/export/starexec/sandbox2/solver/bin/starexec_run_Transition /export/starexec/sandbox2/benchmark/theBenchmark.smt2 /export/starexec/sandbox2/output/output_files -------------------------------------------------------------------------------- YES DP problem for innermost termination. P = f24#(x1, x2, x3, x4, x5, x6, x7, x8) -> f8#(x1, x2, x3, x4, x5, x6, x7, x8) f8#(I0, I1, I2, I3, I4, I5, I6, I7) -> f1#(I0, I1, I2, I3, I4, I5, I6, I7) f4#(I8, I9, I10, I11, I12, I13, I14, I15) -> f23#(I8, I9, I10, I11, rnd5, rnd6, I14, I15) [rnd6 = rnd6 /\ rnd5 = rnd5 /\ I9 <= I11] f4#(I16, I17, I18, I19, I20, I21, I22, I23) -> f21#(I16, I17, I18, I19, I20, I21, I22, I23) [1 + I19 <= I17] f23#(I24, I25, I26, I27, I28, I29, I30, I31) -> f22#(I25, I25, I26, I27, I28, I29, rnd7, I31) [rnd7 = rnd7 /\ 1 + I29 <= I28] f23#(I32, I33, I34, I35, I36, I37, I38, I39) -> f22#(I32, I33, I34, I35, I36, I37, I38, I39) [I36 <= I37] f22#(I40, I41, I42, I43, I44, I45, I46, I47) -> f3#(I40, 1 + I41, I42, I43, I44, I45, I46, I47) f14#(I48, I49, I50, I51, I52, I53, I54, I55) -> f13#(I48, I49, I50, I51, I52, I53, I54, I55) f21#(I56, I57, I58, I59, I60, I61, I62, I63) -> f5#(I56, I57, I58, I59, I60, I61, I62, I63) [1 + I56 <= I58] f21#(I64, I65, I66, I67, I68, I69, I70, I71) -> f5#(I64, I65, I66, I67, I68, I69, I70, I71) [1 + I66 <= I64] f21#(I72, I73, I74, I75, I76, I77, I78, I79) -> f20#(I72, I73, I74, I75, I76, I77, I78, I79) [I74 <= I72 /\ I72 <= I74] f6#(I80, I81, I82, I83, I84, I85, I86, I87) -> f5#(I80, 1 + I81, I82, I83, I84, I85, I86, rnd8) [rnd8 = rnd8 /\ I81 <= I83] f6#(I88, I89, I90, I91, I92, I93, I94, I95) -> f11#(I88, I89, I90, I91, I92, I93, I94, I95) [1 + I91 <= I89] f12#(I96, I97, I98, I99, I100, I101, I102, I103) -> f11#(I96, 1 + I97, I98, I99, I100, I101, I102, I104) [I104 = I104 /\ I97 <= I99] f12#(I105, I106, I107, I108, I109, I110, I111, I112) -> f20#(I105, I106, I107, I108, I109, I110, I111, I112) [1 + I108 <= I106] f16#(I113, I114, I115, I116, I117, I118, I119, I120) -> f15#(I113, I114, I115, I116, I117, I118, I119, I120) f20#(I121, I122, I123, I124, I125, I126, I127, I128) -> f10#(I121, I122, I123, I124, I125, I126, I127, I128) [1 + I127 <= 0] f20#(I129, I130, I131, I132, I133, I134, I135, I136) -> f10#(I129, I130, I131, I132, I133, I134, I135, I136) [1 <= I135] f20#(I137, I138, I139, I140, I141, I142, I143, I144) -> f7#(I137, I138, I139, I140, I141, I142, I143, I144) [0 <= I143 /\ I143 <= 0] f19#(I145, I146, I147, I148, I149, I150, I151, I152) -> f18#(I145, I146, I147, I148, I149, I150, I151, I153) [I153 = I153 /\ I145 <= I148] f19#(I154, I155, I156, I157, I158, I159, I160, I161) -> f7#(I154, I155, I156, I157, I158, I159, I160, I161) [1 + I157 <= I154] f10#(I162, I163, I164, I165, I166, I167, I168, I169) -> f19#(I162, I163, I164, I165, I166, I167, I168, I169) f18#(I170, I171, I172, I173, I174, I175, I176, I177) -> f17#(I170, I171, I172, I173, I174, I175, I176, I177) [1 + I177 <= 0] f18#(I178, I179, I180, I181, I182, I183, I184, I185) -> f17#(I178, I179, I180, I181, I182, I183, I184, I185) [1 <= I185] f18#(I186, I187, I188, I189, I190, I191, I192, I193) -> f9#(I186, I187, I188, I189, I190, I191, I192, I193) [0 <= I193 /\ I193 <= 0] f17#(I194, I195, I196, I197, I198, I199, I200, I201) -> f16#(I194, I195, I196, I197, I198, I199, I200, I202) [I202 = I202] f15#(I203, I204, I205, I206, I207, I208, I209, I210) -> f16#(I203, 1 + I204, I205, I206, I207, I208, I209, I210) [I204 <= I206] f15#(I211, I212, I213, I214, I215, I216, I217, I218) -> f14#(I211, I212, I213, I214, I215, I216, I217, I218) [1 + I214 <= I212] f13#(I219, I220, I221, I222, I223, I224, I225, I226) -> f14#(I219, 1 + I220, I221, I222, I223, I224, I225, I226) [I220 <= I222] f13#(I227, I228, I229, I230, I231, I232, I233, I234) -> f9#(I227, I228, I229, I230, I231, I232, I233, I234) [1 + I230 <= I228] f11#(I235, I236, I237, I238, I239, I240, I241, I242) -> f12#(I235, I236, I237, I238, I239, I240, I241, I242) f9#(I243, I244, I245, I246, I247, I248, I249, I250) -> f10#(1 + I243, I244, I245, I246, I247, I248, I249, I250) f7#(I251, I252, I253, I254, I255, I256, I257, I258) -> f8#(I251, I252, 1 + I253, I254, I255, I256, I257, I258) f5#(I259, I260, I261, I262, I263, I264, I265, I266) -> f6#(I259, I260, I261, I262, I263, I264, I265, I266) f3#(I267, I268, I269, I270, I271, I272, I273, I274) -> f4#(I267, I268, I269, I270, I271, I272, I273, I274) f1#(I275, I276, I277, I278, I279, I280, I281, I282) -> f3#(I277, I276, I277, I278, I279, I280, 0, I282) [1 + I277 <= I278] R = f24(x1, x2, x3, x4, x5, x6, x7, x8) -> f8(x1, x2, x3, x4, x5, x6, x7, x8) f8(I0, I1, I2, I3, I4, I5, I6, I7) -> f1(I0, I1, I2, I3, I4, I5, I6, I7) f4(I8, I9, I10, I11, I12, I13, I14, I15) -> f23(I8, I9, I10, I11, rnd5, rnd6, I14, I15) [rnd6 = rnd6 /\ rnd5 = rnd5 /\ I9 <= I11] f4(I16, I17, I18, I19, I20, I21, I22, I23) -> f21(I16, I17, I18, I19, I20, I21, I22, I23) [1 + I19 <= I17] f23(I24, I25, I26, I27, I28, I29, I30, I31) -> f22(I25, I25, I26, I27, I28, I29, rnd7, I31) [rnd7 = rnd7 /\ 1 + I29 <= I28] f23(I32, I33, I34, I35, I36, I37, I38, I39) -> f22(I32, I33, I34, I35, I36, I37, I38, I39) [I36 <= I37] f22(I40, I41, I42, I43, I44, I45, I46, I47) -> f3(I40, 1 + I41, I42, I43, I44, I45, I46, I47) f14(I48, I49, I50, I51, I52, I53, I54, I55) -> f13(I48, I49, I50, I51, I52, I53, I54, I55) f21(I56, I57, I58, I59, I60, I61, I62, I63) -> f5(I56, I57, I58, I59, I60, I61, I62, I63) [1 + I56 <= I58] f21(I64, I65, I66, I67, I68, I69, I70, I71) -> f5(I64, I65, I66, I67, I68, I69, I70, I71) [1 + I66 <= I64] f21(I72, I73, I74, I75, I76, I77, I78, I79) -> f20(I72, I73, I74, I75, I76, I77, I78, I79) [I74 <= I72 /\ I72 <= I74] f6(I80, I81, I82, I83, I84, I85, I86, I87) -> f5(I80, 1 + I81, I82, I83, I84, I85, I86, rnd8) [rnd8 = rnd8 /\ I81 <= I83] f6(I88, I89, I90, I91, I92, I93, I94, I95) -> f11(I88, I89, I90, I91, I92, I93, I94, I95) [1 + I91 <= I89] f12(I96, I97, I98, I99, I100, I101, I102, I103) -> f11(I96, 1 + I97, I98, I99, I100, I101, I102, I104) [I104 = I104 /\ I97 <= I99] f12(I105, I106, I107, I108, I109, I110, I111, I112) -> f20(I105, I106, I107, I108, I109, I110, I111, I112) [1 + I108 <= I106] f16(I113, I114, I115, I116, I117, I118, I119, I120) -> f15(I113, I114, I115, I116, I117, I118, I119, I120) f20(I121, I122, I123, I124, I125, I126, I127, I128) -> f10(I121, I122, I123, I124, I125, I126, I127, I128) [1 + I127 <= 0] f20(I129, I130, I131, I132, I133, I134, I135, I136) -> f10(I129, I130, I131, I132, I133, I134, I135, I136) [1 <= I135] f20(I137, I138, I139, I140, I141, I142, I143, I144) -> f7(I137, I138, I139, I140, I141, I142, I143, I144) [0 <= I143 /\ I143 <= 0] f19(I145, I146, I147, I148, I149, I150, I151, I152) -> f18(I145, I146, I147, I148, I149, I150, I151, I153) [I153 = I153 /\ I145 <= I148] f19(I154, I155, I156, I157, I158, I159, I160, I161) -> f7(I154, I155, I156, I157, I158, I159, I160, I161) [1 + I157 <= I154] f10(I162, I163, I164, I165, I166, I167, I168, I169) -> f19(I162, I163, I164, I165, I166, I167, I168, I169) f18(I170, I171, I172, I173, I174, I175, I176, I177) -> f17(I170, I171, I172, I173, I174, I175, I176, I177) [1 + I177 <= 0] f18(I178, I179, I180, I181, I182, I183, I184, I185) -> f17(I178, I179, I180, I181, I182, I183, I184, I185) [1 <= I185] f18(I186, I187, I188, I189, I190, I191, I192, I193) -> f9(I186, I187, I188, I189, I190, I191, I192, I193) [0 <= I193 /\ I193 <= 0] f17(I194, I195, I196, I197, I198, I199, I200, I201) -> f16(I194, I195, I196, I197, I198, I199, I200, I202) [I202 = I202] f15(I203, I204, I205, I206, I207, I208, I209, I210) -> f16(I203, 1 + I204, I205, I206, I207, I208, I209, I210) [I204 <= I206] f15(I211, I212, I213, I214, I215, I216, I217, I218) -> f14(I211, I212, I213, I214, I215, I216, I217, I218) [1 + I214 <= I212] f13(I219, I220, I221, I222, I223, I224, I225, I226) -> f14(I219, 1 + I220, I221, I222, I223, I224, I225, I226) [I220 <= I222] f13(I227, I228, I229, I230, I231, I232, I233, I234) -> f9(I227, I228, I229, I230, I231, I232, I233, I234) [1 + I230 <= I228] f11(I235, I236, I237, I238, I239, I240, I241, I242) -> f12(I235, I236, I237, I238, I239, I240, I241, I242) f9(I243, I244, I245, I246, I247, I248, I249, I250) -> f10(1 + I243, I244, I245, I246, I247, I248, I249, I250) f7(I251, I252, I253, I254, I255, I256, I257, I258) -> f8(I251, I252, 1 + I253, I254, I255, I256, I257, I258) f5(I259, I260, I261, I262, I263, I264, I265, I266) -> f6(I259, I260, I261, I262, I263, I264, I265, I266) f3(I267, I268, I269, I270, I271, I272, I273, I274) -> f4(I267, I268, I269, I270, I271, I272, I273, I274) f1(I275, I276, I277, I278, I279, I280, I281, I282) -> f3(I277, I276, I277, I278, I279, I280, 0, I282) [1 + I277 <= I278] f1(I283, I284, I285, I286, I287, I288, I289, I290) -> f2(I283, I284, I285, I286, I287, I288, I289, I290) [I286 <= I285] The dependency graph for this problem is: 0 -> 1 1 -> 35 2 -> 4, 5 3 -> 8, 9, 10 4 -> 6 5 -> 6 6 -> 34 7 -> 28, 29 8 -> 33 9 -> 33 10 -> 16, 17, 18 11 -> 33 12 -> 30 13 -> 30 14 -> 16, 17, 18 15 -> 26, 27 16 -> 21 17 -> 21 18 -> 32 19 -> 22, 23, 24 20 -> 32 21 -> 19, 20 22 -> 25 23 -> 25 24 -> 31 25 -> 15 26 -> 15 27 -> 7 28 -> 7 29 -> 31 30 -> 13, 14 31 -> 21 32 -> 1 33 -> 11, 12 34 -> 2, 3 35 -> 34 Where: 0) f24#(x1, x2, x3, x4, x5, x6, x7, x8) -> f8#(x1, x2, x3, x4, x5, x6, x7, x8) 1) f8#(I0, I1, I2, I3, I4, I5, I6, I7) -> f1#(I0, I1, I2, I3, I4, I5, I6, I7) 2) f4#(I8, I9, I10, I11, I12, I13, I14, I15) -> f23#(I8, I9, I10, I11, rnd5, rnd6, I14, I15) [rnd6 = rnd6 /\ rnd5 = rnd5 /\ I9 <= I11] 3) f4#(I16, I17, I18, I19, I20, I21, I22, I23) -> f21#(I16, I17, I18, I19, I20, I21, I22, I23) [1 + I19 <= I17] 4) f23#(I24, I25, I26, I27, I28, I29, I30, I31) -> f22#(I25, I25, I26, I27, I28, I29, rnd7, I31) [rnd7 = rnd7 /\ 1 + I29 <= I28] 5) f23#(I32, I33, I34, I35, I36, I37, I38, I39) -> f22#(I32, I33, I34, I35, I36, I37, I38, I39) [I36 <= I37] 6) f22#(I40, I41, I42, I43, I44, I45, I46, I47) -> f3#(I40, 1 + I41, I42, I43, I44, I45, I46, I47) 7) f14#(I48, I49, I50, I51, I52, I53, I54, I55) -> f13#(I48, I49, I50, I51, I52, I53, I54, I55) 8) f21#(I56, I57, I58, I59, I60, I61, I62, I63) -> f5#(I56, I57, I58, I59, I60, I61, I62, I63) [1 + I56 <= I58] 9) f21#(I64, I65, I66, I67, I68, I69, I70, I71) -> f5#(I64, I65, I66, I67, I68, I69, I70, I71) [1 + I66 <= I64] 10) f21#(I72, I73, I74, I75, I76, I77, I78, I79) -> f20#(I72, I73, I74, I75, I76, I77, I78, I79) [I74 <= I72 /\ I72 <= I74] 11) f6#(I80, I81, I82, I83, I84, I85, I86, I87) -> f5#(I80, 1 + I81, I82, I83, I84, I85, I86, rnd8) [rnd8 = rnd8 /\ I81 <= I83] 12) f6#(I88, I89, I90, I91, I92, I93, I94, I95) -> f11#(I88, I89, I90, I91, I92, I93, I94, I95) [1 + I91 <= I89] 13) f12#(I96, I97, I98, I99, I100, I101, I102, I103) -> f11#(I96, 1 + I97, I98, I99, I100, I101, I102, I104) [I104 = I104 /\ I97 <= I99] 14) f12#(I105, I106, I107, I108, I109, I110, I111, I112) -> f20#(I105, I106, I107, I108, I109, I110, I111, I112) [1 + I108 <= I106] 15) f16#(I113, I114, I115, I116, I117, I118, I119, I120) -> f15#(I113, I114, I115, I116, I117, I118, I119, I120) 16) f20#(I121, I122, I123, I124, I125, I126, I127, I128) -> f10#(I121, I122, I123, I124, I125, I126, I127, I128) [1 + I127 <= 0] 17) f20#(I129, I130, I131, I132, I133, I134, I135, I136) -> f10#(I129, I130, I131, I132, I133, I134, I135, I136) [1 <= I135] 18) f20#(I137, I138, I139, I140, I141, I142, I143, I144) -> f7#(I137, I138, I139, I140, I141, I142, I143, I144) [0 <= I143 /\ I143 <= 0] 19) f19#(I145, I146, I147, I148, I149, I150, I151, I152) -> f18#(I145, I146, I147, I148, I149, I150, I151, I153) [I153 = I153 /\ I145 <= I148] 20) f19#(I154, I155, I156, I157, I158, I159, I160, I161) -> f7#(I154, I155, I156, I157, I158, I159, I160, I161) [1 + I157 <= I154] 21) f10#(I162, I163, I164, I165, I166, I167, I168, I169) -> f19#(I162, I163, I164, I165, I166, I167, I168, I169) 22) f18#(I170, I171, I172, I173, I174, I175, I176, I177) -> f17#(I170, I171, I172, I173, I174, I175, I176, I177) [1 + I177 <= 0] 23) f18#(I178, I179, I180, I181, I182, I183, I184, I185) -> f17#(I178, I179, I180, I181, I182, I183, I184, I185) [1 <= I185] 24) f18#(I186, I187, I188, I189, I190, I191, I192, I193) -> f9#(I186, I187, I188, I189, I190, I191, I192, I193) [0 <= I193 /\ I193 <= 0] 25) f17#(I194, I195, I196, I197, I198, I199, I200, I201) -> f16#(I194, I195, I196, I197, I198, I199, I200, I202) [I202 = I202] 26) f15#(I203, I204, I205, I206, I207, I208, I209, I210) -> f16#(I203, 