Current library: Alpha v34, 4,223 checked-use theorems; Stable remains 432. Historical first admissions, original proof editions, and non-admitted aliases are preserved.
Signed one is code 2. Positive-only table graphs preserve arbitrary zeroth values, including N=0. The unit and divisor-sum identities are proved, never embedded in definitions. The separate inverse family proves the general unit-at-one criterion; multiplicative-function closure remains open.
Exact theorem in conservative defined notation
∀ N. ∀ F. ∀ G. ConstantOneTable(N,F) → ConstantOneTable(N,G) → ArithPositiveEqual(F,G,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_constant_oneunique_firsttable dst_positive_scale_constant_oneunique_firsttable dst_negative_code_constant_oneunique_firsttable dst_negative_scale_constant_oneunique_firsttable. (((F) = (((((dst_positive_code_constant_oneunique_firsttable) + (dst_positive_scale_constant_oneunique_firsttable)) * S ((dst_positive_code_constant_oneunique_firsttable) + (dst_positive_scale_constant_oneunique_firsttable)) + ((dst_positive_scale_constant_oneunique_firsttable) + (dst_positive_scale_constant_oneunique_firsttable))) + (((dst_negative_code_constant_oneunique_firsttable) + (dst_negative_scale_constant_oneunique_firsttable)) * S ((dst_negative_code_constant_oneunique_firsttable) + (dst_negative_scale_constant_oneunique_firsttable)) + ((dst_negative_scale_constant_oneunique_firsttable) + (dst_negative_scale_constant_oneunique_firsttable)))) * S ((((dst_positive_code_constant_oneunique_firsttable) + (dst_positive_scale_constant_oneunique_firsttable)) * S ((dst_positive_code_constant_oneunique_firsttable) + (dst_positive_scale_constant_oneunique_firsttable)) + ((dst_positive_scale_constant_oneunique_firsttable) + (dst_positive_scale_constant_oneunique_firsttable))) + (((dst_negative_code_constant_oneunique_firsttable) + (dst_negative_scale_constant_oneunique_firsttable)) * S ((dst_negative_code_constant_oneunique_firsttable) + (dst_negative_scale_constant_oneunique_firsttable)) + ((dst_negative_scale_constant_oneunique_firsttable) + (dst_negative_scale_constant_oneunique_firsttable)))) + ((((dst_negative_code_constant_oneunique_firsttable) + (dst_negative_scale_constant_oneunique_firsttable)) * S ((dst_negative_code_constant_oneunique_firsttable) + (dst_negative_scale_constant_oneunique_firsttable)) + ((dst_negative_scale_constant_oneunique_firsttable) + (dst_negative_scale_constant_oneunique_firsttable))) + (((dst_negative_code_constant_oneunique_firsttable) + (dst_negative_scale_constant_oneunique_firsttable)) * S ((dst_negative_code_constant_oneunique_firsttable) + (dst_negative_scale_constant_oneunique_firsttable)) + ((dst_negative_scale_constant_oneunique_firsttable) + (dst_negative_scale_constant_oneunique_firsttable)))))) /\ (forall dst_index_constant_oneunique_firsttable. (exists pvs_le_gap_constant_oneunique_firsttabledomain. pvs_le_gap_constant_oneunique_firsttabledomain + (dst_index_constant_oneunique_firsttable) = (N)) -> exists dst_positive_constant_oneunique_firsttable dst_negative_constant_oneunique_firsttable dst_value_constant_oneunique_firsttable. ((((exists ff_h_pvs_constant_oneunique_firsttableentrypositive. ff_h_pvs_constant_oneunique_firsttableentrypositive + S (dst_positive_constant_oneunique_firsttable) = S ((S (dst_index_constant_oneunique_firsttable)) * dst_positive_scale_constant_oneunique_firsttable)) /\ exists ff_q_pvs_constant_oneunique_firsttableentrypositive. dst_positive_code_constant_oneunique_firsttable = ff_q_pvs_constant_oneunique_firsttableentrypositive * S ((S (dst_index_constant_oneunique_firsttable)) * dst_positive_scale_constant_oneunique_firsttable) + (dst_positive_constant_oneunique_firsttable))) /\ (((((exists ff_h_pvs_constant_oneunique_firsttableentrynegative. ff_h_pvs_constant_oneunique_firsttableentrynegative + S (dst_negative_constant_oneunique_firsttable) = S ((S (dst_index_constant_oneunique_firsttable)) * dst_negative_scale_constant_oneunique_firsttable)) /\ exists ff_q_pvs_constant_oneunique_firsttableentrynegative. dst_negative_code_constant_oneunique_firsttable = ff_q_pvs_constant_oneunique_firsttableentrynegative * S ((S (dst_index_constant_oneunique_firsttable)) * dst_negative_scale_constant_oneunique_firsttable) + (dst_negative_constant_oneunique_firsttable))) /\ (exists ge_balance_positive_constant_oneunique_firsttableentryvalue ge_balance_negative_constant_oneunique_firsttableentryvalue. (((((dst_value_constant_oneunique_firsttable) = 2 * (ge_balance_positive_constant_oneunique_firsttableentryvalue) /\ (ge_balance_negative_constant_oneunique_firsttableentryvalue) = 0) \/ exists ge_signed_half_constant_oneunique_firsttableentryvaluedecode. (((dst_value_constant_oneunique_firsttable) = 2 * ge_signed_half_constant_oneunique_firsttableentryvaluedecode + 1 /\ (ge_balance_positive_constant_oneunique_firsttableentryvalue) = 0) /\ (ge_balance_negative_constant_oneunique_firsttableentryvalue) = S ge_signed_half_constant_oneunique_firsttableentryvaluedecode))) /\ ((dst_positive_constant_oneunique_firsttable) + ge_balance_negative_constant_oneunique_firsttableentryvalue = (dst_negative_constant_oneunique_firsttable) + ge_balance_positive_constant_oneunique_firsttableentryvalue))))))))) /\ (forall du_index_constant_oneunique_first du_value_constant_oneunique_first. ~(du_index_constant_oneunique_first=0) -> (exists pvs_le_gap_constant_oneunique_firstbound. pvs_le_gap_constant_oneunique_firstbound + (du_index_constant_oneunique_first) = (N)) -> (exists dst_positive_code_constant_oneunique_firstentry dst_positive_scale_constant_oneunique_firstentry dst_negative_code_constant_oneunique_firstentry dst_negative_scale_constant_oneunique_firstentry dst_positive_constant_oneunique_firstentry dst_negative_constant_oneunique_firstentry. (((F) = (((((dst_positive_code_constant_oneunique_firstentry) + (dst_positive_scale_constant_oneunique_firstentry)) * S ((dst_positive_code_constant_oneunique_firstentry) + (dst_positive_scale_constant_oneunique_firstentry)) + ((dst_positive_scale_constant_oneunique_firstentry) + (dst_positive_scale_constant_oneunique_firstentry))) + (((dst_negative_code_constant_oneunique_firstentry) + (dst_negative_scale_constant_oneunique_firstentry)) * S ((dst_negative_code_constant_oneunique_firstentry) + (dst_negative_scale_constant_oneunique_firstentry)) + ((dst_negative_scale_constant_oneunique_firstentry) + (dst_negative_scale_constant_oneunique_firstentry)))) * S ((((dst_positive_code_constant_oneunique_firstentry) + (dst_positive_scale_constant_oneunique_firstentry)) * S ((dst_positive_code_constant_oneunique_firstentry) + (dst_positive_scale_constant_oneunique_firstentry)) + ((dst_positive_scale_constant_oneunique_firstentry) + (dst_positive_scale_constant_oneunique_firstentry))) + (((dst_negative_code_constant_oneunique_firstentry) + (dst_negative_scale_constant_oneunique_firstentry)) * S ((dst_negative_code_constant_oneunique_firstentry) + (dst_negative_scale_constant_oneunique_firstentry)) + ((dst_negative_scale_constant_oneunique_firstentry) + (dst_negative_scale_constant_oneunique_firstentry)))) + ((((dst_negative_code_constant_oneunique_firstentry) + (dst_negative_scale_constant_oneunique_firstentry)) * S ((dst_negative_code_constant_oneunique_firstentry) + (dst_negative_scale_constant_oneunique_firstentry)) + ((dst_negative_scale_constant_oneunique_firstentry) + (dst_negative_scale_constant_oneunique_firstentry))) + (((dst_negative_code_constant_oneunique_firstentry) + (dst_negative_scale_constant_oneunique_firstentry)) * S ((dst_negative_code_constant_oneunique_firstentry) + (dst_negative_scale_constant_oneunique_firstentry)) + ((dst_negative_scale_constant_oneunique_firstentry) + (dst_negative_scale_constant_oneunique_firstentry)))))) /\ (((((exists ff_h_pvs_constant_oneunique_firstentrypositive. ff_h_pvs_constant_oneunique_firstentrypositive + S (dst_positive_constant_oneunique_firstentry) = S ((S (du_index_constant_oneunique_first)) * dst_positive_scale_constant_oneunique_firstentry)) /\ exists ff_q_pvs_constant_oneunique_firstentrypositive. dst_positive_code_constant_oneunique_firstentry = ff_q_pvs_constant_oneunique_firstentrypositive * S ((S (du_index_constant_oneunique_first)) * dst_positive_scale_constant_oneunique_firstentry) + (dst_positive_constant_oneunique_firstentry))) /\ (((((exists ff_h_pvs_constant_oneunique_firstentrynegative. ff_h_pvs_constant_oneunique_firstentrynegative + S (dst_negative_constant_oneunique_firstentry) = S ((S (du_index_constant_oneunique_first)) * dst_negative_scale_constant_oneunique_firstentry)) /\ exists ff_q_pvs_constant_oneunique_firstentrynegative. dst_negative_code_constant_oneunique_firstentry = ff_q_pvs_constant_oneunique_firstentrynegative * S ((S (du_index_constant_oneunique_first)) * dst_negative_scale_constant_oneunique_firstentry) + (dst_negative_constant_oneunique_firstentry))) /\ (exists ge_balance_positive_constant_oneunique_firstentryvalue ge_balance_negative_constant_oneunique_firstentryvalue. (((((du_value_constant_oneunique_first) = 2 * (ge_balance_positive_constant_oneunique_firstentryvalue) /\ (ge_balance_negative_constant_oneunique_firstentryvalue) = 0) \/ exists ge_signed_half_constant_oneunique_firstentryvaluedecode. (((du_value_constant_oneunique_first) = 2 * ge_signed_half_constant_oneunique_firstentryvaluedecode + 1 /\ (ge_balance_positive_constant_oneunique_firstentryvalue) = 0) /\ (ge_balance_negative_constant_oneunique_firstentryvalue) = S ge_signed_half_constant_oneunique_firstentryvaluedecode))) /\ ((dst_positive_constant_oneunique_firstentry) + ge_balance_negative_constant_oneunique_firstentryvalue = (dst_negative_constant_oneunique_firstentry) + ge_balance_positive_constant_oneunique_firstentryvalue))))))))) -> du_value_constant_oneunique_first=2))) -> (((exists dst_positive_code_constant_oneunique_secondtable dst_positive_scale_constant_oneunique_secondtable dst_negative_code_constant_oneunique_secondtable dst_negative_scale_constant_oneunique_secondtable. (((G) = (((((dst_positive_code_constant_oneunique_secondtable) + (dst_positive_scale_constant_oneunique_secondtable)) * S ((dst_positive_code_constant_oneunique_secondtable) + (dst_positive_scale_constant_oneunique_secondtable)) + ((dst_positive_scale_constant_oneunique_secondtable) + (dst_positive_scale_constant_oneunique_secondtable))) + (((dst_negative_code_constant_oneunique_secondtable) + (dst_negative_scale_constant_oneunique_secondtable)) * S ((dst_negative_code_constant_oneunique_secondtable) + (dst_negative_scale_constant_oneunique_secondtable)) + ((dst_negative_scale_constant_oneunique_secondtable) + (dst_negative_scale_constant_oneunique_secondtable)))) * S ((((dst_positive_code_constant_oneunique_secondtable) + (dst_positive_scale_constant_oneunique_secondtable)) * S ((dst_positive_code_constant_oneunique_secondtable) + (dst_positive_scale_constant_oneunique_secondtable)) + ((dst_positive_scale_constant_oneunique_secondtable) + (dst_positive_scale_constant_oneunique_secondtable))) + (((dst_negative_code_constant_oneunique_secondtable) + (dst_negative_scale_constant_oneunique_secondtable)) * S ((dst_negative_code_constant_oneunique_secondtable) + (dst_negative_scale_constant_oneunique_secondtable)) + ((dst_negative_scale_constant_oneunique_secondtable) + (dst_negative_scale_constant_oneunique_secondtable)))) + ((((dst_negative_code_constant_oneunique_secondtable) + (dst_negative_scale_constant_oneunique_secondtable)) * S ((dst_negative_code_constant_oneunique_secondtable) + (dst_negative_scale_constant_oneunique_secondtable)) + ((dst_negative_scale_constant_oneunique_secondtable) + (dst_negative_scale_constant_oneunique_secondtable))) + (((dst_negative_code_constant_oneunique_secondtable) + (dst_negative_scale_constant_oneunique_secondtable)) * S ((dst_negative_code_constant_oneunique_secondtable) + (dst_negative_scale_constant_oneunique_secondtable)) + ((dst_negative_scale_constant_oneunique_secondtable) + (dst_negative_scale_constant_oneunique_secondtable)))))) /\ (forall dst_index_constant_oneunique_secondtable. (exists pvs_le_gap_constant_oneunique_secondtabledomain. pvs_le_gap_constant_oneunique_secondtabledomain + (dst_index_constant_oneunique_secondtable) = (N)) -> exists dst_positive_constant_oneunique_secondtable dst_negative_constant_oneunique_secondtable dst_value_constant_oneunique_secondtable. ((((exists ff_h_pvs_constant_oneunique_secondtableentrypositive. ff_h_pvs_constant_oneunique_secondtableentrypositive + S (dst_positive_constant_oneunique_secondtable) = S ((S (dst_index_constant_oneunique_secondtable)) * dst_positive_scale_constant_oneunique_secondtable)) /\ exists ff_q_pvs_constant_oneunique_secondtableentrypositive. dst_positive_code_constant_oneunique_secondtable = ff_q_pvs_constant_oneunique_secondtableentrypositive * S ((S (dst_index_constant_oneunique_secondtable)) * dst_positive_scale_constant_oneunique_secondtable) + (dst_positive_constant_oneunique_secondtable))) /\ (((((exists ff_h_pvs_constant_oneunique_secondtableentrynegative. ff_h_pvs_constant_oneunique_secondtableentrynegative + S (dst_negative_constant_oneunique_secondtable) = S ((S (dst_index_constant_oneunique_secondtable)) * dst_negative_scale_constant_oneunique_secondtable)) /\ exists ff_q_pvs_constant_oneunique_secondtableentrynegative. dst_negative_code_constant_oneunique_secondtable = ff_q_pvs_constant_oneunique_secondtableentrynegative * S ((S (dst_index_constant_oneunique_secondtable)) * dst_negative_scale_constant_oneunique_secondtable) + (dst_negative_constant_oneunique_secondtable))) /\ (exists ge_balance_positive_constant_oneunique_secondtableentryvalue ge_balance_negative_constant_oneunique_secondtableentryvalue. (((((dst_value_constant_oneunique_secondtable) = 2 * (ge_balance_positive_constant_oneunique_secondtableentryvalue) /\ (ge_balance_negative_constant_oneunique_secondtableentryvalue) = 0) \/ exists ge_signed_half_constant_oneunique_secondtableentryvaluedecode. (((dst_value_constant_oneunique_secondtable) = 2 * ge_signed_half_constant_oneunique_secondtableentryvaluedecode + 1 /\ (ge_balance_positive_constant_oneunique_secondtableentryvalue) = 0) /\ (ge_balance_negative_constant_oneunique_secondtableentryvalue) = S ge_signed_half_constant_oneunique_secondtableentryvaluedecode))) /\ ((dst_positive_constant_oneunique_secondtable) + ge_balance_negative_constant_oneunique_secondtableentryvalue = (dst_negative_constant_oneunique_secondtable) + ge_balance_positive_constant_oneunique_secondtableentryvalue))))))))) /\ (forall du_index_constant_oneunique_second du_value_constant_oneunique_second. ~(du_index_constant_oneunique_second=0) -> (exists pvs_le_gap_constant_oneunique_secondbound. pvs_le_gap_constant_oneunique_secondbound + (du_index_constant_oneunique_second) = (N)) -> (exists dst_positive_code_constant_oneunique_secondentry dst_positive_scale_constant_oneunique_secondentry dst_negative_code_constant_oneunique_secondentry dst_negative_scale_constant_oneunique_secondentry dst_positive_constant_oneunique_secondentry dst_negative_constant_oneunique_secondentry. (((G) = (((((dst_positive_code_constant_oneunique_secondentry) + (dst_positive_scale_constant_oneunique_secondentry)) * S ((dst_positive_code_constant_oneunique_secondentry) + (dst_positive_scale_constant_oneunique_secondentry)) + ((dst_positive_scale_constant_oneunique_secondentry) + (dst_positive_scale_constant_oneunique_secondentry))) + (((dst_negative_code_constant_oneunique_secondentry) + (dst_negative_scale_constant_oneunique_secondentry)) * S ((dst_negative_code_constant_oneunique_secondentry) + (dst_negative_scale_constant_oneunique_secondentry)) + ((dst_negative_scale_constant_oneunique_secondentry) + (dst_negative_scale_constant_oneunique_secondentry)))) * S ((((dst_positive_code_constant_oneunique_secondentry) + (dst_positive_scale_constant_oneunique_secondentry)) * S ((dst_positive_code_constant_oneunique_secondentry) + (dst_positive_scale_constant_oneunique_secondentry)) + ((dst_positive_scale_constant_oneunique_secondentry) + (dst_positive_scale_constant_oneunique_secondentry))) + (((dst_negative_code_constant_oneunique_secondentry) + (dst_negative_scale_constant_oneunique_secondentry)) * S ((dst_negative_code_constant_oneunique_secondentry) + (dst_negative_scale_constant_oneunique_secondentry)) + ((dst_negative_scale_constant_oneunique_secondentry) + (dst_negative_scale_constant_oneunique_secondentry)))) + ((((dst_negative_code_constant_oneunique_secondentry) + (dst_negative_scale_constant_oneunique_secondentry)) * S ((dst_negative_code_constant_oneunique_secondentry) + (dst_negative_scale_constant_oneunique_secondentry)) + ((dst_negative_scale_constant_oneunique_secondentry) + (dst_negative_scale_constant_oneunique_secondentry))) + (((dst_negative_code_constant_oneunique_secondentry) + (dst_negative_scale_constant_oneunique_secondentry)) * S ((dst_negative_code_constant_oneunique_secondentry) + (dst_negative_scale_constant_oneunique_secondentry)) + ((dst_negative_scale_constant_oneunique_secondentry) + (dst_negative_scale_constant_oneunique_secondentry)))))) /\ (((((exists ff_h_pvs_constant_oneunique_secondentrypositive. ff_h_pvs_constant_oneunique_secondentrypositive + S (dst_positive_constant_oneunique_secondentry) = S ((S (du_index_constant_oneunique_second)) * dst_positive_scale_constant_oneunique_secondentry)) /\ exists ff_q_pvs_constant_oneunique_secondentrypositive. dst_positive_code_constant_oneunique_secondentry = ff_q_pvs_constant_oneunique_secondentrypositive * S ((S (du_index_constant_oneunique_second)) * dst_positive_scale_constant_oneunique_secondentry) + (dst_positive_constant_oneunique_secondentry))) /\ (((((exists ff_h_pvs_constant_oneunique_secondentrynegative. ff_h_pvs_constant_oneunique_secondentrynegative + S (dst_negative_constant_oneunique_secondentry) = S ((S (du_index_constant_oneunique_second)) * dst_negative_scale_constant_oneunique_secondentry)) /\ exists ff_q_pvs_constant_oneunique_secondentrynegative. dst_negative_code_constant_oneunique_secondentry = ff_q_pvs_constant_oneunique_secondentrynegative * S ((S (du_index_constant_oneunique_second)) * dst_negative_scale_constant_oneunique_secondentry) + (dst_negative_constant_oneunique_secondentry))) /\ (exists ge_balance_positive_constant_oneunique_secondentryvalue ge_balance_negative_constant_oneunique_secondentryvalue. (((((du_value_constant_oneunique_second) = 2 * (ge_balance_positive_constant_oneunique_secondentryvalue) /\ (ge_balance_negative_constant_oneunique_secondentryvalue) = 0) \/ exists ge_signed_half_constant_oneunique_secondentryvaluedecode. (((du_value_constant_oneunique_second) = 2 * ge_signed_half_constant_oneunique_secondentryvaluedecode + 1 /\ (ge_balance_positive_constant_oneunique_secondentryvalue) = 0) /\ (ge_balance_negative_constant_oneunique_secondentryvalue) = S ge_signed_half_constant_oneunique_secondentryvaluedecode))) /\ ((dst_positive_constant_oneunique_secondentry) + ge_balance_negative_constant_oneunique_secondentryvalue = (dst_negative_constant_oneunique_secondentry) + ge_balance_positive_constant_oneunique_secondentryvalue))))))))) -> du_value_constant_oneunique_second=2))) -> (forall dm_index_constant_oneunique_result dm_first_value_constant_oneunique_result dm_second_value_constant_oneunique_result. ~(dm_index_constant_oneunique_result=0) -> (exists pvs_le_gap_constant_oneunique_resultdomain. pvs_le_gap_constant_oneunique_resultdomain + (dm_index_constant_oneunique_result) = (N)) -> (exists dst_positive_code_constant_oneunique_resultfirst dst_positive_scale_constant_oneunique_resultfirst dst_negative_code_constant_oneunique_resultfirst dst_negative_scale_constant_oneunique_resultfirst dst_positive_constant_oneunique_resultfirst dst_negative_constant_oneunique_resultfirst. (((F) = (((((dst_positive_code_constant_oneunique_resultfirst) + (dst_positive_scale_constant_oneunique_resultfirst)) * S ((dst_positive_code_constant_oneunique_resultfirst) + (dst_positive_scale_constant_oneunique_resultfirst)) + ((dst_positive_scale_constant_oneunique_resultfirst) + (dst_positive_scale_constant_oneunique_resultfirst))) + (((dst_negative_code_constant_oneunique_resultfirst) + (dst_negative_scale_constant_oneunique_resultfirst)) * S ((dst_negative_code_constant_oneunique_resultfirst) + (dst_negative_scale_constant_oneunique_resultfirst)) + ((dst_negative_scale_constant_oneunique_resultfirst) + (dst_negative_scale_constant_oneunique_resultfirst)))) * S ((((dst_positive_code_constant_oneunique_resultfirst) + (dst_positive_scale_constant_oneunique_resultfirst)) * S ((dst_positive_code_constant_oneunique_resultfirst) + (dst_positive_scale_constant_oneunique_resultfirst)) + ((dst_positive_scale_constant_oneunique_resultfirst) + (dst_positive_scale_constant_oneunique_resultfirst))) + (((dst_negative_code_constant_oneunique_resultfirst) + (dst_negative_scale_constant_oneunique_resultfirst)) * S ((dst_negative_code_constant_oneunique_resultfirst) + (dst_negative_scale_constant_oneunique_resultfirst)) + ((dst_negative_scale_constant_oneunique_resultfirst) + (dst_negative_scale_constant_oneunique_resultfirst)))) + ((((dst_negative_code_constant_oneunique_resultfirst) + (dst_negative_scale_constant_oneunique_resultfirst)) * S ((dst_negative_code_constant_oneunique_resultfirst) + (dst_negative_scale_constant_oneunique_resultfirst)) + ((dst_negative_scale_constant_oneunique_resultfirst) + (dst_negative_scale_constant_oneunique_resultfirst))) + (((dst_negative_code_constant_oneunique_resultfirst) + (dst_negative_scale_constant_oneunique_resultfirst)) * S ((dst_negative_code_constant_oneunique_resultfirst) + (dst_negative_scale_constant_oneunique_resultfirst)) + ((dst_negative_scale_constant_oneunique_resultfirst) + (dst_negative_scale_constant_oneunique_resultfirst)))))) /\ (((((exists ff_h_pvs_constant_oneunique_resultfirstpositive. ff_h_pvs_constant_oneunique_resultfirstpositive + S (dst_positive_constant_oneunique_resultfirst) = S ((S (dm_index_constant_oneunique_result)) * dst_positive_scale_constant_oneunique_resultfirst)) /\ exists