DC0018

dirichlet_convolution_table_zero_constructor

At bound zero any actual output table is a valid empty positive-window convolution table; no zero-entry value is prescribed.

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.

Each retained summand has a witnessed n=d*q and actual signed multiplication. Zero and nondivisors contribute zero. Input and output values at zero are unrestricted; uniqueness is for positive represented values. The separate inverse family proves the unit-at-one criterion. Full G009 multiplicative-function closure is now admitted in the separate Alpha-v32 multiplicative-convolution family.

Exact theorem in conservative defined notation

∀ F. ∀ G. ∀ H. ArithTable(0,F)ArithTable(0,G)ArithTable(0,H)DirichletTable(0,F,G,H)

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

Definition DAG

Actual proof prerequisites

Original expanded first-order statement
forall F G H. (exists dst_positive_code_table_zero_left dst_positive_scale_table_zero_left dst_negative_code_table_zero_left dst_negative_scale_table_zero_left. (((F) = (((((dst_positive_code_table_zero_left) + (dst_positive_scale_table_zero_left)) * S ((dst_positive_code_table_zero_left) + (dst_positive_scale_table_zero_left)) + ((dst_positive_scale_table_zero_left) + (dst_positive_scale_table_zero_left))) + (((dst_negative_code_table_zero_left) + (dst_negative_scale_table_zero_left)) * S ((dst_negative_code_table_zero_left) + (dst_negative_scale_table_zero_left)) + ((dst_negative_scale_table_zero_left) + (dst_negative_scale_table_zero_left)))) * S ((((dst_positive_code_table_zero_left) + (dst_positive_scale_table_zero_left)) * S ((dst_positive_code_table_zero_left) + (dst_positive_scale_table_zero_left)) + ((dst_positive_scale_table_zero_left) + (dst_positive_scale_table_zero_left))) + (((dst_negative_code_table_zero_left) + (dst_negative_scale_table_zero_left)) * S ((dst_negative_code_table_zero_left) + (dst_negative_scale_table_zero_left)) + ((dst_negative_scale_table_zero_left) + (dst_negative_scale_table_zero_left)))) + ((((dst_negative_code_table_zero_left) + (dst_negative_scale_table_zero_left)) * S ((dst_negative_code_table_zero_left) + (dst_negative_scale_table_zero_left)) + ((dst_negative_scale_table_zero_left) + (dst_negative_scale_table_zero_left))) + (((dst_negative_code_table_zero_left) + (dst_negative_scale_table_zero_left)) * S ((dst_negative_code_table_zero_left) + (dst_negative_scale_table_zero_left)) + ((dst_negative_scale_table_zero_left) + (dst_negative_scale_table_zero_left)))))) /\ (forall dst_index_table_zero_left. (exists pvs_le_gap_table_zero_leftdomain. pvs_le_gap_table_zero_leftdomain + (dst_index_table_zero_left) = (0)) -> exists dst_positive_table_zero_left dst_negative_table_zero_left dst_value_table_zero_left. ((((exists ff_h_pvs_table_zero_leftentrypositive. ff_h_pvs_table_zero_leftentrypositive + S (dst_positive_table_zero_left) = S ((S (dst_index_table_zero_left)) * dst_positive_scale_table_zero_left)) /\ exists ff_q_pvs_table_zero_leftentrypositive. dst_positive_code_table_zero_left = ff_q_pvs_table_zero_leftentrypositive * S ((S (dst_index_table_zero_left)) * dst_positive_scale_table_zero_left) + (dst_positive_table_zero_left))) /\ (((((exists ff_h_pvs_table_zero_leftentrynegative. ff_h_pvs_table_zero_leftentrynegative + S (dst_negative_table_zero_left) = S ((S (dst_index_table_zero_left)) * dst_negative_scale_table_zero_left)) /\ exists ff_q_pvs_table_zero_leftentrynegative. dst_negative_code_table_zero_left = ff_q_pvs_table_zero_leftentrynegative * S ((S (dst_index_table_zero_left)) * dst_negative_scale_table_zero_left) + (dst_negative_table_zero_left))) /\ (exists ge_balance_positive_table_zero_leftentryvalue ge_balance_negative_table_zero_leftentryvalue. (((((dst_value_table_zero_left) = 2 * (ge_balance_positive_table_zero_leftentryvalue) /\ (ge_balance_negative_table_zero_leftentryvalue) = 0) \/ exists ge_signed_half_table_zero_leftentryvaluedecode. (((dst_value_table_zero_left) = 2 * ge_signed_half_table_zero_leftentryvaluedecode + 1 /\ (ge_balance_positive_table_zero_leftentryvalue) = 0) /\ (ge_balance_negative_table_zero_leftentryvalue) = S ge_signed_half_table_zero_leftentryvaluedecode))) /\ ((dst_positive_table_zero_left) + ge_balance_negative_table_zero_leftentryvalue = (dst_negative_table_zero_left) + ge_balance_positive_table_zero_leftentryvalue))))))))) -> (exists dst_positive_code_table_zero_right dst_positive_scale_table_zero_right dst_negative_code_table_zero_right dst_negative_scale_table_zero_right. (((G) = (((((dst_positive_code_table_zero_right) + (dst_positive_scale_table_zero_right)) * S ((dst_positive_code_table_zero_right) + (dst_positive_scale_table_zero_right)) + ((dst_positive_scale_table_zero_right) + (dst_positive_scale_table_zero_right))) + (((dst_negative_code_table_zero_right) + (dst_negative_scale_table_zero_right)) * S ((dst_negative_code_table_zero_right) + (dst_negative_scale_table_zero_right)) + ((dst_negative_scale_table_zero_right) + (dst_negative_scale_table_zero_right)))) * S ((((dst_positive_code_table_zero_right) + (dst_positive_scale_table_zero_right)) * S ((dst_positive_code_table_zero_right) + (dst_positive_scale_table_zero_right)) + ((dst_positive_scale_table_zero_right) + (dst_positive_scale_table_zero_right))) + (((dst_negative_code_table_zero_right) + (dst_negative_scale_table_zero_right)) * S ((dst_negative_code_table_zero_right) + (dst_negative_scale_table_zero_right)) + ((dst_negative_scale_table_zero_right) + (dst_negative_scale_table_zero_right)))) + ((((dst_negative_code_table_zero_right) + (dst_negative_scale_table_zero_right)) * S ((dst_negative_code_table_zero_right) + (dst_negative_scale_table_zero_right)) + ((dst_negative_scale_table_zero_right) + (dst_negative_scale_table_zero_right))) + (((dst_negative_code_table_zero_right) + (dst_negative_scale_table_zero_right)) * S ((dst_negative_code_table_zero_right) + (dst_negative_scale_table_zero_right)) + ((dst_negative_scale_table_zero_right) + (dst_negative_scale_table_zero_right)))))) /\ (forall dst_index_table_zero_right. (exists pvs_le_gap_table_zero_rightdomain. pvs_le_gap_table_zero_rightdomain + (dst_index_table_zero_right) = (0)) -> exists dst_positive_table_zero_right dst_negative_table_zero_right dst_value_table_zero_right. ((((exists ff_h_pvs_table_zero_rightentrypositive. ff_h_pvs_table_zero_rightentrypositive + S (dst_positive_table_zero_right) = S ((S (dst_index_table_zero_right)) * dst_positive_scale_table_zero_right)) /\ exists ff_q_pvs_table_zero_rightentrypositive. dst_positive_code_table_zero_right = ff_q_pvs_table_zero_rightentrypositive * S ((S (dst_index_table_zero_right)) * dst_positive_scale_table_zero_right) + (dst_positive_table_zero_right))) /\ (((((exists ff_h_pvs_table_zero_rightentrynegative. ff_h_pvs_table_zero_rightentrynegative + S (dst_negative_table_zero_right) = S ((S (dst_index_table_zero_right)) * dst_negative_scale_table_zero_right)) /\ exists ff_q_pvs_table_zero_rightentrynegative. dst_negative_code_table_zero_right = ff_q_pvs_table_zero_rightentrynegative * S ((S (dst_index_table_zero_right)) * dst_negative_scale_table_zero_right) + (dst_negative_table_zero_right))) /\ (exists ge_balance_positive_table_zero_rightentryvalue ge_balance_negative_table_zero_rightentryvalue. (((((dst_value_table_zero_right) = 2 * (ge_balance_positive_table_zero_rightentryvalue) /\ (ge_balance_negative_table_zero_rightentryvalue) = 0) \/ exists ge_signed_half_table_zero_rightentryvaluedecode. (((dst_value_table_zero_right) = 2 * ge_signed_half_table_zero_rightentryvaluedecode + 1 /\ (ge_balance_positive_table_zero_rightentryvalue) = 0) /\ (ge_balance_negative_table_zero_rightentryvalue) = S ge_signed_half_table_zero_rightentryvaluedecode))) /\ ((dst_positive_table_zero_right) + ge_balance_negative_table_zero_rightentryvalue = (dst_negative_table_zero_right) + ge_balance_positive_table_zero_rightentryvalue))))))))) -> (exists dst_positive_code_table_zero_output dst_positive_scale_table_zero_output dst_negative_code_table_zero_output dst_negative_scale_table_zero_output. (((H) = (((((dst_positive_code_table_zero_output) + (dst_positive_scale_table_zero_output)) * S ((dst_positive_code_table_zero_output) + (dst_positive_scale_table_zero_output)) + ((dst_positive_scale_table_zero_output) + (dst_positive_scale_table_zero_output))) + (((dst_negative_code_table_zero_output) + (dst_negative_scale_table_zero_output)) * S ((dst_negative_code_table_zero_output) + (dst_negative_scale_table_zero_output)) + ((dst_negative_scale_table_zero_output) + (dst_negative_scale_table_zero_output)))) * S ((((dst_positive_code_table_zero_output) + (dst_positive_scale_table_zero_output)) * S ((dst_positive_code_table_zero_output) + (dst_positive_scale_table_zero_output)) + ((dst_positive_scale_table_zero_output) + (dst_positive_scale_table_zero_output))) + (((dst_negative_code_table_zero_output) + (dst_negative_scale_table_zero_output)) * S ((dst_negative_code_table_zero_output) + (dst_negative_scale_table_zero_output)) + ((dst_negative_scale_table_zero_output) + (dst_negative_scale_table_zero_output)))) + ((((dst_negative_code_table_zero_output) + (dst_negative_scale_table_zero_output)) * S ((dst_negative_code_table_zero_output) + (dst_negative_scale_table_zero_output)) + ((dst_negative_scale_table_zero_output) + (dst_negative_scale_table_zero_output))) + (((dst_negative_code_table_zero_output) + (dst_negative_scale_table_zero_output)) * S ((dst_negative_code_table_zero_output) + (dst_negative_scale_table_zero_output)) + ((dst_negative_scale_table_zero_output) + (dst_negative_scale_table_zero_output)))))) /\ (forall dst_index_table_zero_output. (exists pvs_le_gap_table_zero_outputdomain. pvs_le_gap_table_zero_outputdomain + (dst_index_table_zero_output) = (0)) -> exists dst_positive_table_zero_output dst_negative_table_zero_output dst_value_table_zero_output. ((((exists ff_h_pvs_table_zero_outputentrypositive. ff_h_pvs_table_zero_outputentrypositive + S (dst_positive_table_zero_output) = S ((S (dst_index_table_zero_output)) * dst_positive_scale_table_zero_output)) /\ exists ff_q_pvs_table_zero_outputentrypositive. dst_positive_code_table_zero_output = ff_q_pvs_table_zero_outputentrypositive * S ((S (dst_index_table_zero_output)) * dst_positive_scale_table_zero_output) + (dst_positive_table_zero_output))) /\ (((((exists ff_h_pvs_table_zero_outputentrynegative. ff_h_pvs_table_zero_outputentrynegative + S (dst_negative_table_zero_output) = S ((S (dst_index_table_zero_output)) * dst_negative_scale_table_zero_output)) /\ exists ff_q_pvs_table_zero_outputentrynegative. dst_negative_code_table_zero_output = ff_q_pvs_table_zero_outputentrynegative * S ((S (dst_index_table_zero_output)) * dst_negative_scale_table_zero_output) + (dst_negative_table_zero_output))) /\ (exists ge_balance_positive_table_zero_outputentryvalue ge_balance_negative_table_zero_outputentryvalue. (((((dst_value_table_zero_output) = 2 * (ge_balance_positive_table_zero_outputentryvalue) /\ (ge_balance_negative_table_zero_outputentryvalue) = 0) \/ exists ge_signed_half_table_zero_outputentryvaluedecode. (((dst_value_table_zero_output) = 2 * ge_signed_half_table_zero_outputentryvaluedecode + 1 /\ (ge_balance_positive_table_zero_outputentryvalue) = 0) /\ (ge_balance_negative_table_zero_outputentryvalue) = S ge_signed_half_table_zero_outputentryvaluedecode))) /\ ((dst_positive_table_zero_output) + ge_balance_negative_table_zero_outputentryvalue = (dst_negative_table_zero_output) + ge_balance_positive_table_zero_outputentryvalue))))))))) -> (((exists dst_positive_code_table_zero_resultleft dst_positive_scale_table_zero_resultleft dst_negative_code_table_zero_resultleft dst_negative_scale_table_zero_resultleft. (((F) = (((((dst_positive_code_table_zero_resultleft) + (dst_positive_scale_table_zero_resultleft)) * S ((dst_positive_code_table_zero_resultleft) + (dst_positive_scale_table_zero_resultleft)) + ((dst_positive_scale_table_zero_resultleft) + (dst_positive_scale_table_zero_resultleft))) + (((dst_negative_code_table_zero_resultleft) + (dst_negative_scale_table_zero_resultleft)) * S ((dst_negative_code_table_zero_resultleft) + (dst_negative_scale_table_zero_resultleft)) + ((dst_negative_scale_table_zero_resultleft) + (dst_negative_scale_table_zero_resultleft)))) * S ((((dst_positive_code_table_zero_resultleft) + (dst_positive_scale_table_zero_resultleft)) * S ((dst_positive_code_table_zero_resultleft) + (dst_positive_scale_table_zero_resultleft)) + ((dst_positive_scale_table_zero_resultleft) + (dst_positive_scale_table_zero_resultleft))) + (((dst_negative_code_table_zero_resultleft) + (dst_negative_scale_table_zero_resultleft)) * S ((dst_negative_code_table_zero_resultleft) + (dst_negative_scale_table_zero_resultleft)) + ((dst_negative_scale_table_zero_resultleft) + (dst_negative_scale_table_zero_resultleft)))) + ((((dst_negative_code_table_zero_resultleft) + (dst_negative_scale_table_zero_resultleft)) * S ((dst_negative_code_table_zero_resultleft) + (dst_negative_scale_table_zero_resultleft)) + ((dst_negative_scale_table_zero_resultleft) + (dst_negative_scale_table_zero_resultleft))) + (((dst_negative_code_table_zero_resultleft) + (dst_negative_scale_table_zero_resultleft)) * S ((dst_negative_code_table_zero_resultleft) + (dst_negative_scale_table_zero_resultleft)) + ((dst_negative_scale_table_zero_resultleft) + (dst_negative_scale_table_zero_resultleft)))))) /\ (forall dst_index_table_zero_resultleft. (exists pvs_le_gap_table_zero_resultleftdomain. pvs_le_gap_table_zero_resultleftdomain + (dst_index_table_zero_resultleft) = (0)) -> exists dst_positive_table_zero_resultleft dst_negative_table_zero_resultleft dst_value_table_zero_resultleft. ((((exists ff_h_pvs_table_zero_resultleftentrypositive. ff_h_pvs_table_zero_resultleftentrypositive + S (dst_positive_table_zero_resultleft) = S ((S (dst_index_table_zero_resultleft)) * dst_positive_scale_table_zero_resultleft)) /\ exists ff_q_pvs_table_zero_resultleftentrypositive. dst_positive_code_table_zero_resultleft = ff_q_pvs_table_zero_resultleftentrypositive * S ((S (dst_index_table_zero_resultleft)) * dst_positive_scale_table_zero_resultleft) + (dst_positive_table_zero_resultleft))) /\ (((((exists ff_h_pvs_table_zero_resultleftentrynegative. ff_h_pvs_table_zero_resultleftentrynegative + S (dst_negative_table_zero_resultleft) = S ((S (dst_index_table_zero_resultleft)) * dst_negative_scale_table_zero_resultleft)) /\ exists ff_q_pvs_table_zero_resultleftentrynegative. dst_negative_code_table_zero_resultleft = ff_q_pvs_table_zero_resultleftentrynegative * S ((S (dst_index_table_zero_resultleft)) * dst_negative_scale_table_zero_resultleft) + (dst_negative_table_zero_resultleft))) /\ (exists ge_balance_positive_table_zero_resultleftentryvalue ge_balance_negative_table_zero_resultleftentryvalue. (((((dst_value_table_zero_resultleft) = 2 * (ge_balance_positive_table_zero_resultleftentryvalue) /\ (ge_balance_negative_table_zero_resultleftentryvalue) = 0) \/ exists ge_signed_half_table_zero_resultleftentryvaluedecode. (((dst_value_table_zero_resultleft) = 2 * ge_signed_half_table_zero_resultleftentryvaluedecode + 1 /\ (ge_balance_positive_table_zero_resultleftentryvalue) = 0) /\ (ge_balance_negative_table_zero_resultleftentryvalue) = S ge_signed_half_table_zero_resultleftentryvaluedecode))) /\ ((dst_positive_table_zero_resultleft) + ge_balance_negative_table_zero_resultleftentryvalue = (dst_negative_table_zero_resultleft) + ge_balance_positive_table_zero_resultleftentryvalue))))))))) /\ (((exists dst_positive_code_table_zero_resultright dst_positive_scale_table_zero_resultright dst_negative_code_table_zero_resultright dst_negative_scale_table_zero_resultright. (((G) = (((((dst_positive_code_table_zero_resultright) + (dst_positive_scale_table_zero_resultright)) * S ((dst_positive_code_table_zero_resultright) + (dst_positive_scale_table_zero_resultright)) + ((dst_positive_scale_table_zero_resultright) + (dst_positive_scale_table_zero_resultright))) + (((dst_negative_code_table_zero_resultright) + (dst_negative_scale_table_zero_resultright)) * S ((dst_negative_code_table_zero_resultright) + (dst_negative_scale_table_zero_resultright)) + ((dst_negative_scale_table_zero_resultright) + (dst_negative_scale_table_zero_resultright)))) * S ((((dst_positive_code_table_zero_resultright) + (dst_positive_scale_table_zero_resultright)) * S ((dst_positive_code_table_zero_resultright) + (dst_positive_scale_table_zero_resultright)) + ((dst_positive_scale_table_zero_resultright) + (dst_positive_scale_table_zero_resultright))) + (((dst_negative_code_table_zero_resultright) + (dst_negative_scale_table_zero_resultright)) * S ((dst_negative_code_table_zero_resultright) + (dst_negative_scale_table_zero_resultright)) + ((dst_negative_scale_table_zero_resultright) + (dst_negative_scale_table_zero_resultright)))) + ((((dst_negative_code_table_zero_resultright) + (dst_negative_scale_table_zero_resultright)) * S ((dst_negative_code_table_zero_resultright) + (dst_negative_scale_table_zero_resultright)) + ((dst_negative_scale_table_zero_resultright) + (dst_negative_scale_table_zero_resultright))) + (((dst_negative_code_table_zero_resultright) + (dst_negative_scale_table_zero_resultright)) * S ((dst_negative_code_table_zero_resultright) + (dst_negative_scale_table_zero_resultright)) + ((dst_negative_scale_table_zero_resultright) + (dst_negative_scale_table_zero_resultright)))))) /\ (forall dst_index_table_zero_resultright. (exists pvs_le_gap_table_zero_resultrightdomain. pvs_le_gap_table_zero_resultrightdomain + (dst_index_table_zero_resultright) = (0)) -> exists dst_positive_table_zero_resultright dst_negative_table_zero_resultright dst_value_table_zero_resultright. ((((exists ff_h_pvs_table_zero_resultrightentrypositive. ff_h_pvs_table_zero_resultrightentrypositive + S (dst_positive_table_zero_resultright) = S ((S (dst_index_table_zero_resultright)) * dst_positive_scale_table_zero_resultright)) /\ exists ff_q_pvs_table_zero_resultrightentrypositive. dst_positive_code_table_zero_resultright = ff_q_pvs_table_zero_resultrightentrypositive * S ((S (dst_index_table_zero_resultright)) * dst_positive_scale_table_zero_resultright) + (dst_positive_table_zero_resultright))) /\ (((((exists ff_h_pvs_table_zero_resultrightentrynegative. ff_h_pvs_table_zero_resultrightentrynegative + S (dst_negative_table_zero_resultright) = S ((S (dst_index_table_zero_resultright)) * dst_negative_scale_table_zero_resultright)) /\ exists ff_q_pvs_table_zero_resultrightentrynegative. dst_negative_code_table_zero_resultright = ff_q_pvs_table_zero_resultrightentrynegative * S ((S (dst_index_table_zero_resultright)) * dst_negative_scale_table_zero_resultright) + (dst_negative_table_zero_resultright))) /\ (exists ge_balance_positive_table_zero_resultrightentryvalue ge_balance_negative_table_zero_resultrightentryvalue. (((((dst_value_table_zero_resultright) = 2 * (ge_balance_positive_table_zero_resultrightentryvalue) /\ (ge_balance_negative_table_zero_resultrightentryvalue) = 0) \/ exists ge_signed_half_table_zero_resultrightentryvaluedecode. (((dst_value_table_zero_resultright) = 2 * ge_signed_half_table_zero_resultrightentryvaluedecode + 1 /\ (ge_balance_positive_table_zero_resultrightentryvalue) = 0) /\ (ge_balance_negative_table_zero_resultrightentryvalue) = S ge_signed_half_table_zero_resultrightentryvaluedecode))) /\ ((dst_positive_table_zero_resultright) + ge_balance_negative_table_zero_resultrightentryvalue = (dst_negative_table_zero_resultright) + ge_balance_positive_table_zero_resultrightentryvalue))))))))) /\ (((exists dst_positive_code_table_zero_resulttable dst_positive_scale_table_zero_resulttable dst_negative_code_table_zero_resulttable dst_negative_scale_table_zero_resulttable. (((H) = (((((dst_positive_code_table_zero_resulttable) + (dst_positive_scale_table_zero_resulttable)) * S ((dst_positive_code_table_zero_resulttable) + (dst_positive_scale_table_zero_resulttable)) + ((dst_positive_scale_table_zero_resulttable) + (dst_positive_scale_table_zero_resulttable))) + (((dst_negative_code_table_zero_resulttable) + (dst_negative_scale_table_zero_resulttable)) * S ((dst_negative_code_table_zero_resulttable) + (dst_negative_scale_table_zero_resulttable)) + ((dst_negative_scale_table_zero_resulttable) + (dst_negative_scale_table_zero_resulttable)))) * S ((((dst_positive_code_table_zero_resulttable) + (dst_positive_scale_table_zero_resulttable)) * S ((dst_positive_code_table_zero_resulttable) + (dst_positive_scale_table_zero_resulttable)) + ((dst_positive_scale_table_zero_resulttable) + (dst_positive_scale_table_zero_resulttable))) + (((dst_negative_code_table_zero_resulttable) + (dst_negative_scale_table_zero_resulttable)) * S ((dst_negative_code_table_zero_resulttable) + (dst_negative_scale_table_zero_resulttable)) + ((dst_negative_scale_table_zero_resulttable) + (dst_negative_scale_table_zero_resulttable)))) + ((((dst_negative_code_table_zero_resulttable) + (dst_negative_scale_table_zero_resulttable)) * S ((dst_negative_code_table_zero_resulttable) + (dst_negative_scale_table_zero_resulttable)) + ((dst_negative_scale_table_zero_resulttable) + (dst_negative_scale_table_zero_resulttable))) + (((dst_negative_code_table_zero_resulttable) + (dst_negative_scale_table_zero_resulttable)) * S ((dst_negative_code_table_zero_resulttable) + (dst_negative_scale_table_zero_resulttable)) + ((dst_negative_scale_table_zero_resulttable) + (dst_negative_scale_table_zero_resulttable)))))) /\ (forall dst_index_table_zero_resulttable. (exists pvs_le_gap_table_zero_resulttabledomain. pvs_le_gap_table_zero_resulttabledomain + (dst_index_table_zero_resulttable) = (0)) -> exists dst_positive_table_zero_resulttable dst_negative_table_zero_resulttable dst_value_table_zero_resulttable. ((((exists ff_h_pvs_table_zero_resulttableentrypositive. ff_h_pvs_table_zero_resulttableentrypositive + S (dst_positive_table_zero_resulttable) = S ((S (dst_index_table_zero_resulttable)) * dst_positive_scale_table_zero_resulttable)) /\ exists ff_q_pvs_table_zero_resulttableentrypositive. dst_positive_code_table_zero_resulttable = ff_q_pvs_table_zero_resulttableentrypositive * S ((S (dst_index_table_zero_resulttable)) * dst_positive_scale_table_zero_resulttable) + (dst_positive_table_zero_resulttable))) /\ (((((exists ff_h_pvs_table_zero_resulttableentrynegative. ff_h_pvs_table_zero_resulttableentrynegative + S (dst_negative_table_zero_resulttable) = S ((S (dst_index_table_zero_resulttable)) * dst_negative_scale_table_zero_resulttable)) /\ exists ff_q_pvs_table_zero_resulttableentrynegative. dst_negative_code_table_zero_resulttable = ff_q_pvs_table_zero_resulttableentrynegative * S ((S (dst_index_table_zero_resulttable)) * dst_negative_scale_table_zero_resulttable) + (dst_negative_table_zero_resulttable))) /\ (exists ge_balance_positive_table_zero_resulttableentryvalue ge_balance_negative_table_zero_resulttableentryvalue. (((((dst_value_table_zero_resulttable) = 2 * (ge_balance_positive_table_zero_resulttableentryvalue) /\ (ge_balance_negative_table_zero_resulttableentryvalue) = 0) \/ exists ge_signed_half_table_zero_resulttableentryvaluedecode. (((dst_value_table_zero_resulttable) = 2 * ge_signed_half_table_zero_resulttableentryvaluedecode + 1 /\ (ge_balance_positive_table_zero_resulttableentryvalue) = 0) /\ (ge_balance_negative_table_zero_resulttableentryvalue) = S ge_signed_half_table_zero_resulttableentryvaluedecode))) /\ ((dst_positive_table_zero_resulttable) + ge_balance_negative_table_zero_resulttableentryvalue = (dst_negative_table_zero_resulttable) + ge_balance_positive_table_zero_resulttableentryvalue))))))))) /\ (forall dc_input_table_zero_result dc_output_table_zero_result. ~(dc_input_table_zero_result=0) -> (exists pvs_le_gap_table_zero_resultdomain. pvs_le_gap_table_zero_resultdomain + (dc_input_table_zero_result) = (0)) -> (exists dst_positive_code_table_zero_resultlookup dst_positive_scale_table_zero_resultlookup dst_negative_code_table_zero_resultlookup dst_negative_scale_table_zero_resultlookup dst_positive_table_zero_resultlookup dst_negative_table_zero_resultlookup. (((H) = (((((dst_positive_code_table_zero_resultlookup) + (dst_positive_scale_table_zero_resultlookup)) * S ((dst_positive_code_table_zero_resultlookup) + (dst_positive_scale_table_zero_resultlookup)) + ((dst_positive_scale_table_zero_resultlookup) + (dst_positive_scale_table_zero_resultlookup))) + (((dst_negative_code_table_zero_resultlookup) + (dst_negative_scale_table_zero_resultlookup)) * S ((dst_negative_code_table_zero_resultlookup) + (dst_negative_scale_table_zero_resultlookup)) + ((dst_negative_scale_table_zero_resultlookup) + (dst_negative_scale_table_zero_resultlookup)))) * S ((((dst_positive_code_table_zero_resultlookup) + (dst_positive_scale_table_zero_resultlookup)) * S ((dst_positive_code_table_zero_resultlookup) + (dst_positive_scale_table_zero_resultlookup)) + ((dst_positive_scale_table_zero_resultlookup) + (dst_positive_scale_table_zero_resultlookup))) + (((dst_negative_code_table_zero_resultlookup) + (dst_negative_scale_table_zero_resultlookup)) * S ((dst_negative_code_table_zero_resultlookup) + (dst_negative_scale_table_zero_resultlookup)) + ((dst_negative_scale_table_zero_resultlookup) + (dst_negative_scale_table_zero_resultlookup)))) + ((((dst_negative_code_table_zero_resultlookup) + (dst_negative_scale_table_zero_resultlookup)) * S ((dst_negative_code_table_zero_resultlookup) + (dst_negative_scale_table_zero_resultlookup)) + ((dst_negative_scale_table_zero_resultlookup) + (dst_negative_scale_table_zero_resultlookup))) + (((dst_negative_code_table_zero_resultlookup) + (dst_negative_scale_table_zero_resultlookup)) * S ((dst_negative_code_table_zero_resultlookup) + (dst_negative_scale_table_zero_resultlookup)) + ((dst_negative_scale_table_zero_resultlookup) + (dst_negative_scale_table_zero_resultlookup)))))) /\ (((((exists