CD000A

finite_bit_member_count_nonzero

An actual bounded member rules out zero for the witnessed set cardinality.

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. ∀ n. ∀ i. BitCount(b,c,l,n) → ModularSetMember(b,c,l,i) → ¬n = 0

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

Definition DAG

ModularSetMember(b,c,p,x) · 1BitCount(b,c,l,z) · 1

Actual proof prerequisites

finite_sum_entry_lene_zero_of_one_le · checked external prerequisite
Original expanded first-order statement 
forall b c l n i. (((exists ff_u_fms_count ff_v_fms_count. ((((exists ff_h_fms_count_start. ff_h_fms_count_start + S (0) = S ((S (0)) * ff_v_fms_count)) /\ exists ff_q_fms_count_start. ff_u_fms_count = ff_q_fms_count_start * S ((S (0)) * ff_v_fms_count) + (0))) /\ ((((exists ff_h_fms_count_terminal. ff_h_fms_count_terminal + S ((n)) = S ((S ((l))) * ff_v_fms_count)) /\ exists ff_q_fms_count_terminal. ff_u_fms_count = ff_q_fms_count_terminal * S ((S ((l))) * ff_v_fms_count) + ((n)))) /\ forall ff_i_fms_count. (exists ff_lt_fms_count_bound. ff_lt_fms_count_bound + S ff_i_fms_count = (l)) -> exists ff_a_fms_count ff_r_fms_count ff_s_fms_count. ((((exists ff_h_fms_count_summand. ff_h_fms_count_summand + S (ff_a_fms_count) = S ((S (ff_i_fms_count)) * (c))) /\ exists ff_q_fms_count_summand. (b) = ff_q_fms_count_summand * S ((S (ff_i_fms_count)) * (c)) + (ff_a_fms_count))) /\ ((((exists ff_h_fms_count_partial. ff_h_fms_count_partial + S (ff_r_fms_count) = S ((S (ff_i_fms_count)) * ff_v_fms_count)) /\ exists ff_q_fms_count_partial. ff_u_fms_count = ff_q_fms_count_partial * S ((S (ff_i_fms_count)) * ff_v_fms_count) + (ff_r_fms_count))) /\ ((((exists ff_h_fms_count_successor. ff_h_fms_count_successor + S (ff_s_fms_count) = S ((S (S ff_i_fms_count)) * ff_v_fms_count)) /\ exists ff_q_fms_count_successor. ff_u_fms_count = ff_q_fms_count_successor * S ((S (S ff_i_fms_count)) * ff_v_fms_count) + (ff_s_fms_count))) /\ ff_s_fms_count = ff_r_fms_count + ff_a_fms_count)))))) /\ (forall ff_i_fms_count. (exists ff_lt_fms_count_bound. ff_lt_fms_count_bound + S ff_i_fms_count = (l)) -> exists ff_bit_fms_count. ((((exists ff_h_fms_count_decoded. ff_h_fms_count_decoded + S (ff_bit_fms_count) = S ((S (ff_i_fms_count)) * (c))) /\ exists ff_q_fms_count_decoded. (b) = ff_q_fms_count_decoded * S ((S (ff_i_fms_count)) * (c)) + (ff_bit_fms_count))) /\ (ff_bit_fms_count = 0 \/ ff_bit_fms_count = 1))))) -> (((exists fms_gap_member. fms_gap_member + S (i) = (l)) /\ (((exists fs_h_fms_member. fs_h_fms_member + S (1) = S ((S (i)) * c)) /\ exists fs_q_fms_member. b = fs_q_fms_member * S ((S (i)) * c) + (1))))) -> ~(n=0)

Complete tactic proof in conservative notation

All 23 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

23 script commands · 4 reading checkpoints · 0 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 sum entry le
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 n
  5. L5
    intro i
  6. L6
    intro hcount
  7. L7
    intro hmember
  8. L8
    intro hz
02Separate the logical casesL9–10

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

  1. L9
    cases hcount
  2. L10
    cases hmember
03Use earlier factsL11–20

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

  1. L11
    specialize ne_zero_of_one_le n
  2. L12
    apply ne_zero_of_one_le
  3. L13
    specialize finite_sum_entry_le b
  4. L14
    specialize finite_sum_entry_le c
  5. L15
    specialize finite_sum_entry_le l
  6. L16
    specialize finite_sum_entry_le n
  7. L17
    specialize finite_sum_entry_le i
  8. L18
    specialize finite_sum_entry_le 1
  9. L19
    apply finite_sum_entry_le
  10. L20
    exact hcount_left
04Use earlier factsL21–23

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

  1. L21
    exact hmember_left
  2. L22
    exact hmember_right
  3. L23
    exact hz

Library-wide reading audit

Original defined command ledger · 23 lines
  1. 0001intro b
  2. 0002intro c
  3. 0003intro l
  4. 0004intro n
  5. 0005intro i
  6. 0006intro hcount
  7. 0007intro hmember
  8. 0008intro hz
  9. 0009cases hcount
  10. 0010cases hmember
  11. 0011specialize ne_zero_of_one_le n
  12. 0012apply ne_zero_of_one_le
  13. 0013specialize finite_sum_entry_le b
  14. 0014specialize finite_sum_entry_le c
  15. 0015specialize finite_sum_entry_le l
  16. 0016specialize finite_sum_entry_le n
  17. 0017specialize finite_sum_entry_le i
  18. 0018specialize finite_sum_entry_le 1
  19. 0019apply finite_sum_entry_le
  20. 0020exact hcount_left
  21. 0021exact hmember_left
  22. 0022exact hmember_right
  23. 0023exact hz