YES proof of prog.inttrs # AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty Termination of the given IRSwT could be proven: (0) IRSwT (1) IRSFormatTransformerProof [EQUIVALENT, 0 ms] (2) IRSwT (3) IRSwTTerminationDigraphProof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, 879 ms] (4) IRSwT (5) IntTRSCompressionProof [EQUIVALENT, 46 ms] (6) IRSwT (7) IntTRSUnneededArgumentFilterProof [EQUIVALENT, 0 ms] (8) IRSwT (9) TempFilterProof [SOUND, 34 ms] (10) IntTRS (11) PolynomialOrderProcessor [EQUIVALENT, 14 ms] (12) YES ---------------------------------------- (0) Obligation: Rules: l0(nI6HAT0, nX4HAT0, nX9HAT0, nXHAT0, res10HAT0, res5HAT0, ret_nBC18HAT0, ret_nBC211HAT0, tmp7HAT0, tmpHAT0, tmp___0HAT0) -> l1(nI6HATpost, nX4HATpost, nX9HATpost, nXHATpost, res10HATpost, res5HATpost, ret_nBC18HATpost, ret_nBC211HATpost, tmp7HATpost, tmpHATpost, tmp___0HATpost) :|: tmp___0HAT0 = tmp___0HATpost && tmp7HAT0 = tmp7HATpost && tmpHAT0 = tmpHATpost && ret_nBC211HAT0 = ret_nBC211HATpost && ret_nBC18HAT0 = ret_nBC18HATpost && res10HAT0 = res10HATpost && nX9HAT0 = nX9HATpost && nX4HAT0 = nX4HATpost && nXHAT0 = nXHATpost && nI6HATpost = 1 + nI6HAT0 && res5HATpost = res5HAT0 + tmp7HAT0 l2(x, x1, x2, x3, x4, x5, x6, x7, x8, x9, x10) -> l0(x11, x12, x13, x14, x15, x16, x17, x18, x19, x20, x21) :|: x10 = x21 && x9 = x20 && x7 = x18 && x6 = x17 && x5 = x16 && x4 = x15 && x2 = x13 && x1 = x12 && x3 = x14 && x = x11 && x19 = 0 l2(x22, x23, x24, x25, x26, x27, x28, x29, x30, x31, x32) -> l0(x33, x34, x35, x36, x37, x38, x39, x40, x41, x42, x43) :|: x32 = x43 && x31 = x42 && x29 = x40 && x28 = x39 && x27 = x38 && x26 = x37 && x24 = x35 && x23 = x34 && x25 = x36 && x22 = x33 && x41 = 1 l3(x44, x45, x46, x47, x48, x49, x50, x51, x52, x53, x54) -> l4(x55, x56, x57, x58, x59, x60, x61, x62, x63, x64, x65) :|: 32 <= x44 && x61 = x49 && x64 = x61 && x57 = x47 && x66 = x57 && x67 = x67 && x68 = x68 && x69 = x69 && x70 = x70 && x59 = x59 && x62 = x59 && x65 = x62 && x44 = x55 && x47 = x58 && x45 = x56 && x49 = x60 && x52 = x63 l3(x71, x72, x73, x74, x75, x76, x77, x78, x79, x80, x81) -> l2(x82, x83, x84, x85, x86, x87, x88, x89, x90, x91, x92) :|: x81 = x92 && x79 = x90 && x80 = x91 && x78 = x89 && x77 = x88 && x76 = x87 && x75 = x86 && x73 = x84 && x72 = x83 && x74 = x85 && x71 = x82 && 1 + x71 <= 32 l1(x93, x94, x95, x96, x97, x98, x99, x100, x101, x102, x103) -> l3(x104, x105, x106, x107, x108, x109, x110, x111, x112, x113, x114) :|: x103 = x114 && x101 = x112 && x102 = x113 && x100 = x111 && x99 = x110 && x98 = x109 && x97 = x108 && x95 = x106 && x94 = x105 && x96 = x107 && x93 = x104 l5(x115, x116, x117, x118, x119, x120, x121, x122, x123, x124, x125) -> l1(x126, x127, x128, x129, x130, x131, x132, x133, x134, x135, x136) :|: x125 = x136 && x123 = x134 && x124 = x135 && x122 = x133 && x121 = x132 && x119 = x130 && x117 = x128 && x118 = x129 && x126 = 0 && x131 = 0 && x127 = x118 l6(x137, x138, x139, x140, x141, x142, x143, x144, x145, x146, x147) -> l5(x148, x149, x150, x151, x152, x153, x154, x155, x156, x157, x158) :|: x147 = x158 && x145 = x156 && x146 = x157 && x144 = x155 && x143 = x154 && x142 = x153 && x141 = x152 && x139 = x150 && x138 = x149 && x140 = x151 && x137 = x148 Start term: l6(nI6HAT0, nX4HAT0, nX9HAT0, nXHAT0, res10HAT0, res5HAT0, ret_nBC18HAT0, ret_nBC211HAT0, tmp7HAT0, tmpHAT0, tmp___0HAT0) ---------------------------------------- (1) IRSFormatTransformerProof (EQUIVALENT) Reformatted IRS to match normalized format (transformed away non-linear left-hand sides, !