YES proof of prog.inttrs # AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty Termination of the given IRSwT could be proven: (0) IRSwT (1) IRSFormatTransformerProof [EQUIVALENT, 0 ms] (2) IRSwT (3) IRSwTTerminationDigraphProof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, 3304 ms] (4) IRSwT (5) IntTRSCompressionProof [EQUIVALENT, 0 ms] (6) IRSwT (7) IntTRSUnneededArgumentFilterProof [EQUIVALENT, 3 ms] (8) IRSwT (9) TempFilterProof [SOUND, 91 ms] (10) IntTRS (11) PolynomialOrderProcessor [EQUIVALENT, 0 ms] (12) AND (13) IntTRS (14) RankingReductionPairProof [EQUIVALENT, 5 ms] (15) YES (16) IntTRS (17) PolynomialOrderProcessor [EQUIVALENT, 3 ms] (18) YES ---------------------------------------- (0) Obligation: Rules: l0(oldX0HAT0, oldX1HAT0, oldX2HAT0, oldX3HAT0, x0HAT0, x1HAT0) -> l1(oldX0HATpost, oldX1HATpost, oldX2HATpost, oldX3HATpost, x0HATpost, x1HATpost) :|: x1HATpost = oldX3HATpost && x0HATpost = oldX2HATpost && oldX3HATpost = oldX3HATpost && oldX2HATpost = oldX2HATpost && oldX1HATpost = x1HAT0 && oldX0HATpost = x0HAT0 l2(x, x1, x2, x3, x4, x5) -> l1(x6, x7, x8, x9, x10, x11) :|: x11 = x9 && x10 = x8 && x9 = x9 && x8 = x8 && x7 = x5 && x6 = x4 l2(x12, x13, x14, x15, x16, x17) -> l3(x18, x19, x20, x21, x22, x23) :|: x15 = x21 && x14 = x20 && x23 = -2 + x19 && x22 = 1 + x18 && x19 = x17 && x18 = x16 l4(x24, x25, x26, x27, x28, x29) -> l0(x30, x31, x32, x33, x34, x35) :|: x27 = x33 && x26 = x32 && x35 = x31 && x34 = x30 && x31 <= 2 && x31 = x29 && x30 = x28 l4(x36, x37, x38, x39, x40, x41) -> l2(x42, x43, x44, x45, x46, x47) :|: x39 = x45 && x38 = x44 && x47 = x43 && x46 = x42 && 3 <= x43 && x43 = x41 && x42 = x40 l5(x48, x49, x50, x51, x52, x53) -> l1(x54, x55, x56, x57, x58, x59) :|: x59 = x57 && x58 = x56 && x57 = x57 && x56 = x56 && x55 = x53 && x54 = x52 l5(x60, x61, x62, x63, x64, x65) -> l3(x66, x67, x68, x69, x70, x71) :|: x63 = x69 && x62 = x68 && x71 = 1 + x67 && x70 = -1 + x66 && x67 = x65 && x66 = x64 l6(x72, x73, x74, x75, x76, x77) -> l4(x78, x79, x80, x81, x82, x83) :|: x75 = x81 && x74 = x80 && x83 = x79 && x82 = x78 && x78 <= 0 && x79 = x77 && x78 = x76 l6(x84, x85, x86, x87, x88, x89) -> l5(x90, x91, x92, x93, x94, x95) :|: x87 = x93 && x86 = x92 && x95 = x91 && x94 = x90 && 1 <= x90 && x91 = x89 && x90 = x88 l3(x96, x97, x98, x99, x100, x101) -> l6(x102, x103, x104, x105, x106, x107) :|: x99 = x105 && x98 = x104 && x107 = x103 && x106 = x102 && x103 = x101 && x102 = x100 l7(x108, x109, x110, x111, x112, x113) -> l0(x114, x115, x116, x117, x118, x119) :|: x113 = x119 && x112 = x118 && x111 = x117 && x110 = x116 && x109 = x115 && x108 = x114 l7(x120, x121, x122, x123, x124, x125) -> l2(x126, x127, x128, x129, x130, x131) :|: x125 = x131 && x124 = x130 && x123 = x129 && x122 = x128 && x121 = x127 && x120 = x126 l7(x132, x133, x134, x135, x136, x137) -> l1(x138, x139, x140, x141, x142, x143) :|: x137 = x143 && x136 = x142 && x135 = x141 && x134 = x140 && x133 = x139 && x132 = x138 