24.11/24.19 YES 24.11/24.19 24.11/24.19 DP problem for innermost termination. 24.11/24.19 P = 24.11/24.19 f9#(x1, x2, x3, x4, x5, x6) -> f8#(x1, x2, x3, x4, x5, x6) 24.11/24.19 f8#(I0, I1, I2, I3, I4, I5) -> f7#(I0, I1, I2, I3, I4, I5) 24.11/24.19 f8#(I6, I7, I8, I9, I10, I11) -> f2#(I6, I7, I8, I9, I10, I11) 24.11/24.19 f8#(I12, I13, I14, I15, I16, I17) -> f4#(I12, I13, I14, I15, I16, I17) 24.11/24.19 f8#(I18, I19, I20, I21, I22, I23) -> f5#(I18, I19, I20, I21, I22, I23) 24.11/24.19 f8#(I24, I25, I26, I27, I28, I29) -> f3#(I24, I25, I26, I27, I28, I29) 24.11/24.19 f8#(I30, I31, I32, I33, I34, I35) -> f1#(I30, I31, I32, I33, I34, I35) 24.11/24.19 f8#(I42, I43, I44, I45, I46, I47) -> f7#(I46, I47, rnd3, rnd4, rnd5, rnd6) [rnd6 = rnd4 /\ rnd5 = rnd3 /\ rnd4 = rnd4 /\ rnd3 = rnd3] 24.11/24.19 f7#(I48, I49, I50, I51, I52, I53) -> f2#(I52, I53, I54, I51, 0, I55) [I55 = I54 /\ I54 = I54] 24.11/24.19 f2#(I56, I57, I58, I59, I60, I61) -> f4#(I60, I61, I58, I59, I60, 3) [I60 <= 9] 24.11/24.19 f2#(I62, I63, I64, I65, I66, I67) -> f5#(I66, I67, I68, I65, I66, I69) [I69 = I68 /\ 10 <= I66 /\ I68 = I68] 24.11/24.19 f4#(I70, I71, I72, I73, I74, I75) -> f3#(I74, I75, I72, I73, I74, I75) [I75 <= 11] 24.11/24.19 f4#(I76, I77, I78, I79, I80, I81) -> f1#(I80, I81, I78, I79, I80, I81) [12 <= I81] 24.11/24.19 f3#(I92, I93, I94, I95, I96, I97) -> f4#(I96, I97, I94, I95, I96, 1 + I97) 24.11/24.19 f1#(I98, I99, I100, I101, I102, I103) -> f2#(I102, I103, I104, I101, 1 + I102, I105) [I105 = I104 /\ I104 = I104] 24.11/24.19 R = 24.11/24.19 f9(x1, x2, x3, x4, x5, x6) -> f8(x1, x2, x3, x4, x5, x6) 24.11/24.19 f8(I0, I1, I2, I3, I4, I5) -> f7(I0, I1, I2, I3, I4, I5) 24.11/24.19 f8(I6, I7, I8, I9, I10, I11) -> f2(I6, I7, I8, I9, I10, I11) 24.11/24.19 f8(I12, I13, I14, I15, I16, I17) -> f4(I12, I13, I14, I15, I16, I17) 24.11/24.19 f8(I18, I19, I20, I21, I22, I23) -> f5(I18, I19, I20, I21, I22, I23) 24.11/24.19 f8(I24, I25, I26, I27, I28, I29) -> f3(I24, I25, I26, I27, I28, I29) 24.11/24.19 f8(I30, I31, I32, I33, I34, I35) -> f1(I30, I31, I32, I33, I34, I35) 24.11/24.19 f8(I36, I37, I38, I39, I40, I41) -> f6(I36, I37, I38, I39, I40, I41) 24.11/24.19 f8(I42, I43, I44, I45, I46, I47) -> f7(I46, I47, rnd3, rnd4, rnd5, rnd6) [rnd6 = rnd4 /\ rnd5 = rnd3 /\ rnd4 = rnd4 /\ rnd3 = rnd3] 24.11/24.19 f7(I48, I49, I50, I51, I52, I53) -> f2(I52, I53, I54, I51, 0, I55) [I55 = I54 /\ I54 = I54] 24.11/24.19 f2(I56, I57, I58, I59, I60, I61) -> f4(I60, I61, I58, I59, I60, 3) [I60 <= 9] 24.11/24.19 