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
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 = zConstructive 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.
Alpha v34 checked-use · first admitted v31 · independently kernel and Lean verified; not Stable
Proof neighborhood
Direct dependencies
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
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)
01Fix variables and assumptionsL1–2
02Establish hzL3–6
03Separate the logical casesL7–7
Follow the explicit conjunction, disjunction, witness, or contradiction step recorded below.
- L7
cases hz
04Construct an explicit witnessL8–8
Supply the displayed value, then prove that it has the required property.
- L8
exists x
05Separate the logical casesL9–9
Follow the explicit conjunction, disjunction, witness, or contradiction step recorded below.
- L9
split
06Use earlier factsL10–10
Instantiate or apply named facts and discharge the corresponding proof obligations.
- L10
exact hz_witness
07Fix variables and assumptionsL11–12
Original exact command ledger · 18 lines
- 0001
intro n - 0002
intro hn - 0003
have 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))))))))))) - 0004
specialize mobius_value_exists (n) - 0005
apply mobius_value_exists - 0006
exact hn - 0007
cases hz - 0008
exists x - 0009
split - 0010
exact hz_witness - 0011
intro w - 0012
intro hw - 0013
specialize mobius_value_functional (n) - 0014
specialize mobius_value_functional (w) - 0015
specialize mobius_value_functional (x) - 0016
apply mobius_value_functional - 0017
exact hw - 0018
exact hz_witness