MV0004

mobius_prime_factor_count_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

Actual unordered prime-factor uniqueness proves literal equality of factor counts; no canonical list is assumed.

Exact expanded first-order arithmetic statement

forall n b c l d e m. ((~(n = 0) /\ ((exists ff_u_fsat_mv_count_first_product ff_v_fsat_mv_count_first_product. ((((exists ff_h_fsat_mv_count_first_product_start. ff_h_fsat_mv_count_first_product_start + S (1) = S ((S (0)) * ff_v_fsat_mv_count_first_product)) /\ exists ff_q_fsat_mv_count_first_product_start. ff_u_fsat_mv_count_first_product = ff_q_fsat_mv_count_first_product_start * S ((S (0)) * ff_v_fsat_mv_count_first_product) + (1))) /\ ((((exists ff_h_fsat_mv_count_first_product_terminal. ff_h_fsat_mv_count_first_product_terminal + S (n) = S ((S (l)) * ff_v_fsat_mv_count_first_product)) /\ exists ff_q_fsat_mv_count_first_product_terminal. ff_u_fsat_mv_count_first_product = ff_q_fsat_mv_count_first_product_terminal * S ((S (l)) * ff_v_fsat_mv_count_first_product) + (n))) /\ forall ff_i_fsat_mv_count_first_product. (exists ff_lt_fsat_mv_count_first_product_bound. ff_lt_fsat_mv_count_first_product_bound + S ff_i_fsat_mv_count_first_product = l) -> exists ff_p_fsat_mv_count_first_product ff_r_fsat_mv_count_first_product ff_s_fsat_mv_count_first_product. ((((exists ff_h_fsat_mv_count_first_product_factor. ff_h_fsat_mv_count_first_product_factor + S (ff_p_fsat_mv_count_first_product) = S ((S (ff_i_fsat_mv_count_first_product)) * c)) /\ exists ff_q_fsat_mv_count_first_product_factor. b = ff_q_fsat_mv_count_first_product_factor * S ((S (ff_i_fsat_mv_count_first_product)) * c) + (ff_p_fsat_mv_count_first_product))) /\ ((((exists ff_h_fsat_mv_count_first_product_partial. ff_h_fsat_mv_count_first_product_partial + S (ff_r_fsat_mv_count_first_product) = S ((S (ff_i_fsat_mv_count_first_product)) * ff_v_fsat_mv_count_first_product)) /\ exists ff_q_fsat_mv_count_first_product_partial. ff_u_fsat_mv_count_first_product = ff_q_fsat_mv_count_first_product_partial * S ((S (ff_i_fsat_mv_count_first_product)) * ff_v_fsat_mv_count_first_product) + (ff_r_fsat_mv_count_first_product))) /\ ((((exists ff_h_fsat_mv_count_first_product_successor. ff_h_fsat_mv_count_first_product_successor + S (ff_s_fsat_mv_count_first_product) = S ((S (S ff_i_fsat_mv_count_first_product)) * ff_v_fsat_mv_count_first_product)) /\ exists ff_q_fsat_mv_count_first_product_successor. ff_u_fsat_mv_count_first_product = ff_q_fsat_mv_count_first_product_successor * S ((S (S ff_i_fsat_mv_count_first_product)) * ff_v_fsat_mv_count_first_product) + (ff_s_fsat_mv_count_first_product))) /\ ff_s_fsat_mv_count_first_product = ff_r_fsat_mv_count_first_product * ff_p_fsat_mv_count_first_product)))))) /\ (forall ftsf_index_fsat_mv_count_first_primes. (exists ftsf_gap_fsat_mv_count_first_primes_bound. ftsf_gap_fsat_mv_count_first_primes_bound + S ftsf_index_fsat_mv_count_first_primes = (l)) -> exists ftsf_factor_fsat_mv_count_first_primes. ((((exists ff_h_ftsf_fsat_mv_count_first_primes_entry. ff_h_ftsf_fsat_mv_count_first_primes_entry + S (ftsf_factor_fsat_mv_count_first_primes) = S ((S (ftsf_index_fsat_mv_count_first_primes)) * c)) /\ exists ff_q_ftsf_fsat_mv_count_first_primes_entry. b = ff_q_ftsf_fsat_mv_count_first_primes_entry * S ((S (ftsf_index_fsat_mv_count_first_primes)) * c) + (ftsf_factor_fsat_mv_count_first_primes))) /\ ((~(ftsf_factor_fsat_mv_count_first_primes = 1) /\ forall frm_prime_left_ftsf_fsat_mv_count_first_primes_prime frm_prime_right_ftsf_fsat_mv_count_first_primes_prime. ftsf_factor_fsat_mv_count_first_primes = frm_prime_left_ftsf_fsat_mv_count_first_primes_prime * frm_prime_right_ftsf_fsat_mv_count_first_primes_prime -> frm_prime_left_ftsf_fsat_mv_count_first_primes_prime = 1 \/ frm_prime_right_ftsf_fsat_mv_count_first_primes_prime = 1))))))) -> ((~(n = 0) /\ ((exists ff_u_fsat_mv_count_second_product ff_v_fsat_mv_count_second_product. ((((exists ff_h_fsat_mv_count_second_product_start. ff_h_fsat_mv_count_second_product_start + S (1) = S ((S (0)) * ff_v_fsat_mv_count_second_product)) /\ exists ff_q_fsat_mv_count_second_product_start. ff_u_fsat_mv_count_second_product = ff_q_fsat_mv_count_second_product_start * S ((S (0)) * ff_v_fsat_mv_count_second_product) + (1))) /\ ((((exists ff_h_fsat_mv_count_second_product_terminal. ff_h_fsat_mv_count_second_product_terminal + S (n) = S ((S (m)) * ff_v_fsat_mv_count_second_product)) /\ exists ff_q_fsat_mv_count_second_product_terminal. ff_u_fsat_mv_count_second_product = ff_q_fsat_mv_count_second_product_terminal * S ((S (m)) * ff_v_fsat_mv_count_second_product) + (n))) /\ forall ff_i_fsat_mv_count_second_product. (exists ff_lt_fsat_mv_count_second_product_bound. ff_lt_fsat_mv_count_second_product_bound + S ff_i_fsat_mv_count_second_product = m) -> exists ff_p_fsat_mv_count_second_product ff_r_fsat_mv_count_second_product ff_s_fsat_mv_count_second_product. ((((exists ff_h_fsat_mv_count_second_product_factor. ff_h_fsat_mv_count_second_product_factor + S (ff_p_fsat_mv_count_second_product) = S ((S (ff_i_fsat_mv_count_second_product)) * e)) /\ exists ff_q_fsat_mv_count_second_product_factor. d = ff_q_fsat_mv_count_second_product_factor * S ((S (ff_i_fsat_mv_count_second_product)) * e) + (ff_p_fsat_mv_count_second_product))) /\ ((((exists ff_h_fsat_mv_count_second_product_partial. ff_h_fsat_mv_count_second_product_partial + S (ff_r_fsat_mv_count_second_product) = S ((S (ff_i_fsat_mv_count_second_product)) * ff_v_fsat_mv_count_second_product)) /\ exists ff_q_fsat_mv_count_second_product_partial. ff_u_fsat_mv_count_second_product = ff_q_fsat_mv_count_second_product_partial * S ((S (ff_i_fsat_mv_count_second_product)) * ff_v_fsat_mv_count_second_product) + (ff_r_fsat_mv_count_second_product))) /\ ((((exists ff_h_fsat_mv_count_second_product_successor. ff_h_fsat_mv_count_second_product_successor + S (ff_s_fsat_mv_count_second_product) = S ((S (S ff_i_fsat_mv_count_second_product)) * ff_v_fsat_mv_count_second_product)) /\ exists ff_q_fsat_mv_count_second_product_successor. ff_u_fsat_mv_count_second_product = ff_q_fsat_mv_count_second_product_successor * S ((S (S ff_i_fsat_mv_count_second_product)) * ff_v_fsat_mv_count_second_product) + (ff_s_fsat_mv_count_second_product))) /\ ff_s_fsat_mv_count_second_product = ff_r_fsat_mv_count_second_product * ff_p_fsat_mv_count_second_product)))))) /\ (forall ftsf_index_fsat_mv_count_second_primes. (exists ftsf_gap_fsat_mv_count_second_primes_bound. ftsf_gap_fsat_mv_count_second_primes_bound + S ftsf_index_fsat_mv_count_second_primes = (m)) -> exists ftsf_factor_fsat_mv_count_second_primes. ((((exists ff_h_ftsf_fsat_mv_count_second_primes_entry. ff_h_ftsf_fsat_mv_count_second_primes_entry + S (ftsf_factor_fsat_mv_count_second_primes) = S ((S (ftsf_index_fsat_mv_count_second_primes)) * e)) /\ exists ff_q_ftsf_fsat_mv_count_second_primes_entry. d = ff_q_ftsf_fsat_mv_count_second_primes_entry * S ((S (ftsf_index_fsat_mv_count_second_primes)) * e) + (ftsf_factor_fsat_mv_count_second_primes))) /\ ((~(ftsf_factor_fsat_mv_count_second_primes = 1) /\ forall frm_prime_left_ftsf_fsat_mv_count_second_primes_prime frm_prime_right_ftsf_fsat_mv_count_second_primes_prime. ftsf_factor_fsat_mv_count_second_primes = frm_prime_left_ftsf_fsat_mv_count_second_primes_prime * frm_prime_right_ftsf_fsat_mv_count_second_primes_prime -> frm_prime_left_ftsf_fsat_mv_count_second_primes_prime = 1 \/ frm_prime_right_ftsf_fsat_mv_count_second_primes_prime = 1))))))) -> l = m

