MC0011

mobius_divisor_mask_actual_value

Every retained entry of an actual Möbius divisor mask is the independent positive-input Möbius value at that divisor.

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

Current library: Alpha v34, 4,223 checked-use theorems; Stable remains 432. Historical first admissions, original proof editions, and non-admitted aliases are preserved.

The input n is positive. Signed code 2 denotes +1, so the actual sum is +1 at n=1 and zero for n>1. The positive-values result permits arbitrary F(0), which the divisor mask excludes. Prime-square multiples contribute zero. Full G007 inversion is established in its separate family.

Exact theorem in conservative defined notation

∀ N. ∀ M. ∀ n. ∀ K. ∀ d. ∀ z. MobiusTable(N,M)Le(n,N)DivisorMask(M,n,n,K) → ¬d = 0 → Dvd(d,n)Le(d,n)ArithAt(K,d,z)Mobius(d,z)

Every linked abbreviation expands hygienically to the identical original native formula.

Definition DAG

Actual proof prerequisites

Original expanded first-order statement
forall N M n K d z. (((exists dst_positive_code_mask_mu_tabletable dst_positive_scale_mask_mu_tabletable dst_negative_code_mask_mu_tabletable dst_negative_scale_mask_mu_tabletable. (((M) = (((((dst_positive_code_mask_mu_tabletable) + (dst_positive_scale_mask_mu_tabletable)) * S ((dst_positive_code_mask_mu_tabletable) + (dst_positive_scale_mask_mu_tabletable)) + ((dst_positive_scale_mask_mu_tabletable) + (dst_positive_scale_mask_mu_tabletable))) + (((dst_negative_code_mask_mu_tabletable) + (dst_negative_scale_mask_mu_tabletable)) * S ((dst_negative_code_mask_mu_tabletable) + (dst_negative_scale_mask_mu_tabletable)) + ((dst_negative_scale_mask_mu_tabletable) + (dst_negative_scale_mask_mu_tabletable)))) * S ((((dst_positive_code_mask_mu_tabletable) + (dst_positive_scale_mask_mu_tabletable)) * S ((dst_positive_code_mask_mu_tabletable) + (dst_positive_scale_mask_mu_tabletable)) + ((dst_positive_scale_mask_mu_tabletable) + (dst_positive_scale_mask_mu_tabletable))) + (((dst_negative_code_mask_mu_tabletable) + (dst_negative_scale_mask_mu_tabletable)) * S ((dst_negative_code_mask_mu_tabletable) + (dst_negative_scale_mask_mu_tabletable)) + ((dst_negative_scale_mask_mu_tabletable) + (dst_negative_scale_mask_mu_tabletable)))) + ((((dst_negative_code_mask_mu_tabletable) + (dst_negative_scale_mask_mu_tabletable)) * S ((dst_negative_code_mask_mu_tabletable) + (dst_negative_scale_mask_mu_tabletable)) + ((dst_negative_scale_mask_mu_tabletable) + (dst_negative_scale_mask_mu_tabletable))) + (((dst_negative_code_mask_mu_tabletable) + (dst_negative_scale_mask_mu_tabletable)) * S ((dst_negative_code_mask_mu_tabletable) + (dst_negative_scale_mask_mu_tabletable)) + ((dst_negative_scale_mask_mu_tabletable) + (dst_negative_scale_mask_mu_tabletable)))))) /\ (forall dst_index_mask_mu_tabletable. (exists pvs_le_gap_mask_mu_tabletabledomain. pvs_le_gap_mask_mu_tabletabledomain + (dst_index_mask_mu_tabletable) = (N)) -> exists dst_positive_mask_mu_tabletable dst_negative_mask_mu_tabletable dst_value_mask_mu_tabletable. ((((exists ff_h_pvs_mask_mu_tabletableentrypositive. ff_h_pvs_mask_mu_tabletableentrypositive + S (dst_positive_mask_mu_tabletable) = S ((S (dst_index_mask_mu_tabletable)) * dst_positive_scale_mask_mu_tabletable)) /\ exists ff_q_pvs_mask_mu_tabletableentrypositive. dst_positive_code_mask_mu_tabletable = ff_q_pvs_mask_mu_tabletableentrypositive * S ((S (dst_index_mask_mu_tabletable)) * dst_positive_scale_mask_mu_tabletable) + (dst_positive_mask_mu_tabletable))) /\ (((((exists ff_h_pvs_mask_mu_tabletableentrynegative. ff_h_pvs_mask_mu_tabletableentrynegative + S (dst_negative_mask_mu_tabletable) = S ((S (dst_index_mask_mu_tabletable)) * dst_negative_scale_mask_mu_tabletable)) /\ exists ff_q_pvs_mask_mu_tabletableentrynegative. dst_negative_code_mask_mu_tabletable = ff_q_pvs_mask_mu_tabletableentrynegative * S ((S (dst_index_mask_mu_tabletable)) * dst_negative_scale_mask_mu_tabletable) + (dst_negative_mask_mu_tabletable))) /\ (exists ge_balance_positive_mask_mu_tabletableentryvalue ge_balance_negative_mask_mu_tabletableentryvalue. (((((dst_value_mask_mu_tabletable) = 2 * (ge_balance_positive_mask_mu_tabletableentryvalue) /\ (ge_balance_negative_mask_mu_tabletableentryvalue) = 0) \/ exists ge_signed_half_mask_mu_tabletableentryvaluedecode. (((dst_value_mask_mu_tabletable) = 2 * ge_signed_half_mask_mu_tabletableentryvaluedecode + 1 /\ (ge_balance_positive_mask_mu_tabletableentryvalue) = 0) /\ (ge_balance_negative_mask_mu_tabletableentryvalue) = S ge_signed_half_mask_mu_tabletableentryvaluedecode))) /\ ((dst_positive_mask_mu_tabletable) + ge_balance_negative_mask_mu_tabletableentryvalue = (dst_negative_mask_mu_tabletable) + ge_balance_positive_mask_mu_tabletableentryvalue))))))))) /\ (((exists dst_positive_code_mask_mu_tablezero dst_positive_scale_mask_mu_tablezero dst_negative_code_mask_mu_tablezero dst_negative_scale_mask_mu_tablezero dst_positive_mask_mu_tablezero dst_negative_mask_mu_tablezero. (((M) = (((((dst_positive_code_mask_mu_tablezero) + (dst_positive_scale_mask_mu_tablezero)) * S ((dst_positive_code_mask_mu_tablezero) + (dst_positive_scale_mask_mu_tablezero)) + ((dst_positive_scale_mask_mu_tablezero) + (dst_positive_scale_mask_mu_tablezero))) + (((dst_negative_code_mask_mu_tablezero) + (dst_negative_scale_mask_mu_tablezero)) * S ((dst_negative_code_mask_mu_tablezero) + (dst_negative_scale_mask_mu_tablezero)) + ((dst_negative_scale_mask_mu_tablezero) + (dst_negative_scale_mask_mu_tablezero)))) * S ((((dst_positive_code_mask_mu_tablezero) + (dst_positive_scale_mask_mu_tablezero)) * S ((dst_positive_code_mask_mu_tablezero) + (dst_positive_scale_mask_mu_tablezero)) + ((dst_positive_scale_mask_mu_tablezero) + (dst_positive_scale_mask_mu_tablezero))) + (((dst_negative_code_mask_mu_tablezero) + (dst_negative_scale_mask_mu_tablezero)) * S ((dst_negative_code_mask_mu_tablezero) + (dst_negative_scale_mask_mu_tablezero)) + ((dst_negative_scale_mask_mu_tablezero) + (dst_negative_scale_mask_mu_tablezero)))) + ((((dst_negative_code_mask_mu_tablezero) + (dst_negative_scale_mask_mu_tablezero)) * S ((dst_negative_code_mask_mu_tablezero) + (dst_negative_scale_mask_mu_tablezero)) + ((dst_negative_scale_mask_mu_tablezero) + (dst_negative_scale_mask_mu_tablezero))) + (((dst_negative_code_mask_mu_tablezero) + (dst_negative_scale_mask_mu_tablezero)) * S ((dst_negative_code_mask_mu_tablezero) + (dst_negative_scale_mask_mu_tablezero)) + ((dst_negative_scale_mask_mu_tablezero) + (dst_negative_scale_mask_mu_tablezero)))))) /\ (((((exists ff_h_pvs_mask_mu_tablezeropositive. ff_h_pvs_mask_mu_tablezeropositive + S (dst_positive_mask_mu_tablezero) = S ((S (0)) * dst_positive_scale_mask_mu_tablezero)) /\ exists ff_q_pvs_mask_mu_tablezeropositive. dst_positive_code_mask_mu_tablezero = ff_q_pvs_mask_mu_tablezeropositive * S ((S (0)) * dst_positive_scale_mask_mu_tablezero) + (dst_positive_mask_mu_tablezero))) /\ (((((exists ff_h_pvs_mask_mu_tablezeronegative. ff_h_pvs_mask_mu_tablezeronegative + S (dst_negative_mask_mu_tablezero) = S ((S (0)) * dst_negative_scale_mask_mu_tablezero)) /\ exists ff_q_pvs_mask_mu_tablezeronegative. dst_negative_code_mask_mu_tablezero = ff_q_pvs_mask_mu_tablezeronegative * S ((S (0)) * dst_negative_scale_mask_mu_tablezero) + (dst_negative_mask_mu_tablezero))) /\ (exists ge_balance_positive_mask_mu_tablezerovalue ge_balance_negative_mask_mu_tablezerovalue. (((((0) = 2 * (ge_balance_positive_mask_mu_tablezerovalue) /\ (ge_balance_negative_mask_mu_tablezerovalue) = 0) \/ exists ge_signed_half_mask_mu_tablezerovaluedecode. (((0) = 2 * ge_signed_half_mask_mu_tablezerovaluedecode + 1 /\ (ge_balance_positive_mask_mu_tablezerovalue) = 0) /\ (ge_balance_negative_mask_mu_tablezerovalue) = S ge_signed_half_mask_mu_tablezerovaluedecode))) /\ ((dst_positive_mask_mu_tablezero) + ge_balance_negative_mask_mu_tablezerovalue = (dst_negative_mask_mu_tablezero) + ge_balance_positive_mask_mu_tablezerovalue))))))))) /\ (forall mt_index_mask_mu_table mt_value_mask_mu_table. ~(mt_index_mask_mu_table=0) -> (exists pvs_le_gap_mask_mu_tabledomain. pvs_le_gap_mask_mu_tabledomain + (mt_index_mask_mu_table) = (N)) -> (exists dst_positive_code_mask_mu_tableentry dst_positive_scale_mask_mu_tableentry dst_negative_code_mask_mu_tableentry dst_negative_scale_mask_mu_tableentry dst_positive_mask_mu_tableentry dst_negative_mask_mu_tableentry. (((M) = (((((dst_positive_code_mask_mu_tableentry) + (dst_positive_scale_mask_mu_tableentry)) * S ((dst_positive_code_mask_mu_tableentry) + (dst_positive_scale_mask_mu_tableentry)) + ((dst_positive_scale_mask_mu_tableentry) + (dst_positive_scale_mask_mu_tableentry))) + (((dst_negative_code_mask_mu_tableentry) + (dst_negative_scale_mask_mu_tableentry)) * S ((dst_negative_code_mask_mu_tableentry) + (dst_negative_scale_mask_mu_tableentry)) + ((dst_negative_scale_mask_mu_tableentry) + (dst_negative_scale_mask_mu_tableentry)))) * S ((((dst_positive_code_mask_mu_tableentry) + (dst_positive_scale_mask_mu_tableentry)) * S ((dst_positive_code_mask_mu_tableentry) + (dst_positive_scale_mask_mu_tableentry)) + ((dst_positive_scale_mask_mu_tableentry) + (dst_positive_scale_mask_mu_tableentry))) + (((dst_negative_code_mask_mu_tableentry) + (dst_negative_scale_mask_mu_tableentry)) * S ((dst_negative_code_mask_mu_tableentry) + (dst_negative_scale_mask_mu_tableentry)) + ((dst_negative_scale_mask_mu_tableentry) + (dst_negative_scale_mask_mu_tableentry)))) + ((((dst_negative_code_mask_mu_tableentry) + (dst_negative_scale_mask_mu_tableentry)) * S ((dst_negative_code_mask_mu_tableentry) + (dst_negative_scale_mask_mu_tableentry)) + ((dst_negative_scale_mask_mu_tableentry) + (dst_negative_scale_mask_mu_tableentry))) + (((dst_negative_code_mask_mu_tableentry) + (dst_negative_scale_mask_mu_tableentry)) * S ((dst_negative_code_mask_mu_tableentry) + (dst_negative_scale_mask_mu_tableentry)) + ((dst_negative_scale_mask_mu_tableentry) + (dst_negative_scale_mask_mu_tableentry)))))) /\ (((((exists ff_h_pvs_mask_mu_tableentrypositive. ff_h_pvs_mask_mu_tableentrypositive + S (dst_positive_mask_mu_tableentry) = S ((S (mt_index_mask_mu_table)) * dst_positive_scale_mask_mu_tableentry)) /\ exists ff_q_pvs_mask_mu_tableentrypositive. dst_positive_code_mask_mu_tableentry = ff_q_pvs_mask_mu_tableentrypositive * S ((S (mt_index_mask_mu_table)) * dst_positive_scale_mask_mu_tableentry) + (dst_positive_mask_mu_tableentry))) /\ (((((exists ff_h_pvs_mask_mu_tableentrynegative. ff_h_pvs_mask_mu_tableentrynegative + S (dst_negative_mask_mu_tableentry) = S ((S (mt_index_mask_mu_table)) * dst_negative_scale_mask_mu_tableentry)) /\ exists ff_q_pvs_mask_mu_tableentrynegative. dst_negative_code_mask_mu_tableentry = ff_q_pvs_mask_mu_tableentrynegative * S ((S (mt_index_mask_mu_table)) * dst_negative_scale_mask_mu_tableentry) + (dst_negative_mask_mu_tableentry))) /\ (exists ge_balance_positive_mask_mu_tableentryvalue ge_balance_negative_mask_mu_tableentryvalue. (((((mt_value_mask_mu_table) = 2 * (ge_balance_positive_mask_mu_tableentryvalue) /\ (ge_balance_negative_mask_mu_tableentryvalue) = 0) \/ exists ge_signed_half_mask_mu_tableentryvaluedecode. (((mt_value_mask_mu_table) = 2 * ge_signed_half_mask_mu_tableentryvaluedecode + 1 /\ (ge_balance_positive_mask_mu_tableentryvalue) = 0) /\ (ge_balance_negative_mask_mu_tableentryvalue) = S ge_signed_half_mask_mu_tableentryvaluedecode))) /\ ((dst_positive_mask_mu_tableentry) + ge_balance_negative_mask_mu_tableentryvalue = (dst_negative_mask_mu_tableentry) + ge_balance_positive_mask_mu_tableentryvalue))))))))) -> (((~((mt_index_mask_mu_table) = 0)) /\ ((((exists mv_square_prime_mask_mu_tablevaluesquare. ((~((mv_square_prime_mask_mu_tablevaluesquare) = 1) /\ forall pvs_left_mask_mu_tablevaluesquareprime