ff_h_pvs_table_zero_resultlookuppositive. ff_h_pvs_table_zero_resultlookuppositive + S (dst_positive_table_zero_resultlookup) = S ((S (dc_input_table_zero_result)) * dst_positive_scale_table_zero_resultlookup)) /\ exists ff_q_pvs_table_zero_resultlookuppositive. dst_positive_code_table_zero_resultlookup = ff_q_pvs_table_zero_resultlookuppositive * S ((S (dc_input_table_zero_result)) * dst_positive_scale_table_zero_resultlookup) + (dst_positive_table_zero_resultlookup))) /\ (((((exists ff_h_pvs_table_zero_resultlookupnegative. ff_h_pvs_table_zero_resultlookupnegative + S (dst_negative_table_zero_resultlookup) = S ((S (dc_input_table_zero_result)) * dst_negative_scale_table_zero_resultlookup)) /\ exists ff_q_pvs_table_zero_resultlookupnegative. dst_negative_code_table_zero_resultlookup = ff_q_pvs_table_zero_resultlookupnegative * S ((S (dc_input_table_zero_result)) * dst_negative_scale_table_zero_resultlookup) + (dst_negative_table_zero_resultlookup))) /\ (exists ge_balance_positive_table_zero_resultlookupvalue ge_balance_negative_table_zero_resultlookupvalue. (((((dc_output_table_zero_result) = 2 * (ge_balance_positive_table_zero_resultlookupvalue) /\ (ge_balance_negative_table_zero_resultlookupvalue) = 0) \/ exists ge_signed_half_table_zero_resultlookupvaluedecode. (((dc_output_table_zero_result) = 2 * ge_signed_half_table_zero_resultlookupvaluedecode + 1 /\ (ge_balance_positive_table_zero_resultlookupvalue) = 0) /\ (ge_balance_negative_table_zero_resultlookupvalue) = S ge_signed_half_table_zero_resultlookupvaluedecode))) /\ ((dst_positive_table_zero_resultlookup) + ge_balance_negative_table_zero_resultlookupvalue = (dst_negative_table_zero_resultlookup) + ge_balance_positive_table_zero_resultlookupvalue))))))))) -> (((~((dc_input_table_zero_result)=0)) /\ (exists dc_mask_table_zero_resultvalue. ((((exists dst_positive_code_table_zero_resultvaluemasktable dst_positive_scale_table_zero_resultvaluemasktable dst_negative_code_table_zero_resultvaluemasktable dst_negative_scale_table_zero_resultvaluemasktable. (((dc_mask_table_zero_resultvalue) = (((((dst_positive_code_table_zero_resultvaluemasktable) + (dst_positive_scale_table_zero_resultvaluemasktable)) * S ((dst_positive_code_table_zero_resultvaluemasktable) + (dst_positive_scale_table_zero_resultvaluemasktable)) + ((dst_positive_scale_table_zero_resultvaluemasktable) + (dst_positive_scale_table_zero_resultvaluemasktable))) + (((dst_negative_code_table_zero_resultvaluemasktable) + (dst_negative_scale_table_zero_resultvaluemasktable)) * S ((dst_negative_code_table_zero_resultvaluemasktable) + (dst_negative_scale_table_zero_resultvaluemasktable)) + ((dst_negative_scale_table_zero_resultvaluemasktable) + (dst_negative_scale_table_zero_resultvaluemasktable)))) * S ((((dst_positive_code_table_zero_resultvaluemasktable) + (dst_positive_scale_table_zero_resultvaluemasktable)) * S ((dst_positive_code_table_zero_resultvaluemasktable) + (dst_positive_scale_table_zero_resultvaluemasktable)) + ((dst_positive_scale_table_zero_resultvaluemasktable) + (dst_positive_scale_table_zero_resultvaluemasktable))) + (((dst_negative_code_table_zero_resultvaluemasktable) + (dst_negative_scale_table_zero_resultvaluemasktable)) * S ((dst_negative_code_table_zero_resultvaluemasktable) + (dst_negative_scale_table_zero_resultvaluemasktable)) + ((dst_negative_scale_table_zero_resultvaluemasktable) + (dst_negative_scale_table_zero_resultvaluemasktable)))) + ((((dst_negative_code_table_zero_resultvaluemasktable) + (dst_negative_scale_table_zero_resultvaluemasktable)) * S ((dst_negative_code_table_zero_resultvaluemasktable) + (dst_negative_scale_table_zero_resultvaluemasktable)) + ((dst_negative_scale_table_zero_resultvaluemasktable) + (dst_negative_scale_table_zero_resultvaluemasktable))) + (((dst_negative_code_table_zero_resultvaluemasktable) + (dst_negative_scale_table_zero_resultvaluemasktable)) * S ((dst_negative_code_table_zero_resultvaluemasktable) + (dst_negative_scale_table_zero_resultvaluemasktable)) + ((dst_negative_scale_table_zero_resultvaluemasktable) + (dst_negative_scale_table_zero_resultvaluemasktable)))))) /\ (forall dst_index_table_zero_resultvaluemasktable. (exists pvs_le_gap_table_zero_resultvaluemasktabledomain. pvs_le_gap_table_zero_resultvaluemasktabledomain + (dst_index_table_zero_resultvaluemasktable) = (dc_input_table_zero_result)) -> exists dst_positive_table_zero_resultvaluemasktable dst_negative_table_zero_resultvaluemasktable dst_value_table_zero_resultvaluemasktable. ((((exists ff_h_pvs_table_zero_resultvaluemasktableentrypositive. ff_h_pvs_table_zero_resultvaluemasktableentrypositive + S (dst_positive_table_zero_resultvaluemasktable) = S ((S (dst_index_table_zero_resultvaluemasktable)) * dst_positive_scale_table_zero_resultvaluemasktable)) /\ exists ff_q_pvs_table_zero_resultvaluemasktableentrypositive. dst_positive_code_table_zero_resultvaluemasktable = ff_q_pvs_table_zero_resultvaluemasktableentrypositive * S ((S (dst_index_table_zero_resultvaluemasktable)) * dst_positive_scale_table_zero_resultvaluemasktable) + (dst_positive_table_zero_resultvaluemasktable))) /\ (((((exists ff_h_pvs_table_zero_resultvaluemasktableentrynegative. ff_h_pvs_table_zero_resultvaluemasktableentrynegative + S (dst_negative_table_zero_resultvaluemasktable) = S ((S (dst_index_table_zero_resultvaluemasktable)) * dst_negative_scale_table_zero_resultvaluemasktable)) /\ exists ff_q_pvs_table_zero_resultvaluemasktableentrynegative. dst_negative_code_table_zero_resultvaluemasktable = ff_q_pvs_table_zero_resultvaluemasktableentrynegative * S ((S (dst_index_table_zero_resultvaluemasktable)) * dst_negative_scale_table_zero_resultvaluemasktable) + (dst_negative_table_zero_resultvaluemasktable))) /\ (exists ge_balance_positive_table_zero_resultvaluemasktableentryvalue ge_balance_negative_table_zero_resultvaluemasktableentryvalue. (((((dst_value_table_zero_resultvaluemasktable) = 2 * (ge_balance_positive_table_zero_resultvaluemasktableentryvalue) /\ (ge_balance_negative_table_zero_resultvaluemasktableentryvalue) = 0) \/ exists ge_signed_half_table_zero_resultvaluemasktableentryvaluedecode. (((dst_value_table_zero_resultvaluemasktable) = 2 * ge_signed_half_table_zero_resultvaluemasktableentryvaluedecode + 1 /\ (ge_balance_positive_table_zero_resultvaluemasktableentryvalue) = 0) /\ (ge_balance_negative_table_zero_resultvaluemasktableentryvalue) = S ge_signed_half_table_zero_resultvaluemasktableentryvaluedecode))) /\ ((dst_positive_table_zero_resultvaluemasktable) + ge_balance_negative_table_zero_resultvaluemasktableentryvalue = (dst_negative_table_zero_resultvaluemasktable) + ge_balance_positive_table_zero_resultvaluemasktableentryvalue))))))))) /\ (forall dc_index_table_zero_resultvaluemask dc_value_table_zero_resultvaluemask. (exists pvs_le_gap_table_zero_resultvaluemaskdomain. pvs_le_gap_table_zero_resultvaluemaskdomain + (dc_index_table_zero_resultvaluemask) = (dc_input_table_zero_result)) -> (exists dst_positive_code_table_zero_resultvaluemasklookup dst_positive_scale_table_zero_resultvaluemasklookup dst_negative_code_table_zero_resultvaluemasklookup dst_negative_scale_table_zero_resultvaluemasklookup dst_positive_table_zero_resultvaluemasklookup dst_negative_table_zero_resultvaluemasklookup. (((dc_mask_table_zero_resultvalue) = (((((dst_positive_code_table_zero_resultvaluemasklookup) + (dst_positive_scale_table_zero_resultvaluemasklookup)) * S ((dst_positive_code_table_zero_resultvaluemasklookup) + (dst_positive_scale_table_zero_resultvaluemasklookup)) + ((dst_positive_scale_table_zero_resultvaluemasklookup) + (dst_positive_scale_table_zero_resultvaluemasklookup))) + (((dst_negative_code_table_zero_resultvaluemasklookup) + (dst_negative_scale_table_zero_resultvaluemasklookup)) * S ((dst_negative_code_table_zero_resultvaluemasklookup) + (dst_negative_scale_table_zero_resultvaluemasklookup)) + ((dst_negative_scale_table_zero_resultvaluemasklookup) + (dst_negative_scale_table_zero_resultvaluemasklookup)))) * S ((((dst_positive_code_table_zero_resultvaluemasklookup) + (dst_positive_scale_table_zero_resultvaluemasklookup)) * S ((dst_positive_code_table_zero_resultvaluemasklookup) + (dst_positive_scale_table_zero_resultvaluemasklookup)) + ((dst_positive_scale_table_zero_resultvaluemasklookup) + (dst_positive_scale_table_zero_resultvaluemasklookup))) + (((dst_negative_code_table_zero_resultvaluemasklookup) + (dst_negative_scale_table_zero_resultvaluemasklookup)) * S ((dst_negative_code_table_zero_resultvaluemasklookup) + (dst_negative_scale_table_zero_resultvaluemasklookup)) + ((dst_negative_scale_table_zero_resultvaluemasklookup) + (dst_negative_scale_table_zero_resultvaluemasklookup)))) + ((((dst_negative_code_table_zero_resultvaluemasklookup) + (dst_negative_scale_table_zero_resultvaluemasklookup)) * S ((dst_negative_code_table_zero_resultvaluemasklookup) + (dst_negative_scale_table_zero_resultvaluemasklookup)) + ((dst_negative_scale_table_zero_resultvaluemasklookup) + (dst_negative_scale_table_zero_resultvaluemasklookup))) + (((dst_negative_code_table_zero_resultvaluemasklookup) + (dst_negative_scale_table_zero_resultvaluemasklookup)) * S ((dst_negative_code_table_zero_resultvaluemasklookup) + (dst_negative_scale_table_zero_resultvaluemasklookup)) + ((dst_negative_scale_table_zero_resultvaluemasklookup) + (dst_negative_scale_table_zero_resultvaluemasklookup)))))) /\ (((((exists ff_h_pvs_table_zero_resultvaluemasklookuppositive. ff_h_pvs_table_zero_resultvaluemasklookuppositive + S (dst_positive_table_zero_resultvaluemasklookup) = S ((S (dc_index_table_zero_resultvaluemask)) * dst_positive_scale_table_zero_resultvaluemasklookup)) /\ exists ff_q_pvs_table_zero_resultvaluemasklookuppositive. dst_positive_code_table_zero_resultvaluemasklookup = ff_q_pvs_table_zero_resultvaluemasklookuppositive * S ((S (dc_index_table_zero_resultvaluemask)) * dst_positive_scale_table_zero_resultvaluemasklookup) + (dst_positive_table_zero_resultvaluemasklookup))) /\ (((((exists ff_h_pvs_table_zero_resultvaluemasklookupnegative. ff_h_pvs_table_zero_resultvaluemasklookupnegative + S (dst_negative_table_zero_resultvaluemasklookup) = S ((S (dc_index_table_zero_resultvaluemask)) * dst_negative_scale_table_zero_resultvaluemasklookup)) /\ exists ff_q_pvs_table_zero_resultvaluemasklookupnegative. dst_negative_code_table_zero_resultvaluemasklookup = ff_q_pvs_table_zero_resultvaluemasklookupnegative * S ((S (dc_index_table_zero_resultvaluemask)) * dst_negative_scale_table_zero_resultvaluemasklookup) + (dst_negative_table_zero_resultvaluemasklookup))) /\ (exists ge_balance_positive_table_zero_resultvaluemasklookupvalue ge_balance_negative_table_zero_resultvaluemasklookupvalue. (((((dc_value_table_zero_resultvaluemask) = 2 * (ge_balance_positive_table_zero_resultvaluemasklookupvalue) /\ (ge_balance_negative_table_zero_resultvaluemasklookupvalue) = 0) \/ exists ge_signed_half_table_zero_resultvaluemasklookupvaluedecode. (((dc_value_table_zero_resultvaluemask) = 2 * ge_signed_half_table_zero_resultvaluemasklookupvaluedecode + 1 /\ (ge_balance_positive_table_zero_resultvaluemasklookupvalue) = 0) /\ (ge_balance_negative_table_zero_resultvaluemasklookupvalue) = S ge_signed_half_table_zero_resultvaluemasklookupvaluedecode))) /\ ((dst_positive_table_zero_resultvaluemasklookup) + ge_balance_negative_table_zero_resultvaluemasklookupvalue = (dst_negative_table_zero_resultvaluemasklookup) + ge_balance_positive_table_zero_resultvaluemasklookupvalue))))))))) -> ((((~((dc_index_table_zero_resultvaluemask)=0)) /\ (exists dc_quotient_table_zero_resultvaluemaskentry dc_left_table_zero_resultvaluemaskentry dc_right_table_zero_resultvaluemaskentry. (((dc_input_table_zero_result)=(dc_index_table_zero_resultvaluemask)*dc_quotient_table_zero_resultvaluemaskentry) /\ (((exists dst_positive_code_table_zero_resultvaluemaskentryleft dst_positive_scale_table_zero_resultvaluemaskentryleft dst_negative_code_table_zero_resultvaluemaskentryleft dst_negative_scale_table_zero_resultvaluemaskentryleft dst_positive_table_zero_resultvaluemaskentryleft dst_negative_table_zero_resultvaluemaskentryleft. (((F) = (((((dst_positive_code_table_zero_resultvaluemaskentryleft) + (dst_positive_scale_table_zero_resultvaluemaskentryleft)) * S ((dst_positive_code_table_zero_resultvaluemaskentryleft) + (dst_positive_scale_table_zero_resultvaluemaskentryleft)) + ((dst_positive_scale_table_zero_resultvaluemaskentryleft) + (dst_positive_scale_table_zero_resultvaluemaskentryleft))) + (((dst_negative_code_table_zero_resultvaluemaskentryleft) + (dst_negative_scale_table_zero_resultvaluemaskentryleft)) * S ((dst_negative_code_table_zero_resultvaluemaskentryleft) + (dst_negative_scale_table_zero_resultvaluemaskentryleft)) + ((dst_negative_scale_table_zero_resultvaluemaskentryleft) + (dst_negative_scale_table_zero_resultvaluemaskentryleft)))) * S ((((dst_positive_code_table_zero_resultvaluemaskentryleft) + (dst_positive_scale_table_zero_resultvaluemaskentryleft)) * S ((dst_positive_code_table_zero_resultvaluemaskentryleft) + (dst_positive_scale_table_zero_resultvaluemaskentryleft)) + ((dst_positive_scale_table_zero_resultvaluemaskentryleft) + (dst_positive_scale_table_zero_resultvaluemaskentryleft))) + (((dst_negative_code_table_zero_resultvaluemaskentryleft) + (dst_negative_scale_table_zero_resultvaluemaskentryleft)) * S ((dst_negative_code_table_zero_resultvaluemaskentryleft) + (dst_negative_scale_table_zero_resultvaluemaskentryleft)) + ((dst_negative_scale_table_zero_resultvaluemaskentryleft) + (dst_negative_scale_table_zero_resultvaluemaskentryleft)))) + ((((dst_negative_code_table_zero_resultvaluemaskentryleft) + (dst_negative_scale_table_zero_resultvaluemaskentryleft)) * S ((dst_negative_code_table_zero_resultvaluemaskentryleft) + (dst_negative_scale_table_zero_resultvaluemaskentryleft)) + ((dst_negative_scale_table_zero_resultvaluemaskentryleft) + (dst_negative_scale_table_zero_resultvaluemaskentryleft))) + (((dst_negative_code_table_zero_resultvaluemaskentryleft) + (dst_negative_scale_table_zero_resultvaluemaskentryleft)) * S ((dst_negative_code_table_zero_resultvaluemaskentryleft) + (dst_negative_scale_table_zero_resultvaluemaskentryleft)) + ((dst_negative_scale_table_zero_resultvaluemaskentryleft) + (dst_negative_scale_table_zero_resultvaluemaskentryleft)))))) /\ (((((exists ff_h_pvs_table_zero_resultvaluemaskentryleftpositive. ff_h_pvs_table_zero_resultvaluemaskentryleftpositive + S (dst_positive_table_zero_resultvaluemaskentryleft) = S ((S (dc_index_table_zero_resultvaluemask)) * dst_positive_scale_table_zero_resultvaluemaskentryleft)) /\ exists ff_q_pvs_table_zero_resultvaluemaskentryleftpositive. dst_positive_code_table_zero_resultvaluemaskentryleft = ff_q_pvs_table_zero_resultvaluemaskentryleftpositive * S ((S (dc_index_table_zero_resultvaluemask)) * dst_positive_scale_table_zero_resultvaluemaskentryleft) + (dst_positive_table_zero_resultvaluemaskentryleft))) /\ (((((exists ff_h_pvs_table_zero_resultvaluemaskentryleftnegative. ff_h_pvs_table_zero_resultvaluemaskentryleftnegative + S (dst_negative_table_zero_resultvaluemaskentryleft) = S ((S (dc_index_table_zero_resultvaluemask)) * dst_negative_scale_table_zero_resultvaluemaskentryleft)) /\ exists ff_q_pvs_table_zero_resultvaluemaskentryleftnegative. dst_negative_code_table_zero_resultvaluemaskentryleft = ff_q_pvs_table_zero_resultvaluemaskentryleftnegative * S ((S (dc_index_table_zero_resultvaluemask)) * dst_negative_scale_table_zero_resultvaluemaskentryleft) + (dst_negative_table_zero_resultvaluemaskentryleft))) /\ (exists ge_balance_positive_table_zero_resultvaluemaskentryleftvalue ge_balance_negative_table_zero_resultvaluemaskentryleftvalue. (((((dc_left_table_zero_resultvaluemaskentry) = 2 * (ge_balance_positive_table_zero_resultvaluemaskentryleftvalue) /\ (ge_balance_negative_table_zero_resultvaluemaskentryleftvalue) = 0) \/ exists ge_signed_half_table_zero_resultvaluemaskentryleftvaluedecode. (((dc_left_table_zero_resultvaluemaskentry) = 2 * ge_signed_half_table_zero_resultvaluemaskentryleftvaluedecode + 1 /\ (ge_balance_positive_table_zero_resultvaluemaskentryleftvalue) = 0) /\ (ge_balance_negative_table_zero_resultvaluemaskentryleftvalue) = S ge_signed_half_table_zero_resultvaluemaskentryleftvaluedecode))) /\ ((dst_positive_table_zero_resultvaluemaskentryleft) + ge_balance_negative_table_zero_resultvaluemaskentryleftvalue = (dst_negative_table_zero_resultvaluemaskentryleft) + ge_balance_positive_table_zero_resultvaluemaskentryleftvalue))))))))) /\ (((exists dst_positive_code_table_zero_resultvaluemaskentryright dst_positive_scale_table_zero_resultvaluemaskentryright dst_negative_code_table_zero_resultvaluemaskentryright dst_negative_scale_table_zero_resultvaluemaskentryright dst_positive_table_zero_resultvaluemaskentryright dst_negative_table_zero_resultvaluemaskentryright. (((G) = (((((dst_positive_code_table_zero_resultvaluemaskentryright) + (dst_positive_scale_table_zero_resultvaluemaskentryright)) * S ((dst_positive_code_table_zero_resultvaluemaskentryright) + (dst_positive_scale_table_zero_resultvaluemaskentryright)) + ((dst_positive_scale_table_zero_resultvaluemaskentryright) + (dst_positive_scale_table_zero_resultvaluemaskentryright))) + (((dst_negative_code_table_zero_resultvaluemaskentryright) + (dst_negative_scale_table_zero_resultvaluemaskentryright)) * S ((dst_negative_code_table_zero_resultvaluemaskentryright) + (dst_negative_scale_table_zero_resultvaluemaskentryright)) + ((dst_negative_scale_table_zero_resultvaluemaskentryright) + (dst_negative_scale_table_zero_resultvaluemaskentryright)))) * S ((((dst_positive_code_table_zero_resultvaluemaskentryright) + (dst_positive_scale_table_zero_resultvaluemaskentryright)) * S ((dst_positive_code_table_zero_resultvaluemaskentryright) + (dst_positive_scale_table_zero_resultvaluemaskentryright)) + ((dst_positive_scale_table_zero_resultvaluemaskentryright) + (dst_positive_scale_table_zero_resultvaluemaskentryright))) + (((dst_negative_code_table_zero_resultvaluemaskentryright) + (dst_negative_scale_table_zero_resultvaluemaskentryright)) * S ((dst_negative_code_table_zero_resultvaluemaskentryright) + (dst_negative_scale_table_zero_resultvaluemaskentryright)) + ((dst_negative_scale_table_zero_resultvaluemaskentryright) + (dst_negative_scale_table_zero_resultvaluemaskentryright)))) + ((((dst_negative_code_table_zero_resultvaluemaskentryright) + (dst_negative_scale_table_zero_resultvaluemaskentryright)) * S ((dst_negative_code_table_zero_resultvaluemaskentryright) + (dst_negative_scale_table_zero_resultvaluemaskentryright)) + ((dst_negative_scale_table_zero_resultvaluemaskentryright) + (dst_negative_scale_table_zero_resultvaluemaskentryright))) + (((dst_negative_code_table_zero_resultvaluemaskentryright) + (dst_negative_scale_table_zero_resultvaluemaskentryright)) * S ((dst_negative_code_table_zero_resultvaluemaskentryright) + (dst_negative_scale_table_zero_resultvaluemaskentryright)) + ((dst_negative_scale_table_zero_resultvaluemaskentryright) + (dst_negative_scale_table_zero_resultvaluemaskentryright)))))) /\ (((((exists ff_h_pvs_table_zero_resultvaluemaskentryrightpositive. ff_h_pvs_table_zero_resultvaluemaskentryrightpositive + S (dst_positive_table_zero_resultvaluemaskentryright) = S ((S (dc_quotient_table_zero_resultvaluemaskentry)) * dst_positive_scale_table_zero_resultvaluemaskentryright)) /\ exists ff_q_pvs_table_zero_resultvaluemaskentryrightpositive. dst_positive_code_table_zero_resultvaluemaskentryright = ff_q_pvs_table_zero_resultvaluemaskentryrightpositive * S ((S (dc_quotient_table_zero_resultvaluemaskentry)) * dst_positive_scale_table_zero_resultvaluemaskentryright) + (dst_positive_table_zero_resultvaluemaskentryright))) /\ (((((exists ff_h_pvs_table_zero_resultvaluemaskentryrightnegative. ff_h_pvs_table_zero_resultvaluemaskentryrightnegative + S (dst_negative_table_zero_resultvaluemaskentryright) = S ((S (dc_quotient_table_zero_resultvaluemaskentry)) * dst_negative_scale_table_zero_resultvaluemaskentryright)) /\ exists ff_q_pvs_table_zero_resultvaluemaskentryrightnegative. dst_negative_code_table_zero_resultvaluemaskentryright = ff_q_pvs_table_zero_resultvaluemaskentryrightnegative * S ((S (dc_quotient_table_zero_resultvaluemaskentry)) * dst_negative_scale_table_zero_resultvaluemaskentryright) + (dst_negative_table_zero_resultvaluemaskentryright))) /\ (exists ge_balance_positive_table_zero_resultvaluemaskentryrightvalue ge_balance_negative_table_zero_resultvaluemaskentryrightvalue. (((((dc_right_table_zero_resultvaluemaskentry) = 2 * (ge_balance_positive_table_zero_resultvaluemaskentryrightvalue) /\ (ge_balance_negative_table_zero_resultvaluemaskentryrightvalue) = 0) \/ exists ge_signed_half_table_zero_resultvaluemaskentryrightvaluedecode. (((dc_right_table_zero_resultvaluemaskentry) = 2 * ge_signed_half_table_zero_resultvaluemaskentryrightvaluedecode + 1 /\ (ge_balance_positive_table_zero_resultvaluemaskentryrightvalue) = 0) /\ (ge_balance_negative_table_zero_resultvaluemaskentryrightvalue) = S ge_signed_half_table_zero_resultvaluemaskentryrightvaluedecode))) /\ ((dst_positive_table_zero_resultvaluemaskentryright) + ge_balance_negative_table_zero_resultvaluemaskentryrightvalue = (dst_negative_table_zero_resultvaluemaskentryright) + ge_balance_positive_table_zero_resultvaluemaskentryrightvalue))))))))) /\ (exists sto_ap_table_zero_resultvaluemaskentryproduct sto_an_table_zero_resultvaluemaskentryproduct sto_bp_table_zero_resultvaluemaskentryproduct sto_bn_table_zero_resultvaluemaskentryproduct sto_cp_table_zero_resultvaluemaskentryproduct sto_cn_table_zero_resultvaluemaskentryproduct. (((((dc_left_table_zero_resultvaluemaskentry) = 2 * (sto_ap_table_zero_resultvaluemaskentryproduct) /\ (sto_an_table_zero_resultvaluemaskentryproduct) = 0) \/ exists ge_signed_half_table_zero_resultvaluemaskentryproductleft. (((dc_left_table_zero_resultvaluemaskentry) = 2 * ge_signed_half_table_zero_resultvaluemaskentryproductleft + 1 /\ (sto_ap_table_zero_resultvaluemaskentryproduct) = 0) /\ (sto_an_table_zero_resultvaluemaskentryproduct) = S ge_signed_half_table_zero_resultvaluemaskentryproductleft))) /\ ((((((dc_right_table_zero_resultvaluemaskentry) = 2 * (sto_bp_table_zero_resultvaluemaskentryproduct) /\ (sto_bn_table_zero_resultvaluemaskentryproduct) = 0) \/ exists ge_signed_half_table_zero_resultvaluemaskentryproductright. (((dc_right_table_zero_resultvaluemaskentry) = 2 * ge_signed_half_table_zero_resultvaluemaskentryproductright + 1 /\ (sto_bp_table_zero_resultvaluemaskentryproduct) = 0) /\ (sto_bn_table_zero_resultvaluemaskentryproduct) = S ge_signed_half_table_zero_resultvaluemaskentryproductright))) /\ ((((((dc_value_table_zero_resultvaluemask) = 2 * (sto_cp_table_zero_resultvaluemaskentryproduct) /\ (sto_cn_table_zero_resultvaluemaskentryproduct) = 0) \/ exists ge_signed_half_table_zero_resultvaluemaskentryproductoutput. (((dc_value_table_zero_resultvaluemask) = 2 * ge_signed_half_table_zero_resultvaluemaskentryproductoutput + 1 /\ (sto_cp_table_zero_resultvaluemaskentryproduct) = 0) /\ (sto_cn_table_zero_resultvaluemaskentryproduct) = S ge_signed_half_table_zero_resultvaluemaskentryproductoutput))) /\ ((sto_ap_table_zero_resultvaluemaskentryproduct * sto_bp_table_zero_resultvaluemaskentryproduct + sto_an_table_zero_resultvaluemaskentryproduct * sto_bn_table_zero_resultvaluemaskentryproduct) + sto_cn_table_zero_resultvaluemaskentryproduct = (sto_ap_table_zero_resultvaluemaskentryproduct * sto_bn_table_zero_resultvaluemaskentryproduct + sto_an_table_zero_resultvaluemaskentryproduct * sto_bp_table_zero_resultvaluemaskentryproduct) + sto_cp_table_zero_resultvaluemaskentryproduct))))))))))))))) \/ ((((dc_index_table_zero_resultvaluemask)=0 \/ ~(exists pvs_factor_table_zero_resultvaluemaskentrynondivisor. (dc_input_table_zero_result) = (dc_index_table_zero_resultvaluemask) * pvs_factor_table_zero_resultvaluemaskentrynondivisor)) /\ ((dc_value_table_zero_resultvaluemask)=0))))))) /\ (exists dst_positive_code_table_zero_resultvaluefold dst_positive_scale_table_zero_resultvaluefold dst_negative_code_table_zero_resultvaluefold dst_negative_scale_table_zero_resultvaluefold dst_positive_sum_table_zero_resultvaluefold dst_negative_sum_table_zero_resultvaluefold. (((dc_mask_table_zero_resultvalue) = (((((dst_positive_code_table_zero_resultvaluefold) + (dst_positive_scale_table_zero_resultvaluefold)) * S ((dst_positive_code_table_zero_resultvaluefold) + (dst_positive_scale_table_zero_resultvaluefold)) + ((dst_positive_scale_table_zero_resultvaluefold) + (dst_positive_scale_table_zero_resultvaluefold))) + (((dst_negative_code_table_zero_resultvaluefold) + (dst_negative_scale_table_zero_resultvaluefold)) * S ((dst_negative_code_table_zero_resultvaluefold) + (dst_negative_scale_table_zero_resultvaluefold)) + ((dst_negative_scale_table_zero_resultvaluefold) + (dst_negative_scale_table_zero_resultvaluefold)))) * S ((((dst_positive_code_table_zero_resultvaluefold) + (dst_positive_scale_table_zero_resultvaluefold)) * S ((dst_positive_code_table_zero_resultvaluefold) + (dst_positive_scale_table_zero_resultvaluefold)) + ((dst_positive_scale_table_zero_resultvaluefold) + (dst_positive_scale_table_zero_resultvaluefold))) + (((dst_negative_code_table_zero_resultvaluefold) + (dst_negative_scale_table_zero_resultvaluefold)) * S ((dst_negative_code_table_zero_resultvaluefold) + (dst_negative_scale_table_zero_resultvaluefold)) + ((dst_negative_scale_table_zero_resultvaluefold) + (dst_negative_scale_table_zero_resultvaluefold)))) + ((((dst_negative_code_table_zero_resultvaluefold) + (dst_negative_scale_table_zero_resultvaluefold)) * S ((dst_negative_code_table_zero_resultvaluefold) + (dst_negative_scale_table_zero_resultvaluefold)) + ((dst_negative_scale_table_zero_resultvaluefold) + (dst_negative_scale_table_zero_resultvaluefold))) + (((dst_negative_code_table_zero_resultvaluefold) + (dst_negative_scale_table_zero_resultvaluefold)) * S ((dst_negative_code_table_zero_resultvaluefold) + (dst_negative_scale_table_zero_resultvaluefold)) + ((dst_negative_scale_table_zero_resultvaluefold) + (dst_negative_scale_table_zero_resultvaluefold)))))) /\ (((exists fs_u_dst_table_zero_resultvaluefoldpositive fs_v_dst_table_zero_resultvaluefoldpositive. ((((exists fs_h_dst_table_zero_resultvaluefoldpositive_body_start. fs_h_dst_table_zero_resultvaluefoldpositive_body_start + S (0) = S ((S (0)) * fs_v_dst_table_zero_resultvaluefoldpositive)) /\ exists fs_q_dst_table_zero_resultvaluefoldpositive_body_start. fs_u_dst_table_zero_resultvaluefoldpositive = fs_q_dst_table_zero_resultvaluefoldpositive_body_start * S ((S (0)) * fs_v_dst_table_zero_resultvaluefoldpositive) + (0))) /\ ((((exists fs_h_dst_table_zero_resultvaluefoldpositive_body_terminal. fs_h_dst_table_zero_resultvaluefoldpositive_body_terminal + S (dst_positive_sum_table_zero_resultvaluefold) = S ((S (S (dc_input_table_zero_result))) * fs_v_dst_table_zero_resultvaluefoldpositive)) /\ exists fs_q_dst_table_zero_resultvaluefoldpositive_body_terminal. fs_u_dst_table_zero_resultvaluefoldpositive = fs_q_dst_table_zero_resultvaluefoldpositive_body_terminal * S ((S (S (dc_input_table_zero_result))) * fs_v_dst_table_zero_resultvaluefoldpositive) + (dst_positive_sum_table_zero_resultvaluefold))) /\ forall fs_i_dst_table_zero_resultvaluefoldpositive_body_steps. (exists fs_lt_dst_table_zero_resultvaluefoldpositive_body_steps_bound. fs_lt_dst_table_zero_resultvaluefoldpositive_body_steps_bound + S fs_i_dst_table_zero_resultvaluefoldpositive_body_steps = S (dc_input_table_zero_result)) -> exists fs_a_dst_table_zero_resultvaluefoldpositive_body_steps fs_r_dst_table_zero_resultvaluefoldpositive_body_steps fs_s_dst_table_zero_resultvaluefoldpositive_body_steps. ((((exists fs_h_dst_table_zero_resultvaluefoldpositive_body_steps_summand. fs_h_dst_table_zero_resultvaluefoldpositive_body_steps_summand + S (fs_a_dst_table_zero_resultvaluefoldpositive_body_steps) = S ((S (fs_i_dst_table_zero_resultvaluefoldpositive_body_steps)) * dst_positive_scale_table_zero_resultvaluefold)) /\ exists fs_q_dst_table_zero_resultvaluefoldpositive_body_steps_summand. dst_positive_code_table_zero_resultvaluefold = fs_q_dst_table_zero_resultvaluefoldpositive_body_steps_summand * S ((S (fs_i_dst_table_zero_resultvaluefoldpositive_body_steps)) * dst_positive_scale_table_zero_resultvaluefold) + (fs_a_dst_table_zero_resultvaluefoldpositive_body_steps))) /\ ((((exists fs_h_dst_table_zero_resultvaluefoldpositive_body_steps_partial. fs_h_dst_table_zero_resultvaluefoldpositive_body_steps_partial + S (fs_r_dst_table_zero_resultvaluefoldpositive_body_steps) = S ((S (fs_i_dst_table_zero_resultvaluefoldpositive_body_steps)) * fs_v_dst_table_zero_resultvaluefoldpositive)) /\ exists fs_q_dst_table_zero_resultvaluefoldpositive_body_steps_partial. fs_u_dst_table_zero_resultvaluefoldpositive = fs_q_dst_table_zero_resultvaluefoldpositive_body_steps_partial * S ((S (fs_i_dst_table_zero_resultvaluefoldpositive_body_steps)) * fs_v_dst_table_zero_resultvaluefoldpositive) + (fs_r_dst_table_zero_resultvaluefoldpositive_body_steps))) /\ ((((exists fs_h_dst_table_zero_resultvaluefoldpositive_body_steps_successor. fs_h_dst_table_zero_resultvaluefoldpositive_body_steps_successor + S (fs_s_dst_table_zero_resultvaluefoldpositive_body_steps) = S ((S (S fs_i_dst_table_zero_resultvaluefoldpositive_body_steps)) * fs_v_dst_table_zero_resultvaluefoldpositive)) /\ exists fs_q_dst_table_zero_resultvaluefoldpositive_body_steps_successor. fs_u_dst_table_zero_resultvaluefoldpositive = fs_q_dst_table_zero_resultvaluefoldpositive_body_steps_successor * S ((S (S fs_i_dst_table_zero_resultvaluefoldpositive_body_steps)) * fs_v_dst_table_zero_resultvaluefoldpositive) + (fs_s_dst_table_zero_resultvaluefoldpositive_body_steps))) /\ fs_s_dst_table_zero_resultvaluefoldpositive_body_steps = fs_r_dst_table_zero_resultvaluefoldpositive_body_steps + fs_a_dst_table_zero_resultvaluefoldpositive_body_steps)))))) /\ (((exists fs_u_dst_table_zero_resultvaluefoldnegative fs_v_dst_table_zero_resultvaluefoldnegative. ((((exists fs_h_dst_table_zero_resultvaluefoldnegative_body_start. fs_h_dst_table_zero_resultvaluefoldnegative_body_start + S (0) = S ((S (0)) * fs_v_dst_table_zero_resultvaluefoldnegative)) /\ exists fs_q_dst_table_zero_resultvaluefoldnegative_body_start. fs_u_dst_table_zero_resultvaluefoldnegative = fs_q_dst_table_zero_resultvaluefoldnegative_body_start * S ((S (0)) * fs_v_dst_table_zero_resultvaluefoldnegative) + (0))) /\ ((((exists fs_h_dst_table_zero_resultvaluefoldnegative_body_terminal. fs_h_dst_table_zero_resultvaluefoldnegative_body_terminal + S (dst_negative_sum_table_zero_resultvaluefold) = S ((S (S (dc_input_table_zero_result))) * fs_v_dst_table_zero_resultvaluefoldnegative)) /\ exists fs_q_dst_table_zero_resultvaluefoldnegative_body_terminal. fs_u_dst_table_zero_resultvaluefoldnegative = fs_q_dst_table_zero_resultvaluefoldnegative_body_terminal * S ((S (S (dc_input_table_zero_result))) * fs_v_dst_table_zero_resultvaluefoldnegative) + (dst_negative_sum_table_zero_resultvaluefold))) /\ forall fs_i_dst_table_zero_resultvaluefoldnegative_body_steps. (exists fs_lt_dst_table_zero_resultvaluefoldnegative_body_steps_bound. fs_lt_dst_table_zero_resultvaluefoldnegative_body_steps_bound + S fs_i_dst_table_zero_resultvaluefoldnegative_body_steps = S (dc_input_table_zero_result)) -> exists fs_a_dst_table_zero_resultvaluefoldnegative_body_steps fs_r_dst_table_zero_resultvaluefoldnegative_body_steps fs_s_dst_table_zero_resultvaluefoldnegative_body_steps. ((((exists fs_h_dst_table_zero_resultvaluefoldnegative_body_steps_summand. fs_h_dst_table_zero_resultvaluefoldnegative_body_steps_summand + S (fs_a_dst_table_zero_resultvaluefoldnegative_body_steps) = S ((S (fs_i_dst_table_zero_resultvaluefoldnegative_body_steps)) * dst_negative_scale_table_zero_resultvaluefold)) /\ exists fs_q_dst_table_zero_resultvaluefoldnegative_body_steps_summand. dst_negative_code_table_zero_resultvaluefold = fs_q_dst_table_zero_resultvaluefoldnegative_body_steps_summand * S ((S (fs_i_dst_table_zero_resultvaluefoldnegative_body_steps)) * dst_negative_scale_table_zero_resultvaluefold) + (fs_a_dst_table_zero_resultvaluefoldnegative_body_steps))) /\ ((((exists fs_h_dst_table_zero_resultvaluefoldnegative_body_steps_partial. fs_h_dst_table_zero_resultvaluefoldnegative_body_steps_partial + S (fs_r_dst_table_zero_resultvaluefoldnegative_body_steps) = S ((S (fs_i_dst_table_zero_resultvaluefoldnegative_body_steps)) * fs_v_dst_table_zero_resultvaluefoldnegative)) /\ exists fs_q_dst_table_zero_resultvaluefoldnegative_body_steps_partial. fs_u_dst_table_zero_resultvaluefoldnegative = fs_q_dst_table_zero_resultvaluefoldnegative_body_steps_partial * S ((S (fs_i_dst_table_zero_resultvaluefoldnegative_body_steps)) * fs_v_dst_table_zero_resultvaluefoldnegative) + (fs_r_dst_table_zero_resultvaluefoldnegative_body_steps))) /\ ((((exists fs_h_dst_table_zero_resultvaluefoldnegative_body_steps_successor. fs_h_dst_table_zero_resultvaluefoldnegative_body_steps_successor + S (fs_s_dst_table_zero_resultvaluefoldnegative_body_steps) = S ((S (S fs_i_dst_table_zero_resultvaluefoldnegative_body_steps)) * fs_v_dst_table_zero_resultvaluefoldnegative)) /\ exists fs_q_dst_table_zero_resultvaluefoldnegative_body_steps_successor. fs_u_dst_table_zero_resultvaluefoldnegative = fs_q_dst_table_zero_resultvaluefoldnegative_body_steps_successor * S ((S (S fs_i_dst_table_zero_resultvaluefoldnegative_body_steps)) * fs_v_dst_table_zero_resultvaluefoldnegative) + (fs_s_dst_table_zero_resultvaluefoldnegative_body_steps))) /\ fs_s_dst_table_zero_resultvaluefoldnegative_body_steps = fs_r_dst_table_zero_resultvaluefoldnegative_body_steps + fs_a_dst_table_zero_resultvaluefoldnegative_body_steps)))))) /\ (exists ge_balance_positive_table_zero_resultvaluefoldresult ge_balance_negative_table_zero_resultvaluefoldresult. (((((dc_output_table_zero_result) = 2 * (ge_balance_positive_table_zero_resultvaluefoldresult) /\ (ge_balance_negative_table_zero_resultvaluefoldresult) = 0) \/ exists ge_signed_half_table_zero_resultvaluefoldresultdecode. (((dc_output_table_zero_result) = 2 * ge_signed_half_table_zero_resultvaluefoldresultdecode + 1 /\ (ge_balance_positive_table_zero_resultvaluefoldresult) = 0) /\ (ge_balance_negative_table_zero_resultvaluefoldresult) = S ge_signed_half_table_zero_resultvaluefoldresultdecode))) /\ ((dst_positive_sum_table_zero_resultvaluefold) + ge_balance_negative_table_zero_resultvaluefoldresult = (dst_negative_sum_table_zero_resultvaluefold) + ge_balance_positive_table_zero_resultvaluefoldresult))))))))))))))))))))

