Question 26524: Finite Clopen Components Do Not Supply a General Nonopen Witness or a Cardinality Definition

Original question and givens

Use mode=cardinality_not_definition and complete model={"points":[0,1,2,3,4,5,6,7],"topology":[[],[0,1,2,3,4,5,6,7],[0],[1,2,3,4,5,6,7]],"basepoint":0}. Give (mode, basepoint, own_component, larger_other_component, own_cardinality, other_cardinality, global_cardinality_definition_claim, boundary).

• G001 — Use mode=cardinality_not_definition and complete model={"points":[0,1,2,3,4,5,6,7],"topology":[[],[0,1,2,3,4,5,6,7],[0],[1,2,3,4,5,6,7]],"basepoint":0}. Give (mode, basepoint, own_component, larger_other_component, own_cardinality, other_cardinality, global_cardinality_definition_claim, boundary).

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-topological-connected-component

Step-by-step derivation

source roles = [component-boundary]

derived = [[mode, basepoint, own_component, larger_component, own_size, larger_size, global_maximum_definition, certificate]]

source roles = [component ORIGINAL space, [0, 1, 2, 3, 4, 5, 6, 7], [[], [0, 1, 2, 3, 4, 5, 6, 7], [0], [1, 2, 3, 4, 5, 6, 7]], No]

derived = [[0, 1, 2, 3, 4, 5, 6, 7], [[], [0, 1, 2, 3, 4, 5, 6, 7], [0], [1, 2, 3, 4, 5, 6, 7]], [[0, 0, 0, 0, Yes, Yes], [0, 1, 255, 0, Yes, Yes], [0, 2, 1, 0, Yes, Yes], [0, 3, 254, 0, Yes, Yes], [1, 0, 255, 0, Yes, Yes], [1, 1, 255, 255, Yes, Yes], [1, 2, 255, 1, Yes, Yes], [1, 3, 255, 254, Yes, Yes], [2, 0, 1, 0, Yes, Yes], [2, 1, 255, 1, Yes, Yes], [2, 2, 1, 1, Yes, Yes], [2, 3, 255, 0, Yes, Yes], [3, 0, 254, 0, Yes, Yes], [3, 1, 255, 254, Yes, Yes], [3, 2, 255, 0, Yes, Yes], [3, 3, 254, 254, Yes, Yes]]]

source roles = [[0, 1, 2, 3, 4, 5, 6, 7], [[], [0, 1, 2, 3, 4, 5, 6, 7], [0], [1, 2, 3, 4, 5, 6, 7]]]

