central_binom_succ_double_middle

Proof neighborhood

Direct dependencies

Direct dependents

Formal native tactic body

Dependencies are hypotheses of this body receipt. The focused endpoint audits separately check the complete empty-context certificates.

1. 0001intro n
2. 0002intro d
3. 0003intro hsuccessor
4. 0004have hupper : S n + S n = S (S (n + n))
5. 0005trans S (n + S n)
6. 0006specialize add_succ_left n
7. 0007specialize add_succ_left (S n)
8. 0008apply add_succ_left
9. 0009congr
10. 0010apply PA4
11. 0011have hnormalized : ((exists bcf_lt_gap_bcbsdm_normalized_out_of_range. bcf_lt_gap_bcbsdm_normalized_out_of_range + S (S (S (n + n))) = S n) /\ d = 0) \/ ((exists bcf_le_gap_bcbsdm_normalized_in_range. bcf_le_gap_bcbsdm_normalized_in_range + (S n) = S (S (n + n))) /\ (exists bcf_row_code_code_bcbsdm_normalized bcf_row_code_scale_bcbsdm_normalized bcf_row_scale_code_bcbsdm_normalized bcf_row_scale_scale_bcbsdm_normalized bcf_row_code_bcbsdm_normalized bcf_row_scale_bcbsdm_normalized. ((forall bcf_row_index_bcbsdm_normalized_table. (exists bcf_lt_gap_bcbsdm_normalized_table_row_bound. bcf_lt_gap_bcbsdm_normalized_table_row_bound + S (bcf_row_index_bcbsdm_normalized_table) = S (S (S (n + n)))) -> exists bcf_row_code_bcbsdm_normalized_table bcf_row_scale_bcbsdm_normalized_table. ((((exists bcf_height_bcbsdm_normalized_table_decoded_row_code. bcf_height_bcbsdm_normalized_table_decoded_row_code + S (bcf_row_code_bcbsdm_normalized_table) = S ((S (bcf_row_index_bcbsdm_normalized_table)) * bcf_row_code_scale_bcbsdm_normalized)) /\ exists bcf_quotient_bcbsdm_normalized_table_decoded_row_code. bcf_row_code_code_bcbsdm_normalized = bcf_quotient_bcbsdm_normalized_table_decoded_row_code * S ((S (bcf_row_index_bcbsdm_normalized_table)) * bcf_row_code_scale_bcbsdm_normalized) + (bcf_row_code_bcbsdm_normalized_table))) /\ ((((exists bcf_height_bcbsdm_normalized_table_decoded_row_scale. bcf_height_bcbsdm_normalized_table_decoded_row_scale + S (bcf_row_scale_bcbsdm_normalized_table) = S ((S (bcf_row_index_bcbsdm_normalized_table)) * bcf_row_scale_scale_bcbsdm_normalized)) /\ exists bcf_quotient_bcbsdm_normalized_table_decoded_row_scale. bcf_row_scale_code_bcbsdm_normalized = bcf_quotient_bcbsdm_normalized_table_decoded_row_scale * S ((S (bcf_row_index_bcbsdm_normalized_table)) * bcf_row_scale_scale_bcbsdm_normalized) + (bcf_row_scale_bcbsdm_normalized_table))) /\ ((bcf_row_index_bcbsdm_normalized_table = 0 /\ (forall bcf_index_bcbsdm_normalized_table_zero_row. (exists bcf_lt_gap_bcbsdm_normalized_table_zero_row_bound. bcf_lt_gap_bcbsdm_normalized_table_zero_row_bound + S (bcf_index_bcbsdm_normalized_table_zero_row) = S (S (S (n + n)))) -> exists bcf_value_bcbsdm_normalized_table_zero_row. ((((exists bcf_height_bcbsdm_normalized_table_zero_row_entry. bcf_height_bcbsdm_normalized_table_zero_row_entry + S (bcf_value_bcbsdm_normalized_table_zero_row) = S ((S (bcf_index_bcbsdm_normalized_table_zero_row)) * bcf_row_scale_bcbsdm_normalized_table)) /\ exists bcf_quotient_bcbsdm_normalized_table_zero_row_entry. bcf_row_code_bcbsdm_normalized_table = bcf_quotient_bcbsdm_normalized_table_zero_row_entry * S ((S (bcf_index_bcbsdm_normalized_table_zero_row)) * bcf_row_scale_bcbsdm_normalized_table) + (bcf_value_bcbsdm_normalized_table_zero_row))) /\ ((bcf_index_bcbsdm_normalized_table_zero_row = 0 /\ bcf_value_bcbsdm_normalized_table_zero_row = 1) \/ exists bcf_predecessor_bcbsdm_normalized_table_zero_row. bcf_index_bcbsdm_normalized_table_zero_row = S bcf_predecessor_bcbsdm_normalized_table_zero_row /\ bcf_value_bcbsdm_normalized_table_zero_row = 0)))) \/ exists bcf_predecessor_bcbsdm_normalized_table bcf_previous_code_bcbsdm_normalized_table bcf_previous_scale_bcbsdm_normalized_table. bcf_row_index_bcbsdm_normalized_table = S bcf_predecessor_bcbsdm_normalized_table /\ ((((exists bcf_height_bcbsdm_normalized_table_decoded_previous_code. bcf_height_bcbsdm_normalized_table_decoded_previous_code + S (bcf_previous_code_bcbsdm_normalized_table) = S ((S (bcf_predecessor_bcbsdm_normalized_table)) * bcf_row_code_scale_bcbsdm_normalized)) /\ exists bcf_quotient_bcbsdm_normalized_table_decoded_previous_code. bcf_row_code_code_bcbsdm_normalized = bcf_quotient_bcbsdm_normalized_table_decoded_previous_code * S ((S (bcf_predecessor_bcbsdm_normalized_table)) * bcf_row_code_scale_bcbsdm_normalized) + (bcf_previous_code_bcbsdm_normalized_table))) /\ ((((exists bcf_height_bcbsdm_normalized_table_decoded_previous_scale. bcf_height_bcbsdm_normalized_table_decoded_previous_scale + S (bcf_previous_scale_bcbsdm_normalized_table) = S ((S (bcf_predecessor_bcbsdm_normalized_table)) * bcf_row_scale_scale_bcbsdm_normalized)) /\ exists bcf_quotient_bcbsdm_normalized_table_decoded_previous_scale. bcf_row_scale_code_bcbsdm_normalized = bcf_quotient_bcbsdm_normalized_table_decoded_previous_scale * S ((S (bcf_predecessor_bcbsdm_normalized_table)) * bcf_row_scale_scale_bcbsdm_normalized) + (bcf_previous_scale_bcbsdm_normalized_table))) /\ (forall bcf_index_bcbsdm_normalized_table_row_step. (exists bcf_lt_gap_bcbsdm_normalized_table_row_step_bound. bcf_lt_gap_bcbsdm_normalized_table_row_step_bound + S (bcf_index_bcbsdm_normalized_table_row_step) = S (S (S (n + n)))) -> exists bcf_value_bcbsdm_normalized_table_row_step. ((((exists bcf_height_bcbsdm_normalized_table_row_step_entry. bcf_height_bcbsdm_normalized_table_row_step_entry + S (bcf_value_bcbsdm_normalized_table_row_step) = S ((S (bcf_index_bcbsdm_normalized_table_row_step)) * bcf_row_scale_bcbsdm_normalized_table)) /\ exists bcf_quotient_bcbsdm_normalized_table_row_step_entry. bcf_row_code_bcbsdm_normalized_table = bcf_quotient_bcbsdm_normalized_table_row_step_entry * S ((S (bcf_index_bcbsdm_normalized_table_row_step)) * bcf_row_scale_bcbsdm_normalized_table) + (bcf_value_bcbsdm_normalized_table_row_step))) /\ ((bcf_index_bcbsdm_normalized_table_row_step = 0 /\ bcf_value_bcbsdm_normalized_table_row_step = 1) \/ exists bcf_predecessor_bcbsdm_normalized_table_row_step bcf_left_bcbsdm_normalized_table_row_step bcf_right_bcbsdm_normalized_table_row_step. bcf_index_bcbsdm_normalized_table_row_step = S bcf_predecessor_bcbsdm_normalized_table_row_step /\ ((((exists bcf_height_bcbsdm_normalized_table_row_step_previous_left. bcf_height_bcbsdm_normalized_table_row_step_previous_left + S (bcf_left_bcbsdm_normalized_table_row_step) = S ((S (bcf_predecessor_bcbsdm_normalized_table_row_step)) * bcf_previous_scale_bcbsdm_normalized_table)) /\ exists bcf_quotient_bcbsdm_normalized_table_row_step_previous_left. bcf_previous_code_bcbsdm_normalized_table = bcf_quotient_bcbsdm_normalized_table_row_step_previous_left * S ((S (bcf_predecessor_bcbsdm_normalized_table_row_step)) * bcf_previous_scale_bcbsdm_normalized_table) + (bcf_left_bcbsdm_normalized_table_row_step))) /\ ((((exists bcf_height_bcbsdm_normalized_table_row_step_previous_right. bcf_height_bcbsdm_normalized_table_row_step_previous_right + S (bcf_right_bcbsdm_normalized_table_row_step) = S ((S (S (bcf_predecessor_bcbsdm_normalized_table_row_step))) * bcf_previous_scale_bcbsdm_normalized_table)) /\ exists bcf_quotient_bcbsdm_normalized_table_row_step_previous_right. bcf_previous_code_bcbsdm_normalized_table = bcf_quotient_bcbsdm_normalized_table_row_step_previous_right * S ((S (S (bcf_predecessor_bcbsdm_normalized_table_row_step))) * bcf_previous_scale_bcbsdm_normalized_table) + (bcf_right_bcbsdm_normalized_table_row_step))) /\ bcf_value_bcbsdm_normalized_table_row_step = bcf_left_bcbsdm_normalized_table_row_step + bcf_right_bcbsdm_normalized_table_row_step))))))))))) /\ ((((exists bcf_height_bcbsdm_normalized_decoded_row_code. bcf_height_bcbsdm_normalized_decoded_row_code + S (bcf_row_code_bcbsdm_normalized) = S ((S (S (S (n + n)))) * bcf_row_code_scale_bcbsdm_normalized)) /\ exists bcf_quotient_bcbsdm_normalized_decoded_row_code. bcf_row_code_code_bcbsdm_normalized = bcf_quotient_bcbsdm_normalized_decoded_row_code * S ((S (S (S (n + n)))) * bcf_row_code_scale_bcbsdm_normalized) + (bcf_row_code_bcbsdm_normalized))) /\ ((((exists bcf_height_bcbsdm_normalized_decoded_row_scale. bcf_height_bcbsdm_normalized_decoded_row_scale + S (bcf_row_scale_bcbsdm_normalized) = S ((S (S (S (n + n)))) * bcf_row_scale_scale_bcbsdm_normalized)) /\ exists bcf_quotient_bcbsdm_normalized_decoded_row_scale. bcf_row_scale_code_bcbsdm_normalized = bcf_quotient_bcbsdm_normalized_decoded_row_scale * S ((S (S (S (n + n)))) * bcf_row_scale_scale_bcbsdm_normalized) + (bcf_row_scale_bcbsdm_normalized))) /\ (((exists bcf_height_bcbsdm_normalized_decoded_value. bcf_height_bcbsdm_normalized_decoded_value + S (d) = S ((S (S n)) * bcf_row_scale_bcbsdm_normalized)) /\ exists bcf_quotient_bcbsdm_normalized_decoded_value. bcf_row_code_bcbsdm_normalized = bcf_quotient_bcbsdm_normalized_decoded_value * S ((S (S n)) * bcf_row_scale_bcbsdm_normalized) + (d))))))))
12. 0012specialize choose_upper_eq_transport (S n + S n)
13. 0013specialize choose_upper_eq_transport (S (S (n + n)))
14. 0014specialize choose_upper_eq_transport (S n)
15. 0015specialize choose_upper_eq_transport d
16. 0016apply choose_upper_eq_transport
17. 0017exact hupper
18. 0018exact hsuccessor