f2(I62, I63, I64, I65, I66, I67) -> f5(I66, I67, I68, I65, I66, I69) [I69 = I68 /\ 10 <= I66 /\ I68 = I68] 24.11/24.19 f4(I70, I71, I72, I73, I74, I75) -> f3(I74, I75, I72, I73, I74, I75) [I75 <= 11] 24.11/24.19 f4(I76, I77, I78, I79, I80, I81) -> f1(I80, I81, I78, I79, I80, I81) [12 <= I81] 24.11/24.19 f5(I82, I83, I84, I85, I86, I87) -> f6(I86, I87, I88, I89, I90, I91) [I91 = I89 /\ I90 = I88 /\ I89 = I89 /\ I88 = I88] 24.11/24.19 f3(I92, I93, I94, I95, I96, I97) -> f4(I96, I97, I94, I95, I96, 1 + I97) 24.11/24.19 f1(I98, I99, I100, I101, I102, I103) -> f2(I102, I103, I104, I101, 1 + I102, I105) [I105 = I104 /\ I104 = I104] 24.11/24.19 24.11/24.19 The dependency graph for this problem is: 24.11/24.19 0 -> 1, 2, 3, 4, 5, 6, 7 24.11/24.19 1 -> 8 24.11/24.19 2 -> 9, 10 24.11/24.19 3 -> 11, 12 24.11/24.19 4 -> 24.11/24.19 5 -> 13 24.11/24.19 6 -> 14 24.11/24.19 7 -> 8 24.11/24.19 8 -> 9 24.11/24.19 9 -> 11 24.11/24.19 10 -> 24.11/24.19 11 -> 13 24.11/24.19 12 -> 14 24.11/24.19 13 -> 11, 12 24.11/24.19 14 -> 9, 10 24.11/24.19 Where: 24.11/24.19 0) f9#(x1, x2, x3, x4, x5, x6) -> f8#(x1, x2, x3, x4, x5, x6) 24.11/24.19 1) f8#(I0, I1, I2, I3, I4, I5) -> f7#(I0, I1, I2, I3, I4, I5) 24.11/24.19 2) f8#(I6, I7, I8, I9, I10, I11) -> f2#(I6, I7, I8, I9, I10, I11) 24.11/24.19 3) f8#(I12, I13, I14, I15, I16, I17) -> f4#(I12, I13, I14, I15, I16, I17) 24.11/24.19 4) f8#(I18, I19, I20, I21, I22, I23) -> f5#(I18, I19, I20, I21, I22, I23) 24.11/24.19 5) f8#(I24, I25, I26, I27, I28, I29) -> f3#(I24, I25, I26, I27, I28, I29) 24.11/24.19 6) f8#(I30, I31, I32, I33, I34, I35) -> f1#(I30, I31, I32, I33, I34, I35) 24.11/24.19 7) f8#(I42, I43, I44, I45, I46, I47) -> f7#(I46, I47, rnd3, rnd4, rnd5, rnd6) [rnd6 = rnd4 /\ rnd5 = rnd3 /\ rnd4 = rnd4 /\ rnd3 = rnd3] 24.11/24.19 8) f7#(I48, I49, I50, I51, I52, I53) -> f2#(I52, I53, I54, I51, 0, I55) [I55 = I54 /\ I54 = I54] 24.11/24.19 9) f2#(I56, I57, I58, I59, I60, I61) -> f4#(I60, I61, I58, I59, I60, 3) [I60 <= 9] 24.11/24.19 10) f2#(I62, I63, I64, I65, I66, I67) -> f5#(I66, I67, I68, I65, I66, I69) [I69 = I68 /\ 10 <= I66 /\ I68 = I68] 24.11/24.19 11) f4#(I70, I71, I72, I73, I74, I75) -> f3#(I74, I75, I72, I73, I74, I75) [I75 <= 11] 24.11/24.19 12) f4#(I76, I77, I78, I79, I80, I81) -> f1#(I80, I81, I78, I79, I80, I81) [12 <= I81] 24.11/24.19 13) f3#(I92, I93, I94, I95, I96, I97) -> f4#(I96, I97, I94, I95, I96, 1 + I97) 24.11/24.19 14) f1#(I98, I99, I100, I101, I102, I103) -> f2#(I102, I103, I104, I101, 1 + I102, I105) [I105 = I104 /\ I104 = I104] 24.11/24.19 24.11/24.19 We have the following SCCs. 24.11/24.19 { 9, 11, 12, 13, 14 } 24.11/24.19 24.11/24.19 DP problem for innermost termination. 