CD0019

finite_bit_nonmember_zero

Constructive nonmembership at a bounded index exposes its actual zero bit.

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,0)

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_cases
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_not_member. fs_h_fms_not_member + S (1) = S ((S (i)) * c)) /\ exists fs_q_fms_not_member. b = fs_q_fms_not_member * S ((S (i)) * c) + (1))) -> (((exists fs_h_fms_zero_member. fs_h_fms_zero_member + S (0) = S ((S (i)) * c)) /\ exists fs_q_fms_zero_member. b = fs_q_fms_zero_member * S ((S (i)) * c) + (0)))

Complete tactic proof in conservative notation

All 30 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

30 script commands · 11 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.

Named ingredients (1)
finite bit entry cases
01Fix variables and assumptionsL1–7

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
  7. L7
    intro hnot
02Use earlier factsL8–10

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

  1. L8
    specialize beta_at_exists b
  2. L9
    specialize beta_at_exists c
  3. L10
    specialize beta_at_exists i
03Separate the logical casesL11–11

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

  1. L11
    cases beta_at_exists
04Establish hcL12–21

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

  1. L12
    have hc : x=0 \/ x=1
  2. L13
    specialize finite_bit_entry_cases b
  3. L14
    specialize finite_bit_entry_cases c
  4. L15
    specialize finite_bit_entry_cases l
  5. L16
    specialize finite_bit_entry_cases i
  6. L17
    specialize finite_bit_entry_cases x
  7. L18
    apply finite_bit_entry_cases
  8. L19
    exact hbits
  9. L20
    exact hi
  10. L21
    exact beta_at_exists_witness
05Separate the logical casesL22–22

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

  1. L22
    cases hc
06Calculate and transport equalitiesL23–24

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

  1. L23
    rewrite hc_left at beta_at_exists_witness
  2. L24
    rewrite hc_left at beta_at_exists_witness
07Use earlier factsL25–25

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

  1. L25
    exact beta_at_exists_witness
08Separate the logical casesL26–26

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

  1. L26
    exfalso
09Use earlier factsL27–27

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

  1. L27
    apply hnot
10Calculate and transport equalitiesL28–29

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

  1. L28
    rewrite hc_right at beta_at_exists_witness
  2. L29
    rewrite hc_right at beta_at_exists_witness
11Use earlier factsL30–30

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

  1. L30
    exact beta_at_exists_witness

Library-wide reading audit

Original defined command ledger · 30 lines
  1. 0001intro b
  2. 0002intro c
  3. 0003intro l
  4. 0004intro i
  5. 0005intro hbits
  6. 0006intro hi
  7. 0007intro hnot
  8. 0008specialize beta_at_exists b
  9. 0009specialize beta_at_exists c
  10. 0010specialize beta_at_exists i
  11. 0011cases beta_at_exists
  12. 0012have hc : x=0 \/ x=1
  13. 0013specialize finite_bit_entry_cases b
  14. 0014specialize finite_bit_entry_cases c
  15. 0015specialize finite_bit_entry_cases l
  16. 0016specialize finite_bit_entry_cases i
  17. 0017specialize finite_bit_entry_cases x
  18. 0018apply finite_bit_entry_cases
  19. 0019exact hbits
  20. 0020exact hi
  21. 0021exact beta_at_exists_witness
  22. 0022cases hc
  23. 0023rewrite hc_left at beta_at_exists_witness
  24. 0024rewrite hc_left at beta_at_exists_witness
  25. 0025exact beta_at_exists_witness
  26. 0026exfalso
  27. 0027apply hnot
  28. 0028rewrite hc_right at beta_at_exists_witness
  29. 0029rewrite hc_right at beta_at_exists_witness
  30. 0030exact beta_at_exists_witness