MV000D

mobius_one

Alpha v34 independently verified · alpha_closed; checked-use authorized; not Stable

The unit boundary has Möbius value positive one (signed code two), proved from its actual empty prime factorization.

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 expanded first-order arithmetic statement

((~((1) = 0)) /\ ((((exists mv_square_prime_one_valuesquare. ((~((mv_square_prime_one_valuesquare) = 1) /\ forall pvs_left_one_valuesquareprime pvs_right_one_valuesquareprime. (mv_square_prime_one_valuesquare) = pvs_left_one_valuesquareprime * pvs_right_one_valuesquareprime -> pvs_left_one_valuesquareprime = 1 \/ pvs_right_one_valuesquareprime = 1) /\ (exists pvs_factor_one_valuesquaredivisor. (1) = (mv_square_prime_one_valuesquare * mv_square_prime_one_valuesquare) * pvs_factor_one_valuesquaredivisor))) /\ ((2) = 0))) \/ (((((~((1) = 0)) /\ (forall sfd_prime_one_valuesquarefree. (~((sfd_prime_one_valuesquarefree) = 1) /\ forall pvs_left_one_valuesquarefreedomain pvs_right_one_valuesquarefreedomain. (sfd_prime_one_valuesquarefree) = pvs_left_one_valuesquarefreedomain * pvs_right_one_valuesquarefreedomain -> pvs_left_one_valuesquarefreedomain = 1 \/ pvs_right_one_valuesquarefreedomain = 1) -> (exists pvs_le_gap_one_valuesquarefreebound. pvs_le_gap_one_valuesquarefreebound + (sfd_prime_one_valuesquarefree) = (1)) -> ~(exists pvs_factor_one_valuesquarefreesquare. (1) = (sfd_prime_one_valuesquarefree * sfd_prime_one_valuesquarefree) * pvs_factor_one_valuesquarefreesquare)))) /\ (exists mv_factor_code_one_valuefactors mv_factor_scale_one_valuefactors mv_factor_count_one_valuefactors. (((~(1 = 0) /\ ((exists ff_u_fsat_one_valuefactorsfactorization_product ff_v_fsat_one_valuefactorsfactorization_product. ((((exists ff_h_fsat_one_valuefactorsfactorization_product_start. ff_h_fsat_one_valuefactorsfactorization_product_start + S (1) = S ((S (0)) * ff_v_fsat_one_valuefactorsfactorization_product)) /\ exists ff_q_fsat_one_valuefactorsfactorization_product_start. ff_u_fsat_one_valuefactorsfactorization_product = ff_q_fsat_one_valuefactorsfactorization_product_start * S ((S (0)) * ff_v_fsat_one_valuefactorsfactorization_product) + (1))) /\ ((((exists ff_h_fsat_one_valuefactorsfactorization_product_terminal. ff_h_fsat_one_valuefactorsfactorization_product_terminal + S (1) = S ((S (mv_factor_count_one_valuefactors)) * ff_v_fsat_one_valuefactorsfactorization_product)) /\ exists ff_q_fsat_one_valuefactorsfactorization_product_terminal. ff_u_fsat_one_valuefactorsfactorization_product = ff_q_fsat_one_valuefactorsfactorization_product_terminal * S ((S (mv_factor_count_one_valuefactors)) * ff_v_fsat_one_valuefactorsfactorization_product) + (1))) /\ forall ff_i_fsat_one_valuefactorsfactorization_product. (exists ff_lt_fsat_one_valuefactorsfactorization_product_bound. ff_lt_fsat_one_valuefactorsfactorization_product_bound + S ff_i_fsat_one_valuefactorsfactorization_product = mv_factor_count_one_valuefactors) -> exists ff_p_fsat_one_valuefactorsfactorization_product ff_r_fsat_one_valuefactorsfactorization_product ff_s_fsat_one_valuefactorsfactorization_product. ((((exists ff_h_fsat_one_valuefactorsfactorization_product_factor. ff_h_fsat_one_valuefactorsfactorization_product_factor + S (ff_p_fsat_one_valuefactorsfactorization_product) = S ((S (ff_i_fsat_one_valuefactorsfactorization_product)) * mv_factor_scale_one_valuefactors)) /\ exists ff_q_fsat_one_valuefactorsfactorization_product_factor. mv_factor_code_one_valuefactors = ff_q_fsat_one_valuefactorsfactorization_product_factor * S ((S (ff_i_fsat_one_valuefactorsfactorization_product)) * mv_factor_scale_one_valuefactors) + (ff_p_fsat_one_valuefactorsfactorization_product))) /\ ((((exists ff_h_fsat_one_valuefactorsfactorization_product_partial. ff_h_fsat_one_valuefactorsfactorization_product_partial + S (ff_r_fsat_one_valuefactorsfactorization_product) = S ((S (ff_i_fsat_one_valuefactorsfactorization_product)) * ff_v_fsat_one_valuefactorsfactorization_product)) /\ exists ff_q_fsat_one_valuefactorsfactorization_product_partial. ff_u_fsat_one_valuefactorsfactorization_product = ff_q_fsat_one_valuefactorsfactorization_product_partial * S ((S (ff_i_fsat_one_valuefactorsfactorization_product)) * ff_v_fsat_one_valuefactorsfactorization_product) + (ff_r_fsat_one_valuefactorsfactorization_product))) /\ ((((exists ff_h_fsat_one_valuefactorsfactorization_product_successor. ff_h_fsat_one_valuefactorsfactorization_product_successor + S (ff_s_fsat_one_valuefactorsfactorization_product) = S ((S (S ff_i_fsat_one_valuefactorsfactorization_product)) * ff_v_fsat_one_valuefactorsfactorization_product)) /\ exists ff_q_fsat_one_valuefactorsfactorization_product_successor. ff_u_fsat_one_valuefactorsfactorization_product = ff_q_fsat_one_valuefactorsfactorization_product_successor * S ((S (S ff_i_fsat_one_valuefactorsfactorization_product)) * ff_v_fsat_one_valuefactorsfactorization_product) + (ff_s_fsat_one_valuefactorsfactorization_product))) /\ ff_s_fsat_one_valuefactorsfactorization_product = ff_r_fsat_one_valuefactorsfactorization_product * ff_p_fsat_one_valuefactorsfactorization_product)))))) /\ (forall ftsf_index_fsat_one_valuefactorsfactorization_primes. (exists ftsf_gap_fsat_one_valuefactorsfactorization_primes_bound. ftsf_gap_fsat_one_valuefactorsfactorization_primes_bound + S ftsf_index_fsat_one_valuefactorsfactorization_primes = (mv_factor_count_one_valuefactors)) -> exists ftsf_factor_fsat_one_valuefactorsfactorization_primes. ((((exists ff_h_ftsf_fsat_one_valuefactorsfactorization_primes_entry. ff_h_ftsf_fsat_one_valuefactorsfactorization_primes_entry + S (ftsf_factor_fsat_one_valuefactorsfactorization_primes) = S ((S (ftsf_index_fsat_one_valuefactorsfactorization_primes)) * mv_factor_scale_one_valuefactors)) /\ exists ff_q_ftsf_fsat_one_valuefactorsfactorization_primes_entry. mv_factor_code_one_valuefactors = ff_q_ftsf_fsat_one_valuefactorsfactorization_primes_entry * S ((S (ftsf_index_fsat_one_valuefactorsfactorization_primes)) * mv_factor_scale_one_valuefactors) + (ftsf_factor_fsat_one_valuefactorsfactorization_primes))) /\ ((~(ftsf_factor_fsat_one_valuefactorsfactorization_primes = 1) /\ forall frm_prime_left_ftsf_fsat_one_valuefactorsfactorization_primes_prime frm_prime_right_ftsf_fsat_one_valuefactorsfactorization_primes_prime. ftsf_factor_fsat_one_valuefactorsfactorization_primes = frm_prime_left_ftsf_fsat_one_valuefactorsfactorization_primes_prime * frm_prime_right_ftsf_fsat_one_valuefactorsfactorization_primes_prime -> frm_prime_left_ftsf_fsat_one_valuefactorsfactorization_primes_prime = 1 \/ frm_prime_right_ftsf_fsat_one_valuefactorsfactorization_primes_prime = 1))))))) /\ ((((exists mv_even_half_one_valuefactorsparityeven. (mv_factor_count_one_valuefactors) = 2 * mv_even_half_one_valuefactorsparityeven) /\ ((2) = 2))) \/ (((exists mv_odd_half_one_valuefactorsparityodd. (mv_factor_count_one_valuefactors) = 2 * mv_odd_half_one_valuefactorsparityodd + 1) /\ ((2) = 1))))))))))

