CD0002

finite_bit_membership_decidable

Membership in a genuine finite characteristic set is constructively decidable.

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. AllBits(b,c,l) → Lt(i,l) → BetaAt(b,c,i,1) ∨ ¬BetaAt(b,c,i,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_exists · checked external prerequisitefinite_bit_entry_casesbeta_at_unique · checked external prerequisitesucc_ne_zero · checked external prerequisite
Original expanded first-order statement 
forall b c l i. (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_member_yes. fs_h_fms_member_yes + S (1) = S ((S (i)) * c)) /\ exists fs_q_fms_member_yes. b = fs_q_fms_member_yes * S ((S (i)) * c) + (1))) \/ ~(((exists fs_h_fms_member_no. fs_h_fms_member_no + S (1) = S ((S (i)) * c)) /\ exists fs_q_fms_member_no. b = fs_q_fms_member_no * S ((S (i)) * c) + (1)))

Complete tactic proof in conservative notation

All 40 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

40 script commands · 11 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.

Named ingredients (1)
finite bit entry cases
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 l
  4. L4
    intro i
  5. L5
    intro hbits
  6. L6
    intro hi
02Use earlier factsL7–9

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

  1. L7
    specialize beta_at_exists b
  2. L8
    specialize beta_at_exists c
  3. L9
    specialize beta_at_exists i
03Separate the logical casesL10–10

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

  1. L10
    cases beta_at_exists
04Establish hcaseL11–20

Establish this local claim before using it. It is not an additional assumption. The following proof commands apply finite bit entry cases.

  1. L11
    have hcase : x=0 \/ x=1
  2. L12
    specialize finite_bit_entry_cases b
  3. L13
    specialize finite_bit_entry_cases c
  4. L14
    specialize finite_bit_entry_cases l
  5. L15
    specialize finite_bit_entry_cases i
  6. L16
    specialize finite_bit_entry_cases x
  7. L17
    apply finite_bit_entry_cases
  8. L18
    exact hbits
  9. L19
    exact hi
  10. L20
    exact beta_at_exists_witness
05Separate the logical casesL21–22

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

  1. L21
    cases hcase
  2. L22
    right
06Fix variables and assumptionsL23–23

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

  1. L23
    intro hone
07Establish heqL24–33

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

  1. L24
    have heq : 1=x
  2. L25
    specialize beta_at_unique b
  3. L26
    specialize beta_at_unique c
  4. L27
    specialize beta_at_unique i
  5. L28
    specialize beta_at_unique 1
  6. L29
    specialize beta_at_unique x
  7. L30
    apply beta_at_unique
  8. L31
    exact hone
  9. L32
    exact beta_at_exists_witness
  10. L33
    rewrite hcase_left at heq
08Use earlier factsL34–36

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

  1. L34
    specialize succ_ne_zero 0
  2. L35
    apply succ_ne_zero
  3. L36
    exact heq
09Separate the logical casesL37–37

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

  1. L37
    left
10Calculate and transport equalitiesL38–39

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

  1. L38
    rewrite hcase_right at beta_at_exists_witness
  2. L39
    rewrite hcase_right at beta_at_exists_witness
11Use earlier factsL40–40

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

  1. L40
    exact beta_at_exists_witness

Library-wide reading audit

Original defined command ledger · 40 lines
  1. 0001intro b
  2. 0002intro c
  3. 0003intro l
  4. 0004intro i
  5. 0005intro hbits
  6. 0006intro hi
  7. 0007specialize beta_at_exists b
  8. 0008specialize beta_at_exists c
  9. 0009specialize beta_at_exists i
  10. 0010cases beta_at_exists
  11. 0011have hcase : x=0 \/ x=1
  12. 0012specialize finite_bit_entry_cases b
  13. 0013specialize finite_bit_entry_cases c
  14. 0014specialize finite_bit_entry_cases l
  15. 0015specialize finite_bit_entry_cases i
  16. 0016specialize finite_bit_entry_cases x
  17. 0017apply finite_bit_entry_cases
  18. 0018exact hbits
  19. 0019exact hi
  20. 0020exact beta_at_exists_witness
  21. 0021cases hcase
  22. 0022right
  23. 0023intro hone
  24. 0024have heq : 1=x
  25. 0025specialize beta_at_unique b
  26. 0026specialize beta_at_unique c
  27. 0027specialize beta_at_unique i
  28. 0028specialize beta_at_unique 1
  29. 0029specialize beta_at_unique x
  30. 0030apply beta_at_unique
  31. 0031exact hone
  32. 0032exact beta_at_exists_witness
  33. 0033rewrite hcase_left at heq
  34. 0034specialize succ_ne_zero 0
  35. 0035apply succ_ne_zero
  36. 0036exact heq
  37. 0037left
  38. 0038rewrite hcase_right at beta_at_exists_witness
  39. 0039rewrite hcase_right at beta_at_exists_witness
  40. 0040exact beta_at_exists_witness