19. 0019have hmiddle_exists : exists m. (((exists bcf_lt_gap_bcbsdm_middle_out_of_range. bcf_lt_gap_bcbsdm_middle_out_of_range + S (S (n + n)) = n) /\ m = 0) \/ ((exists bcf_le_gap_bcbsdm_middle_in_range. bcf_le_gap_bcbsdm_middle_in_range + (n) = S (n + n)) /\ (exists bcf_row_code_code_bcbsdm_middle bcf_row_code_scale_bcbsdm_middle bcf_row_scale_code_bcbsdm_middle bcf_row_scale_scale_bcbsdm_middle bcf_row_code_bcbsdm_middle bcf_row_scale_bcbsdm_middle. ((forall bcf_row_index_bcbsdm_middle_table. (exists bcf_lt_gap_bcbsdm_middle_table_row_bound. bcf_lt_gap_bcbsdm_middle_table_row_bound + S (bcf_row_index_bcbsdm_middle_table) = S (S (n + n))) -> exists bcf_row_code_bcbsdm_middle_table bcf_row_scale_bcbsdm_middle_table. ((((exists bcf_height_bcbsdm_middle_table_decoded_row_code. bcf_height_bcbsdm_middle_table_decoded_row_code + S (bcf_row_code_bcbsdm_middle_table) = S ((S (bcf_row_index_bcbsdm_middle_table)) * bcf_row_code_scale_bcbsdm_middle)) /\ exists bcf_quotient_bcbsdm_middle_table_decoded_row_code. bcf_row_code_code_bcbsdm_middle = bcf_quotient_bcbsdm_middle_table_decoded_row_code * S ((S (bcf_row_index_bcbsdm_middle_table)) * bcf_row_code_scale_bcbsdm_middle) + (bcf_row_code_bcbsdm_middle_table))) /\ ((((exists bcf_height_bcbsdm_middle_table_decoded_row_scale. bcf_height_bcbsdm_middle_table_decoded_row_scale + S (bcf_row_scale_bcbsdm_middle_table) = S ((S (bcf_row_index_bcbsdm_middle_table)) * bcf_row_scale_scale_bcbsdm_middle)) /\ exists bcf_quotient_bcbsdm_middle_table_decoded_row_scale. bcf_row_scale_code_bcbsdm_middle = bcf_quotient_bcbsdm_middle_table_decoded_row_scale * S ((S (bcf_row_index_bcbsdm_middle_table)) * bcf_row_scale_scale_bcbsdm_middle) + (bcf_row_scale_bcbsdm_middle_table))) /\ ((bcf_row_index_bcbsdm_middle_table = 0 /\ (forall bcf_index_bcbsdm_middle_table_zero_row. (exists bcf_lt_gap_bcbsdm_middle_table_zero_row_bound. bcf_lt_gap_bcbsdm_middle_table_zero_row_bound + S (bcf_index_bcbsdm_middle_table_zero_row) = S (S (n + n))) -> exists bcf_value_bcbsdm_middle_table_zero_row. ((((exists bcf_height_bcbsdm_middle_table_zero_row_entry. bcf_height_bcbsdm_middle_table_zero_row_entry + S (bcf_value_bcbsdm_middle_table_zero_row) = S ((S (bcf_index_bcbsdm_middle_table_zero_row)) * bcf_row_scale_bcbsdm_middle_table)) /\ exists bcf_quotient_bcbsdm_middle_table_zero_row_entry. bcf_row_code_bcbsdm_middle_table = bcf_quotient_bcbsdm_middle_table_zero_row_entry * S ((S (bcf_index_bcbsdm_middle_table_zero_row)) * bcf_row_scale_bcbsdm_middle_table) + (bcf_value_bcbsdm_middle_table_zero_row))) /\ ((bcf_index_bcbsdm_middle_table_zero_row = 0 /\ bcf_value_bcbsdm_middle_table_zero_row = 1) \/ exists bcf_predecessor_bcbsdm_middle_table_zero_row. bcf_index_bcbsdm_middle_table_zero_row = S bcf_predecessor_bcbsdm_middle_table_zero_row /\ bcf_value_bcbsdm_middle_table_zero_row = 0)))) \/ exists bcf_predecessor_bcbsdm_middle_table bcf_previous_code_bcbsdm_middle_table bcf_previous_scale_bcbsdm_middle_table. bcf_row_index_bcbsdm_middle_table = S bcf_predecessor_bcbsdm_middle_table /\ ((((exists bcf_height_bcbsdm_middle_table_decoded_previous_code. bcf_height_bcbsdm_middle_table_decoded_previous_code + S (bcf_previous_code_bcbsdm_middle_table) = S ((S (bcf_predecessor_bcbsdm_middle_table)) * bcf_row_code_scale_bcbsdm_middle)) /\ exists bcf_quotient_bcbsdm_middle_table_decoded_previous_code. bcf_row_code_code_bcbsdm_middle = bcf_quotient_bcbsdm_middle_table_decoded_previous_code * S ((S (bcf_predecessor_bcbsdm_middle_table)) * bcf_row_code_scale_bcbsdm_middle) + (bcf_previous_code_bcbsdm_middle_table))) /\ ((((exists bcf_height_bcbsdm_middle_table_decoded_previous_scale. bcf_height_bcbsdm_middle_table_decoded_previous_scale + S (bcf_previous_scale_bcbsdm_middle_table) = S ((S (bcf_predecessor_bcbsdm_middle_table)) * bcf_row_scale_scale_bcbsdm_middle)) /\ exists bcf_quotient_bcbsdm_middle_table_decoded_previous_scale. bcf_row_scale_code_bcbsdm_middle = bcf_quotient_bcbsdm_middle_table_decoded_previous_scale * S ((S (bcf_predecessor_bcbsdm_middle_table)) * bcf_row_scale_scale_bcbsdm_middle) + (bcf_previous_scale_bcbsdm_middle_table))) /\ (forall bcf_index_bcbsdm_middle_table_row_step. (exists bcf_lt_gap_bcbsdm_middle_table_row_step_bound. bcf_lt_gap_bcbsdm_middle_table_row_step_bound + S (bcf_index_bcbsdm_middle_table_row_step) = S (S (n + n))) -> exists bcf_value_bcbsdm_middle_table_row_step. ((((exists bcf_height_bcbsdm_middle_table_row_step_entry. bcf_height_bcbsdm_middle_table_row_step_entry + S (bcf_value_bcbsdm_middle_table_row_step) = S ((S (bcf_index_bcbsdm_middle_table_row_step)) * bcf_row_scale_bcbsdm_middle_table)) /\ exists bcf_quotient_bcbsdm_middle_table_row_step_entry. bcf_row_code_bcbsdm_middle_table = bcf_quotient_bcbsdm_middle_table_row_step_entry * S ((S (bcf_index_bcbsdm_middle_table_row_step)) * bcf_row_scale_bcbsdm_middle_table) + (bcf_value_bcbsdm_middle_table_row_step))) /\ ((bcf_index_bcbsdm_middle_table_row_step = 0 /\ bcf_value_bcbsdm_middle_table_row_step = 1) \/ exists bcf_predecessor_bcbsdm_middle_table_row_step bcf_left_bcbsdm_middle_table_row_step bcf_right_bcbsdm_middle_table_row_step. bcf_index_bcbsdm_middle_table_row_step = S bcf_predecessor_bcbsdm_middle_table_row_step /\ ((((exists bcf_height_bcbsdm_middle_table_row_step_previous_left. bcf_height_bcbsdm_middle_table_row_step_previous_left + S (bcf_left_bcbsdm_middle_table_row_step) = S ((S (bcf_predecessor_bcbsdm_middle_table_row_step)) * bcf_previous_scale_bcbsdm_middle_table)) /\ exists bcf_quotient_bcbsdm_middle_table_row_step_previous_left. bcf_previous_code_bcbsdm_middle_table = bcf_quotient_bcbsdm_middle_table_row_step_previous_left * S ((S (bcf_predecessor_bcbsdm_middle_table_row_step)) * bcf_previous_scale_bcbsdm_middle_table) + (bcf_left_bcbsdm_middle_table_row_step))) /\ ((((exists bcf_height_bcbsdm_middle_table_row_step_previous_right. bcf_height_bcbsdm_middle_table_row_step_previous_right + S (bcf_right_bcbsdm_middle_table_row_step) = S ((S (S (bcf_predecessor_bcbsdm_middle_table_row_step))) * bcf_previous_scale_bcbsdm_middle_table)) /\ exists bcf_quotient_bcbsdm_middle_table_row_step_previous_right. bcf_previous_code_bcbsdm_middle_table = bcf_quotient_bcbsdm_middle_table_row_step_previous_right * S ((S (S (bcf_predecessor_bcbsdm_middle_table_row_step))) * bcf_previous_scale_bcbsdm_middle_table) + (bcf_right_bcbsdm_middle_table_row_step))) /\ bcf_value_bcbsdm_middle_table_row_step = bcf_left_bcbsdm_middle_table_row_step + bcf_right_bcbsdm_middle_table_row_step))))))))))) /\ ((((exists bcf_height_bcbsdm_middle_decoded_row_code. bcf_height_bcbsdm_middle_decoded_row_code + S (bcf_row_code_bcbsdm_middle) = S ((S (S (n + n))) * bcf_row_code_scale_bcbsdm_middle)) /\ exists bcf_quotient_bcbsdm_middle_decoded_row_code. bcf_row_code_code_bcbsdm_middle = bcf_quotient_bcbsdm_middle_decoded_row_code * S ((S (S (n + n))) * bcf_row_code_scale_bcbsdm_middle) + (bcf_row_code_bcbsdm_middle))) /\ ((((exists bcf_height_bcbsdm_middle_decoded_row_scale. bcf_height_bcbsdm_middle_decoded_row_scale + S (bcf_row_scale_bcbsdm_middle) = S ((S (S (n + n))) * bcf_row_scale_scale_bcbsdm_middle)) /\ exists bcf_quotient_bcbsdm_middle_decoded_row_scale. bcf_row_scale_code_bcbsdm_middle = bcf_quotient_bcbsdm_middle_decoded_row_scale * S ((S (S (n + n))) * bcf_row_scale_scale_bcbsdm_middle) + (bcf_row_scale_bcbsdm_middle))) /\ (((exists bcf_height_bcbsdm_middle_decoded_value. bcf_height_bcbsdm_middle_decoded_value + S (m) = S ((S (n)) * bcf_row_scale_bcbsdm_middle)) /\ exists bcf_quotient_bcbsdm_middle_decoded_value. bcf_row_code_bcbsdm_middle = bcf_quotient_bcbsdm_middle_decoded_value * S ((S (n)) * bcf_row_scale_bcbsdm_middle) + (m)))))))))
20. 0020specialize choose_exists (S (n + n))
21. 0021specialize choose_exists n
22. 0022exact choose_exists
23. 0023cases hmiddle_exists
24. 0024have hmirror_exists : exists r. (((exists bcf_lt_gap_bcbsdm_mirror_out_of_range. bcf_lt_gap_bcbsdm_mirror_out_of_range + S (S (n + n)) = S n) /\ r = 0) \/ ((exists bcf_le_gap_bcbsdm_mirror_in_range. bcf_le_gap_bcbsdm_mirror_in_range + (S n) = S (n + n)) /\ (exists bcf_row_code_code_bcbsdm_mirror bcf_row_code_scale_bcbsdm_mirror bcf_row_scale_code_bcbsdm_mirror bcf_row_scale_scale_bcbsdm_mirror bcf_row_code_bcbsdm_mirror bcf_row_scale_bcbsdm_mirror. ((forall bcf_row_index_bcbsdm_mirror_table. (exists bcf_lt_gap_bcbsdm_mirror_table_row_bound. bcf_lt_gap_bcbsdm_mirror_table_row_bound + S (bcf_row_index_bcbsdm_mirror_table) = S (S (n + n))) -> exists bcf_row_code_bcbsdm_mirror_table bcf_row_scale_bcbsdm_mirror_table. ((((exists bcf_height_bcbsdm_mirror_table_decoded_row_code. bcf_height_bcbsdm_mirror_table_decoded_row_code + S (bcf_row_code_bcbsdm_mirror_table) = S ((S (bcf_row_index_bcbsdm_mirror_table)) * bcf_row_code_scale_bcbsdm_mirror)) /\ exists bcf_quotient_bcbsdm_mirror_table_decoded_row_code. bcf_row_code_code_bcbsdm_mirror = bcf_quotient_bcbsdm_mirror_table_decoded_row_code * S ((S (bcf_row_index_bcbsdm_mirror_table)) * bcf_row_code_scale_bcbsdm_mirror) + (bcf_row_code_bcbsdm_mirror_table))) /\ ((((exists bcf_height_bcbsdm_mirror_table_decoded_row_scale. bcf_height_bcbsdm_mirror_table_decoded_row_scale + S (bcf_row_scale_bcbsdm_mirror_table) = S ((S (bcf_row_index_bcbsdm_mirror_table)) * bcf_row_scale_scale_bcbsdm_mirror)) /\ exists bcf_quotient_bcbsdm_mirror_table_decoded_row_scale. bcf_row_scale_code_bcbsdm_mirror = bcf_quotient_bcbsdm_mirror_table_decoded_row_scale * S ((S (bcf_row_index_bcbsdm_mirror_table)) * bcf_row_scale_scale_bcbsdm_mirror) + (bcf_row_scale_bcbsdm_mirror_table))) /\ ((bcf_row_index_bcbsdm_mirror_table = 0 /\ (forall bcf_index_bcbsdm_mirror_table_zero_row. (exists bcf_lt_gap_bcbsdm_mirror_table_zero_row_bound. bcf_lt_gap_bcbsdm_mirror_table_zero_row_bound + S (bcf_index_bcbsdm_mirror_table_zero_row) = S (S (n + n))) -> exists bcf_value_bcbsdm_mirror_table_zero_row. ((((exists bcf_height_bcbsdm_mirror_table_zero_row_entry. bcf_height_bcbsdm_mirror_table_zero_row_entry + S (bcf_value_bcbsdm_mirror_table_zero_row) = S ((S (bcf_index_bcbsdm_mirror_table_zero_row)) * bcf_row_scale_bcbsdm_mirror_table)) /\ exists bcf_quotient_bcbsdm_mirror_table_zero_row_entry. bcf_row_code_bcbsdm_mirror_table = bcf_quotient_bcbsdm_mirror_table_zero_row_entry * S ((S (bcf_index_bcbsdm_mirror_table_zero_row)) * bcf_row_scale_bcbsdm_mirror_table) + (bcf_value_bcbsdm_mirror_table_zero_row))) /\ ((bcf_index_bcbsdm_mirror_table_zero_row = 0 /\ bcf_value_bcbsdm_mirror_table_zero_row = 1) \/ exists bcf_predecessor_bcbsdm_mirror_table_zero_row. bcf_index_bcbsdm_mirror_table_zero_row = S bcf_predecessor_bcbsdm_mirror_table_zero_row /\ bcf_value_bcbsdm_mirror_table_zero_row = 0)))) \/ exists bcf_predecessor_bcbsdm_mirror_table bcf_previous_code_bcbsdm_mirror_table bcf_previous_scale_bcbsdm_mirror_table. bcf_row_index_bcbsdm_mirror_table = S bcf_predecessor_bcbsdm_mirror_table /\ ((((exists bcf_height_bcbsdm_mirror_table_decoded_previous_code. bcf_height_bcbsdm_mirror_table_decoded_previous_code + S (bcf_previous_code_bcbsdm_mirror_table) = S ((S (bcf_predecessor_bcbsdm_mirror_table)) * bcf_row_code_scale_bcbsdm_mirror)) /\ exists bcf_quotient_bcbsdm_mirror_table_decoded_previous_code. bcf_row_code_code_bcbsdm_mirror = bcf_quotient_bcbsdm_mirror_table_decoded_previous_code * S ((S (bcf_predecessor_bcbsdm_mirror_table)) * bcf_row_code_scale_bcbsdm_mirror) + (bcf_previous_code_bcbsdm_mirror_table))) /\ ((((exists bcf_height_bcbsdm_mirror_table_decoded_previous_scale. bcf_height_bcbsdm_mirror_table_decoded_previous_scale + S (bcf_previous_scale_bcbsdm_mirror_table) = S ((S (bcf_predecessor_bcbsdm_mirror_table)) * bcf_row_scale_scale_bcbsdm_mirror)) /\ exists bcf_quotient_bcbsdm_mirror_table_decoded_previous_scale. bcf_row_scale_code_bcbsdm_mirror = bcf_quotient_bcbsdm_mirror_table_decoded_previous_scale * S ((S (bcf_predecessor_bcbsdm_mirror_table)) * bcf_row_scale_scale_bcbsdm_mirror) + (bcf_previous_scale_bcbsdm_mirror_table))) /\ (forall bcf_index_bcbsdm_mirror_table_row_step. (exists bcf_lt_gap_bcbsdm_mirror_table_row_step_bound. bcf_lt_gap_bcbsdm_mirror_table_row_step_bound + S (bcf_index_bcbsdm_mirror_table_row_step) = S (S (n + n))) -> exists bcf_value_bcbsdm_mirror_table_row_step. ((((exists bcf_height_bcbsdm_mirror_table_row_step_entry. bcf_height_bcbsdm_mirror_table_row_step_entry + S (bcf_value_bcbsdm_mirror_table_row_step) = S ((S (bcf_index_bcbsdm_mirror_table_row_step)) * bcf_row_scale_bcbsdm_mirror_table)) /\ exists bcf_quotient_bcbsdm_mirror_table_row_step_entry. bcf_row_code_bcbsdm_mirror_table = bcf_quotient_bcbsdm_mirror_table_row_step_entry * S ((S (bcf_index_bcbsdm_mirror_table_row_step)) * bcf_row_scale_bcbsdm_mirror_table) + (bcf_value_bcbsdm_mirror_table_row_step))) /\ ((bcf_index_bcbsdm_mirror_table_row_step = 0 /\ bcf_value_bcbsdm_mirror_table_row_step = 1) \/ exists bcf_predecessor_bcbsdm_mirror_table_row_step bcf_left_bcbsdm_mirror_table_row_step bcf_right_bcbsdm_mirror_table_row_step. bcf_index_bcbsdm_mirror_table_row_step = S bcf_predecessor_bcbsdm_mirror_table_row_step /\ ((((exists bcf_height_bcbsdm_mirror_table_row_step_previous_left. bcf_height_bcbsdm_mirror_table_row_step_previous_left + S (bcf_left_bcbsdm_mirror_table_row_step) = S ((S (bcf_predecessor_bcbsdm_mirror_table_row_step)) * bcf_previous_scale_bcbsdm_mirror_table)) /\ exists bcf_quotient_bcbsdm_mirror_table_row_step_previous_left. bcf_previous_code_bcbsdm_mirror_table = bcf_quotient_bcbsdm_mirror_table_row_step_previous_left * S ((S (bcf_predecessor_bcbsdm_mirror_table_row_step)) * bcf_previous_scale_bcbsdm_mirror_table) + (bcf_left_bcbsdm_mirror_table_row_step))) /\ ((((exists bcf_height_bcbsdm_mirror_table_row_step_previous_right. bcf_height_bcbsdm_mirror_table_row_step_previous_right + S (bcf_right_bcbsdm_mirror_table_row_step) = S ((S (S (bcf_predecessor_bcbsdm_mirror_table_row_step))) * bcf_previous_scale_bcbsdm_mirror_table)) /\ exists bcf_quotient_bcbsdm_mirror_table_row_step_previous_right. bcf_previous_code_bcbsdm_mirror_table = bcf_quotient_bcbsdm_mirror_table_row_step_previous_right * S ((S (S (bcf_predecessor_bcbsdm_mirror_table_row_step))) * bcf_previous_scale_bcbsdm_mirror_table) + (bcf_right_bcbsdm_mirror_table_row_step))) /\ bcf_value_bcbsdm_mirror_table_row_step = bcf_left_bcbsdm_mirror_table_row_step + bcf_right_bcbsdm_mirror_table_row_step))))))))))) /\ ((((exists bcf_height_bcbsdm_mirror_decoded_row_code. bcf_height_bcbsdm_mirror_decoded_row_code + S (bcf_row_code_bcbsdm_mirror) = S ((S (S (n + n))) * bcf_row_code_scale_bcbsdm_mirror)) /\ exists bcf_quotient_bcbsdm_mirror_decoded_row_code. bcf_row_code_code_bcbsdm_mirror = bcf_quotient_bcbsdm_mirror_decoded_row_code * S ((S (S (n + n))) * bcf_row_code_scale_bcbsdm_mirror) + (bcf_row_code_bcbsdm_mirror))) /\ ((((exists bcf_height_bcbsdm_mirror_decoded_row_scale. bcf_height_bcbsdm_mirror_decoded_row_scale + S (bcf_row_scale_bcbsdm_mirror) = S ((S (S (n + n))) * bcf_row_scale_scale_bcbsdm_mirror)) /\ exists bcf_quotient_bcbsdm_mirror_decoded_row_scale. bcf_row_scale_code_bcbsdm_mirror = bcf_quotient_bcbsdm_mirror_decoded_row_scale * S ((S (S (n + n))) * bcf_row_scale_scale_bcbsdm_mirror) + (bcf_row_scale_bcbsdm_mirror))) /\ (((exists bcf_height_bcbsdm_mirror_decoded_value. bcf_height_bcbsdm_mirror_decoded_value + S (r) = S ((S (S n)) * bcf_row_scale_bcbsdm_mirror)) /\ exists bcf_quotient_bcbsdm_mirror_decoded_value. bcf_row_code_bcbsdm_mirror = bcf_quotient_bcbsdm_mirror_decoded_value * S ((S (S n)) * bcf_row_scale_bcbsdm_mirror) + (r)))))))))
25. 0025specialize choose_exists (S (n + n))
26. 0026specialize choose_exists (S n)
27. 0027exact choose_exists
28. 0028cases hmirror_exists
29. 0029have hsym : x = x1
30. 0030specialize choose_symmetry (S (n + n))
31. 0031specialize choose_symmetry n
32. 0032specialize choose_symmetry (S n)
33. 0033specialize choose_symmetry x
34. 0034specialize choose_symmetry x1
35. 0035apply choose_symmetry
36. 0036apply PA4
37. 0037exact hmiddle_exists_witness
38. 0038exact hmirror_exists_witness
39. 0039have hsum : d = x + x1
40. 0040specialize choose_succ_succ (S (n + n))
41. 0041specialize choose_succ_succ n
42. 0042specialize choose_succ_succ x
43. 0043specialize choose_succ_succ x1
44. 0044specialize choose_succ_succ d
45. 0045apply choose_succ_succ
46. 0046exact hmiddle_exists_witness
47. 0047exact hmirror_exists_witness
48. 0048exact hnormalized
49. 0049exists x
50. 0050split
51. 0051exact hmiddle_exists_witness
52. 0052trans x + x1
53. 0053exact hsum
54. 0054rewrite <- hsym
55. 0055refl