YES proof of prog.inttrs # AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty Termination of the given IRSwT could be proven: (0) IRSwT (1) IRSFormatTransformerProof [EQUIVALENT, 0 ms] (2) IRSwT (3) IRSwTTerminationDigraphProof [EQUIVALENT, 2361 ms] (4) IRSwT (5) IntTRSCompressionProof [EQUIVALENT, 57 ms] (6) IRSwT (7) IntTRSUnneededArgumentFilterProof [EQUIVALENT, 0 ms] (8) IRSwT (9) TempFilterProof [SOUND, 157 ms] (10) IntTRS (11) PolynomialOrderProcessor [EQUIVALENT, 0 ms] (12) IntTRS (13) RankingReductionPairProof [EQUIVALENT, 18 ms] (14) IntTRS (15) PolynomialOrderProcessor [EQUIVALENT, 0 ms] (16) IntTRS (17) RankingReductionPairProof [EQUIVALENT, 0 ms] (18) IntTRS (19) PolynomialOrderProcessor [EQUIVALENT, 0 ms] (20) IntTRS (21) PolynomialOrderProcessor [EQUIVALENT, 2 ms] (22) YES ---------------------------------------- (0) Obligation: Rules: l0(i2HAT0, j3HAT0, k4HAT0, l5HAT0, x1HAT0) -> l1(i2HATpost, j3HATpost, k4HATpost, l5HATpost, x1HATpost) :|: x1HAT0 = x1HATpost && l5HAT0 = l5HATpost && k4HAT0 = k4HATpost && j3HAT0 = j3HATpost && i2HAT0 = i2HATpost && 5 <= i2HAT0 l0(x, x1, x2, x3, x4) -> l2(x5, x6, x7, x8, x9) :|: x4 = x9 && x3 = x8 && x2 = x7 && x = x5 && x6 = 0 && 1 + x <= 5 l3(x10, x11, x12, x13, x14) -> l0(x15, x16, x17, x18, x19) :|: x14 = x19 && x13 = x18 && x12 = x17 && x11 = x16 && x10 = x15 l2(x20, x21, x22, x23, x24) -> l4(x25, x26, x27, x28, x29) :|: x24 = x29 && x23 = x28 && x22 = x27 && x21 = x26 && x20 = x25 l1(x30, x31, x32, x33, x34) -> l5(x35, x36, x37, x38, x39) :|: x34 = x39 && x33 = x38 && x32 = x37 && x31 = x36 && x30 = x35 l6(x40, x41, x42, x43, x44) -> l7(x45, x46, x47, x48, x49) :|: x44 = x49 && x43 = x48 && x42 = x47 && x41 = x46 && x40 = x45 l8(x50, x51, x52, x53, x54) -> l9(x55, x56, x57, x58, x59) :|: x54 = x59 && x52 = x57 && x51 = x56 && x50 = x55 && x58 = 1 + x53 l9(x60, x61, x62, x63, x64) -> l10(x65, x66, x67, x68, x69) :|: x64 = x69 && x63 = x68 && x62 = x67 && x61 = x66 && x60 = x65 l11(x70, x71, x72, x73, x74) -> l8(x75, x76, x77, x78, x79) :|: x74 = x79 && x73 = x78 && x72 = x77 && x71 = x76 && x70 = x75 l11(x80, x81, x82, x83, x84) -> l1(x85, x86, x87, x88, x89) :|: x84 = x89 && x83 = x88 && x82 = x87 && x81 = x86 && x80 = x85 l10(x90, x91, x92, x93, x94) -> l6(x95, x96, x97, x98, x99) :|: x94 = x99 && x93 = x98 && x91 = x96 && x90 = x95 && x97 = 1 + x92 && 5 <= x93 l10(x100, x101, x102, x103, x104) -> l11(x105, x106, x107, x108, x109) :|: x104 = x109 && x103 = x108 && x102 = x107 && x101 = x106 && x100 = x105 && 1 + x103 <= 5 l7(x110, x111, x112, x113, x114) -> l2(x115, x116, x117, x118, x119) :|: x114 = x119 && x113 = x118 && x112 = x117 && x110 = x115 && x116 = 1 + x111 && 5 <= x112 