24.11/24.19 P = 24.11/24.19 f2#(I56, I57, I58, I59, I60, I61) -> f4#(I60, I61, I58, I59, I60, 3) [I60 <= 9] 24.11/24.19 f4#(I70, I71, I72, I73, I74, I75) -> f3#(I74, I75, I72, I73, I74, I75) [I75 <= 11] 24.11/24.19 f4#(I76, I77, I78, I79, I80, I81) -> f1#(I80, I81, I78, I79, I80, I81) [12 <= I81] 24.11/24.19 f3#(I92, I93, I94, I95, I96, I97) -> f4#(I96, I97, I94, I95, I96, 1 + I97) 24.11/24.19 f1#(I98, I99, I100, I101, I102, I103) -> f2#(I102, I103, I104, I101, 1 + I102, I105) [I105 = I104 /\ I104 = I104] 24.11/24.19 R = 24.11/24.19 f9(x1, x2, x3, x4, x5, x6) -> f8(x1, x2, x3, x4, x5, x6) 24.11/24.19 f8(I0, I1, I2, I3, I4, I5) -> f7(I0, I1, I2, I3, I4, I5) 24.11/24.19 f8(I6, I7, I8, I9, I10, I11) -> f2(I6, I7, I8, I9, I10, I11) 24.11/24.19 f8(I12, I13, I14, I15, I16, I17) -> f4(I12, I13, I14, I15, I16, I17) 24.11/24.19 f8(I18, I19, I20, I21, I22, I23) -> f5(I18, I19, I20, I21, I22, I23) 24.11/24.19 f8(I24, I25, I26, I27, I28, I29) -> f3(I24, I25, I26, I27, I28, I29) 24.11/24.19 f8(I30, I31, I32, I33, I34, I35) -> f1(I30, I31, I32, I33, I34, I35) 24.11/24.19 f8(I36, I37, I38, I39, I40, I41) -> f6(I36, I37, I38, I39, I40, I41) 24.11/24.19 f8(I42, I43, I44, I45, I46, I47) -> f7(I46, I47, rnd3, rnd4, rnd5, rnd6) [rnd6 = rnd4 /\ rnd5 = rnd3 /\ rnd4 = rnd4 /\ rnd3 = rnd3] 24.11/24.19 f7(I48, I49, I50, I51, I52, I53) -> f2(I52, I53, I54, I51, 0, I55) [I55 = I54 /\ I54 = I54] 24.11/24.19 f2(I56, I57, I58, I59, I60, I61) -> f4(I60, I61, I58, I59, I60, 3) [I60 <= 9] 24.11/24.19 f2(I62, I63, I64, I65, I66, I67) -> f5(I66, I67, I68, I65, I66, I69) [I69 = I68 /\ 10 <= I66 /\ I68 = I68] 24.11/24.19 f4(I70, I71, I72, I73, I74, I75) -> f3(I74, I75, I72, I73, I74, I75) [I75 <= 11] 24.11/24.19 f4(I76, I77, I78, I79, I80, I81) -> f1(I80, I81, I78, I79, I80, I81) [12 <= I81] 24.11/24.19 f5(I82, I83, I84, I85, I86, I87) -> f6(I86, I87, I88, I89, I90, I91) [I91 = I89 /\ I90 = I88 /\ I89 = I89 /\ I88 = I88] 24.11/24.19 f3(I92, I93, I94, I95, I96, I97) -> f4(I96, I97, I94, I95, I96, 1 + I97) 24.11/24.19 f1(I98, I99, I100, I101, I102, I103) -> f2(I102, I103, I104, I101, 1 + I102, I105) [I105 = I104 /\ I104 = I104] 24.11/24.19 24.11/24.19 We use the extended value criterion with the projection function NU: 24.11/24.19 NU[f1#(x0,x1,x2,x3,x4,x5)] = -x4 + 8 24.11/24.19 NU[f3#(x0,x1,x2,x3,x4,x5)] = -x4 + 8 24.11/24.19 NU[f4#(x0,x1,x2,x3,x4,x5)] = -x4 + 8 24.11/24.19 NU[f2#(x0,x1,x2,x3,x4,x5)] = -x4 + 9 24.11/24.19 