27.09/26.66 YES 27.09/26.66 27.09/26.66 DP problem for innermost termination. 27.09/26.66 P = 27.09/26.66 f19#(x1, x2, x3, x4) -> f18#(x1, x2, x3, x4) 27.09/26.66 f18#(I0, I1, I2, I3) -> f7#(I0, I1, 0, I3) [y1 = 0] 27.09/26.66 f3#(I4, I5, I6, I7) -> f17#(I4, I5, I6, I7) 27.09/26.66 f3#(I8, I9, I10, I11) -> f13#(I8, I9, I10, I11) 27.09/26.66 f3#(I12, I13, I14, I15) -> f17#(I12, I13, I14, I15) 27.09/26.66 f17#(I16, I17, I18, I19) -> f16#(I16, I17, I18, I19) 27.09/26.66 f16#(I20, I21, I22, I23) -> f15#(I20, I21, I22, I23) 27.09/26.66 f16#(I24, I25, I26, I27) -> f14#(I24, I25, I26, I27) 27.09/26.66 f16#(I28, I29, I30, I31) -> f14#(I28, I29, I30, I31) 27.09/26.66 f15#(I32, I33, I34, I35) -> f13#(I32, I33, I34, I35) 27.09/26.66 f14#(I36, I37, I38, I39) -> f15#(I36, I37, I38, I39) 27.09/26.66 f2#(I40, I41, I42, I43) -> f11#(I40, I41, I42, I43) 27.09/26.66 f13#(I44, I45, I46, I47) -> f7#(I44, I45, 1 + I46, I47) 27.09/26.66 f11#(I48, I49, I50, I51) -> f10#(I48, I49, I50, I51) [1 + I50 <= I48] 27.09/26.66 f10#(I56, I57, I58, I59) -> f9#(I56, I57, I58, I59) 27.09/26.66 f10#(I60, I61, I62, I63) -> f4#(I60, I61, I62, I63) 27.09/26.66 f10#(I64, I65, I66, I67) -> f9#(I64, I65, I66, I67) 27.09/26.66 f9#(I68, I69, I70, I71) -> f8#(I68, I69, I70, rnd4) [rnd4 = rnd4] 27.09/26.66 f8#(I72, I73, I74, I75) -> f6#(I72, I73, I74, I75) 27.09/26.66 f8#(I76, I77, I78, I79) -> f5#(I76, I77, I78, I79) 27.09/26.66 f8#(I80, I81, I82, I83) -> f5#(I80, I81, I82, I83) 27.09/26.66 f7#(I84, I85, I86, I87) -> f1#(I84, I85, I86, I87) 27.09/26.66 f6#(I88, I89, I90, I91) -> f4#(I88, I89, I90, I91) 27.09/26.66 f5#(I92, I93, I94, I95) -> f6#(I92, I93, I94, I95) 27.09/26.66 f4#(I96, I97, I98, I99) -> f2#(I96, I97, 1 + I98, I99) 27.09/26.66 f1#(I100, I101, I102, I103) -> f3#(I100, I102, I102, I103) [1 + I102 <= I100] 27.09/26.66 f1#(I104, I105, I106, I107) -> f2#(I104, I105, 0, I107) [I104 <= I106] 27.09/26.66 R = 27.09/26.66 f19(x1, x2, x3, x4) -> f18(x1, x2, x3, x4) 27.09/26.66 f18(I0, I1, I2, I3) -> f7(I0, I1, 0, I3) [y1 = 0] 27.09/26.66 f3(I4, I5, I6, I7) -> f17(I4, I5, I6, I7) 27.09/26.66 f3(I8, I9, I10, I11) -> f13(I8, I9, I10, I11) 27.09/26.66 f3(I12, I13, I14, I15) -> f17(I12, I13, I14, I15) 27.09/26.66 f17(I16, I17, I18, I19) -> f16(I16, I17, I18, I19) 27.09/26.66 f16(I20, I21, I22, I23) -> f15(I20, I21, I22, I23) 27.09/26.66 f16(I24, I25, I26, I27) -> f14(I24, I25, I26, I27) 27.09/26.66 f16(I28, I29, I30, I31) -> f14(I28, I29, I30, I31) 27.09/26.66 f15(I32, I33, I34, I35) -> f13(I32, I33, I34, I35) 27.09/26.66 f14(I36, I37, I38, I39) -> f15(I36, I37, I38, I39) 27.09/26.66 f2(I40, I41, I42, I43) -> f11(I40, I41, I42, I43) 27.09/26.66 f13(I44, I45, I46, I47) -> f7(I44, I45, 1 + I46, I47) 27.09/26.66 