Constructive proof overview

Generated structural guide

The unit boundary has Möbius value positive one (signed code two), proved from its actual empty prime factorization.

The unchanged tactic script uses 5 declared prerequisites and contains 32 exact native proof lines.

Alpha v34 checked-use · first admitted v31 · independently kernel and Lean verified; not Stable

Proof neighborhood

Direct dependencies

foundation_prime_factor_list_exists Alpha theorem; checked-use authorized factor_permutation_unit_length_zero Alpha theorem; checked-use authorized MV0003 alternating_signed_unit_zero MV0008 mobius_from_squarefree_factor_count squarefree_one Alpha theorem; checked-use authorized

Direct dependents

none

Formal native tactic body

Dependencies are introduced as named hypotheses before line 1. Local theorem links identify exact declared prerequisites. This exact body belongs to a complete independently kernel-checked constructive proof bundle and has Alpha checked-use authority; it does not imply Stable membership.

Read the argument

Proof checkpoints

32 script commands · 6 reading checkpoints · 4 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)
alternating signed unit zeromobius from squarefree factor count

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.

01Establish hnL1–4

Establish this local claim before using it. It is not an additional assumption. The following proof commands apply PA1.

  1. L1
    have hn : ~(1 = 0)
  2. L2
    intro hzero
  3. L3
    apply PA1
  4. L4
    exact hzero
02Establish hfL5–8

Establish this local claim before using it. It is not an additional assumption. The following proof commands apply foundation prime factor list exists.

  1. L5
    have hf : ∃ l. ∃ b. ∃ c. PrimeFactorList(1,b,c,l)Definitions: PrimeFactorList
  2. L6
    specialize foundation_prime_factor_list_exists (1)
  3. L7
    apply foundation_prime_factor_list_exists
  4. L8
    exact hn
03Separate the logical casesL9–11

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

  1. L9
    cases hf
  2. L10
    cases hf_witness
  3. L11
    cases hf_witness_witness
04Establish heqL12–19

Establish this local claim before using it. It is not an additional assumption. The following proof commands apply factor permutation unit length zero.

  1. L12
    have heq : x = 0
  2. L13
    specialize factor_permutation_unit_length_zero (1)
  3. L14
    specialize factor_permutation_unit_length_zero (x1)
  4. L15
    specialize factor_permutation_unit_length_zero (x2)
  5. L16
    specialize factor_permutation_unit_length_zero (x)
  6. L17
    apply factor_permutation_unit_length_zero
  7. L18
    exact hf_witness_witness_witness
  8. L19
    refl
05Establish hsL20–29

Establish this local claim before using it. It is not an additional assumption. The following proof commands apply alternating signed unit zero.