pvs_right_mask_mu_tablevaluesquareprime. (mv_square_prime_mask_mu_tablevaluesquare) = pvs_left_mask_mu_tablevaluesquareprime * pvs_right_mask_mu_tablevaluesquareprime -> pvs_left_mask_mu_tablevaluesquareprime = 1 \/ pvs_right_mask_mu_tablevaluesquareprime = 1) /\ (exists pvs_factor_mask_mu_tablevaluesquaredivisor. (mt_index_mask_mu_table) = (mv_square_prime_mask_mu_tablevaluesquare * mv_square_prime_mask_mu_tablevaluesquare) * pvs_factor_mask_mu_tablevaluesquaredivisor))) /\ ((mt_value_mask_mu_table) = 0))) \/ (((((~((mt_index_mask_mu_table) = 0)) /\ (forall sfd_prime_mask_mu_tablevaluesquarefree. (~((sfd_prime_mask_mu_tablevaluesquarefree) = 1) /\ forall pvs_left_mask_mu_tablevaluesquarefreedomain pvs_right_mask_mu_tablevaluesquarefreedomain. (sfd_prime_mask_mu_tablevaluesquarefree) = pvs_left_mask_mu_tablevaluesquarefreedomain * pvs_right_mask_mu_tablevaluesquarefreedomain -> pvs_left_mask_mu_tablevaluesquarefreedomain = 1 \/ pvs_right_mask_mu_tablevaluesquarefreedomain = 1) -> (exists pvs_le_gap_mask_mu_tablevaluesquarefreebound. pvs_le_gap_mask_mu_tablevaluesquarefreebound + (sfd_prime_mask_mu_tablevaluesquarefree) = (mt_index_mask_mu_table)) -> ~(exists pvs_factor_mask_mu_tablevaluesquarefreesquare. (mt_index_mask_mu_table) = (sfd_prime_mask_mu_tablevaluesquarefree * sfd_prime_mask_mu_tablevaluesquarefree) * pvs_factor_mask_mu_tablevaluesquarefreesquare)))) /\ (exists mv_factor_code_mask_mu_tablevaluefactors mv_factor_scale_mask_mu_tablevaluefactors mv_factor_count_mask_mu_tablevaluefactors. (((~(mt_index_mask_mu_table = 0) /\ ((exists ff_u_fsat_mask_mu_tablevaluefactorsfactorization_product ff_v_fsat_mask_mu_tablevaluefactorsfactorization_product. ((((exists ff_h_fsat_mask_mu_tablevaluefactorsfactorization_product_start. ff_h_fsat_mask_mu_tablevaluefactorsfactorization_product_start + S (1) = S ((S (0)) * ff_v_fsat_mask_mu_tablevaluefactorsfactorization_product)) /\ exists ff_q_fsat_mask_mu_tablevaluefactorsfactorization_product_start. ff_u_fsat_mask_mu_tablevaluefactorsfactorization_product = ff_q_fsat_mask_mu_tablevaluefactorsfactorization_product_start * S ((S (0)) * ff_v_fsat_mask_mu_tablevaluefactorsfactorization_product) + (1))) /\ ((((exists ff_h_fsat_mask_mu_tablevaluefactorsfactorization_product_terminal. ff_h_fsat_mask_mu_tablevaluefactorsfactorization_product_terminal + S (mt_index_mask_mu_table) = S ((S (mv_factor_count_mask_mu_tablevaluefactors)) * ff_v_fsat_mask_mu_tablevaluefactorsfactorization_product)) /\ exists ff_q_fsat_mask_mu_tablevaluefactorsfactorization_product_terminal. ff_u_fsat_mask_mu_tablevaluefactorsfactorization_product = ff_q_fsat_mask_mu_tablevaluefactorsfactorization_product_terminal * S ((S (mv_factor_count_mask_mu_tablevaluefactors)) * ff_v_fsat_mask_mu_tablevaluefactorsfactorization_product) + (mt_index_mask_mu_table))) /\ forall ff_i_fsat_mask_mu_tablevaluefactorsfactorization_product. (exists ff_lt_fsat_mask_mu_tablevaluefactorsfactorization_product_bound. ff_lt_fsat_mask_mu_tablevaluefactorsfactorization_product_bound + S ff_i_fsat_mask_mu_tablevaluefactorsfactorization_product = mv_factor_count_mask_mu_tablevaluefactors) -> exists ff_p_fsat_mask_mu_tablevaluefactorsfactorization_product ff_r_fsat_mask_mu_tablevaluefactorsfactorization_product ff_s_fsat_mask_mu_tablevaluefactorsfactorization_product. ((((exists ff_h_fsat_mask_mu_tablevaluefactorsfactorization_product_factor. ff_h_fsat_mask_mu_tablevaluefactorsfactorization_product_factor + S (ff_p_fsat_mask_mu_tablevaluefactorsfactorization_product) = S ((S (ff_i_fsat_mask_mu_tablevaluefactorsfactorization_product)) * mv_factor_scale_mask_mu_tablevaluefactors)) /\ exists ff_q_fsat_mask_mu_tablevaluefactorsfactorization_product_factor. mv_factor_code_mask_mu_tablevaluefactors = ff_q_fsat_mask_mu_tablevaluefactorsfactorization_product_factor * S ((S (ff_i_fsat_mask_mu_tablevaluefactorsfactorization_product)) * mv_factor_scale_mask_mu_tablevaluefactors) + (ff_p_fsat_mask_mu_tablevaluefactorsfactorization_product))) /\ ((((exists ff_h_fsat_mask_mu_tablevaluefactorsfactorization_product_partial. ff_h_fsat_mask_mu_tablevaluefactorsfactorization_product_partial + S (ff_r_fsat_mask_mu_tablevaluefactorsfactorization_product) = S ((S (ff_i_fsat_mask_mu_tablevaluefactorsfactorization_product)) * ff_v_fsat_mask_mu_tablevaluefactorsfactorization_product)) /\ exists ff_q_fsat_mask_mu_tablevaluefactorsfactorization_product_partial. ff_u_fsat_mask_mu_tablevaluefactorsfactorization_product = ff_q_fsat_mask_mu_tablevaluefactorsfactorization_product_partial * S ((S (ff_i_fsat_mask_mu_tablevaluefactorsfactorization_product)) * ff_v_fsat_mask_mu_tablevaluefactorsfactorization_product) + (ff_r_fsat_mask_mu_tablevaluefactorsfactorization_product))) /\ ((((exists ff_h_fsat_mask_mu_tablevaluefactorsfactorization_product_successor. ff_h_fsat_mask_mu_tablevaluefactorsfactorization_product_successor + S (ff_s_fsat_mask_mu_tablevaluefactorsfactorization_product) = S ((S (S ff_i_fsat_mask_mu_tablevaluefactorsfactorization_product)) * ff_v_fsat_mask_mu_tablevaluefactorsfactorization_product)) /\ exists ff_q_fsat_mask_mu_tablevaluefactorsfactorization_product_successor. ff_u_fsat_mask_mu_tablevaluefactorsfactorization_product = ff_q_fsat_mask_mu_tablevaluefactorsfactorization_product_successor * S ((S (S ff_i_fsat_mask_mu_tablevaluefactorsfactorization_product)) * ff_v_fsat_mask_mu_tablevaluefactorsfactorization_product) + (ff_s_fsat_mask_mu_tablevaluefactorsfactorization_product))) /\ ff_s_fsat_mask_mu_tablevaluefactorsfactorization_product = ff_r_fsat_mask_mu_tablevaluefactorsfactorization_product * ff_p_fsat_mask_mu_tablevaluefactorsfactorization_product)))))) /\ (forall ftsf_index_fsat_mask_mu_tablevaluefactorsfactorization_primes. (exists ftsf_gap_fsat_mask_mu_tablevaluefactorsfactorization_primes_bound. ftsf_gap_fsat_mask_mu_tablevaluefactorsfactorization_primes_bound + S ftsf_index_fsat_mask_mu_tablevaluefactorsfactorization_primes = (mv_factor_count_mask_mu_tablevaluefactors)) -> exists ftsf_factor_fsat_mask_mu_tablevaluefactorsfactorization_primes. ((((exists ff_h_ftsf_fsat_mask_mu_tablevaluefactorsfactorization_primes_entry. ff_h_ftsf_fsat_mask_mu_tablevaluefactorsfactorization_primes_entry + S (ftsf_factor_fsat_mask_mu_tablevaluefactorsfactorization_primes) = S ((S (ftsf_index_fsat_mask_mu_tablevaluefactorsfactorization_primes)) * mv_factor_scale_mask_mu_tablevaluefactors)) /\ exists ff_q_ftsf_fsat_mask_mu_tablevaluefactorsfactorization_primes_entry. mv_factor_code_mask_mu_tablevaluefactors = ff_q_ftsf_fsat_mask_mu_tablevaluefactorsfactorization_primes_entry * S ((S (ftsf_index_fsat_mask_mu_tablevaluefactorsfactorization_primes)) * mv_factor_scale_mask_mu_tablevaluefactors) + (ftsf_factor_fsat_mask_mu_tablevaluefactorsfactorization_primes))) /\ ((~(ftsf_factor_fsat_mask_mu_tablevaluefactorsfactorization_primes = 1) /\ forall frm_prime_left_ftsf_fsat_mask_mu_tablevaluefactorsfactorization_primes_prime frm_prime_right_ftsf_fsat_mask_mu_tablevaluefactorsfactorization_primes_prime. ftsf_factor_fsat_mask_mu_tablevaluefactorsfactorization_primes = frm_prime_left_ftsf_fsat_mask_mu_tablevaluefactorsfactorization_primes_prime * frm_prime_right_ftsf_fsat_mask_mu_tablevaluefactorsfactorization_primes_prime -> frm_prime_left_ftsf_fsat_mask_mu_tablevaluefactorsfactorization_primes_prime = 1 \/ frm_prime_right_ftsf_fsat_mask_mu_tablevaluefactorsfactorization_primes_prime = 1))))))) /\ ((((exists mv_even_half_mask_mu_tablevaluefactorsparityeven. (mv_factor_count_mask_mu_tablevaluefactors) = 2 * mv_even_half_mask_mu_tablevaluefactorsparityeven) /\ ((mt_value_mask_mu_table) = 2))) \/ (((exists mv_odd_half_mask_mu_tablevaluefactorsparityodd. (mv_factor_count_mask_mu_tablevaluefactors) = 2 * mv_odd_half_mask_mu_tablevaluefactorsparityodd + 1) /\ ((mt_value_mask_mu_table) = 1)))))))))))))))) -> (exists pvs_le_gap_mask_input_bound. pvs_le_gap_mask_input_bound + (n) = (N)) -> (((exists dst_positive_code_mask_sourcetable dst_positive_scale_mask_sourcetable dst_negative_code_mask_sourcetable dst_negative_scale_mask_sourcetable. (((K) = (((((dst_positive_code_mask_sourcetable) + (dst_positive_scale_mask_sourcetable)) * S ((dst_positive_code_mask_sourcetable) + (dst_positive_scale_mask_sourcetable)) + ((dst_positive_scale_mask_sourcetable) + (dst_positive_scale_mask_sourcetable))) + (((dst_negative_code_mask_sourcetable) + (dst_negative_scale_mask_sourcetable)) * S ((dst_negative_code_mask_sourcetable) + (dst_negative_scale_mask_sourcetable)) + ((dst_negative_scale_mask_sourcetable) + (dst_negative_scale_mask_sourcetable)))) * S ((((dst_positive_code_mask_sourcetable) + (dst_positive_scale_mask_sourcetable)) * S ((dst_positive_code_mask_sourcetable) + (dst_positive_scale_mask_sourcetable)) + ((dst_positive_scale_mask_sourcetable) + (dst_positive_scale_mask_sourcetable))) + (((dst_negative_code_mask_sourcetable) + (dst_negative_scale_mask_sourcetable)) * S ((dst_negative_code_mask_sourcetable) + (dst_negative_scale_mask_sourcetable)) + ((dst_negative_scale_mask_sourcetable) + (dst_negative_scale_mask_sourcetable)))) + ((((dst_negative_code_mask_sourcetable) + (dst_negative_scale_mask_sourcetable)) * S ((dst_negative_code_mask_sourcetable) + (dst_negative_scale_mask_sourcetable)) + ((dst_negative_scale_mask_sourcetable) + (dst_negative_scale_mask_sourcetable))) + (((dst_negative_code_mask_sourcetable) + (dst_negative_scale_mask_sourcetable)) * S ((dst_negative_code_mask_sourcetable) + (dst_negative_scale_mask_sourcetable)) + ((dst_negative_scale_mask_sourcetable) + (dst_negative_scale_mask_sourcetable)))))) /\ (forall dst_index_mask_sourcetable. (exists pvs_le_gap_mask_sourcetabledomain. pvs_le_gap_mask_sourcetabledomain + (dst_index_mask_sourcetable) = (n)) -> exists dst_positive_mask_sourcetable dst_negative_mask_sourcetable dst_value_mask_sourcetable. ((((exists ff_h_pvs_mask_sourcetableentrypositive. ff_h_pvs_mask_sourcetableentrypositive + S (dst_positive_mask_sourcetable) = S ((S (dst_index_mask_sourcetable)) * dst_positive_scale_mask_sourcetable)) /\ exists ff_q_pvs_mask_sourcetableentrypositive. dst_positive_code_mask_sourcetable = ff_q_pvs_mask_sourcetableentrypositive * S ((S (dst_index_mask_sourcetable)) * dst_positive_scale_mask_sourcetable) + (dst_positive_mask_sourcetable))) /\ (((((exists ff_h_pvs_mask_sourcetableentrynegative. ff_h_pvs_mask_sourcetableentrynegative + S (dst_negative_mask_sourcetable) = S ((S (dst_index_mask_sourcetable)) * dst_negative_scale_mask_sourcetable)) /\ exists ff_q_pvs_mask_sourcetableentrynegative. dst_negative_code_mask_sourcetable = ff_q_pvs_mask_sourcetableentrynegative * S ((S (dst_index_mask_sourcetable)) * dst_negative_scale_mask_sourcetable) + (dst_negative_mask_sourcetable))) /\ (exists ge_balance_positive_mask_sourcetableentryvalue ge_balance_negative_mask_sourcetableentryvalue. (((((dst_value_mask_sourcetable) = 2 * (ge_balance_positive_mask_sourcetableentryvalue) /\ (ge_balance_negative_mask_sourcetableentryvalue) = 0) \/ exists ge_signed_half_mask_sourcetableentryvaluedecode. (((dst_value_mask_sourcetable) = 2 * ge_signed_half_mask_sourcetableentryvaluedecode + 1 /\ (ge_balance_positive_mask_sourcetableentryvalue) = 0) /\ (ge_balance_negative_mask_sourcetableentryvalue) = S ge_signed_half_mask_sourcetableentryvaluedecode))) /\ ((dst_positive_mask_sourcetable) + ge_balance_negative_mask_sourcetableentryvalue = (dst_negative_mask_sourcetable) + ge_balance_positive_mask_sourcetableentryvalue))))))))) /\ (forall dm_index_mask_source dm_value_mask_source. (exists pvs_le_gap_mask_sourcedomain. pvs_le_gap_mask_sourcedomain + (dm_index_mask_source) = (n)) -> (exists dst_positive_code_mask_sourcelookup dst_positive_scale_mask_sourcelookup dst_negative_code_mask_sourcelookup dst_negative_scale_mask_sourcelookup dst_positive_mask_sourcelookup