1 + I204, I205, I206, I207, I208, I209, I210) [I204 <= I206] 27) f15#(I211, I212, I213, I214, I215, I216, I217, I218) -> f14#(I211, I212, I213, I214, I215, I216, I217, I218) [1 + I214 <= I212] 28) f13#(I219, I220, I221, I222, I223, I224, I225, I226) -> f14#(I219, 1 + I220, I221, I222, I223, I224, I225, I226) [I220 <= I222] 29) f13#(I227, I228, I229, I230, I231, I232, I233, I234) -> f9#(I227, I228, I229, I230, I231, I232, I233, I234) [1 + I230 <= I228] 30) f11#(I235, I236, I237, I238, I239, I240, I241, I242) -> f12#(I235, I236, I237, I238, I239, I240, I241, I242) 31) f9#(I243, I244, I245, I246, I247, I248, I249, I250) -> f10#(1 + I243, I244, I245, I246, I247, I248, I249, I250) 32) f7#(I251, I252, I253, I254, I255, I256, I257, I258) -> f8#(I251, I252, 1 + I253, I254, I255, I256, I257, I258) 33) f5#(I259, I260, I261, I262, I263, I264, I265, I266) -> f6#(I259, I260, I261, I262, I263, I264, I265, I266) 34) f3#(I267, I268, I269, I270, I271, I272, I273, I274) -> f4#(I267, I268, I269, I270, I271, I272, I273, I274) 35) f1#(I275, I276, I277, I278, I279, I280, I281, I282) -> f3#(I277, I276, I277, I278, I279, I280, 0, I282) [1 + I277 <= I278] We have the following SCCs. { 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35 } DP problem for innermost termination. P = f8#(I0, I1, I2, I3, I4, I5, I6, I7) -> f1#(I0, I1, I2, I3, I4, I5, I6, I7) f4#(I8, I9, I10, I11, I12, I13, I14, I15) -> f23#(I8, I9, I10, I11, rnd5, rnd6, I14, I15) [rnd6 = rnd6 /\ rnd5 = rnd5 /\ I9 <= I11] f4#(I16, I17, I18, I19, I20, I21, I22, I23) -> f21#(I16, I17, I18, I19, I20, I21, I22, I23) [1 + I19 <= I17] f23#(I24, I25, I26, I27, I28, I29, I30, I31) -> f22#(I25, I25, I26, I27, I28, I29, rnd7, I31) [rnd7 = rnd7 /\ 1 + I29 <= I28] f23#(I32, I33, I34, I35, I36, I37, I38, I39) -> f22#(I32, I33, I34, I35, I36, I37, I38, I39) [I36 <= I37] f22#(I40, I41, I42, I43, I44, I45, I46, I47) -> f3#(I40, 1 + I41, I42, I43, I44, I45, I46, I47) f14#(I48, I49, I50, I51, I52, I53, I54, I55) -> f13#(I48, I49, I50, I51, I52, I53, I54, I55) f21#(I56, I57, I58, I59, I60, I61, I62, I63) -> f5#(I56, I57, I58, I59, I60, I61, I62, I63) [1 + I56 <= I58] f21#(I64, I65, I66, I67, I68, I69, I70, I71) -> f5#(I64, I65, I66, I67, I68, I69, I70, I71) [1 + I66 <= I64] f21#(I72, I73, I74, I75, I76, I77, I78, I79) -> f20#(I72, I73, I74, I75, I76, I77, I78, I79) [I74 <= I72 /\ I72 <= I74] f6#(I80, I81, I82, I83, I84, I85, I86, I87) -> f5#(I80, 1 + I81, I82, I83, I84, I85, I86, rnd8) [rnd8 = rnd8 /\ I81 <= I83] f6#(I88, I89, I90, I91, I92, I93, I94, I95) -> f11#(I88, I89, I90, I91, I92, I93, I94, I95) [1 + I91 <= I89] f12#(I96, I97, I98, I99, I100, I101, I102, I103) -> f11#(I96, 1 + I97, I98, I99, I100, I101, I102, I104) [I104 = I104 /\ I97 <= I99] f12#(I105, I106, I107, I108, I109, I110, I111, I112) -> f20#(I105, I106, I107, I108, I109, I110, I111, I112) [1 + I108 <= I106] f16#(I113, I114, I115, I116, I117, I118, I119, I120) -> f15#(I113, I114, I115, I116, I117, I118, I119, I120) f20#(I121, I122, I123, I124, I125, I126, I127, I128) -> f10#(I121, I122, I123, I124, I125, I126, I127, I128) [1 + I127 <= 0] f20#(I129, I130, I131, I132, I133, I134, I135, I136) -> f10#(I129, I130, I131, I132, I133, I134, I135, I136) [1 <= I135] f20#(I137, I138, I139, I140, I141, I142, I143, I144) -> f7#(I137, I138, I139, I140, I141, I142, I143, I144) [0 <= I143 /\ I143 <= 0] f19#(I145, I146, I147, I148, I149, I150, I151, I152) -> f18#(I145, I146, I147, I148, I149, I150, I151, I153) [I153 = I153 /\ I145 <= I148] f19#(I154, I155, I156, I157, I158, I159, I160, I161) -> f7#(I154, I155, I156, I157, I158, I159, I160, I161) [1 + I157 <= I154] f10#(I162, I163, I164, I165, I166, I167, I168, I169) -> f19#(I162, I163, I164, I165, I166, I167, I168, I169) f18#(I170, I171, I172, I173, I174, I175, I176, I177) -> f17#(I170, I171, I172, I173, I174, I175, I176, I177) [1 + I177 <= 0] f18#(I178, I179, I180, I181, I182, I183, I184, I185) -> f17#(I178, I179, I180, I181, I182, I183, I184, I185) [1 <= I185] f18#(I186, I187, I188, I189, I190, I191, I192, I193) -> f9#(I186, I187, I188, I189, I190, I191, I192, I193) [0 <= I193 /\ I193 <= 0] f17#(I194, I195, I196, I197, I198, I199, I200, I201) -> f16#(I194, I195, I196, I197, I198, I199, I200, I202) [I202 = I202] f15#(I203, I204, I205, I206, I207, I208, I209, I210) -> f16#(I203, 1 + I204, I205, I206, I207, I208, I209, I210) [I204 <= I206] f15#(I211, I212, I213, I214, I215, I216, I217, I218) -> f14#(I211, I212, I213, I214, I215, I216, I217, I218) [1 + I214 <= I212] f13#(I219, I220, I221, I222, I223, I224, I225, I226) -> f14#(I219, 1 + I220, I221, I222, I223, I224, I225, I226) [I220 <= I222] f13#(I227, I228, I229, I230, I231, I232, I233, I234) -> f9#(I227, I228, I229, I230, I231, I232, I233, I234) [1 + I230 <= I228] f11#(I235, I236, I237, I238, I239, I240, I241, I242) -> f12#(I235, I236, I237, I238, I239, I240, I241, I242) f9#(I243, I244, I245, I246, I247, I248, I249, I250) -> f10#(1 + I243, I244, I245, I246, I247, I248, I249, I250) f7#(I251, I252, I253, I254, I255, I256, I257, I258) -> f8#(I251, I252, 1 + I253, I254, I255, I256, I257, I258) f5#(I259, I260, I261, I262, I263, I264, I265, I266) -> f6#(I259, I260, I261, I262, I263, I264, I265, I266) f3#(I267, I268, I269, I270, I271, I272, I273, I274) -> f4#(I267, I268, I269, I270, I271, I272, I273, I274) f1#(I275, I276, I277, I278, I279, I280, I281, I282) -> f3#(I277, I276, I277, I278, I279, I280, 0, I282) [1 + I277 <= I278] R = f24(x1, x2, x3, x4, x5, x6, x7, x8) -> f8(x1, x2, x3, x4, x5, x6, x7, x8) f8(I0, I1, I2, I3, I4, I5, I6, I7) -> f1(I0, I1, I2, I3, I4, I5, I6, I7) f4(I8, I9, I10, I11, I12, I13, I14, I15) -> f23(I8, I9, I10, I11, rnd5, rnd6, I14, I15) [rnd6 = rnd6 /\ rnd5 = rnd5 /\ I9 <= I11] f4(I16, I17, I18, I19, I20, I21, I22, I23) -> f21(I16, I17, I18, I19, I20, I21, I22, I23) [1 + I19 <= I17] f23(I24, I25, I26, I27, I28, I29, I30, I31) -> f22(I25, I25, I26, I27, I28, I29, rnd7, I31) [rnd7 = rnd7 /\ 1 + I29 <= I28] f23(I32, I33, I34, I35, I36, I37, I38, I39) -> f22(I32, I33, I34, I35, I36, I37, I38, I39) [I36 <= I37] f22(I40, I41, I42, I43, I44, I45, I46, I47) -> f3(I40, 1 + I41, I42, I43, I44, I45, I46, I47) f14(I48, I49, I50, I51, I52, I53, I54, I55) -> f13(I48, I49, I50, I51, I52, I53, I54, I55) f21(I56, I57, I58, I59, I60, I61, I62, I63) -> f5(I56, I57, I58, I59, I60, I61, I62, I63) [1 + I56 <= I58] f21(I64, I65, I66, I67, I68, I69, I70, I71) -> f5(I64, I65, I66, I67, I68, I69, I70, I71) [1 + I66 <= I64] f21(I72, I73, I74, I75, I76, I77, I78, I79) -> f20(I72, I73, I74, I75, I76, I77, I78, I79) [I74 <= I72 /\ I72 <= I74] f6(I80, I81, I82, I83, I84, I85, I86, I87) -> f5(I80, 1 + I81, I82, I83, I84, I85, I86, rnd8) [rnd8 = rnd8 /\ I81 <= I83] f6(I88, I89, I90, I91, I92, I93, I94, I95) -> f11(I88, I89, I90, I91, I92, I93, I94, I95) [1 + I91 <= I89] f12(I96, I97, I98, I99, I100, I101, I102, I103) -> f11(I96, 1 + I97, I98, I99, I100, I101, I102, I104) [I104 = I104 /\ I97 <= I99] f12(I105, I106, I107, I108, I109, I110, I111, I112) -> f20(I105, I106, I107, I108, I109, I110, I111, I112) [1 + I108 <= I106] f16(I113, I114, I115, I116, I117, I118, I119, I120) -> f15(I113, I114, I115, I116, I117, I118, I119, I120) f20(I121, I122, I123, I124, I125, I126, I127, I128) -> f10(I121, I122, I123, I124, I125, I126, I127, I128) [1 + I127 <= 0] f20(I129, I130, I131, I132, I133, I134, I135, I136) -> f10(I129, I130, I131, I132, I133, I134, I135, I136) [1 <= I135] f20(I137, I138, I139, I140, I141, I142, I143, I144) -> f7(I137, I138, I139, I140, I141, I142, I143, I144) [0 <= I143 /\ I143 <= 0] f19(I145, I146, I147, I148, I149, I150, I151, I152) -> f18(I145, I146, I147, I148, I149, I150, I151, I153) [I153 = I153 /\ I145 <= I148] f19(I154, I155, I156, I157, I158, I159, I160, I161) -> f7(I154, I155, I156, I157, I158, I159, I160, I161) [1 + I157 <= I154] f10(I162, I163, I164, I165, I166, I167, I168, I169) -> f19(I162, I163, I164, I165, I166, I167, I168, I169) f18(I170, I171, I172, I173, I174, I175, I176, I177) -> f17(I170, I171, I172, I173, I174, I175, I176, I177) [1 + I177 <= 0] f18(I178, I179, I180, I181, I182, I183, I184, I185) -> f17(I178, I179, I180, I181, I182, I183, I184, I185) [1 <= I185] f18(I186, I187, I188, I189, I190, I191, I192, I193) -> f9(I186, I187, I188, I189, I190, I191, I192, I193) [0 <= I193 /\ I193 <= 0] f17(I194, I195, I196, I197, I198, I199, I200, I201) -> f16(I194, I195, I196, I197, I198, I199, I200, I202) [I202 = I202] f15(I203, I204, I205, I206, I207, I208, I209, I210) -> f16(I203, 1 + I204, I205, I206, I207, I208, I209, I210) [I204 <= I206] f15(I211, I212, I213, I214, I215, I216, I217, I218) -> f14(I211, I212, I213, I214, I215, I216, I217, I218) [1 + I214 <= I212] f13(I219, I220, I221, I222, I223, I224, I225, I226) -> f14(I219, 1 + I220, I221, I222, I223, I224, I225, I226) [I220 <= I222] f13(I227, I228, I229, I230, I231, I232, I233, I234) -> f9(I227, I228, I229, I230, I231, I232, I233, I234) [1 + I230 <= I228] f11(I235, I236, I237, I238, I239, I240, I241, I242) -> f12(I235, I236, I237, I238, I239, I240, I241, I242) f9(I243, I244, I245, I246, I247, I248, I249, I250) -> f10(1 + I243, I244, I245, I246, I247, I248, I249, I250) f7(I251, I252, I253, I254, I255, I256, I257, I258) -> f8(I251, I252, 1 + I253, I254, I255, I256, I257, I258) f5(I259, I260, I261, I262, I263, I264, I265, I266) -> f6(I259, I260, I261, I262, I263, I264, I265, I266) f3(I267, I268, I269, I270, I271, I272, I273, I274) -> f4(I267, I268, I269, I270, I271, I272, I273, I274) f1(I275, I276, I277, I278, I279, I280, I281, I282) -> f3(I277, I276, I277, I278, I279, I280, 0, I282) [1 + I277 <= I278] f1(I283, I284, I285, I286, I287, I288, I289, I290) -> f2(I283, I284, I285, I286, I287, I288, I289, I290) [I286 <= I285] We use the extended value criterion with the projection function NU: NU[f9#(x0,x1,x2,x3,x4,x5,x6,x7)] = -x2 + x3 - 1 NU[f17#(x0,x1,x2,x3,x4,x5,x6,x7)] = -x2 + x3 - 1 NU[f18#(x0,x1,x2,x3,x4,x5,x6,x7)] = -x2 + x3 - 1 NU[f19#(x0,x1,x2,x3,x4,x5,x6,x7)] = -x2 + x3 - 1 NU[f7#(x0,x1,x2,x3,x4,x5,x6,x7)] = -x2 + x3 - 1 NU[f10#(x0,x1,x2,x3,x4,x5,x6,x7)] = -x2 + x3 - 1 NU[f15#(x0,x1,x2,x3,x4,x5,x6,x7)] = -x2 + x3 - 1 NU[f16#(x0,x1,x2,x3,x4,x5,x6,x7)] = -x2 + x3 - 1 NU[f12#(x0,x1,x2,x3,x4,x5,x6,x7)] = -x2 + x3 - 1 NU[f11#(x0,x1,x2,x3,x4,x5,x6,x7)] = -x2 + x3 - 1 NU[f6#(x0,x1,x2,x3,x4,x5,x6,x7)] = -x2 + x3 - 1 NU[f20#(x0,x1,x2,x3,x4,x5,x6,x7)] = -x2 + x3 - 1 NU[f5#(x0,x1,x2,x3,x4,x5,x6,x7)] = -x2 + x3 - 1 NU[f13#(x0,x1,x2,x3,x4,x5,x6,x7)] = -x2 + x3 - 1 NU[f14#(x0,x1,x2,x3,x4,x5,x6,x7)] = -x2 + x3 - 1 NU[f3#(x0,x1,x2,x3,x4,x5,x6,x7)] = -x2 + x3 - 1 NU[f22#(x0,x1,x2,x3,x4,x5,x6,x7)] = -x2 + x3 - 1 NU[f21#(x0,x1,x2,x3,x4,x5,x6,x7)] = -x2 + x3 - 1 NU[f23#(x0,x1,x2,x3,x4,x5,x6,x7)] = -x2 + x3 - 1 NU[f4#(x0,x1,x2,x3,x4,x5,x6,x7)] = -x2 + x3 - 1 NU[f1#(x0,x1,x2,x3,x4,x5,x6,x7)] = -x2 + x3 NU[f8#(x0,x1,x2,x3,x4,x5,x6,x7)] = -x2 + x3 This gives the following inequalities: ==> -I2 + I3 >= -I2 + I3 rnd6 = rnd6 /\ rnd5 = rnd5 /\ I9 <= I11 ==> -I10 + I11 - 1 >= -I10 + I11 - 1 1 + I19 <= I17 ==> -I18 + I19 - 1 >= -I18 + I19 - 1 rnd7 = rnd7 /\ 1 + I29 <= I28 ==> -I26 + I27 - 1 >= -I26 + I27 - 1 I36 <= I37 ==> -I34 + I35 - 1 >= -I34 + I35 - 1 ==> -I42 + I43 - 1 >= -I42 + I43 - 1 ==> -I50 + I51 - 1 >= -I50 + I51 - 1 1 + I56 <= I58 ==> -I58 + I59 - 1 >= -I58 + I59 - 1 1 + I66 <= I64 ==> -I66 + I67 - 1 >= -I66 + I67 - 1 I74 <= I72 /\ I72 <= I74 ==> -I74 + I75 - 1 >= -I74 + I75 - 1 rnd8 = rnd8 /\ I81 <= I83 ==> -I82 + I83 - 1 >= -I82 + I83 - 1 1 + I91 <= I89 ==> -I90 + I91 - 1 >= -I90 + I91 - 1 I104 = I104 /\ I97 <= I99 ==> -I98 + I99 - 1 >= -I98 + I99 - 1 1 + I108 <= I106 ==> -I107 + I108 - 1 >= -I107 + I108 - 1 ==> -I115 + I116 - 1 >= -I115 + I116 - 1 1 + I127 <= 0 ==> -I123 + I124 - 1 >= -I123 + I124 - 1 1 <= I135 ==> -I131 + I132 - 1 >= -I131 + I132 - 1 0 <= I143 /\ I143 <= 0 ==> -I139 + I140 - 1 >= -I139 + I140 - 1 I153 = I153 /\ I145 <= I148 ==> -I147 + I148 - 1 >= -I147 + I148 - 1 1 + I157 <= I154 ==> -I156 + I157 - 1 >= -I156 + I157 - 1 ==> -I164 + I165 - 1 >= -I164 + I165 - 1 1 + I177 <= 0 ==> -I172 + I173 - 1 >= -I172 + I173 - 1 1 <= I185 ==> -I180 + I181 - 1 >= -I180 + I181 - 1 0 <= I193 /\ I193 <= 0 ==> -I188 + I189 - 1 >= -I188 + I189 - 1 I202 = I202 ==> -I196 + I197 - 1 >= -I196 + I197 - 1 I204 <= I206 ==> -I205 + I206 - 1 >= -I205 + I206 - 1 1 + I214 <= I212 ==> -I213 + I214 - 1 >= -I213 + I214 - 1 I220 <= I222 ==> -I221 + I222 - 1 >= -I221 + I222 - 1 1 + I230 <= I228 ==> -I229 + I230 - 1 >= -I229 + I230 - 1 ==> -I237 + I238 - 1 >= -I237 + I238 - 1 ==> -I245 + I246 - 1 >= -I245 + I246 - 1 ==> -I253 + I254 - 1 >= -(1 + I253) + I254 ==> -I261 + I262 - 1 >= -I261 + I262 - 1 ==> -I269 + I270 - 1 >= -I269 + I270 - 1 1 + I277 <= I278 ==> -I277 + I278 > -I277 + I278 - 1 with -I277 + I278 >= 0 We remove all the strictly oriented dependency pairs. DP problem for innermost termination. P = f8#(I0, I1, I2, I3, I4, I5, I6, I7) -> f1#(I0, I1, I2, I3, I4, I5, I6, I7) f4#(I8, I9, I10, I11, I12, I13, I14, I15) -> f23#(I8, I9, I10, I11, rnd5, rnd6, I14, I15) [rnd6 = rnd6 /\ rnd5 = rnd5 /\ I9 <= I11] f4#(I16, I17, I18, I19, I20, I21, I22, I23) -> f21#(I16, I17, I18, I19, I20, I21, I22, I23) [1 + I19 <= I17] f23#(I24, I25, I26, I27, I28, I29, I30, I31) -> f22#(I25, I25, I26, I27, I28, I29, rnd7, I31) [rnd7 = rnd7 /\ 1 + I29 <= I28] f23#(I32, I33, I34, I35, I36, I37, I38, I39) -> f22#(I32, I33, I34, I35, I36, I37, I38, I39) [I36 <= I37] f22#(I40, I41, I42, I43, I44, I45, I46, I47) -> f3#(I40, 1 + I41, I42, I43, I44, I45, I46, I47) f14#(I48, I49, I50, I51, I52, I53, I54, I55) -> f13#(I48, I49, I50, I51, I52, I53, I54, I55) f21#(I56, I57, I58, I59, I60, I61, I62, I63) -> f5#(I56, I57, I58, I59, I60, I61, I62, I63) [1 + I56 <= I58] f21#(I64, I65, I66, I67, I68, I69, I70, I71) -> f5#(I64, I65, I66, I67, I68, I69, I70, I71) [1 + I66 <= I64] f21#(I72, I73, I74, I75, I76, I77, I78, I79) -> f20#(I72, I73, I74, I75, I76, I77, I78, I79) [I74 <= I72 /\ I72 <= I74] f6#(I80, I81, I82, I83, I84, I85, I86, I87) -> f5#(I80, 1 + I81, I82, I83, I84, I85, I86, rnd8) [rnd8 = rnd8 /\ I81 <= I83] f6#(I88, I89, I90, I91, I92, I93, I94, I95) -> f11#(I88, I89, I90, I91, I92, I93, I94, I95) [1 + I91 <= I89] f12#(I96, I97, I98, I99, I100, I101, I102, I103) -> f11#(I96, 1 + I97, I98, I99, I100, I101, I102, I104) [I104 = I104 /\ I97 <= I99] f12#(I105, I106, I107, I108, I109, I110, I111, I112) -> f20#(I105, I106, I107, I108, I109, I110, I111, I112) [1 + I108 <= I106] f16#(I113, I114, I115, I116, I117, I118, I119, I120) -> f15#(I113, I114, I115, I116, I117, I118, I119, I120) f20#(I121, I122, I123, I124, I125, I126, I127, I128) -> f10#(I121, I122, I123, I124, I125, I126, I127, I128) [1 + I127 <= 0] f20#(I129, I130, I131, I132, I133, I134, I135, I136) -> f10#(I129, I130, I131, I132, I133, I134, I135, I136) [1 <= I135] f20#(I137, I138, I139, I140, I141, I142, I143, I144) -> f7#(I137, I138, I139, I140, I141, I142, I143, I144) [0 <= I143 /\ I143 <= 0] f19#(I145, I146, I147, I148, I149, I150, I151, I152) -> f18#(I145, I146, I147, I148, I149, I150, I151, I153) [I153 = I153 /\ I145 <= I148] f19#(I154, I155, I156, I157, I158, I159, I160, I161) -> f7#(I154, I155, I156, I157, I158, I159, I160, I161) [1 + I157 <= I154] f10#(I162, I163, I164, I165, I166, I167, I168, I169) -> f19#(I162, I163, I164, I165, I166, I167, I168, I169) f18#(I170, I171, I172, I173, I174, I175, I176, I177) -> f17#(I170, I171, I172, I173, I174, I175, I176, I177) [1 + I177 <= 0] f18#(I178, I179, I180, I181, I182, I183, I184, I185) -> f17#(I178, I179, I180, I181, I182, I183, I184, I185) [1 <= I185] f18#(I186, I187, I188, I189, I190, I191, I192, I193) -> f9#(I186, I187, I188, I189, I190, I191, I192, I193) [0 <= I193 /\ I193 <= 0] f17#(I194, I195, I196, I197, I198, I199, I200, I201) -> f16#(I194, I195, I196, I197, I198, I199, I200, I202) [I202 = I202] f15#(I203, I204, I205, I206, I207, I208, I209, I210) -> f16#(I203, 1 + I204, I205, I206, I207, I208, I209, I210) [I204 <= I206] f15#(I211, I212, I213, I214, I215, I216, I217, I218) -> f14#(I211, I212, I213, I214, I215, I216, I217, I218) [1 + I214 <= I212] f13#(I219, I220, I221, I222, I223, I224, I225, I226) -> f14#(I219, 1 + I220, I221, I222, I223, I224, I225, I226) [I220 <= I222] f13#(I227, I228, I229, I230, I231, I232, I233, I234) -> f9#(I227, I228, I229, I230, I231, I232, I233, I234) [1 + I230 <= I228] f11#(I235, I236, I237, I238, I239, I240, I241, I242) -> f12#(I235, I236, I237, I238, I239, I240, I241, I242) f9#(I243, I244, I245, I246, I247, I248, I249, I250) -> f10#(1 + I243, I244, I245, I246, I247, I248, I249, I250) f7#(I251, I252, I253, I254, I255, I256, I257, I258) -> f8#(I251, I252, 1 + I253, I254, I255, I256, I257, I258) f5#(I259, I260, I261, I262, I263, I264, I265, I266) -> f6#(I259, I260, I261, I262, I263, I264, I265, I266) f3#(I267, I268, I269, I270, I271, I272, I273, I274) -> f4#(I267, I268, I269, I270, I271, I272, I273, I274) R = f24(x1, x2, x3, x4, x5, x6, x7, x8) -> f8(x1, x2, x3, x4, x5, x6, x7, x8) f8(I0, I1, I2, I3, I4, I5, I6, I7) -> f1(I0, I1, I2, I3, I4, I5, I6, I7) f4(I8, I9, I10, I11, I12, I13, I14, I15) -> f23(I8, I9, I10, I11, rnd5, rnd6, I14, I15) [rnd6 = rnd6 /\ rnd5 = rnd5 /\ I9 <= I11] f4(I16, I17, I18, I19, I20, I21, I22, I23) -> f21(I16, I17, I18, I19, I20, I21, I22, I23) [1 + I19 <= I17] f23(I24, I25, I26, I27, I28, I29, I30, I31) -> f22(I25, I25, I26, I27, I28, I29, rnd7, I31) [rnd7 = rnd7 /\ 1 + I29 <= I28] f23(I32, I33, I34, I35, I36, I37, I38, I39) -> f22(I32, I33, I34, I35, I36, I37, I38, I39) [I36 <= I37] f22(I40, I41, I42, I43, I44, I45, I46, I47) -> f3(I40, 1 + I41, I42, I43, I44, I45, I46, I47) f14(I48, I49, I50, I51, I52, I53, I54, I55) -> f13(I48, I49, I50, I51, I52, I53, I54, I55) f21(I56, I57, I58, I59, I60, I61, I62, I63) -> f5(I56, I57, I58, I59, I60, I61, I62, I63) [1 + I56 <= I58] f21(I64, I65, I66, I67, I68, I69, I70, I71) -> f5(I64, I65, I66, I67, I68, I69, I70, I71) [1 + I66 <= I64] f21(I72, I73, I74, I75, I76, I77, I78, I79) -> f20(I72, I73, I74, I75, I76, I77, I78, I79) [I74 <= I72 /\ I72 <= I74] f6(I80, I81, I82, I83, I84, I85, I86, I87) -> f5(I80, 1 + I81, I82, I83, I84, I85, I86, rnd8) [rnd8 = rnd8 /\ I81 <= I83] f6(I88, I89, I90, I91, I92, I93, I94, I95) -> f11(I88, I89, I90, I91, I92, I93, I94, I95) [1 + I91 <= I89] f12(I96, I97, I98, I99, I100, I101, I102, I103) -> f11(I96, 1 + I97, I98, I99, I100, I101, I102, I104) [I104 = I104 /\ I97 <= I99] f12(I105, I106, I107, I108, I109, I110, I111, I112) -> f20(I105, I106, I107, I108, I109, I110, I111, I112) [1 + I108 <= I106] f16(I113, I114, I115, I116, I117, I118, I119, I120) -> f15(I113, I114, I115, I116, I117, I118, I119, I120) f20(I121, I122, I123, I124, I125, I126, I127, I128) -> f10(I121, I122, I123, I124, I125, I126, I127, I128) [1 + I127 <= 0] f20(I129, I130, I131, I132, I133, I134, I135, I136) -> f10(I129, I130, I131, I132, I133, I134, I135, I136) [1 <= I135] f20(I137, I138, I139, I140, I141, I142, I143, I144) -> f7(I137, I138, I139, I140, I141, I142, I143, I144) [0 <= I143 /\ I143 <= 0] f19(I145, I146, I147, I148, I149, I150, I151, I152) -> f18(I145, I146, I147, I148, I149, I150, I151, I153) [I153 = I153 /\ I145 <= I148] f19(I154, I155, I156, I157, I158, I159, I160, I161) -> f7(I154, I155, I156, I157, I158, I159, I160, I161) [1 + I157 <= I154] f10(I162, I163, I164, I165, I166, I167, I168, I169) -> f19(I162, I163, I164, I165, I166, I167, I168, I169) f18(I170, I171, I172, I173, I174, I175, I176, I177) -> f17(I170, I171, I172, I173, I174, I175, I176, I177) [1 + I177 <= 0] f18(I178, I179, I180, I181, I182, I183, I184, I185) -> f17(I178, I179, I180, I181, I182, I183, I184, I185) [1 <= I185] f18(I186, I187, I188, I189, I190, I191, I192, I193) -> f9(I186, I187, I188, I189, I190, I191, I192, I193) [0 <= I193 /\ I193 <= 0] f17(I194, I195, I196, I197, I198, I199, I200, I201) -> f16(I194, I195, I196, I197, I198, I199, I200, I202) [I202 = I202] f15(I203, I204, I205, I206, I207, I208, I209, I210) -> f16(I203, 1 + I204, I205, I206, I207, I208, I209, I210) [I204 <= I206] f15(I211, I212, I213, I214, I215, I216, I217, I218) -> f14(I211, I212, I213, I214, I215, I216, I217, I218) [1 + I214 <= I212] f13(I219, I220, I221, I222, I223, I224, I225, I226) -> f14(I219, 1 + I220, I221, I222, I223, I224, I225, I226) [I220 <= I222] f13(I227, I228, I229, I230, I231, I232, I233, I234) -> f9(I227, I228, I229, I230, I231, I232, I233, I234) [1 + I230 <= I228] f11(I235, I236, I237, I238, I239, I240, I241, I242) -> f12(I235, I236, I237, I238, I239, I240, I241, I242) f9(I243, I244, I245, I246, I247, I248, I249, I250) -> f10(1 + I243, I244, I245, I246, I247, I248, I249, I250) f7(I251, I252, I253, I254, I255, I256, I257, I258) -> f8(I251, I252, 1 + I253, I254, I255, I256, I257, I258) f5(I259, I260, I261, I262, I263, I264, I265, I266) -> f6(I259, I260, I261, I262, I263, I264, I265, I266) f3(I267, I268, I269, I270, I271, I272, I273, I274) -> f4(I267, I268, I269, I270, I271, I272, I273, I274) f1(I275, I276, I277, I278, I279, I280, I281, I282) -> f3(I277, I276, I277, I278, I279, I280, 0, I282) [1 + I277 <= I278] f1(I283, I284, I285, I286, I287, I288, I289, I290) -> f2(I283, I284, I285, I286, I287, I288, I289, I290) [I286 <= I285] The dependency graph for this problem is: 1 -> 2 -> 4, 5 3 -> 8, 9, 10 4 -> 6 5 -> 6 6 -> 34 7 -> 28, 29 8 -> 33 9 -> 33 10 -> 16, 17, 18 11 -> 33 12 -> 30 13 -> 30 14 -> 16, 17, 18 15 -> 26, 27 16 -> 21 17 -> 21 18 -> 32 19 -> 22, 23, 24 20 -> 32 21 -> 19, 20 22 -> 25 23 -> 25 24 -> 31 25 -> 15 26 -> 15 27 -> 7 28 -> 7 29 -> 31 30 -> 13, 14 31 -> 21 32 -> 1 33 -> 11, 12 34 -> 2, 3 Where: 1) f8#(I0, I1, I2, I3, I4, I5, I6, I7) -> f1#(I0, I1, I2, I3, I4, I5, I6, I7) 2) f4#(I8, I9, I10, I11, I12, I13, I14, I15) -> f23#(I8, I9, I10, I11, rnd5, rnd6, I14, I15) [rnd6 = rnd6 /\ rnd5 = rnd5 /\ I9 <= I11] 3) f4#(I16, I17, I18, I19, I20, I21, I22, I23) -> f21#(I16, I17, I18, I19, I20, I21, I22, I23) [1 + I19 <= I17] 4) f23#(I24, I25, I26, I27, I28, I29, I30, I31) -> f22#(I25, I25, I26, I27, I28, I29, rnd7, I31) [rnd7 = rnd7 /\ 1 + I29 <= I28] 5) f23#(I32, I33, I34, I35, I36, I37, I38, I39) -> f22#(I32, I33, I34, I35, I36, I37, I38, I39) [I36 <= I37] 6) f22#(I40, I41, I42, I43, I44, I45, I46, I47) -> f3#(I40, 1 + I41, I42, I43, I44, I45, I46, I47) 7) f14#(I48, I49, I50, I51, I52, I53, I54, I55) -> f13#(I48, I49, I50, I51, I52, I53, I54, I55) 8) f21#(I56, I57, I58, I59, I60, I61, I62, I63) -> f5#(I56, I57, I58, I59, I60, I61, I62, I63) [1 + I56 <= I58] 9) f21#(I64, I65, I66, I67, I68, I69, I70, I71) -> f5#(I64, I65, I66, I67, I68, I69, I70, I71) [1 + I66 <= I64] 10) f21#(I72, I73, I74, I75, I76, I77, I78, I79) -> f20#(I72, I73, I74, I75, I76, I77, I78, I79) [I74 <= I72 /\ I72 <= I74] 11) f6#(I80, I81, I82, I83, I84, I85, I86, I87) -> f5#(I80, 1 + I81, I82, I83, I84, I85, I86, rnd8) [rnd8 = rnd8 /\ I81 <= I83] 12) f6#(I88, I89, I90, I91, I92, I93, I94, I95) -> f11#(I88, I89, I90, I91, I92, I93, I94, I95) [1 + I91 <= I89] 13) f12#(I96, I97, I98, I99, I100, I101, I102, I103) -> f11#(I96, 1 + I97, I98, I99, I100, I101, I102, I104) [I104 = I104 /\ I97 <= I99] 14) f12#(I105, I106, I107, I108, I109, I110, I111, I112) -> f20#(I105, I106, I107, I108, I109, I110, I111, I112) [1 + I108 <= I106] 15) f16#(I113, I114, I115, I116, I117, I118, I119, I120) -> f15#(I113, I114, I115, I116, I117, I118, I119, I120) 16) f20#(I121, I122, I123, I124, I125, I126, I127, I128) -> f10#(I121, I122, I123, I124, I125, I126, I127, I128) [1 + I127 <= 0] 17) f20#(I129, I130, I131, I132, I133, I134, I135, I136) -> f10#(I129, I130, I131, I132, I133, I134, I135, I136) [1 <= I135] 18) f20#(I137, I138, I139, I140, I141, I142, I143, I144) -> f7#(I137, I138, I139, I140, I141, I142, I143, I144) [0 <= I143 /\ I143 <= 0] 19) f19#(I145, I146, I147, I148, I149, I150, I151, I152) -> f18#(I145, I146, I147, I148, I149, I150, I151, I153) [I153 = I153 /\ I145 <= I148] 20) f19#(I154, I155, I156, I157, I158, I159, I160, I161) -> f7#(I154, I155, I156, I157, I158, I159, I160, I161) [1 + I157 <= I154] 21) f10#(I162, I163, I164, I165, I166, I167, I168, I169) -> f19#(I162, I163, I164, I165, I166, I167, I168, I169) 22) f18#(I170, I171, I172, I173, I174, I175, I176, I177) -> f17#(I170, I171, I172, I173, I174, I175, I176, I177) [1 + I177 <= 0] 23) f18#(I178, I179, I180, I181, I182, I183, I184, I185) -> f17#(I178, I179, I180, I181, I182, I183, I184, I185) [1 <= I185] 24) f18#(I186, I187, I188, I189, I190, I191, I192, I193) -> f9#(I186, I187, I188, I189, I190, I191, I192, I193) [0 <= I193 /\ I193 <= 0] 25) f17#(I194, I195, I196, I197, I198, I199, I200, I201) -> f16#(I194, I195, I196, I197, I198, I199, I200, I202) [I202 = I202] 26) f15#(I203, I204, I205, I206, I207, I208, I209, I210) -> f16#(I203, 1 + I204, I205, I206, I207, I208, I209, I210) [I204 <= I206] 