I37, I38) -> f7(I39, I40, I34, I35, I41, I42) [I36 + 6 <= I33 /\ -1 <= I41 - 1 /\ -1 <= I40 - 1 /\ 5 <= I39 - 1 /\ -1 <= I37 - 1 /\ 5 <= I33 - 1 /\ I41 <= I37 /\ I41 + 2 <= I33 /\ I40 <= I37 /\ I40 + 2 <= I33 /\ I39 <= I33] 13.52/13.79 f5(I43, I44, I45, I46, I47, I48) -> f7(I49, I50, I51, I44, I52, I53) [-1 <= I52 - 1 /\ -1 <= I50 - 1 /\ 4 <= I49 - 1 /\ 0 <= I43 - 1 /\ I52 + 1 <= I43 /\ 1 <= I44 - 1 /\ I50 + 1 <= I43] 13.52/13.79 f5(I54, I55, I56, I57, I58, I59) -> f5(I60, I55, I61, I62, I63, I64) [-1 <= I60 - 1 /\ 1 <= I54 - 1 /\ 1 <= I55 - 1 /\ I60 + 2 <= I54] 13.52/13.79 f6(I65, I66, I67, I68, I69, I70) -> f6(I65, I66, I67 + 1, I68, I69, I71) [I66 - 1 <= I66 - 1 /\ -1 <= I66 - 1 /\ I66 - 1 <= I65 - 1 /\ I66 <= I65 - 1 /\ 1 <= I70 - 1 /\ 0 <= I65 - 1 /\ I67 <= I68 - 1 /\ 0 <= I68 - 1] 13.52/13.79 f6(I72, I73, I74, I75, I76, I77) -> f6(I72, I73, I74 + 1, I75, I76, I77) [I73 - 1 <= I73 - 1 /\ -1 <= I73 - 1 /\ I73 - 1 <= I72 - 1 /\ I73 <= I72 - 1 /\ 1 <= I77 - 1 /\ 0 <= I72 - 1 /\ I74 <= I75 - 1 /\ 0 <= I75 - 1] 13.52/13.79 f6(I78, I79, I80, I81, I82, I83) -> f2(I79 - 1, I82, I83, I84, I85, I86) [I79 - 1 <= I79 - 1 /\ -1 <= I79 - 1 /\ I79 - 1 <= I78 - 1 /\ I79 <= I78 - 1 /\ 1 <= I83 - 1 /\ 0 <= I78 - 1 /\ I80 <= I81 - 1 /\ 0 <= I81 - 1] 13.52/13.79 f2(I87, I88, I89, I90, I91, I92) -> f6(I87, I87 - 1, 0, I93, I88, I89 + 1) [-1 <= I93 - 1 /\ I89 <= I88 - 1 /\ 0 <= I89 - 1 /\ 0 <= I87 - 1 /\ -1 <= I88 - 1] 13.52/13.79 f3(I94, I95, I96, I97, I98, I99) -> f5(I100, I96, I101, I102, I103, I104) [0 <= y1 - 1 /\ 1 <= I96 - 1 /\ I100 <= I95 /\ 0 <= I94 - 1 /\ -1 <= I95 - 1 /\ -1 <= I100 - 1] 13.52/13.79 f4(I105, I106, I107, I108, I109, I110) -> f3(I111, I112, I106, I113, I114, I115) [-1 <= I112 - 1 /\ 0 <= I111 - 1 /\ 0 <= I105 - 1 /\ I112 + 1 <= I105 /\ I111 <= I105] 13.52/13.79 f1(I116, I117, I118, I119, I120, I121) -> f3(I122, I123, I124, I125, I126, I127) [-1 <= I128 - 1 /\ 0 <= I117 - 1 /\ I122 <= I116 /\ 0 <= I116 - 1 /\ 0 <= I122 - 1 /\ -1 <= I123 - 1] 13.52/13.79 f1(I129, I130, I131, I132, I133, I134) -> f2(I135, I130, 1, I136, I137, I138) [0 <= I129 - 1 /\ 0 <= I130 - 1 /\ -1 <= I135 - 1] 13.52/13.79 13.52/13.79 We use the reverse value criterion with the projection function NU: 13.52/13.79 