/export/starexec/sandbox2/solver/bin/starexec_run_Transition /export/starexec/sandbox2/benchmark/theBenchmark.smt2 /export/starexec/sandbox2/output/output_files -------------------------------------------------------------------------------- YES DP problem for innermost termination. P = f18#(x1, x2, x3, x4, x5, x6, x7, x8) -> f17#(x1, x2, x3, x4, x5, x6, x7, x8) f17#(I0, I1, I2, I3, I4, I5, I6, I7) -> f1#(I0, I0, 0, I3, I0, 0, I6, I7) f5#(I8, I9, I10, I11, I12, I13, I14, I15) -> f3#(I8, I9, I10, I11, I12, I13, I14, I15) f2#(I16, I17, I18, I19, I20, I21, I22, I23) -> f16#(I16, I17, I18, I19, I20, I21, I22, I23) [1 + I18 <= I20] f2#(I24, I25, I26, I27, I28, I29, I30, I31) -> f8#(I24, I25, 0, I27, I28, I29, I30, I31) [I28 <= I26] f16#(I32, I33, I34, I35, I36, I37, I38, I39) -> f15#(I32, I33, I34, I35, I36, I37, I38, I39) [I37 <= I34 /\ I34 <= I37] f16#(I40, I41, I42, I43, I44, I45, I46, I47) -> f14#(I40, I41, I42, I43, I44, I45, I46, I47) [1 + I42 <= I45] f16#(I48, I49, I50, I51, I52, I53, I54, I55) -> f14#(I48, I49, I50, I51, I52, I53, I54, I55) [1 + I53 <= I50] f15#(I56, I57, I58, I59, I60, I61, I62, I63) -> f1#(I56, I57, 1 + I58, I59, I60, I61, I62, I63) f7#(I64, I65, I66, I67, I68, I69, I70, I71) -> f10#(I64, I65, I66, I67, I68, I69, I70, I71) f14#(I72, I73, I74, I75, I76, I77, I78, I79) -> f15#(I72, I73, I74, I75, I76, I77, I78, I79) f9#(I80, I81, I82, I83, I84, I85, I86, I87) -> f12#(I80, I81, I82, 0, I84, I85, I86, I87) [1 + I82 <= I84] f9#(I88, I89, I90, I91, I92, I93, I94, I95) -> f7#(I88, I89, 0, I91, I92, I93, I94, I95) [I92 <= I90] f13#(I96, I97, I98, I99, I100, I101, I102, I103) -> f11#(I96, I97, I98, I99, I100, I101, rnd7, rnd8) [rnd8 = rnd8 /\ rnd7 = rnd7 /\ 1 + I99 <= I97] f13#(I104, I105, I106, I107, I108, I109, I110, I111) -> f8#(I104, I105, 1 + I106, I107, I108, I109, I110, I111) [I105 <= I107] f12#(I112, I113, I114, I115, I116, I117, I118, I119) -> f13#(I112, I113, I114, I115, I116, I117, I118, I119) f11#(I120, I121, I122, I123, I124, I125, I126, I127) -> f12#(I120, I121, I122, 1 + I123, I124, I125, I126, I127) f10#(I128, I129, I130, I131, I132, I133, I134, I135) -> f6#(I128, I129, I130, I131, I132, I133, I136, I137) [I137 = I137 /\ I136 = I136 /\ 1 + I130 <= I129] f10#(I138, I139, I140, I141, I142, I143, I144, I145) -> f5#(I138, I139, 0, I141, I142, I143, I144, I145) [I139 <= I140] f8#(I146, I147, I148, I149, I150, I151, I152, I153) -> f9#(I146, I147, I148, I149, I150, I151, I152, I153) f6#(I162, I163, I164, I165, I166, I167, I168, I169) -> f7#(I162, I163, 1 + I164, I165, I166, I167, I168, I169) f3#(I170, I171, I172, I173, I174, I175, I176, I177) -> f5#(I170, I171, 1 + I172, I173, I174, I175, I176, I177) [1 + I172 <= I174] f1#(I186, I187, I188, I189, I190, I191, I192, I193) -> f2#(I186, I187, I188, I189, I190, I191, I192, I193) R = f18(x1, x2, x3, x4, x5, x6, x7, x8) -> f17(x1, x2, x3, x4, x5, x6, x7, x8) f17(I0, I1, I2, I3, I4, I5, I6, I7) -> f1(I0, I0, 0, I3, I0, 0, I6, I7) f5(I8, I9, I10, I11, I12, I13, I14, I15) -> f3(I8, I9, I10, I11, I12, I13, I14, I15) f2(I16, I17, I18, I19, I20, I21, I22, I23) -> f16(I16, I17, I18, I19, I20, I21, I22, I23) [1 + I18 <= I20] f2(I24, I25, I26, I27, I28, I29, I30, I31) -> f8(I24, I25, 0, I27, I28, I29, I30, I31) [I28 <= I26] f16(I32, I33, I34, I35, I36, I37, I38, I39) -> f15(I32, I33, I34, I35, I36, I37, I38, I39) [I37 <= I34 /\ I34 <= I37] f16(I40, I41, I42, I43, I44, I45, I46, I47) -> f14(I40, I41, I42, I43, I44, I45, I46, I47) [1 + I42 <= I45] f16(I48, I49, I50, I51, I52, I53, I54, I55) -> f14(I48, I49, I50, I51, I52, I53, I54, I55) [1 + I53 <= I50] f15(I56, I57, I58, I59, I60, I61, I62, I63) -> f1(I56, I57, 1 + I58, I59, I60, I61, I62, I63) f7(I64, I65, I66, I67, I68, I69, I70, I71) -> f10(I64, I65, I66, I67, I68, I69, I70, I71) f14(I72, I73, I74, I75, I76, I77, I78, I79) -> f15(I72, I73, I74, I75, I76, I77, I78, I79) f9(I80, I81, I82, I83, I84, I85, I86, I87) -> f12(I80, I81, I82, 0, I84, I85, I86, I87) [1 + I82 <= I84] f9(I88, I89, I90, I91, I92, I93, I94, I95) -> f7(I88, I89, 0, I91, I92, I93, I94, I95) [I92 <= I90] f13(I96, I97, I98, I99, I100, I101, I102, I103) -> f11(I96, I97, I98, I99, I100, I101, rnd7, rnd8) [rnd8 = rnd8 /\ rnd7 = rnd7 /\ 1 + I99 <= I97] f13(I104, I105, I106, I107, I108, I109, I110, I111) -> f8(I104, I105, 1 + I106, I107, I108, I109, I110, I111) [I105 <= I107] f12(I112, I113, I114, I115, I116, I117, I118, I119) -> f13(I112, I113, I114, I115, I116, I117, I118, I119) f11(I120, I121, I122, I123, I124, I125, I126, I127) -> f12(I120, I121, I122, 1 + I123, I124, I125, I126, I127) f10(I128, I129, I130, I131, I132, I133, I134, I135) -> f6(I128, I129, I130, I131, I132, I133, I136, I137) [I137 = I137 /\ I136 = I136 /\ 1 + I130 <= I129] f10(I138, I139, I140, I141, I142, I143, I144, I145) -> f5(I138, I139, 0, I141, I142, I143, I144, I145) [I139 <= I140] f8(I146, I147, I148, I149, I150, I151, I152, I153) -> f9(I146, I147, I148, I149, I150, I151, I152, I153) f6(I154, I155, I156, I157, I158, I159, I160, I161) -> f4(I154, I155, I156, I157, I158, I159, I160, I161) f6(I162, I163, I164, I165, I166, I167, I168, I169) -> f7(I162, I163, 1 + I164, I165, I166, I167, I168, I169) f3(I170, I171, I172, I173, I174, I175, I176, I177) -> f5(I170, I171, 1 + I172, I173, I174, I175, I176, I177) [1 + I172 <= I174] f3(I178, I179, I180, I181, I182, I183, I184, I185) -> f4(I178, I179, I180, I181, I182, I183, I184, I185) [I182 <= I180] f1(I186, I187, I188, I189, I190, I191, I192, I193) -> f2(I186, I187, I188, I189, I190, I191, I192, I193) The dependency graph for this problem is: 0 -> 1 1 -> 22 2 -> 21 3 -> 5, 6, 7 4 -> 19 5 -> 8 6 -> 10 7 -> 10 8 -> 22 9 -> 17, 18 10 -> 8 11 -> 15 12 -> 9 13 -> 