BL0002

binary_length_digit_split_exists

Every natural number has an exact binary quotient and zero-or-one digit.

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)

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

Definition DAG

BinaryExponentSplit(e,h,b) · 1

Actual proof prerequisites

binary_exponent_split_exists · checked external prerequisite
Original expanded first-order statement 
forall n. exists h b. ((((b = 0) \/ (b = 1)) /\ n = (h + h) + b))

Complete unchanged native tactic proof

All 3 lines are the exact independently kernel-checked original script.

Read the argument

Proof checkpoints

3 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.

01Fix variables and assumptionsL1–1

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

  1. L1
    intro n
02Use earlier factsL2–3

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

  1. L2
    specialize binary_exponent_split_exists n
  2. L3
    exact binary_exponent_split_exists

Library-wide reading audit

Original defined command ledger · 3 lines
  1. 0001intro n
  2. 0002specialize binary_exponent_split_exists n
  3. 0003exact binary_exponent_split_exists