DV000D

mobius_table_one_entry

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

Whenever index one lies in the table, it contains canonical +1 (code two), while the unrelated zero-index convention stays separate.

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 M. (((exists dst_positive_code_unit_tabletable dst_positive_scale_unit_tabletable dst_negative_code_unit_tabletable dst_negative_scale_unit_tabletable. (((M) = (((((dst_positive_code_unit_tabletable) + (dst_positive_scale_unit_tabletable)) * S ((dst_positive_code_unit_tabletable) + (dst_positive_scale_unit_tabletable)) + ((dst_positive_scale_unit_tabletable) + (dst_positive_scale_unit_tabletable))) + (((dst_negative_code_unit_tabletable) + (dst_negative_scale_unit_tabletable)) * S ((dst_negative_code_unit_tabletable) + (dst_negative_scale_unit_tabletable)) + ((dst_negative_scale_unit_tabletable) + (dst_negative_scale_unit_tabletable)))) * S ((((dst_positive_code_unit_tabletable) + (dst_positive_scale_unit_tabletable)) * S ((dst_positive_code_unit_tabletable) + (dst_positive_scale_unit_tabletable)) + ((dst_positive_scale_unit_tabletable) + (dst_positive_scale_unit_tabletable))) + (((dst_negative_code_unit_tabletable) + (dst_negative_scale_unit_tabletable)) * S ((dst_negative_code_unit_tabletable) + (dst_negative_scale_unit_tabletable)) + ((dst_negative_scale_unit_tabletable) + (dst_negative_scale_unit_tabletable)))) + ((((dst_negative_code_unit_tabletable) + (dst_negative_scale_unit_tabletable)) * S ((dst_negative_code_unit_tabletable) + (dst_negative_scale_unit_tabletable)) + ((dst_negative_scale_unit_tabletable) + (dst_negative_scale_unit_tabletable))) + (((dst_negative_code_unit_tabletable) + (dst_negative_scale_unit_tabletable)) * S ((dst_negative_code_unit_tabletable) + (dst_negative_scale_unit_tabletable)) + ((dst_negative_scale_unit_tabletable) + (dst_negative_scale_unit_tabletable)))))) /\ (forall dst_index_unit_tabletable. (exists pvs_le_gap_unit_tabletabledomain. pvs_le_gap_unit_tabletabledomain + (dst_index_unit_tabletable) = (N)) -> exists dst_positive_unit_tabletable dst_negative_unit_tabletable dst_value_unit_tabletable. ((((exists ff_h_pvs_unit_tabletableentrypositive. ff_h_pvs_unit_tabletableentrypositive + S (dst_positive_unit_tabletable) = S ((S (dst_index_unit_tabletable)) * dst_positive_scale_unit_tabletable)) /\ exists ff_q_pvs_unit_tabletableentrypositive. dst_positive_code_unit_tabletable = ff_q_pvs_unit_tabletableentrypositive * S ((S (dst_index_unit_tabletable)) * dst_positive_scale_unit_tabletable) + (dst_positive_unit_tabletable))) /\ (((((exists ff_h_pvs_unit_tabletableentrynegative. ff_h_pvs_unit_tabletableentrynegative + S (dst_negative_unit_tabletable) = S ((S (dst_index_unit_tabletable)) * dst_negative_scale_unit_tabletable)) /\ exists ff_q_pvs_unit_tabletableentrynegative. dst_negative_code_unit_tabletable = ff_q_pvs_unit_tabletableentrynegative * S ((S (dst_index_unit_tabletable)) * dst_negative_scale_unit_tabletable) + (dst_negative_unit_tabletable))) /\ (exists ge_balance_positive_unit_tabletableentryvalue ge_balance_negative_unit_tabletableentryvalue. (((((dst_value_unit_tabletable) = 2 * (ge_balance_positive_unit_tabletableentryvalue) /\ (ge_balance_negative_unit_tabletableentryvalue) = 0) \/ exists ge_signed_half_unit_tabletableentryvaluedecode. (((dst_value_unit_tabletable) = 2 * ge_signed_half_unit_tabletableentryvaluedecode + 1 /\ (ge_balance_positive_unit_tabletableentryvalue) = 0) /\ (ge_balance_negative_unit_tabletableentryvalue) = S ge_signed_half_unit_tabletableentryvaluedecode))) /\ ((dst_positive_unit_tabletable) + ge_balance_negative_unit_tabletableentryvalue = (dst_negative_unit_tabletable) + ge_balance_positive_unit_tabletableentryvalue))))))))) /\ (((exists dst_positive_code_unit_tablezero dst_positive_scale_unit_tablezero dst_negative_code_unit_tablezero dst_negative_scale_unit_tablezero dst_positive_unit_tablezero dst_negative_unit_tablezero. (((M) = (((((dst_positive_code_unit_tablezero) + (dst_positive_scale_unit_tablezero)) * S ((dst_positive_code_unit_tablezero) + (dst_positive_scale_unit_tablezero)) + ((dst_positive_scale_unit_tablezero) + (dst_positive_scale_unit_tablezero))) + (((dst_negative_code_unit_tablezero) + (dst_negative_scale_unit_tablezero)) * S ((dst_negative_code_unit_tablezero) + (dst_negative_scale_unit_tablezero)) + ((dst_negative_scale_unit_tablezero) + (dst_negative_scale_unit_tablezero)))) * S ((((dst_positive_code_unit_tablezero) + (dst_positive_scale_unit_tablezero)) * S ((dst_positive_code_unit_tablezero) + (dst_positive_scale_unit_tablezero)) + ((dst_positive_scale_unit_tablezero) + (dst_positive_scale_unit_tablezero))) + (((dst_negative_code_unit_tablezero) + (dst_negative_scale_unit_tablezero)) * S ((dst_negative_code_unit_tablezero) + (dst_negative_scale_unit_tablezero)) + ((dst_negative_scale_unit_tablezero) + (dst_negative_scale_unit_tablezero)))) + ((((dst_negative_code_unit_tablezero) + (dst_negative_scale_unit_tablezero)) * S ((dst_negative_code_unit_tablezero) + (dst_negative_scale_unit_tablezero)) + ((dst_negative_scale_unit_tablezero) + (dst_negative_scale_unit_tablezero))) + (((dst_negative_code_unit_tablezero) + (dst_negative_scale_unit_tablezero)) * S ((dst_negative_code_unit_tablezero) + (dst_negative_scale_unit_tablezero)) + ((dst_negative_scale_unit_tablezero) + (dst_negative_scale_unit_tablezero)))))) /\ (((((exists ff_h_pvs_unit_tablezeropositive. ff_h_pvs_unit_tablezeropositive + S (dst_positive_unit_tablezero) = S ((S (0)) * dst_positive_scale_unit_tablezero)) /\ exists ff_q_pvs_unit_tablezeropositive. dst_positive_code_unit_tablezero = ff_q_pvs_unit_tablezeropositive * S ((S (0)) * dst_positive_scale_unit_tablezero) + (dst_positive_unit_tablezero))) /\ (((((exists ff_h_pvs_unit_tablezeronegative. ff_h_pvs_unit_tablezeronegative + S (dst_negative_unit_tablezero) = S ((S (0)) * dst_negative_scale_unit_tablezero)) /\ exists ff_q_pvs_unit_tablezeronegative. dst_negative_code_unit_tablezero = ff_q_pvs_unit_tablezeronegative * S ((S (0)) * dst_negative_scale_unit_tablezero) + (dst_negative_unit_tablezero))) /\ (exists ge_balance_positive_unit_tablezerovalue ge_balance_negative_unit_tablezerovalue. (((((0) = 2 * (ge_balance_positive_unit_tablezerovalue) /\ (ge_balance_negative_unit_tablezerovalue) = 0) \/ exists ge_signed_half_unit_tablezerovaluedecode. (((0) = 2 * ge_signed_half_unit_tablezerovaluedecode + 1 /\ (ge_balance_positive_unit_tablezerovalue) = 0) /\ (ge_balance_negative_unit_tablezerovalue) = S ge_signed_half_unit_tablezerovaluedecode))) /\ ((dst_positive_unit_tablezero) + ge_balance_negative_unit_tablezerovalue = (dst_negative_unit_tablezero) + ge_balance_positive_unit_tablezerovalue))))))))) /\ (forall mt_index_unit_table mt_value_unit_table. ~(mt_index_unit_table=0) -> (exists pvs_le_gap_unit_tabledomain. pvs_le_gap_unit_tabledomain + (mt_index_unit_table) = (N)) -> (exists dst_positive_code_unit_tableentry dst_positive_scale_unit_tableentry dst_negative_code_unit_tableentry dst_negative_scale_unit_tableentry dst_positive_unit_tableentry dst_negative_unit_tableentry. (((M) = (((((dst_positive_code_unit_tableentry) + (dst_positive_scale_unit_tableentry)) * S ((dst_positive_code_unit_tableentry) + (dst_positive_scale_unit_tableentry)) + ((dst_positive_scale_unit_tableentry) + (dst_positive_scale_unit_tableentry))) + (((dst_negative_code_unit_tableentry) + (dst_negative_scale_unit_tableentry)) * S ((dst_negative_code_unit_tableentry) + (dst_negative_scale_unit_tableentry)) + ((dst_negative_scale_unit_tableentry) + (dst_negative_scale_unit_tableentry)))) * S ((((dst_positive_code_unit_tableentry) + (dst_positive_scale_unit_tableentry)) * S ((dst_positive_code_unit_tableentry) + (dst_positive_scale_unit_tableentry)) + ((dst_positive_scale_unit_tableentry) + (dst_positive_scale_unit_tableentry))) + (((dst_negative_code_unit_tableentry) + (dst_negative_scale_unit_tableentry)) * S ((dst_negative_code_unit_tableentry) + (dst_negative_scale_unit_tableentry)) + ((dst_negative_scale_unit_tableentry) + (dst_negative_scale_unit_tableentry)))) + ((((dst_negative_code_unit_tableentry) + (dst_negative_scale_unit_tableentry)) * S ((dst_negative_code_unit_tableentry) + (dst_negative_scale_unit_tableentry)) + ((dst_negative_scale_unit_tableentry) + (dst_negative_scale_unit_tableentry))) + (((dst_negative_code_unit_tableentry) + (dst_negative_scale_unit_tableentry)) * S ((dst_negative_code_unit_tableentry) + (dst_negative_scale_unit_tableentry)) + ((dst_negative_scale_unit_tableentry) + (dst_negative_scale_unit_tableentry)))))) /\ (((((exists ff_h_pvs_unit_tableentrypositive. ff_h_pvs_unit_tableentrypositive + S (dst_positive_unit_tableentry) = S ((S (mt_index_unit_table)) * dst_positive_scale_unit_tableentry)) /\ exists ff_q_pvs_unit_tableentrypositive. dst_positive_code_unit_tableentry = ff_q_pvs_unit_tableentrypositive * S ((S (mt_index_unit_table)) * dst_positive_scale_unit_tableentry) + (dst_positive_unit_tableentry))) /\ (((((exists ff_h_pvs_unit_tableentrynegative. ff_h_pvs_unit_tableentrynegative + S (dst_negative_unit_tableentry) = S ((S (mt_index_unit_table)) * dst_negative_scale_unit_tableentry)) /\ exists ff_q_pvs_unit_tableentrynegative. dst_negative_code_unit_tableentry = ff_q_pvs_unit_tableentrynegative * S ((S (mt_index_unit_table)) * dst_negative_scale_unit_tableentry) + (dst_negative_unit_tableentry))) /\ (exists ge_balance_positive_unit_tableentryvalue ge_balance_negative_unit_tableentryvalue. (((((mt_value_unit_table) = 2 * (ge_balance_positive_unit_tableentryvalue) /\ (ge_balance_negative_unit_tableentryvalue) = 0) \/ exists ge_signed_half_unit_tableentryvaluedecode. (((mt_value_unit_table) = 2 * ge_signed_half_unit_tableentryvaluedecode + 1 /\ (ge_balance_positive_unit_tableentryvalue) = 0) /\ (ge_balance_negative_unit_tableentryvalue) = S ge_signed_half_unit_tableentryvaluedecode))) /\ ((dst_positive_unit_tableentry) + ge_balance_negative_unit_tableentryvalue = (dst_negative_unit_tableentry) + ge_balance_positive_unit_tableentryvalue))))))))) -> (((~((mt_index_unit_table) = 0)) /\ ((((exists mv_square_prime_unit_tablevaluesquare. ((~((mv_square_prime_unit_tablevaluesquare) = 1) /\ forall