BE0001

binary_digit_prefix_empty

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

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

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 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))

Constructive proof overview

Generated structural guide

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

The unchanged tactic script uses 2 declared prerequisites and contains 16 exact native proof lines.

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

Proof neighborhood

Direct dependencies

add_eq_zero_right Stable theorem; checked-use authorized succ_ne_zero Stable theorem; checked-use authorized

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

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.

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 exact 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