derived = [[[1, [0], [[], [0]], null, Yes], [2, [1], [[], [1]], null, Yes], [3, [0, 1], [[], [0, 1], [0], [1]], [0], No], [4, [2], [[], [2]], null, Yes], [5, [0, 2], [[], [0, 2], [0], [2]], [0], No], [6, [1, 2], [[], [1, 2]], null, Yes], [7, [0, 1, 2], [[], [0, 1, 2], [0], [1, 2]], [0], No], [8, [3], [[], [3]], null, Yes], [9, [0, 3], [[], [0, 3], [0], [3]], [0], No], [10, [1, 3], [[], [1, 3]], null, Yes], [11, [0, 1, 3], [[], [0, 1, 3], [0], [1, 3]], [0], No], [12, [2, 3], [[], [2, 3]], null, Yes], [13, [0, 2, 3], [[], [0, 2, 3], [0], [2, 3]], [0], No], [14, [1, 2, 3], [[], [1, 2, 3]], null, Yes], [15, [0, 1, 2, 3], [[], [0, 1, 2, 3], [0], [1, 2, 3]], [0], No], [16, [4], [[], [4]], null, Yes], [17, [0, 4], [[], [0, 4], [0], [4]], [0], No], [18, [1, 4], [[], [1, 4]], null, Yes], [19, [0, 1, 4], [[], [0, 1, 4], [0], [1, 4]], [0], No], [20, [2, 4], [[], [2, 4]], null, Yes], [21, [0, 2, 4], [[], [0, 2, 4], [0], [2, 4]], [0], No], [22, [1, 2, 4], [[], [1, 2, 4]], null, Yes], [23, [0, 1, 2, 4], [[], [0, 1, 2, 4], [0], [1, 2, 4]], [0], No], [24, [3, 4], [[], [3, 4]], null, Yes], [25, [0, 3, 4], [[], [0, 3, 4], [0], [3, 4]], [0], No], [26, [1, 3, 4], [[], [1, 3, 4]], null, Yes], [27, [0, 1, 3, 4], [[], [0, 1, 3, 4], [0], [1, 3, 4]], [0], No], [28, [2, 3, 4], [[], [2, 3, 4]], null, Yes], [29, [0, 2, 3, 4], [[], [0, 2, 3, 4], [0], [2, 3, 4]], [0], No], [30, [1, 2, 3, 4], [[], [1, 2, 3, 4]], null, Yes], [31, [0, 1, 2, 3, 4], [[], [0, 1, 2, 3, 4], [0], [1, 2, 3, 4]], [0], No], [32, [5], [[], [5]], null, Yes], [33, [0, 5], [[], [0, 5], [0], [5]], [0], No], [34, [1, 5], [[], [1, 5]], null, Yes], [35, [0, 1, 5], [[], [0, 1, 5], [0], [1, 5]], [0], No], [36, [2, 5], [[], [2, 5]], null, Yes], [37, [0, 2, 5], [[], [0, 2, 5], [0], [2, 5]], [0], No], [38, [1, 2, 5], [[], [1, 2, 5]], null, Yes], [39, [0, 1, 2, 5], [[], [0, 1, 2, 5], [0], [1, 2, 5]], [0], No], [40, [3, 5], [[], [3, 5]], null, Yes], [41, [0, 3, 5], [[], [0, 3, 5], [0], [3, 5]], [0], No], [42, [1, 3, 5], [[], [1, 3, 5]], null, Yes], [43, [0, 1, 3, 5], [[], [0, 1, 3, 5], [0], [1, 3, 5]], [0], No], [44, [2, 3, 5], [[], [2, 3, 5]], null, Yes], [45, [0, 2, 3, 5], [[], [0, 2, 3, 5], [0], [2, 3, 5]], [0], No], [46, [1, 2, 3, 5], [[], [1, 2, 3, 5]], null, Yes], [47, [0, 1, 2, 3, 5], [[], [0, 1, 2, 3, 5], [0], [1, 2, 3, 5]], [0], No], [48, [4, 5], [[], [4, 5]], null, Yes], [49, [0, 4, 5], [[], [0, 4, 5], [0], [4, 5]], [0], No], [50, [1, 4, 5], [[], [1, 4, 5]], null, Yes], [51, [0, 1, 4, 5], [[], [0, 1, 4, 5], [0], [1, 4, 5]], [0], No], [52, [2, 4, 5], [[], [2, 4, 5]], null, Yes], [53, [0, 2, 4, 5], [[], [0, 2, 4, 5], [0], [2, 4, 5]], [0], No], [54, [1, 2, 4, 5], [[], [1, 2, 4, 5]], null, Yes], [55, [0, 1, 2, 4, 5], [[], [0, 1, 2, 4, 5], [0], [1, 2, 4, 5]], [0], No], [56, [3, 4, 5], [[], [3, 4, 5]], null, Yes], [57, [0, 3, 4, 5], [[], [0, 3, 4, 5], [0], [3, 4, 5]], [0], No], [58, [1, 3, 4, 5], [[], [1, 3, 4, 5]], null, Yes], [59, [0, 1, 3, 4, 5], [[], [0, 1, 3, 4, 5], [0], [1, 3, 4, 5]], [0], No], [60, [2, 3, 4, 5], [[], [2, 3, 4, 5]], null, Yes], [61, [0, 2, 3, 4, 5], [[], [0, 2, 3, 4, 5], [0], [2, 3, 4, 5]], [0], No], [62, [1, 2, 3, 4, 5], [[], [1, 2, 3, 4, 5]], null, Yes], [63, [0, 1, 2, 3, 4, 5], [[], [0, 1, 2, 3, 4, 5], [0], [1, 2, 3, 4, 5]], [0], No], [64, [6], [[], [6]], null, Yes], [65, [0, 6], [[], [0, 6], [0], [6]], [0], No], [66, [1, 6], [[], [1, 