pvs_left_unit_tablevaluesquareprime pvs_right_unit_tablevaluesquareprime. (mv_square_prime_unit_tablevaluesquare) = pvs_left_unit_tablevaluesquareprime * pvs_right_unit_tablevaluesquareprime -> pvs_left_unit_tablevaluesquareprime = 1 \/ pvs_right_unit_tablevaluesquareprime = 1) /\ (exists pvs_factor_unit_tablevaluesquaredivisor. (mt_index_unit_table) = (mv_square_prime_unit_tablevaluesquare * mv_square_prime_unit_tablevaluesquare) * pvs_factor_unit_tablevaluesquaredivisor))) /\ ((mt_value_unit_table) = 0))) \/ (((((~((mt_index_unit_table) = 0)) /\ (forall sfd_prime_unit_tablevaluesquarefree. (~((sfd_prime_unit_tablevaluesquarefree) = 1) /\ forall pvs_left_unit_tablevaluesquarefreedomain pvs_right_unit_tablevaluesquarefreedomain. (sfd_prime_unit_tablevaluesquarefree) = pvs_left_unit_tablevaluesquarefreedomain * pvs_right_unit_tablevaluesquarefreedomain -> pvs_left_unit_tablevaluesquarefreedomain = 1 \/ pvs_right_unit_tablevaluesquarefreedomain = 1) -> (exists pvs_le_gap_unit_tablevaluesquarefreebound. pvs_le_gap_unit_tablevaluesquarefreebound + (sfd_prime_unit_tablevaluesquarefree) = (mt_index_unit_table)) -> ~(exists pvs_factor_unit_tablevaluesquarefreesquare. (mt_index_unit_table) = (sfd_prime_unit_tablevaluesquarefree * sfd_prime_unit_tablevaluesquarefree) * pvs_factor_unit_tablevaluesquarefreesquare)))) /\ (exists mv_factor_code_unit_tablevaluefactors mv_factor_scale_unit_tablevaluefactors mv_factor_count_unit_tablevaluefactors. (((~(mt_index_unit_table = 0) /\ ((exists ff_u_fsat_unit_tablevaluefactorsfactorization_product ff_v_fsat_unit_tablevaluefactorsfactorization_product. ((((exists ff_h_fsat_unit_tablevaluefactorsfactorization_product_start. ff_h_fsat_unit_tablevaluefactorsfactorization_product_start + S (1) = S ((S (0)) * ff_v_fsat_unit_tablevaluefactorsfactorization_product)) /\ exists ff_q_fsat_unit_tablevaluefactorsfactorization_product_start. ff_u_fsat_unit_tablevaluefactorsfactorization_product = ff_q_fsat_unit_tablevaluefactorsfactorization_product_start * S ((S (0)) * ff_v_fsat_unit_tablevaluefactorsfactorization_product) + (1))) /\ ((((exists ff_h_fsat_unit_tablevaluefactorsfactorization_product_terminal. ff_h_fsat_unit_tablevaluefactorsfactorization_product_terminal + S (mt_index_unit_table) = S ((S (mv_factor_count_unit_tablevaluefactors)) * ff_v_fsat_unit_tablevaluefactorsfactorization_product)) /\ exists ff_q_fsat_unit_tablevaluefactorsfactorization_product_terminal. ff_u_fsat_unit_tablevaluefactorsfactorization_product = ff_q_fsat_unit_tablevaluefactorsfactorization_product_terminal * S ((S (mv_factor_count_unit_tablevaluefactors)) * ff_v_fsat_unit_tablevaluefactorsfactorization_product) + (mt_index_unit_table))) /\ forall ff_i_fsat_unit_tablevaluefactorsfactorization_product. (exists ff_lt_fsat_unit_tablevaluefactorsfactorization_product_bound. ff_lt_fsat_unit_tablevaluefactorsfactorization_product_bound + S ff_i_fsat_unit_tablevaluefactorsfactorization_product = mv_factor_count_unit_tablevaluefactors) -> exists ff_p_fsat_unit_tablevaluefactorsfactorization_product ff_r_fsat_unit_tablevaluefactorsfactorization_product ff_s_fsat_unit_tablevaluefactorsfactorization_product. ((((exists ff_h_fsat_unit_tablevaluefactorsfactorization_product_factor. ff_h_fsat_unit_tablevaluefactorsfactorization_product_factor + S (ff_p_fsat_unit_tablevaluefactorsfactorization_product) = S ((S (ff_i_fsat_unit_tablevaluefactorsfactorization_product)) * mv_factor_scale_unit_tablevaluefactors)) /\ exists ff_q_fsat_unit_tablevaluefactorsfactorization_product_factor. mv_factor_code_unit_tablevaluefactors = ff_q_fsat_unit_tablevaluefactorsfactorization_product_factor * S ((S (ff_i_fsat_unit_tablevaluefactorsfactorization_product)) * mv_factor_scale_unit_tablevaluefactors) + (ff_p_fsat_unit_tablevaluefactorsfactorization_product))) /\ ((((exists ff_h_fsat_unit_tablevaluefactorsfactorization_product_partial. ff_h_fsat_unit_tablevaluefactorsfactorization_product_partial + S (ff_r_fsat_unit_tablevaluefactorsfactorization_product) = S ((S (ff_i_fsat_unit_tablevaluefactorsfactorization_product)) * ff_v_fsat_unit_tablevaluefactorsfactorization_product)) /\ exists ff_q_fsat_unit_tablevaluefactorsfactorization_product_partial. ff_u_fsat_unit_tablevaluefactorsfactorization_product = ff_q_fsat_unit_tablevaluefactorsfactorization_product_partial * S ((S (ff_i_fsat_unit_tablevaluefactorsfactorization_product)) * ff_v_fsat_unit_tablevaluefactorsfactorization_product) + (ff_r_fsat_unit_tablevaluefactorsfactorization_product))) /\ ((((exists ff_h_fsat_unit_tablevaluefactorsfactorization_product_successor. ff_h_fsat_unit_tablevaluefactorsfactorization_product_successor + S (ff_s_fsat_unit_tablevaluefactorsfactorization_product) = S ((S (S ff_i_fsat_unit_tablevaluefactorsfactorization_product)) * ff_v_fsat_unit_tablevaluefactorsfactorization_product)) /\ exists ff_q_fsat_unit_tablevaluefactorsfactorization_product_successor. ff_u_fsat_unit_tablevaluefactorsfactorization_product = ff_q_fsat_unit_tablevaluefactorsfactorization_product_successor * S ((S (S