MV0008

mobius_from_squarefree_factor_count

Public research checkpoint, not admitted to Alpha or Stable. Alpha v30 remains 3222 checked-use theorems; Stable remains 432. The on-demand Alpha Lean service does not yet expose these checkpoint theorems; their independently checked literal bundles and unchanged sources are available below.

Squarefreeness, a real prime-factor list and its actual length parity construct the nonzero Möbius value.

Public research checkpoint: original HA and independently compiled Lean verified; not Alpha-enrolled, no Alpha checked-use authority; not Stable

Mobius(n,z) is defined only for positive n. The canonical signed codes are 0 for zero, 2 for +1, and 1 for −1. The function is defined independently of any divisor-sum identity. These values and prime-step lemmas are prerequisites for G007; divisor-sum cancellation and full Möbius inversion remain open in this checkpoint. No signed-table proof is included here.

Exact theorem in conservative defined notation

∀ n. ∀ b. ∀ c. ∀ l. ∀ z. Squarefree(n) → PrimeFactorList(n,b,c,l) → AlternatingSignedUnit(l,z) → Mobius(n,z)

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

Definition DAG

PrimeFactorList(n,b,c,l) · 1Squarefree(n) · 1AlternatingSignedUnit(n,z) · 1Mobius(n,z) · 1

