MV000C

mobius_value_exists_unique

Every positive natural has one actual, uniquely determined, factorization-defined canonical Möbius value.

Alpha v34 checked-use · first admitted v31 · 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.

Mobius(n,z) is positive-domain only, independently defined from squarefreeness and actual prime-factor parity. Signed codes 0, 2 and 1 represent zero, +1 and -1. This family proves values and prime-adjunction laws; the separate Möbius-inversion family supplies the complete G007 endpoint.

Exact theorem in conservative defined notation

∀ n. ¬n = 0 → ∃ x. Mobius(n,x) ∧ (∀ y. Mobius(n,y) → y = x)

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

Definition DAG

Mobius(n,z) · 3

Actual proof prerequisites

mobius_value_existsmobius_value_functional
Original expanded first-order statement 
forall n. ~(n = 0) -> exists z. (((~((n) = 0)) /\ ((((exists mv_square_prime_unique_chosensquare. ((~((mv_square_prime_unique_chosensquare) = 1) /\ forall pvs_left_unique_chosensquareprime pvs_right_unique_chosensquareprime. (mv_square_prime_unique_chosensquare) = pvs_left_unique_chosensquareprime * pvs_right_unique_chosensquareprime -> pvs_left_unique_chosensquareprime = 1 \/ pvs_right_unique_chosensquareprime = 1) /\ (exists pvs_factor_unique_chosensquaredivisor. (n) = (mv_square_prime_unique_chosensquare * mv_square_prime_unique_chosensquare) * pvs_factor_unique_chosensquaredivisor))) /\ ((z) = 0))) \/ (((((~((n) = 0)) /\ (forall sfd_prime_unique_chosensquarefree. (~((sfd_prime_unique_chosensquarefree) = 1) /\ forall pvs_left_unique_chosensquarefreedomain pvs_right_unique_chosensquarefreedomain. (sfd_prime_unique_chosensquarefree) = pvs_left_unique_chosensquarefreedomain * pvs_right_unique_chosensquarefreedomain -> pvs_left_unique_chosensquarefreedomain = 1 \/ pvs_right_unique_chosensquarefreedomain = 1) -> (exists pvs_le_gap_unique_chosensquarefreebound. pvs_le_gap_unique_chosensquarefreebound + (sfd_prime_unique_chosensquarefree) = (n)) -> ~(exists pvs_factor_unique_chosensquarefreesquare. (n) = (sfd_prime_unique_chosensquarefree * sfd_prime_unique_chosensquarefree) * pvs_factor_unique_chosensquarefreesquare)))) /\ (exists mv_factor_code_unique_chosenfactors mv_factor_scale_unique_chosenfactors mv_factor_count_unique_chosenfactors. (((~(n = 0) /\ ((exists ff_u_fsat_unique_chosenfactorsfactorization_product ff_v_fsat_unique_chosenfactorsfactorization_product. ((((exists ff_h_fsat_unique_chosenfactorsfactorization_product_start. ff_h_fsat_unique_chosenfactorsfactorization_product_start + S (1) = S ((S (0)) * ff_v_fsat_unique_chosenfactorsfactorization_product)) /\ exists ff_q_fsat_unique_chosenfactorsfactorization_product_start. ff_u_fsat_unique_chosenfactorsfactorization_product = ff_q_fsat_unique_chosenfactorsfactorization_product_start * S ((S (0)) * ff_v_fsat_unique_chosenfactorsfactorization_product) + (1))) /\ ((((exists ff_h_fsat_unique_chosenfactorsfactorization_product_terminal. ff_h_fsat_unique_chosenfactorsfactorization_product_terminal + S (n) = S ((S (mv_factor_count_unique_chosenfactors)) * ff_v_fsat_unique_chosenfactorsfactorization_product)) /\ exists ff_q_fsat_unique_chosenfactorsfactorization_product_terminal. ff_u_fsat_unique_chosenfactorsfactorization_product = ff_q_fsat_unique_chosenfactorsfactorization_product_terminal * S ((S (mv_factor_count_unique_chosenfactors)) * ff_v_fsat_unique_chosenfactorsfactorization_product) + (n))) /\ forall ff_i_fsat_unique_chosenfactorsfactorization_product. (exists ff_lt_fsat_unique_chosenfactorsfactorization_product_bound. ff_lt_fsat_unique_chosenfactorsfactorization_product_bound + S ff_i_fsat_unique_chosenfactorsfactorization_product = mv_factor_count_unique_chosenfactors) -> exists