Constructive proof overview

Generated structural guide

Actual unordered prime-factor uniqueness proves literal equality of factor counts; no canonical list is assumed.

The unchanged tactic script uses 1 declared prerequisite and contains 22 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

prime_factor_lists_matching_by_length Alpha theorem; checked-use authorized

Direct dependents

MV000A mobius_squarefree_evaluation

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

22 script commands · 5 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.

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 assumptionsL1–9

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 d
  6. L6
    intro e
  7. L7
    intro m
  8. L8
    intro ha
  9. L9
    intro hb
02Establish hL10–19

Establish this local claim before using it. It is not an additional assumption. The following proof commands apply prime factor lists matching by length.

  1. L10
    have h : l = m ∧ (∃ x. ∃ y. BoundedPrefix(x,y,l) ∧ (InjectivePrefix(x,y,l) ∧ SurjectivePrefix(x,y,l)) ∧ (∀ z. ∀ n. ∀ k. Lt(z,l) → BetaAt(x,y,z,n) → BetaAt(b,c,z,k) → BetaAt(d,e,n,k)))Definitions: LtBetaAtBoundedPrefixInjectivePrefixSurjectivePrefix
  2. L11
    specialize prime_factor_lists_matching_by_length (l)
  3. L12
    specialize prime_factor_lists_matching_by_length (n)
  4. L13
    specialize prime_factor_lists_matching_by_length (b)
  5. L14
    specialize prime_factor_lists_matching_by_length (c)
  6. L15
    specialize prime_factor_lists_matching_by_length (m)
  7. L16
    specialize prime_factor_lists_matching_by_length (d)
  8. L17
    specialize prime_factor_lists_matching_by_length (e)
  9. L18
    apply prime_factor_lists_matching_by_length
  10. L19
    exact ha
03Use earlier factsL20–20