Actual proof prerequisites

none
Original expanded first-order statement 
forall n b c l z. (((~((n) = 0)) /\ (forall sfd_prime_constructor_sf. (~((sfd_prime_constructor_sf) = 1) /\ forall pvs_left_constructor_sfdomain pvs_right_constructor_sfdomain. (sfd_prime_constructor_sf) = pvs_left_constructor_sfdomain * pvs_right_constructor_sfdomain -> pvs_left_constructor_sfdomain = 1 \/ pvs_right_constructor_sfdomain = 1) -> (exists pvs_le_gap_constructor_sfbound. pvs_le_gap_constructor_sfbound + (sfd_prime_constructor_sf) = (n)) -> ~(exists pvs_factor_constructor_sfsquare. (n) = (sfd_prime_constructor_sf * sfd_prime_constructor_sf) * pvs_factor_constructor_sfsquare)))) -> ((~(n = 0) /\ ((exists ff_u_fsat_mv_constructor_factors_product ff_v_fsat_mv_constructor_factors_product. ((((exists ff_h_fsat_mv_constructor_factors_product_start. ff_h_fsat_mv_constructor_factors_product_start + S (1) = S ((S (0)) * ff_v_fsat_mv_constructor_factors_product)) /\ exists ff_q_fsat_mv_constructor_factors_product_start. ff_u_fsat_mv_constructor_factors_product = ff_q_fsat_mv_constructor_factors_product_start * S ((S (0)) * ff_v_fsat_mv_constructor_factors_product) + (1))) /\ ((((exists ff_h_fsat_mv_constructor_factors_product_terminal. ff_h_fsat_mv_constructor_factors_product_terminal + S (n) = S ((S (l)) * ff_v_fsat_mv_constructor_factors_product)) /\ exists ff_q_fsat_mv_constructor_factors_product_terminal. ff_u_fsat_mv_constructor_factors_product = ff_q_fsat_mv_constructor_factors_product_terminal * S ((S (l)) * ff_v_fsat_mv_constructor_factors_product) + (n))) /\ forall ff_i_fsat_mv_constructor_factors_product. (exists ff_lt_fsat_mv_constructor_factors_product_bound. ff_lt_fsat_mv_constructor_factors_product_bound + S ff_i_fsat_mv_constructor_factors_product = l) -> exists ff_p_fsat_mv_constructor_factors_product ff_r_fsat_mv_constructor_factors_product ff_s_fsat_mv_constructor_factors_product. ((((exists ff_h_fsat_mv_constructor_factors_product_factor. ff_h_fsat_mv_constructor_factors_product_factor + S (ff_p_fsat_mv_constructor_factors_product) = S ((S (ff_i_fsat_mv_constructor_factors_product)) * c)) /\ exists ff_q_fsat_mv_constructor_factors_product_factor. b = ff_q_fsat_mv_constructor_factors_product_factor * S ((S (ff_i_fsat_mv_constructor_factors_product)) * c) + (ff_p_fsat_mv_constructor_factors_product))) /\ ((((exists ff_h_fsat_mv_constructor_factors_product_partial. ff_h_fsat_mv_constructor_factors_product_partial + S (ff_r_fsat_mv_constructor_factors_product) = S ((S (ff_i_fsat_mv_constructor_factors_product)) * ff_v_fsat_mv_constructor_factors_product)) /\ exists ff_q_fsat_mv_constructor_factors_product_partial. ff_u_fsat_mv_constructor_factors_product = ff_q_fsat_mv_constructor_factors_product_partial * S ((S (ff_i_fsat_mv_constructor_factors_product)) * ff_v_fsat_mv_constructor_factors_product) + (ff_r_fsat_mv_constructor_factors_product))) /\ ((((exists ff_h_fsat_mv_constructor_factors_product_successor. ff_h_fsat_mv_constructor_factors_product_successor + S (ff_s_fsat_mv_constructor_factors_product) = S ((S (S ff_i_fsat_mv_constructor_factors_product)) * ff_v_fsat_mv_constructor_factors_product)) /\ exists ff_q_fsat_mv_constructor_factors_product_successor. ff_u_fsat_mv_constructor_factors_product = ff_q_fsat_mv_constructor_factors_product_successor * S ((S (S ff_i_fsat_mv_constructor_factors_product)) * ff_v_fsat_mv_constructor_factors_product) + (ff_s_fsat_mv_constructor_factors_product))) /\ ff_s_fsat_mv_constructor_factors_product = ff_r_fsat_mv_constructor_factors_product * ff_p_fsat_mv_constructor_factors_product)))))) /\ (forall ftsf_index_fsat_mv_constructor_factors_primes. (exists ftsf_gap_fsat_mv_constructor_factors_primes_bound. ftsf_gap_fsat_mv_constructor_factors_primes_bound + S ftsf_index_fsat_mv_constructor_factors_primes = (l)) -> exists ftsf_factor_fsat_mv_constructor_factors_primes. ((((exists ff_h_ftsf_fsat_mv_constructor_factors_primes_entry. ff_h_ftsf_fsat_mv_constructor_factors_primes_entry + S (ftsf_factor_fsat_mv_constructor_factors_primes) = S ((S (ftsf_index_fsat_mv_constructor_factors_primes)) * c)) /\ exists ff_q_ftsf_fsat_mv_constructor_factors_primes_entry. b = ff_q_ftsf_fsat_mv_constructor_factors_primes_entry * S ((S (ftsf_index_fsat_mv_constructor_factors_primes)) * c) + (ftsf_factor_fsat_mv_constructor_factors_primes))) /\ ((~(ftsf_factor_fsat_mv_constructor_factors_primes = 1) /\ forall frm_prime_left_ftsf_fsat_mv_constructor_factors_primes_prime frm_prime_right_ftsf_fsat_mv_constructor_factors_primes_prime. ftsf_factor_fsat_mv_constructor_factors_primes = frm_prime_left_ftsf_fsat_mv_constructor_factors_primes_prime * frm_prime_right_ftsf_fsat_mv_constructor_factors_primes_prime -> frm_prime_left_ftsf_fsat_mv_constructor_factors_primes_prime = 1 \/ frm_prime_right_ftsf_fsat_mv_constructor_factors_primes_prime = 1))))))) -> ((((exists mv_even_half_constructor_signeven. (l) = 2 * mv_even_half_constructor_signeven) /\ ((z) = 2))) \/ (((exists mv_odd_half_constructor_signodd. (l) = 2 * mv_odd_half_constructor_signodd + 1) /\ ((z) = 1)))) -> (((~((n) = 0)) /\ ((((exists mv_square_prime_constructor_valuesquare. ((~((mv_square_prime_constructor_valuesquare) = 1) /\ forall pvs_left_constructor_valuesquareprime pvs_right_constructor_valuesquareprime. (mv_square_prime_constructor_valuesquare) = pvs_left_constructor_valuesquareprime * pvs_right_constructor_valuesquareprime -> pvs_left_constructor_valuesquareprime = 1 \/ pvs_right_constructor_valuesquareprime = 1) /\ (exists pvs_factor_constructor_valuesquaredivisor. (n) = (mv_square_prime_constructor_valuesquare * mv_square_prime_constructor_valuesquare) * pvs_factor_constructor_valuesquaredivisor))) /\ ((z) = 0))) \/ (((((~((n) = 0)) /\ (forall sfd_prime_constructor_valuesquarefree. (~((sfd_prime_constructor_valuesquarefree) = 1) /\ forall pvs_left_constructor_valuesquarefreedomain pvs_right_constructor_valuesquarefreedomain. (sfd_prime_constructor_valuesquarefree) = pvs_left_constructor_valuesquarefreedomain * pvs_right_constructor_valuesquarefreedomain -> pvs_left_constructor_valuesquarefreedomain = 1 \/ pvs_right_constructor_valuesquarefreedomain = 1) -> (exists pvs_le_gap_constructor_valuesquarefreebound. pvs_le_gap_constructor_valuesquarefreebound + (sfd_prime_constructor_valuesquarefree) = (n)) -> ~(exists