Question 26071: A Cover Can Resist Every Single Deletion yet Be Larger than Another Cover

Original question and givens

Use complete unique vertices=[c0, l0_0, l0_1, l0_2, c1, l1_0, l1_1, l1_2, l1_3, l1_4] and loop-free duplicate-free undirected edges=[[c0, l0_0], [c0, l0_1], [c0, l0_2], [c1, l1_0], [c1, l1_1], [c1, l1_2], [c1, l1_3], [c1, l1_4]], candidate cover=[l0_0, l0_1, l0_2, l1_0, l1_1, l1_2, l1_3, l1_4]. An edge needs at least one selected endpoint. Preserve source canonical subset order. Give (candidate, deletion_rows, inclusion_minimal, minimum_size, smaller_cover, minimum, certificate).

• G001 — Use complete unique vertices=[c0, l0_0, l0_1, l0_2, c1, l1_0, l1_1, l1_2, l1_3, l1_4] and loop-free duplicate-free undirected edges=[[c0, l0_0], [c0, l0_1], [c0, l0_2], [c1, l1_0], [c1, l1_1], [c1, l1_2], [c1, l1_3], [c1, l1_4]], candidate cover=[l0_0, l0_1, l0_2, l1_0, l1_1, l1_2, l1_3, l1_4]. An edge needs at least one selected endpoint. Preserve source canonical subset order. Give (candidate, deletion_rows, inclusion_minimal, minimum_size, smaller_cover, minimum, certificate).

Definition and scope

Use the original question domain, assumptions and cited MKO definitions. This item has second-context mathematical and bilingual review, with the model, answer and every arithmetic step independently recomputed from the original parameters; the source question remains a candidate.

• mko-minimum-vertex-cover

Step-by-step derivation

source roles = [mincover-minimal-not-minimum]

derived = [[candidate, deletion_rows, inclusion_minimal, minimum_size, smaller_cover, minimum, certificate]]

source roles = [[c0, l0_0, l0_1, l0_2, c1, l1_0, l1_1, l1_2, l1_3, l1_4], [[c0, l0_0], [c0, l0_1], [c0, l0_2], [c1, l1_0], [c1, l1_1], [c1, l1_2], [c1, l1_3], [c1, l1_4]], [l0_0, l0_1, l0_2, l1_0, l1_1, l1_2, l1_3, l1_4]]

derived = [[c0, c1, l0_0, l0_1, l0_2, l1_0, l1_1, l1_2, l1_3, l1_4], [[c0, l0_0], [c0, l0_1], [c0, l0_2], [c1, l1_0], [c1, l1_1], [c1, l1_2], [c1, l1_3], [c1, l1_4]], [l0_0, l0_1, l0_2, l1_0, l1_1, l1_2, l1_3, l1_4]]

source roles = [[c0, c1, l0_0, l0_1, l0_2, l1_0, l1_1, l1_2, l1_3, l1_4], [[c0, l0_0], [c0, l0_1], [c0, l0_2], [c1, l1_0], [c1, l1_1], [c1, l1_2], [c1, l1_3], [c1, l1_4]], 1024]

