BT00TK · Bertrand theorem

choose_self_of_eq

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

A column equal to its row has Choose value one.

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. ∀ k. ∀ z. k = n → Choose(n,k,z) → z = 1

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

PD0041 Choose

1 occurrences

In local proof propositions

none

0 occurrences

Exact expanded native-PA statement
forall n k z. k = n -> (((exists bcf_lt_gap_bcse_source_out_of_range. bcf_lt_gap_bcse_source_out_of_range + S (n) = k) /\ z = 0) \/ ((exists bcf_le_gap_bcse_source_in_range. bcf_le_gap_bcse_source_in_range + (k) = n) /\ (exists bcf_row_code_code_bcse_source bcf_row_code_scale_bcse_source bcf_row_scale_code_bcse_source bcf_row_scale_scale_bcse_source bcf_row_code_bcse_source bcf_row_scale_bcse_source. ((forall bcf_row_index_bcse_source_table. (exists bcf_lt_gap_bcse_source_table_row_bound. bcf_lt_gap_bcse_source_table_row_bound + S (bcf_row_index_bcse_source_table) = S (n)) -> exists bcf_row_code_bcse_source_table bcf_row_scale_bcse_source_table. ((((exists bcf_height_bcse_source_table_decoded_row_code. bcf_height_bcse_source_table_decoded_row_code + S (bcf_row_code_bcse_source_table) = S ((S (bcf_row_index_bcse_source_table)) * bcf_row_code_scale_bcse_source)) /\ exists bcf_quotient_bcse_source_table_decoded_row_code. bcf_row_code_code_bcse_source = bcf_quotient_bcse_source_table_decoded_row_code * S ((S (bcf_row_index_bcse_source_table)) * bcf_row_code_scale_bcse_source) + (bcf_row_code_bcse_source_table))) /\ ((((exists bcf_height_bcse_source_table_decoded_row_scale. bcf_height_bcse_source_table_decoded_row_scale + S (bcf_row_scale_bcse_source_table) = S ((S (bcf_row_index_bcse_source_table)) * bcf_row_scale_scale_bcse_source)) /\ exists bcf_quotient_bcse_source_table_decoded_row_scale. bcf_row_scale_code_bcse_source = bcf_quotient_bcse_source_table_decoded_row_scale * S ((S (bcf_row_index_bcse_source_table)) * bcf_row_scale_scale_bcse_source) + (bcf_row_scale_bcse_source_table))) /\ ((bcf_row_index_bcse_source_table = 0 /\ (forall bcf_index_bcse_source_table_zero_row. (exists bcf_lt_gap_bcse_source_table_zero_row_bound. bcf_lt_gap_bcse_source_table_zero_row_bound + S (bcf_index_bcse_source_table_zero_row) = S (n)) -> exists bcf_value_bcse_source_table_zero_row. ((((exists bcf_height_bcse_source_table_zero_row_entry. bcf_height_bcse_source_table_zero_row_entry + S (bcf_value_bcse_source_table_zero_row) = S ((S (bcf_index_bcse_source_table_zero_row)) * bcf_row_scale_bcse_source_table)) /\ exists bcf_quotient_bcse_source_table_zero_row_entry. bcf_row_code_bcse_source_table = bcf_quotient_bcse_source_table_zero_row_entry * S ((S (bcf_index_bcse_source_table_zero_row)) * bcf_row_scale_bcse_source_table) + (bcf_value_bcse_source_table_zero_row))) /\ ((bcf_index_bcse_source_table_zero_row = 0 /\ bcf_value_bcse_source_table_zero_row = 1) \/ exists bcf_predecessor_bcse_source_table_zero_row. bcf_index_bcse_source_table_zero_row = S bcf_predecessor_bcse_source_table_zero_row /\ bcf_value_bcse_source_table_zero_row = 0)))) \/ exists bcf_predecessor_bcse_source_table bcf_previous_code_bcse_source_table bcf_previous_scale_bcse_source_table. bcf_row_index_bcse_source_table = S bcf_predecessor_bcse_source_table /\ ((((exists bcf_height_bcse_source_table_decoded_previous_code. bcf_height_bcse_source_table_decoded_previous_code + S (bcf_previous_code_bcse_source_table) = S ((S (bcf_predecessor_bcse_source_table)) * bcf_row_code_scale_bcse_source)) /\ exists bcf_quotient_bcse_source_table_decoded_previous_code. bcf_row_code_code_bcse_source = bcf_quotient_bcse_source_table_decoded_previous_code * S ((S (bcf_predecessor_bcse_source_table)) * bcf_row_code_scale_bcse_source) + (bcf_previous_code_bcse_source_table))) /\ ((((exists