NU[f6#(z1,z2,z3,z4,z5,z6)] = z4 - 1 + -1 * z3 13.52/13.79 13.52/13.79 This gives the following inequalities: 13.52/13.79 I66 - 1 <= I66 - 1 /\ -1 <= I66 - 1 /\ I66 - 1 <= I65 - 1 /\ I66 <= I65 - 1 /\ 1 <= I70 - 1 /\ 0 <= I65 - 1 /\ I67 <= I68 - 1 /\ 0 <= I68 - 1 ==> I68 - 1 + -1 * I67 > I68 - 1 + -1 * (I67 + 1) with I68 - 1 + -1 * I67 >= 0 13.52/13.79 I73 - 1 <= I73 - 1 /\ -1 <= I73 - 1 /\ I73 - 1 <= I72 - 1 /\ I73 <= I72 - 1 /\ 1 <= I77 - 1 /\ 0 <= I72 - 1 /\ I74 <= I75 - 1 /\ 0 <= I75 - 1 ==> I75 - 1 + -1 * I74 > I75 - 1 + -1 * (I74 + 1) with I75 - 1 + -1 * I74 >= 0 13.52/13.79 13.52/13.79 All dependency pairs are strictly oriented, so the entire dependency pair problem may be removed. 13.52/13.79 13.52/13.79 DP problem for innermost termination. 13.52/13.79 P = 13.52/13.79 f5#(I0, I1, I2, I3, I4, I5) -> f5#(I6, I1, I7, I8, I9, I10) [0 <= I6 - 1 /\ 4 <= I0 - 1 /\ 1 <= I1 - 1 /\ I6 + 4 <= I0] 13.52/13.79 f7#(I11, I12, I13, I14, I15, I16) -> f5#(I17, I14, I18, I19, I20, I21) [-1 <= I17 - 1 /\ -1 <= I15 - 1 /\ -1 <= I12 - 1 /\ 4 <= I11 - 1 /\ I17 <= I15 /\ I17 <= I12 /\ 1 <= I14 - 1 /\ I17 + 2 <= I11] 13.52/13.79 f5#(I22, I23, I24, I25, I26, I27) -> f7#(I28, I29, I30, I23, I31, I32) [-1 <= I31 - 1 /\ -1 <= I29 - 1 /\ 7 <= I28 - 1 /\ 0 <= I22 - 1 /\ I31 + 1 <= I22 /\ I29 + 1 <= I22 /\ 1 <= I23 - 1 /\ I28 - 7 <= I22] 13.52/13.79 f5#(I43, I44, I45, I46, I47, I48) -> f7#(I49, I50, I51, I44, I52, I53) [-1 <= I52 - 1 /\ -1 <= I50 - 1 /\ 4 <= I49 - 1 /\ 0 <= I43 - 1 /\ I52 + 1 <= I43 /\ 1 <= I44 - 1 /\ I50 + 1 <= I43] 13.52/13.79 f5#(I54, I55, I56, I57, I58, I59) -> f5#(I60, I55, I61, I62, I63, I64) [-1 <= I60 - 1 /\ 1 <= I54 - 1 /\ 1 <= I55 - 1 /\ I60 + 2 <= I54] 13.52/13.79 R = 13.52/13.79 init(x1, x2, x3, x4, x5, x6) -> f1(rnd1, rnd2, rnd3, rnd4, rnd5, rnd6) 13.52/13.79 f5(I0, I1, I2, I3, I4, I5) -> f5(I6, I1, I7, I8, I9, I10) [0 <= I6 - 1 /\ 4 <= I0 - 1 /\ 1 <= I1 - 1 /\ I6 + 4 <= I0] 13.52/13.79 f7(I11, I12, I13, I14, I15, I16) -> f5(I17, I14, I18, I19, I20, I21) [-1 <= I17 - 1 /\ -1 <= I15 - 1 /\ -1 <= I12 - 1 /\ 4 <= I11 - 1 /\ I17 <= I15 /\ I17 <= I12 /\ 1 <= I14 - 1 /\ I17 + 2 <= I11] 13.52/13.79 f5(I22, I23, I24, I25, I26, I27) -> f7(I28, I29, I30, I23, I31, I32) [-1 <= I31 - 1 /\ -1 <= I29 - 1 /\ 7 <= I28 - 1 /\ 0 <= I22 - 1 /\ I31 + 1 <= I22 /\ I29 + 1 <= I22 /\ 1 <= I23 - 1 /\ I28 - 7 <= I22] 13.52/13.79 f8(I33, I34, I35, I36, I37, I38) -> f7(I39, I40, I34, I35, I41, I42) [I36 + 6 <= I33 /\ -1 <= I41 - 1 /\ -1 <= I40 - 1 /\ 5 <= I39 - 1 /\ -1 <= I37 - 1 /\ 5 <= I33 - 1 /\ I41 <= I37 /\ I41 + 2 <= I33 /\ I40 <= I37 /\ I40 + 2 <= I33 /\ I39 <= I33] 13.52/13.79 f5(I43, I44, I45, I46, I47, I48) -> f7(I49, I50, I51, I44, I52, I53) [-1 <= I52 - 1 /\ -1 <= I50 - 1 /\ 4 <= I49 - 1 /\ 0 <= I43 - 1 /\ I52 + 1 <= I43 /\ 1 <= I44 - 1 /\ I50 + 1 <= I43] 13.52/13.79 f5(I54, I55, I56, I57, I58, I59) -> f5(I60, I55, I61, I62, I63, I64) [-1 <= I60 - 1 /\ 1 <= I54 - 1 /\ 1 <= I55 - 1 /\ I60 + 2 <= I54] 13.52/13.79 f6(I65, I66, I67, I68, I69, I70) -> f6(I65, I66, I67 + 1, I68, I69, I71) [I66 - 1 <= I66 - 1 /\ -1 <= I66 - 1 /\ I66 - 1 <= I65 - 1 /\ I66 <= I65 - 1 /\ 1 <= I70 - 1 /\ 0 <= I65 - 1 /\ I67 <= I68 - 1 /\ 0 <= I68 - 1] 13.52/13.79 f6(I72, I73, I74, I75, I76, I77) -> f6(I72, I73, I74 + 1, I75, I76, I77) [I73 - 1 <= I73 - 1 /\ -1 <= I73 - 1 /\ I73 - 1 <= I72 - 1 /\ I73 <= I72 - 1 /\ 1 <= I77 - 1 /\ 0 <= I72 - 1 /\ I74 <= I75 - 1 /\ 0 <= I75 - 1] 13.52/13.79 f6(I78, I79, I80, I81, I82, I83) -> f2(I79 - 1, I82, I83, I84, I85, I86) [I79 - 1 <= I79 - 1 /\ -1 <= I79 - 1 /\ I79 - 1 <= I78 - 1 /\ I79 <= I78 - 1 /\ 1 <= I83 - 1 /\ 0 <= I78 - 1 /\ I80 <= I81 - 1 /\ 0 <= I81 - 1] 13.52/13.79 f2(I87, I88, I89, I90, I91, I92) -> f6(I87, I87 - 1, 0, I93, I88, I89 + 1) [-1 <= I93 - 1 /\ I89 <= I88 - 1 /\ 0 <= I89 - 1 /\ 0 <= I87 - 1 /\ -1 <= I88 - 1] 13.52/13.79 f3(I94, I95, I96, I97, I98, I99) -> f5(I100, I96, I101, I102, I103, I104) [0 <= y1 - 1 /\ 1 <= I96 - 1 /\ I100 <= I95 /\ 0 <= I94 - 1 /\ -1 <= I95 - 1 /\ -1 <= I100 - 1] 13.52/13.79 f4(I105, I106, I107, I108, I109, I110) -> f3(I111, I112, I106, I113, I114, I115) [-1 <= I112 - 1 /\ 0 <= I111 - 1 /\ 0 <= I105 - 1 /\ I112 + 1 <= I105 /\ I111 <= I105] 13.52/13.79 f1(I116, I117, I118, I119, I120, I121) -> f3(I122, I123, I124, I125, I126, I127) [-1 <= I128 - 1 /\ 0 <= I117 - 1 /\ I122 <= I116 /\ 0 <= I116 - 1 /\ 0 <= I122 - 1 /\ -1 <= I123 - 1] 13.52/13.79 f1(I129, I130, I131, I132, I133, I134) -> f2(I135, I130, 1, I136, I137, I138) [0 <= I129 - 1 /\ 0 <= I130 - 1 /\ -1 <= I135 - 1] 13.52/13.79 13.52/13.79 We use the basic value criterion with the projection function NU: 13.52/13.79 NU[f7#(z1,z2,z3,z4,z5,z6)] = z5 13.52/13.79 NU[f5#(z1,z2,z3,z4,z5,z6)] = z1 13.52/13.79 13.52/13.79 This gives the following inequalities: 13.52/13.79 0 <= I6 - 1 /\ 4 <= I0 - 1 /\ 1 <= I1 - 1 /\ I6 + 4 <= I0 ==> I0 >! I6 13.52/13.79 -1 <= I17 - 1 /\ -1 <= I15 - 1 /\ -1 <= I12 - 1 /\ 4 <= I11 - 1 /\ I17 <= I15 /\ I17 <= I12 /\ 1 <= I14 - 1 /\ I17 + 2 <= I11 ==> I15 (>! \union =) I17 13.52/13.79 -1 <= I31 - 1 /\ -1 <= I29 - 1 /\ 7 <= I28 - 1 /\ 0 <= I22 - 1 /\ I31 + 1 <= I22 /\ I29 + 1 <= I22 /\ 1 <= I23 - 1 /\ I28 - 7 <= I22 ==> I22 >! I31 13.52/13.79 -1 <= I52 - 1 /\ -1 <= I50 - 1 /\ 4 <= I49 - 1 /\ 0 <= I43 - 1 /\ I52 + 1 <= I43 /\ 1 <= I44 - 1 /\ I50 + 1 <= I43 ==> I43 >! I52 13.52/13.79 -1 <= I60 - 1 /\ 1 <= I54 - 1 /\ 1 <= I55 - 1 /\ I60 + 2 <= I54 ==> I54 >! I60 13.52/13.79 13.52/13.79 We remove all the strictly oriented dependency pairs. 13.52/13.79 13.52/13.79 DP problem for innermost termination. 13.52/13.79 P = 13.52/13.79 f7#(I11, I12, I13, I14, I15, I16) -> f5#(I17, I14, I18, I19, I20, I21) [-1 <= I17 - 1 /\ -1 <= I15 - 1 /\ -1 <= I12 - 1 /\ 4 <= I11 - 1 /\ I17 <= I15 /\ I17 <= I12 /\ 1 <= I14 - 1 /\ I17 + 2 <= I11] 13.52/13.79 R = 13.52/13.79 init(x1, x2, x3, x4, x5, x6) -> f1(rnd1, rnd2, rnd3, rnd4, rnd5, rnd6) 13.52/13.79 f5(I0, I1, I2, I3, I4, I5) -> f5(I6, I1, I7, I8, I9, I10) [0 <= I6 - 1 /\ 4 <= I0 - 1 /\ 1 <= I1 - 1 /\ I6 + 4 <= I0] 13.52/13.79 f7(I11, I12, I13, I14, I15, I16) -> f5(I17, I14, I18, I19, I20, I21) [-1 <= I17 - 1 /\ -1 <= I15 - 1 /\ -1 <= I12 - 1 /\ 4 <= I11 - 1 /\ I17 <= I15 /\ I17 <= I12 /\ 1 <= I14 - 1 /\ I17 + 2 <= I11] 13.52/13.79 f5(I22, I23, I24, I25, I26, I27) -> f7(I28, I29, I30, I23, I31, I32) [-1 <= I31 - 1 /\ -1 <= I29 - 1 /\ 7 <= I28 - 1 /\ 0 <= I22 - 1 /\ I31 + 1 <= I22 /\ I29 + 1 <= I22 /\ 1 <= I23 - 1 /\ I28 - 7 <= I22] 13.52/13.79 f8(I33, I34, I35, I36, I37, I38) -> f7(I39, I40, I34, I35, I41, I42) [I36 + 6 <= I33 /\ -1 <= I41 - 1 /\ -1 <= I40 - 1 /\ 5 <= I39 - 1 /\ -1 <= I37 - 1 /\ 5 <= I33 - 1 /\ I41 <= I37 /\ I41 + 2 <= I33 /\ I40 <= I37 /\ I40 + 2 <= I33 /\ I39 <= I33] 13.52/13.79 f5(I43, I44, I45, I46, I47, I48) -> f7(I49, I50, I51, I44, I52, I53) [-1 <= I52 - 1 /\ -1 <= I50 - 1 /\ 4 <= I49 - 1 /\ 0 <= I43 - 1 /\ I52 + 1 <= I43 /\ 1 <= I44 - 1 /\ I50 + 1 <= I43] 13.52/13.79 f5(I54, I55, I56, I57, I58, I59) -> f5(I60, I55, I61, I62, I63, I64) [-1 <= I60 - 1 /\ 1 <= I54 - 1 /\ 1 <= I55 - 1 /\ I60 + 2 <= I54] 13.52/13.79 f6(I65, I66, I67, I68, I69, I70) -> f6(I65, I66, I67 + 1, I68, I69, I71) [I66 - 1 <= I66 - 1 /\ -1 <= I66 - 1 /\ I66 - 1 <= I65 - 1 /\ I66 <= I65 - 1 /\ 1 <= I70 - 1 /\ 0 <= I65 - 1 /\ I67 <= I68 - 1 /\ 0 <= I68 - 1] 13.52/13.79 f6(I72, I73, I74, I75, I76, I77) -> f6(I72, I73, I74 + 1, I75, I76, I77) [I73 - 1 <= I73 - 1 /\ -1 <= I73 - 1 /\ I73 - 1 <= I72 - 1 /\ I73 <= I72 - 1 /\ 1 <= I77 - 1 /\ 0 <= I72 - 1 /\ I74 <= I75 - 1 /\ 0 <= I75 - 1] 13.52/13.79 f6(I78, I79, I80, I81, I82, I83) -> f2(I79 - 1, I82, I83, I84, I85, I86) [I79 - 1 <= I79 - 1 /\ -1 <= I79 - 1 /\ I79 - 1 <= I78 - 1 /\ I79 <= I78 - 1 /\ 1 <= I83 - 1 /\ 0 <= I78 - 1 /\ I80 <= I81 - 1 /\ 0 <= I81 - 1] 13.52/13.79 f2(I87, I88, I89, I90, I91, I92) -> f6(I87, I87 - 1, 0, I93, I88, I89 + 1) [-1 <= I93 - 1 /\ I89 <= I88 - 1 /\ 0 <= I89 - 1 /\ 0 <= I87 - 1 /\ -1 <= I88 - 1] 13.52/13.79 f3(I94, I95, I96, I97, I98, I99) -> f5(I100, I96, I101, I102, I103, I104) [0 <= y1 - 1 /\ 1 <= I96 - 1 /\ I100 <= I95 /\ 0 <= I94 - 1 /\ -1 <= I95 - 1 /\ -1 <= I100 - 1] 13.52/13.79 f4(I105, I106, I107, I108, I109, I110) -> f3(I111, I112, I106, I113, I114, I115) [-1 <= I112 - 1 /\ 0 <= I111 - 1 /\ 0 <= I105 - 1 /\ I112 + 1 <= I105 /\ I111 <= I105] 13.52/13.79 f1(I116, I117, I118, I119, I120, I121) -> f3(I122, I123, I124, I125, I126, I127) [-1 <= I128 - 1 /\ 0 <= I117 - 1 /\ I122 <= I116 /\ 0 <= I116 - 1 /\ 0 <= I122 - 1 /\ -1 <= I123 - 1] 13.52/13.79 f1(I129, I130, I131, I132, I133, I134) -> f2(I135, I130, 1, I136, I137, I138) [0 <= I129 - 1 /\ 0 <= I130 - 1 /\ -1 <= I135 - 1] 13.52/13.79 13.52/13.79 The dependency graph for this problem is: 13.52/13.79 2 -> 13.52/13.79 Where: 13.52/13.79 2) f7#(I11, I12, I13, I14, I15, I16) -> f5#(I17, I14, I18, I19, I20, I21) [-1 <= I17 - 1 /\ -1 <= I15 - 1 /\ -1 <= I12 - 1 /\ 4 <= I11 - 1 /\ I17 <= I15 /\ I17 <= I12 /\ 1 <= I14 - 1 /\ I17 + 2 <= I11] 13.52/13.79 13.52/13.79 We have the following SCCs. 13.52/13.79 13.52/16.77 EOF