27) f15#(I211, I212, I213, I214, I215, I216, I217, I218) -> f14#(I211, I212, I213, I214, I215, I216, I217, I218) [1 + I214 <= I212] 28) f13#(I219, I220, I221, I222, I223, I224, I225, I226) -> f14#(I219, 1 + I220, I221, I222, I223, I224, I225, I226) [I220 <= I222] 29) f13#(I227, I228, I229, I230, I231, I232, I233, I234) -> f9#(I227, I228, I229, I230, I231, I232, I233, I234) [1 + I230 <= I228] 30) f11#(I235, I236, I237, I238, I239, I240, I241, I242) -> f12#(I235, I236, I237, I238, I239, I240, I241, I242) 31) f9#(I243, I244, I245, I246, I247, I248, I249, I250) -> f10#(1 + I243, I244, I245, I246, I247, I248, I249, I250) 32) f7#(I251, I252, I253, I254, I255, I256, I257, I258) -> f8#(I251, I252, 1 + I253, I254, I255, I256, I257, I258) 33) f5#(I259, I260, I261, I262, I263, I264, I265, I266) -> f6#(I259, I260, I261, I262, I263, I264, I265, I266) 34) f3#(I267, I268, I269, I270, I271, I272, I273, I274) -> f4#(I267, I268, I269, I270, I271, I272, I273, I274) We have the following SCCs. { 2, 4, 5, 6, 34 } { 11, 33 } { 13, 30 } { 7, 15, 19, 21, 22, 23, 24, 25, 26, 27, 28, 29, 31 } DP problem for innermost termination. P = f14#(I48, I49, I50, I51, I52, I53, I54, I55) -> f13#(I48, I49, I50, I51, I52, I53, I54, I55) f16#(I113, I114, I115, I116, I117, I118, I119, I120) -> f15#(I113, I114, I115, I116, I117, I118, I119, I120) f19#(I145, I146, I147, I148, I149, I150, I151, I152) -> f18#(I145, I146, I147, I148, I149, I150, I151, I153) [I153 = I153 /\ I145 <= I148] f10#(I162, I163, I164, I165, I166, I167, I168, I169) -> f19#(I162, I163, I164, I165, I166, I167, I168, I169) f18#(I170, I171, I172, I173, I174, I175, I176, I177) -> f17#(I170, I171, I172, I173, I174, I175, I176, I177) [1 + I177 <= 0] f18#(I178, I179, I180, I181, I182, I183, I184, I185) -> f17#(I178, I179, I180, I181, I182, I183, I184, I185) [1 <= I185] f18#(I186, I187, I188, I189, I190, I191, I192, I193) -> f9#(I186, I187, I188, I189, I190, I191, I192, I193) [0 <= I193 /\ I193 <= 0] f17#(I194, I195, I196, I197, I198, I199, I200, I201) -> f16#(I194, I195, I196, I197, I198, I199, I200, I202) [I202 = I202] f15#(I203, I204, I205, I206, I207, I208, I209, I210) -> f16#(I203, 1 + I204, I205, I206, I207, I208, I209, I210) [I204 <= I206] f15#(I211, I212, I213, I214, I215, I216, I217, I218) -> f14#(I211, I212, I213, I214, I215, I216, I217, I218) [1 + I214 <= I212] f13#(I219, I220, I221, I222, I223, I224, I225, I226) -> f14#(I219, 1 + I220, I221, I222, I223, I224, I225, I226) [I220 <= I222] f13#(I227, I228, I229, I230, I231, I232, I233, I234) -> f9#(I227, I228, I229, I230, I231, I232, I233, I234) [1 + I230 <= I228] f9#(I243, I244, I245, I246, I247, I248, I249, I250) -> f10#(1 + I243, I244, I245, I246, I247, I248, I249, I250) R = f24(x1, x2, x3, x4, x5, x6, x7, x8) -> f8(x1, x2, x3, x4, x5, x6, x7, x8) f8(I0, I1, I2, I3, I4, I5, I6, I7) -> f1(I0, I1, I2, I3, I4, I5, I6, I7) f4(I8, I9, I10, I11, I12, I13, I14, I15) -> f23(I8, I9, I10, I11, rnd5, rnd6, I14, I15) [rnd6 = rnd6 /\ rnd5 = rnd5 /\ I9 <= I11] f4(I16, I17, I18, I19, I20, I21, I22, I23) -> f21(I16, I17, I18, I19, I20, I21, I22, I23) [1 + I19 <= I17] f23(I24, I25, I26, I27, I28, I29, I30, I31) -> f22(I25, I25, I26, I27, I28, I29, rnd7, I31) [rnd7 = rnd7 /\ 1 + I29 <= I28] f23(I32, I33, I34, I35, I36, I37, I38, I39) -> f22(I32, I33, I34, I35, I36, I37, I38, I39) [I36 <= I37] f22(I40, I41, I42, I43, I44, I45, I46, I47) -> f3(I40, 1 + I41, I42, I43, I44, I45, I46, I47) f14(I48, I49, I50, I51, I52, I53, I54, I55) -> f13(I48, I49, I50, I51, I52, I53, I54, I55) f21(I56, I57, I58, I59, I60, I61, I62, I63) -> f5(I56, I57, I58, I59, I60, I61, I62, I63) [1 + I56 <= I58] f21(I64, I65, I66, I67, I68, I69, I70, I71) -> f5(I64, I65, I66, I67, I68, I69, I70, I71) [1 + I66 <= I64] f21(I72, I73, I74, I75, I76, I77, I78, I79) -> f20(I72, I73, I74, I75, I76, I77, I78, I79) [I74 <= I72 /\ I72 <= I74] f6(I80, I81, I82, I83, I84, I85, I86, I87) -> f5(I80, 1 + I81, I82, I83, I84, I85, I86, rnd8) [rnd8 = rnd8 /\ I81 <= I83] f6(I88, I89, I90, I91, I92, I93, I94, I95) -> f11(I88, I89, I90, I91, I92, I93, I94, I95) [1 + I91 <= I89] f12(I96, I97, I98, I99, I100, I101, I102, I103) -> f11(I96, 1 + I97, I98, I99, I100, I101, I102, I104) [I104 = I104 /\ I97 <= I99] f12(I105, I106, I107, I108, I109, I110, I111, I112) -> f20(I105, I106, I107, I108, I109, I110, I111, I112) [1 + I108 <= I106] f16(I113, I114, I115, I116, I117, I118, I119, I120) -> f15(I113, I114, I115, I116, I117, I118, I119, I120) f20(I121, I122, I123, I124, I125, I126, I127, I128) -> f10(I121, I122, I123, I124, I125, I126, I127, I128) [1 + I127 <= 0] f20(I129, I130, I131, I132, I133, I134, I135, I136) -> f10(I129, I130, I131, I132, I133, I134, I135, I136) [1 <= I135] f20(I137, I138, I139, I140, I141, I142, I143, I144) -> f7(I137, I138, I139, I140, I141, I142, I143, I144) [0 <= I143 /\ I143 <= 0] f19(I145, I146, I147, I148, I149, I150, I151, I152) -> f18(I145, I146, I147, I148, I149, I150, I151, I153) [I153 = I153 /\ I145 <= I148] f19(I154, I155, I156, I157, I158, I159, I160, I161) -> f7(I154, I155, I156, I157, I158, I159, I160, I161) [1 + I157 <= I154] f10(I162, I163, I164, I165, I166, I167, I168, I169) -> f19(I162, I163, I164, I165, I166, I167, I168, I169) f18(I170, I171, I172, I173, I174, I175, I176, I177) -> f17(I170, I171, I172, I173, I174, I175, I176, I177) [1 + I177 <= 0] f18(I178, I179, I180, I181, I182, I183, I184, I185) -> f17(I178, I179, I180, I181, I182, I183, I184, I185) [1 <= I185] f18(I186, I187, I188, I189, I190, I191, I192, I193) -> f9(I186, I187, I188, I189, I190, I191, I192, I193) [0 <= I193 /\ I193 <= 0] f17(I194, I195, I196, I197, I198, I199, I200, I201) -> f16(I194, I195, I196, I197, I198, I199, I200, I202) [I202 = I202] f15(I203, I204, I205, I206, I207, I208, I209, I210) -> f16(I203, 1 + I204, I205, I206, I207, I208, I209, I210) [I204 <= I206] f15(I211, I212, I213, I214, I215, I216, I217, I218) -> f14(I211, I212, I213, I214, I215, I216, I217, I218) [1 + I214 <= I212] f13(I219, I220, I221, I222, I223, I224, I225, I226) -> f14(I219, 1 + I220, I221, I222, I223, I224, I225, I226) [I220 <= I222] f13(I227, I228, I229, I230, I231, I232, I233, I234) -> f9(I227, I228, I229, I230, I231, I232, I233, I234) [1 + I230 <= I228] f11(I235, I236, I237, I238, I239, I240, I241, I242) -> f12(I235, I236, I237, I238, I239, I240, I241, I242) f9(I243, I244, I245, I246, I247, I248, I249, I250) -> f10(1 + I243, I244, I245, I246, I247, I248, I249, I250) f7(I251, I252, I253, I254, I255, I256, I257, I258) -> f8(I251, I252, 1 + I253, I254, I255, I256, I257, I258) f5(I259, I260, I261, I262, I263, I264, I265, I266) -> f6(I259, I260, I261, I262, I263, I264, I265, I266) f3(I267, I268, I269, I270, I271, I272, I273, I274) -> f4(I267, I268, I269, I270, I271, I272, I273, I274) f1(I275, I276, I277, I278, I279, I280, I281, I282) -> f3(I277, I276, I277, I278, I279, I280, 0, I282) [1 + I277 <= I278] f1(I283, I284, I285, I286, I287, I288, I289, I290) -> f2(I283, I284, I285, I286, I287, I288, I289, I290) [I286 <= I285] We use the extended value criterion with the projection function NU: NU[f9#(x0,x1,x2,x3,x4,x5,x6,x7)] = -x1 + x3 NU[f17#(x0,x1,x2,x3,x4,x5,x6,x7)] = -x1 + x3 NU[f10#(x0,x1,x2,x3,x4,x5,x6,x7)] = -x1 + x3 NU[f18#(x0,x1,x2,x3,x4,x5,x6,x7)] = -x1 + x3 NU[f19#(x0,x1,x2,x3,x4,x5,x6,x7)] = -x1 + x3 NU[f15#(x0,x1,x2,x3,x4,x5,x6,x7)] = -x1 + x3 NU[f16#(x0,x1,x2,x3,x4,x5,x6,x7)] = -x1 + x3 NU[f13#(x0,x1,x2,x3,x4,x5,x6,x7)] = -x1 + x3 NU[f14#(x0,x1,x2,x3,x4,x5,x6,x7)] = -x1 + x3 This gives the following inequalities: ==> -I49 + I51 >= -I49 + I51 ==> -I114 + I116 >= -I114 + I116 I153 = I153 /\ I145 <= I148 ==> -I146 + I148 >= -I146 + I148 ==> -I163 + I165 >= -I163 + I165 1 + I177 <= 0 ==> -I171 + I173 >= -I171 + I173 1 <= I185 ==> -I179 + I181 >= -I179 + I181 0 <= I193 /\ I193 <= 0 ==> -I187 + I189 >= -I187 + I189 I202 = I202 ==> -I195 + I197 >= -I195 + I197 I204 <= I206 ==> -I204 + I206 >= -(1 + I204) + I206 1 + I214 <= I212 ==> -I212 + I214 >= -I212 + I214 I220 <= I222 ==> -I220 + I222 > -(1 + I220) + I222 with -I220 + I222 >= 0 1 + I230 <= I228 ==> -I228 + I230 >= -I228 + I230 ==> -I244 + I246 >= -I244 + I246 We remove all the strictly oriented dependency pairs. DP problem for innermost termination. P = f14#(I48, I49, I50, I51, I52, I53, I54, I55) -> f13#(I48, I49, I50, I51, I52, I53, I54, I55) f16#(I113, I114, I115, I116, I117, I118, I119, I120) -> f15#(I113, I114, I115, I116, I117, I118, I119, I120) f19#(I145, I146, I147, I148, I149, I150, I151, I152) -> f18#(I145, I146, I147, I148, I149, I150, I151, I153) [I153 = I153 /\ I145 <= I148] f10#(I162, I163, I164, I165, I166, I167, I168, I169) -> f19#(I162, I163, I164, I165, I166, I167, I168, I169) f18#(I170, I171, I172, I173, I174, I175, I176, I177) -> f17#(I170, I171, I172, I173, I174, I175, I176, I177) [1 + I177 <= 0] f18#(I178, I179, I180, I181, I182, I183, I184, I185) -> f17#(I178, I179, I180, I181, I182, I183, I184, I185) [1 <= I185] f18#(I186, I187, I188, I189, I190, I191, I192, I193) -> f9#(I186, I187, I188, I189, I190, I191, I192, I193) [0 <= I193 /\ I193 <= 0] f17#(I194, I195, I196, I197, I198, I199, I200, I201) -> f16#(I194, I195, I196, I197, I198, I199, I200, I202) [I202 = I202] f15#(I203, I204, I205, I206, I207, I208, I209, I210) -> f16#(I203, 1 + I204, I205, I206, I207, I208, I209, I210) [I204 <= I206] f15#(I211, I212, I213, I214, I215, I216, I217, I218) -> f14#(I211, I212, I213, I214, I215, I216, I217, I218) [1 + I214 <= I212] f13#(I227, I228, I229, I230, I231, I232, I233, I234) -> f9#(I227, I228, I229, I230, I231, I232, I233, I234) [1 + I230 <= I228] f9#(I243, I244, I245, I246, I247, I248, I249, I250) -> f10#(1 + I243, I244, I245, I246, I247, I248, I249, I250) R = f24(x1, x2, x3, x4, x5, x6, x7, x8) -> f8(x1, x2, x3, x4, x5, x6, x7, x8) f8(I0, I1, I2, I3, I4, I5, I6, I7) -> f1(I0, I1, I2, I3, I4, I5, I6, I7) f4(I8, I9, I10, I11, I12, I13, I14, I15) -> f23(I8, I9, I10, I11, rnd5, rnd6, I14, I15) [rnd6 = rnd6 /\ rnd5 = rnd5 /\ I9 <= I11] f4(I16, I17, I18, I19, I20, I21, I22, I23) -> f21(I16, I17, I18, I19, I20, I21, I22, I23) [1 + I19 <= I17] f23(I24, I25, I26, I27, I28, I29, I30, I31) -> f22(I25, I25, I26, I27, I28, I29, rnd7, I31) [rnd7 = rnd7 /\ 1 + I29 <= I28] f23(I32, I33, I34, I35, I36, I37, I38, I39) -> f22(I32, I33, I34, I35, I36, I37, I38, I39) [I36 <= I37] f22(I40, I41, I42, I43, I44, I45, I46, I47) -> f3(I40, 1 + I41, I42, I43, I44, I45, I46, I47) f14(I48, I49, I50, I51, I52, I53, I54, I55) -> f13(I48, I49, I50, I51, I52, I53, I54, I55) f21(I56, I57, I58, I59, I60, I61, I62, I63) -> f5(I56, I57, I58, I59, I60, I61, I62, I63) [1 + I56 <= I58] f21(I64, I65, I66, I67, I68, I69, I70, I71) -> f5(I64, I65, I66, I67, I68, I69, I70, I71) [1 + I66 <= I64] f21(I72, I73, I74, I75, I76, I77, I78, I79) -> f20(I72, I73, I74, I75, I76, I77, I78, I79) [I74 <= I72 /\ I72 <= I74] f6(I80, I81, I82, I83, I84, I85, I86, I87) -> f5(I80, 1 + I81, I82, I83, I84, I85, I86, rnd8) [rnd8 = rnd8 /\ I81 <= I83] f6(I88, I89, I90, I91, I92, I93, I94, I95) -> f11(I88, I89, I90, I91, I92, I93, I94, I95) [1 + I91 <= I89] f12(I96, I97, I98, I99, I100, I101, I102, I103) -> f11(I96, 1 + I97, I98, I99, I100, I101, I102, I104) [I104 = I104 /\ I97 <= I99] f12(I105, I106, I107, I108, I109, I110, I111, I112) -> f20(I105, I106, I107, I108, I109, I110, I111, I112) [1 + I108 <= I106] f16(I113, I114, I115, I116, I117, I118, I119, I120) -> f15(I113, I114, I115, I116, I117, I118, I119, I120) f20(I121, I122, I123, I124, I125, I126, I127, I128) -> f10(I121, I122, I123, I124, I125, I126, I127, I128) [1 + I127 <= 0] f20(I129, I130, I131, I132, I133, I134, I135, I136) -> f10(I129, I130, I131, I132, I133, I134, I135, I136) [1 <= I135] f20(I137, I138, I139, I140, I141, I142, I143, I144) -> f7(I137, I138, I139, I140, I141, I142, I143, I144) [0 <= I143 /\ I143 <= 0] f19(I145, I146, I147, I148, I149, I150, I151, I152) -> f18(I145, I146, I147, I148, I149, I150, I151, I153) [I153 = I153 /\ I145 <= I148] f19(I154, I155, I156, I157, I158, I159, I160, I161) -> f7(I154, I155, I156, I157, I158, I159, I160, I161) [1 + I157 <= I154] f10(I162, I163, I164, I165, I166, I167, I168, I169) -> f19(I162, I163, I164, I165, I166, I167, I168, I169) f18(I170, I171, I172, I173, I174, I175, I176, I177) -> f17(I170, I171, I172, I173, I174, I175, I176, I177) [1 + I177 <= 0] f18(I178, I179, I180, I181, I182, I183, I184, I185) -> f17(I178, I179, I180, I181, I182, I183, I184, I185) [1 <= I185] f18(I186, I187, I188, I189, I190, I191, I192, I193) -> f9(I186, I187, I188, I189, I190, I191, I192, I193) [0 <= I193 /\ I193 <= 0] f17(I194, I195, I196, I197, I198, I199, I200, I201) -> f16(I194, I195, I196, I197, I198, I199, I200, I202) [I202 = I202] f15(I203, I204, I205, I206, I207, I208, I209, I210) -> f16(I203, 1 + I204, I205, I206, I207, I208, I209, I210) [I204 <= I206] f15(I211, I212, I213, I214, I215, I216, I217, I218) -> f14(I211, I212, I213, I214, I215, I216, I217, I218) [1 + I214 <= I212] f13(I219, I220, I221, I222, I223, I224, I225, I226) -> f14(I219, 1 + I220, I221, I222, I223, I224, I225, I226) [I220 <= I222] f13(I227, I228, I229, I230, I231, I232, I233, I234) -> f9(I227, I228, I229, I230, I231, I232, I233, I234) [1 + I230 <= I228] f11(I235, I236, I237, I238, I239, I240, I241, I242) -> f12(I235, I236, I237, I238, I239, I240, I241, I242) f9(I243, I244, I245, I246, I247, I248, I249, I250) -> f10(1 + I243, I244, I245, I246, I247, I248, I249, I250) f7(I251, I252, I253, I254, I255, I256, I257, I258) -> f8(I251, I252, 1 + I253, I254, I255, I256, I257, I258) f5(I259, I260, I261, I262, I263, I264, I265, I266) -> f6(I259, I260, I261, I262, I263, I264, I265, I266) f3(I267, I268, I269, I270, I271, I272, I273, I274) -> f4(I267, I268, I269, I270, I271, I272, I273, I274) f1(I275, I276, I277, I278, I279, I280, I281, I282) -> f3(I277, I276, I277, I278, I279, I280, 0, I282) [1 + I277 <= I278] f1(I283, I284, I285, I286, I287, I288, I289, I290) -> f2(I283, I284, I285, I286, I287, I288, I289, I290) [I286 <= I285] We use the extended value criterion with the projection function NU: NU[f9#(x0,x1,x2,x3,x4,x5,x6,x7)] = -x0 + x3 + 1 NU[f17#(x0,x1,x2,x3,x4,x5,x6,x7)] = -x0 + x3 + 1 NU[f10#(x0,x1,x2,x3,x4,x5,x6,x7)] = -x0 + x3 + 2 NU[f18#(x0,x1,x2,x3,x4,x5,x6,x7)] = -x0 + x3 + 1 NU[f19#(x0,x1,x2,x3,x4,x5,x6,x7)] = -x0 + x3 + 2 NU[f15#(x0,x1,x2,x3,x4,x5,x6,x7)] = -x0 + x3 + 1 NU[f16#(x0,x1,x2,x3,x4,x5,x6,x7)] = -x0 + x3 + 1 NU[f13#(x0,x1,x2,x3,x4,x5,x6,x7)] = -x0 + x3 + 1 NU[f14#(x0,x1,x2,x3,x4,x5,x6,x7)] = -x0 + x3 + 1 This gives the following inequalities: ==> -I48 + I51 + 1 >= -I48 + I51 + 1 ==> -I113 + I116 + 1 >= -I113 + I116 + 1 I153 = I153 /\ I145 <= I148 ==> -I145 + I148 + 2 > -I145 + I148 + 1 with -I145 + I148 + 2 >= 0 ==> -I162 + I165 + 2 >= -I162 + I165 + 2 1 + I177 <= 0 ==> -I170 + I173 + 1 >= -I170 + I173 + 1 1 <= I185 ==> -I178 + I181 + 1 >= -I178 + I181 + 1 0 <= I193 /\ I193 <= 0 ==> -I186 + I189 + 1 >= -I186 + I189 + 1 I202 = I202 ==> -I194 + I197 + 1 >= -I194 + I197 + 1 I204 <= I206 ==> -I203 + I206 + 1 >= -I203 + I206 + 1 1 + I214 <= I212 ==> -I211 + I214 + 1 >= -I211 + I214 + 1 1 + I230 <= I228 ==> -I227 + I230 + 1 >= -I227 + I230 + 1 ==> -I243 + I246 + 1 >= -(1 + I243) + I246 + 2 We remove all the strictly oriented dependency pairs. DP problem for innermost termination. P = f14#(I48, I49, I50, I51, I52, I53, I54, I55) -> f13#(I48, I49, I50, I51, I52, I53, I54, I55) f16#(I113, I114, I115, I116, I117, I118, I119, I120) -> f15#(I113, I114, I115, I116, I117, I118, I119, I120) f10#(I162, I163, I164, I165, I166, I167, I168, I169) -> f19#(I162, I163, I164, I165, I166, I167, I168, I169) f18#(I170, I171, I172, I173, I174, I175, I176, I177) -> f17#(I170, I171, I172, I173, I174, I175, I176, I177) [1 + I177 <= 0] f18#(I178, I179, I180, I181, I182, I183, I184, I185) -> f17#(I178, I179, I180, I181, I182, I183, I184, I185) [1 <= I185] f18#(I186, I187, I188, I189, I190, I191, I192, I193) -> f9#(I186, I187, I188, I189, I190, I191, I192, I193) [0 <= I193 /\ I193 <= 0] f17#(I194, I195, I196, I197, I198, I199, I200, I201) -> f16#(I194, I195, I196, I197, I198, I199, I200, I202) [I202 = I202] f15#(I203, I204, I205, I206, I207, I208, I209, I210) -> f16#(I203, 1 + I204, I205, I206, I207, I208, I209, I210) [I204 <= I206] f15#(I211, I212, I213, I214, I215, I216, I217, I218) -> f14#(I211, I212, I213, I214, I215, I216, I217, I218) [1 + I214 <= I212] f13#(I227, I228, I229, I230, I231, I232, I233, I234) -> f9#(I227, I228, I229, I230, I231, I232, I233, I234) [1 + I230 <= I228] f9#(I243, I244, I245, I246, I247, I248, I249, I250) -> f10#(1 + I243, I244, I245, I246, I247, I248, I249, I250) R = f24(x1, x2, x3, x4, x5, x6, x7, x8) -> f8(x1, x2, x3, x4, x5, x6, x7, x8) f8(I0, I1, I2, I3, I4, I5, I6, I7) -> f1(I0, I1, I2, I3, I4, I5, I6, I7) f4(I8, I9, I10, I11, I12, I13, I14, I15) -> f23(I8, I9, I10, I11, rnd5, rnd6, I14, I15) [rnd6 = rnd6 /\ rnd5 = rnd5 /\ I9 <= I11] f4(I16, I17, I18, I19, I20, I21, I22, I23) -> f21(I16, I17, I18, I19, I20, I21, I22, I23) [1 + I19 <= I17] f23(I24, I25, I26, I27, I28, I29, I30, I31) -> f22(I25, I25, I26, I27, I28, I29, rnd7, I31) [rnd7 = rnd7 /\ 1 + I29 <= I28] f23(I32, I33, I34, I35, I36, I37, I38, I39) -> f22(I32, I33, I34, I35, I36, I37, I38, I39) [I36 <= I37] f22(I40, I41, I42, I43, I44, I45, I46, I47) -> f3(I40, 1 + I41, I42, I43, I44, I45, I46, I47) f14(I48, I49, I50, I51, I52, I53, I54, I55) -> f13(I48, I49, I50, I51, I52, I53, I54, I55) f21(I56, I57, I58, I59, I60, I61, I62, I63) -> f5(I56, I57, I58, I59, I60, I61, I62, I63) [1 + I56 <= I58] f21(I64, I65, I66, I67, I68, I69, I70, I71) -> f5(I64, I65, I66, I67, I68, I69, I70, I71) [1 + I66 <= I64] f21(I72, I73, I74, I75, I76, I77, I78, I79) -> f20(I72, I73, I74, I75, I76, I77, I78, I79) [I74 <= I72 /\ I72 <= I74] f6(I80, I81, I82, I83, I84, I85, I86, I87) -> f5(I80, 1 + I81, I82, I83, I84, I85, I86, rnd8) [rnd8 = rnd8 /\ I81 <= I83] f6(I88, I89, I90, I91, I92, I93, I94, I95) -> f11(I88, I89, I90, I91, I92, I93, I94, I95) [1 + I91 <= I89] f12(I96, I97, I98, I99, I100, I101, I102, I103) -> f11(I96, 1 + I97, I98, I99, I100, I101, I102, I104) [I104 = I104 /\ I97 <= I99] f12(I105, I106, I107, I108, I109, I110, I111, I112) -> f20(I105, I106, I107, I108, I109, I110, I111, I112) [1 + I108 <= I106] f16(I113, I114, I115, I116, I117, I118, I119, I120) -> f15(I113, I114, I115, I116, I117, I118, I119, I120) f20(I121, I122, I123, I124, I125, I126, I127, I128) -> f10(I121, I122, I123, I124, I125, I126, I127, I128) [1 + I127 <= 0] f20(I129, I130, I131, I132, I133, I134, I135, I136) -> f10(I129, I130, I131, I132, I133, I134, I135, I136) [1 <= I135] f20(I137, I138, I139, I140, I141, I142, I143, I144) -> f7(I137, I138, I139, I140, I141, I142, I143, I144) [0 <= I143 /\ I143 <= 0] f19(I145, I146, I147, I148, I149, I150, I151, I152) -> f18(I145, I146, I147, I148, I149, I150, I151, I153) [I153 = I153 /\ I145 <= I148] f19(I154, I155, I156, I157, I158, I159, I160, I161) -> f7(I154, I155, I156, I157, I158, I159, I160, I161) [1 + I157 <= I154] f10(I162, I163, I164, I165, I166, I167, I168, I169) -> f19(I162, I163, I164, I165, I166, I167, I168, I169) f18(I170, I171, I172, I173, I174, I175, I176, I177) -> f17(I170, I171, I172, I173, I174, I175, I176, I177) [1 + I177 <= 0] f18(I178, I179, I180, I181, I182, I183, I184, I185) -> f17(I178, I179, I180, I181, I182, I183, I184, I185) [1 <= I185] f18(I186, I187, I188, I189, I190, I191, I192, I193) -> f9(I186, I187, I188, I189, I190, I191, I192, I193) [0 <= I193 /\ I193 <= 0] f17(I194, I195, I196, I197, I198, I199, I200, I201) -> f16(I194, I195, I196, I197, I198, I199, I200, I202) [I202 = I202] f15(I203, I204, I205, I206, I207, I208, I209, I210) -> f16(I203, 1 + I204, I205, I206, I207, I208, I209, I210) [I204 <= I206] f15(I211, I212, I213, I214, I215, I216, I217, I218) -> f14(I211, I212, I213, I214, I215, I216, I217, I218) [1 + I214 <= I212] f13(I219, I220, I221, I222, I223, I224, I225, I226) -> f14(I219, 1 + I220, I221, I222, I223, I224, I225, I226) [I220 <= I222] f13(I227, I228, I229, I230, I231, I232, I233, I234) -> f9(I227, I228, I229, I230, I231, I232, I233, I234) [1 + I230 <= I228] f11(I235, I236, I237, I238, I239, I240, I241, I242) -> f12(I235, I236, I237, I238, I239, I240, I241, I242) f9(I243, I244, I245, I246, I247, I248, I249, I250) -> f10(1 + I243, I244, I245, I246, I247, I248, I249, I250) f7(I251, I252, I253, I254, I255, I256, I257, I258) -> f8(I251, I252, 1 + I253, I254, I255, I256, I257, I258) f5(I259, I260, I261, I262, I263, I264, I265, I266) -> f6(I259, I260, I261, I262, I263, I264, I265, I266) f3(I267, I268, I269, I270, I271, I272, I273, I274) -> f4(I267, I268, I269, I270, I271, I272, I273, I274) f1(I275, I276, I277, I278, I279, I280, I281, I282) -> f3(I277, I276, I277, I278, I279, I280, 0, I282) [1 + I277 <= I278] f1(I283, I284, I285, I286, I287, I288, I289, I290) -> f2(I283, I284, I285, I286, I287, I288, I289, I290) [I286 <= I285] The dependency graph for this problem is: 7 -> 29 15 -> 26, 27 21 -> 22 -> 25 23 -> 25 24 -> 31 25 -> 15 26 -> 15 27 -> 7 29 -> 31 31 -> 21 Where: 7) f14#(I48, I49, I50, I51, I52, I53, I54, I55) -> f13#(I48, I49, I50, I51, I52, I53, I54, I55) 15) f16#(I113, I114, I115, I116, I117, I118, I119, I120) -> f15#(I113, I114, I115, I116, I117, I118, I119, I120) 21) f10#(I162, I163, I164, I165, I166, I167, I168, I169) -> f19#(I162, I163, I164, I165, I166, I167, I168, I169) 22) f18#(I170, I171, I172, I173, I174, I175, I176, I177) -> f17#(I170, I171, I172, I173, I174, I175, I176, I177) [1 + I177 <= 0] 23) f18#(I178, I179, I180, I181, I182, I183, I184, I185) -> f17#(I178, I179, I180, I181, I182, I183, I184, I185) [1 <= I185] 24) f18#(I186, I187, I188, I189, I190, I191, I192, I193) -> f9#(I186, I187, I188, I189, I190, I191, I192, I193) [0 <= I193 /\ I193 <= 0] 25) f17#(I194, I195, I196, I197, I198, I199, I200, I201) -> f16#(I194, I195, I196, I197, I198, I199, I200, I202) [I202 = I202] 26) f15#(I203, I204, I205, I206, I207, I208, I209, I210) -> f16#(I203, 1 + I204, I205, I206, I207, I208, I209, I210) [I204 <= I206] 27) f15#(I211, I212, I213, I214, I215, I216, I217, I218) -> f14#(I211, I212, I213, I214, I215, I216, I217, I218) [1 + I214 <= I212] 29) f13#(I227, I228, I229, I230, I231, I232, I233, I234) -> f9#(I227, I228, I229, I230, I231, I232, I233, I234) [1 + I230 <= I228] 31) f9#(I243, I244, I245, I246, I247, I248, I249, I250) -> f10#(1 + I243, I244, I245, I246, I247, I248, I249, I250) We have the following SCCs. { 15, 26 } DP problem for innermost termination. P = f16#(I113, I114, I115, I116, I117, I118, I119, I120) -> f15#(I113, I114, I115, I116, I117, I118, I119, I120) f15#(I203, I204, I205, I206, I207, I208, I209, I210) -> f16#(I203, 1 + I204, I205, I206, I207, I208, I209, I210) [I204 <= I206] R = f24(x1, x2, x3, x4, x5, x6, x7, x8) -> f8(x1, x2, x3, x4, x5, x6, x7, x8) f8(I0, I1, I2, I3, I4, I5, I6, I7) -> f1(I0, I1, I2, I3, I4, I5, I6, I7) f4(I8, I9, I10, I11, I12, I13, I14, I15) -> f23(I8, I9, I10, I11, rnd5, rnd6, I14, I15) [rnd6 = rnd6 /\ rnd5 = rnd5 /\ I9 <= I11] f4(I16, I17, I18, I19, I20, I21, I22, I23) -> f21(I16, I17, I18, I19, I20, I21, I22, I23) [1 + I19 <= I17] f23(I24, I25, I26, I27, I28, I29, I30, I31) -> f22(I25, I25, I26, I27, I28, I29, rnd7, I31) [rnd7 = rnd7 /\ 1 + I29 <= I28] f23(I32, I33, I34, I35, I36, I37, I38, I39) -> f22(I32, I33, I34, I35, I36, I37, I38, I39) [I36 <= I37] f22(I40, I41, I42, I43, I44, I45, I46, I47) -> f3(I40, 1 + I41, I42, I43, I44, I45, I46, I47) f14(I48, I49, I50, I51, I52, I53, I54, I55) -> f13(I48, I49, I50, I51, I52, I53, I54, I55) f21(I56, I57, I58, I59, I60, I61, I62, I63) -> f5(I56, I57, I58, I59, I60, I61, I62, I63) [1 + I56 <= I58] f21(I64, I65, I66, I67, I68, I69, I70, I71) -> f5(I64, I65, I66, I67, I68, I69, I70, I71) [1 + I66 <= I64] f21(I72, I73, I74, I75, I76, I77, I78, I79) -> f20(I72, I73, I74, I75, I76, I77, I78, I79) [I74 <= I72 /\ I72 <= I74] f6(I80, I81, I82, I83, I84, I85, I86, I87) -> f5(I80, 1 + I81, I82, I83, I84, I85, I86, rnd8) [rnd8 = rnd8 /\ I81 <= I83] f6(I88, I89, I90, I91, I92, I93, I94, I95) -> f11(I88, I89, I90, I91, I92, I93, I94, I95) [1 + I91 <= I89] f12(I96, I97, I98, I99, I100, I101, I102, I103) -> f11(I96, 1 + I97, I98, I99, I100, I101, I102, I104) [I104 = I104 /\ I97 <= I99] f12(I105, I106, I107, I108, I109, I110, I111, I112) -> f20(I105, I106, I107, I108, I109, I110, I111, I112) [1 + I108 <= I106] f16(I113, I114, I115, I116, I117, I118, I119, I120) -> f15(I113, I114, I115, I116, I117, I118, I119, I120) f20(I121, I122, I123, I124, I125, I126, I127, I128) -> f10(I121, I122, I123, I124, I125, I126, I127, I128) [1 + I127 <= 0] f20(I129, I130, I131, I132, I133, I134, I135, I136) -> f10(I129, I130, I131, I132, I133, I134, I135, I136) [1 <= I135] f20(I137, I138, I139, I140, I141, I142, I143, I144) -> f7(I137, I138, I139, I140, I141, I142, I143, I144) [0 <= I143 /\ I143 <= 0] f19(I145, I146, I147, I148, I149, I150, I151, I152) -> f18(I145, I146, I147, I148, I149, I150, I151, I153) [I153 = I153 /\ I145 <= I148] f19(I154, I155, I156, I157, I158, I159, I160, I161) -> f7(I154, I155, I156, I157, I158, I159, I160, I161) [1 + I157 <= I154] f10(I162, I163, I164, I165, I166, I167, I168, I169) -> f19(I162, I163, I164, I165, I166, I167, I168, I169) f18(I170, I171, I172, I173, I174, I175, I176, I177) -> f17(I170, I171, I172, I173, I174, I175, I176, I177) [1 + I177 <= 0] f18(I178, I179, I180, I181, I182, I183, I184, I185) -> f17(I178, I179, I180, I181, I182, I183, I184, I185) [1 <= I185] f18(I186, I187, I188, I189, I190, I191, I192, I193) -> f9(I186, I187, I188, I189, I190, I191, I192, I193) [0 <= I193 /\ I193 <= 0] f17(I194, I195, I196, I197, I198, I199, I200, I201) -> f16(I194, I195, I196, I197, I198, I199, I200, I202) [I202 = I202] f15(I203, I204, I205, I206, I207, I208, I209, I210) -> f16(I203, 1 + I204, I205, I206, I207, I208, I209, I210) [I204 <= I206] f15(I211, I212, I213, I214, I215, I216, I217, I218) -> f14(I211, I212, I213, I214, I215, I216, I217, I218) [1 + I214 <= I212] f13(I219, I220, I221, I222, I223, I224, I225, I226) -> f14(I219, 1 + I220, I221, I222, I223, I224, I225, I226) [I220 <= I222] f13(I227, I228, I229, I230, I231, I232, I233, I234) -> f9(I227, I228, I229, I230, I231, I232, I233, I234) [1 + I230 <= I228] f11(I235, I236, I237, I238, I239, I240, I241, I242) -> f12(I235, I236, I237, I238, I239, I240, I241, I242) f9(I243, I244, I245, I246, I247, I248, I249, I250) -> f10(1 + I243, I244, I245, I246, I247, I248, I249, I250) f7(I251, I252, I253, I254, I255, I256, I257, I258) -> f8(I251, I252, 1 + I253, I254, I255, I256, I257, I258) f5(I259, I260, I261, I262, I263, I264, I265, I266) -> f6(I259, I260, I261, I262, I263, I264, I265, I266) f3(I267, I268, I269, I270, I271, I272, I273, I274) -> f4(I267, I268, I269, I270, I271, I272, I273, I274) f1(I275, I276, I277, I278, I279, I280, I281, I282) -> f3(I277, I276, I277, I278, I279, I280, 0, I282) [1 + I277 <= I278] f1(I283, I284, I285, I286, I287, I288, I289, I290) -> f2(I283, I284, I285, I286, I287, I288, I289, I290) [I286 <= I285] We use the reverse value criterion with the projection function NU: NU[f15#(z1,z2,z3,z4,z5,z6,z7,z8)] = z4 + -1 * z2 NU[f16#(z1,z2,z3,z4,z5,z6,z7,z8)] = z4 + -1 * z2 This gives the following inequalities: ==> I116 + -1 * I114 >= I116 + -1 * I114 I204 <= I206 ==> I206 + -1 * I204 > I206 + -1 * (1 + I204) with I206 + -1 * I204 >= 0 We remove all the strictly oriented dependency pairs. DP problem for innermost termination. P = f16#(I113, I114, I115, I116, I117, I118, I119, I120) -> f15#(I113, I114, I115, I116, I117, I118, I119, I120) R = f24(x1, x2, x3, x4, x5, x6, x7, x8) -> f8(x1, x2, x3, x4, x5, x6, x7, x8) f8(I0, I1, I2, I3, I4, I5, I6, I7) -> f1(I0, I1, I2, I3, I4, I5, I6, I7) f4(I8, I9, I10, I11, I12, I13, I14, I15) -> f23(I8, I9, I10, I11, rnd5, rnd6, I14, I15) [rnd6 = rnd6 /\ rnd5 = rnd5 /\ I9 <= I11] f4(I16, I17, I18, I19, I20, I21, I22, I23) -> f21(I16, I17, I18, I19, I20, I21, I22, I23) [1 + I19 <= I17] f23(I24, I25, I26, I27, I28, I29, I30, I31) -> f22(I25, I25, I26, I27, I28, I29, rnd7, I31) [rnd7 = rnd7 /\ 1 + I29 <= I28] f23(I32, I33, I34, I35, I36, I37, I38, I39) -> f22(I32, I33, I34, I35, I36, I37, I38, I39) [I36 <= I37] f22(I40, I41, I42, I43, I44, I45, I46, I47) -> f3(I40, 1 + I41, I42, I43, I44, I45, I46, I47) f14(I48, I49, I50, I51, I52, I53, I54, I55) -> f13(I48, I49, I50, I51, I52, I53, I54, I55) f21(I56, I57, I58, I59, I60, I61, I62, I63) -> f5(I56, I57, I58, I59, I60, I61, I62, I63) [1 + I56 <= I58] f21(I64, I65, I66, I67, I68, I69, I70, I71) -> f5(I64, I65, I66, I67, I68, I69, I70, I71) [1 + I66 <= I64] f21(I72, I73, I74, I75, I76, I77, I78, I79) -> f20(I72, I73, I74, I75, I76, I77, I78, I79) [I74 <= I72 /\ I72 <= I74] f6(I80, I81, I82, I83, I84, I85, I86, I87) -> f5(I80, 1 + I81, I82, I83, I84, I85, I86, rnd8) [rnd8 = rnd8 /\ I81 <= I83] f6(I88, I89, I90, I91, I92, I93, I94, I95) -> f11(I88, I89, I90, I91, I92, I93, I94, I95) [1 + I91 <= I89] f12(I96, I97, I98, I99, I100, I101, I102, I103) -> f11(I96, 1 + I97, I98, I99, I100, I101, I102, I104) [I104 = I104 /\ I97 <= I99] f12(I105, I106, I107, I108, I109, I110, I111, I112) -> f20(I105, I106, I107, I108, I109, I110, I111, I112) [1 + I108 <= I106] f16(I113, I114, I115, I116, I117, I118, I119, I120) -> f15(I113, I114, I115, I116, I117, I118, I119, I120) f20(I121, I122, I123, I124, I125, I126, I127, I128) -> f10(I121, I122, I123, I124, I125, I126, I127, I128) [1 + I127 <= 0] f20(I129, I130, I131, I132, I133, I134, I135, I136) -> f10(I129, I130, I131, I132, I133, I134, I135, I136) [1 <= I135] f20(I137, I138, I139, I140, I141, I142, I143, I144) -> f7(I137, I138, I139, I140, I141, I142, I143, I144) [0 <= I143 /\ I143 <= 0] f19(I145, I146, I147, I148, I149, I150, I151, I152) -> f18(I145, I146, I147, I148, I149, I150, I151, I153) [I153 = I153 /\ I145 <= I148] f19(I154, I155, I156, I157, I158, I159, I160, I161) -> f7(I154, I155, I156, I157, I158, I159, I160, I161) [1 + I157 <= I154] f10(I162, I163, I164, I165, I166, I167, I168, I169) -> f19(I162, I163, I164, I165, I166, I167, I168, I169) f18(I170, I171, I172, I173, I174, I175, I176, I177) -> f17(I170, I171, I172, I173, I174, I175, I176, I177) [1 + I177 <= 0] f18(I178, I179, I180, I181, I182, I183, I184, I185) -> f17(I178, I179, I180, I181, I182, I183, I184, I185) [1 <= I185] f18(I186, I187, I188, I189, I190, I191, I192, I193) -> f9(I186, I187, I188, I189, I190, I191, I192, I193) [0 <= I193 /\ I193 <= 0] f17(I194, I195, I196, I197, I198, I199, I200, I201) -> f16(I194, I195, I196, I197, I198, I199, I200, I202) [I202 = I202] f15(I203, I204, I205, I206, I207, I208, I209, I210) -> f16(I203, 1 + I204, I205, I206, I207, I208, I209, I210) [I204 <= I206] f15(I211, I212, I213, I214, I215, I216, I217, I218) -> f14(I211, I212, I213, I214, I215, I216, I217, I218) [1 + I214 <= I212] f13(I219, I220, I221, I222, I223, I224, I225, I226) -> f14(I219, 1 + I220, I221, I222, I223, I224, I225, I226) [I220 <= I222] f13(I227, I228, I229, I230, I231, I232, I233, I234) -> f9(I227, I228, I229, I230, I231, I232, I233, I234) [1 + I230 <= I228] f11(I235, I236, I237, I238, I239, I240, I241, I242) -> f12(I235, I236, I237, I238, I239, I240, I241, I242) f9(I243, I244, I245, I246, I247, I248, I249, I250) -> f10(1 + I243, I244, I245, I246, I247, I248, I249, I250) f7(I251, I252, I253, I254, I255, I256, I257, I258) -> f8(I251, I252, 1 + I253, I254, I255, I256, I257, I258) f5(I259, I260, I261, I262, I263, I264, I265, I266) -> f6(I259, I260, I261, I262, I263, I264, I265, I266) f3(I267, I268, I269, I270, I271, I272, I273, I274) -> f4(I267, I268, I269, I270, I271, I272, I273, I274) f1(I275, I276, I277, I278, I279, I280, I281, I282) -> f3(I277, I276, I277, I278, I279, I280, 0, I282) [1 + I277 <= I278] f1(I283, I284, I285, I286, I287, I288, I289, I290) -> f2(I283, I284, I285, I286, I287, I288, I289, I290) [I286 <= I285] The dependency graph for this problem is: 15 -> Where: 15) f16#(I113, I114, I115, I116, I117, I118, I119, I120) -> f15#(I113, I114, I115, I116, I117, I118, I119, I120) We have the following SCCs. DP problem for innermost termination. P = f12#(I96, I97, I98, I99, I100, I101, I102, I103) -> f11#(I96, 1 + I97, I98, I99, I100, I101, I102, I104) [I104 = I104 /\ I97 <= I99] f11#(I235, I236, I237, I238, I239, I240, I241, I242) -> f12#(I235, I236, I237, I238, I239, I240, I241, I242) R = f24(x1, x2, x3, x4, x5, x6, x7, x8) -> f8(x1, x2, x3, x4, x5, x6, x7, x8) f8(I0, I1, I2, I3, I4, I5, I6, I7) -> f1(I0, I1, I2, I3, I4, I5, I6, I7) f4(I8, I9, I10, I11, I12, I13, I14, I15) -> f23(I8, I9, I10, I11, rnd5, rnd6, I14, I15) [rnd6 = rnd6 /\ rnd5 = rnd5 /\ I9 <= I11] f4(I16, I17, I18, I19, I20, I21, I22, I23) -> f21(I16, I17, I18, I19, I20, I21, I22, I23) [1 + I19 <= I17] f23(I24, I25, I26, I27, I28, I29, I30, I31) -> f22(I25, I25, I26, I27, I28, I29, rnd7, I31) [rnd7 = rnd7 /\ 1 + I29 <= I28] f23(I32, I33, I34, I35, I36, I37, I38, I39) -> f22(I32, I33, I34, I35, I36, I37, I38, I39) [I36 <= I37] f22(I40, I41, I42, I43, I44, I45, I46, I47) -> f3(I40, 1 + I41, I42, I43, I44, I45, I46, I47) f14(I48, I49, I50, I51, I52, I53, I54, I55) -> f13(I48, I49, I50, I51, I52, I53, I54, I55) f21(I56, I57, I58, I59, I60, I61, I62, I63) -> f5(I56, I57, I58, I59, I60, I61, I62, I63) [1 + I56 <= I58] f21(I64, I65, I66, I67, I68, I69, I70, I71) -> f5(I64, I65, I66, I67, I68, I69, I70, I71) [1 + I66 <= I64] f21(I72, I73, I74, I75, I76, I77, I78, I79) -> f20(I72, I73, I74, I75, I76, I77, I78, I79) [I74 <= I72 /\ I72 <= I74] f6(I80, I81, I82, I83, I84, I85, I86, I87) -> f5(I80, 1 + I81, I82, I83, I84, I85, I86, rnd8) [rnd8 = rnd8 /\ I81 <= I83] f6(I88, I89, I90, I91, I92, I93, I94, I95) -> f11(I88, I89, I90, I91, I92, I93, I94, I95) [1 + I91 <= I89] f12(I96, I97, I98, I99, I100, I101, I102, I103) -> f11(I96, 1 + I97, I98, I99, I100, I101, I102, I104) [I104 = I104 /\ I97 <= I99] f12(I105, I106, I107, I108, I109, I110, I111, I112) -> f20(I105, I106, I107, I108, I109, I110, I111, I112) [1 + I108 <= I106] f16(I113, I114, I115, I116, I117, I118, I119, I120) -> f15(I113, I114, I115, I116, I117, I118, I119, I120) f20(I121, I122, I123, I124, I125, I126, I127, I128) -> f10(I121, I122, I123, I124, I125, I126, I127, I128) [1 + I127 <= 0] f20(I129, I130, I131, I132, I133, I134, I135, I136) -> f10(I129, I130, I131, I132, I133, I134, I135, I136) [1 <= I135] f20(I137, I138, I139, I140, I141, I142, I143, I144) -> f7(I137, I138, I139, I140, I141, I142, I143, I144) [0 <= I143 /\ I143 <= 0] f19(I145, I146, I147, I148, I149, I150, I151, I152) -> f18(I145, I146, I147, I148, I149, I150, I151, I153) [I153 = I153 /\ I145 <= I148] f19(I154, I155, I156, I157, I158, I159, I160, I161) -> f7(I154, I155, I156, I157, I158, I159, I160, I161) [1 + I157 <= I154] f10(I162, I163, I164, I165, I166, I167, I168, I169) -> f19(I162, I163, I164, I165, I166, I167, I168, I169) f18(I170, I171, I172, I173, I174, I175, I176, I177) -> f17(I170, I171, I172, I173, I174, I175, I176, I177) [1 + I177 <= 0] f18(I178, I179, I180, I181, I182, I183, I184, I185) -> f17(I178, I179, I180, I181, I182, I183, I184, I185) [1 <= I185] f18(I186, I187, I188, I189, I190, I191, I192, I193) -> f9(I186, I187, I188, I189, I190, I191, I192, I193) [0 <= I193 /\ I193 <= 0] f17(I194, I195, I196, I197, I198, I199, I200, I201) -> f16(I194, I195, I196, I197, I198, I199, I200, I202) [I202 = I202] f15(I203, I204, I205, I206, I207, I208, I209, I210) -> f16(I203, 1 + I204, I205, I206, I207, I208, I209, I210) [I204 <= I206] f15(I211, I212, I213, I214, I215, I216, I217, I218) -> f14(I211, I212, I213, I214, I215, I216, I217, I218) [1 + I214 <= I212] f13(I219, I220, I221, I222, I223, I224, I225, I226) -> f14(I219, 1 + I220, I221, I222, I223, I224, I225, I226) [I220 <= I222] f13(I227, I228, I229, I230, I231, I232, I233, I234) -> f9(I227, I228, I229, I230, I231, I232, I233, I234) [1 + I230 <= I228] f11(I235, I236, I237, I238, I239, I240, I241, I242) -> f12(I235, I236, I237, I238, I239, I240, I241, I242) f9(I243, I244, I245, I246, I247, I248, I249, I250) -> f10(1 + I243, I244, I245, I246, I247, I248, I249, I250) f7(I251, I252, I253, I254, I255, I256, I257, I258) -> f8(I251, I252, 1 + I253, I254, I255, I256, I257, I258) f5(I259, I260, I261, I262, I263, I264, I265, I266) -> f6(I259, I260, I261, I262, I263, I264, I265, I266) f3(I267, I268, I269, I270, I271, I272, I273, I274) -> f4(I267, I268, I269, I270, I271, I272, I273, I274) f1(I275, I276, I277, I278, I279, I280, I281, I282) -> f3(I277, I276, I277, I278, I279, I280, 0, I282) [1 + I277 <= I278] f1(I283, I284, I285, I286, I287, I288, I289, I290) -> f2(I283, I284, I285, I286, I287, I288, I289, I290) [I286 <= I285] We use the reverse value criterion with the projection function NU: NU[f11#(z1,z2,z3,z4,z5,z6,z7,z8)] = z4 + -1 * z2 NU[f12#(z1,z2,z3,z4,z5,z6,z7,z8)] = z4 + -1 * z2 This gives the following inequalities: I104 = I104 /\ I97 <= I99 ==> I99 + -1 * I97 > I99 + -1 * (1 + I97) with I99 + -1 * I97 >= 0 ==> I238 + -1 * I236 >= I238 + -1 * I236 We remove all the strictly oriented dependency pairs. DP problem for innermost termination. P = f11#(I235, I236, I237, I238, I239, I240, I241, I242) -> f12#(I235, I236, I237, I238, I239, I240, I241, I242) R = f24(x1, x2, x3, x4, x5, x6, x7, x8) -> f8(x1, x2, x3, x4, x5, x6, x7, x8) f8(I0, I1, I2, I3, I4, I5, I6, I7) -> f1(I0, I1, I2, I3, I4, I5, I6, I7) f4(I8, I9, I10, I11, I12, I13, I14, I15) -> f23(I8, I9, I10, I11, rnd5, rnd6, I14, I15) [rnd6 = rnd6 /\ rnd5 = rnd5 /\ I9 <= I11] f4(I16, I17, I18, I19, I20, I21, I22, I23) -> f21(I16, I17, I18, I19, I20, I21, I22, I23) [1 + I19 <= I17] f23(I24, I25, I26, I27, I28, I29, I30, I31) -> f22(I25, I25, I26, I27, I28, I29, rnd7, I31) [rnd7 = rnd7 /\ 1 + I29 <= I28] f23(I32, I33, I34, I35, I36, I37, I38, I39) -> f22(I32, I33, I34, I35, I36, I37, I38, I39) [I36 <= I37] f22(I40, I41, I42, I43, I44, I45, I46, I47) -> f3(I40, 1 + I41, I42, I43, I44, I45, I46, I47) f14(I48, I49, I50, I51, I52, I53, I54, I55) -> f13(I48, I49, I50, I51, I52, I53, I54, I55) f21(I56, I57, I58, I59, I60, I61, I62, I63) -> f5(I56, I57, I58, I59, I60, I61, I62, I63) [1 + I56 <= I58] f21(I64, I65, I66, I67, I68, I69, I70, I71) -> f5(I64, I65, I66, I67, I68, I69, I70, I71) [1 + I66 <= I64] f21(I72, I73, I74, I75, I76, I77, I78, I79) -> f20(I72, I73, I74, I75, I76, I77, I78, I79) [I74 <= I72 /\ I72 <= I74] f6(I80, I81, I82, I83, I84, I85, I86, I87) -> f5(I80, 1 + I81, I82, I83, I84, I85, I86, rnd8) [rnd8 = rnd8 /\ I81 <= I83] f6(I88, I89, I90, I91, I92, I93, I94, I95) -> f11(I88, I89, I90, I91, I92, I93, I94, I95) [1 + I91 <= I89] f12(I96, I97, I98, I99, I100, I101, I102, I103) -> f11(I96, 1 + I97, I98, I99, I100, I101, I102, I104) [I104 = I104 /\ I97 <= I99] f12(I105, I106, I107, I108, I109, I110, I111, I112) -> f20(I105, I106, I107, I108, I109, I110, I111, I112) [1 + I108 <= I106] f16(I113, I114, I115, I116, I117, I118, I119, I120) -> f15(I113, I114, I115, I116, I117, I118, I119, I120) f20(I121, I122, I123, I124, I125, I126, I127, I128) -> f10(I121, I122, I123, I124, I125, I126, I127, I128) [1 + I127 <= 0] f20(I129, I130, I131, I132, I133, I134, I135, I136) -> f10(I129, I130, I131, I132, I133, I134, I135, I136) [1 <= I135] f20(I137, I138, I139, I140, I141, I142, I143, I144) -> f7(I137, I138, I139, I140, I141, I142, I143, I144) [0 <= I143 /\ I143 <= 0] f19(I145, I146, I147, I148, I149, I150, I151, I152) -> f18(I145, I146, I147, I148, I149, I150, I151, I153) [I153 = I153 /\ I145 <= I148] f19(I154, I155, I156, I157, I158, I159, I160, I161) -> f7(I154, I155, I156, I157, I158, I159, I160, I161) [1 + I157 <= I154] f10(I162, I163, I164, I165, I166, I167, I168, I169) -> f19(I162, I163, I164, I165, I166, I167, I168, I169) f18(I170, I171, I172, I173, I174, I175, I176, I177) -> f17(I170, I171, I172, I173, I174, I175, I176, I177) [1 + I177 <= 0] f18(I178, I179, I180, I181, I182, I183, I184, I185) -> f17(I178, I179, I180, I181, I182, I183, I184, I185) [1 <= I185] f18(I186, I187, I188, I189, I190, I191, I192, I193) -> f9(I186, I187, I188, I189, I190, I191, I192, I193) [0 <= I193 /\ I193 <= 0] f17(I194, I195, I196, I197, I198, I199, I200, I201) -> f16(I194, I195, I196, I197, I198, I199, I200, I202) [I202 = I202] f15(I203, I204, I205, I206, I207, I208, I209, I210) -> f16(I203, 1 + I204, I205, I206, I207, I208, I209, I210) [I204 <= I206] f15(I211, I212, I213, I214, I215, I216, I217, I218) -> f14(I211, I212, I213, I214, I215, I216, I217, I218) [1 + I214 <= I212] f13(I219, I220, I221, I222, I223, I224, I225, I226) -> f14(I219, 1 + I220, I221, I222, I223, I224, I225, I226) [I220 <= I222] f13(I227, I228, I229, I230, I231, I232, I233, I234) -> f9(I227, I228, I229, I230, I231, I232, I233, I234) [1 + I230 <= I228] f11(I235, I236, I237, I238, I239, I240, I241, I242) -> f12(I235, I236, I237, I238, I239, I240, I241, I242) f9(I243, I244, I245, I246, I247, I248, I249, I250) -> f10(1 + I243, I244, I245, I246, I247, I248, I249, I250) f7(I251, I252, I253, I254, I255, I256, I257, I258) -> f8(I251, I252, 1 + I253, I254, I255, I256, I257, I258) f5(I259, I260, I261, I262, I263, I264, I265, I266) -> f6(I259, I260, I261, I262, I263, I264, I265, I266) f3(I267, I268, I269, I270, I271, I272, I273, I274) -> f4(I267, I268, I269, I270, I271, I272, I273, I274) f1(I275, I276, I277, I278, I279, I280, I281, I282) -> f3(I277, I276, I277, I278, I279, I280, 0, I282) [1 + I277 <= I278] f1(I283, I284, I285, I286, I287, I288, I289, I290) -> f2(I283, I284, I285, I286, I287, I288, I289, I290) [I286 <= I285] The dependency graph for this problem is: 30 -> Where: 30) f11#(I235, I236, I237, I238, I239, I240, I241, I242) -> f12#(I235, I236, I237, I238, I239, I240, I241, I242) We have the following SCCs. DP problem for innermost termination. P = f6#(I80, I81, I82, I83, I84, I85, I86, I87) -> f5#(I80, 1 + I81, I82, I83, I84, I85, I86, rnd8) [rnd8 = rnd8 /\ I81 <= I83] f5#(I259, I260, I261, I262, I263, I264, I265, I266) -> f6#(I259, I260, I261, I262, I263, I264, I265, I266) R = f24(x1, x2, x3, x4, x5, x6, x7, x8) -> f8(x1, x2, x3, x4, x5, x6, x7, x8) f8(I0, I1, I2, I3, I4, I5, I6, I7) -> f1(I0, I1, I2, I3, I4, I5, I6, I7) f4(I8, I9, I10, I11, I12, I13, I14, I15) -> f23(I8, I9, I10, I11, rnd5, rnd6, I14, I15) [rnd6 = rnd6 /\ rnd5 = rnd5 /\ I9 <= I11] f4(I16, I17, I18, I19, I20, I21, I22, I23) -> f21(I16, I17, I18, I19, I20, I21, I22, I23) [1 + I19 <= I17] f23(I24, I25, I26, I27, I28, I29, I30, I31) -> f22(I25, I25, I26, I27, I28, I29, rnd7, I31) [rnd7 = rnd7 /\ 1 + I29 <= I28] f23(I32, I33, I34, I35, I36, I37, I38, I39) -> f22(I32, I33, I34, I35, I36, I37, I38, I39) [I36 <= I37] f22(I40, I41, I42, I43, I44, I45, I46, I47) -> f3(I40, 1 + I41, I42, I43, I44, I45, I46, I47) f14(I48, I49, I50, I51, I52, I53, I54, I55) -> f13(I48, I49, I50, I51, I52, I53, I54, I55) f21(I56, I57, I58, I59, I60, I61, I62, I63) -> f5(I56, I57, I58, I59, I60, I61, I62, I63) [1 + I56 <= I58] f21(I64, I65, I66, I67, I68, I69, I70, I71) -> f5(I64, I65, I66, I67, I68, I69, I70, I71) [1 + I66 <= I64] f21(I72, I73, I74, I75, I76, I77, I78, I79) -> f20(I72, I73, I74, I75, I76, I77, I78, I79) [I74 <= I72 /\ I72 <= I74] f6(I80, I81, I82, I83, I84, I85, I86, I87) -> f5(I80, 1 + I81, I82, I83, I84, I85, I86, rnd8) [rnd8 = rnd8 /\ I81 <= I83] f6(I88, I89, I90, I91, I92, I93, I94, I95) -> f11(I88, I89, I90, I91, I92, I93, I94, I95) [1 + I91 <= I89] f12(I96, I97, I98, I99, I100, I101, I102, I103) -> f11(I96, 1 + I97, I98, I99, I100, I101, I102, I104) [I104 = I104 /\ I97 <= I99] f12(I105, I106, I107, I108, I109, I110, I111, I112) -> f20(I105, I106, I107, I108, I109, I110, I111, I112) [1 + I108 <= I106] f16(I113, I114, I115, I116, I117, I118, I119, I120) -> f15(I113, I114, I115, I116, I117, I118, I119, I120) f20(I121, I122, I123, I124, I125, I126, I127, I128) -> f10(I121, I122, I123, I124, I125, I126, I127, I128) [1 + I127 <= 0] f20(I129, I130, I131, I132, I133, I134, I135, I136) -> f10(I129, I130, I131, I132, I133, I134, I135, I136) [1 <= I135] f20(I137, I138, I139, I140, I141, I142, I143, I144) -> f7(I137, I138, I139, I140, I141, I142, I143, I144) [0 <= I143 /\ I143 <= 0] f19(I145, I146, I147, I148, I149, I150, I151, I152) -> f18(I145, I146, I147, I148, I149, I150, I151, I153) [I153 = I153 /\ I145 <= I148] f19(I154, I155, I156, I157, I158, I159, I160, I161) -> f7(I154, I155, I156, I157, I158, I159, I160, I161) [1 + I157 <= I154] f10(I162, I163, I164, I165, I166, I167, I168, I169) -> f19(I162, I163, I164, I165, I166, I167, I168, I169) f18(I170, I171, I172, I173, I174, I175, I176, I177) -> f17(I170, I171, I172, I173, I174, I175, I176, I177) [1 + I177 <= 0] f18(I178, I179, I180, I181, I182, I183, I184, I185) -> f17(I178, I179, I180, I181, I182, I183, I184, I185) [1 <= I185] f18(I186, I187, I188, I189, I190, I191, I192, I193) -> f9(I186, I187, I188, I189, I190, I191, I192, I193) [0 <= I193 /\ I193 <= 0] f17(I194, I195, I196, I197, I198, I199, I200, I201) -> f16(I194, I195, I196, I197, I198, I199, I200, I202) [I202 = I202] f15(I203, I204, I205, I206, I207, I208, I209, I210) -> f16(I203, 1 + I204, I205, I206, I207, I208, I209, I210) [I204 <= I206] f15(I211, I212, I213, I214, I215, I216, I217, I218) -> f14(I211, I212, I213, I214, I215, I216, I217, I218) [1 + I214 <= I212] f13(I219, I220, I221, I222, I223, I224, I225, I226) -> f14(I219, 1 + I220, I221, I222, I223, I224, I225, I226) [I220 <= I222] f13(I227, I228, I229, I230, I231, I232, I233, I234) -> f9(I227, I228, I229, I230, I231, I232, I233, I234) [1 + I230 <= I228] f11(I235, I236, I237, I238, I239, I240, I241, I242) -> f12(I235, I236, I237, I238, I239, I240, I241, I242) f9(I243, I244, I245, I246, I247, I248, I249, I250) -> f10(1 + I243, I244, I245, I246, I247, I248, I249, I250) f7(I251, I252, I253, I254, I255, I256, I257, I258) -> f8(I251, I252, 1 + I253, I254, I255, I256, I257, I258) f5(I259, I260, I261, I262, I263, I264, I265, I266) -> f6(I259, I260, I261, I262, I263, I264, I265, I266) f3(I267, I268, I269, I270, I271, I272, I273, I274) -> f4(I267, I268, I269, I270, I271, I272, I273, I274) f1(I275, I276, I277, I278, I279, I280, I281, I282) -> f3(I277, I276, I277, I278, I279, I280, 0, I282) [1 + I277 <= I278] f1(I283, I284, I285, I286, I287, I288, I289, I290) -> f2(I283, I284, I285, I286, I287, I288, I289, I290) [I286 <= I285] We use the reverse value criterion with the projection function NU: NU[f5#(z1,z2,z3,z4,z5,z6,z7,z8)] = z4 + -1 * z2 NU[f6#(z1,z2,z3,z4,z5,z6,z7,z8)] = z4 + -1 * z2 This gives the following inequalities: rnd8 = rnd8 /\ I81 <= I83 ==> I83 + -1 * I81 > I83 + -1 * (1 + I81) with I83 + -1 * I81 >= 0 ==> I262 + -1 * I260 >= I262 + -1 * I260 We remove all the strictly oriented dependency pairs. DP problem for innermost termination. P = f5#(I259, I260, I261, I262, I263, I264, I265, I266) -> f6#(I259, I260, I261, I262, I263, I264, I265, I266) R = f24(x1, x2, x3, x4, x5, x6, x7, x8) -> f8(x1, x2, x3, x4, x5, x6, x7, x8) f8(I0, I1, I2, I3, I4, I5, I6, I7) -> f1(I0, I1, I2, I3, I4, I5, I6, I7) f4(I8, I9, I10, I11, I12, I13, I14, I15) -> f23(I8, I9, I10, I11, rnd5, rnd6, I14, I15) [rnd6 = rnd6 /\ rnd5 = rnd5 /\ I9 <= I11] f4(I16, I17, I18, I19, I20, I21, I22, I23) -> f21(I16, I17, I18, I19, I20, I21, I22, I23) [1 + I19 <= I17] f23(I24, I25, I26, I27, I28, I29, I30, I31) -> f22(I25, I25, I26, I27, I28, I29, rnd7, I31) [rnd7 = rnd7 /\ 1 + I29 <= I28] f23(I32, I33, I34, I35, I36, I37, I38, I39) -> f22(I32, I33, I34, I35, I36, I37, I38, I39) [I36 <= I37] f22(I40, I41, I42, I43, I44, I45, I46, I47) -> f3(I40, 1 + I41, I42, I43, I44, I45, I46, I47) f14(I48, I49, I50, I51, I52, I53, I54, I55) -> f13(I48, I49, I50, I51, I52, I53, I54, I55) f21(I56, I57, I58, I59, I60, I61, I62, I63) -> f5(I56, I57, I58, I59, I60, I61, I62, I63) [1 + I56 <= I58] f21(I64, I65, I66, I67, I68, I69, I70, I71) -> f5(I64, I65, I66, I67, I68, I69, I70, I71) [1 + I66 <= I64] f21(I72, I73, I74, I75, I76, I77, I78, I79) -> f20(I72, I73, I74, I75, I76, I77, I78, I79) [I74 <= I72 /\ I72 <= I74] f6(I80, I81, I82, I83, I84, I85, I86, I87) -> f5(I80, 1 + I81, I82, I83, I84, I85, I86, rnd8) [rnd8 = rnd8 /\ I81 <= I83] f6(I88, I89, I90, I91, I92, I93, I94, I95) -> f11(I88, I89, I90, I91, I92, I93, I94, I95) [1 + I91 <= I89] f12(I96, I97, I98, I99, I100, I101, I102, I103) -> f11(I96, 1 + I97, I98, I99, I100, I101, I102, I104) [I104 = I104 /\ I97 <= I99] f12(I105, I106, I107, I108, I109, I110, I111, I112) -> f20(I105, I106, I107, I108, I109, I110, I111, I112) [1 + I108 <= I106] f16(I113, I114, I115, I116, I117, I118, I119, I120) -> f15(I113, I114, I115, I116, I117, I118, I119, I120) f20(I121, I122, I123, I124, I125, I126, I127, I128) -> f10(I121, I122, I123, I124, I125, I126, I127, I128) [1 + I127 <= 0] f20(I129, I130, I131, I132, I133, I134, I135, I136) -> f10(I129, I130, I131, I132, I133, I134, I135, I136) [1 <= I135] f20(I137, I138, I139, I140, I141, I142, I143, I144) -> f7(I137, I138, I139, I140, I141, I142, I143, I144) [0 <= I143 /\ I143 <= 0] f19(I145, I146, I147, I148, I149, I150, I151, I152) -> f18(I145, I146, I147, I148, I149, I150, I151, I153) [I153 = I153 /\ I145 <= I148] f19(I154, I155, I156, I157, I158, I159, I160, I161) -> f7(I154, I155, I156, I157, I158, I159, I160, I161) [1 + I157 <= I154] f10(I162, I163, I164, I165, I166, I167, I168, I169) -> f19(I162, I163, I164, I165, I166, I167, I168, I169) f18(I170, I171, I172, I173, I174, I175, I176, I177) -> f17(I170, I171, I172, I173, I174, I175, I176, I177) [1 + I177 <= 0] f18(I178, I179, I180, I181, I182, I183, I184, I185) -> f17(I178, I179, I180, I181, I182, I183, I184, I185) [1 <= I185] f18(I186, I187, I188, I189, I190, I191, I192, I193) -> f9(I186, I187, I188, I189, I190, I191, I192, I193) [0 <= I193 /\ I193 <= 