pvs_factor_constructor_valuesquarefreesquare. (n) = (sfd_prime_constructor_valuesquarefree * sfd_prime_constructor_valuesquarefree) * pvs_factor_constructor_valuesquarefreesquare)))) /\ (exists mv_factor_code_constructor_valuefactors mv_factor_scale_constructor_valuefactors mv_factor_count_constructor_valuefactors. (((~(n = 0) /\ ((exists ff_u_fsat_constructor_valuefactorsfactorization_product ff_v_fsat_constructor_valuefactorsfactorization_product. ((((exists ff_h_fsat_constructor_valuefactorsfactorization_product_start. ff_h_fsat_constructor_valuefactorsfactorization_product_start + S (1) = S ((S (0)) * ff_v_fsat_constructor_valuefactorsfactorization_product)) /\ exists ff_q_fsat_constructor_valuefactorsfactorization_product_start. ff_u_fsat_constructor_valuefactorsfactorization_product = ff_q_fsat_constructor_valuefactorsfactorization_product_start * S ((S (0)) * ff_v_fsat_constructor_valuefactorsfactorization_product) + (1))) /\ ((((exists ff_h_fsat_constructor_valuefactorsfactorization_product_terminal. ff_h_fsat_constructor_valuefactorsfactorization_product_terminal + S (n) = S ((S (mv_factor_count_constructor_valuefactors)) * ff_v_fsat_constructor_valuefactorsfactorization_product)) /\ exists ff_q_fsat_constructor_valuefactorsfactorization_product_terminal. ff_u_fsat_constructor_valuefactorsfactorization_product = ff_q_fsat_constructor_valuefactorsfactorization_product_terminal * S ((S (mv_factor_count_constructor_valuefactors)) * ff_v_fsat_constructor_valuefactorsfactorization_product) + (n))) /\ forall ff_i_fsat_constructor_valuefactorsfactorization_product. (exists ff_lt_fsat_constructor_valuefactorsfactorization_product_bound. ff_lt_fsat_constructor_valuefactorsfactorization_product_bound + S ff_i_fsat_constructor_valuefactorsfactorization_product = mv_factor_count_constructor_valuefactors) -> exists ff_p_fsat_constructor_valuefactorsfactorization_product ff_r_fsat_constructor_valuefactorsfactorization_product ff_s_fsat_constructor_valuefactorsfactorization_product. ((((exists ff_h_fsat_constructor_valuefactorsfactorization_product_factor. ff_h_fsat_constructor_valuefactorsfactorization_product_factor + S (ff_p_fsat_constructor_valuefactorsfactorization_product) = S ((S (ff_i_fsat_constructor_valuefactorsfactorization_product)) * mv_factor_scale_constructor_valuefactors)) /\ exists ff_q_fsat_constructor_valuefactorsfactorization_product_factor. mv_factor_code_constructor_valuefactors = ff_q_fsat_constructor_valuefactorsfactorization_product_factor * S ((S (ff_i_fsat_constructor_valuefactorsfactorization_product)) * mv_factor_scale_constructor_valuefactors) + (ff_p_fsat_constructor_valuefactorsfactorization_product))) /\ ((((exists ff_h_fsat_constructor_valuefactorsfactorization_product_partial. ff_h_fsat_constructor_valuefactorsfactorization_product_partial + S (ff_r_fsat_constructor_valuefactorsfactorization_product) = S ((S (ff_i_fsat_constructor_valuefactorsfactorization_product)) * ff_v_fsat_constructor_valuefactorsfactorization_product)) /\ exists ff_q_fsat_constructor_valuefactorsfactorization_product_partial. ff_u_fsat_constructor_valuefactorsfactorization_product = ff_q_fsat_constructor_valuefactorsfactorization_product_partial * S ((S (ff_i_fsat_constructor_valuefactorsfactorization_product)) * ff_v_fsat_constructor_valuefactorsfactorization_product) + (ff_r_fsat_constructor_valuefactorsfactorization_product))) /\ ((((exists ff_h_fsat_constructor_valuefactorsfactorization_product_successor. ff_h_fsat_constructor_valuefactorsfactorization_product_successor + S (ff_s_fsat_constructor_valuefactorsfactorization_product) = S ((S (S ff_i_fsat_constructor_valuefactorsfactorization_product)) * ff_v_fsat_constructor_valuefactorsfactorization_product)) /\ exists ff_q_fsat_constructor_valuefactorsfactorization_product_successor. ff_u_fsat_constructor_valuefactorsfactorization_product = ff_q_fsat_constructor_valuefactorsfactorization_product_successor * S ((S (S ff_i_fsat_constructor_valuefactorsfactorization_product)) * ff_v_fsat_constructor_valuefactorsfactorization_product) + (ff_s_fsat_constructor_valuefactorsfactorization_product))) /\ ff_s_fsat_constructor_valuefactorsfactorization_product = ff_r_fsat_constructor_valuefactorsfactorization_product * ff_p_fsat_constructor_valuefactorsfactorization_product)))))) /\ (forall ftsf_index_fsat_constructor_valuefactorsfactorization_primes. (exists ftsf_gap_fsat_constructor_valuefactorsfactorization_primes_bound. ftsf_gap_fsat_constructor_valuefactorsfactorization_primes_bound + S ftsf_index_fsat_constructor_valuefactorsfactorization_primes = (mv_factor_count_constructor_valuefactors)) -> exists ftsf_factor_fsat_constructor_valuefactorsfactorization_primes. ((((exists ff_h_ftsf_fsat_constructor_valuefactorsfactorization_primes_entry. ff_h_ftsf_fsat_constructor_valuefactorsfactorization_primes_entry + S (ftsf_factor_fsat_constructor_valuefactorsfactorization_primes) = S ((S (ftsf_index_fsat_constructor_valuefactorsfactorization_primes)) * mv_factor_scale_constructor_valuefactors)) /\ exists ff_q_ftsf_fsat_constructor_valuefactorsfactorization_primes_entry. mv_factor_code_constructor_valuefactors = ff_q_ftsf_fsat_constructor_valuefactorsfactorization_primes_entry * S ((S (ftsf_index_fsat_constructor_valuefactorsfactorization_primes)) * mv_factor_scale_constructor_valuefactors) + (ftsf_factor_fsat_constructor_valuefactorsfactorization_primes))) /\ ((~(ftsf_factor_fsat_constructor_valuefactorsfactorization_primes = 1) /\ forall frm_prime_left_ftsf_fsat_constructor_valuefactorsfactorization_primes_prime frm_prime_right_ftsf_fsat_constructor_valuefactorsfactorization_primes_prime. ftsf_factor_fsat_constructor_valuefactorsfactorization_primes = frm_prime_left_ftsf_fsat_constructor_valuefactorsfactorization_primes_prime * frm_prime_right_ftsf_fsat_constructor_valuefactorsfactorization_primes_prime -> frm_prime_left_ftsf_fsat_constructor_valuefactorsfactorization_primes_prime = 1 \/ frm_prime_right_ftsf_fsat_constructor_valuefactorsfactorization_primes_prime = 1))))))) /\ ((((exists mv_even_half_constructor_valuefactorsparityeven. (mv_factor_count_constructor_valuefactors) = 2 * mv_even_half_constructor_valuefactorsparityeven) /\ ((z) = 2))) \/ (((exists mv_odd_half_constructor_valuefactorsparityodd. (mv_factor_count_constructor_valuefactors) = 2 * mv_odd_half_constructor_valuefactorsparityodd + 1) /\ ((z) = 1)))))))))))