l7(x144, x145, x146, x147, x148, x149) -> l4(x150, x151, x152, x153, x154, x155) :|: x149 = x155 && x148 = x154 && x147 = x153 && x146 = x152 && x145 = x151 && x144 = x150 l7(x156, x157, x158, x159, x160, x161) -> l5(x162, x163, x164, x165, x166, x167) :|: x161 = x167 && x160 = x166 && x159 = x165 && x158 = x164 && x157 = x163 && x156 = x162 l7(x168, x169, x170, x171, x172, x173) -> l6(x174, x175, x176, x177, x178, x179) :|: x173 = x179 && x172 = x178 && x171 = x177 && x170 = x176 && x169 = x175 && x168 = x174 l7(x180, x181, x182, x183, x184, x185) -> l3(x186, x187, x188, x189, x190, x191) :|: x185 = x191 && x184 = x190 && x183 = x189 && x182 = x188 && x181 = x187 && x180 = x186 l8(x192, x193, x194, x195, x196, x197) -> l7(x198, x199, x200, x201, x202, x203) :|: x197 = x203 && x196 = x202 && x195 = x201 && x194 = x200 && x193 = x199 && x192 = x198 Start term: l8(oldX0HAT0, oldX1HAT0, oldX2HAT0, oldX3HAT0, x0HAT0, x1HAT0) ---------------------------------------- (1) IRSFormatTransformerProof (EQUIVALENT) Reformatted IRS to match normalized format (transformed away non-linear left-hand sides, !=, / and %). ---------------------------------------- (2) Obligation: Rules: l0(oldX0HAT0, oldX1HAT0, oldX2HAT0, oldX3HAT0, x0HAT0, x1HAT0) -> l1(oldX0HATpost, oldX1HATpost, oldX2HATpost, oldX3HATpost, x0HATpost, x1HATpost) :|: x1HATpost = oldX3HATpost && x0HATpost = oldX2HATpost && oldX3HATpost = oldX3HATpost && oldX2HATpost = oldX2HATpost && oldX1HATpost = x1HAT0 && oldX0HATpost = x0HAT0 l2(x, x1, x2, x3, x4, x5) -> l1(x6, x7, x8, x9, x10, x11) :|: x11 = x9 && x10 = x8 && x9 = x9 && x8 = x8 && x7 = x5 && x6 = x4 l2(x12, x13, x14, x15, x16, x17) -> l3(x18, x19, x20, x21, x22, x23) :|: x15 = x21 && x14 = x20 && x23 = -2 + x19 && x22 = 1 + x18 && x19 = x17 && x18 = x16 l4(x24, x25, x26, x27, x28, x29) -> l0(x30, x31, x32, x33, x34, x35) :|: x27 = x33 && x26 = x32 && x35 = x31 && x34 = x30 && x31 <= 2 && x31 = x29 && x30 = x28 l4(x36, x37, x38, x39, x40, x41) -> l2(x42, x43, x44, x45, x46, x47) :|: x39 = x45 && x38 = x44 && x47 = x43 && x46 = x42 && 3 <= x43 && x43 = x41 && x42 = x40 l5(x48, x49, x50, x51, x52, x53) -> l1(x54, x55, x56, x57, x58, x59) :|: x59 = x57 && x58 = x56 && x57 = x57 && x56 = x56 && x55 = x53 && x54 = x52 l5(x60, x61, x62, x63, x64, x65) -> l3(x66, x67, x68, x69, x70, x71) :|: x63 = x69 && x62 = x68 && x71 = 1 + x67 && x70 = -1 + x66 && x67 = x65 && x66 = x64 l6(x72, x73, x74, x75, x76, x77) -> l4(x78, x79, x80, x81, x82, x83) :|: x75 = x81 && x74 = x80 && x83 = x79 && x82 = x78 && x78 <= 0 && x79 = x77 && x78 = x76 l6(x84, x85, x86, x87, x88, x89) -> l5(x90, x91, x92, x93, x94, x95) :|: x87 = x93 && x86 = x92 && x95 = x91 && x94 = x90 && 1 <= x90 && x91 = x89 && x90 = x88 l3(x96, x97, x98, x99, x100, x101) -> l6(x102, x103, x104, x105, x106, x107) :|: x99 = x105 && x98 = x104 && x107 = x103 && x106 = x102 && x103 = x101 && x102 = x100 l7(x108, x109, x110, x111, x112, x113) -> l0(x114, x115, x116, x117, x118, x119) :|: x113 = x119 && x112 = x118 && x111 = x117 && x110 = x116 && x109 = x115 && x108 = x114 l7(x120, x121, x122, x123, x124, x125) -> l2(x126, x127, x128, x129, x130, x131) :|: x125 = x131 && x124 = x130 && x123 = x129 && x122 = x128 && x121 = x127 && x120 = x126 l7(x132, x133, x134, x135, x136, x137) -> l1(x138, x139, x140, x141, x142, x143) :|: x137 = x143 && x136 = x142 && x135 = x141 && x134 = x140 && x133 = x139 && x132 = x138 l7(x144, x145, x146, x147, x148, x149) -> l4(x150, x151, x152, x153, x154, x155) :|: x149 = x155 && x148 = x154 && x147 = x153 && x146 = x152 && x145 = x151 && x144 = x150 l7(x156, x157, x158, x159, x160, x161) -> l5(x162, x163, x164, x165, x166, x167) :|: x161 = x167 && x160 = x166 && x159 = x165 && x158 = x164 && x157 = x163 && x156 = x162 l7(x168, x169, x170, x171, x172, x173) -> l6(x174, x175, x176, x177, x178, x179) :|: x173 = x179 && x172 = x178 && x171 = x177 && x170 = x176 && x169 = x175 && x168 = x174 l7(x180, x181, x182, x183, x184, x185) -> l3(x186, x187, x188, x189, x190, x191) :|: x185 = x191 && x184 = x190 && x183 = x189 && x182 = x188 && x181 = x187 && x180 = x186 l8(x192, x193, x194, x195, x196, x197) -> l7(x198, x199, x200, x201, x202, x203) :|: x197 = x203 && x196 = x202 && x195 = x201 && x194 = x200 && x193 = x199 && x192 = x198 Start term: l8(oldX0HAT0, oldX1HAT0, oldX2HAT0, oldX3HAT0, x0HAT0, x1HAT0) ---------------------------------------- (3) IRSwTTerminationDigraphProof (EQUIVALENT) Constructed termination digraph! Nodes: (1) l0(oldX0HAT0, oldX1HAT0, oldX2HAT0, oldX3HAT0, x0HAT0, x1HAT0) -> l1(oldX0HATpost, oldX1HATpost, oldX2HATpost, oldX3HATpost, x0HATpost, x1HATpost) :|: x1HATpost = oldX3HATpost && x0HATpost = oldX2HATpost && oldX3HATpost = oldX3HATpost && oldX2HATpost = oldX2HATpost && oldX1HATpost = x1HAT0 && oldX0HATpost = x0HAT0 (2) l2(x, x1, x2, x3, x4, x5) -> l1(x6, x7, x8, x9, x10, x11) :|: x11 = x9 && x10 = x8 && x9 = x9 && x8 = x8 && x7 = x5 && x6 = x4 (3) l2(x12, x13, x14, x15, x16, x17) -> l3(x18, x19, x20, x21, x22, x23) :|: x15 = x21 && x14 = x20 && x23 = -2 + x19 && x22 = 1 + x18 && x19 = x17 && x18 = x16 (4) l4(x24, x25, x26, x27, x28, x29) -> l0(x30, x31, x32, x33, x34, x35) :|: x27 = x33 && x26 = x32 && x35 = x31 && x34 = x30 && x31 <= 2 && x31 = x29 && x30 = x28 (5) l4(x36, x37, x38, x39, x40, x41) -> l2(x42, x43, x44, x45, x46, x47) :|: x39 = x45 && x38 = x44 && x47 = x43 && x46 = x42 && 3 <= x43 && x43 = x41 && x42 = x40 (6) l5(x48, x49, x50, x51, x52, x53) -> l1(x54, x55, x56, x57, x58, x59) :|: x59 = x57 && x58 = x56 && x57 = x57 && x56 = x56 && x55 = x53 && x54 = x52 (7) l5(x60, x61, x62, x63, x64, x65) -> l3(x66, x67, x68, x69, x70, x71) :|: x63 = x69 && x62 = x68 && x71 = 1 + x67 && x70 = -1 + x66 && x67 = x65 && x66 = x64 (8) l6(x72, x73, x74, x75, x76, x77) -> l4(x78, x79, x80, x81, x82, x83) :|: x75 = x81 && x74 = x80 && x83 = x79 && x82 = x78 && x78 <= 0 && x79 = x77 && x78 = x76 (9) l6(x84, x85, x86, x87, x88, x89) -> l5(x90, x91, x92, x93, x94, x95) :|: x87 = x93 && x86 = x92 && x95 = x91 && x94 = x90 && 1 <= x90 && x91 = x89 && x90 = x88 (10) l3(x96, x97, x98, x99, x100, x101) -> l6(x102, x103, x104, x105, x106, x107) :|: x99 = x105 && x98 = x104 && x107 = x103 && x106 = x102 && x103 = x101 && x102 = x100 (11) l7(x108, x109, x110, x111, x112, x113) -> l0(x114, x115, x116, x117, x118, x119) :|: x113 = x119 && x112 = x118 && x111 = x117 && x110 = x116 && x109 = x115 && x108 = x114 (12) l7(x120, x121, x122, x123, x124, x125) -> l2(x126, x127, x128, x129, x130, x131) :|: x125 = x131 && x124 = x130 && x123 = x129 && x122 = x128 && x121 = x127 && x120 = x126 (13) l7(x132, x133, x134, x135, x136, x137) -> l1(x138, x139, x140, x141, x142, x143) :|: x137 = x143 && x136 = x142 && x135 = x141 && x134 = x140 && x133 = x139 && x132 = x138 (14) l7(x144, x145, x146, x147, x148, x149) -> l4(x150, x151, x152, x153, x154, x155) :|: x149 = x155 && x148 = x154 && x147 = x153 && x146 = x152 && x145 = x151 && x144 = x150 (15) l7(x156, x157, x158, x159, x160, x161) -> l5(x162, x163, x164, x165, x166, x167) :|: x161 = x167 && x160 = x166 && x159 = x165 && x158 = x164 && x157 = x163 && x156 = x162 (16) l7(x168, x169, x170, x171, x172, x173) -> l6(x174, x175, x176, x177, x178, x179) :|: x173 = x179 && x172 = x178 && x171 = x177 && x170 = x176 && x169 = x175 && x168 = x174 (17) l7(x180, x181, x182, x183, x184, x185) -> l3(x186, x187, x188, x189, x190, x191) :|: x185 = x191 && x184 = x190 && x183 = x189 && x182 = x188 && x181 = x187 && x180 = x186 (18) l8(x192, x193, x194, x195, x196, x197) -> l7(x198, x199, x200, x201, x202, x203) :|: x197 = x203 && x196 = x202 && x195 = x201 && x194 = x200 && x193 = x199 && x192 = x198 Arcs: (3) -> (10) (4) -> (1) (5) -> (2), (3) (7) -> (10) (8) -> (4), (5) (9) -> (6), (7) (10) -> (8), (9) (11) -> (1) (12) -> (2), (3) (14) -> (4), (5) (15) -> (6), (7) (16) -> (8), (9) (17) -> (10) (18) -> (11), (12), (13), (14), (15), (16), (17) This digraph is fully evaluated! ---------------------------------------- (4) Obligation: Termination digraph: Nodes: (1) l2(x12, x13, x14, x15, x16, x17) -> l3(x18, x19, x20, x21, x22, x23) :|: x15 = x21 && x14 = x20 && x23 = -2 + x19 && x22 = 1 + x18 && x19 = x17 && x18 = x16 (2) l4(x36, x37, x38, x39, x40, x41) -> l2(x42, x43, x44, x45, x46, x47) :|: x39 = x45 && x38 = x44 && x47 = x43 && x46 = x42 && 3 <= x43 && x43 = x41 && x42 = x40 (3) l6(x72, x73, x74, x75, x76, x77) -> l4(x78, x79, x80, x81, x82, x83) :|: x75 = x81 && x74 = x80 && x83 = x79 && x82 = x78 && x78 <= 0 && x79 = x77 && x78 = x76 (4) l3(x96, x97, x98, x99, x100, x101) -> l6(x102, x103, x104, x105, x106, x107) :|: x99 = x105 && x98 = x104 && x107 = x103 && x106 = x102 && x103 = x101 && x102 = x100 (5) l5(x60, x61, x62, x63, x64, x65) -> l3(x66, x67, x68, x69, x70, x71) :|: x63 = x69 && x62 = x68 && x71 = 1 + x67 && x70 = -1 + x66 && x67 = x65 && x66 = x64 (6) l6(x84, x85, x86, x87, x88, x89) -> l5(x90, x91, x92, x93, x94, x95) :|: x87 = x93 && x86 = x92 && x95 = x91 && x94 = x90 && 