ff_q_pvs_constant_oneunique_resultfirstpositive. dst_positive_code_constant_oneunique_resultfirst = ff_q_pvs_constant_oneunique_resultfirstpositive * S ((S (dm_index_constant_oneunique_result)) * dst_positive_scale_constant_oneunique_resultfirst) + (dst_positive_constant_oneunique_resultfirst))) /\ (((((exists ff_h_pvs_constant_oneunique_resultfirstnegative. ff_h_pvs_constant_oneunique_resultfirstnegative + S (dst_negative_constant_oneunique_resultfirst) = S ((S (dm_index_constant_oneunique_result)) * dst_negative_scale_constant_oneunique_resultfirst)) /\ exists ff_q_pvs_constant_oneunique_resultfirstnegative. dst_negative_code_constant_oneunique_resultfirst = ff_q_pvs_constant_oneunique_resultfirstnegative * S ((S (dm_index_constant_oneunique_result)) * dst_negative_scale_constant_oneunique_resultfirst) + (dst_negative_constant_oneunique_resultfirst))) /\ (exists ge_balance_positive_constant_oneunique_resultfirstvalue ge_balance_negative_constant_oneunique_resultfirstvalue. (((((dm_first_value_constant_oneunique_result) = 2 * (ge_balance_positive_constant_oneunique_resultfirstvalue) /\ (ge_balance_negative_constant_oneunique_resultfirstvalue) = 0) \/ exists ge_signed_half_constant_oneunique_resultfirstvaluedecode. (((dm_first_value_constant_oneunique_result) = 2 * ge_signed_half_constant_oneunique_resultfirstvaluedecode + 1 /\ (ge_balance_positive_constant_oneunique_resultfirstvalue) = 0) /\ (ge_balance_negative_constant_oneunique_resultfirstvalue) = S ge_signed_half_constant_oneunique_resultfirstvaluedecode))) /\ ((dst_positive_constant_oneunique_resultfirst) + ge_balance_negative_constant_oneunique_resultfirstvalue = (dst_negative_constant_oneunique_resultfirst) + ge_balance_positive_constant_oneunique_resultfirstvalue))))))))) -> (exists dst_positive_code_constant_oneunique_resultsecond dst_positive_scale_constant_oneunique_resultsecond dst_negative_code_constant_oneunique_resultsecond dst_negative_scale_constant_oneunique_resultsecond dst_positive_constant_oneunique_resultsecond dst_negative_constant_oneunique_resultsecond. (((G) = (((((dst_positive_code_constant_oneunique_resultsecond) + (dst_positive_scale_constant_oneunique_resultsecond)) * S ((dst_positive_code_constant_oneunique_resultsecond) + (dst_positive_scale_constant_oneunique_resultsecond)) + ((dst_positive_scale_constant_oneunique_resultsecond) + (dst_positive_scale_constant_oneunique_resultsecond))) + (((dst_negative_code_constant_oneunique_resultsecond) + (dst_negative_scale_constant_oneunique_resultsecond)) * S ((dst_negative_code_constant_oneunique_resultsecond) + (dst_negative_scale_constant_oneunique_resultsecond)) + ((dst_negative_scale_constant_oneunique_resultsecond) + (dst_negative_scale_constant_oneunique_resultsecond)))) * S ((((dst_positive_code_constant_oneunique_resultsecond) + (dst_positive_scale_constant_oneunique_resultsecond)) * S ((dst_positive_code_constant_oneunique_resultsecond) + (dst_positive_scale_constant_oneunique_resultsecond)) + ((dst_positive_scale_constant_oneunique_resultsecond) + (dst_positive_scale_constant_oneunique_resultsecond))) + (((dst_negative_code_constant_oneunique_resultsecond) + (dst_negative_scale_constant_oneunique_resultsecond)) * S ((dst_negative_code_constant_oneunique_resultsecond) + (dst_negative_scale_constant_oneunique_resultsecond)) + ((dst_negative_scale_constant_oneunique_resultsecond) + (dst_negative_scale_constant_oneunique_resultsecond)))) + ((((dst_negative_code_constant_oneunique_resultsecond) + (dst_negative_scale_constant_oneunique_resultsecond)) * S ((dst_negative_code_constant_oneunique_resultsecond) + (dst_negative_scale_constant_oneunique_resultsecond)) + ((dst_negative_scale_constant_oneunique_resultsecond) + (dst_negative_scale_constant_oneunique_resultsecond))) + (((dst_negative_code_constant_oneunique_resultsecond) + (dst_negative_scale_constant_oneunique_resultsecond)) * S ((dst_negative_code_constant_oneunique_resultsecond) + (dst_negative_scale_constant_oneunique_resultsecond)) + ((dst_negative_scale_constant_oneunique_resultsecond) + (dst_negative_scale_constant_oneunique_resultsecond)))))) /\ (((((exists ff_h_pvs_constant_oneunique_resultsecondpositive. ff_h_pvs_constant_oneunique_resultsecondpositive + S (dst_positive_constant_oneunique_resultsecond) = S ((S (dm_index_constant_oneunique_result)) * dst_positive_scale_constant_oneunique_resultsecond)) /\ exists ff_q_pvs_constant_oneunique_resultsecondpositive. dst_positive_code_constant_oneunique_resultsecond = ff_q_pvs_constant_oneunique_resultsecondpositive * S ((S (dm_index_constant_oneunique_result)) * dst_positive_scale_constant_oneunique_resultsecond) + (dst_positive_constant_oneunique_resultsecond))) /\ (((((exists ff_h_pvs_constant_oneunique_resultsecondnegative. ff_h_pvs_constant_oneunique_resultsecondnegative + S (dst_negative_constant_oneunique_resultsecond) = S ((S (dm_index_constant_oneunique_result)) * dst_negative_scale_constant_oneunique_resultsecond)) /\ exists ff_q_pvs_constant_oneunique_resultsecondnegative. dst_negative_code_constant_oneunique_resultsecond = ff_q_pvs_constant_oneunique_resultsecondnegative * S ((S (dm_index_constant_oneunique_result)) * dst_negative_scale_constant_oneunique_resultsecond) + (dst_negative_constant_oneunique_resultsecond))) /\ (exists ge_balance_positive_constant_oneunique_resultsecondvalue ge_balance_negative_constant_oneunique_resultsecondvalue. (((((dm_second_value_constant_oneunique_result) = 2 * (ge_balance_positive_constant_oneunique_resultsecondvalue) /\ (ge_balance_negative_constant_oneunique_resultsecondvalue) = 0) \/ exists ge_signed_half_constant_oneunique_resultsecondvaluedecode. (((dm_second_value_constant_oneunique_result) = 2 * ge_signed_half_constant_oneunique_resultsecondvaluedecode + 1 /\ (ge_balance_positive_constant_oneunique_resultsecondvalue) = 0) /\ (ge_balance_negative_constant_oneunique_resultsecondvalue) = S ge_signed_half_constant_oneunique_resultsecondvaluedecode))) /\ ((dst_positive_constant_oneunique_resultsecond) + ge_balance_negative_constant_oneunique_resultsecondvalue = (dst_negative_constant_oneunique_resultsecond) + ge_balance_positive_constant_oneunique_resultsecondvalue))))))))) -> dm_first_value_constant_oneunique_result=dm_second_value_constant_oneunique_result)Complete tactic proof in conservative notation
All 24 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
24 script commands · 6 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.
Named ingredients (1)
01Fix variables and assumptionsL1–10
02Fix variables and assumptionsL11–12
03Calculate and transport equalitiesL13–13
Carry out the recorded arithmetic or equality steps; inspect the exact commands for their direction and premises.
- L13
trans 2
04Use earlier factsL14–18
05Calculate and transport equalitiesL19–19
Carry out the recorded arithmetic or equality steps; inspect the exact commands for their direction and premises.
- L19
symm
Original defined command ledger · 24 lines
- 0001
intro N - 0002
intro F - 0003
intro G - 0004
intro hf - 0005
intro hg - 0006
intro i - 0007
intro a - 0008
intro b - 0009
intro hi - 0010
intro hb - 0011
intro ha - 0012
intro hbv - 0013
trans 2 - 0014
apply dirichlet_constant_one_table_value - 0015
exact hf - 0016
exact hi - 0017
exact hb - 0018
exact ha - 0019
symm - 0020
apply dirichlet_constant_one_table_value - 0021
exact hg - 0022
exact hi - 0023
exact hb - 0024
exact hbv