DV001B

divisor_mask_prefix_restrict

The same actual mask code restricts to any smaller inclusive prefix for its fixed divisibility target.

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.

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

∀ F. ∀ n. ∀ l. ∀ k. ∀ M. DivisorMask(F,n,l,M)Le(k,l)DivisorMask(F,n,k,M)

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

Definition DAG

Actual proof prerequisites

Original expanded first-order statement
forall F n l k M. (((exists dst_positive_code_restrict_masktable dst_positive_scale_restrict_masktable dst_negative_code_restrict_masktable dst_negative_scale_restrict_masktable. (((M) = (((((dst_positive_code_restrict_masktable) + (dst_positive_scale_restrict_masktable)) * S ((dst_positive_code_restrict_masktable) + (dst_positive_scale_restrict_masktable)) + ((dst_positive_scale_restrict_masktable) + (dst_positive_scale_restrict_masktable))) + (((dst_negative_code_restrict_masktable) + (dst_negative_scale_restrict_masktable)) * S ((dst_negative_code_restrict_masktable) + (dst_negative_scale_restrict_masktable)) + ((dst_negative_scale_restrict_masktable) + (dst_negative_scale_restrict_masktable)))) * S ((((dst_positive_code_restrict_masktable) + (dst_positive_scale_restrict_masktable)) * S ((dst_positive_code_restrict_masktable) + (dst_positive_scale_restrict_masktable)) + ((dst_positive_scale_restrict_masktable) + (dst_positive_scale_restrict_masktable))) + (((dst_negative_code_restrict_masktable) + (dst_negative_scale_restrict_masktable)) * S ((dst_negative_code_restrict_masktable) + (dst_negative_scale_restrict_masktable)) + ((dst_negative_scale_restrict_masktable) + (dst_negative_scale_restrict_masktable)))) + ((((dst_negative_code_restrict_masktable) + (dst_negative_scale_restrict_masktable)) * S ((dst_negative_code_restrict_masktable) + (dst_negative_scale_restrict_masktable)) + ((dst_negative_scale_restrict_masktable) + (dst_negative_scale_restrict_masktable))) + (((dst_negative_code_restrict_masktable) + (dst_negative_scale_restrict_masktable)) * S ((dst_negative_code_restrict_masktable) + (dst_negative_scale_restrict_masktable)) + ((dst_negative_scale_restrict_masktable) + (dst_negative_scale_restrict_masktable)))))) /\ (forall dst_index_restrict_masktable. (exists pvs_le_gap_restrict_masktabledomain. pvs_le_gap_restrict_masktabledomain + (dst_index_restrict_masktable) = (l)) -> exists dst_positive_restrict_masktable dst_negative_restrict_masktable dst_value_restrict_masktable. ((((exists ff_h_pvs_restrict_masktableentrypositive. ff_h_pvs_restrict_masktableentrypositive + S (dst_positive_restrict_masktable) = S ((S (dst_index_restrict_masktable)) * dst_positive_scale_restrict_masktable)) /\ exists ff_q_pvs_restrict_masktableentrypositive. dst_positive_code_restrict_masktable = ff_q_pvs_restrict_masktableentrypositive * S ((S (dst_index_restrict_masktable)) * dst_positive_scale_restrict_masktable) + (dst_positive_restrict_masktable))) /\ (((((exists ff_h_pvs_restrict_masktableentrynegative. ff_h_pvs_restrict_masktableentrynegative + S (dst_negative_restrict_masktable) = S ((S (dst_index_restrict_masktable)) * dst_negative_scale_restrict_masktable)) /\ exists ff_q_pvs_restrict_masktableentrynegative. dst_negative_code_restrict_masktable = ff_q_pvs_restrict_masktableentrynegative * S ((S (dst_index_restrict_masktable)) * dst_negative_scale_restrict_masktable) + (dst_negative_restrict_masktable))) /\ (exists ge_balance_positive_restrict_masktableentryvalue ge_balance_negative_restrict_masktableentryvalue. (((((dst_value_restrict_masktable) = 2 * (ge_balance_positive_restrict_masktableentryvalue) /\ (ge_balance_negative_restrict_masktableentryvalue) = 0) \/ exists ge_signed_half_restrict_masktableentryvaluedecode. (((dst_value_restrict_masktable) = 2 * ge_signed_half_restrict_masktableentryvaluedecode + 1 /\ (ge_balance_positive_restrict_masktableentryvalue) = 0) /\ (ge_balance_negative_restrict_masktableentryvalue) = S ge_signed_half_restrict_masktableentryvaluedecode))) /\ ((dst_positive_restrict_masktable) + ge_balance_negative_restrict_masktableentryvalue = (dst_negative_restrict_masktable) + ge_balance_positive_restrict_masktableentryvalue))))))))) /\ (forall dm_index_restrict_mask dm_value_restrict_mask. (exists pvs_le_gap_restrict_maskdomain. pvs_le_gap_restrict_maskdomain + (dm_index_restrict_mask) = (l)) -> (exists dst_positive_code_restrict_masklookup dst_positive_scale_restrict_masklookup dst_negative_code_restrict_masklookup dst_negative_scale_restrict_masklookup dst_positive_restrict_masklookup dst_negative_restrict_masklookup. (((M) = (((((dst_positive_code_restrict_masklookup) + (dst_positive_scale_restrict_masklookup)) * S ((dst_positive_code_restrict_masklookup) + (dst_positive_scale_restrict_masklookup)) + ((dst_positive_scale_restrict_masklookup) + (dst_positive_scale_restrict_masklookup))) + (((dst_negative_code_restrict_masklookup) + (dst_negative_scale_restrict_masklookup)) * S ((dst_negative_code_restrict_masklookup) + (dst_negative_scale_restrict_masklookup)) + ((dst_negative_scale_restrict_masklookup) + (dst_negative_scale_restrict_masklookup)))) * S ((((dst_positive_code_restrict_masklookup) + (dst_positive_scale_restrict_masklookup)) * S ((dst_positive_code_restrict_masklookup) + (dst_positive_scale_restrict_masklookup)) + ((dst_positive_scale_restrict_masklookup) + (dst_positive_scale_restrict_masklookup))) + (((dst_negative_code_restrict_masklookup) + (dst_negative_scale_restrict_masklookup)) * S ((dst_negative_code_restrict_masklookup) + (dst_negative_scale_restrict_masklookup)) + ((dst_negative_scale_restrict_masklookup) + (dst_negative_scale_restrict_masklookup)))) + ((((dst_negative_code_restrict_masklookup) + (dst_negative_scale_restrict_masklookup)) * S ((dst_negative_code_restrict_masklookup) + (dst_negative_scale_restrict_masklookup)) + ((dst_negative_scale_restrict_masklookup) + (dst_negative_scale_restrict_masklookup))) + (((dst_negative_code_restrict_masklookup) + (dst_negative_scale_restrict_masklookup)) * S ((dst_negative_code_restrict_masklookup) + (dst_negative_scale_restrict_masklookup)) + ((dst_negative_scale_restrict_masklookup) + (dst_negative_scale_restrict_masklookup)))))) /\ (((((exists ff_h_pvs_restrict_masklookuppositive. ff_h_pvs_restrict_masklookuppositive + S (dst_positive_restrict_masklookup) = S ((S (dm_index_restrict_mask)) * dst_positive_scale_restrict_masklookup)) /\ exists ff_q_pvs_restrict_masklookuppositive. dst_positive_code_restrict_masklookup = ff_q_pvs_restrict_masklookuppositive * S ((S (dm_index_restrict_mask)) * dst_positive_scale_restrict_masklookup) + (dst_positive_restrict_masklookup))) /\ (((((exists ff_h_pvs_restrict_masklookupnegative. ff_h_pvs_restrict_masklookupnegative + S (dst_negative_restrict_masklookup) = S ((S (dm_index_restrict_mask)) * dst_negative_scale_restrict_masklookup)) /\ exists ff_q_pvs_restrict_masklookupnegative. dst_negative_code_restrict_masklookup = ff_q_pvs_restrict_masklookupnegative * S ((S (dm_index_restrict_mask)) * dst_negative_scale_restrict_masklookup) + (dst_negative_restrict_masklookup))) /\ (exists ge_balance_positive_restrict_masklookupvalue ge_balance_negative_restrict_masklookupvalue. (((((dm_value_restrict_mask) = 2 * (ge_balance_positive_restrict_masklookupvalue) /\ (ge_balance_negative_restrict_masklookupvalue) = 0) \/ exists ge_signed_half_restrict_masklookupvaluedecode. (((dm_value_restrict_mask) = 2 * ge_signed_half_restrict_masklookupvaluedecode + 1 /\ (ge_balance_positive_restrict_masklookupvalue) = 0) /\ (ge_balance_negative_restrict_masklookupvalue) = S ge_signed_half_restrict_masklookupvaluedecode))) /\ ((dst_positive_restrict_masklookup) + ge_balance_negative_restrict_masklookupvalue = (dst_negative_restrict_masklookup) + ge_balance_positive_restrict_masklookupvalue))))))))) -> ((((~((dm_index_restrict_mask)=0)) /\ (exists dm_quotient_restrict_maskentry. (((n)=(dm_index_restrict_mask)*dm_quotient_restrict_maskentry) /\ (exists dst_positive_code_restrict_maskentryinput dst_positive_scale_restrict_maskentryinput dst_negative_code_restrict_maskentryinput dst_negative_scale_restrict_maskentryinput dst_positive_restrict_maskentryinput dst_negative_restrict_maskentryinput. (((F) = (((((dst_positive_code_restrict_maskentryinput) + (dst_positive_scale_restrict_maskentryinput)) * S ((dst_positive_code_restrict_maskentryinput) + (dst_positive_scale_restrict_maskentryinput)) + ((dst_positive_scale_restrict_maskentryinput) + (dst_positive_scale_restrict_maskentryinput))) + (((dst_negative_code_restrict_maskentryinput) + (dst_negative_scale_restrict_maskentryinput)) * S ((dst_negative_code_restrict_maskentryinput) + (dst_negative_scale_restrict_maskentryinput)) + ((dst_negative_scale_restrict_maskentryinput) + (dst_negative_scale_restrict_maskentryinput)))) * S ((((dst_positive_code_restrict_maskentryinput) + (dst_positive_scale_restrict_maskentryinput)) * S ((dst_positive_code_restrict_maskentryinput) + (dst_positive_scale_restrict_maskentryinput)) + ((dst_positive_scale_restrict_maskentryinput) + (dst_positive_scale_restrict_maskentryinput))) + (((dst_negative_code_restrict_maskentryinput) + (dst_negative_scale_restrict_maskentryinput)) * S ((dst_negative_code_restrict_maskentryinput) + (dst_negative_scale_restrict_maskentryinput)) + ((dst_negative_scale_restrict_maskentryinput) + (dst_negative_scale_restrict_maskentryinput)))) + ((((dst_negative_code_restrict_maskentryinput) + (dst_negative_scale_restrict_maskentryinput)) * S ((dst_negative_code_restrict_maskentryinput) + (dst_negative_scale_restrict_maskentryinput)) + ((dst_negative_scale_restrict_maskentryinput) + (dst_negative_scale_restrict_maskentryinput))) + (((dst_negative_code_restrict_maskentryinput) + (dst_negative_scale_restrict_maskentryinput)) * S ((dst_negative_code_restrict_maskentryinput) + (dst_negative_scale_restrict_maskentryinput)) + ((dst_negative_scale_restrict_maskentryinput) + (dst_negative_scale_restrict_maskentryinput)))))) /\ (((((exists ff_h_pvs_restrict_maskentryinputpositive. ff_h_pvs_restrict_maskentryinputpositive + S (dst_positive_restrict_maskentryinput) = S ((S (dm_index_restrict_mask)) * dst_positive_scale_restrict_maskentryinput)) /\ exists ff_q_pvs_restrict_maskentryinputpositive. dst_positive_code_restrict_maskentryinput = ff_q_pvs_restrict_maskentryinputpositive * S ((S (dm_index_restrict_mask)) * dst_positive_scale_restrict_maskentryinput) + (dst_positive_restrict_maskentryinput))) /\ (((((exists ff_h_pvs_restrict_maskentryinputnegative. ff_h_pvs_restrict_maskentryinputnegative + S (dst_negative_restrict_maskentryinput) = S ((S (dm_index_restrict_mask)) * dst_negative_scale_restrict_maskentryinput)) /\ exists ff_q_pvs_restrict_maskentryinputnegative. dst_negative_code_restrict_maskentryinput = ff_q_pvs_restrict_maskentryinputnegative * S ((S (dm_index_restrict_mask)) * dst_negative_scale_restrict_maskentryinput) + (dst_negative_restrict_maskentryinput))) /\ (exists ge_balance_positive_restrict_maskentryinputvalue ge_balance_negative_restrict_maskentryinputvalue. (((((dm_value_restrict_mask) = 2 * (ge_balance_positive_restrict_maskentryinputvalue) /\ (ge_balance_negative_restrict_maskentryinputvalue) = 0) \/ exists ge_signed_half_restrict_maskentryinputvaluedecode. (((dm_value_restrict_mask) = 2 * ge_signed_half_restrict_maskentryinputvaluedecode + 1 /\ (ge_balance_positive_restrict_maskentryinputvalue) = 0) /\ (ge_balance_negative_restrict_maskentryinputvalue) = S ge_signed_half_restrict_maskentryinputvaluedecode))) /\ ((dst_positive_restrict_maskentryinput) + ge_balance_negative_restrict_maskentryinputvalue = (dst_negative_restrict_maskentryinput) + ge_balance_positive_restrict_maskentryinputvalue))))))))))))) \/ ((((dm_index_restrict_mask)=0 \/ ~(exists pvs_factor_restrict_maskentrynondivisor. (n) = (dm_index_restrict_mask) * pvs_factor_restrict_maskentrynondivisor)) /\ ((dm_value_restrict_mask)=0))))))) -> (exists pvs_le_gap_restrict_bound. pvs_le_gap_restrict_bound + (k) = (l)) -> (((exists dst_positive_code_restrict_resulttable dst_positive_scale_restrict_resulttable dst_negative_code_restrict_resulttable dst_negative_scale_restrict_resulttable. (((M) = (((((dst_positive_code_restrict_resulttable) + (dst_positive_scale_restrict_resulttable)) * S ((dst_positive_code_restrict_resulttable) + (dst_positive_scale_restrict_resulttable)) + ((dst_positive_scale_restrict_resulttable) + (dst_positive_scale_restrict_resulttable))) + (((dst_negative_code_restrict_resulttable) + (dst_negative_scale_restrict_resulttable)) * S ((dst_negative_code_restrict_resulttable) + (dst_negative_scale_restrict_resulttable)) + ((dst_negative_scale_restrict_resulttable) + (dst_negative_scale_restrict_resulttable)))) * S ((((dst_positive_code_restrict_resulttable) + (dst_positive_scale_restrict_resulttable)) * S ((dst_positive_code_restrict_resulttable) + (dst_positive_scale_restrict_resulttable)) + ((dst_positive_scale_restrict_resulttable) + (dst_positive_scale_restrict_resulttable))) + (((dst_negative_code_restrict_resulttable) + (dst_negative_scale_restrict_resulttable)) * S ((dst_negative_code_restrict_resulttable) + (dst_negative_scale_restrict_resulttable)) + ((dst_negative_scale_restrict_resulttable) + (dst_negative_scale_restrict_resulttable)))) + ((((dst_negative_code_restrict_resulttable) + (dst_negative_scale_restrict_resulttable)) * S ((dst_negative_code_restrict_resulttable) + (dst_negative_scale_restrict_resulttable)) + ((dst_negative_scale_restrict_resulttable) + (dst_negative_scale_restrict_resulttable))) + (((dst_negative_code_restrict_resulttable) + (dst_negative_scale_restrict_resulttable)) * S ((dst_negative_code_restrict_resulttable) + (dst_negative_scale_restrict_resulttable)) + ((dst_negative_scale_restrict_resulttable) + (dst_negative_scale_restrict_resulttable)))))) /\ (forall dst_index_restrict_resulttable. (exists pvs_le_gap_restrict_resulttabledomain. pvs_le_gap_restrict_resulttabledomain + (dst_index_restrict_resulttable) = (k)) -> exists dst_positive_restrict_resulttable dst_negative_restrict_resulttable dst_value_restrict_resulttable. ((((exists ff_h_pvs_restrict_resulttableentrypositive. ff_h_pvs_restrict_resulttableentrypositive + S (dst_positive_restrict_resulttable) = S ((S (dst_index_restrict_resulttable)) * dst_positive_scale_restrict_resulttable)) /\ exists ff_q_pvs_restrict_resulttableentrypositive. dst_positive_code_restrict_resulttable = ff_q_pvs_restrict_resulttableentrypositive * S ((S (dst_index_restrict_resulttable)) * dst_positive_scale_restrict_resulttable) + (dst_positive_restrict_resulttable))) /\ (((((exists ff_h_pvs_restrict_resulttableentrynegative. ff_h_pvs_restrict_resulttableentrynegative + S (dst_negative_restrict_resulttable) = S ((S (dst_index_restrict_resulttable)) * dst_negative_scale_restrict_resulttable)) /\ exists ff_q_pvs_restrict_resulttableentrynegative. dst_negative_code_restrict_resulttable = ff_q_pvs_restrict_resulttableentrynegative * S ((S (dst_index_restrict_resulttable)) * dst_negative_scale_restrict_resulttable) + (dst_negative_restrict_resulttable))) /\ (exists ge_balance_positive_restrict_resulttableentryvalue ge_balance_negative_restrict_resulttableentryvalue. (((((dst_value_restrict_resulttable) = 2 * (ge_balance_positive_restrict_resulttableentryvalue) /\ (ge_balance_negative_restrict_resulttableentryvalue) = 0) \/ exists ge_signed_half_restrict_resulttableentryvaluedecode. (((dst_value_restrict_resulttable) = 2 * ge_signed_half_restrict_resulttableentryvaluedecode + 1 /\ (ge_balance_positive_restrict_resulttableentryvalue) = 0) /\ (ge_balance_negative_restrict_resulttableentryvalue) = S ge_signed_half_restrict_resulttableentryvaluedecode))) /\ ((dst_positive_restrict_resulttable) + ge_balance_negative_restrict_resulttableentryvalue = (dst_negative_restrict_resulttable) + ge_balance_positive_restrict_resulttableentryvalue))))))))) /\ (forall dm_index_restrict_result dm_value_restrict_result. (exists pvs_le_gap_restrict_resultdomain. pvs_le_gap_restrict_resultdomain + (dm_index_restrict_result) = (k)) -> (exists dst_positive_code_restrict_resultlookup dst_positive_scale_restrict_resultlookup dst_negative_code_restrict_resultlookup dst_negative_scale_restrict_resultlookup dst_positive_restrict_resultlookup dst_negative_restrict_resultlookup. (((M) = (((((dst_positive_code_restrict_resultlookup) + (dst_positive_scale_restrict_resultlookup)) * S ((dst_positive_code_restrict_resultlookup) + (dst_positive_scale_restrict_resultlookup)) + ((dst_positive_scale_restrict_resultlookup) + (dst_positive_scale_restrict_resultlookup))) + (((dst_negative_code_restrict_resultlookup) + (dst_negative_scale_restrict_resultlookup)) * S ((dst_negative_code_restrict_resultlookup) + (dst_negative_scale_restrict_resultlookup)) + ((dst_negative_scale_restrict_resultlookup) + (dst_negative_scale_restrict_resultlookup)))) * S ((((dst_positive_code_restrict_resultlookup) + (dst_positive_scale_restrict_resultlookup)) * S ((dst_positive_code_restrict_resultlookup) + (dst_positive_scale_restrict_resultlookup)) + ((dst_positive_scale_restrict_resultlookup) + (dst_positive_scale_restrict_resultlookup))) + (((dst_negative_code_restrict_resultlookup) + (dst_negative_scale_restrict_resultlookup)) * S ((dst_negative_code_restrict_resultlookup) + (dst_negative_scale_restrict_resultlookup)) + ((dst_negative_scale_restrict_resultlookup) + (dst_negative_scale_restrict_resultlookup)))) + ((((dst_negative_code_restrict_resultlookup) + (dst_negative_scale_restrict_resultlookup)) * S ((dst_negative_code_restrict_resultlookup) + (dst_negative_scale_restrict_resultlookup)) + ((dst_negative_scale_restrict_resultlookup) + (dst_negative_scale_restrict_resultlookup))) + (((dst_negative_code_restrict_resultlookup) + (dst_negative_scale_restrict_resultlookup)) * S ((dst_negative_code_restrict_resultlookup) + (dst_negative_scale_restrict_resultlookup)) + ((dst_negative_scale_restrict_resultlookup) + (dst_negative_scale_restrict_resultlookup)))))) /\ (((((exists ff_h_pvs_restrict_resultlookuppositive. ff_h_pvs_restrict_resultlookuppositive + S (dst_positive_restrict_resultlookup) = S ((S (dm_index_restrict_result)) * dst_positive_scale_restrict_resultlookup)) /\ exists ff_q_pvs_restrict_resultlookuppositive. dst_positive_code_restrict_resultlookup = ff_q_pvs_restrict_resultlookuppositive * S ((S (dm_index_restrict_result)) * dst_positive_scale_restrict_resultlookup) + (dst_positive_restrict_resultlookup))) /\ (((((exists ff_h_pvs_restrict_resultlookupnegative. ff_h_pvs_restrict_resultlookupnegative + S (dst_negative_restrict_resultlookup) = S ((S (dm_index_restrict_result)) * dst_negative_scale_restrict_resultlookup)) /\ exists ff_q_pvs_restrict_resultlookupnegative. dst_negative_code_restrict_resultlookup = ff_q_pvs_restrict_resultlookupnegative * S ((S (dm_index_restrict_result)) * dst_negative_scale_restrict_resultlookup) + (dst_negative_restrict_resultlookup))) /\ (exists ge_balance_positive_restrict_resultlookupvalue ge_balance_negative_restrict_resultlookupvalue. (((((dm_value_restrict_result) = 2 * (ge_balance_positive_restrict_resultlookupvalue) /\ (ge_balance_negative_restrict_resultlookupvalue) = 0) \/ exists ge_signed_half_restrict_resultlookupvaluedecode. (((dm_value_restrict_result) = 2 * ge_signed_half_restrict_resultlookupvaluedecode + 1 /\ (ge_balance_positive_restrict_resultlookupvalue) = 0) /\ (ge_balance_negative_restrict_resultlookupvalue) = S ge_signed_half_restrict_resultlookupvaluedecode))) /\ ((dst_positive_restrict_resultlookup) + ge_balance_negative_restrict_resultlookupvalue = (dst_negative_restrict_resultlookup) + ge_balance_positive_restrict_resultlookupvalue))))))))) -> ((((~((dm_index_restrict_result)=0)) /\ (exists dm_quotient_restrict_resultentry. (((n)=(dm_index_restrict_result)*dm_quotient_restrict_resultentry) /\ (exists dst_positive_code_restrict_resultentryinput dst_positive_scale_restrict_resultentryinput dst_negative_code_restrict_resultentryinput dst_negative_scale_restrict_resultentryinput dst_positive_restrict_resultentryinput dst_negative_restrict_resultentryinput. (((F) = (((((dst_positive_code_restrict_resultentryinput) + (dst_positive_scale_restrict_resultentryinput)) * S ((dst_positive_code_restrict_resultentryinput) + (dst_positive_scale_restrict_resultentryinput)) + ((dst_positive_scale_restrict_resultentryinput) + (dst_positive_scale_restrict_resultentryinput))) + (((dst_negative_code_restrict_resultentryinput) + (dst_negative_scale_restrict_resultentryinput)) * S ((dst_negative_code_restrict_resultentryinput) + (dst_negative_scale_restrict_resultentryinput)) + ((dst_negative_scale_restrict_resultentryinput) + (dst_negative_scale_restrict_resultentryinput)))) * S ((((dst_positive_code_restrict_resultentryinput) + (dst_positive_scale_restrict_resultentryinput)) * S ((dst_positive_code_restrict_resultentryinput) + (dst_positive_scale_restrict_resultentryinput)) + ((dst_positive_scale_restrict_resultentryinput) + (dst_positive_scale_restrict_resultentryinput))) + (((dst_negative_code_restrict_resultentryinput) + (dst_negative_scale_restrict_resultentryinput)) * S ((dst_negative_code_restrict_resultentryinput) + (dst_negative_scale_restrict_resultentryinput)) + ((dst_negative_scale_restrict_resultentryinput) + (dst_negative_scale_restrict_resultentryinput)))) + ((((dst_negative_code_restrict_resultentryinput) + (dst_negative_scale_restrict_resultentryinput)) * S ((dst_negative_code_restrict_resultentryinput) + (dst_negative_scale_restrict_resultentryinput)) + ((dst_negative_scale_restrict_resultentryinput) + (dst_negative_scale_restrict_resultentryinput))) + (((dst_negative_code_restrict_resultentryinput) + (dst_negative_scale_restrict_resultentryinput)) * S ((dst_negative_code_restrict_resultentryinput) + (dst_negative_scale_restrict_resultentryinput)) + ((dst_negative_scale_restrict_resultentryinput) + (dst_negative_scale_restrict_resultentryinput)))))) /\ (((((exists ff_h_pvs_restrict_resultentryinputpositive. ff_h_pvs_restrict_resultentryinputpositive + S (dst_positive_restrict_resultentryinput) = S ((S (dm_index_restrict_result)) * dst_positive_scale_restrict_resultentryinput)) /\ exists ff_q_pvs_restrict_resultentryinputpositive. dst_positive_code_restrict_resultentryinput = ff_q_pvs_restrict_resultentryinputpositive * S ((S (dm_index_restrict_result)) * dst_positive_scale_restrict_resultentryinput) + (dst_positive_restrict_resultentryinput))) /\ (((((exists ff_h_pvs_restrict_resultentryinputnegative. ff_h_pvs_restrict_resultentryinputnegative + S (dst_negative_restrict_resultentryinput) = S ((S (dm_index_restrict_result)) * dst_negative_scale_restrict_resultentryinput)) /\ exists ff_q_pvs_restrict_resultentryinputnegative. dst_negative_code_restrict_resultentryinput = ff_q_pvs_restrict_resultentryinputnegative * S ((S (dm_index_restrict_result)) * dst_negative_scale_restrict_resultentryinput) + (dst_negative_restrict_resultentryinput))) /\ (exists ge_balance_positive_restrict_resultentryinputvalue ge_balance_negative_restrict_resultentryinputvalue. (((((dm_value_restrict_result) = 2 * (ge_balance_positive_restrict_resultentryinputvalue) /\ (ge_balance_negative_restrict_resultentryinputvalue) = 0) \/ exists ge_signed_half_restrict_resultentryinputvaluedecode. (((dm_value_restrict_result) = 2 * ge_signed_half_restrict_resultentryinputvaluedecode + 1 /\ (ge_balance_positive_restrict_resultentryinputvalue) = 0) /\ (ge_balance_negative_restrict_resultentryinputvalue) = S ge_signed_half_restrict_resultentryinputvaluedecode))) /\ ((dst_positive_restrict_resultentryinput) + ge_balance_negative_restrict_resultentryinputvalue = (dst_negative_restrict_resultentryinput) + ge_balance_positive_restrict_resultentryinputvalue))))))))))))) \/ ((((dm_index_restrict_result)=0 \/ ~(exists pvs_factor_restrict_resultentrynondivisor. (n) = (dm_index_restrict_result) * pvs_factor_restrict_resultentrynondivisor)) /\ ((dm_value_restrict_result)=0)))))))

