S001 · implies
Before
source roles = [pump-dfa-loop-witness]
After
derived = [[modulus, accepted_unary_word_length, first_modulus_plus_one_states, repeated_state_positions_and_state, x_length, y_length, z_length, all_pump_counts_invariant, loop_qualified]]
Conditions: none
List the complete source field order and mode-specific roles without JavaScript or answer-shape inference;No,undefined and infinity retain their distinct meanings.
Step identifiers and dependencies
Rule: source_ordered_fields_without_JS · Uses: G001
S003 · implies
Before
source roles = [23, 69]
After
derived = [[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22], [[0, 1], [1, 2], [2, 3], [3, 4], [4, 5], [5, 6], [6, 7], [7, 8], [8, 9], [9, 10], [10, 11], [11, 12], [12, 13], [13, 14], [14, 15], [15, 16], [16, 17], [17, 18], [18, 19], [19, 20], [20, 21], [21, 22], [22, 0]], [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 0], [0, 23, 0], 0, 23, 46]
Conditions: none
The DFA state is length mod m;every unary transition adds1 modulo m.The first m+1 states repeat0 at positions0,m.The ORIGINAL word length is divisible by m,so x empty,y length m and the remaining z form a legal accepted word.
Step identifiers and dependencies
Rule: Pumping_complete_unary_DFA_cycle_and_actual_accepted_length · Uses: G001, S002
S004 · implies
Before
source roles = [23, 69, 46]
After
derived = [EVERY integer i>=0: pumped length=i*modulus+z, z is divisible by modulus, state remains0 after each loop;accepted, Yes]
Conditions: none
For arbitrary i>=0,traverse the same complete loop i times and then z.Since z is also divisible by m,the final state remains0.Full transitions and this quantified invariant support EVERY i,not sampled pump counts.
Step identifiers and dependencies
Rule: Pumping_ALL_i_loop_invariant_NOT_finite_pump_tests · Uses: G001, S003