f11(I48, I49, I50, I51) -> f10(I48, I49, I50, I51) [1 + I50 <= I48] 27.09/26.66 f11(I52, I53, I54, I55) -> f12(I52, I53, I54, I55) [I52 <= I54] 27.09/26.66 f10(I56, I57, I58, I59) -> f9(I56, I57, I58, I59) 27.09/26.66 f10(I60, I61, I62, I63) -> f4(I60, I61, I62, I63) 27.09/26.66 f10(I64, I65, I66, I67) -> f9(I64, I65, I66, I67) 27.09/26.66 f9(I68, I69, I70, I71) -> f8(I68, I69, I70, rnd4) [rnd4 = rnd4] 27.09/26.66 f8(I72, I73, I74, I75) -> f6(I72, I73, I74, I75) 27.09/26.66 f8(I76, I77, I78, I79) -> f5(I76, I77, I78, I79) 27.09/26.66 f8(I80, I81, I82, I83) -> f5(I80, I81, I82, I83) 27.09/26.66 f7(I84, I85, I86, I87) -> f1(I84, I85, I86, I87) 27.09/26.66 f6(I88, I89, I90, I91) -> f4(I88, I89, I90, I91) 27.09/26.66 f5(I92, I93, I94, I95) -> f6(I92, I93, I94, I95) 27.09/26.66 f4(I96, I97, I98, I99) -> f2(I96, I97, 1 + I98, I99) 27.09/26.66 f1(I100, I101, I102, I103) -> f3(I100, I102, I102, I103) [1 + I102 <= I100] 27.09/26.66 f1(I104, I105, I106, I107) -> f2(I104, I105, 0, I107) [I104 <= I106] 27.09/26.66 27.09/26.66 The dependency graph for this problem is: 27.09/26.66 0 -> 1 27.09/26.66 1 -> 21 27.09/26.66 2 -> 5 27.09/26.66 3 -> 12 27.09/26.66 4 -> 5 27.09/26.66 5 -> 6, 7, 8 27.09/26.66 6 -> 9 27.09/26.66 7 -> 10 27.09/26.66 8 -> 10 27.09/26.66 9 -> 12 27.09/26.66 10 -> 9 27.09/26.66 11 -> 13 27.09/26.66 12 -> 21 27.09/26.66 13 -> 14, 15, 16 27.09/26.66 14 -> 17 27.09/26.66 15 -> 24 27.09/26.66 16 -> 17 27.09/26.66 17 -> 18, 19, 20 27.09/26.66 18 -> 22 27.09/26.66 19 -> 23 27.09/26.66 20 -> 23 27.09/26.66 21 -> 25, 26 27.09/26.66 22 -> 24 27.09/26.66 23 -> 22 27.09/26.66 24 -> 11 27.09/26.66 25 -> 2, 3, 4 27.09/26.66 26 -> 11 27.09/26.66 Where: 27.09/26.66 0) f19#(x1, x2, x3, x4) -> f18#(x1, x2, x3, x4) 27.09/26.66 1) f18#(I0, I1, I2, I3) -> f7#(I0, I1, 0, I3) [y1 = 0] 27.09/26.66 2) f3#(I4, I5, I6, I7) -> f17#(I4, I5, I6, I7) 27.09/26.66 3) f3#(I8, I9, I10, I11) -> f13#(I8, I9, I10, I11) 27.09/26.66 4) f3#(I12, I13, I14, I15) -> f17#(I12, I13, I14, I15) 27.09/26.66 5) f17#(I16, I17, I18, I19) -> f16#(I16, I17, I18, I19) 27.09/26.66 6) f16#(I20, I21, I22, I23) -> f15#(I20, I21, I22, I23) 27.09/26.66 7) f16#(I24, I25, I26, I27) -> f14#(I24, I25, I26, I27) 27.09/26.66 8) f16#(I28, I29, I30, I31) -> f14#(I28, I29, I30, I31) 27.09/26.66 9) f15#(I32, I33, I34, I35) -> f13#(I32, I33, I34, I35) 27.09/26.66 10) f14#(I36, I37, I38, I39) -> f15#(I36, I37, I38, I39) 27.09/26.66 11) f2#(I40, I41, I42, I43) -> f11#(I40, I41, I42, I43) 27.09/26.66 12) f13#(I44, I45, I46, I47) -> f7#(I44, I45, 1 + I46, I47) 27.09/26.66 13) f11#(I48, I49, I50, I51) -> f10#(I48, I49, I50, I51) [1 + I50 <= I48] 27.09/26.66 14) f10#(I56, I57, I58, I59) -> f9#(I56, I57, I58, I59) 27.09/26.66 15) f10#(I60, I61, I62, I63) -> f4#(I60, I61, I62, I63) 