dst_negative_mask_sourcelookup. (((K) = (((((dst_positive_code_mask_sourcelookup) + (dst_positive_scale_mask_sourcelookup)) * S ((dst_positive_code_mask_sourcelookup) + (dst_positive_scale_mask_sourcelookup)) + ((dst_positive_scale_mask_sourcelookup) + (dst_positive_scale_mask_sourcelookup))) + (((dst_negative_code_mask_sourcelookup) + (dst_negative_scale_mask_sourcelookup)) * S ((dst_negative_code_mask_sourcelookup) + (dst_negative_scale_mask_sourcelookup)) + ((dst_negative_scale_mask_sourcelookup) + (dst_negative_scale_mask_sourcelookup)))) * S ((((dst_positive_code_mask_sourcelookup) + (dst_positive_scale_mask_sourcelookup)) * S ((dst_positive_code_mask_sourcelookup) + (dst_positive_scale_mask_sourcelookup)) + ((dst_positive_scale_mask_sourcelookup) + (dst_positive_scale_mask_sourcelookup))) + (((dst_negative_code_mask_sourcelookup) + (dst_negative_scale_mask_sourcelookup)) * S ((dst_negative_code_mask_sourcelookup) + (dst_negative_scale_mask_sourcelookup)) + ((dst_negative_scale_mask_sourcelookup) + (dst_negative_scale_mask_sourcelookup)))) + ((((dst_negative_code_mask_sourcelookup) + (dst_negative_scale_mask_sourcelookup)) * S ((dst_negative_code_mask_sourcelookup) + (dst_negative_scale_mask_sourcelookup)) + ((dst_negative_scale_mask_sourcelookup) + (dst_negative_scale_mask_sourcelookup))) + (((dst_negative_code_mask_sourcelookup) + (dst_negative_scale_mask_sourcelookup)) * S ((dst_negative_code_mask_sourcelookup) + (dst_negative_scale_mask_sourcelookup)) + ((dst_negative_scale_mask_sourcelookup) + (dst_negative_scale_mask_sourcelookup)))))) /\ (((((exists ff_h_pvs_mask_sourcelookuppositive. ff_h_pvs_mask_sourcelookuppositive + S (dst_positive_mask_sourcelookup) = S ((S (dm_index_mask_source)) * dst_positive_scale_mask_sourcelookup)) /\ exists ff_q_pvs_mask_sourcelookuppositive. dst_positive_code_mask_sourcelookup = ff_q_pvs_mask_sourcelookuppositive * S ((S (dm_index_mask_source)) * dst_positive_scale_mask_sourcelookup) + (dst_positive_mask_sourcelookup))) /\ (((((exists ff_h_pvs_mask_sourcelookupnegative. ff_h_pvs_mask_sourcelookupnegative + S (dst_negative_mask_sourcelookup) = S ((S (dm_index_mask_source)) * dst_negative_scale_mask_sourcelookup)) /\ exists ff_q_pvs_mask_sourcelookupnegative. dst_negative_code_mask_sourcelookup = ff_q_pvs_mask_sourcelookupnegative * S ((S (dm_index_mask_source)) * dst_negative_scale_mask_sourcelookup) + (dst_negative_mask_sourcelookup))) /\ (exists ge_balance_positive_mask_sourcelookupvalue ge_balance_negative_mask_sourcelookupvalue. (((((dm_value_mask_source) = 2 * (ge_balance_positive_mask_sourcelookupvalue) /\ (ge_balance_negative_mask_sourcelookupvalue) = 0) \/ exists ge_signed_half_mask_sourcelookupvaluedecode. (((dm_value_mask_source) = 2 * ge_signed_half_mask_sourcelookupvaluedecode + 1 /\ (ge_balance_positive_mask_sourcelookupvalue) = 0) /\ (ge_balance_negative_mask_sourcelookupvalue) = S ge_signed_half_mask_sourcelookupvaluedecode))) /\ ((dst_positive_mask_sourcelookup) + ge_balance_negative_mask_sourcelookupvalue = (dst_negative_mask_sourcelookup) + ge_balance_positive_mask_sourcelookupvalue))))))))) -> ((((~((dm_index_mask_source)=0)) /\ (exists dm_quotient_mask_sourceentry. (((n)=(dm_index_mask_source)*dm_quotient_mask_sourceentry) /\ (exists dst_positive_code_mask_sourceentryinput dst_positive_scale_mask_sourceentryinput dst_negative_code_mask_sourceentryinput dst_negative_scale_mask_sourceentryinput dst_positive_mask_sourceentryinput dst_negative_mask_sourceentryinput. (((M) = (((((dst_positive_code_mask_sourceentryinput) + (dst_positive_scale_mask_sourceentryinput)) * S ((dst_positive_code_mask_sourceentryinput) + (dst_positive_scale_mask_sourceentryinput)) + ((dst_positive_scale_mask_sourceentryinput) + (dst_positive_scale_mask_sourceentryinput))) + (((dst_negative_code_mask_sourceentryinput) + (dst_negative_scale_mask_sourceentryinput)) * S ((dst_negative_code_mask_sourceentryinput) + (dst_negative_scale_mask_sourceentryinput)) + ((dst_negative_scale_mask_sourceentryinput) + (dst_negative_scale_mask_sourceentryinput)))) * S ((((dst_positive_code_mask_sourceentryinput) + (dst_positive_scale_mask_sourceentryinput)) * S ((dst_positive_code_mask_sourceentryinput) + (dst_positive_scale_mask_sourceentryinput)) + ((dst_positive_scale_mask_sourceentryinput) + (dst_positive_scale_mask_sourceentryinput))) + (((dst_negative_code_mask_sourceentryinput) + (dst_negative_scale_mask_sourceentryinput)) * S ((dst_negative_code_mask_sourceentryinput) + (dst_negative_scale_mask_sourceentryinput)) + ((dst_negative_scale_mask_sourceentryinput) + (dst_negative_scale_mask_sourceentryinput)))) + ((((dst_negative_code_mask_sourceentryinput) + (dst_negative_scale_mask_sourceentryinput)) * S ((dst_negative_code_mask_sourceentryinput) + (dst_negative_scale_mask_sourceentryinput)) + ((dst_negative_scale_mask_sourceentryinput) + (dst_negative_scale_mask_sourceentryinput))) + (((dst_negative_code_mask_sourceentryinput) + (dst_negative_scale_mask_sourceentryinput)) * S ((dst_negative_code_mask_sourceentryinput) + (dst_negative_scale_mask_sourceentryinput)) + ((dst_negative_scale_mask_sourceentryinput) + (dst_negative_scale_mask_sourceentryinput)))))) /\ (((((exists ff_h_pvs_mask_sourceentryinputpositive. ff_h_pvs_mask_sourceentryinputpositive + S (dst_positive_mask_sourceentryinput) = S ((S (dm_index_mask_source)) * dst_positive_scale_mask_sourceentryinput)) /\ exists ff_q_pvs_mask_sourceentryinputpositive. dst_positive_code_mask_sourceentryinput = ff_q_pvs_mask_sourceentryinputpositive * S ((S (dm_index_mask_source)) * dst_positive_scale_mask_sourceentryinput) + (dst_positive_mask_sourceentryinput))) /\ (((((exists ff_h_pvs_mask_sourceentryinputnegative. ff_h_pvs_mask_sourceentryinputnegative + S (dst_negative_mask_sourceentryinput) = S ((S (dm_index_mask_source)) * dst_negative_scale_mask_sourceentryinput)) /\ exists