Complete tactic proof in conservative notation

All 29 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

29 script commands · 5 reading checkpoints · 0 local claims

This is a reading aid, not a new proof or a proof-tree certificate. Checkpoint groups are consecutive commands, not inferred branch boundaries. Every step links to the preserved script.

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

01Fix variables and assumptionsL1–7

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

  1. L1
    intro F
  2. L2
    intro n
  3. L3
    intro l
  4. L4
    intro k
  5. L5
    intro M
  6. L6
    intro hm
  7. L7
    intro hkl
02Separate the logical casesL8–9

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

  1. L8
    cases hm
  2. L9
    split
03Use earlier factsL10–15

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

  1. L10
    specialize divisor_signed_table_restrict (l)
  2. L11
    specialize divisor_signed_table_restrict (k)
  3. L12
    specialize divisor_signed_table_restrict (M)
  4. L13
    apply divisor_signed_table_restrict
  5. L14
    exact hm_left
  6. L15
    exact hkl
04Fix variables and assumptionsL16–19

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

  1. L16
    intro d
  2. L17
    intro z
  3. L18
    intro hd
  4. L19
    intro hz
05Use earlier factsL20–29

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

  1. L20
    specialize hm_right (d)
  2. L21
    specialize hm_right (z)
  3. L22
    apply hm_right
  4. L23
    specialize le_trans (d)
  5. L24
    specialize le_trans (k)
  6. L25
    specialize le_trans (l)
  7. L26
    apply le_trans
  8. L27
    exact hd
  9. L28
    exact hkl
  10. L29
    exact hz

Library-wide reading audit

Original defined command ledger · 29 lines
  1. 0001intro F
  2. 0002intro n
  3. 0003intro l
  4. 0004intro k
  5. 0005intro M
  6. 0006intro hm
  7. 0007intro hkl
  8. 0008cases hm
  9. 0009split
  10. 0010specialize divisor_signed_table_restrict (l)
  11. 0011specialize divisor_signed_table_restrict (k)
  12. 0012specialize divisor_signed_table_restrict (M)
  13. 0013apply divisor_signed_table_restrict
  14. 0014exact hm_left
  15. 0015exact hkl
  16. 0016intro d
  17. 0017intro z
  18. 0018intro hd
  19. 0019intro hz
  20. 0020specialize hm_right (d)
  21. 0021specialize hm_right (z)
  22. 0022apply hm_right
  23. 0023specialize le_trans (d)
  24. 0024specialize le_trans (k)
  25. 0025specialize le_trans (l)
  26. 0026apply le_trans
  27. 0027exact hd
  28. 0028exact hkl
  29. 0029exact hz