24.11/24.19 This gives the following inequalities: 24.11/24.19 I60 <= 9 ==> -I60 + 9 > -I60 + 8 with -I60 + 9 >= 0 24.11/24.19 I75 <= 11 ==> -I74 + 8 >= -I74 + 8 24.11/24.19 12 <= I81 ==> -I80 + 8 >= -I80 + 8 24.11/24.19 ==> -I96 + 8 >= -I96 + 8 24.11/24.19 I105 = I104 /\ I104 = I104 ==> -I102 + 8 >= -(1 + I102) + 9 24.11/24.19 24.11/24.19 We remove all the strictly oriented dependency pairs. 24.11/24.19 24.11/24.19 DP problem for innermost termination. 24.11/24.19 P = 24.11/24.19 f4#(I70, I71, I72, I73, I74, I75) -> f3#(I74, I75, I72, I73, I74, I75) [I75 <= 11] 24.11/24.19 f4#(I76, I77, I78, I79, I80, I81) -> f1#(I80, I81, I78, I79, I80, I81) [12 <= I81] 24.11/24.19 f3#(I92, I93, I94, I95, I96, I97) -> f4#(I96, I97, I94, I95, I96, 1 + I97) 24.11/24.19 f1#(I98, I99, I100, I101, I102, I103) -> f2#(I102, I103, I104, I101, 1 + I102, I105) [I105 = I104 /\ I104 = I104] 24.11/24.19 R = 24.11/24.19 f9(x1, x2, x3, x4, x5, x6) -> f8(x1, x2, x3, x4, x5, x6) 24.11/24.19 f8(I0, I1, I2, I3, I4, I5) -> f7(I0, I1, I2, I3, I4, I5) 24.11/24.19 f8(I6, I7, I8, I9, I10, I11) -> f2(I6, I7, I8, I9, I10, I11) 24.11/24.19 f8(I12, I13, I14, I15, I16, I17) -> f4(I12, I13, I14, I15, I16, I17) 24.11/24.19 f8(I18, I19, I20, I21, I22, I23) -> f5(I18, I19, I20, I21, I22, I23) 24.11/24.19 f8(I24, I25, I26, I27, I28, I29) -> f3(I24, I25, I26, I27, I28, I29) 24.11/24.19 f8(I30, I31, I32, I33, I34, I35) -> f1(I30, I31, I32, I33, I34, I35) 24.11/24.19 f8(I36, I37, I38, I39, I40, I41) -> f6(I36, I37, I38, I39, I40, I41) 24.11/24.19 f8(I42, I43, I44, I45, I46, I47) -> f7(I46, I47, rnd3, rnd4, rnd5, rnd6) [rnd6 = rnd4 /\ rnd5 = rnd3 /\ rnd4 = rnd4 /\ rnd3 = rnd3] 24.11/24.19 f7(I48, I49, I50, I51, I52, I53) -> f2(I52, I53, I54, I51, 0, I55) [I55 = I54 /\ I54 = I54] 24.11/24.19 f2(I56, I57, I58, I59, I60, I61) -> f4(I60, I61, I58, I59, I60, 3) [I60 <= 9] 24.11/24.19 f2(I62, I63, I64, I65, I66, I67) -> f5(I66, I67, I68, I65, I66, I69) [I69 = I68 /\ 10 <= I66 /\ I68 = I68] 24.11/24.19 f4(I70, I71, I72, I73, I74, I75) -> f3(I74, I75, I72, I73, I74, I75) [I75 <= 11] 24.11/24.19 f4(I76, I77, I78, I79, I80, I81) -> f1(I80, I81, I78, I79, I80, I81) [12 <= I81] 24.11/24.19 f5(I82, I83, I84, I85, I86, I87) -> f6(I86, I87, I88, I89, I90, I91) [I91 = I89 /\ I90 = I88 /\ I89 = I89 /\ I88 = I88] 24.11/24.19 f3(I92, I93, I94, I95, I96, I97) -> f4(I96, I97, I94, I95, I96, 1 + I97) 24.11/24.19 f1(I98, I99, I100, I101, I102, I103) -> f2(I102, I103, I104, I101, 1 + I102, I105) [I105 = I104 /\ I104 = I104] 24.11/24.19 24.11/24.19 The