27.09/26.66 16) f10#(I64, I65, I66, I67) -> f9#(I64, I65, I66, I67) 27.09/26.66 17) f9#(I68, I69, I70, I71) -> f8#(I68, I69, I70, rnd4) [rnd4 = rnd4] 27.09/26.66 18) f8#(I72, I73, I74, I75) -> f6#(I72, I73, I74, I75) 27.09/26.66 19) f8#(I76, I77, I78, I79) -> f5#(I76, I77, I78, I79) 27.09/26.66 20) f8#(I80, I81, I82, I83) -> f5#(I80, I81, I82, I83) 27.09/26.66 21) f7#(I84, I85, I86, I87) -> f1#(I84, I85, I86, I87) 27.09/26.66 22) f6#(I88, I89, I90, I91) -> f4#(I88, I89, I90, I91) 27.09/26.66 23) f5#(I92, I93, I94, I95) -> f6#(I92, I93, I94, I95) 27.09/26.66 24) f4#(I96, I97, I98, I99) -> f2#(I96, I97, 1 + I98, I99) 27.09/26.66 25) f1#(I100, I101, I102, I103) -> f3#(I100, I102, I102, I103) [1 + I102 <= I100] 27.09/26.66 26) f1#(I104, I105, I106, I107) -> f2#(I104, I105, 0, I107) [I104 <= I106] 27.09/26.66 27.09/26.66 We have the following SCCs. 27.09/26.66 { 2, 3, 4, 5, 6, 7, 8, 9, 10, 12, 21, 25 } 27.09/26.66 { 11, 13, 14, 15, 16, 17, 18, 19, 20, 22, 23, 24 } 27.09/26.66 27.09/26.66 DP problem for innermost termination. 27.09/26.66 P = 27.09/26.66 f2#(I40, I41, I42, I43) -> f11#(I40, I41, I42, I43) 27.09/26.66 f11#(I48, I49, I50, I51) -> f10#(I48, I49, I50, I51) [1 + I50 <= I48] 27.09/26.66 f10#(I56, I57, I58, I59) -> f9#(I56, I57, I58, I59) 27.09/26.66 f10#(I60, I61, I62, I63) -> f4#(I60, I61, I62, I63) 27.09/26.66 f10#(I64, I65, I66, I67) -> f9#(I64, I65, I66, I67) 27.09/26.66 f9#(I68, I69, I70, I71) -> f8#(I68, I69, I70, rnd4) [rnd4 = rnd4] 27.09/26.66 f8#(I72, I73, I74, I75) -> f6#(I72, I73, I74, I75) 27.09/26.66 f8#(I76, I77, I78, I79) -> f5#(I76, I77, I78, I79) 27.09/26.66 f8#(I80, I81, I82, I83) -> f5#(I80, I81, I82, I83) 27.09/26.66 f6#(I88, I89, I90, I91) -> f4#(I88, I89, I90, I91) 27.09/26.66 f5#(I92, I93, I94, I95) -> f6#(I92, I93, I94, I95) 27.09/26.66 f4#(I96, I97, I98, I99) -> f2#(I96, I97, 1 + I98, I99) 27.09/26.66 R = 27.09/26.66 f19(x1, x2, x3, x4) -> f18(x1, x2, x3, x4) 27.09/26.66 f18(I0, I1, I2, I3) -> f7(I0, I1, 0, I3) [y1 = 0] 27.09/26.66 f3(I4, I5, I6, I7) -> f17(I4, I5, I6, I7) 27.09/26.66 f3(I8, I9, I10, I11) -> f13(I8, I9, I10, I11) 27.09/26.66 f3(I12, I13, I14, I15) -> f17(I12, I13, I14, I15) 27.09/26.66 f17(I16, I17, I18, I19) -> f16(I16, I17, I18, I19) 27.09/26.66 f16(I20, I21, I22, I23) -> f15(I20, I21, I22, I23) 27.09/26.66 f16(I24, I25, I26, I27) -> f14(I24, I25, I26, I27) 27.09/26.66 f16(I28, I29, I30, I31) -> f14(I28, I29, I30, I31) 27.09/26.66 f15(I32, I33, I34, I35) -> f13(I32, I33, I34, I35) 27.09/26.66 f14(I36, I37, I38, I39) -> f15(I36, I37, I38, I39) 27.09/26.66 f2(I40, I41, I42, I43) -> f11(I40, I41, I42, I43) 27.09/26.66 f13(I44, I45, I46, I47) -> f7(I44, I45, 1 + I46, I47) 27.09/26.66 f11(I48, I49, I50, I51) -> f10(I48, I49, I50, I51) [1 + I50 <= I48] 27.09/26.66 f11(I52, I53, I54, I55) -> f12(I52, I53, I54, I55) [I52 <= I54] 27.09/26.66 f10(I56, I57, I58, I59) -> f9(I56, I57, I58, I59) 27.09/26.66 f10(I60, I61, I62, I63) -> f4(I60, I61, I62, I63) 27.09/26.66 f10(I64, I65, I66, I67) -> f9(I64, I65, I66, I67) 27.09/26.66 f9(I68, I69, I70, I71) -> f8(I68, I69, I70, rnd4) [rnd4 = rnd4] 27.09/26.66 f8(I72, I73, I74, I75) -> f6(I72, I73, I74, I75) 27.09/26.66 f8(I76, I77, I78, I79) -> f5(I76, I77, I78, I79) 27.09/26.66 f8(I80, I81, I82, I83) -> f5(I80, I81, I82, I83) 27.09/26.66 f7(I84, I85, I86, I87) -> f1(I84, I85, I86, I87) 27.09/26.66 f6(I88, I89, I90, I91) -> f4(I88, I89, I90, I91) 27.09/26.66 f5(I92, I93, I94, I95) -> f6(I92, I93, I94, I95) 27.09/26.66 f4(I96, I97, I98, I99) -> f2(I96, I97, 1 + I98, I99) 27.09/26.66 f1(I100, I101, I102, I103) -> f3(I100, I102, I102, I103) [1 + I102 <= I100] 27.09/26.66 f1(I104, I105, I106, I107) -> f2(I104, I105, 0, I107) [I104 <= I106] 27.09/26.66 27.09/26.66 We use the extended value criterion with the projection function NU: 27.09/26.66 NU[f5#(x0,x1,x2,x3)] = x0 - x2 - 2 27.09/26.66 NU[f6#(x0,x1,x2,x3)] = x0 - x2 - 2 27.09/26.66 NU[f8#(x0,x1,x2,x3)] = x0 - x2 - 2 27.09/26.66 NU[f4#(x0,x1,x2,x3)] = x0 - x2 - 2 27.09/26.66 NU[f9#(x0,x1,x2,x3)] = x0 - x2 - 2 27.09/26.66 NU[f10#(x0,x1,x2,x3)] = x0 - x2 - 2 27.09/26.66 NU[f11#(x0,x1,x2,x3)] = x0 - x2 - 1 27.09/26.66 NU[f2#(x0,x1,x2,x3)] = x0 - x2 - 1 27.09/26.66 27.09/26.66 This gives the following inequalities: 27.09/26.66 ==> I40 - I42 - 1 >= I40 - I42 - 1 27.09/26.66 1 + I50 <= I48 ==> I48 - I50 - 1 > I48 - I50 - 2 with I48 - I50 - 1 >= 0 27.09/26.66 ==> I56 - I58 - 2 >= I56 - I58 - 2 27.09/26.66 ==> I60 - I62 - 2 >= I60 - I62 - 2 27.09/26.66 ==> I64 - I66 - 2 >= I64 - I66 - 2 27.09/26.66 rnd4 = rnd4 ==> I68 - I70 - 2 >= I68 - I70 - 2 27.09/26.66 ==> I72 - I74 - 2 >= I72 - I74 - 2 27.09/26.66 ==> I76 - I78 - 2 >= I76 - I78 - 2 27.09/26.66 ==> I80 - I82 - 2 >= I80 - I82 - 2 27.09/26.66 ==> I88 - I90 - 2 >= I88 - I90 - 2 27.09/26.66 ==> I92 - I94 - 2 >= I92 - I94 - 2 27.09/26.66 ==> I96 - I98 - 2 >= I96 - (1 + I98) - 1 27.09/26.66 27.09/26.66 We remove all the strictly oriented dependency pairs. 27.09/26.66 27.09/26.66 DP problem for innermost termination. 27.09/26.66 P = 27.09/26.66 f2#(I40, I41, I42, I43) -> f11#(I40, I41, I42, I43) 27.09/26.66 f10#(I56, I57, I58, I59) -> f9#(I56, I57, I58, I59) 27.09/26.66 f10#(I60, I61, I62, I63) -> f4#(I60, I61, I62, I63) 27.09/26.66 f10#(I64, I65, I66, I67) -> f9#(I64, I65, I66, I67) 27.09/26.66 f9#(I68, I69, I70, I71) -> f8#(I68, I69, I70, rnd4) [rnd4 = rnd4] 27.09/26.66 f8#(I72, I73, I74, I75) -> f6#(I72, I73, I74, I75) 27.09/26.66 f8#(I76, I77, I78, I79) -> f5#(I76, I77, I78, I79) 