l7(x120, x121, x122, x123, x124) -> l9(x125, x126, x127, x128, x129) :|: x124 = x129 && x122 = x127 && x121 = x126 && x120 = x125 && x128 = 0 && 1 + x122 <= 5 l4(x130, x131, x132, x133, x134) -> l3(x135, x136, x137, x138, x139) :|: x134 = x139 && x133 = x138 && x132 = x137 && x131 = x136 && x135 = 1 + x130 && 5 <= x131 l4(x140, x141, x142, x143, x144) -> l6(x145, x146, x147, x148, x149) :|: x144 = x149 && x143 = x148 && x141 = x146 && x140 = x145 && x147 = 0 && 1 + x141 <= 5 l12(x150, x151, x152, x153, x154) -> l3(x155, x156, x157, x158, x159) :|: x153 = x158 && x152 = x157 && x151 = x156 && x155 = 0 && x159 = 400 l13(x160, x161, x162, x163, x164) -> l12(x165, x166, x167, x168, x169) :|: x164 = x169 && x163 = x168 && x162 = x167 && x161 = x166 && x160 = x165 Start term: l13(i2HAT0, j3HAT0, k4HAT0, l5HAT0, x1HAT0) ---------------------------------------- (1) IRSFormatTransformerProof (EQUIVALENT) Reformatted IRS to match normalized format (transformed away non-linear left-hand sides, !=, / and %). ---------------------------------------- (2) Obligation: Rules: l0(i2HAT0, j3HAT0, k4HAT0, l5HAT0, x1HAT0) -> l1(i2HATpost, j3HATpost, k4HATpost, l5HATpost, x1HATpost) :|: x1HAT0 = x1HATpost && l5HAT0 = l5HATpost && k4HAT0 = k4HATpost && j3HAT0 = j3HATpost && i2HAT0 = i2HATpost && 5 <= i2HAT0 l0(x, x1, x2, x3, x4) -> l2(x5, x6, x7, x8, x9) :|: x4 = x9 && x3 = x8 && x2 = x7 && x = x5 && x6 = 0 && 1 + x <= 5 l3(x10, x11, x12, x13, x14) -> l0(x15, x16, x17, x18, x19) :|: x14 = x19 && x13 = x18 && x12 = x17 && x11 = x16 && x10 = x15 l2(x20, x21, x22, x23, x24) -> l4(x25, x26, x27, x28, x29) :|: x24 = x29 && x23 = x28 && x22 = x27 && x21 = x26 && x20 = x25 l1(x30, x31, x32, x33, x34) -> l5(x35, x36, x37, x38, x39) :|: x34 = x39 && x33 = x38 && x32 = x37 && x31 = x36 && x30 = x35 l6(x40, x41, x42, x43, x44) -> l7(x45, x46, x47, x48, x49) :|: x44 = x49 && x43 = x48 && x42 = x47 && x41 = x46 && x40 = x45 l8(x50, x51, x52, x53, x54) -> l9(x55, x56, x57, x58, x59) :|: x54 = x59 && x52 = x57 && x51 = x56 && x50 = x55 && x58 = 1 + x53 l9(x60, x61, x62, x63, x64) -> l10(x65, x66, x67, x68, x69) :|: x64 = x69 && x63 = x68 && x62 = x67 && x61 = x66 && x60 = x65 l11(x70, x71, x72, x73, x74) -> l8(x75, x76, x77, x78, x79) :|: x74 = x79 && x73 = x78 && x72 = x77 && x71 = x76 && x70 = x75 l11(x80, x81, x82, x83, x84) -> l1(x85, x86, x87, x88, x89) :|: x84 = x89 && x83 = x88 && x82 = x87 && x81 = x86 && x80 = x85 l10(x90, x91, x92, x93, x94) -> l6(x95, x96, x97, x98, x99) :|: x94 = x99 && x93 = x98 && x91 = x96 && x90 = x95 && x97 = 1 + x92 && 5 <= x93 l10(x100, x101, x102, x103, x104) -> l11(x105, x106, x107, x108, x109) :|: x104 = x109 && x103 = x108 && x102 = x107 && x101 = x106 && x100 = x105 && 1 + x103 <= 5 l7(x110, x111, x112, x113, x114) -> l2(x115, x116, x117, x118, x119) :|: x114 = x119 && x113 = x118 && x112 = x117 && x110 = x115 && x116 = 1 + x111 && 5 <= x112 l7(x120, x121, x122, x123, x124) -> l9(x125, x126, x127, x128, x129) :|: x124 = x129 && x122 = x127 && x121 = x126 && x120 = x125 && x128 = 0 && 1 + x122 <= 5 l4(x130, x131, x132, x133, x134) -> l3(x135, x136, x137, x138, x139) :|: x134 = x139 && x133 = x138 && x132 = x137 && x131 = x136 && x135 = 1 + x130 && 5 <= x131 l4(x140, x141, x142, x143, x144) -> l6(x145, x146, x147, x148, x149) :|: x144 = x149 && x143 = x148 && x141 = x146 && x140 = x145 && x147 = 0 && 1 + x141 <= 5 l12(x150, x151, x152, x153, x154) -> l3(x155, x156, x157, x158, x159) :|: x153 = x158 && x152 = x157 && x151 = x156 && x155 = 0 && x159 = 400 l13(x160, x161, x162, x163, x164) -> l12(x165, x166, x167, x168, x169) :|: x164 = x169 && x163 = x168 && x162 = x167 && x161 = x166 && x160 = x165 Start term: l13(i2HAT0, j3HAT0, k4HAT0, l5HAT0, x1HAT0) ---------------------------------------- (3) IRSwTTerminationDigraphProof (EQUIVALENT) Constructed termination digraph! Nodes: (1) l0(i2HAT0, j3HAT0, k4HAT0, l5HAT0, x1HAT0) -> l1(i2HATpost, j3HATpost, k4HATpost, l5HATpost, x1HATpost) :|: x1HAT0 = x1HATpost && l5HAT0 = l5HATpost && k4HAT0 = k4HATpost && j3HAT0 = j3HATpost && i2HAT0 = i2HATpost && 5 <= i2HAT0 (2) l0(x, x1, x2, x3, x4) -> l2(x5, x6, x7, x8, x9) :|: x4 = x9 && x3 = x8 && x2 = x7 && x = x5 && x6 = 0 && 1 + x <= 5 (3) l3(x10, x11, x12, x13, x14) -> l0(x15, x16, x17, x18, x19) :|: x14 = x19 && x13 = x18 && x12 = x17 && x11 = x16 && x10 = x15 (4) l2(x20, x21, x22, x23, x24) -> l4(x25, x26, x27, x28, x29) :|: x24 = x29 && x23 = x28 && x22 = x27 && x21 = x26 && x20 = x25 (5) l1(x30, x31, x32, x33, x34) -> l5(x35, x36, x37, x38, x39) :|: x34 = x39 && x33 = x38 && x32 = x37 && x31 = x36 && x30 = x35 (6) l6(x40, x41, x42, x43, x44) -> l7(x45, x46, x47, x48, x49) :|: x44 = x49 && x43 = x48 && x42 = x47 && x41 = x46 && x40 = x45 (7) l8(x50, x51, x52, x53, x54) -> l9(x55, x56, x57, x58, x59) :|: x54 = x59 && x52 = x57 && x51 = x56 && x50 = x55 && x58 = 1 + x53 (8) l9(x60, x61, x62, x63, x64) -> l10(x65, x66, x67, x68, x69) :|: x64 = x69 && x63 = x68 && x62 = x67 && x61 = x66 && x60 = x65 (9) l11(x70, x71, x72, x73, x74) -> l8(x75, x76, x77, x78, x79) :|: x74 = x79 && x73 = x78 && x72 = x77 && x71 = x76 && x70 = x75 (10) l11(x80, x81, x82, x83, x84) -> l1(x85, x86, x87, x88, x89) :|: x84 = x89 && x83 = x88 && x82 = x87 && x81 = x86 && x80 = x85 (11) l10(x90, x91, x92, x93, x94) -> l6(x95, x96, x97, x98, x99) :|: x94 = x99 && x93 = x98 && x91 = x96 && x90 = x95 && x97 = 1 + x92 && 5 <= x93 (12) l10(x100, x101, x102, x103, x104) -> l11(x105, x106, x107, x108, x109) :|: x104 = x109 && x103 = x108 && x102 = x107 && x101 = x106 && x100 = x105 && 1 + x103 <= 5 (13) l7(x110, x111, x112, x113, x114) -> l2(x115, x116, x117, x118, x119) :|: x114 = x119 && x113 = x118 && x112 = x117 && x110 = x115 && x116 = 1 + x111 && 5 <= x112 (14) l7(x120, x121, x122, x123, x124) -> l9(x125, x126, x127, x128, x129) :|: x124 = x129 && x122 = x127 && x121 = x126 && x120 = x125 && x128 = 0 && 1 + x122 <= 5 (15) l4(x130, x131, x132, x133, x134) -> l3(x135, x136, x137, x138, x139) :|: x134 = x139 && x133 = x138 && x132 = x137 && x131 = x136 && x135 = 1 + x130 && 5 <= x131 (16) l4(x140, x141, x142, x143, x144) -> l6(x145, x146, x147, x148, x149) :|: x144 = x149 && x143 = x148 && x141 = x146 && x140 = x145 && x147 = 0 && 1 + x141 <= 5 (17) l12(x150, x151, x152, x153, x154) -> l3(x155, x156, x157, x158, x159) :|: x153 = x158 && x152 = x157 && x151 = x156 && x155 = 0 && x159 = 400 (18) l13(x160, x161, x162, x163, x164) -> l12(x165, x166, x167, x168, x169) :|: x164 = x169 && x163 = x168 && x162 = x167 && x161 = x166 && x160 = x165 Arcs: (1) -> (5) (2) -> (4) (3) -> (1), (2) (4) -> (15), (16) (6) -> (13), (14) (7) -> (8) (8) -> (11), (12) (9) -> (7) (10) -> (5) (11) -> (6) (12) -> (9), (10) (13) -> (4) (14) -> (8) (15) -> (3) (16) -> (6) (17) -> (3) (18) -> (17) This digraph is fully evaluated! ---------------------------------------- (4) Obligation: Termination digraph: Nodes: (1) l0(x, x1, x2, x3, x4) -> l2(x5, x6, x7, x8, x9) :|: x4 = x9 && x3 = x8 && x2 = x7 && x = x5 && x6 = 0 && 1 + x <= 5 (2) l3(x10, x11, x12, x13, x14) -> l0(x15, x16, x17, x18, x19) :|: x14 = x19 && x13 = x18 && x12 = x17 && x11 = x16 && x10 = x15 (3) l4(x130, x131, x132, x133, x134) -> l3(x135, x136, x137, x138, x139) :|: x134 = x139 && x133 = x138 && x132 = x137 && x131 = x136 && x135 = 1 + x130 && 5 <= x131 (4) l2(x20, x21, x22, x23, x24) -> l4(x25, x26, x27, x28, x29) :|: x24 = x29 && x23 = x28 && x22 = x27 && x21 = x26 && x20 = x25 (5) l7(x110, x111, x112, x113, x114) -> l2(x115, x116, x117, x118, x119) :|: x114 = x119 && x113 = x118 && x112 = x117 && x110 = x115 && x116 = 1 + x111 && 5 <= x112 (6) l6(x40, x41, x42, x43, x44) -> l7(x45, x46, x47, x48, x49) :|: x44 = x49 && x43 = x48 && x42 = x47 && x41 = x46 && x40 = x45 (7) l4(x140, x141, x142, x143, x144) -> l6(x145, x146, x147, x148, x149) :|: x144 = x149 && x143 = x148 && x141 = x146 && x140 = x145 && x147 = 0 && 1 + x141 <= 5 (8) l10(x90, x91, x92, x93, x94) -> l6(x95, x96, x97, x98, x99) :|: x94 = x99 && x93 = x98 && x91 = x96 && x90 = x95 && x97 = 1 + x92 && 5 <= x93 (9) l9(x60, x61, x62, x63, x64) -> l10(x65, x66, x67, x68, x69) :|: x64 = x69 && x63 = x68 && x62 = x67 && x61 = x66 && x60 = x65 (10) l7(x120, x121, x122, x123, x124) -> l9(x125, x126, x127, x128, x129) :|: x124 = x129 && x122 = x127 && x121 = x126 && x120 = x125 && x128 = 0 && 1 + x122 <= 5 (11) l8(x50, x51, x52, x53, x54) -> l9(x55, x56, x57, x58, x59) :|: x54 = x59 && x52 = x57 && x51 = x56 && x50 = x55 && x58 = 1 + x53 (12) l11(x70, x71, x72, x73, x74) -> l8(x75, x76, x77, x78, x79) :|: x74 = x79 && x73 = x78 && x72 = x77 && x71 = x76 && x70 = x75 (13) l10(x100, x101, x102, x103, x104) -> l11(x105, x106, x107, x108, x109) :|: x104 = x109 && x103 = x108 && x102 = x107 && x101 = x106 && x100 = x105 && 1 + x103 <= 5 Arcs: (1) -> (4) (2) -> (1) (3) -> (2) (4) -> (3), (7) (5) -> (4) (6) -> (5), (10) (7) -> (6) (8) -> (6) (9) -> (8), (13) (10) -> (9) (11) -> (9) (12) -> (11) (13) -> (12) This digraph is fully evaluated! ---------------------------------------- (5) IntTRSCompressionProof (EQUIVALENT) Compressed rules. ---------------------------------------- (6) Obligation: Rules: l4(x140:0, x141:0, x142:0, x143:0, x144:0) -> l6(x140:0, x141:0, 0, x143:0, x144:0) :|: x141:0 < 5 l6(x125:0, x126:0, x127:0, x43:0, x129:0) -> l9(x125:0, x126:0, x127:0, 0, x129:0) :|: x127:0 < 5 l6(x115:0, x41:0, x117:0, x118:0, x119:0) -> l4(x115:0, 1 + x41:0, x117:0, x118:0, x119:0) :|: x117:0 > 4 l9(x105:0, x106:0, x107:0, x108:0, x109:0) -> l9(x105:0, x106:0, x107:0, 1 + x108:0, x109:0) :|: x108:0 < 5 l9(x60:0, x61:0, x62:0, x63:0, x64:0) -> l6(x60:0, x61:0, 1 + x62:0, x63:0, x64:0) :|: x63:0 > 4 l4(x130:0, x131:0, x132:0, x133:0, x134:0) -> l4(1 + x130:0, 0, x132:0, x133:0, x134:0) :|: x131:0 > 4 && x130:0 < 4 ---------------------------------------- (7) IntTRSUnneededArgumentFilterProof (EQUIVALENT) Some arguments are removed because they cannot influence termination. We removed arguments according to the following replacements: l4(x1, x2, x3, x4, x5) -> l4(x1, x2) l6(x1, x2, x3, x4, x5) -> l6(x1, x2, x3) l9(x1, x2, x3, x4, x5) -> l9(x1, x2, x3, x4) ---------------------------------------- (8) Obligation: Rules: l4(x140:0, x141:0) -> l6(x140:0, x141:0, 0) :|: x141:0 < 5 l6(x125:0, x126:0, x127:0) -> l9(x125:0, x126:0, x127:0, 0) :|: x127:0 < 5 l6(x115:0, x41:0, x117:0) -> l4(x115:0, 1 + x41:0) :|: x117:0 > 4 l9(x105:0, x106:0, x107:0, x108:0) -> l9(x105:0, x106:0, x107:0, 1 + x108:0) :|: x108:0 < 5 l9(x60:0, x61:0, x62:0, x63:0) -> l6(x60:0, x61:0, 1 + x62:0) :|: x63:0 > 4 l4(x130:0, x131:0) -> l4(1 + x130:0, 0) :|: x131:0 > 4 && x130:0 < 4 ---------------------------------------- (9) TempFilterProof (SOUND) Used the following sort dictionary for filtering: l4(VARIABLE, VARIABLE) l6(VARIABLE, VARIABLE, VARIABLE) l9(VARIABLE, VARIABLE, VARIABLE, VARIABLE) Replaced