YES proof of prog.inttrs # AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty Termination of the given IRSwT could be proven: (0) IRSwT (1) IRSFormatTransformerProof [EQUIVALENT, 0 ms] (2) IRSwT (3) IRSwTTerminationDigraphProof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, 2668 ms] (4) IRSwT (5) IntTRSCompressionProof [EQUIVALENT, 0 ms] (6) IRSwT (7) IntTRSUnneededArgumentFilterProof [EQUIVALENT, 0 ms] (8) IRSwT (9) TempFilterProof [SOUND, 37 ms] (10) IntTRS (11) PolynomialOrderProcessor [EQUIVALENT, 15 ms] (12) 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 l0(x, x1, x2, x3, x4, x5) -> l2(x6, x7, x8, x9, x10, x11) :|: x3 = x9 && x2 = x8 && x11 = -1 + x7 && x10 = -1 + x6 && x7 = x5 && x6 = x4 l3(x12, x13, x14, x15, x16, x17) -> l1(x18, x19, x20, x21, x22, x23) :|: x23 = x21 && x22 = x20 && 0 <= x19 && x19 <= 0 && 0 <= x18 && x18 <= 0 && x21 = x21 && x20 = x20 && x19 = x17 && x18 = x16 l3(x24, x25, x26, x27, x28, x29) -> l1(x30, x31, x32, x33, x34, x35) :|: x35 = x33 && x34 = x32 && 1 <= x31 && x33 = x33 && x32 = x32 && x31 = x29 && x30 = x28 l3(x36, x37, x38, x39, x40, x41) -> l1(x42, x43, x44, x45, x46, x47) :|: x47 = x45 && x46 = x44 && 1 + x43 <= 0 && x45 = x45 && x44 = x44 && x43 = x41 && x42 = x40 l3(x48, x49, x50, x51, x52, x53) -> l1(x54, x55, x56, x57, x58, x59) :|: x59 = x57 && x58 = x56 && 1 <= x54 && x57 = x57 && x56 = x56 && x55 = x53 && x54 = x52 l3(x60, x61, x62, x63, x64, x65) -> l1(x66, x67, x68, x69, x70, x71) :|: x71 = x69 && x70 = x68 && 1 + x66 <= 0 && x69 = x69 && x68 = x68 && x67 = x65 && x66 = x64 l4(x72, x73, x74, x75, x76, x77) -> l0(x78, x79, x80, x81, x82, x83) :|: x75 = x81 && x74 = x80 && x83 = x79 && x82 = x78 && 1 <= x79 && 1 <= x78 && x79 = x77 && x78 = x76 l4(x84, x85, x86, x87, x88, x89) -> l3(x90, x91, x92, x93, x94, x95) :|: x87 = x93 && x86 = x92 && x95 = x91 && x94 = x90 && x91 <= 0 && x91 = x89 && x90 = x88 l4(x96, x97, x98, x99, x100, x101) -> l3(x102, x103, x104, x105, x106, x107) :|: x99 = x105 && x98 = x104 && x107 = x103 && x106 = x102 && x102 <= 0 && x103 = x101 && x102 = x100 l2(x108, x109, x110, x111, x112, x113) -> l4(x114, x115, x116, x117, x118, x119) :|: x111 = x117 && x110 = x116 && x119 = x115 && x118 = x114 && x115 = x113 && x114 = x112 l5(x120, x121, x122, x123, x124, x125) -> l1(x126, x127, x128, x129, x130, x131) :|: x125 = x131 && x124 = x130 && x123 = x129 && x122 = x128 && x121 = x127 && x120 = x126 l5(x132, x133, x134, x135, x136, x137) -> l0(x138, x139, x140, x141, x142, x143) :|: x137 = x143 && x136 = x142 && x135 = x141 && x134 = x140 && x133 = x139 && x132 = x138 l5(x144, x145, x146, x147, x148, x149) -> l3(x150, x151, x152, x153, x154, x155) :|: x149 = x155 && x148 = x154 && x147 = x153 && x146 = x152 && x145 = x151 && x144 = x150 