ff_i_fsat_unit_tablevaluefactorsfactorization_product)) * ff_v_fsat_unit_tablevaluefactorsfactorization_product) + (ff_s_fsat_unit_tablevaluefactorsfactorization_product))) /\ ff_s_fsat_unit_tablevaluefactorsfactorization_product = ff_r_fsat_unit_tablevaluefactorsfactorization_product * ff_p_fsat_unit_tablevaluefactorsfactorization_product)))))) /\ (forall ftsf_index_fsat_unit_tablevaluefactorsfactorization_primes. (exists ftsf_gap_fsat_unit_tablevaluefactorsfactorization_primes_bound. ftsf_gap_fsat_unit_tablevaluefactorsfactorization_primes_bound + S ftsf_index_fsat_unit_tablevaluefactorsfactorization_primes = (mv_factor_count_unit_tablevaluefactors)) -> exists ftsf_factor_fsat_unit_tablevaluefactorsfactorization_primes. ((((exists ff_h_ftsf_fsat_unit_tablevaluefactorsfactorization_primes_entry. ff_h_ftsf_fsat_unit_tablevaluefactorsfactorization_primes_entry + S (ftsf_factor_fsat_unit_tablevaluefactorsfactorization_primes) = S ((S (ftsf_index_fsat_unit_tablevaluefactorsfactorization_primes)) * mv_factor_scale_unit_tablevaluefactors)) /\ exists ff_q_ftsf_fsat_unit_tablevaluefactorsfactorization_primes_entry. mv_factor_code_unit_tablevaluefactors = ff_q_ftsf_fsat_unit_tablevaluefactorsfactorization_primes_entry * S ((S (ftsf_index_fsat_unit_tablevaluefactorsfactorization_primes)) * mv_factor_scale_unit_tablevaluefactors) + (ftsf_factor_fsat_unit_tablevaluefactorsfactorization_primes))) /\ ((~(ftsf_factor_fsat_unit_tablevaluefactorsfactorization_primes = 1) /\ forall frm_prime_left_ftsf_fsat_unit_tablevaluefactorsfactorization_primes_prime frm_prime_right_ftsf_fsat_unit_tablevaluefactorsfactorization_primes_prime. ftsf_factor_fsat_unit_tablevaluefactorsfactorization_primes = frm_prime_left_ftsf_fsat_unit_tablevaluefactorsfactorization_primes_prime * frm_prime_right_ftsf_fsat_unit_tablevaluefactorsfactorization_primes_prime -> frm_prime_left_ftsf_fsat_unit_tablevaluefactorsfactorization_primes_prime = 1 \/ frm_prime_right_ftsf_fsat_unit_tablevaluefactorsfactorization_primes_prime = 1))))))) /\ ((((exists mv_even_half_unit_tablevaluefactorsparityeven. (mv_factor_count_unit_tablevaluefactors) = 2 * mv_even_half_unit_tablevaluefactorsparityeven) /\ ((mt_value_unit_table) = 2))) \/ (((exists mv_odd_half_unit_tablevaluefactorsparityodd. (mv_factor_count_unit_tablevaluefactors) = 2 * mv_odd_half_unit_tablevaluefactorsparityodd + 1) /\ ((mt_value_unit_table) = 1)))))))))))))))) -> (exists pvs_le_gap_unit_bound. pvs_le_gap_unit_bound + (1) = (N)) -> (exists dst_positive_code_unit_entry dst_positive_scale_unit_entry dst_negative_code_unit_entry dst_negative_scale_unit_entry dst_positive_unit_entry dst_negative_unit_entry. (((M) = (((((dst_positive_code_unit_entry) + (dst_positive_scale_unit_entry)) * S ((dst_positive_code_unit_entry) + (dst_positive_scale_unit_entry)) + ((dst_positive_scale_unit_entry) + (dst_positive_scale_unit_entry))) + (((dst_negative_code_unit_entry) + (dst_negative_scale_unit_entry)) * S ((dst_negative_code_unit_entry) + (dst_negative_scale_unit_entry)) + ((dst_negative_scale_unit_entry) + (dst_negative_scale_unit_entry)))) * S ((((dst_positive_code_unit_entry) + (dst_positive_scale_unit_entry)) * S ((dst_positive_code_unit_entry) + (dst_positive_scale_unit_entry)) + ((dst_positive_scale_unit_entry) + (dst_positive_scale_unit_entry))) + (((dst_negative_code_unit_entry) + (dst_negative_scale_unit_entry)) * S ((dst_negative_code_unit_entry) + (dst_negative_scale_unit_entry)) + ((dst_negative_scale_unit_entry) + (dst_negative_scale_unit_entry)))) + ((((dst_negative_code_unit_entry) + (dst_negative_scale_unit_entry)) * S ((dst_negative_code_unit_entry) + (dst_negative_scale_unit_entry)) + ((dst_negative_scale_unit_entry) + (dst_negative_scale_unit_entry))) + (((dst_negative_code_unit_entry) + (dst_negative_scale_unit_entry)) * S ((dst_negative_code_unit_entry) + (dst_negative_scale_unit_entry)) + ((dst_negative_scale_unit_entry) + (dst_negative_scale_unit_entry)))))) /\ (((((exists ff_h_pvs_unit_entrypositive. ff_h_pvs_unit_entrypositive + S (dst_positive_unit_entry) = S ((S (1)) * dst_positive_scale_unit_entry)) /\ exists ff_q_pvs_unit_entrypositive. dst_positive_code_unit_entry = ff_q_pvs_unit_entrypositive * S ((S (1)) * dst_positive_scale_unit_entry) + (dst_positive_unit_entry))) /\ (((((exists ff_h_pvs_unit_entrynegative. ff_h_pvs_unit_entrynegative + S (dst_negative_unit_entry) = S ((S (1)) * dst_negative_scale_unit_entry)) /\ exists ff_q_pvs_unit_entrynegative. dst_negative_code_unit_entry = ff_q_pvs_unit_entrynegative * S ((S (1)) * dst_negative_scale_unit_entry) + (dst_negative_unit_entry))) /\ (exists ge_balance_positive_unit_entryvalue ge_balance_negative_unit_entryvalue. (((((2) = 2 * (ge_balance_positive_unit_entryvalue) /\ (ge_balance_negative_unit_entryvalue) = 0) \/ exists ge_signed_half_unit_entryvaluedecode. (((2) = 2 * ge_signed_half_unit_entryvaluedecode + 1 /\ (ge_balance_positive_unit_entryvalue) = 0) /\ (ge_balance_negative_unit_entryvalue) = S ge_signed_half_unit_entryvaluedecode))) /\ ((dst_positive_unit_entry) + ge_balance_negative_unit_entryvalue = (dst_negative_unit_entry) + ge_balance_positive_unit_entryvalue)))))))))

