BL0004

binary_length_digit_split_exists_unique

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

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

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.

G101 and G102 were OPEN when these BitLen foundations were first admitted in Alpha v22. Both are now CLOSED in Alpha v23: complete canonical exponent digits and both exact logarithmic execution bounds are proved.

Exact theorem in conservative defined notation

∀ n. ∃ h. ∃ b. BinaryExponentSplit(n,h,b) ∧ (∀ x. ∀ y. BinaryExponentSplit(n,x,y) → h = x ∧ b = y)

Every linked abbreviation expands hygienically to the identical original native formula.

Definition DAG

BinaryExponentSplit(e,h,b) · 2

Actual proof prerequisites

binary_length_digit_split_existsbinary_length_digit_split_functional
Original expanded first-order 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))

Complete unchanged native tactic proof

All 19 lines are the exact independently kernel-checked original script.

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.

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 (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 defined 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