YES proof of prog.inttrs # AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty Termination of the given IRSwT could be proven: (0) IRSwT (1) IRSFormatTransformerProof [EQUIVALENT, 0 ms] (2) IRSwT (3) IRSwTTerminationDigraphProof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, 1243 ms] (4) IRSwT (5) IntTRSCompressionProof [EQUIVALENT, 0 ms] (6) IRSwT (7) IntTRSUnneededArgumentFilterProof [EQUIVALENT, 0 ms] (8) IRSwT (9) TempFilterProof [SOUND, 41 ms] (10) IntTRS (11) RankingReductionPairProof [EQUIVALENT, 18 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 l2(x, x1, x2, x3, x4, x5) -> l3(x6, x7, x8, x9, x10, x11) :|: x3 = x9 && x2 = x8 && x11 = 1 + x7 && x10 = 2 * x6 && x7 = x5 && x6 = x4 l3(x12, x13, x14, x15, x16, x17) -> l0(x18, x19, x20, x21, x22, x23) :|: x15 = x21 && x14 = x20 && x23 = x19 && x22 = x18 && x18 <= 2 && x19 = x17 && x18 = x16 l3(x24, x25, x26, x27, x28, x29) -> l0(x30, x31, x32, x33, x34, x35) :|: x27 = x33 && x26 = x32 && x35 = x31 && x34 = x30 && x31 <= x30 && x31 = x29 && x30 = x28 l3(x36, x37, x38, x39, x40, x41) -> l2(x42, x43, x44, x45, x46, x47) :|: x39 = x45 && x38 = x44 && x47 = x43 && x46 = x42 && 3 <= x42 && 1 + x42 <= x43 && x43 = x41 && x42 = x40 l4(x48, x49, x50, x51, x52, x53) -> l3(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) -> l4(x66, x67, x68, x69, x70, x71) :|: x71 = x69 && x70 = x68 && x69 = x69 && x68 = x68 && x67 = x65 && x66 = x64 l5(x72, x73, x74, x75, x76, x77) -> l1(x78, x79, x80, x81, x82, x83) :|: x77 = x83 && x76 = x82 && x75 = x81 && x74 = x80 && x73 = x79 && x72 = x78 l5(x84, x85, x86, x87, x88, x89) -> l0(x90, x91, x92, x93, x94, x95) :|: x89 = x95 && x88 = x94 && x87 = x93 && x86 = x92 && x85 = x91 && x84 = x90 l5(x96, x97, x98, x99, x100, x101) -> l2(x102, x103, x104, x105, x106, x107) :|: x101 = x107 && x100 = x106 && x99 = x105 && x98 = x104 && x97 = x103 && x96 = x102 l5(x108, x109, x110, x111, x112, x113) -> l3(x114, x115, x116, x117, x118, x119) :|: x113 = x119 && x112 = x118 && x111 = x117 && x110 = x116 && x109 = x115 && x108 = x114 l5(x120, x121, x122, x123, x124, x125) -> l4(x126, x127, x128, x129, x130, x131) :|: x125 = x131 && x124 = x130 && x123 = x129 && x122 = x128 && x121 = x127 && x120 = x126 l6(x132, x133, x134, x135, x136, x137) -> l5(x138, x139, x140, x141, x142, x143) :|: x137 = x143 && x136 = x142 && x135 = x141 && x134 = x140 && x133 = x139 && x132 = x138 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 l2(x, x1, x2, x3, x4, x5) -> l3(x6, x7, x8, x9, x10, x11) :|: x3 = x9 && x2 = x8 && x11 = 1 + x7 && x10 = 2 * x6 && x7 = x5 && x6 = x4 l3(x12, x13, x14, x15, x16, x17) -> l0(x18, x19, x20, x21, x22, x23) :|: x15 = x21 && x14 = x20 && x23 = x19 && x22 = x18 && x18 <= 2 && x19 = x17 && x18 = x16 l3(x24, x25, x26, x27, x28, x29) -> l0(x30, x31, x32, x33, x34, x35) :|: x27 = x33 && x26 = x32 && x35 = x31 && x34 = x30 && x31 <= x30 && x31 = x29 && x30 = x28 l3(x36, x37, x38, x39, x40, x41) -> l2(x42, x43, x44, x45, x46, x47) :|: x39 = x45 && x38 = x44 && x47 = x43 && x46 = x42 && 3 <= x42 && 1 + x42 <= x43 && x43 = x41 && x42 = x40 l4(x48, x49, x50, x51, x52, x53) -> l3(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) -> l4(x66, x67, x68, x69, x70, x71) :|: x71 = x69 && x70 = x68 && x69 = x69 && x68 = x68 && x67 = x65 && x66 = x64 l5(x72, x73, x74, x75, x76, x77) -> l1(x78, x79, x80, x81, x82, x83) :|: x77 = x83 && x76 = x82 && x75 = x81 && x74 = x80 && x73 = x79 && x72 = x78 l5(x84, x85, x86, x87, x88, x89) -> l0(x90, x91, x92, x93, x94, x95) :|: x89 = x95 && x88 = x94 && x87 = x93 && x86 = x92 && x85 = x91 && x84 = x90 l5(x96, x97, x98, x99, x100, x101) -> l2(x102, x103, x104, x105, x106, x107) :|: x101 = x107 && x100 = x106 && x99 = x105 && x98 = x104 && x97 = x103 && x96 = x102 l5(x108, x109, x110, x111, x112, x113) -> l3(x114, x115, x116, x117, x118, x119) :|: x113 = x119 && x112 = x118 && x111 = x117 && x110 = x116 && x109 = x115 && x108 = x114 l5(x120, x121, x122, x123, x124, x125) -> l4(x126, x127, x128, x129, x130, x131) :|: x125 = x131 && x124 = x130 && x123 = x129 && x122 = x128 && x121 = x127 && x120 = x126 l6(x132, x133, x134, x135, x136, x137) -> l5(x138, x139, x140, x141, x142, x143) :|: x137 = x143 && x136 = x142 && x135 = x141 && x134 = x140 && x133 = x139 && x132 = x138 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) l2(x, x1, x2, x3, x4, x5) -> l3(x6, x7, x8, x9, x10, x11) :|: x3 = x9 && x2 = x8 && x11 = 1 + x7 && x10 = 2 * x6 && x7 = x5 && x6 = x4 (3) l3(x12, x13, x14, x15, x16, x17) -> l0(x18, x19, x20, x21, x22, x23) :|: x15 = x21 && x14 = x20 && x23 = x19 && x22 = x18 && x18 <= 2 && x19 = x17 && x18 = x16 (4) l3(x24, x25, x26, x27, x28, x29) -> l0(x30, x31, x32, x33, x34, x35) :|: x27 = x33 && x26 = x32 && x35 = x31 && x34 = x30 && x31 <= x30 && x31 = x29 && x30 = x28 (5) l3(x36, x37, x38, x39, x40, x41) -> l2(x42, x43, x44, x45, x46, x47) :|: x39 = x45 && x38 = x44 && x47 = x43 && x46 = x42 && 3 <= x42 && 1 + x42 <= x43 && x43 = x41 && x42 = x40 (6) l4(x48, x49, x50, x51, x52, x53) -> l3(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) -> l4(x66, x67, x68, x69, x70, x71) :|: x71 = x69 && x70 = x68 && x69 = x69 && x68 = x68 && x67 = x65 && x66 = x64 (8) l5(x72, x73, x74, x75, x76, x77) -> l1(x78, x79, x80, x81, x82, x83) :|: x77 = x83 && x76 = x82 && x75 = x81 && x74 = x80 && x73 = x79 && x72 = x78 (9) l5(x84, x85, x86, x87, x88, x89) -> l0(x90, x91, x92, x93, x94, x95) :|: x89 = x95 && x88 = x94 && x87 = x93 && x86 = x92 && x85 = x91 && x84 = x90 (10) l5(x96, x97, x98, x99, x100, x101) -> l2(x102, x103, x104, x105, x106, x107) :|: x101 = x107 && x100 = x106 && x99 = x105 && x98 = x104 && x97 = x103 && x96 = x102 (11) l5(x108, x109, x110, x111, x112, x113) -> l3(x114, x115, x116, x117, x118, x119) :|: x113 = x119 && x112 = x118 && x111 = x117 && x110 = x116 && x109 = x115 && x108 = x114 (12) l5(x120, x121, x122, x123, x124, x125) -> l4(x126, x127, x128, x129, x130, x131) :|: x125 = x131 && x124 = x130 && x123 = x129 && x122 = x128 && x121 = x127 && x120 = x126 (13) l6(x132, x133, x134, x135, x136, x137) -> l5(x138, x139, x140, x141, x142, x143) :|: x137 = x143 && x136 = x142 && x135 = x141 && x134 = x140 && x133 = x139 && x132 = x138 Arcs: (2) -> (3), (4), (5) (3) -> (1) (4) -> (1) (5) -> (2) (6) -> (3), (4), (5) (7) -> (6) (9) -> (1) (10) -> (2) (11) -> (3), (4), (5) (12) -> (6) (13) -> (7), (8), (9), (10), (11), (12) This digraph is fully evaluated! ---------------------------------------- (4) Obligation: Termination digraph: Nodes: (1) l2(x, x1, x2, x3, x4, x5) -> l3(x6, x7, x8, x9, x10, x11) :|: x3 = x9 && x2 = x8 && x11 = 1 + x7 && x10 = 2 * x6 && x7 = x5 && x6 = x4 (2) l3(x36, x37, x38, x39, x40, x41) -> l2(x42, x43, x44, x45, x46, x47) :|: x39 = x45 && x38 = x44 && x47 = x43 && x46 = x42 && 3 <= x42 && 1 + x42 <= x43 && x43 = x41 && x42 = x40 Arcs: (1) -> (2) (2) -> (1) This digraph is fully evaluated! ---------------------------------------- (5) IntTRSCompressionProof (EQUIVALENT) Compressed rules. ---------------------------------------- (6) Obligation: Rules: l2(x:0, x1:0, x2:0, x3:0, x4:0, x5:0) -> l2(2 * x4:0, 1 + x5:0, x2:0, x3:0, 2 * x4:0, 1 + x5:0) :|: 1 + x5:0 >= 1 + 2 * x4:0 && 2 * x4:0 > 2 ---------------------------------------- (7) IntTRSUnneededArgumentFilterProof (EQUIVALENT) Some arguments are removed because they cannot influence termination. We removed arguments according to the following replacements: l2(x1, x2, x3, x4, x5, x6) -> l2(x5, x6) ---------------------------------------- (8) Obligation: Rules: l2(x4:0, x5:0) -> l2(2 * x4:0, 1 + x5:0) :|: 1 + x5:0 >= 1 + 2 * x4:0 && 2 * x4:0 > 2 ---------------------------------------- (9) TempFilterProof (SOUND) Used the following sort dictionary for filtering: l2(INTEGER, INTEGER) Replaced non-predefined constructor symbols by 0. ---------------------------------------- (10) Obligation: Rules: l2(x4:0, x5:0) -> l2(c, c1) :|: c1 = 1 + x5:0 && c = 2 * x4:0 && (1 + x5:0 >= 1 + 2 * x4:0 && 2 * x4:0 > 2) ---------------------------------------- (11) RankingReductionPairProof (EQUIVALENT) Interpretation: [ l2 ] = 1/3*l2_2 + -2/3*l2_1 The following rules are decreasing: l2(x4:0, x5:0) -> l2(c, c1) :|: c1 = 1 + x5:0 && c = 2 * x4:0 && (1 + x5:0 >= 1 + 2 * x4:0 && 2 * x4:0 > 2) The following rules are bounded: l2(x4:0, x5:0) -> l2(c, c1) :|: c1 = 1 + x5:0 && c = 2 * x4:0 && (1 + x5:0 >= 1 + 2 * x4:0 && 2 * x4:0 > 2) ---------------------------------------- (12) YES