BT00TN · Bertrand theorem

central_binom_exists

Alpha v34 checked-use theorem · independently kernel and Lean verified; not Stable

Every row has a relational central-binomial value.

Current library: Alpha v34, 4,223 checked-use theorems; Stable remains 432. Historical first admissions, original proof editions, and non-admitted aliases are preserved. Exact original first-admission records.

Statement with defined notation

∀ n. ∃ z. CentralBinom(n,z)

Every purple notation token opens its conservative definition. Expanding the displayed statement recovers the exact first-order Peano-arithmetic formula checked by the unchanged kernel.

Definitions used by this theorem

In the theorem statement

PD0042 CentralBinom

1 occurrences

In local proof propositions

none

0 occurrences

Exact expanded native-PA statement
forall n. exists z. (((exists bcf_lt_gap_bcbe_result_out_of_range. bcf_lt_gap_bcbe_result_out_of_range + S (n + n) = n) /\ z = 0) \/ ((exists bcf_le_gap_bcbe_result_in_range. bcf_le_gap_bcbe_result_in_range + (n) = n + n) /\ (exists bcf_row_code_code_bcbe_result bcf_row_code_scale_bcbe_result bcf_row_scale_code_bcbe_result bcf_row_scale_scale_bcbe_result bcf_row_code_bcbe_result bcf_row_scale_bcbe_result. ((forall bcf_row_index_bcbe_result_table. (exists bcf_lt_gap_bcbe_result_table_row_bound. bcf_lt_gap_bcbe_result_table_row_bound + S (bcf_row_index_bcbe_result_table) = S (n + n)) -> exists bcf_row_code_bcbe_result_table bcf_row_scale_bcbe_result_table. ((((exists bcf_height_bcbe_result_table_decoded_row_code. bcf_height_bcbe_result_table_decoded_row_code + S (bcf_row_code_bcbe_result_table) = S ((S (bcf_row_index_bcbe_result_table)) * bcf_row_code_scale_bcbe_result)) /\ exists bcf_quotient_bcbe_result_table_decoded_row_code. bcf_row_code_code_bcbe_result = bcf_quotient_bcbe_result_table_decoded_row_code * S ((S (bcf_row_index_bcbe_result_table)) * bcf_row_code_scale_bcbe_result) + (bcf_row_code_bcbe_result_table))) /\ ((((exists bcf_height_bcbe_result_table_decoded_row_scale. bcf_height_bcbe_result_table_decoded_row_scale + S (bcf_row_scale_bcbe_result_table) = S ((S (bcf_row_index_bcbe_result_table)) * bcf_row_scale_scale_bcbe_result)) /\ exists bcf_quotient_bcbe_result_table_decoded_row_scale. bcf_row_scale_code_bcbe_result = bcf_quotient_bcbe_result_table_decoded_row_scale * S ((S (bcf_row_index_bcbe_result_table)) * bcf_row_scale_scale_bcbe_result) + (bcf_row_scale_bcbe_result_table))) /\ ((bcf_row_index_bcbe_result_table = 0 /\ (forall bcf_index_bcbe_result_table_zero_row. (exists bcf_lt_gap_bcbe_result_table_zero_row_bound. bcf_lt_gap_bcbe_result_table_zero_row_bound + S (bcf_index_bcbe_result_table_zero_row) = S (n + n)) -> exists bcf_value_bcbe_result_table_zero_row. ((((exists bcf_height_bcbe_result_table_zero_row_entry. bcf_height_bcbe_result_table_zero_row_entry + S (bcf_value_bcbe_result_table_zero_row) = S ((S (bcf_index_bcbe_result_table_zero_row)) * bcf_row_scale_bcbe_result_table)) /\ exists bcf_quotient_bcbe_result_table_zero_row_entry. bcf_row_code_bcbe_result_table = bcf_quotient_bcbe_result_table_zero_row_entry * S ((S (bcf_index_bcbe_result_table_zero_row)) * bcf_row_scale_bcbe_result_table) + (bcf_value_bcbe_result_table_zero_row))) /\ ((bcf_index_bcbe_result_table_zero_row = 0 /\ bcf_value_bcbe_result_table_zero_row = 1) \/ exists bcf_predecessor_bcbe_result_table_zero_row. bcf_index_bcbe_result_table_zero_row = S bcf_predecessor_bcbe_result_table_zero_row /\ bcf_value_bcbe_result_table_zero_row = 0)))) \/ exists bcf_predecessor_bcbe_result_table bcf_previous_code_bcbe_result_table bcf_previous_scale_bcbe_result_table. bcf_row_index_bcbe_result_table = S bcf_predecessor_bcbe_result_table /\ ((((exists bcf_height_bcbe_result_table_decoded_previous_code. bcf_height_bcbe_result_table_decoded_previous_code + S (bcf_previous_code_bcbe_result_table) = S ((S (bcf_predecessor_bcbe_result_table)) * bcf_row_code_scale_bcbe_result)) /\ exists bcf_quotient_bcbe_result_table_decoded_previous_code. bcf_row_code_code_bcbe_result = bcf_quotient_bcbe_result_table_decoded_previous_code * S ((S (bcf_predecessor_bcbe_result_table)) * bcf_row_code_scale_bcbe_result) + (bcf_previous_code_bcbe_result_table))) /\ ((((exists bcf_height_bcbe_result_table_decoded_previous_scale. bcf_height_bcbe_result_table_decoded_previous_scale + S (bcf_previous_scale_bcbe_result_table) = S ((S (bcf_predecessor_bcbe_result_table)) * bcf_row_scale_scale_bcbe_result)) /\ exists bcf_quotient_bcbe_result_table_decoded_previous_scale. bcf_row_scale_code_bcbe_result = bcf_quotient_bcbe_result_table_decoded_previous_scale * S ((S (bcf_predecessor_bcbe_result_table)) * bcf_row_scale_scale_bcbe_result) + (bcf_previous_scale_bcbe_result_table))) /\ (forall bcf_index_bcbe_result_table_row_step. (exists bcf_lt_gap_bcbe_result_table_row_step_bound. bcf_lt_gap_bcbe_result_table_row_step_bound + S (bcf_index_bcbe_result_table_row_step) = S (n + n)) -> exists bcf_value_bcbe_result_table_row_step. ((((exists bcf_height_bcbe_result_table_row_step_entry. bcf_height_bcbe_result_table_row_step_entry + S (bcf_value_bcbe_result_table_row_step) = S ((S (bcf_index_bcbe_result_table_row_step)) * bcf_row_scale_bcbe_result_table)) /\ exists bcf_quotient_bcbe_result_table_row_step_entry. bcf_row_code_bcbe_result_table = bcf_quotient_bcbe_result_table_row_step_entry * S ((S (bcf_index_bcbe_result_table_row_step)) * bcf_row_scale_bcbe_result_table) + (bcf_value_bcbe_result_table_row_step))) /\ ((bcf_index_bcbe_result_table_row_step = 0 /\ bcf_value_bcbe_result_table_row_step = 1) \/ exists bcf_predecessor_bcbe_result_table_row_step bcf_left_bcbe_result_table_row_step bcf_right_bcbe_result_table_row_step. bcf_index_bcbe_result_table_row_step = S bcf_predecessor_bcbe_result_table_row_step /\ ((((exists bcf_height_bcbe_result_table_row_step_previous_left. bcf_height_bcbe_result_table_row_step_previous_left + S (bcf_left_bcbe_result_table_row_step) = S ((S (bcf_predecessor_bcbe_result_table_row_step)) * bcf_previous_scale_bcbe_result_table)) /\ exists bcf_quotient_bcbe_result_table_row_step_previous_left. bcf_previous_code_bcbe_result_table = bcf_quotient_bcbe_result_table_row_step_previous_left * S ((S (bcf_predecessor_bcbe_result_table_row_step)) * bcf_previous_scale_bcbe_result_table) + (bcf_left_bcbe_result_table_row_step))) /\ ((((exists bcf_height_bcbe_result_table_row_step_previous_right. bcf_height_bcbe_result_table_row_step_previous_right + S (bcf_right_bcbe_result_table_row_step) = S ((S (S (bcf_predecessor_bcbe_result_table_row_step))) * bcf_previous_scale_bcbe_result_table)) /\ exists bcf_quotient_bcbe_result_table_row_step_previous_right. bcf_previous_code_bcbe_result_table = bcf_quotient_bcbe_result_table_row_step_previous_right * S ((S (S (bcf_predecessor_bcbe_result_table_row_step))) * bcf_previous_scale_bcbe_result_table) + (bcf_right_bcbe_result_table_row_step))) /\ bcf_value_bcbe_result_table_row_step = bcf_left_bcbe_result_table_row_step + bcf_right_bcbe_result_table_row_step))))))))))) /\ ((((exists bcf_height_bcbe_result_decoded_row_code. bcf_height_bcbe_result_decoded_row_code + S (bcf_row_code_bcbe_result) = S ((S (n + n)) * bcf_row_code_scale_bcbe_result)) /\ exists bcf_quotient_bcbe_result_decoded_row_code. bcf_row_code_code_bcbe_result = bcf_quotient_bcbe_result_decoded_row_code * S ((S (n + n)) * bcf_row_code_scale_bcbe_result) + (bcf_row_code_bcbe_result))) /\ ((((exists bcf_height_bcbe_result_decoded_row_scale. bcf_height_bcbe_result_decoded_row_scale + S (bcf_row_scale_bcbe_result) = S ((S (n + n)) * bcf_row_scale_scale_bcbe_result)) /\ exists bcf_quotient_bcbe_result_decoded_row_scale. bcf_row_scale_code_bcbe_result = bcf_quotient_bcbe_result_decoded_row_scale * S ((S (n + n)) * bcf_row_scale_scale_bcbe_result) + (bcf_row_scale_bcbe_result))) /\ (((exists bcf_height_bcbe_result_decoded_value. bcf_height_bcbe_result_decoded_value + S (z) = S ((S (n)) * bcf_row_scale_bcbe_result)) /\ exists bcf_quotient_bcbe_result_decoded_value. bcf_row_code_bcbe_result = bcf_quotient_bcbe_result_decoded_value * S ((S (n)) * bcf_row_scale_bcbe_result) + (z)))))))))

