MAYBE proof of prog.inttrs # AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty Termination of the given IRSwT could not be shown: (0) IRSwT (1) IRSFormatTransformerProof [EQUIVALENT, 0 ms] (2) IRSwT (3) IRSwTTerminationDigraphProof [EQUIVALENT, 1624 ms] (4) AND (5) IRSwT (6) IntTRSCompressionProof [EQUIVALENT, 0 ms] (7) IRSwT (8) IntTRSUnneededArgumentFilterProof [EQUIVALENT, 0 ms] (9) IRSwT (10) TempFilterProof [SOUND, 97 ms] (11) IRSwT (12) IRSwTTerminationDigraphProof [EQUIVALENT, 0 ms] (13) IRSwT (14) IRSwT (15) IntTRSCompressionProof [EQUIVALENT, 0 ms] (16) IRSwT (17) IntTRSUnneededArgumentFilterProof [EQUIVALENT, 0 ms] (18) IRSwT (19) TempFilterProof [SOUND, 17 ms] (20) IntTRS (21) RankingReductionPairProof [EQUIVALENT, 0 ms] (22) YES ---------------------------------------- (0) Obligation: Rules: l0(iHAT0, jHAT0, kHAT0, tmpHAT0, tmp___0HAT0) -> l1(iHATpost, jHATpost, kHATpost, tmpHATpost, tmp___0HATpost) :|: tmpHAT0 = tmpHATpost && kHAT0 = kHATpost && jHAT0 = jHATpost && iHATpost = 0 && tmp___0HATpost = tmp___0HATpost && kHAT0 <= 100 l0(x, x1, x2, x3, x4) -> l2(x5, x6, x7, x8, x9) :|: x4 = x9 && x3 = x8 && x2 = x7 && x1 = x6 && x = x5 && 101 <= x2 l3(x10, x11, x12, x13, x14) -> l0(x15, x16, x17, x18, x19) :|: x14 = x19 && x13 = x18 && x12 = x17 && x11 = x16 && x10 = x15 && 1 <= x12 l3(x20, x21, x22, x23, x24) -> l2(x25, x26, x27, x28, x29) :|: x24 = x29 && x23 = x28 && x22 = x27 && x21 = x26 && x20 = x25 && x22 <= 0 l1(x30, x31, x32, x33, x34) -> l4(x35, x36, x37, x38, x39) :|: x34 = x39 && x33 = x38 && x32 = x37 && x31 = x36 && x30 = x35 l5(x40, x41, x42, x43, x44) -> l2(x45, x46, x47, x48, x49) :|: x44 = x49 && x43 = x48 && x42 = x47 && x41 = x46 && x40 = x45 && x40 <= x41 l5(x50, x51, x52, x53, x54) -> l6(x55, x56, x57, x58, x59) :|: x54 = x59 && x53 = x58 && x52 = x57 && x50 = x55 && x56 = 1 + x51 && 1 + x51 <= x50 l7(x60, x61, x62, x63, x64) -> l6(x65, x66, x67, x68, x69) :|: x64 = x69 && x63 = x68 && x62 = x67 && x66 = 0 && x65 = -1 + x60 l6(x70, x71, x72, x73, x74) -> l5(x75, x76, x77, x78, x79) :|: x74 = x79 && x73 = x78 && x72 = x77 && x71 = x76 && x70 = x75 l8(x80, x81, x82, x83, x84) -> l1(x85, x86, x87, x88, x89) :|: x84 = x89 && x83 = x88 && x82 = x87 && x81 = x86 && x85 = 1 + x80 l8(x90, x91, x92, x93, x94) -> l7(x95, x96, x97, x98, x99) :|: x94 = x99 && x93 = x98 && x92 = x97 && x91 = x96 && x90 = x95 l4(x100, x101, x102, x103, x104) -> l7(x105, x106, x107, x108, x109) :|: x104 = x109 && x103 = x108 && x102 = x107 && x101 = x106 && x100 = x105 && x102 <= x100 && x100 <= x102 l4(x110, x111, x112, x113, x114) -> l8(x115, x116, x117, x118, x119) :|: x114 = x119 && x113 = x118 && x112 = x117 && x111 = x116 && x110 = x115 && 1 + x112 <= x110 l4(x120, x121, x122, x123, x124) -> l8(x125, x126, x127, x128, x129) :|: x124 = x129 && x123 = x128 && x122 = x127 && x121 = x126 && x120 = x125 && 1 + x120 <= x122 l9(x130, x131, x132, x133, x134) -> l3(x135, x136, x137, x138, x139) :|: x134 = x139 && x131 = x136 && x130 = x135 && x137 = x138 && x138 = x138 l10(x140, x141, x142, x143, x144) -> l9(x145, x146, x147, x148, x149) :|: x144 = x149 && x143 = x148 && x142 = x147 && x141 = x146 && x140 = x145 Start term: l10(iHAT0, jHAT0, kHAT0, tmpHAT0, tmp___0HAT0) ---------------------------------------- (1) IRSFormatTransformerProof (EQUIVALENT) Reformatted IRS to match normalized format (transformed away non-linear left-hand sides, !=, / and %). ---------------------------------------- (2) Obligation: Rules: l0(iHAT0, jHAT0, kHAT0, tmpHAT0, tmp___0HAT0) -> l1(iHATpost, jHATpost, kHATpost, tmpHATpost, tmp___0HATpost) :|: tmpHAT0 = tmpHATpost && kHAT0 = kHATpost && jHAT0 = jHATpost && iHATpost = 0 && tmp___0HATpost = tmp___0HATpost && kHAT0 <= 100 l0(x, x1, x2, x3, x4) -> l2(x5, x6, x7, x8, x9) :|: x4 = x9 && x3 = x8 && x2 = x7 && x1 = x6 && x = x5 && 101 <= x2 l3(x10, x11, x12, x13, x14) -> l0(x15, x16, x17, x18, x19) :|: x14 = x19 && x13 = x18 && x12 = x17 && x11 = x16 && x10 = x15 && 1 <= x12 l3(x20, x21, x22, x23, x24) -> l2(x25, x26, x27, x28, x29) :|: x24 = x29 && x23 = x28 && x22 = x27 && x21 = x26 && x20 = x25 && x22 <= 0 l1(x30, x31, x32, x33, x34) -> l4(x35, x36, x37, x38, x39) :|: x34 = x39 && x33 = x38 && x32 = x37 && x31 = x36 && x30 = x35 l5(x40, x41, x42, x43, x44) -> l2(x45, x46, x47, x48, x49) :|: x44 = x49 && x43 = x48 && x42 = x47 && x41 = x46 && x40 = x45 && x40 <= x41 l5(x50, x51, x52, x53, x54) -> l6(x55, x56, x57, x58, x59) :|: x54 = x59 && x53 = x58 && x52 = x57 && x50 = x55 && x56 = 1 + x51 && 1 + x51 <= x50 l7(x60, x61, x62, x63, x64) -> l6(x65, x66, x67, x68, x69) :|: x64 = x69 && x63 = x68 && x62 = x67 && x66 = 0 && x65 = -1 + x60 l6(x70, x71, x72, x73, x74) -> l5(x75, x76, x77, x78, x79) :|: x74 = x79 && x73 = x78 && x72 = x77 && x71 = x76 && x70 = x75 l8(x80, x81, x82, x83, x84) -> l1(x85, x86, x87, x88, x89) :|: x84 = x89 && x83 = x88 && x82 = x87 && x81 = x86 && x85 = 1 + x80 l8(x90, x91, x92, x93, x94) -> l7(x95, x96, x97, x98, x99) :|: x94 = x99 && x93 = x98 && x92 = x97 && x91 = x96 && x90 = x95 l4(x100, x101, x102, x103, x104) -> l7(x105, x106, x107, x108, x109) :|: x104 = x109 && x103 = x108 && x102 = x107 && x101 = x106 && x100 = x105 && x102 <= x100 && x100 <= x102 l4(x110, x111, x112, x113, x114) -> l8(x115, x116, x117, x118, x119) :|: x114 = x119 && x113 = x118 && x112 = x117 && x111 = x116 && x110 = x115 && 1 + x112 <= x110 l4(x120, x121, x122, x123, x124) -> l8(x125, x126, x127, x128, x129) :|: