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
02Fix variables and assumptionsL11–13
03Separate the logical casesL14–16
04Use earlier factsL17–26
Instantiate or apply named facts and discharge the corresponding proof obligations.
05Separate the logical casesL27–27
Follow the explicit conjunction, disjunction, witness, or contradiction step recorded below.
- L27
cases hdiv
06Use earlier factsL28–37
Instantiate or apply named facts and discharge the corresponding proof obligations.
- L28
specialize divisor_mask_entry_quotient_input (M) - L29
specialize divisor_mask_entry_quotient_input (n) - L30
specialize divisor_mask_entry_quotient_input (d) - L31
specialize divisor_mask_entry_quotient_input (x) - L32
specialize divisor_mask_entry_quotient_input (z) - L33
apply divisor_mask_entry_quotient_input - L34
exact hd - L35
exact hdiv_witness - L36
specialize hm_right (d) - L37
specialize hm_right (z)
Original defined command ledger · 40 lines
- 0001
intro N - 0002
intro M - 0003
intro n - 0004
intro K - 0005
intro d - 0006
intro z - 0007
intro hmu - 0008
intro hnN - 0009
intro hm - 0010
intro hd - 0011
intro hdiv - 0012
intro hbound - 0013
intro hz - 0014
cases hmu - 0015
cases hmu_right - 0016
cases hm - 0017
specialize hmu_right_right (d) - 0018
specialize hmu_right_right (z) - 0019
apply hmu_right_right - 0020
exact hd - 0021
specialize le_trans (d) - 0022
specialize le_trans (n) - 0023
specialize le_trans (N) - 0024
apply le_trans - 0025
exact hbound - 0026
exact hnN - 0027
cases hdiv - 0028
specialize divisor_mask_entry_quotient_input (M) - 0029
specialize divisor_mask_entry_quotient_input (n) - 0030
specialize divisor_mask_entry_quotient_input (d) - 0031
specialize divisor_mask_entry_quotient_input (x) - 0032
specialize divisor_mask_entry_quotient_input (z) - 0033
apply divisor_mask_entry_quotient_input - 0034
exact hd - 0035
exact hdiv_witness - 0036
specialize hm_right (d) - 0037
specialize hm_right (z) - 0038
apply hm_right - 0039
exact hbound - 0040
exact hz