BE0001

binary_digit_prefix_empty

Every empty beta-coded prefix is constructively a valid binary digit sequence.

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.

G102 was OPEN at this family's Alpha-v22 first admission: complete execution was proved only for a supplied valid beta-coded digit prefix. G102 is now CLOSED in Alpha v23 for every arbitrary exponent, with actual canonical digits and operations≤3*BitLen(e)+2.

Exact theorem in conservative defined notation

∀ b. ∀ c. BinaryDigitPrefix(b,c,0)

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

Definition DAG

BinaryDigitPrefix(d,s,n) · 1

Actual proof prerequisites

add_eq_zero_right · checked external prerequisitesucc_ne_zero · checked external prerequisite
Original expanded first-order statement 
forall b c. (forall ff_index_be_empty ff_digit_be_empty. (exists ff_lt_be_empty_bound. ff_lt_be_empty_bound + S ff_index_be_empty = 0) -> (((exists ff_h_be_empty_digit. ff_h_be_empty_digit + S (ff_digit_be_empty) = S ((S (ff_index_be_empty)) * c)) /\ exists ff_q_be_empty_digit. b = ff_q_be_empty_digit * S ((S (ff_index_be_empty)) * c) + (ff_digit_be_empty))) -> (ff_digit_be_empty = 0 \/ ff_digit_be_empty = 1))

Complete unchanged native tactic proof

All 16 lines are the exact independently kernel-checked original script.

Read the argument

Proof checkpoints

16 script commands · 3 reading checkpoints · 1 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 assumptionsL1–6

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

  1. L1
    intro b
  2. L2
    intro c
  3. L3
    intro i
  4. L4
    intro d
  5. L5
    intro hi
  6. L6
    intro hat
02Separate the logical casesL7–8

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

  1. L7
    exfalso
  2. L8
    cases hi
03Establish hzeroL9–16

Establish this local claim before using it. It is not an additional assumption. The following proof commands apply add eq zero right.

  1. L9
    have hzero : S i = 0
  2. L10
    specialize add_eq_zero_right x
  3. L11
    specialize add_eq_zero_right (S i)
  4. L12
    apply add_eq_zero_right
  5. L13
    exact hi_witness
  6. L14
    specialize succ_ne_zero i
  7. L15
    apply succ_ne_zero
  8. L16
    exact hzero

Library-wide reading audit

Original defined command ledger · 16 lines
  1. 0001intro b
  2. 0002intro c
  3. 0003intro i
  4. 0004intro d
  5. 0005intro hi
  6. 0006intro hat
  7. 0007exfalso
  8. 0008cases hi
  9. 0009have hzero : S i = 0
  10. 0010specialize add_eq_zero_right x
  11. 0011specialize add_eq_zero_right (S i)
  12. 0012apply add_eq_zero_right
  13. 0013exact hi_witness
  14. 0014specialize succ_ne_zero i
  15. 0015apply succ_ne_zero
  16. 0016exact hzero