Constructive proof overview

Generated structural guide

Whenever index one lies in the table, it contains canonical +1 (code two), while the unrelated zero-index convention stays separate.

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

DV000C mobius_table_entry_iff mobius_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

18 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.

Named ingredients (1)
mobius table entry iff

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 assumptionsL1–4

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

  1. L1
    intro N
  2. L2
    intro M
  3. L3
    intro hm
  4. L4
    intro hN
02Establish hiffL5–14

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

  1. L5
    have hiff : (ArithAt(M,1,2) → Mobius(1,2)) ∧ (Mobius(1,2) → ArithAt(M,1,2))Definitions: MobiusArithAt
  2. L6
    specialize mobius_table_entry_iff (N)
  3. L7
    specialize mobius_table_entry_iff (M)
  4. L8
    specialize mobius_table_entry_iff (1)
  5. L9
    specialize mobius_table_entry_iff (2)
  6. L10
    apply mobius_table_entry_iff
  7. L11
    exact hm
  8. L12
    intro hzero
  9. L13
    apply PA1
  10. L14
    exact hzero
03Use earlier factsL15–15

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

  1. L15
    exact hN
04Separate the logical casesL16–16

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

  1. L16
    cases hiff
05Use earlier factsL17–18

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

  1. L17
    apply hiff_right
  2. L18
    exact mobius_one

Library-wide reading audit

Original exact command ledger · 18 lines
  1. 0001intro N
  2. 0002intro M
  3. 0003intro hm
  4. 0004intro hN
  5. 0005have hiff : ((exists dst_positive_code_unit_iff_entry dst_positive_scale_unit_iff_entry dst_negative_code_unit_iff_entry dst_negative_scale_unit_iff_entry dst_positive_unit_iff_entry dst_negative_unit_iff_entry. (((M) = (((((dst_positive_code_unit_iff_entry) + (dst_positive_scale_unit_iff_entry)) * S ((dst_positive_code_unit_iff_entry) + (dst_positive_scale_unit_iff_entry)) + ((dst_positive_scale_unit_iff_entry) + (dst_positive_scale_unit_iff_entry))) + (((dst_negative_code_unit_iff_entry) + (dst_negative_scale_unit_iff_entry)) * S ((dst_negative_code_unit_iff_entry) + (dst_negative_scale_unit_iff_entry)) + ((dst_negative_scale_unit_iff_entry) + (dst_negative_scale_unit_iff_entry)))) * S ((((dst_positive_code_unit_iff_entry) + (dst_positive_scale_unit_iff_entry)) * S ((dst_positive_code_unit_iff_entry) + (dst_positive_scale_unit_iff_entry)) + ((dst_positive_scale_unit_iff_entry) + (dst_positive_scale_unit_iff_entry))) + (((dst_negative_code_unit_iff_entry) + (dst_negative_scale_unit_iff_entry)) * S ((dst_negative_code_unit_iff_entry) + (dst_negative_scale_unit_iff_entry)) + ((dst_negative_scale_unit_iff_entry) + (dst_negative_scale_unit_iff_entry)))) + ((((dst_negative_code_unit_iff_entry) + (dst_negative_scale_unit_iff_entry)) * S ((dst_negative_code_unit_iff_entry) + (dst_negative_scale_unit_iff_entry)) + ((dst_negative_scale_unit_iff_entry) + (dst_negative_scale_unit_iff_entry))) + (((dst_negative_code_unit_iff_entry) + (dst_negative_scale_unit_iff_entry)) * S ((dst_negative_code_unit_iff_entry) + (dst_negative_scale_unit_iff_entry)) + ((dst_negative_scale_unit_iff_entry) + (dst_negative_scale_unit_iff_entry)))))) /\ (((((exists ff_h_pvs_unit_iff_entrypositive. ff_h_pvs_unit_iff_entrypositive + S (dst_positive_unit_iff_entry) = S ((S (1)) * dst_positive_scale_unit_iff_entry)) /\ exists ff_q_pvs_unit_iff_entrypositive. dst_positive_code_unit_iff_entry = ff_q_pvs_unit_iff_entrypositive * S ((S (1)) * dst_positive_scale_unit_iff_entry) + (dst_positive_unit_iff_entry))) /\ (((((exists ff_h_pvs_unit_iff_entrynegative. ff_h_pvs_unit_iff_entrynegative + S (dst_negative_unit_iff_entry) = S ((S (1)) * dst_negative_scale_unit_iff_entry)) /\ exists ff_q_pvs_unit_iff_entrynegative. dst_negative_code_unit_iff_entry = ff_q_pvs_unit_iff_entrynegative * S ((S (1)) * dst_negative_scale_unit_iff_entry) + (dst_negative_unit_iff_entry))) /\ (exists ge_balance_positive_unit_iff_entryvalue ge_balance_negative_unit_iff_entryvalue. (((((2) = 2 * (ge_balance_positive_unit_iff_entryvalue) /\ (ge_balance_negative_unit_iff_entryvalue) = 0) \/ exists ge_signed_half_unit_iff_entryvaluedecode. (((2) = 2 * ge_signed_half_unit_iff_entryvaluedecode + 1 /\ (ge_balance_positive_unit_iff_entryvalue) = 0) /\ (ge_balance_negative_unit_iff_entryvalue) = S ge_signed_half_unit_iff_entryvaluedecode))) /\ ((dst_positive_unit_iff_entry) + ge_balance_negative_unit_iff_entryvalue = (dst_negative_unit_iff_entry) + ge_balance_positive_unit_iff_entryvalue))))))))) -> (((~((1) = 0)) /\ ((((exists mv_square_prime_unit_iff_valuesquare. ((~((mv_square_prime_unit_iff_valuesquare) = 1) /\ forall pvs_left_unit_iff_valuesquareprime pvs_right_unit_iff_valuesquareprime. (mv_square_prime_unit_iff_valuesquare) = pvs_left_unit_iff_valuesquareprime * pvs_right_unit_iff_valuesquareprime -> pvs_left_unit_iff_valuesquareprime = 1 \/ pvs_right_unit_iff_valuesquareprime = 1) /\ (exists pvs_factor_unit_iff_valuesquaredivisor. (1) = (mv_square_prime_unit_iff_valuesquare * mv_square_prime_unit_iff_valuesquare) * pvs_factor_unit_iff_valuesquaredivisor))) /\ ((2) = 0))) \/ (((((~((1) = 0)) /\ (forall sfd_prime_unit_iff_valuesquarefree. (~((sfd_prime_unit_iff_valuesquarefree) = 1) /\ forall pvs_left_unit_iff_valuesquarefreedomain pvs_right_unit_iff_valuesquarefreedomain. (sfd_prime_unit_iff_valuesquarefree) = pvs_left_unit_iff_valuesquarefreedomain * pvs_right_unit_iff_valuesquarefreedomain -> pvs_left_unit_iff_valuesquarefreedomain = 1 \/ pvs_right_unit_iff_valuesquarefreedomain = 1) -> (exists pvs_le_gap_unit_iff_valuesquarefreebound. pvs_le_gap_unit_iff_valuesquarefreebound + (sfd_prime_unit_iff_valuesquarefree) = (1)) -> ~(exists pvs_factor_unit_iff_valuesquarefreesquare. (1) = (sfd_prime_unit_iff_valuesquarefree * sfd_prime_unit_iff_valuesquarefree) * pvs_factor_unit_iff_valuesquarefreesquare)))) /\ (exists