16 14 -> 19 15 -> 13, 14 16 -> 15 17 -> 20 18 -> 2 19 -> 11, 12 20 -> 9 21 -> 2 22 -> 3, 4 Where: 0) f18#(x1, x2, x3, x4, x5, x6, x7, x8) -> f17#(x1, x2, x3, x4, x5, x6, x7, x8) 1) f17#(I0, I1, I2, I3, I4, I5, I6, I7) -> f1#(I0, I0, 0, I3, I0, 0, I6, I7) 2) f5#(I8, I9, I10, I11, I12, I13, I14, I15) -> f3#(I8, I9, I10, I11, I12, I13, I14, I15) 3) f2#(I16, I17, I18, I19, I20, I21, I22, I23) -> f16#(I16, I17, I18, I19, I20, I21, I22, I23) [1 + I18 <= I20] 4) f2#(I24, I25, I26, I27, I28, I29, I30, I31) -> f8#(I24, I25, 0, I27, I28, I29, I30, I31) [I28 <= I26] 5) f16#(I32, I33, I34, I35, I36, I37, I38, I39) -> f15#(I32, I33, I34, I35, I36, I37, I38, I39) [I37 <= I34 /\ I34 <= I37] 6) f16#(I40, I41, I42, I43, I44, I45, I46, I47) -> f14#(I40, I41, I42, I43, I44, I45, I46, I47) [1 + I42 <= I45] 7) f16#(I48, I49, I50, I51, I52, I53, I54, I55) -> f14#(I48, I49, I50, I51, I52, I53, I54, I55) [1 + I53 <= I50] 8) f15#(I56, I57, I58, I59, I60, I61, I62, I63) -> f1#(I56, I57, 1 + I58, I59, I60, I61, I62, I63) 9) f7#(I64, I65, I66, I67, I68, I69, I70, I71) -> f10#(I64, I65, I66, I67, I68, I69, I70, I71) 10) f14#(I72, I73, I74, I75, I76, I77, I78, I79) -> f15#(I72, I73, I74, I75, I76, I77, I78, I79) 11) f9#(I80, I81, I82, I83, I84, I85, I86, I87) -> f12#(I80, I81, I82, 0, I84, I85, I86, I87) [1 + I82 <= I84] 12) f9#(I88, I89, I90, I91, I92, I93, I94, I95) -> f7#(I88, I89, 0, I91, I92, I93, I94, I95) [I92 <= I90] 13) f13#(I96, I97, I98, I99, I100, I101, I102, I103) -> f11#(I96, I97, I98, I99, I100, I101, rnd7, rnd8) [rnd8 = rnd8 /\ rnd7 = rnd7 /\ 1 + I99 <= I97] 14) f13#(I104, I105, I106, I107, I108, I109, I110, I111) -> f8#(I104, I105, 1 + I106, I107, I108, I109, I110, I111) [I105 <= I107] 15) f12#(I112, I113, I114, I115, I116, I117, I118, I119) -> f13#(I112, I113, I114, I115, I116, I117, I118, I119) 16) f11#(I120, I121, I122, I123, I124, I125, I126, I127) -> f12#(I120, I121, I122, 1 + I123, I124, I125, I126, I127) 17) f10#(I128, I129, I130, I131, I132, I133, I134, I135) -> f6#(I128, I129, I130, I131, I132, I133, I136, I137) [I137 = I137 /\ I136 = I136 /\ 1 + I130 <= I129] 18) f10#(I138, I139, I140, I141, I142, I143, I144, I145) -> f5#(I138, I139, 0, I141, I142, I143, I144, I145) [I139 <= I140] 19) f8#(I146, I147, I148, I149, I150, I151, I152, I153) -> f9#(I146, I147, I148, I149, I150, I151, I152, I153) 20) f6#(I162, I163, I164, I165, I166, I167, I168, I169) -> f7#(I162, I163, 1 + I164, I165, I166, I167, I168, I169) 21) f3#(I170, I171, I172, I173, I174, I175, I176, I177) -> f5#(I170, I171, 1 + I172, I173, I174, I175, I176, I177) [1 + I172 <= I174] 22) f1#(I186, I187, I188, I189, I190, I191, I192, I193) -> f2#(I186, I187, I188, I189, I190, I191, I192, I193) We have the following SCCs. { 3, 5, 6, 7, 8, 10, 22 } { 11, 13, 14, 15, 16, 19 } { 9, 17, 20 } { 2, 21 } DP problem for innermost termination. P = f5#(I8, I9, I10, I11, I12, I13, I14, I15) -> f3#(I8, I9, I10, I11, I12, I13, I14, I15) f3#(I170, I171, I172, I173, I174, I175, I176, I177) -> f5#(I170, I171, 1 + I172, I173, I174, I175, I176, I177) [1 + I172 <= I174] R = f18(x1, x2, x3, x4, x5, x6, x7, x8) -> f17(x1, x2, x3, x4, x5, x6, x7, x8) f17(I0, I1, I2, I3, I4, I5, I6, I7) -> f1(I0, I0, 0, I3, I0, 0, I6, I7) f5(I8, I9, I10, I11, I12, I13, I14, I15) -> f3(I8, I9, I10, I11, I12, I13, I14, I15) f2(I16, I17, I18, I19, I20, I21, I22, I23) -> f16(I16, I17, I18, I19, I20, I21, I22, I23) [1 + I18 <= I20] f2(I24, I25, I26, I27, I28, I29, I30, I31) -> f8(I24, I25, 0, I27, I28, I29, I30, I31) [I28 <= I26] f16(I32, I33, I34, I35, I36, I37, I38, I39) -> f15(I32, I33, I34, I35, I36, I37, I38, I39) [I37 <= I34 /\ I34 <= I37] f16(I40, I41, I42, I43, I44, I45, I46, I47) -> f14(I40, I41, I42, I43, I44, I45, I46, I47) [1 + I42 <= I45] f16(I48, I49, I50, I51, I52, I53, I54, I55) -> f14(I48, I49, I50, I51, I52, I53, I54, I55) [1 + I53 <= I50] f15(I56, I57, I58, I59, I60, I61, I62, I63) -> f1(I56, I57, 1 + I58, I59, I60, I61, I62, I63) f7(I64, I65, I66, I67, I68, I69, I70, I71) -> f10(I64, I65, I66, I67, I68, I69, I70, I71) f14(I72, I73, I74, I75, I76, I77, I78, I79) -> f15(I72, I73, I74, I75, I76, I77, I78, I79) f9(I80, I81, I82, I83, I84, I85, I86, I87) -> f12(I80, I81, I82, 0, I84, I85, I86, I87) [1 + I82 <= I84] f9(I88, I89, I90, I91, I92, I93, I94, I95) -> f7(I88, I89, 0, I91, I92, I93, I94, I95) [I92 <= I90] f13(I96, I97, I98, I99, I100, I101, I102, I103) -> f11(I96, I97, I98, I99, I100, I101, rnd7, rnd8) [rnd8 = rnd8 /\ rnd7 = rnd7 /\ 1 + I99 <= I97] f13(I104, I105, I106, I107, I108, I109, I110, I111) -> f8(I104, I105, 1 + I106, I107, I108, I109, I110, I111) [I105 <= I107] f12(I112, I113, I114, I115, I116, I117, I118, I119) -> f13(I112, I113, I114, I115, I116, I117, I118, I119) f11(I120, I121, I122, I123, I124, I125, I126, I127) -> f12(I120, I121, I122, 1 + I123, I124, I125, I126, I127) f10(I128, I129, I130, I131, I132, I133, I134, I135) -> f6(I128, I129, I130, I131, I132, I133, I136, I137) [I137 = I137 /\ I136 = I136 /\ 1 + I130 <= I129] f10(I138, I139, I140, I141, I142, I143, I144, I145) -> f5(I138, I139, 0, I141, I142, I143, I144, I145) [I139 <= I140] f8(I146, I147, I148, I149, I150, I151, I152, I153) -> f9(I146, I147, I148, I149, I150, I151, I152, I153) f6(I154, I155, I156, I157, I158, I159, I160, I161) -> f4(I154, I155, I156, I157, I158, I159, I160, I161) f6(I162, I163, I164, I165, I166, I167, I168, I169) -> f7(I162, I163, 1 + I164, I165, I166, I167, I168, I169) f3(I170, I171, I172, I173, I174, I175, I176, I177) -> f5(I170, I171, 1 + I172, I173, I174, I175, I176, I177) [1 + I172 <= I174] f3(I178, I179, I180, I181, I182, I183, I184, I185) -> f4(I178, I179, I180, I181, I182, I183, I184, I185) [I182 <= I180] f1(I186, I187, I188, I189, I190, I191, I192, I193) -> f2(I186, I187, I188, I189, I190, I191, I192, I193) We use the reverse value criterion with the projection function NU: NU[f3#(z1,z2,z3,z4,z5,z6,z7,z8)] = z5 + -1 * (1 + z3) NU[f5#(z1,z2,z3,z4,z5,z6,z7,z8)] = z5 + -1 * (1 + z3) This gives the following inequalities: ==> I12 + -1 * (1 + I10) >= I12 + -1 * (1 + I10) 1 + I172 <= I174 ==> I174 + -1 * (1 + I172) > I174 + -1 * (1 + (1 + I172)) with I174 + -1 * (1 + I172) >= 0 We remove all the strictly oriented dependency pairs. DP problem for innermost termination. P = f5#(I8, I9, I10, I11, I12, I13, I14, I15) -> f3#(I8, I9, I10, I11, I12, I13, I14, I15) R = f18(x1, x2, x3, x4, x5, x6, x7, x8) -> f17(x1, x2, x3, x4, x5, x6, x7, x8) f17(I0, I1, I2, I3, I4, I5, I6, I7) -> f1(I0, I0, 0, I3, I0, 0, I6, I7) f5(I8, I9, I10, I11, I12, I13, I14, I15) -> f3(I8, I9, I10, I11, I12, I13, I14, I15) f2(I16, I17, I18, I19, I20, I21, I22, I23) -> f16(I16, I17, I18, I19, I20, I21, I22, I23) [1 + I18 <= I20] f2(I24, I25, I26, I27, I28, I29, I30, I31) -> f8(I24, I25, 0, I27, I28, I29, I30, I31) [I28 <= I26] f16(I32, I33, I34, I35, I36, I37, I38, I39) -> f15(I32, I33, I34, I35, I36, I37, I38, I39) [I37 <= I34 /\ I34 <= I37] f16(I40, I41, I42, I43, I44, I45, I46, I47) -> f14(I40, I41, I42, I43, I44, I45, I46, I47) [1 + I42 <= I45] f16(I48, I49, I50, I51, I52, I53, I54, I55) -> f14(I48, I49, I50, I51, I52, I53, I54, I55) [1 + I53 <= I50] f15(I56, I57, I58, I59, I60, I61, I62, I63) -> f1(I56, I57, 1 + I58, I59, I60, I61, I62, I63) f7(I64, I65, I66, I67, I68, I69, I70, I71) -> f10(I64, I65, I66, I67, I68, I69, I70, I71) f14(I72, I73, I74, I75, I76, I77, I78, I79) -> f15(I72, I73, I74, I75, I76, I77, I78, I79) f9(I80, I81, I82, I83, I84, I85, I86, I87) -> f12(I80, I81, I82, 0, I84, I85, I86, I87) [1 + I82 <= I84] f9(I88, I89, I90, I91, I92, I93, I94, I95) -> f7(I88, I89, 0, I91, I92, I93, I94, I95) [I92 <= I90] f13(I96, I97, I98, I99, I100, I101, I102, I103) -> f11(I96, I97, I98, I99, I100, I101, rnd7, rnd8) [rnd8 = rnd8 /\ rnd7 = rnd7 /\ 1 + I99 <= I97] f13(I104, I105, I106, I107, I108, I109, I110, I111) -> f8(I104, I105, 1 + I106, I107, I108, I109, I110, I111) [I105 <= I107] f12(I112, I113, I114, I115, I116, I117, I118, I119) -> f13(I112, I113, I114, I115, I116, I117, I118, I119) f11(I120, I121, I122, I123, I124, I125, I126, I127) -> f12(I120, I121, I122, 1 + I123, I124, I125, I126, I127) f10(I128, I129, I130, I131, I132, I133, I134, I135) -> f6(I128, I129, I130, I131, I132, I133, I136, I137) [I137 = I137 /\ I136 = I136 /\ 1 + I130 <= I129] f10(I138, I139, I140, I141, I142, I143, I144, I145) -> f5(I138, I139, 0, I141, I142, I143, I144, I145) [I139 <= I140] f8(I146, I147, I148, I149, I150, I151, I152, I153) -> f9(I146, I147, I148, I149, I150, I151, I152, I153) f6(I154, I155, I156, I157, I158, I159, I160, I161) -> f4(I154, I155, I156, I157, I158, I159, I160, I161) f6(I162, I163, I164, I165, I166, I167, I168, I169) -> f7(I162, I163, 1 + I164, I165, I166, I167, I168, I169) f3(I170, I171, I172, I173, I174, I175, I176, I177) -> f5(I170, I171, 1 + I172, I173, I174, I175, I176, I177) [1 + I172 <= I174] f3(I178, I179, I180, I181, I182, I183, I184, I185) -> f4(I178, I179, I180, I181, I182, I183, I184, I185) [I182 <= I180] f1(I186, I187, I188, I189, I190, I191, I192, I193) -> f2(I186, I187, I188, I189, I190, I191, I192, I193) The dependency graph for this problem is: 2 -> Where: 2) f5#(I8, I9, I10, I11, I12, I13, I14, I15) -> f3#(I8, I9, I10, I11, I12, I13, I14, I15) We have the following SCCs. DP problem for innermost termination. P = f7#(I64, I65, I66, I67, I68, I69, I70, I71) -> f10#(I64, I65, I66, I67, I68, I69, I70, I71) f10#(I128, I129, I130, I131, I132, I133, I134, I135) -> f6#(I128, I129, I130, I131, I132, I133, I136, I137) [I137 = I137 /\ I136 = I136 /\ 1 + I130 <= I129] f6#(I162, I163, I164, I165, I166, I167, I168, I169) -> f7#(I162, I163, 1 + I164, I165, I166, I167, I168, I169) R = f18(x1, x2, x3, x4, x5, x6, x7, x8) -> f17(x1, x2, x3, x4, x5, x6, x7, x8) f17(I0, I1, I2, I3, I4, I5, I6, I7) -> f1(I0, I0, 0, I3, I0, 0, I6, I7) f5(I8, I9, I10, I11, I12, I13, I14, I15) -> f3(I8, I9, I10, I11, I12, I13, I14, I15) f2(I16, I17, I18, I19, I20, I21, I22, I23) -> f16(I16, I17, I18, I19, I20, I21, I22, I23) [1 + I18 <= I20] f2(I24, I25, I26, I27, I28, I29, I30, I31) -> f8(I24, I25, 0, I27, I28, I29, I30, I31) [I28 <= I26] f16(I32, I33, I34, I35, I36, I37, I38, I39) -> f15(I32, I33, I34, I35, I36, I37, I38, I39) [I37 <= I34 /\ I34 <= I37] f16(I40, I41, I42, I43, I44, I45, I46, I47) -> f14(I40, I41, I42, I43, I44, I45, I46, I47) [1 + I42 <= I45] f16(I48, I49, I50, I51, I52, I53, I54, I55) -> f14(I48, I49, I50, I51, I52, I53, I54, I55) [1 + I53 <= I50] f15(I56, I57, I58, I59, I60, I61, I62, I63) -> f1(I56, I57, 1 + I58, I59, I60, I61, I62, I63) f7(I64, I65, I66, I67, I68, I69, I70, I71) -> f10(I64, I65, I66, I67, I68, I69, I70, I71) f14(I72, I73, I74, I75, I76, I77, I78, I79) -> f15(I72, I73, I74, I75, I76, I77, I78, I79) f9(I80, I81, I82, I83, I84, I85, I86, I87) -> f12(I80, I81, I82, 0, I84, I85, I86, I87) [1 + I82 <= I84] f9(I88, I89, I90, I91, I92, I93, I94, I95) -> f7(I88, I89, 0, I91, I92, I93, I94, I95) [I92 <= I90] f13(I96, I97, I98, I99, I100, I101, I102, I103) -> f11(I96, I97, I98, I99, I100, I101, rnd7, rnd8) [rnd8 = rnd8 /\ rnd7 = rnd7 /\ 1 + I99 <= I97] f13(I104, I105, I106, I107, I108, I109, I110, I111) -> f8(I104, I105, 1 + I106, I107, I108, I109, I110, I111) [I105 <= I107] f12(I112, I113, I114, I115, I116, I117, I118, I119) -> f13(I112, I113, I114, I115, I116, I117, I118, I119) f11(I120, I121, I122, I123, I124, I125, I126, I127) -> f12(I120, I121, I122, 1 + I123, I124, I125, I126, I127) f10(I128, I129, I130, I131, I132, I133, I134, I135) -> f6(I128, I129, I130, I131, I132, I133, I136, I137) [I137 = I137 /\ I136 = I136 /\ 1 + I130 <= I129] f10(I138, I139, I140, I141, I142, I143, I144, I145) -> f5(I138, I139, 0, I141, I142, I143, I144, I145) [I139 <= I140] f8(I146, I147, I148, I149, I150, I151, I152, I153) -> f9(I146, I147, I148, I149, I150, I151, I152, I153) f6(I154, I155, I156, I157, I158, I159, I160, I161) -> f4(I154, I155, I156, I157, I158, I159, I160, I161) f6(I162, I163, I164, I165, I166, I167, I168, I169) -> f7(I162, I163, 1 + I164, I165, I166, I167, I168, I169) f3(I170, I171, I172, I173, I174, I175, I176, I177) -> f5(I170, I171, 1 + I172, I173, I174, I175, I176, I177) [1 + I172 <= I174] f3(I178, I179, I180, I181, I182, I183, I184, I185) -> f4(I178, I179, I180, I181, I182, I183, I184, I185) [I182 <= I180] f1(I186, I187, I188, I189, I190, I191, I192, I193) -> f2(I186, I187, I188, I189, I190, I191, I192, I193) We use the extended value criterion with the projection function NU: NU[f6#(x0,x1,x2,x3,x4,x5,x6,x7)] = x1 - x2 - 2 NU[f10#(x0,x1,x2,x3,x4,x5,x6,x7)] = x1 - x2 - 1 NU[f7#(x0,x1,x2,x3,x4,x5,x6,x7)] = x1 - x2 - 1 This gives the following inequalities: ==> I65 - I66 - 1 >= I65 - I66 - 1 I137 = I137 /\ I136 = I136 /\ 1 + I130 <= I129 ==> I129 - I130 - 1 > I129 - I130 - 2 with I129 - I130 - 1 >= 0 ==> I163 - I164 - 2 >= I163 - (1 + I164) - 1 We remove all the strictly oriented dependency pairs. DP problem for innermost termination. P = f7#(I64, I65, I66, I67, I68, I69, I70, I71) -> f10#(I64, I65, I66, I67, I68, I69, I70, I71) f6#(I162, I163, I164, I165, I166, I167, I168, I169) -> f7#(I162, I163, 1 + I164, I165, I166, I167, I168, I169) R = f18(x1, x2, x3, x4, x5, x6, x7, x8) -> f17(x1, x2, x3, x4, x5, x6, x7, x8) f17(I0, I1, I2, I3, I4, I5, I6, I7) -> f1(I0, I0, 0, I3, I0, 0, I6, I7) f5(I8, I9, I10, I11, I12, I13, I14, I15) -> f3(I8, I9, I10, I11, I12, I13, I14, I15) f2(I16, I17, I18, I19, I20, I21, I22, I23) -> f16(I16, I17, I18, I19, I20, I21, I22, I23) [1 + I18 <= I20] f2(I24, I25, I26, I27, I28, I29, I30, I31) -> f8(I24, I25, 0, I27, I28, I29, I30, I31) [I28 <= I26] f16(I32, I33, I34, I35, I36, I37, I38, I39) -> f15(I32, I33, I34, I35, I36, I37, I38, I39) [I37 <= I34 /\ I34 <= I37] f16(I40, I41, I42, I43, I44, I45, I46, I47) -> f14(I40, I41, I42, I43, I44, I45, I46, I47) [1 + I42 <= I45] f16(I48, I49, I50, I51, I52, I53, I54, I55) -> f14(I48, I49, I50, I51, I52, I53, I54, I55) [1 + I53 <= I50] f15(I56, I57, I58, I59, I60, I61, I62, I63) -> f1(I56, I57, 1 + I58, I59, I60, I61, I62, I63) f7(I64, I65, I66, I67, I68, I69, I70, I71) -> f10(I64, I65, I66, I67, I68, I69, I70, I71) f14(I72, I73, I74, I75, I76, I77, I78, I79) -> f15(I72, I73, I74, I75, I76, I77, I78, I79) f9(I80, I81, I82, I83, I84, I85, I86, I87) -> f12(I80, I81, I82, 0, I84, I85, I86, I87) [1 + I82 <= I84] f9(I88, I89, I90, I91, I92, I93, I94, I95) -> f7(I88, I89, 0, I91, I92, I93, I94, I95) [I92 <= I90] f13(I96, I97, I98, I99, I100, I101, I102, I103) -> f11(I96, I97, I98, I99, I100, I101, rnd7, rnd8) [rnd8 = rnd8 /\ rnd7 = rnd7 /\ 1 + I99 <= I97] f13(I104, I105, I106, I107, I108, I109, I110, I111) -> f8(I104, I105, 1 + I106, I107, I108, I109, I110, I111) [I105 <= I107] f12(I112, I113, I114, I115, I116, I117, I118, I119) -> f13(I112, I113, I114, I115, I116, I117, I118, I119) f11(I120, I121, I122, I123, I124, I125, I126, I127) -> f12(I120, I121, I122, 1 + I123, I124, I125, I126, I127) f10(I128, I129, I130, I131, I132, I133, I134, I135) -> f6(I128, I129, I130, I131, I132, I133, I136, I137) [I137 = I137 /\ I136 = I136 /\ 1 + I130 <= I129] f10(I138, I139, I140, I141, I142, I143, I144, I145) -> f5(I138, I139, 0, I141, I142, I143, I144, I145) [I139 <= I140] f8(I146, I147, I148, I149, I150, I151, I152, I153) -> f9(I146, I147, I148, I149, I150, I151, I152, I153) f6(I154, I155, I156, I157, I158, I159, I160, I161) -> f4(I154, I155, I156, I157, I158, I159, I160, I161) f6(I162, I163, I164, I165, I166, I167, I168, I169) -> f7(I162, I163, 1 + I164, I165, I166, I167, I168, I169) f3(I170, I171, I172, I173, I174, I175, I176, I177) -> f5(I170, I171, 1 + I172, I173, I174, I175, I176, I177) [1 + I172 <= I174] f3(I178, I179, I180, I181, I182, I183, I184, I185) -> f4(I178, I179, I180, I181, I182, I183, I184, I185) [I182 <= I180] f1(I186, I187, I188, I189, I190, I191, I192, I193) -> f2(I186, I187, I188, I189, I190, I191, I192, I193) The dependency graph for this problem is: 9 -> 20 -> 9 Where: 9) f7#(I64, I65, I66, I67, I68, I69, I70, I71) -> f10#(I64, I65, I66, I67, I68, I69, I70, I71) 20) f6#(I162, I163, I164, I165, I166, I167, I168, I169) -> f7#(I162, I163, 1 + I164, I165, I166, I167, I168, I169) We have the following SCCs. DP problem for innermost termination. P = f9#(I80, I81, I82, I83, I84, I85, I86, I87) -> f12#(I80, I81, I82, 0, I84, I85, I86, I87) [1 + I82 <= I84] f13#(I96, I97, I98, I99, I100, I101, I102, I103) -> f11#(I96, I97, I98, I99, I100, I101, rnd7, rnd8) [rnd8 = rnd8 /\ rnd7 = rnd7 /\ 1 + I99 <= I97] f13#(I104, I105, I106, I107, I108, I109, I110, I111) -> f8#(I104, I105, 1 + I106, I107, I108, I109, I110, I111) [I105 <= I107] f12#(I112, I113, I114, I115, I116, I117, I118, I119) -> f13#(I112, I113, I114, I115, I116, I117, I118, I119) f11#(I120, I121, I122, I123, I124, I125, I126, I127) -> f12#(I120, I121, I122, 1 + I123, I124, I125, I126, I127) f8#(I146, I147, I148, I149, I150, I151, I152, I153) -> f9#(I146, I147, I148, I149, I150, I151, I152, I153) R = f18(x1, x2, x3, x4, x5, x6, x7, x8) -> f17(x1, x2, x3, x4, x5, x6, x7, x8) f17(I0, I1, I2, I3, I4, I5, I6, I7) -> f1(I0, I0, 0, I3, I0, 