=, / and %). ---------------------------------------- (2) Obligation: Rules: l0(nI6HAT0, nX4HAT0, nX9HAT0, nXHAT0, res10HAT0, res5HAT0, ret_nBC18HAT0, ret_nBC211HAT0, tmp7HAT0, tmpHAT0, tmp___0HAT0) -> l1(nI6HATpost, nX4HATpost, nX9HATpost, nXHATpost, res10HATpost, res5HATpost, ret_nBC18HATpost, ret_nBC211HATpost, tmp7HATpost, tmpHATpost, tmp___0HATpost) :|: tmp___0HAT0 = tmp___0HATpost && tmp7HAT0 = tmp7HATpost && tmpHAT0 = tmpHATpost && ret_nBC211HAT0 = ret_nBC211HATpost && ret_nBC18HAT0 = ret_nBC18HATpost && res10HAT0 = res10HATpost && nX9HAT0 = nX9HATpost && nX4HAT0 = nX4HATpost && nXHAT0 = nXHATpost && nI6HATpost = 1 + nI6HAT0 && res5HATpost = res5HAT0 + tmp7HAT0 l2(x, x1, x2, x3, x4, x5, x6, x7, x8, x9, x10) -> l0(x11, x12, x13, x14, x15, x16, x17, x18, x19, x20, x21) :|: x10 = x21 && x9 = x20 && x7 = x18 && x6 = x17 && x5 = x16 && x4 = x15 && x2 = x13 && x1 = x12 && x3 = x14 && x = x11 && x19 = 0 l2(x22, x23, x24, x25, x26, x27, x28, x29, x30, x31, x32) -> l0(x33, x34, x35, x36, x37, x38, x39, x40, x41, x42, x43) :|: x32 = x43 && x31 = x42 && x29 = x40 && x28 = x39 && x27 = x38 && x26 = x37 && x24 = x35 && x23 = x34 && x25 = x36 && x22 = x33 && x41 = 1 l3(x44, x45, x46, x47, x48, x49, x50, x51, x52, x53, x54) -> l4(x55, x56, x57, x58, x59, x60, x61, x62, x63, x64, x65) :|: 32 <= x44 && x61 = x49 && x64 = x61 && x57 = x47 && x66 = x57 && x67 = x67 && x68 = x68 && x69 = x69 && x70 = x70 && x59 = x59 && x62 = x59 && x65 = x62 && x44 = x55 && x47 = x58 && x45 = x56 && x49 = x60 && x52 = x63 l3(x71, x72, x73, x74, x75, x76, x77, x78, x79, x80, x81) -> l2(x82, x83, x84, x85, x86, x87, x88, x89, x90, x91, x92) :|: x81 = x92 && x79 = x90 && x80 = x91 && x78 = x89 && x77 = x88 && x76 = x87 && x75 = x86 && x73 = x84 && x72 = x83 && x74 = x85 && x71 = x82 && 1 + x71 <= 32 l1(x93, x94, x95, x96, x97, x98, x99, x100, x101, x102, x103) -> l3(x104, x105, x106, x107, x108, x109, x110, x111, x112, x113, x114) :|: x103 = x114 && x101 = x112 && x102 = x113 && x100 = x111 && x99 = x110 && x98 = x109 && x97 = x108 && x95 = x106 && x94 = x105 && x96 = x107 && x93 = x104 l5(x115, x116, x117, x118, x119, x120, x121, x122, x123, x124, x125) -> l1(x126, x127, x128, x129, x130, x131, x132, x133, x134, x135, x136) :|: x125 = x136 && x123 = x134 && x124 = x135 && x122 = x133 && x121 = x132 && x119 = x130 && x117 = x128 && x118 = x129 && x126 = 0 && x131 = 0 && x127 = x118 l6(x137, x138, x139, x140, x141, x142, x143, x144, x145, x146, x147) -> l5(x148, x149, x150, x151, x152, x153, x154, x155, x156, x157, x158) :|: x147 = x158 && x145 = x156 && x146 = x157 && x144 = x155 && x143 = x154 && x142 = x153 && x141 = x152 && x139 = x150 && x138 = x149 && x140 = x151 && x137 = x148 Start term: l6(nI6HAT0, nX4HAT0, nX9HAT0, nXHAT0, res10HAT0, res5HAT0, ret_nBC18HAT0, ret_nBC211HAT0, tmp7HAT0, tmpHAT0, tmp___0HAT0) ---------------------------------------- (3) IRSwTTerminationDigraphProof (EQUIVALENT) Constructed termination digraph! Nodes: (1) l0(nI6HAT0, nX4HAT0, nX9HAT0, nXHAT0, res10HAT0, res5HAT0, ret_nBC18HAT0, ret_nBC211HAT0, tmp7HAT0, tmpHAT0, tmp___0HAT0) -> l1(nI6HATpost, nX4HATpost, nX9HATpost, nXHATpost, res10HATpost, res5HATpost, ret_nBC18HATpost, ret_nBC211HATpost, tmp7HATpost, tmpHATpost, tmp___0HATpost) :|: tmp___0HAT0 = tmp___0HATpost && tmp7HAT0 = tmp7HATpost && tmpHAT0 = tmpHATpost && ret_nBC211HAT0 = ret_nBC211HATpost && ret_nBC18HAT0 = ret_nBC18HATpost && res10HAT0 = res10HATpost && nX9HAT0 = nX9HATpost && nX4HAT0 = nX4HATpost && nXHAT0 = nXHATpost && nI6HATpost = 1 + nI6HAT0 && res5HATpost = res5HAT0 + tmp7HAT0 (2) l2(x, x1, x2, x3, x4, x5, x6, x7, x8, x9, x10) -> l0(x11, x12, x13, x14, x15, x16, x17, x18, x19, x20, x21) :|: x10 = x21 && x9 = x20 && x7 = x18 && x6 = x17 && x5 = x16 && x4 = x15 && x2 = x13 && x1 = x12 && x3 = x14 && x = x11 && x19 = 0 (3) l2(x22, x23, x24, x25, x26, x27, x28, x29, x30, x31, x32) -> l0(x33, x34, x35, x36, x37, x38, x39, x40, x41, x42, x43) :|: x32 = x43 && x31 = x42 && x29 = x40 && x28 = x39 && x27 = x38 && x26 = x37 && x24 = x35 && x23 = x34 && x25 = x36 && x22 = x33 && x41 = 1 (4) l3(x44, x45, x46, x47, x48, x49, x50, x51, x52, x53, x54) -> l4(x55, x56, x57, x58, x59, x60, x61, x62, x63, x64, x65) :|: 32 <= x44 && x61 = x49 && x64 = x61 && x57 = x47 && x66 = x57 && x67 = x67 && x68 = x68 && x69 = x69 && x70 = x70 && x59 = x59 && x62 = x59 && x65 = x62 && x44 = x55 && x47 = x58 && x45 = x56 && x49 = x60 && x52 = x63 (5) l3(x71, x72, x73, x74, x75, x76, x77, x78, x79, x80, x81) -> l2(x82, x83, x84, x85, x86, x87, x88, x89, x90, x91, x92) :|: x81 = x92 && x79 = x90 && x80 = x91 && x78 = x89 && x77 = x88 && x76 = x87 && x75 = x86 && x73 = x84 && x72 = x83 && x74 = x85 && x71 = x82 && 1 + x71 <= 32 (6) l1(x93, x94, x95, x96, x97, x98, x99, x100, x101, x102, x103) -> l3(x104, x105, x106, x107, x108, x109, x110, x111, x112, x113, x114) :|: x103 = x114 && x101 = x112 && x102 = x113 && x100 = x111 && x99 = x110 && x98 = x109 && x97 = x108 && x95 = x106 && x94 = x105 && x96 = x107 && x93 = x104 (7) l5(x115, x116, x117, x118, x119, x120, x121, x122, x123, x124, x125) -> l1(x126, x127, x128, x129, x130, x131, x132, x133, x134, x135, x136) :|: x125 = x136 && x123 = x134 && x124 = x135 && x122 = x133 && x121 = x132 && x119 = x130 && x117 = x128 && x118 = x129 && x126 = 0 && x131 = 0 && x127 = x118 (8) l6(x137, x138, x139, x140, x141, x142, x143, x144, x145, x146, x147) -> l5(x148, x149, x150, x151, x152, x153, x154, x155, x156, x157, x158) :|: x147 = x158 && x145 = x156 && x146 = x157 && x144 = x155 && x143 = x154 && x142 = x153 && x141 = x152 && x139 = x150 && x138 = x149 && x140 = x151 && x137 = x148 Arcs: (1) -> (6) (2) -> (1) (3) -> (1) (5) -> (2), (3) (6) -> (4), (5) (7) -> (6) (8) -> (7) This digraph is fully evaluated! ---------------------------------------- (4) Obligation: Termination digraph: Nodes: (1) l0(nI6HAT0, nX4HAT0, nX9HAT0, nXHAT0, res10HAT0, res5HAT0, ret_nBC18HAT0, ret_nBC211HAT0, tmp7HAT0, tmpHAT0, tmp___0HAT0) -> l1(nI6HATpost, nX4HATpost, nX9HATpost, nXHATpost, res10HATpost, res5HATpost, ret_nBC18HATpost, ret_nBC211HATpost, tmp7HATpost, tmpHATpost, tmp___0HATpost) :|: tmp___0HAT0 = tmp___0HATpost && tmp7HAT0 = tmp7HATpost && tmpHAT0 = tmpHATpost && ret_nBC211HAT0 = ret_nBC211HATpost && ret_nBC18HAT0 = ret_nBC18HATpost && res10HAT0 = res10HATpost && nX9HAT0 = nX9HATpost && nX4HAT0 = nX4HATpost && nXHAT0 = nXHATpost && nI6HATpost = 1 + nI6HAT0 && res5HATpost = res5HAT0 + tmp7HAT0 (2) l2(x22, x23, x24, x25, x26, x27, x28, x29, x30, x31, x32) -> l0(x33, x34, x35, x36, x37, x38, x39, x40, x41, x42, x43) :|: x32 = x43 && x31 = x42 && x29 = x40 && x28 = x39 && x27 = x38 && x26 = x37 && x24 = x35 && x23 = x34 && x25 = x36 && x22 = x33 && x41 = 1 (3) l2(x, x1, x2, x3, x4, x5, x6, x7, x8, x9, x10) -> l0(x11, x12, x13, x14, x15, x16, x17, x18, x19, x20, x21) :|: x10 = x21 && x9 = x20 && x7 = x18 && x6 = x17 && x5 = x16 && x4 = x15 && x2 = x13 && x1 = x12 && x3 = x14 && x = x11 && x19 = 0 (4) l3(x71, x72, x73, x74, x75, x76, x77, x78, x79, x80, x81) -> l2(x82, x83, x84, x85, x86, x87, x88, x89, x90, x91, x92) :|: x81 = x92 && x79 = x90 && x80 = x91 && x78 = x89 && x77 = x88 && x76 = x87 && x75 = x86 && x73 = x84 && x72 = x83 && x74 = x85 && x71 = x82 && 1 + x71 <= 32 (5) l1(x93, x94, x95, x96, x97, x98, x99, x100, x101, x102, x103) -> l3(x104, x105, x106, x107, x108, x109, x110, x111, x112, x113, x114) :|: x103 = x114 && x101 = x112 && x102 = x113 && x100 = x111 && x99 = x110 && x98 = x109 && x97 = x108 && x95 = x106 && x94 = x105 && x96 = x107 && x93 = x104 Arcs: (1) -> (5) (2) -> (1) (3) -> (1) (4) -> (2), (3) (5) -> (4) This digraph is fully evaluated! ---------------------------------------- (5) IntTRSCompressionProof (EQUIVALENT) Compressed rules. ---------------------------------------- (6) Obligation: Rules: l3(x11:0, nX4HATpost:0, nX9HATpost:0, nXHATpost:0, res10HATpost:0, res5HATpost:0, ret_nBC18HATpost:0, ret_nBC211HATpost:0, x79:0, tmpHATpost:0, tmp___0HATpost:0) -> l3(1 + x11:0, nX4HATpost:0, nX9HATpost:0, nXHATpost:0, res10HATpost:0, res5HATpost:0, ret_nBC18HATpost:0, ret_nBC211HATpost:0, 0, tmpHATpost:0, tmp___0HATpost:0) :|: x11:0 < 32 l3(x, x1, x2, x3, x4, x5, x6, x7, x8, x9, x10) -> l3(1 + x, x1, x2, x3, x4, x5 + 1, x6, x7, 1, x9, x10) :|: x < 32 ---------------------------------------- (7) IntTRSUnneededArgumentFilterProof (EQUIVALENT) Some arguments are removed because they cannot influence termination. We removed arguments according to the following replacements: l3(x1, x2, x3, x4, x5, x6, x7, x8, x9, x10, x11) -> l3(x1) ---------------------------------------- (8) Obligation: Rules: l3(x11:0) -> l3(1 + x11:0) :|: x11:0 < 32 ---------------------------------------- (9) TempFilterProof (SOUND) Used the following sort dictionary for filtering: l3(INTEGER) Replaced non-predefined constructor symbols by 0. ---------------------------------------- (10) Obligation: Rules: l3(x11:0) -> l3(c) :|: c = 1 + x11:0 && x11:0 < 32 ---------------------------------------- (11) PolynomialOrderProcessor (EQUIVALENT) Found the following polynomial interpretation: [l3(x)] = 31 - x The following rules are decreasing: l3(x11:0) -> l3(c) :|: c = 1 + x11:0 && x11:0 < 32 The following rules are bounded: l3(x11:0) -> l3(c) :|: c = 1 + x11:0 && x11:0 < 32 ---------------------------------------- (12) YES