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