/export/starexec/sandbox/solver/bin/starexec_run_Transition /export/starexec/sandbox/benchmark/theBenchmark.smt2 /export/starexec/sandbox/output/output_files -------------------------------------------------------------------------------- YES DP problem for innermost termination. P = f18#(x1, x2, x3, x4, x5, x6, x7, x8, x9) -> f17#(x1, x2, x3, x4, x5, x6, x7, x8, x9) f17#(I0, I1, I2, I3, I4, I5, I6, I7, I8) -> f4#(I0, I1, I1, 0, I4, I0, 0, I7, I8) f3#(I9, I10, I11, I12, I13, I14, I15, I16, I17) -> f1#(I9, I10, I11, I12, I13, I14, I15, I16, I17) f7#(I18, I19, I20, I21, I22, I23, I24, I25, I26) -> f8#(I18, I19, I20, I21, I22, I23, I24, I25, I26) f5#(I27, I28, I29, I30, I31, I32, I33, I34, I35) -> f16#(I27, I28, I29, I30, I31, I32, I33, I34, I35) [1 + I30 <= I32] f5#(I36, I37, I38, I39, I40, I41, I42, I43, I44) -> f9#(I36, I37, I38, 0, I40, I41, I42, I43, I44) [I41 <= I39] f16#(I45, I46, I47, I48, I49, I50, I51, I52, I53) -> f15#(I45, I46, I47, I48, I49, I50, I51, I52, I53) [I51 <= I48 /\ I48 <= I51] f16#(I54, I55, I56, I57, I58, I59, I60, I61, I62) -> f14#(I54, I55, I56, I57, I58, I59, I60, I61, I62) [1 + I57 <= I60] f16#(I63, I64, I65, I66, I67, I68, I69, I70, I71) -> f14#(I63, I64, I65, I66, I67, I68, I69, I70, I71) [1 + I69 <= I66] f15#(I72, I73, I74, I75, I76, I77, I78, I79, I80) -> f4#(I72, I73, I74, 1 + I75, I76, I77, I78, I79, I80) f14#(I81, I82, I83, I84, I85, I86, I87, I88, I89) -> f15#(I81, I82, I83, I84, I85, I86, I87, I88, I89) f10#(I90, I91, I92, I93, I94, I95, I96, I97, I98) -> f12#(I90, I91, I92, I93, 0, I95, I96, I97, I98) [1 + I93 <= I95] f10#(I99, I100, I101, I102, I103, I104, I105, I106, I107) -> f7#(I99, I100, I101, 0, I103, I104, I105, I106, I107) [I104 <= I102] f12#(I108, I109, I110, I111, I112, I113, I114, I115, I116) -> f13#(I108, I109, I110, I111, I112, I113, I114, I115, I116) f13#(I117, I118, I119, I120, I121, I122, I123, I124, I125) -> f11#(I117, I118, I119, I120, I121, I122, I123, rnd8, rnd9) [rnd9 = rnd9 /\ rnd8 = rnd8 /\ 1 + I121 <= I119] f13#(I126, I127, I128, I129, I130, I131, I132, I133, I134) -> f9#(I126, I127, I128, 1 + I129, I130, I131, I132, I133, I134) [I128 <= I130] f11#(I135, I136, I137, I138, I139, I140, I141, I142, I143) -> f12#(I135, I136, I137, I138, 1 + I139, I140, I141, I142, I143) f9#(I144, I145, I146, I147, I148, I149, I150, I151, I152) -> f10#(I144, I145, I146, I147, I148, I149, I150, I151, I152) f8#(I153, I154, I155, I156, I157, I158, I159, I160, I161) -> f6#(I153, I154, I155, I156, I157, I158, I159, I162, I163) [I163 = I163 /\ I162 = I162 /\ 1 + I156 <= I155] f8#(I164, I165, I166, I167, I168, I169, I170, I171, I172) -> f3#(I164, I165, I166, 0, I168, I169, I170, I171, I172) [I166 <= I167] f6#(I182, I183, I184, I185, I186, I187, I188, I189, I190) -> f7#(I182, I183, I184, 1 + I185, I186, I187, I188, I189, I190) f4#(I191, I192, I193, I194, I195, I196, I197, I198, I199) -> f5#(I191, I192, I193, I194, I195, I196, I197, I198, I199) f1#(I200, I201, I202, I203, I204, I205, I206, I207, I208) -> f3#(I200, I201, I202, 1 + I203, I204, I205, I206, I207, I208) [1 + I203 <= I205] R = f18(x1, x2, x3, x4, x5, x6, x7, x8, x9) -> f17(x1, x2, x3, x4, x5, x6, x7, x8, x9) f17(I0, I1, I2, I3, I4, I5, I6, I7, I8) -> f4(I0, I1, I1, 0, I4, I0, 0, I7, I8) f3(I9, I10, I11, I12, I13, I14, I15, I16, I17) -> f1(I9, I10, I11, I12, I13, I14, I15, I16, I17) f7(I18, I19, I20, I21, I22, I23, I24, I25, I26) -> f8(I18, I19, I20, I21, I22, I23, I24, I25, I26) f5(I27, I28, I29, I30, I31, I32, I33, I34, I35) -> f16(I27, I28, I29, I30, I31, I32, I33, I34, I35) [1 + I30 <= I32] f5(I36, I37, I38, I39, I40, I41, I42, I43, I44) -> f9(I36, I37, I38, 0, I40, I41, I42, I43, I44) [I41 <= I39] f16(I45, I46, I47, I48, I49, I50, I51, I52, I53) -> f15(I45, I46, I47, I48, I49, I50, I51, I52, I53) [I51 <= I48 /\ I48 <= I51] f16(I54, I55, I56, I57, I58, I59, I60, I61, I62) -> f14(I54, I55, I56, I57, I58, I59, I60, I61, I62) [1 + I57 <= I60] f16(I63, I64, I65, I66, I67, I68, I69, I70, I71) -> f14(I63, I64, I65, I66, I67, I68, I69, I70, I71) [1 + I69 <= I66] f15(I72, I73, I74, I75, I76, I77, I78, I79, I80) -> f4(I72, I73, I74, 1 + I75, I76, I77, I78, I79, I80) f14(I81, I82, I83, I84, I85, I86, I87, I88, I89) -> f15(I81, I82, I83, I84, I85, I86, I87, I88, I89) f10(I90, I91, I92, I93, I94, I95, I96, I97, I98) -> f12(I90, I91, I92, I93, 0, I95, I96, I97, I98) [1 + I93 <= I95] f10(I99, I100, I101, I102, I103, I104, I105, I106, I107) -> f7(I99, I100, I101, 0, I103, I104, I105, I106, I107) [I104 <= I102] f12(I108, I109, I110, I111, I112, I113, I114, I115, I116) -> f13(I108, I109, I110, I111, I112, I113, I114, I115, I116) f13(I117, I118, I119, I120, I121, I122, I123, I124, I125) -> f11(I117, I118, I119, I120, I121, I122, I123, rnd8, rnd9) [rnd9 = rnd9 /\ rnd8 = rnd8 /\ 1 + I121 <= I119] f13(I126, I127, I128, I129, I130, I131, I132, I133, I134) -> f9(I126, I127, I128, 1 + I129, I130, I131, I132, I133, I134) [I128 <= I130] f11(I135, I136, I137, I138, I139, I140, I141, I142, I143) -> f12(I135, I136, I137, I138, 1 + I139, I140, I141, I142, I143) f9(I144, I145, I146, I147, I148, I149, I150, I151, I152) -> f10(I144, I145, I146, I147, I148, I149, I150, I151, I152) f8(I153, I154, I155, I156, I157, I158, I159, I160, I161) -> f6(I153, I154, I155, I156, I157, I158, I159, I162, I163) [I163 = I163 /\ I162 = I162 /\ 1 + I156 <= I155] f8(I164, I165, I166, I167, I168, I169, I170, I171, I172) -> f3(I164, I165, I166, 0, I168, I169, I170, I171, I172) [I166 <= I167] f6(I173, I174, I175, I176, I177, I178, I179, I180, I181) -> f2(I173, I174, I175, I176, I177, I178, I179, I180, I181) f6(I182, I183, I184, I185, I186, I187, I188, I189, I190) -> f7(I182, I183, I184, 1 + I185, I186, I187, I188, I189, I190) f4(I191, I192, I193, I194, I195, I196, I197, I198, I199) -> f5(I191, I192, I193, I194, I195, I196, I197, I198, I199) f1(I200, I201, I202, I203, I204, I205, I206, I207, I208) -> f3(I200, I201, I202, 1 + I203, I204, I205, I206, I207, I208) [1 + I203 <= I205] f1(I209, I210, I211, I212, I213, I214, I215, I216, I217) -> f2(I209, I210, I211, I212, I213, I214, I215, I216, I217) [I214 <= I212] The dependency graph for this problem is: 0 -> 1 1 -> 21 2 -> 22 3 -> 18, 19 4 -> 6, 7, 8 5 -> 17 6 -> 9 7 -> 10 8 -> 10 9 -> 21 10 -> 9 11 -> 13 12 -> 3 13 -> 14, 15 14 -> 16 15 -> 17 16 -> 13 17 -> 11, 12 18 -> 20 19 -> 2 20 -> 3 21 -> 4, 5 22 -> 2 Where: 0) f18#(x1, x2, x3, x4, x5, x6, x7, x8, x9) -> f17#(x1, x2, x3, x4, x5, x6, x7, x8, x9) 1) f17#(I0, I1, I2, I3, I4, I5, I6, I7, I8) -> f4#(I0, I1, I1, 0, I4, I0, 0, I7, I8) 2) f3#(I9, I10, I11, I12, I13, I14, I15, I16, I17) -> f1#(I9, I10, I11, I12, I13, I14, I15, I16, I17) 3) f7#(I18, I19, I20, I21, I22, I23, I24, I25, I26) -> f8#(I18, I19, I20, I21, I22, I23, I24, I25, I26) 4) f5#(I27, I28, I29, I30, I31, I32, I33, I34, I35) -> f16#(I27, I28, I29, I30, I31, I32, I33, I34, I35) [1 + I30 <= I32] 5) f5#(I36, I37, I38, I39, I40, I41, I42, I43, I44) -> f9#(I36, I37, I38, 0, I40, I41, I42, I43, I44) [I41 <= I39] 6) f16#(I45, I46, I47, I48, I49, I50, I51, I52, I53) -> f15#(I45, I46, I47, I48, I49, I50, I51, I52, I53) [I51 <= I48 /\ I48 <= I51] 7) f16#(I54, I55, I56, I57, I58, I59, I60, I61, I62) -> f14#(I54, I55, I56, I57, I58, I59, I60, I61, I62) [1 + I57 <= I60] 8) f16#(I63, I64, I65, I66, I67, I68, I69, I70, I71) -> f14#(I63, I64, I65, I66, I67, I68, I69, I70, I71) [1 + I69 <= I66] 9) f15#(I72, I73, I74, I75, I76, I77, I78, I79, I80) -> f4#(I72, I73, I74, 1 + I75, I76, I77, I78, I79, I80) 10) f14#(I81, I82, I83, I84, I85, I86, I87, I88, I89) -> f15#(I81, I82, I83, I84, I85, I86, I87, I88, I89) 11) f10#(I90, I91, I92, I93, I94, I95, I96, I97, I98) -> f12#(I90, I91, I92, I93, 0, I95, I96, I97, I98) [1 + I93 <= I95] 12) f10#(I99, I100, I101, I102, I103, I104, I105, I106, I107) -> f7#(I99, I100, I101, 0, I103, I104, I105, I106, I107) [I104 <= I102] 13) f12#(I108, I109, I110, I111, I112, I113, I114, I115, I116) -> f13#(I108, I109, I110, I111, I112, I113, I114, I115, I116) 14) f13#(I117, I118, I119, I120, I121, I122, I123, I124, I125) -> f11#(I117, I118, I119, I120, I121, I122, I123, rnd8, rnd9) [rnd9 = rnd9 /\ rnd8 = rnd8 /\ 1 + I121 <= I119] 15) f13#(I126, I127, I128, I129, I130, I131, I132, I133, I134) -> f9#(I126, I127, I128, 1 + I129, I130, I131, I132, I133, I134) [I128 <= I130] 16) f11#(I135, I136, I137, I138, I139, I140, I141, I142, I143) -> f12#(I135, I136, I137, I138, 1 + I139, I140, I141, I142, I143) 17) f9#(I144, I145, I146, I147, I148, I149, I150, I151, I152) -> f10#(I144, I145, I146, I147, I148, I149, I150, I151, I152) 18) f8#(I153, I154, I155, I156, I157, I158, I159, I160, I161) -> f6#(I153, I154, I155, I156, I157, I158, I159, I162, I163) [I163 = I163 /\ I162 = I162 /\ 1 + I156 <= I155] 19) f8#(I164, I165, I166, I167, I168, I169, I170, I171, I172) -> f3#(I164, I165, I166, 0, I168, I169, I170, I171, I172) [I166 <= I167] 20) f6#(I182, I183, I184, I185, I186, I187, I188, I189, I190) -> f7#(I182, I183, I184, 1 + I185, I186, I187, I188, I189, I190) 21) f4#(I191, I192, I193, I194, I195, I196, I197, I198, I199) -> f5#(I191, I192, I193, I194, I195, I196, I197, I198, I199) 22) f1#(I200, I201, I202, I203, I204, I205, I206, I207, I208) -> f3#(I200, I201, I202, 1 + I203, I204, I205, I206, I207, I208) [1 + I203 <= I205] We have the following SCCs. { 4, 6, 7, 8, 9, 10, 21 } { 11, 13, 14, 15, 16, 17 } { 3, 18, 20 } { 2, 22 } DP problem for innermost termination. P = f3#(I9, I10, I11, I12, I13, I14, I15, I16, I17) -> f1#(I9, I10, I11, I12, I13, I14, I15, I16, I17) f1#(I200, I201, I202, I203, I204, I205, I206, I207, I208) -> f3#(I200, I201, I202, 1 + I203, I204, I205, I206, I207, I208) [1 + I203 <= I205] R = f18(x1, x2, x3, x4, x5, x6, x7, x8, x9) -> f17(x1, x2, x3, x4, x5, x6, x7, x8, x9) f17(I0, I1, I2, I3, I4, I5, I6, I7, I8) -> f4(I0, I1, I1, 0, I4, I0, 0, I7, I8) f3(I9, I10, I11, I12, I13, I14, I15, I16, I17) -> f1(I9, I10, I11, I12, I13, I14, I15, I16, I17) f7(I18, I19, I20, I21, I22, I23, I24, I25, I26) -> f8(I18, I19, I20, I21, I22, I23, I24, I25, I26) f5(I27, I28, I29, I30, I31, I32, I33, I34, I35) -> f16(I27, I28, I29, I30, I31, I32, I33, I34, I35) [1 + I30 <= I32] f5(I36, I37, I38, I39, I40, I41, I42, I43, I44) -> f9(I36, I37, I38, 0, I40, I41, I42, I43, I44) [I41 <= I39] f16(I45, I46, I47, I48, I49, I50, I51, I52, I53) -> f15(I45, I46, I47, I48, I49, I50, I51, I52, I53) [I51 <= I48 /\ I48 <= I51] f16(I54, I55, I56, I57, I58, I59, I60, I61, I62) -> f14(I54, I55, I56, I57, I58, I59, I60, I61, I62) [1 + I57 <= I60] f16(I63, I64, I65, I66, I67, I68, I69, I70, I71) -> f14(I63, I64, I65, I66, I67, I68, I69, I70, I71) [1 + I69 <= I66] f15(I72, I73, I74, I75, I76, I77, I78, I79, I80) -> f4(I72, I73, I74, 1 + I75, I76, I77, I78, I79, I80) f14(I81, I82, I83, I84, I85, I86, I87, I88, I89) -> f15(I81, I82, I83, I84, I85, I86, I87, I88, I89) f10(I90, I91, I92, I93, I94, I95, I96, I97, I98) -> f12(I90, I91, I92, I93, 0, I95, I96, I97, I98) [1 + I93 <= I95] f10(I99, I100, I101, I102, I103, I104, I105, I106, I107) -> f7(I99, I100, I101, 0, I103, I104, I105, I106, I107) [I104 <= I102] f12(I108, I109, I110, I111, I112, I113, I114, I115, I116) -> f13(I108, I109, I110, I111, I112, I113, I114, I115, I116) f13(I117, I118, I119, I120, I121, I122, I123, I124, I125) -> f11(I117, I118, I119, I120, I121, I122, I123, rnd8, rnd9) [rnd9 = rnd9 /\ rnd8 = rnd8 /\ 1 + I121 <= I119] f13(I126, I127, I128, I129, I130, I131, I132, I133, I134) -> f9(I126, I127, I128, 1 + I129, I130, I131, I132, I133, I134) [I128 <= I130] f11(I135, I136, I137, I138, I139, I140, I141, I142, I143) -> f12(I135, I136, I137, I138, 1 + I139, I140, I141, I142, I143) f9(I144, I145, I146, I147, I148, I149, I150, I151, I152) -> f10(I144, I145, I146, I147, I148, I149, I150, I151, I152) f8(I153, I154, I155, I156, I157, I158, I159, I160, I161) -> f6(I153, I154, I155, I156, I157, I158, I159, I162, I163) [I163 = I163 /\ I162 = I162 /\ 1 + I156 <= I155] f8(I164, I165, I166, I167, I168, I169, I170, I171, I172) -> f3(I164, I165, I166, 0, I168, I169, I170, I171, I172) [I166 <= I167] f6(I173, I174, I175, I176, I177, I178, I179, I180, I181) -> f2(I173, I174, I175, I176, I177, I178, I179, I180, I181) f6(I182, I183, I184, I185, I186, I187, I188, I189, I190) -> f7(I182, I183, I184, 1 + I185, I186, I187, I188, I189, I190) f4(I191, I192, I193, I194, I195, I196, I197, I198, I199) -> f5(I191, I192, I193, I194, I195, I196, I197, I198, I199) f1(I200, I201, I202, I203, I204, I205, I206, I207, I208) -> f3(I200, I201, I202, 1 + I203, I204, I205, I206, I207, I208) [1 + I203 <= I205] f1(I209, I210, I211, I212, I213, I214, I215, I216, I217) -> f2(I209, I210, I211, I212, I213, I214, I215, I216, I217) [I214 <= I212] We use the reverse value criterion with the projection function NU: NU[f1#(z1,z2,z3,z4,z5,z6,z7,z8,z9)] = z6 + -1 * (1 + z4) NU[f3#(z1,z2,z3,z4,z5,z6,z7,z8,z9)] = z6 + -1 * (1 + z4) This gives the following inequalities: ==> I14 + -1 * (1 + I12) >= I14 + -1 * (1 + I12) 1 + I203 <= I205 ==> I205 + -1 * (1 + I203) > I205 + -1 * (1 + (1 + I203)) with I205 + -1 * (1 + I203) >= 0 We remove all the strictly oriented dependency pairs. DP problem for innermost termination. P = f3#(I9, I10, I11, I12, I13, I14, I15, I16, I17) -> f1#(I9, I10, I11, I12, I13, I14, I15, I16, I17) R = f18(x1, x2, x3, x4, x5, x6, x7, x8, x9) -> f17(x1, x2, x3, x4, x5, x6, x7, x8, x9) f17(I0, I1, I2, I3, I4, I5, I6, I7, I8) -> f4(I0, I1, I1, 0, I4, I0, 0, I7, I8) f3(I9, I10, I11, I12, I13, I14, I15, I16, I17) -> f1(I9, I10, I11, I12, I13, I14, I15, I16, I17) f7(I18, I19, I20, I21, I22, I23, I24, I25, I26) -> f8(I18, I19, I20, I21, I22, I23, I24, I25, I26) f5(I27, I28, I29, I30, I31, I32, I33, I34, I35) -> f16(I27, I28, I29, I30, I31, I32, I33, I34, I35) [1 + I30 <= I32] f5(I36, I37, I38, I39, I40, I41, I42, I43, I44) -> f9(I36, I37, I38, 0, I40, I41, I42, I43, I44) [I41 <= I39] f16(I45, I46, I47, I48, I49, I50, I51, I52, I53) -> f15(I45, I46, I47, I48, I49, I50, I51, I52, I53) [I51 <= I48 /\ I48 <= I51] f16(I54, I55, I56, I57, I58, I59, I60, I61, I62) -> f14(I54, I55, I56, I57, I58, I59, I60, I61, I62) [1 + I57 <= I60] f16(I63, I64, I65, I66, I67, I68, I69, I70, I71) -> f14(I63, I64, I65, I66, I67, I68, I69, I70, I71) [1 + I69 <= I66] f15(I72, I73, I74, I75, I76, I77, I78, I79, I80) -> f4(I72, I73, I74, 1 + I75, I76, I77, I78, I79, I80) f14(I81, I82, I83, I84, I85, I86, I87, I88, I89) -> f15(I81, I82, I83, I84, I85, I86, I87, I88, I89) f10(I90, I91, I92, I93, I94, I95, I96, I97, I98) -> f12(I90, I91, I92, I93, 0, I95, I96, I97, I98) [1 + I93 <= I95] f10(I99, I100, I101, I102, I103, I104, I105, I106, I107) -> f7(I99, I100, I101, 0, I103, I104, I105, I106, I107) [I104 <= I102] f12(I108, I109, I110, I111, I112, I113, I114, I115, I116) -> f13(I108, I109, I110, I111, I112, I113, I114, I115, I116) f13(I117, I118, I119, I120, I121, I122, I123, I124, I125) -> f11(I117, I118, I119, I120, I121, I122, I123, rnd8, rnd9) [rnd9 = rnd9 /\ rnd8 = rnd8 /\ 1 + I121 <= I119] f13(I126, I127, I128, I129, I130, I131, I132, I133, I134) -> f9(I126, I127, I128, 1 + I129, I130, I131, I132, I133, I134) [I128 <= I130] f11(I135, I136, I137, I138, I139, I140, I141, I142, I143) -> f12(I135, I136, I137, I138, 1 + I139, I140, I141, I142, I143) f9(I144, I145, I146, I147, I148, I149, I150, I151, I152) -> f10(I144, I145, I146, I147, I148, I149, I150, I151, I152) f8(I153, I154, I155, I156, I157, I158, I159, I160, I161) -> f6(I153, I154, I155, I156, I157, I158, I159, I162, I163) [I163 = I163 /\ I162 = I162 /\ 1 + I156 <= I155] f8(I164, I165, I166, I167, I168, I169, I170, I171, I172) -> f3(I164, I165, I166, 0, I168, I169, I170, I171, I172) [I166 <= I167] f6(I173, I174, I175, I176, I177, I178, I179, I180, I181) -> f2(I173, I174, I175, I176, I177, I178, I179, I180, I181) f6(I182, I183, I184, I185, I186, I187, I188, I189, I190) -> f7(I182, I183, I184, 1 + I185, I186, I187, I188, I189, I190) f4(I191, I192, I193, I194, I195, I196, I197, I198, I199) -> f5(I191, I192, I193, I194, I195, I196, I197, I198, I199) f1(I200, I201, I202, I203, I204, I205, I206, I207, I208) -> f3(I200, I201, I202, 1 + I203, I204, I205, I206, I207, I208) [1 + I203 <= I205] f1(I209, I210, I211, I212, I213, I214, I215, I216, I217) -> f2(I209, I210, I211, I212, I213, I214, I215, I216, I217) [I214 <= I212] The dependency graph for this problem is: 2 -> Where: 2) f3#(I9, I10, I11, I12, I13, I14, I15, I16, I17) -> f1#(I9, I10, I11, I12, I13, I14, I15, I16, I17) We have the following SCCs. DP problem for innermost termination. P = f7#(I18, I19, I20, I21, I22, I23, I24, I25, I26) -> f8#(I18, I19, I20, I21, I22, I23, I24, I25, I26) f8#(I153, I154, I155, I156, I157, I158, I159, I160, I161) -> f6#(I153, I154, I155, I156, I157, I158, I159, I162, I163) [I163 = I163 /\ I162 = I162 /\ 1 + I156 <= I155] f6#(I182, I183, I184, I185, I186, I187, I188, I189, I190) -> f7#(I182, I183, I184, 1 + I185, I186, I187, I188, I189, I190) R = f18(x1, x2, x3, x4, x5, x6, x7, x8, x9) -> f17(x1, x2, x3, x4, x5, x6, x7, x8, x9) f17(I0, I1, I2, I3, I4, I5, I6, I7, I8) -> f4(I0, I1, I1, 0, I4, I0, 0, I7, I8) f3(I9, I10, I11, I12, I13, I14, I15, I16, I17) -> f1(I9, I10, I11, I12, I13, I14, I15, I16, I17) f7(I18, I19, I20, I21, I22, I23, I24, I25, I26) -> f8(I18, I19, I20, I21, I22, I23, I24, I25, I26) f5(I27, I28, I29, I30, I31, I32, I33, I34, I35) -> f16(I27, I28, I29, I30, I31, I32, I33, I34, I35) [1 + I30 <= I32] f5(I36, I37, I38, I39, I40, I41, I42, I43, I44) -> f9(I36, I37, I38, 0, I40, I41, I42, I43, I44) [I41 <= I39] f16(I45, I46, I47, I48, I49, I50, I51, I52, I53) -> f15(I45, I46, I47, I48, I49, I50, I51, I52, I53) [I51 <= I48 /\ I48 <= I51] f16(I54, I55, I56, I57, I58, I59, I60, I61, I62) -> f14(I54, I55, I56, I57, I58, I59, I60, I61, I62) [1 + I57 <= I60] f16(I63, I64, I65, I66, I67, I68, I69, I70, I71) -> f14(I63, I64, I65, I66, I67, I68, I69, I70, I71) [1 + I69 <= I66] f15(I72, I73, I74, I75, I76, I77, I78, I79, I80) -> f4(I72, I73, I74, 1 + I75, I76, I77, I78, I79, I80) f14(I81, I82, I83, I84, I85, I86, I87, I88, I89) -> f15(I81, I82, I83, I84, I85, I86, I87, I88, I89) f10(I90, I91, I92, I93, I94, I95, I96, I97, I98) -> f12(I90, I91, I92, I93, 0, I95, I96, I97, I98) [1 + I93 <= I95] f10(I99, I100, I101, I102, I103, I104, I105, I106, I107) -> f7(I99, I100, I101, 0, I103, I104, I105, I106, I107) [I104 <= I102] f12(I108, I109, I110, I111, I112, I113, I114, I115, I116) -> f13(I108, I109, I110, I111, I112, I113, I114, I115, I116) f13(I117, I118, I119, I120, I121, I122, I123, I124, I125) -> f11(I117, I118, I119, I120, I121, I122, I123, rnd8, rnd9) [rnd9 = rnd9 /\ rnd8 = rnd8 /\ 1 + I121 <= I119] f13(I126, I127, I128, I129, I130, I131, I132, I133, I134) -> f9(I126, I127, I128, 1 + I129, I130, I131, I132, I133, I134) [I128 <= I130] f11(I135, I136, I137, I138, I139, I140, I141, I142, I143) -> f12(I135, I136, I137, I138, 