0, I6, I7) f5(I8, I9, I10, I11, I12, I13, I14, I15) -> f3(I8, I9, I10, I11, I12, I13, I14, I15) f2(I16, I17, I18, I19, I20, I21, I22, I23) -> f16(I16, I17, I18, I19, I20, I21, I22, I23) [1 + I18 <= I20] f2(I24, I25, I26, I27, I28, I29, I30, I31) -> f8(I24, I25, 0, I27, I28, I29, I30, I31) [I28 <= I26] f16(I32, I33, I34, I35, I36, I37, I38, I39) -> f15(I32, I33, I34, I35, I36, I37, I38, I39) [I37 <= I34 /\ I34 <= I37] f16(I40, I41, I42, I43, I44, I45, I46, I47) -> f14(I40, I41, I42, I43, I44, I45, I46, I47) [1 + I42 <= I45] f16(I48, I49, I50, I51, I52, I53, I54, I55) -> f14(I48, I49, I50, I51, I52, I53, I54, I55) [1 + I53 <= I50] f15(I56, I57, I58, I59, I60, I61, I62, I63) -> f1(I56, I57, 1 + I58, I59, I60, I61, I62, I63) f7(I64, I65, I66, I67, I68, I69, I70, I71) -> f10(I64, I65, I66, I67, I68, I69, I70, I71) f14(I72, I73, I74, I75, I76, I77, I78, I79) -> f15(I72, I73, I74, I75, I76, I77, I78, I79) f9(I80, I81, I82, I83, I84, I85, I86, I87) -> f12(I80, I81, I82, 0, I84, I85, I86, I87) [1 + I82 <= I84] f9(I88, I89, I90, I91, I92, I93, I94, I95) -> f7(I88, I89, 0, I91, I92, I93, I94, I95) [I92 <= I90] f13(I96, I97, I98, I99, I100, I101, I102, I103) -> f11(I96, I97, I98, I99, I100, I101, rnd7, rnd8) [rnd8 = rnd8 /\ rnd7 = rnd7 /\ 1 + I99 <= I97] f13(I104, I105, I106, I107, I108, I109, I110, I111) -> f8(I104, I105, 1 + I106, I107, I108, I109, I110, I111) [I105 <= I107] f12(I112, I113, I114, I115, I116, I117, I118, I119) -> f13(I112, I113, I114, I115, I116, I117, I118, I119) f11(I120, I121, I122, I123, I124, I125, I126, I127) -> f12(I120, I121, I122, 1 + I123, I124, I125, I126, I127) f10(I128, I129, I130, I131, I132, I133, I134, I135) -> f6(I128, I129, I130, I131, I132, I133, I136, I137) [I137 = I137 /\ I136 = I136 /\ 1 + I130 <= I129] f10(I138, I139, I140, I141, I142, I143, I144, I145) -> f5(I138, I139, 0, I141, I142, I143, I144, I145) [I139 <= I140] f8(I146, I147, I148, I149, I150, I151, I152, I153) -> f9(I146, I147, I148, I149, I150, I151, I152, I153) f6(I154, I155, I156, I157, I158, I159, I160, I161) -> f4(I154, I155, I156, I157, I158, I159, I160, I161) f6(I162, I163, I164, I165, I166, I167, I168, I169) -> f7(I162, I163, 1 + I164, I165, I166, I167, I168, I169) f3(I170, I171, I172, I173, I174, I175, I176, I177) -> f5(I170, I171, 1 + I172, I173, I174, I175, I176, I177) [1 + I172 <= I174] f3(I178, I179, I180, I181, I182, I183, I184, I185) -> f4(I178, I179, I180, I181, I182, I183, I184, I185) [I182 <= I180] f1(I186, I187, I188, I189, I190, I191, I192, I193) -> f2(I186, I187, I188, I189, I190, I191, I192, I193) We use the extended value criterion with the projection function NU: NU[f8#(x0,x1,x2,x3,x4,x5,x6,x7)] = -x2 + x4 - 1 NU[f11#(x0,x1,x2,x3,x4,x5,x6,x7)] = -x2 + x4 - 2 NU[f13#(x0,x1,x2,x3,x4,x5,x6,x7)] = -x2 + x4 - 2 NU[f12#(x0,x1,x2,x3,x4,x5,x6,x7)] = -x2 + x4 - 2 NU[f9#(x0,x1,x2,x3,x4,x5,x6,x7)] = -x2 + x4 - 1 This gives the following inequalities: 1 + I82 <= I84 ==> -I82 + I84 - 1 > -I82 + I84 - 2 with -I82 + I84 - 1 >= 0 rnd8 = rnd8 /\ rnd7 = rnd7 /\ 1 + I99 <= I97 ==> -I98 + I100 - 2 >= -I98 + I100 - 2 I105 <= I107 ==> -I106 + I108 - 2 >= -(1 + I106) + I108 - 1 ==> -I114 + I116 - 2 >= -I114 + I116 - 2 ==> -I122 + I124 - 2 >= -I122 + I124 - 2 ==> -I148 + I150 - 1 >= -I148 + I150 - 1 We remove all the strictly oriented dependency pairs. DP problem for innermost termination. P = f13#(I96, I97, I98, I99, I100, I101, I102, I103) -> f11#(I96, I97, I98, I99, I100, I101, rnd7, rnd8) [rnd8 = rnd8 /\ rnd7 = rnd7 /\ 1 + I99 <= I97] f13#(I104, I105, I106, I107, I108, I109, I110, I111) -> f8#(I104, I105, 1 + I106, I107, I108, I109, I110, I111) [I105 <= I107] f12#(I112, I113, I114, I115, I116, I117, I118, I119) -> f13#(I112, I113, I114, I115, I116, I117, I118, I119) f11#(I120, I121, I122, I123, I124, I125, I126, I127) -> f12#(I120, I121, I122, 1 + I123, I124, I125, I126, I127) f8#(I146, I147, I148, I149, I150, I151, I152, I153) -> f9#(I146, I147, I148, I149, I150, I151, I152, I153) R = f18(x1, x2, x3, x4, x5, x6, x7, x8) -> f17(x1, x2, x3, x4, x5, x6, x7, x8) f17(I0, I1, I2, I3, I4, I5, I6, I7) -> f1(I0, I0, 0, I3, I0, 0, I6, I7) f5(I8, I9, I10, I11, I12, I13, I14, I15) -> f3(I8, I9, I10, I11, I12, I13, I14, I15) f2(I16, I17, I18, I19, I20, I21, I22, I23) -> f16(I16, I17, I18, I19, I20, I21, I22, I23) [1 + I18 <= I20] f2(I24, I25, I26, I27, I28, I29, I30, I31) -> f8(I24, I25, 0, I27, I28, I29, I30, I31) [I28 <= I26] f16(I32, I33, I34, I35, I36, I37, I38, I39) -> f15(I32, I33, I34, I35, I36, I37, I38, I39) [I37 <= I34 /\ I34 <= I37] f16(I40, I41, I42, I43, I44, I45, I46, I47) -> f14(I40, I41, I42, I43, I44, I45, I46, I47) [1 + I42 <= I45] f16(I48, I49, I50, I51, I52, I53, I54, I55) -> f14(I48, I49, I50, I51, I52, I53, I54, I55) [1 + I53 <= I50] f15(I56, I57, I58, I59, I60, I61, I62, I63) -> f1(I56, I57, 1 + I58, I59, I60, I61, I62, I63) f7(I64, I65, I66, I67, I68, I69, I70, I71) -> f10(I64, I65, I66, I67, I68, I69, I70, I71) f14(I72, I73, I74, I75, I76, I77, I78, I79) -> f15(I72, I73, I74, I75, I76, I77, I78, I79) f9(I80, I81, I82, I83, I84, I85, I86, I87) -> f12(I80, I81, I82, 0, I84, I85, I86, I87) [1 + I82 <= I84] f9(I88, I89, I90, I91, I92, I93, I94, I95) -> f7(I88, I89, 0, I91, I92, I93, I94, I95) [I92 <= I90] f13(I96, I97, I98, I99, I100, I101, I102, I103) -> f11(I96, I97, I98, I99, I100, I101, rnd7, rnd8) [rnd8 = rnd8 /\ rnd7 = rnd7 /\ 1 + I99 <= I97] f13(I104, I105, I106, I107, I108, I109, I110, I111) -> f8(I104, I105, 1 + I106, I107, I108, I109, I110, I111) [I105 <= I107] f12(I112, I113, I114, I115, I116, I117, I118, I119) -> f13(I112, I113, I114, I115, I116, I117, I118, I119) f11(I120, I121, I122, I123, I124, I125, I126, I127) -> f12(I120, I121, I122, 1 + I123, I124, I125, I126, I127) f10(I128, I129, I130, I131, I132, I133, I134, I135) -> f6(I128, I129, I130, I131, I132, I133, I136, I137) [I137 = I137 /\ I136 = I136 /\ 1 + I130 <= I129] f10(I138, I139, I140, I141, I142, I143, I144, I145) -> f5(I138, I139, 0, I141, I142, I143, I144, I145) [I139 <= I140] f8(I146, I147, I148, I149, I150, I151, I152, I153) -> f9(I146, I147, I148, I149, I150, I151, I152, I153) f6(I154, I155, I156, I157, I158, I159, I160, I161) -> f4(I154, I155, I156, I157, I158, I159, I160, I161) f6(I162, I163, I164, I165, I166, I167, I168, I169) -> f7(I162, I163, 1 + I164, I165, I166, I167, I168, I169) f3(I170, I171, I172, I173, I174, I175, I176, I177) -> f5(I170, I171, 1 + I172, I173, I174, I175, I176, I177) [1 + I172 <= I174] f3(I178, I179, I180, I181, I182, I183, I184, I185) -> f4(I178, I179, I180, I181, I182, I183, I184, I185) [I182 <= I180] f1(I186, I187, I188, I189, I190, I191, I192, I193) -> f2(I186, I187, I188, I189, I190, I191, I192, I193) The dependency graph for this problem is: 13 -> 16 14 -> 19 15 -> 13, 14 16 -> 15 19 -> Where: 13) f13#(I96, I97, I98, I99, I100, I101, I102, I103) -> f11#(I96, I97, I98, I99, I100, I101, rnd7, rnd8) [rnd8 = rnd8 /\ rnd7 = rnd7 /\ 1 + I99 <= I97] 14) f13#(I104, I105, I106, I107, I108, I109, I110, I111) -> f8#(I104, I105, 1 + I106, I107, I108, I109, I110, I111) [I105 <= I107] 15) f12#(I112, I113, I114, I115, I116, I117, I118, I119) -> f13#(I112, I113, I114, I115, I116, I117, I118, I119) 16) f11#(I120, I121, I122, I123, I124, I125, I126, I127) -> f12#(I120, I121, I122, 1 + I123, I124, I125, I126, I127) 19) f8#(I146, I147, I148, I149, I150, I151, I152, I153) -> f9#(I146, I147, I148, I149, I150, I151, I152, I153) We have the following SCCs. { 13, 15, 16 } DP problem for innermost termination. P = f13#(I96, I97, I98, I99, I100, I101, I102, I103) -> f11#(I96, I97, I98, I99, I100, I101, rnd7, rnd8) [rnd8 = rnd8 /\ rnd7 = rnd7 /\ 1 + I99 <= I97] f12#(I112, I113, I114, I115, I116, I117, I118, I119) -> f13#(I112, I113, I114, I115, I116, I117, I118, I119) f11#(I120, I121, I122, I123, I124, I125, I126, I127) -> f12#(I120, I121, I122, 1 + I123, I124, I125, I126, I127) R = f18(x1, x2, x3, x4, x5, x6, x7, x8) -> f17(x1, x2, x3, x4, x5, x6, x7, x8) f17(I0, I1, I2, I3, I4, I5, I6, I7) -> f1(I0, I0, 0, I3, I0, 0, I6, I7) f5(I8, I9, I10, I11, I12, I13, I14, I15) -> f3(I8, I9, I10, I11, I12, I13, I14, I15) f2(I16, I17, I18, I19, I20, I21, I22, I23) -> f16(I16, I17, I18, I19, I20, I21, I22, I23) [1 + I18 <= I20] f2(I24, I25, I26, I27, I28, I29, I30, I31) -> f8(I24, I25, 0, I27, I28, I29, I30, I31) [I28 <= I26] f16(I32, I33, I34, I35, I36, I37, I38, I39) -> f15(I32, I33, I34, I35, I36, I37, I38, I39) [I37 <= I34 /\ I34 <= I37] f16(I40, I41, I42, I43, I44, I45, I46, I47) -> f14(I40, I41, I42, I43, I44, I45, I46, I47) [1 + I42 <= I45] f16(I48, I49, I50, I51, I52, I53, I54, I55) -> f14(I48, I49, I50, I51, I52, I53, I54, I55) [1 + I53 <= I50] f15(I56, I57, I58, I59, I60, I61, I62, I63) -> f1(I56, I57, 1 + I58, I59, I60, I61, I62, I63) f7(I64, I65, I66, I67, I68, I69, I70, I71) -> f10(I64, I65, I66, I67, I68, I69, I70, I71) f14(I72, I73, I74, I75, I76, I77, I78, I79) -> f15(I72, I73, I74, I75, I76, I77, I78, I79) f9(I80, I81, I82, I83, I84, I85, I86, I87) -> f12(I80, I81, I82, 0, I84, I85, I86, I87) [1 + I82 <= I84] f9(I88, I89, I90, I91, I92, I93, I94, I95) -> f7(I88, I89, 0, I91, I92, I93, I94, I95) [I92 <= I90] f13(I96, I97, I98, I99, I100, I101, I102, I103) -> f11(I96, I97, I98, I99, I100, I101, rnd7, rnd8) [rnd8 = rnd8 /\ rnd7 = rnd7 /\ 1 + I99 <= I97] f13(I104, I105, I106, I107, I108, I109, I110, I111) -> f8(I104, I105, 1 + I106, I107, I108, I109, I110, I111) [I105 <= I107] f12(I112, I113, I114, I115, I116, I117, I118, I119) -> f13(I112, I113, I114, I115, I116, I117, I118, I119) f11(I120, I121, I122, I123, I124, I125, I126, I127) -> f12(I120, I121, I122, 1 + I123, I124, I125, I126, I127) f10(I128, I129, I130, I131, I132, I133, I134, I135) -> f6(I128, I129, I130, I131, I132, I133, I136, I137) [I137 = I137 /\ I136 = I136 /\ 1 + I130 <= I129] f10(I138, I139, I140, I141, I142, I143, I144, I145) -> f5(I138, I139, 0, I141, I142, I143, I144, I145) [I139 <= I140] f8(I146, I147, I148, I149, I150, I151, I152, I153) -> f9(I146, I147, I148, I149, I150, I151, I152, I153) f6(I154, I155, I156, I157, I158, I159, I160, I161) -> f4(I154, I155, I156, I157, I158, I159, I160, I161) f6(I162, I163, I164, I165, I166, I167, I168, I169) -> f7(I162, I163, 1 + I164, I165, I166, I167, I168, I169) f3(I170, I171, I172, I173, I174, I175, I176, I177) -> f5(I170, I171, 1 + I172, I173, I174, I175, I176, I177) [1 + I172 <= I174] f3(I178, I179, I180, I181, I182, I183, I184, I185) -> f4(I178, I179, I180, I181, I182, I183, I184, I185) [I182 <= I180] f1(I186, I187, I188, I189, I190, I191, I192, I193) -> f2(I186, I187, I188, I189, I190, I191, I192, I193) We use the extended value criterion with the projection function NU: NU[f12#(x0,x1,x2,x3,x4,x5,x6,x7)] = x1 - x3 - 1 NU[f11#(x0,x1,x2,x3,x4,x5,x6,x7)] = x1 - x3 - 2 NU[f13#(x0,x1,x2,x3,x4,x5,x6,x7)] = x1 - x3 - 1 This gives the following inequalities: rnd8 = rnd8 /\ rnd7 = rnd7 /\ 1 + I99 <= I97 ==> I97 - I99 - 1 > I97 - I99 - 2 with I97 - I99 - 1 >= 0 ==> I113 - I115 - 1 >= I113 - I115 - 1 ==> I121 - I123 - 2 >= I121 - (1 + I123) - 1 We remove all the strictly oriented dependency pairs. DP problem for innermost termination. P = f12#(I112, I113, I114, I115, I116, I117, I118, I119) -> f13#(I112, I113, I114, I115, I116, I117, I118, I119) f11#(I120, I121, I122, I123, I124, I125, I126, I127) -> f12#(I120, I121, I122, 1 + I123, I124, I125, I126, I127) R = f18(x1, x2, x3, x4, x5, x6, x7, x8) -> f17(x1, x2, x3, x4, x5, x6, x7, x8) f17(I0, I1, I2, I3, I4, I5, I6, I7) -> f1(I0, I0, 