Proof neighborhood

Direct theorem prerequisites

BT00T8 choose_exists

Direct theorem dependents

BT00U3 four_pow_central_seed_package BT00VN central_binom_upper_support_package BT00VU primorial_four_power_support_package BT0115 bertrand_eventually_closed_upper

Definition-aware tactic body

Only local propositions introduced by have or suffices are compacted. Every changed line has an exact-AST conservative-expansion receipt; the kernel still receives the immutable original tactic script.

Read the argument

Proof checkpoints

4 script commands · 2 reading checkpoints · 0 local claims

This is a reading aid, not a new proof or a proof-tree certificate. Checkpoint groups are consecutive commands, not inferred branch boundaries. Every step links to the preserved script.

Definition notation is shown below. Open the paired exact edition for the original native formulas. Source pairing is not a new equivalence certificate.

Named ingredients (1)
choose exists
01Fix variables and assumptionsL1–1

Work with arbitrary variables or the premises of the current implication.

  1. L1
    intro n
02Use earlier factsL2–4

Instantiate or apply named facts and discharge the corresponding proof obligations.

  1. L2
    specialize choose_exists (n + n)
  2. L3
    specialize choose_exists n
  3. L4
    exact choose_exists

Library-wide reading audit

Original defined command ledger · 4 lines
  1. 0001intro n
  2. 0002specialize choose_exists (n + n)
  3. 0003specialize choose_exists n
  4. 0004exact choose_exists