6]], null, Yes], [67, [0, 1, 6], [[], [0, 1, 6], [0], [1, 6]], [0], No], [68, [2, 6], [[], [2, 6]], null, Yes], [69, [0, 2, 6], [[], [0, 2, 6], [0], [2, 6]], [0], No], [70, [1, 2, 6], [[], [1, 2, 6]], null, Yes], [71, [0, 1, 2, 6], [[], [0, 1, 2, 6], [0], [1, 2, 6]], [0], No], [72, [3, 6], [[], [3, 6]], null, Yes], [73, [0, 3, 6], [[], [0, 3, 6], [0], [3, 6]], [0], No], [74, [1, 3, 6], [[], [1, 3, 6]], null, Yes], [75, [0, 1, 3, 6], [[], [0, 1, 3, 6], [0], [1, 3, 6]], [0], No], [76, [2, 3, 6], [[], [2, 3, 6]], null, Yes], [77, [0, 2, 3, 6], [[], [0, 2, 3, 6], [0], [2, 3, 6]], [0], No], [78, [1, 2, 3, 6], [[], [1, 2, 3, 6]], null, Yes], [79, [0, 1, 2, 3, 6], [[], [0, 1, 2, 3, 6], [0], [1, 2, 3, 6]], [0], No], [80, [4, 6], [[], [4, 6]], null, Yes], [81, [0, 4, 6], [[], [0, 4, 6], [0], [4, 6]], [0], No], [82, [1, 4, 6], [[], [1, 4, 6]], null, Yes], [83, [0, 1, 4, 6], [[], [0, 1, 4, 6], [0], [1, 4, 6]], [0], No], [84, [2, 4, 6], [[], [2, 4, 6]], null, Yes], [85, [0, 2, 4, 6], [[], [0, 2, 4, 6], [0], [2, 4, 6]], [0], No], [86, [1, 2, 4, 6], [[], [1, 2, 4, 6]], null, Yes], [87, [0, 1, 2, 4, 6], [[], [0, 1, 2, 4, 6], [0], [1, 2, 4, 6]], [0], No], [88, [3, 4, 6], [[], [3, 4, 6]], null, Yes], [89, [0, 3, 4, 6], [[], [0, 3, 4, 6], [0], [3, 4, 6]], [0], No], [90, [1, 3, 4, 6], [[], [1, 3, 4, 6]], null, Yes], [91, [0, 1, 3, 4, 6], [[], [0, 1, 3, 4, 6], [0], [1, 3, 4, 6]], [0], No], [92, [2, 3, 4, 6], [[], [2, 3, 4, 6]], null, Yes], [93, [0, 2, 3, 4, 6], [[], [0, 2, 3, 4, 6], [0], [2, 3, 4, 6]], [0], No], [94, [1, 2, 3, 4, 6], [[], [1, 2, 3, 4, 6]], null, Yes], [95, [0, 1, 2, 3, 4, 6], [[], [0, 1, 2, 3, 4, 6], [0], [1, 2, 3, 4, 6]], [0], No], [96, [5, 6], [[], [5, 6]], null, Yes], [97, [0, 5, 6], [[], [0, 5, 6], [0], [5, 6]], [0], No], [98, [1, 5, 6], [[], [1, 5, 6]], null, Yes], [99, [0, 1, 5, 6], [[], [0, 1, 5, 6], [0], [1, 5, 6]], [0], No], [100, [2, 5, 6], [[], [2, 5, 6]], null, Yes], [101, [0, 2, 5, 6], [[], [0, 2, 5, 6], [0], [2, 5, 6]], [0], No], [102, [1, 2, 5, 6], [[], [1, 2, 5, 6]], null, Yes], [103, [0, 1, 2, 5, 6], [[], [0, 1, 2, 5, 6], [0], [1, 2, 5, 6]], [0], No], [104, [3, 5, 6], [[], [3, 5, 6]], null, Yes], [105, [0, 3, 5, 6], [[], [0, 3, 5, 6], [0], [3, 5, 6]], [0], No], [106, [1, 3, 5, 6], [[], [1, 3, 5, 6]], null, Yes], [107, [0, 1, 3, 5, 6], [[], [0, 1, 3, 5, 6], [0], [1, 3, 5, 6]], [0], No], [108, [2, 3, 5, 6], [[], [2, 3, 5, 6]], null, Yes], [109, [0, 2, 3, 5, 6], [[], [0, 2, 3, 5, 6], [0], [2, 3, 5, 6]], [0], No], [110, [1, 2, 3, 5, 6], [[], [1, 2, 3, 5, 6]], null, Yes], [111, [0, 1, 2, 3, 5, 6], [[], [0, 1, 2, 3, 5, 6], [0], [1, 2, 3, 5, 6]], [0], No], [112, [4, 5, 6], [[], [4, 5, 6]], null, Yes], [113, [0, 4, 5, 6], [[], [0, 4, 5, 6], [0], [4, 5, 6]], [0], No], [114, [1, 4, 5, 6], [[], [1, 4, 5, 6]], null, Yes], [115, [0, 1, 4, 5, 6], [[], [0, 1, 4, 5, 6], [0], [1, 4, 5, 6]], [0], No], [116, [2, 4, 5, 6], [[], [2, 4, 5, 6]], null, Yes], [117, [0, 2, 4, 5, 6], [[], [0, 2, 4, 5, 6], [0], [2, 4, 5, 6]], [0], No], [118, [1, 2, 4, 5, 6], [[], [1, 2, 4, 5, 6]], null, Yes], [119, [0, 1, 2, 4, 5, 6], [[], [0, 1, 2, 4, 5, 6], [0], [1, 2, 4, 5, 6]], [0], No], [120, [3, 4, 5, 6], [[], [3, 4, 5, 6]], null, Yes], [121, [0, 3, 4, 5, 6], [[], [0, 3, 4, 5, 6], [0], [3, 4, 5, 6]], [0], No], [122, [1, 3, 4, 5, 6], [[], [1, 3, 4, 5, 6]], null, Yes], [123, [0, 1, 3, 4, 5, 6], [[], [0, 1, 3, 4, 5, 6], [0], [1, 