0, I3, I0, 0, I6, I7) f5(I8, I9, I10, I11, I12, I13, I14, I15) -> f3(I8, I9, I10, I11, I12, I13, I14, I15) f2(I16, I17, I18, I19, I20, I21, I22, I23) -> f16(I16, I17, I18, I19, I20, I21, I22, I23) [1 + I18 <= I20] f2(I24, I25, I26, I27, I28, I29, I30, I31) -> f8(I24, I25, 0, I27, I28, I29, I30, I31) [I28 <= I26] f16(I32, I33, I34, I35, I36, I37, I38, I39) -> f15(I32, I33, I34, I35, I36, I37, I38, I39) [I37 <= I34 /\ I34 <= I37] f16(I40, I41, I42, I43, I44, I45, I46, I47) -> f14(I40, I41, I42, I43, I44, I45, I46, I47) [1 + I42 <= I45] f16(I48, I49, I50, I51, I52, I53, I54, I55) -> f14(I48, I49, I50, I51, I52, I53, I54, I55) [1 + I53 <= I50] f15(I56, I57, I58, I59, I60, I61, I62, I63) -> f1(I56, I57, 1 + I58, I59, I60, I61, I62, I63) f7(I64, I65, I66, I67, I68, I69, I70, I71) -> f10(I64, I65, I66, I67, I68, I69, I70, I71) f14(I72, I73, I74, I75, I76, I77, I78, I79) -> f15(I72, I73, I74, I75, I76, I77, I78, I79) f9(I80, I81, I82, I83, I84, I85, I86, I87) -> f12(I80, I81, I82, 0, I84, I85, I86, I87) [1 + I82 <= I84] f9(I88, I89, I90, I91, I92, I93, I94, I95) -> f7(I88, I89, 0, I91, I92, I93, I94, I95) [I92 <= I90] f13(I96, I97, I98, I99, I100, I101, I102, I103) -> f11(I96, I97, I98, I99, I100, I101, rnd7, rnd8) [rnd8 = rnd8 /\ rnd7 = rnd7 /\ 1 + I99 <= I97] f13(I104, I105, I106, I107, I108, I109, I110, I111) -> f8(I104, I105, 1 + I106, I107, I108, I109, I110, I111) [I105 <= I107] f12(I112, I113, I114, I115, I116, I117, I118, I119) -> f13(I112, I113, I114, I115, I116, I117, I118, I119) f11(I120, I121, I122, I123, I124, I125, I126, I127) -> f12(I120, I121, I122, 1 + I123, I124, I125, I126, I127) f10(I128, I129, I130, I131, I132, I133, I134, I135) -> f6(I128, I129, I130, I131, I132, I133, I136, I137) [I137 = I137 /\ I136 = I136 /\ 1 + I130 <= I129] f10(I138, I139, I140, I141, I142, I143, I144, I145) -> f5(I138, I139, 0, I141, I142, I143, I144, I145) [I139 <= I140] f8(I146, I147, I148, I149, I150, I151, I152, I153) -> f9(I146, I147, I148, I149, I150, I151, I152, I153) f6(I154, I155, I156, I157, I158, I159, I160, I161) -> f4(I154, I155, I156, I157, I158, I159, I160, I161) f6(I162, I163, I164, I165, I166, I167, I168, I169) -> f7(I162, I163, 1 + I164, I165, I166, I167, I168, I169) f3(I170, I171, I172, I173, I174, I175, I176, I177) -> f5(I170, I171, 1 + I172, I173, I174, I175, I176, I177) [1 + I172 <= I174] f3(I178, I179, I180, I181, I182, I183, I184, I185) -> f4(I178, I179, I180, I181, I182, I183, I184, I185) [I182 <= I180] f1(I186, I187, I188, I189, I190, I191, I192, I193) -> f2(I186, I187, I188, I189, I190, I191, I192, I193) The dependency graph for this problem is: 15 -> 16 -> 15 Where: 15) f12#(I112, I113, I114, I115, I116, I117, I118, I119) -> f13#(I112, I113, I114, I115, I116, I117, I118, I119) 16) f11#(I120, I121, I122, I123, I124, I125, I126, I127) -> f12#(I120, I121, I122, 1 + I123, I124, I125, I126, I127) We have the following SCCs. DP problem for innermost termination. P = f2#(I16, I17, I18, I19, I20, I21, I22, I23) -> f16#(I16, I17, I18, I19, I20, I21, I22, I23) [1 + I18 <= I20] f16#(I32, I33, I34, I35, I36, I37, I38, I39) -> f15#(I32, I33, I34, I35, I36, I37, I38, I39) [I37 <= I34 /\ I34 <= I37] f16#(I40, I41, I42, I43, I44, I45, I46, I47) -> f14#(I40, I41, I42, I43, I44, I45, I46, I47) [1 + I42 <= I45] f16#(I48, I49, I50, I51, I52, I53, I54, I55) -> f14#(I48, I49, I50, I51, I52, I53, I54, I55) [1 + I53 <= I50] f15#(I56, I57, I58, I59, I60, I61, I62, I63) -> f1#(I56, I57, 1 + I58, I59, I60, I61, I62, I63) f14#(I72, I73, I74, I75, I76, I77, I78, I79) -> f15#(I72, I73, I74, I75, I76, I77, I78, I79) f1#(I186, I187, I188, I189, I190, I191, I192, I193) -> f2#(I186, I187, I188, I189, I190, I191, I192, I193) R = f18(x1, x2, x3, x4, x5, x6, x7, x8) -> f17(x1, x2, x3, x4, x5, x6, x7, x8) f17(I0, I1, I2, I3, I4, I5, I6, I7) -> f1(I0, I0, 0, I3, I0, 0, I6, I7) f5(I8, I9, I10, I11, I12, I13, I14, I15) -> f3(I8, I9, I10, I11, I12, I13, I14, I15) f2(I16, I17, I18, I19, I20, I21, I22, I23) -> f16(I16, I17, I18, I19, I20, I21, I22, I23) [1 + I18 <= I20] f2(I24, I25, I26, I27, I28, I29, I30, I31) -> f8(I24, I25, 0, I27, I28, I29, I30, I31) [I28 <= I26] f16(I32, I33, I34, I35, I36, I37, I38, I39) -> f15(I32, I33, I34, I35, I36, I37, I38, I39) [I37 <= I34 /\ I34 <= I37] f16(I40, I41, I42, I43, I44, I45, I46, I47) -> f14(I40, I41, I42, I43, I44, I45, I46, I47) [1 + I42 <= I45] f16(I48, I49, I50, I51, I52, I53, I54, I55) -> f14(I48, I49, I50, I51, I52, I53, I54, I55) [1 + I53 <= I50] f15(I56, I57, I58, I59, I60, I61, I62, I63) -> f1(I56, I57, 1 + I58, I59, I60, I61, I62, I63) f7(I64, I65, I66, I67, I68, I69, I70, I71) -> f10(I64, I65, I66, I67, I68, I69, I70, I71) f14(I72, I73, I74, I75, I76, I77, I78, I79) -> f15(I72, I73, I74, I75, I76, I77, I78, I79) f9(I80, I81, I82, I83, I84, I85, I86, I87) -> f12(I80, I81, I82, 0, I84, I85, I86, I87) [1 + I82 <= I84] f9(I88, I89, I90, I91, I92, I93, I94, I95) -> f7(I88, I89, 0, I91, I92, I93, I94, I95) [I92 <= I90] f13(I96, I97, I98, I99, I100, I101, I102, I103) -> f11(I96, I97, I98, I99, I100, I101, rnd7, rnd8) [rnd8 = rnd8 /\ rnd7 = rnd7 /\ 1 + I99 <= I97] f13(I104, I105, I106, I107, I108, I109, I110, I111) -> f8(I104, I105, 1 + I106, I107, I108, I109, I110, I111) [I105 <= I107] f12(I112, I113, I114, I115, I116, I117, I118, I119) -> f13(I112, I113, I114, I115, I116, I117, I118, I119) f11(I120, I121, I122, I123, I124, I125, I126, I127) -> f12(I120, I121, I122, 1 + I123, I124, I125, I126, I127) f10(I128, I129, I130, I131, I132, I133, I134, I135) -> f6(I128, I129, I130, I131, I132, I133, I136, I137) [I137 = I137 /\ I136 = I136 /\ 1 + I130 <= I129] f10(I138, I139, I140, I141, I142, I143, I144, I145) -> f5(I138, I139, 