YES proof of prog.inttrs # AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty Termination of the given IRSwT could be proven: (0) IRSwT (1) IRSFormatTransformerProof [EQUIVALENT, 0 ms] (2) IRSwT (3) IRSwTTerminationDigraphProof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, 1780 ms] (4) IRSwT (5) IntTRSCompressionProof [EQUIVALENT, 11 ms] (6) IRSwT (7) IntTRSUnneededArgumentFilterProof [EQUIVALENT, 0 ms] (8) IRSwT (9) TempFilterProof [SOUND, 16 ms] (10) IntTRS (11) PolynomialOrderProcessor [EQUIVALENT, 0 ms] (12) YES ---------------------------------------- (0) Obligation: Rules: l0(__const_1023HAT0, bHAT0, iHAT0, jHAT0, nHAT0, tmpHAT0) -> l1(__const_1023HATpost, bHATpost, iHATpost, jHATpost, nHATpost, tmpHATpost) :|: tmpHAT0 = tmpHATpost && nHAT0 = nHATpost && jHAT0 = jHATpost && iHAT0 = iHATpost && bHAT0 = bHATpost && __const_1023HAT0 = __const_1023HATpost && 1 + nHAT0 <= iHAT0 l0(x, x1, x2, x3, x4, x5) -> l2(x6, x7, x8, x9, x10, x11) :|: x5 = x11 && x4 = x10 && x1 = x7 && x = x6 && x9 = 2 + x3 && x8 = 1 + x2 && x2 <= x4 l3(x12, x13, x14, x15, x16, x17) -> l2(x18, x19, x20, x21, x22, x23) :|: x17 = x23 && x16 = x22 && x15 = x21 && x13 = x19 && x12 = x18 && x20 = 0 l4(x24, x25, x26, x27, x28, x29) -> l3(x30, x31, x32, x33, x34, x35) :|: x29 = x35 && x27 = x33 && x26 = x32 && x25 = x31 && x24 = x30 && x34 = 0 l5(x36, x37, x38, x39, x40, x41) -> l3(x42, x43, x44, x45, x46, x47) :|: x41 = x47 && x39 = x45 && x38 = x44 && x37 = x43 && x36 = x42 && x46 = x36 && 0 <= x41 && x41 <= 0 l5(x48, x49, x50, x51, x52, x53) -> l4(x54, x55, x56, x57, x58, x59) :|: x53 = x59 && x52 = x58 && x51 = x57 && x50 = x56 && x49 = x55 && x48 = x54 && 1 <= x53 l5(x60, x61, x62, x63, x64, x65) -> l4(x66, x67, x68, x69, x70, x71) :|: x65 = x71 && x64 = x70 && x63 = x69 && x62 = x68 && x61 = x67 && x60 = x66 && 1 + x65 <= 0 l2(x72, x73, x74, x75, x76, x77) -> l0(x78, x79, x80, x81, x82, x83) :|: x77 = x83 && x76 = x82 && x75 = x81 && x74 = x80 && x73 = x79 && x72 = x78 l6(x84, x85, x86, x87, x88, x89) -> l7(x90, x91, x92, x93, x94, x95) :|: x89 = x95 && x88 = x94 && x87 = x93 && x86 = x92 && x85 = x91 && x84 = x90 l8(x96, x97, x98, x99, x100, x101) -> l6(x102, x103, x104, x105, x106, x107) :|: x101 = x107 && x100 = x106 && x99 = x105 && x98 = x104 && x97 = x103 && x96 = x102 && x96 <= x97 l8(x108, x109, x110, x111, x112, x113) -> l6(x114, x115, x116, x117, x118, x119) :|: x113 = x119 && x112 = x118 && x111 = x117 && x110 = x116 && x109 = x115 && x108 = x114 && 1 + x109 <= x108 l1(x120, x121, x122, x123, x124, x125) -> l6(x126, x127, x128, x129, x130, x131) :|: x125 = x131 && x124 = x130 && x123 = x129 && x122 = x128 && x121 = x127 && x120 = x126 && 1 + x121 <= 0 l1(x132, x133, x134, x135, x136, x137) -> l8(x138, x139, x140, x141, x142, x143) :|: x137 = x143 && x136 = x142 && x135 = x141 && x134 = x140 && x133 = x139 && x132 = x138 && 0 <= x133 l9(x144, x145, x146, x147, x148, x149) -> l5(x150, x151, x152, x153, x154, x155) :|: x148 = x154 && x147 = x153 && x146 = x152 && x145 = x151 && x144 = x150 && x155 = x155 l10(x156, x157, x158, x159, x160, x161) -> l9(x162, x163, x164, x165, x166, x167) :|: x161 = x167 && x160 = x166 && x159 = x165 && x158 = x164 && x157 = x163 && x156 = x162 Start term: l10(__const_1023HAT0, bHAT0, iHAT0, jHAT0, nHAT0, tmpHAT0) ---------------------------------------- (1) IRSFormatTransformerProof (EQUIVALENT) Reformatted IRS to match normalized format (transformed away non-linear left-hand sides, !=, / and %). ---------------------------------------- (2) Obligation: Rules: l0(__const_1023HAT0, bHAT0, iHAT0, jHAT0, nHAT0, tmpHAT0) -> l1(__const_1023HATpost, bHATpost, iHATpost, jHATpost, nHATpost, tmpHATpost) :|: tmpHAT0 = tmpHATpost && nHAT0 = nHATpost && jHAT0 = jHATpost && iHAT0 = iHATpost && bHAT0 = bHATpost && __const_1023HAT0 = __const_1023HATpost && 1 + nHAT0 <= iHAT0 l0(x, x1, x2, x3, x4, x5) -> l2(x6, x7, x8, x9, x10, x11) :|: x5 = x11 && x4 = x10 && x1 = x7 && x = x6 && x9 = 2 + x3 && x8 = 1 + x2 && x2 <= x4 l3(x12, x13, x14, x15, x16, x17) -> l2(x18, x19, x20, x21, x22, x23) :|: x17 = x23 && x16 = x22 && x15 = x21 && x13 = x19 && x12 = x18 && x20 = 0 l4(x24, x25, x26, x27, x28, x29) -> l3(x30, x31, x32, x33, x34, x35) :|: x29 = x35 && x27 = x33 && x26 = x32 && x25 = x31 && x24 = x30 && x34 = 0 l5(x36, x37, x38, x39, x40, x41) -> l3(x42, x43, x44, x45, x46, x47) :|: x41 = x47 && x39 = x45 && x38 = x44 && x37 = x43 && x36 = x42 && x46 = x36 && 0 <= x41 && x41 <= 0 l5(x48, x49, x50, x51, x52, x53) -> l4(x54, x55, x56, x57, x58, x59) :|: x53 = x59 && x52 = x58 && x51 = x57 && x50 = x56 && x49 = x55 && x48 = x54 && 1 <= x53 l5(x60, x61, x62, x63, x64, x65) -> l4(x66, x67, x68, x69, x70, x71) :|: x65 = x71 && x64 = x70 && x63 = x69 && x62 = x68 && x61 = x67 && x60 = x66 && 1 + x65 <= 0 l2(x72, x73, x74, x75, x76, x77) -> l0(x78, x79, x80, x81, x82, x83) :|: x77 = x83 && x76 = x82 && x75 = x81 && x74 = x80 && x73 = x79 && x72 = x78 l6(x84, x85, x86, x87, x88, x89) -> l7(x90, x91, x92, x93, x94, x95) :|: x89 = x95 && x88 = x94 && x87 = x93 && x86 = x92 && x85 = x91 && x84 = x90 l8(x96, x97, x98, x99, x100, x101) -> l6(x102, x103, x104, x105, x106, x107) :|: x101 = x107 && x100 = x106 && x99 = x105 && x98 = x104 && x97 = x103 && x96 = x102 && x96 <= x97 l8(x108, x109, x110, x111, x112, x113) -> l6(x114, x115, x116, x117, x118, x119) :|: x113 = x119 && x112 = x118 && x111 = x117 && x110 = x116 && x109 = x115 && x108 = x114 && 1 + x109 <= x108 l1(x120, x121, x122, x123, x124, x125) -> l6(x126, x127, x128, x129, x130, x131) :|: x125 = x131 && x124 = x130 && x123 = x129 && x122 = x128 && x121 = x127 && x120 = x126 && 1 + x121 <= 0 l1(x132, x133, x134, x135, x136, x137) -> l8(x138, x139, x140, x141, x142, x143) :|: x137 = x143 && x136 = x142 && x135 = x141 && x134 = x140 && x133 = x139 && x132 = x138 && 0 <= x133 l9(x144, x145, x146, x147, x148, x149) -> l5(x150, x151, x152, x153, x154, x155) :|: x148 = x154 && x147 = x153 && x146 = x152 && x145 = x151 && x144 = x150 && x155 = x155 l10(x156, x157, x158, x159, x160, x161) -> l9(x162, x163, x164, x165, x166, x167) :|: x161 = x167 && x160 = x166 && x159 = x165 && x158 = x164 && x157 = x163 && x156 = x162 Start term: l10(__const_1023HAT0, bHAT0, iHAT0, jHAT0, nHAT0, tmpHAT0) ---------------------------------------- (3) IRSwTTerminationDigraphProof (EQUIVALENT) Constructed termination digraph! Nodes: (1) l0(__const_1023HAT0, bHAT0, iHAT0, jHAT0, nHAT0, tmpHAT0) -> l1(__const_1023HATpost, bHATpost, iHATpost, jHATpost, nHATpost, tmpHATpost) :|: tmpHAT0 = tmpHATpost && nHAT0 = nHATpost && jHAT0 = jHATpost && iHAT0 = iHATpost && bHAT0 = bHATpost && __const_1023HAT0 = __const_1023HATpost && 1 + nHAT0 <= iHAT0 (2) l0(x, x1, x2, x3, x4, x5) -> l2(x6, x7, x8, x9, x10, x11) :|: x5 = x11 && x4 = x10 && x1 = x7 && x = x6 && x9 = 2 + x3 && x8 = 1 + x2 && x2 <= x4 (3) l3(x12, x13, x14, x15, x16, x17) -> l2(x18, x19, x20, x21, x22, x23) :|: x17 = x23 && x16 = x22 && x15 = x21 && x13 = x19 && x12 = x18 && x20 = 0 (4) l4(x24, x25, x26, x27, x28, x29) -> l3(x30, x31, x32, x33, x34, x35) :|: x29 = x35 && x27 = x33 && x26 = x32 && x25 = x31 && x24 = x30 && x34 = 0 (5) l5(x36, x37, x38, x39, x40, x41) -> l3(x42, x43, x44, x45, x46, x47) :|: x41 = x47 && x39 = x45 && x38 = x44 && x37 = x43 && x36 = x42 && x46 = x36 && 0 <= x41 && x41 <= 0 (6) l5(x48, x49, x50, x51, x52, x53) -> l4(x54, x55, x56, x57, x58, x59) :|: x53 = x59 && x52 = x58 && x51 = x57 && x50 = x56 && x49 = x55 && x48 = x54 && 1 <= x53 (7) l5(x60, x61, x62, x63, x64, x65) -> l4(x66, x67, x68, x69, x70, x71) :|: x65 = x71 && x64 = x70 && x63 = x69 && x62 = x68 && x61 = x67 && x60 = x66 && 1 + x65 <= 0 (8) l2(x72, x73, x74, x75, x76, x77) -> l0(x78, x79, x80, x81, x82, x83) :|: x77 = x83 && x76 = x82 && x75 = x81 && x74 = x80 && x73 = x79 && x72 = x78 (9) l6(x84, x85, x86, x87, x88, x89) -> l7(x90, x91, x92, x93, x94, x95) :|: x89 = x95 && x88 = x94 && x87 = x93 && x86 = x92 && x85 = x91 && x84 = x90 (10) l8(x96, x97, x98, x99, x100, x101) -> l6(x102, x103, x104, x105, x106, x107) :|: x101 = x107 && x100 = x106 && x99 = x105 && x98 = x104 && x97 = x103 && x96 = x102 && x96 <= x97 (11) l8(x108, x109, x110, x111, x112, x113) -> l6(x114, x115, x116, x117, x118, x119) :|: x113 = x119 && x112 = x118 && x111 = x117 && x110 = x116 && x109 = x115 && x108 = x114 && 1 + x109 <= x108 (12) l1(x120, x121, x122, x123, x124, x125) -> l6(x126, x127, x128, x129, x130, x131) :|: x125 = x131 && x124 = x130 && x123 = x129 && x122 = x128 && x121 = x127 && x120 = x126 && 1 + x121 <= 0 (13) l1(x132, x133, x134, x135, x136, x137) -> l8(x138, x139, x140, x141, x142, x143) :|: x137 = x143 && x136 = x142 && x135 = x141 && x134 = x140 && x133 = x139 && x132 = x138 && 0 <= x133 (14) l9(x144, x145, x146, x147, x148, x149) -> l5(x150, x151, x152, x153, x154, x155) :|: x148 = x154 && x147 = x153 && x146 = x152 && x145 = x151 && x144 = x150 && x155 = x155 (15) l10(x156, x157, x158, x159, x160, x161) -> l9(x162, x163, x164, x165, x166, x167) :|: x161 = x167 && x160 = x166 && x159 = x165 && x158 = x164 && x157 = x163 && x156 = x162 Arcs: (1) -> (12), (13) (2) -> (8) (3) -> (8) (4) -> (3) (5) -> (3) (6) -> (4) (7) -> (4) (8) -> (1), (2) (10) -> (9) (11) -> (9) (12) -> (9) (13) -> (10), (11) (14) -> (5), (6), (7) (15) -> (14) 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) :|: x5 = x11 && x4 = x10 && x1 = x7 && x = x6 && x9 = 2 + x3 && x8 = 1 + x2 && x2 <= x4 (2) l2(x72, x73, x74, x75, x76, x77) -> l0(x78, x79, x80, x81, x82, x83) :|: x77 = x83 && x76 = x82 && x75 = x81 && x74 = x80 && x73 = x79 && x72 = x78 Arcs: (1) -> (2) (2) -> (1) This digraph is fully evaluated! ---------------------------------------- (5) IntTRSCompressionProof (EQUIVALENT) Compressed rules. ---------------------------------------- (6) Obligation: Rules: l0(x6:0, x1:0, x2:0, x3:0, x10:0, x11:0) -> l0(x6:0, x1:0, 1 + x2:0, 2 + x3:0, x10:0, x11:0) :|: x2:0 <= x10:0 ---------------------------------------- (7) IntTRSUnneededArgumentFilterProof (EQUIVALENT) Some arguments are removed because they cannot influence termination. We removed arguments according to the following replacements: l0(x1, x2, x3, x4, x5, x6) -> l0(x3, x5) ---------------------------------------- (8) Obligation: Rules: l0(x2:0, x10:0) -> l0(1 + x2:0, x10:0) :|: x2:0 <= x10:0 ---------------------------------------- (9) TempFilterProof (SOUND) Used the following sort dictionary for filtering: l0(INTEGER, INTEGER) Replaced non-predefined constructor symbols by 0. ---------------------------------------- (10) Obligation: Rules: l0(x2:0, x10:0) -> l0(c, x10:0) :|: c = 1 + x2:0 && x2:0 <= x10:0 ---------------------------------------- (11) PolynomialOrderProcessor (EQUIVALENT) Found the following polynomial interpretation: [l0(x, x1)] = -x + x1 The following rules are decreasing: l0(x2:0, x10:0) -> l0(c, x10:0) :|: c = 1 + x2:0 && x2:0 <= x10:0 The following rules are bounded: l0(x2:0, x10:0) -> l0(c, x10:0) :|: c = 1 + x2:0 && x2:0 <= x10:0 ---------------------------------------- (12) YES