27.09/26.66 f8#(I80, I81, I82, I83) -> f5#(I80, I81, I82, I83) 27.09/26.66 f6#(I88, I89, I90, I91) -> f4#(I88, I89, I90, I91) 27.09/26.66 f5#(I92, I93, I94, I95) -> f6#(I92, I93, I94, I95) 27.09/26.66 f4#(I96, I97, I98, I99) -> f2#(I96, I97, 1 + I98, I99) 27.09/26.66 R = 27.09/26.66 f19(x1, x2, x3, x4) -> f18(x1, x2, x3, x4) 27.09/26.66 f18(I0, I1, I2, I3) -> f7(I0, I1, 0, I3) [y1 = 0] 27.09/26.66 f3(I4, I5, I6, I7) -> f17(I4, I5, I6, I7) 27.09/26.66 f3(I8, I9, I10, I11) -> f13(I8, I9, I10, I11) 27.09/26.66 f3(I12, I13, I14, I15) -> f17(I12, I13, I14, I15) 27.09/26.66 f17(I16, I17, I18, I19) -> f16(I16, I17, I18, I19) 27.09/26.66 f16(I20, I21, I22, I23) -> f15(I20, I21, I22, I23) 27.09/26.66 f16(I24, I25, I26, I27) -> f14(I24, I25, I26, I27) 27.09/26.66 f16(I28, I29, I30, I31) -> f14(I28, I29, I30, I31) 27.09/26.66 f15(I32, I33, I34, I35) -> f13(I32, I33, I34, I35) 27.09/26.66 f14(I36, I37, I38, I39) -> f15(I36, I37, I38, I39) 27.09/26.66 f2(I40, I41, I42, I43) -> f11(I40, I41, I42, I43) 27.09/26.66 f13(I44, I45, I46, I47) -> f7(I44, I45, 1 + I46, I47) 27.09/26.66 f11(I48, I49, I50, I51) -> f10(I48, I49, I50, I51) [1 + I50 <= I48] 27.09/26.66 f11(I52, I53, I54, I55) -> f12(I52, I53, I54, I55) [I52 <= I54] 27.09/26.66 f10(I56, I57, I58, I59) -> f9(I56, I57, I58, I59) 27.09/26.66 f10(I60, I61, I62, I63) -> f4(I60, I61, I62, I63) 27.09/26.66 f10(I64, I65, I66, I67) -> f9(I64, I65, I66, I67) 27.09/26.66 f9(I68, I69, I70, I71) -> f8(I68, I69, I70, rnd4) [rnd4 = rnd4] 27.09/26.66 f8(I72, I73, I74, I75) -> f6(I72, I73, I74, I75) 27.09/26.66 f8(I76, I77, I78, I79) -> f5(I76, I77, I78, I79) 27.09/26.66 f8(I80, I81, I82, I83) -> f5(I80, I81, I82, I83) 27.09/26.66 f7(I84, I85, I86, I87) -> f1(I84, I85, I86, I87) 27.09/26.66 f6(I88, I89, I90, I91) -> f4(I88, I89, I90, I91) 27.09/26.66 f5(I92, I93, I94, I95) -> f6(I92, I93, I94, I95) 27.09/26.66 f4(I96, I97, I98, I99) -> f2(I96, I97, 1 + I98, I99) 27.09/26.66 f1(I100, I101, I102, I103) -> f3(I100, I102, I102, I103) [1 + I102 <= I100] 27.09/26.66 f1(I104, I105, I106, I107) -> f2(I104, I105, 0, I107) [I104 <= I106] 27.09/26.66 27.09/26.66 The dependency graph for this problem is: 27.09/26.66 11 -> 27.09/26.66 14 -> 17 27.09/26.66 15 -> 24 27.09/26.66 16 -> 17 27.09/26.66 17 -> 18, 19, 20 27.09/26.66 18 -> 22 27.09/26.66 19 -> 23 27.09/26.66 20 -> 23 27.09/26.66 22 -> 24 27.09/26.66 23 -> 22 27.09/26.66 24 -> 11 27.09/26.66 Where: 27.09/26.66 11) f2#(I40, I41, I42, I43) -> f11#(I40, I41, I42, I43) 27.09/26.66 14) f10#(I56, I57, I58, I59) -> f9#(I56, I57, I58, I59) 27.09/26.66 15) f10#(I60, I61, I62, I63) -> f4#(I60, I61, I62, I63) 27.09/26.66 16) f10#(I64, I65, I66, I67) -> f9#(I64, I65, I66, I67) 27.09/26.66 