1 + I139, I140, I141, I142, I143) f9(I144, I145, I146, I147, I148, I149, I150, I151, I152) -> f10(I144, I145, I146, I147, I148, I149, I150, I151, I152) f8(I153, I154, I155, I156, I157, I158, I159, I160, I161) -> f6(I153, I154, I155, I156, I157, I158, I159, I162, I163) [I163 = I163 /\ I162 = I162 /\ 1 + I156 <= I155] f8(I164, I165, I166, I167, I168, I169, I170, I171, I172) -> f3(I164, I165, I166, 0, I168, I169, I170, I171, I172) [I166 <= I167] f6(I173, I174, I175, I176, I177, I178, I179, I180, I181) -> f2(I173, I174, I175, I176, I177, I178, I179, I180, I181) f6(I182, I183, I184, I185, I186, I187, I188, I189, I190) -> f7(I182, I183, I184, 1 + I185, I186, I187, I188, I189, I190) f4(I191, I192, I193, I194, I195, I196, I197, I198, I199) -> f5(I191, I192, I193, I194, I195, I196, I197, I198, I199) f1(I200, I201, I202, I203, I204, I205, I206, I207, I208) -> f3(I200, I201, I202, 1 + I203, I204, I205, I206, I207, I208) [1 + I203 <= I205] f1(I209, I210, I211, I212, I213, I214, I215, I216, I217) -> f2(I209, I210, I211, I212, I213, I214, I215, I216, I217) [I214 <= I212] We use the extended value criterion with the projection function NU: NU[f6#(x0,x1,x2,x3,x4,x5,x6,x7,x8)] = x2 - x3 - 2 NU[f8#(x0,x1,x2,x3,x4,x5,x6,x7,x8)] = x2 - x3 - 1 NU[f7#(x0,x1,x2,x3,x4,x5,x6,x7,x8)] = x2 - x3 - 1 This gives the following inequalities: ==> I20 - I21 - 1 >= I20 - I21 - 1 I163 = I163 /\ I162 = I162 /\ 1 + I156 <= I155 ==> I155 - I156 - 1 > I155 - I156 - 2 with I155 - I156 - 1 >= 0 ==> I184 - I185 - 2 >= I184 - (1 + I185) - 1 We remove all the strictly oriented dependency pairs. DP problem for innermost termination. P = f7#(I18, I19, I20, I21, I22, I23, I24, I25, I26) -> f8#(I18, I19, I20, I21, I22, I23, I24, I25, I26) f6#(I182, I183, I184, I185, I186, I187, I188, I189, I190) -> f7#(I182, I183, I184, 1 + I185, I186, I187, I188, I189, I190) R = f18(x1, x2, x3, x4, x5, x6, x7, x8, x9) -> f17(x1, x2, x3, x4, x5, x6, x7, x8, x9) f17(I0, I1, I2, I3, I4, I5, I6, I7, I8) -> f4(I0, I1, I1, 0, I4, I0, 0, I7, I8) f3(I9, I10, I11, I12, I13, I14, I15, I16, I17) -> f1(I9, I10, I11, I12, I13, I14, I15, I16, I17) f7(I18, I19, I20, I21, I22, I23, I24, I25, I26) -> f8(I18, I19, I20, I21, I22, I23, I24, I25, I26) f5(I27, I28, I29, I30, I31, I32, I33, I34, I35) -> f16(I27, I28, I29, I30, I31, I32, I33, I34, I35) [1 + I30 <= I32] f5(I36, I37, I38, I39, I40, I41, I42, I43, I44) -> f9(I36, I37, I38, 0, I40, I41, I42, I43, I44) [I41 <= I39] f16(I45, I46, I47, I48, I49, I50, I51, I52, I53) -> f15(I45, I46, I47, I48, I49, I50, I51, I52, I53) [I51 <= I48 /\ I48 <= I51] f16(I54, I55, I56, I57, I58, I59, I60, I61, I62) -> f14(I54, I55, I56, I57, I58, I59, I60, I61, I62) [1 + I57 <= I60] f16(I63, I64, I65, I66, I67, I68, I69, I70, I71) -> f14(I63, I64, I65, I66, I67, I68, I69, I70, I71) [1 + I69 <= I66] f15(I72, I73, I74, I75, I76, I77, I78, I79, I80) -> f4(I72, I73, I74, 1 + I75, I76, I77, I78, I79, I80) f14(I81, I82, I83, I84, I85, I86, I87, I88, I89) -> f15(I81, I82, I83, I84, I85, I86, I87, I88, I89) f10(I90, I91, I92, I93, I94, I95, I96, I97, I98) -> f12(I90, I91, I92, I93, 0, I95, I96, I97, I98) [1 + I93 <= I95] f10(I99, I100, I101, I102, I103, I104, I105, I106, I107) -> f7(I99, I100, I101, 0, I103, I104, I105, I106, I107) [I104 <= I102] f12(I108, I109, I110, I111, I112, I113, I114, I115, I116) -> f13(I108, I109, I110, I111, I112, I113, I114, I115, I116) f13(I117, I118, I119, I120, I121, I122, I123, I124, I125) -> f11(I117, I118, I119, I120, I121, I122, I123, rnd8, rnd9) [rnd9 = rnd9 /\ rnd8 = rnd8 /\ 1 + I121 <= I119] f13(I126, I127, I128, I129, I130, I131, I132, I133, I134) -> f9(I126, I127, I128, 1 + I129, I130, I131, I132, I133, I134) [I128 <= I130] f11(I135, I136, I137, I138, I139, I140, I141, I142, I143) -> f12(I135, I136, I137, I138, 1 + I139, I140, I141, I142, I143) f9(I144, I145, I146, I147, I148, I149, I150, I151, I152) -> f10(I144, I145, I146, I147, I148, I149, I150, I151, I152) f8(I153, I154, I155, I156, I157, I158, I159, I160, I161) -> f6(I153, I154, I155, I156, I157, I158, I159, I162, I163) [I163 = I163 /\ I162 = I162 /\ 1 + I156 <= I155] f8(I164, I165, I166, I167, I168, I169, I170, I171, I172) -> f3(I164, I165, I166, 0, I168, I169, I170, I171, I172) [I166 <= I167] f6(I173, I174, I175, I176, I177, I178, I179, I180, I181) -> f2(I173, I174, I175, I176, I177, I178, I179, I180, I181) f6(I182, I183, I184, I185, I186, I187, I188, I189, I190) -> f7(I182, I183, I184, 1 + I185, I186, I187, I188, I189, I190) f4(I191, I192, I193, I194, I195, I196, I197, I198, I199) -> f5(I191, I192, I193, I194, I195, I196, I197, I198, I199) f1(I200, I201, I202, I203, I204, I205, I206, I207, I208) -> f3(I200, I201, I202, 1 + I203, I204, I205, I206, I207, I208) [1 + I203 <= I205] f1(I209, I210, I211, I212, I213, I214, I215, I216, I217) -> f2(I209, I210, I211, I212, I213, I214, I215, I216, I217) [I214 <= I212] The dependency graph for this problem is: 3 -> 20 -> 3 Where: 3) f7#(I18, I19, I20, I21, I22, I23, I24, I25, I26) -> f8#(I18, I19, I20, I21, I22, I23, I24, I25, I26) 20) f6#(I182, I183, I184, I185, I186, I187, I188, I189, I190) -> f7#(I182, I183, I184, 1 + I185, I186, I187, I188, I189, I190) We have the following SCCs. DP problem for innermost termination. P = f10#(I90, I91, I92, I93, I94, I95, I96, I97, I98) -> f12#(I90, I91, I92, I93, 0, I95, I96, I97, I98) [1 + I93 <= I95] f12#(I108, I109, I110, I111, I112, I113, I114, I115, I116) -> f13#(I108, I109, I110, I111, I112, I113, I114, I115, I116) f13#(I117, I118, I119, I120, I121, I122, I123, I124, I125) -> f11#(I117, I118, I119, I120, I121, I122, I123, rnd8, rnd9) [rnd9 = rnd9 /\ rnd8 = rnd8 /\ 1 + I121 <= I119] f13#(I126, I127, I128, I129, I130, I131, I132, I133, I134) -> f9#(I126, I127, I128, 1 + I129, I130, I131, I132, I133, I134) [I128 <= I130] f11#(I135, I136, I137, I138, I139, I140, I141, I142, I143) -> f12#(I135, I136, I137, I138, 1 + I139, I140, I141, I142, I143) f9#(I144, I145, I146, I147, I148, I149, I150, I151, I152) -> f10#(I144, I145, I146, I147, I148, I149, I150, I151, I152) R = f18(x1, x2, x3, x4, x5, x6, x7, x8, x9) -> f17(x1, x2, x3, x4, x5, x6, x7, x8, x9) f17(I0, I1, I2, I3, I4, I5, I6, I7, I8) -> f4(I0, I1, I1, 0, I4, I0, 