  1. L20
    have hs : (((exists mv_even_half_one_signeven. (x) = 2 * mv_even_half_one_signeven) /\ ((2) = 2))) \/ (((exists mv_odd_half_one_signodd. (x) = 2 * mv_odd_half_one_signodd + 1) /\ ((2) = 1)))
  2. L21
    rewrite heq
  3. L22
    rewrite heq
  4. L23
    apply alternating_signed_unit_zero
  5. L24
    specialize mobius_from_squarefree_factor_count (1)
  6. L25
    specialize mobius_from_squarefree_factor_count (x1)
  7. L26
    specialize mobius_from_squarefree_factor_count (x2)
  8. L27
    specialize mobius_from_squarefree_factor_count (x)
  9. L28
    specialize mobius_from_squarefree_factor_count (2)
  10. L29
    apply mobius_from_squarefree_factor_count
06Use earlier factsL30–32

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

  1. L30
    apply squarefree_one
  2. L31
    exact hf_witness_witness_witness
  3. L32
    exact hs

Library-wide reading audit

Original exact command ledger · 32 lines
  1. 0001have hn : ~(1 = 0)
  2. 0002intro hzero
  3. 0003apply PA1
  4. 0004exact hzero
  5. 0005have hf : exists l b c. ((~(1 = 0) /\ ((exists ff_u_fsat_mv_one_factors_product ff_v_fsat_mv_one_factors_product. ((((exists ff_h_fsat_mv_one_factors_product_start. ff_h_fsat_mv_one_factors_product_start + S (1) = S ((S (0)) * ff_v_fsat_mv_one_factors_product)) /\ exists ff_q_fsat_mv_one_factors_product_start. ff_u_fsat_mv_one_factors_product = ff_q_fsat_mv_one_factors_product_start * S ((S (0)) * ff_v_fsat_mv_one_factors_product) + (1))) /\ ((((exists ff_h_fsat_mv_one_factors_product_terminal. ff_h_fsat_mv_one_factors_product_terminal + S (1) = S ((S (l)) * ff_v_fsat_mv_one_factors_product)) /\ exists ff_q_fsat_mv_one_factors_product_terminal. ff_u_fsat_mv_one_factors_product = ff_q_fsat_mv_one_factors_product_terminal * S ((S (l)) * ff_v_fsat_mv_one_factors_product) + (1))) /\ forall ff_i_fsat_mv_one_factors_product. (exists ff_lt_fsat_mv_one_factors_product_bound. ff_lt_fsat_mv_one_factors_product_bound + S ff_i_fsat_mv_one_factors_product = l) -> exists ff_p_fsat_mv_one_factors_product ff_r_fsat_mv_one_factors_product ff_s_fsat_mv_one_factors_product. ((((exists ff_h_fsat_mv_one_factors_product_factor. ff_h_fsat_mv_one_factors_product_factor + S (ff_p_fsat_mv_one_factors_product) = S ((S (ff_i_fsat_mv_one_factors_product)) * c)) /\ exists ff_q_fsat_mv_one_factors_product_factor. b = ff_q_fsat_mv_one_factors_product_factor * S ((S (ff_i_fsat_mv_one_factors_product)) * c) + (ff_p_fsat_mv_one_factors_product))) /\ ((((exists ff_h_fsat_mv_one_factors_product_partial. ff_h_fsat_mv_one_factors_product_partial + S (ff_r_fsat_mv_one_factors_product) = S ((S (ff_i_fsat_mv_one_factors_product)) * ff_v_fsat_mv_one_factors_product)) /\ exists ff_q_fsat_mv_one_factors_product_partial. ff_u_fsat_mv_one_factors_product = ff_q_fsat_mv_one_factors_product_partial * S ((S (ff_i_fsat_mv_one_factors_product)) * ff_v_fsat_mv_one_factors_product) + (ff_r_fsat_mv_one_factors_product))) /\ ((((exists