17) f9#(I68, I69, I70, I71) -> f8#(I68, I69, I70, rnd4) [rnd4 = rnd4] 27.09/26.66 18) f8#(I72, I73, I74, I75) -> f6#(I72, I73, I74, I75) 27.09/26.66 19) f8#(I76, I77, I78, I79) -> f5#(I76, I77, I78, I79) 27.09/26.66 20) f8#(I80, I81, I82, I83) -> f5#(I80, I81, I82, I83) 27.09/26.66 22) f6#(I88, I89, I90, I91) -> f4#(I88, I89, I90, I91) 27.09/26.66 23) f5#(I92, I93, I94, I95) -> f6#(I92, I93, I94, I95) 27.09/26.66 24) f4#(I96, I97, I98, I99) -> f2#(I96, I97, 1 + I98, I99) 27.09/26.66 27.09/26.66 We have the following SCCs. 27.09/26.66 27.09/26.66 27.09/26.66 DP problem for innermost termination. 27.09/26.66 P = 27.09/26.66 f3#(I4, I5, I6, I7) -> f17#(I4, I5, I6, I7) 27.09/26.66 f3#(I8, I9, I10, I11) -> f13#(I8, I9, I10, I11) 27.09/26.66 f3#(I12, I13, I14, I15) -> f17#(I12, I13, I14, I15) 27.09/26.66 f17#(I16, I17, I18, I19) -> f16#(I16, I17, I18, I19) 27.09/26.66 f16#(I20, I21, I22, I23) -> f15#(I20, I21, I22, I23) 27.09/26.66 f16#(I24, I25, I26, I27) -> f14#(I24, I25, I26, I27) 27.09/26.66 f16#(I28, I29, I30, I31) -> f14#(I28, I29, I30, I31) 27.09/26.66 f15#(I32, I33, I34, I35) -> f13#(I32, I33, I34, I35) 27.09/26.66 f14#(I36, I37, I38, I39) -> f15#(I36, I37, I38, I39) 27.09/26.66 f13#(I44, I45, I46, I47) -> f7#(I44, I45, 1 + I46, I47) 27.09/26.66 f7#(I84, I85, I86, I87) -> f1#(I84, I85, I86, I87) 27.09/26.66 f1#(I100, I101, I102, I103) -> f3#(I100, I102, I102, I103) [1 + I102 <= I100] 27.09/26.66 R = 27.09/26.66 f19(x1, x2, x3, x4) -> f18(x1, x2, x3, x4) 27.09/26.66 f18(I0, I1, I2, I3) -> f7(I0, I1, 0, I3) [y1 = 0] 27.09/26.66 f3(I4, I5, I6, I7) -> f17(I4, I5, I6, I7) 27.09/26.66 f3(I8, I9, I10, I11) -> f13(I8, I9, I10, I11) 27.09/26.66 f3(I12, I13, I14, I15) -> f17(I12, I13, I14, I15) 27.09/26.66 f17(I16, I17, I18, I19) -> f16(I16, I17, I18, I19) 27.09/26.66 f16(I20, I21, I22, I23) -> f15(I20, I21, I22, I23) 27.09/26.66 f16(I24, I25, I26, I27) -> f14(I24, I25, I26, I27) 27.09/26.66 f16(I28, I29, I30, I31) -> f14(I28, I29, I30, I31) 27.09/26.66 f15(I32, I33, I34, I35) -> f13(I32, I33, I34, I35) 27.09/26.66 f14(I36, I37, I38, I39) -> f15(I36, I37, I38, I39) 27.09/26.66 f2(I40, I41, I42, I43) -> f11(I40, I41, I42, I43) 27.09/26.66 f13(I44, I45, I46, I47) -> f7(I44, I45, 1 + I46, I47) 27.09/26.66 f11(I48, I49, I50, I51) -> f10(I48, I49, I50, I51) [1 + I50 <= I48] 27.09/26.66 f11(I52, I53, I54, I55) -> f12(I52, I53, I54, I55) [I52 <= I54] 27.09/26.66 f10(I56, I57, I58, I59) -> f9(I56, I57, I58, I59) 27.09/26.66 f10(I60, I61, I62, I63) -> f4(I60, I61, I62, I63) 27.09/26.66 f10(I64, I65, I66, I67) -> f9(I64, I65, I66, I67) 27.09/26.66 f9(I68, I69, I70, I71) -> f8(I68, I69, I70, rnd4) [rnd4 = rnd4] 27.09/26.66 f8(I72, I73, I74, I75) -> f6(I72, I73, I74, I75) 27.09/26.66 f8(I76, I77, I78, I79) -> f5(I76, I77, I78, I79) 