Complete tactic proof in conservative notation

All 22 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

22 script commands · 10 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 assumptionsL1–6

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

  1. L1
    intro F
  2. L2
    intro G
  3. L3
    intro H
  4. L4
    intro hF
  5. L5
    intro hG
  6. L6
    intro hH
02Separate the logical casesL7–7

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

  1. L7
    split
03Use earlier factsL8–8

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

  1. L8
    exact hF
04Separate the logical casesL9–9

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

  1. L9
    split
05Use earlier factsL10–10

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

  1. L10
    exact hG
06Separate the logical casesL11–11

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

  1. L11
    split
07Use earlier factsL12–12

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

  1. L12
    exact hH
08Fix variables and assumptionsL13–17

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

  1. L13
    intro n
  2. L14
    intro z
  3. L15
    intro hn
  4. L16
    intro hbound
  5. L17
    intro hz
09Separate the logical casesL18–18

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

  1. L18
    exfalso
10Use earlier factsL19–22

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

  1. L19
    apply hn
  2. L20
    specialize le_zero (n)
  3. L21
    apply le_zero
  4. L22
    exact hbound

Library-wide reading audit

Original defined command ledger · 22 lines
  1. 0001intro F
  2. 0002intro G
  3. 0003intro H
  4. 0004intro hF
  5. 0005intro hG
  6. 0006intro hH
  7. 0007split
  8. 0008exact hF
  9. 0009split
  10. 0010exact hG
  11. 0011split
  12. 0012exact hH
  13. 0013intro n
  14. 0014intro z
  15. 0015intro hn
  16. 0016intro hbound
  17. 0017intro hz
  18. 0018exfalso
  19. 0019apply hn
  20. 0020specialize le_zero (n)
  21. 0021apply le_zero
  22. 0022exact hbound