0, I7, I8) f3(I9, I10, I11, I12, I13, I14, I15, I16, I17) -> f1(I9, I10, I11, I12, I13, I14, I15, I16, I17) f7(I18, I19, I20, I21, I22, I23, I24, I25, I26) -> f8(I18, I19, I20, I21, I22, I23, I24, I25, I26) f5(I27, I28, I29, I30, I31, I32, I33, I34, I35) -> f16(I27, I28, I29, I30, I31, I32, I33, I34, I35) [1 + I30 <= I32] f5(I36, I37, I38, I39, I40, I41, I42, I43, I44) -> f9(I36, I37, I38, 0, I40, I41, I42, I43, I44) [I41 <= I39] f16(I45, I46, I47, I48, I49, I50, I51, I52, I53) -> f15(I45, I46, I47, I48, I49, I50, I51, I52, I53) [I51 <= I48 /\ I48 <= I51] f16(I54, I55, I56, I57, I58, I59, I60, I61, I62) -> f14(I54, I55, I56, I57, I58, I59, I60, I61, I62) [1 + I57 <= I60] f16(I63, I64, I65, I66, I67, I68, I69, I70, I71) -> f14(I63, I64, I65, I66, I67, I68, I69, I70, I71) [1 + I69 <= I66] f15(I72, I73, I74, I75, I76, I77, I78, I79, I80) -> f4(I72, I73, I74, 1 + I75, I76, I77, I78, I79, I80) f14(I81, I82, I83, I84, I85, I86, I87, I88, I89) -> f15(I81, I82, I83, I84, I85, I86, I87, I88, I89) f10(I90, I91, I92, I93, I94, I95, I96, I97, I98) -> f12(I90, I91, I92, I93, 0, I95, I96, I97, I98) [1 + I93 <= I95] f10(I99, I100, I101, I102, I103, I104, I105, I106, I107) -> f7(I99, I100, I101, 0, I103, I104, I105, I106, I107) [I104 <= I102] f12(I108, I109, I110, I111, I112, I113, I114, I115, I116) -> f13(I108, I109, I110, I111, I112, I113, I114, I115, I116) f13(I117, I118, I119, I120, I121, I122, I123, I124, I125) -> f11(I117, I118, I119, I120, I121, I122, I123, rnd8, rnd9) [rnd9 = rnd9 /\ rnd8 = rnd8 /\ 1 + I121 <= I119] f13(I126, I127, I128, I129, I130, I131, I132, I133, I134) -> f9(I126, I127, I128, 1 + I129, I130, I131, I132, I133, I134) [I128 <= I130] f11(I135, I136, I137, I138, I139, I140, I141, I142, I143) -> f12(I135, I136, I137, I138, 1 + I139, I140, I141, I142, I143) f9(I144, I145, I146, I147, I148, I149, I150, I151, I152) -> f10(I144, I145, I146, I147, I148, I149, I150, I151, I152) f8(I153, I154, I155, I156, I157, I158, I159, I160, I161) -> f6(I153, I154, I155, I156, I157, I158, I159, I162, I163) [I163 = I163 /\ I162 = I162 /\ 1 + I156 <= I155] f8(I164, I165, I166, I167, I168, I169, I170, I171, I172) -> f3(I164, I165, I166, 0, I168, I169, I170, I171, I172) [I166 <= I167] f6(I173, I174, I175, I176, I177, I178, I179, I180, I181) -> f2(I173, I174, I175, I176, I177, I178, I179, I180, I181) f6(I182, I183, I184, I185, I186, I187, I188, I189, I190) -> f7(I182, I183, I184, 1 + I185, I186, I187, I188, I189, I190) f4(I191, I192, I193, I194, I195, I196, I197, I198, I199) -> f5(I191, I192, I193, I194, I195, I196, I197, I198, I199) f1(I200, I201, I202, I203, I204, I205, I206, I207, I208) -> f3(I200, I201, I202, 1 + I203, I204, I205, I206, I207, I208) [1 + I203 <= I205] f1(I209, I210, I211, I212, I213, I214, I215, I216, I217) -> f2(I209, I210, I211, I212, I213, I214, I215, I216, I217) [I214 <= I212] We use the extended value criterion with the projection function NU: NU[f9#(x0,x1,x2,x3,x4,x5,x6,x7,x8)] = -x3 + x5 - 1 NU[f11#(x0,x1,x2,x3,x4,x5,x6,x7,x8)] = -x3 + x5 - 2 NU[f13#(x0,x1,x2,x3,x4,x5,x6,x7,x8)] = -x3 + x5 - 2 NU[f12#(x0,x1,x2,x3,x4,x5,x6,x7,x8)] = -x3 + x5 - 2 NU[f10#(x0,x1,x2,x3,x4,x5,x6,x7,x8)] = -x3 + x5 - 1 This gives the following inequalities: 1 + I93 <= I95 ==> -I93 + I95 - 1 > -I93 + I95 - 2 with -I93 + I95 - 1 >= 0 ==> -I111 + I113 - 2 >= -I111 + I113 - 2 rnd9 = rnd9 /\ rnd8 = rnd8 /\ 1 + I121 <= I119 ==> -I120 + I122 - 2 >= -I120 + I122 - 2 I128 <= I130 ==> -I129 + I131 - 2 >= -(1 + I129) + I131 - 1 ==> -I138 + I140 - 2 >= -I138 + I140 - 2 ==> -I147 + I149 - 1 >= -I147 + I149 - 1 We remove all the strictly oriented dependency pairs. DP problem for innermost termination. P = f12#(I108, I109, I110, I111, I112, I113, I114, I115, I116) -> f13#(I108, I109, I110, I111, I112, I113, I114, I115, I116) f13#(I117, I118, I119, I120, I121, I122, I123, I124, I125) -> f11#(I117, I118, I119, I120, I121, I122, I123, rnd8, rnd9) [rnd9 = rnd9 /\ rnd8 = rnd8 /\ 1 + I121 <= I119] f13#(I126, I127, I128, I129, I130, I131, I132, I133, I134) -> f9#(I126, I127, I128, 1 + I129, I130, I131, I132, I133, I134) [I128 <= I130] f11#(I135, I136, I137, I138, I139, I140, I141, I142, I143) -> f12#(I135, I136, I137, I138, 1 + I139, I140, I141, I142, I143) f9#(I144, I145, I146, I147, I148, I149, I150, I151, I152) -> f10#(I144, I145, I146, I147, I148, I149, I150, I151, I152) R = f18(x1, x2, x3, x4, x5, x6, x7, x8, x9) -> f17(x1, x2, x3, x4, x5, x6, x7, x8, x9) f17(I0, I1, I2, I3, I4, I5, I6, I7, I8) -> f4(I0, I1, I1, 0, I4, I0, 0, I7, I8) f3(I9, I10, I11, I12, I13, I14, I15, I16, I17) -> f1(I9, I10, I11, I12, I13, I14, I15, I16, I17) f7(I18, I19, I20, I21, I22, I23, I24, I25, I26) -> f8(I18, I19, I20, I21, I22, I23, I24, I25, I26) f5(I27, I28, I29, I30, I31, I32, I33, I34, I35) -> f16(I27, I28, I29, I30, I31, I32, I33, I34, I35) [1 + I30 <= I32] f5(I36, I37, I38, I39, I40, I41, I42, I43, I44) -> f9(I36, I37, I38, 0, I40, I41, I42, I43, I44) [I41 <= I39] f16(I45, I46, I47, I48, I49, I50, I51, I52, I53) -> f15(I45, I46, I47, I48, I49, I50, I51, I52, I53) [I51 <= I48 /\ I48 <= I51] f16(I54, I55, I56, I57, I58, I59, I60, I61, I62) -> f14(I54, I55, I56, I57, I58, I59, I60, I61, I62) [1 + I57 <= I60] f16(I63, I64, I65, I66, I67, I68, I69, I70, I71) -> f14(I63, I64, I65, I66, I67, I68, I69, I70, I71) [1 + I69 <= I66] f15(I72, I73, I74, I75, I76, I77, I78, I79, I80) -> f4(I72, I73, I74, 1 + I75, I76, I77, I78, I79, I80) f14(I81, I82, I83, I84, I85, I86, I87, I88, I89) -> f15(I81, I82, I83, I84, I85, I86, I87, I88, I89) f10(I90, I91, I92, I93, I94, I95, I96, I97, I98) -> f12(I90, I91, I92, I93, 0, I95, I96, I97, I98) [1 + I93 <= I95] f10(I99, I100, I101, I102, I103, I104, I105, I106, I107) -> f7(I99, I100, I101, 0, I103, I104, I105, I106, I107) [I104 <= I102] f12(I108, I109, I110, I111, I112, I113, I114, I115, I116) -> f13(I108, I109, I110, I111, I112, I113, I114, I115, I116) f13(I117, I118, I119, I120, I121, I122, I123, I124, I125) -> f11(I117, I118, I119, I120, I121, I122, I123, rnd8, rnd9) [rnd9 = rnd9 /\ rnd8 = rnd8 /\ 1 + I121 <= I119] f13(I126, I127, I128, I129, I130, I131, I132, I133, I134) -> f9(I126, I127, I128, 1 + I129, I130, I131, I132, I133, I134) [I128 <= I130] f11(I135, I136, I137, I138, I139, I140, I141, I142, I143) -> f12(I135, I136, I137, I138, 1 + I139, I140, I141, I142, I143) f9(I144, I145, I146, I147, I148, I149, I150, I151, I152) -> f10(I144, I145, I146, I147, I148, I149, I150, I151, I152) f8(I153, I154, I155, I156, I157, I158, I159, I160, I161) -> f6(I153, I154, I155, I156, I157, I158, I159, I162, I163) [I163 = I163 /\ I162 = I162 /\ 1 + I156 <= I155] f8(I164, I165, I166, I167, I168, I169, I170, I171, I172) -> f3(I164, I165, I166, 0, I168, I169, I170, I171, I172) [I166 <= I167] f6(I173, I174, I175, I176, I177, I178, I179, I180, I181) -> f2(I173, I174, I175, I176, I177, I178, I179, I180, I181) f6(I182, I183, I184, I185, I186, I187, I188, I189, I190) -> f7(I182, I183, I184, 1 + I185, I186, I187, I188, I189, I190) f4(I191, I192, I193, I194, I195, I196, I197, I198, I199) -> f5(I191, I192, I193, I194, I195, I196, I197, I198, I199) f1(I200, I201, I202, I203, I204, I205, I206, I207, I208) -> f3(I200, I201, I202, 1 + I203, I204, I205, I206, I207, I208) [1 + I203 <= I205] f1(I209, I210, I211, I212, I213, I214, I215, I216, I217) -> f2(I209, I210, I211, I212, I213, I214, I215, I216, I217) [I214 <= I212] The dependency graph for this problem is: 13 -> 14, 15 14 -> 16 15 -> 17 16 -> 13 17 -> Where: 13) f12#(I108, I109, I110, I111, I112, I113, I114, I115, I116) -> f13#(I108, I109, I110, I111, I112, I113, I114, I115, I116) 14) f13#(I117, I118, I119, I120, I121, I122, I123, I124, I125) -> f11#(I117, I118, I119, I120, I121, I122, I123, rnd8, rnd9) [rnd9 = rnd9 /\ rnd8 = rnd8 /\ 1 + I121 <= I119] 15) f13#(I126, I127, I128, I129, I130, I131, I132, I133, I134) -> f9#(I126, I127, I128, 1 + I129, I130, I131, I132, I133, I134) [I128 <= I130] 16) f11#(I135, I136, I137, I138, I139, I140, I141, I142, I143) -> f12#(I135, I136, I137, I138, 1 + I139, I140, I141, I142, I143) 17) f9#(I144, I145, I146, I147, I148, I149, I150, I151, I152) -> f10#(I144, I145, I146, I147, I148, I149, I150, I151, I152) We have the following SCCs. { 13, 14, 16 } DP problem for innermost termination. P = f12#(I108, I109, I110, I111, I112, I113, I114, I115, I116) -> f13#(I108, I109, I110, I111, I112, I113, I114, I115, I116) f13#(I117, I118, I119, I120, I121, I122, I123, I124, I125) -> f11#(I117, I118, I119, I120, I121, I122, I123, rnd8, rnd9) [rnd9 = rnd9 /\ rnd8 = rnd8 /\ 1 + I121 <= I119] f11#(I135, I136, I137, I138, I139, I140, I141, I142, I143) -> f12#(I135, I136, I137, I138, 1 + I139, I140, I141, I142, I143) R = f18(x1, x2, x3, x4, x5, x6, x7, x8, x9) -> f17(x1, x2, x3, x4, x5, x6, x7, x8, x9) f17(I0, I1, I2, I3, I4, I5, I6, I7, I8) -> f4(I0, I1, I1, 0, I4, I0, 0, I7, I8) f3(I9, I10, I11, I12, I13, I14, I15, I16, I17) -> f1(I9, I10, I11, I12, I13, I14, I15, I16, I17) f7(I18, I19, I20, I21, I22, I23, I24, I25, I26) -> f8(I18, I19, I20, I21, I22, I23, I24, I25, I26) f5(I27, I28, I29, I30, I31, I32, I33, I34, I35) -> f16(I27, I28, I29, I30, I31, I32, I33, I34, I35) [1 + I30 <= I32] f5(I36, I37, I38, I39, I40, I41, I42, I43, I44) -> f9(I36, I37, I38, 0, I40, I41, I42, I43, I44) [I41 <= I39] f16(I45, I46, I47, I48, I49, I50, I51, I52, I53) -> f15(I45, I46, I47, I48, I49, I50, I51, I52, I53) [I51 <= I48 /\ I48 <= I51] f16(I54, I55, I56, I57, I58, I59, I60, I61, I62) -> f14(I54, I55, I56, I57, I58, I59, I60, I61, I62) [1 + I57 <= I60] f16(I63, I64, I65, I66, I67, I68, I69, I70, I71) -> f14(I63, I64, I65, I66, I67, I68, I69, I70, I71) [1 + I69 <= I66] f15(I72, I73, I74, I75, I76, I77, I78, I79, I80) -> f4(I72, I73, I74, 1 + I75, I76, I77, I78, I79, I80) f14(I81, I82, I83, I84, I85, I86, I87, I88, I89) -> f15(I81, I82, I83, I84, I85, I86, I87, I88, I89) f10(I90, I91, I92, I93, I94, I95, I96, I97, I98) -> f12(I90, I91, I92, I93, 0, I95, I96, I97, I98) [1 + I93 <= I95] f10(I99, I100, I101, I102, I103, I104, I105, I106, I107) -> f7(I99, I100, I101, 0, I103, I104, I105, I106, I107) [I104 <= I102] f12(I108, I109, I110, I111, I112, I113, I114, I115, I116) -> f13(I108, I109, I110, I111, I112, I113, I114, I115, I116) f13(I117, I118, I119, I120, I121, I122, I123, I124, I125) -> f11(I117, I118, I119, I120, I121, I122, I123, rnd8, rnd9) [rnd9 = rnd9 /\ rnd8 = rnd8 /\ 1 + I121 <= I119] f13(I126, I127, I128, I129, I130, I131, I132, I133, I134) -> f9(I126, I127, I128, 1 + I129, I130, I131, I132, I133, I134) [I128 <= I130] f11(I135, I136, I137, I138, I139, I140, I141, I142, I143) -> f12(I135, I136, I137, I138, 1 + I139, I140, I141, I142, I143) f9(I144, I145, I146, I147, I148, I149, I150, I151, I152) -> f10(I144, I145, I146, I147, I148, I149, I150, I151, I152) f8(I153, I154, I155, I156, I157, I158, I159, I160, I161) -> f6(I153, I154, I155, I156, I157, I158, I159, I162, I163) [I163 = I163 /\ I162 = I162 /\ 1 + I156 <= I155] f8(I164, I165, I166, I167, I168, I169, I170, I171, I172) -> f3(I164, I165, I166, 0, I168, I169, I170, I171, I172) [I166 <= I167] f6(I173, I174, I175, I176, I177, I178, I179, I180, I181) -> f2(I173, I174, I175, I176, I177, I178, I179, I180, I181) f6(I182, I183, I184, I185, I186, I187, I188, I189, I190) -> f7(I182, I183, I184, 1 + I185, I186, I187, I188, I189, I190) f4(I191, I192, I193, I194, I195, I196, I197, I198, I199) -> f5(I191, I192, I193, I194, I195, I196, I197, I198, I199) f1(I200, I201, I202, I203, I204, I205, I206, I207, I208) -> f3(I200, I201, I202, 1 + I203, I204, I205, I206, I207, I208) [1 + I203 <= I205] f1(I209, I210, I211, I212, I213, I214, I215, I216, I217) -> f2(I209, I210, I211, I212, I213, I214, I215, I216, I217) [I214 <= I212] We use the extended value criterion with the projection function NU: NU[f11#(x0,x1,x2,x3,x4,x5,x6,x7,x8)] = x2 - x4 - 2 NU[f13#(x0,x1,x2,x3,x4,x5,x6,x7,x8)] = x2 - x4 - 1 NU[f12#(x0,x1,x2,x3,x4,x5,x6,x7,x8)] = x2 - x4 - 1 This gives the following inequalities: ==> I110 - I112 - 1 >= I110 - I112 - 1 rnd9 = rnd9 /\ rnd8 = rnd8 /\ 1 + I121 <= I119 ==> I119 - I121 - 1 > I119 - I121 - 2 with I119 - I121 - 1 >= 0 ==> I137 - I139 - 2 >= I137 - (1 + I139) - 1 We remove all the strictly oriented dependency pairs. DP problem for innermost termination. P = f12#(I108, I109, I110, I111, I112, I113, I114, I115, I116) -> f13#(I108, I109, I110, I111, I112, I113, I114, I115, I116) f11#(I135, I136, I137, I138, I139, I140, I141, I142, I143) -> f12#(I135, I136, I137, I138, 1 + I139, I140, I141, I142, I143) R = f18(x1, x2, x3, x4, x5, x6, x7, x8, x9) -> f17(x1, x2, x3, x4, x5, x6, x7, x8, x9) f17(I0, I1, I2, I3, I4, I5, I6, I7, I8) -> f4(I0, I1, I1, 