3, 4, 5, 6]], [0], No], [124, [2, 3, 4, 5, 6], [[], [2, 3, 4, 5, 6]], null, Yes], [125, [0, 2, 3, 4, 5, 6], [[], [0, 2, 3, 4, 5, 6], [0], [2, 3, 4, 5, 6]], [0], No], [126, [1, 2, 3, 4, 5, 6], [[], [1, 2, 3, 4, 5, 6]], null, Yes], [127, [0, 1, 2, 3, 4, 5, 6], [[], [0, 1, 2, 3, 4, 5, 6], [0], [1, 2, 3, 4, 5, 6]], [0], No], [128, [7], [[], [7]], null, Yes], [129, [0, 7], [[], [0, 7], [0], [7]], [0], No], [130, [1, 7], [[], [1, 7]], null, Yes], [131, [0, 1, 7], [[], [0, 1, 7], [0], [1, 7]], [0], No], [132, [2, 7], [[], [2, 7]], null, Yes], [133, [0, 2, 7], [[], [0, 2, 7], [0], [2, 7]], [0], No], [134, [1, 2, 7], [[], [1, 2, 7]], null, Yes], [135, [0, 1, 2, 7], [[], [0, 1, 2, 7], [0], [1, 2, 7]], [0], No], [136, [3, 7], [[], [3, 7]], null, Yes], [137, [0, 3, 7], [[], [0, 3, 7], [0], [3, 7]], [0], No], [138, [1, 3, 7], [[], [1, 3, 7]], null, Yes], [139, [0, 1, 3, 7], [[], [0, 1, 3, 7], [0], [1, 3, 7]], [0], No], [140, [2, 3, 7], [[], [2, 3, 7]], null, Yes], [141, [0, 2, 3, 7], [[], [0, 2, 3, 7], [0], [2, 3, 7]], [0], No], [142, [1, 2, 3, 7], [[], [1, 2, 3, 7]], null, Yes], [143, [0, 1, 2, 3, 7], [[], [0, 1, 2, 3, 7], [0], [1, 2, 3, 7]], [0], No], [144, [4, 7], [[], [4, 7]], null, Yes], [145, [0, 4, 7], [[], [0, 4, 7], [0], [4, 7]], [0], No], [146, [1, 4, 7], [[], [1, 4, 7]], null, Yes], [147, [0, 1, 4, 7], [[], [0, 1, 4, 7], [0], [1, 4, 7]], [0], No], [148, [2, 4, 7], [[], [2, 4, 7]], null, Yes], [149, [0, 2, 4, 7], [[], [0, 2, 4, 7], [0], [2, 4, 7]], [0], No], [150, [1, 2, 4, 7], [[], [1, 2, 4, 7]], null, Yes], [151, [0, 1, 2, 4, 7], [[], [0, 1, 2, 4, 7], [0], [1, 2, 4, 7]], [0], No], [152, [3, 4, 7], [[], [3, 4, 7]], null, Yes], [153, [0, 3, 4, 7], [[], [0, 3, 4, 7], [0], [3, 4, 7]], [0], No], [154, [1, 3, 4, 7], [[], [1, 3, 4, 7]], null, Yes], [155, [0, 1, 3, 4, 7], [[], [0, 1, 3, 4, 7], [0], [1, 3, 4, 7]], [0], No], [156, [2, 3, 4, 7], [[], [2, 3, 4, 7]], null, Yes], [157, [0, 2, 3, 4, 7], [[], [0, 2, 3, 4, 7], [0], [2, 3, 4, 7]], [0], No], [158, [1, 2, 3, 4, 7], [[], [1, 2, 3, 4, 7]], null, Yes], [159, [0, 1, 2, 3, 4, 7], [[], [0, 1, 2, 3, 4, 7], [0], [1, 2, 3, 4, 7]], [0], No], [160, [5, 7], [[], [5, 7]], null, Yes], [161, [0, 5, 7], [[], [0, 5, 7], [0], [5, 7]], [0], No], [162, [1, 5, 7], [[], [1, 5, 7]], null, Yes], [163, [0, 1, 5, 7], [[], [0, 1, 5, 7], [0], [1, 5, 7]], [0], No], [164, [2, 5, 7], [[], [2, 5, 7]], null, Yes], [165, [0, 2, 5, 7], [[], [0, 2, 5, 7], [0], [2, 5, 7]], [0], No], [166, [1, 2, 5, 7], [[], [1, 2, 5, 7]], null, Yes], [167, [0, 1, 2, 5, 7], [[], [0, 1, 2, 5, 7], [0], [1, 2, 5, 7]], [0], No], [168, [3, 5, 7], [[], [3, 5, 7]], null, Yes], [169, [0, 3, 5, 7], [[], [0, 3, 5, 7], [0], [3, 5, 7]], [0], No], [170, [1, 3, 5, 7], [[], [1, 3, 5, 7]], null, Yes], [171, [0, 1, 3, 5, 7], [[], [0, 1, 3, 5, 7], [0], [1, 3, 5, 7]], [0], No], [172, [2, 3, 5, 7], [[], [2, 3, 5, 7]], null, Yes], [173, [0, 2, 3, 5, 7], [[], [0, 2, 3, 5, 7], [0], [2, 3, 5, 7]], [0], No], [174, [1, 2, 3, 5, 7], [[], [1, 2, 3, 5, 7]], null, Yes], [175, [0, 1, 2, 3, 5, 7], [[], [0, 1, 2, 3, 5, 7], [0], [1, 2, 3, 5, 7]], [0], No], [176, [4, 5, 7], [[], [4, 5, 7]], null, Yes], [177, [0, 4, 5, 7], [[], [0, 4, 5, 7], [0], [4, 5, 7]], [0], No], [178, [1, 4, 5, 7], [[], [1, 4, 5, 7]], null, Yes], [179, [0, 1, 4, 5, 7], [[], [0, 1, 4, 5, 7], [0], [1, 4, 5, 7]], [0], No], [180, [2, 4, 5, 7], [[], [2, 4, 5, 