bcf_height_bcse_source_table_decoded_previous_scale. bcf_height_bcse_source_table_decoded_previous_scale + S (bcf_previous_scale_bcse_source_table) = S ((S (bcf_predecessor_bcse_source_table)) * bcf_row_scale_scale_bcse_source)) /\ exists bcf_quotient_bcse_source_table_decoded_previous_scale. bcf_row_scale_code_bcse_source = bcf_quotient_bcse_source_table_decoded_previous_scale * S ((S (bcf_predecessor_bcse_source_table)) * bcf_row_scale_scale_bcse_source) + (bcf_previous_scale_bcse_source_table))) /\ (forall bcf_index_bcse_source_table_row_step. (exists bcf_lt_gap_bcse_source_table_row_step_bound. bcf_lt_gap_bcse_source_table_row_step_bound + S (bcf_index_bcse_source_table_row_step) = S (n)) -> exists bcf_value_bcse_source_table_row_step. ((((exists bcf_height_bcse_source_table_row_step_entry. bcf_height_bcse_source_table_row_step_entry + S (bcf_value_bcse_source_table_row_step) = S ((S (bcf_index_bcse_source_table_row_step)) * bcf_row_scale_bcse_source_table)) /\ exists bcf_quotient_bcse_source_table_row_step_entry. bcf_row_code_bcse_source_table = bcf_quotient_bcse_source_table_row_step_entry * S ((S (bcf_index_bcse_source_table_row_step)) * bcf_row_scale_bcse_source_table) + (bcf_value_bcse_source_table_row_step))) /\ ((bcf_index_bcse_source_table_row_step = 0 /\ bcf_value_bcse_source_table_row_step = 1) \/ exists bcf_predecessor_bcse_source_table_row_step bcf_left_bcse_source_table_row_step bcf_right_bcse_source_table_row_step. bcf_index_bcse_source_table_row_step = S bcf_predecessor_bcse_source_table_row_step /\ ((((exists bcf_height_bcse_source_table_row_step_previous_left. bcf_height_bcse_source_table_row_step_previous_left + S (bcf_left_bcse_source_table_row_step) = S ((S (bcf_predecessor_bcse_source_table_row_step)) * bcf_previous_scale_bcse_source_table)) /\ exists bcf_quotient_bcse_source_table_row_step_previous_left. bcf_previous_code_bcse_source_table = bcf_quotient_bcse_source_table_row_step_previous_left * S ((S (bcf_predecessor_bcse_source_table_row_step)) * bcf_previous_scale_bcse_source_table) + (bcf_left_bcse_source_table_row_step))) /\ ((((exists bcf_height_bcse_source_table_row_step_previous_right. bcf_height_bcse_source_table_row_step_previous_right + S (bcf_right_bcse_source_table_row_step) = S ((S (S (bcf_predecessor_bcse_source_table_row_step))) * bcf_previous_scale_bcse_source_table)) /\ exists bcf_quotient_bcse_source_table_row_step_previous_right. bcf_previous_code_bcse_source_table = bcf_quotient_bcse_source_table_row_step_previous_right * S ((S (S (bcf_predecessor_bcse_source_table_row_step))) * bcf_previous_scale_bcse_source_table) + (bcf_right_bcse_source_table_row_step))) /\ bcf_value_bcse_source_table_row_step = bcf_left_bcse_source_table_row_step + bcf_right_bcse_source_table_row_step))))))))))) /\ ((((exists bcf_height_bcse_source_decoded_row_code. bcf_height_bcse_source_decoded_row_code + S (bcf_row_code_bcse_source) = S ((S (n)) * bcf_row_code_scale_bcse_source)) /\ exists bcf_quotient_bcse_source_decoded_row_code. bcf_row_code_code_bcse_source = bcf_quotient_bcse_source_decoded_row_code * S ((S (n)) * bcf_row_code_scale_bcse_source) + (bcf_row_code_bcse_source))) /\ ((((exists bcf_height_bcse_source_decoded_row_scale. bcf_height_bcse_source_decoded_row_scale + S (bcf_row_scale_bcse_source) = S ((S (n)) * bcf_row_scale_scale_bcse_source)) /\ exists bcf_quotient_bcse_source_decoded_row_scale. bcf_row_scale_code_bcse_source = bcf_quotient_bcse_source_decoded_row_scale * S ((S (n)) * bcf_row_scale_scale_bcse_source) + (bcf_row_scale_bcse_source))) /\ (((exists bcf_height_bcse_source_decoded_value. bcf_height_bcse_source_decoded_value + S (z) = S ((S (k)) * bcf_row_scale_bcse_source)) /\ exists bcf_quotient_bcse_source_decoded_value. bcf_row_code_bcse_source = bcf_quotient_bcse_source_decoded_value * S ((S (k)) * bcf_row_scale_bcse_source) + (z))))))))) -> z = 1

