CD0001

finite_bit_entry_cases

Every explicitly decoded entry of a genuine bit prefix is zero or one.

Alpha v34 checked-use · first admitted v27 · 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.

Sets are complete characteristic-bit codes with actual finite cardinality witnesses. The proof constructs translations and the sumset; no finite-choice oracle, supplied cardinality conclusion, or unproved polynomial-method premise is used.

Exact theorem in conservative defined notation

∀ b. ∀ c. ∀ l. ∀ i. ∀ a. AllBits(b,c,l) → Lt(i,l) → BetaAt(b,c,i,a) → a = 0 ∨ a = 1

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

Definition DAG

Lt(a,b) · 1BetaAt(b,c,i,x) · 2AllBits(b,c,l) · 1

Actual proof prerequisites

beta_at_unique · checked external prerequisite
Original expanded first-order statement 
forall b c l i a. (forall ff_i_fms_bits. (exists ff_lt_fms_bits_bound. ff_lt_fms_bits_bound + S ff_i_fms_bits = (l)) -> exists ff_bit_fms_bits. ((((exists ff_h_fms_bits_decoded. ff_h_fms_bits_decoded + S (ff_bit_fms_bits) = S ((S (ff_i_fms_bits)) * (c))) /\ exists ff_q_fms_bits_decoded. (b) = ff_q_fms_bits_decoded * S ((S (ff_i_fms_bits)) * (c)) + (ff_bit_fms_bits))) /\ (ff_bit_fms_bits = 0 \/ ff_bit_fms_bits = 1))) -> (exists fms_gap_lt. fms_gap_lt + S (i) = (l)) -> (((exists fs_h_fms_entry. fs_h_fms_entry + S (a) = S ((S (i)) * c)) /\ exists fs_q_fms_entry. b = fs_q_fms_entry * S ((S (i)) * c) + (a))) -> a=0 \/ a=1

Complete tactic proof in conservative notation

All 26 original proof lines are preserved. Only local proposition formulas are abbreviated; every abbreviation has an exact binder-safe expansion check. The linked exact edition contains the unchanged replay script.

Read the argument

Proof checkpoints

26 script commands · 6 reading checkpoints · 2 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 assumptionsL1–8

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 i
  5. L5
    intro a
  6. L6
    intro hbits
  7. L7
    intro hi
  8. L8
    intro ha
02Establish hbitL9–12

Establish this local claim before using it. It is not an additional assumption. The following proof commands apply hbits.

  1. L9
    have hbit : ∃ v. BetaAt(b,c,i,v) ∧ (v = 0 ∨ v = 1)Definitions: BetaAt(b,c,i,v)Original native command in the exact edition
  2. L10
    specialize hbits i
  3. L11
    apply hbits
  4. L12
    exact hi
03Separate the logical casesL13–14

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

  1. L13
    cases hbit
  2. L14
    cases hbit_witness
04Establish heqL15–24

Establish this local claim before using it. It is not an additional assumption. The following proof commands apply beta at unique.

  1. L15
    have heq : a=x
  2. L16
    specialize beta_at_unique b
  3. L17
    specialize beta_at_unique c
  4. L18
    specialize beta_at_unique i
  5. L19
    specialize beta_at_unique a
  6. L20
    specialize beta_at_unique x
  7. L21
    apply beta_at_unique
  8. L22
    exact ha
  9. L23
    exact hbit_witness_left
  10. L24
    rewrite heq
05Calculate and transport equalitiesL25–25

Carry out the recorded arithmetic or equality steps; inspect the exact commands for their direction and premises.

  1. L25
    rewrite heq
06Use earlier factsL26–26

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

  1. L26
    exact hbit_witness_right

Library-wide reading audit

Original defined command ledger · 26 lines
  1. 0001intro b
  2. 0002intro c
  3. 0003intro l
  4. 0004intro i
  5. 0005intro a
  6. 0006intro hbits
  7. 0007intro hi
  8. 0008intro ha
  9. 0009have hbit : ∃ v. BetaAt(b,c,i,v) ∧ (v = 0 ∨ v = 1)
  10. 0010specialize hbits i
  11. 0011apply hbits
  12. 0012exact hi
  13. 0013cases hbit
  14. 0014cases hbit_witness
  15. 0015have heq : a=x
  16. 0016specialize beta_at_unique b
  17. 0017specialize beta_at_unique c
  18. 0018specialize beta_at_unique i
  19. 0019specialize beta_at_unique a
  20. 0020specialize beta_at_unique x
  21. 0021apply beta_at_unique
  22. 0022exact ha
  23. 0023exact hbit_witness_left
  24. 0024rewrite heq
  25. 0025rewrite heq
  26. 0026exact hbit_witness_right