BT00TP · Bertrand theorem

central_binom_positive

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

Every relational central-binomial value is a successor.

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) → ∃ x. z = S x

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 z. (((exists bcf_lt_gap_bcbp_source_out_of_range. bcf_lt_gap_bcbp_source_out_of_range + S (n + n) = n) /\ z = 0) \/ ((exists bcf_le_gap_bcbp_source_in_range. bcf_le_gap_bcbp_source_in_range + (n) = n + n) /\ (exists bcf_row_code_code_bcbp_source bcf_row_code_scale_bcbp_source bcf_row_scale_code_bcbp_source bcf_row_scale_scale_bcbp_source bcf_row_code_bcbp_source bcf_row_scale_bcbp_source. ((forall bcf_row_index_bcbp_source_table. (exists bcf_lt_gap_bcbp_source_table_row_bound. bcf_lt_gap_bcbp_source_table_row_bound + S (bcf_row_index_bcbp_source_table) = S (n + n)) -> exists bcf_row_code_bcbp_source_table bcf_row_scale_bcbp_source_table. ((((exists bcf_height_bcbp_source_table_decoded_row_code. bcf_height_bcbp_source_table_decoded_row_code + S (bcf_row_code_bcbp_source_table) = S ((S (bcf_row_index_bcbp_source_table)) * bcf_row_code_scale_bcbp_source)) /\ exists bcf_quotient_bcbp_source_table_decoded_row_code. bcf_row_code_code_bcbp_source = bcf_quotient_bcbp_source_table_decoded_row_code * S ((S (bcf_row_index_bcbp_source_table)) * bcf_row_code_scale_bcbp_source) + (bcf_row_code_bcbp_source_table))) /\ ((((exists bcf_height_bcbp_source_table_decoded_row_scale. bcf_height_bcbp_source_table_decoded_row_scale + S (bcf_row_scale_bcbp_source_table) = S ((S (bcf_row_index_bcbp_source_table)) * bcf_row_scale_scale_bcbp_source)) /\ exists bcf_quotient_bcbp_source_table_decoded_row_scale. bcf_row_scale_code_bcbp_source = bcf_quotient_bcbp_source_table_decoded_row_scale * S ((S (bcf_row_index_bcbp_source_table)) * bcf_row_scale_scale_bcbp_source) + (bcf_row_scale_bcbp_source_table))) /\ ((bcf_row_index_bcbp_source_table = 0 /\ (forall bcf_index_bcbp_source_table_zero_row. (exists bcf_lt_gap_bcbp_source_table_zero_row_bound. bcf_lt_gap_bcbp_source_table_zero_row_bound + S (bcf_index_bcbp_source_table_zero_row) = S (n + n)) -> exists bcf_value_bcbp_source_table_zero_row. ((((exists bcf_height_bcbp_source_table_zero_row_entry. bcf_height_bcbp_source_table_zero_row_entry + S (bcf_value_bcbp_source_table_zero_row) = S ((S (bcf_index_bcbp_source_table_zero_row)) * bcf_row_scale_bcbp_source_table)) /\ exists bcf_quotient_bcbp_source_table_zero_row_entry. bcf_row_code_bcbp_source_table = bcf_quotient_bcbp_source_table_zero_row_entry * S ((S (bcf_index_bcbp_source_table_zero_row)) * bcf_row_scale_bcbp_source_table) + (bcf_value_bcbp_source_table_zero_row))) /\ ((bcf_index_bcbp_source_table_zero_row = 0 /\ bcf_value_bcbp_source_table_zero_row = 1) \/ exists bcf_predecessor_bcbp_source_table_zero_row. bcf_index_bcbp_source_table_zero_row = S bcf_predecessor_bcbp_source_table_zero_row /\ bcf_value_bcbp_source_table_zero_row = 0)))) \/ exists bcf_predecessor_bcbp_source_table bcf_previous_code_bcbp_source_table bcf_previous_scale_bcbp_source_table. bcf_row_index_bcbp_source_table = S bcf_predecessor_bcbp_source_table /\ ((((exists bcf_height_bcbp_source_table_decoded_previous_code. bcf_height_bcbp_source_table_decoded_previous_code + S (bcf_previous_code_bcbp_source_table) = S ((S (bcf_predecessor_bcbp_source_table)) * bcf_row_code_scale_bcbp_source)) /\ exists bcf_quotient_bcbp_source_table_decoded_previous_code. bcf_row_code_code_bcbp_source = bcf_quotient_bcbp_source_table_decoded_previous_code * S ((S (bcf_predecessor_bcbp_source_table)) * bcf_row_code_scale_bcbp_source) + (bcf_previous_code_bcbp_source_table))) /\ ((((exists bcf_height_bcbp_source_table_decoded_previous_scale. bcf_height_bcbp_source_table_decoded_previous_scale + S (bcf_previous_scale_bcbp_source_table) = S ((S (bcf_predecessor_bcbp_source_table)) * bcf_row_scale_scale_bcbp_source)) /\ exists bcf_quotient_bcbp_source_table_decoded_previous_scale. bcf_row_scale_code_bcbp_source = bcf_quotient_bcbp_source_table_decoded_previous_scale * S ((S (bcf_predecessor_bcbp_source_table)) * bcf_row_scale_scale_bcbp_source) + (bcf_previous_scale_bcbp_source_table))) /\ (forall bcf_index_bcbp_source_table_row_step. (exists bcf_lt_gap_bcbp_source_table_row_step_bound. bcf_lt_gap_bcbp_source_table_row_step_bound + S (bcf_index_bcbp_source_table_row_step) = S (n + n)) -> exists bcf_value_bcbp_source_table_row_step. ((((exists bcf_height_bcbp_source_table_row_step_entry. bcf_height_bcbp_source_table_row_step_entry + S (bcf_value_bcbp_source_table_row_step) = S ((S (bcf_index_bcbp_source_table_row_step)) * bcf_row_scale_bcbp_source_table)) /\ exists bcf_quotient_bcbp_source_table_row_step_entry. bcf_row_code_bcbp_source_table = bcf_quotient_bcbp_source_table_row_step_entry * S ((S (bcf_index_bcbp_source_table_row_step)) * bcf_row_scale_bcbp_source_table) + (bcf_value_bcbp_source_table_row_step))) /\ ((bcf_index_bcbp_source_table_row_step = 0 /\ bcf_value_bcbp_source_table_row_step = 1) \/ exists bcf_predecessor_bcbp_source_table_row_step bcf_left_bcbp_source_table_row_step bcf_right_bcbp_source_table_row_step. bcf_index_bcbp_source_table_row_step = S bcf_predecessor_bcbp_source_table_row_step /\ ((((exists bcf_height_bcbp_source_table_row_step_previous_left. bcf_height_bcbp_source_table_row_step_previous_left + S (bcf_left_bcbp_source_table_row_step) = S ((S (bcf_predecessor_bcbp_source_table_row_step)) * bcf_previous_scale_bcbp_source_table)) /\ exists bcf_quotient_bcbp_source_table_row_step_previous_left. bcf_previous_code_bcbp_source_table = bcf_quotient_bcbp_source_table_row_step_previous_left * S ((S (bcf_predecessor_bcbp_source_table_row_step)) * bcf_previous_scale_bcbp_source_table) + (bcf_left_bcbp_source_table_row_step))) /\ ((((exists bcf_height_bcbp_source_table_row_step_previous_right. bcf_height_bcbp_source_table_row_step_previous_right + S (bcf_right_bcbp_source_table_row_step) = S ((S (S (bcf_predecessor_bcbp_source_table_row_step))) * bcf_previous_scale_bcbp_source_table)) /\ exists bcf_quotient_bcbp_source_table_row_step_previous_right. bcf_previous_code_bcbp_source_table = bcf_quotient_bcbp_source_table_row_step_previous_right * S ((S (S (bcf_predecessor_bcbp_source_table_row_step))) * bcf_previous_scale_bcbp_source_table) + (bcf_right_bcbp_source_table_row_step))) /\ bcf_value_bcbp_source_table_row_step = bcf_left_bcbp_source_table_row_step + bcf_right_bcbp_source_table_row_step))))))))))) /\ ((((exists bcf_height_bcbp_source_decoded_row_code. bcf_height_bcbp_source_decoded_row_code + S (bcf_row_code_bcbp_source) = S ((S (n + n)) * bcf_row_code_scale_bcbp_source)) /\ exists bcf_quotient_bcbp_source_decoded_row_code. bcf_row_code_code_bcbp_source = bcf_quotient_bcbp_source_decoded_row_code * S ((S (n + n)) * bcf_row_code_scale_bcbp_source) + (bcf_row_code_bcbp_source))) /\ ((((exists bcf_height_bcbp_source_decoded_row_scale. bcf_height_bcbp_source_decoded_row_scale + S (bcf_row_scale_bcbp_source) = S ((S (n + n)) * bcf_row_scale_scale_bcbp_source)) /\ exists bcf_quotient_bcbp_source_decoded_row_scale. bcf_row_scale_code_bcbp_source = bcf_quotient_bcbp_source_decoded_row_scale * S ((S (n + n)) * bcf_row_scale_scale_bcbp_source) + (bcf_row_scale_bcbp_source))) /\ (((exists bcf_height_bcbp_source_decoded_value. bcf_height_bcbp_source_decoded_value + S (z) = S ((S (n)) * bcf_row_scale_bcbp_source)) /\ exists bcf_quotient_bcbp_source_decoded_value. bcf_row_code_bcbp_source = bcf_quotient_bcbp_source_decoded_value * S ((S (n)) * bcf_row_scale_bcbp_source) + (z))))))))) -> exists p. z = S p