1 <= x90 && x91 = x89 && x90 = x88 Arcs: (1) -> (4) (2) -> (1) (3) -> (2) (4) -> (3), (6) (5) -> (4) (6) -> (5) This digraph is fully evaluated! ---------------------------------------- (5) IntTRSCompressionProof (EQUIVALENT) Compressed rules. ---------------------------------------- (6) Obligation: Rules: l6(x72:0, x73:0, x104:0, x105:0, x18:0, x19:0) -> l6(1 + x18:0, -2 + x19:0, x104:0, x105:0, 1 + x18:0, -2 + x19:0) :|: x18:0 < 1 && x19:0 > 2 l6(x, x1, x2, x3, x4, x5) -> l6(-1 + x4, 1 + x5, x2, x3, -1 + x4, 1 + x5) :|: x4 > 0 ---------------------------------------- (7) IntTRSUnneededArgumentFilterProof (EQUIVALENT) Some arguments are removed because they cannot influence termination. We removed arguments according to the following replacements: l6(x1, x2, x3, x4, x5, x6) -> l6(x5, x6) ---------------------------------------- (8) Obligation: Rules: l6(x18:0, x19:0) -> l6(1 + x18:0, -2 + x19:0) :|: x18:0 < 1 && x19:0 > 2 l6(x4, x5) -> l6(-1 + x4, 1 + x5) :|: x4 > 0 ---------------------------------------- (9) TempFilterProof (SOUND) Used the following sort dictionary for filtering: l6(INTEGER, VARIABLE) Replaced non-predefined constructor symbols by 0. ---------------------------------------- (10) Obligation: Rules: l6(x18:0, x19:0) -> l6(c, c1) :|: c1 = -2 + x19:0 && c = 1 + x18:0 && (x18:0 < 1 && x19:0 > 2) l6(x4, x5) -> l6(c2, c3) :|: c3 = 1 + x5 && c2 = -1 + x4 && x4 > 0 ---------------------------------------- (11) PolynomialOrderProcessor (EQUIVALENT) Found the following polynomial interpretation: [l6(x, x1)] = -3 + 2*x^2 + x1 The following rules are decreasing: l6(x4, x5) -> l6(c2, c3) :|: c3 = 1 + x5 && c2 = -1 + x4 && x4 > 0 The following rules are bounded: l6(x18:0, x19:0) -> l6(c, c1) :|: c1 = -2 + x19:0 && c = 1 + x18:0 && (x18:0 < 1 && x19:0 > 2) ---------------------------------------- (12) Complex Obligation (AND) ---------------------------------------- (13) Obligation: Rules: l6(x18:0, x19:0) -> l6(c, c1) :|: c1 = -2 + x19:0 && c = 1 + x18:0 && (x18:0 < 1 && x19:0 > 2) ---------------------------------------- (14) RankingReductionPairProof (EQUIVALENT) Interpretation: [ l6 ] = 1/2*l6_2 The following rules are decreasing: l6(x18:0, x19:0) -> l6(c, c1) :|: c1 = -2 + x19:0 && c = 1 + x18:0 && (x18:0 < 1 && x19:0 > 2) The following rules are bounded: l6(x18:0, x19:0) -> l6(c, c1) :|: c1 = -2 + x19:0 && c = 1 + x18:0 && (x18:0 < 1 && x19:0 > 2) ---------------------------------------- (15) YES ---------------------------------------- (16) Obligation: Rules: l6(x4, x5) -> l6(c2, c3) :|: c3 = 1 + x5 && c2 = -1 + x4 && x4 > 0 ---------------------------------------- (17) PolynomialOrderProcessor (EQUIVALENT) Found the following polynomial interpretation: [l6(x, x1)] = x The following rules are decreasing: l6(x4, x5) -> l6(c2, c3) :|: c3 = 1 + x5 && c2 = -1 + x4 && x4 > 0 The following rules are bounded: l6(x4, x5) -> l6(c2, c3) :|: c3 = 1 + x5 && c2 = -1 + x4 && x4 > 0 ---------------------------------------- (18) YES