7]], null, Yes], [181, [0, 2, 4, 5, 7], [[], [0, 2, 4, 5, 7], [0], [2, 4, 5, 7]], [0], No], [182, [1, 2, 4, 5, 7], [[], [1, 2, 4, 5, 7]], null, Yes], [183, [0, 1, 2, 4, 5, 7], [[], [0, 1, 2, 4, 5, 7], [0], [1, 2, 4, 5, 7]], [0], No], [184, [3, 4, 5, 7], [[], [3, 4, 5, 7]], null, Yes], [185, [0, 3, 4, 5, 7], [[], [0, 3, 4, 5, 7], [0], [3, 4, 5, 7]], [0], No], [186, [1, 3, 4, 5, 7], [[], [1, 3, 4, 5, 7]], null, Yes], [187, [0, 1, 3, 4, 5, 7], [[], [0, 1, 3, 4, 5, 7], [0], [1, 3, 4, 5, 7]], [0], No], [188, [2, 3, 4, 5, 7], [[], [2, 3, 4, 5, 7]], null, Yes], [189, [0, 2, 3, 4, 5, 7], [[], [0, 2, 3, 4, 5, 7], [0], [2, 3, 4, 5, 7]], [0], No], [190, [1, 2, 3, 4, 5, 7], [[], [1, 2, 3, 4, 5, 7]], null, Yes], [191, [0, 1, 2, 3, 4, 5, 7], [[], [0, 1, 2, 3, 4, 5, 7], [0], [1, 2, 3, 4, 5, 7]], [0], No], [192, [6, 7], [[], [6, 7]], null, Yes], [193, [0, 6, 7], [[], [0, 6, 7], [0], [6, 7]], [0], No], [194, [1, 6, 7], [[], [1, 6, 7]], null, Yes], [195, [0, 1, 6, 7], [[], [0, 1, 6, 7], [0], [1, 6, 7]], [0], No], [196, [2, 6, 7], [[], [2, 6, 7]], null, Yes], [197, [0, 2, 6, 7], [[], [0, 2, 6, 7], [0], [2, 6, 7]], [0], No], [198, [1, 2, 6, 7], [[], [1, 2, 6, 7]], null, Yes], [199, [0, 1, 2, 6, 7], [[], [0, 1, 2, 6, 7], [0], [1, 2, 6, 7]], [0], No], [200, [3, 6, 7], [[], [3, 6, 7]], null, Yes], [201, [0, 3, 6, 7], [[], [0, 3, 6, 7], [0], [3, 6, 7]], [0], No], [202, [1, 3, 6, 7], [[], [1, 3, 6, 7]], null, Yes], [203, [0, 1, 3, 6, 7], [[], [0, 1, 3, 6, 7], [0], [1, 3, 6, 7]], [0], No], [204, [2, 3, 6, 7], [[], [2, 3, 6, 7]], null, Yes], [205, [0, 2, 3, 6, 7], [[], [0, 2, 3, 6, 7], [0], [2, 3, 6, 7]], [0], No], [206, [1, 2, 3, 6, 7], [[], [1, 2, 3, 6, 7]], null, Yes], [207, [0, 1, 2, 3, 6, 7], [[], [0, 1, 2, 3, 6, 7], [0], [1, 2, 3, 6, 7]], [0], No], [208, [4, 6, 7], [[], [4, 6, 7]], null, Yes], [209, [0, 4, 6, 7], [[], [0, 4, 6, 7], [0], [4, 6, 7]], [0], No], [210, [1, 4, 6, 7], [[], [1, 4, 6, 7]], null, Yes], [211, [0, 1, 4, 6, 7], [[], [0, 1, 4, 6, 7], [0], [1, 4, 6, 7]], [0], No], [212, [2, 4, 6, 7], [[], [2, 4, 6, 7]], null, Yes], [213, [0, 2, 4, 6, 7], [[], [0, 2, 4, 6, 7], [0], [2, 4, 6, 7]], [0], No], [214, [1, 2, 4, 6, 7], [[], [1, 2, 4, 6, 7]], null, Yes], [215, [0, 1, 2, 4, 6, 7], [[], [0, 1, 2, 4, 6, 7], [0], [1, 2, 4, 6, 7]], [0], No], [216, [3, 4, 6, 7], [[], [3, 4, 6, 7]], null, Yes], [217, [0, 3, 4, 6, 7], [[], [0, 3, 4, 6, 7], [0], [3, 4, 6, 7]], [0], No], [218, [1, 3, 4, 6, 7], [[], [1, 3, 4, 6, 7]], null, Yes], [219, [0, 1, 3, 4, 6, 7], [[], [0, 1, 3, 4, 6, 7], [0], [1, 3, 4, 6, 7]], [0], No], [220, [2, 3, 4, 6, 7], [[], [2, 3, 4, 6, 7]], null, Yes], [221, [0, 2, 3, 4, 6, 7], [[], [0, 2, 3, 4, 6, 7], [0], [2, 3, 4, 6, 7]], [0], No], [222, [1, 2, 3, 4, 6, 7], [[], [1, 2, 3, 4, 6, 7]], null, Yes], [223, [0, 1, 2, 3, 4, 6, 7], [[], [0, 1, 2, 3, 4, 6, 7], [0], [1, 2, 3, 4, 6, 7]], [0], No], [224, [5, 6, 7], [[], [5, 6, 7]], null, Yes], [225, [0, 5, 6, 7], [[], [0, 5, 6, 7], [0], [5, 6, 7]], [0], No], [226, [1, 5, 6, 7], [[], [1, 5, 6, 7]], null, Yes], [227, [0, 1, 5, 6, 7], [[], [0, 1, 5, 6, 7], [0], [1, 5, 6, 7]], [0], No], [228, [2, 5, 6, 7], [[], [2, 5, 6, 7]], null, Yes], [229, [0, 2, 5, 6, 7], [[], [0, 2, 5, 6, 7], [0], [2, 5, 6, 7]], [0], No], [230, [1, 2, 5, 6, 7], [[], [1, 2, 5, 6, 7]], null, Yes], [231, [0, 1, 2, 5, 6, 7], [[], [0, 1, 2, 5, 6, 7], [0], [1, 2, 5, 6, 7]], [0], No], [232, [3, 5, 6, 