Proof neighborhood

Direct theorem prerequisites

BT00TG choose_self

Direct theorem dependents

BT00TL choose_symmetry BT00TT choose_weighted_vertical BT00TX choose_factorial_bridge

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

44 script commands · 18 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 self
01Fix variables and assumptionsL1–5

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

  1. L1
    intro n
  2. L2
    intro k
  3. L3
    intro z
  4. L4
    intro heq
  5. L5
    intro hchoose
02Use earlier factsL6–8

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

  1. L6
    specialize choose_self n
  2. L7
    specialize choose_self z
  3. L8
    apply choose_self
03Separate the logical casesL9–12

Follow the explicit conjunction, disjunction, witness, or contradiction step recorded below.

  1. L9
    cases hchoose
  2. L10
    cases hchoose_left
  3. L11
    left
  4. L12
    split
04Calculate and transport equalitiesL13–13

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

  1. L13
    rewrite heq at hchoose_left_left
05Use earlier factsL14–15

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

  1. L14
    exact hchoose_left_left
  2. L15
    exact hchoose_left_right
06Separate the logical casesL16–18

Follow the explicit conjunction, disjunction, witness, or contradiction step recorded below.

  1. L16
    cases hchoose_right
  2. L17
    right
  3. L18
    split
07Calculate and transport equalitiesL19–19

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

  1. L19
    rewrite heq at hchoose_right_left
08Use earlier factsL20–20

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

  1. L20
    exact hchoose_right_left
09Separate the logical casesL21–29

Follow the explicit conjunction, disjunction, witness, or contradiction step recorded below.

  1. L21
    cases hchoose_right_right
  2. L22
    cases hchoose_right_right_witness
  3. L23
    cases hchoose_right_right_witness_witness
  4. L24
    cases hchoose_right_right_witness_witness_witness
  5. L25
    cases hchoose_right_right_witness_witness_witness_witness
  6. L26
    cases hchoose_right_right_witness_witness_witness_witness_witness
  7. L27
    cases hchoose_right_right_witness_witness_witness_witness_witness_witness
  8. L28
    cases hchoose_right_right_witness_witness_witness_witness_witness_witness_right
  9. L29
    cases hchoose_right_right_witness_witness_witness_witness_witness_witness_right_right
10Construct an explicit witnessL30–35

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

  1. L30
    exists x
  2. L31
    exists x1
  3. L32
    exists x2
  4. L33
    exists x3
  5. L34
    exists x4
  6. L35
    exists x5
11Separate the logical casesL36–36

Follow the explicit conjunction, disjunction, witness, or contradiction step recorded below.