27.09/26.66 f8(I80, I81, I82, I83) -> f5(I80, I81, I82, I83) 27.09/26.66 f7(I84, I85, I86, I87) -> f1(I84, I85, I86, I87) 27.09/26.66 f6(I88, I89, I90, I91) -> f4(I88, I89, I90, I91) 27.09/26.66 f5(I92, I93, I94, I95) -> f6(I92, I93, I94, I95) 27.09/26.66 f4(I96, I97, I98, I99) -> f2(I96, I97, 1 + I98, I99) 27.09/26.66 f1(I100, I101, I102, I103) -> f3(I100, I102, I102, I103) [1 + I102 <= I100] 27.09/26.66 f1(I104, I105, I106, I107) -> f2(I104, I105, 0, I107) [I104 <= I106] 27.09/26.66 27.09/26.66 We use the extended value criterion with the projection function NU: 27.09/26.66 NU[f1#(x0,x1,x2,x3)] = x0 - x2 - 1 27.09/26.66 NU[f7#(x0,x1,x2,x3)] = x0 - x2 - 1 27.09/26.66 NU[f14#(x0,x1,x2,x3)] = x0 - x2 - 2 27.09/26.66 NU[f15#(x0,x1,x2,x3)] = x0 - x2 - 2 27.09/26.66 NU[f16#(x0,x1,x2,x3)] = x0 - x2 - 2 27.09/26.66 NU[f13#(x0,x1,x2,x3)] = x0 - x2 - 2 27.09/26.66 NU[f17#(x0,x1,x2,x3)] = x0 - x2 - 2 27.09/26.66 NU[f3#(x0,x1,x2,x3)] = x0 - x2 - 2 27.09/26.66 27.09/26.66 This gives the following inequalities: 27.09/26.66 ==> I4 - I6 - 2 >= I4 - I6 - 2 27.09/26.66 ==> I8 - I10 - 2 >= I8 - I10 - 2 27.09/26.66 ==> I12 - I14 - 2 >= I12 - I14 - 2 27.09/26.66 ==> I16 - I18 - 2 >= I16 - I18 - 2 27.09/26.66 ==> I20 - I22 - 2 >= I20 - I22 - 2 27.09/26.66 ==> I24 - I26 - 2 >= I24 - I26 - 2 27.09/26.66 ==> I28 - I30 - 2 >= I28 - I30 - 2 27.09/26.66 ==> I32 - I34 - 2 >= I32 - I34 - 2 27.09/26.66 ==> I36 - I38 - 2 >= I36 - I38 - 2 27.09/26.66 ==> I44 - I46 - 2 >= I44 - (1 + I46) - 1 27.09/26.66 ==> I84 - I86 - 1 >= I84 - I86 - 1 27.09/26.66 1 + I102 <= I100 ==> I100 - I102 - 1 > I100 - I102 - 2 with I100 - I102 - 1 >= 0 27.09/26.66 27.09/26.66 We remove all the strictly oriented dependency pairs. 27.09/26.66 27.09/26.66 DP problem for innermost termination. 27.09/26.66 P = 27.09/26.66 f3#(I4, I5, I6, I7) -> f17#(I4, I5, I6, I7) 27.09/26.66 f3#(I8, I9, I10, I11) -> f13#(I8, I9, I10, I11) 27.09/26.66 f3#(I12, I13, I14, I15) -> f17#(I12, I13, I14, I15) 27.09/26.66 f17#(I16, I17, I18, I19) -> f16#(I16, I17, I18, I19) 27.09/26.66 f16#(I20, I21, I22, I23) -> f15#(I20, I21, I22, I23) 27.09/26.66 f16#(I24, I25, I26, I27) -> f14#(I24, I25, I26, I27) 27.09/26.66 f16#(I28, I29, I30, I31) -> f14#(I28, I29, I30, I31) 27.09/26.66 f15#(I32, I33, I34, I35) -> f13#(I32, I33, I34, I35) 27.09/26.66 f14#(I36, I37, I38, I39) -> f15#(I36, I37, I38, I39) 27.09/26.66 f13#(I44, I45, I46, I47) -> f7#(I44, I45, 1 + I46, I47) 27.09/26.66 f7#(I84, I85, I86, I87) -> f1#(I84, I85, I86, I87) 27.09/26.66 R = 27.09/26.66 f19(x1, x2, x3, x4) -> f18(x1, x2, x3, x4) 27.09/26.66 f18(I0, I1, I2, I3) -> f7(I0, I1, 0, I3) [y1 = 0] 27.09/26.66 f3(I4, I5, I6, I7) -> f17(I4, I5, I6, I7) 27.09/26.66 f3(I8, I9, I10, I11) -> f13(I8, I9, I10, I11) 