ff_q_pvs_mask_sourceentryinputnegative. dst_negative_code_mask_sourceentryinput = ff_q_pvs_mask_sourceentryinputnegative * S ((S (dm_index_mask_source)) * dst_negative_scale_mask_sourceentryinput) + (dst_negative_mask_sourceentryinput))) /\ (exists ge_balance_positive_mask_sourceentryinputvalue ge_balance_negative_mask_sourceentryinputvalue. (((((dm_value_mask_source) = 2 * (ge_balance_positive_mask_sourceentryinputvalue) /\ (ge_balance_negative_mask_sourceentryinputvalue) = 0) \/ exists ge_signed_half_mask_sourceentryinputvaluedecode. (((dm_value_mask_source) = 2 * ge_signed_half_mask_sourceentryinputvaluedecode + 1 /\ (ge_balance_positive_mask_sourceentryinputvalue) = 0) /\ (ge_balance_negative_mask_sourceentryinputvalue) = S ge_signed_half_mask_sourceentryinputvaluedecode))) /\ ((dst_positive_mask_sourceentryinput) + ge_balance_negative_mask_sourceentryinputvalue = (dst_negative_mask_sourceentryinput) + ge_balance_positive_mask_sourceentryinputvalue))))))))))))) \/ ((((dm_index_mask_source)=0 \/ ~(exists pvs_factor_mask_sourceentrynondivisor. (n) = (dm_index_mask_source) * pvs_factor_mask_sourceentrynondivisor)) /\ ((dm_value_mask_source)=0))))))) -> ~(d=0) -> (exists pvs_factor_mask_positive_divisor. (n) = (d) * pvs_factor_mask_positive_divisor) -> (exists pvs_le_gap_mask_index_bound. pvs_le_gap_mask_index_bound + (d) = (n)) -> (exists dst_positive_code_mask_lookup dst_positive_scale_mask_lookup dst_negative_code_mask_lookup dst_negative_scale_mask_lookup dst_positive_mask_lookup dst_negative_mask_lookup. (((K) = (((((dst_positive_code_mask_lookup) + (dst_positive_scale_mask_lookup)) * S ((dst_positive_code_mask_lookup) + (dst_positive_scale_mask_lookup)) + ((dst_positive_scale_mask_lookup) + (dst_positive_scale_mask_lookup))) + (((dst_negative_code_mask_lookup) + (dst_negative_scale_mask_lookup)) * S ((dst_negative_code_mask_lookup) + (dst_negative_scale_mask_lookup)) + ((dst_negative_scale_mask_lookup) + (dst_negative_scale_mask_lookup)))) * S ((((dst_positive_code_mask_lookup) + (dst_positive_scale_mask_lookup)) * S ((dst_positive_code_mask_lookup) + (dst_positive_scale_mask_lookup)) + ((dst_positive_scale_mask_lookup) + (dst_positive_scale_mask_lookup))) + (((dst_negative_code_mask_lookup) + (dst_negative_scale_mask_lookup)) * S ((dst_negative_code_mask_lookup) + (dst_negative_scale_mask_lookup)) + ((dst_negative_scale_mask_lookup) + (dst_negative_scale_mask_lookup)))) + ((((dst_negative_code_mask_lookup) + (dst_negative_scale_mask_lookup)) * S ((dst_negative_code_mask_lookup) + (dst_negative_scale_mask_lookup)) + ((dst_negative_scale_mask_lookup) + (dst_negative_scale_mask_lookup))) + (((dst_negative_code_mask_lookup) + (dst_negative_scale_mask_lookup)) * S ((dst_negative_code_mask_lookup) + (dst_negative_scale_mask_lookup)) + ((dst_negative_scale_mask_lookup) + (dst_negative_scale_mask_lookup)))))) /\ (((((exists ff_h_pvs_mask_lookuppositive. ff_h_pvs_mask_lookuppositive + S (dst_positive_mask_lookup) = S ((S (d)) * dst_positive_scale_mask_lookup)) /\ exists ff_q_pvs_mask_lookuppositive. dst_positive_code_mask_lookup = ff_q_pvs_mask_lookuppositive * S ((S (d)) * dst_positive_scale_mask_lookup) + (dst_positive_mask_lookup))) /\ (((((exists ff_h_pvs_mask_lookupnegative. ff_h_pvs_mask_lookupnegative + S (dst_negative_mask_lookup) = S ((S (d)) * dst_negative_scale_mask_lookup)) /\ exists ff_q_pvs_mask_lookupnegative. dst_negative_code_mask_lookup = ff_q_pvs_mask_lookupnegative * S ((S (d)) * dst_negative_scale_mask_lookup) + (dst_negative_mask_lookup))) /\ (exists ge_balance_positive_mask_lookupvalue ge_balance_negative_mask_lookupvalue. (((((z) = 2 * (ge_balance_positive_mask_lookupvalue) /\ (ge_balance_negative_mask_lookupvalue) = 0) \/ exists ge_signed_half_mask_lookupvaluedecode. (((z) = 2 * ge_signed_half_mask_lookupvaluedecode + 1 /\ (ge_balance_positive_mask_lookupvalue) = 0) /\ (ge_balance_negative_mask_lookupvalue) = S ge_signed_half_mask_lookupvaluedecode))) /\ ((dst_positive_mask_lookup) + ge_balance_negative_mask_lookupvalue = (dst_negative_mask_lookup) + ge_balance_positive_mask_lookupvalue))))))))) -> (((~((d) = 0)) /\ ((((exists mv_square_prime_mask_actual_valuesquare. ((~((mv_square_prime_mask_actual_valuesquare) = 1) /\ forall pvs_left_mask_actual_valuesquareprime pvs_right_mask_actual_valuesquareprime. (mv_square_prime_mask_actual_valuesquare) = pvs_left_mask_actual_valuesquareprime * pvs_right_mask_actual_valuesquareprime -> pvs_left_mask_actual_valuesquareprime = 1 \/ pvs_right_mask_actual_valuesquareprime = 1) /\ (exists pvs_factor_mask_actual_valuesquaredivisor. (d) = (mv_square_prime_mask_actual_valuesquare * mv_square_prime_mask_actual_valuesquare) * pvs_factor_mask_actual_valuesquaredivisor))) /\ ((z) = 0))) \/ (((((~((d) = 0)) /\ (forall sfd_prime_mask_actual_valuesquarefree. (~((sfd_prime_mask_actual_valuesquarefree) = 1) /\ forall pvs_left_mask_actual_valuesquarefreedomain pvs_right_mask_actual_valuesquarefreedomain. (sfd_prime_mask_actual_valuesquarefree) = pvs_left_mask_actual_valuesquarefreedomain * pvs_right_mask_actual_valuesquarefreedomain -> pvs_left_mask_actual_valuesquarefreedomain = 1 \/ pvs_right_mask_actual_valuesquarefreedomain = 1) -> (exists pvs_le_gap_mask_actual_valuesquarefreebound. pvs_le_gap_mask_actual_valuesquarefreebound + (sfd_prime_mask_actual_valuesquarefree) = (d)) -> ~(exists pvs_factor_mask_actual_valuesquarefreesquare. (d) = (sfd_prime_mask_actual_valuesquarefree * sfd_prime_mask_actual_valuesquarefree) * pvs_factor_mask_actual_valuesquarefreesquare)))) /\ (exists mv_factor_code_mask_actual_valuefactors mv_factor_scale_mask_actual_valuefactors mv_factor_count_mask_actual_valuefactors. (((~(d = 0) /\ ((exists ff_u_fsat_mask_actual_valuefactorsfactorization_product ff_v_fsat_mask_actual_valuefactorsfactorization_product. ((((exists