0, I141, I142, I143, I144, I145) [I139 <= I140] f8(I146, I147, I148, I149, I150, I151, I152, I153) -> f9(I146, I147, I148, I149, I150, I151, I152, I153) f6(I154, I155, I156, I157, I158, I159, I160, I161) -> f4(I154, I155, I156, I157, I158, I159, I160, I161) f6(I162, I163, I164, I165, I166, I167, I168, I169) -> f7(I162, I163, 1 + I164, I165, I166, I167, I168, I169) f3(I170, I171, I172, I173, I174, I175, I176, I177) -> f5(I170, I171, 1 + I172, I173, I174, I175, I176, I177) [1 + I172 <= I174] f3(I178, I179, I180, I181, I182, I183, I184, I185) -> f4(I178, I179, I180, I181, I182, I183, I184, I185) [I182 <= I180] f1(I186, I187, I188, I189, I190, I191, I192, I193) -> f2(I186, I187, I188, I189, I190, I191, I192, I193) We use the extended value criterion with the projection function NU: NU[f1#(x0,x1,x2,x3,x4,x5,x6,x7)] = -x2 + x4 - 1 NU[f14#(x0,x1,x2,x3,x4,x5,x6,x7)] = -x2 + x4 - 2 NU[f15#(x0,x1,x2,x3,x4,x5,x6,x7)] = -x2 + x4 - 2 NU[f16#(x0,x1,x2,x3,x4,x5,x6,x7)] = -x2 + x4 - 2 NU[f2#(x0,x1,x2,x3,x4,x5,x6,x7)] = -x2 + x4 - 1 This gives the following inequalities: 1 + I18 <= I20 ==> -I18 + I20 - 1 > -I18 + I20 - 2 with -I18 + I20 - 1 >= 0 I37 <= I34 /\ I34 <= I37 ==> -I34 + I36 - 2 >= -I34 + I36 - 2 1 + I42 <= I45 ==> -I42 + I44 - 2 >= -I42 + I44 - 2 1 + I53 <= I50 ==> -I50 + I52 - 2 >= -I50 + I52 - 2 ==> -I58 + I60 - 2 >= -(1 + I58) + I60 - 1 ==> -I74 + I76 - 2 >= -I74 + I76 - 2 ==> -I188 + I190 - 1 >= -I188 + I190 - 1 We remove all the strictly oriented dependency pairs. DP problem for innermost termination. P = f16#(I32, I33, I34, I35, I36, I37, I38, I39) -> f15#(I32, I33, I34, I35, I36, I37, I38, I39) [I37 <= I34 /\ I34 <= I37] f16#(I40, I41, I42, I43, I44, I45, I46, I47) -> f14#(I40, I41, I42, I43, I44, I45, I46, I47) [1 + I42 <= I45] f16#(I48, I49, I50, I51, I52, I53, I54, I55) -> f14#(I48, I49, I50, I51, I52, I53, I54, I55) [1 + I53 <= I50] f15#(I56, I57, I58, I59, I60, I61, I62, I63) -> f1#(I56, I57, 1 + I58, I59, I60, I61, I62, I63) f14#(I72, I73, I74, I75, I76, I77, I78, I79) -> f15#(I72, I73, I74, I75, I76, I77, I78, I79) f1#(I186, I187, I188, I189, I190, I191, I192, I193) -> f2#(I186, I187, I188, I189, I190, I191, I192, I193) R = f18(x1, x2, x3, x4, x5, x6, x7, x8) -> f17(x1, x2, x3, x4, x5, x6, x7, x8) f17(I0, I1, I2, I3, I4, I5, I6, I7) -> f1(I0, I0, 0, I3, I0, 0, I6, I7) f5(I8, I9, I10, I11, I12, I13, I14, I15) -> f3(I8, I9, I10, I11, I12, I13, I14, I15) f2(I16, I17, I18, I19, I20, I21, I22, I23) -> f16(I16, I17, I18, I19, I20, I21, I22, I23) [1 + I18 <= I20] f2(I24, I25, I26, I27, I28, I29, I30, I31) -> f8(I24, I25, 0, I27, I28, I29, I30, I31) [I28 <= I26] f16(I32, I33, I34, I35, I36, I37, I38, I39) -> f15(I32, I33, I34, I35, I36, I37, I38, I39) [I37 <= I34 /\ I34 <= I37] f16(I40, I41, I42, I43, I44, I45, I46, I47) -> f14(I40, I41, I42, I43, I44, I45, I46, I47) [1 + I42 <= I45] f16(I48, I49, I50, I51, I52, I53, I54, I55) -> f14(I48, I49, I50, I51, I52, I53, I54, I55) [1 + I53 <= I50] f15(I56, I57, I58, I59, I60, I61, I62, I63) -> f1(I56, I57, 1 + I58, I59, I60, I61, I62, I63) f7(I64, I65, I66, I67, I68, I69, I70, I71) -> f10(I64, I65, I66, I67, I68, I69, I70, I71) f14(I72, I73, I74, I75, I76, I77, I78, I79) -> f15(I72, I73, I74, I75, I76, I77, I78, I79) f9(I80, I81, I82, I83, I84, I85, I86, I87) -> f12(I80, I81, I82, 0, I84, I85, I86, I87) [1 + I82 <= I84] f9(I88, I89, I90, I91, I92, I93, I94, I95) -> f7(I88, I89, 0, I91, I92, I93, I94, I95) [I92 <= I90] f13(I96, I97, I98, I99, I100, I101, I102, I103) -> f11(I96, I97, I98, I99, I100, I101, rnd7, rnd8) [rnd8 = rnd8 /\ rnd7 = rnd7 /\ 1 + I99 <= I97] f13(I104, I105, I106, I107, I108, I109, I110, I111) -> f8(I104, I105, 1 + I106, I107, I108, I109, I110, I111) [I105 <= I107] f12(I112, I113, I114, I115, I116, I117, I118, I119) -> f13(I112, I113, I114, I115, I116, I117, I118, I119) f11(I120, I121, I122, I123, I124, I125, I126, I127) -> f12(I120, I121, I122, 1 + I123, I124, I125, I126, I127) f10(I128, I129, I130, I131, I132, I133, I134, I135) -> f6(I128, I129, I130, I131, I132, I133, I136, I137) [I137 = I137 /\ I136 = I136 /\ 1 + I130 <= I129] f10(I138, I139, I140, I141, I142, I143, I144, I145) -> f5(I138, I139, 0, I141, I142, I143, I144, I145) [I139 <= I140] f8(I146, I147, I148, I149, I150, I151, I152, I153) -> f9(I146, I147, I148, I149, I150, I151, I152, I153) f6(I154, I155, I156, I157, I158, I159, I160, I161) -> f4(I154, I155, I156, I157, I158, I159, I160, I161) f6(I162, I163, I164, I165, I166, I167, I168, I169) -> f7(I162, I163, 1 + I164, I165, I166, I167, I168, I169) f3(I170, I171, I172, I173, I174, I175, I176, I177) -> f5(I170, I171, 1 + I172, I173, I174, I175, I176, I177) [1 + I172 <= I174] f3(I178, I179, I180, I181, I182, I183, I184, I185) -> f4(I178, I179, I180, I181, I182, I183, I184, I185) [I182 <= I180] f1(I186, I187, I188, I189, I190, I191, I192, I193) -> f2(I186, I187, I188, I189, I190, I191, I192, I193) The dependency graph for this problem is: 5 -> 8 6 -> 10 7 -> 10 8 -> 22 10 -> 8 22 -> Where: 5) f16#(I32, I33, I34, I35, I36, I37, I38, I39) -> f15#(I32, I33, I34, I35, I36, I37, I38, I39) [I37 <= I34 /\ I34 <= I37] 6) f16#(I40, I41, I42, I43, I44, I45, I46, I47) -> f14#(I40, I41, I42, I43, I44, I45, I46, I47) [1 + I42 <= I45] 7) f16#(I48, I49, I50, I51, I52, I53, I54, I55) -> f14#(I48, I49, I50, I51, I52, I53, I54, I55) [1 + I53 <= I50] 8) f15#(I56, I57, I58, I59, I60, I61, I62, I63) -> f1#(I56, I57, 1 + I58, I59, I60, I61, I62, I63) 10) f14#(I72, I73, I74, I75, I76, I77, I78, I79) -> f15#(I72, I73, I74, I75, I76, I77, I78, I79) 22) f1#(I186, I187, I188, I189, I190, I191, I192, I193) -> f2#(I186, I187, I188, I189, I190, I191, I192, I193) We have the following SCCs.