DU000A

dirichlet_constant_one_table_positive_unique

All constructed representations agree at every positive in-domain entry; neither equality at zero nor equality of arbitrary table encodings is asserted.

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

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

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

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

  1. L1
    intro N
  2. L2
    intro F
  3. L3
    intro G
  4. L4
    intro hf
  5. L5
    intro hg
  6. L6
    intro i
  7. L7
    intro a
  8. L8
    intro b
  9. L9
    intro hi
  10. L10
    intro hb
02Fix variables and assumptionsL11–12

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

  1. L11
    intro ha
  2. L12
    intro hbv
03Calculate and transport equalitiesL13–13

Carry out the recorded arithmetic or equality steps; inspect the exact commands for their direction and premises.

  1. L13
    trans 2
04Use earlier factsL14–18

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

  1. L14
    apply dirichlet_constant_one_table_value
  2. L15
    exact hf
  3. L16
    exact hi
  4. L17
    exact hb
  5. L18
    exact ha
05Calculate and transport equalitiesL19–19

Carry out the recorded arithmetic or equality steps; inspect the exact commands for their direction and premises.

  1. L19
    symm
06Use earlier factsL20–24

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

  1. L20
    apply dirichlet_constant_one_table_value
  2. L21
    exact hg
  3. L22
    exact hi
  4. L23
    exact hb
  5. L24
    exact hbv

Library-wide reading audit

Original defined command ledger · 24 lines
  1. 0001intro N
  2. 0002intro F
  3. 0003intro G
  4. 0004intro hf
  5. 0005intro hg
  6. 0006intro i
  7. 0007intro a
  8. 0008intro b
  9. 0009intro hi
  10. 0010intro hb
  11. 0011intro ha
  12. 0012intro hbv
  13. 0013trans 2
  14. 0014apply dirichlet_constant_one_table_value
  15. 0015exact hf
  16. 0016exact hi
  17. 0017exact hb
  18. 0018exact ha
  19. 0019symm
  20. 0020apply dirichlet_constant_one_table_value
  21. 0021exact hg
  22. 0022exact hi
  23. 0023exact hb
  24. 0024exact hbv