ff_h_fsat_mask_actual_valuefactorsfactorization_product_start. ff_h_fsat_mask_actual_valuefactorsfactorization_product_start + S (1) = S ((S (0)) * ff_v_fsat_mask_actual_valuefactorsfactorization_product)) /\ exists ff_q_fsat_mask_actual_valuefactorsfactorization_product_start. ff_u_fsat_mask_actual_valuefactorsfactorization_product = ff_q_fsat_mask_actual_valuefactorsfactorization_product_start * S ((S (0)) * ff_v_fsat_mask_actual_valuefactorsfactorization_product) + (1))) /\ ((((exists ff_h_fsat_mask_actual_valuefactorsfactorization_product_terminal. ff_h_fsat_mask_actual_valuefactorsfactorization_product_terminal + S (d) = S ((S (mv_factor_count_mask_actual_valuefactors)) * ff_v_fsat_mask_actual_valuefactorsfactorization_product)) /\ exists ff_q_fsat_mask_actual_valuefactorsfactorization_product_terminal. ff_u_fsat_mask_actual_valuefactorsfactorization_product = ff_q_fsat_mask_actual_valuefactorsfactorization_product_terminal * S ((S (mv_factor_count_mask_actual_valuefactors)) * ff_v_fsat_mask_actual_valuefactorsfactorization_product) + (d))) /\ forall ff_i_fsat_mask_actual_valuefactorsfactorization_product. (exists ff_lt_fsat_mask_actual_valuefactorsfactorization_product_bound. ff_lt_fsat_mask_actual_valuefactorsfactorization_product_bound + S ff_i_fsat_mask_actual_valuefactorsfactorization_product = mv_factor_count_mask_actual_valuefactors) -> exists ff_p_fsat_mask_actual_valuefactorsfactorization_product ff_r_fsat_mask_actual_valuefactorsfactorization_product ff_s_fsat_mask_actual_valuefactorsfactorization_product. ((((exists ff_h_fsat_mask_actual_valuefactorsfactorization_product_factor. ff_h_fsat_mask_actual_valuefactorsfactorization_product_factor + S (ff_p_fsat_mask_actual_valuefactorsfactorization_product) = S ((S (ff_i_fsat_mask_actual_valuefactorsfactorization_product)) * mv_factor_scale_mask_actual_valuefactors)) /\ exists ff_q_fsat_mask_actual_valuefactorsfactorization_product_factor. mv_factor_code_mask_actual_valuefactors = ff_q_fsat_mask_actual_valuefactorsfactorization_product_factor * S ((S (ff_i_fsat_mask_actual_valuefactorsfactorization_product)) * mv_factor_scale_mask_actual_valuefactors) + (ff_p_fsat_mask_actual_valuefactorsfactorization_product))) /\ ((((exists ff_h_fsat_mask_actual_valuefactorsfactorization_product_partial. ff_h_fsat_mask_actual_valuefactorsfactorization_product_partial + S (ff_r_fsat_mask_actual_valuefactorsfactorization_product) = S ((S (ff_i_fsat_mask_actual_valuefactorsfactorization_product)) * ff_v_fsat_mask_actual_valuefactorsfactorization_product)) /\ exists ff_q_fsat_mask_actual_valuefactorsfactorization_product_partial. ff_u_fsat_mask_actual_valuefactorsfactorization_product = ff_q_fsat_mask_actual_valuefactorsfactorization_product_partial * S ((S (ff_i_fsat_mask_actual_valuefactorsfactorization_product)) * ff_v_fsat_mask_actual_valuefactorsfactorization_product) + (ff_r_fsat_mask_actual_valuefactorsfactorization_product))) /\ ((((exists ff_h_fsat_mask_actual_valuefactorsfactorization_product_successor. ff_h_fsat_mask_actual_valuefactorsfactorization_product_successor + S (ff_s_fsat_mask_actual_valuefactorsfactorization_product) = S ((S (S ff_i_fsat_mask_actual_valuefactorsfactorization_product)) * ff_v_fsat_mask_actual_valuefactorsfactorization_product)) /\ exists ff_q_fsat_mask_actual_valuefactorsfactorization_product_successor. ff_u_fsat_mask_actual_valuefactorsfactorization_product = ff_q_fsat_mask_actual_valuefactorsfactorization_product_successor * S ((S (S ff_i_fsat_mask_actual_valuefactorsfactorization_product)) * ff_v_fsat_mask_actual_valuefactorsfactorization_product) + (ff_s_fsat_mask_actual_valuefactorsfactorization_product))) /\ ff_s_fsat_mask_actual_valuefactorsfactorization_product = ff_r_fsat_mask_actual_valuefactorsfactorization_product * ff_p_fsat_mask_actual_valuefactorsfactorization_product)))))) /\ (forall ftsf_index_fsat_mask_actual_valuefactorsfactorization_primes. (exists ftsf_gap_fsat_mask_actual_valuefactorsfactorization_primes_bound. ftsf_gap_fsat_mask_actual_valuefactorsfactorization_primes_bound + S ftsf_index_fsat_mask_actual_valuefactorsfactorization_primes = (mv_factor_count_mask_actual_valuefactors)) -> exists ftsf_factor_fsat_mask_actual_valuefactorsfactorization_primes. ((((exists ff_h_ftsf_fsat_mask_actual_valuefactorsfactorization_primes_entry. ff_h_ftsf_fsat_mask_actual_valuefactorsfactorization_primes_entry + S (ftsf_factor_fsat_mask_actual_valuefactorsfactorization_primes) = S ((S (ftsf_index_fsat_mask_actual_valuefactorsfactorization_primes)) * mv_factor_scale_mask_actual_valuefactors)) /\ exists ff_q_ftsf_fsat_mask_actual_valuefactorsfactorization_primes_entry. mv_factor_code_mask_actual_valuefactors = ff_q_ftsf_fsat_mask_actual_valuefactorsfactorization_primes_entry * S ((S (ftsf_index_fsat_mask_actual_valuefactorsfactorization_primes)) * mv_factor_scale_mask_actual_valuefactors) + (ftsf_factor_fsat_mask_actual_valuefactorsfactorization_primes))) /\ ((~(ftsf_factor_fsat_mask_actual_valuefactorsfactorization_primes = 1) /\ forall frm_prime_left_ftsf_fsat_mask_actual_valuefactorsfactorization_primes_prime frm_prime_right_ftsf_fsat_mask_actual_valuefactorsfactorization_primes_prime. ftsf_factor_fsat_mask_actual_valuefactorsfactorization_primes = frm_prime_left_ftsf_fsat_mask_actual_valuefactorsfactorization_primes_prime * frm_prime_right_ftsf_fsat_mask_actual_valuefactorsfactorization_primes_prime -> frm_prime_left_ftsf_fsat_mask_actual_valuefactorsfactorization_primes_prime = 1 \/ frm_prime_right_ftsf_fsat_mask_actual_valuefactorsfactorization_primes_prime = 1))))))) /\ ((((exists mv_even_half_mask_actual_valuefactorsparityeven. (mv_factor_count_mask_actual_valuefactors) = 2 * mv_even_half_mask_actual_valuefactorsparityeven) /\ ((z) = 2))) \/ (((exists mv_odd_half_mask_actual_valuefactorsparityodd. (mv_factor_count_mask_actual_valuefactors) = 2 * mv_odd_half_mask_actual_valuefactorsparityodd + 1) /\ ((z) = 1)))))))))))