derived = [[3, 7, 11, 15, 19, 23, 27, 30, 31, 35, 39, 43, 47, 51, 55, 59, 62, 63, 67, 71, 75, 79, 83, 87, 91, 94, 95, 99, 103, 107, 111, 115, 119, 123, 126, 127, 131, 135, 139, 143, 147, 151, 155, 158, 159, 163, 167, 171, 175, 179, 183, 187, 190, 191, 195, 199, 203, 207, 211, 215, 219, 222, 223, 227, 231, 235, 239, 243, 247, 251, 254, 255, 259, 263, 267, 271, 275, 279, 283, 286, 287, 291, 295, 299, 303, 307, 311, 315, 318, 319, 323, 327, 331, 335, 339, 343, 347, 350, 351, 355, 359, 363, 367, 371, 375, 379, 382, 383, 387, 391, 395, 399, 403, 407, 411, 414, 415, 419, 423, 427, 431, 435, 439, 443, 446, 447, 451, 455, 459, 463, 467, 471, 475, 478, 479, 483, 487, 491, 495, 499, 503, 507, 510, 511, 515, 519, 523, 527, 531, 535, 539, 542, 543, 547, 551, 555, 559, 563, 567, 571, 574, 575, 579, 583, 587, 591, 595, 599, 603, 606, 607, 611, 615, 619, 623, 627, 631, 635, 638, 639, 643, 647, 651, 655, 659, 663, 667, 670, 671, 675, 679, 683, 687, 691, 695, 699, 702, 703, 707, 711, 715, 719, 723, 727, 731, 734, 735, 739, 743, 747, 751, 755, 759, 763, 766, 767, 771, 775, 779, 783, 787, 791, 795, 798, 799, 803, 807, 811, 815, 819, 823, 827, 830, 831, 835, 839, 843, 847, 851, 855, 859, 862, 863, 867, 871, 875, 879, 883, 887, 891, 894, 895, 899, 903, 907, 911, 915, 919, 923, 926, 927, 931, 935, 939, 943, 947, 951, 955, 958, 959, 963, 967, 971, 975, 979, 983, 987, 990, 991, 993, 995, 997, 999, 1001, 1003, 1005, 1007, 1009, 1011, 1013, 1015, 1017, 1019, 1020, 1021, 1022, 1023], [[0, 0], [1, 3], [2, 0], [4, 1], [5, 3], [6, 1], [8, 0], [9, 3], [10, 0], [12, 2], [13, 3], [14, 2], [16, 0], [17, 3], [18, 0], [20, 1], [21, 3], [22, 1], [24, 0], [25, 3], [26, 0], [28, 3], [29, 3], [32, 0], [33, 4], [34, 0], [36, 1], [37, 4], [38, 1], [40, 0], [41, 4], [42, 0], [44, 2], [45, 4], [46, 2], [48, 0], [49, 4], [50, 0], [52, 1], [53, 4], [54, 1], [56, 0], [57, 4], [58, 0], [60, 4], [61, 4], [64, 0], [65, 3], [66, 0], [68, 1], [69, 3], [70, 1], [72, 0], [73, 3], [74, 0], [76, 2], [77, 3], [78, 2], [80, 0], [81, 3], [82, 0], [84, 1], [85, 3], [86, 1], [88, 0], [89, 3], [90, 0], [92, 3], [93, 3], [96, 0], [97, 5], [98, 0], [100, 1], [101, 5], [102, 1], [104, 0], [105, 5], [106, 0], [108, 2], [109, 5], [110, 2], [112, 0], [113, 5], [114, 0], [116, 1], [117, 5], [118, 1], [120, 0], [121, 5], [122, 0], [124, 5], [125, 5], [128, 0], [129, 3], [130, 0], [132, 1], [133, 3], [134, 1], [136, 0], [137, 3], [138, 0], [140, 2], [141, 3], [142, 2], [144, 0], [145, 3], [146, 0], [148, 1], [149, 3], [150, 1], [152, 0], [153, 3], [154, 0], [156, 3], [157, 3], [160, 0], [161, 4], [162, 0], [164, 1], [165, 4], [166, 1], [168, 0], [169, 4], [170, 0], [172, 2], [173, 4], [174, 2], [176, 0], [177, 4], [178, 0], [180, 1], [181, 4], [182, 1], [184, 0], [185, 4], [186, 0], [188, 4], [189, 4], [192, 0], [193, 3], [194, 0], [196, 1], [197, 3], [198, 1], [200, 0], [201, 3], [202, 0], [204, 2], [205, 3], [206, 2], [208, 0], [209, 3], [210, 0], [212, 1], [213, 3], [214, 1], [216, 0], [217, 3], [218, 0], [220, 3], [221, 3], [224, 0], [225, 6], [226, 0], [228, 1], [229, 6], [230, 1], [232, 0], [233, 6], [234, 0], [236, 2], [237, 6], [238, 2], [240, 0], [241, 6], [242, 0], [244, 1], [245, 6], [246, 1], [248, 0], [249, 6], [250, 0], [252, 6], [253, 6], [256, 0], [257, 3], [258, 0], [260, 1], [261, 3], [262, 1], [264, 0], [265, 3], [266, 0], [268, 2], [269, 3], [270, 2], [272, 0], [273, 3], [274, 0], [276, 1], [277, 3], [278, 1], [280, 0], [281, 3], [282, 0], [284, 3], [285, 3], [288, 0], [289, 4], [290, 0], [292, 1], [293, 4], [294, 1], [296, 0], [297, 4], [298, 0], [300, 2], [301, 4], [302, 2], [304, 0], [305, 4], [306, 0], [308, 1], [309, 4], [310, 1], [312, 0], [313, 4], [314, 0], [316, 4], [317, 4], [320, 0], [321, 3], [322, 0], [324, 1], [325, 3], [326, 1], [328, 0], [329, 3], [330, 0], [332, 2], [333, 3], [334, 