l5(x156, x157, x158, x159, x160, x161) -> l4(x162, x163, x164, x165, x166, x167) :|: x161 = x167 && x160 = x166 && x159 = x165 && x158 = x164 && x157 = x163 && x156 = x162 l5(x168, x169, x170, x171, x172, x173) -> l2(x174, x175, x176, x177, x178, x179) :|: x173 = x179 && x172 = x178 && x171 = x177 && x170 = x176 && x169 = x175 && x168 = x174 l6(x180, x181, x182, x183, x184, x185) -> l5(x186, x187, x188, x189, x190, x191) :|: x185 = x191 && x184 = x190 && x183 = x189 && x182 = x188 && x181 = x187 && x180 = x186 Start term: l6(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 l0(x, x1, x2, x3, x4, x5) -> l2(x6, x7, x8, x9, x10, x11) :|: x3 = x9 && x2 = x8 && x11 = -1 + x7 && x10 = -1 + x6 && x7 = x5 && x6 = x4 l3(x12, x13, x14, x15, x16, x17) -> l1(x18, x19, x20, x21, x22, x23) :|: x23 = x21 && x22 = x20 && 0 <= x19 && x19 <= 0 && 0 <= x18 && x18 <= 0 && x21 = x21 && x20 = x20 && x19 = x17 && x18 = x16 l3(x24, x25, x26, x27, x28, x29) -> l1(x30, x31, x32, x33, x34, x35) :|: x35 = x33 && x34 = x32 && 1 <= x31 && x33 = x33 && x32 = x32 && x31 = x29 && x30 = x28 l3(x36, x37, x38, x39, x40, x41) -> l1(x42, x43, x44, x45, x46, x47) :|: x47 = x45 && x46 = x44 && 1 + x43 <= 0 && x45 = x45 && x44 = x44 && x43 = x41 && x42 = x40 l3(x48, x49, x50, x51, x52, x53) -> l1(x54, x55, x56, x57, x58, x59) :|: x59 = x57 && x58 = x56 && 1 <= x54 && x57 = x57 && x56 = x56 && x55 = x53 && x54 = x52 l3(x60, x61, x62, x63, x64, x65) -> l1(x66, x67, x68, x69, x70, x71) :|: x71 = x69 && x70 = x68 && 1 + x66 <= 0 && x69 = x69 && x68 = x68 && x67 = x65 && x66 = x64 l4(x72, x73, x74, x75, x76, x77) -> l0(x78, x79, x80, x81, x82, x83) :|: x75 = x81 && x74 = x80 && x83 = x79 && x82 = x78 && 1 <= x79 && 1 <= x78 && x79 = x77 && x78 = x76 l4(x84, x85, x86, x87, x88, x89) -> l3(x90, x91, x92, x93, x94, x95) :|: x87 = x93 && x86 = x92 && x95 = x91 && x94 = x90 && x91 <= 0 && x91 = x89 && x90 = x88 l4(x96, x97, x98, x99, x100, x101) -> l3(x102, x103, x104, x105, x106, x107) :|: x99 = x105 && x98 = x104 && x107 = x103 && x106 = x102 && x102 <= 0 && x103 = x101 && x102 = x100 l2(x108, x109, x110, x111, x112, x113) -> l4(x114, x115, x116, x117, x118, x119) :|: x111 = x117 && x110 = x116 && x119 = x115 && x118 = x114 && x115 = x113 && x114 = x112 l5(x120, x121, x122, x123, x124, x125) -> l1(x126, x127, x128, x129, x130, x131) :|: x125 = x131 && x124 = x130 && x123 = x129 && x122 = x128 && x121 = x127 && x120 = x126 l5(x132, x133, x134, x135, x136, x137) -> l0(x138, x139, x140, x141, x142, x143) :|: x137 = x143 && x136 = x142 && x135 = x141 && x134 = x140 && x133 = x139 && x132 = x138 l5(x144, x145, x146, x147, x148, x149) -> l3(x150, x151, x152, x153, x154, x155) :|: x149 = x155 && x148 = x154 && x147 = x153 && x146 = x152 && x145 = x151 && x144 = x150 l5(x156, x157, x158, x159, x160, x161) -> l4(x162, x163, x164, x165, x166, x167) :|: x161 = x167 && x160 = x166 && x159 = x165 && x158 = x164 && x157 = x163 && x156 = x162 l5(x168, x169, x170, x171, x172, x173) -> l2(x174, x175, x176, x177, x178, x179) :|: x173 = x179 && x172 = x178 && x171 = x177 && x170 = x176 && x169 = x175 && x168 = x174 l6(x180, x181, x182, x183, x184, x185) -> l5(x186, x187, x188, x189, x190, x191) :|: x185 = x191 && x184 = x190 && x183 = x189 && x182 = x188 && x181 = x187 && x180 = x186 Start term: l6(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) l0(x, x1, x2, x3, x4, x5) -> l2(x6, x7, x8, x9, x10, x11) :|: x3 = x9 && x2 = x8 && x11 = -1 + x7 && x10 = -1 + x6 && x7 = x5 && x6 = x4 (3) l3(x12, x13, x14, x15, x16, x17) -> l1(x18, x19, x20, x21, x22, x23) :|: x23 = x21 && x22 = x20 && 0 <= x19 && x19 <= 0 && 0 <= x18 && x18 <= 0 && x21 = x21 && x20 = x20 && x19 = x17 && x18 = x16 (4) l3(x24, x25, x26, x27, x28, x29) -> l1(x30, x31, x32, x33, x34, x35) :|: x35 = x33 && x34 = x32 && 1 <= x31 && x33 = x33 && x32 = x32 && x31 = x29 && x30 = x28 (5) l3(x36, x37, x38, x39, x40, x41) -> l1(x42, x43, x44, x45, x46, x47) :|: x47 = x45 && x46 = x44 && 1 + x43 <= 0 && x45 = x45 && x44 = x44 && x43 = x41 && x42 = x40 (6) l3(x48, x49, x50, x51, x52, x53) -> l1(x54, x55, x56, x57, x58, x59) :|: x59 = x57 && x58 = x56 && 1 <= x54 && x57 = x57 && x56 = x56 && x55 = x53 && x54 = x52 (7) l3(x60, x61, x62, x63, x64, x65) -> l1(x66, x67, x68, x69, x70, x71) :|: x71 = x69 && x70 = x68 && 1 + x66 <= 0 && x69 = x69 && x68 = x68 && x67 = x65 && x66 = x64 (8) l4(x72, x73, x74, x75, x76, x77) -> l0(x78, x79, x80, x81, x82, x83) :|: x75 = x81 && x74 = x80 && x83 = x79 && x82 = x78 && 1 <= x79 && 1 <= x78 && x79 = x77 && x78 = x76 (9) l4(x84, x85, x86, x87, x88, x89) -> l3(x90, x91, x92, x93, x94, x95) :|: x87 = x93 && x86 = x92 && x95 = x91 && x94 = x90 && x91 <= 0 && x91 = x89 && x90 = x88 (10) l4(x96, x97, x98, x99, x100, x101) -> l3(x102, x103, x104, x105, x106, x107) :|: x99 = x105 && x98 = x104 && x107 = x103 && x106 = x102 && x102 <= 0 && x103 = x101 && x102 = x100 (11) l2(x108, x109, x110, x111, x112, x113) -> l4(x114, x115, x116, x117, x118, x119) :|: x111 = x117 && x110 = x116 && x119 = x115 && x118 = x114 && x115 = x113 && x114 = x112 (12) l5(x120, x121, x122, x123, x124, x125) -> l1(x126, x127, x128, x129, x130, x131) :|: x125 = x131 && x124 = x130 && x123 = x129 && x122 = x128 && x121 = x127 && x120 = x126 (13) l5(x132, x133, x134, x135, x136, x137) -> l0(x138, x139, x140, x141, x142, x143) :|: x137 = x143 && x136 = x142 && x135 = x141 && x134 = x140 && x133 = x139 && x132 = x138 (14) l5(x144, x145, x146, x147, x148, x149) -> l3(x150, x151, x152, x153, x154, x155) :|: x149 = x155 && x148 = x154 && x147 = x153 && x146 = x152 && x145 = x151 && x144 = x150 (15) l5(x156, x157, x158, x159, x160, x161) -> l4(x162, x163, x164, x165, x166, x167) :|: x161 = x167 && x160 = x166 && x159 = x165 && x158 = x164 && x157 = x163 && x156 = x162 (16) l5(x168, x169, x170, x171, x172, x173) -> l2(x174, x175, x176, x177, x178, x179) :|: x173 = x179 && x172 = x178 && x171 = x177 && x170 = x176 && x169 = x175 && x168 = x174 (17) l6(x180, x181, x182, x183, x184, x185) -> l5(x186, x187, x188, x189, x190, x191) :|: x185 = x191 && x184 = x190 && x183 = x189 && x182 = x188 && x181 = x187 && x180 = x186 Arcs: (2) -> (11) (8) -> (1), (2) (9) -> (3), (5), (6), (7) (10) -> (3), (4), (5), (7) (11) -> (8), (9), (10) (13) -> (1), (2) (14) -> (3), (4), (5), (6), (7) (15) -> (8), (9), (10) (16) -> (11) (17) -> (12), (13), (14), (15), (16) This digraph is fully evaluated! ---------------------------------------- (4) Obligation: Termination digraph: Nodes: (1) l0(x, x1, x2, x3, x4, x5) -> l2(x6, x7, x8, x9, x10, x11) :|: x3 = x9 && x2 = x8 && x11 = -1 + x7 && x10 = -1 + x6 && x7 = x5 && x6 = x4 (2) l4(x72, x73, x74, x75, x76, x77) -> l0(x78, x79, x80, x81, x82, x83) :|: x75 = x81 && x74 = x80 && x83 = x79 && x82 = x78 && 1 <= x79 && 1 <= x78 && x79 = x77 && x78 = x76 (3) l2(x108, x109, x110, x111, x112, x113) -> l4(x114, x115, x116, x117, x118, x119) :|: x111 = x117 && x110 = x116 && x119 = x115 && x118 = x114 && x115 = x113 && x114 = x112 Arcs: (1) -> (3) (2) -> (1) (3) -> (2) This digraph is fully evaluated! ---------------------------------------- (5) IntTRSCompressionProof (EQUIVALENT) Compressed rules. ---------------------------------------- (6) Obligation: Rules: l4(x72:0, x73:0, x116:0, x117:0, x6:0, x77:0) -> l4(-1 + x6:0, -1 + x77:0, x116:0, x117:0, -1 + x6:0, -1 + x77:0) :|: x6:0 > 0 && x77:0 > 0 ---------------------------------------- (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, x6) -> l4(x5, x6) ---------------------------------------- (8) Obligation: Rules: l4(x6:0, x77:0) -> l4(-1 + x6:0, -1 + x77:0) :|: x6:0 > 0 && x77:0 > 0 ---------------------------------------- (9) TempFilterProof (SOUND) Used the following sort dictionary for filtering: l4(INTEGER, INTEGER) Replaced non-predefined constructor symbols by 0. ---------------------------------------- (10) Obligation: Rules: l4(x6:0, x77:0) -> l4(c, c1) :|: c1 = -1 + x77:0 && c = -1 + x6:0 && (x6:0 > 0 && x77:0 > 0) ---------------------------------------- (11) PolynomialOrderProcessor (EQUIVALENT) Found the following polynomial interpretation: [l4(x, x1)] = x1 The following rules are decreasing: l4(x6:0, x77:0) -> l4(c, c1) :|: c1 = -1 + x77:0 && c = -1 + x6:0 && (x6:0 > 0 && x77:0 > 0) The following rules are bounded: l4(x6:0, x77:0) -> l4(c, c1) :|: c1 = -1 + x77:0 && c = -1 + x6:0 && (x6:0 > 0 && x77:0 > 0) ---------------------------------------- (12) YES