0] f17(I194, I195, I196, I197, I198, I199, I200, I201) -> f16(I194, I195, I196, I197, I198, I199, I200, I202) [I202 = I202] f15(I203, I204, I205, I206, I207, I208, I209, I210) -> f16(I203, 1 + I204, I205, I206, I207, I208, I209, I210) [I204 <= I206] f15(I211, I212, I213, I214, I215, I216, I217, I218) -> f14(I211, I212, I213, I214, I215, I216, I217, I218) [1 + I214 <= I212] f13(I219, I220, I221, I222, I223, I224, I225, I226) -> f14(I219, 1 + I220, I221, I222, I223, I224, I225, I226) [I220 <= I222] f13(I227, I228, I229, I230, I231, I232, I233, I234) -> f9(I227, I228, I229, I230, I231, I232, I233, I234) [1 + I230 <= I228] f11(I235, I236, I237, I238, I239, I240, I241, I242) -> f12(I235, I236, I237, I238, I239, I240, I241, I242) f9(I243, I244, I245, I246, I247, I248, I249, I250) -> f10(1 + I243, I244, I245, I246, I247, I248, I249, I250) f7(I251, I252, I253, I254, I255, I256, I257, I258) -> f8(I251, I252, 1 + I253, I254, I255, I256, I257, I258) f5(I259, I260, I261, I262, I263, I264, I265, I266) -> f6(I259, I260, I261, I262, I263, I264, I265, I266) f3(I267, I268, I269, I270, I271, I272, I273, I274) -> f4(I267, I268, I269, I270, I271, I272, I273, I274) f1(I275, I276, I277, I278, I279, I280, I281, I282) -> f3(I277, I276, I277, I278, I279, I280, 0, I282) [1 + I277 <= I278] f1(I283, I284, I285, I286, I287, I288, I289, I290) -> f2(I283, I284, I285, I286, I287, I288, I289, I290) [I286 <= I285] The dependency graph for this problem is: 33 -> Where: 33) f5#(I259, I260, I261, I262, I263, I264, I265, I266) -> f6#(I259, I260, I261, I262, I263, I264, I265, I266) We have the following SCCs. DP problem for innermost termination. P = f4#(I8, I9, I10, I11, I12, I13, I14, I15) -> f23#(I8, I9, I10, I11, rnd5, rnd6, I14, I15) [rnd6 = rnd6 /\ rnd5 = rnd5 /\ I9 <= I11] f23#(I24, I25, I26, I27, I28, I29, I30, I31) -> f22#(I25, I25, I26, I27, I28, I29, rnd7, I31) [rnd7 = rnd7 /\ 1 + I29 <= I28] f23#(I32, I33, I34, I35, I36, I37, I38, I39) -> f22#(I32, I33, I34, I35, I36, I37, I38, I39) [I36 <= I37] f22#(I40, I41, I42, I43, I44, I45, I46, I47) -> f3#(I40, 1 + I41, I42, I43, I44, I45, I46, I47) f3#(I267, I268, I269, I270, I271, I272, I273, I274) -> f4#(I267, I268, I269, I270, I271, I272, I273, I274) R = f24(x1, x2, x3, x4, x5, x6, x7, x8) -> f8(x1, x2, x3, x4, x5, x6, x7, x8) f8(I0, I1, I2, I3, I4, I5, I6, I7) -> f1(I0, I1, I2, I3, I4, I5, I6, I7) f4(I8, I9, I10, I11, I12, I13, I14, I15) -> f23(I8, I9, I10, I11, rnd5, rnd6, I14, I15) [rnd6 = rnd6 /\ rnd5 = rnd5 /\ I9 <= I11] f4(I16, I17, I18, I19, I20, I21, I22, I23) -> f21(I16, I17, I18, I19, I20, I21, I22, I23) [1 + I19 <= I17] f23(I24, I25, I26, I27, I28, I29, I30, I31) -> f22(I25, I25, I26, I27, I28, I29, rnd7, I31) [rnd7 = rnd7 /\ 1 + I29 <= I28] f23(I32, I33, I34, I35, I36, I37, I38, I39) -> f22(I32, I33, I34, I35, I36, I37, I38, I39) [I36 <= I37] f22(I40, I41, I42, I43, I44, I45, I46, I47) -> f3(I40, 1 + I41, I42, I43, I44, I45, I46, I47) f14(I48, I49, I50, I51, I52, I53, I54, I55) -> f13(I48, I49, I50, I51, I52, I53, I54, I55) f21(I56, I57, I58, I59, I60, I61, I62, I63) -> f5(I56, I57, I58, I59, I60, I61, I62, I63) [1 + I56 <= I58] f21(I64, I65, I66, I67, I68, I69, I70, I71) -> f5(I64, I65, I66, I67, I68, I69, I70, I71) [1 + I66 <= I64] f21(I72, I73, I74, I75, I76, I77, I78, I79) -> f20(I72, I73, I74, I75, I76, I77, I78, I79) [I74 <= I72 /\ I72 <= I74] f6(I80, I81, I82, I83, I84, I85, I86, I87) -> f5(I80, 1 + I81, I82, I83, I84, I85, I86, rnd8) [rnd8 = rnd8 /\ I81 <= I83] f6(I88, I89, I90, I91, I92, I93, I94, I95) -> f11(I88, I89, I90, I91, I92, I93, I94, I95) [1 + I91 <= I89] f12(I96, I97, I98, I99, I100, I101, I102, I103) -> f11(I96, 1 + I97, I98, I99, I100, I101, I102, I104) [I104 = I104 /\ I97 <= I99] f12(I105, I106, I107, I108, I109, I110, I111, I112) -> f20(I105, I106, I107, I108, I109, I110, I111, I112) [1 + I108 <= I106] f16(I113, I114, I115, I116, I117, I118, I119, I120) -> f15(I113, I114, I115, I116, I117, I118, I119, I120) f20(I121, I122, I123, I124, I125, I126, I127, I128) -> f10(I121, I122, I123, I124, I125, I126, I127, I128) [1 + I127 <= 0] f20(I129, I130, I131, I132, I133, I134, I135, I136) -> f10(I129, I130, I131, I132, I133, I134, I135, I136) [1 <= I135] f20(I137, I138, I139, I140, I141, I142, I143, I144) -> f7(I137, I138, I139, I140, I141, I142, I143, I144) [0 <= I143 /\ I143 <= 0] f19(I145, I146, I147, I148, I149, I150, I151, I152) -> f18(I145, I146, I147, I148, I149, I150, I151, I153) [I153 = I153 /\ I145 <= I148] f19(I154, I155, I156, I157, I158, I159, I160, I161) -> f7(I154, I155, I156, I157, I158, I159, I160, I161) [1 + I157 <= I154] f10(I162, I163, I164, I165, I166, I167, I168, I169) -> f19(I162, I163, I164, I165, I166, I167, I168, I169) f18(I170, I171, I172, I173, I174, I175, I176, I177) -> f17(I170, I171, I172, I173, I174, I175, I176, I177) [1 + I177 <= 0] f18(I178, I179, I180, I181, I182, I183, I184, I185) -> f17(I178, I179, I180, I181, I182, I183, I184, I185) [1 <= I185] f18(I186, I187, I188, I189, I190, I191, I192, I193) -> f9(I186, I187, I188, I189, I190, I191, I192, I193) [0 <= I193 /\ I193 <= 0] f17(I194, I195, I196, I197, I198, I199, I200, I201) -> f16(I194, I195, I196, I197, I198, I199, I200, I202) [I202 = I202] f15(I203, I204, I205, I206, I207, I208, I209, I210) -> f16(I203, 1 + I204, I205, I206, I207, I208, I209, I210) [I204 <= I206] f15(I211, I212, I213, I214, I215, I216, I217, I218) -> f14(I211, I212, I213, I214, I215, I216, I217, I218) [1 + I214 <= I212] f13(I219, I220, I221, I222, I223, I224, I225, I226) -> f14(I219, 1 + I220, I221, I222, I223, I224, I225, I226) [I220 <= I222] f13(I227, I228, I229, I230, I231, I232, I233, I234) -> f9(I227, I228, I229, I230, I231, I232, I233, I234) [1 + I230 <= I228] f11(I235, I236, I237, I238, I239, I240, I241, I242) -> f12(I235, I236, I237, I238, I239, I240, I241, I242) f9(I243, I244, I245, I246, I247, I248, I249, I250) -> f10(1 + I243, I244, I245, I246, I247, I248, I249, I250) f7(I251, I252, I253, I254, I255, I256, I257, I258) -> f8(I251, I252, 1 + I253, I254, I255, I256, I257, I258) f5(I259, I260, I261, I262, I263, I264, I265, I266) -> f6(I259, I260, I261, I262, I263, I264, I265, I266) f3(I267, I268, I269, I270, I271, I272, I273, I274) -> f4(I267, I268, I269, I270, I271, I272, I273, I274) f1(I275, I276, I277, I278, I279, I280, I281, I282) -> f3(I277, I276, I277, I278, I279, I280, 0, I282) [1 + I277 <= I278] f1(I283, I284, I285, I286, I287, I288, I289, I290) -> f2(I283, I284, I285, I286, I287, I288, I289, I290) [I286 <= I285] We use the extended value criterion with the projection function NU: NU[f3#(x0,x1,x2,x3,x4,x5,x6,x7)] = -x1 + x3 NU[f22#(x0,x1,x2,x3,x4,x5,x6,x7)] = -x1 + x3 - 1 NU[f23#(x0,x1,x2,x3,x4,x5,x6,x7)] = -x1 + x3 - 1 NU[f4#(x0,x1,x2,x3,x4,x5,x6,x7)] = -x1 + x3 This gives the following inequalities: rnd6 = rnd6 /\ rnd5 = rnd5 /\ I9 <= I11 ==> -I9 + I11 > -I9 + I11 - 1 with -I9 + I11 >= 0 rnd7 = rnd7 /\ 1 + I29 <= I28 ==> -I25 + I27 - 1 >= -I25 + I27 - 1 I36 <= I37 ==> -I33 + I35 - 1 >= -I33 + I35 - 1 ==> -I41 + I43 - 1 >= -(1 + I41) + I43 ==> -I268 + I270 >= -I268 + I270 We remove all the strictly oriented dependency pairs. DP problem for innermost termination. P = f23#(I24, I25, I26, I27, I28, I29, I30, I31) -> f22#(I25, I25, I26, I27, I28, I29, rnd7, I31) [rnd7 = rnd7 /\ 1 + I29 <= I28] f23#(I32, I33, I34, I35, I36, I37, I38, I39) -> f22#(I32, I33, I34, I35, I36, I37, I38, I39) [I36 <= I37] f22#(I40, I41, I42, I43, I44, I45, I46, I47) -> f3#(I40, 1 + I41, I42, I43, I44, I45, I46, I47) f3#(I267, I268, I269, I270, I271, I272, I273, I274) -> f4#(I267, I268, I269, I270, I271, I272, I273, I274) R = f24(x1, x2, x3, x4, x5, x6, x7, x8) -> f8(x1, x2, x3, x4, x5, x6, x7, x8) f8(I0, I1, I2, I3, I4, I5, I6, I7) -> f1(I0, I1, I2, I3, I4, I5, I6, I7) f4(I8, I9, I10, I11, I12, I13, I14, I15) -> f23(I8, I9, I10, I11, rnd5, rnd6, I14, I15) [rnd6 = rnd6 /\ rnd5 = rnd5 /\ I9 <= I11] f4(I16, I17, I18, I19, I20, I21, I22, I23) -> f21(I16, I17, I18, I19, I20, I21, I22, I23) [1 + I19 <= I17] f23(I24, I25, I26, I27, I28, I29, I30, I31) -> f22(I25, I25, I26, I27, I28, I29, rnd7, I31) [rnd7 = rnd7 /\ 1 + I29 <= I28] f23(I32, I33, I34, I35, I36, I37, I38, I39) -> f22(I32, I33, I34, I35, I36, I37, I38, I39) [I36 <= I37] f22(I40, I41, I42, I43, I44, I45, I46, I47) -> f3(I40, 1 + I41, I42, I43, I44, I45, I46, I47) f14(I48, I49, I50, I51, I52, I53, I54, I55) -> f13(I48, I49, I50, I51, I52, I53, I54, I55) f21(I56, I57, I58, I59, I60, I61, I62, I63) -> f5(I56, I57, I58, I59, I60, I61, I62, I63) [1 + I56 <= I58] f21(I64, I65, I66, I67, I68, I69, I70, I71) -> f5(I64, I65, I66, I67, I68, I69, I70, I71) [1 + I66 <= I64] f21(I72, I73, I74, I75, I76, I77, I78, I79) -> f20(I72, I73, I74, I75, I76, I77, I78, I79) [I74 <= I72 /\ I72 <= I74] f6(I80, I81, I82, I83, I84, I85, I86, I87) -> f5(I80, 1 + I81, I82, I83, I84, I85, I86, rnd8) [rnd8 = rnd8 /\ I81 <= I83] f6(I88, I89, I90, I91, I92, I93, I94, I95) -> f11(I88, I89, I90, I91, I92, I93, I94, I95) [1 + I91 <= I89] f12(I96, I97, I98, I99, I100, I101, I102, I103) -> f11(I96, 1 + I97, I98, I99, I100, I101, I102, I104) [I104 = I104 /\ I97 <= I99] f12(I105, I106, I107, I108, I109, I110, I111, I112) -> f20(I105, I106, I107, I108, I109, I110, I111, I112) [1 + I108 <= I106] f16(I113, I114, I115, I116, I117, I118, I119, I120) -> f15(I113, I114, I115, I116, I117, I118, I119, I120) f20(I121, I122, I123, I124, I125, I126, I127, I128) -> f10(I121, I122, I123, I124, I125, I126, I127, I128) [1 + I127 <= 0] f20(I129, I130, I131, I132, I133, I134, I135, I136) -> f10(I129, I130, I131, I132, I133, I134, I135, I136) [1 <= I135] f20(I137, I138, I139, I140, I141, I142, I143, I144) -> f7(I137, I138, I139, I140, I141, I142, I143, I144) [0 <= I143 /\ I143 <= 0] f19(I145, I146, I147, I148, I149, I150, I151, I152) -> f18(I145, I146, I147, I148, I149, I150, I151, I153) [I153 = I153 /\ I145 <= I148] f19(I154, I155, I156, I157, I158, I159, I160, I161) -> f7(I154, I155, I156, I157, I158, I159, I160, I161) [1 + I157 <= I154] f10(I162, I163, I164, I165, I166, I167, I168, I169) -> f19(I162, I163, I164, I165, I166, I167, I168, I169) f18(I170, I171, I172, I173, I174, I175, I176, I177) -> f17(I170, I171, I172, I173, I174, I175, I176, I177) [1 + I177 <= 0] f18(I178, I179, I180, I181, I182, I183, I184, I185) -> f17(I178, I179, I180, I181, I182, I183, I184, I185) [1 <= I185] f18(I186, I187, I188, I189, I190, I191, I192, I193) -> f9(I186, I187, I188, I189, I190, I191, I192, I193) [0 <= I193 /\ I193 <= 0] f17(I194, I195, I196, I197, I198, I199, I200, I201) -> f16(I194, I195, I196, I197, I198, I199, I200, I202) [I202 = I202] f15(I203, I204, I205, I206, I207, I208, I209, I210) -> f16(I203, 1 + I204, I205, I206, I207, I208, I209, I210) [I204 <= I206] f15(I211, I212, I213, I214, I215, I216, I217, I218) -> f14(I211, I212, I213, I214, I215, I216, I217, I218) [1 + I214 <= I212] f13(I219, I220, I221, I222, I223, I224, I225, I226) -> f14(I219, 1 + I220, I221, I222, I223, I224, I225, I226) [I220 <= I222] f13(I227, I228, I229, I230, I231, I232, I233, I234) -> f9(I227, I228, I229, I230, I231, I232, I233, I234) [1 + I230 <= I228] f11(I235, I236, I237, I238, I239, I240, I241, I242) -> f12(I235, I236, I237, I238, I239, I240, I241, I242) f9(I243, I244, I245, I246, I247, I248, I249, I250) -> f10(1 + I243, I244, I245, I246, I247, I248, I249, I250) f7(I251, I252, I253, I254, I255, I256, I257, I258) -> f8(I251, I252, 1 + I253, I254, I255, I256, I257, I258) f5(I259, I260, I261, I262, I263, I264, I265, I266) -> f6(I259, I260, I261, I262, I263, I264, I265, I266) f3(I267, I268, I269, I270, I271, I272, I273, I274) -> f4(I267, I268, I269, I270, I271, I272, I273, I274) f1(I275, I276, I277, I278, I279, I280, I281, I282) -> f3(I277, I276, I277, I278, I279, I280, 0, I282) [1 + I277 <= I278] f1(I283, I284, I285, I286, I287, I288, I289, I290) -> f2(I283, I284, I285, I286, I287, I288, I289, I290) [I286 <= I285] The dependency graph for this problem is: 4 -> 6 5 -> 6 6 -> 34 34 -> Where: 4) f23#(I24, I25, I26, I27, I28, I29, I30, I31) -> f22#(I25, I25, I26, I27, I28, I29, rnd7, I31) [rnd7 = rnd7 /\ 1 + I29 <= I28] 5) f23#(I32, I33, I34, I35, I36, I37, I38, I39) -> f22#(I32, I33, I34, I35, I36, I37, I38, I39) [I36 <= I37] 6) f22#(I40, I41, I42, I43, I44, I45, I46, I47) -> f3#(I40, 1 + I41, I42, I43, I44, I45, I46, I47) 34) f3#(I267, I268, I269, I270, I271, I272, I273, I274) -> f4#(I267, I268, I269, I270, I271, I272, I273, I274) We have the following SCCs.