2], [336, 0], [337, 3], [338, 0], [340, 1], [341, 3], [342, 1], [344, 0], [345, 3], [346, 0], [348, 3], [349, 3], [352, 0], [353, 5], [354, 0], [356, 1], [357, 5], [358, 1], [360, 0], [361, 5], [362, 0], [364, 2], [365, 5], [366, 2], [368, 0], [369, 5], [370, 0], [372, 1], [373, 5], [374, 1], [376, 0], [377, 5], [378, 0], [380, 5], [381, 5], [384, 0], [385, 3], [386, 0], [388, 1], [389, 3], [390, 1], [392, 0], [393, 3], [394, 0], [396, 2], [397, 3], [398, 2], [400, 0], [401, 3], [402, 0], [404, 1], [405, 3], [406, 1], [408, 0], [409, 3], [410, 0], [412, 3], [413, 3], [416, 0], [417, 4], [418, 0], [420, 1], [421, 4], [422, 1], [424, 0], [425, 4], [426, 0], [428, 2], [429, 4], [430, 2], [432, 0], [433, 4], [434, 0], [436, 1], [437, 4], [438, 1], [440, 0], [441, 4], [442, 0], [444, 4], [445, 4], [448, 0], [449, 3], [450, 0], [452, 1], [453, 3], [454, 1], [456, 0], [457, 3], [458, 0], [460, 2], [461, 3], [462, 2], [464, 0], [465, 3], [466, 0], [468, 1], [469, 3], [470, 1], [472, 0], [473, 3], [474, 0], [476, 3], [477, 3], [480, 0], [481, 7], [482, 0], [484, 1], [485, 7], [486, 1], [488, 0], [489, 7], [490, 0], [492, 2], [493, 7], [494, 2], [496, 0], [497, 7], [498, 0], [500, 1], [501, 7], [502, 1], [504, 0], [505, 7], [506, 0], [508, 7], [509, 7], [512, 0], [513, 3], [514, 0], [516, 1], [517, 3], [518, 1], [520, 0], [521, 3], [522, 0], [524, 2], [525, 3], [526, 2], [528, 0], [529, 3], [530, 0], [532, 1], [533, 3], [534, 1], [536, 0], [537, 3], [538, 0], [540, 3], [541, 3], [544, 0], [545, 4], [546, 0], [548, 1], [549, 4], [550, 1], [552, 0], [553, 4], [554, 0], [556, 2], [557, 4], [558, 2], [560, 0], [561, 4], [562, 0], [564, 1], [565, 4], [566, 1], [568, 0], [569, 4], [570, 0], [572, 4], [573, 4], [576, 0], [577, 3], [578, 0], [580, 1], [581, 3], [582, 1], [584, 0], [585, 3], [586, 0], [588, 2], [589, 3], [590, 2], [592, 0], [593, 3], [594, 0], [596, 1], [597, 3], [598, 1], [600, 0], [601, 3], [602, 0], [604, 3], [605, 3], [608, 0], [609, 5], [610, 0], [612, 1], [613, 5], [614, 1], [616, 0], [617, 5], [618, 0], [620, 2], [621, 5], [622, 2], [624, 0], [625, 5], [626, 0], [628, 1], [629, 5], [630, 1], [632, 0], [633, 5], [634, 0], [636, 5], [637, 5], [640, 0], [641, 3], [642, 0], [644, 1], [645, 3], [646, 1], [648, 0], [649, 3], [650, 0], [652, 2], [653, 3], [654, 2], [656, 0], [657, 3], [658, 0], [660, 1], [661, 3], [662, 1], [664, 0], [665, 3], [666, 0], [668, 3], [669, 3], [672, 0], [673, 4], [674, 0], [676, 1], [677, 4], [678, 1], [680, 0], [681, 4], [682, 0], [684, 2], [685, 4], [686, 2], [688, 0], [689, 4], [690, 0], [692, 1], [693, 4], [694, 1], [696, 0], [697, 4], [698, 0], [700, 4], [701, 4], [704, 0], [705, 3], [706, 0], [708, 1], [709, 3], [710, 1], [712, 0], [713, 3], [714, 0], [716, 2], [717, 3], [718, 2], [720, 0], [721, 3], [722, 0], [724, 1], [725, 3], [726, 1], [728, 0], [729, 3], [730, 0], [732, 3], [733, 3], [736, 0], [737, 6], [738, 0], [740, 1], [741, 6], [742, 1], [744, 0], [745, 6], [746, 0], [748, 2], [749, 6], [750, 2], [752, 0], [753, 6], [754, 0], [756, 1], [757, 6], [758, 1], [760, 0], [761, 6], [762, 0], [764, 6], [765, 6], [768, 0], [769, 3], [770, 0], [772, 1], [773, 3], [774, 1], [776, 0], [777, 3], [778, 0], [780, 2], [781, 3], [782, 2], [784, 0], [785, 3], [786, 0], [788, 1], [789, 3], [790, 1], [792, 0], [793, 3], [794, 0], [796, 3], [797, 3], [800, 0], [801, 4], [802, 0], [804, 1], [805, 4], [806, 1], [808, 0], [809, 4], [810, 0], [812, 2], [813, 4], [814, 2], [816, 0], [817, 4], [818, 0], [820, 1], [821, 4], [822, 1], [824, 0], [825, 4], [826, 0], [828, 4], [829, 4], [832, 0], [833, 3], [834, 0], [836, 1], [837, 3], [838, 1], [840, 0], [841, 3], [842, 0], [844, 2], [845, 3], [846, 2], [848, 0], [849, 3], [850, 0], [852, 1], [853, 3], [854, 1], [856, 0], [857, 3], [858, 0], [860, 3], [861, 3], [864, 0], [865, 5], [866, 