7], [[], [3, 5, 6, 7]], null, Yes], [233, [0, 3, 5, 6, 7], [[], [0, 3, 5, 6, 7], [0], [3, 5, 6, 7]], [0], No], [234, [1, 3, 5, 6, 7], [[], [1, 3, 5, 6, 7]], null, Yes], [235, [0, 1, 3, 5, 6, 7], [[], [0, 1, 3, 5, 6, 7], [0], [1, 3, 5, 6, 7]], [0], No], [236, [2, 3, 5, 6, 7], [[], [2, 3, 5, 6, 7]], null, Yes], [237, [0, 2, 3, 5, 6, 7], [[], [0, 2, 3, 5, 6, 7], [0], [2, 3, 5, 6, 7]], [0], No], [238, [1, 2, 3, 5, 6, 7], [[], [1, 2, 3, 5, 6, 7]], null, Yes], [239, [0, 1, 2, 3, 5, 6, 7], [[], [0, 1, 2, 3, 5, 6, 7], [0], [1, 2, 3, 5, 6, 7]], [0], No], [240, [4, 5, 6, 7], [[], [4, 5, 6, 7]], null, Yes], [241, [0, 4, 5, 6, 7], [[], [0, 4, 5, 6, 7], [0], [4, 5, 6, 7]], [0], No], [242, [1, 4, 5, 6, 7], [[], [1, 4, 5, 6, 7]], null, Yes], [243, [0, 1, 4, 5, 6, 7], [[], [0, 1, 4, 5, 6, 7], [0], [1, 4, 5, 6, 7]], [0], No], [244, [2, 4, 5, 6, 7], [[], [2, 4, 5, 6, 7]], null, Yes], [245, [0, 2, 4, 5, 6, 7], [[], [0, 2, 4, 5, 6, 7], [0], [2, 4, 5, 6, 7]], [0], No], [246, [1, 2, 4, 5, 6, 7], [[], [1, 2, 4, 5, 6, 7]], null, Yes], [247, [0, 1, 2, 4, 5, 6, 7], [[], [0, 1, 2, 4, 5, 6, 7], [0], [1, 2, 4, 5, 6, 7]], [0], No], [248, [3, 4, 5, 6, 7], [[], [3, 4, 5, 6, 7]], null, Yes], [249, [0, 3, 4, 5, 6, 7], [[], [0, 3, 4, 5, 6, 7], [0], [3, 4, 5, 6, 7]], [0], No], [250, [1, 3, 4, 5, 6, 7], [[], [1, 3, 4, 5, 6, 7]], null, Yes], [251, [0, 1, 3, 4, 5, 6, 7], [[], [0, 1, 3, 4, 5, 6, 7], [0], [1, 3, 4, 5, 6, 7]], [0], No], [252, [2, 3, 4, 5, 6, 7], [[], [2, 3, 4, 5, 6, 7]], null, Yes], [253, [0, 2, 3, 4, 5, 6, 7], [[], [0, 2, 3, 4, 5, 6, 7], [0], [2, 3, 4, 5, 6, 7]], [0], No], [254, [1, 2, 3, 4, 5, 6, 7], [[], [1, 2, 3, 4, 5, 6, 7]], null, Yes], [255, [0, 1, 2, 3, 4, 5, 6, 7], [[], [0, 1, 2, 3, 4, 5, 6, 7], [0], [1, 2, 3, 4, 5, 6, 7]], [0], No]], [[0], [1], [2], [1, 2], [3], [1, 3], [2, 3], [1, 2, 3], [4], [1, 4], [2, 4], [1, 2, 4], [3, 4], [1, 3, 4], [2, 3, 4], [1, 2, 3, 4], [5], [1, 5], [2, 5], [1, 2, 5], [3, 5], [1, 3, 5], [2, 3, 5], [1, 2, 3, 5], [4, 5], [1, 4, 5], [2, 4, 5], [1, 2, 4, 5], [3, 4, 5], [1, 3, 4, 5], [2, 3, 4, 5], [1, 2, 3, 4, 5], [6], [1, 6], [2, 6], [1, 2, 6], [3, 6], [1, 3, 6], [2, 3, 6], [1, 2, 3, 6], [4, 6], [1, 4, 6], [2, 4, 6], [1, 2, 4, 6], [3, 4, 6], [1, 3, 4, 6], [2, 3, 4, 6], [1, 2, 3, 4, 6], [5, 6], [1, 5, 6], [2, 5, 6], [1, 2, 5, 6], [3, 5, 6], [1, 3, 5, 6], [2, 3, 5, 6], [1, 2, 3, 5, 6], [4, 5, 6], [1, 4, 5, 6], [2, 4, 5, 6], [1, 2, 4, 5, 6], [3, 4, 5, 6], [1, 3, 4, 5, 6], [2, 3, 4, 5, 6], [1, 2, 3, 4, 5, 6], [7], [1, 7], [2, 7], [1, 2, 7], [3, 7], [1, 3, 7], [2, 3, 7], [1, 2, 3, 7], [4, 7], [1, 4, 7], [2, 4, 7], [1, 2, 4, 7], [3, 4, 7], [1, 3, 4, 7], [2, 3, 4, 7], [1, 2, 3, 4, 7], [5, 7], [1, 5, 7], [2, 5, 7], [1, 2, 5, 7], [3, 5, 7], [1, 3, 5, 7], [2, 3, 5, 7], [1, 2, 3, 5, 7], [4, 5, 7], [1, 4, 5, 7], [2, 4, 5, 7], [1, 2, 4, 5, 7], [3, 4, 5, 7], [1, 3, 4, 5, 7], [2, 3, 4, 5, 7], [1, 2, 3, 4, 5, 7], [6, 7], [1, 6, 7], [2, 6, 7], [1, 2, 6, 7], [3, 6, 7], [1, 3, 6, 7], [2, 3, 6, 7], [1, 2, 3, 6, 7], [4, 6, 7], [1, 4, 6, 7], [2, 4, 6, 7], [1, 2, 4, 6, 7], [3, 4, 6, 7], [1, 3, 4, 6, 7], [2, 3, 4, 6, 7], [1, 2, 3, 4, 6, 7], [5, 6, 7], [1, 5, 6, 7], [2, 5, 6, 7], [1, 2, 5, 6, 7], [3, 5, 6, 7], [1, 3, 5, 6, 7], [2, 3, 5, 6, 7], [1, 2, 3, 5, 6, 7], [4, 5, 6, 7], [1, 4, 5, 6, 7], [2, 4, 5, 6, 7], [1, 2, 4, 5, 6, 7], [3, 4, 5, 6, 7], [1, 3, 4, 5, 6, 7], [2, 3, 4, 5, 6, 7], [1, 2, 3, 4, 5, 6, 7]]]