Complete tactic proof in conservative notation

All 40 original proof lines are preserved. Only local proposition formulas are abbreviated; every abbreviation has an exact binder-safe expansion check. The linked exact edition contains the unchanged replay script.

Read the argument

Proof checkpoints

40 script commands · 7 reading checkpoints · 0 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.

Definition notation is shown below. Open the paired exact edition for the original native formulas. Source pairing is not a new equivalence certificate.

01Fix variables and assumptionsL1–10

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

  1. L1
    intro N
  2. L2
    intro M
  3. L3
    intro n
  4. L4
    intro K
  5. L5
    intro d
  6. L6
    intro z
  7. L7
    intro hmu
  8. L8
    intro hnN
  9. L9
    intro hm
  10. L10
    intro hd
02Fix variables and assumptionsL11–13

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

  1. L11
    intro hdiv
  2. L12
    intro hbound
  3. L13
    intro hz
03Separate the logical casesL14–16

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

  1. L14
    cases hmu
  2. L15
    cases hmu_right
  3. L16
    cases hm
04Use earlier factsL17–26

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

  1. L17
    specialize hmu_right_right (d)
  2. L18
    specialize hmu_right_right (z)
  3. L19
    apply hmu_right_right
  4. L20
    exact hd
  5. L21
    specialize le_trans (d)
  6. L22
    specialize le_trans (n)
  7. L23
    specialize le_trans (N)
  8. L24
    apply le_trans
  9. L25
    exact hbound
  10. L26
    exact hnN
05Separate the logical casesL27–27

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

  1. L27
    cases hdiv
06Use earlier factsL28–37

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

  1. L28
    specialize divisor_mask_entry_quotient_input (M)
  2. L29
    specialize divisor_mask_entry_quotient_input (n)
  3. L30
    specialize divisor_mask_entry_quotient_input (d)
  4. L31
    specialize divisor_mask_entry_quotient_input (x)
  5. L32
    specialize divisor_mask_entry_quotient_input (z)
  6. L33
    apply divisor_mask_entry_quotient_input
  7. L34
    exact hd
  8. L35
    exact hdiv_witness
  9. L36
    specialize hm_right (d)
  10. L37
    specialize hm_right (z)
07Use earlier factsL38–40

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

  1. L38
    apply hm_right
  2. L39
    exact hbound
  3. L40
    exact hz

Library-wide reading audit

Original defined command ledger · 40 lines
  1. 0001intro N
  2. 0002intro M
  3. 0003intro n
  4. 0004intro K
  5. 0005intro d
  6. 0006intro z
  7. 0007intro hmu
  8. 0008intro hnN
  9. 0009intro hm
  10. 0010intro hd
  11. 0011intro hdiv
  12. 0012intro hbound
  13. 0013intro hz
  14. 0014cases hmu
  15. 0015cases hmu_right
  16. 0016cases hm
  17. 0017specialize hmu_right_right (d)
  18. 0018specialize hmu_right_right (z)
  19. 0019apply hmu_right_right
  20. 0020exact hd
  21. 0021specialize le_trans (d)
  22. 0022specialize le_trans (n)
  23. 0023specialize le_trans (N)
  24. 0024apply le_trans
  25. 0025exact hbound
  26. 0026exact hnN
  27. 0027cases hdiv
  28. 0028specialize divisor_mask_entry_quotient_input (M)
  29. 0029specialize divisor_mask_entry_quotient_input (n)
  30. 0030specialize divisor_mask_entry_quotient_input (d)
  31. 0031specialize divisor_mask_entry_quotient_input (x)
  32. 0032specialize divisor_mask_entry_quotient_input (z)
  33. 0033apply divisor_mask_entry_quotient_input
  34. 0034exact hd
  35. 0035exact hdiv_witness
  36. 0036specialize hm_right (d)
  37. 0037specialize hm_right (z)
  38. 0038apply hm_right
  39. 0039exact hbound
  40. 0040exact hz