non-predefined constructor symbols by 0. ---------------------------------------- (10) Obligation: Rules: l4(x140:0, x141:0) -> l6(x140:0, x141:0, c) :|: c = 0 && x141:0 < 5 l6(x125:0, x126:0, x127:0) -> l9(x125:0, x126:0, x127:0, c1) :|: c1 = 0 && x127:0 < 5 l6(x115:0, x41:0, x117:0) -> l4(x115:0, c2) :|: c2 = 1 + x41:0 && x117:0 > 4 l9(x105:0, x106:0, x107:0, x108:0) -> l9(x105:0, x106:0, x107:0, c3) :|: c3 = 1 + x108:0 && x108:0 < 5 l9(x60:0, x61:0, x62:0, x63:0) -> l6(x60:0, x61:0, c4) :|: c4 = 1 + x62:0 && x63:0 > 4 l4(x130:0, x131:0) -> l4(c5, c6) :|: c6 = 0 && c5 = 1 + x130:0 && (x131:0 > 4 && x130:0 < 4) ---------------------------------------- (11) PolynomialOrderProcessor (EQUIVALENT) Found the following polynomial interpretation: [l4(x, x1)] = 3 - x [l6(x2, x3, x4)] = 3 - x2 [l9(x5, x6, x7, x8)] = 3 - x5 The following rules are decreasing: l4(x130:0, x131:0) -> l4(c5, c6) :|: c6 = 0 && c5 = 1 + x130:0 && (x131:0 > 4 && x130:0 < 4) The following rules are bounded: l4(x130:0, x131:0) -> l4(c5, c6) :|: c6 = 0 && c5 = 1 + x130:0 && (x131:0 > 4 && x130:0 < 4) ---------------------------------------- (12) Obligation: Rules: l4(x140:0, x141:0) -> l6(x140:0, x141:0, c) :|: c = 0 && x141:0 < 5 l6(x125:0, x126:0, x127:0) -> l9(x125:0, x126:0, x127:0, c1) :|: c1 = 0 && x127:0 < 5 l6(x115:0, x41:0, x117:0) -> l4(x115:0, c2) :|: c2 = 1 + x41:0 && x117:0 > 4 l9(x105:0, x106:0, x107:0, x108:0) -> l9(x105:0, x106:0, x107:0, c3) :|: c3 = 1 + x108:0 && x108:0 < 5 l9(x60:0, x61:0, x62:0, x63:0) -> l6(x60:0, x61:0, c4) :|: c4 = 1 + x62:0 && x63:0 > 4 ---------------------------------------- (13) RankingReductionPairProof (EQUIVALENT) Interpretation: [ l4 ] = -2*l4_2 [ l6 ] = -2*l6_2 + -1 [ l9 ] = -2*l9_2 + -1 The following rules are decreasing: l4(x140:0, x141:0) -> l6(x140:0, x141:0, c) :|: c = 0 && x141:0 < 5 l6(x115:0, x41:0, x117:0) -> l4(x115:0, c2) :|: c2 = 1 + x41:0 && x117:0 > 4 The following rules are bounded: l4(x140:0, x141:0) -> l6(x140:0, x141:0, c) :|: c = 0 && x141:0 < 5 ---------------------------------------- (14) Obligation: Rules: l6(x125:0, x126:0, x127:0) -> l9(x125:0, x126:0, x127:0, c1) :|: c1 = 0 && x127:0 < 5 l6(x115:0, x41:0, x117:0) -> l4(x115:0, c2) :|: c2 = 1 + x41:0 && x117:0 > 4 l9(x105:0, x106:0, x107:0, x108:0) -> l9(x105:0, x106:0, x107:0, c3) :|: c3 = 1 + x108:0 && x108:0 < 5 l9(x60:0, x61:0, x62:0, x63:0) -> l6(x60:0, x61:0, c4) :|: c4 = 1 + x62:0 && x63:0 > 4 ---------------------------------------- (15) PolynomialOrderProcessor (EQUIVALENT) Found the following polynomial interpretation: [l6(x, x1, x2)] = 1 [l9(x3, x4, x5, x6)] = 1 [l4(x7, x8)] = 0 The following rules are decreasing: l6(x115:0, x41:0, x117:0) -> l4(x115:0, c2) :|: c2 = 1 + x41:0 && x117:0 > 4 The following rules are bounded: l6(x125:0, x126:0, x127:0) -> l9(x125:0, x126:0, x127:0, c1) :|: c1 = 0 && x127:0 < 5 l6(x115:0, x41:0, x117:0) -> l4(x115:0, c2) :|: c2 = 1 + x41:0 && x117:0 > 4 l9(x105:0, x106:0, x107:0, x108:0) -> l9(x105:0, x106:0, x107:0, c3) :|: c3 = 1 + x108:0 && x108:0 < 5 l9(x60:0, x61:0, x62:0, x63:0) -> l6(x60:0, x61:0, c4) :|: c4 = 1 + x62:0 && x63:0 > 4 ---------------------------------------- (16) Obligation: Rules: l6(x125:0, x126:0, x127:0) -> l9(x125:0, x126:0, x127:0, c1) :|: c1 = 0 && x127:0 < 5 l9(x105:0, x106:0, x107:0, x108:0) -> l9(x105:0, x106:0, x107:0, c3) :|: c3 = 1 + x108:0 && x108:0 < 5 l9(x60:0, x61:0, x62:0, x63:0) -> l6(x60:0, x61:0, c4) :|: c4 = 1 + x62:0 && x63:0 > 4 ---------------------------------------- (17) RankingReductionPairProof (EQUIVALENT) Interpretation: [ l6 ] = -2*l6_3 [ l9 ] = -2*l9_3 + -1 The following rules are decreasing: l6(x125:0, x126:0, x127:0) -> l9(x125:0, x126:0, x127:0, c1) :|: c1 = 0 && x127:0 < 5 l9(x60:0, x61:0, x62:0, x63:0) -> l6(x60:0, x61:0, c4) :|: c4 = 1 + x62:0 && x63:0 > 4 The following rules are bounded: l6(x125:0, x126:0, x127:0) -> l9(x125:0, x126:0, x127:0, c1) :|: c1 = 0 && x127:0 < 5 ---------------------------------------- (18) Obligation: Rules: l9(x105:0, x106:0, x107:0, x108:0) -> l9(x105:0, x106:0, x107:0, c3) :|: c3 = 1 + x108:0 && x108:0 < 5 l9(x60:0, x61:0, x62:0, x63:0) -> l6(x60:0, x61:0, c4) :|: c4 = 1 + x62:0 && x63:0 > 4 ---------------------------------------- (19) PolynomialOrderProcessor (EQUIVALENT) Found the following polynomial interpretation: [l9(x, x1, x2, x3)] = 1 [l6(x4, x5, x6)] = 0 The following rules are decreasing: l9(x60:0, x61:0, x62:0, x63:0) -> l6(x60:0, x61:0, c4) :|: c4 = 1 + x62:0 && x63:0 > 4 The following rules are bounded: l9(x105:0, x106:0, x107:0, x108:0) -> l9(x105:0, x106:0, x107:0, c3) :|: c3 = 1 + x108:0 && x108:0 < 5 l9(x60:0, x61:0, x62:0, x63:0) -> l6(x60:0, x61:0, c4) :|: c4 = 1 + x62:0 && x63:0 > 4 ---------------------------------------- (20) Obligation: Rules: l9(x105:0, x106:0, x107:0, x108:0) -> l9(x105:0, x106:0, x107:0, c3) :|: c3 = 1 + x108:0 && x108:0 < 5 ---------------------------------------- (21) PolynomialOrderProcessor (EQUIVALENT) Found the following polynomial interpretation: [l9(x, x1, x2, x3)] = 4 - x3 The following rules are decreasing: l9(x105:0, x106:0, x107:0, x108:0) -> l9(x105:0, x106:0, x107:0, c3) :|: c3 = 1 + x108:0 && x108:0 < 5 The following rules are bounded: l9(x105:0, x106:0, x107:0, x108:0) -> l9(x105:0, x106:0, x107:0, c3) :|: c3 = 1 + x108:0 && x108:0 < 5 ---------------------------------------- (22) YES