source roles = [[[0], [1], [2], [1, 2], [3], [1, 3], [2, 3], [1, 2, 3], [4], [1, 4], [2, 4], [1, 2, 4], [3, 4], [1, 3, 4], [2, 3, 4], [1, 2, 3, 4], [5], [1, 5], [2, 5], [1, 2, 5], [3, 5], [1, 3, 5], [2, 3, 5], [1, 2, 3, 5], [4, 5], [1, 4, 5], [2, 4, 5], [1, 2, 4, 5], [3, 4, 5], [1, 3, 4, 5], [2, 3, 4, 5], [1, 2, 3, 4, 5], [6], [1, 6], [2, 6], [1, 2, 6], [3, 6], [1, 3, 6], [2, 3, 6], [1, 2, 3, 6], [4, 6], [1, 4, 6], [2, 4, 6], [1, 2, 4, 6], [3, 4, 6], [1, 3, 4, 6], [2, 3, 4, 6], [1, 2, 3, 4, 6], [5, 6], [1, 5, 6], [2, 5, 6], [1, 2, 5, 6], [3, 5, 6], [1, 3, 5, 6], [2, 3, 5, 6], [1, 2, 3, 5, 6], [4, 5, 6], [1, 4, 5, 6], [2, 4, 5, 6], [1, 2, 4, 5, 6], [3, 4, 5, 6], [1, 3, 4, 5, 6], [2, 3, 4, 5, 6], [1, 2, 3, 4, 5, 6], [7], [1, 7], [2, 7], [1, 2, 7], [3, 7], [1, 3, 7], [2, 3, 7], [1, 2, 3, 7], [4, 7], [1, 4, 7], [2, 4, 7], [1, 2, 4, 7], [3, 4, 7], [1, 3, 4, 7], [2, 3, 4, 7], [1, 2, 3, 4, 7], [5, 7], [1, 5, 7], [2, 5, 7], [1, 2, 5, 7], [3, 5, 7], [1, 3, 5, 7], [2, 3, 5, 7], [1, 2, 3, 5, 7], [4, 5, 7], [1, 4, 5, 7], [2, 4, 5, 7], [1, 2, 4, 5, 7], [3, 4, 5, 7], [1, 3, 4, 5, 7], [2, 3, 4, 5, 7], [1, 2, 3, 4, 5, 7], [6, 7], [1, 6, 7], [2, 6, 7], [1, 2, 6, 7], [3, 6, 7], [1, 3, 6, 7], [2, 3, 6, 7], [1, 2, 3, 6, 7], [4, 6, 7], [1, 4, 6, 7], [2, 4, 6, 7], [1, 2, 4, 6, 7], [3, 4, 6, 7], [1, 3, 4, 6, 7], [2, 3, 4, 6, 7], [1, 2, 3, 4, 6, 7], [5, 6, 7], [1, 5, 6, 7], [2, 5, 6, 7], [1, 2, 5, 6, 7], [3, 5, 6, 7], [1, 3, 5, 6, 7], [2, 3, 5, 6, 7], [1, 2, 3, 5, 6, 7], [4, 5, 6, 7], [1, 4, 5, 6, 7], [2, 4, 5, 6, 7], [1, 2, 4, 5, 6, 7], [3, 4, 5, 6, 7], [1, 3, 4, 5, 6, 7], [2, 3, 4, 5, 6, 7], [1, 2, 3, 4, 5, 6, 7]]]

