Current library: Alpha v34, 4,223 checked-use theorems; Stable remains 432. Historical first admissions, original proof editions, and non-admitted aliases are preserved.
A genuine divisor mask has S n entries, indexed zero through n, and forces its zeroth entry to zero regardless of F(0). Möbius values remain positive-domain only. This family constructs divisor sums and Möbius tables; cancellation and the full G007 endpoint are separately proved later in the same release.
Exact theorem in conservative defined notation
∀ N. ∀ F. ∀ G. MobiusTable(N,F) → MobiusTable(N,G) → ArithTableEqual(F,G,S N)
Every linked abbreviation expands hygienically to the identical original native formula.
Definition DAG
Actual proof prerequisites
Original expanded first-order statement
forall N F G. (((exists dst_positive_code_unique_firsttable dst_positive_scale_unique_firsttable dst_negative_code_unique_firsttable dst_negative_scale_unique_firsttable. (((F) = (((((dst_positive_code_unique_firsttable) + (dst_positive_scale_unique_firsttable)) * S ((dst_positive_code_unique_firsttable) + (dst_positive_scale_unique_firsttable)) + ((dst_positive_scale_unique_firsttable) + (dst_positive_scale_unique_firsttable))) + (((dst_negative_code_unique_firsttable) + (dst_negative_scale_unique_firsttable)) * S ((dst_negative_code_unique_firsttable) + (dst_negative_scale_unique_firsttable)) + ((dst_negative_scale_unique_firsttable) + (dst_negative_scale_unique_firsttable)))) * S ((((dst_positive_code_unique_firsttable) + (dst_positive_scale_unique_firsttable)) * S ((dst_positive_code_unique_firsttable) + (dst_positive_scale_unique_firsttable)) + ((dst_positive_scale_unique_firsttable) + (dst_positive_scale_unique_firsttable))) + (((dst_negative_code_unique_firsttable) + (dst_negative_scale_unique_firsttable)) * S ((dst_negative_code_unique_firsttable) + (dst_negative_scale_unique_firsttable)) + ((dst_negative_scale_unique_firsttable) + (dst_negative_scale_unique_firsttable)))) + ((((dst_negative_code_unique_firsttable) + (dst_negative_scale_unique_firsttable)) * S ((dst_negative_code_unique_firsttable) + (dst_negative_scale_unique_firsttable)) + ((dst_negative_scale_unique_firsttable) + (dst_negative_scale_unique_firsttable))) + (((dst_negative_code_unique_firsttable) + (dst_negative_scale_unique_firsttable)) * S ((dst_negative_code_unique_firsttable) + (dst_negative_scale_unique_firsttable)) + ((dst_negative_scale_unique_firsttable) + (dst_negative_scale_unique_firsttable)))))) /\ (forall dst_index_unique_firsttable. (exists pvs_le_gap_unique_firsttabledomain. pvs_le_gap_unique_firsttabledomain + (dst_index_unique_firsttable) = (N)) -> exists dst_positive_unique_firsttable dst_negative_unique_firsttable dst_value_unique_firsttable. ((((exists ff_h_pvs_unique_firsttableentrypositive. ff_h_pvs_unique_firsttableentrypositive + S (dst_positive_unique_firsttable) = S ((S (dst_index_unique_firsttable)) * dst_positive_scale_unique_firsttable)) /\ exists ff_q_pvs_unique_firsttableentrypositive. dst_positive_code_unique_firsttable = ff_q_pvs_unique_firsttableentrypositive * S ((S (dst_index_unique_firsttable)) * dst_positive_scale_unique_firsttable) + (dst_positive_unique_firsttable))) /\ (((((exists ff_h_pvs_unique_firsttableentrynegative. ff_h_pvs_unique_firsttableentrynegative + S (dst_negative_unique_firsttable) = S ((S (dst_index_unique_firsttable)) * dst_negative_scale_unique_firsttable)) /\ exists ff_q_pvs_unique_firsttableentrynegative. dst_negative_code_unique_firsttable = ff_q_pvs_unique_firsttableentrynegative * S ((S (dst_index_unique_firsttable)) * dst_negative_scale_unique_firsttable) + (dst_negative_unique_firsttable))) /\ (exists ge_balance_positive_unique_firsttableentryvalue ge_balance_negative_unique_firsttableentryvalue. (((((dst_value_unique_firsttable) = 2 * (ge_balance_positive_unique_firsttableentryvalue) /\ (ge_balance_negative_unique_firsttableentryvalue) = 0) \/ exists ge_signed_half_unique_firsttableentryvaluedecode. (((dst_value_unique_firsttable) = 2 * ge_signed_half_unique_firsttableentryvaluedecode + 1 /\ (ge_balance_positive_unique_firsttableentryvalue) = 0) /\ (ge_balance_negative_unique_firsttableentryvalue) = S ge_signed_half_unique_firsttableentryvaluedecode))) /\ ((dst_positive_unique_firsttable) + ge_balance_negative_unique_firsttableentryvalue = (dst_negative_unique_firsttable) + ge_balance_positive_unique_firsttableentryvalue))))))))) /\ (((exists dst_positive_code_unique_firstzero dst_positive_scale_unique_firstzero dst_negative_code_unique_firstzero dst_negative_scale_unique_firstzero dst_positive_unique_firstzero dst_negative_unique_firstzero. (((F) = (((((dst_positive_code_unique_firstzero) + (dst_positive_scale_unique_firstzero)) * S ((dst_positive_code_unique_firstzero) + (dst_positive_scale_unique_firstzero)) + ((dst_positive_scale_unique_firstzero) + (dst_positive_scale_unique_firstzero))) + (((dst_negative_code_unique_firstzero) + (dst_negative_scale_unique_firstzero)) * S ((dst_negative_code_unique_firstzero) + (dst_negative_scale_unique_firstzero)) + ((dst_negative_scale_unique_firstzero) + (dst_negative_scale_unique_firstzero)))) * S ((((dst_positive_code_unique_firstzero) + (dst_positive_scale_unique_firstzero)) * S ((dst_positive_code_unique_firstzero) + (dst_positive_scale_unique_firstzero)) + ((dst_positive_scale_unique_firstzero) + (dst_positive_scale_unique_firstzero))) + (((dst_negative_code_unique_firstzero) + (dst_negative_scale_unique_firstzero)) * S ((dst_negative_code_unique_firstzero) + (dst_negative_scale_unique_firstzero)) + ((dst_negative_scale_unique_firstzero) + (dst_negative_scale_unique_firstzero)))) + ((((dst_negative_code_unique_firstzero) + (dst_negative_scale_unique_firstzero)) * S ((dst_negative_code_unique_firstzero) + (dst_negative_scale_unique_firstzero)) + ((dst_negative_scale_unique_firstzero) + (dst_negative_scale_unique_firstzero))) + (((dst_negative_code_unique_firstzero) + (dst_negative_scale_unique_firstzero)) * S ((dst_negative_code_unique_firstzero) + (dst_negative_scale_unique_firstzero)) + ((dst_negative_scale_unique_firstzero) + (dst_negative_scale_unique_firstzero)))))) /\ (((((exists ff_h_pvs_unique_firstzeropositive. ff_h_pvs_unique_firstzeropositive + S (dst_positive_unique_firstzero) = S ((S (0)) * dst_positive_scale_unique_firstzero)) /\ exists ff_q_pvs_unique_firstzeropositive. dst_positive_code_unique_firstzero = ff_q_pvs_unique_firstzeropositive * S ((S (0)) * dst_positive_scale_unique_firstzero) + (dst_positive_unique_firstzero))) /\ (((((exists ff_h_pvs_unique_firstzeronegative. ff_h_pvs_unique_firstzeronegative + S (dst_negative_unique_firstzero) = S ((S (0)) * dst_negative_scale_unique_firstzero)) /\ exists ff_q_pvs_unique_firstzeronegative. dst_negative_code_unique_firstzero = ff_q_pvs_unique_firstzeronegative * S ((S (0)) * dst_negative_scale_unique_firstzero) + (dst_negative_unique_firstzero))) /\ (exists ge_balance_positive_unique_firstzerovalue ge_balance_negative_unique_firstzerovalue. (((((0) = 2 * (ge_balance_positive_unique_firstzerovalue) /\ (ge_balance_negative_unique_firstzerovalue) = 0) \/ exists ge_signed_half_unique_firstzerovaluedecode. (((0) = 2 * ge_signed_half_unique_firstzerovaluedecode + 1 /\ (ge_balance_positive_unique_firstzerovalue) = 0) /\ (ge_balance_negative_unique_firstzerovalue) = S ge_signed_half_unique_firstzerovaluedecode))) /\ ((dst_positive_unique_firstzero) + ge_balance_negative_unique_firstzerovalue = (dst_negative_unique_firstzero) + ge_balance_positive_unique_firstzerovalue))))))))) /\ (forall mt_index_unique_first mt_value_unique_first. ~(mt_index_unique_first=0) -> (exists pvs_le_gap_unique_firstdomain. pvs_le_gap_unique_firstdomain + (mt_index_unique_first) = (N)) -> (exists dst_positive_code_unique_firstentry dst_positive_scale_unique_firstentry dst_negative_code_unique_firstentry dst_negative_scale_unique_firstentry dst_positive_unique_firstentry dst_negative_unique_firstentry. (((F) = (((((dst_positive_code_unique_firstentry) + (dst_positive_scale_unique_firstentry)) * S ((dst_positive_code_unique_firstentry) + (dst_positive_scale_unique_firstentry)) + ((dst_positive_scale_unique_firstentry) + (dst_positive_scale_unique_firstentry))) + (((dst_negative_code_unique_firstentry) + (dst_negative_scale_unique_firstentry)) * S ((dst_negative_code_unique_firstentry) + (dst_negative_scale_unique_firstentry)) + ((dst_negative_scale_unique_firstentry) + (dst_negative_scale_unique_firstentry)))) * S ((((dst_positive_code_unique_firstentry) + (dst_positive_scale_unique_firstentry)) * S ((dst_positive_code_unique_firstentry) + (dst_positive_scale_unique_firstentry)) + ((dst_positive_scale_unique_firstentry) + (dst_positive_scale_unique_firstentry))) + (((dst_negative_code_unique_firstentry) + (dst_negative_scale_unique_firstentry)) * S ((dst_negative_code_unique_firstentry) + (dst_negative_scale_unique_firstentry)) + ((dst_negative_scale_unique_firstentry) + (dst_negative_scale_unique_firstentry)))) + ((((dst_negative_code_unique_firstentry) + (dst_negative_scale_unique_firstentry)) * S ((dst_negative_code_unique_firstentry) + (dst_negative_scale_unique_firstentry)) + ((dst_negative_scale_unique_firstentry) + (dst_negative_scale_unique_firstentry))) + (((dst_negative_code_unique_firstentry) + (dst_negative_scale_unique_firstentry)) * S ((dst_negative_code_unique_firstentry) + (dst_negative_scale_unique_firstentry)) + ((dst_negative_scale_unique_firstentry) + (dst_negative_scale_unique_firstentry)))))) /\ (((((exists ff_h_pvs_unique_firstentrypositive. ff_h_pvs_unique_firstentrypositive + S (dst_positive_unique_firstentry) = S ((S (mt_index_unique_first)) * dst_positive_scale_unique_firstentry)) /\ exists ff_q_pvs_unique_firstentrypositive. dst_positive_code_unique_firstentry = ff_q_pvs_unique_firstentrypositive * S ((S (mt_index_unique_first)) * dst_positive_scale_unique_firstentry) + (dst_positive_unique_firstentry))) /\ (((((exists ff_h_pvs_unique_firstentrynegative. ff_h_pvs_unique_firstentrynegative + S (dst_negative_unique_firstentry) = S ((S (mt_index_unique_first)) * dst_negative_scale_unique_firstentry)) /\ exists ff_q_pvs_unique_firstentrynegative. dst_negative_code_unique_firstentry = ff_q_pvs_unique_firstentrynegative * S ((S (mt_index_unique_first)) * dst_negative_scale_unique_firstentry) + (dst_negative_unique_firstentry))) /\ (exists ge_balance_positive_unique_firstentryvalue ge_balance_negative_unique_firstentryvalue. (((((mt_value_unique_first) = 2 * (ge_balance_positive_unique_firstentryvalue) /\ (ge_balance_negative_unique_firstentryvalue) = 0) \/ exists ge_signed_half_unique_firstentryvaluedecode. (((mt_value_unique_first) = 2 * ge_signed_half_unique_firstentryvaluedecode + 1 /\ (ge_balance_positive_unique_firstentryvalue) = 0) /\ (ge_balance_negative_unique_firstentryvalue) = S ge_signed_half_unique_firstentryvaluedecode))) /\ ((dst_positive_unique_firstentry) + ge_balance_negative_unique_firstentryvalue = (dst_negative_unique_firstentry) + ge_balance_positive_unique_firstentryvalue))))))))) -> (((~((mt_index_unique_first) = 0)) /\ ((((exists mv_square_prime_unique_firstvaluesquare. ((~((mv_square_prime_unique_firstvaluesquare) = 1) /\ forall pvs_left_unique_firstvaluesquareprime pvs_right_unique_firstvaluesquareprime. (mv_square_prime_unique_firstvaluesquare) = pvs_left_unique_firstvaluesquareprime * pvs_right_unique_firstvaluesquareprime -> pvs_left_unique_firstvaluesquareprime = 1 \/ pvs_right_unique_firstvaluesquareprime = 1) /\ (exists pvs_factor_unique_firstvaluesquaredivisor. (mt_index_unique_first) = (mv_square_prime_unique_firstvaluesquare * mv_square_prime_unique_firstvaluesquare) * pvs_factor_unique_firstvaluesquaredivisor))) /\ ((mt_value_unique_first) = 0))) \/ (((((~((mt_index_unique_first) = 0)) /\ (forall sfd_prime_unique_firstvaluesquarefree. (~((sfd_prime_unique_firstvaluesquarefree) = 1) /\ forall pvs_left_unique_firstvaluesquarefreedomain pvs_right_unique_firstvaluesquarefreedomain. (sfd_prime_unique_firstvaluesquarefree) = pvs_left_unique_firstvaluesquarefreedomain * pvs_right_unique_firstvaluesquarefreedomain -> pvs_left_unique_firstvaluesquarefreedomain = 1 \/ pvs_right_unique_firstvaluesquarefreedomain = 1) -> (exists pvs_le_gap_unique_firstvaluesquarefreebound. pvs_le_gap_unique_firstvaluesquarefreebound + (sfd_prime_unique_firstvaluesquarefree) = (mt_index_unique_first)) -> ~(exists pvs_factor_unique_firstvaluesquarefreesquare. (mt_index_unique_first) = (sfd_prime_unique_firstvaluesquarefree * sfd_prime_unique_firstvaluesquarefree) * pvs_factor_unique_firstvaluesquarefreesquare)))) /\ (exists mv_factor_code_unique_firstvaluefactors mv_factor_scale_unique_firstvaluefactors mv_factor_count_unique_firstvaluefactors. (((~(mt_index_unique_first = 0) /\ ((exists ff_u_fsat_unique_firstvaluefactorsfactorization_product ff_v_fsat_unique_firstvaluefactorsfactorization_product. ((((exists ff_h_fsat_unique_firstvaluefactorsfactorization_product_start. ff_h_fsat_unique_firstvaluefactorsfactorization_product_start + S (1) = S ((S (0)) * ff_v_fsat_unique_firstvaluefactorsfactorization_product)) /\ exists ff_q_fsat_unique_firstvaluefactorsfactorization_product_start. ff_u_fsat_unique_firstvaluefactorsfactorization_product = ff_q_fsat_unique_firstvaluefactorsfactorization_product_start * S ((S (0)) * ff_v_fsat_unique_firstvaluefactorsfactorization_product) + (1))) /\ ((((exists ff_h_fsat_unique_firstvaluefactorsfactorization_product_terminal. ff_h_fsat_unique_firstvaluefactorsfactorization_product_terminal + S (mt_index_unique_first) = S ((S (mv_factor_count_unique_firstvaluefactors)) * ff_v_fsat_unique_firstvaluefactorsfactorization_product)) /\ exists ff_q_fsat_unique_firstvaluefactorsfactorization_product_terminal. ff_u_fsat_unique_firstvaluefactorsfactorization_product = ff_q_fsat_unique_firstvaluefactorsfactorization_product_terminal * S ((S (mv_factor_count_unique_firstvaluefactors)) * ff_v_fsat_unique_firstvaluefactorsfactorization_product) + (mt_index_unique_first))) /\ forall ff_i_fsat_unique_firstvaluefactorsfactorization_product. (exists ff_lt_fsat_unique_firstvaluefactorsfactorization_product_bound. ff_lt_fsat_unique_firstvaluefactorsfactorization_product_bound + S ff_i_fsat_unique_firstvaluefactorsfactorization_product = mv_factor_count_unique_firstvaluefactors) -> exists ff_p_fsat_unique_firstvaluefactorsfactorization_product ff_r_fsat_unique_firstvaluefactorsfactorization_product ff_s_fsat_unique_firstvaluefactorsfactorization_product. ((((exists ff_h_fsat_unique_firstvaluefactorsfactorization_product_factor. ff_h_fsat_unique_firstvaluefactorsfactorization_product_factor + S (ff_p_fsat_unique_firstvaluefactorsfactorization_product) = S ((S (ff_i_fsat_unique_firstvaluefactorsfactorization_product)) * mv_factor_scale_unique_firstvaluefactors)) /\ exists ff_q_fsat_unique_firstvaluefactorsfactorization_product_factor. mv_factor_code_unique_firstvaluefactors = ff_q_fsat_unique_firstvaluefactorsfactorization_product_factor * S ((S (ff_i_fsat_unique_firstvaluefactorsfactorization_product)) * mv_factor_scale_unique_firstvaluefactors) + (ff_p_fsat_unique_firstvaluefactorsfactorization_product))) /\ ((((exists ff_h_fsat_unique_firstvaluefactorsfactorization_product_partial. ff_h_fsat_unique_firstvaluefactorsfactorization_product_partial + S (ff_r_fsat_unique_firstvaluefactorsfactorization_product) = S ((S (ff_i_fsat_unique_firstvaluefactorsfactorization_product)) * ff_v_fsat_unique_firstvaluefactorsfactorization_product)) /\ exists ff_q_fsat_unique_firstvaluefactorsfactorization_product_partial. ff_u_fsat_unique_firstvaluefactorsfactorization_product = ff_q_fsat_unique_firstvaluefactorsfactorization_product_partial * S ((S (ff_i_fsat_unique_firstvaluefactorsfactorization_product)) * ff_v_fsat_unique_firstvaluefactorsfactorization_product) + (ff_r_fsat_unique_firstvaluefactorsfactorization_product))) /\ ((((exists ff_h_fsat_unique_firstvaluefactorsfactorization_product_successor. ff_h_fsat_unique_firstvaluefactorsfactorization_product_successor + S (ff_s_fsat_unique_firstvaluefactorsfactorization_product) = S ((S (S ff_i_fsat_unique_firstvaluefactorsfactorization_product)) * ff_v_fsat_unique_firstvaluefactorsfactorization_product)) /\ exists ff_q_fsat_unique_firstvaluefactorsfactorization_product_successor. ff_u_fsat_unique_firstvaluefactorsfactorization_product = ff_q_fsat_unique_firstvaluefactorsfactorization_product_successor * S ((S (S ff_i_fsat_unique_firstvaluefactorsfactorization_product)) * ff_v_fsat_unique_firstvaluefactorsfactorization_product) + (ff_s_fsat_unique_firstvaluefactorsfactorization_product))) /\ ff_s_fsat_unique_firstvaluefactorsfactorization_product = ff_r_fsat_unique_firstvaluefactorsfactorization_product * ff_p_fsat_unique_firstvaluefactorsfactorization_product)))))) /\ (forall ftsf_index_fsat_unique_firstvaluefactorsfactorization_primes. (exists ftsf_gap_fsat_unique_firstvaluefactorsfactorization_primes_bound. ftsf_gap_fsat_unique_firstvaluefactorsfactorization_primes_bound + S ftsf_index_fsat_unique_firstvaluefactorsfactorization_primes = (mv_factor_count_unique_firstvaluefactors)) -> exists ftsf_factor_fsat_unique_firstvaluefactorsfactorization_primes. ((((exists ff_h_ftsf_fsat_unique_firstvaluefactorsfactorization_primes_entry. ff_h_ftsf_fsat_unique_firstvaluefactorsfactorization_primes_entry + S (ftsf_factor_fsat_unique_firstvaluefactorsfactorization_primes) = S ((S (ftsf_index_fsat_unique_firstvaluefactorsfactorization_primes)) * mv_factor_scale_unique_firstvaluefactors)) /\ exists ff_q_ftsf_fsat_unique_firstvaluefactorsfactorization_primes_entry. mv_factor_code_unique_firstvaluefactors = ff_q_ftsf_fsat_unique_firstvaluefactorsfactorization_primes_entry * S ((S (ftsf_index_fsat_unique_firstvaluefactorsfactorization_primes)) * mv_factor_scale_unique_firstvaluefactors) + (ftsf_factor_fsat_unique_firstvaluefactorsfactorization_primes))) /\ ((~(ftsf_factor_fsat_unique_firstvaluefactorsfactorization_primes = 1) /\ forall frm_prime_left_ftsf_fsat_unique_firstvaluefactorsfactorization_primes_prime