0, I4, I0, 0, I7, I8) f3(I9, I10, I11, I12, I13, I14, I15, I16, I17) -> f1(I9, I10, I11, I12, I13, I14, I15, I16, I17) f7(I18, I19, I20, I21, I22, I23, I24, I25, I26) -> f8(I18, I19, I20, I21, I22, I23, I24, I25, I26) f5(I27, I28, I29, I30, I31, I32, I33, I34, I35) -> f16(I27, I28, I29, I30, I31, I32, I33, I34, I35) [1 + I30 <= I32] f5(I36, I37, I38, I39, I40, I41, I42, I43, I44) -> f9(I36, I37, I38, 0, I40, I41, I42, I43, I44) [I41 <= I39] f16(I45, I46, I47, I48, I49, I50, I51, I52, I53) -> f15(I45, I46, I47, I48, I49, I50, I51, I52, I53) [I51 <= I48 /\ I48 <= I51] f16(I54, I55, I56, I57, I58, I59, I60, I61, I62) -> f14(I54, I55, I56, I57, I58, I59, I60, I61, I62) [1 + I57 <= I60] f16(I63, I64, I65, I66, I67, I68, I69, I70, I71) -> f14(I63, I64, I65, I66, I67, I68, I69, I70, I71) [1 + I69 <= I66] f15(I72, I73, I74, I75, I76, I77, I78, I79, I80) -> f4(I72, I73, I74, 1 + I75, I76, I77, I78, I79, I80) f14(I81, I82, I83, I84, I85, I86, I87, I88, I89) -> f15(I81, I82, I83, I84, I85, I86, I87, I88, I89) f10(I90, I91, I92, I93, I94, I95, I96, I97, I98) -> f12(I90, I91, I92, I93, 0, I95, I96, I97, I98) [1 + I93 <= I95] f10(I99, I100, I101, I102, I103, I104, I105, I106, I107) -> f7(I99, I100, I101, 0, I103, I104, I105, I106, I107) [I104 <= I102] f12(I108, I109, I110, I111, I112, I113, I114, I115, I116) -> f13(I108, I109, I110, I111, I112, I113, I114, I115, I116) f13(I117, I118, I119, I120, I121, I122, I123, I124, I125) -> f11(I117, I118, I119, I120, I121, I122, I123, rnd8, rnd9) [rnd9 = rnd9 /\ rnd8 = rnd8 /\ 1 + I121 <= I119] f13(I126, I127, I128, I129, I130, I131, I132, I133, I134) -> f9(I126, I127, I128, 1 + I129, I130, I131, I132, I133, I134) [I128 <= I130] f11(I135, I136, I137, I138, I139, I140, I141, I142, I143) -> f12(I135, I136, I137, I138, 1 + I139, I140, I141, I142, I143) f9(I144, I145, I146, I147, I148, I149, I150, I151, I152) -> f10(I144, I145, I146, I147, I148, I149, I150, I151, I152) f8(I153, I154, I155, I156, I157, I158, I159, I160, I161) -> f6(I153, I154, I155, I156, I157, I158, I159, I162, I163) [I163 = I163 /\ I162 = I162 /\ 1 + I156 <= I155] f8(I164, I165, I166, I167, I168, I169, I170, I171, I172) -> f3(I164, I165, I166, 0, I168, I169, I170, I171, I172) [I166 <= I167] f6(I173, I174, I175, I176, I177, I178, I179, I180, I181) -> f2(I173, I174, I175, I176, I177, I178, I179, I180, I181) f6(I182, I183, I184, I185, I186, I187, I188, I189, I190) -> f7(I182, I183, I184, 1 + I185, I186, I187, I188, I189, I190) f4(I191, I192, I193, I194, I195, I196, I197, I198, I199) -> f5(I191, I192, I193, I194, I195, I196, I197, I198, I199) f1(I200, I201, I202, I203, I204, I205, I206, I207, I208) -> f3(I200, I201, I202, 1 + I203, I204, I205, I206, I207, I208) [1 + I203 <= I205] f1(I209, I210, I211, I212, I213, I214, I215, I216, I217) -> f2(I209, I210, I211, I212, I213, I214, I215, I216, I217) [I214 <= I212] The dependency graph for this problem is: 13 -> 16 -> 13 Where: 13) f12#(I108, I109, I110, I111, I112, I113, I114, I115, I116) -> f13#(I108, I109, I110, I111, I112, I113, I114, I115, I116) 16) f11#(I135, I136, I137, I138, I139, I140, I141, I142, I143) -> f12#(I135, I136, I137, I138, 1 + I139, I140, I141, I142, I143) We have the following SCCs. DP problem for innermost termination. P = f5#(I27, I28, I29, I30, I31, I32, I33, I34, I35) -> f16#(I27, I28, I29, I30, I31, I32, I33, I34, I35) [1 + I30 <= I32] f16#(I45, I46, I47, I48, I49, I50, I51, I52, I53) -> f15#(I45, I46, I47, I48, I49, I50, I51, I52, I53) [I51 <= I48 /\ I48 <= I51] f16#(I54, I55, I56, I57, I58, I59, I60, I61, I62) -> f14#(I54, I55, I56, I57, I58, I59, I60, I61, I62) [1 + I57 <= I60] f16#(I63, I64, I65, I66, I67, I68, I69, I70, I71) -> f14#(I63, I64, I65, I66, I67, I68, I69, I70, I71) [1 + I69 <= I66] f15#(I72, I73, I74, I75, I76, I77, I78, I79, I80) -> f4#(I72, I73, I74, 1 + I75, I76, I77, I78, I79, I80) f14#(I81, I82, I83, I84, I85, I86, I87, I88, I89) -> f15#(I81, I82, I83, I84, I85, I86, I87, I88, I89) f4#(I191, I192, I193, I194, I195, I196, I197, I198, I199) -> f5#(I191, I192, I193, I194, I195, I196, I197, I198, I199) R = f18(x1, x2, x3, x4, x5, x6, x7, x8, x9) -> f17(x1, x2, x3, x4, x5, x6, x7, x8, x9) f17(I0, I1, I2, I3, I4, I5, I6, I7, I8) -> f4(I0, I1, I1, 0, I4, I0, 0, I7, I8) f3(I9, I10, I11, I12, I13, I14, I15, I16, I17) -> f1(I9, I10, I11, I12, I13, I14, I15, I16, I17) f7(I18, I19, I20, I21, I22, I23, I24, I25, I26) -> f8(I18, I19, I20, I21, I22, I23, I24, I25, I26) f5(I27, I28, I29, I30, I31, I32, I33, I34, I35) -> f16(I27, I28, I29, I30, I31, I32, I33, I34, I35) [1 + I30 <= I32] f5(I36, I37, I38, I39, I40, I41, I42, I43, I44) -> f9(I36, I37, I38, 0, I40, I41, I42, I43, I44) [I41 <= I39] f16(I45, I46, I47, I48, I49, I50, I51, I52, I53) -> f15(I45, I46, I47, I48, I49, I50, I51, I52, I53) [I51 <= I48 /\ I48 <= I51] f16(I54, I55, I56, I57, I58, I59, I60, I61, I62) -> f14(I54, I55, I56, I57, I58, I59, I60, I61, I62) [1 + I57 <= I60] f16(I63, I64, I65, I66, I67, I68, I69, I70, I71) -> f14(I63, I64, I65, I66, I67, I68, I69, I70, I71) [1 + I69 <= I66] f15(I72, I73, I74, I75, I76, I77, I78, I79, I80) -> f4(I72, I73, I74, 1 + I75, I76, I77, I78, I79, I80) f14(I81, I82, I83, I84, I85, I86, I87, I88, I89) -> f15(I81, I82, I83, I84, I85, I86, I87, I88, I89) f10(I90, I91, I92, I93, I94, I95, I96, I97, I98) -> f12(I90, I91, I92, I93, 0, I95, I96, I97, I98) [1 + I93 <= I95] f10(I99, I100, I101, I102, I103, I104, I105, I106, I107) -> f7(I99, I100, I101, 0, I103, I104, I105, I106, I107) [I104 <= I102] f12(I108, I109, I110, I111, I112, I113, I114, I115, I116) -> f13(I108, I109, I110, I111, I112, I113, I114, I115, I116) f13(I117, I118, I119, I120, I121, I122, I123, I124, I125) -> f11(I117, I118, I119, I120, I121, I122, I123, rnd8, rnd9) [rnd9 = rnd9 /\ rnd8 = rnd8 /\ 1 + I121 <= I119] f13(I126, I127, I128, I129, I130, I131, I132, I133, I134) -> f9(I126, I127, I128, 1 + I129, I130, I131, I132, I133, I134) [I128 <= I130] f11(I135, I136, I137, I138, I139, I140, I141, I142, I143) -> f12(I135, I136, I137, I138, 1 + I139, I140, I141, I142, I143) f9(I144, I145, I146, I147, I148, I149, I150, I151, I152) -> f10(I144, I145, I146, I147, I148, I149, I150, I151, I152) f8(I153, I154, I155, I156, I157, I158, I159, I160, I161) -> f6(I153, I154, I155, I156, I157, I158, I159, I162, I163) [I163 = I163 /\ I162 = I162 /\ 1 + I156 <= I155] f8(I164, I165, I166, I167, I168, I169, I170, I171, I172) -> f3(I164, I165, I166, 0, I168, I169, I170, I171, I172) [I166 <= I167] f6(I173, I174, I175, I176, I177, I178, I179, I180, I181) -> f2(I173, I174, I175, I176, I177, I178, I179, I180, I181) f6(I182, I183, I184, I185, I186, I187, I188, I189, I190) -> f7(I182, I183, I184, 1 + I185, I186, I187, I188, I189, I190) f4(I191, I192, I193, I194, I195, I196, I197, I198, I199) -> f5(I191, I192, I193, I194, I195, I196, I197, I198, I199) f1(I200, I201, I202, I203, I204, I205, I206, I207, I208) -> f3(I200, I201, I202, 1 + I203, I204, I205, I206, I207, I208) [1 + I203 <= I205] f1(I209, I210, I211, I212, I213, I214, I215, I216, I217) -> f2(I209, I210, I211, I212, I213, I214, I215, I216, I217) [I214 <= I212] We use the extended value criterion with the projection function NU: NU[f4#(x0,x1,x2,x3,x4,x5,x6,x7,x8)] = -x3 + x5 - 1 NU[f14#(x0,x1,x2,x3,x4,x5,x6,x7,x8)] = -x3 + x5 - 2 NU[f15#(x0,x1,x2,x3,x4,x5,x6,x7,x8)] = -x3 + x5 - 2 NU[f16#(x0,x1,x2,x3,x4,x5,x6,x7,x8)] = -x3 + x5 - 2 NU[f5#(x0,x1,x2,x3,x4,x5,x6,x7,x8)] = -x3 + x5 - 1 This gives the following inequalities: 1 + I30 <= I32 ==> -I30 + I32 - 1 > -I30 + I32 - 2 with -I30 + I32 - 1 >= 0 I51 <= I48 /\ I48 <= I51 ==> -I48 + I50 - 2 >= -I48 + I50 - 2 1 + I57 <= I60 ==> -I57 + I59 - 2 >= -I57 + I59 - 2 1 + I69 <= I66 ==> -I66 + I68 - 2 >= -I66 + I68 - 2 ==> -I75 + I77 - 2 >= -(1 + I75) + I77 - 1 ==> -I84 + I86 - 2 >= -I84 + I86 - 2 ==> -I194 + I196 - 1 >= -I194 + I196 - 1 We remove all the strictly oriented dependency pairs. DP problem for innermost termination. P = f16#(I45, I46, I47, I48, I49, I50, I51, I52, I53) -> f15#(I45, I46, I47, I48, I49, I50, I51, I52, I53) [I51 <= I48 /\ I48 <= I51] f16#(I54, I55, I56, I57, I58, I59, I60, I61, I62) -> f14#(I54, I55, I56, I57, I58, I59, I60, I61, I62) [1 + I57 <= I60] f16#(I63, I64, I65, I66, I67, I68, I69, I70, I71) -> f14#(I63, I64, I65, I66, I67, I68, I69, I70, I71) [1 + I69 <= I66] f15#(I72, I73, I74, I75, I76, I77, I78, I79, I80) -> f4#(I72, I73, I74, 1 + I75, I76, I77, I78, I79, I80) f14#(I81, I82, I83, I84, I85, I86, I87, I88, I89) -> f15#(I81, I82, I83, I84, I85, I86, I87, I88, I89) f4#(I191, I192, I193, I194, I195, I196, I197, I198, I199) -> f5#(I191, I192, I193, I194, I195, I196, I197, I198, I199) R = f18(x1, x2, x3, x4, x5, x6, x7, x8, x9) -> f17(x1, x2, x3, x4, x5, x6, x7, x8, x9) f17(I0, I1, I2, I3, I4, I5, I6, I7, I8) -> f4(I0, I1, I1, 0, I4, I0, 0, I7, I8) f3(I9, I10, I11, I12, I13, I14, I15, I16, I17) -> f1(I9, I10, I11, I12, I13, I14, I15, I16, I17) f7(I18, I19, I20, I21, I22, I23, I24, I25, I26) -> f8(I18, I19, I20, I21, I22, I23, I24, I25, I26) f5(I27, I28, I29, I30, I31, I32, I33, I34, I35) -> f16(I27, I28, I29, I30, I31, I32, I33, I34, I35) [1 + I30 <= I32] f5(I36, I37, I38, I39, I40, I41, I42, I43, I44) -> f9(I36, I37, I38, 0, I40, I41, I42, I43, I44) [I41 <= I39] f16(I45, I46, I47, I48, I49, I50, I51, I52, I53) -> f15(I45, I46, I47, I48, I49, I50, I51, I52, I53) [I51 <= I48 /\ I48 <= I51] f16(I54, I55, I56, I57, I58, I59, I60, I61, I62) -> f14(I54, I55, I56, I57, I58, I59, I60, I61, I62) [1 + I57 <= I60] f16(I63, I64, I65, I66, I67, I68, I69, I70, I71) -> f14(I63, I64, I65, I66, I67, I68, I69, I70, I71) [1 + I69 <= I66] f15(I72, I73, I74, I75, I76, I77, I78, I79, I80) -> f4(I72, I73, I74, 1 + I75, I76, I77, I78, I79, I80) f14(I81, I82, I83, I84, I85, I86, I87, I88, I89) -> f15(I81, I82, I83, I84, I85, I86, I87, I88, I89) f10(I90, I91, I92, I93, I94, I95, I96, I97, I98) -> f12(I90, I91, I92, I93, 0, I95, I96, I97, I98) [1 + I93 <= I95] f10(I99, I100, I101, I102, I103, I104, I105, I106, I107) -> f7(I99, I100, I101, 0, I103, I104, I105, I106, I107) [I104 <= I102] f12(I108, I109, I110, I111, I112, I113, I114, I115, I116) -> f13(I108, I109, I110, I111, I112, I113, I114, I115, I116) f13(I117, I118, I119, I120, I121, I122, I123, I124, I125) -> f11(I117, I118, I119, I120, I121, I122, I123, rnd8, rnd9) [rnd9 = rnd9 /\ rnd8 = rnd8 /\ 1 + I121 <= I119] f13(I126, I127, I128, I129, I130, I131, I132, I133, I134) -> f9(I126, I127, I128, 1 + I129, I130, I131, I132, I133, I134) [I128 <= I130] f11(I135, I136, I137, I138, I139, I140, I141, I142, I143) -> f12(I135, I136, I137, I138, 1 + I139, I140, I141, I142, I143) f9(I144, I145, I146, I147, I148, I149, I150, I151, I152) -> f10(I144, I145, I146, I147, I148, I149, I150, I151, I152) f8(I153, I154, I155, I156, I157, I158, I159, I160, I161) -> f6(I153, I154, I155, I156, I157, I158, I159, I162, I163) [I163 = I163 /\ I162 = I162 /\ 1 + I156 <= I155] f8(I164, I165, I166, I167, I168, I169, I170, I171, I172) -> f3(I164, I165, I166, 0, I168, I169, I170, I171, I172) [I166 <= I167] f6(I173, I174, I175, I176, I177, I178, I179, I180, I181) -> f2(I173, I174, I175, I176, I177, I178, I179, I180, I181) f6(I182, I183, I184, I185, I186, I187, I188, I189, I190) -> f7(I182, I183, I184, 1 + I185, I186, I187, I188, I189, I190) f4(I191, I192, I193, I194, I195, I196, I197, I198, I199) -> f5(I191, I192, I193, I194, I195, I196, I197, I198, I199) f1(I200, I201, I202, I203, I204, I205, I206, I207, I208) -> f3(I200, I201, I202, 1 + I203, I204, I205, I206, I207, I208) [1 + I203 <= I205] f1(I209, I210, I211, I212, I213, I214, I215, I216, I217) -> f2(I209, I210, I211, I212, I213, I214, I215, I216, I217) [I214 <= I212] The dependency graph for this problem is: 6 -> 9 7 -> 10 8 -> 10 9 -> 21 10 -> 9 21 -> Where: 6) f16#(I45, I46, I47, I48, I49, I50, I51, I52, I53) -> f15#(I45, I46, I47, I48, I49, I50, I51, I52, I53) [I51 <= I48 /\ I48 <= I51] 7) f16#(I54, I55, I56, I57, I58, I59, I60, I61, I62) -> f14#(I54, I55, I56, I57, I58, I59, I60, I61, I62) [1 + I57 <= I60] 8) f16#(I63, I64, I65, I66, I67, I68, I69, I70, I71) -> f14#(I63, I64, I65, I66, I67, I68, I69, I70, I71) [1 + I69 <= I66] 9) f15#(I72, I73, I74, I75, I76, I77, I78, I79, I80) -> f4#(I72, I73, I74, 1 + I75, I76, I77, I78, I79, I80) 10) f14#(I81, I82, I83, I84, I85, I86, I87, I88, I89) -> f15#(I81, I82, I83, I84, I85, I86, I87, I88, I89) 21) f4#(I191, I192, I193, I194, I195, I196, I197, I198, I199) -> f5#(I191, I192, I193, I194, I195, I196, I197, I198, I199) We have the following SCCs.