derived = [[[0], [1, 2, 3, 4, 5, 6, 7]], Yes, Yes, [Yes, Yes], [Yes, Yes]]

source roles = [0, [[0]]]

derived = [[0], [0], Yes, Yes]

source roles = [0, [0], [[0], [1, 2, 3, 4, 5, 6, 7]]]

derived = [[1, 2, 3, 4, 5, 6, 7], 1, 7, No]

Reverse checks

• S006 · premise_and_result_audit — Reverse-check the complete original data,types,field order and premises;finite audits are not formal proofs of infinite theorems.
• S005 · premise_and_result_audit — Reverse-check the complete original data,types,field order and premises;finite audits are not formal proofs of infinite theorems.
• S004 · premise_and_result_audit — Reverse-check the complete original data,types,field order and premises;finite audits are not formal proofs of infinite theorems.
• S003 · premise_and_result_audit — Reverse-check the complete original data,types,field order and premises;finite audits are not formal proofs of infinite theorems.
• S002 · premise_and_result_audit — Reverse-check the complete original data,types,field order and premises;finite audits are not formal proofs of infinite theorems.
• S001 · premise_and_result_audit — Reverse-check the complete original data,types,field order and premises;finite audits are not formal proofs of infinite theorems.

Result

(cardinality_not_definition, 0, [0], [1, 2, 3, 4, 5, 6, 7], 1, 7, No, component_is_inclusion_maximal_not_global_maximum_cardinality)

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.