27.09/26.66 f3(I12, I13, I14, I15) -> f17(I12, I13, I14, I15) 27.09/26.66 f17(I16, I17, I18, I19) -> f16(I16, I17, I18, I19) 27.09/26.66 f16(I20, I21, I22, I23) -> f15(I20, I21, I22, I23) 27.09/26.66 f16(I24, I25, I26, I27) -> f14(I24, I25, I26, I27) 27.09/26.66 f16(I28, I29, I30, I31) -> f14(I28, I29, I30, I31) 27.09/26.66 f15(I32, I33, I34, I35) -> f13(I32, I33, I34, I35) 27.09/26.66 f14(I36, I37, I38, I39) -> f15(I36, I37, I38, I39) 27.09/26.66 f2(I40, I41, I42, I43) -> f11(I40, I41, I42, I43) 27.09/26.66 f13(I44, I45, I46, I47) -> f7(I44, I45, 1 + I46, I47) 27.09/26.66 f11(I48, I49, I50, I51) -> f10(I48, I49, I50, I51) [1 + I50 <= I48] 27.09/26.66 f11(I52, I53, I54, I55) -> f12(I52, I53, I54, I55) [I52 <= I54] 27.09/26.66 f10(I56, I57, I58, I59) -> f9(I56, I57, I58, I59) 27.09/26.66 f10(I60, I61, I62, I63) -> f4(I60, I61, I62, I63) 27.09/26.66 f10(I64, I65, I66, I67) -> f9(I64, I65, I66, I67) 27.09/26.66 f9(I68, I69, I70, I71) -> f8(I68, I69, I70, rnd4) [rnd4 = rnd4] 27.09/26.66 f8(I72, I73, I74, I75) -> f6(I72, I73, I74, I75) 27.09/26.66 f8(I76, I77, I78, I79) -> f5(I76, I77, I78, I79) 27.09/26.66 f8(I80, I81, I82, I83) -> f5(I80, I81, I82, I83) 27.09/26.66 f7(I84, I85, I86, I87) -> f1(I84, I85, I86, I87) 27.09/26.66 f6(I88, I89, I90, I91) -> f4(I88, I89, I90, I91) 27.09/26.66 f5(I92, I93, I94, I95) -> f6(I92, I93, I94, I95) 27.09/26.66 f4(I96, I97, I98, I99) -> f2(I96, I97, 1 + I98, I99) 27.09/26.66 f1(I100, I101, I102, I103) -> f3(I100, I102, I102, I103) [1 + I102 <= I100] 27.09/26.66 f1(I104, I105, I106, I107) -> f2(I104, I105, 0, I107) [I104 <= I106] 27.09/26.66 27.09/26.66 The dependency graph for this problem is: 27.09/26.66 2 -> 5 27.09/26.66 3 -> 12 27.09/26.66 4 -> 5 27.09/26.66 5 -> 6, 7, 8 27.09/26.66 6 -> 9 27.09/26.66 7 -> 10 27.09/26.66 8 -> 10 27.09/26.66 9 -> 12 27.09/26.66 10 -> 9 27.09/26.66 12 -> 21 27.09/26.66 21 -> 27.09/26.66 Where: 27.09/26.66 2) f3#(I4, I5, I6, I7) -> f17#(I4, I5, I6, I7) 27.09/26.66 3) f3#(I8, I9, I10, I11) -> f13#(I8, I9, I10, I11) 27.09/26.66 4) f3#(I12, I13, I14, I15) -> f17#(I12, I13, I14, I15) 27.09/26.66 5) f17#(I16, I17, I18, I19) -> f16#(I16, I17, I18, I19) 27.09/26.66 6) f16#(I20, I21, I22, I23) -> f15#(I20, I21, I22, I23) 27.09/26.66 7) f16#(I24, I25, I26, I27) -> f14#(I24, I25, I26, I27) 27.09/26.66 8) f16#(I28, I29, I30, I31) -> f14#(I28, I29, I30, I31) 27.09/26.66 9) f15#(I32, I33, I34, I35) -> f13#(I32, I33, I34, I35) 27.09/26.66 10) f14#(I36, I37, I38, I39) -> f15#(I36, I37, I38, I39) 27.09/26.66 12) f13#(I44, I45, I46, I47) -> f7#(I44, I45, 1 + I46, I47) 27.09/26.66 21) f7#(I84, I85, I86, I87) -> f1#(I84, I85, I86, I87) 27.09/26.66 27.09/26.66 We have the following SCCs. 27.09/26.66 27.09/29.63 EOF