BE0003

binary_digit_prefix_terminal_bit

Every nonempty valid beta-coded binary prefix has a genuine final 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.

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. ∀ l. BinaryDigitPrefix(b,c,S l) → ∃ x. Beta(b,c,l,x) ∧ (x = 0 ∨ x = 1)

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

Definition DAG

Beta(b,c,i,x) · 1BinaryDigitPrefix(d,s,n) · 1

Actual proof prerequisites

beta_at_exists · checked external prerequisitele_refl · checked external prerequisite
Original expanded first-order statement 
forall b c l. (forall ff_index_be_prefix_next ff_digit_be_prefix_next. (exists ff_lt_be_prefix_next_bound. ff_lt_be_prefix_next_bound + S ff_index_be_prefix_next = S l) -> (((exists ff_h_be_prefix_next_digit. ff_h_be_prefix_next_digit + S (ff_digit_be_prefix_next) = S ((S (ff_index_be_prefix_next)) * c)) /\ exists ff_q_be_prefix_next_digit. b = ff_q_be_prefix_next_digit * S ((S (ff_index_be_prefix_next)) * c) + (ff_digit_be_prefix_next))) -> (ff_digit_be_prefix_next = 0 \/ ff_digit_be_prefix_next = 1)) -> exists x. ((((exists ff_h_be_terminal_digit. ff_h_be_terminal_digit + S (x) = S ((S (l)) * c)) /\ exists ff_q_be_terminal_digit. b = ff_q_be_terminal_digit * S ((S (l)) * c) + (x))) /\ (x = 0 \/ x = 1))

Complete unchanged native tactic proof

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

Read the argument

Proof checkpoints

19 script commands · 6 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 assumptionsL1–4

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

  1. L1
    intro b
  2. L2
    intro c
  3. L3
    intro l
  4. L4
    intro hdigits
02Establish hdigitL5–9

Establish this local claim before using it. It is not an additional assumption.

  1. L5
    have hdigit : exists x. (((exists ff_h_be_terminal_digit. ff_h_be_terminal_digit + S (x) = S ((S (l)) * c)) /\ exists ff_q_be_terminal_digit. b = ff_q_be_terminal_digit * S ((S (l)) * c) + (x)))
  2. L6
    specialize beta_at_exists b
  3. L7
    specialize beta_at_exists c
  4. L8
    specialize beta_at_exists l
  5. L9
    exact beta_at_exists
03Separate the logical casesL10–10

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

  1. L10
    cases hdigit
04Construct an explicit witnessL11–11

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

  1. L11
    exists x
05Separate the logical casesL12–12

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

  1. L12
    split
06Use earlier factsL13–19

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

  1. L13
    exact hdigit_witness
  2. L14
    specialize hdigits l
  3. L15
    specialize hdigits x
  4. L16
    apply hdigits
  5. L17
    specialize le_refl (S l)
  6. L18
    exact le_refl
  7. L19
    exact hdigit_witness

Library-wide reading audit

Original defined command ledger · 19 lines
  1. 0001intro b
  2. 0002intro c
  3. 0003intro l
  4. 0004intro hdigits
  5. 0005have hdigit : exists x. (((exists ff_h_be_terminal_digit. ff_h_be_terminal_digit + S (x) = S ((S (l)) * c)) /\ exists ff_q_be_terminal_digit. b = ff_q_be_terminal_digit * S ((S (l)) * c) + (x)))
  6. 0006specialize beta_at_exists b
  7. 0007specialize beta_at_exists c
  8. 0008specialize beta_at_exists l
  9. 0009exact beta_at_exists
  10. 0010cases hdigit
  11. 0011exists x
  12. 0012split
  13. 0013exact hdigit_witness
  14. 0014specialize hdigits l
  15. 0015specialize hdigits x
  16. 0016apply hdigits
  17. 0017specialize le_refl (S l)
  18. 0018exact le_refl
  19. 0019exact hdigit_witness