mv_factor_code_unit_iff_valuefactors mv_factor_scale_unit_iff_valuefactors mv_factor_count_unit_iff_valuefactors. (((~(1 = 0) /\ ((exists ff_u_fsat_unit_iff_valuefactorsfactorization_product ff_v_fsat_unit_iff_valuefactorsfactorization_product. ((((exists ff_h_fsat_unit_iff_valuefactorsfactorization_product_start. ff_h_fsat_unit_iff_valuefactorsfactorization_product_start + S (1) = S ((S (0)) * ff_v_fsat_unit_iff_valuefactorsfactorization_product)) /\ exists ff_q_fsat_unit_iff_valuefactorsfactorization_product_start. ff_u_fsat_unit_iff_valuefactorsfactorization_product = ff_q_fsat_unit_iff_valuefactorsfactorization_product_start * S ((S (0)) * ff_v_fsat_unit_iff_valuefactorsfactorization_product) + (1))) /\ ((((exists ff_h_fsat_unit_iff_valuefactorsfactorization_product_terminal. ff_h_fsat_unit_iff_valuefactorsfactorization_product_terminal + S (1) = S ((S (mv_factor_count_unit_iff_valuefactors)) * ff_v_fsat_unit_iff_valuefactorsfactorization_product)) /\ exists ff_q_fsat_unit_iff_valuefactorsfactorization_product_terminal. ff_u_fsat_unit_iff_valuefactorsfactorization_product = ff_q_fsat_unit_iff_valuefactorsfactorization_product_terminal * S ((S (mv_factor_count_unit_iff_valuefactors)) * ff_v_fsat_unit_iff_valuefactorsfactorization_product) + (1))) /\ forall ff_i_fsat_unit_iff_valuefactorsfactorization_product. (exists ff_lt_fsat_unit_iff_valuefactorsfactorization_product_bound. ff_lt_fsat_unit_iff_valuefactorsfactorization_product_bound + S ff_i_fsat_unit_iff_valuefactorsfactorization_product = mv_factor_count_unit_iff_valuefactors) -> exists ff_p_fsat_unit_iff_valuefactorsfactorization_product ff_r_fsat_unit_iff_valuefactorsfactorization_product ff_s_fsat_unit_iff_valuefactorsfactorization_product. ((((exists ff_h_fsat_unit_iff_valuefactorsfactorization_product_factor. ff_h_fsat_unit_iff_valuefactorsfactorization_product_factor + S (ff_p_fsat_unit_iff_valuefactorsfactorization_product) = S ((S (ff_i_fsat_unit_iff_valuefactorsfactorization_product)) * mv_factor_scale_unit_iff_valuefactors)) /\ exists ff_q_fsat_unit_iff_valuefactorsfactorization_product_factor. mv_factor_code_unit_iff_valuefactors = ff_q_fsat_unit_iff_valuefactorsfactorization_product_factor * S ((S (ff_i_fsat_unit_iff_valuefactorsfactorization_product)) * mv_factor_scale_unit_iff_valuefactors) + (ff_p_fsat_unit_iff_valuefactorsfactorization_product))) /\ ((((exists ff_h_fsat_unit_iff_valuefactorsfactorization_product_partial. ff_h_fsat_unit_iff_valuefactorsfactorization_product_partial + S (ff_r_fsat_unit_iff_valuefactorsfactorization_product) = S ((S (ff_i_fsat_unit_iff_valuefactorsfactorization_product)) * ff_v_fsat_unit_iff_valuefactorsfactorization_product)) /\ exists ff_q_fsat_unit_iff_valuefactorsfactorization_product_partial. ff_u_fsat_unit_iff_valuefactorsfactorization_product = ff_q_fsat_unit_iff_valuefactorsfactorization_product_partial * S ((S (ff_i_fsat_unit_iff_valuefactorsfactorization_product)) * ff_v_fsat_unit_iff_valuefactorsfactorization_product) + (ff_r_fsat_unit_iff_valuefactorsfactorization_product))) /\ ((((exists ff_h_fsat_unit_iff_valuefactorsfactorization_product_successor. ff_h_fsat_unit_iff_valuefactorsfactorization_product_successor + S (ff_s_fsat_unit_iff_valuefactorsfactorization_product) = S ((S (S ff_i_fsat_unit_iff_valuefactorsfactorization_product)) * ff_v_fsat_unit_iff_valuefactorsfactorization_product)) /\ exists ff_q_fsat_unit_iff_valuefactorsfactorization_product_successor. ff_u_fsat_unit_iff_valuefactorsfactorization_product = ff_q_fsat_unit_iff_valuefactorsfactorization_product_successor * S ((S (S ff_i_fsat_unit_iff_valuefactorsfactorization_product)) * ff_v_fsat_unit_iff_valuefactorsfactorization_product) + (ff_s_fsat_unit_iff_valuefactorsfactorization_product))) /\ ff_s_fsat_unit_iff_valuefactorsfactorization_product = ff_r_fsat_unit_iff_valuefactorsfactorization_product * ff_p_fsat_unit_iff_valuefactorsfactorization_product)))))) /\ (forall ftsf_index_fsat_unit_iff_valuefactorsfactorization_primes. (exists ftsf_gap_fsat_unit_iff_valuefactorsfactorization_primes_bound. ftsf_gap_fsat_unit_iff_valuefactorsfactorization_primes_bound + S ftsf_index_fsat_unit_iff_valuefactorsfactorization_primes = (mv_factor_count_unit_iff_valuefactors)) -> exists ftsf_factor_fsat_unit_iff_valuefactorsfactorization_primes. ((((exists ff_h_ftsf_fsat_unit_iff_valuefactorsfactorization_primes_entry. ff_h_ftsf_fsat_unit_iff_valuefactorsfactorization_primes_entry + S (ftsf_factor_fsat_unit_iff_valuefactorsfactorization_primes) = S ((S (ftsf_index_fsat_unit_iff_valuefactorsfactorization_primes)) * mv_factor_scale_unit_iff_valuefactors)) /\ exists ff_q_ftsf_fsat_unit_iff_valuefactorsfactorization_primes_entry. mv_factor_code_unit_iff_valuefactors = ff_q_ftsf_fsat_unit_iff_valuefactorsfactorization_primes_entry * S ((S (ftsf_index_fsat_unit_iff_valuefactorsfactorization_primes)) * mv_factor_scale_unit_iff_valuefactors) + (ftsf_factor_fsat_unit_iff_valuefactorsfactorization_primes))) /\ ((~(ftsf_factor_fsat_unit_iff_valuefactorsfactorization_primes = 1) /\ forall frm_prime_left_ftsf_fsat_unit_iff_valuefactorsfactorization_primes_prime frm_prime_right_ftsf_fsat_unit_iff_valuefactorsfactorization_primes_prime. ftsf_factor_fsat_unit_iff_valuefactorsfactorization_primes = frm_prime_left_ftsf_fsat_unit_iff_valuefactorsfactorization_primes_prime * frm_prime_right_ftsf_fsat_unit_iff_valuefactorsfactorization_primes_prime -> frm_prime_left_ftsf_fsat_unit_iff_valuefactorsfactorization_primes_prime = 1 \/ frm_prime_right_ftsf_fsat_unit_iff_valuefactorsfactorization_primes_prime = 1))))))) /\ ((((exists mv_even_half_unit_iff_valuefactorsparityeven. (mv_factor_count_unit_iff_valuefactors) = 2 * mv_even_half_unit_iff_valuefactorsparityeven) /\ ((2) = 2))) \/ (((exists mv_odd_half_unit_iff_valuefactorsparityodd. (mv_factor_count_unit_iff_valuefactors) = 2 * mv_odd_half_unit_iff_valuefactorsparityodd + 1) /\ ((2) = 1)))))))))))) /\ ((((~((1) = 0)) /\ ((((exists mv_square_prime_unit_iff_value_reversesquare. ((~((mv_square_prime_unit_iff_value_reversesquare) = 1) /\ forall