MV000C

mobius_value_exists_unique

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.

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

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

Exact expanded first-order arithmetic 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

Constructive proof overview

Generated structural guide

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

The unchanged tactic script uses 2 declared prerequisites and contains 18 exact native proof lines.

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

Proof neighborhood

Direct dependencies

MV0009 mobius_value_exists MV000B mobius_value_functional

Direct dependents

none

Formal native tactic body

Dependencies are introduced as named hypotheses before line 1. Local theorem links identify exact declared prerequisites. The literal dependency-closed bundle is checked by original HA and the independently compiled Lean verifier. Public delivery grants no Alpha checked-use authority or Stable membership.

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.

Named ingredients (2)
mobius value existsmobius value functional

Long local formulas use this family’s existing definitions. Each new abbreviation was expanded back to the identical native formula, including its free-variable context. The original edition is preserved below.

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
  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 exact command ledger · 18 lines
  1. 0001intro n
  2. 0002intro hn
  3. 0003have hz : exists z. (((~((n) = 0)) /\ ((((exists mv_square_prime_unique_existssquare. ((~((mv_square_prime_unique_existssquare) = 1) /\ forall pvs_left_unique_existssquareprime pvs_right_unique_existssquareprime. (mv_square_prime_unique_existssquare) = pvs_left_unique_existssquareprime * pvs_right_unique_existssquareprime -> pvs_left_unique_existssquareprime = 1 \/ pvs_right_unique_existssquareprime = 1) /\ (exists pvs_factor_unique_existssquaredivisor. (n) = (mv_square_prime_unique_existssquare * mv_square_prime_unique_existssquare) * pvs_factor_unique_existssquaredivisor))) /\ ((z) = 0))) \/ (((((~((n) = 0)) /\ (forall sfd_prime_unique_existssquarefree. (~((sfd_prime_unique_existssquarefree) = 1) /\ forall pvs_left_unique_existssquarefreedomain pvs_right_unique_existssquarefreedomain. (sfd_prime_unique_existssquarefree) = pvs_left_unique_existssquarefreedomain * pvs_right_unique_existssquarefreedomain -> pvs_left_unique_existssquarefreedomain = 1 \/ pvs_right_unique_existssquarefreedomain = 1) -> (exists pvs_le_gap_unique_existssquarefreebound. pvs_le_gap_unique_existssquarefreebound + (sfd_prime_unique_existssquarefree) = (n)) -> ~(exists pvs_factor_unique_existssquarefreesquare. (n) = (sfd_prime_unique_existssquarefree * sfd_prime_unique_existssquarefree) * pvs_factor_unique_existssquarefreesquare)))) /\ (exists mv_factor_code_unique_existsfactors mv_factor_scale_unique_existsfactors mv_factor_count_unique_existsfactors. (((~(n = 0) /\ ((exists ff_u_fsat_unique_existsfactorsfactorization_product ff_v_fsat_unique_existsfactorsfactorization_product. ((((exists ff_h_fsat_unique_existsfactorsfactorization_product_start. ff_h_fsat_unique_existsfactorsfactorization_product_start + S (1) = S ((S (0)) * ff_v_fsat_unique_existsfactorsfactorization_product)) /\ exists ff_q_fsat_unique_existsfactorsfactorization_product_start. ff_u_fsat_unique_existsfactorsfactorization_product = ff_q_fsat_unique_existsfactorsfactorization_product_start * S ((S (0)) * ff_v_fsat_unique_existsfactorsfactorization_product) + (1))) /\ ((((exists ff_h_fsat_unique_existsfactorsfactorization_product_terminal. ff_h_fsat_unique_existsfactorsfactorization_product_terminal + S (n) = S ((S (mv_factor_count_unique_existsfactors)) * ff_v_fsat_unique_existsfactorsfactorization_product)) /\ exists ff_q_fsat_unique_existsfactorsfactorization_product_terminal. ff_u_fsat_unique_existsfactorsfactorization_product = ff_q_fsat_unique_existsfactorsfactorization_product_terminal * S ((S (mv_factor_count_unique_existsfactors)) * ff_v_fsat_unique_existsfactorsfactorization_product) + (n))) /\ forall ff_i_fsat_unique_existsfactorsfactorization_product. (exists ff_lt_fsat_unique_existsfactorsfactorization_product_bound. ff_lt_fsat_unique_existsfactorsfactorization_product_bound + S ff_i_fsat_unique_existsfactorsfactorization_product = mv_factor_count_unique_existsfactors) -> exists ff_p_fsat_unique_existsfactorsfactorization_product ff_r_fsat_unique_existsfactorsfactorization_product ff_s_fsat_unique_existsfactorsfactorization_product. ((((exists ff_h_fsat_unique_existsfactorsfactorization_product_factor. ff_h_fsat_unique_existsfactorsfactorization_product_factor + S (ff_p_fsat_unique_existsfactorsfactorization_product) = S ((S (ff_i_fsat_unique_existsfactorsfactorization_product)) * mv_factor_scale_unique_existsfactors)) /\ exists ff_q_fsat_unique_existsfactorsfactorization_product_factor. mv_factor_code_unique_existsfactors = ff_q_fsat_unique_existsfactorsfactorization_product_factor * S ((S (ff_i_fsat_unique_existsfactorsfactorization_product)) * mv_factor_scale_unique_existsfactors) + (ff_p_fsat_unique_existsfactorsfactorization_product))) /\ ((((exists ff_h_fsat_unique_existsfactorsfactorization_product_partial. ff_h_fsat_unique_existsfactorsfactorization_product_partial + S (ff_r_fsat_unique_existsfactorsfactorization_product) = S ((S (ff_i_fsat_unique_existsfactorsfactorization_product)) * ff_v_fsat_unique_existsfactorsfactorization_product)) /\ exists ff_q_fsat_unique_existsfactorsfactorization_product_partial. ff_u_fsat_unique_existsfactorsfactorization_product = ff_q_fsat_unique_existsfactorsfactorization_product_partial * S ((S (ff_i_fsat_unique_existsfactorsfactorization_product)) * ff_v_fsat_unique_existsfactorsfactorization_product) + (ff_r_fsat_unique_existsfactorsfactorization_product))) /\ ((((exists ff_h_fsat_unique_existsfactorsfactorization_product_successor. ff_h_fsat_unique_existsfactorsfactorization_product_successor + S (ff_s_fsat_unique_existsfactorsfactorization_product) = S ((S (S ff_i_fsat_unique_existsfactorsfactorization_product)) * ff_v_fsat_unique_existsfactorsfactorization_product)) /\ exists ff_q_fsat_unique_existsfactorsfactorization_product_successor. ff_u_fsat_unique_existsfactorsfactorization_product = ff_q_fsat_unique_existsfactorsfactorization_product_successor * S ((S (S ff_i_fsat_unique_existsfactorsfactorization_product)) * ff_v_fsat_unique_existsfactorsfactorization_product) + (ff_s_fsat_unique_existsfactorsfactorization_product))) /\ ff_s_fsat_unique_existsfactorsfactorization_product = ff_r_fsat_unique_existsfactorsfactorization_product * ff_p_fsat_unique_existsfactorsfactorization_product)))))) /\ (forall ftsf_index_fsat_unique_existsfactorsfactorization_primes. (exists ftsf_gap_fsat_unique_existsfactorsfactorization_primes_bound. ftsf_gap_fsat_unique_existsfactorsfactorization_primes_bound + S ftsf_index_fsat_unique_existsfactorsfactorization_primes = (mv_factor_count_unique_existsfactors)) -> exists ftsf_factor_fsat_unique_existsfactorsfactorization_primes. ((((exists ff_h_ftsf_fsat_unique_existsfactorsfactorization_primes_entry. ff_h_ftsf_fsat_unique_existsfactorsfactorization_primes_entry + S (ftsf_factor_fsat_unique_existsfactorsfactorization_primes) = S ((S (ftsf_index_fsat_unique_existsfactorsfactorization_primes)) * mv_factor_scale_unique_existsfactors)) /\ exists ff_q_ftsf_fsat_unique_existsfactorsfactorization_primes_entry. mv_factor_code_unique_existsfactors = ff_q_ftsf_fsat_unique_existsfactorsfactorization_primes_entry * S ((S (ftsf_index_fsat_unique_existsfactorsfactorization_primes)) * mv_factor_scale_unique_existsfactors) + (ftsf_factor_fsat_unique_existsfactorsfactorization_primes))) /\ ((~(ftsf_factor_fsat_unique_existsfactorsfactorization_primes = 1) /\ forall frm_prime_left_ftsf_fsat_unique_existsfactorsfactorization_primes_prime frm_prime_right_ftsf_fsat_unique_existsfactorsfactorization_primes_prime. ftsf_factor_fsat_unique_existsfactorsfactorization_primes = frm_prime_left_ftsf_fsat_unique_existsfactorsfactorization_primes_prime * frm_prime_right_ftsf_fsat_unique_existsfactorsfactorization_primes_prime -> frm_prime_left_ftsf_fsat_unique_existsfactorsfactorization_primes_prime = 1 \/ frm_prime_right_ftsf_fsat_unique_existsfactorsfactorization_primes_prime = 1))))))) /\ ((((exists mv_even_half_unique_existsfactorsparityeven. (mv_factor_count_unique_existsfactors) = 2 * mv_even_half_unique_existsfactorsparityeven) /\ ((z) = 2))) \/ (((exists mv_odd_half_unique_existsfactorsparityodd. (mv_factor_count_unique_existsfactors) = 2 * mv_odd_half_unique_existsfactorsparityodd + 1) /\ ((z) = 1)))))))))))
  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