  1. L36
    split
12Use earlier factsL37–37

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

  1. L37
    exact hchoose_right_right_witness_witness_witness_witness_witness_witness_left
13Separate the logical casesL38–38

Follow the explicit conjunction, disjunction, witness, or contradiction step recorded below.

  1. L38
    split
14Use earlier factsL39–39

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

  1. L39
    exact hchoose_right_right_witness_witness_witness_witness_witness_witness_right_left
15Separate the logical casesL40–40

Follow the explicit conjunction, disjunction, witness, or contradiction step recorded below.

  1. L40
    split
16Use earlier factsL41–41

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

  1. L41
    exact hchoose_right_right_witness_witness_witness_witness_witness_witness_right_right_left
17Calculate and transport equalitiesL42–43

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

  1. L42
    rewrite heq at hchoose_right_right_witness_witness_witness_witness_witness_witness_right_right_right
  2. L43
    rewrite heq at hchoose_right_right_witness_witness_witness_witness_witness_witness_right_right_right
18Use earlier factsL44–44

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

  1. L44
    exact hchoose_right_right_witness_witness_witness_witness_witness_witness_right_right_right

Library-wide reading audit

Original defined command ledger · 44 lines
  1. 0001intro n
  2. 0002intro k
  3. 0003intro z
  4. 0004intro heq
  5. 0005intro hchoose
  6. 0006specialize choose_self n
  7. 0007specialize choose_self z
  8. 0008apply choose_self
  9. 0009cases hchoose
  10. 0010cases hchoose_left
  11. 0011left
  12. 0012split
  13. 0013rewrite heq at hchoose_left_left
  14. 0014exact hchoose_left_left
  15. 0015exact hchoose_left_right
  16. 0016cases hchoose_right
  17. 0017right
  18. 0018split
  19. 0019rewrite heq at hchoose_right_left
  20. 0020exact hchoose_right_left
  21. 0021cases hchoose_right_right
  22. 0022cases hchoose_right_right_witness
  23. 0023cases hchoose_right_right_witness_witness
  24. 0024cases hchoose_right_right_witness_witness_witness
  25. 0025cases hchoose_right_right_witness_witness_witness_witness
  26. 0026cases hchoose_right_right_witness_witness_witness_witness_witness
  27. 0027cases hchoose_right_right_witness_witness_witness_witness_witness_witness
  28. 0028cases hchoose_right_right_witness_witness_witness_witness_witness_witness_right
  29. 0029cases hchoose_right_right_witness_witness_witness_witness_witness_witness_right_right
  30. 0030exists x
  31. 0031exists x1
  32. 0032exists x2
  33. 0033exists x3
  34. 0034exists x4
  35. 0035exists x5
  36. 0036split
  37. 0037exact hchoose_right_right_witness_witness_witness_witness_witness_witness_left
  38. 0038split
  39. 0039exact hchoose_right_right_witness_witness_witness_witness_witness_witness_right_left
  40. 0040split
  41. 0041exact hchoose_right_right_witness_witness_witness_witness_witness_witness_right_right_left
  42. 0042rewrite heq at hchoose_right_right_witness_witness_witness_witness_witness_witness_right_right_right
  43. 0043rewrite heq at hchoose_right_right_witness_witness_witness_witness_witness_witness_right_right_right
  44. 0044exact hchoose_right_right_witness_witness_witness_witness_witness_witness_right_right_right