0], [868, 1], [869, 5], [870, 1], [872, 0], [873, 5], [874, 0], [876, 2], [877, 5], [878, 2], [880, 0], [881, 5], [882, 0], [884, 1], [885, 5], [886, 1], [888, 0], [889, 5], [890, 0], [892, 5], [893, 5], [896, 0], [897, 3], [898, 0], [900, 1], [901, 3], [902, 1], [904, 0], [905, 3], [906, 0], [908, 2], [909, 3], [910, 2], [912, 0], [913, 3], [914, 0], [916, 1], [917, 3], [918, 1], [920, 0], [921, 3], [922, 0], [924, 3], [925, 3], [928, 0], [929, 4], [930, 0], [932, 1], [933, 4], [934, 1], [936, 0], [937, 4], [938, 0], [940, 2], [941, 4], [942, 2], [944, 0], [945, 4], [946, 0], [948, 1], [949, 4], [950, 1], [952, 0], [953, 4], [954, 0], [956, 4], [957, 4], [960, 0], [961, 3], [962, 0], [964, 1], [965, 3], [966, 1], [968, 0], [969, 3], [970, 0], [972, 2], [973, 3], [974, 2], [976, 0], [977, 3], [978, 0], [980, 1], [981, 3], [982, 1], [984, 0], [985, 3], [986, 0], [988, 3], [989, 3], [992, 0], [994, 0], [996, 1], [998, 1], [1000, 0], [1002, 0], [1004, 2], [1006, 2], [1008, 0], [1010, 0], [1012, 1], [1014, 1], [1016, 0], [1018, 0]], [[c0, c1], [c0, c1, l0_0], [c0, c1, l0_1], [c0, c1, l0_0, l0_1], [c0, c1, l0_2], [c0, c1, l0_0, l0_2], [c0, c1, l0_1, l0_2], [c1, l0_0, l0_1, l0_2], [c0, c1, l0_0, l0_1, l0_2], [c0, c1, l1_0], [c0, c1, l0_0, l1_0], [c0, c1, l0_1, l1_0], [c0, c1, l0_0, l0_1, l1_0], [c0, c1, l0_2, l1_0], [c0, c1, l0_0, l0_2, l1_0], [c0, c1, l0_1, l0_2, l1_0], [c1, l0_0, l0_1, l0_2, l1_0], [c0, c1, l0_0, l0_1, l0_2, l1_0], [c0, c1, l1_1], [c0, c1, l0_0, l1_1], [c0, c1, l0_1, l1_1], [c0, c1, l0_0, l0_1, l1_1], [c0, c1, l0_2, l1_1], [c0, c1, l0_0, l0_2, l1_1], [c0, c1, l0_1, l0_2, l1_1], [c1, l0_0, l0_1, l0_2, l1_1], [c0, c1, l0_0, l0_1, l0_2, l1_1], [c0, c1, l1_0, l1_1], [c0, c1, l0_0, l1_0, l1_1], [c0, c1, l0_1, l1_0, l1_1], [c0, c1, l0_0, l0_1, l1_0, l1_1], [c0, c1, l0_2, l1_0, l1_1], [c0, c1, l0_0, l0_2, l1_0, l1_1], [c0, c1, l0_1, l0_2, l1_0, l1_1], [c1, l0_0, l0_1, l0_2, l1_0, l1_1], [c0, c1, l0_0, l0_1, l0_2, l1_0, l1_1], [c0, c1, l1_2], [c0, c1, l0_0, l1_2], [c0, c1, l0_1, l1_2], [c0, c1, l0_0, l0_1, l1_2], [c0, c1, l0_2, l1_2], [c0, c1, l0_0, l0_2, l1_2], [c0, c1, l0_1, l0_2, l1_2], [c1, l0_0, l0_1, l0_2, l1_2], [c0, c1, l0_0, l0_1, l0_2, l1_2], [c0, c1, l1_0, l1_2], [c0, c1, l0_0, l1_0, l1_2], [c0, c1, l0_1, l1_0, l1_2], [c0, c1, l0_0, l0_1, l1_0, l1_2], [c0, c1, l0_2, l1_0, l1_2], [c0, c1, l0_0, l0_2, l1_0, l1_2], [c0, c1, l0_1, l0_2, l1_0, l1_2], [c1, l0_0, l0_1, l0_2, l1_0, l1_2], [c0, c1, l0_0, l0_1, l0_2, l1_0, l1_2], [c0, c1, l1_1, l1_2], [c0, c1, l0_0, l1_1, l1_2], [c0, c1, l0_1, l1_1, l1_2], [c0, c1, l0_0, l0_1, l1_1, l1_2], [c0, c1, l0_2, l1_1, l1_2], [c0, c1, l0_0, l0_2, l1_1, l1_2], [c0, c1, l0_1, l0_2, l1_1, l1_2], [c1, l0_0, l0_1, l0_2, l1_1, l1_2], [c0, c1, l0_0, l0_1, l0_2, l1_1, l1_2], [c0, c1, l1_0, l1_1, l1_2], [c0, c1, l0_0, l1_0, l1_1, l1_2], [c0, c1, l0_1, l1_0, l1_1, l1_2], [c0, c1, l0_0, l0_1, l1_0, l1_1, l1_2], [c0, c1, l0_2, l1_0, l1_1, l1_2], [c0, c1, l0_0, l0_2, l1_0, l1_1, l1_2], [c0, c1, l0_1, l0_2, l1_0, l1_1, l1_2], [c1, l0_0, l0_1, l0_2, l1_0, l1_1, l1_2], [c0, c1, l0_0, l0_1, l0_2, l1_0, l1_1, l1_2], [c0, c1, l1_3], [c0, c1, l0_0, l1_3], [c0, c1, l0_1, l1_3], [c0, c1, l0_0, l0_1, l1_3], [c0, c1, l0_2, l1_3], [c0, c1, l0_0, l0_2, l1_3], [c0, c1, l0_1, l0_2, l1_3], [c1, l0_0, l0_1, l0_2, l1_3], [c0, c1, l0_0, l0_1, l0_2, l1_3], [c0, c1, l1_0, l1_3], [c0, c1, l0_0, l1_0, l1_3], [c0, c1, l0_1, l1_0, l1_3], [c0, c1, l0_0, l0_1, l1_0, l1_3], [c0, c1, l0_2, l1_0, l1_3], [c0, c1, l0_0, l0_2, l1_0, l1_3], [c0, c1, l0_1, l0_2, l1_0, l1_3], [c1, l0_0, l0_1, l0_2, l1_0, l1_3], [c0, c1, l0_0, l0_1, l0_2, l1_0, l1_3], [c0, c1, l1_1, l1_3], [c0, c1, l0_0, l1_1, l1_3], [c0, c1, l0_1, l1_1, l1_3], [c0, c1, l0_0, l0_1, l1_1, l1_3], [c0, c1, l0_2, l1_1, l1_3], [c0, c1, l0_0, l0_2, l1_1, l1_3], [c0, c1, l0_1, l0_2, l1_1, l1_3], [c1, l0_0, l0_1, l0_2, l1_1, l1_3], [c0, c1, l0_0, l0_1, l0_2, l1_1, l1_3], [c0, c1, l1_0, l1_1, l1_3], [c0, c1, l0_0, l1_0, l1_1, l1_3], [c0, c1, l0_1, l1_0, l1_1, l1_3], [c0, c1, l0_0, l0_1, l1_0, l1_1, l1_3], [c0, c1, l0_2, l1_0, l1_1, l1_3], [c0, c1, l0_0, l0_2, l1_0, l1_1, l1_3], [c0, c1, l0_1, l0_2, l1_0, l1_1, l1_3], [c1, l0_0, l0_1, l0_2, l1_0, l1_1, l1_3], [c0, c1, l0_0, l0_1, l0_2, l1_0, l1_1, l1_3], [c0, c1, l1_2, l1_3], [c0, c1, l0_0, l1_2, l1_3], [c0, c1, l0_1, l1_2, l1_3], [c0, c1, l0_0, l0_1, l1_2, l1_3], [c0, c1, l0_2, l1_2, l1_3], [c0, c1, l0_0, l0_2, l1_2, l1_3], [c0, c1, l0_1, l0_2, l1_2, l1_3], [c1, l0_0, l0_1, l0_2, l1_2, l1_3], [c0, c1, l0_0, l0_1, l0_2, l1_2, l1_3], [c0, c1, l1_0, l1_2, l1_3], [c0, c1, l0_0, l1_0, l1_2, l1_3], [c0, c1, l0_1, l1_0, l1_2, l1_3], [c0, c1, l0_0, l0_1, l1_0, l1_2, l1_3], [c0, c1, l0_2, l1_0, l1_2, l1_3], [c0, c1, l0_0, l0_2, l1_0, l1_2, l1_3], [c0, c1, l0_1, l0_2, l1_0, l1_2, l1_3], [c1, l0_0, l0_1, l0_2, l1_0, l1_2, l1_3], [c0, c1, l0_0, l0_1, l0_2, l1_0, l1_2, l1_3], [c0, c1, l1_1, l1_2, l1_3], [c0, c1, l0_0, l1_1, l1_2, l1_3], [c0, c1, l0_1, l1_1, l1_2, l1_3], [c0, c1, l0_0, l0_1, l1_1, l1_2, l1_3], [c0, c1, l0_2, l1_1, l1_2, l1_3], [c0, c1, l0_0, l0_2, l1_1, l1_2, l1_3], [c0, c1, l0_1, l0_2, l1_1, l1_2, l1_3], [c1, l0_0, l0_1, l0_2, l1_1, l1_2, l1_3], [c0, c1, l0_0, l0_1, l0_2, l1_1, l1_2, l1_3], [c0, c1, l1_0, l1_1, l1_2, l1_3], [c0, c1, l0_0, l1_0, l1_1, l1_2, l1_3], [c0, c1, l0_1, l1_0, l1_1, l1_2, l1_3], [c0, c1, l0_0, l0_1, l1_0, l1_1, l1_2, l1_3], [c0, c1, l0_2, l1_0, l1_1, l1_2, l1_3], [c0, c1, l0_0, l0_2, l1_0, l1_1, l1_2, l1_3], [c0, c1, l0_1, l0_2, l1_0, l1_1, l1_2, l1_3], [c1, l0_0, l0_1, l0_2, l1_0, l1_1, l1_2, l1_3], [c0, c1, l0_0, l0_1, l0_2, l1_0, l1_1, l1_2, l1_3], [c0, c1, l1_4], [c0, c1, l0_0, l1_4], [c0, c1, l0_1, l1_4], [c0, c1, l0_0, l0_1, l1_4], [c0, c1, l0_2, l1_4], [c0, c1, l0_0, l0_2, l1_4], [c0, c1, l0_1, l0_2, l1_4], [c1, l0_0, l0_1, l0_2, l1_4], [c0, c1, l0_0, l0_1, l0_2, l1_4], [c0, c1, l1_0, l1_4], [c0, c1, l0_0, l1_0, l1_4], [c0, c1, l0_1, l1_0, l1_4], [c0, c1, l0_0, l0_1, l1_0, l1_4], [c0, c1, l0_2, l1_0, l1_4], [c0, c1, l0_0, l0_2, l1_0, l1_4], [c0, c1, l0_1, l0_2, l1_0, l1_4], [c1, l0_0, l0_1, l0_2, l1_0, l1_4], [c0, c1, l0_0, l0_1, l0_2, l1_0, l1_4], [c0, c1, l1_1, l1_4], [c0, c1, l0_0, l1_1, l1_4], [c0, c1, l0_1, l1_1, l1_4], [c0, c1, l0_0, l0_1, l1_1, l1_4], [c0, c1, l0_2, l1_1, l1_4], [c0, c1, l0_0, l0_2, l1_1, l1_4], [c0, c1, l0_1, l0_2, l1_1, l1_4], [c1, l0_0, l0_1, l0_2, l1_1, l1_4], [c0, c1, l0_0, l0_1, l0_2, l1_1, l1_4], [c0, c1, l1_0, l1_1, l1_4], [c0, c1, l0_0, l1_0, l1_1, l1_4], [c0, c1, l0_1, l1_0, l1_1, l1_4], [c0, c1, l0_0, l0_1, l1_0, l1_1, l1_4], [c0, c1, l0_2, l1_0, l1_1, l1_4], [c0, c1, l0_0, l0_2, l1_0, l1_1, l1_4], [c0, c1, l0_1, l0_2, l1_0, l1_1, l1_4], [c1, l0_0, l0_1, l0_2, l1_0, l1_1, l1_4], [c0, c1, l0_0, l0_1, l0_2, l1_0, l1_1, l1_4], [c0, c1, l1_2, l1_4], [c0, c1, l0_0, l1_2, l1_4], [c0, c1, l0_1, l1_2, l1_4], [c0, c1, l0_0, l0_1, l1_2, l1_4], [c0, c1, l0_2, l1_2, l1_4], [c0, c1, l0_0, l0_2, l1_2, l1_4], [c0, c1, l0_1, l0_2, l1_2, l1_4], [c1, l0_0, l0_1, l0_2, l1_2, l1_4], [c0, c1, l0_0, l0_1, l0_2, l1_2, l1_4], [c0, c1, l1_0, l1_2, l1_4], [c0, c1, l0_0, l1_0, l1_2, l1_4], [c0, c1, l0_1, l1_0, l1_2, l1_4], [c0, c1, l0_0, l0_1, l1_0, l1_2, l1_4], [c0, c1, l0_2, l1_0, l1_2, l1_4], [c0, c1, l0_0, l0_2, l1_0, l1_2, l1_4], [c0, c1, l0_1, l0_2, l1_0, l1_2, l1_4], [c1, l0_0, l0_1, l0_2, l1_0, l1_2, l1_4], [c0, c1, l0_0, l0_1, l0_2, l1_0, l1_2, l1_4], [c0, c1, l1_1, l1_2, l1_4], [c0, c1, l0_0, l1_1, l1_2, l1_4], [c0, c1, l0_1, l1_1, l1_2, l1_4], [c0, c1, l0_0, l0_1, l1_1, l1_2, l1_4], [c0, c1, l0_2, l1_1, l1_2, l1_4], [c0, c1, l0_0, l0_2, l1_1, l1_2, l1_4], [c0, c1, l0_1, l0_2, l1_1, l1_2, l1_4], [c1, l0_0, l0_1, l0_2, l1_1, l1_2, l1_4], [c0, c1, l0_0, l0_1, l0_2, l1_1, l1_2, l1_4], [c0, c1, l1_0, l1_1, l1_2, l1_4], [c0, c1, l0_0, l1_0, l1_1, l1_2, l1_4], [c0, c1, l0_1, l1_0, l1_1, l1_2, l1_4], [c0, c1, l0_0, l0_1, l1_0, l1_1, l1_2, l1_4], [c0, c1, l0_2, l1_0, l1_1, l1_2, l1_4], [c0, c1, l0_0, l0_2, l1_0, l1_1, l1_2, l1_4], [c0, c1, l0_1, l0_2, l1_0, l1_1, l1_2, l1_4], [c1, l0_0, l0_1, l0_2, l1_0, l1_1, l1_2, l1_4], [c0, c1, l0_0, l0_1, l0_2, l1_0, l1_1, l1_2, l1_4], [c0, c1, l1_3, l1_4], [c0, c1, l0_0, l1_3, l1_4], [c0, c1, l0_1, l1_3, l1_4], [c0, c1, l0_0, l0_1, l1_3, l1_4], [c0, c1, l0_2, l1_3, l1_4], [c0, c1, l0_0, l0_2, l1_3, l1_4], [c0, c1, l0_1, l0_2, l1_3, l1_4], [c1, l0_0, l0_1, l0_2, l1_3, l1_4], [c0, c1, l0_0, l0_1, l0_2, l1_3, l1_4], [c0, c1, l1_0, l1_3, l1_4], [c0, c1, l0_0, l1_0, l1_3, l1_4], [c0, c1, l0_1, l1_0, l1_3, l1_4], [c0, c1, l0_0, l0_1, l1_0, l1_3, l1_4], [c0, c1, l0_2, l1_0, l1_3, l1_4], [c0, c1, l0_0, l0_2, l1_0, l1_3, l1_4], [c0, c1, l0_1, l0_2, l1_0, l1_3, l1_4], [c1, l0_0, l0_1, l0_2, l1_0, l1_3, l1_4], [c0, c1, l0_0, l0_1, l0_2, l1_0, l1_3, l1_4], [c0, c1, l1_1, l1_3, l1_4], [c0, c1, l0_0, l1_1, l1_3, l1_4], [c0, c1, l0_1, l1_1, l1_3, l1_4], [c0, c1, l0_0, l0_1, l1_1, l1_3, l1_4], [c0, c1, l0_2, l1_1, l1_3, l1_4], [c0, c1, l0_0, l0_2, l1_1, l1_3, l1_4], [c0, c1, l0_1, l0_2, l1_1, l1_3, l1_4], [c1, l0_0, l0_1, l0_2, l1_1, l1_3, l1_4], [c0, c1, l0_0, l0_1, l0_2, l1_1, l1_3, l1_4], [c0, c1, l1_0, l1_1, l1_3, l1_4], [c0, c1, l0_0, l1_0, l1_1, l1_3, l1_4], [c0, c1, l0_1, l1_0, l1_1, l1_3, l1_4], [c0, c1, l0_0, l0_1, l1_0, l1_1, l1_3, l1_4], [c0, c1, l0_2, l1_0, l1_1, l1_3, l1_4], [c0, c1, l0_0, l0_2, l1_0, l1_1, l1_3, l1_4], [c0, c1, l0_1, l0_2, l1_0, l1_1, l1_3, l1_4], [c1, l0_0, l0_1, l0_2, l1_0, l1_1, l1_3, l1_4], [c0, c1, l0_0, l0_1, l0_2, l1_0, l1_1, l1_3, l1_4], [c0, c1, l1_2, l1_3, l1_4], [c0, c1, l0_0, l1_2, l1_3, l1_4], [c0, c1, l0_1, l1_2, l1_3, l1_4], [c0, c1, l0_0, l0_1, l1_2, l1_3, l1_4], [c0, c1, l0_2, l1_2, l1_3, l1_4], [c0, c1, l0_0, l0_2, l1_2, l1_3, l1_4], [c0, c1, l0_1, l0_2, l1_2, l1_3, l1_4], [c1, l0_0, l0_1, l0_2, l1_2, l1_3, l1_4], [c0, c1, l0_0, l0_1, l0_2, l1_2, l1_3, l1_4], [c0, c1, l1_0, l1_2, l1_3, l1_4], [c0, c1, l0_0, l1_0, l1_2, l1_3, l1_4], [c0, c1, l0_1, l1_0, l1_2, l1_3, l1_4], [c0, c1, l0_0, l0_1, l1_0, l1_2, l1_3, l1_4], [c0, c1, l0_2, l1_0, l1_2, l1_3, l1_4], [c0, c1, l0_0, l0_2, l1_0, l1_2, l1_3, l1_4], [c0, c1, l0_1, l0_2, l1_0, l1_2, l1_3, l1_4], [c1, l0_0, l0_1, l0_2, l1_0, l1_2, l1_3, l1_4], [c0, c1, l0_0, l0_1, l0_2, l1_0, l1_2, l1_3, l1_4], [c0, c1, l1_1, l1_2, l1_3, l1_4], [c0, c1, l0_0, l1_1, l1_2, l1_3, l1_4], [c0, c1, l0_1, l1_1, l1_2, l1_3, l1_4], [c0, c1, l0_0, l0_1, l1_1, l1_2, l1_3, l1_4], [c0, c1, l0_2, l1_1, l1_2, l1_3, l1_4], [c0, c1, l0_0, l0_2, l1_1, l1_2, l1_3, l1_4], [c0, c1, l0_1, l0_2, l1_1, l1_2, l1_3, l1_4], [c1, l0_0, l0_1, l0_2, l1_1, l1_2, l1_3, l1_4], [c0, c1, l0_0, l0_1, l0_2, l1_1, l1_2, l1_3, l1_4], [c0, l1_0, l1_1, l1_2, l1_3, l1_4], [c0, c1, l1_0, l1_1, l1_2, l1_3, l1_4], [c0, l0_0, l1_0, l1_1, l1_2, l1_3, l1_4], [c0, c1, l0_0, l1_0, l1_1, l1_2, l1_3, l1_4], [c0, l0_1, l1_0, l1_1, l1_2, l1_3, l1_4], [c0, c1, l0_1, l1_0, l1_1, l1_2, l1_3, l1_4], [c0, l0_0, l0_1, l1_0, l1_1, l1_2, l1_3, l1_4], [c0, c1, l0_0, l0_1, l1_0, l1_1, l1_2, l1_3, l1_4], [c0, l0_2, l1_0, l1_1, l1_2, l1_3, l1_4], [c0, c1, l0_2, l1_0, l1_1, l1_2, l1_3, l1_4], [c0, l0_0, l0_2, l1_0, l1_1, l1_2, l1_3, l1_4], [c0, c1, l0_0, l0_2, l1_0, l1_1, l1_2, l1_3, l1_4], [c0, l0_1, l0_2, l1_0, l1_1, l1_2, l1_3, l1_4], [c0, c1, l0_1, l0_2, l1_0, l1_1, l1_2, l1_3, l1_4], [l0_0, l0_1, l0_2, l1_0, l1_1, l1_2, l1_3, l1_4], [c0, l0_0, l0_1, l0_2, l1_0, l1_1, l1_2, l1_3, l1_4], [c1, l0_0, l0_1, l0_2, l1_0, l1_1, l1_2, l1_3, l1_4], [c0, c1, l0_0, l0_1, l0_2, l1_0, l1_1, l1_2, l1_3, l1_4]], [0, 0, 1, 8, 29, 61, 81, 69, 37, 10, 1], 2, [[c0, c1]]]