x124 = x129 && x123 = x128 && x122 = x127 && x121 = x126 && x120 = x125 && 1 + x120 <= x122 l9(x130, x131, x132, x133, x134) -> l3(x135, x136, x137, x138, x139) :|: x134 = x139 && x131 = x136 && x130 = x135 && x137 = x138 && x138 = x138 l10(x140, x141, x142, x143, x144) -> l9(x145, x146, x147, x148, x149) :|: x144 = x149 && x143 = x148 && x142 = x147 && x141 = x146 && x140 = x145 Start term: l10(iHAT0, jHAT0, kHAT0, tmpHAT0, tmp___0HAT0) ---------------------------------------- (3) IRSwTTerminationDigraphProof (EQUIVALENT) Constructed termination digraph! Nodes: (1) l0(iHAT0, jHAT0, kHAT0, tmpHAT0, tmp___0HAT0) -> l1(iHATpost, jHATpost, kHATpost, tmpHATpost, tmp___0HATpost) :|: tmpHAT0 = tmpHATpost && kHAT0 = kHATpost && jHAT0 = jHATpost && iHATpost = 0 && tmp___0HATpost = tmp___0HATpost && kHAT0 <= 100 (2) l0(x, x1, x2, x3, x4) -> l2(x5, x6, x7, x8, x9) :|: x4 = x9 && x3 = x8 && x2 = x7 && x1 = x6 && x = x5 && 101 <= x2 (3) l3(x10, x11, x12, x13, x14) -> l0(x15, x16, x17, x18, x19) :|: x14 = x19 && x13 = x18 && x12 = x17 && x11 = x16 && x10 = x15 && 1 <= x12 (4) l3(x20, x21, x22, x23, x24) -> l2(x25, x26, x27, x28, x29) :|: x24 = x29 && x23 = x28 && x22 = x27 && x21 = x26 && x20 = x25 && x22 <= 0 (5) l1(x30, x31, x32, x33, x34) -> l4(x35, x36, x37, x38, x39) :|: x34 = x39 && x33 = x38 && x32 = x37 && x31 = x36 && x30 = x35 (6) l5(x40, x41, x42, x43, x44) -> l2(x45, x46, x47, x48, x49) :|: x44 = x49 && x43 = x48 && x42 = x47 && x41 = x46 && x40 = x45 && x40 <= x41 (7) l5(x50, x51, x52, x53, x54) -> l6(x55, x56, x57, x58, x59) :|: x54 = x59 && x53 = x58 && x52 = x57 && x50 = x55 && x56 = 1 + x51 && 1 + x51 <= x50 (8) l7(x60, x61, x62, x63, x64) -> l6(x65, x66, x67, x68, x69) :|: x64 = x69 && x63 = x68 && x62 = x67 && x66 = 0 && x65 = -1 + x60 (9) l6(x70, x71, x72, x73, x74) -> l5(x75, x76, x77, x78, x79) :|: x74 = x79 && x73 = x78 && x72 = x77 && x71 = x76 && x70 = x75 (10) l8(x80, x81, x82, x83, x84) -> l1(x85, x86, x87, x88, x89) :|: x84 = x89 && x83 = x88 && x82 = x87 && x81 = x86 && x85 = 1 + x80 (11) l8(x90, x91, x92, x93, x94) -> l7(x95, x96, x97, x98, x99) :|: x94 = x99 && x93 = x98 && x92 = x97 && x91 = x96 && x90 = x95 (12) l4(x100, x101, x102, x103, x104) -> l7(x105, x106, x107, x108, x109) :|: x104 = x109 && x103 = x108 && x102 = x107 && x101 = x106 && x100 = x105 && x102 <= x100 && x100 <= x102 (13) l4(x110, x111, x112, x113, x114) -> l8(x115, x116, x117, x118, x119) :|: x114 = x119 && x113 = x118 && x112 = x117 && x111 = x116 && x110 = x115 && 1 + x112 <= x110 (14) l4(x120, x121, x122, x123, x124) -> l8(x125, x126, x127, x128, x129) :|: x124 = x129 && x123 = x128 && x122 = x127 && x121 = x126 && x120 = x125 && 1 + x120 <= x122 (15) l9(x130, x131, x132, x133, x134) -> l3(x135, x136, x137, x138, x139) :|: x134 = x139 && x131 = x136 && x130 = x135 && x137 = x138 && x138 = x138 (16) l10(x140, x141, x142, x143, x144) -> l9(x145, x146, x147, x148, x149) :|: x144 = x149 && x143 = x148 && x142 = x147 && x141 = x146 && x140 = x145 Arcs: (1) -> (5) (3) -> (1), (2) (5) -> (12), (13), (14) (7) -> (9) (8) -> (9) (9) -> (6), (7) (10) -> (5) (11) -> (8) (12) -> (8) (13) -> (10), (11) (14) -> (10), (11) (15) -> (3), (4) (16) -> (15) This digraph is fully evaluated! ---------------------------------------- (4) Complex Obligation (AND) ---------------------------------------- (5) Obligation: Termination digraph: Nodes: (1) l1(x30, x31, x32, x33, x34) -> l4(x35, x36, x37, x38, x39) :|: x34 = x39 && x33 = x38 && x32 = x37 && x31 = x36 && x30 = x35 (2) l8(x80, x81, x82, x83, x84) -> l1(x85, x86, x87, x88, x89) :|: x84 = x89 && x83 = x88 && x82 = x87 && x81 = x86 && x85 = 1 + x80 (3) l4(x120, x121, x122, x123, x124) -> l8(x125, x126, x127, x128, x129) :|: x124 = x129 && x123 = x128 && x122 = x127 && x121 = x126 && x120 = x125 && 1 + x120 <= x122 (4) l4(x110, x111, x112, x113, x114) -> l8(x115, x116, x117, x118, x119) :|: x114 = x119 && x113 = x118 && x112 = x117 && x111 = x116 && x110 = x115 && 1 + x112 <= x110 Arcs: (1) -> (3), (4) (2) -> (1) (3) -> (2) (4) -> (2) This digraph is fully evaluated! ---------------------------------------- (6) IntTRSCompressionProof (EQUIVALENT) Compressed rules. ---------------------------------------- (7) Obligation: Rules: l8(x80:0, x116:0, x117:0, x118:0, x119:0) -> l8(1 + x80:0, x116:0, x117:0, x118:0, x119:0) :|: 1 + x80:0 >= 1 + x117:0 l8(x, x1, x2, x3, x4) -> l8(1 + x, x1, x2, x3, x4) :|: x2 >= 1 + (1 + x) ---------------------------------------- (8) IntTRSUnneededArgumentFilterProof (EQUIVALENT) Some arguments are removed because they cannot influence termination. We removed arguments according to the following replacements: l8(x1, x2, x3, x4, x5) -> l8(x1, x3) ---------------------------------------- (9) Obligation: Rules: l8(x80:0, x117:0) -> l8(1 + x80:0, x117:0) :|: 1 + x80:0 >= 1 + x117:0 l8(x, x2) -> l8(1 + x, x2) :|: x2 >= 1 + (1 + x) ---------------------------------------- (10) TempFilterProof (SOUND) Used the following sort dictionary for filtering: l8(INTEGER, INTEGER) Replaced non-predefined constructor symbols by 0.The following proof was generated: # AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty Termination of the given IntTRS could not be shown: - IntTRS - RankingReductionPairProof Rules: l8(x80:0, x117:0) -> l8(c, x117:0) :|: c = 1 + x80:0 && 1 + x80:0 >= 1 + x117:0 l8(x, x2) -> l8(c1, x2) :|: c1 = 1 + x && x2 >= 1 + (1 + x) Interpretation: [ l8 ] = -3*l8_1 + 3*l8_2 The following rules are decreasing: l8(x80:0, x117:0) -> l8(c, x117:0) :|: c = 1 + x80:0 && 1 + x80:0 >= 1 + x117:0 l8(x, x2) -> l8(c1, x2) :|: c1 = 1 + x && x2 >= 1 + (1 + x) The following rules are bounded: l8(x, x2) -> l8(c1, x2) :|: c1 = 1 + x && x2 >= 1 + (1 + x) - IntTRS - RankingReductionPairProof - IntTRS Rules: l8(x80:0, x117:0) -> l8(c, x117:0) :|: c = 1 + x80:0 && 1 + x80:0 >= 1 + x117:0 ---------------------------------------- (11) Obligation: Rules: l8(x80:0, x117:0) -> l8(1 + x80:0, x117:0) :|: 1 + x80:0 >= 1 + x117:0 ---------------------------------------- (12) IRSwTTerminationDigraphProof (EQUIVALENT) Constructed termination digraph! Nodes: (1) l8(x80:0, x117:0) -> l8(1 + x80:0, x117:0) :|: 1 + x80:0 >= 1 + x117:0 Arcs: (1) -> (1) This digraph is fully evaluated! ---------------------------------------- (13) Obligation: Termination digraph: Nodes: (1) l8(x80:0, x117:0) -> l8(1 + x80:0, x117:0) :|: 1 + x80:0 >= 1 + x117:0 Arcs: (1) -> (1) This digraph is fully evaluated! ---------------------------------------- (14) Obligation: Termination digraph: Nodes: (1) l6(x70, x71, x72, x73, x74) -> l5(x75, x76, x77, x78, x79) :|: x74 = x79 && x73 = x78 && x72 = x77 && x71 = x76 && x70 = x75 (2) l5(x50, x51, x52, x53, x54) -> l6(x55, x56, x57, x58, x59) :|: x54 = x59 && x53 = x58 && x52 = x57 && x50 = x55 && x56 = 1 + x51 && 1 + x51 <= x50 Arcs: (1) -> (2) (2) -> (1) This digraph is fully evaluated! ---------------------------------------- (15) IntTRSCompressionProof (EQUIVALENT) Compressed rules. ---------------------------------------- (16) Obligation: Rules: l6(x55:0, x71:0, x57:0, x58:0, x59:0) -> l6(x55:0, 1 + x71:0, x57:0, x58:0, x59:0) :|: x55:0 >= 1 + x71:0 ---------------------------------------- (17) 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) -> l6(x1, x2) ---------------------------------------- (18) Obligation: Rules: l6(x55:0, x71:0) -> l6(x55:0, 1 + x71:0) :|: x55:0 >= 1 + x71:0 ---------------------------------------- (19) TempFilterProof (SOUND) Used the following sort dictionary for filtering: l6(INTEGER, INTEGER) Replaced non-predefined constructor symbols by 0. ---------------------------------------- (20) Obligation: Rules: l6(x55:0, x71:0) -> l6(x55:0, c) :|: c = 1 + x71:0 && x55:0 >= 1 + x71:0 ---------------------------------------- (21) RankingReductionPairProof (EQUIVALENT) Interpretation: [ l6 ] = l6_1 + -1*l6_2 The following rules are decreasing: l6(x55:0, x71:0) -> l6(x55:0, c) :|: c = 1 + x71:0 && x55:0 >= 1 + x71:0 The following rules are bounded: l6(x55:0, x71:0) -> l6(x55:0, c) :|: c = 1 + x71:0 && x55:0 >= 1 + x71:0 ---------------------------------------- (22) YES