ff_p_fsat_unique_chosenfactorsfactorization_product ff_r_fsat_unique_chosenfactorsfactorization_product ff_s_fsat_unique_chosenfactorsfactorization_product. ((((exists ff_h_fsat_unique_chosenfactorsfactorization_product_factor. ff_h_fsat_unique_chosenfactorsfactorization_product_factor + S (ff_p_fsat_unique_chosenfactorsfactorization_product) = S ((S (ff_i_fsat_unique_chosenfactorsfactorization_product)) * mv_factor_scale_unique_chosenfactors)) /\ exists ff_q_fsat_unique_chosenfactorsfactorization_product_factor. mv_factor_code_unique_chosenfactors = ff_q_fsat_unique_chosenfactorsfactorization_product_factor * S ((S (ff_i_fsat_unique_chosenfactorsfactorization_product)) * mv_factor_scale_unique_chosenfactors) + (ff_p_fsat_unique_chosenfactorsfactorization_product))) /\ ((((exists ff_h_fsat_unique_chosenfactorsfactorization_product_partial. ff_h_fsat_unique_chosenfactorsfactorization_product_partial + S (ff_r_fsat_unique_chosenfactorsfactorization_product) = S ((S (ff_i_fsat_unique_chosenfactorsfactorization_product)) * ff_v_fsat_unique_chosenfactorsfactorization_product)) /\ exists ff_q_fsat_unique_chosenfactorsfactorization_product_partial. ff_u_fsat_unique_chosenfactorsfactorization_product = ff_q_fsat_unique_chosenfactorsfactorization_product_partial * S ((S (ff_i_fsat_unique_chosenfactorsfactorization_product)) * ff_v_fsat_unique_chosenfactorsfactorization_product) + (ff_r_fsat_unique_chosenfactorsfactorization_product))) /\ ((((exists ff_h_fsat_unique_chosenfactorsfactorization_product_successor. ff_h_fsat_unique_chosenfactorsfactorization_product_successor + S (ff_s_fsat_unique_chosenfactorsfactorization_product) = S ((S (S ff_i_fsat_unique_chosenfactorsfactorization_product)) * ff_v_fsat_unique_chosenfactorsfactorization_product)) /\ exists ff_q_fsat_unique_chosenfactorsfactorization_product_successor. ff_u_fsat_unique_chosenfactorsfactorization_product = ff_q_fsat_unique_chosenfactorsfactorization_product_successor * S ((S (S ff_i_fsat_unique_chosenfactorsfactorization_product)) * ff_v_fsat_unique_chosenfactorsfactorization_product) + (ff_s_fsat_unique_chosenfactorsfactorization_product))) /\ ff_s_fsat_unique_chosenfactorsfactorization_product = ff_r_fsat_unique_chosenfactorsfactorization_product * ff_p_fsat_unique_chosenfactorsfactorization_product)))))) /\ (forall ftsf_index_fsat_unique_chosenfactorsfactorization_primes. (exists ftsf_gap_fsat_unique_chosenfactorsfactorization_primes_bound. ftsf_gap_fsat_unique_chosenfactorsfactorization_primes_bound + S ftsf_index_fsat_unique_chosenfactorsfactorization_primes = (mv_factor_count_unique_chosenfactors)) -> exists ftsf_factor_fsat_unique_chosenfactorsfactorization_primes. ((((exists ff_h_ftsf_fsat_unique_chosenfactorsfactorization_primes_entry. ff_h_ftsf_fsat_unique_chosenfactorsfactorization_primes_entry + S (ftsf_factor_fsat_unique_chosenfactorsfactorization_primes) = S ((S (ftsf_index_fsat_unique_chosenfactorsfactorization_primes)) * mv_factor_scale_unique_chosenfactors)) /\ exists ff_q_ftsf_fsat_unique_chosenfactorsfactorization_primes_entry. mv_factor_code_unique_chosenfactors = ff_q_ftsf_fsat_unique_chosenfactorsfactorization_primes_entry * S ((S (ftsf_index_fsat_unique_chosenfactorsfactorization_primes)) * mv_factor_scale_unique_chosenfactors) + (ftsf_factor_fsat_unique_chosenfactorsfactorization_primes))) /\ ((~(ftsf_factor_fsat_unique_chosenfactorsfactorization_primes = 1) /\ forall frm_prime_left_ftsf_fsat_unique_chosenfactorsfactorization_primes_prime frm_prime_right_ftsf_fsat_unique_chosenfactorsfactorization_primes_prime. ftsf_factor_fsat_unique_chosenfactorsfactorization_primes = frm_prime_left_ftsf_fsat_unique_chosenfactorsfactorization_primes_prime * frm_prime_right_ftsf_fsat_unique_chosenfactorsfactorization_primes_prime -> frm_prime_left_ftsf_fsat_unique_chosenfactorsfactorization_primes_prime = 1 \/ frm_prime_right_ftsf_fsat_unique_chosenfactorsfactorization_primes_prime = 1))))))) /\ ((((exists mv_even_half_unique_chosenfactorsparityeven. (mv_factor_count_unique_chosenfactors) = 2 * mv_even_half_unique_chosenfactorsparityeven) /\ ((z) = 2))) \/ (((exists mv_odd_half_unique_chosenfactorsparityodd. (mv_factor_count_unique_chosenfactors) = 2 * mv_odd_half_unique_chosenfactorsparityodd + 1) /\ ((z) = 1))))))))))) /\ forall w. (((~((n) = 0)) /\ ((((exists mv_square_prime_unique_othersquare. ((~((mv_square_prime_unique_othersquare) = 1) /\ forall pvs_left_unique_othersquareprime pvs_right_unique_othersquareprime. (mv_square_prime_unique_othersquare) = pvs_left_unique_othersquareprime * pvs_right_unique_othersquareprime -> pvs_left_unique_othersquareprime = 1 \/ pvs_right_unique_othersquareprime = 1) /\ (exists pvs_factor_unique_othersquaredivisor. (n) = (mv_square_prime_unique_othersquare * mv_square_prime_unique_othersquare) * pvs_factor_unique_othersquaredivisor))) /\ ((w) = 0))) \/ (((((~((n) = 0)) /\ (forall sfd_prime_unique_othersquarefree. (~((sfd_prime_unique_othersquarefree) = 1) /\ forall pvs_left_unique_othersquarefreedomain pvs_right_unique_othersquarefreedomain. (sfd_prime_unique_othersquarefree) = pvs_left_unique_othersquarefreedomain * pvs_right_unique_othersquarefreedomain -> pvs_left_unique_othersquarefreedomain = 1 \/ pvs_right_unique_othersquarefreedomain = 1) -> (exists pvs_le_gap_unique_othersquarefreebound. pvs_le_gap_unique_othersquarefreebound + (sfd_prime_unique_othersquarefree) = (n)) -> ~(exists pvs_factor_unique_othersquarefreesquare. (n) = (sfd_prime_unique_othersquarefree * sfd_prime_unique_othersquarefree) * pvs_factor_unique_othersquarefreesquare)))) /\ (exists mv_factor_code_unique_otherfactors mv_factor_scale_unique_otherfactors mv_factor_count_unique_otherfactors. (((~(n = 0) /\ ((exists ff_u_fsat_unique_otherfactorsfactorization_product ff_v_fsat_unique_otherfactorsfactorization_product. ((((exists ff_h_fsat_unique_otherfactorsfactorization_product_start. ff_h_fsat_unique_otherfactorsfactorization_product_start + S (1) = S ((S (0)) * ff_v_fsat_unique_otherfactorsfactorization_product)) /\ exists ff_q_fsat_unique_otherfactorsfactorization_product_start. ff_u_fsat_unique_otherfactorsfactorization_product = ff_q_fsat_unique_otherfactorsfactorization_product_start * S ((S (0)) * ff_v_fsat_unique_otherfactorsfactorization_product) + (1))) /\ ((((exists ff_h_fsat_unique_otherfactorsfactorization_product_terminal. ff_h_fsat_unique_otherfactorsfactorization_product_terminal + S (n) = S ((S (mv_factor_count_unique_otherfactors)) * ff_v_fsat_unique_otherfactorsfactorization_product)) /\ exists ff_q_fsat_unique_otherfactorsfactorization_product_terminal. ff_u_fsat_unique_otherfactorsfactorization_product = ff_q_fsat_unique_otherfactorsfactorization_product_terminal * S ((S (mv_factor_count_unique_otherfactors)) * ff_v_fsat_unique_otherfactorsfactorization_product) + (n))) /\ forall ff_i_fsat_unique_otherfactorsfactorization_product. (exists ff_lt_fsat_unique_otherfactorsfactorization_product_bound. ff_lt_fsat_unique_otherfactorsfactorization_product_bound + S ff_i_fsat_unique_otherfactorsfactorization_product = mv_factor_count_unique_otherfactors) -> exists ff_p_fsat_unique_otherfactorsfactorization_product ff_r_fsat_unique_otherfactorsfactorization_product ff_s_fsat_unique_otherfactorsfactorization_product. ((((exists ff_h_fsat_unique_otherfactorsfactorization_product_factor. ff_h_fsat_unique_otherfactorsfactorization_product_factor + S (ff_p_fsat_unique_otherfactorsfactorization_product) = S ((S (ff_i_fsat_unique_otherfactorsfactorization_product)) * mv_factor_scale_unique_otherfactors)) /\ exists ff_q_fsat_unique_otherfactorsfactorization_product_factor. mv_factor_code_unique_otherfactors = ff_q_fsat_unique_otherfactorsfactorization_product_factor * S ((S (ff_i_fsat_unique_otherfactorsfactorization_product)) * mv_factor_scale_unique_otherfactors) + (ff_p_fsat_unique_otherfactorsfactorization_product))) /\ ((((exists ff_h_fsat_unique_otherfactorsfactorization_product_partial. ff_h_fsat_unique_otherfactorsfactorization_product_partial + S (ff_r_fsat_unique_otherfactorsfactorization_product) = S ((S (ff_i_fsat_unique_otherfactorsfactorization_product)) * ff_v_fsat_unique_otherfactorsfactorization_product)) /\ exists ff_q_fsat_unique_otherfactorsfactorization_product_partial. ff_u_fsat_unique_otherfactorsfactorization_product = ff_q_fsat_unique_otherfactorsfactorization_product_partial * S ((S (ff_i_fsat_unique_otherfactorsfactorization_product)) * ff_v_fsat_unique_otherfactorsfactorization_product) + (ff_r_fsat_unique_otherfactorsfactorization_product))) /\ ((((exists ff_h_fsat_unique_otherfactorsfactorization_product_successor. ff_h_fsat_unique_otherfactorsfactorization_product_successor + S (ff_s_fsat_unique_otherfactorsfactorization_product) = S ((S (S ff_i_fsat_unique_otherfactorsfactorization_product)) * ff_v_fsat_unique_otherfactorsfactorization_product)) /\ exists ff_q_fsat_unique_otherfactorsfactorization_product_successor. ff_u_fsat_unique_otherfactorsfactorization_product = ff_q_fsat_unique_otherfactorsfactorization_product_successor * S ((S (S ff_i_fsat_unique_otherfactorsfactorization_product)) * ff_v_fsat_unique_otherfactorsfactorization_product) + (ff_s_fsat_unique_otherfactorsfactorization_product))) /\ ff_s_fsat_unique_otherfactorsfactorization_product = ff_r_fsat_unique_otherfactorsfactorization_product * ff_p_fsat_unique_otherfactorsfactorization_product)))))) /\ (forall ftsf_index_fsat_unique_otherfactorsfactorization_primes. (exists ftsf_gap_fsat_unique_otherfactorsfactorization_primes_bound. ftsf_gap_fsat_unique_otherfactorsfactorization_primes_bound + S ftsf_index_fsat_unique_otherfactorsfactorization_primes = (mv_factor_count_unique_otherfactors)) -> exists ftsf_factor_fsat_unique_otherfactorsfactorization_primes. ((((exists ff_h_ftsf_fsat_unique_otherfactorsfactorization_primes_entry. ff_h_ftsf_fsat_unique_otherfactorsfactorization_primes_entry + S (ftsf_factor_fsat_unique_otherfactorsfactorization_primes) = S ((S (ftsf_index_fsat_unique_otherfactorsfactorization_primes)) * mv_factor_scale_unique_otherfactors)) /\ exists ff_q_ftsf_fsat_unique_otherfactorsfactorization_primes_entry. mv_factor_code_unique_otherfactors = ff_q_ftsf_fsat_unique_otherfactorsfactorization_primes_entry * S ((S (ftsf_index_fsat_unique_otherfactorsfactorization_primes)) * mv_factor_scale_unique_otherfactors) + (ftsf_factor_fsat_unique_otherfactorsfactorization_primes))) /\ ((~(ftsf_factor_fsat_unique_otherfactorsfactorization_primes = 1) /\ forall frm_prime_left_ftsf_fsat_unique_otherfactorsfactorization_primes_prime frm_prime_right_ftsf_fsat_unique_otherfactorsfactorization_primes_prime. ftsf_factor_fsat_unique_otherfactorsfactorization_primes = frm_prime_left_ftsf_fsat_unique_otherfactorsfactorization_primes_prime * frm_prime_right_ftsf_fsat_unique_otherfactorsfactorization_primes_prime -> frm_prime_left_ftsf_fsat_unique_otherfactorsfactorization_primes_prime = 1 \/ frm_prime_right_ftsf_fsat_unique_otherfactorsfactorization_primes_prime = 1))))))) /\ ((((exists mv_even_half_unique_otherfactorsparityeven. (mv_factor_count_unique_otherfactors) = 2 * mv_even_half_unique_otherfactorsparityeven) /\ ((w) = 2))) \/ (((exists mv_odd_half_unique_otherfactorsparityodd. (mv_factor_count_unique_otherfactors) = 2 * mv_odd_half_unique_otherfactorsparityodd + 1) /\ ((w) = 1))))))))))) -> w = z