pvs_left_unit_iff_value_reversesquareprime pvs_right_unit_iff_value_reversesquareprime. (mv_square_prime_unit_iff_value_reversesquare) = pvs_left_unit_iff_value_reversesquareprime * pvs_right_unit_iff_value_reversesquareprime -> pvs_left_unit_iff_value_reversesquareprime = 1 \/ pvs_right_unit_iff_value_reversesquareprime = 1) /\ (exists pvs_factor_unit_iff_value_reversesquaredivisor. (1) = (mv_square_prime_unit_iff_value_reversesquare * mv_square_prime_unit_iff_value_reversesquare) * pvs_factor_unit_iff_value_reversesquaredivisor))) /\ ((2) = 0))) \/ (((((~((1) = 0)) /\ (forall sfd_prime_unit_iff_value_reversesquarefree. (~((sfd_prime_unit_iff_value_reversesquarefree) = 1) /\ forall pvs_left_unit_iff_value_reversesquarefreedomain pvs_right_unit_iff_value_reversesquarefreedomain. (sfd_prime_unit_iff_value_reversesquarefree) = pvs_left_unit_iff_value_reversesquarefreedomain * pvs_right_unit_iff_value_reversesquarefreedomain -> pvs_left_unit_iff_value_reversesquarefreedomain = 1 \/ pvs_right_unit_iff_value_reversesquarefreedomain = 1) -> (exists pvs_le_gap_unit_iff_value_reversesquarefreebound. pvs_le_gap_unit_iff_value_reversesquarefreebound + (sfd_prime_unit_iff_value_reversesquarefree) = (1)) -> ~(exists pvs_factor_unit_iff_value_reversesquarefreesquare. (1) = (sfd_prime_unit_iff_value_reversesquarefree * sfd_prime_unit_iff_value_reversesquarefree) * pvs_factor_unit_iff_value_reversesquarefreesquare)))) /\ (exists mv_factor_code_unit_iff_value_reversefactors mv_factor_scale_unit_iff_value_reversefactors mv_factor_count_unit_iff_value_reversefactors. (((~(1 = 0) /\ ((exists ff_u_fsat_unit_iff_value_reversefactorsfactorization_product ff_v_fsat_unit_iff_value_reversefactorsfactorization_product. ((((exists ff_h_fsat_unit_iff_value_reversefactorsfactorization_product_start. ff_h_fsat_unit_iff_value_reversefactorsfactorization_product_start + S (1) = S ((S (0)) * ff_v_fsat_unit_iff_value_reversefactorsfactorization_product)) /\ exists ff_q_fsat_unit_iff_value_reversefactorsfactorization_product_start. ff_u_fsat_unit_iff_value_reversefactorsfactorization_product = ff_q_fsat_unit_iff_value_reversefactorsfactorization_product_start * S ((S (0)) * ff_v_fsat_unit_iff_value_reversefactorsfactorization_product) + (1))) /\ ((((exists ff_h_fsat_unit_iff_value_reversefactorsfactorization_product_terminal. ff_h_fsat_unit_iff_value_reversefactorsfactorization_product_terminal + S (1) = S ((S (mv_factor_count_unit_iff_value_reversefactors)) * ff_v_fsat_unit_iff_value_reversefactorsfactorization_product)) /\ exists ff_q_fsat_unit_iff_value_reversefactorsfactorization_product_terminal. ff_u_fsat_unit_iff_value_reversefactorsfactorization_product = ff_q_fsat_unit_iff_value_reversefactorsfactorization_product_terminal * S ((S (mv_factor_count_unit_iff_value_reversefactors)) * ff_v_fsat_unit_iff_value_reversefactorsfactorization_product) + (1))) /\ forall ff_i_fsat_unit_iff_value_reversefactorsfactorization_product. (exists ff_lt_fsat_unit_iff_value_reversefactorsfactorization_product_bound. ff_lt_fsat_unit_iff_value_reversefactorsfactorization_product_bound + S ff_i_fsat_unit_iff_value_reversefactorsfactorization_product = mv_factor_count_unit_iff_value_reversefactors) -> exists ff_p_fsat_unit_iff_value_reversefactorsfactorization_product ff_r_fsat_unit_iff_value_reversefactorsfactorization_product ff_s_fsat_unit_iff_value_reversefactorsfactorization_product. ((((exists ff_h_fsat_unit_iff_value_reversefactorsfactorization_product_factor. ff_h_fsat_unit_iff_value_reversefactorsfactorization_product_factor + S (ff_p_fsat_unit_iff_value_reversefactorsfactorization_product) = S ((S (ff_i_fsat_unit_iff_value_reversefactorsfactorization_product)) * mv_factor_scale_unit_iff_value_reversefactors)) /\ exists ff_q_fsat_unit_iff_value_reversefactorsfactorization_product_factor. mv_factor_code_unit_iff_value_reversefactors = ff_q_fsat_unit_iff_value_reversefactorsfactorization_product_factor * S ((S (ff_i_fsat_unit_iff_value_reversefactorsfactorization_product)) * mv_factor_scale_unit_iff_value_reversefactors) + (ff_p_fsat_unit_iff_value_reversefactorsfactorization_product))) /\ ((((exists ff_h_fsat_unit_iff_value_reversefactorsfactorization_product_partial. ff_h_fsat_unit_iff_value_reversefactorsfactorization_product_partial + S (ff_r_fsat_unit_iff_value_reversefactorsfactorization_product) = S ((S (ff_i_fsat_unit_iff_value_reversefactorsfactorization_product)) * ff_v_fsat_unit_iff_value_reversefactorsfactorization_product)) /\ exists ff_q_fsat_unit_iff_value_reversefactorsfactorization_product_partial. ff_u_fsat_unit_iff_value_reversefactorsfactorization_product = ff_q_fsat_unit_iff_value_reversefactorsfactorization_product_partial * S ((S (ff_i_fsat_unit_iff_value_reversefactorsfactorization_product)) * ff_v_fsat_unit_iff_value_reversefactorsfactorization_product) + (ff_r_fsat_unit_iff_value_reversefactorsfactorization_product))) /\ ((((exists ff_h_fsat_unit_iff_value_reversefactorsfactorization_product_successor. ff_h_fsat_unit_iff_value_reversefactorsfactorization_product_successor + S (ff_s_fsat_unit_iff_value_reversefactorsfactorization_product) = S ((S (S ff_i_fsat_unit_iff_value_reversefactorsfactorization_product)) * ff_v_fsat_unit_iff_value_reversefactorsfactorization_product)) /\ exists ff_q_fsat_unit_iff_value_reversefactorsfactorization_product_successor. ff_u_fsat_unit_iff_value_reversefactorsfactorization_product = ff_q_fsat_unit_iff_value_reversefactorsfactorization_product_successor * S ((S (S ff_i_fsat_unit_iff_value_reversefactorsfactorization_product)) * ff_v_fsat_unit_iff_value_reversefactorsfactorization_product) + (ff_s_fsat_unit_iff_value_reversefactorsfactorization_product))) /\ ff_s_fsat_unit_iff_value_reversefactorsfactorization_product = ff_r_fsat_unit_iff_value_reversefactorsfactorization_product * ff_p_fsat_unit_iff_value_reversefactorsfactorization_product)))))) /\ (forall ftsf_index_fsat_unit_iff_value_reversefactorsfactorization_primes. (exists ftsf_gap_fsat_unit_iff_value_reversefactorsfactorization_primes_bound. ftsf_gap_fsat_unit_iff_value_reversefactorsfactorization_primes_bound + S ftsf_index_fsat_unit_iff_value_reversefactorsfactorization_primes = (mv_factor_count_unit_iff_value_reversefactors)) -> exists ftsf_factor_fsat_unit_iff_value_reversefactorsfactorization_primes. ((((exists ff_h_ftsf_fsat_unit_iff_value_reversefactorsfactorization_primes_entry. ff_h_ftsf_fsat_unit_iff_value_reversefactorsfactorization_primes_entry + S (ftsf_factor_fsat_unit_iff_value_reversefactorsfactorization_primes) = S ((S (ftsf_index_fsat_unit_iff_value_reversefactorsfactorization_primes)) * mv_factor_scale_unit_iff_value_reversefactors)) /\ exists ff_q_ftsf_fsat_unit_iff_value_reversefactorsfactorization_primes_entry. mv_factor_code_unit_iff_value_reversefactors = ff_q_ftsf_fsat_unit_iff_value_reversefactorsfactorization_primes_entry * S ((S (ftsf_index_fsat_unit_iff_value_reversefactorsfactorization_primes)) * mv_factor_scale_unit_iff_value_reversefactors) + (ftsf_factor_fsat_unit_iff_value_reversefactorsfactorization_primes))) /\ ((~(ftsf_factor_fsat_unit_iff_value_reversefactorsfactorization_primes = 1) /\ forall frm_prime_left_ftsf_fsat_unit_iff_value_reversefactorsfactorization_primes_prime frm_prime_right_ftsf_fsat_unit_iff_value_reversefactorsfactorization_primes_prime. ftsf_factor_fsat_unit_iff_value_reversefactorsfactorization_primes = frm_prime_left_ftsf_fsat_unit_iff_value_reversefactorsfactorization_primes_prime * frm_prime_right_ftsf_fsat_unit_iff_value_reversefactorsfactorization_primes_prime -> frm_prime_left_ftsf_fsat_unit_iff_value_reversefactorsfactorization_primes_prime = 1 \/ frm_prime_right_ftsf_fsat_unit_iff_value_reversefactorsfactorization_primes_prime = 1))))))) /\ ((((exists mv_even_half_unit_iff_value_reversefactorsparityeven. (mv_factor_count_unit_iff_value_reversefactors) = 2 * mv_even_half_unit_iff_value_reversefactorsparityeven) /\ ((2) = 2))) \/ (((exists mv_odd_half_unit_iff_value_reversefactorsparityodd. (mv_factor_count_unit_iff_value_reversefactors) = 2 * mv_odd_half_unit_iff_value_reversefactorsparityodd + 1) /\ ((2) = 1))))))))))) -> (exists dst_positive_code_unit_iff_entry_reverse dst_positive_scale_unit_iff_entry_reverse dst_negative_code_unit_iff_entry_reverse dst_negative_scale_unit_iff_entry_reverse dst_positive_unit_iff_entry_reverse dst_negative_unit_iff_entry_reverse. (((M) = (((((dst_positive_code_unit_iff_entry_reverse) + (dst_positive_scale_unit_iff_entry_reverse)) * S ((dst_positive_code_unit_iff_entry_reverse) + (dst_positive_scale_unit_iff_entry_reverse)) + ((dst_positive_scale_unit_iff_entry_reverse) + (dst_positive_scale_unit_iff_entry_reverse))) + (((dst_negative_code_unit_iff_entry_reverse) + (dst_negative_scale_unit_iff_entry_reverse)) * S ((dst_negative_code_unit_iff_entry_reverse) + (dst_negative_scale_unit_iff_entry_reverse)) + ((dst_negative_scale_unit_iff_entry_reverse) + (dst_negative_scale_unit_iff_entry_reverse)))) * S ((((dst_positive_code_unit_iff_entry_reverse) + (dst_positive_scale_unit_iff_entry_reverse)) * S ((dst_positive_code_unit_iff_entry_reverse) + (dst_positive_scale_unit_iff_entry_reverse)) + ((dst_positive_scale_unit_iff_entry_reverse) + (dst_positive_scale_unit_iff_entry_reverse))) + (((dst_negative_code_unit_iff_entry_reverse) + (dst_negative_scale_unit_iff_entry_reverse)) * S ((dst_negative_code_unit_iff_entry_reverse) + (dst_negative_scale_unit_iff_entry_reverse)) + ((dst_negative_scale_unit_iff_entry_reverse) + (dst_negative_scale_unit_iff_entry_reverse)))) + ((((dst_negative_code_unit_iff_entry_reverse) + (dst_negative_scale_unit_iff_entry_reverse)) * S ((dst_negative_code_unit_iff_entry_reverse) + (dst_negative_scale_unit_iff_entry_reverse)) + ((dst_negative_scale_unit_iff_entry_reverse) + (dst_negative_scale_unit_iff_entry_reverse))) + (((dst_negative_code_unit_iff_entry_reverse) + (dst_negative_scale_unit_iff_entry_reverse)) * S ((dst_negative_code_unit_iff_entry_reverse) + (dst_negative_scale_unit_iff_entry_reverse)) + ((dst_negative_scale_unit_iff_entry_reverse) + (dst_negative_scale_unit_iff_entry_reverse)))))) /\ (((((exists ff_h_pvs_unit_iff_entry_reversepositive. ff_h_pvs_unit_iff_entry_reversepositive + S (dst_positive_unit_iff_entry_reverse) = S ((S (1)) * dst_positive_scale_unit_iff_entry_reverse)) /\ exists ff_q_pvs_unit_iff_entry_reversepositive. dst_positive_code_unit_iff_entry_reverse = ff_q_pvs_unit_iff_entry_reversepositive * S ((S (1)) * dst_positive_scale_unit_iff_entry_reverse) + (dst_positive_unit_iff_entry_reverse))) /\ (((((exists ff_h_pvs_unit_iff_entry_reversenegative. ff_h_pvs_unit_iff_entry_reversenegative + S (dst_negative_unit_iff_entry_reverse) = S ((S (1)) * dst_negative_scale_unit_iff_entry_reverse)) /\ exists ff_q_pvs_unit_iff_entry_reversenegative. dst_negative_code_unit_iff_entry_reverse = ff_q_pvs_unit_iff_entry_reversenegative * S ((S (1)) * dst_negative_scale_unit_iff_entry_reverse) + (dst_negative_unit_iff_entry_reverse))) /\ (exists ge_balance_positive_unit_iff_entry_reversevalue ge_balance_negative_unit_iff_entry_reversevalue. (((((2) = 2 * (ge_balance_positive_unit_iff_entry_reversevalue) /\ (ge_balance_negative_unit_iff_entry_reversevalue) = 0) \/ exists ge_signed_half_unit_iff_entry_reversevaluedecode. (((2) = 2 * ge_signed_half_unit_iff_entry_reversevaluedecode + 1 /\ (ge_balance_positive_unit_iff_entry_reversevalue) = 0) /\ (ge_balance_negative_unit_iff_entry_reversevalue) = S ge_signed_half_unit_iff_entry_reversevaluedecode))) /\ ((dst_positive_unit_iff_entry_reverse) + ge_balance_negative_unit_iff_entry_reversevalue = (dst_negative_unit_iff_entry_reverse) + ge_balance_positive_unit_iff_entry_reversevalue))))))))))
  6. 0006specialize mobius_table_entry_iff (N)
  7. 0007specialize mobius_table_entry_iff (M)
  8. 0008specialize mobius_table_entry_iff (1)
  9. 0009specialize mobius_table_entry_iff (2)
  10. 0010apply mobius_table_entry_iff
  11. 0011exact hm
  12. 0012intro hzero
  13. 0013apply PA1
  14. 0014exact hzero
  15. 0015exact hN
  16. 0016cases hiff
  17. 0017apply hiff_right
  18. 0018exact mobius_one