ff_h_fsat_mv_one_factors_product_successor. ff_h_fsat_mv_one_factors_product_successor + S (ff_s_fsat_mv_one_factors_product) = S ((S (S ff_i_fsat_mv_one_factors_product)) * ff_v_fsat_mv_one_factors_product)) /\ exists ff_q_fsat_mv_one_factors_product_successor. ff_u_fsat_mv_one_factors_product = ff_q_fsat_mv_one_factors_product_successor * S ((S (S ff_i_fsat_mv_one_factors_product)) * ff_v_fsat_mv_one_factors_product) + (ff_s_fsat_mv_one_factors_product))) /\ ff_s_fsat_mv_one_factors_product = ff_r_fsat_mv_one_factors_product * ff_p_fsat_mv_one_factors_product)))))) /\ (forall ftsf_index_fsat_mv_one_factors_primes. (exists ftsf_gap_fsat_mv_one_factors_primes_bound. ftsf_gap_fsat_mv_one_factors_primes_bound + S ftsf_index_fsat_mv_one_factors_primes = (l)) -> exists ftsf_factor_fsat_mv_one_factors_primes. ((((exists ff_h_ftsf_fsat_mv_one_factors_primes_entry. ff_h_ftsf_fsat_mv_one_factors_primes_entry + S (ftsf_factor_fsat_mv_one_factors_primes) = S ((S (ftsf_index_fsat_mv_one_factors_primes)) * c)) /\ exists ff_q_ftsf_fsat_mv_one_factors_primes_entry. b = ff_q_ftsf_fsat_mv_one_factors_primes_entry * S ((S (ftsf_index_fsat_mv_one_factors_primes)) * c) + (ftsf_factor_fsat_mv_one_factors_primes))) /\ ((~(ftsf_factor_fsat_mv_one_factors_primes = 1) /\ forall frm_prime_left_ftsf_fsat_mv_one_factors_primes_prime frm_prime_right_ftsf_fsat_mv_one_factors_primes_prime. ftsf_factor_fsat_mv_one_factors_primes = frm_prime_left_ftsf_fsat_mv_one_factors_primes_prime * frm_prime_right_ftsf_fsat_mv_one_factors_primes_prime -> frm_prime_left_ftsf_fsat_mv_one_factors_primes_prime = 1 \/ frm_prime_right_ftsf_fsat_mv_one_factors_primes_prime = 1)))))))
  6. 0006specialize foundation_prime_factor_list_exists (1)
  7. 0007apply foundation_prime_factor_list_exists
  8. 0008exact hn
  9. 0009cases hf
  10. 0010cases hf_witness
  11. 0011cases hf_witness_witness
  12. 0012have heq : x = 0
  13. 0013specialize factor_permutation_unit_length_zero (1)
  14. 0014specialize factor_permutation_unit_length_zero (x1)
  15. 0015specialize factor_permutation_unit_length_zero (x2)
  16. 0016specialize factor_permutation_unit_length_zero (x)
  17. 0017apply factor_permutation_unit_length_zero
  18. 0018exact hf_witness_witness_witness
  19. 0019refl
  20. 0020have hs : (((exists mv_even_half_one_signeven. (x) = 2 * mv_even_half_one_signeven) /\ ((2) = 2))) \/ (((exists mv_odd_half_one_signodd. (x) = 2 * mv_odd_half_one_signodd + 1) /\ ((2) = 1)))
  21. 0021rewrite heq
  22. 0022rewrite heq
  23. 0023apply alternating_signed_unit_zero
  24. 0024specialize mobius_from_squarefree_factor_count (1)
  25. 0025specialize mobius_from_squarefree_factor_count (x1)
  26. 0026specialize mobius_from_squarefree_factor_count (x2)
  27. 0027specialize mobius_from_squarefree_factor_count (x)
  28. 0028specialize mobius_from_squarefree_factor_count (2)
  29. 0029apply mobius_from_squarefree_factor_count
  30. 0030apply squarefree_one
  31. 0031exact hf_witness_witness_witness
  32. 0032exact hs