Proof neighborhood

Direct theorem prerequisites

BT00TM choose_positive

Direct theorem dependents

BT00VI primorial_even_interval_le_central BT00XM central_binom_factorial_valuation_balance BT0107 central_binom_prime_contribution_product_exists

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

10 script commands · 5 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 positive
01Fix variables and assumptionsL1–3

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

  1. L1
    intro n
  2. L2
    intro z
  3. L3
    intro hcentral
02Use earlier factsL4–7

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

  1. L4
    specialize choose_positive (n + n)
  2. L5
    specialize choose_positive n
  3. L6
    specialize choose_positive z
  4. L7
    apply choose_positive
03Construct an explicit witnessL8–8

Supply the displayed value, then prove that it has the required property.

  1. L8
    exists n
04Calculate and transport equalitiesL9–9

Carry out the recorded arithmetic or equality steps; inspect the exact commands for their direction and premises.

  1. L9
    refl
05Use earlier factsL10–10

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

  1. L10
    exact hcentral

Library-wide reading audit

Original defined command ledger · 10 lines
  1. 0001intro n
  2. 0002intro z
  3. 0003intro hcentral
  4. 0004specialize choose_positive (n + n)
  5. 0005specialize choose_positive n
  6. 0006specialize choose_positive z
  7. 0007apply choose_positive
  8. 0008exists n
  9. 0009refl
  10. 0010exact hcentral