dependency graph for this problem is: 24.11/24.19 11 -> 13 24.11/24.19 12 -> 14 24.11/24.19 13 -> 11, 12 24.11/24.19 14 -> 24.11/24.19 Where: 24.11/24.19 11) f4#(I70, I71, I72, I73, I74, I75) -> f3#(I74, I75, I72, I73, I74, I75) [I75 <= 11] 24.11/24.19 12) f4#(I76, I77, I78, I79, I80, I81) -> f1#(I80, I81, I78, I79, I80, I81) [12 <= I81] 24.11/24.19 13) f3#(I92, I93, I94, I95, I96, I97) -> f4#(I96, I97, I94, I95, I96, 1 + I97) 24.11/24.19 14) f1#(I98, I99, I100, I101, I102, I103) -> f2#(I102, I103, I104, I101, 1 + I102, I105) [I105 = I104 /\ I104 = I104] 24.11/24.19 24.11/24.19 We have the following SCCs. 24.11/24.19 { 11, 13 } 24.11/24.19 24.11/24.19 DP problem for innermost termination. 24.11/24.19 P = 24.11/24.19 f4#(I70, I71, I72, I73, I74, I75) -> f3#(I74, I75, I72, I73, I74, I75) [I75 <= 11] 24.11/24.19 f3#(I92, I93, I94, I95, I96, I97) -> f4#(I96, I97, I94, I95, I96, 1 + I97) 24.11/24.19 R = 24.11/24.19 f9(x1, x2, x3, x4, x5, x6) -> f8(x1, x2, x3, x4, x5, x6) 24.11/24.19 f8(I0, I1, I2, I3, I4, I5) -> f7(I0, I1, I2, I3, I4, I5) 24.11/24.19 f8(I6, I7, I8, I9, I10, I11) -> f2(I6, I7, I8, I9, I10, I11) 24.11/24.19 f8(I12, I13, I14, I15, I16, I17) -> f4(I12, I13, I14, I15, I16, I17) 24.11/24.19 f8(I18, I19, I20, I21, I22, I23) -> f5(I18, I19, I20, I21, I22, I23) 24.11/24.19 f8(I24, I25, I26, I27, I28, I29) -> f3(I24, I25, I26, I27, I28, I29) 24.11/24.19 f8(I30, I31, I32, I33, I34, I35) -> f1(I30, I31, I32, I33, I34, I35) 24.11/24.19 f8(I36, I37, I38, I39, I40, I41) -> f6(I36, I37, I38, I39, I40, I41) 24.11/24.19 f8(I42, I43, I44, I45, I46, I47) -> f7(I46, I47, rnd3, rnd4, rnd5, rnd6) [rnd6 = rnd4 /\ rnd5 = rnd3 /\ rnd4 = rnd4 /\ rnd3 = rnd3] 24.11/24.19 f7(I48, I49, I50, I51, I52, I53) -> f2(I52, I53, I54, I51, 0, I55) [I55 = I54 /\ I54 = I54] 24.11/24.19 f2(I56, I57, I58, I59, I60, I61) -> f4(I60, I61, I58, I59, I60, 3) [I60 <= 9] 24.11/24.20 f2(I62, I63, I64, I65, I66, I67) -> f5(I66, I67, I68, I65, I66, I69) [I69 = I68 /\ 10 <= I66 /\ I68 = I68] 24.11/24.20 f4(I70, I71, I72, I73, I74, I75) -> f3(I74, I75, I72, I73, I74, I75) [I75 <= 11] 24.11/24.20 f4(I76, I77, I78, I79, I80, I81) -> f1(I80, I81, I78, I79, I80, I81) [12 <= I81] 24.11/24.20 f5(I82, I83, I84, I85, I86, I87) -> f6(I86, I87, I88, I89, I90, I91) [I91 = I89 /\ I90 = I88 /\ I89 = I89 /\ I88 = I88] 24.11/24.20 f3(I92, I93, I94, I95, I96, I97) -> f4(I96, I97, I94, I95, I96, 1 + I97) 24.11/24.20 f1(I98, I99, I100, I101, I102, I103) -> f2(I102, I103, I104, I101, 1 + I102, I105) [I105 = I104 /\ I104 = I104] 24.11/24.20 24.11/24.20 