source roles = [[l0_0, l0_1, l0_2, l1_0, l1_1, l1_2, l1_3, l1_4], [[c0, l0_0], [c0, l0_1], [c0, l0_2], [c1, l1_0], [c1, l1_1], [c1, l1_2], [c1, l1_3], [c1, l1_4]]]

derived = [[[[c0, l0_0], Yes], [[c0, l0_1], Yes], [[c0, l0_2], Yes], [[c1, l1_0], Yes], [[c1, l1_1], Yes], [[c1, l1_2], Yes], [[c1, l1_3], Yes], [[c1, l1_4], Yes]], 8, 2, [c0, c1]]

source roles = [[l0_0, l0_1, l0_2, l1_0, l1_1, l1_2, l1_3, l1_4], [[l0_0, [l0_1, l0_2, l1_0, l1_1, l1_2, l1_3, l1_4], No], [l0_1, [l0_0, l0_2, l1_0, l1_1, l1_2, l1_3, l1_4], No], [l0_2, [l0_0, l0_1, l1_0, l1_1, l1_2, l1_3, l1_4], No], [l1_0, [l0_0, l0_1, l0_2, l1_1, l1_2, l1_3, l1_4], No], [l1_1, [l0_0, l0_1, l0_2, l1_0, l1_2, l1_3, l1_4], No], [l1_2, [l0_0, l0_1, l0_2, l1_0, l1_1, l1_3, l1_4], No], [l1_3, [l0_0, l0_1, l0_2, l1_0, l1_1, l1_2, l1_4], No], [l1_4, [l0_0, l0_1, l0_2, l1_0, l1_1, l1_2, l1_3], No]]]

