BL0004

binary_length_digit_split_exists_unique

Alpha v34 independently verified · alpha_closed; checked-use authorized; not Stable

Every natural has exactly one fully witnessed binary quotient/digit pair.

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.

Exact expanded first-order arithmetic statement

forall n. exists h b. (((((b = 0) \/ (b = 1)) /\ n = (h + h) + b)) /\ forall k c. ((((c = 0) \/ (c = 1)) /\ n = (k + k) + c)) -> (h = k /\ b = c))

Constructive proof overview

Generated structural guide

Every natural has exactly one fully witnessed binary quotient/digit pair.

The unchanged tactic script uses 2 declared prerequisites and contains 19 exact native proof lines.

Alpha v34 checked-use · first admitted v22 · independently kernel and Lean verified; not Stable

Proof neighborhood

Direct dependencies

BL0002 binary_length_digit_split_exists BL0003 binary_length_digit_split_functional

Direct dependents

none

Formal native tactic body

Dependencies are introduced as named hypotheses before line 1. Local theorem links identify exact declared prerequisites. This exact body belongs to a complete independently kernel-checked constructive proof bundle and has Alpha checked-use authority; it does not imply Stable membership.

Read the argument

Proof checkpoints

19 script commands · 8 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.

Named ingredients (2)
binary length digit split existsbinary length digit split functional
01Fix variables and assumptionsL1–1

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

  1. L1
    intro n
02Use earlier factsL2–2

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

  1. L2
    specialize binary_length_digit_split_exists n
03Separate the logical casesL3–4

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

  1. L3
    cases binary_length_digit_split_exists
  2. L4
    cases binary_length_digit_split_exists_witness
04Construct an explicit witnessL5–6

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

  1. L5
    exists x
  2. L6
    exists x1
05Separate the logical casesL7–7

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

  1. L7
    split
06Use earlier factsL8–8

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

  1. L8
    exact binary_length_digit_split_exists_witness_witness
07Fix variables and assumptionsL9–11

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

  1. L9
    intro k
  2. L10
    intro c
  3. L11
    intro hother
08Use earlier factsL12–19

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

  1. L12
    specialize binary_length_digit_split_functional n
  2. L13
    specialize binary_length_digit_split_functional x
  3. L14
    specialize binary_length_digit_split_functional x1
  4. L15
    specialize binary_length_digit_split_functional k
  5. L16
    specialize binary_length_digit_split_functional c
  6. L17
    apply binary_length_digit_split_functional
  7. L18
    exact binary_length_digit_split_exists_witness_witness
  8. L19
    exact hother

Library-wide reading audit

Original exact command ledger · 19 lines
  1. 0001intro n
  2. 0002specialize binary_length_digit_split_exists n
  3. 0003cases binary_length_digit_split_exists
  4. 0004cases binary_length_digit_split_exists_witness
  5. 0005exists x
  6. 0006exists x1
  7. 0007split
  8. 0008exact binary_length_digit_split_exists_witness_witness
  9. 0009intro k
  10. 0010intro c
  11. 0011intro hother
  12. 0012specialize binary_length_digit_split_functional n
  13. 0013specialize binary_length_digit_split_functional x
  14. 0014specialize binary_length_digit_split_functional x1
  15. 0015specialize binary_length_digit_split_functional k
  16. 0016specialize binary_length_digit_split_functional c
  17. 0017apply binary_length_digit_split_functional
  18. 0018exact binary_length_digit_split_exists_witness_witness
  19. 0019exact hother