We use the reverse value criterion with the projection function NU: 24.11/24.20 NU[f3#(z1,z2,z3,z4,z5,z6)] = 11 + -1 * (1 + z6) 24.11/24.20 NU[f4#(z1,z2,z3,z4,z5,z6)] = 11 + -1 * z6 24.11/24.20 24.11/24.20 This gives the following inequalities: 24.11/24.20 I75 <= 11 ==> 11 + -1 * I75 > 11 + -1 * (1 + I75) with 11 + -1 * I75 >= 0 24.11/24.20 ==> 11 + -1 * (1 + I97) >= 11 + -1 * (1 + I97) 24.11/24.20 24.11/24.20 We remove all the strictly oriented dependency pairs. 24.11/24.20 24.11/24.20 DP problem for innermost termination. 24.11/24.20 P = 24.11/24.20 f3#(I92, I93, I94, I95, I96, I97) -> f4#(I96, I97, I94, I95, I96, 1 + I97) 24.11/24.20 R = 24.11/24.20 f9(x1, x2, x3, x4, x5, x6) -> f8(x1, x2, x3, x4, x5, x6) 24.11/24.20 f8(I0, I1, I2, I3, I4, I5) -> f7(I0, I1, I2, I3, I4, I5) 24.11/24.20 f8(I6, I7, I8, I9, I10, I11) -> f2(I6, I7, I8, I9, I10, I11) 24.11/24.20 f8(I12, I13, I14, I15, I16, I17) -> f4(I12, I13, I14, I15, I16, I17) 24.11/24.20 f8(I18, I19, I20, I21, I22, I23) -> f5(I18, I19, I20, I21, I22, I23) 24.11/24.20 f8(I24, I25, I26, I27, I28, I29) -> f3(I24, I25, I26, I27, I28, I29) 24.11/24.20 f8(I30, I31, I32, I33, I34, I35) -> f1(I30, I31, I32, I33, I34, I35) 24.11/24.20 f8(I36, I37, I38, I39, I40, I41) -> f6(I36, I37, I38, I39, I40, I41) 24.11/24.20 f8(I42, I43, I44, I45, I46, I47) -> f7(I46, I47, rnd3, rnd4, rnd5, rnd6) [rnd6 = rnd4 /\ rnd5 = rnd3 /\ rnd4 = rnd4 /\ rnd3 = rnd3] 24.11/24.20 f7(I48, I49, I50, I51, I52, I53) -> f2(I52, I53, I54, I51, 0, I55) [I55 = I54 /\ I54 = I54] 24.11/24.20 f2(I56, I57, I58, I59, I60, I61) -> f4(I60, I61, I58, I59, I60, 3) [I60 <= 9] 24.11/24.20 f2(I62, I63, I64, I65, I66, I67) -> f5(I66, I67, I68, I65, I66, I69) [I69 = I68 /\ 10 <= I66 /\ I68 = I68] 24.11/24.20 f4(I70, I71, I72, I73, I74, I75) -> f3(I74, I75, I72, I73, I74, I75) [I75 <= 11] 24.11/24.20 f4(I76, I77, I78, I79, I80, I81) -> f1(I80, I81, I78, I79, I80, I81) [12 <= I81] 24.11/24.20 f5(I82, I83, I84, I85, I86, I87) -> f6(I86, I87, I88, I89, I90, I91) [I91 = I89 /\ I90 = I88 /\ I89 = I89 /\ I88 = I88] 24.11/24.20 f3(I92, I93, I94, I95, I96, I97) -> f4(I96, I97, I94, I95, I96, 1 + I97) 24.11/24.20 f1(I98, I99, I100, I101, I102, I103) -> f2(I102, I103, I104, I101, 1 + I102, I105) [I105 = I104 /\ I104 = I104] 24.11/24.20 24.11/24.20 The dependency graph for this problem is: 24.11/24.20 13 -> 24.11/24.20 Where: 24.11/24.20 13) f3#(I92, I93, I94, I95, I96, I97) -> f4#(I96, I97, I94, I95, I96, 1 + I97) 24.11/24.20 24.11/24.20 We have the following SCCs. 24.11/24.20 24.11/27.17 EOF