frm_prime_right_ftsf_fsat_unique_firstvaluefactorsfactorization_primes_prime. ftsf_factor_fsat_unique_firstvaluefactorsfactorization_primes = frm_prime_left_ftsf_fsat_unique_firstvaluefactorsfactorization_primes_prime * frm_prime_right_ftsf_fsat_unique_firstvaluefactorsfactorization_primes_prime -> frm_prime_left_ftsf_fsat_unique_firstvaluefactorsfactorization_primes_prime = 1 \/ frm_prime_right_ftsf_fsat_unique_firstvaluefactorsfactorization_primes_prime = 1))))))) /\ ((((exists mv_even_half_unique_firstvaluefactorsparityeven. (mv_factor_count_unique_firstvaluefactors) = 2 * mv_even_half_unique_firstvaluefactorsparityeven) /\ ((mt_value_unique_first) = 2))) \/ (((exists mv_odd_half_unique_firstvaluefactorsparityodd. (mv_factor_count_unique_firstvaluefactors) = 2 * mv_odd_half_unique_firstvaluefactorsparityodd + 1) /\ ((mt_value_unique_first) = 1)))))))))))))))) -> (((exists dst_positive_code_unique_secondtable dst_positive_scale_unique_secondtable dst_negative_code_unique_secondtable dst_negative_scale_unique_secondtable. (((G) = (((((dst_positive_code_unique_secondtable) + (dst_positive_scale_unique_secondtable)) * S ((dst_positive_code_unique_secondtable) + (dst_positive_scale_unique_secondtable)) + ((dst_positive_scale_unique_secondtable) + (dst_positive_scale_unique_secondtable))) + (((dst_negative_code_unique_secondtable) + (dst_negative_scale_unique_secondtable)) * S ((dst_negative_code_unique_secondtable) + (dst_negative_scale_unique_secondtable)) + ((dst_negative_scale_unique_secondtable) + (dst_negative_scale_unique_secondtable)))) * S ((((dst_positive_code_unique_secondtable) + (dst_positive_scale_unique_secondtable)) * S ((dst_positive_code_unique_secondtable) + (dst_positive_scale_unique_secondtable)) + ((dst_positive_scale_unique_secondtable) + (dst_positive_scale_unique_secondtable))) + (((dst_negative_code_unique_secondtable) + (dst_negative_scale_unique_secondtable)) * S ((dst_negative_code_unique_secondtable) + (dst_negative_scale_unique_secondtable)) + ((dst_negative_scale_unique_secondtable) + (dst_negative_scale_unique_secondtable)))) + ((((dst_negative_code_unique_secondtable) + (dst_negative_scale_unique_secondtable)) * S ((dst_negative_code_unique_secondtable) + (dst_negative_scale_unique_secondtable)) + ((dst_negative_scale_unique_secondtable) + (dst_negative_scale_unique_secondtable))) + (((dst_negative_code_unique_secondtable) + (dst_negative_scale_unique_secondtable)) * S ((dst_negative_code_unique_secondtable) + (dst_negative_scale_unique_secondtable)) + ((dst_negative_scale_unique_secondtable) + (dst_negative_scale_unique_secondtable)))))) /\ (forall dst_index_unique_secondtable. (exists pvs_le_gap_unique_secondtabledomain. pvs_le_gap_unique_secondtabledomain + (dst_index_unique_secondtable) = (N)) -> exists dst_positive_unique_secondtable dst_negative_unique_secondtable dst_value_unique_secondtable. ((((exists ff_h_pvs_unique_secondtableentrypositive. ff_h_pvs_unique_secondtableentrypositive + S (dst_positive_unique_secondtable) = S ((S (dst_index_unique_secondtable)) * dst_positive_scale_unique_secondtable)) /\ exists ff_q_pvs_unique_secondtableentrypositive. dst_positive_code_unique_secondtable = ff_q_pvs_unique_secondtableentrypositive * S ((S (dst_index_unique_secondtable)) * dst_positive_scale_unique_secondtable) + (dst_positive_unique_secondtable))) /\ (((((exists ff_h_pvs_unique_secondtableentrynegative. ff_h_pvs_unique_secondtableentrynegative + S (dst_negative_unique_secondtable) = S ((S (dst_index_unique_secondtable)) * dst_negative_scale_unique_secondtable)) /\ exists ff_q_pvs_unique_secondtableentrynegative. dst_negative_code_unique_secondtable = ff_q_pvs_unique_secondtableentrynegative * S ((S (dst_index_unique_secondtable)) * dst_negative_scale_unique_secondtable) + (dst_negative_unique_secondtable))) /\ (exists ge_balance_positive_unique_secondtableentryvalue ge_balance_negative_unique_secondtableentryvalue. (((((dst_value_unique_secondtable) = 2 * (ge_balance_positive_unique_secondtableentryvalue) /\ (ge_balance_negative_unique_secondtableentryvalue) = 0) \/ exists ge_signed_half_unique_secondtableentryvaluedecode. (((dst_value_unique_secondtable) = 2 * ge_signed_half_unique_secondtableentryvaluedecode + 1 /\ (ge_balance_positive_unique_secondtableentryvalue) = 0) /\ (ge_balance_negative_unique_secondtableentryvalue) = S ge_signed_half_unique_secondtableentryvaluedecode))) /\ ((dst_positive_unique_secondtable) + ge_balance_negative_unique_secondtableentryvalue = (dst_negative_unique_secondtable) + ge_balance_positive_unique_secondtableentryvalue))))))))) /\ (((exists dst_positive_code_unique_secondzero dst_positive_scale_unique_secondzero dst_negative_code_unique_secondzero dst_negative_scale_unique_secondzero dst_positive_unique_secondzero dst_negative_unique_secondzero. (((G) = (((((dst_positive_code_unique_secondzero) + (dst_positive_scale_unique_secondzero)) * S ((dst_positive_code_unique_secondzero) + (dst_positive_scale_unique_secondzero)) + ((dst_positive_scale_unique_secondzero) + (dst_positive_scale_unique_secondzero))) + (((dst_negative_code_unique_secondzero) + (dst_negative_scale_unique_secondzero)) * S ((dst_negative_code_unique_secondzero) + (dst_negative_scale_unique_secondzero)) + ((dst_negative_scale_unique_secondzero) + (dst_negative_scale_unique_secondzero)))) * S ((((dst_positive_code_unique_secondzero) + (dst_positive_scale_unique_secondzero)) * S ((dst_positive_code_unique_secondzero) + (dst_positive_scale_unique_secondzero)) + ((dst_positive_scale_unique_secondzero) + (dst_positive_scale_unique_secondzero))) + (((dst_negative_code_unique_secondzero) + (dst_negative_scale_unique_secondzero)) * S ((dst_negative_code_unique_secondzero) + (dst_negative_scale_unique_secondzero)) + ((dst_negative_scale_unique_secondzero) + (dst_negative_scale_unique_secondzero)))) + ((((dst_negative_code_unique_secondzero) + (dst_negative_scale_unique_secondzero)) * S ((dst_negative_code_unique_secondzero) + (dst_negative_scale_unique_secondzero)) + ((dst_negative_scale_unique_secondzero) + (dst_negative_scale_unique_secondzero))) + (((dst_negative_code_unique_secondzero) + (dst_negative_scale_unique_secondzero)) * S ((dst_negative_code_unique_secondzero) + (dst_negative_scale_unique_secondzero)) + ((dst_negative_scale_unique_secondzero) + (dst_negative_scale_unique_secondzero)))))) /\ (((((exists ff_h_pvs_unique_secondzeropositive. ff_h_pvs_unique_secondzeropositive + S (dst_positive_unique_secondzero) = S ((S (0)) * dst_positive_scale_unique_secondzero)) /\ exists ff_q_pvs_unique_secondzeropositive. dst_positive_code_unique_secondzero = ff_q_pvs_unique_secondzeropositive * S ((S (0)) * dst_positive_scale_unique_secondzero) + (dst_positive_unique_secondzero))) /\ (((((exists ff_h_pvs_unique_secondzeronegative. ff_h_pvs_unique_secondzeronegative + S (dst_negative_unique_secondzero) = S ((S (0)) * dst_negative_scale_unique_secondzero)) /\ exists ff_q_pvs_unique_secondzeronegative. dst_negative_code_unique_secondzero = ff_q_pvs_unique_secondzeronegative * S ((S (0)) * dst_negative_scale_unique_secondzero) + (dst_negative_unique_secondzero))) /\ (exists ge_balance_positive_unique_secondzerovalue ge_balance_negative_unique_secondzerovalue. (((((0) = 2 * (ge_balance_positive_unique_secondzerovalue) /\ (ge_balance_negative_unique_secondzerovalue) = 0) \/ exists ge_signed_half_unique_secondzerovaluedecode. (((0) = 2 * ge_signed_half_unique_secondzerovaluedecode + 1 /\ (ge_balance_positive_unique_secondzerovalue) = 0) /\ (ge_balance_negative_unique_secondzerovalue) = S ge_signed_half_unique_secondzerovaluedecode))) /\ ((dst_positive_unique_secondzero) + ge_balance_negative_unique_secondzerovalue = (dst_negative_unique_secondzero) + ge_balance_positive_unique_secondzerovalue))))))))) /\ (forall mt_index_unique_second mt_value_unique_second. ~(mt_index_unique_second=0) -> (exists pvs_le_gap_unique_seconddomain. pvs_le_gap_unique_seconddomain + (mt_index_unique_second) = (N)) -> (exists dst_positive_code_unique_secondentry dst_positive_scale_unique_secondentry dst_negative_code_unique_secondentry dst_negative_scale_unique_secondentry dst_positive_unique_secondentry dst_negative_unique_secondentry. (((G) = (((((dst_positive_code_unique_secondentry) + (dst_positive_scale_unique_secondentry)) * S ((dst_positive_code_unique_secondentry) + (dst_positive_scale_unique_secondentry)) + ((dst_positive_scale_unique_secondentry) + (dst_positive_scale_unique_secondentry))) + (((dst_negative_code_unique_secondentry) + (dst_negative_scale_unique_secondentry)) * S ((dst_negative_code_unique_secondentry) + (dst_negative_scale_unique_secondentry)) + ((dst_negative_scale_unique_secondentry) + (dst_negative_scale_unique_secondentry)))) * S ((((dst_positive_code_unique_secondentry) + (dst_positive_scale_unique_secondentry)) * S ((dst_positive_code_unique_secondentry) + (dst_positive_scale_unique_secondentry)) + ((dst_positive_scale_unique_secondentry) + (dst_positive_scale_unique_secondentry))) + (((dst_negative_code_unique_secondentry) + (dst_negative_scale_unique_secondentry)) * S ((dst_negative_code_unique_secondentry) + (dst_negative_scale_unique_secondentry)) + ((dst_negative_scale_unique_secondentry) + (dst_negative_scale_unique_secondentry)))) + ((((dst_negative_code_unique_secondentry) + (dst_negative_scale_unique_secondentry)) * S ((dst_negative_code_unique_secondentry) + (dst_negative_scale_unique_secondentry)) + ((dst_negative_scale_unique_secondentry) + (dst_negative_scale_unique_secondentry))) + (((dst_negative_code_unique_secondentry) + (dst_negative_scale_unique_secondentry)) * S ((dst_negative_code_unique_secondentry) + (dst_negative_scale_unique_secondentry)) + ((dst_negative_scale_unique_secondentry) + (dst_negative_scale_unique_secondentry)))))) /\ (((((exists ff_h_pvs_unique_secondentrypositive. ff_h_pvs_unique_secondentrypositive + S (dst_positive_unique_secondentry) = S ((S (mt_index_unique_second)) * dst_positive_scale_unique_secondentry)) /\ exists ff_q_pvs_unique_secondentrypositive. dst_positive_code_unique_secondentry = ff_q_pvs_unique_secondentrypositive * S ((S (mt_index_unique_second)) * dst_positive_scale_unique_secondentry) + (dst_positive_unique_secondentry))) /\ (((((exists ff_h_pvs_unique_secondentrynegative. ff_h_pvs_unique_secondentrynegative + S (dst_negative_unique_secondentry) = S ((S (mt_index_unique_second)) * dst_negative_scale_unique_secondentry)) /\ exists ff_q_pvs_unique_secondentrynegative. dst_negative_code_unique_secondentry = ff_q_pvs_unique_secondentrynegative * S ((S (mt_index_unique_second)) * dst_negative_scale_unique_secondentry) + (dst_negative_unique_secondentry))) /\ (exists ge_balance_positive_unique_secondentryvalue ge_balance_negative_unique_secondentryvalue. (((((mt_value_unique_second) = 2 * (ge_balance_positive_unique_secondentryvalue) /\ (ge_balance_negative_unique_secondentryvalue) = 0) \/ exists ge_signed_half_unique_secondentryvaluedecode. (((mt_value_unique_second) = 2 * ge_signed_half_unique_secondentryvaluedecode + 1 /\ (ge_balance_positive_unique_secondentryvalue) = 0) /\ (ge_balance_negative_unique_secondentryvalue) = S ge_signed_half_unique_secondentryvaluedecode))) /\ ((dst_positive_unique_secondentry) + ge_balance_negative_unique_secondentryvalue = (dst_negative_unique_secondentry) + ge_balance_positive_unique_secondentryvalue))))))))) -> (((~((mt_index_unique_second) = 0)) /\ ((((exists mv_square_prime_unique_secondvaluesquare. ((~((mv_square_prime_unique_secondvaluesquare) = 1) /\ forall pvs_left_unique_secondvaluesquareprime pvs_right_unique_secondvaluesquareprime. (mv_square_prime_unique_secondvaluesquare) = pvs_left_unique_secondvaluesquareprime * pvs_right_unique_secondvaluesquareprime -> pvs_left_unique_secondvaluesquareprime = 1 \/ pvs_right_unique_secondvaluesquareprime = 1) /\ (exists pvs_factor_unique_secondvaluesquaredivisor. (mt_index_unique_second) = (mv_square_prime_unique_secondvaluesquare * mv_square_prime_unique_secondvaluesquare) * pvs_factor_unique_secondvaluesquaredivisor))) /\ ((mt_value_unique_second) = 0))) \/ (((((~((mt_index_unique_second) = 0)) /\ (forall