Complete tactic proof in conservative notation

All 18 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

18 script commands · 8 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 (2)
mobius value existsmobius value functional
01Fix variables and assumptionsL1–2

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

  1. L1
    intro n
  2. L2
    intro hn
02Establish hzL3–6

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

  1. L3
    have hz : ∃ z. Mobius(n,z)Definitions: Mobius(n,z)Original native command in the exact edition
  2. L4
    specialize mobius_value_exists (n)
  3. L5
    apply mobius_value_exists
  4. L6
    exact hn
03Separate the logical casesL7–7

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

  1. L7
    cases hz
04Construct an explicit witnessL8–8

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

  1. L8
    exists x
05Separate the logical casesL9–9

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

  1. L9
    split
06Use earlier factsL10–10

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

  1. L10
    exact hz_witness
07Fix variables and assumptionsL11–12

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

  1. L11
    intro w
  2. L12
    intro hw
08Use earlier factsL13–18

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

  1. L13
    specialize mobius_value_functional (n)
  2. L14
    specialize mobius_value_functional (w)
  3. L15
    specialize mobius_value_functional (x)
  4. L16
    apply mobius_value_functional
  5. L17
    exact hw
  6. L18
    exact hz_witness

Library-wide reading audit

Original defined command ledger · 18 lines
  1. 0001intro n
  2. 0002intro hn
  3. 0003have hz : ∃ z. Mobius(n,z)
  4. 0004specialize mobius_value_exists (n)
  5. 0005apply mobius_value_exists
  6. 0006exact hn
  7. 0007cases hz
  8. 0008exists x
  9. 0009split
  10. 0010exact hz_witness
  11. 0011intro w
  12. 0012intro hw
  13. 0013specialize mobius_value_functional (n)
  14. 0014specialize mobius_value_functional (w)
  15. 0015specialize mobius_value_functional (x)
  16. 0016apply mobius_value_functional
  17. 0017exact hw
  18. 0018exact hz_witness