derived = [Yes, 2, [c0, c1], No]

Reverse checks

• S005 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S004 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S003 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S002 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S001 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.

Result

([l0_0, l0_1, l0_2, l1_0, l1_1, l1_2, l1_3, l1_4], [[l0_0, [l0_1, l0_2, l1_0, l1_1, l1_2, l1_3, l1_4], No], [l0_1, [l0_0, l0_2, l1_0, l1_1, l1_2, l1_3, l1_4], No], [l0_2, [l0_0, l0_1, l1_0, l1_1, l1_2, l1_3, l1_4], No], [l1_0, [l0_0, l0_1, l0_2, l1_1, l1_2, l1_3, l1_4], No], [l1_1, [l0_0, l0_1, l0_2, l1_0, l1_2, l1_3, l1_4], No], [l1_2, [l0_0, l0_1, l0_2, l1_0, l1_1, l1_3, l1_4], No], [l1_3, [l0_0, l0_1, l0_2, l1_0, l1_1, l1_2, l1_4], No], [l1_4, [l0_0, l0_1, l0_2, l1_0, l1_1, l1_2, l1_3], No]], Yes, 2, [c0, c1], No, inclusion_minimal_is_not_minimum)

The full substitutions and premise checks give the original answer; source identity,format and status are unchanged.

This question review and source binding

Review date: 2026-10-26

Original questions, answers, step expressions, numeric ASTs and historical records are retained. Source candidate labels remain historical and are not overwritten as formal proof.