sfd_prime_unique_secondvaluesquarefree. (~((sfd_prime_unique_secondvaluesquarefree) = 1) /\ forall pvs_left_unique_secondvaluesquarefreedomain pvs_right_unique_secondvaluesquarefreedomain. (sfd_prime_unique_secondvaluesquarefree) = pvs_left_unique_secondvaluesquarefreedomain * pvs_right_unique_secondvaluesquarefreedomain -> pvs_left_unique_secondvaluesquarefreedomain = 1 \/ pvs_right_unique_secondvaluesquarefreedomain = 1) -> (exists pvs_le_gap_unique_secondvaluesquarefreebound. pvs_le_gap_unique_secondvaluesquarefreebound + (sfd_prime_unique_secondvaluesquarefree) = (mt_index_unique_second)) -> ~(exists pvs_factor_unique_secondvaluesquarefreesquare. (mt_index_unique_second) = (sfd_prime_unique_secondvaluesquarefree * sfd_prime_unique_secondvaluesquarefree) * pvs_factor_unique_secondvaluesquarefreesquare)))) /\ (exists mv_factor_code_unique_secondvaluefactors mv_factor_scale_unique_secondvaluefactors mv_factor_count_unique_secondvaluefactors. (((~(mt_index_unique_second = 0) /\ ((exists ff_u_fsat_unique_secondvaluefactorsfactorization_product ff_v_fsat_unique_secondvaluefactorsfactorization_product. ((((exists ff_h_fsat_unique_secondvaluefactorsfactorization_product_start. ff_h_fsat_unique_secondvaluefactorsfactorization_product_start + S (1) = S ((S (0)) * ff_v_fsat_unique_secondvaluefactorsfactorization_product)) /\ exists ff_q_fsat_unique_secondvaluefactorsfactorization_product_start. ff_u_fsat_unique_secondvaluefactorsfactorization_product = ff_q_fsat_unique_secondvaluefactorsfactorization_product_start * S ((S (0)) * ff_v_fsat_unique_secondvaluefactorsfactorization_product) + (1))) /\ ((((exists ff_h_fsat_unique_secondvaluefactorsfactorization_product_terminal. ff_h_fsat_unique_secondvaluefactorsfactorization_product_terminal + S (mt_index_unique_second) = S ((S (mv_factor_count_unique_secondvaluefactors)) * ff_v_fsat_unique_secondvaluefactorsfactorization_product)) /\ exists ff_q_fsat_unique_secondvaluefactorsfactorization_product_terminal. ff_u_fsat_unique_secondvaluefactorsfactorization_product = ff_q_fsat_unique_secondvaluefactorsfactorization_product_terminal * S ((S (mv_factor_count_unique_secondvaluefactors)) * ff_v_fsat_unique_secondvaluefactorsfactorization_product) + (mt_index_unique_second))) /\ forall ff_i_fsat_unique_secondvaluefactorsfactorization_product. (exists ff_lt_fsat_unique_secondvaluefactorsfactorization_product_bound. ff_lt_fsat_unique_secondvaluefactorsfactorization_product_bound + S ff_i_fsat_unique_secondvaluefactorsfactorization_product = mv_factor_count_unique_secondvaluefactors) -> exists ff_p_fsat_unique_secondvaluefactorsfactorization_product ff_r_fsat_unique_secondvaluefactorsfactorization_product ff_s_fsat_unique_secondvaluefactorsfactorization_product. ((((exists ff_h_fsat_unique_secondvaluefactorsfactorization_product_factor. ff_h_fsat_unique_secondvaluefactorsfactorization_product_factor + S (ff_p_fsat_unique_secondvaluefactorsfactorization_product) = S ((S (ff_i_fsat_unique_secondvaluefactorsfactorization_product)) * mv_factor_scale_unique_secondvaluefactors)) /\ exists ff_q_fsat_unique_secondvaluefactorsfactorization_product_factor. mv_factor_code_unique_secondvaluefactors = ff_q_fsat_unique_secondvaluefactorsfactorization_product_factor * S ((S (ff_i_fsat_unique_secondvaluefactorsfactorization_product)) * mv_factor_scale_unique_secondvaluefactors) + (ff_p_fsat_unique_secondvaluefactorsfactorization_product))) /\ ((((exists ff_h_fsat_unique_secondvaluefactorsfactorization_product_partial. ff_h_fsat_unique_secondvaluefactorsfactorization_product_partial + S (ff_r_fsat_unique_secondvaluefactorsfactorization_product) = S ((S (ff_i_fsat_unique_secondvaluefactorsfactorization_product)) * ff_v_fsat_unique_secondvaluefactorsfactorization_product)) /\ exists ff_q_fsat_unique_secondvaluefactorsfactorization_product_partial. ff_u_fsat_unique_secondvaluefactorsfactorization_product = ff_q_fsat_unique_secondvaluefactorsfactorization_product_partial * S ((S (ff_i_fsat_unique_secondvaluefactorsfactorization_product)) * ff_v_fsat_unique_secondvaluefactorsfactorization_product) + (ff_r_fsat_unique_secondvaluefactorsfactorization_product))) /\ ((((exists ff_h_fsat_unique_secondvaluefactorsfactorization_product_successor. ff_h_fsat_unique_secondvaluefactorsfactorization_product_successor + S (ff_s_fsat_unique_secondvaluefactorsfactorization_product) = S ((S (S ff_i_fsat_unique_secondvaluefactorsfactorization_product)) * ff_v_fsat_unique_secondvaluefactorsfactorization_product)) /\ exists ff_q_fsat_unique_secondvaluefactorsfactorization_product_successor. ff_u_fsat_unique_secondvaluefactorsfactorization_product = ff_q_fsat_unique_secondvaluefactorsfactorization_product_successor * S ((S (S ff_i_fsat_unique_secondvaluefactorsfactorization_product)) * ff_v_fsat_unique_secondvaluefactorsfactorization_product) + (ff_s_fsat_unique_secondvaluefactorsfactorization_product))) /\ ff_s_fsat_unique_secondvaluefactorsfactorization_product = ff_r_fsat_unique_secondvaluefactorsfactorization_product * ff_p_fsat_unique_secondvaluefactorsfactorization_product)))))) /\ (forall ftsf_index_fsat_unique_secondvaluefactorsfactorization_primes. (exists ftsf_gap_fsat_unique_secondvaluefactorsfactorization_primes_bound. ftsf_gap_fsat_unique_secondvaluefactorsfactorization_primes_bound + S ftsf_index_fsat_unique_secondvaluefactorsfactorization_primes = (mv_factor_count_unique_secondvaluefactors)) -> exists ftsf_factor_fsat_unique_secondvaluefactorsfactorization_primes. ((((exists ff_h_ftsf_fsat_unique_secondvaluefactorsfactorization_primes_entry. ff_h_ftsf_fsat_unique_secondvaluefactorsfactorization_primes_entry + S (ftsf_factor_fsat_unique_secondvaluefactorsfactorization_primes) = S ((S (ftsf_index_fsat_unique_secondvaluefactorsfactorization_primes)) * mv_factor_scale_unique_secondvaluefactors)) /\ exists ff_q_ftsf_fsat_unique_secondvaluefactorsfactorization_primes_entry. mv_factor_code_unique_secondvaluefactors = ff_q_ftsf_fsat_unique_secondvaluefactorsfactorization_primes_entry * S ((S (ftsf_index_fsat_unique_secondvaluefactorsfactorization_primes)) * mv_factor_scale_unique_secondvaluefactors) + (ftsf_factor_fsat_unique_secondvaluefactorsfactorization_primes))) /\ ((~(ftsf_factor_fsat_unique_secondvaluefactorsfactorization_primes = 1) /\ forall frm_prime_left_ftsf_fsat_unique_secondvaluefactorsfactorization_primes_prime frm_prime_right_ftsf_fsat_unique_secondvaluefactorsfactorization_primes_prime. ftsf_factor_fsat_unique_secondvaluefactorsfactorization_primes = frm_prime_left_ftsf_fsat_unique_secondvaluefactorsfactorization_primes_prime * frm_prime_right_ftsf_fsat_unique_secondvaluefactorsfactorization_primes_prime -> frm_prime_left_ftsf_fsat_unique_secondvaluefactorsfactorization_primes_prime = 1 \/ frm_prime_right_ftsf_fsat_unique_secondvaluefactorsfactorization_primes_prime = 1))))))) /\ ((((exists mv_even_half_unique_secondvaluefactorsparityeven. (mv_factor_count_unique_secondvaluefactors) = 2 * mv_even_half_unique_secondvaluefactorsparityeven) /\ ((mt_value_unique_second) = 2))) \/ (((exists mv_odd_half_unique_secondvaluefactorsparityodd. (mv_factor_count_unique_secondvaluefactors) = 2 * mv_odd_half_unique_secondvaluefactorsparityodd + 1) /\ ((mt_value_unique_second) = 1)))))))))))))))) -> (forall dst_index_unique_values dst_first_unique_values dst_second_unique_values. (exists pvs_gap_unique_valuesbound. pvs_gap_unique_valuesbound + S (dst_index_unique_values) = (S N)) -> (exists dst_positive_code_unique_valuesfirst dst_positive_scale_unique_valuesfirst dst_negative_code_unique_valuesfirst dst_negative_scale_unique_valuesfirst dst_positive_unique_valuesfirst dst_negative_unique_valuesfirst. (((F) = (((((dst_positive_code_unique_valuesfirst) + (dst_positive_scale_unique_valuesfirst)) * S ((dst_positive_code_unique_valuesfirst) + (dst_positive_scale_unique_valuesfirst)) + ((dst_positive_scale_unique_valuesfirst) + (dst_positive_scale_unique_valuesfirst))) + (((dst_negative_code_unique_valuesfirst) + (dst_negative_scale_unique_valuesfirst)) * S ((dst_negative_code_unique_valuesfirst) + (dst_negative_scale_unique_valuesfirst)) + ((dst_negative_scale_unique_valuesfirst) + (dst_negative_scale_unique_valuesfirst)))) * S ((((dst_positive_code_unique_valuesfirst) + (dst_positive_scale_unique_valuesfirst)) * S ((dst_positive_code_unique_valuesfirst) + (dst_positive_scale_unique_valuesfirst)) + ((dst_positive_scale_unique_valuesfirst) + (dst_positive_scale_unique_valuesfirst))) + (((dst_negative_code_unique_valuesfirst) + (dst_negative_scale_unique_valuesfirst)) * S ((dst_negative_code_unique_valuesfirst) + (dst_negative_scale_unique_valuesfirst)) + ((dst_negative_scale_unique_valuesfirst) + (dst_negative_scale_unique_valuesfirst)))) + ((((dst_negative_code_unique_valuesfirst) + (dst_negative_scale_unique_valuesfirst)) * S ((dst_negative_code_unique_valuesfirst) + (dst_negative_scale_unique_valuesfirst)) + ((dst_negative_scale_unique_valuesfirst) + (dst_negative_scale_unique_valuesfirst))) + (((dst_negative_code_unique_valuesfirst) + (dst_negative_scale_unique_valuesfirst)) * S ((dst_negative_code_unique_valuesfirst) + (dst_negative_scale_unique_valuesfirst)) + ((dst_negative_scale_unique_valuesfirst) + (dst_negative_scale_unique_valuesfirst)))))) /\ (((((exists ff_h_pvs_unique_valuesfirstpositive. ff_h_pvs_unique_valuesfirstpositive + S (dst_positive_unique_valuesfirst) = S ((S (dst_index_unique_values)) * dst_positive_scale_unique_valuesfirst)) /\ exists ff_q_pvs_unique_valuesfirstpositive. dst_positive_code_unique_valuesfirst = ff_q_pvs_unique_valuesfirstpositive * S ((S (dst_index_unique_values)) * dst_positive_scale_unique_valuesfirst) + (dst_positive_unique_valuesfirst))) /\ (((((exists ff_h_pvs_unique_valuesfirstnegative. ff_h_pvs_unique_valuesfirstnegative + S (dst_negative_unique_valuesfirst) = S ((S (dst_index_unique_values)) * dst_negative_scale_unique_valuesfirst)) /\ exists ff_q_pvs_unique_valuesfirstnegative. dst_negative_code_unique_valuesfirst = ff_q_pvs_unique_valuesfirstnegative * S ((S (dst_index_unique_values)) * dst_negative_scale_unique_valuesfirst) + (dst_negative_unique_valuesfirst))) /\ (exists ge_balance_positive_unique_valuesfirstvalue ge_balance_negative_unique_valuesfirstvalue. (((((dst_first_unique_values) = 2 * (ge_balance_positive_unique_valuesfirstvalue) /\ (ge_balance_negative_unique_valuesfirstvalue) = 0) \/ exists ge_signed_half_unique_valuesfirstvaluedecode. (((dst_first_unique_values) = 2 * ge_signed_half_unique_valuesfirstvaluedecode + 1 /\ (ge_balance_positive_unique_valuesfirstvalue) = 0) /\ (ge_balance_negative_unique_valuesfirstvalue) = S ge_signed_half_unique_valuesfirstvaluedecode))) /\ ((dst_positive_unique_valuesfirst) + ge_balance_negative_unique_valuesfirstvalue = (dst_negative_unique_valuesfirst) + ge_balance_positive_unique_valuesfirstvalue))))))))) -> (exists dst_positive_code_unique_valuessecond dst_positive_scale_unique_valuessecond dst_negative_code_unique_valuessecond dst_negative_scale_unique_valuessecond dst_positive_unique_valuessecond dst_negative_unique_valuessecond. (((G) = (((((dst_positive_code_unique_valuessecond) + (dst_positive_scale_unique_valuessecond)) * S ((dst_positive_code_unique_valuessecond) + (dst_positive_scale_unique_valuessecond)) + ((dst_positive_scale_unique_valuessecond) + (dst_positive_scale_unique_valuessecond))) + (((dst_negative_code_unique_valuessecond) + (dst_negative_scale_unique_valuessecond)) * S ((dst_negative_code_unique_valuessecond) + (dst_negative_scale_unique_valuessecond)) + ((dst_negative_scale_unique_valuessecond) + (dst_negative_scale_unique_valuessecond)))) * S ((((dst_positive_code_unique_valuessecond) + (dst_positive_scale_unique_valuessecond)) * S ((dst_positive_code_unique_valuessecond) + (dst_positive_scale_unique_valuessecond)) + ((dst_positive_scale_unique_valuessecond) + (dst_positive_scale_unique_valuessecond))) + (((dst_negative_code_unique_valuessecond) + (dst_negative_scale_unique_valuessecond)) * S ((dst_negative_code_unique_valuessecond) + (dst_negative_scale_unique_valuessecond)) + ((dst_negative_scale_unique_valuessecond) + (dst_negative_scale_unique_valuessecond)))) + ((((dst_negative_code_unique_valuessecond) + (dst_negative_scale_unique_valuessecond)) * S ((dst_negative_code_unique_valuessecond) + (dst_negative_scale_unique_valuessecond)) + ((dst_negative_scale_unique_valuessecond) + (dst_negative_scale_unique_valuessecond))) + (((dst_negative_code_unique_valuessecond) + (dst_negative_scale_unique_valuessecond)) * S ((dst_negative_code_unique_valuessecond) + (dst_negative_scale_unique_valuessecond)) + ((dst_negative_scale_unique_valuessecond) + (dst_negative_scale_unique_valuessecond)))))) /\ (((((exists ff_h_pvs_unique_valuessecondpositive. ff_h_pvs_unique_valuessecondpositive + S (dst_positive_unique_valuessecond) = S ((S (dst_index_unique_values)) * dst_positive_scale_unique_valuessecond)) /\ exists ff_q_pvs_unique_valuessecondpositive. dst_positive_code_unique_valuessecond = ff_q_pvs_unique_valuessecondpositive * S ((S (dst_index_unique_values)) * dst_positive_scale_unique_valuessecond) + (dst_positive_unique_valuessecond))) /\ (((((exists ff_h_pvs_unique_valuessecondnegative. ff_h_pvs_unique_valuessecondnegative + S (dst_negative_unique_valuessecond) = S ((S (dst_index_unique_values)) * dst_negative_scale_unique_valuessecond)) /\ exists ff_q_pvs_unique_valuessecondnegative. dst_negative_code_unique_valuessecond = ff_q_pvs_unique_valuessecondnegative * S ((S (dst_index_unique_values)) * dst_negative_scale_unique_valuessecond) + (dst_negative_unique_valuessecond))) /\ (exists ge_balance_positive_unique_valuessecondvalue ge_balance_negative_unique_valuessecondvalue. (((((dst_second_unique_values) = 2 * (ge_balance_positive_unique_valuessecondvalue) /\ (ge_balance_negative_unique_valuessecondvalue) = 0) \/ exists ge_signed_half_unique_valuessecondvaluedecode. (((dst_second_unique_values) = 2 * ge_signed_half_unique_valuessecondvaluedecode + 1 /\ (ge_balance_positive_unique_valuessecondvalue) = 0) /\ (ge_balance_negative_unique_valuessecondvalue) = S ge_signed_half_unique_valuessecondvaluedecode))) /\ ((dst_positive_unique_valuessecond) + ge_balance_negative_unique_valuessecondvalue = (dst_negative_unique_valuessecond) + ge_balance_positive_unique_valuessecondvalue))))))))) -> dst_first_unique_values = dst_second_unique_values)Complete tactic proof in conservative notation
All 65 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
65 script commands · 12 reading checkpoints · 2 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 assumptionsL1–5
02Separate the logical casesL6–9
03Fix variables and assumptionsL10–15
04Establish hibL16–20
Establish this local claim before using it. It is not an additional assumption. The following proof commands apply le of succ le succ.
05Establish hcaseL21–24
06Separate the logical casesL25–25
Follow the explicit conjunction, disjunction, witness, or contradiction step recorded below.
- L25
cases hcase
07Calculate and transport equalitiesL26–34
Carry out the recorded arithmetic or equality steps; inspect the exact commands for their direction and premises.
08Use earlier factsL35–41
Instantiate or apply named facts and discharge the corresponding proof obligations.
09Calculate and transport equalitiesL42–42
Carry out the recorded arithmetic or equality steps; inspect the exact commands for their direction and premises.
- L42
symm
10Use earlier factsL43–52
Instantiate or apply named facts and discharge the corresponding proof obligations.
- L43
specialize divisor_signed_table_at_functional (G) - L44
specialize divisor_signed_table_at_functional (0) - L45
specialize divisor_signed_table_at_functional (b) - L46
specialize divisor_signed_table_at_functional (0) - L47
apply divisor_signed_table_at_functional - L48
exact hb - L49
exact hg_right_left - L50
specialize mobius_value_functional (i) - L51
specialize mobius_value_functional (a) - L52
specialize mobius_value_functional (b)
11Use earlier factsL53–62
Instantiate or apply named facts and discharge the corresponding proof obligations.
Original defined command ledger · 65 lines
- 0001
intro N - 0002
intro F - 0003
intro G - 0004
intro hf - 0005
intro hg - 0006
cases hf - 0007
cases hf_right - 0008
cases hg - 0009
cases hg_right - 0010
intro i - 0011
intro a - 0012
intro b - 0013
intro hi - 0014
intro ha - 0015
intro hb - 0016
have hib : Le(i,N) - 0017
specialize le_of_succ_le_succ (i) - 0018
specialize le_of_succ_le_succ (N) - 0019
apply le_of_succ_le_succ - 0020
exact hi - 0021
have hcase : i=0 \/ ~(i=0) - 0022
specialize eq_decidable (i) - 0023
specialize eq_decidable (0) - 0024
apply eq_decidable - 0025
cases hcase - 0026
rewrite hcase_left at ha - 0027
rewrite hcase_left at ha - 0028
rewrite hcase_left at ha - 0029
rewrite hcase_left at ha - 0030
rewrite hcase_left at hb - 0031
rewrite hcase_left at hb - 0032
rewrite hcase_left at hb - 0033
rewrite hcase_left at hb - 0034
trans 0 - 0035
specialize divisor_signed_table_at_functional (F) - 0036
specialize divisor_signed_table_at_functional (0) - 0037
specialize divisor_signed_table_at_functional (a) - 0038
specialize divisor_signed_table_at_functional (0) - 0039
apply divisor_signed_table_at_functional - 0040
exact ha - 0041
exact hf_right_left - 0042
symm - 0043
specialize divisor_signed_table_at_functional (G) - 0044
specialize divisor_signed_table_at_functional (0) - 0045
specialize divisor_signed_table_at_functional (b) - 0046
specialize divisor_signed_table_at_functional (0) - 0047
apply divisor_signed_table_at_functional - 0048
exact hb - 0049
exact hg_right_left - 0050
specialize mobius_value_functional (i) - 0051
specialize mobius_value_functional (a) - 0052
specialize mobius_value_functional (b) - 0053
apply mobius_value_functional - 0054
specialize hf_right_right (i) - 0055
specialize hf_right_right (a) - 0056
apply hf_right_right - 0057
exact hcase_right - 0058
exact hib - 0059
exact ha - 0060
specialize hg_right_right (i) - 0061
specialize hg_right_right (b) - 0062
apply hg_right_right - 0063
exact hcase_right - 0064
exact hib - 0065
exact hb