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

  1. L20
    exact hb
04Separate the logical casesL21–21

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

  1. L21
    cases h
05Use earlier factsL22–22

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

  1. L22
    exact h_left

Library-wide reading audit

Original exact command ledger · 22 lines
  1. 0001intro n
  2. 0002intro b
  3. 0003intro c
  4. 0004intro l
  5. 0005intro d
  6. 0006intro e
  7. 0007intro m
  8. 0008intro ha
  9. 0009intro hb
  10. 0010have h : ((l = m) /\ (exists pfp_u_mv_count_matching pfp_v_mv_count_matching. (((((forall pfp_i_mv_count_matchingmatchingpermutationbounded. (exists pfp_gap_mv_count_matchingmatchingpermutationboundedindex. pfp_gap_mv_count_matchingmatchingpermutationboundedindex + S (pfp_i_mv_count_matchingmatchingpermutationbounded) = (l)) -> exists pfp_a_mv_count_matchingmatchingpermutationbounded. (((exists ff_h_pfp_mv_count_matchingmatchingpermutationboundedentry. ff_h_pfp_mv_count_matchingmatchingpermutationboundedentry + S (pfp_a_mv_count_matchingmatchingpermutationbounded) = S ((S (pfp_i_mv_count_matchingmatchingpermutationbounded)) * pfp_v_mv_count_matching)) /\ exists ff_q_pfp_mv_count_matchingmatchingpermutationboundedentry. pfp_u_mv_count_matching = ff_q_pfp_mv_count_matchingmatchingpermutationboundedentry * S ((S (pfp_i_mv_count_matchingmatchingpermutationbounded)) * pfp_v_mv_count_matching) + (pfp_a_mv_count_matchingmatchingpermutationbounded))) /\ (exists pfp_gap_mv_count_matchingmatchingpermutationboundedvalue. pfp_gap_mv_count_matchingmatchingpermutationboundedvalue + S (pfp_a_mv_count_matchingmatchingpermutationbounded) = (l))) /\ (((forall pfp_i_mv_count_matchingmatchingpermutationinjective pfp_j_mv_count_matchingmatchingpermutationinjective pfp_a_mv_count_matchingmatchingpermutationinjective. (exists pfp_gap_mv_count_matchingmatchingpermutationinjectivefirst. pfp_gap_mv_count_matchingmatchingpermutationinjectivefirst + S (pfp_i_mv_count_matchingmatchingpermutationinjective) = (l)) -> (exists pfp_gap_mv_count_matchingmatchingpermutationinjectivesecond. pfp_gap_mv_count_matchingmatchingpermutationinjectivesecond + S (pfp_j_mv_count_matchingmatchingpermutationinjective) = (l)) -> (((exists ff_h_pfp_mv_count_matchingmatchingpermutationinjectiveleft. ff_h_pfp_mv_count_matchingmatchingpermutationinjectiveleft + S (pfp_a_mv_count_matchingmatchingpermutationinjective) = S ((S (pfp_i_mv_count_matchingmatchingpermutationinjective)) * pfp_v_mv_count_matching)) /\ exists ff_q_pfp_mv_count_matchingmatchingpermutationinjectiveleft. pfp_u_mv_count_matching = ff_q_pfp_mv_count_matchingmatchingpermutationinjectiveleft * S ((S (pfp_i_mv_count_matchingmatchingpermutationinjective)) * pfp_v_mv_count_matching) + (pfp_a_mv_count_matchingmatchingpermutationinjective))) -> (((exists ff_h_pfp_mv_count_matchingmatchingpermutationinjectiveright. ff_h_pfp_mv_count_matchingmatchingpermutationinjectiveright + S (pfp_a_mv_count_matchingmatchingpermutationinjective) = S ((S (pfp_j_mv_count_matchingmatchingpermutationinjective)) * pfp_v_mv_count_matching)) /\ exists ff_q_pfp_mv_count_matchingmatchingpermutationinjectiveright. pfp_u_mv_count_matching = ff_q_pfp_mv_count_matchingmatchingpermutationinjectiveright * S ((S (pfp_j_mv_count_matchingmatchingpermutationinjective)) * pfp_v_mv_count_matching) + (pfp_a_mv_count_matchingmatchingpermutationinjective))) -> pfp_i_mv_count_matchingmatchingpermutationinjective = pfp_j_mv_count_matchingmatchingpermutationinjective) /\ (forall pfp_a_mv_count_matchingmatchingpermutationsurjective. (exists pfp_gap_mv_count_matchingmatchingpermutationsurjectivevalue. pfp_gap_mv_count_matchingmatchingpermutationsurjectivevalue + S (pfp_a_mv_count_matchingmatchingpermutationsurjective) = (l)) -> exists pfp_i_mv_count_matchingmatchingpermutationsurjective. (exists pfp_gap_mv_count_matchingmatchingpermutationsurjectiveindex. pfp_gap_mv_count_matchingmatchingpermutationsurjectiveindex + S (pfp_i_mv_count_matchingmatchingpermutationsurjective) = (l)) /\ (((exists ff_h_pfp_mv_count_matchingmatchingpermutationsurjectiveentry. ff_h_pfp_mv_count_matchingmatchingpermutationsurjectiveentry + S (pfp_a_mv_count_matchingmatchingpermutationsurjective) = S ((S (pfp_i_mv_count_matchingmatchingpermutationsurjective)) * pfp_v_mv_count_matching)) /\ exists ff_q_pfp_mv_count_matchingmatchingpermutationsurjectiveentry. pfp_u_mv_count_matching = ff_q_pfp_mv_count_matchingmatchingpermutationsurjectiveentry * S ((S (pfp_i_mv_count_matchingmatchingpermutationsurjective)) * pfp_v_mv_count_matching) + (pfp_a_mv_count_matchingmatchingpermutationsurjective)))))))) /\ (forall pfp_i_mv_count_matchingmatchingmatching pfp_j_mv_count_matchingmatchingmatching pfp_a_mv_count_matchingmatchingmatching. (exists pfp_gap_mv_count_matchingmatchingmatchingbound. pfp_gap_mv_count_matchingmatchingmatchingbound + S (pfp_i_mv_count_matchingmatchingmatching) = (l)) -> (((exists ff_h_pfp_mv_count_matchingmatchingmatchingmap. ff_h_pfp_mv_count_matchingmatchingmatchingmap + S (pfp_j_mv_count_matchingmatchingmatching) = S ((S (pfp_i_mv_count_matchingmatchingmatching)) * pfp_v_mv_count_matching)) /\ exists ff_q_pfp_mv_count_matchingmatchingmatchingmap. pfp_u_mv_count_matching = ff_q_pfp_mv_count_matchingmatchingmatchingmap * S ((S (pfp_i_mv_count_matchingmatchingmatching)) * pfp_v_mv_count_matching) + (pfp_j_mv_count_matchingmatchingmatching))) -> (((exists ff_h_pfp_mv_count_matchingmatchingmatchingsource. ff_h_pfp_mv_count_matchingmatchingmatchingsource + S (pfp_a_mv_count_matchingmatchingmatching) = S ((S (pfp_i_mv_count_matchingmatchingmatching)) * c)) /\ exists ff_q_pfp_mv_count_matchingmatchingmatchingsource. b = ff_q_pfp_mv_count_matchingmatchingmatchingsource * S ((S (pfp_i_mv_count_matchingmatchingmatching)) * c) + (pfp_a_mv_count_matchingmatchingmatching))) -> (((exists ff_h_pfp_mv_count_matchingmatchingmatchingtarget. ff_h_pfp_mv_count_matchingmatchingmatchingtarget + S (pfp_a_mv_count_matchingmatchingmatching) = S ((S (pfp_j_mv_count_matchingmatchingmatching)) * e)) /\ exists ff_q_pfp_mv_count_matchingmatchingmatchingtarget. d = ff_q_pfp_mv_count_matchingmatchingmatchingtarget * S ((S (pfp_j_mv_count_matchingmatchingmatching)) * e) + (pfp_a_mv_count_matchingmatchingmatching))))))))
  11. 0011specialize prime_factor_lists_matching_by_length (l)
  12. 0012specialize prime_factor_lists_matching_by_length (n)
  13. 0013specialize prime_factor_lists_matching_by_length (b)
  14. 0014specialize prime_factor_lists_matching_by_length (c)
  15. 0015specialize prime_factor_lists_matching_by_length (m)
  16. 0016specialize prime_factor_lists_matching_by_length (d)
  17. 0017specialize prime_factor_lists_matching_by_length (e)
  18. 0018apply prime_factor_lists_matching_by_length
  19. 0019exact ha
  20. 0020exact hb
  21. 0021cases h
  22. 0022exact h_left