Complete tactic proof in conservative notation

All 20 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

20 script commands · 8 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.

01Fix variables and assumptionsL1–8

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

  1. L1
    intro n
  2. L2
    intro b
  3. L3
    intro c
  4. L4
    intro l
  5. L5
    intro z
  6. L6
    intro hsf
  7. L7
    intro hf
  8. L8
    intro hs
02Separate the logical casesL9–10

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

  1. L9
    cases hsf
  2. L10
    split
03Use earlier factsL11–11

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

  1. L11
    exact hsf_left
04Separate the logical casesL12–13

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

  1. L12
    right
  2. L13
    split
05Use earlier factsL14–14

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

  1. L14
    exact hsf
06Construct an explicit witnessL15–17

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

  1. L15
    exists b
  2. L16
    exists c
  3. L17
    exists l
07Separate the logical casesL18–18

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

  1. L18
    split
08Use earlier factsL19–20

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

  1. L19
    exact hf
  2. L20
    exact hs

Library-wide reading audit

Original defined command ledger · 20 lines
  1. 0001intro n
  2. 0002intro b
  3. 0003intro c
  4. 0004intro l
  5. 0005intro z
  6. 0006intro hsf
  7. 0007intro hf
  8. 0008intro hs
  9. 0009cases hsf
  10. 0010split
  11. 0011exact hsf_left
  12. 0012right
  13. 0013split
  14. 0014exact hsf
  15. 0015exists b
  16. 0016exists c
  17. 0017exists l
  18. 0018split
  19. 0019exact hf
  20. 0020exact hs